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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2411.07970v2/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S1" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">1 Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2 MUltiplexed Survey Telescope</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS1" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.1 A 6.5-m Telescope for Spectroscopic Surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS2" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.2 Multiplexed Focal Plane & Instruments</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS3" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.3 Site and Observing Conditions</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS4" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.4 Survey Capability and Overall Scientific Potential</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3 Scientific Motivations of the Stage-V Cosmological Surveys</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS1" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.1 Nature & Evolution of Dark Energy</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS2" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.2 Growth of Structure & Nature of Gravity</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS3" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.3 Inflation & Primordial Physics</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS4" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.4 Neutrinos & Light Relics</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS5" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.5 Dark Matter</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6 Synergy with Other Probes</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS1" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.1 Imaging surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS2" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.2 CMB S4</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS3" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.3 Radio surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS4" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.4 Gravitational waves and Fast Radio Bursts</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4 Target Selection</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS1" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.1 Challenges of Target Selection for Stage-V Spectroscopic Surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2 Low-Redshift Tracers</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS1" title="In 4.2 Low-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2.1 Bright Galaxy Sample (BGS)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS2" title="In 4.2 Low-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2.2 Luminous Red Galaxies (LRG)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS3" title="In 4.2 Low-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2.3 Emission Line Galaxies (ELG)</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3 High-Redshift Tracers</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1" title="In 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3.1 Lyman-Break Galaxies (LBG)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS2" title="In 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3.2 Lyman-<math alttext="\alpha" class="ltx_Math" display="inline"><semantics><mi>α</mi><annotation-xml encoding="MathML-Content"><ci>𝛼</ci></annotation-xml><annotation encoding="application/x-tex">\alpha</annotation><annotation encoding="application/x-llamapun">italic_α</annotation></semantics></math> Emitters (LAE)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS3" title="In 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3.3 Quasi-Stellar Objects (QSO)</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS4" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.4 Summary of the Targets Selection</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS5" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.5 Conceptual Survey Design</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS5.SSS1" title="In 4.5 Conceptual Survey Design ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.5.1 Dark Time Survey</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS5.SSS2" title="In 4.5 Conceptual Survey Design ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.5.2 Grey Time Survey</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5 Cosmological Forecasts</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS1" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.1 Methodology</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS2" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.2 Dark Energy</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS3" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.3 Structure Growth & Gravity</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS4" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.4 Primordial Non-Gaussianity</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS5" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.5 Neutrinos</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS6" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.6 Warm Dark Matter</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S6" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">6 Conclusions</a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> \ensubject subject </div> <div class="ltx_para" id="p2"> \ArticleType Article \SpecialTopicSPECIAL TOPIC: \Year2023 \MonthJanuary \Vol66 \No1 \DOI?? \ArtNo000000 \ReceiveDateJanuary 11, 2023 \AcceptDateApril 6, 2023 </div> <div class="ltx_para" id="p3"> \AuthorMark Zhao </div> <div class="ltx_para" id="p4"> \AuthorCitation Zhao, et al </div> <h1 class="ltx_title ltx_title_document">MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey</h1> <div class="ltx_authors"> Cheng Zhao Email: <a class="ltx_ref ltx_href" href="mailto:czhao@tsinghua.edu.cn" title="">czhao@tsinghua.edu.cn</a> Song Huang Email: <a class="ltx_ref ltx_href" href="mailto:shuang@tsinghua.edu.cn" title="">shuang@tsinghua.edu.cn</a> Mengfan He Paulo Montero-Camacho Yu Liu Pablo Renard Yunyi Tang Aurélien Verdier Wenshuo Xu Xiaorui Yang Jiaxi Yu Yao Zhang Siyi Zhao Xingchen Zhou Sheng-Yu He Jean-Paul Kneib Jiayi Li Zhuoyang Li Wen-Ting Wang Zhong-Zhi Xianyu Yidian Zhang Rafaela Gsponer Xiao-Dong Li Antoine Rocher Siwei Zou Ting Tan Zhiqi Huang Zhuoxiao Wang Pei Li Maxime Rombach Chenxing Dong Daniel Forero-Sánchez Huanyuan Shan Tao Wang Yin Li Zhongxu Zhai Yuting Wang Gong-Bo Zhao Yong Shi Shude Mao Lei Huang Liquan Guo Zheng Cai Department of Astronomy, Tsinghua University, Beijing 100084, China Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China Pengcheng Laboratory, Nanshan District, Shenzhen, Guangdong 518000, China Institute of Physics, Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, CH-1290 Versoix, Switzerland National Astronomical Observatories, CAS, Beijing 100101, China Science Center for China Space Station Telescope, National Astronomical Observatories, Chinese Academy of Science, 20A Datun Road, Beijing 100101, China Zhili College, Tsinghua University, Beijing 100084, China Department of Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240, China Department of Physics, Tsinghua University, Beijing 100084, China School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China Chinese Academy of Sciences South America Center for Astronomy (CASSACA), National Astronomical Observatories of China, Beijing 100101, China IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France School of Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, China Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, 163 Xianlin Avenue, Nanjing 210023, China Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> The MUltiplexed Survey Telescope (MUST) is a 6.5-meter telescope under development. Dedicated to highly-multiplexed, wide-field spectroscopic surveys, MUST observes over 20,000 targets simultaneously using 6.2-mm pitch positioning robots within a <math alttext="\sim 5\,{\rm deg}^{2}" class="ltx_Math" display="inline" id="1.m1.1"><semantics id="1.m1.1a"><mrow id="1.m1.1.1" xref="1.m1.1.1.cmml"><mi id="1.m1.1.1.2" xref="1.m1.1.1.2.cmml"></mi><mo id="1.m1.1.1.1" xref="1.m1.1.1.1.cmml">∼</mo><mrow id="1.m1.1.1.3" xref="1.m1.1.1.3.cmml"><mn id="1.m1.1.1.3.2" xref="1.m1.1.1.3.2.cmml">5</mn><mo id="1.m1.1.1.3.1" lspace="0.170em" xref="1.m1.1.1.3.1.cmml">⁢</mo><msup id="1.m1.1.1.3.3" xref="1.m1.1.1.3.3.cmml"><mi id="1.m1.1.1.3.3.2" xref="1.m1.1.1.3.3.2.cmml">deg</mi><mn id="1.m1.1.1.3.3.3" xref="1.m1.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="1.m1.1b"><apply id="1.m1.1.1.cmml" xref="1.m1.1.1"><csymbol cd="latexml" id="1.m1.1.1.1.cmml" xref="1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="1.m1.1.1.2.cmml" xref="1.m1.1.1.2">absent</csymbol><apply id="1.m1.1.1.3.cmml" xref="1.m1.1.1.3"><times id="1.m1.1.1.3.1.cmml" xref="1.m1.1.1.3.1"></times><cn id="1.m1.1.1.3.2.cmml" type="integer" xref="1.m1.1.1.3.2">5</cn><apply id="1.m1.1.1.3.3.cmml" xref="1.m1.1.1.3.3"><csymbol cd="ambiguous" id="1.m1.1.1.3.3.1.cmml" xref="1.m1.1.1.3.3">superscript</csymbol><ci id="1.m1.1.1.3.3.2.cmml" xref="1.m1.1.1.3.3.2">deg</ci><cn id="1.m1.1.1.3.3.3.cmml" type="integer" xref="1.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="1.m1.1c">\sim 5\,{\rm deg}^{2}</annotation><annotation encoding="application/x-llamapun" id="1.m1.1d">∼ 5 roman_deg start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> field of view. MUST aims to carry out the first Stage-V spectroscopic survey in the 2030s to map the 3D Universe with over 100 million galaxies and quasars, spanning from the nearby Universe to redshift <math alttext="z\sim 5.5" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mrow id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml"><mi id="id2.2.m2.1.1.2" xref="id2.2.m2.1.1.2.cmml">z</mi><mo id="id2.2.m2.1.1.1" xref="id2.2.m2.1.1.1.cmml">∼</mo><mn id="id2.2.m2.1.1.3" xref="id2.2.m2.1.1.3.cmml">5.5</mn></mrow><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><apply id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1"><csymbol cd="latexml" id="id2.2.m2.1.1.1.cmml" xref="id2.2.m2.1.1.1">similar-to</csymbol><ci id="id2.2.m2.1.1.2.cmml" xref="id2.2.m2.1.1.2">𝑧</ci><cn id="id2.2.m2.1.1.3.cmml" type="float" xref="id2.2.m2.1.1.3">5.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">z\sim 5.5</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_z ∼ 5.5</annotation></semantics></math>, corresponding to around 1 billion years after the Big Bang. To cover this extensive redshift range, we present an initial conceptual target selection algorithm for different types of galaxies, from local bright galaxies, luminous red galaxies, and emission line galaxies to high-redshift (<math alttext="2<z<5.5" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mrow id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml"><mn id="id3.3.m3.1.1.2" xref="id3.3.m3.1.1.2.cmml">2</mn><mo id="id3.3.m3.1.1.3" xref="id3.3.m3.1.1.3.cmml"><</mo><mi id="id3.3.m3.1.1.4" xref="id3.3.m3.1.1.4.cmml">z</mi><mo id="id3.3.m3.1.1.5" xref="id3.3.m3.1.1.5.cmml"><</mo><mn id="id3.3.m3.1.1.6" xref="id3.3.m3.1.1.6.cmml">5.5</mn></mrow><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><apply id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1"><and id="id3.3.m3.1.1a.cmml" xref="id3.3.m3.1.1"></and><apply id="id3.3.m3.1.1b.cmml" xref="id3.3.m3.1.1"><lt id="id3.3.m3.1.1.3.cmml" xref="id3.3.m3.1.1.3"></lt><cn id="id3.3.m3.1.1.2.cmml" type="integer" xref="id3.3.m3.1.1.2">2</cn><ci id="id3.3.m3.1.1.4.cmml" xref="id3.3.m3.1.1.4">𝑧</ci></apply><apply id="id3.3.m3.1.1c.cmml" xref="id3.3.m3.1.1"><lt id="id3.3.m3.1.1.5.cmml" xref="id3.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#id3.3.m3.1.1.4.cmml" id="id3.3.m3.1.1d.cmml" xref="id3.3.m3.1.1"></share><cn id="id3.3.m3.1.1.6.cmml" type="float" xref="id3.3.m3.1.1.6">5.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">2<z<5.5</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">2 < italic_z < 5.5</annotation></semantics></math>) Lyman-break galaxies. Using Fisher forecasts, we demonstrate that MUST can address fundamental questions in cosmology, including the nature of dark energy, test of gravity theories, and investigations into primordial physics. This is the first paper in the series of science white papers for MUST, with subsequent developments focusing on additional scientific cases such as galaxy and quasar evolution, Milky Way physics, and dynamic phenomena in the time-domain Universe. </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">keywords: </h6>Optical instruments, Sky surveys, Cosmology </div> <div class="ltx_para" id="p5"> \PACS 07.60.-j, 95.80.+p, 98.80.-k </div> <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S1" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">1 Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2 MUltiplexed Survey Telescope</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS1" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.1 A 6.5-m Telescope for Spectroscopic Surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS2" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.2 Multiplexed Focal Plane & Instruments</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS3" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.3 Site and Observing Conditions</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS4" title="In 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.4 Survey Capability and Overall Scientific Potential</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3 Scientific Motivations of the Stage-V Cosmological Surveys</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS1" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.1 Nature & Evolution of Dark Energy</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS2" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.2 Growth of Structure & Nature of Gravity</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS3" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.3 Inflation & Primordial Physics</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS4" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.4 Neutrinos & Light Relics</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS5" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.5 Dark Matter</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6" title="In 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6 Synergy with Other Probes</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS1" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.1 Imaging surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS2" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.2 CMB S4</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS3" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.3 Radio surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS6.SSS4" title="In 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.6.4 Gravitational waves and Fast Radio Bursts</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4 Target Selection</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS1" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.1 Challenges of Target Selection for Stage-V Spectroscopic Surveys</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2 Low-Redshift Tracers</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS1" title="In 4.2 Low-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2.1 Bright Galaxy Sample (BGS)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS2" title="In 4.2 Low-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2.2 Luminous Red Galaxies (LRG)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS3" title="In 4.2 Low-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.2.3 Emission Line Galaxies (ELG)</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3 High-Redshift Tracers</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1" title="In 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3.1 Lyman-Break Galaxies (LBG)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS2" title="In 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3.2 Lyman-<math alttext="\alpha" class="ltx_Math" display="inline"><semantics><mi>α</mi><annotation-xml encoding="MathML-Content"><ci>𝛼</ci></annotation-xml><annotation encoding="application/x-tex">\alpha</annotation><annotation encoding="application/x-llamapun">italic_α</annotation></semantics></math> Emitters (LAE)</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS3" title="In 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.3.3 Quasi-Stellar Objects (QSO)</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS4" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.4 Summary of the Targets Selection</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS5" title="In 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.5 Conceptual Survey Design</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS5.SSS1" title="In 4.5 Conceptual Survey Design ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.5.1 Dark Time Survey</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.SS5.SSS2" title="In 4.5 Conceptual Survey Design ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4.5.2 Grey Time Survey</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5 Cosmological Forecasts</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS1" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.1 Methodology</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS2" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.2 Dark Energy</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS3" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.3 Structure Growth & Gravity</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS4" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.4 Primordial Non-Gaussianity</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS5" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.5 Neutrinos</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS6" title="In 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.6 Warm Dark Matter</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S6" title="In MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">6 Conclusions</a></li> </ol></nav> <div class="ltx_pagination ltx_role_start_2_columns"></div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> 1 Introduction</h2> <div class="ltx_para" id="S1.p1"> Over the past four decades, beginning with the “Stick Man” from the CfA Redshift Survey in 1982 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib2" title="">2</a>]</cite>, spectroscopic mapping of large-scale structures (LSS) has accumulated more than 30 million redshifts of nearby and distant galaxies (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.F2" title="Figure 2 ‣ 2.4 Survey Capability and Overall Scientific Potential ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2</a>). This monumental achievement has contributed significantly to the establishment of the current cosmological model <math alttext="\Lambda" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" mathvariant="normal" xref="S1.p1.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">roman_Λ</annotation></semantics></math>CDM, along with other cosmological probes, i.e., cosmic microwave background (CMB; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib4" title="">4</a>]</cite>), Type-Ia supernovae <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib7" title="">7</a>]</cite>, or measurements of weak lensing (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib9" title="">9</a>]</cite>). For two decades (2000–2020), major experiments such as the Sloan Digital Sky Survey (SDSS; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib10" title="">10</a>]</cite>), followed by the ongoing (2021–2026) survey from the Dark Energy Spectroscopic Instrument111<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://www.desi.lbl.gov/the-desi-survey/" title="">https://www.desi.lbl.gov/the-desi-survey/</a> (DESI; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib11" title="">11</a>]</cite>), have mapped the 3D universe at low and intermediate redshift (<math alttext="z\lesssim 3" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mrow id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml"><mi id="S1.p1.2.m2.1.1.2" xref="S1.p1.2.m2.1.1.2.cmml">z</mi><mo id="S1.p1.2.m2.1.1.1" xref="S1.p1.2.m2.1.1.1.cmml">≲</mo><mn id="S1.p1.2.m2.1.1.3" xref="S1.p1.2.m2.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><apply id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1"><csymbol cd="latexml" id="S1.p1.2.m2.1.1.1.cmml" xref="S1.p1.2.m2.1.1.1">less-than-or-similar-to</csymbol><ci id="S1.p1.2.m2.1.1.2.cmml" xref="S1.p1.2.m2.1.1.2">𝑧</ci><cn id="S1.p1.2.m2.1.1.3.cmml" type="integer" xref="S1.p1.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">z\lesssim 3</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_z ≲ 3</annotation></semantics></math>). Clustering measurements from spectroscopic surveys of galaxies and quasars have become a key probe of cosmology. They provide precise measurements on the baryon acoustic oscillations (BAO) scale <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib12" title="">12</a>]</cite> and the linear growth rate of structure <math alttext="f\sigma_{8}" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mrow id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml"><mi id="S1.p1.3.m3.1.1.2" xref="S1.p1.3.m3.1.1.2.cmml">f</mi><mo id="S1.p1.3.m3.1.1.1" xref="S1.p1.3.m3.1.1.1.cmml">⁢</mo><msub id="S1.p1.3.m3.1.1.3" xref="S1.p1.3.m3.1.1.3.cmml"><mi id="S1.p1.3.m3.1.1.3.2" xref="S1.p1.3.m3.1.1.3.2.cmml">σ</mi><mn id="S1.p1.3.m3.1.1.3.3" xref="S1.p1.3.m3.1.1.3.3.cmml">8</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><apply id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1"><times id="S1.p1.3.m3.1.1.1.cmml" xref="S1.p1.3.m3.1.1.1"></times><ci id="S1.p1.3.m3.1.1.2.cmml" xref="S1.p1.3.m3.1.1.2">𝑓</ci><apply id="S1.p1.3.m3.1.1.3.cmml" xref="S1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.p1.3.m3.1.1.3.1.cmml" xref="S1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S1.p1.3.m3.1.1.3.2.cmml" xref="S1.p1.3.m3.1.1.3.2">𝜎</ci><cn id="S1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S1.p1.3.m3.1.1.3.3">8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">f\sigma_{8}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_f italic_σ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT</annotation></semantics></math> through redshift space distortions (RSD) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib13" title="">13</a>]</cite>. Recent results from the DESI collaboration <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib14" title="">14</a>]</cite> suggest a potential deviation from the cosmological constant to time-varying dark energy. By the end of this decade, we expect to have sub-percent-level constraints on dark energy and gravity from galaxy clustering up to redshift <math alttext="z\sim 2" class="ltx_Math" display="inline" id="S1.p1.4.m4.1"><semantics id="S1.p1.4.m4.1a"><mrow id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml"><mi id="S1.p1.4.m4.1.1.2" xref="S1.p1.4.m4.1.1.2.cmml">z</mi><mo id="S1.p1.4.m4.1.1.1" xref="S1.p1.4.m4.1.1.1.cmml">∼</mo><mn id="S1.p1.4.m4.1.1.3" xref="S1.p1.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.1b"><apply id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1"><csymbol cd="latexml" id="S1.p1.4.m4.1.1.1.cmml" xref="S1.p1.4.m4.1.1.1">similar-to</csymbol><ci id="S1.p1.4.m4.1.1.2.cmml" xref="S1.p1.4.m4.1.1.2">𝑧</ci><cn id="S1.p1.4.m4.1.1.3.cmml" type="integer" xref="S1.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">z\sim 2</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">italic_z ∼ 2</annotation></semantics></math>. When the DESI project wraps up, we have finished the spectroscopic survey component of the four stages envisioned by the Dark Energy Task Force (DETF) report <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib15" title="">15</a>]</cite>. Yet, many fundamental questions remain unanswered, calling for a new era of cosmological experiments in the 2030s. </div> <div class="ltx_para" id="S1.p2"> Going one step further, 3D maps of the universe at high redshift (<math alttext="z>2" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml"><mi id="S1.p2.1.m1.1.1.2" xref="S1.p2.1.m1.1.1.2.cmml">z</mi><mo id="S1.p2.1.m1.1.1.1" xref="S1.p2.1.m1.1.1.1.cmml">></mo><mn id="S1.p2.1.m1.1.1.3" xref="S1.p2.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1"><gt id="S1.p2.1.m1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1"></gt><ci id="S1.p2.1.m1.1.1.2.cmml" xref="S1.p2.1.m1.1.1.2">𝑧</ci><cn id="S1.p2.1.m1.1.1.3.cmml" type="integer" xref="S1.p2.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">z>2</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_z > 2</annotation></semantics></math>) will enable the observation of linear modes in the primordial universe, significantly enhancing our ability to constrain dark energy and inflation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib16" title="">16</a>]</cite>. In the next decade, a series of ground- and space-based photometric surveys CSST <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib17" title="">17</a>]</cite>, Euclid <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib18" title="">18</a>]</cite>, Nancy Grace Roman Space Telescope <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib19" title="">19</a>]</cite>, LSST <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib20" title="">20</a>]</cite> will provide deep and high-quality images for future galaxy spectroscopic surveys allowing to target galaxies such as Lyman Break Galaxy (LBG) or Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_α</annotation></semantics></math> emitter (LAE) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib22" title="">22</a>]</cite> as tracer of matter at high redshift <math alttext="2<z<5" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mrow id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mn id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">2</mn><mo id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml"><</mo><mi id="S1.p2.3.m3.1.1.4" xref="S1.p2.3.m3.1.1.4.cmml">z</mi><mo id="S1.p2.3.m3.1.1.5" xref="S1.p2.3.m3.1.1.5.cmml"><</mo><mn id="S1.p2.3.m3.1.1.6" xref="S1.p2.3.m3.1.1.6.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><and id="S1.p2.3.m3.1.1a.cmml" xref="S1.p2.3.m3.1.1"></and><apply id="S1.p2.3.m3.1.1b.cmml" xref="S1.p2.3.m3.1.1"><lt id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3"></lt><cn id="S1.p2.3.m3.1.1.2.cmml" type="integer" xref="S1.p2.3.m3.1.1.2">2</cn><ci id="S1.p2.3.m3.1.1.4.cmml" xref="S1.p2.3.m3.1.1.4">𝑧</ci></apply><apply id="S1.p2.3.m3.1.1c.cmml" xref="S1.p2.3.m3.1.1"><lt id="S1.p2.3.m3.1.1.5.cmml" xref="S1.p2.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S1.p2.3.m3.1.1.4.cmml" id="S1.p2.3.m3.1.1d.cmml" xref="S1.p2.3.m3.1.1"></share><cn id="S1.p2.3.m3.1.1.6.cmml" type="integer" xref="S1.p2.3.m3.1.1.6">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">2<z<5</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">2 < italic_z < 5</annotation></semantics></math>. Large-volume high-redshift redshift surveys using these new tracers have the unprecedented potential to help us test primordial non-Gaussianity, probe dynamic dark energy, and reveal possible new features in the primordial power spectrum, uncovering tantalizing hints for new physics. </div> <div class="ltx_para" id="S1.p3"> At the same time, a high-density spectroscopic survey of low-redshift (<math alttext="z<1.5" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mi id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">z</mi><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml"><</mo><mn id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><lt id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1.1"></lt><ci id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2">𝑧</ci><cn id="S1.p3.1.m1.1.1.3.cmml" type="float" xref="S1.p3.1.m1.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">z<1.5</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_z < 1.5</annotation></semantics></math>) can provide a high-fidelity 3D map of the cosmic web and trace the matter distributions into the non-linear regimes, opening doors to unexplored scientific opportunities. Such a multi-purpose dataset can also enhance the scientific performance of other cosmological probes, such as calibrating the photometric redshift and intrinsic alignment models for weak gravitational lensing surveys (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib23" title="">23</a>]</cite>; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib25" title="">25</a>]</cite>), providing spectroscopic follow-up and host galaxy properties for supernova surveys (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib27" title="">27</a>]</cite>), or exploring new approaches to map the low-redshift large-scale structures (LSS) such as a peculiar velocities survey (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib29" title="">29</a>]</cite>). More importantly, it will help maximize the potential for synergies between spectroscopic surveys and other cosmological probes, such as CMB (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib30" title="">30</a>]</cite>), weak lensing (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib34" title="">34</a>]</cite>), and intensity mapping (IM, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib35" title="">35</a>]</cite>) experiments. </div> <div class="ltx_para" id="S1.p4"> Motivated by these two promising directions, the cosmological & high-energy physics community has recently coined the concept for a Stage-V spectroscopic experiment (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib36" title="">36</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib37" title="">37</a>]</cite>) to fulfill these high expectations. By definition, a Stage-V spectroscopic facility should utilize a telescope with high etendue value (<math alttext="A\Omega" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mi id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml">A</mi><mo id="S1.p4.1.m1.1.1.1" xref="S1.p4.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.p4.1.m1.1.1.3" mathvariant="normal" xref="S1.p4.1.m1.1.1.3.cmml">Ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><times id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1.1"></times><ci id="S1.p4.1.m1.1.1.2.cmml" xref="S1.p4.1.m1.1.1.2">𝐴</ci><ci id="S1.p4.1.m1.1.1.3.cmml" xref="S1.p4.1.m1.1.1.3">Ω</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">A\Omega</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_A roman_Ω</annotation></semantics></math>; <math alttext="A" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">italic_A</annotation></semantics></math> and <math alttext="\Omega" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" mathvariant="normal" xref="S1.p4.3.m3.1.1.cmml">Ω</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">Ω</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">\Omega</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">roman_Ω</annotation></semantics></math> are the collecting area and the field-of-view of the telescope) to ensure a high survey speed. More importantly, the facility should have significantly improved multiplexed capability (number of targets that can be observed simultaneously) compared to the Stage-IV survey (e.g., 5,000 fibers for DESI). Conceptually, a Stage-V facility demands a minimum of 10,000 fibers that could be easily reconfigured to target different objects. This is the primary technical challenge now for such an ambitious vision. At the same time, a Stage-V facility should also have excellent optical performance, high-performance multi-object spectrographs that at least cover the whole optical wavelength range, and a site with good observing conditions. Building on these requirements, multiple ground-based concepts have been proposed, including the 6.5 m MegaMapper telescope <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib39" title="">39</a>]</cite>, the dual-6 m & dual-hemisphere Spec-S5 project222<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://www.spec-s5.org/" title="">https://www.spec-s5.org/</a> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib40" title="">40</a>]</cite>, the 11 m Maunakea Spectroscopic Explorer333<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://mse.cfht.hawaii.edu/" title="">https://mse.cfht.hawaii.edu/</a> (MSE; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib41" title="">41</a>]</cite>), the 12 m Wide-field Spectroscopic Telescope444<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://www.wstelescope.com/" title="">https://www.wstelescope.com/</a> (WST; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib42" title="">42</a>]</cite>), and the <math alttext="\sim" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mo id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><csymbol cd="latexml" id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">∼</annotation></semantics></math>12 m Extremely Large Spectroscopic Survey Telescope (ESST; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib43" title="">43</a>]</cite>). </div> <div class="ltx_para" id="S1.p5"> The MUltiplexed Survey Telescope555<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://must.astro.tsinghua.edu.cn/en" title="">https://must.astro.tsinghua.edu.cn/en</a> (MUST) is a 6.5-meter telescope <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib44" title="">44</a>]</cite> under active development. MUST will be located at the 4358 m Peak A of Saishiteng Mountain in Qinghai, China. Equipped with over 20,000 fibers over a <math alttext="\sim 5~{}\rm{deg}^{2}" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mrow id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml"></mi><mo id="S1.p5.1.m1.1.1.1" xref="S1.p5.1.m1.1.1.1.cmml">∼</mo><mrow id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml"><mn id="S1.p5.1.m1.1.1.3.2" xref="S1.p5.1.m1.1.1.3.2.cmml">5</mn><mo id="S1.p5.1.m1.1.1.3.1" lspace="0.330em" xref="S1.p5.1.m1.1.1.3.1.cmml">⁢</mo><msup id="S1.p5.1.m1.1.1.3.3" xref="S1.p5.1.m1.1.1.3.3.cmml"><mi id="S1.p5.1.m1.1.1.3.3.2" xref="S1.p5.1.m1.1.1.3.3.2.cmml">deg</mi><mn id="S1.p5.1.m1.1.1.3.3.3" xref="S1.p5.1.m1.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="latexml" id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2">absent</csymbol><apply id="S1.p5.1.m1.1.1.3.cmml" xref="S1.p5.1.m1.1.1.3"><times id="S1.p5.1.m1.1.1.3.1.cmml" xref="S1.p5.1.m1.1.1.3.1"></times><cn id="S1.p5.1.m1.1.1.3.2.cmml" type="integer" xref="S1.p5.1.m1.1.1.3.2">5</cn><apply id="S1.p5.1.m1.1.1.3.3.cmml" xref="S1.p5.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.p5.1.m1.1.1.3.3.1.cmml" xref="S1.p5.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.p5.1.m1.1.1.3.3.2.cmml" xref="S1.p5.1.m1.1.1.3.3.2">deg</ci><cn id="S1.p5.1.m1.1.1.3.3.3.cmml" type="integer" xref="S1.p5.1.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">\sim 5~{}\rm{deg}^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">∼ 5 roman_deg start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> field of view (FoV), it features three-channel spectrographs covering wavelengths from 370-960 nm, with spectral resolution between <math alttext="R=2,000" class="ltx_Math" display="inline" id="S1.p5.2.m2.2"><semantics id="S1.p5.2.m2.2a"><mrow id="S1.p5.2.m2.2.3" xref="S1.p5.2.m2.2.3.cmml"><mi id="S1.p5.2.m2.2.3.2" xref="S1.p5.2.m2.2.3.2.cmml">R</mi><mo id="S1.p5.2.m2.2.3.1" xref="S1.p5.2.m2.2.3.1.cmml">=</mo><mrow id="S1.p5.2.m2.2.3.3.2" xref="S1.p5.2.m2.2.3.3.1.cmml"><mn id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml">2</mn><mo id="S1.p5.2.m2.2.3.3.2.1" xref="S1.p5.2.m2.2.3.3.1.cmml">,</mo><mn id="S1.p5.2.m2.2.2" xref="S1.p5.2.m2.2.2.cmml">000</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.2b"><apply id="S1.p5.2.m2.2.3.cmml" xref="S1.p5.2.m2.2.3"><eq id="S1.p5.2.m2.2.3.1.cmml" xref="S1.p5.2.m2.2.3.1"></eq><ci id="S1.p5.2.m2.2.3.2.cmml" xref="S1.p5.2.m2.2.3.2">𝑅</ci><list id="S1.p5.2.m2.2.3.3.1.cmml" xref="S1.p5.2.m2.2.3.3.2"><cn id="S1.p5.2.m2.1.1.cmml" type="integer" xref="S1.p5.2.m2.1.1">2</cn><cn id="S1.p5.2.m2.2.2.cmml" type="integer" xref="S1.p5.2.m2.2.2">000</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.2c">R=2,000</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.2d">italic_R = 2 , 000</annotation></semantics></math> and <math alttext="4,500" class="ltx_Math" display="inline" id="S1.p5.3.m3.2"><semantics id="S1.p5.3.m3.2a"><mrow id="S1.p5.3.m3.2.3.2" xref="S1.p5.3.m3.2.3.1.cmml"><mn id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml">4</mn><mo id="S1.p5.3.m3.2.3.2.1" xref="S1.p5.3.m3.2.3.1.cmml">,</mo><mn id="S1.p5.3.m3.2.2" xref="S1.p5.3.m3.2.2.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.2b"><list id="S1.p5.3.m3.2.3.1.cmml" xref="S1.p5.3.m3.2.3.2"><cn id="S1.p5.3.m3.1.1.cmml" type="integer" xref="S1.p5.3.m3.1.1">4</cn><cn id="S1.p5.3.m3.2.2.cmml" type="integer" xref="S1.p5.3.m3.2.2">500</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.2c">4,500</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.2d">4 , 500</annotation></semantics></math>. MUST is designed to conduct an ambitious Stage-V cosmological spectroscopic survey, aiming to precisely measure key cosmological parameters and improve our understanding of dark energy and cosmic evolution. With the first light scheduled for 2031, MUST expect to conduct the first Stage-V spectroscopic survey, targeting Lyman Break Galaxy (LBG) and Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mi id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><ci id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">italic_α</annotation></semantics></math> emitter (LAE) at high redshift across <math alttext="\sim 13,000" class="ltx_Math" display="inline" id="S1.p5.5.m5.2"><semantics id="S1.p5.5.m5.2a"><mrow id="S1.p5.5.m5.2.3" xref="S1.p5.5.m5.2.3.cmml"><mi id="S1.p5.5.m5.2.3.2" xref="S1.p5.5.m5.2.3.2.cmml"></mi><mo id="S1.p5.5.m5.2.3.1" xref="S1.p5.5.m5.2.3.1.cmml">∼</mo><mrow id="S1.p5.5.m5.2.3.3.2" xref="S1.p5.5.m5.2.3.3.1.cmml"><mn id="S1.p5.5.m5.1.1" xref="S1.p5.5.m5.1.1.cmml">13</mn><mo id="S1.p5.5.m5.2.3.3.2.1" xref="S1.p5.5.m5.2.3.3.1.cmml">,</mo><mn id="S1.p5.5.m5.2.2" xref="S1.p5.5.m5.2.2.cmml">000</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.2b"><apply id="S1.p5.5.m5.2.3.cmml" xref="S1.p5.5.m5.2.3"><csymbol cd="latexml" id="S1.p5.5.m5.2.3.1.cmml" xref="S1.p5.5.m5.2.3.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p5.5.m5.2.3.2.cmml" xref="S1.p5.5.m5.2.3.2">absent</csymbol><list id="S1.p5.5.m5.2.3.3.1.cmml" xref="S1.p5.5.m5.2.3.3.2"><cn id="S1.p5.5.m5.1.1.cmml" type="integer" xref="S1.p5.5.m5.1.1">13</cn><cn id="S1.p5.5.m5.2.2.cmml" type="integer" xref="S1.p5.5.m5.2.2">000</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m5.2c">\sim 13,000</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m5.2d">∼ 13 , 000</annotation></semantics></math> deg2 of the northern sky. Clustering analysis of these tracers will provide sub-percent precision measurements on BAO parameters – <math alttext="D_{A}(z)/r_{d}" class="ltx_Math" display="inline" id="S1.p5.7.m7.1"><semantics id="S1.p5.7.m7.1a"><mrow id="S1.p5.7.m7.1.2" xref="S1.p5.7.m7.1.2.cmml"><mrow id="S1.p5.7.m7.1.2.2" xref="S1.p5.7.m7.1.2.2.cmml"><msub id="S1.p5.7.m7.1.2.2.2" xref="S1.p5.7.m7.1.2.2.2.cmml"><mi id="S1.p5.7.m7.1.2.2.2.2" xref="S1.p5.7.m7.1.2.2.2.2.cmml">D</mi><mi id="S1.p5.7.m7.1.2.2.2.3" xref="S1.p5.7.m7.1.2.2.2.3.cmml">A</mi></msub><mo id="S1.p5.7.m7.1.2.2.1" xref="S1.p5.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S1.p5.7.m7.1.2.2.3.2" xref="S1.p5.7.m7.1.2.2.cmml"><mo id="S1.p5.7.m7.1.2.2.3.2.1" stretchy="false" xref="S1.p5.7.m7.1.2.2.cmml">(</mo><mi id="S1.p5.7.m7.1.1" xref="S1.p5.7.m7.1.1.cmml">z</mi><mo id="S1.p5.7.m7.1.2.2.3.2.2" stretchy="false" xref="S1.p5.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S1.p5.7.m7.1.2.1" xref="S1.p5.7.m7.1.2.1.cmml">/</mo><msub id="S1.p5.7.m7.1.2.3" xref="S1.p5.7.m7.1.2.3.cmml"><mi id="S1.p5.7.m7.1.2.3.2" xref="S1.p5.7.m7.1.2.3.2.cmml">r</mi><mi id="S1.p5.7.m7.1.2.3.3" xref="S1.p5.7.m7.1.2.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.7.m7.1b"><apply id="S1.p5.7.m7.1.2.cmml" xref="S1.p5.7.m7.1.2"><divide id="S1.p5.7.m7.1.2.1.cmml" xref="S1.p5.7.m7.1.2.1"></divide><apply id="S1.p5.7.m7.1.2.2.cmml" xref="S1.p5.7.m7.1.2.2"><times id="S1.p5.7.m7.1.2.2.1.cmml" xref="S1.p5.7.m7.1.2.2.1"></times><apply id="S1.p5.7.m7.1.2.2.2.cmml" xref="S1.p5.7.m7.1.2.2.2"><csymbol cd="ambiguous" id="S1.p5.7.m7.1.2.2.2.1.cmml" xref="S1.p5.7.m7.1.2.2.2">subscript</csymbol><ci id="S1.p5.7.m7.1.2.2.2.2.cmml" xref="S1.p5.7.m7.1.2.2.2.2">𝐷</ci><ci id="S1.p5.7.m7.1.2.2.2.3.cmml" xref="S1.p5.7.m7.1.2.2.2.3">𝐴</ci></apply><ci id="S1.p5.7.m7.1.1.cmml" xref="S1.p5.7.m7.1.1">𝑧</ci></apply><apply id="S1.p5.7.m7.1.2.3.cmml" xref="S1.p5.7.m7.1.2.3"><csymbol cd="ambiguous" id="S1.p5.7.m7.1.2.3.1.cmml" xref="S1.p5.7.m7.1.2.3">subscript</csymbol><ci id="S1.p5.7.m7.1.2.3.2.cmml" xref="S1.p5.7.m7.1.2.3.2">𝑟</ci><ci id="S1.p5.7.m7.1.2.3.3.cmml" xref="S1.p5.7.m7.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.7.m7.1c">D_{A}(z)/r_{d}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.7.m7.1d">italic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_z ) / italic_r start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H(z)r_{d}" class="ltx_Math" display="inline" id="S1.p5.8.m8.1"><semantics id="S1.p5.8.m8.1a"><mrow id="S1.p5.8.m8.1.2" xref="S1.p5.8.m8.1.2.cmml"><mi id="S1.p5.8.m8.1.2.2" xref="S1.p5.8.m8.1.2.2.cmml">H</mi><mo id="S1.p5.8.m8.1.2.1" xref="S1.p5.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.p5.8.m8.1.2.3.2" xref="S1.p5.8.m8.1.2.cmml"><mo id="S1.p5.8.m8.1.2.3.2.1" stretchy="false" xref="S1.p5.8.m8.1.2.cmml">(</mo><mi id="S1.p5.8.m8.1.1" xref="S1.p5.8.m8.1.1.cmml">z</mi><mo id="S1.p5.8.m8.1.2.3.2.2" stretchy="false" xref="S1.p5.8.m8.1.2.cmml">)</mo></mrow><mo id="S1.p5.8.m8.1.2.1a" xref="S1.p5.8.m8.1.2.1.cmml">⁢</mo><msub id="S1.p5.8.m8.1.2.4" xref="S1.p5.8.m8.1.2.4.cmml"><mi id="S1.p5.8.m8.1.2.4.2" xref="S1.p5.8.m8.1.2.4.2.cmml">r</mi><mi id="S1.p5.8.m8.1.2.4.3" xref="S1.p5.8.m8.1.2.4.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.8.m8.1b"><apply id="S1.p5.8.m8.1.2.cmml" xref="S1.p5.8.m8.1.2"><times id="S1.p5.8.m8.1.2.1.cmml" xref="S1.p5.8.m8.1.2.1"></times><ci id="S1.p5.8.m8.1.2.2.cmml" xref="S1.p5.8.m8.1.2.2">𝐻</ci><ci id="S1.p5.8.m8.1.1.cmml" xref="S1.p5.8.m8.1.1">𝑧</ci><apply id="S1.p5.8.m8.1.2.4.cmml" xref="S1.p5.8.m8.1.2.4"><csymbol cd="ambiguous" id="S1.p5.8.m8.1.2.4.1.cmml" xref="S1.p5.8.m8.1.2.4">subscript</csymbol><ci id="S1.p5.8.m8.1.2.4.2.cmml" xref="S1.p5.8.m8.1.2.4.2">𝑟</ci><ci id="S1.p5.8.m8.1.2.4.3.cmml" xref="S1.p5.8.m8.1.2.4.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.8.m8.1c">H(z)r_{d}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.8.m8.1d">italic_H ( italic_z ) italic_r start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> – and the linear growth rate of structure, <math alttext="f\sigma_{8}" class="ltx_Math" display="inline" id="S1.p5.9.m9.1"><semantics id="S1.p5.9.m9.1a"><mrow id="S1.p5.9.m9.1.1" xref="S1.p5.9.m9.1.1.cmml"><mi id="S1.p5.9.m9.1.1.2" xref="S1.p5.9.m9.1.1.2.cmml">f</mi><mo id="S1.p5.9.m9.1.1.1" xref="S1.p5.9.m9.1.1.1.cmml">⁢</mo><msub id="S1.p5.9.m9.1.1.3" xref="S1.p5.9.m9.1.1.3.cmml"><mi id="S1.p5.9.m9.1.1.3.2" xref="S1.p5.9.m9.1.1.3.2.cmml">σ</mi><mn id="S1.p5.9.m9.1.1.3.3" xref="S1.p5.9.m9.1.1.3.3.cmml">8</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.9.m9.1b"><apply id="S1.p5.9.m9.1.1.cmml" xref="S1.p5.9.m9.1.1"><times id="S1.p5.9.m9.1.1.1.cmml" xref="S1.p5.9.m9.1.1.1"></times><ci id="S1.p5.9.m9.1.1.2.cmml" xref="S1.p5.9.m9.1.1.2">𝑓</ci><apply id="S1.p5.9.m9.1.1.3.cmml" xref="S1.p5.9.m9.1.1.3"><csymbol cd="ambiguous" id="S1.p5.9.m9.1.1.3.1.cmml" xref="S1.p5.9.m9.1.1.3">subscript</csymbol><ci id="S1.p5.9.m9.1.1.3.2.cmml" xref="S1.p5.9.m9.1.1.3.2">𝜎</ci><cn id="S1.p5.9.m9.1.1.3.3.cmml" type="integer" xref="S1.p5.9.m9.1.1.3.3">8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.9.m9.1c">f\sigma_{8}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.9.m9.1d">italic_f italic_σ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT</annotation></semantics></math>, at redshift <math alttext="2<z<5" class="ltx_Math" display="inline" id="S1.p5.10.m10.1"><semantics id="S1.p5.10.m10.1a"><mrow id="S1.p5.10.m10.1.1" xref="S1.p5.10.m10.1.1.cmml"><mn id="S1.p5.10.m10.1.1.2" xref="S1.p5.10.m10.1.1.2.cmml">2</mn><mo id="S1.p5.10.m10.1.1.3" xref="S1.p5.10.m10.1.1.3.cmml"><</mo><mi id="S1.p5.10.m10.1.1.4" xref="S1.p5.10.m10.1.1.4.cmml">z</mi><mo id="S1.p5.10.m10.1.1.5" xref="S1.p5.10.m10.1.1.5.cmml"><</mo><mn id="S1.p5.10.m10.1.1.6" xref="S1.p5.10.m10.1.1.6.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.10.m10.1b"><apply id="S1.p5.10.m10.1.1.cmml" xref="S1.p5.10.m10.1.1"><and id="S1.p5.10.m10.1.1a.cmml" xref="S1.p5.10.m10.1.1"></and><apply id="S1.p5.10.m10.1.1b.cmml" xref="S1.p5.10.m10.1.1"><lt id="S1.p5.10.m10.1.1.3.cmml" xref="S1.p5.10.m10.1.1.3"></lt><cn id="S1.p5.10.m10.1.1.2.cmml" type="integer" xref="S1.p5.10.m10.1.1.2">2</cn><ci id="S1.p5.10.m10.1.1.4.cmml" xref="S1.p5.10.m10.1.1.4">𝑧</ci></apply><apply id="S1.p5.10.m10.1.1c.cmml" xref="S1.p5.10.m10.1.1"><lt id="S1.p5.10.m10.1.1.5.cmml" xref="S1.p5.10.m10.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S1.p5.10.m10.1.1.4.cmml" id="S1.p5.10.m10.1.1d.cmml" xref="S1.p5.10.m10.1.1"></share><cn id="S1.p5.10.m10.1.1.6.cmml" type="integer" xref="S1.p5.10.m10.1.1.6">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.10.m10.1c">2<z<5</annotation><annotation encoding="application/x-llamapun" id="S1.p5.10.m10.1d">2 < italic_z < 5</annotation></semantics></math>, a region not yet covered by current spectroscopic galaxy surveys. Additionally, MUST will provide stringent constraints on primordial non-Gaussianities (PNG, local type) through the parameter <math alttext="f_{\mathrm{NL}}^{\mathrm{local}}" class="ltx_Math" display="inline" id="S1.p5.11.m11.1"><semantics id="S1.p5.11.m11.1a"><msubsup id="S1.p5.11.m11.1.1" xref="S1.p5.11.m11.1.1.cmml"><mi id="S1.p5.11.m11.1.1.2.2" xref="S1.p5.11.m11.1.1.2.2.cmml">f</mi><mi id="S1.p5.11.m11.1.1.2.3" xref="S1.p5.11.m11.1.1.2.3.cmml">NL</mi><mi id="S1.p5.11.m11.1.1.3" xref="S1.p5.11.m11.1.1.3.cmml">local</mi></msubsup><annotation-xml encoding="MathML-Content" id="S1.p5.11.m11.1b"><apply id="S1.p5.11.m11.1.1.cmml" xref="S1.p5.11.m11.1.1"><csymbol cd="ambiguous" id="S1.p5.11.m11.1.1.1.cmml" xref="S1.p5.11.m11.1.1">superscript</csymbol><apply id="S1.p5.11.m11.1.1.2.cmml" xref="S1.p5.11.m11.1.1"><csymbol cd="ambiguous" id="S1.p5.11.m11.1.1.2.1.cmml" xref="S1.p5.11.m11.1.1">subscript</csymbol><ci id="S1.p5.11.m11.1.1.2.2.cmml" xref="S1.p5.11.m11.1.1.2.2">𝑓</ci><ci id="S1.p5.11.m11.1.1.2.3.cmml" xref="S1.p5.11.m11.1.1.2.3">NL</ci></apply><ci id="S1.p5.11.m11.1.1.3.cmml" xref="S1.p5.11.m11.1.1.3">local</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.11.m11.1c">f_{\mathrm{NL}}^{\mathrm{local}}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.11.m11.1d">italic_f start_POSTSUBSCRIPT roman_NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_local end_POSTSUPERSCRIPT</annotation></semantics></math> with a precision of <math alttext="\sigma(f_{\mathrm{NL}}^{\mathrm{local}})\sim 1" class="ltx_Math" display="inline" id="S1.p5.12.m12.1"><semantics id="S1.p5.12.m12.1a"><mrow id="S1.p5.12.m12.1.1" xref="S1.p5.12.m12.1.1.cmml"><mrow id="S1.p5.12.m12.1.1.1" xref="S1.p5.12.m12.1.1.1.cmml"><mi id="S1.p5.12.m12.1.1.1.3" xref="S1.p5.12.m12.1.1.1.3.cmml">σ</mi><mo id="S1.p5.12.m12.1.1.1.2" xref="S1.p5.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S1.p5.12.m12.1.1.1.1.1" xref="S1.p5.12.m12.1.1.1.1.1.1.cmml"><mo id="S1.p5.12.m12.1.1.1.1.1.2" stretchy="false" xref="S1.p5.12.m12.1.1.1.1.1.1.cmml">(</mo><msubsup id="S1.p5.12.m12.1.1.1.1.1.1" xref="S1.p5.12.m12.1.1.1.1.1.1.cmml"><mi id="S1.p5.12.m12.1.1.1.1.1.1.2.2" xref="S1.p5.12.m12.1.1.1.1.1.1.2.2.cmml">f</mi><mi id="S1.p5.12.m12.1.1.1.1.1.1.2.3" xref="S1.p5.12.m12.1.1.1.1.1.1.2.3.cmml">NL</mi><mi id="S1.p5.12.m12.1.1.1.1.1.1.3" xref="S1.p5.12.m12.1.1.1.1.1.1.3.cmml">local</mi></msubsup><mo id="S1.p5.12.m12.1.1.1.1.1.3" stretchy="false" xref="S1.p5.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p5.12.m12.1.1.2" xref="S1.p5.12.m12.1.1.2.cmml">∼</mo><mn id="S1.p5.12.m12.1.1.3" xref="S1.p5.12.m12.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.12.m12.1b"><apply id="S1.p5.12.m12.1.1.cmml" xref="S1.p5.12.m12.1.1"><csymbol cd="latexml" id="S1.p5.12.m12.1.1.2.cmml" xref="S1.p5.12.m12.1.1.2">similar-to</csymbol><apply id="S1.p5.12.m12.1.1.1.cmml" xref="S1.p5.12.m12.1.1.1"><times id="S1.p5.12.m12.1.1.1.2.cmml" xref="S1.p5.12.m12.1.1.1.2"></times><ci id="S1.p5.12.m12.1.1.1.3.cmml" xref="S1.p5.12.m12.1.1.1.3">𝜎</ci><apply id="S1.p5.12.m12.1.1.1.1.1.1.cmml" xref="S1.p5.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p5.12.m12.1.1.1.1.1.1.1.cmml" xref="S1.p5.12.m12.1.1.1.1.1">superscript</csymbol><apply id="S1.p5.12.m12.1.1.1.1.1.1.2.cmml" xref="S1.p5.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p5.12.m12.1.1.1.1.1.1.2.1.cmml" xref="S1.p5.12.m12.1.1.1.1.1">subscript</csymbol><ci id="S1.p5.12.m12.1.1.1.1.1.1.2.2.cmml" xref="S1.p5.12.m12.1.1.1.1.1.1.2.2">𝑓</ci><ci id="S1.p5.12.m12.1.1.1.1.1.1.2.3.cmml" xref="S1.p5.12.m12.1.1.1.1.1.1.2.3">NL</ci></apply><ci id="S1.p5.12.m12.1.1.1.1.1.1.3.cmml" xref="S1.p5.12.m12.1.1.1.1.1.1.3">local</ci></apply></apply><cn id="S1.p5.12.m12.1.1.3.cmml" type="integer" xref="S1.p5.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.12.m12.1c">\sigma(f_{\mathrm{NL}}^{\mathrm{local}})\sim 1</annotation><annotation encoding="application/x-llamapun" id="S1.p5.12.m12.1d">italic_σ ( italic_f start_POSTSUBSCRIPT roman_NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_local end_POSTSUPERSCRIPT ) ∼ 1</annotation></semantics></math>. This will enable stringent testing of a wide range of inflationary models. MUST is expected to provide sufficient precision (<math alttext="\sim 0.03\,{\rm eV}" class="ltx_Math" display="inline" id="S1.p5.13.m13.1"><semantics id="S1.p5.13.m13.1a"><mrow id="S1.p5.13.m13.1.1" xref="S1.p5.13.m13.1.1.cmml"><mi id="S1.p5.13.m13.1.1.2" xref="S1.p5.13.m13.1.1.2.cmml"></mi><mo id="S1.p5.13.m13.1.1.1" xref="S1.p5.13.m13.1.1.1.cmml">∼</mo><mrow id="S1.p5.13.m13.1.1.3" xref="S1.p5.13.m13.1.1.3.cmml"><mn id="S1.p5.13.m13.1.1.3.2" xref="S1.p5.13.m13.1.1.3.2.cmml">0.03</mn><mo id="S1.p5.13.m13.1.1.3.1" lspace="0.170em" xref="S1.p5.13.m13.1.1.3.1.cmml">⁢</mo><mi id="S1.p5.13.m13.1.1.3.3" xref="S1.p5.13.m13.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.13.m13.1b"><apply id="S1.p5.13.m13.1.1.cmml" xref="S1.p5.13.m13.1.1"><csymbol cd="latexml" id="S1.p5.13.m13.1.1.1.cmml" xref="S1.p5.13.m13.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p5.13.m13.1.1.2.cmml" xref="S1.p5.13.m13.1.1.2">absent</csymbol><apply id="S1.p5.13.m13.1.1.3.cmml" xref="S1.p5.13.m13.1.1.3"><times id="S1.p5.13.m13.1.1.3.1.cmml" xref="S1.p5.13.m13.1.1.3.1"></times><cn id="S1.p5.13.m13.1.1.3.2.cmml" type="float" xref="S1.p5.13.m13.1.1.3.2">0.03</cn><ci id="S1.p5.13.m13.1.1.3.3.cmml" xref="S1.p5.13.m13.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.13.m13.1c">\sim 0.03\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.13.m13.1d">∼ 0.03 roman_eV</annotation></semantics></math> when combined with CMB data) on the sum of neutrino masses to constrain a nonzero neutrino mass with <math alttext="\sim 2\sigma" class="ltx_Math" display="inline" id="S1.p5.14.m14.1"><semantics id="S1.p5.14.m14.1a"><mrow id="S1.p5.14.m14.1.1" xref="S1.p5.14.m14.1.1.cmml"><mi id="S1.p5.14.m14.1.1.2" xref="S1.p5.14.m14.1.1.2.cmml"></mi><mo id="S1.p5.14.m14.1.1.1" xref="S1.p5.14.m14.1.1.1.cmml">∼</mo><mrow id="S1.p5.14.m14.1.1.3" xref="S1.p5.14.m14.1.1.3.cmml"><mn id="S1.p5.14.m14.1.1.3.2" xref="S1.p5.14.m14.1.1.3.2.cmml">2</mn><mo id="S1.p5.14.m14.1.1.3.1" xref="S1.p5.14.m14.1.1.3.1.cmml">⁢</mo><mi id="S1.p5.14.m14.1.1.3.3" xref="S1.p5.14.m14.1.1.3.3.cmml">σ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.14.m14.1b"><apply id="S1.p5.14.m14.1.1.cmml" xref="S1.p5.14.m14.1.1"><csymbol cd="latexml" id="S1.p5.14.m14.1.1.1.cmml" xref="S1.p5.14.m14.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p5.14.m14.1.1.2.cmml" xref="S1.p5.14.m14.1.1.2">absent</csymbol><apply id="S1.p5.14.m14.1.1.3.cmml" xref="S1.p5.14.m14.1.1.3"><times id="S1.p5.14.m14.1.1.3.1.cmml" xref="S1.p5.14.m14.1.1.3.1"></times><cn id="S1.p5.14.m14.1.1.3.2.cmml" type="integer" xref="S1.p5.14.m14.1.1.3.2">2</cn><ci id="S1.p5.14.m14.1.1.3.3.cmml" xref="S1.p5.14.m14.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.14.m14.1c">\sim 2\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p5.14.m14.1d">∼ 2 italic_σ</annotation></semantics></math> significance, assuming normal hierarchy. The inverted mass hierarchy can be tested with a significance <math alttext="\sim 1.3\sigma" class="ltx_Math" display="inline" id="S1.p5.15.m15.1"><semantics id="S1.p5.15.m15.1a"><mrow id="S1.p5.15.m15.1.1" xref="S1.p5.15.m15.1.1.cmml"><mi id="S1.p5.15.m15.1.1.2" xref="S1.p5.15.m15.1.1.2.cmml"></mi><mo id="S1.p5.15.m15.1.1.1" xref="S1.p5.15.m15.1.1.1.cmml">∼</mo><mrow id="S1.p5.15.m15.1.1.3" xref="S1.p5.15.m15.1.1.3.cmml"><mn id="S1.p5.15.m15.1.1.3.2" xref="S1.p5.15.m15.1.1.3.2.cmml">1.3</mn><mo id="S1.p5.15.m15.1.1.3.1" xref="S1.p5.15.m15.1.1.3.1.cmml">⁢</mo><mi id="S1.p5.15.m15.1.1.3.3" xref="S1.p5.15.m15.1.1.3.3.cmml">σ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.15.m15.1b"><apply id="S1.p5.15.m15.1.1.cmml" xref="S1.p5.15.m15.1.1"><csymbol cd="latexml" id="S1.p5.15.m15.1.1.1.cmml" xref="S1.p5.15.m15.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S1.p5.15.m15.1.1.2.cmml" xref="S1.p5.15.m15.1.1.2">absent</csymbol><apply id="S1.p5.15.m15.1.1.3.cmml" xref="S1.p5.15.m15.1.1.3"><times id="S1.p5.15.m15.1.1.3.1.cmml" xref="S1.p5.15.m15.1.1.3.1"></times><cn id="S1.p5.15.m15.1.1.3.2.cmml" type="float" xref="S1.p5.15.m15.1.1.3.2">1.3</cn><ci id="S1.p5.15.m15.1.1.3.3.cmml" xref="S1.p5.15.m15.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.15.m15.1c">\sim 1.3\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p5.15.m15.1d">∼ 1.3 italic_σ</annotation></semantics></math>. Finally, from power spectrum measurements of the Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p5.16.m16.1"><semantics id="S1.p5.16.m16.1a"><mi id="S1.p5.16.m16.1.1" xref="S1.p5.16.m16.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p5.16.m16.1b"><ci id="S1.p5.16.m16.1.1.cmml" xref="S1.p5.16.m16.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.16.m16.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p5.16.m16.1d">italic_α</annotation></semantics></math> forest, MUST will yield the most precise constraint on warm dark matter mass to date <math alttext="m_{X}>10.5" class="ltx_Math" display="inline" id="S1.p5.17.m17.1"><semantics id="S1.p5.17.m17.1a"><mrow id="S1.p5.17.m17.1.1" xref="S1.p5.17.m17.1.1.cmml"><msub id="S1.p5.17.m17.1.1.2" xref="S1.p5.17.m17.1.1.2.cmml"><mi id="S1.p5.17.m17.1.1.2.2" xref="S1.p5.17.m17.1.1.2.2.cmml">m</mi><mi id="S1.p5.17.m17.1.1.2.3" xref="S1.p5.17.m17.1.1.2.3.cmml">X</mi></msub><mo id="S1.p5.17.m17.1.1.1" xref="S1.p5.17.m17.1.1.1.cmml">></mo><mn id="S1.p5.17.m17.1.1.3" xref="S1.p5.17.m17.1.1.3.cmml">10.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.17.m17.1b"><apply id="S1.p5.17.m17.1.1.cmml" xref="S1.p5.17.m17.1.1"><gt id="S1.p5.17.m17.1.1.1.cmml" xref="S1.p5.17.m17.1.1.1"></gt><apply id="S1.p5.17.m17.1.1.2.cmml" xref="S1.p5.17.m17.1.1.2"><csymbol cd="ambiguous" id="S1.p5.17.m17.1.1.2.1.cmml" xref="S1.p5.17.m17.1.1.2">subscript</csymbol><ci id="S1.p5.17.m17.1.1.2.2.cmml" xref="S1.p5.17.m17.1.1.2.2">𝑚</ci><ci id="S1.p5.17.m17.1.1.2.3.cmml" xref="S1.p5.17.m17.1.1.2.3">𝑋</ci></apply><cn id="S1.p5.17.m17.1.1.3.cmml" type="float" xref="S1.p5.17.m17.1.1.3">10.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.17.m17.1c">m_{X}>10.5</annotation><annotation encoding="application/x-llamapun" id="S1.p5.17.m17.1d">italic_m start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT > 10.5</annotation></semantics></math> keV at 95% confidence level (assuming 14,000 deg2 and <math alttext="k_{\rm max}=0.67" class="ltx_Math" display="inline" id="S1.p5.19.m19.1"><semantics id="S1.p5.19.m19.1a"><mrow id="S1.p5.19.m19.1.1" xref="S1.p5.19.m19.1.1.cmml"><msub id="S1.p5.19.m19.1.1.2" xref="S1.p5.19.m19.1.1.2.cmml"><mi id="S1.p5.19.m19.1.1.2.2" xref="S1.p5.19.m19.1.1.2.2.cmml">k</mi><mi id="S1.p5.19.m19.1.1.2.3" xref="S1.p5.19.m19.1.1.2.3.cmml">max</mi></msub><mo id="S1.p5.19.m19.1.1.1" xref="S1.p5.19.m19.1.1.1.cmml">=</mo><mn id="S1.p5.19.m19.1.1.3" xref="S1.p5.19.m19.1.1.3.cmml">0.67</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.19.m19.1b"><apply id="S1.p5.19.m19.1.1.cmml" xref="S1.p5.19.m19.1.1"><eq id="S1.p5.19.m19.1.1.1.cmml" xref="S1.p5.19.m19.1.1.1"></eq><apply id="S1.p5.19.m19.1.1.2.cmml" xref="S1.p5.19.m19.1.1.2"><csymbol cd="ambiguous" id="S1.p5.19.m19.1.1.2.1.cmml" xref="S1.p5.19.m19.1.1.2">subscript</csymbol><ci id="S1.p5.19.m19.1.1.2.2.cmml" xref="S1.p5.19.m19.1.1.2.2">𝑘</ci><ci id="S1.p5.19.m19.1.1.2.3.cmml" xref="S1.p5.19.m19.1.1.2.3">max</ci></apply><cn id="S1.p5.19.m19.1.1.3.cmml" type="float" xref="S1.p5.19.m19.1.1.3">0.67</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.19.m19.1c">k_{\rm max}=0.67</annotation><annotation encoding="application/x-llamapun" id="S1.p5.19.m19.1d">italic_k start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT = 0.67</annotation></semantics></math> <math alttext="h" class="ltx_Math" display="inline" id="S1.p5.20.m20.1"><semantics id="S1.p5.20.m20.1a"><mi id="S1.p5.20.m20.1.1" xref="S1.p5.20.m20.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S1.p5.20.m20.1b"><ci id="S1.p5.20.m20.1.1.cmml" xref="S1.p5.20.m20.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.20.m20.1c">h</annotation><annotation encoding="application/x-llamapun" id="S1.p5.20.m20.1d">italic_h</annotation></semantics></math> Mpc-1). </div> <div class="ltx_para" id="S1.p6"> Besides the unique potential of the MUST project, synergies with other cosmological surveys, i.e., future imaging surveys (e.g. CSST, Euclid, LSST), CMB experiments (e.g., Simon Observatory <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib45" title="">45</a>]</cite>, CMB-S4 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib46" title="">46</a>]</cite>, LiteBird <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib47" title="">47</a>]</cite>) or radio surveys (e.g., SKAO <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib48" title="">48</a>]</cite>) will enhance the constraining power of MUST and will result in a better understanding of our Universe. As a dedicated spectroscopic survey facility, MUST can also carry out spectroscopic surveys supporting a wide range of scientific topics outside of LSS cosmology, such as the study of galaxy evolution, super-massive black hole (SMBH), the structure of the Milky Way, and time-domain astrophysics. </div> <div class="ltx_para" id="S1.p7"> This paper describes the MUST instrument and the scientific objectives of the cosmological survey that will be conducted over 5 years of observation. Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2" title="2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2</a> provides an overview of the MUST project, including the current status of the whole project, the design of the telescope (Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS1" title="2.1 A 6.5-m Telescope for Spectroscopic Surveys ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.1</a>), the focal plane system & the spectrograph (Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS2" title="2.2 Multiplexed Focal Plane & Instruments ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.2</a>), the site & observing condition (Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS3" title="2.3 Site and Observing Conditions ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.3</a>), and the overall scientific capabilities (Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.SS4" title="2.4 Survey Capability and Overall Scientific Potential ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2.4</a>). Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3" title="3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3</a> describes the key scientific motivations of MUST for the Stage-V cosmological surveys. The primary scientific goals are covered in detail while briefly summarizing the potential for new probes and the potential synergies with other cosmological surveys. Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4" title="4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4</a> presents the current target selection strategy and provides the redshift distribution and target density estimations for the cosmological forecast. We will also introduce the conceptual survey design for MUST. Finally, Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5" title="5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5</a> describes the method to forecast the cosmological potential of MUST theoretically and summarizes the forecasted results on dark energy, structure growth, primordial non-Gaussianity, neutrino mass, and warm dark matter constraints. Discussions and main conclusions of this project and future directions are described in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S6" title="6 Conclusions ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">6</a>. </div> <div class="ltx_para" id="S1.p8"> Throughout this work, we adopt as fiducial baseline <math alttext="\Lambda" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><mi id="S1.p8.1.m1.1.1" mathvariant="normal" xref="S1.p8.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><ci id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">roman_Λ</annotation></semantics></math>CDM cosmology with parameters <math alttext="H_{0}=67.6" class="ltx_Math" display="inline" id="S1.p8.2.m2.1"><semantics id="S1.p8.2.m2.1a"><mrow id="S1.p8.2.m2.1.1" xref="S1.p8.2.m2.1.1.cmml"><msub id="S1.p8.2.m2.1.1.2" xref="S1.p8.2.m2.1.1.2.cmml"><mi id="S1.p8.2.m2.1.1.2.2" xref="S1.p8.2.m2.1.1.2.2.cmml">H</mi><mn id="S1.p8.2.m2.1.1.2.3" xref="S1.p8.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S1.p8.2.m2.1.1.1" xref="S1.p8.2.m2.1.1.1.cmml">=</mo><mn id="S1.p8.2.m2.1.1.3" xref="S1.p8.2.m2.1.1.3.cmml">67.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.2.m2.1b"><apply id="S1.p8.2.m2.1.1.cmml" xref="S1.p8.2.m2.1.1"><eq id="S1.p8.2.m2.1.1.1.cmml" xref="S1.p8.2.m2.1.1.1"></eq><apply id="S1.p8.2.m2.1.1.2.cmml" xref="S1.p8.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p8.2.m2.1.1.2.1.cmml" xref="S1.p8.2.m2.1.1.2">subscript</csymbol><ci id="S1.p8.2.m2.1.1.2.2.cmml" xref="S1.p8.2.m2.1.1.2.2">𝐻</ci><cn id="S1.p8.2.m2.1.1.2.3.cmml" type="integer" xref="S1.p8.2.m2.1.1.2.3">0</cn></apply><cn id="S1.p8.2.m2.1.1.3.cmml" type="float" xref="S1.p8.2.m2.1.1.3">67.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.2.m2.1c">H_{0}=67.6</annotation><annotation encoding="application/x-llamapun" id="S1.p8.2.m2.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 67.6</annotation></semantics></math> km s-1 Mpc-1, <math alttext="\Omega_{b}=0.046" class="ltx_Math" display="inline" id="S1.p8.5.m5.1"><semantics id="S1.p8.5.m5.1a"><mrow id="S1.p8.5.m5.1.1" xref="S1.p8.5.m5.1.1.cmml"><msub id="S1.p8.5.m5.1.1.2" xref="S1.p8.5.m5.1.1.2.cmml"><mi id="S1.p8.5.m5.1.1.2.2" mathvariant="normal" xref="S1.p8.5.m5.1.1.2.2.cmml">Ω</mi><mi id="S1.p8.5.m5.1.1.2.3" xref="S1.p8.5.m5.1.1.2.3.cmml">b</mi></msub><mo id="S1.p8.5.m5.1.1.1" xref="S1.p8.5.m5.1.1.1.cmml">=</mo><mn id="S1.p8.5.m5.1.1.3" xref="S1.p8.5.m5.1.1.3.cmml">0.046</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.5.m5.1b"><apply id="S1.p8.5.m5.1.1.cmml" xref="S1.p8.5.m5.1.1"><eq id="S1.p8.5.m5.1.1.1.cmml" xref="S1.p8.5.m5.1.1.1"></eq><apply id="S1.p8.5.m5.1.1.2.cmml" xref="S1.p8.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.p8.5.m5.1.1.2.1.cmml" xref="S1.p8.5.m5.1.1.2">subscript</csymbol><ci id="S1.p8.5.m5.1.1.2.2.cmml" xref="S1.p8.5.m5.1.1.2.2">Ω</ci><ci id="S1.p8.5.m5.1.1.2.3.cmml" xref="S1.p8.5.m5.1.1.2.3">𝑏</ci></apply><cn id="S1.p8.5.m5.1.1.3.cmml" type="float" xref="S1.p8.5.m5.1.1.3">0.046</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.5.m5.1c">\Omega_{b}=0.046</annotation><annotation encoding="application/x-llamapun" id="S1.p8.5.m5.1d">roman_Ω start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT = 0.046</annotation></semantics></math> and <math alttext="\Omega_{m}=0.31" class="ltx_Math" display="inline" id="S1.p8.6.m6.1"><semantics id="S1.p8.6.m6.1a"><mrow id="S1.p8.6.m6.1.1" xref="S1.p8.6.m6.1.1.cmml"><msub id="S1.p8.6.m6.1.1.2" xref="S1.p8.6.m6.1.1.2.cmml"><mi id="S1.p8.6.m6.1.1.2.2" mathvariant="normal" xref="S1.p8.6.m6.1.1.2.2.cmml">Ω</mi><mi id="S1.p8.6.m6.1.1.2.3" xref="S1.p8.6.m6.1.1.2.3.cmml">m</mi></msub><mo id="S1.p8.6.m6.1.1.1" xref="S1.p8.6.m6.1.1.1.cmml">=</mo><mn id="S1.p8.6.m6.1.1.3" xref="S1.p8.6.m6.1.1.3.cmml">0.31</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.6.m6.1b"><apply id="S1.p8.6.m6.1.1.cmml" xref="S1.p8.6.m6.1.1"><eq id="S1.p8.6.m6.1.1.1.cmml" xref="S1.p8.6.m6.1.1.1"></eq><apply id="S1.p8.6.m6.1.1.2.cmml" xref="S1.p8.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.p8.6.m6.1.1.2.1.cmml" xref="S1.p8.6.m6.1.1.2">subscript</csymbol><ci id="S1.p8.6.m6.1.1.2.2.cmml" xref="S1.p8.6.m6.1.1.2.2">Ω</ci><ci id="S1.p8.6.m6.1.1.2.3.cmml" xref="S1.p8.6.m6.1.1.2.3">𝑚</ci></apply><cn id="S1.p8.6.m6.1.1.3.cmml" type="float" xref="S1.p8.6.m6.1.1.3">0.31</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.6.m6.1c">\Omega_{m}=0.31</annotation><annotation encoding="application/x-llamapun" id="S1.p8.6.m6.1d">roman_Ω start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0.31</annotation></semantics></math>. All magnitudes in this work are defined in the AB magnitude system <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib49" title="">49</a>]</cite>. </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="337" id="S1.F1.g1" src="extracted/5997959/figures/fig1.png" width="598"/> <figcaption class="ltx_caption ltx_centering">Figure 1: Overview of the MUST project. The top right panel shows the location of the MUST site currently selected in Qinghai province of China. The top left panel is a picture of the Saishiteng mountain near Lenghu. We highlight the peaks that have been or are being developed for astronomy. The highest peak – Peak A – with an altitude of 4358 m, was selected as the site for MUST. From left to right, the bottom panels illustrate the preliminary design of the dome of MUST and the telescope, a sketch of the conceptual design of the focal plane of MUST, fiber, and spectrograph systems, and the fiducial design of the modular focal plane of MUST. By current design, MUST will host 21,168 robotic fiber positioners using 336 triangular modules. </figcaption> </figure> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> 2 MUltiplexed Survey Telescope</h2> <div class="ltx_para" id="S2.p1"> The MUltiplexed Survey Telescope (MUST) is a dedicated spectroscopic survey facility proposed and led by the Department of Astronomy at Tsinghua University and co-founded with École Polytechnique Fédérale de Lausanne (EPFL). </div> <div class="ltx_para" id="S2.p2"> The MUST project aims to build a 6.5-meter wide-field telescope with multiplexed spectroscopic observation capability by 2030. While being designed as a flexible platform for various spectroscopic surveys, the primary scientific drive of MUST is to become the first Stage-V spectroscopic survey <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib50" title="">50</a>]</cite> to answer fundamental questions in cosmology and physics. The concept and early design of MUST drew inspiration from the MegaMapper project <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib39" title="">39</a>]</cite>. Now, the MUST collaboration has independently finished the preliminary design of the optical system <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib44" title="">44</a>]</cite>, structure, and dome <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib51" title="">51</a>]</cite> of the telescope and is working with collaborators and vendors to design the modular focal plane, fiber, and spectrograph systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib52" title="">52</a>]</cite>. Here, we briefly introduce the design and the technical capabilities of MUST. A summary of the key specifications is available in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.T1" title="Table 1 ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">1</a>. An upcoming paper will summarize a more detailed description and analysis of the MUST project’s technical aspects. </div> <figure class="ltx_table" id="S2.T1"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S2.T1.12" style="width:341.4pt;height:295.1pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S2.T1.12.12"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S2.T1.12.12.13.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="4" id="S2.T1.12.12.13.1.1" style="background-color:#DAE8FC;">Optical System</th> </tr> <tr class="ltx_tr" id="S2.T1.12.12.14.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.14.2.1" style="background-color:#EFEFEF;">Optical Design</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" colspan="3" id="S2.T1.12.12.14.2.2">R-C with WFC</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T1.12.12.15.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.15.1.1" style="background-color:#EFEFEF;">Primary Mirror Diameter</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.15.1.2">6.5 m Hyperboloid</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.15.1.3">With 2 m central hole</td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.16.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.16.2.1" style="background-color:#EFEFEF;">Secondary Mirror Diameter</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.16.2.2">2.4 m Hyperboloid</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.16.2.3">Convex</td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.17.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.17.3.1" style="background-color:#EFEFEF;">Wide Field Corrector</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.17.3.2">Five-Lens Design</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.17.3.3">Largest lens diameter: 1.6 m</td> </tr> <tr class="ltx_tr" id="S2.T1.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.1.1.1.2" style="background-color:#EFEFEF;">Throughput Requirement</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.1.1.1.1"> <math alttext=">50" class="ltx_Math" display="inline" id="S2.T1.1.1.1.1.m1.1"><semantics id="S2.T1.1.1.1.1.m1.1a"><mrow id="S2.T1.1.1.1.1.m1.1.1" xref="S2.T1.1.1.1.1.m1.1.1.cmml"><mi id="S2.T1.1.1.1.1.m1.1.1.2" xref="S2.T1.1.1.1.1.m1.1.1.2.cmml"></mi><mo id="S2.T1.1.1.1.1.m1.1.1.1" xref="S2.T1.1.1.1.1.m1.1.1.1.cmml">></mo><mn id="S2.T1.1.1.1.1.m1.1.1.3" xref="S2.T1.1.1.1.1.m1.1.1.3.cmml">50</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.1.1.1.1.m1.1b"><apply id="S2.T1.1.1.1.1.m1.1.1.cmml" xref="S2.T1.1.1.1.1.m1.1.1"><gt id="S2.T1.1.1.1.1.m1.1.1.1.cmml" xref="S2.T1.1.1.1.1.m1.1.1.1"></gt><csymbol cd="latexml" id="S2.T1.1.1.1.1.m1.1.1.2.cmml" xref="S2.T1.1.1.1.1.m1.1.1.2">absent</csymbol><cn id="S2.T1.1.1.1.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.1.1.1.1.m1.1.1.3">50</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.1.1.1.1.m1.1c">>50</annotation><annotation encoding="application/x-llamapun" id="S2.T1.1.1.1.1.m1.1d">> 50</annotation></semantics></math>% from 370-960 nm</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.1.1.1.3"> <table class="ltx_tabular ltx_align_middle" id="S2.T1.1.1.1.3.1"> <tr class="ltx_tr" id="S2.T1.1.1.1.3.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S2.T1.1.1.1.3.1.1.1">Considering the reflectance of the primary & secondary mirror,</td> </tr> <tr class="ltx_tr" id="S2.T1.1.1.1.3.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S2.T1.1.1.1.3.1.2.1">WFC throughput, and the vignetting of the secondary & corrector</td> </tr> </table> </td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.18.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.18.4.1" style="background-color:#EFEFEF;">Focal Ratio</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.18.4.2">F/3.7</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.18.4.3">Cassegrain Focus</td> </tr> <tr class="ltx_tr" id="S2.T1.2.2.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.2.2.2.2" style="background-color:#EFEFEF;">Field of View</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.2.2.2.1">2.8∘ in Diameter</th> <td class="ltx_td ltx_border_r ltx_border_t" colspan="2" id="S2.T1.2.2.2.3"></td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.19.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="4" id="S2.T1.12.12.19.5.1" style="background-color:#CBCEFB;">Focal Plane & Fiber System (Preliminery)</th> </tr> <tr class="ltx_tr" id="S2.T1.12.12.20.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.20.6.1" style="background-color:#EFEFEF;">Focal Plane Diameter</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.20.6.2">1.2 m</th> <td class="ltx_td ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.20.6.3"></td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.21.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.21.7.1" style="background-color:#EFEFEF;">Total # of Modules</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.21.7.2">336</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.21.7.3">Semi-frameless design</td> </tr> <tr class="ltx_tr" id="S2.T1.3.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.3.3.3.2" style="background-color:#EFEFEF;"># of Positioners per Module</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.3.3.3.3">63</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.3.3.3.1">21 <math alttext="\times" class="ltx_Math" display="inline" id="S2.T1.3.3.3.1.m1.1"><semantics id="S2.T1.3.3.3.1.m1.1a"><mo id="S2.T1.3.3.3.1.m1.1.1" xref="S2.T1.3.3.3.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.T1.3.3.3.1.m1.1b"><times id="S2.T1.3.3.3.1.m1.1.1.cmml" xref="S2.T1.3.3.3.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.3.3.3.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.T1.3.3.3.1.m1.1d">×</annotation></semantics></math> 3 groups</td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.22.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.22.8.1" style="background-color:#EFEFEF;">Total # of Fiber Positioners</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.22.8.2">21,168</th> <td class="ltx_td ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.22.8.3"></td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.23.9"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.23.9.1" style="background-color:#EFEFEF;">Effective Positioner Coverage</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.23.9.2">74.0%</th> <td class="ltx_td ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.23.9.3"></td> </tr> <tr class="ltx_tr" id="S2.T1.5.5.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.5.5.5.3" style="background-color:#EFEFEF;">Pitch Distance between Positioners</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.5.5.5.4">6.2 mm</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.5.5.5.2">Using a <math alttext="\theta" class="ltx_Math" display="inline" id="S2.T1.4.4.4.1.m1.1"><semantics id="S2.T1.4.4.4.1.m1.1a"><mi id="S2.T1.4.4.4.1.m1.1.1" xref="S2.T1.4.4.4.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.T1.4.4.4.1.m1.1b"><ci id="S2.T1.4.4.4.1.m1.1.1.cmml" xref="S2.T1.4.4.4.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.4.4.4.1.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.T1.4.4.4.1.m1.1d">italic_θ</annotation></semantics></math>-<math alttext="\phi" class="ltx_Math" display="inline" id="S2.T1.5.5.5.2.m2.1"><semantics id="S2.T1.5.5.5.2.m2.1a"><mi id="S2.T1.5.5.5.2.m2.1.1" xref="S2.T1.5.5.5.2.m2.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.T1.5.5.5.2.m2.1b"><ci id="S2.T1.5.5.5.2.m2.1.1.cmml" xref="S2.T1.5.5.5.2.m2.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.5.5.5.2.m2.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.T1.5.5.5.2.m2.1d">italic_ϕ</annotation></semantics></math> design</td> </tr> <tr class="ltx_tr" id="S2.T1.6.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.6.6.6.2" style="background-color:#EFEFEF;">Fiber Core Diameter</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.6.6.6.1">150 <math alttext="\mu" class="ltx_Math" display="inline" id="S2.T1.6.6.6.1.m1.1"><semantics id="S2.T1.6.6.6.1.m1.1a"><mi id="S2.T1.6.6.6.1.m1.1.1" xref="S2.T1.6.6.6.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.T1.6.6.6.1.m1.1b"><ci id="S2.T1.6.6.6.1.m1.1.1.cmml" xref="S2.T1.6.6.6.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.6.6.6.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.T1.6.6.6.1.m1.1d">italic_μ</annotation></semantics></math>m</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.6.6.6.3">1.3 arcsec on the sky</td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.24.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.24.10.1" style="background-color:#EFEFEF;">Fiber Route Length</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.24.10.2">45-50 m</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="2" id="S2.T1.12.12.24.10.3">From the fiber tip to the slithead</td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.25.11"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="4" id="S2.T1.12.12.25.11.1" style="background-color:#FFFFC7;">Spectrograph (Preliminery)</th> </tr> <tr class="ltx_tr" id="S2.T1.12.12.26.12"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.26.12.1" style="background-color:#EFEFEF;">Channel</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.12.12.26.12.2" style="background-color:#EFEFEF;">Wavelength Coverage</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.12.12.26.12.3" style="background-color:#EFEFEF;">Spectral Resolution</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.12.12.26.12.4" style="background-color:#EFEFEF;">Average Throughput</td> </tr> <tr class="ltx_tr" id="S2.T1.8.8.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.8.8.8.3" style="background-color:#96FFFB;">Blue</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.8.8.8.4">370-590 nm</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.7.7.7.1">R<math alttext="\sim 2513" class="ltx_Math" display="inline" id="S2.T1.7.7.7.1.m1.1"><semantics id="S2.T1.7.7.7.1.m1.1a"><mrow id="S2.T1.7.7.7.1.m1.1.1" xref="S2.T1.7.7.7.1.m1.1.1.cmml"><mi id="S2.T1.7.7.7.1.m1.1.1.2" xref="S2.T1.7.7.7.1.m1.1.1.2.cmml"></mi><mo id="S2.T1.7.7.7.1.m1.1.1.1" xref="S2.T1.7.7.7.1.m1.1.1.1.cmml">∼</mo><mn id="S2.T1.7.7.7.1.m1.1.1.3" xref="S2.T1.7.7.7.1.m1.1.1.3.cmml">2513</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.7.7.7.1.m1.1b"><apply id="S2.T1.7.7.7.1.m1.1.1.cmml" xref="S2.T1.7.7.7.1.m1.1.1"><csymbol cd="latexml" id="S2.T1.7.7.7.1.m1.1.1.1.cmml" xref="S2.T1.7.7.7.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.T1.7.7.7.1.m1.1.1.2.cmml" xref="S2.T1.7.7.7.1.m1.1.1.2">absent</csymbol><cn id="S2.T1.7.7.7.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.7.7.7.1.m1.1.1.3">2513</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.7.7.7.1.m1.1c">\sim 2513</annotation><annotation encoding="application/x-llamapun" id="S2.T1.7.7.7.1.m1.1d">∼ 2513</annotation></semantics></math>-3668</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.8.8.8.2"> <math alttext="\geq 55" class="ltx_Math" display="inline" id="S2.T1.8.8.8.2.m1.1"><semantics id="S2.T1.8.8.8.2.m1.1a"><mrow id="S2.T1.8.8.8.2.m1.1.1" xref="S2.T1.8.8.8.2.m1.1.1.cmml"><mi id="S2.T1.8.8.8.2.m1.1.1.2" xref="S2.T1.8.8.8.2.m1.1.1.2.cmml"></mi><mo id="S2.T1.8.8.8.2.m1.1.1.1" xref="S2.T1.8.8.8.2.m1.1.1.1.cmml">≥</mo><mn id="S2.T1.8.8.8.2.m1.1.1.3" xref="S2.T1.8.8.8.2.m1.1.1.3.cmml">55</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.8.8.8.2.m1.1b"><apply id="S2.T1.8.8.8.2.m1.1.1.cmml" xref="S2.T1.8.8.8.2.m1.1.1"><geq id="S2.T1.8.8.8.2.m1.1.1.1.cmml" xref="S2.T1.8.8.8.2.m1.1.1.1"></geq><csymbol cd="latexml" id="S2.T1.8.8.8.2.m1.1.1.2.cmml" xref="S2.T1.8.8.8.2.m1.1.1.2">absent</csymbol><cn id="S2.T1.8.8.8.2.m1.1.1.3.cmml" type="integer" xref="S2.T1.8.8.8.2.m1.1.1.3">55</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.8.8.8.2.m1.1c">\geq 55</annotation><annotation encoding="application/x-llamapun" id="S2.T1.8.8.8.2.m1.1d">≥ 55</annotation></semantics></math>%</td> </tr> <tr class="ltx_tr" id="S2.T1.10.10.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.10.10.10.3" style="background-color:#FFCE93;">Red</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S2.T1.10.10.10.4">565-775 nm</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.9.9.9.1">R<math alttext="\sim 2287" class="ltx_Math" display="inline" id="S2.T1.9.9.9.1.m1.1"><semantics id="S2.T1.9.9.9.1.m1.1a"><mrow id="S2.T1.9.9.9.1.m1.1.1" xref="S2.T1.9.9.9.1.m1.1.1.cmml"><mi id="S2.T1.9.9.9.1.m1.1.1.2" xref="S2.T1.9.9.9.1.m1.1.1.2.cmml"></mi><mo id="S2.T1.9.9.9.1.m1.1.1.1" xref="S2.T1.9.9.9.1.m1.1.1.1.cmml">∼</mo><mn id="S2.T1.9.9.9.1.m1.1.1.3" xref="S2.T1.9.9.9.1.m1.1.1.3.cmml">2287</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.9.9.9.1.m1.1b"><apply id="S2.T1.9.9.9.1.m1.1.1.cmml" xref="S2.T1.9.9.9.1.m1.1.1"><csymbol cd="latexml" id="S2.T1.9.9.9.1.m1.1.1.1.cmml" xref="S2.T1.9.9.9.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.T1.9.9.9.1.m1.1.1.2.cmml" xref="S2.T1.9.9.9.1.m1.1.1.2">absent</csymbol><cn id="S2.T1.9.9.9.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.9.9.9.1.m1.1.1.3">2287</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.9.9.9.1.m1.1c">\sim 2287</annotation><annotation encoding="application/x-llamapun" id="S2.T1.9.9.9.1.m1.1d">∼ 2287</annotation></semantics></math>-3442</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.T1.10.10.10.2"> <math alttext="\geq 60" class="ltx_Math" display="inline" id="S2.T1.10.10.10.2.m1.1"><semantics id="S2.T1.10.10.10.2.m1.1a"><mrow id="S2.T1.10.10.10.2.m1.1.1" xref="S2.T1.10.10.10.2.m1.1.1.cmml"><mi id="S2.T1.10.10.10.2.m1.1.1.2" xref="S2.T1.10.10.10.2.m1.1.1.2.cmml"></mi><mo id="S2.T1.10.10.10.2.m1.1.1.1" xref="S2.T1.10.10.10.2.m1.1.1.1.cmml">≥</mo><mn id="S2.T1.10.10.10.2.m1.1.1.3" xref="S2.T1.10.10.10.2.m1.1.1.3.cmml">60</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.10.10.10.2.m1.1b"><apply id="S2.T1.10.10.10.2.m1.1.1.cmml" xref="S2.T1.10.10.10.2.m1.1.1"><geq id="S2.T1.10.10.10.2.m1.1.1.1.cmml" xref="S2.T1.10.10.10.2.m1.1.1.1"></geq><csymbol cd="latexml" id="S2.T1.10.10.10.2.m1.1.1.2.cmml" xref="S2.T1.10.10.10.2.m1.1.1.2">absent</csymbol><cn id="S2.T1.10.10.10.2.m1.1.1.3.cmml" type="integer" xref="S2.T1.10.10.10.2.m1.1.1.3">60</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.10.10.10.2.m1.1c">\geq 60</annotation><annotation encoding="application/x-llamapun" id="S2.T1.10.10.10.2.m1.1d">≥ 60</annotation></semantics></math>%</td> </tr> <tr class="ltx_tr" id="S2.T1.12.12.12"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S2.T1.12.12.12.3" style="background-color:#FFCCC9;">NIR</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_r ltx_border_t" id="S2.T1.12.12.12.4">750-960 nm</th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S2.T1.11.11.11.1">R<math alttext="\sim 4124" class="ltx_Math" display="inline" id="S2.T1.11.11.11.1.m1.1"><semantics id="S2.T1.11.11.11.1.m1.1a"><mrow id="S2.T1.11.11.11.1.m1.1.1" xref="S2.T1.11.11.11.1.m1.1.1.cmml"><mi id="S2.T1.11.11.11.1.m1.1.1.2" xref="S2.T1.11.11.11.1.m1.1.1.2.cmml"></mi><mo id="S2.T1.11.11.11.1.m1.1.1.1" xref="S2.T1.11.11.11.1.m1.1.1.1.cmml">∼</mo><mn id="S2.T1.11.11.11.1.m1.1.1.3" xref="S2.T1.11.11.11.1.m1.1.1.3.cmml">4124</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.11.11.11.1.m1.1b"><apply id="S2.T1.11.11.11.1.m1.1.1.cmml" xref="S2.T1.11.11.11.1.m1.1.1"><csymbol cd="latexml" id="S2.T1.11.11.11.1.m1.1.1.1.cmml" xref="S2.T1.11.11.11.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.T1.11.11.11.1.m1.1.1.2.cmml" xref="S2.T1.11.11.11.1.m1.1.1.2">absent</csymbol><cn id="S2.T1.11.11.11.1.m1.1.1.3.cmml" type="integer" xref="S2.T1.11.11.11.1.m1.1.1.3">4124</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.11.11.11.1.m1.1c">\sim 4124</annotation><annotation encoding="application/x-llamapun" id="S2.T1.11.11.11.1.m1.1d">∼ 4124</annotation></semantics></math>-5279</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S2.T1.12.12.12.2"> <math alttext="\geq 60" class="ltx_Math" display="inline" id="S2.T1.12.12.12.2.m1.1"><semantics id="S2.T1.12.12.12.2.m1.1a"><mrow id="S2.T1.12.12.12.2.m1.1.1" xref="S2.T1.12.12.12.2.m1.1.1.cmml"><mi id="S2.T1.12.12.12.2.m1.1.1.2" xref="S2.T1.12.12.12.2.m1.1.1.2.cmml"></mi><mo id="S2.T1.12.12.12.2.m1.1.1.1" xref="S2.T1.12.12.12.2.m1.1.1.1.cmml">≥</mo><mn id="S2.T1.12.12.12.2.m1.1.1.3" xref="S2.T1.12.12.12.2.m1.1.1.3.cmml">60</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.T1.12.12.12.2.m1.1b"><apply id="S2.T1.12.12.12.2.m1.1.1.cmml" xref="S2.T1.12.12.12.2.m1.1.1"><geq id="S2.T1.12.12.12.2.m1.1.1.1.cmml" xref="S2.T1.12.12.12.2.m1.1.1.1"></geq><csymbol cd="latexml" id="S2.T1.12.12.12.2.m1.1.1.2.cmml" xref="S2.T1.12.12.12.2.m1.1.1.2">absent</csymbol><cn id="S2.T1.12.12.12.2.m1.1.1.3.cmml" type="integer" xref="S2.T1.12.12.12.2.m1.1.1.3">60</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.12.12.12.2.m1.1c">\geq 60</annotation><annotation encoding="application/x-llamapun" id="S2.T1.12.12.12.2.m1.1d">≥ 60</annotation></semantics></math>%</td> </tr> </tbody> </table> </div> <figcaption class="ltx_caption ltx_centering">Table 1: Summary of the key specification of the optical, focal plane & fiber, and spectrograph systems of MUST. Note that the design specifications for the focal plane and spectrograph systems are still very preliminary. </figcaption> </figure> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> 2.1 A 6.5-m Telescope for Spectroscopic Surveys</h3> <div class="ltx_para" id="S2.SS1.p1"> Driven by the demanding scientific requirements of the next-generation surveys for cosmology, a Stage-V spectroscopic telescope requires excellent optical quality over a large Field-of-View (FoV) on a <math alttext=">" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mo id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">></mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><gt id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">></annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">></annotation></semantics></math>6-meter telescope. The optical design of MUST is driven by solving these technical challenges. Currently, MUST adopts a compact Ritchey-Chretien (R-C) design with a multi-element Wide Field Corrector (WFC). The hyperboloid primary and secondary mirrors of MUST are 6.5 m and 2.4 m in diameter. The primary mirror has a 2 m diameter central hole to facilitate the installation of a five-lens WFC that ensures excellent imaging quality of MUST. The largest lens for the WFC of MUST is 1.6 m in diameter, even slightly larger than the largest lens of the camera of LSST <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib20" title="">20</a>]</cite>. MUST published the conceptual optical design in 2023 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib44" title="">44</a>]</cite>. Since then, the collaboration has made a series of modifications, primarily to the configuration of the WFC, to improve the engineering feasibility of the telescope and reduce the risk during the manufacture and assembly of critical sub-systems (Zhang et al. in prep.). Under the updated design, MUST imaging quality over the entire 2.8∘ diameter FoV is excellent: up to a 50∘ zenith angle and within the wavelength range of 0.365 to 1.0 <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mi id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><ci id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_μ</annotation></semantics></math>m, the 80% Encircled Energy (EE80) size of the image spot is <math alttext="<0.6" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mrow id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml"><mi id="S2.SS1.p1.5.m5.1.1.2" xref="S2.SS1.p1.5.m5.1.1.2.cmml"></mi><mo id="S2.SS1.p1.5.m5.1.1.1" xref="S2.SS1.p1.5.m5.1.1.1.cmml"><</mo><mn id="S2.SS1.p1.5.m5.1.1.3" xref="S2.SS1.p1.5.m5.1.1.3.cmml">0.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"><lt id="S2.SS1.p1.5.m5.1.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1.1"></lt><csymbol cd="latexml" id="S2.SS1.p1.5.m5.1.1.2.cmml" xref="S2.SS1.p1.5.m5.1.1.2">absent</csymbol><cn id="S2.SS1.p1.5.m5.1.1.3.cmml" type="float" xref="S2.SS1.p1.5.m5.1.1.3">0.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c"><0.6</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">< 0.6</annotation></semantics></math> arcsec. At the Cassegrain focus that hosts the modular focal plane system of MUST, the optical system has an F/3.7 focal ratio. </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> 2.2 Multiplexed Focal Plane & Instruments</h3> <div class="ltx_para" id="S2.SS2.p1"> Enabled by the optimized optical design, MUST can achieve a much more improved multiplexed capability than Stage-IV spectroscopic facilities of today using a novel modular focal plane design (See Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">1</a>). Starting from the LAMOST survey, modern spectroscopic surveys have adopted different types of robotic fiber positioners (<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib55" title="">55</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib56" title="">56</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib57" title="">57</a>]</cite>) to guide the light from distant targets to the scientific instruments and enable efficient & flexible survey design. The largest multiplexed survey facility, DESI, currently hosts 5,000 fiber positioners, each with a 10 mm outer diameter <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib58" title="">58</a>]</cite>. Each positioner was installed and operated individually and independently on the focal plate of DESI. However, as a Stage-V facility MUST have a much more demanding requirement for multiplexed capability, individually managing <math alttext=">" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mo id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">></mo><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><gt id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">></annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">></annotation></semantics></math>10,000 fiber positioners significantly increases the complexity and the risk of operating the instruments. The MUST collaboration is working with industrial partners to develop a novel modular focal plane system (<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib59" title="">59</a>]</cite>). As shown in the bottom-right panel of Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">1</a>, a 1.2 m diameter focal plate will host 336 triangular fiber positioner modules at the Cassegrain focus under the current preliminary design. Each module will integrate 63 fiber positioning robots in three groups. </div> <div class="ltx_para" id="S2.SS2.p2"> MUST plans to adopt a miniaturized fiber positioning robot with a 6.2 mm outer diameter and optical fiber with a 150 <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_μ</annotation></semantics></math>m core diameter corresponding to 1.3 arcsecs on the sky to ensure the fiber density satisfies the requirement of future cosmology surveys. To optimize the effective coverage of the fibers, the focal plate assembles four modules into a group. The gaps between modules are 1 mm and 3 mm within the group and between adjacent groups. Altogether, this allows MUST to equip 21,168 fiber positioners over the focal plane with a 74% coverage, which results in a <math alttext="\sim 5" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><mrow id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml"><mi id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml"></mi><mo id="S2.SS2.p2.2.m2.1.1.1" xref="S2.SS2.p2.2.m2.1.1.1.cmml">∼</mo><mn id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.1b"><apply id="S2.SS2.p2.2.m2.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1"><csymbol cd="latexml" id="S2.SS2.p2.2.m2.1.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p2.2.m2.1.1.2.cmml" xref="S2.SS2.p2.2.m2.1.1.2">absent</csymbol><cn id="S2.SS2.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p2.2.m2.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">\sim 5</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">∼ 5</annotation></semantics></math> deg2 FoV covered by fibers with a <math alttext="\sim" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.1"><semantics id="S2.SS2.p2.4.m4.1a"><mo id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.1b"><csymbol cd="latexml" id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">∼</annotation></semantics></math>4,000 deg-2 fiber density. It is worth noting that, given the aspheric shape of the focal plane and the requirements of the fiber throughput & focal ratio degradation (FRD) budgets of MUST, the fiber tips of all the positioners should be located within <math alttext="\pm 50~{}\mu{\rm m}" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m6.1"><semantics id="S2.SS2.p2.6.m6.1a"><mrow id="S2.SS2.p2.6.m6.1.1" xref="S2.SS2.p2.6.m6.1.1.cmml"><mo id="S2.SS2.p2.6.m6.1.1a" xref="S2.SS2.p2.6.m6.1.1.cmml">±</mo><mrow id="S2.SS2.p2.6.m6.1.1.2" xref="S2.SS2.p2.6.m6.1.1.2.cmml"><mn id="S2.SS2.p2.6.m6.1.1.2.2" xref="S2.SS2.p2.6.m6.1.1.2.2.cmml">50</mn><mo id="S2.SS2.p2.6.m6.1.1.2.1" lspace="0.330em" xref="S2.SS2.p2.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S2.SS2.p2.6.m6.1.1.2.3" xref="S2.SS2.p2.6.m6.1.1.2.3.cmml">μ</mi><mo id="S2.SS2.p2.6.m6.1.1.2.1a" xref="S2.SS2.p2.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S2.SS2.p2.6.m6.1.1.2.4" mathvariant="normal" xref="S2.SS2.p2.6.m6.1.1.2.4.cmml">m</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m6.1b"><apply id="S2.SS2.p2.6.m6.1.1.cmml" xref="S2.SS2.p2.6.m6.1.1"><csymbol cd="latexml" id="S2.SS2.p2.6.m6.1.1.1.cmml" xref="S2.SS2.p2.6.m6.1.1">plus-or-minus</csymbol><apply id="S2.SS2.p2.6.m6.1.1.2.cmml" xref="S2.SS2.p2.6.m6.1.1.2"><times id="S2.SS2.p2.6.m6.1.1.2.1.cmml" xref="S2.SS2.p2.6.m6.1.1.2.1"></times><cn id="S2.SS2.p2.6.m6.1.1.2.2.cmml" type="integer" xref="S2.SS2.p2.6.m6.1.1.2.2">50</cn><ci id="S2.SS2.p2.6.m6.1.1.2.3.cmml" xref="S2.SS2.p2.6.m6.1.1.2.3">𝜇</ci><ci id="S2.SS2.p2.6.m6.1.1.2.4.cmml" xref="S2.SS2.p2.6.m6.1.1.2.4">m</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m6.1c">\pm 50~{}\mu{\rm m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m6.1d">± 50 italic_μ roman_m</annotation></semantics></math> from the theoretical focal plane. The design of MUST achieves this goal by approximating the focal plane with a best-fit spherical surface that meets the requirement. In this way, all the modules can have an identical configuration, and within each module, the tips of the 63 positioners will match the shape of the spherical surface. This challenging and ambitious design is essential to the overall scientific capability of MUST, especially for the LSS survey discussed here. </div> <div class="ltx_para" id="S2.SS2.p3"> The first-generation instruments of MUST consist of <math alttext="\sim 42" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mrow id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml"></mi><mo id="S2.SS2.p3.1.m1.1.1.1" xref="S2.SS2.p3.1.m1.1.1.1.cmml">∼</mo><mn id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml">42</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><csymbol cd="latexml" id="S2.SS2.p3.1.m1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2">absent</csymbol><cn id="S2.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p3.1.m1.1.1.3">42</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">\sim 42</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">∼ 42</annotation></semantics></math> multi-object spectrographs; each will host <math alttext="\sim 500" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><mrow id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml"><mi id="S2.SS2.p3.2.m2.1.1.2" xref="S2.SS2.p3.2.m2.1.1.2.cmml"></mi><mo id="S2.SS2.p3.2.m2.1.1.1" xref="S2.SS2.p3.2.m2.1.1.1.cmml">∼</mo><mn id="S2.SS2.p3.2.m2.1.1.3" xref="S2.SS2.p3.2.m2.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.1b"><apply id="S2.SS2.p3.2.m2.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1"><csymbol cd="latexml" id="S2.SS2.p3.2.m2.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p3.2.m2.1.1.2.cmml" xref="S2.SS2.p3.2.m2.1.1.2">absent</csymbol><cn id="S2.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p3.2.m2.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">\sim 500</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">∼ 500</annotation></semantics></math> optical fibers on a 4k<math alttext="\times" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.1"><semantics id="S2.SS2.p3.3.m3.1a"><mo id="S2.SS2.p3.3.m3.1.1" xref="S2.SS2.p3.3.m3.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m3.1b"><times id="S2.SS2.p3.3.m3.1.1.cmml" xref="S2.SS2.p3.3.m3.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m3.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m3.1d">×</annotation></semantics></math>4k-pixels CCD detector. The preliminary design of the spectrograph of MUST is still underway. Regardless, given the very similar scientific goals to the DESI project, the current concept adopts a similar three-channel design as the spectrograph of DESI, covering the wavelength range between <math alttext="\sim 3700" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m4.1"><semantics id="S2.SS2.p3.4.m4.1a"><mrow id="S2.SS2.p3.4.m4.1.1" xref="S2.SS2.p3.4.m4.1.1.cmml"><mi id="S2.SS2.p3.4.m4.1.1.2" xref="S2.SS2.p3.4.m4.1.1.2.cmml"></mi><mo id="S2.SS2.p3.4.m4.1.1.1" xref="S2.SS2.p3.4.m4.1.1.1.cmml">∼</mo><mn id="S2.SS2.p3.4.m4.1.1.3" xref="S2.SS2.p3.4.m4.1.1.3.cmml">3700</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m4.1b"><apply id="S2.SS2.p3.4.m4.1.1.cmml" xref="S2.SS2.p3.4.m4.1.1"><csymbol cd="latexml" id="S2.SS2.p3.4.m4.1.1.1.cmml" xref="S2.SS2.p3.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p3.4.m4.1.1.2.cmml" xref="S2.SS2.p3.4.m4.1.1.2">absent</csymbol><cn id="S2.SS2.p3.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.p3.4.m4.1.1.3">3700</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m4.1c">\sim 3700</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m4.1d">∼ 3700</annotation></semantics></math> to <math alttext="\sim 9600~{}\AA" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m5.1"><semantics id="S2.SS2.p3.5.m5.1a"><mrow id="S2.SS2.p3.5.m5.1.1" xref="S2.SS2.p3.5.m5.1.1.cmml"><mi id="S2.SS2.p3.5.m5.1.1.2" xref="S2.SS2.p3.5.m5.1.1.2.cmml"></mi><mo id="S2.SS2.p3.5.m5.1.1.1" xref="S2.SS2.p3.5.m5.1.1.1.cmml">∼</mo><mrow id="S2.SS2.p3.5.m5.1.1.3" xref="S2.SS2.p3.5.m5.1.1.3.cmml"><mn id="S2.SS2.p3.5.m5.1.1.3.2" xref="S2.SS2.p3.5.m5.1.1.3.2.cmml">9600</mn><mo id="S2.SS2.p3.5.m5.1.1.3.1" lspace="0.330em" xref="S2.SS2.p3.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p3.5.m5.1.1.3.3" xref="S2.SS2.p3.5.m5.1.1.3.3.cmml">Å</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m5.1b"><apply id="S2.SS2.p3.5.m5.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1"><csymbol cd="latexml" id="S2.SS2.p3.5.m5.1.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p3.5.m5.1.1.2.cmml" xref="S2.SS2.p3.5.m5.1.1.2">absent</csymbol><apply id="S2.SS2.p3.5.m5.1.1.3.cmml" xref="S2.SS2.p3.5.m5.1.1.3"><times id="S2.SS2.p3.5.m5.1.1.3.1.cmml" xref="S2.SS2.p3.5.m5.1.1.3.1"></times><cn id="S2.SS2.p3.5.m5.1.1.3.2.cmml" type="integer" xref="S2.SS2.p3.5.m5.1.1.3.2">9600</cn><ci id="S2.SS2.p3.5.m5.1.1.3.3.cmml" xref="S2.SS2.p3.5.m5.1.1.3.3">italic-Å</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m5.1c">\sim 9600~{}\AA</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m5.1d">∼ 9600 italic_Å</annotation></semantics></math> with a <math alttext="R\sim 2000" class="ltx_Math" display="inline" id="S2.SS2.p3.6.m6.1"><semantics id="S2.SS2.p3.6.m6.1a"><mrow id="S2.SS2.p3.6.m6.1.1" xref="S2.SS2.p3.6.m6.1.1.cmml"><mi id="S2.SS2.p3.6.m6.1.1.2" xref="S2.SS2.p3.6.m6.1.1.2.cmml">R</mi><mo id="S2.SS2.p3.6.m6.1.1.1" xref="S2.SS2.p3.6.m6.1.1.1.cmml">∼</mo><mn id="S2.SS2.p3.6.m6.1.1.3" xref="S2.SS2.p3.6.m6.1.1.3.cmml">2000</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.6.m6.1b"><apply id="S2.SS2.p3.6.m6.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1"><csymbol cd="latexml" id="S2.SS2.p3.6.m6.1.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1.1">similar-to</csymbol><ci id="S2.SS2.p3.6.m6.1.1.2.cmml" xref="S2.SS2.p3.6.m6.1.1.2">𝑅</ci><cn id="S2.SS2.p3.6.m6.1.1.3.cmml" type="integer" xref="S2.SS2.p3.6.m6.1.1.3">2000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.6.m6.1c">R\sim 2000</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.6.m6.1d">italic_R ∼ 2000</annotation></semantics></math> to 4000 spectroscopic resolution. We currently aim to achieve a <math alttext=">60" class="ltx_Math" display="inline" id="S2.SS2.p3.7.m7.1"><semantics id="S2.SS2.p3.7.m7.1a"><mrow id="S2.SS2.p3.7.m7.1.1" xref="S2.SS2.p3.7.m7.1.1.cmml"><mi id="S2.SS2.p3.7.m7.1.1.2" xref="S2.SS2.p3.7.m7.1.1.2.cmml"></mi><mo id="S2.SS2.p3.7.m7.1.1.1" xref="S2.SS2.p3.7.m7.1.1.1.cmml">></mo><mn id="S2.SS2.p3.7.m7.1.1.3" xref="S2.SS2.p3.7.m7.1.1.3.cmml">60</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.7.m7.1b"><apply id="S2.SS2.p3.7.m7.1.1.cmml" xref="S2.SS2.p3.7.m7.1.1"><gt id="S2.SS2.p3.7.m7.1.1.1.cmml" xref="S2.SS2.p3.7.m7.1.1.1"></gt><csymbol cd="latexml" id="S2.SS2.p3.7.m7.1.1.2.cmml" xref="S2.SS2.p3.7.m7.1.1.2">absent</csymbol><cn id="S2.SS2.p3.7.m7.1.1.3.cmml" type="integer" xref="S2.SS2.p3.7.m7.1.1.3">60</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.7.m7.1c">>60</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.7.m7.1d">> 60</annotation></semantics></math>% average throughput in all three channels while exploring different approaches to further improve it to an average <math alttext="\sim 70" class="ltx_Math" display="inline" id="S2.SS2.p3.8.m8.1"><semantics id="S2.SS2.p3.8.m8.1a"><mrow id="S2.SS2.p3.8.m8.1.1" xref="S2.SS2.p3.8.m8.1.1.cmml"><mi id="S2.SS2.p3.8.m8.1.1.2" xref="S2.SS2.p3.8.m8.1.1.2.cmml"></mi><mo id="S2.SS2.p3.8.m8.1.1.1" xref="S2.SS2.p3.8.m8.1.1.1.cmml">∼</mo><mn id="S2.SS2.p3.8.m8.1.1.3" xref="S2.SS2.p3.8.m8.1.1.3.cmml">70</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.8.m8.1b"><apply id="S2.SS2.p3.8.m8.1.1.cmml" xref="S2.SS2.p3.8.m8.1.1"><csymbol cd="latexml" id="S2.SS2.p3.8.m8.1.1.1.cmml" xref="S2.SS2.p3.8.m8.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS2.p3.8.m8.1.1.2.cmml" xref="S2.SS2.p3.8.m8.1.1.2">absent</csymbol><cn id="S2.SS2.p3.8.m8.1.1.3.cmml" type="integer" xref="S2.SS2.p3.8.m8.1.1.3">70</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.8.m8.1c">\sim 70</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.8.m8.1d">∼ 70</annotation></semantics></math>% efficiency. </div> <div class="ltx_para" id="S2.SS2.p4"> In addition to the fiber positioning system and the spectrographs, the scientific instrument system of MUST includes other crucial sub-systems, such as the fiber view camera (FVC), fiducial fibers, focal plate adjustment & derotation mechanism, and a complex fiber route (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">1</a>). Altogether, they will enable the Stage-V LSS survey potential of MUST. In the upcoming publications, we will describe the technical design of the whole focal plane system and the spectrographs. </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> 2.3 Site and Observing Conditions</h3> <div class="ltx_para" id="S2.SS3.p1"> MUST has selected the Peak A of the Saishiteng Mountain near the Lenghu Town in Haixi Mongol and Tibetan Autonomous Prefecture, Qinghai Province, China, as its candidate site (referred to as the “Lenghu” site). The Lenghu site was first reported in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib60" title="">60</a>]</cite> and has been selected by a series of domestic astronomical projects in China, including the Mozi Survey Telescope666<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://wfst.ustc.edu.cn/" title="">https://wfst.ustc.edu.cn/</a> (a 2.5 m wide field imaging survey telescope in operation) and the Jiao-tong University Spectroscopic Telescope (JUST; a planned 4 m segmented mirror telescope; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib61" title="">61</a>]</cite>), and more. Along with several other peaks (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">1</a>), Peak A, at 4,358 m, has been flattened for construction. Starting from Oct. 2023, the MUST collaboration has begun to monitor the weather and seeing condition on Peak A. While we are still accumulating data for a more precise site condition assessment, combining our data with the public data collected by the NAOC team at the nearby Peak C777<a class="ltx_ref ltx_url ltx_font_typewriter" href="http://lenghu.china-vo.org/" title="">http://lenghu.china-vo.org/</a>, we estimate that the Lenghu site has an annual clear night fraction between 60% to 70%, with worse observing conditions during the summer, and a median DIMM seeing FWHM between 0.8 to 1.0 arcsec. These specifications satisfy the site requirement to carry out a fiber-spectroscopic survey. While the light pollution from the development of nearby towns and mining businesses is a concern, the local government has passed legislation to protect the dark night conditions within a 50 km area around the Lenghu site. </div> <div class="ltx_para" id="S2.SS3.p2"> At 4,358 m, Peak A enjoys significantly less atmospheric attenuation in the shorter wavelength range. Preliminary photometric observations from the Mozi telescope point to the promising result that the atmospheric attenuation level at <math alttext="\lambda<450" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mrow id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml"><mi id="S2.SS3.p2.1.m1.1.1.2" xref="S2.SS3.p2.1.m1.1.1.2.cmml">λ</mi><mo id="S2.SS3.p2.1.m1.1.1.1" xref="S2.SS3.p2.1.m1.1.1.1.cmml"><</mo><mn id="S2.SS3.p2.1.m1.1.1.3" xref="S2.SS3.p2.1.m1.1.1.3.cmml">450</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><apply id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1"><lt id="S2.SS3.p2.1.m1.1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1.1"></lt><ci id="S2.SS3.p2.1.m1.1.1.2.cmml" xref="S2.SS3.p2.1.m1.1.1.2">𝜆</ci><cn id="S2.SS3.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS3.p2.1.m1.1.1.3">450</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">\lambda<450</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_λ < 450</annotation></semantics></math> nm of Lenghu is comparable with that of Mauna Kea and is much better than lower altitude sites such as Kitt Peak. This presents MUST the opportunity to improve the overall throughput in the blue wavelength end, which is crucial for identifying the Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mi id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">italic_α</annotation></semantics></math> (Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><mi id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><ci id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">italic_α</annotation></semantics></math>) emission line at <math alttext="z>2" class="ltx_Math" display="inline" id="S2.SS3.p2.4.m4.1"><semantics id="S2.SS3.p2.4.m4.1a"><mrow id="S2.SS3.p2.4.m4.1.1" xref="S2.SS3.p2.4.m4.1.1.cmml"><mi id="S2.SS3.p2.4.m4.1.1.2" xref="S2.SS3.p2.4.m4.1.1.2.cmml">z</mi><mo id="S2.SS3.p2.4.m4.1.1.1" xref="S2.SS3.p2.4.m4.1.1.1.cmml">></mo><mn id="S2.SS3.p2.4.m4.1.1.3" xref="S2.SS3.p2.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.4.m4.1b"><apply id="S2.SS3.p2.4.m4.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1"><gt id="S2.SS3.p2.4.m4.1.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1.1"></gt><ci id="S2.SS3.p2.4.m4.1.1.2.cmml" xref="S2.SS3.p2.4.m4.1.1.2">𝑧</ci><cn id="S2.SS3.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS3.p2.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.4.m4.1c">z>2</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.4.m4.1d">italic_z > 2</annotation></semantics></math>. Meanwhile, the high altitude also results in a median nighttime air temperature of -8.1∘C (minimum & maximum temperatures are -25∘C and 15.2∘C), which increases the challenges for the construction and maintenance of MUST. At the same time, the median nighttime humidity is <math alttext="\sim 30" class="ltx_Math" display="inline" id="S2.SS3.p2.8.m8.1"><semantics id="S2.SS3.p2.8.m8.1a"><mrow id="S2.SS3.p2.8.m8.1.1" xref="S2.SS3.p2.8.m8.1.1.cmml"><mi id="S2.SS3.p2.8.m8.1.1.2" xref="S2.SS3.p2.8.m8.1.1.2.cmml"></mi><mo id="S2.SS3.p2.8.m8.1.1.1" xref="S2.SS3.p2.8.m8.1.1.1.cmml">∼</mo><mn id="S2.SS3.p2.8.m8.1.1.3" xref="S2.SS3.p2.8.m8.1.1.3.cmml">30</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.8.m8.1b"><apply id="S2.SS3.p2.8.m8.1.1.cmml" xref="S2.SS3.p2.8.m8.1.1"><csymbol cd="latexml" id="S2.SS3.p2.8.m8.1.1.1.cmml" xref="S2.SS3.p2.8.m8.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S2.SS3.p2.8.m8.1.1.2.cmml" xref="S2.SS3.p2.8.m8.1.1.2">absent</csymbol><cn id="S2.SS3.p2.8.m8.1.1.3.cmml" type="integer" xref="S2.SS3.p2.8.m8.1.1.3">30</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.8.m8.1c">\sim 30</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.8.m8.1d">∼ 30</annotation></semantics></math>% at Peak A. </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> 2.4 Survey Capability and Overall Scientific Potential</h3> <div class="ltx_para" id="S2.SS4.p1"> Given the current monthly statistics of clear night fraction, the Lenghu site could provide <math alttext="\sim" class="ltx_Math" display="inline" id="S2.SS4.p1.1.m1.1"><semantics id="S2.SS4.p1.1.m1.1a"><mo id="S2.SS4.p1.1.m1.1.1" xref="S2.SS4.p1.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.m1.1b"><csymbol cd="latexml" id="S2.SS4.p1.1.m1.1.1.cmml" xref="S2.SS4.p1.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.m1.1d">∼</annotation></semantics></math> 2,400 and 2,800 observing hours per year. Assuming a configuration with 20,000 working fibers and a <math alttext=">90" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m2.1"><semantics id="S2.SS4.p1.2.m2.1a"><mrow id="S2.SS4.p1.2.m2.1.1" xref="S2.SS4.p1.2.m2.1.1.cmml"><mi id="S2.SS4.p1.2.m2.1.1.2" xref="S2.SS4.p1.2.m2.1.1.2.cmml"></mi><mo id="S2.SS4.p1.2.m2.1.1.1" xref="S2.SS4.p1.2.m2.1.1.1.cmml">></mo><mn id="S2.SS4.p1.2.m2.1.1.3" xref="S2.SS4.p1.2.m2.1.1.3.cmml">90</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.2.m2.1b"><apply id="S2.SS4.p1.2.m2.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1"><gt id="S2.SS4.p1.2.m2.1.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1.1"></gt><csymbol cd="latexml" id="S2.SS4.p1.2.m2.1.1.2.cmml" xref="S2.SS4.p1.2.m2.1.1.2">absent</csymbol><cn id="S2.SS4.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS4.p1.2.m2.1.1.3">90</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m2.1c">>90</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m2.1d">> 90</annotation></semantics></math>% up time, MUST will have 210-270 million fiber hours in a five-year survey, allowing MUST to conduct Stage-V LSS surveys. As a dedicated survey facility, the long-term commitment of MUST to spectroscopic surveys is another unique strength in fulfilling the Stage-V cosmological goals. Compared to DESI, MUST has a 3.4<math alttext="\times" class="ltx_Math" display="inline" id="S2.SS4.p1.3.m3.1"><semantics id="S2.SS4.p1.3.m3.1a"><mo id="S2.SS4.p1.3.m3.1.1" xref="S2.SS4.p1.3.m3.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.3.m3.1b"><times id="S2.SS4.p1.3.m3.1.1.cmml" xref="S2.SS4.p1.3.m3.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m3.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m3.1d">×</annotation></semantics></math> higher light collecting capability and 4<math alttext="\times" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m4.1"><semantics id="S2.SS4.p1.4.m4.1a"><mo id="S2.SS4.p1.4.m4.1.1" xref="S2.SS4.p1.4.m4.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.4.m4.1b"><times id="S2.SS4.p1.4.m4.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.4.m4.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.4.m4.1d">×</annotation></semantics></math> more fibers. Even considering the throughput costs of the secondary mirror, the more complex WFC design, the longer fiber route, and challenges in the spectrograph design, MUST can still improve the spectroscopic survey efficiency – the number of redshifts measures for the same set of targets during the same time – by <math alttext="\sim 10\times" class="ltx_math_unparsed" display="inline" id="S2.SS4.p1.5.m5.1"><semantics id="S2.SS4.p1.5.m5.1a"><mrow id="S2.SS4.p1.5.m5.1b"><mo id="S2.SS4.p1.5.m5.1.1">∼</mo><mn id="S2.SS4.p1.5.m5.1.2">10</mn><mo id="S2.SS4.p1.5.m5.1.3" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="S2.SS4.p1.5.m5.1c">\sim 10\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.5.m5.1d">∼ 10 ×</annotation></semantics></math>. </div> <div class="ltx_para" id="S2.SS4.p2"> While the Stage-V cosmological survey will be the highest priority of MUST during the first phase of its operation, the significant fiber-hour budget will allow us to design versatile programs with a wide range of scientific goals. In addition to mapping the large-scale structures for cosmology, the same dataset will enable robust statistical studies of galaxies and Active Galactic Nuclei (AGN) near and far. These data can help us understand the rise and fall of star formation in galaxies in the last 10 Gyrs and the assembly of different populations of galaxies. It will also enable us to measure the accretion rate and mass of supermassive black holes (SMBHs) in a galaxy sample that is one order of magnitude larger than the existing one and significantly improve our understanding of the growth of SMBH and its impact on galaxy evolution. At the small, non-linear scale, the detailed picture of galaxy clustering will enable better modeling of galaxy-halo connections, which will help cosmology in return. During bright nights, MUST can measure the radial velocities and chemical abundances of many halo stars in the Milky Way to shed light on its assembly history and constrain the nature of dark matter. Located at a longitude less populated with large ground-based telescopes, MUST is also poised to play an exciting role in the age of time-domain spectroscopic surveys. The list can go on. </div> <div class="ltx_para" id="S2.SS4.p3"> In addition, the modular focal plane of MUST secures future opportunities for instrument upgrades: high-resolution fiber spectrograph and integrated field spectrograph (IFS) can be straightforwardly added to the focal plane as long as their front-end optics fit into the current fiber positioner module. In the following publications from this series, we will discuss these topics’ scientific potential and strategy in detail. </div> <div class="ltx_para" id="S2.SS4.p4"> For the rest of this work, we will focus on the first and foremost scientific goal of MUST: the first Stage-V LSS spectroscopic survey to deepen our understanding of several fundamental questions in cosmology and physics. </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="405" id="S2.F2.g1" src="extracted/5997959/figures/fig2.png" width="598"/> <figcaption class="ltx_caption ltx_centering">Figure 2: (a) Full sky map displaying relevant imaging (CSST red, UNIONS yellow/purple, LSST green, and Euclid light blue) and spectroscopic (DESI black, 4MOST light yellow) surveys on top of the dust opacity map from the Planck mission <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib62" title="">62</a>]</cite>. The survey area reachable by MUST is highlighted in blue (dec <math alttext=">-10^{\circ}" class="ltx_Math" display="inline" id="S2.F2.3.m1.1"><semantics id="S2.F2.3.m1.1b"><mrow id="S2.F2.3.m1.1.1" xref="S2.F2.3.m1.1.1.cmml"><mi id="S2.F2.3.m1.1.1.2" xref="S2.F2.3.m1.1.1.2.cmml"></mi><mo id="S2.F2.3.m1.1.1.1" xref="S2.F2.3.m1.1.1.1.cmml">></mo><mrow id="S2.F2.3.m1.1.1.3" xref="S2.F2.3.m1.1.1.3.cmml"><mo id="S2.F2.3.m1.1.1.3b" xref="S2.F2.3.m1.1.1.3.cmml">−</mo><msup id="S2.F2.3.m1.1.1.3.2" xref="S2.F2.3.m1.1.1.3.2.cmml"><mn id="S2.F2.3.m1.1.1.3.2.2" xref="S2.F2.3.m1.1.1.3.2.2.cmml">10</mn><mo id="S2.F2.3.m1.1.1.3.2.3" xref="S2.F2.3.m1.1.1.3.2.3.cmml">∘</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.3.m1.1c"><apply id="S2.F2.3.m1.1.1.cmml" xref="S2.F2.3.m1.1.1"><gt id="S2.F2.3.m1.1.1.1.cmml" xref="S2.F2.3.m1.1.1.1"></gt><csymbol cd="latexml" id="S2.F2.3.m1.1.1.2.cmml" xref="S2.F2.3.m1.1.1.2">absent</csymbol><apply id="S2.F2.3.m1.1.1.3.cmml" xref="S2.F2.3.m1.1.1.3"><minus id="S2.F2.3.m1.1.1.3.1.cmml" xref="S2.F2.3.m1.1.1.3"></minus><apply id="S2.F2.3.m1.1.1.3.2.cmml" xref="S2.F2.3.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.F2.3.m1.1.1.3.2.1.cmml" xref="S2.F2.3.m1.1.1.3.2">superscript</csymbol><cn id="S2.F2.3.m1.1.1.3.2.2.cmml" type="integer" xref="S2.F2.3.m1.1.1.3.2.2">10</cn><compose id="S2.F2.3.m1.1.1.3.2.3.cmml" xref="S2.F2.3.m1.1.1.3.2.3"></compose></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.3.m1.1d">>-10^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.3.m1.1e">> - 10 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>). CSST, UNIONS, LSST, and Euclid will be critical for the target selection of the cosmological survey of MUST. (b) Evolution of the number of galaxy redshifts from redshift surveys starting from the CfA1 survey in the early 1980s. The shaded regions with different colors represent the rough separations between various stages of large-scale structure surveys for cosmology. The star symbols highlight a few signature surveys for the previous four stages of spectroscopic surveys. MUST strives to become the first Stage-V (red background) survey in operation in the early 2030s. (c) Apparent magnitude limit in <math alttext="i" class="ltx_Math" display="inline" id="S2.F2.4.m2.1"><semantics id="S2.F2.4.m2.1b"><mi id="S2.F2.4.m2.1.1" xref="S2.F2.4.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.F2.4.m2.1c"><ci id="S2.F2.4.m2.1.1.cmml" xref="S2.F2.4.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.4.m2.1d">i</annotation><annotation encoding="application/x-llamapun" id="S2.F2.4.m2.1e">italic_i</annotation></semantics></math>-band reachable by different redshift surveys as a function of the survey area. (d) Number of redshifts as a function of the comoving volume of redshift galaxy surveys. In the two bottom panels, the colors of the symbols reflect the different cosmology stages. These different metrics highlight the unique scientific potential of MUST. </figcaption> </figure> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> 3 Scientific Motivations of the Stage-V Cosmological Surveys</h2> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="787" id="S3.F3.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 3: Top panel: A schematic diagram illustrating power spectra of various cosmological models, with error bars predicted from the current survey design of MUST. Middle panel: A ratio plot highlighting MUST’s ability to distinguish between different cosmological models. Bottom panel: Key science cases for different scales and redshifts. The fiducial <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.F3.5.m1.1"><semantics id="S3.F3.5.m1.1b"><mi id="S3.F3.5.m1.1.1" mathvariant="normal" xref="S3.F3.5.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.F3.5.m1.1c"><ci id="S3.F3.5.m1.1.1.cmml" xref="S3.F3.5.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.5.m1.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.F3.5.m1.1e">roman_Λ</annotation></semantics></math>CDM power spectrum is based on Planck 2015 results <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib63" title="">63</a>]</cite>; error bars and power spectra of different cosmological models are rescaled to the fiducial power spectrum accordingly. Error bars for <math alttext="k<0.4\,h\,{\rm Mpc^{-1}}" class="ltx_Math" display="inline" id="S3.F3.6.m2.1"><semantics id="S3.F3.6.m2.1b"><mrow id="S3.F3.6.m2.1.1" xref="S3.F3.6.m2.1.1.cmml"><mi id="S3.F3.6.m2.1.1.2" xref="S3.F3.6.m2.1.1.2.cmml">k</mi><mo id="S3.F3.6.m2.1.1.1" xref="S3.F3.6.m2.1.1.1.cmml"><</mo><mrow id="S3.F3.6.m2.1.1.3" xref="S3.F3.6.m2.1.1.3.cmml"><mn id="S3.F3.6.m2.1.1.3.2" xref="S3.F3.6.m2.1.1.3.2.cmml">0.4</mn><mo id="S3.F3.6.m2.1.1.3.1" lspace="0.170em" xref="S3.F3.6.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.F3.6.m2.1.1.3.3" xref="S3.F3.6.m2.1.1.3.3.cmml">h</mi><mo id="S3.F3.6.m2.1.1.3.1b" lspace="0.170em" xref="S3.F3.6.m2.1.1.3.1.cmml">⁢</mo><msup id="S3.F3.6.m2.1.1.3.4" xref="S3.F3.6.m2.1.1.3.4.cmml"><mi id="S3.F3.6.m2.1.1.3.4.2" xref="S3.F3.6.m2.1.1.3.4.2.cmml">Mpc</mi><mrow id="S3.F3.6.m2.1.1.3.4.3" xref="S3.F3.6.m2.1.1.3.4.3.cmml"><mo id="S3.F3.6.m2.1.1.3.4.3b" xref="S3.F3.6.m2.1.1.3.4.3.cmml">−</mo><mn id="S3.F3.6.m2.1.1.3.4.3.2" xref="S3.F3.6.m2.1.1.3.4.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.6.m2.1c"><apply id="S3.F3.6.m2.1.1.cmml" xref="S3.F3.6.m2.1.1"><lt id="S3.F3.6.m2.1.1.1.cmml" xref="S3.F3.6.m2.1.1.1"></lt><ci id="S3.F3.6.m2.1.1.2.cmml" xref="S3.F3.6.m2.1.1.2">𝑘</ci><apply id="S3.F3.6.m2.1.1.3.cmml" xref="S3.F3.6.m2.1.1.3"><times id="S3.F3.6.m2.1.1.3.1.cmml" xref="S3.F3.6.m2.1.1.3.1"></times><cn id="S3.F3.6.m2.1.1.3.2.cmml" type="float" xref="S3.F3.6.m2.1.1.3.2">0.4</cn><ci id="S3.F3.6.m2.1.1.3.3.cmml" xref="S3.F3.6.m2.1.1.3.3">ℎ</ci><apply id="S3.F3.6.m2.1.1.3.4.cmml" xref="S3.F3.6.m2.1.1.3.4"><csymbol cd="ambiguous" id="S3.F3.6.m2.1.1.3.4.1.cmml" xref="S3.F3.6.m2.1.1.3.4">superscript</csymbol><ci id="S3.F3.6.m2.1.1.3.4.2.cmml" xref="S3.F3.6.m2.1.1.3.4.2">Mpc</ci><apply id="S3.F3.6.m2.1.1.3.4.3.cmml" xref="S3.F3.6.m2.1.1.3.4.3"><minus id="S3.F3.6.m2.1.1.3.4.3.1.cmml" xref="S3.F3.6.m2.1.1.3.4.3"></minus><cn id="S3.F3.6.m2.1.1.3.4.3.2.cmml" type="integer" xref="S3.F3.6.m2.1.1.3.4.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.6.m2.1d">k<0.4\,h\,{\rm Mpc^{-1}}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.6.m2.1e">italic_k < 0.4 italic_h roman_Mpc start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> are estimated using Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.E15" title="Equation 15 ‣ 5.1 Methodology ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">15</a>) with the expected bias and number density of MUST’s LBG sample, while those for <math alttext="k>0.4\,h\,{\rm Mpc^{-1}}" class="ltx_Math" display="inline" id="S3.F3.7.m3.1"><semantics id="S3.F3.7.m3.1b"><mrow id="S3.F3.7.m3.1.1" xref="S3.F3.7.m3.1.1.cmml"><mi id="S3.F3.7.m3.1.1.2" xref="S3.F3.7.m3.1.1.2.cmml">k</mi><mo id="S3.F3.7.m3.1.1.1" xref="S3.F3.7.m3.1.1.1.cmml">></mo><mrow id="S3.F3.7.m3.1.1.3" xref="S3.F3.7.m3.1.1.3.cmml"><mn id="S3.F3.7.m3.1.1.3.2" xref="S3.F3.7.m3.1.1.3.2.cmml">0.4</mn><mo id="S3.F3.7.m3.1.1.3.1" lspace="0.170em" xref="S3.F3.7.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.F3.7.m3.1.1.3.3" xref="S3.F3.7.m3.1.1.3.3.cmml">h</mi><mo id="S3.F3.7.m3.1.1.3.1b" lspace="0.170em" xref="S3.F3.7.m3.1.1.3.1.cmml">⁢</mo><msup id="S3.F3.7.m3.1.1.3.4" xref="S3.F3.7.m3.1.1.3.4.cmml"><mi id="S3.F3.7.m3.1.1.3.4.2" xref="S3.F3.7.m3.1.1.3.4.2.cmml">Mpc</mi><mrow id="S3.F3.7.m3.1.1.3.4.3" xref="S3.F3.7.m3.1.1.3.4.3.cmml"><mo id="S3.F3.7.m3.1.1.3.4.3b" xref="S3.F3.7.m3.1.1.3.4.3.cmml">−</mo><mn id="S3.F3.7.m3.1.1.3.4.3.2" xref="S3.F3.7.m3.1.1.3.4.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.7.m3.1c"><apply id="S3.F3.7.m3.1.1.cmml" xref="S3.F3.7.m3.1.1"><gt id="S3.F3.7.m3.1.1.1.cmml" xref="S3.F3.7.m3.1.1.1"></gt><ci id="S3.F3.7.m3.1.1.2.cmml" xref="S3.F3.7.m3.1.1.2">𝑘</ci><apply id="S3.F3.7.m3.1.1.3.cmml" xref="S3.F3.7.m3.1.1.3"><times id="S3.F3.7.m3.1.1.3.1.cmml" xref="S3.F3.7.m3.1.1.3.1"></times><cn id="S3.F3.7.m3.1.1.3.2.cmml" type="float" xref="S3.F3.7.m3.1.1.3.2">0.4</cn><ci id="S3.F3.7.m3.1.1.3.3.cmml" xref="S3.F3.7.m3.1.1.3.3">ℎ</ci><apply id="S3.F3.7.m3.1.1.3.4.cmml" xref="S3.F3.7.m3.1.1.3.4"><csymbol cd="ambiguous" id="S3.F3.7.m3.1.1.3.4.1.cmml" xref="S3.F3.7.m3.1.1.3.4">superscript</csymbol><ci id="S3.F3.7.m3.1.1.3.4.2.cmml" xref="S3.F3.7.m3.1.1.3.4.2">Mpc</ci><apply id="S3.F3.7.m3.1.1.3.4.3.cmml" xref="S3.F3.7.m3.1.1.3.4.3"><minus id="S3.F3.7.m3.1.1.3.4.3.1.cmml" xref="S3.F3.7.m3.1.1.3.4.3"></minus><cn id="S3.F3.7.m3.1.1.3.4.3.2.cmml" type="integer" xref="S3.F3.7.m3.1.1.3.4.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.7.m3.1d">k>0.4\,h\,{\rm Mpc^{-1}}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.7.m3.1e">italic_k > 0.4 italic_h roman_Mpc start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> are obtained from Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.F3.8.m4.1"><semantics id="S3.F3.8.m4.1b"><mi id="S3.F3.8.m4.1.1" xref="S3.F3.8.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.F3.8.m4.1c"><ci id="S3.F3.8.m4.1.1.cmml" xref="S3.F3.8.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.8.m4.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.F3.8.m4.1e">italic_α</annotation></semantics></math> 3D power spectrum forecasts, taking into account the effects of spectroscopic systematics and the quasar luminosity function. MUST’s unprecedentedly large survey volume enables probing of the large-scale power spectrum with high accuracy, which is crucial for distinguishing between different inflation models. Moreover, MUST’s high sample number density allows for precise measurements power spectrum at small scales, which is sensitive to science cases such as dark matter properties and modified gravity models. </figcaption> </figure> <div class="ltx_para" id="S3.p1"> Ever since the discovery of the accelerated expansion of the Universe using type Ia supernova data <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib64" title="">64</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib65" title="">65</a>]</cite>, <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" mathvariant="normal" xref="S3.p1.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">roman_Λ</annotation></semantics></math>CDM has been widely accepted as the standard cosmology model (cf., however, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib66" title="">66</a>]</cite>, for earlier observational evidence of a nonzero <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mi id="S3.p1.2.m2.1.1" mathvariant="normal" xref="S3.p1.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><ci id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">roman_Λ</annotation></semantics></math>). However, several key components of the <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mi id="S3.p1.3.m3.1.1" mathvariant="normal" xref="S3.p1.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">roman_Λ</annotation></semantics></math>CDM model remain unknown, such as the physical origin of <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mi id="S3.p1.4.m4.1.1" mathvariant="normal" xref="S3.p1.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><ci id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">roman_Λ</annotation></semantics></math> (or, more generally, dark energy), the nature of dark matter, and the establishment of initial conditions of cosmic structures. Meanwhile, several observational challenges to <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><mi id="S3.p1.5.m5.1.1" mathvariant="normal" xref="S3.p1.5.m5.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><ci id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">roman_Λ</annotation></semantics></math>CDM have emerged as the precision of cosmological measurements improves (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib67" title="">67</a>]</cite>). Massive spectroscopic surveys are expected to help address these issues. By analyzing the clustering of large-scale structures (LSS) from spectroscopic data, we can investigate the properties of dark energy and dark matter through the dynamic evolution of the Universe governed by these competing components and extract signatures of fundamental physical processes at extremely high energy scale in the primordial Universe. </div> <div class="ltx_para" id="S3.p2"> In this section, we review the scientific motivations of the next-generation (Stage-V) spectroscopic survey to be carried out by MUST. Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.F3" title="Figure 3 ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3</a> shows some of the key scientific cases that can be tested by MUST with a high significance level. </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> 3.1 Nature & Evolution of Dark Energy</h3> <div class="ltx_para" id="S3.SS1.p1"> Dark energy is one of the key scientific problems of our time and poses a significant challenge to the Standard Model of particle physics. While CMB observations suggest that dark energy constitutes about 70 % of the energy density of the Universe in the <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mi id="S3.SS1.p1.1.m1.1.1" mathvariant="normal" xref="S3.SS1.p1.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">roman_Λ</annotation></semantics></math>CDM model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib4" title="">4</a>]</cite>, its physical properties are largely unexplored. One exception is the equation of state (EoS) parameter <math alttext="w" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_w</annotation></semantics></math>, which influences the dynamics of the Universe and can be probed through geometrical measurements. The most straightforward interpretation of dark energy is the cosmological constant (<math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mi id="S3.SS1.p1.3.m3.1.1" mathvariant="normal" xref="S3.SS1.p1.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">roman_Λ</annotation></semantics></math>), corresponding to a constant <math alttext="w=-1" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mrow id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml"><mi id="S3.SS1.p1.4.m4.1.1.2" xref="S3.SS1.p1.4.m4.1.1.2.cmml">w</mi><mo id="S3.SS1.p1.4.m4.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S3.SS1.p1.4.m4.1.1.3" xref="S3.SS1.p1.4.m4.1.1.3.cmml"><mo id="S3.SS1.p1.4.m4.1.1.3a" xref="S3.SS1.p1.4.m4.1.1.3.cmml">−</mo><mn id="S3.SS1.p1.4.m4.1.1.3.2" xref="S3.SS1.p1.4.m4.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><apply id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1"><eq id="S3.SS1.p1.4.m4.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"></eq><ci id="S3.SS1.p1.4.m4.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.2">𝑤</ci><apply id="S3.SS1.p1.4.m4.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.3"><minus id="S3.SS1.p1.4.m4.1.1.3.1.cmml" xref="S3.SS1.p1.4.m4.1.1.3"></minus><cn id="S3.SS1.p1.4.m4.1.1.3.2.cmml" type="integer" xref="S3.SS1.p1.4.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">w=-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">italic_w = - 1</annotation></semantics></math>, which is well consistent with most observations to date (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib10" title="">10</a>]</cite>). A natural extension beyond <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mi id="S3.SS1.p1.5.m5.1.1" mathvariant="normal" xref="S3.SS1.p1.5.m5.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><ci id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">roman_Λ</annotation></semantics></math> involves introducing dynamic dark energy, which potentially originates from various physical mechanisms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib68" title="">68</a>]</cite> and leads to a time-evolving EoS parameter typically expressed as </div> <div class="ltx_para" id="S3.SS1.p2"> <table class="ltx_equation ltx_eqn_table" id="S3.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w(a)=w_{0}+w_{a}(1-a)," class="ltx_Math" display="block" id="S3.E1.m1.2"><semantics id="S3.E1.m1.2a"><mrow id="S3.E1.m1.2.2.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1.3" xref="S3.E1.m1.2.2.1.1.3.cmml"><mi id="S3.E1.m1.2.2.1.1.3.2" xref="S3.E1.m1.2.2.1.1.3.2.cmml">w</mi><mo id="S3.E1.m1.2.2.1.1.3.1" xref="S3.E1.m1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S3.E1.m1.2.2.1.1.3.3.2" xref="S3.E1.m1.2.2.1.1.3.cmml"><mo id="S3.E1.m1.2.2.1.1.3.3.2.1" stretchy="false" xref="S3.E1.m1.2.2.1.1.3.cmml">(</mo><mi id="S3.E1.m1.1.1" xref="S3.E1.m1.1.1.cmml">a</mi><mo id="S3.E1.m1.2.2.1.1.3.3.2.2" stretchy="false" xref="S3.E1.m1.2.2.1.1.3.cmml">)</mo></mrow></mrow><mo id="S3.E1.m1.2.2.1.1.2" xref="S3.E1.m1.2.2.1.1.2.cmml">=</mo><mrow id="S3.E1.m1.2.2.1.1.1" xref="S3.E1.m1.2.2.1.1.1.cmml"><msub id="S3.E1.m1.2.2.1.1.1.3" xref="S3.E1.m1.2.2.1.1.1.3.cmml"><mi id="S3.E1.m1.2.2.1.1.1.3.2" xref="S3.E1.m1.2.2.1.1.1.3.2.cmml">w</mi><mn id="S3.E1.m1.2.2.1.1.1.3.3" xref="S3.E1.m1.2.2.1.1.1.3.3.cmml">0</mn></msub><mo id="S3.E1.m1.2.2.1.1.1.2" xref="S3.E1.m1.2.2.1.1.1.2.cmml">+</mo><mrow id="S3.E1.m1.2.2.1.1.1.1" xref="S3.E1.m1.2.2.1.1.1.1.cmml"><msub id="S3.E1.m1.2.2.1.1.1.1.3" xref="S3.E1.m1.2.2.1.1.1.1.3.cmml"><mi id="S3.E1.m1.2.2.1.1.1.1.3.2" xref="S3.E1.m1.2.2.1.1.1.1.3.2.cmml">w</mi><mi id="S3.E1.m1.2.2.1.1.1.1.3.3" xref="S3.E1.m1.2.2.1.1.1.1.3.3.cmml">a</mi></msub><mo id="S3.E1.m1.2.2.1.1.1.1.2" xref="S3.E1.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.2.2.1.1.1.1.1.1" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E1.m1.2.2.1.1.1.1.1.1.1" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.cmml"><mn id="S3.E1.m1.2.2.1.1.1.1.1.1.1.2" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.E1.m1.2.2.1.1.1.1.1.1.1.1" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S3.E1.m1.2.2.1.1.1.1.1.1.1.3" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.3.cmml">a</mi></mrow><mo id="S3.E1.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.E1.m1.2.2.1.2" xref="S3.E1.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E1.m1.2b"><apply id="S3.E1.m1.2.2.1.1.cmml" xref="S3.E1.m1.2.2.1"><eq id="S3.E1.m1.2.2.1.1.2.cmml" xref="S3.E1.m1.2.2.1.1.2"></eq><apply id="S3.E1.m1.2.2.1.1.3.cmml" xref="S3.E1.m1.2.2.1.1.3"><times id="S3.E1.m1.2.2.1.1.3.1.cmml" xref="S3.E1.m1.2.2.1.1.3.1"></times><ci id="S3.E1.m1.2.2.1.1.3.2.cmml" xref="S3.E1.m1.2.2.1.1.3.2">𝑤</ci><ci id="S3.E1.m1.1.1.cmml" xref="S3.E1.m1.1.1">𝑎</ci></apply><apply id="S3.E1.m1.2.2.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.1"><plus id="S3.E1.m1.2.2.1.1.1.2.cmml" xref="S3.E1.m1.2.2.1.1.1.2"></plus><apply id="S3.E1.m1.2.2.1.1.1.3.cmml" xref="S3.E1.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.1.3.1.cmml" xref="S3.E1.m1.2.2.1.1.1.3">subscript</csymbol><ci id="S3.E1.m1.2.2.1.1.1.3.2.cmml" xref="S3.E1.m1.2.2.1.1.1.3.2">𝑤</ci><cn id="S3.E1.m1.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.E1.m1.2.2.1.1.1.3.3">0</cn></apply><apply id="S3.E1.m1.2.2.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1"><times id="S3.E1.m1.2.2.1.1.1.1.2.cmml" xref="S3.E1.m1.2.2.1.1.1.1.2"></times><apply id="S3.E1.m1.2.2.1.1.1.1.3.cmml" xref="S3.E1.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.1.1.3.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.3">subscript</csymbol><ci id="S3.E1.m1.2.2.1.1.1.1.3.2.cmml" xref="S3.E1.m1.2.2.1.1.1.1.3.2">𝑤</ci><ci id="S3.E1.m1.2.2.1.1.1.1.3.3.cmml" xref="S3.E1.m1.2.2.1.1.1.1.3.3">𝑎</ci></apply><apply id="S3.E1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.1"><minus id="S3.E1.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.1"></minus><cn id="S3.E1.m1.2.2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.2">1</cn><ci id="S3.E1.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.1.1.3">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1.m1.2c">w(a)=w_{0}+w_{a}(1-a),</annotation><annotation encoding="application/x-llamapun" id="S3.E1.m1.2d">italic_w ( italic_a ) = italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_w start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( 1 - italic_a ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(1)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.p3"> where <math alttext="w_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><msub id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml"><mi id="S3.SS1.p3.1.m1.1.1.2" xref="S3.SS1.p3.1.m1.1.1.2.cmml">w</mi><mn id="S3.SS1.p3.1.m1.1.1.3" xref="S3.SS1.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><apply id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.p3.1.m1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.1.1.2">𝑤</ci><cn id="S3.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">w_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the current value of <math alttext="w" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mi id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">italic_w</annotation></semantics></math> and <math alttext="w_{a}" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><msub id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><mi id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml">w</mi><mi id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.3.m3.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2">𝑤</ci><ci id="S3.SS1.p3.3.m3.1.1.3.cmml" xref="S3.SS1.p3.3.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">w_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">italic_w start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> quantifies its time evolution. For instance, the quintessence model, describing a dynamical scalar field minimally coupled to gravity, typically yields <math alttext="w\gtrsim-1" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mrow id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml"><mi id="S3.SS1.p3.4.m4.1.1.2" xref="S3.SS1.p3.4.m4.1.1.2.cmml">w</mi><mo id="S3.SS1.p3.4.m4.1.1.1" xref="S3.SS1.p3.4.m4.1.1.1.cmml">≳</mo><mrow id="S3.SS1.p3.4.m4.1.1.3" xref="S3.SS1.p3.4.m4.1.1.3.cmml"><mo id="S3.SS1.p3.4.m4.1.1.3a" xref="S3.SS1.p3.4.m4.1.1.3.cmml">−</mo><mn id="S3.SS1.p3.4.m4.1.1.3.2" xref="S3.SS1.p3.4.m4.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><apply id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1"><csymbol cd="latexml" id="S3.SS1.p3.4.m4.1.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S3.SS1.p3.4.m4.1.1.2.cmml" xref="S3.SS1.p3.4.m4.1.1.2">𝑤</ci><apply id="S3.SS1.p3.4.m4.1.1.3.cmml" xref="S3.SS1.p3.4.m4.1.1.3"><minus id="S3.SS1.p3.4.m4.1.1.3.1.cmml" xref="S3.SS1.p3.4.m4.1.1.3"></minus><cn id="S3.SS1.p3.4.m4.1.1.3.2.cmml" type="integer" xref="S3.SS1.p3.4.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">w\gtrsim-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">italic_w ≳ - 1</annotation></semantics></math>, while the phantom energy, a scalar field with negative kinematic energy, predicts <math alttext="w<-1" class="ltx_Math" display="inline" id="S3.SS1.p3.5.m5.1"><semantics id="S3.SS1.p3.5.m5.1a"><mrow id="S3.SS1.p3.5.m5.1.1" xref="S3.SS1.p3.5.m5.1.1.cmml"><mi id="S3.SS1.p3.5.m5.1.1.2" xref="S3.SS1.p3.5.m5.1.1.2.cmml">w</mi><mo id="S3.SS1.p3.5.m5.1.1.1" xref="S3.SS1.p3.5.m5.1.1.1.cmml"><</mo><mrow id="S3.SS1.p3.5.m5.1.1.3" xref="S3.SS1.p3.5.m5.1.1.3.cmml"><mo id="S3.SS1.p3.5.m5.1.1.3a" xref="S3.SS1.p3.5.m5.1.1.3.cmml">−</mo><mn id="S3.SS1.p3.5.m5.1.1.3.2" xref="S3.SS1.p3.5.m5.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.5.m5.1b"><apply id="S3.SS1.p3.5.m5.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1"><lt id="S3.SS1.p3.5.m5.1.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1.1"></lt><ci id="S3.SS1.p3.5.m5.1.1.2.cmml" xref="S3.SS1.p3.5.m5.1.1.2">𝑤</ci><apply id="S3.SS1.p3.5.m5.1.1.3.cmml" xref="S3.SS1.p3.5.m5.1.1.3"><minus id="S3.SS1.p3.5.m5.1.1.3.1.cmml" xref="S3.SS1.p3.5.m5.1.1.3"></minus><cn id="S3.SS1.p3.5.m5.1.1.3.2.cmml" type="integer" xref="S3.SS1.p3.5.m5.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.5.m5.1c">w<-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.5.m5.1d">italic_w < - 1</annotation></semantics></math>. Observational constraints on <math alttext="w_{0}" class="ltx_Math" display="inline" id="S3.SS1.p3.6.m6.1"><semantics id="S3.SS1.p3.6.m6.1a"><msub id="S3.SS1.p3.6.m6.1.1" xref="S3.SS1.p3.6.m6.1.1.cmml"><mi id="S3.SS1.p3.6.m6.1.1.2" xref="S3.SS1.p3.6.m6.1.1.2.cmml">w</mi><mn id="S3.SS1.p3.6.m6.1.1.3" xref="S3.SS1.p3.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.6.m6.1b"><apply id="S3.SS1.p3.6.m6.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.6.m6.1.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.p3.6.m6.1.1.2.cmml" xref="S3.SS1.p3.6.m6.1.1.2">𝑤</ci><cn id="S3.SS1.p3.6.m6.1.1.3.cmml" type="integer" xref="S3.SS1.p3.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.6.m6.1c">w_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.6.m6.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="w_{a}" class="ltx_Math" display="inline" id="S3.SS1.p3.7.m7.1"><semantics id="S3.SS1.p3.7.m7.1a"><msub id="S3.SS1.p3.7.m7.1.1" xref="S3.SS1.p3.7.m7.1.1.cmml"><mi id="S3.SS1.p3.7.m7.1.1.2" xref="S3.SS1.p3.7.m7.1.1.2.cmml">w</mi><mi id="S3.SS1.p3.7.m7.1.1.3" xref="S3.SS1.p3.7.m7.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.7.m7.1b"><apply id="S3.SS1.p3.7.m7.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.7.m7.1.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1">subscript</csymbol><ci id="S3.SS1.p3.7.m7.1.1.2.cmml" xref="S3.SS1.p3.7.m7.1.1.2">𝑤</ci><ci id="S3.SS1.p3.7.m7.1.1.3.cmml" xref="S3.SS1.p3.7.m7.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.7.m7.1c">w_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.7.m7.1d">italic_w start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> provide a pathway to distinguishing these models. </div> <div class="ltx_para" id="S3.SS1.p4"> The constraints on EoS parameters rely primarily on geometrical probes, including “standard(-isable) candles” such as Cepheids and type Ia supernovae (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib7" title="">7</a>]</cite>), and “standard rulers” such as Baryon Acoustic Oscillations (BAO; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib69" title="">69</a>]</cite>) and the matter-radiation equality scale (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib70" title="">70</a>]</cite>). BAO arises from acoustic density waves in the primordial Universe and produces an excess of matter correlation amplitude at a characteristic comoving scale of <math alttext="\sim 100\,h^{-1}\,{\rm Mpc}" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mrow id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml"><mi id="S3.SS1.p4.1.m1.1.1.2" xref="S3.SS1.p4.1.m1.1.1.2.cmml"></mi><mo id="S3.SS1.p4.1.m1.1.1.1" xref="S3.SS1.p4.1.m1.1.1.1.cmml">∼</mo><mrow id="S3.SS1.p4.1.m1.1.1.3" xref="S3.SS1.p4.1.m1.1.1.3.cmml"><mn id="S3.SS1.p4.1.m1.1.1.3.2" xref="S3.SS1.p4.1.m1.1.1.3.2.cmml">100</mn><mo id="S3.SS1.p4.1.m1.1.1.3.1" lspace="0.170em" xref="S3.SS1.p4.1.m1.1.1.3.1.cmml">⁢</mo><msup id="S3.SS1.p4.1.m1.1.1.3.3" xref="S3.SS1.p4.1.m1.1.1.3.3.cmml"><mi id="S3.SS1.p4.1.m1.1.1.3.3.2" xref="S3.SS1.p4.1.m1.1.1.3.3.2.cmml">h</mi><mrow id="S3.SS1.p4.1.m1.1.1.3.3.3" xref="S3.SS1.p4.1.m1.1.1.3.3.3.cmml"><mo id="S3.SS1.p4.1.m1.1.1.3.3.3a" xref="S3.SS1.p4.1.m1.1.1.3.3.3.cmml">−</mo><mn id="S3.SS1.p4.1.m1.1.1.3.3.3.2" xref="S3.SS1.p4.1.m1.1.1.3.3.3.2.cmml">1</mn></mrow></msup><mo id="S3.SS1.p4.1.m1.1.1.3.1a" xref="S3.SS1.p4.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.p4.1.m1.1.1.3.4" xref="S3.SS1.p4.1.m1.1.1.3.4.cmml">Mpc</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><apply id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1"><csymbol cd="latexml" id="S3.SS1.p4.1.m1.1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS1.p4.1.m1.1.1.2.cmml" xref="S3.SS1.p4.1.m1.1.1.2">absent</csymbol><apply id="S3.SS1.p4.1.m1.1.1.3.cmml" xref="S3.SS1.p4.1.m1.1.1.3"><times id="S3.SS1.p4.1.m1.1.1.3.1.cmml" xref="S3.SS1.p4.1.m1.1.1.3.1"></times><cn id="S3.SS1.p4.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS1.p4.1.m1.1.1.3.2">100</cn><apply id="S3.SS1.p4.1.m1.1.1.3.3.cmml" xref="S3.SS1.p4.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p4.1.m1.1.1.3.3.1.cmml" xref="S3.SS1.p4.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.SS1.p4.1.m1.1.1.3.3.2.cmml" xref="S3.SS1.p4.1.m1.1.1.3.3.2">ℎ</ci><apply id="S3.SS1.p4.1.m1.1.1.3.3.3.cmml" xref="S3.SS1.p4.1.m1.1.1.3.3.3"><minus id="S3.SS1.p4.1.m1.1.1.3.3.3.1.cmml" xref="S3.SS1.p4.1.m1.1.1.3.3.3"></minus><cn id="S3.SS1.p4.1.m1.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS1.p4.1.m1.1.1.3.3.3.2">1</cn></apply></apply><ci id="S3.SS1.p4.1.m1.1.1.3.4.cmml" xref="S3.SS1.p4.1.m1.1.1.3.4">Mpc</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">\sim 100\,h^{-1}\,{\rm Mpc}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">∼ 100 italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_Mpc</annotation></semantics></math>. By extracting the BAO signal from the clustering of matter tracers, such as galaxies and QSOs, we can probe the expansion rate of our Universe and thus constrain dark energy models. Recently, the tomographic Alcock–Paczynski (AP) test, which performs geometrical measurements based on the response of the apparent shape of galaxy clustering to the adopted cosmological model for redshift-to-distance conversion, has also been shown as a complementary probe of dark energy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib71" title="">71</a>]</cite>. </div> <div class="ltx_para" id="S3.SS1.p5"> Observational programs for measuring the dark energy EoS have been strategically planned to proceed through successive stages of cosmic surveys, as outlined in the Dark Energy Task Force report <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib15" title="">15</a>]</cite>. Since the first detections of the BAO signal from galaxies at <math alttext="z<0.47" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.1"><semantics id="S3.SS1.p5.1.m1.1a"><mrow id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml"><mi id="S3.SS1.p5.1.m1.1.1.2" xref="S3.SS1.p5.1.m1.1.1.2.cmml">z</mi><mo id="S3.SS1.p5.1.m1.1.1.1" xref="S3.SS1.p5.1.m1.1.1.1.cmml"><</mo><mn id="S3.SS1.p5.1.m1.1.1.3" xref="S3.SS1.p5.1.m1.1.1.3.cmml">0.47</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.1.m1.1b"><apply id="S3.SS1.p5.1.m1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1"><lt id="S3.SS1.p5.1.m1.1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1.1"></lt><ci id="S3.SS1.p5.1.m1.1.1.2.cmml" xref="S3.SS1.p5.1.m1.1.1.2">𝑧</ci><cn id="S3.SS1.p5.1.m1.1.1.3.cmml" type="float" xref="S3.SS1.p5.1.m1.1.1.3">0.47</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.1.m1.1c">z<0.47</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.1.m1.1d">italic_z < 0.47</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib72" title="">72</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib12" title="">12</a>]</cite>, subsequent spectroscopic surveys have extended BAO measurements to galaxies across a wide redshift range (up to <math alttext="z<1.5" class="ltx_Math" display="inline" id="S3.SS1.p5.2.m2.1"><semantics id="S3.SS1.p5.2.m2.1a"><mrow id="S3.SS1.p5.2.m2.1.1" xref="S3.SS1.p5.2.m2.1.1.cmml"><mi id="S3.SS1.p5.2.m2.1.1.2" xref="S3.SS1.p5.2.m2.1.1.2.cmml">z</mi><mo id="S3.SS1.p5.2.m2.1.1.1" xref="S3.SS1.p5.2.m2.1.1.1.cmml"><</mo><mn id="S3.SS1.p5.2.m2.1.1.3" xref="S3.SS1.p5.2.m2.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.2.m2.1b"><apply id="S3.SS1.p5.2.m2.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1"><lt id="S3.SS1.p5.2.m2.1.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1.1"></lt><ci id="S3.SS1.p5.2.m2.1.1.2.cmml" xref="S3.SS1.p5.2.m2.1.1.2">𝑧</ci><cn id="S3.SS1.p5.2.m2.1.1.3.cmml" type="float" xref="S3.SS1.p5.2.m2.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.2.m2.1c">z<1.5</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.2.m2.1d">italic_z < 1.5</annotation></semantics></math>), as well as QSOs and Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p5.3.m3.1"><semantics id="S3.SS1.p5.3.m3.1a"><mi id="S3.SS1.p5.3.m3.1.1" xref="S3.SS1.p5.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.3.m3.1b"><ci id="S3.SS1.p5.3.m3.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.3.m3.1d">italic_α</annotation></semantics></math> tracers at even higher redshifts <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib10" title="">10</a>]</cite>. These efforts have substantially improved the precision of <math alttext="w" class="ltx_Math" display="inline" id="S3.SS1.p5.4.m4.1"><semantics id="S3.SS1.p5.4.m4.1a"><mi id="S3.SS1.p5.4.m4.1.1" xref="S3.SS1.p5.4.m4.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.4.m4.1b"><ci id="S3.SS1.p5.4.m4.1.1.cmml" xref="S3.SS1.p5.4.m4.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.4.m4.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.4.m4.1d">italic_w</annotation></semantics></math> measurements. Recently, DESI, a Stage-IV dark energy survey, reported a <math alttext="w" class="ltx_Math" display="inline" id="S3.SS1.p5.5.m5.1"><semantics id="S3.SS1.p5.5.m5.1a"><mi id="S3.SS1.p5.5.m5.1.1" xref="S3.SS1.p5.5.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.5.m5.1b"><ci id="S3.SS1.p5.5.m5.1.1.cmml" xref="S3.SS1.p5.5.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.5.m5.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.5.m5.1d">italic_w</annotation></semantics></math> consistent with <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS1.p5.6.m6.1"><semantics id="S3.SS1.p5.6.m6.1a"><mi id="S3.SS1.p5.6.m6.1.1" mathvariant="normal" xref="S3.SS1.p5.6.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.6.m6.1b"><ci id="S3.SS1.p5.6.m6.1.1.cmml" xref="S3.SS1.p5.6.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.6.m6.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.6.m6.1d">roman_Λ</annotation></semantics></math>CDM with an uncertainty of <math alttext="\sim 2.5\%" class="ltx_Math" display="inline" id="S3.SS1.p5.7.m7.1"><semantics id="S3.SS1.p5.7.m7.1a"><mrow id="S3.SS1.p5.7.m7.1.1" xref="S3.SS1.p5.7.m7.1.1.cmml"><mi id="S3.SS1.p5.7.m7.1.1.2" xref="S3.SS1.p5.7.m7.1.1.2.cmml"></mi><mo id="S3.SS1.p5.7.m7.1.1.1" xref="S3.SS1.p5.7.m7.1.1.1.cmml">∼</mo><mrow id="S3.SS1.p5.7.m7.1.1.3" xref="S3.SS1.p5.7.m7.1.1.3.cmml"><mn id="S3.SS1.p5.7.m7.1.1.3.2" xref="S3.SS1.p5.7.m7.1.1.3.2.cmml">2.5</mn><mo id="S3.SS1.p5.7.m7.1.1.3.1" xref="S3.SS1.p5.7.m7.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.7.m7.1b"><apply id="S3.SS1.p5.7.m7.1.1.cmml" xref="S3.SS1.p5.7.m7.1.1"><csymbol cd="latexml" id="S3.SS1.p5.7.m7.1.1.1.cmml" xref="S3.SS1.p5.7.m7.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS1.p5.7.m7.1.1.2.cmml" xref="S3.SS1.p5.7.m7.1.1.2">absent</csymbol><apply id="S3.SS1.p5.7.m7.1.1.3.cmml" xref="S3.SS1.p5.7.m7.1.1.3"><csymbol cd="latexml" id="S3.SS1.p5.7.m7.1.1.3.1.cmml" xref="S3.SS1.p5.7.m7.1.1.3.1">percent</csymbol><cn id="S3.SS1.p5.7.m7.1.1.3.2.cmml" type="float" xref="S3.SS1.p5.7.m7.1.1.3.2">2.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.7.m7.1c">\sim 2.5\%</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.7.m7.1d">∼ 2.5 %</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib73" title="">73</a>]</cite>. However, when the EoS is allowed to vary over time, current spectroscopic data hint at the potential dynamical behavior of dark energy (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib74" title="">74</a>]</cite>). With its unprecedented survey efficiency, MUST is expected to deliver even more precise BAO measurements, covering a redshift range from the nearby Universe to <math alttext="z\sim 5.5" class="ltx_Math" display="inline" id="S3.SS1.p5.8.m8.1"><semantics id="S3.SS1.p5.8.m8.1a"><mrow id="S3.SS1.p5.8.m8.1.1" xref="S3.SS1.p5.8.m8.1.1.cmml"><mi id="S3.SS1.p5.8.m8.1.1.2" xref="S3.SS1.p5.8.m8.1.1.2.cmml">z</mi><mo id="S3.SS1.p5.8.m8.1.1.1" xref="S3.SS1.p5.8.m8.1.1.1.cmml">∼</mo><mn id="S3.SS1.p5.8.m8.1.1.3" xref="S3.SS1.p5.8.m8.1.1.3.cmml">5.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.8.m8.1b"><apply id="S3.SS1.p5.8.m8.1.1.cmml" xref="S3.SS1.p5.8.m8.1.1"><csymbol cd="latexml" id="S3.SS1.p5.8.m8.1.1.1.cmml" xref="S3.SS1.p5.8.m8.1.1.1">similar-to</csymbol><ci id="S3.SS1.p5.8.m8.1.1.2.cmml" xref="S3.SS1.p5.8.m8.1.1.2">𝑧</ci><cn id="S3.SS1.p5.8.m8.1.1.3.cmml" type="float" xref="S3.SS1.p5.8.m8.1.1.3">5.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.8.m8.1c">z\sim 5.5</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.8.m8.1d">italic_z ∼ 5.5</annotation></semantics></math>. Thus, MUST meets the requirements of a Stage-V dark energy survey suggested by the Snowmass Cosmic Frontier report <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib75" title="">75</a>]</cite> and may offer deeper insights into the potential evolution of dark energy (see Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS2" title="5.2 Dark Energy ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.2</a>). Additionally, we will investigate the possibility of improving dark energy constraints using alternative probes, such as the tomographic AP test and the matter-radiation equality scale. </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> 3.2 Growth of Structure & Nature of Gravity</h3> <div class="ltx_para" id="S3.SS2.p1"> Within the framework of hot Big Bang cosmology, gravity amplifies primordial density perturbations of the order of <math alttext="10^{-5}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><msup id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mn id="S3.SS2.p1.1.m1.1.1.2" xref="S3.SS2.p1.1.m1.1.1.2.cmml">10</mn><mrow id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml"><mo id="S3.SS2.p1.1.m1.1.1.3a" xref="S3.SS2.p1.1.m1.1.1.3.cmml">−</mo><mn id="S3.SS2.p1.1.m1.1.1.3.2" xref="S3.SS2.p1.1.m1.1.1.3.2.cmml">5</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">superscript</csymbol><cn id="S3.SS2.p1.1.m1.1.1.2.cmml" type="integer" xref="S3.SS2.p1.1.m1.1.1.2">10</cn><apply id="S3.SS2.p1.1.m1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.1.1.3"><minus id="S3.SS2.p1.1.m1.1.1.3.1.cmml" xref="S3.SS2.p1.1.m1.1.1.3"></minus><cn id="S3.SS2.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS2.p1.1.m1.1.1.3.2">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math> seeded by inflation in an expanding cosmic background, leading to the formation and growth of LSS. As a result, the growth of structures is a sensitive probe of both gravity theories and dark energy. Notably, in modified gravity theories that deviate from General Relativity (GR), the intrinsic relationship between structure growth and geometrical quantities is generally altered or even disrupted. Thus, monitoring the structure growth over time provides stringent tests of gravity theories <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib76" title="">76</a>]</cite>. </div> <div class="ltx_para" id="S3.SS2.p2"> The growth of cosmic structures largely shapes the spatial correlations and the amplitude of peculiar velocities of galaxies. One prominent effect is redshift-space distortions (RSD; e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib77" title="">77</a>]</cite>), where the observed redshift maps of galaxies are distorted by their peculiar velocities along the line of sight. This distortion leads to the anisotropic clustering of galaxies, which allows for measurements of the autocorrelation of galaxy velocity divergence <math alttext="P_{\theta\theta}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><msub id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">P</mi><mrow id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml"><mi id="S3.SS2.p2.1.m1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.3.2.cmml">θ</mi><mo id="S3.SS2.p2.1.m1.1.1.3.1" xref="S3.SS2.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.1.m1.1.1.3.3" xref="S3.SS2.p2.1.m1.1.1.3.3.cmml">θ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2">𝑃</ci><apply id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3"><times id="S3.SS2.p2.1.m1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3.1"></times><ci id="S3.SS2.p2.1.m1.1.1.3.2.cmml" xref="S3.SS2.p2.1.m1.1.1.3.2">𝜃</ci><ci id="S3.SS2.p2.1.m1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">P_{\theta\theta}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_P start_POSTSUBSCRIPT italic_θ italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> and its cross-correlation with galaxy densities <math alttext="P_{g\theta}" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><msub id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml"><mi id="S3.SS2.p2.2.m2.1.1.2" xref="S3.SS2.p2.2.m2.1.1.2.cmml">P</mi><mrow id="S3.SS2.p2.2.m2.1.1.3" xref="S3.SS2.p2.2.m2.1.1.3.cmml"><mi id="S3.SS2.p2.2.m2.1.1.3.2" xref="S3.SS2.p2.2.m2.1.1.3.2.cmml">g</mi><mo id="S3.SS2.p2.2.m2.1.1.3.1" xref="S3.SS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.2.m2.1.1.3.3" xref="S3.SS2.p2.2.m2.1.1.3.3.cmml">θ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><apply id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p2.2.m2.1.1.2.cmml" xref="S3.SS2.p2.2.m2.1.1.2">𝑃</ci><apply id="S3.SS2.p2.2.m2.1.1.3.cmml" xref="S3.SS2.p2.2.m2.1.1.3"><times id="S3.SS2.p2.2.m2.1.1.3.1.cmml" xref="S3.SS2.p2.2.m2.1.1.3.1"></times><ci id="S3.SS2.p2.2.m2.1.1.3.2.cmml" xref="S3.SS2.p2.2.m2.1.1.3.2">𝑔</ci><ci id="S3.SS2.p2.2.m2.1.1.3.3.cmml" xref="S3.SS2.p2.2.m2.1.1.3.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">P_{g\theta}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_g italic_θ end_POSTSUBSCRIPT</annotation></semantics></math>. In this context, the effect of structure growth can be parameterized through the combination of the linear growth rate and the amplitude of matter fluctuations <math alttext="f\sigma_{8}" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.1"><semantics id="S3.SS2.p2.3.m3.1a"><mrow id="S3.SS2.p2.3.m3.1.1" xref="S3.SS2.p2.3.m3.1.1.cmml"><mi id="S3.SS2.p2.3.m3.1.1.2" xref="S3.SS2.p2.3.m3.1.1.2.cmml">f</mi><mo id="S3.SS2.p2.3.m3.1.1.1" xref="S3.SS2.p2.3.m3.1.1.1.cmml">⁢</mo><msub id="S3.SS2.p2.3.m3.1.1.3" xref="S3.SS2.p2.3.m3.1.1.3.cmml"><mi id="S3.SS2.p2.3.m3.1.1.3.2" xref="S3.SS2.p2.3.m3.1.1.3.2.cmml">σ</mi><mn id="S3.SS2.p2.3.m3.1.1.3.3" xref="S3.SS2.p2.3.m3.1.1.3.3.cmml">8</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.1b"><apply id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1"><times id="S3.SS2.p2.3.m3.1.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1.1"></times><ci id="S3.SS2.p2.3.m3.1.1.2.cmml" xref="S3.SS2.p2.3.m3.1.1.2">𝑓</ci><apply 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xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.2.cmml">σ</mi><mn id="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.3" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.3.cmml">8</mn></msub></mrow><mo id="S3.SS2.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.SS2.p2.4.m4.1.1.1.3" xref="S3.SS2.p2.4.m4.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><apply id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1"><csymbol cd="latexml" id="S3.SS2.p2.4.m4.1.1.2.cmml" xref="S3.SS2.p2.4.m4.1.1.2">proportional-to</csymbol><apply id="S3.SS2.p2.4.m4.1.1.3.cmml" xref="S3.SS2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.1.1.3.1.cmml" xref="S3.SS2.p2.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS2.p2.4.m4.1.1.3.2.cmml" xref="S3.SS2.p2.4.m4.1.1.3.2">𝑃</ci><apply id="S3.SS2.p2.4.m4.1.1.3.3.cmml" xref="S3.SS2.p2.4.m4.1.1.3.3"><times id="S3.SS2.p2.4.m4.1.1.3.3.1.cmml" xref="S3.SS2.p2.4.m4.1.1.3.3.1"></times><ci id="S3.SS2.p2.4.m4.1.1.3.3.2.cmml" xref="S3.SS2.p2.4.m4.1.1.3.3.2">𝜃</ci><ci id="S3.SS2.p2.4.m4.1.1.3.3.3.cmml" xref="S3.SS2.p2.4.m4.1.1.3.3.3">𝜃</ci></apply></apply><apply id="S3.SS2.p2.4.m4.1.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.1.1.1.2.cmml" xref="S3.SS2.p2.4.m4.1.1.1">superscript</csymbol><apply id="S3.SS2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1.1.1.1"><times id="S3.SS2.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.1"></times><ci id="S3.SS2.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.2">𝑓</ci><apply id="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.2">𝜎</ci><cn id="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.p2.4.m4.1.1.1.1.1.1.3.3">8</cn></apply></apply><cn id="S3.SS2.p2.4.m4.1.1.1.3.cmml" type="integer" xref="S3.SS2.p2.4.m4.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">P_{\theta\theta}\propto(f\sigma_{8})^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_θ italic_θ end_POSTSUBSCRIPT ∝ ( italic_f italic_σ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="P_{g\theta}\propto f\sigma_{8}" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m5.1"><semantics id="S3.SS2.p2.5.m5.1a"><mrow id="S3.SS2.p2.5.m5.1.1" xref="S3.SS2.p2.5.m5.1.1.cmml"><msub id="S3.SS2.p2.5.m5.1.1.2" xref="S3.SS2.p2.5.m5.1.1.2.cmml"><mi id="S3.SS2.p2.5.m5.1.1.2.2" xref="S3.SS2.p2.5.m5.1.1.2.2.cmml">P</mi><mrow id="S3.SS2.p2.5.m5.1.1.2.3" xref="S3.SS2.p2.5.m5.1.1.2.3.cmml"><mi id="S3.SS2.p2.5.m5.1.1.2.3.2" xref="S3.SS2.p2.5.m5.1.1.2.3.2.cmml">g</mi><mo id="S3.SS2.p2.5.m5.1.1.2.3.1" xref="S3.SS2.p2.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.5.m5.1.1.2.3.3" xref="S3.SS2.p2.5.m5.1.1.2.3.3.cmml">θ</mi></mrow></msub><mo id="S3.SS2.p2.5.m5.1.1.1" xref="S3.SS2.p2.5.m5.1.1.1.cmml">∝</mo><mrow id="S3.SS2.p2.5.m5.1.1.3" xref="S3.SS2.p2.5.m5.1.1.3.cmml"><mi id="S3.SS2.p2.5.m5.1.1.3.2" xref="S3.SS2.p2.5.m5.1.1.3.2.cmml">f</mi><mo id="S3.SS2.p2.5.m5.1.1.3.1" xref="S3.SS2.p2.5.m5.1.1.3.1.cmml">⁢</mo><msub id="S3.SS2.p2.5.m5.1.1.3.3" xref="S3.SS2.p2.5.m5.1.1.3.3.cmml"><mi id="S3.SS2.p2.5.m5.1.1.3.3.2" xref="S3.SS2.p2.5.m5.1.1.3.3.2.cmml">σ</mi><mn id="S3.SS2.p2.5.m5.1.1.3.3.3" xref="S3.SS2.p2.5.m5.1.1.3.3.3.cmml">8</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m5.1b"><apply id="S3.SS2.p2.5.m5.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1"><csymbol cd="latexml" id="S3.SS2.p2.5.m5.1.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1.1">proportional-to</csymbol><apply id="S3.SS2.p2.5.m5.1.1.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m5.1.1.2.1.cmml" xref="S3.SS2.p2.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS2.p2.5.m5.1.1.2.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2.2">𝑃</ci><apply id="S3.SS2.p2.5.m5.1.1.2.3.cmml" xref="S3.SS2.p2.5.m5.1.1.2.3"><times id="S3.SS2.p2.5.m5.1.1.2.3.1.cmml" xref="S3.SS2.p2.5.m5.1.1.2.3.1"></times><ci id="S3.SS2.p2.5.m5.1.1.2.3.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2.3.2">𝑔</ci><ci id="S3.SS2.p2.5.m5.1.1.2.3.3.cmml" xref="S3.SS2.p2.5.m5.1.1.2.3.3">𝜃</ci></apply></apply><apply id="S3.SS2.p2.5.m5.1.1.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3"><times id="S3.SS2.p2.5.m5.1.1.3.1.cmml" xref="S3.SS2.p2.5.m5.1.1.3.1"></times><ci id="S3.SS2.p2.5.m5.1.1.3.2.cmml" xref="S3.SS2.p2.5.m5.1.1.3.2">𝑓</ci><apply id="S3.SS2.p2.5.m5.1.1.3.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.p2.5.m5.1.1.3.3.1.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3">subscript</csymbol><ci id="S3.SS2.p2.5.m5.1.1.3.3.2.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3.2">𝜎</ci><cn id="S3.SS2.p2.5.m5.1.1.3.3.3.cmml" type="integer" xref="S3.SS2.p2.5.m5.1.1.3.3.3">8</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m5.1c">P_{g\theta}\propto f\sigma_{8}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m5.1d">italic_P start_POSTSUBSCRIPT italic_g italic_θ end_POSTSUBSCRIPT ∝ italic_f italic_σ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT</annotation></semantics></math> (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib78" title="">78</a>]</cite>). </div> <div class="ltx_para" id="S3.SS2.p3"> Another common approach to test gravity theories is to compare the two gravitational potentials <math alttext="\Psi" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mi id="S3.SS2.p3.1.m1.1.1" mathvariant="normal" xref="S3.SS2.p3.1.m1.1.1.cmml">Ψ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><ci id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">Ψ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">\Psi</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">roman_Ψ</annotation></semantics></math> and <math alttext="\Phi" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mi id="S3.SS2.p3.2.m2.1.1" mathvariant="normal" xref="S3.SS2.p3.2.m2.1.1.cmml">Φ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><ci id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1">Φ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">\Phi</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">roman_Φ</annotation></semantics></math> by parameterizing them with phenomenological functions <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mi id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><ci id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">italic_μ</annotation></semantics></math> and <math alttext="\Sigma" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.1"><semantics id="S3.SS2.p3.4.m4.1a"><mi id="S3.SS2.p3.4.m4.1.1" mathvariant="normal" xref="S3.SS2.p3.4.m4.1.1.cmml">Σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.1b"><ci id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1">Σ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.1c">\Sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.1d">roman_Σ</annotation></semantics></math> that linearly perturb GR (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib79" title="">79</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib80" title="">80</a>]</cite>). Many modified gravity theories of cosmological interest can be represented using the following equations, which allow the two potentials to differ: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx1"> <tbody id="S3.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle k^{2}\Psi" class="ltx_Math" display="inline" id="S3.E2.m1.1"><semantics id="S3.E2.m1.1a"><mrow id="S3.E2.m1.1.1" xref="S3.E2.m1.1.1.cmml"><msup id="S3.E2.m1.1.1.2" xref="S3.E2.m1.1.1.2.cmml"><mi id="S3.E2.m1.1.1.2.2" xref="S3.E2.m1.1.1.2.2.cmml">k</mi><mn id="S3.E2.m1.1.1.2.3" xref="S3.E2.m1.1.1.2.3.cmml">2</mn></msup><mo id="S3.E2.m1.1.1.1" xref="S3.E2.m1.1.1.1.cmml">⁢</mo><mi id="S3.E2.m1.1.1.3" mathvariant="normal" xref="S3.E2.m1.1.1.3.cmml">Ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.E2.m1.1b"><apply id="S3.E2.m1.1.1.cmml" xref="S3.E2.m1.1.1"><times id="S3.E2.m1.1.1.1.cmml" xref="S3.E2.m1.1.1.1"></times><apply id="S3.E2.m1.1.1.2.cmml" xref="S3.E2.m1.1.1.2"><csymbol cd="ambiguous" id="S3.E2.m1.1.1.2.1.cmml" xref="S3.E2.m1.1.1.2">superscript</csymbol><ci id="S3.E2.m1.1.1.2.2.cmml" xref="S3.E2.m1.1.1.2.2">𝑘</ci><cn id="S3.E2.m1.1.1.2.3.cmml" type="integer" xref="S3.E2.m1.1.1.2.3">2</cn></apply><ci id="S3.E2.m1.1.1.3.cmml" xref="S3.E2.m1.1.1.3">Ψ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.1c">\displaystyle k^{2}\Psi</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.1d">italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ψ</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-4\pi Ga^{2}(1+\mu(a,k))\rho\delta," class="ltx_Math" display="inline" id="S3.E2.m2.3"><semantics id="S3.E2.m2.3a"><mrow id="S3.E2.m2.3.3.1" xref="S3.E2.m2.3.3.1.1.cmml"><mrow id="S3.E2.m2.3.3.1.1" xref="S3.E2.m2.3.3.1.1.cmml"><mi id="S3.E2.m2.3.3.1.1.3" xref="S3.E2.m2.3.3.1.1.3.cmml"></mi><mo id="S3.E2.m2.3.3.1.1.2" xref="S3.E2.m2.3.3.1.1.2.cmml">=</mo><mrow id="S3.E2.m2.3.3.1.1.1" xref="S3.E2.m2.3.3.1.1.1.cmml"><mo id="S3.E2.m2.3.3.1.1.1a" xref="S3.E2.m2.3.3.1.1.1.cmml">−</mo><mrow id="S3.E2.m2.3.3.1.1.1.1" xref="S3.E2.m2.3.3.1.1.1.1.cmml"><mn id="S3.E2.m2.3.3.1.1.1.1.3" xref="S3.E2.m2.3.3.1.1.1.1.3.cmml">4</mn><mo id="S3.E2.m2.3.3.1.1.1.1.2" xref="S3.E2.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mi id="S3.E2.m2.3.3.1.1.1.1.4" xref="S3.E2.m2.3.3.1.1.1.1.4.cmml">π</mi><mo id="S3.E2.m2.3.3.1.1.1.1.2a" xref="S3.E2.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mi id="S3.E2.m2.3.3.1.1.1.1.5" xref="S3.E2.m2.3.3.1.1.1.1.5.cmml">G</mi><mo id="S3.E2.m2.3.3.1.1.1.1.2b" xref="S3.E2.m2.3.3.1.1.1.1.2.cmml">⁢</mo><msup id="S3.E2.m2.3.3.1.1.1.1.6" xref="S3.E2.m2.3.3.1.1.1.1.6.cmml"><mi id="S3.E2.m2.3.3.1.1.1.1.6.2" xref="S3.E2.m2.3.3.1.1.1.1.6.2.cmml">a</mi><mn id="S3.E2.m2.3.3.1.1.1.1.6.3" 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xref="S3.E2.m2.3.3.1.1.1.1.1.1.1.3.3.1.cmml">(</mo><mi id="S3.E2.m2.1.1" xref="S3.E2.m2.1.1.cmml">a</mi><mo id="S3.E2.m2.3.3.1.1.1.1.1.1.1.3.3.2.2" xref="S3.E2.m2.3.3.1.1.1.1.1.1.1.3.3.1.cmml">,</mo><mi id="S3.E2.m2.2.2" xref="S3.E2.m2.2.2.cmml">k</mi><mo id="S3.E2.m2.3.3.1.1.1.1.1.1.1.3.3.2.3" stretchy="false" xref="S3.E2.m2.3.3.1.1.1.1.1.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E2.m2.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S3.E2.m2.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E2.m2.3.3.1.1.1.1.2d" xref="S3.E2.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mi id="S3.E2.m2.3.3.1.1.1.1.7" xref="S3.E2.m2.3.3.1.1.1.1.7.cmml">ρ</mi><mo id="S3.E2.m2.3.3.1.1.1.1.2e" xref="S3.E2.m2.3.3.1.1.1.1.2.cmml">⁢</mo><mi id="S3.E2.m2.3.3.1.1.1.1.8" xref="S3.E2.m2.3.3.1.1.1.1.8.cmml">δ</mi></mrow></mrow></mrow><mo id="S3.E2.m2.3.3.1.2" xref="S3.E2.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E2.m2.3b"><apply id="S3.E2.m2.3.3.1.1.cmml" xref="S3.E2.m2.3.3.1"><eq id="S3.E2.m2.3.3.1.1.2.cmml" xref="S3.E2.m2.3.3.1.1.2"></eq><csymbol cd="latexml" id="S3.E2.m2.3.3.1.1.3.cmml" xref="S3.E2.m2.3.3.1.1.3">absent</csymbol><apply id="S3.E2.m2.3.3.1.1.1.cmml" xref="S3.E2.m2.3.3.1.1.1"><minus id="S3.E2.m2.3.3.1.1.1.2.cmml" xref="S3.E2.m2.3.3.1.1.1"></minus><apply id="S3.E2.m2.3.3.1.1.1.1.cmml" xref="S3.E2.m2.3.3.1.1.1.1"><times id="S3.E2.m2.3.3.1.1.1.1.2.cmml" xref="S3.E2.m2.3.3.1.1.1.1.2"></times><cn id="S3.E2.m2.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.E2.m2.3.3.1.1.1.1.3">4</cn><ci id="S3.E2.m2.3.3.1.1.1.1.4.cmml" xref="S3.E2.m2.3.3.1.1.1.1.4">𝜋</ci><ci id="S3.E2.m2.3.3.1.1.1.1.5.cmml" xref="S3.E2.m2.3.3.1.1.1.1.5">𝐺</ci><apply id="S3.E2.m2.3.3.1.1.1.1.6.cmml" xref="S3.E2.m2.3.3.1.1.1.1.6"><csymbol cd="ambiguous" id="S3.E2.m2.3.3.1.1.1.1.6.1.cmml" xref="S3.E2.m2.3.3.1.1.1.1.6">superscript</csymbol><ci id="S3.E2.m2.3.3.1.1.1.1.6.2.cmml" xref="S3.E2.m2.3.3.1.1.1.1.6.2">𝑎</ci><cn id="S3.E2.m2.3.3.1.1.1.1.6.3.cmml" type="integer" 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xref="S3.E2.m2.3.3.1.1.1.1.8">𝛿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m2.3c">\displaystyle=-4\pi Ga^{2}(1+\mu(a,k))\rho\delta,</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m2.3d">= - 4 italic_π italic_G italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + italic_μ ( italic_a , italic_k ) ) italic_ρ italic_δ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(2)</td> </tr></tbody> <tbody id="S3.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle k^{2}(\Psi+\Phi)" class="ltx_Math" display="inline" id="S3.E3.m1.1"><semantics id="S3.E3.m1.1a"><mrow id="S3.E3.m1.1.1" xref="S3.E3.m1.1.1.cmml"><msup id="S3.E3.m1.1.1.3" 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id="S3.E3.m2.3b"><apply id="S3.E3.m2.3.3.1.1.cmml" xref="S3.E3.m2.3.3.1"><eq id="S3.E3.m2.3.3.1.1.2.cmml" xref="S3.E3.m2.3.3.1.1.2"></eq><csymbol cd="latexml" id="S3.E3.m2.3.3.1.1.3.cmml" xref="S3.E3.m2.3.3.1.1.3">absent</csymbol><apply id="S3.E3.m2.3.3.1.1.1.cmml" xref="S3.E3.m2.3.3.1.1.1"><minus id="S3.E3.m2.3.3.1.1.1.2.cmml" xref="S3.E3.m2.3.3.1.1.1"></minus><apply id="S3.E3.m2.3.3.1.1.1.1.cmml" xref="S3.E3.m2.3.3.1.1.1.1"><times id="S3.E3.m2.3.3.1.1.1.1.2.cmml" xref="S3.E3.m2.3.3.1.1.1.1.2"></times><cn id="S3.E3.m2.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.E3.m2.3.3.1.1.1.1.3">8</cn><ci id="S3.E3.m2.3.3.1.1.1.1.4.cmml" xref="S3.E3.m2.3.3.1.1.1.1.4">𝜋</ci><ci id="S3.E3.m2.3.3.1.1.1.1.5.cmml" xref="S3.E3.m2.3.3.1.1.1.1.5">𝐺</ci><apply id="S3.E3.m2.3.3.1.1.1.1.6.cmml" xref="S3.E3.m2.3.3.1.1.1.1.6"><csymbol cd="ambiguous" id="S3.E3.m2.3.3.1.1.1.1.6.1.cmml" xref="S3.E3.m2.3.3.1.1.1.1.6">superscript</csymbol><ci id="S3.E3.m2.3.3.1.1.1.1.6.2.cmml" 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id="S3.E3.m2.3.3.1.1.1.1.8.cmml" xref="S3.E3.m2.3.3.1.1.1.1.8">𝛿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m2.3c">\displaystyle=-8\pi Ga^{2}(1+\Sigma(a,k))\rho\delta.</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m2.3d">= - 8 italic_π italic_G italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + roman_Σ ( italic_a , italic_k ) ) italic_ρ italic_δ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(3)</td> </tr></tbody> </table> Here, <math alttext="\rho" class="ltx_Math" display="inline" id="S3.SS2.p3.5.m1.1"><semantics id="S3.SS2.p3.5.m1.1a"><mi id="S3.SS2.p3.5.m1.1.1" xref="S3.SS2.p3.5.m1.1.1.cmml">ρ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.5.m1.1b"><ci id="S3.SS2.p3.5.m1.1.1.cmml" xref="S3.SS2.p3.5.m1.1.1">𝜌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.5.m1.1c">\rho</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.5.m1.1d">italic_ρ</annotation></semantics></math> indicates the background matter density and <math alttext="\delta" class="ltx_Math" display="inline" id="S3.SS2.p3.6.m2.1"><semantics id="S3.SS2.p3.6.m2.1a"><mi id="S3.SS2.p3.6.m2.1.1" xref="S3.SS2.p3.6.m2.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.6.m2.1b"><ci id="S3.SS2.p3.6.m2.1.1.cmml" xref="S3.SS2.p3.6.m2.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.6.m2.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.6.m2.1d">italic_δ</annotation></semantics></math> is the comoving density perturbation. The functions <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS2.p3.7.m3.1"><semantics id="S3.SS2.p3.7.m3.1a"><mi id="S3.SS2.p3.7.m3.1.1" xref="S3.SS2.p3.7.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.7.m3.1b"><ci id="S3.SS2.p3.7.m3.1.1.cmml" xref="S3.SS2.p3.7.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.7.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.7.m3.1d">italic_μ</annotation></semantics></math> and <math alttext="\Sigma" class="ltx_Math" display="inline" id="S3.SS2.p3.8.m4.1"><semantics id="S3.SS2.p3.8.m4.1a"><mi id="S3.SS2.p3.8.m4.1.1" mathvariant="normal" xref="S3.SS2.p3.8.m4.1.1.cmml">Σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.8.m4.1b"><ci id="S3.SS2.p3.8.m4.1.1.cmml" xref="S3.SS2.p3.8.m4.1.1">Σ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.8.m4.1c">\Sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.8.m4.1d">roman_Σ</annotation></semantics></math> effectively describe the variations of matter density field and changes of lensing effect on massless particles, respectively. Comparing measurements of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS2.p3.9.m5.1"><semantics id="S3.SS2.p3.9.m5.1a"><mi id="S3.SS2.p3.9.m5.1.1" xref="S3.SS2.p3.9.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.9.m5.1b"><ci id="S3.SS2.p3.9.m5.1.1.cmml" xref="S3.SS2.p3.9.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.9.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.9.m5.1d">italic_μ</annotation></semantics></math> and <math alttext="\Sigma" class="ltx_Math" display="inline" id="S3.SS2.p3.10.m6.1"><semantics id="S3.SS2.p3.10.m6.1a"><mi id="S3.SS2.p3.10.m6.1.1" mathvariant="normal" xref="S3.SS2.p3.10.m6.1.1.cmml">Σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.10.m6.1b"><ci id="S3.SS2.p3.10.m6.1.1.cmml" xref="S3.SS2.p3.10.m6.1.1">Σ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.10.m6.1c">\Sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.10.m6.1d">roman_Σ</annotation></semantics></math> to their fiducial values of zero thus constitutes a test of deviations from GR. </div> <div class="ltx_para" id="S3.SS2.p4"> Galaxy spectroscopic surveys enable structure growth measurements through both correlations of galaxy velocities and comparisons of the spatial clustering of galaxies across different redshifts. Testing gravity is, therefore, a primary scientific motivation of recent spectroscopic surveys, such as SDSS and DESI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib81" title="">81</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib11" title="">11</a>]</cite>. Currently, measurements of structure growth at <math alttext="z\lesssim 1" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><mrow id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml"><mi id="S3.SS2.p4.1.m1.1.1.2" xref="S3.SS2.p4.1.m1.1.1.2.cmml">z</mi><mo id="S3.SS2.p4.1.m1.1.1.1" xref="S3.SS2.p4.1.m1.1.1.1.cmml">≲</mo><mn id="S3.SS2.p4.1.m1.1.1.3" xref="S3.SS2.p4.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><apply id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1"><csymbol cd="latexml" id="S3.SS2.p4.1.m1.1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1.1">less-than-or-similar-to</csymbol><ci id="S3.SS2.p4.1.m1.1.1.2.cmml" xref="S3.SS2.p4.1.m1.1.1.2">𝑧</ci><cn id="S3.SS2.p4.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">z\lesssim 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">italic_z ≲ 1</annotation></semantics></math> have achieved precision at the <math alttext="\lesssim 10" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.1"><semantics id="S3.SS2.p4.2.m2.1a"><mrow id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml"><mi id="S3.SS2.p4.2.m2.1.1.2" xref="S3.SS2.p4.2.m2.1.1.2.cmml"></mi><mo id="S3.SS2.p4.2.m2.1.1.1" xref="S3.SS2.p4.2.m2.1.1.1.cmml">≲</mo><mn id="S3.SS2.p4.2.m2.1.1.3" xref="S3.SS2.p4.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.1b"><apply id="S3.SS2.p4.2.m2.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1"><csymbol cd="latexml" id="S3.SS2.p4.2.m2.1.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1.1">less-than-or-similar-to</csymbol><csymbol cd="latexml" id="S3.SS2.p4.2.m2.1.1.2.cmml" xref="S3.SS2.p4.2.m2.1.1.2">absent</csymbol><cn id="S3.SS2.p4.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.p4.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.1c">\lesssim 10</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.1d">≲ 10</annotation></semantics></math> % level (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib10" title="">10</a>]</cite>, see also Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS3" title="5.3 Structure Growth & Gravity ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.3</a>). While results from spectroscopic surveys are generally consistent with predictions of the GR-<math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.1"><semantics id="S3.SS2.p4.3.m3.1a"><mi id="S3.SS2.p4.3.m3.1.1" mathvariant="normal" xref="S3.SS2.p4.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.1b"><ci id="S3.SS2.p4.3.m3.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.1d">roman_Λ</annotation></semantics></math>CDM model, the structure growth history at high redshifts remains largely unexplored. Probing this regime is essential for differentiating between various gravity theories. As a Stage-V cosmological survey, MUST is expected to fill in this gap and provide substantial improvements in our understanding of gravity theory, as well as deeper insights into the tension between cosmic shear and matter perturbations from CMB measurements (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib8" title="">8</a>]</cite>). </div> <div class="ltx_para" id="S3.SS2.p5"> MUST also has the potential to measure relativistic effects, such as the gravitational redshifts, that are sensitive to gravity theories but have been too subtle to be detected with current data <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib82" title="">82</a>]</cite>. These effects can introduce additional distortions to galaxy clustering beyond RSD, primarily generating odd Legendre multipoles in the cross-correlation of different galaxy populations that can be measured from galaxy spectroscopic data. </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> 3.3 Inflation & Primordial Physics</h3> <div class="ltx_para" id="S3.SS3.p1"> Large-scale structures (LSS) of the Universe not only trace the evolution of the Universe but also encode rich information about its initial conditions generated during a primordial era. The most prominent scenario for the primordial universe is cosmic inflation, which refers to a nearly exponential expansion of the universe by roughly 60 <math alttext="e" class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><mi id="S3.SS3.p1.1.m1.1.1" xref="S3.SS3.p1.1.m1.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><ci id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.1.m1.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.1.m1.1d">italic_e</annotation></semantics></math>-folds in a blink. To probe the microscopic quantum fluctuations of the spacetime and matter/fields during inflation that created the LSS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib83" title="">83</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib84" title="">84</a>]</cite> is a primary motivation for MUST. </div> <div class="ltx_para" id="S3.SS3.p2"> Through setting the initial conditions of the LSS, inflationary cosmology provides a wealth of observables that reveal the dynamics of the primordial universe and, more interestingly, the fundamental physics at the inflation scale. Current theory and observations suggest that the inflation scale could be up to <math alttext="\mathcal{O}(10^{13}\,\text{GeV})" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><mrow id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.1.m1.1.1.3" xref="S3.SS3.p2.1.m1.1.1.3.cmml">𝒪</mi><mo id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p2.1.m1.1.1.1.1" xref="S3.SS3.p2.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS3.p2.1.m1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p2.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS3.p2.1.m1.1.1.1.1.1" xref="S3.SS3.p2.1.m1.1.1.1.1.1.cmml"><msup id="S3.SS3.p2.1.m1.1.1.1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.1.1.1.2.cmml"><mn id="S3.SS3.p2.1.m1.1.1.1.1.1.2.2" xref="S3.SS3.p2.1.m1.1.1.1.1.1.2.2.cmml">10</mn><mn id="S3.SS3.p2.1.m1.1.1.1.1.1.2.3" xref="S3.SS3.p2.1.m1.1.1.1.1.1.2.3.cmml">13</mn></msup><mo id="S3.SS3.p2.1.m1.1.1.1.1.1.1" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mtext id="S3.SS3.p2.1.m1.1.1.1.1.1.3" xref="S3.SS3.p2.1.m1.1.1.1.1.1.3a.cmml">GeV</mtext></mrow><mo id="S3.SS3.p2.1.m1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p2.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><times id="S3.SS3.p2.1.m1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.2"></times><ci id="S3.SS3.p2.1.m1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3">𝒪</ci><apply id="S3.SS3.p2.1.m1.1.1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1"><times id="S3.SS3.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.1"></times><apply id="S3.SS3.p2.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.1.1.1.2.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.2">superscript</csymbol><cn id="S3.SS3.p2.1.m1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS3.p2.1.m1.1.1.1.1.1.2.2">10</cn><cn id="S3.SS3.p2.1.m1.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.SS3.p2.1.m1.1.1.1.1.1.2.3">13</cn></apply><ci id="S3.SS3.p2.1.m1.1.1.1.1.1.3a.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.3"><mtext id="S3.SS3.p2.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.1.1.1.3">GeV</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\mathcal{O}(10^{13}\,\text{GeV})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">caligraphic_O ( 10 start_POSTSUPERSCRIPT 13 end_POSTSUPERSCRIPT GeV )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib85" title="">85</a>]</cite>, which is much higher than the energy scale of any imaginable terrestrial experiments. Therefore, primordial fluctuations provide a unique window into high-energy fundamental physics, complementing other high-energy experiments. </div> <div class="ltx_para" id="S3.SS3.p3"> The inflation predictions on the primordial scalar power spectrum have been delicately measured at the CMB scale through the overall amplitude <math alttext="A_{s}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><msub id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml">A</mi><mi id="S3.SS3.p3.1.m1.1.1.3" xref="S3.SS3.p3.1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2">𝐴</ci><ci id="S3.SS3.p3.1.m1.1.1.3.cmml" xref="S3.SS3.p3.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">A_{s}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> and the spectral index <math alttext="n_{s}" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.1"><semantics id="S3.SS3.p3.2.m2.1a"><msub id="S3.SS3.p3.2.m2.1.1" xref="S3.SS3.p3.2.m2.1.1.cmml"><mi id="S3.SS3.p3.2.m2.1.1.2" xref="S3.SS3.p3.2.m2.1.1.2.cmml">n</mi><mi id="S3.SS3.p3.2.m2.1.1.3" xref="S3.SS3.p3.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.2.m2.1b"><apply id="S3.SS3.p3.2.m2.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.2.m2.1.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p3.2.m2.1.1.2.cmml" xref="S3.SS3.p3.2.m2.1.1.2">𝑛</ci><ci id="S3.SS3.p3.2.m2.1.1.3.cmml" xref="S3.SS3.p3.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.1c">n_{s}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib85" title="">85</a>]</cite>. These parameters are also expected to be measured by MUST by extracting the primordial scalar power spectrum from the LSS data. However, due to the near-scale invariance of the scalar fluctuation at large scales, the information extractable from the power spectrum is limited, although there have been ongoing efforts in searching and understanding tiny scale-dependent (and very often oscillatory) features in the power spectrum <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib86" title="">86</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib87" title="">87</a>]</cite>. On the other hand, a vast amount of information can be revealed by going to higher <math alttext="n" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><mi id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><ci id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">italic_n</annotation></semantics></math>-point (<math alttext="n\geq 3" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mrow id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml"><mi id="S3.SS3.p3.4.m4.1.1.2" xref="S3.SS3.p3.4.m4.1.1.2.cmml">n</mi><mo id="S3.SS3.p3.4.m4.1.1.1" xref="S3.SS3.p3.4.m4.1.1.1.cmml">≥</mo><mn id="S3.SS3.p3.4.m4.1.1.3" xref="S3.SS3.p3.4.m4.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><apply id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1"><geq id="S3.SS3.p3.4.m4.1.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1.1"></geq><ci id="S3.SS3.p3.4.m4.1.1.2.cmml" xref="S3.SS3.p3.4.m4.1.1.2">𝑛</ci><cn id="S3.SS3.p3.4.m4.1.1.3.cmml" type="integer" xref="S3.SS3.p3.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">n\geq 3</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">italic_n ≥ 3</annotation></semantics></math>) correlations of scalar modes, which are traditionally known as primordial non-Gaussianity (PNG; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib88" title="">88</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib89" title="">89</a>]</cite>). There is in principle fruitful information encoded in the PNG, though historically, many efforts have been made to the search of 3-point statistics, such as bispectra <math alttext="B(k_{1},k_{2},k_{3})" class="ltx_Math" display="inline" id="S3.SS3.p3.5.m5.3"><semantics id="S3.SS3.p3.5.m5.3a"><mrow id="S3.SS3.p3.5.m5.3.3" xref="S3.SS3.p3.5.m5.3.3.cmml"><mi id="S3.SS3.p3.5.m5.3.3.5" xref="S3.SS3.p3.5.m5.3.3.5.cmml">B</mi><mo id="S3.SS3.p3.5.m5.3.3.4" xref="S3.SS3.p3.5.m5.3.3.4.cmml">⁢</mo><mrow id="S3.SS3.p3.5.m5.3.3.3.3" xref="S3.SS3.p3.5.m5.3.3.3.4.cmml"><mo id="S3.SS3.p3.5.m5.3.3.3.3.4" stretchy="false" xref="S3.SS3.p3.5.m5.3.3.3.4.cmml">(</mo><msub id="S3.SS3.p3.5.m5.1.1.1.1.1" xref="S3.SS3.p3.5.m5.1.1.1.1.1.cmml"><mi id="S3.SS3.p3.5.m5.1.1.1.1.1.2" xref="S3.SS3.p3.5.m5.1.1.1.1.1.2.cmml">k</mi><mn id="S3.SS3.p3.5.m5.1.1.1.1.1.3" xref="S3.SS3.p3.5.m5.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS3.p3.5.m5.3.3.3.3.5" xref="S3.SS3.p3.5.m5.3.3.3.4.cmml">,</mo><msub 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id="S3.SS3.p3.6.m6.1b"><apply id="S3.SS3.p3.6.m6.1.1.cmml" xref="S3.SS3.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.6.m6.1.1.1.cmml" xref="S3.SS3.p3.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.p3.6.m6.1.1.2.cmml" xref="S3.SS3.p3.6.m6.1.1.2">𝑘</ci><ci id="S3.SS3.p3.6.m6.1.1.3.cmml" xref="S3.SS3.p3.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.6.m6.1c">k_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.6.m6.1d">italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="i=1,2,3" class="ltx_Math" display="inline" id="S3.SS3.p3.7.m7.3"><semantics id="S3.SS3.p3.7.m7.3a"><mrow id="S3.SS3.p3.7.m7.3.4" xref="S3.SS3.p3.7.m7.3.4.cmml"><mi id="S3.SS3.p3.7.m7.3.4.2" xref="S3.SS3.p3.7.m7.3.4.2.cmml">i</mi><mo id="S3.SS3.p3.7.m7.3.4.1" xref="S3.SS3.p3.7.m7.3.4.1.cmml">=</mo><mrow id="S3.SS3.p3.7.m7.3.4.3.2" xref="S3.SS3.p3.7.m7.3.4.3.1.cmml"><mn id="S3.SS3.p3.7.m7.1.1" xref="S3.SS3.p3.7.m7.1.1.cmml">1</mn><mo id="S3.SS3.p3.7.m7.3.4.3.2.1" xref="S3.SS3.p3.7.m7.3.4.3.1.cmml">,</mo><mn id="S3.SS3.p3.7.m7.2.2" xref="S3.SS3.p3.7.m7.2.2.cmml">2</mn><mo id="S3.SS3.p3.7.m7.3.4.3.2.2" xref="S3.SS3.p3.7.m7.3.4.3.1.cmml">,</mo><mn id="S3.SS3.p3.7.m7.3.3" xref="S3.SS3.p3.7.m7.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.7.m7.3b"><apply id="S3.SS3.p3.7.m7.3.4.cmml" xref="S3.SS3.p3.7.m7.3.4"><eq id="S3.SS3.p3.7.m7.3.4.1.cmml" xref="S3.SS3.p3.7.m7.3.4.1"></eq><ci id="S3.SS3.p3.7.m7.3.4.2.cmml" xref="S3.SS3.p3.7.m7.3.4.2">𝑖</ci><list id="S3.SS3.p3.7.m7.3.4.3.1.cmml" xref="S3.SS3.p3.7.m7.3.4.3.2"><cn id="S3.SS3.p3.7.m7.1.1.cmml" type="integer" xref="S3.SS3.p3.7.m7.1.1">1</cn><cn id="S3.SS3.p3.7.m7.2.2.cmml" type="integer" xref="S3.SS3.p3.7.m7.2.2">2</cn><cn id="S3.SS3.p3.7.m7.3.3.cmml" type="integer" xref="S3.SS3.p3.7.m7.3.3">3</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.7.m7.3c">i=1,2,3</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.7.m7.3d">italic_i = 1 , 2 , 3</annotation></semantics></math>), including the local, equilateral, and orthogonal shapes (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib90" title="">90</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib91" title="">91</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib92" title="">92</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib93" title="">93</a>]</cite>). </div> <div class="ltx_para" id="S3.SS3.p4"> The overall amplitudes of these shapes are parameterized by dimensionless parameters called <math alttext="f_{\text{NL}}" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><msub id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml"><mi id="S3.SS3.p4.1.m1.1.1.2" xref="S3.SS3.p4.1.m1.1.1.2.cmml">f</mi><mtext id="S3.SS3.p4.1.m1.1.1.3" xref="S3.SS3.p4.1.m1.1.1.3a.cmml">NL</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><apply id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.1.m1.1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p4.1.m1.1.1.2.cmml" xref="S3.SS3.p4.1.m1.1.1.2">𝑓</ci><ci id="S3.SS3.p4.1.m1.1.1.3a.cmml" xref="S3.SS3.p4.1.m1.1.1.3"><mtext id="S3.SS3.p4.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p4.1.m1.1.1.3">NL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">f_{\text{NL}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT</annotation></semantics></math>, which are crucial cosmological parameters for the next generation LSS surveys. Within typical single-field slow-roll inflation models, it is worth noting that the minimal signal of equilateral bispectrum coming from the gravitational interaction of the scalar modes is <math alttext="f_{\text{NL}}\sim\mathcal{O}(10^{-2})" class="ltx_Math" display="inline" id="S3.SS3.p4.2.m2.1"><semantics id="S3.SS3.p4.2.m2.1a"><mrow id="S3.SS3.p4.2.m2.1.1" xref="S3.SS3.p4.2.m2.1.1.cmml"><msub id="S3.SS3.p4.2.m2.1.1.3" xref="S3.SS3.p4.2.m2.1.1.3.cmml"><mi id="S3.SS3.p4.2.m2.1.1.3.2" xref="S3.SS3.p4.2.m2.1.1.3.2.cmml">f</mi><mtext id="S3.SS3.p4.2.m2.1.1.3.3" xref="S3.SS3.p4.2.m2.1.1.3.3a.cmml">NL</mtext></msub><mo id="S3.SS3.p4.2.m2.1.1.2" xref="S3.SS3.p4.2.m2.1.1.2.cmml">∼</mo><mrow id="S3.SS3.p4.2.m2.1.1.1" xref="S3.SS3.p4.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p4.2.m2.1.1.1.3" xref="S3.SS3.p4.2.m2.1.1.1.3.cmml">𝒪</mi><mo id="S3.SS3.p4.2.m2.1.1.1.2" xref="S3.SS3.p4.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p4.2.m2.1.1.1.1.1" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p4.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS3.p4.2.m2.1.1.1.1.1.1" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.cmml"><mn id="S3.SS3.p4.2.m2.1.1.1.1.1.1.2" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.2.cmml">10</mn><mrow id="S3.SS3.p4.2.m2.1.1.1.1.1.1.3" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.cmml"><mo id="S3.SS3.p4.2.m2.1.1.1.1.1.1.3a" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.cmml">−</mo><mn id="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.2" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.2.cmml">2</mn></mrow></msup><mo id="S3.SS3.p4.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.2.m2.1b"><apply id="S3.SS3.p4.2.m2.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1"><csymbol cd="latexml" id="S3.SS3.p4.2.m2.1.1.2.cmml" xref="S3.SS3.p4.2.m2.1.1.2">similar-to</csymbol><apply id="S3.SS3.p4.2.m2.1.1.3.cmml" xref="S3.SS3.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p4.2.m2.1.1.3.1.cmml" xref="S3.SS3.p4.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.p4.2.m2.1.1.3.2.cmml" xref="S3.SS3.p4.2.m2.1.1.3.2">𝑓</ci><ci id="S3.SS3.p4.2.m2.1.1.3.3a.cmml" xref="S3.SS3.p4.2.m2.1.1.3.3"><mtext id="S3.SS3.p4.2.m2.1.1.3.3.cmml" mathsize="70%" xref="S3.SS3.p4.2.m2.1.1.3.3">NL</mtext></ci></apply><apply id="S3.SS3.p4.2.m2.1.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1.1"><times id="S3.SS3.p4.2.m2.1.1.1.2.cmml" xref="S3.SS3.p4.2.m2.1.1.1.2"></times><ci id="S3.SS3.p4.2.m2.1.1.1.3.cmml" xref="S3.SS3.p4.2.m2.1.1.1.3">𝒪</ci><apply id="S3.SS3.p4.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1.1.1.1">superscript</csymbol><cn id="S3.SS3.p4.2.m2.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.2">10</cn><apply id="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.3"><minus id="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.3"></minus><cn id="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.SS3.p4.2.m2.1.1.1.1.1.1.3.2">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.2.m2.1c">f_{\text{NL}}\sim\mathcal{O}(10^{-2})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.2.m2.1d">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT ∼ caligraphic_O ( 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, which, if detected, would mark our detection of gravitational interaction <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib94" title="">94</a>]</cite>. However, other fundamental interactions, particles, and fields besides the inflaton field may carry stronger-than-gravity coupling to the scalar mode. Thus, it is possible that more significant signals can be detected, which will revolutionize our understanding of the primordial universe and fundamental physics. Typically, <math alttext="f_{\text{NL}}\sim 1" class="ltx_Math" display="inline" id="S3.SS3.p4.3.m3.1"><semantics id="S3.SS3.p4.3.m3.1a"><mrow id="S3.SS3.p4.3.m3.1.1" xref="S3.SS3.p4.3.m3.1.1.cmml"><msub id="S3.SS3.p4.3.m3.1.1.2" xref="S3.SS3.p4.3.m3.1.1.2.cmml"><mi id="S3.SS3.p4.3.m3.1.1.2.2" xref="S3.SS3.p4.3.m3.1.1.2.2.cmml">f</mi><mtext id="S3.SS3.p4.3.m3.1.1.2.3" xref="S3.SS3.p4.3.m3.1.1.2.3a.cmml">NL</mtext></msub><mo id="S3.SS3.p4.3.m3.1.1.1" xref="S3.SS3.p4.3.m3.1.1.1.cmml">∼</mo><mn id="S3.SS3.p4.3.m3.1.1.3" xref="S3.SS3.p4.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.3.m3.1b"><apply id="S3.SS3.p4.3.m3.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1"><csymbol cd="latexml" id="S3.SS3.p4.3.m3.1.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1.1">similar-to</csymbol><apply id="S3.SS3.p4.3.m3.1.1.2.cmml" xref="S3.SS3.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p4.3.m3.1.1.2.1.cmml" xref="S3.SS3.p4.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS3.p4.3.m3.1.1.2.2.cmml" xref="S3.SS3.p4.3.m3.1.1.2.2">𝑓</ci><ci id="S3.SS3.p4.3.m3.1.1.2.3a.cmml" xref="S3.SS3.p4.3.m3.1.1.2.3"><mtext id="S3.SS3.p4.3.m3.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.p4.3.m3.1.1.2.3">NL</mtext></ci></apply><cn id="S3.SS3.p4.3.m3.1.1.3.cmml" type="integer" xref="S3.SS3.p4.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.3.m3.1c">f_{\text{NL}}\sim 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.3.m3.1d">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT ∼ 1</annotation></semantics></math> is a crucial threshold for the theories with and without the slow-roll condition <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib95" title="">95</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib96" title="">96</a>]</cite>. </div> <div class="ltx_para" id="S3.SS3.p5"> There have been active searches of <math alttext="f_{\text{NL}}" class="ltx_Math" display="inline" id="S3.SS3.p5.1.m1.1"><semantics id="S3.SS3.p5.1.m1.1a"><msub id="S3.SS3.p5.1.m1.1.1" xref="S3.SS3.p5.1.m1.1.1.cmml"><mi id="S3.SS3.p5.1.m1.1.1.2" xref="S3.SS3.p5.1.m1.1.1.2.cmml">f</mi><mtext id="S3.SS3.p5.1.m1.1.1.3" xref="S3.SS3.p5.1.m1.1.1.3a.cmml">NL</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.1.m1.1b"><apply id="S3.SS3.p5.1.m1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.1.m1.1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p5.1.m1.1.1.2.cmml" xref="S3.SS3.p5.1.m1.1.1.2">𝑓</ci><ci id="S3.SS3.p5.1.m1.1.1.3a.cmml" xref="S3.SS3.p5.1.m1.1.1.3"><mtext id="S3.SS3.p5.1.m1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p5.1.m1.1.1.3">NL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.1.m1.1c">f_{\text{NL}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.1.m1.1d">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT</annotation></semantics></math>, especially for the local shape (<math alttext="f_{\text{NL}}^{\text{local}}" class="ltx_Math" display="inline" id="S3.SS3.p5.2.m2.1"><semantics id="S3.SS3.p5.2.m2.1a"><msubsup id="S3.SS3.p5.2.m2.1.1" xref="S3.SS3.p5.2.m2.1.1.cmml"><mi id="S3.SS3.p5.2.m2.1.1.2.2" xref="S3.SS3.p5.2.m2.1.1.2.2.cmml">f</mi><mtext id="S3.SS3.p5.2.m2.1.1.2.3" xref="S3.SS3.p5.2.m2.1.1.2.3a.cmml">NL</mtext><mtext id="S3.SS3.p5.2.m2.1.1.3" xref="S3.SS3.p5.2.m2.1.1.3a.cmml">local</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.2.m2.1b"><apply id="S3.SS3.p5.2.m2.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.2.m2.1.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1">superscript</csymbol><apply id="S3.SS3.p5.2.m2.1.1.2.cmml" xref="S3.SS3.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.2.m2.1.1.2.1.cmml" xref="S3.SS3.p5.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p5.2.m2.1.1.2.2.cmml" xref="S3.SS3.p5.2.m2.1.1.2.2">𝑓</ci><ci id="S3.SS3.p5.2.m2.1.1.2.3a.cmml" xref="S3.SS3.p5.2.m2.1.1.2.3"><mtext id="S3.SS3.p5.2.m2.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.p5.2.m2.1.1.2.3">NL</mtext></ci></apply><ci id="S3.SS3.p5.2.m2.1.1.3a.cmml" xref="S3.SS3.p5.2.m2.1.1.3"><mtext id="S3.SS3.p5.2.m2.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p5.2.m2.1.1.3">local</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.2.m2.1c">f_{\text{NL}}^{\text{local}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.2.m2.1d">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT</annotation></semantics></math>), but a firm detection has not been made yet. The state-of-the-art constraint comes from CMB measurements by Planck, with <math alttext="f_{\text{NL}}^{\text{local}}=-0.9\pm 5.1" class="ltx_Math" display="inline" id="S3.SS3.p5.3.m3.1"><semantics id="S3.SS3.p5.3.m3.1a"><mrow id="S3.SS3.p5.3.m3.1.1" xref="S3.SS3.p5.3.m3.1.1.cmml"><msubsup id="S3.SS3.p5.3.m3.1.1.2" xref="S3.SS3.p5.3.m3.1.1.2.cmml"><mi id="S3.SS3.p5.3.m3.1.1.2.2.2" xref="S3.SS3.p5.3.m3.1.1.2.2.2.cmml">f</mi><mtext id="S3.SS3.p5.3.m3.1.1.2.2.3" xref="S3.SS3.p5.3.m3.1.1.2.2.3a.cmml">NL</mtext><mtext id="S3.SS3.p5.3.m3.1.1.2.3" xref="S3.SS3.p5.3.m3.1.1.2.3a.cmml">local</mtext></msubsup><mo id="S3.SS3.p5.3.m3.1.1.1" xref="S3.SS3.p5.3.m3.1.1.1.cmml">=</mo><mrow id="S3.SS3.p5.3.m3.1.1.3" xref="S3.SS3.p5.3.m3.1.1.3.cmml"><mrow id="S3.SS3.p5.3.m3.1.1.3.2" xref="S3.SS3.p5.3.m3.1.1.3.2.cmml"><mo id="S3.SS3.p5.3.m3.1.1.3.2a" xref="S3.SS3.p5.3.m3.1.1.3.2.cmml">−</mo><mn id="S3.SS3.p5.3.m3.1.1.3.2.2" xref="S3.SS3.p5.3.m3.1.1.3.2.2.cmml">0.9</mn></mrow><mo id="S3.SS3.p5.3.m3.1.1.3.1" xref="S3.SS3.p5.3.m3.1.1.3.1.cmml">±</mo><mn id="S3.SS3.p5.3.m3.1.1.3.3" xref="S3.SS3.p5.3.m3.1.1.3.3.cmml">5.1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.3.m3.1b"><apply id="S3.SS3.p5.3.m3.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1"><eq id="S3.SS3.p5.3.m3.1.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1.1"></eq><apply id="S3.SS3.p5.3.m3.1.1.2.cmml" xref="S3.SS3.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p5.3.m3.1.1.2.1.cmml" xref="S3.SS3.p5.3.m3.1.1.2">superscript</csymbol><apply id="S3.SS3.p5.3.m3.1.1.2.2.cmml" xref="S3.SS3.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p5.3.m3.1.1.2.2.1.cmml" xref="S3.SS3.p5.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS3.p5.3.m3.1.1.2.2.2.cmml" xref="S3.SS3.p5.3.m3.1.1.2.2.2">𝑓</ci><ci id="S3.SS3.p5.3.m3.1.1.2.2.3a.cmml" xref="S3.SS3.p5.3.m3.1.1.2.2.3"><mtext id="S3.SS3.p5.3.m3.1.1.2.2.3.cmml" mathsize="70%" xref="S3.SS3.p5.3.m3.1.1.2.2.3">NL</mtext></ci></apply><ci id="S3.SS3.p5.3.m3.1.1.2.3a.cmml" xref="S3.SS3.p5.3.m3.1.1.2.3"><mtext id="S3.SS3.p5.3.m3.1.1.2.3.cmml" mathsize="70%" xref="S3.SS3.p5.3.m3.1.1.2.3">local</mtext></ci></apply><apply id="S3.SS3.p5.3.m3.1.1.3.cmml" xref="S3.SS3.p5.3.m3.1.1.3"><csymbol cd="latexml" id="S3.SS3.p5.3.m3.1.1.3.1.cmml" xref="S3.SS3.p5.3.m3.1.1.3.1">plus-or-minus</csymbol><apply id="S3.SS3.p5.3.m3.1.1.3.2.cmml" xref="S3.SS3.p5.3.m3.1.1.3.2"><minus id="S3.SS3.p5.3.m3.1.1.3.2.1.cmml" xref="S3.SS3.p5.3.m3.1.1.3.2"></minus><cn id="S3.SS3.p5.3.m3.1.1.3.2.2.cmml" type="float" xref="S3.SS3.p5.3.m3.1.1.3.2.2">0.9</cn></apply><cn id="S3.SS3.p5.3.m3.1.1.3.3.cmml" type="float" xref="S3.SS3.p5.3.m3.1.1.3.3">5.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.3.m3.1c">f_{\text{NL}}^{\text{local}}=-0.9\pm 5.1</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.3.m3.1d">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT = - 0.9 ± 5.1</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib97" title="">97</a>]</cite>. Meanwhile, galaxy spectroscopic surveys are becoming more promising as LSS clustering can be very sensitive to the PNG through the scale-dependent galaxy bias <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib98" title="">98</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib99" title="">99</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib83" title="">83</a>]</cite>. Current bounds on the local shape bispectrum from spectroscopic surveys are <math alttext="\left|f_{\mathrm{NL}}^{\text{local}}\right|\sim\mathcal{O}(20-30)" class="ltx_Math" display="inline" id="S3.SS3.p5.4.m4.2"><semantics id="S3.SS3.p5.4.m4.2a"><mrow id="S3.SS3.p5.4.m4.2.2" xref="S3.SS3.p5.4.m4.2.2.cmml"><mrow id="S3.SS3.p5.4.m4.1.1.1.1" xref="S3.SS3.p5.4.m4.1.1.1.2.cmml"><mo id="S3.SS3.p5.4.m4.1.1.1.1.2" xref="S3.SS3.p5.4.m4.1.1.1.2.1.cmml">|</mo><msubsup id="S3.SS3.p5.4.m4.1.1.1.1.1" xref="S3.SS3.p5.4.m4.1.1.1.1.1.cmml"><mi id="S3.SS3.p5.4.m4.1.1.1.1.1.2.2" xref="S3.SS3.p5.4.m4.1.1.1.1.1.2.2.cmml">f</mi><mi id="S3.SS3.p5.4.m4.1.1.1.1.1.2.3" xref="S3.SS3.p5.4.m4.1.1.1.1.1.2.3.cmml">NL</mi><mtext id="S3.SS3.p5.4.m4.1.1.1.1.1.3" xref="S3.SS3.p5.4.m4.1.1.1.1.1.3a.cmml">local</mtext></msubsup><mo id="S3.SS3.p5.4.m4.1.1.1.1.3" xref="S3.SS3.p5.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS3.p5.4.m4.2.2.3" xref="S3.SS3.p5.4.m4.2.2.3.cmml">∼</mo><mrow id="S3.SS3.p5.4.m4.2.2.2" xref="S3.SS3.p5.4.m4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p5.4.m4.2.2.2.3" xref="S3.SS3.p5.4.m4.2.2.2.3.cmml">𝒪</mi><mo id="S3.SS3.p5.4.m4.2.2.2.2" xref="S3.SS3.p5.4.m4.2.2.2.2.cmml">⁢</mo><mrow id="S3.SS3.p5.4.m4.2.2.2.1.1" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.cmml"><mo id="S3.SS3.p5.4.m4.2.2.2.1.1.2" stretchy="false" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS3.p5.4.m4.2.2.2.1.1.1" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.cmml"><mn id="S3.SS3.p5.4.m4.2.2.2.1.1.1.2" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.2.cmml">20</mn><mo id="S3.SS3.p5.4.m4.2.2.2.1.1.1.1" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.1.cmml">−</mo><mn id="S3.SS3.p5.4.m4.2.2.2.1.1.1.3" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.3.cmml">30</mn></mrow><mo id="S3.SS3.p5.4.m4.2.2.2.1.1.3" stretchy="false" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.4.m4.2b"><apply id="S3.SS3.p5.4.m4.2.2.cmml" xref="S3.SS3.p5.4.m4.2.2"><csymbol cd="latexml" id="S3.SS3.p5.4.m4.2.2.3.cmml" xref="S3.SS3.p5.4.m4.2.2.3">similar-to</csymbol><apply id="S3.SS3.p5.4.m4.1.1.1.2.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1"><abs id="S3.SS3.p5.4.m4.1.1.1.2.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.2"></abs><apply id="S3.SS3.p5.4.m4.1.1.1.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1">superscript</csymbol><apply id="S3.SS3.p5.4.m4.1.1.1.1.1.2.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.4.m4.1.1.1.1.1.2.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p5.4.m4.1.1.1.1.1.2.2.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1.2.2">𝑓</ci><ci id="S3.SS3.p5.4.m4.1.1.1.1.1.2.3.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1.2.3">NL</ci></apply><ci id="S3.SS3.p5.4.m4.1.1.1.1.1.3a.cmml" xref="S3.SS3.p5.4.m4.1.1.1.1.1.3"><mtext id="S3.SS3.p5.4.m4.1.1.1.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p5.4.m4.1.1.1.1.1.3">local</mtext></ci></apply></apply><apply id="S3.SS3.p5.4.m4.2.2.2.cmml" xref="S3.SS3.p5.4.m4.2.2.2"><times id="S3.SS3.p5.4.m4.2.2.2.2.cmml" xref="S3.SS3.p5.4.m4.2.2.2.2"></times><ci id="S3.SS3.p5.4.m4.2.2.2.3.cmml" xref="S3.SS3.p5.4.m4.2.2.2.3">𝒪</ci><apply id="S3.SS3.p5.4.m4.2.2.2.1.1.1.cmml" xref="S3.SS3.p5.4.m4.2.2.2.1.1"><minus id="S3.SS3.p5.4.m4.2.2.2.1.1.1.1.cmml" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.1"></minus><cn id="S3.SS3.p5.4.m4.2.2.2.1.1.1.2.cmml" type="integer" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.2">20</cn><cn id="S3.SS3.p5.4.m4.2.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS3.p5.4.m4.2.2.2.1.1.1.3">30</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.4.m4.2c">\left|f_{\mathrm{NL}}^{\text{local}}\right|\sim\mathcal{O}(20-30)</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.4.m4.2d">| italic_f start_POSTSUBSCRIPT roman_NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT | ∼ caligraphic_O ( 20 - 30 )</annotation></semantics></math> (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib91" title="">91</a>]</cite>). MUST, as a Stage-V spectroscopic survey, is expected to improve the constraints on <math alttext="f_{\text{NL}}" class="ltx_Math" display="inline" id="S3.SS3.p5.5.m5.1"><semantics id="S3.SS3.p5.5.m5.1a"><msub id="S3.SS3.p5.5.m5.1.1" xref="S3.SS3.p5.5.m5.1.1.cmml"><mi id="S3.SS3.p5.5.m5.1.1.2" xref="S3.SS3.p5.5.m5.1.1.2.cmml">f</mi><mtext id="S3.SS3.p5.5.m5.1.1.3" xref="S3.SS3.p5.5.m5.1.1.3a.cmml">NL</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.5.m5.1b"><apply id="S3.SS3.p5.5.m5.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.5.m5.1.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1">subscript</csymbol><ci id="S3.SS3.p5.5.m5.1.1.2.cmml" xref="S3.SS3.p5.5.m5.1.1.2">𝑓</ci><ci id="S3.SS3.p5.5.m5.1.1.3a.cmml" xref="S3.SS3.p5.5.m5.1.1.3"><mtext id="S3.SS3.p5.5.m5.1.1.3.cmml" mathsize="70%" xref="S3.SS3.p5.5.m5.1.1.3">NL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.5.m5.1c">f_{\text{NL}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.5.m5.1d">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT</annotation></semantics></math> significantly with the benefit of the increasing survey volume and redshift range, surpassing current CMB constraints (see Sec. <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS4" title="5.4 Primordial Non-Gaussianity ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.4</a>). </div> <div class="ltx_para" id="S3.SS3.p6"> Besides the traditional “PNG” searches, recent years have witnessed the fast development of new directions, such as cosmological collider (CC) physics. For instance, oscillatory signals of <math alttext="n" class="ltx_Math" display="inline" id="S3.SS3.p6.1.m1.1"><semantics id="S3.SS3.p6.1.m1.1a"><mi id="S3.SS3.p6.1.m1.1.1" xref="S3.SS3.p6.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p6.1.m1.1b"><ci id="S3.SS3.p6.1.m1.1.1.cmml" xref="S3.SS3.p6.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p6.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p6.1.m1.1d">italic_n</annotation></semantics></math>-point correlation functions can be generated by the resonance between heavy states spontaneously created during inflation and scalar (or tensor) modes and encode a wealth of information about the evolution history of spacetime and the dynamical information of the heavy particle, including the mass, spin, sound speed, chemical potential, interaction type, etc. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib100" title="">100</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib101" title="">101</a>]</cite> Meanwhile, parity-odd patterns in galaxy clustering have gained considerable attention recently. If confirmed, they will be a clear signal of new physics. Active efforts are ongoing in search for these signals in current data and forecast for detectability from future observations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib102" title="">102</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib99" title="">99</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib103" title="">103</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib104" title="">104</a>]</cite>. To maximize the potential scientific outcome of MUST, we aim to explore the feasibility of achieving these goals with MUST in subsequent studies. </div> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> 3.4 Neutrinos & Light Relics</h3> <div class="ltx_para" id="S3.SS4.p1"> The existence of neutrino mass, as revealed by atmospheric and solar neutrino experiments, provides strong evidence for new physics beyond the Standard Model. Neutrino oscillation experiments suggest two possible hierarchies in their mass spectrum: the normal (<math alttext="m_{1}<m_{2}<m_{3}" class="ltx_Math" display="inline" id="S3.SS4.p1.1.m1.1"><semantics id="S3.SS4.p1.1.m1.1a"><mrow id="S3.SS4.p1.1.m1.1.1" xref="S3.SS4.p1.1.m1.1.1.cmml"><msub id="S3.SS4.p1.1.m1.1.1.2" xref="S3.SS4.p1.1.m1.1.1.2.cmml"><mi id="S3.SS4.p1.1.m1.1.1.2.2" xref="S3.SS4.p1.1.m1.1.1.2.2.cmml">m</mi><mn id="S3.SS4.p1.1.m1.1.1.2.3" xref="S3.SS4.p1.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS4.p1.1.m1.1.1.3" xref="S3.SS4.p1.1.m1.1.1.3.cmml"><</mo><msub id="S3.SS4.p1.1.m1.1.1.4" xref="S3.SS4.p1.1.m1.1.1.4.cmml"><mi id="S3.SS4.p1.1.m1.1.1.4.2" xref="S3.SS4.p1.1.m1.1.1.4.2.cmml">m</mi><mn id="S3.SS4.p1.1.m1.1.1.4.3" xref="S3.SS4.p1.1.m1.1.1.4.3.cmml">2</mn></msub><mo id="S3.SS4.p1.1.m1.1.1.5" xref="S3.SS4.p1.1.m1.1.1.5.cmml"><</mo><msub id="S3.SS4.p1.1.m1.1.1.6" xref="S3.SS4.p1.1.m1.1.1.6.cmml"><mi id="S3.SS4.p1.1.m1.1.1.6.2" 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xref="S3.SS4.p1.1.m1.1.1.4.2">𝑚</ci><cn id="S3.SS4.p1.1.m1.1.1.4.3.cmml" type="integer" xref="S3.SS4.p1.1.m1.1.1.4.3">2</cn></apply></apply><apply id="S3.SS4.p1.1.m1.1.1c.cmml" xref="S3.SS4.p1.1.m1.1.1"><lt id="S3.SS4.p1.1.m1.1.1.5.cmml" xref="S3.SS4.p1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S3.SS4.p1.1.m1.1.1.4.cmml" id="S3.SS4.p1.1.m1.1.1d.cmml" xref="S3.SS4.p1.1.m1.1.1"></share><apply id="S3.SS4.p1.1.m1.1.1.6.cmml" xref="S3.SS4.p1.1.m1.1.1.6"><csymbol cd="ambiguous" id="S3.SS4.p1.1.m1.1.1.6.1.cmml" xref="S3.SS4.p1.1.m1.1.1.6">subscript</csymbol><ci id="S3.SS4.p1.1.m1.1.1.6.2.cmml" xref="S3.SS4.p1.1.m1.1.1.6.2">𝑚</ci><cn id="S3.SS4.p1.1.m1.1.1.6.3.cmml" type="integer" xref="S3.SS4.p1.1.m1.1.1.6.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.1.m1.1c">m_{1}<m_{2}<m_{3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.1.m1.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="M_{\nu}\gtrsim 0.059\,{\rm eV}" class="ltx_Math" display="inline" id="S3.SS4.p1.2.m2.1"><semantics id="S3.SS4.p1.2.m2.1a"><mrow id="S3.SS4.p1.2.m2.1.1" xref="S3.SS4.p1.2.m2.1.1.cmml"><msub id="S3.SS4.p1.2.m2.1.1.2" xref="S3.SS4.p1.2.m2.1.1.2.cmml"><mi id="S3.SS4.p1.2.m2.1.1.2.2" xref="S3.SS4.p1.2.m2.1.1.2.2.cmml">M</mi><mi id="S3.SS4.p1.2.m2.1.1.2.3" xref="S3.SS4.p1.2.m2.1.1.2.3.cmml">ν</mi></msub><mo id="S3.SS4.p1.2.m2.1.1.1" xref="S3.SS4.p1.2.m2.1.1.1.cmml">≳</mo><mrow id="S3.SS4.p1.2.m2.1.1.3" xref="S3.SS4.p1.2.m2.1.1.3.cmml"><mn id="S3.SS4.p1.2.m2.1.1.3.2" xref="S3.SS4.p1.2.m2.1.1.3.2.cmml">0.059</mn><mo id="S3.SS4.p1.2.m2.1.1.3.1" lspace="0.170em" xref="S3.SS4.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS4.p1.2.m2.1.1.3.3" xref="S3.SS4.p1.2.m2.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.2.m2.1b"><apply id="S3.SS4.p1.2.m2.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.SS4.p1.2.m2.1.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1.1">greater-than-or-equivalent-to</csymbol><apply id="S3.SS4.p1.2.m2.1.1.2.cmml" xref="S3.SS4.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p1.2.m2.1.1.2.1.cmml" xref="S3.SS4.p1.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS4.p1.2.m2.1.1.2.2.cmml" xref="S3.SS4.p1.2.m2.1.1.2.2">𝑀</ci><ci id="S3.SS4.p1.2.m2.1.1.2.3.cmml" xref="S3.SS4.p1.2.m2.1.1.2.3">𝜈</ci></apply><apply id="S3.SS4.p1.2.m2.1.1.3.cmml" xref="S3.SS4.p1.2.m2.1.1.3"><times id="S3.SS4.p1.2.m2.1.1.3.1.cmml" xref="S3.SS4.p1.2.m2.1.1.3.1"></times><cn id="S3.SS4.p1.2.m2.1.1.3.2.cmml" type="float" xref="S3.SS4.p1.2.m2.1.1.3.2">0.059</cn><ci id="S3.SS4.p1.2.m2.1.1.3.3.cmml" xref="S3.SS4.p1.2.m2.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.2.m2.1c">M_{\nu}\gtrsim 0.059\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.2.m2.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ≳ 0.059 roman_eV</annotation></semantics></math>) and the inverted (<math alttext="m_{3}<m_{1}<m_{2}" class="ltx_Math" display="inline" id="S3.SS4.p1.3.m3.1"><semantics id="S3.SS4.p1.3.m3.1a"><mrow id="S3.SS4.p1.3.m3.1.1" xref="S3.SS4.p1.3.m3.1.1.cmml"><msub id="S3.SS4.p1.3.m3.1.1.2" xref="S3.SS4.p1.3.m3.1.1.2.cmml"><mi id="S3.SS4.p1.3.m3.1.1.2.2" xref="S3.SS4.p1.3.m3.1.1.2.2.cmml">m</mi><mn id="S3.SS4.p1.3.m3.1.1.2.3" xref="S3.SS4.p1.3.m3.1.1.2.3.cmml">3</mn></msub><mo id="S3.SS4.p1.3.m3.1.1.3" xref="S3.SS4.p1.3.m3.1.1.3.cmml"><</mo><msub id="S3.SS4.p1.3.m3.1.1.4" xref="S3.SS4.p1.3.m3.1.1.4.cmml"><mi id="S3.SS4.p1.3.m3.1.1.4.2" xref="S3.SS4.p1.3.m3.1.1.4.2.cmml">m</mi><mn id="S3.SS4.p1.3.m3.1.1.4.3" xref="S3.SS4.p1.3.m3.1.1.4.3.cmml">1</mn></msub><mo id="S3.SS4.p1.3.m3.1.1.5" xref="S3.SS4.p1.3.m3.1.1.5.cmml"><</mo><msub id="S3.SS4.p1.3.m3.1.1.6" xref="S3.SS4.p1.3.m3.1.1.6.cmml"><mi id="S3.SS4.p1.3.m3.1.1.6.2" xref="S3.SS4.p1.3.m3.1.1.6.2.cmml">m</mi><mn id="S3.SS4.p1.3.m3.1.1.6.3" xref="S3.SS4.p1.3.m3.1.1.6.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.3.m3.1b"><apply id="S3.SS4.p1.3.m3.1.1.cmml" xref="S3.SS4.p1.3.m3.1.1"><and id="S3.SS4.p1.3.m3.1.1a.cmml" xref="S3.SS4.p1.3.m3.1.1"></and><apply id="S3.SS4.p1.3.m3.1.1b.cmml" xref="S3.SS4.p1.3.m3.1.1"><lt id="S3.SS4.p1.3.m3.1.1.3.cmml" xref="S3.SS4.p1.3.m3.1.1.3"></lt><apply id="S3.SS4.p1.3.m3.1.1.2.cmml" xref="S3.SS4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p1.3.m3.1.1.2.1.cmml" xref="S3.SS4.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS4.p1.3.m3.1.1.2.2.cmml" xref="S3.SS4.p1.3.m3.1.1.2.2">𝑚</ci><cn id="S3.SS4.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S3.SS4.p1.3.m3.1.1.2.3">3</cn></apply><apply id="S3.SS4.p1.3.m3.1.1.4.cmml" xref="S3.SS4.p1.3.m3.1.1.4"><csymbol cd="ambiguous" id="S3.SS4.p1.3.m3.1.1.4.1.cmml" xref="S3.SS4.p1.3.m3.1.1.4">subscript</csymbol><ci id="S3.SS4.p1.3.m3.1.1.4.2.cmml" xref="S3.SS4.p1.3.m3.1.1.4.2">𝑚</ci><cn id="S3.SS4.p1.3.m3.1.1.4.3.cmml" type="integer" xref="S3.SS4.p1.3.m3.1.1.4.3">1</cn></apply></apply><apply id="S3.SS4.p1.3.m3.1.1c.cmml" xref="S3.SS4.p1.3.m3.1.1"><lt id="S3.SS4.p1.3.m3.1.1.5.cmml" xref="S3.SS4.p1.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S3.SS4.p1.3.m3.1.1.4.cmml" id="S3.SS4.p1.3.m3.1.1d.cmml" xref="S3.SS4.p1.3.m3.1.1"></share><apply id="S3.SS4.p1.3.m3.1.1.6.cmml" xref="S3.SS4.p1.3.m3.1.1.6"><csymbol cd="ambiguous" id="S3.SS4.p1.3.m3.1.1.6.1.cmml" xref="S3.SS4.p1.3.m3.1.1.6">subscript</csymbol><ci id="S3.SS4.p1.3.m3.1.1.6.2.cmml" xref="S3.SS4.p1.3.m3.1.1.6.2">𝑚</ci><cn id="S3.SS4.p1.3.m3.1.1.6.3.cmml" type="integer" xref="S3.SS4.p1.3.m3.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.3.m3.1c">m_{3}<m_{1}<m_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.3.m3.1d">italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT < italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="M_{\nu}\gtrsim 0.099\,{\rm eV}" class="ltx_Math" display="inline" id="S3.SS4.p1.4.m4.1"><semantics id="S3.SS4.p1.4.m4.1a"><mrow id="S3.SS4.p1.4.m4.1.1" xref="S3.SS4.p1.4.m4.1.1.cmml"><msub id="S3.SS4.p1.4.m4.1.1.2" xref="S3.SS4.p1.4.m4.1.1.2.cmml"><mi id="S3.SS4.p1.4.m4.1.1.2.2" xref="S3.SS4.p1.4.m4.1.1.2.2.cmml">M</mi><mi id="S3.SS4.p1.4.m4.1.1.2.3" xref="S3.SS4.p1.4.m4.1.1.2.3.cmml">ν</mi></msub><mo id="S3.SS4.p1.4.m4.1.1.1" xref="S3.SS4.p1.4.m4.1.1.1.cmml">≳</mo><mrow id="S3.SS4.p1.4.m4.1.1.3" xref="S3.SS4.p1.4.m4.1.1.3.cmml"><mn id="S3.SS4.p1.4.m4.1.1.3.2" xref="S3.SS4.p1.4.m4.1.1.3.2.cmml">0.099</mn><mo id="S3.SS4.p1.4.m4.1.1.3.1" lspace="0.170em" xref="S3.SS4.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS4.p1.4.m4.1.1.3.3" xref="S3.SS4.p1.4.m4.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.4.m4.1b"><apply id="S3.SS4.p1.4.m4.1.1.cmml" xref="S3.SS4.p1.4.m4.1.1"><csymbol cd="latexml" id="S3.SS4.p1.4.m4.1.1.1.cmml" xref="S3.SS4.p1.4.m4.1.1.1">greater-than-or-equivalent-to</csymbol><apply id="S3.SS4.p1.4.m4.1.1.2.cmml" xref="S3.SS4.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p1.4.m4.1.1.2.1.cmml" xref="S3.SS4.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS4.p1.4.m4.1.1.2.2.cmml" xref="S3.SS4.p1.4.m4.1.1.2.2">𝑀</ci><ci id="S3.SS4.p1.4.m4.1.1.2.3.cmml" xref="S3.SS4.p1.4.m4.1.1.2.3">𝜈</ci></apply><apply id="S3.SS4.p1.4.m4.1.1.3.cmml" xref="S3.SS4.p1.4.m4.1.1.3"><times id="S3.SS4.p1.4.m4.1.1.3.1.cmml" xref="S3.SS4.p1.4.m4.1.1.3.1"></times><cn id="S3.SS4.p1.4.m4.1.1.3.2.cmml" type="float" xref="S3.SS4.p1.4.m4.1.1.3.2">0.099</cn><ci id="S3.SS4.p1.4.m4.1.1.3.3.cmml" xref="S3.SS4.p1.4.m4.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.4.m4.1c">M_{\nu}\gtrsim 0.099\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.4.m4.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ≳ 0.099 roman_eV</annotation></semantics></math>) hierarchies, where <math alttext="m_{1}" class="ltx_Math" display="inline" id="S3.SS4.p1.5.m5.1"><semantics id="S3.SS4.p1.5.m5.1a"><msub id="S3.SS4.p1.5.m5.1.1" xref="S3.SS4.p1.5.m5.1.1.cmml"><mi id="S3.SS4.p1.5.m5.1.1.2" xref="S3.SS4.p1.5.m5.1.1.2.cmml">m</mi><mn id="S3.SS4.p1.5.m5.1.1.3" xref="S3.SS4.p1.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.5.m5.1b"><apply id="S3.SS4.p1.5.m5.1.1.cmml" xref="S3.SS4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.5.m5.1.1.1.cmml" xref="S3.SS4.p1.5.m5.1.1">subscript</csymbol><ci id="S3.SS4.p1.5.m5.1.1.2.cmml" xref="S3.SS4.p1.5.m5.1.1.2">𝑚</ci><cn id="S3.SS4.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.SS4.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.5.m5.1c">m_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.5.m5.1d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="m_{2}" class="ltx_Math" display="inline" id="S3.SS4.p1.6.m6.1"><semantics id="S3.SS4.p1.6.m6.1a"><msub id="S3.SS4.p1.6.m6.1.1" xref="S3.SS4.p1.6.m6.1.1.cmml"><mi id="S3.SS4.p1.6.m6.1.1.2" xref="S3.SS4.p1.6.m6.1.1.2.cmml">m</mi><mn id="S3.SS4.p1.6.m6.1.1.3" xref="S3.SS4.p1.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.6.m6.1b"><apply id="S3.SS4.p1.6.m6.1.1.cmml" xref="S3.SS4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.6.m6.1.1.1.cmml" xref="S3.SS4.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS4.p1.6.m6.1.1.2.cmml" xref="S3.SS4.p1.6.m6.1.1.2">𝑚</ci><cn id="S3.SS4.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.SS4.p1.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.6.m6.1c">m_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.6.m6.1d">italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="m_{3}" class="ltx_Math" display="inline" id="S3.SS4.p1.7.m7.1"><semantics id="S3.SS4.p1.7.m7.1a"><msub id="S3.SS4.p1.7.m7.1.1" xref="S3.SS4.p1.7.m7.1.1.cmml"><mi id="S3.SS4.p1.7.m7.1.1.2" xref="S3.SS4.p1.7.m7.1.1.2.cmml">m</mi><mn id="S3.SS4.p1.7.m7.1.1.3" xref="S3.SS4.p1.7.m7.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.7.m7.1b"><apply id="S3.SS4.p1.7.m7.1.1.cmml" xref="S3.SS4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.7.m7.1.1.1.cmml" xref="S3.SS4.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS4.p1.7.m7.1.1.2.cmml" xref="S3.SS4.p1.7.m7.1.1.2">𝑚</ci><cn id="S3.SS4.p1.7.m7.1.1.3.cmml" type="integer" xref="S3.SS4.p1.7.m7.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.7.m7.1c">m_{3}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.7.m7.1d">italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> indicate the 3 mass eigenstates, and <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S3.SS4.p1.8.m8.1"><semantics id="S3.SS4.p1.8.m8.1a"><msub id="S3.SS4.p1.8.m8.1.1" xref="S3.SS4.p1.8.m8.1.1.cmml"><mi id="S3.SS4.p1.8.m8.1.1.2" xref="S3.SS4.p1.8.m8.1.1.2.cmml">M</mi><mi id="S3.SS4.p1.8.m8.1.1.3" xref="S3.SS4.p1.8.m8.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.8.m8.1b"><apply id="S3.SS4.p1.8.m8.1.1.cmml" xref="S3.SS4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.8.m8.1.1.1.cmml" xref="S3.SS4.p1.8.m8.1.1">subscript</csymbol><ci id="S3.SS4.p1.8.m8.1.1.2.cmml" xref="S3.SS4.p1.8.m8.1.1.2">𝑀</ci><ci id="S3.SS4.p1.8.m8.1.1.3.cmml" xref="S3.SS4.p1.8.m8.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.8.m8.1c">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.8.m8.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> denotes the total neutrino mass <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib105" title="">105</a>]</cite>. Determination of the mass hierarchy is essential for understanding the nature of neutrinos (Dirac or Majorana) and formulating the generalized standard model. However, the most stringent constraint on <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S3.SS4.p1.9.m9.1"><semantics id="S3.SS4.p1.9.m9.1a"><msub id="S3.SS4.p1.9.m9.1.1" xref="S3.SS4.p1.9.m9.1.1.cmml"><mi id="S3.SS4.p1.9.m9.1.1.2" xref="S3.SS4.p1.9.m9.1.1.2.cmml">M</mi><mi id="S3.SS4.p1.9.m9.1.1.3" xref="S3.SS4.p1.9.m9.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.9.m9.1b"><apply id="S3.SS4.p1.9.m9.1.1.cmml" xref="S3.SS4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.9.m9.1.1.1.cmml" xref="S3.SS4.p1.9.m9.1.1">subscript</csymbol><ci id="S3.SS4.p1.9.m9.1.1.2.cmml" xref="S3.SS4.p1.9.m9.1.1.2">𝑀</ci><ci id="S3.SS4.p1.9.m9.1.1.3.cmml" xref="S3.SS4.p1.9.m9.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.9.m9.1c">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.9.m9.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> from current particle physics experiments, set by the KATRIN experiment <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib106" title="">106</a>]</cite>, is only <math alttext="M_{\nu}\lesssim 1.4\,{\rm eV}" class="ltx_Math" display="inline" id="S3.SS4.p1.10.m10.1"><semantics id="S3.SS4.p1.10.m10.1a"><mrow id="S3.SS4.p1.10.m10.1.1" xref="S3.SS4.p1.10.m10.1.1.cmml"><msub id="S3.SS4.p1.10.m10.1.1.2" xref="S3.SS4.p1.10.m10.1.1.2.cmml"><mi id="S3.SS4.p1.10.m10.1.1.2.2" xref="S3.SS4.p1.10.m10.1.1.2.2.cmml">M</mi><mi id="S3.SS4.p1.10.m10.1.1.2.3" xref="S3.SS4.p1.10.m10.1.1.2.3.cmml">ν</mi></msub><mo id="S3.SS4.p1.10.m10.1.1.1" xref="S3.SS4.p1.10.m10.1.1.1.cmml">≲</mo><mrow id="S3.SS4.p1.10.m10.1.1.3" xref="S3.SS4.p1.10.m10.1.1.3.cmml"><mn id="S3.SS4.p1.10.m10.1.1.3.2" xref="S3.SS4.p1.10.m10.1.1.3.2.cmml">1.4</mn><mo id="S3.SS4.p1.10.m10.1.1.3.1" lspace="0.170em" xref="S3.SS4.p1.10.m10.1.1.3.1.cmml">⁢</mo><mi id="S3.SS4.p1.10.m10.1.1.3.3" xref="S3.SS4.p1.10.m10.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.10.m10.1b"><apply id="S3.SS4.p1.10.m10.1.1.cmml" xref="S3.SS4.p1.10.m10.1.1"><csymbol cd="latexml" id="S3.SS4.p1.10.m10.1.1.1.cmml" xref="S3.SS4.p1.10.m10.1.1.1">less-than-or-similar-to</csymbol><apply id="S3.SS4.p1.10.m10.1.1.2.cmml" xref="S3.SS4.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p1.10.m10.1.1.2.1.cmml" xref="S3.SS4.p1.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS4.p1.10.m10.1.1.2.2.cmml" xref="S3.SS4.p1.10.m10.1.1.2.2">𝑀</ci><ci id="S3.SS4.p1.10.m10.1.1.2.3.cmml" xref="S3.SS4.p1.10.m10.1.1.2.3">𝜈</ci></apply><apply id="S3.SS4.p1.10.m10.1.1.3.cmml" xref="S3.SS4.p1.10.m10.1.1.3"><times id="S3.SS4.p1.10.m10.1.1.3.1.cmml" xref="S3.SS4.p1.10.m10.1.1.3.1"></times><cn id="S3.SS4.p1.10.m10.1.1.3.2.cmml" type="float" xref="S3.SS4.p1.10.m10.1.1.3.2">1.4</cn><ci id="S3.SS4.p1.10.m10.1.1.3.3.cmml" xref="S3.SS4.p1.10.m10.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.10.m10.1c">M_{\nu}\lesssim 1.4\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.10.m10.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ≲ 1.4 roman_eV</annotation></semantics></math> at the 90 % credible level (CL). This limit is insufficient to distinguish between the mass hierarchies. </div> <div class="ltx_para" id="S3.SS4.p2"> Neutrinos act as radiation in the early Universe and subsequently contribute matter-energy budget at the late Universe with <math alttext="\Omega_{\nu}=M_{\nu}/(93.14\,h^{2}\,{\rm eV})" class="ltx_Math" display="inline" id="S3.SS4.p2.1.m1.1"><semantics id="S3.SS4.p2.1.m1.1a"><mrow id="S3.SS4.p2.1.m1.1.1" xref="S3.SS4.p2.1.m1.1.1.cmml"><msub id="S3.SS4.p2.1.m1.1.1.3" xref="S3.SS4.p2.1.m1.1.1.3.cmml"><mi id="S3.SS4.p2.1.m1.1.1.3.2" mathvariant="normal" xref="S3.SS4.p2.1.m1.1.1.3.2.cmml">Ω</mi><mi id="S3.SS4.p2.1.m1.1.1.3.3" xref="S3.SS4.p2.1.m1.1.1.3.3.cmml">ν</mi></msub><mo id="S3.SS4.p2.1.m1.1.1.2" xref="S3.SS4.p2.1.m1.1.1.2.cmml">=</mo><mrow id="S3.SS4.p2.1.m1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.cmml"><msub id="S3.SS4.p2.1.m1.1.1.1.3" xref="S3.SS4.p2.1.m1.1.1.1.3.cmml"><mi id="S3.SS4.p2.1.m1.1.1.1.3.2" xref="S3.SS4.p2.1.m1.1.1.1.3.2.cmml">M</mi><mi id="S3.SS4.p2.1.m1.1.1.1.3.3" xref="S3.SS4.p2.1.m1.1.1.1.3.3.cmml">ν</mi></msub><mo id="S3.SS4.p2.1.m1.1.1.1.2" xref="S3.SS4.p2.1.m1.1.1.1.2.cmml">/</mo><mrow id="S3.SS4.p2.1.m1.1.1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p2.1.m1.1.1.1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.cmml"><mn id="S3.SS4.p2.1.m1.1.1.1.1.1.1.2" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.2.cmml">93.14</mn><mo id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1" lspace="0.170em" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.2" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.2.cmml">h</mi><mn id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.3" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.3.cmml">2</mn></msup><mo id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1a" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS4.p2.1.m1.1.1.1.1.1.1.4" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.4.cmml">eV</mi></mrow><mo id="S3.SS4.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.1.m1.1b"><apply id="S3.SS4.p2.1.m1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1"><eq id="S3.SS4.p2.1.m1.1.1.2.cmml" xref="S3.SS4.p2.1.m1.1.1.2"></eq><apply id="S3.SS4.p2.1.m1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.1.m1.1.1.3.1.cmml" xref="S3.SS4.p2.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS4.p2.1.m1.1.1.3.2.cmml" xref="S3.SS4.p2.1.m1.1.1.3.2">Ω</ci><ci id="S3.SS4.p2.1.m1.1.1.3.3.cmml" xref="S3.SS4.p2.1.m1.1.1.3.3">𝜈</ci></apply><apply id="S3.SS4.p2.1.m1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1"><divide id="S3.SS4.p2.1.m1.1.1.1.2.cmml" xref="S3.SS4.p2.1.m1.1.1.1.2"></divide><apply id="S3.SS4.p2.1.m1.1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.1.m1.1.1.1.3.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3">subscript</csymbol><ci id="S3.SS4.p2.1.m1.1.1.1.3.2.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3.2">𝑀</ci><ci id="S3.SS4.p2.1.m1.1.1.1.3.3.cmml" xref="S3.SS4.p2.1.m1.1.1.1.3.3">𝜈</ci></apply><apply id="S3.SS4.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1"><times id="S3.SS4.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.1"></times><cn id="S3.SS4.p2.1.m1.1.1.1.1.1.1.2.cmml" type="float" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.2">93.14</cn><apply id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.2">ℎ</ci><cn id="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.3.3">2</cn></apply><ci id="S3.SS4.p2.1.m1.1.1.1.1.1.1.4.cmml" xref="S3.SS4.p2.1.m1.1.1.1.1.1.1.4">eV</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.1.m1.1c">\Omega_{\nu}=M_{\nu}/(93.14\,h^{2}\,{\rm eV})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.1.m1.1d">roman_Ω start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT / ( 93.14 italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_eV )</annotation></semantics></math>, which alters the expansion rate and shifts the redshift of matter-radiation equality. Besides, due to their significant velocity dispersion, massive neutrinos slow down the growth of structures, resulting in a suppression of the matter power spectrum below the neutrino free-streaming scale <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib107" title="">107</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib108" title="">108</a>]</cite>. As a result, LSS probes are sensitive to the total neutrino mass, providing complementary constraints to particle physics experiments. </div> <div class="ltx_para" id="S3.SS4.p3"> In addition to neutrino mass, the clustering of galaxies also permits constraints on the effective number of neutrino species <math alttext="N_{\rm eff}" class="ltx_Math" display="inline" id="S3.SS4.p3.1.m1.1"><semantics id="S3.SS4.p3.1.m1.1a"><msub id="S3.SS4.p3.1.m1.1.1" xref="S3.SS4.p3.1.m1.1.1.cmml"><mi id="S3.SS4.p3.1.m1.1.1.2" xref="S3.SS4.p3.1.m1.1.1.2.cmml">N</mi><mi id="S3.SS4.p3.1.m1.1.1.3" xref="S3.SS4.p3.1.m1.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.1.m1.1b"><apply id="S3.SS4.p3.1.m1.1.1.cmml" xref="S3.SS4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.1.m1.1.1.1.cmml" xref="S3.SS4.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS4.p3.1.m1.1.1.2.cmml" xref="S3.SS4.p3.1.m1.1.1.2">𝑁</ci><ci id="S3.SS4.p3.1.m1.1.1.3.cmml" xref="S3.SS4.p3.1.m1.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.1.m1.1c">N_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.1.m1.1d">italic_N start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math>, which can be used to probe non-standard neutrino interactions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib109" title="">109</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib110" title="">110</a>]</cite> or extra light thermal relics from the primordial Universe, such as light sterile neutrinos <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib111" title="">111</a>]</cite> and axions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib112" title="">112</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib113" title="">113</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib114" title="">114</a>]</cite>. This is because <math alttext="N_{\rm eff}" class="ltx_Math" display="inline" id="S3.SS4.p3.2.m2.1"><semantics id="S3.SS4.p3.2.m2.1a"><msub id="S3.SS4.p3.2.m2.1.1" xref="S3.SS4.p3.2.m2.1.1.cmml"><mi id="S3.SS4.p3.2.m2.1.1.2" xref="S3.SS4.p3.2.m2.1.1.2.cmml">N</mi><mi id="S3.SS4.p3.2.m2.1.1.3" xref="S3.SS4.p3.2.m2.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.2.m2.1b"><apply id="S3.SS4.p3.2.m2.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.2.m2.1.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1">subscript</csymbol><ci id="S3.SS4.p3.2.m2.1.1.2.cmml" xref="S3.SS4.p3.2.m2.1.1.2">𝑁</ci><ci id="S3.SS4.p3.2.m2.1.1.3.cmml" xref="S3.SS4.p3.2.m2.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.2.m2.1c">N_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.2.m2.1d">italic_N start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math> can be inferred from the phase shift of the BAO spectrum. Any deviation from the value <math alttext="N_{\rm eff}=3.044" class="ltx_Math" display="inline" id="S3.SS4.p3.3.m3.1"><semantics id="S3.SS4.p3.3.m3.1a"><mrow id="S3.SS4.p3.3.m3.1.1" xref="S3.SS4.p3.3.m3.1.1.cmml"><msub id="S3.SS4.p3.3.m3.1.1.2" xref="S3.SS4.p3.3.m3.1.1.2.cmml"><mi id="S3.SS4.p3.3.m3.1.1.2.2" xref="S3.SS4.p3.3.m3.1.1.2.2.cmml">N</mi><mi id="S3.SS4.p3.3.m3.1.1.2.3" xref="S3.SS4.p3.3.m3.1.1.2.3.cmml">eff</mi></msub><mo id="S3.SS4.p3.3.m3.1.1.1" xref="S3.SS4.p3.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS4.p3.3.m3.1.1.3" xref="S3.SS4.p3.3.m3.1.1.3.cmml">3.044</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.3.m3.1b"><apply id="S3.SS4.p3.3.m3.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1"><eq id="S3.SS4.p3.3.m3.1.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1.1"></eq><apply id="S3.SS4.p3.3.m3.1.1.2.cmml" xref="S3.SS4.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p3.3.m3.1.1.2.1.cmml" xref="S3.SS4.p3.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS4.p3.3.m3.1.1.2.2.cmml" xref="S3.SS4.p3.3.m3.1.1.2.2">𝑁</ci><ci id="S3.SS4.p3.3.m3.1.1.2.3.cmml" xref="S3.SS4.p3.3.m3.1.1.2.3">eff</ci></apply><cn id="S3.SS4.p3.3.m3.1.1.3.cmml" type="float" xref="S3.SS4.p3.3.m3.1.1.3">3.044</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.3.m3.1c">N_{\rm eff}=3.044</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.3.m3.1d">italic_N start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 3.044</annotation></semantics></math>, as predicted by the standard cosmological model with 3 massive neutrino species, notably if exceeding 1 %, would indicate new physics beyond the Standard Models of particle physics and cosmology. </div> <div class="ltx_para" id="S3.SS4.p4"> It is worth noting that the constraints on <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S3.SS4.p4.1.m1.1"><semantics id="S3.SS4.p4.1.m1.1a"><msub id="S3.SS4.p4.1.m1.1.1" xref="S3.SS4.p4.1.m1.1.1.cmml"><mi id="S3.SS4.p4.1.m1.1.1.2" xref="S3.SS4.p4.1.m1.1.1.2.cmml">M</mi><mi id="S3.SS4.p4.1.m1.1.1.3" xref="S3.SS4.p4.1.m1.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.1.m1.1b"><apply id="S3.SS4.p4.1.m1.1.1.cmml" xref="S3.SS4.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p4.1.m1.1.1.1.cmml" xref="S3.SS4.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS4.p4.1.m1.1.1.2.cmml" xref="S3.SS4.p4.1.m1.1.1.2">𝑀</ci><ci id="S3.SS4.p4.1.m1.1.1.3.cmml" xref="S3.SS4.p4.1.m1.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.1.m1.1c">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.1.m1.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="N_{\rm eff}" class="ltx_Math" display="inline" id="S3.SS4.p4.2.m2.1"><semantics id="S3.SS4.p4.2.m2.1a"><msub id="S3.SS4.p4.2.m2.1.1" xref="S3.SS4.p4.2.m2.1.1.cmml"><mi id="S3.SS4.p4.2.m2.1.1.2" xref="S3.SS4.p4.2.m2.1.1.2.cmml">N</mi><mi id="S3.SS4.p4.2.m2.1.1.3" xref="S3.SS4.p4.2.m2.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.2.m2.1b"><apply id="S3.SS4.p4.2.m2.1.1.cmml" xref="S3.SS4.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS4.p4.2.m2.1.1.1.cmml" xref="S3.SS4.p4.2.m2.1.1">subscript</csymbol><ci id="S3.SS4.p4.2.m2.1.1.2.cmml" xref="S3.SS4.p4.2.m2.1.1.2">𝑁</ci><ci id="S3.SS4.p4.2.m2.1.1.3.cmml" xref="S3.SS4.p4.2.m2.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.2.m2.1c">N_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.2.m2.1d">italic_N start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math> from CMB alone exhibit a geometrical degeneracy with <math alttext="H_{0}" class="ltx_Math" display="inline" id="S3.SS4.p4.3.m3.1"><semantics id="S3.SS4.p4.3.m3.1a"><msub id="S3.SS4.p4.3.m3.1.1" xref="S3.SS4.p4.3.m3.1.1.cmml"><mi id="S3.SS4.p4.3.m3.1.1.2" xref="S3.SS4.p4.3.m3.1.1.2.cmml">H</mi><mn id="S3.SS4.p4.3.m3.1.1.3" xref="S3.SS4.p4.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.3.m3.1b"><apply id="S3.SS4.p4.3.m3.1.1.cmml" xref="S3.SS4.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p4.3.m3.1.1.1.cmml" xref="S3.SS4.p4.3.m3.1.1">subscript</csymbol><ci id="S3.SS4.p4.3.m3.1.1.2.cmml" xref="S3.SS4.p4.3.m3.1.1.2">𝐻</ci><cn id="S3.SS4.p4.3.m3.1.1.3.cmml" type="integer" xref="S3.SS4.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.3.m3.1c">H_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.3.m3.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> (or <math alttext="\Omega_{m}" class="ltx_Math" display="inline" id="S3.SS4.p4.4.m4.1"><semantics id="S3.SS4.p4.4.m4.1a"><msub id="S3.SS4.p4.4.m4.1.1" xref="S3.SS4.p4.4.m4.1.1.cmml"><mi id="S3.SS4.p4.4.m4.1.1.2" mathvariant="normal" xref="S3.SS4.p4.4.m4.1.1.2.cmml">Ω</mi><mi id="S3.SS4.p4.4.m4.1.1.3" xref="S3.SS4.p4.4.m4.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.4.m4.1b"><apply id="S3.SS4.p4.4.m4.1.1.cmml" xref="S3.SS4.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS4.p4.4.m4.1.1.1.cmml" xref="S3.SS4.p4.4.m4.1.1">subscript</csymbol><ci id="S3.SS4.p4.4.m4.1.1.2.cmml" xref="S3.SS4.p4.4.m4.1.1.2">Ω</ci><ci id="S3.SS4.p4.4.m4.1.1.3.cmml" xref="S3.SS4.p4.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.4.m4.1c">\Omega_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.4.m4.1d">roman_Ω start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>). BAO measurements can help break this degeneracy due to their ability to constrain <math alttext="\Omega_{m}" class="ltx_Math" display="inline" id="S3.SS4.p4.5.m5.1"><semantics id="S3.SS4.p4.5.m5.1a"><msub id="S3.SS4.p4.5.m5.1.1" xref="S3.SS4.p4.5.m5.1.1.cmml"><mi id="S3.SS4.p4.5.m5.1.1.2" mathvariant="normal" xref="S3.SS4.p4.5.m5.1.1.2.cmml">Ω</mi><mi id="S3.SS4.p4.5.m5.1.1.3" xref="S3.SS4.p4.5.m5.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.5.m5.1b"><apply id="S3.SS4.p4.5.m5.1.1.cmml" xref="S3.SS4.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS4.p4.5.m5.1.1.1.cmml" xref="S3.SS4.p4.5.m5.1.1">subscript</csymbol><ci id="S3.SS4.p4.5.m5.1.1.2.cmml" xref="S3.SS4.p4.5.m5.1.1.2">Ω</ci><ci id="S3.SS4.p4.5.m5.1.1.3.cmml" xref="S3.SS4.p4.5.m5.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.5.m5.1c">\Omega_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.5.m5.1d">roman_Ω start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib73" title="">73</a>]</cite>. Combining CMB and LSS data has already led to more stringent constraints on <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S3.SS4.p4.6.m6.1"><semantics id="S3.SS4.p4.6.m6.1a"><msub id="S3.SS4.p4.6.m6.1.1" xref="S3.SS4.p4.6.m6.1.1.cmml"><mi id="S3.SS4.p4.6.m6.1.1.2" xref="S3.SS4.p4.6.m6.1.1.2.cmml">M</mi><mi id="S3.SS4.p4.6.m6.1.1.3" xref="S3.SS4.p4.6.m6.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.6.m6.1b"><apply id="S3.SS4.p4.6.m6.1.1.cmml" xref="S3.SS4.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS4.p4.6.m6.1.1.1.cmml" xref="S3.SS4.p4.6.m6.1.1">subscript</csymbol><ci id="S3.SS4.p4.6.m6.1.1.2.cmml" xref="S3.SS4.p4.6.m6.1.1.2">𝑀</ci><ci id="S3.SS4.p4.6.m6.1.1.3.cmml" xref="S3.SS4.p4.6.m6.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.6.m6.1c">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.6.m6.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> than those from terrestrial experiments, approaching the lower bound in the inverted hierarchical scenario <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib115" title="">115</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib116" title="">116</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib117" title="">117</a>]</cite>. Recently, DESI reported an upper bound of <math alttext="M_{\nu}<0.072\,{\rm eV}" class="ltx_Math" display="inline" id="S3.SS4.p4.7.m7.1"><semantics id="S3.SS4.p4.7.m7.1a"><mrow id="S3.SS4.p4.7.m7.1.1" xref="S3.SS4.p4.7.m7.1.1.cmml"><msub id="S3.SS4.p4.7.m7.1.1.2" xref="S3.SS4.p4.7.m7.1.1.2.cmml"><mi id="S3.SS4.p4.7.m7.1.1.2.2" xref="S3.SS4.p4.7.m7.1.1.2.2.cmml">M</mi><mi id="S3.SS4.p4.7.m7.1.1.2.3" xref="S3.SS4.p4.7.m7.1.1.2.3.cmml">ν</mi></msub><mo id="S3.SS4.p4.7.m7.1.1.1" xref="S3.SS4.p4.7.m7.1.1.1.cmml"><</mo><mrow id="S3.SS4.p4.7.m7.1.1.3" xref="S3.SS4.p4.7.m7.1.1.3.cmml"><mn id="S3.SS4.p4.7.m7.1.1.3.2" xref="S3.SS4.p4.7.m7.1.1.3.2.cmml">0.072</mn><mo id="S3.SS4.p4.7.m7.1.1.3.1" lspace="0.170em" xref="S3.SS4.p4.7.m7.1.1.3.1.cmml">⁢</mo><mi id="S3.SS4.p4.7.m7.1.1.3.3" xref="S3.SS4.p4.7.m7.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.7.m7.1b"><apply id="S3.SS4.p4.7.m7.1.1.cmml" xref="S3.SS4.p4.7.m7.1.1"><lt id="S3.SS4.p4.7.m7.1.1.1.cmml" xref="S3.SS4.p4.7.m7.1.1.1"></lt><apply id="S3.SS4.p4.7.m7.1.1.2.cmml" xref="S3.SS4.p4.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p4.7.m7.1.1.2.1.cmml" xref="S3.SS4.p4.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS4.p4.7.m7.1.1.2.2.cmml" xref="S3.SS4.p4.7.m7.1.1.2.2">𝑀</ci><ci id="S3.SS4.p4.7.m7.1.1.2.3.cmml" xref="S3.SS4.p4.7.m7.1.1.2.3">𝜈</ci></apply><apply id="S3.SS4.p4.7.m7.1.1.3.cmml" xref="S3.SS4.p4.7.m7.1.1.3"><times id="S3.SS4.p4.7.m7.1.1.3.1.cmml" xref="S3.SS4.p4.7.m7.1.1.3.1"></times><cn id="S3.SS4.p4.7.m7.1.1.3.2.cmml" type="float" xref="S3.SS4.p4.7.m7.1.1.3.2">0.072</cn><ci id="S3.SS4.p4.7.m7.1.1.3.3.cmml" xref="S3.SS4.p4.7.m7.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.7.m7.1c">M_{\nu}<0.072\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.7.m7.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT < 0.072 roman_eV</annotation></semantics></math> (<math alttext="95\,\%" class="ltx_Math" display="inline" id="S3.SS4.p4.8.m8.1"><semantics id="S3.SS4.p4.8.m8.1a"><mrow id="S3.SS4.p4.8.m8.1.1" xref="S3.SS4.p4.8.m8.1.1.cmml"><mn id="S3.SS4.p4.8.m8.1.1.2" xref="S3.SS4.p4.8.m8.1.1.2.cmml">95</mn><mo id="S3.SS4.p4.8.m8.1.1.1" lspace="0.170em" xref="S3.SS4.p4.8.m8.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.8.m8.1b"><apply id="S3.SS4.p4.8.m8.1.1.cmml" xref="S3.SS4.p4.8.m8.1.1"><csymbol cd="latexml" id="S3.SS4.p4.8.m8.1.1.1.cmml" xref="S3.SS4.p4.8.m8.1.1.1">percent</csymbol><cn id="S3.SS4.p4.8.m8.1.1.2.cmml" type="integer" xref="S3.SS4.p4.8.m8.1.1.2">95</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.8.m8.1c">95\,\%</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.8.m8.1d">95 %</annotation></semantics></math> CL) with a prior of <math alttext="M_{\nu}>0\,{\rm eV}" class="ltx_Math" display="inline" id="S3.SS4.p4.9.m9.1"><semantics id="S3.SS4.p4.9.m9.1a"><mrow id="S3.SS4.p4.9.m9.1.1" xref="S3.SS4.p4.9.m9.1.1.cmml"><msub id="S3.SS4.p4.9.m9.1.1.2" xref="S3.SS4.p4.9.m9.1.1.2.cmml"><mi id="S3.SS4.p4.9.m9.1.1.2.2" xref="S3.SS4.p4.9.m9.1.1.2.2.cmml">M</mi><mi id="S3.SS4.p4.9.m9.1.1.2.3" xref="S3.SS4.p4.9.m9.1.1.2.3.cmml">ν</mi></msub><mo id="S3.SS4.p4.9.m9.1.1.1" xref="S3.SS4.p4.9.m9.1.1.1.cmml">></mo><mrow id="S3.SS4.p4.9.m9.1.1.3" xref="S3.SS4.p4.9.m9.1.1.3.cmml"><mn id="S3.SS4.p4.9.m9.1.1.3.2" xref="S3.SS4.p4.9.m9.1.1.3.2.cmml">0</mn><mo id="S3.SS4.p4.9.m9.1.1.3.1" lspace="0.170em" xref="S3.SS4.p4.9.m9.1.1.3.1.cmml">⁢</mo><mi id="S3.SS4.p4.9.m9.1.1.3.3" xref="S3.SS4.p4.9.m9.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.9.m9.1b"><apply id="S3.SS4.p4.9.m9.1.1.cmml" xref="S3.SS4.p4.9.m9.1.1"><gt id="S3.SS4.p4.9.m9.1.1.1.cmml" xref="S3.SS4.p4.9.m9.1.1.1"></gt><apply id="S3.SS4.p4.9.m9.1.1.2.cmml" xref="S3.SS4.p4.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p4.9.m9.1.1.2.1.cmml" xref="S3.SS4.p4.9.m9.1.1.2">subscript</csymbol><ci id="S3.SS4.p4.9.m9.1.1.2.2.cmml" xref="S3.SS4.p4.9.m9.1.1.2.2">𝑀</ci><ci id="S3.SS4.p4.9.m9.1.1.2.3.cmml" xref="S3.SS4.p4.9.m9.1.1.2.3">𝜈</ci></apply><apply id="S3.SS4.p4.9.m9.1.1.3.cmml" xref="S3.SS4.p4.9.m9.1.1.3"><times id="S3.SS4.p4.9.m9.1.1.3.1.cmml" xref="S3.SS4.p4.9.m9.1.1.3.1"></times><cn id="S3.SS4.p4.9.m9.1.1.3.2.cmml" type="integer" xref="S3.SS4.p4.9.m9.1.1.3.2">0</cn><ci id="S3.SS4.p4.9.m9.1.1.3.3.cmml" xref="S3.SS4.p4.9.m9.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.9.m9.1c">M_{\nu}>0\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.9.m9.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT > 0 roman_eV</annotation></semantics></math>, and updated the constraint <math alttext="N_{\rm eff}=3.10\pm 0.17" class="ltx_Math" display="inline" id="S3.SS4.p4.10.m10.1"><semantics id="S3.SS4.p4.10.m10.1a"><mrow id="S3.SS4.p4.10.m10.1.1" xref="S3.SS4.p4.10.m10.1.1.cmml"><msub id="S3.SS4.p4.10.m10.1.1.2" xref="S3.SS4.p4.10.m10.1.1.2.cmml"><mi id="S3.SS4.p4.10.m10.1.1.2.2" xref="S3.SS4.p4.10.m10.1.1.2.2.cmml">N</mi><mi id="S3.SS4.p4.10.m10.1.1.2.3" xref="S3.SS4.p4.10.m10.1.1.2.3.cmml">eff</mi></msub><mo id="S3.SS4.p4.10.m10.1.1.1" xref="S3.SS4.p4.10.m10.1.1.1.cmml">=</mo><mrow id="S3.SS4.p4.10.m10.1.1.3" xref="S3.SS4.p4.10.m10.1.1.3.cmml"><mn id="S3.SS4.p4.10.m10.1.1.3.2" xref="S3.SS4.p4.10.m10.1.1.3.2.cmml">3.10</mn><mo id="S3.SS4.p4.10.m10.1.1.3.1" xref="S3.SS4.p4.10.m10.1.1.3.1.cmml">±</mo><mn id="S3.SS4.p4.10.m10.1.1.3.3" xref="S3.SS4.p4.10.m10.1.1.3.3.cmml">0.17</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.10.m10.1b"><apply id="S3.SS4.p4.10.m10.1.1.cmml" xref="S3.SS4.p4.10.m10.1.1"><eq id="S3.SS4.p4.10.m10.1.1.1.cmml" xref="S3.SS4.p4.10.m10.1.1.1"></eq><apply id="S3.SS4.p4.10.m10.1.1.2.cmml" xref="S3.SS4.p4.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.p4.10.m10.1.1.2.1.cmml" xref="S3.SS4.p4.10.m10.1.1.2">subscript</csymbol><ci id="S3.SS4.p4.10.m10.1.1.2.2.cmml" xref="S3.SS4.p4.10.m10.1.1.2.2">𝑁</ci><ci id="S3.SS4.p4.10.m10.1.1.2.3.cmml" xref="S3.SS4.p4.10.m10.1.1.2.3">eff</ci></apply><apply id="S3.SS4.p4.10.m10.1.1.3.cmml" xref="S3.SS4.p4.10.m10.1.1.3"><csymbol cd="latexml" id="S3.SS4.p4.10.m10.1.1.3.1.cmml" xref="S3.SS4.p4.10.m10.1.1.3.1">plus-or-minus</csymbol><cn id="S3.SS4.p4.10.m10.1.1.3.2.cmml" type="float" xref="S3.SS4.p4.10.m10.1.1.3.2">3.10</cn><cn id="S3.SS4.p4.10.m10.1.1.3.3.cmml" type="float" xref="S3.SS4.p4.10.m10.1.1.3.3">0.17</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.10.m10.1c">N_{\rm eff}=3.10\pm 0.17</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.10.m10.1d">italic_N start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 3.10 ± 0.17</annotation></semantics></math>, which is in remarkable agreement with the Standard Model expectation. With its significantly larger survey volume and increased tracer number density, MUST is expected to substantially tighten constraints on both parameters and investigate the possibility of distinguishing between the two mass hierarchies in combination with data from next-generation CMB experiments (e.g., CMB-S4 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib118" title="">118</a>]</cite>). </div> </section> <section class="ltx_subsection" id="S3.SS5"> <h3 class="ltx_title ltx_title_subsection"> 3.5 Dark Matter</h3> <div class="ltx_para" id="S3.SS5.p1"> Although dark matter (DM) – which constitutes <math alttext="\sim 25" class="ltx_Math" display="inline" id="S3.SS5.p1.1.m1.1"><semantics id="S3.SS5.p1.1.m1.1a"><mrow id="S3.SS5.p1.1.m1.1.1" xref="S3.SS5.p1.1.m1.1.1.cmml"><mi id="S3.SS5.p1.1.m1.1.1.2" xref="S3.SS5.p1.1.m1.1.1.2.cmml"></mi><mo id="S3.SS5.p1.1.m1.1.1.1" xref="S3.SS5.p1.1.m1.1.1.1.cmml">∼</mo><mn id="S3.SS5.p1.1.m1.1.1.3" xref="S3.SS5.p1.1.m1.1.1.3.cmml">25</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p1.1.m1.1b"><apply id="S3.SS5.p1.1.m1.1.1.cmml" xref="S3.SS5.p1.1.m1.1.1"><csymbol cd="latexml" id="S3.SS5.p1.1.m1.1.1.1.cmml" xref="S3.SS5.p1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS5.p1.1.m1.1.1.2.cmml" xref="S3.SS5.p1.1.m1.1.1.2">absent</csymbol><cn id="S3.SS5.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS5.p1.1.m1.1.1.3">25</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p1.1.m1.1c">\sim 25</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p1.1.m1.1d">∼ 25</annotation></semantics></math> % of the energy density of the Universe – was first postulated almost 90 years ago <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib119" title="">119</a>]</cite>, its fundamental nature remains unknown. Observational efforts have focused mainly on the cold dark matter (CDM) paradigm, yet its constituents remain elusive, prompting interest in alternative DM models. These alternative models aim to address specific challenges to the CDM model, such as the missing satellite, cusp-core (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib120" title="">120</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib121" title="">121</a>]</cite>), and too-big-to-fail problems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib122" title="">122</a>]</cite> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib123" title="">123</a>]</cite> for a review). </div> <div class="ltx_para" id="S3.SS5.p2"> Popular alternatives to the CDM model include warm dark matter (WDM; see e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib124" title="">124</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib125" title="">125</a>]</cite>), fuzzy dark matter (FDM; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib126" title="">126</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib127" title="">127</a>]</cite>), self-interacting dark matter (SIDM; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib128" title="">128</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib129" title="">129</a>]</cite>), and primordial black holes (PBH; see e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib130" title="">130</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib131" title="">131</a>]</cite>). These models gain traction for their potential to mitigate the challenges of CMB or for their natural origin, such as PBHs, which do not require new physics beyond the standard model. MUST can offer unique insight into discovering the nature of dark matter primarily through two avenues: Milky Way (MW) and Local Group (LG) observations, as well as the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p2.1.m1.1"><semantics id="S3.SS5.p2.1.m1.1a"><mi id="S3.SS5.p2.1.m1.1.1" xref="S3.SS5.p2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p2.1.m1.1b"><ci id="S3.SS5.p2.1.m1.1.1.cmml" xref="S3.SS5.p2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p2.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p2.1.m1.1d">italic_α</annotation></semantics></math> forest. </div> <div class="ltx_para" id="S3.SS5.p3"> In the current Gaia era, the measurement of dark matter distribution around our MW based on stellar kinematics still has significant uncertainties (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib132" title="">132</a>]</cite>). One dominant reason is model extrapolations due to the lack of stellar tracers with full phase-space information in the outer MW halo (<math alttext="\gtrsim 100" class="ltx_Math" display="inline" id="S3.SS5.p3.1.m1.1"><semantics id="S3.SS5.p3.1.m1.1a"><mrow id="S3.SS5.p3.1.m1.1.1" xref="S3.SS5.p3.1.m1.1.1.cmml"><mi id="S3.SS5.p3.1.m1.1.1.2" xref="S3.SS5.p3.1.m1.1.1.2.cmml"></mi><mo id="S3.SS5.p3.1.m1.1.1.1" xref="S3.SS5.p3.1.m1.1.1.1.cmml">≳</mo><mn id="S3.SS5.p3.1.m1.1.1.3" xref="S3.SS5.p3.1.m1.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p3.1.m1.1b"><apply id="S3.SS5.p3.1.m1.1.1.cmml" xref="S3.SS5.p3.1.m1.1.1"><csymbol cd="latexml" id="S3.SS5.p3.1.m1.1.1.1.cmml" xref="S3.SS5.p3.1.m1.1.1.1">greater-than-or-equivalent-to</csymbol><csymbol cd="latexml" id="S3.SS5.p3.1.m1.1.1.2.cmml" xref="S3.SS5.p3.1.m1.1.1.2">absent</csymbol><cn id="S3.SS5.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS5.p3.1.m1.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p3.1.m1.1c">\gtrsim 100</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p3.1.m1.1d">≳ 100</annotation></semantics></math> kpc). MUST can measure the line-of-sight velocities (LOSV) of a plethora of MW halo stars, which, in combination with a deep proper motion survey (e.g., the Roman space telescope high-latitude survey <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib133" title="">133</a>]</cite>), will allow for the mapping of the local distribution of dark matter (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib134" title="">134</a>]</cite>) around our MW, enabling constraints on the radial mass profile and 3D shape of the MW dark halo, which will significantly benefit direct dark matter detection programs (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib135" title="">135</a>]</cite>). </div> <div class="ltx_para" id="S3.SS5.p4"> In particular, the ongoing Stage-IV DESI Milky Way survey (MWS; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib136" title="">136</a>]</cite>) reaches <math alttext="\sim 20" class="ltx_Math" display="inline" id="S3.SS5.p4.1.m1.1"><semantics id="S3.SS5.p4.1.m1.1a"><mrow id="S3.SS5.p4.1.m1.1.1" xref="S3.SS5.p4.1.m1.1.1.cmml"><mi id="S3.SS5.p4.1.m1.1.1.2" xref="S3.SS5.p4.1.m1.1.1.2.cmml"></mi><mo id="S3.SS5.p4.1.m1.1.1.1" xref="S3.SS5.p4.1.m1.1.1.1.cmml">∼</mo><mn id="S3.SS5.p4.1.m1.1.1.3" xref="S3.SS5.p4.1.m1.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p4.1.m1.1b"><apply id="S3.SS5.p4.1.m1.1.1.cmml" xref="S3.SS5.p4.1.m1.1.1"><csymbol cd="latexml" id="S3.SS5.p4.1.m1.1.1.1.cmml" xref="S3.SS5.p4.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S3.SS5.p4.1.m1.1.1.2.cmml" xref="S3.SS5.p4.1.m1.1.1.2">absent</csymbol><cn id="S3.SS5.p4.1.m1.1.1.3.cmml" type="integer" xref="S3.SS5.p4.1.m1.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p4.1.m1.1c">\sim 20</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p4.1.m1.1d">∼ 20</annotation></semantics></math> – 30 % completeness for MW targets in the flux range of <math alttext="16<r<19" class="ltx_Math" display="inline" id="S3.SS5.p4.2.m2.1"><semantics id="S3.SS5.p4.2.m2.1a"><mrow id="S3.SS5.p4.2.m2.1.1" xref="S3.SS5.p4.2.m2.1.1.cmml"><mn id="S3.SS5.p4.2.m2.1.1.2" xref="S3.SS5.p4.2.m2.1.1.2.cmml">16</mn><mo id="S3.SS5.p4.2.m2.1.1.3" xref="S3.SS5.p4.2.m2.1.1.3.cmml"><</mo><mi id="S3.SS5.p4.2.m2.1.1.4" xref="S3.SS5.p4.2.m2.1.1.4.cmml">r</mi><mo id="S3.SS5.p4.2.m2.1.1.5" xref="S3.SS5.p4.2.m2.1.1.5.cmml"><</mo><mn id="S3.SS5.p4.2.m2.1.1.6" xref="S3.SS5.p4.2.m2.1.1.6.cmml">19</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p4.2.m2.1b"><apply id="S3.SS5.p4.2.m2.1.1.cmml" xref="S3.SS5.p4.2.m2.1.1"><and id="S3.SS5.p4.2.m2.1.1a.cmml" xref="S3.SS5.p4.2.m2.1.1"></and><apply id="S3.SS5.p4.2.m2.1.1b.cmml" xref="S3.SS5.p4.2.m2.1.1"><lt id="S3.SS5.p4.2.m2.1.1.3.cmml" xref="S3.SS5.p4.2.m2.1.1.3"></lt><cn id="S3.SS5.p4.2.m2.1.1.2.cmml" type="integer" xref="S3.SS5.p4.2.m2.1.1.2">16</cn><ci id="S3.SS5.p4.2.m2.1.1.4.cmml" xref="S3.SS5.p4.2.m2.1.1.4">𝑟</ci></apply><apply id="S3.SS5.p4.2.m2.1.1c.cmml" xref="S3.SS5.p4.2.m2.1.1"><lt id="S3.SS5.p4.2.m2.1.1.5.cmml" xref="S3.SS5.p4.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S3.SS5.p4.2.m2.1.1.4.cmml" id="S3.SS5.p4.2.m2.1.1d.cmml" xref="S3.SS5.p4.2.m2.1.1"></share><cn id="S3.SS5.p4.2.m2.1.1.6.cmml" type="integer" xref="S3.SS5.p4.2.m2.1.1.6">19</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p4.2.m2.1c">16<r<19</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p4.2.m2.1d">16 < italic_r < 19</annotation></semantics></math>. MUST can potentially reach fainter magnitudes than DESI and thus can provide more distant MW halo star LOSVs and increase the completeness at similar magnitudes as DESI, bringing in better constraints on the dark matter distribution in the outer halo. In addition to halo stars, dwarf galaxies and stellar streams around our MW and in the LG carry valuable information on the nature of dark matter (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib137" title="">137</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib138" title="">138</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib139" title="">139</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib140" title="">140</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib141" title="">141</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib129" title="">129</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib142" title="">142</a>]</cite>), and MUST can potentially measure the LOSVs for member stars in distant dwarf galaxies or streams to better infer their mass content or past interactions with dark matter clumps. </div> <div class="ltx_para" id="S3.SS5.p5"> The Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p5.1.m1.1"><semantics id="S3.SS5.p5.1.m1.1a"><mi id="S3.SS5.p5.1.m1.1.1" xref="S3.SS5.p5.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.1.m1.1b"><ci id="S3.SS5.p5.1.m1.1.1.cmml" xref="S3.SS5.p5.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.1.m1.1d">italic_α</annotation></semantics></math> forest consists of absorption features in the spectra of distant quasars caused by Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p5.2.m2.1"><semantics id="S3.SS5.p5.2.m2.1a"><mi id="S3.SS5.p5.2.m2.1.1" xref="S3.SS5.p5.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.2.m2.1b"><ci id="S3.SS5.p5.2.m2.1.1.cmml" xref="S3.SS5.p5.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.2.m2.1d">italic_α</annotation></semantics></math> transitions of neutral hydrogen in the foreground intergalactic medium (IGM). Thus, the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p5.3.m3.1"><semantics id="S3.SS5.p5.3.m3.1a"><mi id="S3.SS5.p5.3.m3.1.1" xref="S3.SS5.p5.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.3.m3.1b"><ci id="S3.SS5.p5.3.m3.1.1.cmml" xref="S3.SS5.p5.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.3.m3.1d">italic_α</annotation></semantics></math> forest is a biased tracer of the neutral hydrogen distribution, and hence, it probes the underlying dark matter density field. As a dark matter probe covering higher redshifts than traditional galaxy surveys, the forest has been used to measure the BAO feature; nonetheless, its relatively high redshift <math alttext="2\lesssim z\lesssim 5" class="ltx_Math" display="inline" id="S3.SS5.p5.4.m4.1"><semantics id="S3.SS5.p5.4.m4.1a"><mrow id="S3.SS5.p5.4.m4.1.1" xref="S3.SS5.p5.4.m4.1.1.cmml"><mn id="S3.SS5.p5.4.m4.1.1.2" xref="S3.SS5.p5.4.m4.1.1.2.cmml">2</mn><mo id="S3.SS5.p5.4.m4.1.1.3" xref="S3.SS5.p5.4.m4.1.1.3.cmml">≲</mo><mi id="S3.SS5.p5.4.m4.1.1.4" xref="S3.SS5.p5.4.m4.1.1.4.cmml">z</mi><mo id="S3.SS5.p5.4.m4.1.1.5" xref="S3.SS5.p5.4.m4.1.1.5.cmml">≲</mo><mn id="S3.SS5.p5.4.m4.1.1.6" xref="S3.SS5.p5.4.m4.1.1.6.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.4.m4.1b"><apply id="S3.SS5.p5.4.m4.1.1.cmml" xref="S3.SS5.p5.4.m4.1.1"><and id="S3.SS5.p5.4.m4.1.1a.cmml" xref="S3.SS5.p5.4.m4.1.1"></and><apply id="S3.SS5.p5.4.m4.1.1b.cmml" xref="S3.SS5.p5.4.m4.1.1"><csymbol cd="latexml" id="S3.SS5.p5.4.m4.1.1.3.cmml" xref="S3.SS5.p5.4.m4.1.1.3">less-than-or-similar-to</csymbol><cn id="S3.SS5.p5.4.m4.1.1.2.cmml" type="integer" xref="S3.SS5.p5.4.m4.1.1.2">2</cn><ci id="S3.SS5.p5.4.m4.1.1.4.cmml" xref="S3.SS5.p5.4.m4.1.1.4">𝑧</ci></apply><apply id="S3.SS5.p5.4.m4.1.1c.cmml" xref="S3.SS5.p5.4.m4.1.1"><csymbol cd="latexml" id="S3.SS5.p5.4.m4.1.1.5.cmml" xref="S3.SS5.p5.4.m4.1.1.5">less-than-or-similar-to</csymbol><share href="https://arxiv.org/html/2411.07970v2#S3.SS5.p5.4.m4.1.1.4.cmml" id="S3.SS5.p5.4.m4.1.1d.cmml" xref="S3.SS5.p5.4.m4.1.1"></share><cn id="S3.SS5.p5.4.m4.1.1.6.cmml" type="integer" xref="S3.SS5.p5.4.m4.1.1.6">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.4.m4.1c">2\lesssim z\lesssim 5</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.4.m4.1d">2 ≲ italic_z ≲ 5</annotation></semantics></math> and sensitivity to small-scale clustering (<math alttext="\gtrsim" class="ltx_Math" display="inline" id="S3.SS5.p5.5.m5.1"><semantics id="S3.SS5.p5.5.m5.1a"><mo id="S3.SS5.p5.5.m5.1.1" xref="S3.SS5.p5.5.m5.1.1.cmml">≳</mo><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.5.m5.1b"><csymbol cd="latexml" id="S3.SS5.p5.5.m5.1.1.cmml" xref="S3.SS5.p5.5.m5.1.1">greater-than-or-equivalent-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.5.m5.1c">\gtrsim</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.5.m5.1d">≳</annotation></semantics></math> Mpc) make it an ideal probe of dark matter models888However, the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="footnote8.m1.1"><semantics id="footnote8.m1.1b"><mi id="footnote8.m1.1.1" xref="footnote8.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="footnote8.m1.1c"><ci id="footnote8.m1.1.1.cmml" xref="footnote8.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m1.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="footnote8.m1.1e">italic_α</annotation></semantics></math> forest is unlikely to detect subtle differences between dark matter models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib143" title="">143</a>]</cite>, such as the oscillations predicted by FDM at very small scales <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib144" title="">144</a>]</cite>.. For instance, constraints from the 1D Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p5.6.m6.1"><semantics id="S3.SS5.p5.6.m6.1a"><mi id="S3.SS5.p5.6.m6.1.1" xref="S3.SS5.p5.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.6.m6.1b"><ci id="S3.SS5.p5.6.m6.1.1.cmml" xref="S3.SS5.p5.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.6.m6.1d">italic_α</annotation></semantics></math> flux power spectrum, obtained from medium and high-resolution spectra, set a firm lower limit on the WDM particle mass of <math alttext="m_{\rm X}\geq 5.3" class="ltx_Math" display="inline" id="S3.SS5.p5.7.m7.1"><semantics id="S3.SS5.p5.7.m7.1a"><mrow id="S3.SS5.p5.7.m7.1.1" xref="S3.SS5.p5.7.m7.1.1.cmml"><msub id="S3.SS5.p5.7.m7.1.1.2" xref="S3.SS5.p5.7.m7.1.1.2.cmml"><mi id="S3.SS5.p5.7.m7.1.1.2.2" xref="S3.SS5.p5.7.m7.1.1.2.2.cmml">m</mi><mi id="S3.SS5.p5.7.m7.1.1.2.3" mathvariant="normal" xref="S3.SS5.p5.7.m7.1.1.2.3.cmml">X</mi></msub><mo id="S3.SS5.p5.7.m7.1.1.1" xref="S3.SS5.p5.7.m7.1.1.1.cmml">≥</mo><mn id="S3.SS5.p5.7.m7.1.1.3" xref="S3.SS5.p5.7.m7.1.1.3.cmml">5.3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.7.m7.1b"><apply id="S3.SS5.p5.7.m7.1.1.cmml" xref="S3.SS5.p5.7.m7.1.1"><geq id="S3.SS5.p5.7.m7.1.1.1.cmml" xref="S3.SS5.p5.7.m7.1.1.1"></geq><apply id="S3.SS5.p5.7.m7.1.1.2.cmml" xref="S3.SS5.p5.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS5.p5.7.m7.1.1.2.1.cmml" xref="S3.SS5.p5.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS5.p5.7.m7.1.1.2.2.cmml" xref="S3.SS5.p5.7.m7.1.1.2.2">𝑚</ci><ci id="S3.SS5.p5.7.m7.1.1.2.3.cmml" xref="S3.SS5.p5.7.m7.1.1.2.3">X</ci></apply><cn id="S3.SS5.p5.7.m7.1.1.3.cmml" type="float" xref="S3.SS5.p5.7.m7.1.1.3">5.3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.7.m7.1c">m_{\rm X}\geq 5.3</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.7.m7.1d">italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT ≥ 5.3</annotation></semantics></math> – <math alttext="5.7" class="ltx_Math" display="inline" id="S3.SS5.p5.8.m8.1"><semantics id="S3.SS5.p5.8.m8.1a"><mn id="S3.SS5.p5.8.m8.1.1" xref="S3.SS5.p5.8.m8.1.1.cmml">5.7</mn><annotation-xml encoding="MathML-Content" id="S3.SS5.p5.8.m8.1b"><cn id="S3.SS5.p5.8.m8.1.1.cmml" type="float" xref="S3.SS5.p5.8.m8.1.1">5.7</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p5.8.m8.1c">5.7</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p5.8.m8.1d">5.7</annotation></semantics></math> keV <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib145" title="">145</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib146" title="">146</a>]</cite>. </div> <div class="ltx_para" id="S3.SS5.p6"> Although MW astrophysics and the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p6.1.m1.1"><semantics id="S3.SS5.p6.1.m1.1a"><mi id="S3.SS5.p6.1.m1.1.1" xref="S3.SS5.p6.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p6.1.m1.1b"><ci id="S3.SS5.p6.1.m1.1.1.cmml" xref="S3.SS5.p6.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p6.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p6.1.m1.1d">italic_α</annotation></semantics></math> forest can provide robust dark matter constraints, the accuracy of these constraints depends on careful modeling of baryonic physics and nonlinearities (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib143" title="">143</a>]</cite>). In particular, Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p6.2.m2.1"><semantics id="S3.SS5.p6.2.m2.1a"><mi id="S3.SS5.p6.2.m2.1.1" xref="S3.SS5.p6.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p6.2.m2.1b"><ci id="S3.SS5.p6.2.m2.1.1.cmml" xref="S3.SS5.p6.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p6.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p6.2.m2.1d">italic_α</annotation></semantics></math> forest constraints require precise modeling of IGM astrophysics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib147" title="">147</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib148" title="">148</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib149" title="">149</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib150" title="">150</a>]</cite> due to the degeneracies between the response of IGM to cosmic reionization and the impact of non-cold dark matter on the thermal state of the IGM. With a conservative flux limit of <math alttext="r\leq 23.5" class="ltx_Math" display="inline" id="S3.SS5.p6.3.m3.1"><semantics id="S3.SS5.p6.3.m3.1a"><mrow id="S3.SS5.p6.3.m3.1.1" xref="S3.SS5.p6.3.m3.1.1.cmml"><mi id="S3.SS5.p6.3.m3.1.1.2" xref="S3.SS5.p6.3.m3.1.1.2.cmml">r</mi><mo id="S3.SS5.p6.3.m3.1.1.1" xref="S3.SS5.p6.3.m3.1.1.1.cmml">≤</mo><mn id="S3.SS5.p6.3.m3.1.1.3" xref="S3.SS5.p6.3.m3.1.1.3.cmml">23.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p6.3.m3.1b"><apply id="S3.SS5.p6.3.m3.1.1.cmml" xref="S3.SS5.p6.3.m3.1.1"><leq id="S3.SS5.p6.3.m3.1.1.1.cmml" xref="S3.SS5.p6.3.m3.1.1.1"></leq><ci id="S3.SS5.p6.3.m3.1.1.2.cmml" xref="S3.SS5.p6.3.m3.1.1.2">𝑟</ci><cn id="S3.SS5.p6.3.m3.1.1.3.cmml" type="float" xref="S3.SS5.p6.3.m3.1.1.3">23.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p6.3.m3.1c">r\leq 23.5</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p6.3.m3.1d">italic_r ≤ 23.5</annotation></semantics></math> (larger than that of DESI, which is <math alttext="r<23" class="ltx_Math" display="inline" id="S3.SS5.p6.4.m4.1"><semantics id="S3.SS5.p6.4.m4.1a"><mrow id="S3.SS5.p6.4.m4.1.1" xref="S3.SS5.p6.4.m4.1.1.cmml"><mi id="S3.SS5.p6.4.m4.1.1.2" xref="S3.SS5.p6.4.m4.1.1.2.cmml">r</mi><mo id="S3.SS5.p6.4.m4.1.1.1" xref="S3.SS5.p6.4.m4.1.1.1.cmml"><</mo><mn id="S3.SS5.p6.4.m4.1.1.3" xref="S3.SS5.p6.4.m4.1.1.3.cmml">23</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p6.4.m4.1b"><apply id="S3.SS5.p6.4.m4.1.1.cmml" xref="S3.SS5.p6.4.m4.1.1"><lt id="S3.SS5.p6.4.m4.1.1.1.cmml" xref="S3.SS5.p6.4.m4.1.1.1"></lt><ci id="S3.SS5.p6.4.m4.1.1.2.cmml" xref="S3.SS5.p6.4.m4.1.1.2">𝑟</ci><cn id="S3.SS5.p6.4.m4.1.1.3.cmml" type="integer" xref="S3.SS5.p6.4.m4.1.1.3">23</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p6.4.m4.1c">r<23</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p6.4.m4.1d">italic_r < 23</annotation></semantics></math>) and an estimated quasar number density surpassing 80 deg-2 at <math alttext="z>2.1" class="ltx_Math" display="inline" id="S3.SS5.p6.6.m6.1"><semantics id="S3.SS5.p6.6.m6.1a"><mrow id="S3.SS5.p6.6.m6.1.1" xref="S3.SS5.p6.6.m6.1.1.cmml"><mi id="S3.SS5.p6.6.m6.1.1.2" xref="S3.SS5.p6.6.m6.1.1.2.cmml">z</mi><mo id="S3.SS5.p6.6.m6.1.1.1" xref="S3.SS5.p6.6.m6.1.1.1.cmml">></mo><mn id="S3.SS5.p6.6.m6.1.1.3" xref="S3.SS5.p6.6.m6.1.1.3.cmml">2.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS5.p6.6.m6.1b"><apply id="S3.SS5.p6.6.m6.1.1.cmml" xref="S3.SS5.p6.6.m6.1.1"><gt id="S3.SS5.p6.6.m6.1.1.1.cmml" xref="S3.SS5.p6.6.m6.1.1.1"></gt><ci id="S3.SS5.p6.6.m6.1.1.2.cmml" xref="S3.SS5.p6.6.m6.1.1.2">𝑧</ci><cn id="S3.SS5.p6.6.m6.1.1.3.cmml" type="float" xref="S3.SS5.p6.6.m6.1.1.3">2.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p6.6.m6.1c">z>2.1</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p6.6.m6.1d">italic_z > 2.1</annotation></semantics></math>, MUST is expected to refine our ability to constrain dark matter models using the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS5.p6.7.m7.1"><semantics id="S3.SS5.p6.7.m7.1a"><mi id="S3.SS5.p6.7.m7.1.1" xref="S3.SS5.p6.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS5.p6.7.m7.1b"><ci id="S3.SS5.p6.7.m7.1.1.cmml" xref="S3.SS5.p6.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS5.p6.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS5.p6.7.m7.1d">italic_α</annotation></semantics></math> forest (see <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS6" title="5.6 Warm Dark Matter ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.6</a> and Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.F10" title="Figure 10 ‣ 5.6 Warm Dark Matter ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">10</a>). </div> </section> <section class="ltx_subsection" id="S3.SS6"> <h3 class="ltx_title ltx_title_subsection"> 3.6 Synergy with Other Probes</h3> <section class="ltx_subsubsection" id="S3.SS6.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> 3.6.1 Imaging surveys</h4> <div class="ltx_para" id="S3.SS6.SSS1.p1"> Next-generation space-based wide-field imaging surveys such as the China Space Station Telescope (CSST; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib17" title="">17</a>]</cite>), Euclid <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib151" title="">151</a>]</cite>, and Roman <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib152" title="">152</a>]</cite> will measure galaxy shears with unprecedented precision, providing tighter dark energy constraints with weak lensing. The ground-based Legacy Survey of Space and Time (LSST; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib20" title="">20</a>]</cite>) will observe the whole southern sky every few nights, building the most extensive catalog of transient phenomena of cosmological interest such as type Ia supernovae or variable AGN. Other imaging surveys worth noting are the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib153" title="">153</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib154" title="">154</a>]</cite>) and the Mozi Wide Field Survey Telescope (WFST; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib155" title="">155</a>]</cite>), which will provide deeper broad-band sky coverage than currently available data, as well as the Javalambre Physics of the Accelerating Universe Survey (J-PAS; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib156" title="">156</a>]</cite>), a narrow-band imaging survey with higher-precision photometric redshifts. </div> <div class="ltx_para" id="S3.SS6.SSS1.p2"> Apart from using imaging data from these surveys for MUST target selection, we aim to maximize the scientific outcome through joint analyses of the data for cosmological measurements or even dedicated coordinated observations. For instance, the spectroscopic and photometric samples can be cross-correlated to break degeneracies associated with intrinsic alignments and galaxy bias, thus improving the precision of cosmological measurements (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib157" title="">157</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib158" title="">158</a>]</cite>). Additionally, spare fibers of MUST can be used to complete the spectroscopic data of galaxies in color space to expand the training set for the calibration of photometric or slitless spectroscopic redshift measurements (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib159" title="">159</a>]</cite>). The high efficiency and large survey volume of MUST offers opportunities to identify or confirm strong lensing systems, e.g., low-<math alttext="z" class="ltx_Math" display="inline" id="S3.SS6.SSS1.p2.1.m1.1"><semantics id="S3.SS6.SSS1.p2.1.m1.1a"><mi id="S3.SS6.SSS1.p2.1.m1.1.1" xref="S3.SS6.SSS1.p2.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS1.p2.1.m1.1b"><ci id="S3.SS6.SSS1.p2.1.m1.1.1.cmml" xref="S3.SS6.SSS1.p2.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS1.p2.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS1.p2.1.m1.1d">italic_z</annotation></semantics></math> bright galaxies with anomaly high-<math alttext="z" class="ltx_Math" display="inline" id="S3.SS6.SSS1.p2.2.m2.1"><semantics id="S3.SS6.SSS1.p2.2.m2.1a"><mi id="S3.SS6.SSS1.p2.2.m2.1.1" xref="S3.SS6.SSS1.p2.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS1.p2.2.m2.1b"><ci id="S3.SS6.SSS1.p2.2.m2.1.1.cmml" xref="S3.SS6.SSS1.p2.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS1.p2.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS1.p2.2.m2.1d">italic_z</annotation></semantics></math> emission lines <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib160" title="">160</a>]</cite>. Moreover, the cross-correlation between imaging data and Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS6.SSS1.p2.3.m3.1"><semantics id="S3.SS6.SSS1.p2.3.m3.1a"><mi id="S3.SS6.SSS1.p2.3.m3.1.1" xref="S3.SS6.SSS1.p2.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS1.p2.3.m3.1b"><ci id="S3.SS6.SSS1.p2.3.m3.1.1.cmml" xref="S3.SS6.SSS1.p2.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS1.p2.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS1.p2.3.m3.1d">italic_α</annotation></semantics></math> forests from MUST may enable tracing LSS using Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS6.SSS1.p2.4.m4.1"><semantics id="S3.SS6.SSS1.p2.4.m4.1a"><mi id="S3.SS6.SSS1.p2.4.m4.1.1" xref="S3.SS6.SSS1.p2.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS1.p2.4.m4.1b"><ci id="S3.SS6.SSS1.p2.4.m4.1.1.cmml" xref="S3.SS6.SSS1.p2.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS1.p2.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS1.p2.4.m4.1d">italic_α</annotation></semantics></math> intensity mapping <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib161" title="">161</a>]</cite>. </div> </section> <section class="ltx_subsubsection" id="S3.SS6.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> 3.6.2 CMB S4</h4> <div class="ltx_para" id="S3.SS6.SSS2.p1"> The next generation of CMB experiments, such as AliCPT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib162" title="">162</a>]</cite>, Simons Observatory <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib163" title="">163</a>]</cite>, CMB-S4 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib164" title="">164</a>]</cite>, LiteBIRD <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib165" title="">165</a>]</cite>, and PICO <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib166" title="">166</a>]</cite>, holds significant promise for detecting primordial gravitational waves from the early-universe inflationary epoch. Precisely measuring primordial gravitational waves faces two significant obstacles: foreground contamination and gravitational lensing distortions. Consequently, accurate reconstruction of the CMB lensing potential is critical for isolating the primordial signal. Multi-tracer delensing methods, which leverage galaxy survey data, are expected to significantly enhance the capability of CMB experiments to detect the elusive primordial gravitational waves <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib167" title="">167</a>]</cite>. </div> <div class="ltx_para" id="S3.SS6.SSS2.p2"> In addition to enhancing our understanding of inflationary physics, MUST will facilitate synergistic investigations of large-scale structures. This includes the measurement of the kinetic Sunyaev-Zel’dovich (kSZ) effect, which provides insights into the dynamics of baryonic matter (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib168" title="">168</a>]</cite>). Furthermore, cross-correlations with CMB lensing will improve our understanding of structure formation (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib30" title="">30</a>]</cite>). With these complementary approaches, MUST, in combination with next-generation CMB experiments, promises to enrich our understanding of cosmic evolution. </div> </section> <section class="ltx_subsubsection" id="S3.SS6.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> 3.6.3 Radio surveys</h4> <div class="ltx_para" id="S3.SS6.SSS3.p1"> The Square Kilometre Array (SKA; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib48" title="">48</a>]</cite>) will conduct an IM program using the 21 cm hyperfine transition of neutral Hydrogen. Due to the significant disparity in strength between foreground emissions and the cosmological 21 cm signal – differing by several orders of magnitude – confirming the cosmic origin of the signal, particularly at high redshifts (<math alttext="z\gtrsim 1" class="ltx_Math" display="inline" id="S3.SS6.SSS3.p1.1.m1.1"><semantics id="S3.SS6.SSS3.p1.1.m1.1a"><mrow id="S3.SS6.SSS3.p1.1.m1.1.1" xref="S3.SS6.SSS3.p1.1.m1.1.1.cmml"><mi id="S3.SS6.SSS3.p1.1.m1.1.1.2" xref="S3.SS6.SSS3.p1.1.m1.1.1.2.cmml">z</mi><mo id="S3.SS6.SSS3.p1.1.m1.1.1.1" xref="S3.SS6.SSS3.p1.1.m1.1.1.1.cmml">≳</mo><mn id="S3.SS6.SSS3.p1.1.m1.1.1.3" xref="S3.SS6.SSS3.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS3.p1.1.m1.1b"><apply id="S3.SS6.SSS3.p1.1.m1.1.1.cmml" xref="S3.SS6.SSS3.p1.1.m1.1.1"><csymbol cd="latexml" id="S3.SS6.SSS3.p1.1.m1.1.1.1.cmml" xref="S3.SS6.SSS3.p1.1.m1.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S3.SS6.SSS3.p1.1.m1.1.1.2.cmml" xref="S3.SS6.SSS3.p1.1.m1.1.1.2">𝑧</ci><cn id="S3.SS6.SSS3.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS6.SSS3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS3.p1.1.m1.1c">z\gtrsim 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS3.p1.1.m1.1d">italic_z ≳ 1</annotation></semantics></math>), will likely require cross-correlations with other cosmological tracers. Current studies typically focus on cross-correlations with high-redshift galaxies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib169" title="">169</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib170" title="">170</a>]</cite>, LBG <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib171" title="">171</a>]</cite>, and the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS6.SSS3.p1.2.m2.1"><semantics id="S3.SS6.SSS3.p1.2.m2.1a"><mi id="S3.SS6.SSS3.p1.2.m2.1.1" xref="S3.SS6.SSS3.p1.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS3.p1.2.m2.1b"><ci id="S3.SS6.SSS3.p1.2.m2.1.1.cmml" xref="S3.SS6.SSS3.p1.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS3.p1.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS3.p1.2.m2.1d">italic_α</annotation></semantics></math> forest <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib172" title="">172</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib173" title="">173</a>]</cite>. These complementary approaches coupling 21 cm measurements with spectroscopic surveys enhance our ability to discern the cosmological signal from foreground contamination, thus improving the robustness of IM measurements. </div> <div class="ltx_para" id="S3.SS6.SSS3.p2"> MUST will significantly enhance cross-correlation efforts involving 21 cm observations with high-redshift galaxies and the Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS6.SSS3.p2.1.m1.1"><semantics id="S3.SS6.SSS3.p2.1.m1.1a"><mi id="S3.SS6.SSS3.p2.1.m1.1.1" xref="S3.SS6.SSS3.p2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS3.p2.1.m1.1b"><ci id="S3.SS6.SSS3.p2.1.m1.1.1.cmml" xref="S3.SS6.SSS3.p2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS3.p2.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS3.p2.1.m1.1d">italic_α</annotation></semantics></math> forest, enabling measurements with high signal-to-noise. Such cross-correlations will be instrumental in advancing our understanding of various cosmological phenomena, such as dark energy constraints <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib174" title="">174</a>]</cite>, modified gravity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib175" title="">175</a>]</cite>, dark matter <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib176" title="">176</a>]</cite>, and hydrogen reionization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib173" title="">173</a>]</cite>. </div> </section> <section class="ltx_subsubsection" id="S3.SS6.SSS4"> <h4 class="ltx_title ltx_title_subsubsection"> 3.6.4 Gravitational waves and Fast Radio Bursts</h4> <div class="ltx_para" id="S3.SS6.SSS4.p1"> Very high-energy, transient phenomena such as gravitational waves and Fast Radio Bursts (FRBs) may also provide valuable cosmological constraints. Ever since the first detection of a gravitational wave event by LIGO <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib177" title="">177</a>]</cite>, several large observatories are projected, e.g., LISA <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib178" title="">178</a>]</cite>, Taiji <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib179" title="">179</a>]</cite>, TianQin <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib179" title="">179</a>]</cite> and the Einstein Telescope <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib180" title="">180</a>]</cite>. In addition to the detection of singular events, the gravitational wave background has also been recently detected by Pulsar Timing Array collaborations (e.g., NANOGrav, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib181" title="">181</a>]</cite>, CPTA <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib182" title="">182</a>]</cite>, EPTA, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib183" title="">183</a>]</cite>). Regarding FRBs, current radio observatories detect these events regularly, compiling ever-growing catalogs with hundreds of events (e.g., CHIME, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib184" title="">184</a>]</cite>, ASKAP, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib185" title="">185</a>]</cite>, MeerKAT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib186" title="">186</a>]</cite>, FAST <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib187" title="">187</a>]</cite>). </div> <div class="ltx_para" id="S3.SS6.SSS4.p2"> The most direct synergy MUST will have is the measurement of redshifts and peculiar velocities for both gravitational waves and FRBs electromagnetic (EM) counterparts. Gravitational waves from binary inspirals act as standard sirens since their luminosity distance can be inferred from their observation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib188" title="">188</a>]</cite>. Hence, a redshift measurement of its EM counterpart (either a transient event or a host galaxy) can provide an independent measurement of the Hubble constant <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib189" title="">189</a>]</cite>, and a large spectroscopic catalog of gravitational wave EM counterparts may yield independent <math alttext="H_{\rm 0}" class="ltx_Math" display="inline" id="S3.SS6.SSS4.p2.1.m1.1"><semantics id="S3.SS6.SSS4.p2.1.m1.1a"><msub id="S3.SS6.SSS4.p2.1.m1.1.1" xref="S3.SS6.SSS4.p2.1.m1.1.1.cmml"><mi id="S3.SS6.SSS4.p2.1.m1.1.1.2" xref="S3.SS6.SSS4.p2.1.m1.1.1.2.cmml">H</mi><mn id="S3.SS6.SSS4.p2.1.m1.1.1.3" xref="S3.SS6.SSS4.p2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS4.p2.1.m1.1b"><apply id="S3.SS6.SSS4.p2.1.m1.1.1.cmml" xref="S3.SS6.SSS4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS6.SSS4.p2.1.m1.1.1.1.cmml" xref="S3.SS6.SSS4.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS6.SSS4.p2.1.m1.1.1.2.cmml" xref="S3.SS6.SSS4.p2.1.m1.1.1.2">𝐻</ci><cn id="S3.SS6.SSS4.p2.1.m1.1.1.3.cmml" type="integer" xref="S3.SS6.SSS4.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS4.p2.1.m1.1c">H_{\rm 0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS4.p2.1.m1.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> measurements with a precision of few percent <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib190" title="">190</a>]</cite>. In addition to the direct observation of gravitational wave EM counterparts, cross-correlation of gravitational wave events with spectroscopic galaxy catalogs may also yield improved joint constraints on <math alttext="H_{\rm 0}" class="ltx_Math" display="inline" id="S3.SS6.SSS4.p2.2.m2.1"><semantics id="S3.SS6.SSS4.p2.2.m2.1a"><msub id="S3.SS6.SSS4.p2.2.m2.1.1" xref="S3.SS6.SSS4.p2.2.m2.1.1.cmml"><mi id="S3.SS6.SSS4.p2.2.m2.1.1.2" xref="S3.SS6.SSS4.p2.2.m2.1.1.2.cmml">H</mi><mn id="S3.SS6.SSS4.p2.2.m2.1.1.3" xref="S3.SS6.SSS4.p2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS4.p2.2.m2.1b"><apply id="S3.SS6.SSS4.p2.2.m2.1.1.cmml" xref="S3.SS6.SSS4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS6.SSS4.p2.2.m2.1.1.1.cmml" xref="S3.SS6.SSS4.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS6.SSS4.p2.2.m2.1.1.2.cmml" xref="S3.SS6.SSS4.p2.2.m2.1.1.2">𝐻</ci><cn id="S3.SS6.SSS4.p2.2.m2.1.1.3.cmml" type="integer" xref="S3.SS6.SSS4.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS4.p2.2.m2.1c">H_{\rm 0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS4.p2.2.m2.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib191" title="">191</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib192" title="">192</a>]</cite>. </div> <div class="ltx_para" id="S3.SS6.SSS4.p3"> Similarly, the distance to FRBs may also be determined by their dispersion measure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib193" title="">193</a>]</cite>. However, this method presents degeneracies with the baryon distribution in the FRB sightline, which in turn allows FRBs to constrain the baryon content of the Universe <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib194" title="">194</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib195" title="">195</a>]</cite>. Regardless of these degeneracies, measurements of <math alttext="H_{\rm 0}" class="ltx_Math" display="inline" id="S3.SS6.SSS4.p3.1.m1.1"><semantics id="S3.SS6.SSS4.p3.1.m1.1a"><msub id="S3.SS6.SSS4.p3.1.m1.1.1" xref="S3.SS6.SSS4.p3.1.m1.1.1.cmml"><mi id="S3.SS6.SSS4.p3.1.m1.1.1.2" xref="S3.SS6.SSS4.p3.1.m1.1.1.2.cmml">H</mi><mn id="S3.SS6.SSS4.p3.1.m1.1.1.3" xref="S3.SS6.SSS4.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS4.p3.1.m1.1b"><apply id="S3.SS6.SSS4.p3.1.m1.1.1.cmml" xref="S3.SS6.SSS4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS6.SSS4.p3.1.m1.1.1.1.cmml" xref="S3.SS6.SSS4.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS6.SSS4.p3.1.m1.1.1.2.cmml" xref="S3.SS6.SSS4.p3.1.m1.1.1.2">𝐻</ci><cn id="S3.SS6.SSS4.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS6.SSS4.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS4.p3.1.m1.1c">H_{\rm 0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS4.p3.1.m1.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> using FRBs and spectroscopic redshifts of their host galaxies are possible <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib196" title="">196</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib197" title="">197</a>]</cite>. Therefore, MUST will undoubtedly help with these measurements of <math alttext="H_{\rm 0}" class="ltx_Math" display="inline" id="S3.SS6.SSS4.p3.2.m2.1"><semantics id="S3.SS6.SSS4.p3.2.m2.1a"><msub id="S3.SS6.SSS4.p3.2.m2.1.1" xref="S3.SS6.SSS4.p3.2.m2.1.1.cmml"><mi id="S3.SS6.SSS4.p3.2.m2.1.1.2" xref="S3.SS6.SSS4.p3.2.m2.1.1.2.cmml">H</mi><mn id="S3.SS6.SSS4.p3.2.m2.1.1.3" xref="S3.SS6.SSS4.p3.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS6.SSS4.p3.2.m2.1b"><apply id="S3.SS6.SSS4.p3.2.m2.1.1.cmml" xref="S3.SS6.SSS4.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS6.SSS4.p3.2.m2.1.1.1.cmml" xref="S3.SS6.SSS4.p3.2.m2.1.1">subscript</csymbol><ci id="S3.SS6.SSS4.p3.2.m2.1.1.2.cmml" xref="S3.SS6.SSS4.p3.2.m2.1.1.2">𝐻</ci><cn id="S3.SS6.SSS4.p3.2.m2.1.1.3.cmml" type="integer" xref="S3.SS6.SSS4.p3.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS6.SSS4.p3.2.m2.1c">H_{\rm 0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS6.SSS4.p3.2.m2.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> by providing unprecedently large spectroscopic redshift catalogs of potential gravitational wave and FRB hosts. </div> <figure class="ltx_table" id="S3.T2"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S3.T2.23" style="width:433.6pt;height:165.1pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_align_middle" id="S3.T2.23.23"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T2.23.23.24.1"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.23.23.24.1.1"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.23.23.24.1.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.23.23.24.1.3"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="9" id="S3.T2.23.23.24.1.4" style="background-color:#EFEFEF;">Imaging Depth (AB Mag)</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.23.23.24.1.5"></td> </tr> <tr class="ltx_tr" id="S3.T2.6.6.6" style="background-color:#EFEFEF;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S3.T2.6.6.6.7" style="background-color:#EFEFEF;">Project</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T2.6.6.6.8" style="background-color:#EFEFEF;">Survey Name</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T2.1.1.1.1" style="background-color:#EFEFEF;"> Survey Area (deg2) </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.6.6.6.9" style="background-color:#EFEFEF;">NUV</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.2.2.2.2" style="background-color:#EFEFEF;"><math alttext="u" class="ltx_Math" display="inline" id="S3.T2.2.2.2.2.1.m1.1"><semantics id="S3.T2.2.2.2.2.1.m1.1a"><mi id="S3.T2.2.2.2.2.1.m1.1.1" mathbackground="#EFEFEF" mathcolor="#3166FF" xref="S3.T2.2.2.2.2.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S3.T2.2.2.2.2.1.m1.1b"><ci id="S3.T2.2.2.2.2.1.m1.1.1.cmml" xref="S3.T2.2.2.2.2.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.2.2.2.2.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S3.T2.2.2.2.2.1.m1.1d">italic_u</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.3.3.3.3" style="background-color:#EFEFEF;"><math alttext="g" class="ltx_Math" display="inline" id="S3.T2.3.3.3.3.1.m1.1"><semantics id="S3.T2.3.3.3.3.1.m1.1a"><mi id="S3.T2.3.3.3.3.1.m1.1.1" mathbackground="#EFEFEF" mathcolor="#009901" xref="S3.T2.3.3.3.3.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S3.T2.3.3.3.3.1.m1.1b"><ci id="S3.T2.3.3.3.3.1.m1.1.1.cmml" xref="S3.T2.3.3.3.3.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.3.3.3.3.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S3.T2.3.3.3.3.1.m1.1d">italic_g</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.4.4.4.4" style="background-color:#EFEFEF;"><math alttext="r" class="ltx_Math" display="inline" id="S3.T2.4.4.4.4.1.m1.1"><semantics id="S3.T2.4.4.4.4.1.m1.1a"><mi id="S3.T2.4.4.4.4.1.m1.1.1" mathbackground="#EFEFEF" mathcolor="#F56B00" xref="S3.T2.4.4.4.4.1.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S3.T2.4.4.4.4.1.m1.1b"><ci id="S3.T2.4.4.4.4.1.m1.1.1.cmml" xref="S3.T2.4.4.4.4.1.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.4.4.4.4.1.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S3.T2.4.4.4.4.1.m1.1d">italic_r</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.5.5.5.5" style="background-color:#EFEFEF;"><math alttext="i" class="ltx_Math" display="inline" id="S3.T2.5.5.5.5.1.m1.1"><semantics id="S3.T2.5.5.5.5.1.m1.1a"><mi id="S3.T2.5.5.5.5.1.m1.1.1" mathbackground="#EFEFEF" mathcolor="#FE0000" xref="S3.T2.5.5.5.5.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.T2.5.5.5.5.1.m1.1b"><ci id="S3.T2.5.5.5.5.1.m1.1.1.cmml" xref="S3.T2.5.5.5.5.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.5.5.5.5.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.T2.5.5.5.5.1.m1.1d">italic_i</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.6.6.6.6" style="background-color:#EFEFEF;"><math alttext="z" class="ltx_Math" display="inline" id="S3.T2.6.6.6.6.1.m1.1"><semantics id="S3.T2.6.6.6.6.1.m1.1a"><mi id="S3.T2.6.6.6.6.1.m1.1.1" mathbackground="#EFEFEF" mathcolor="#963400" xref="S3.T2.6.6.6.6.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.T2.6.6.6.6.1.m1.1b"><ci id="S3.T2.6.6.6.6.1.m1.1.1.cmml" xref="S3.T2.6.6.6.6.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.6.6.6.6.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.T2.6.6.6.6.1.m1.1d">italic_z</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.6.6.6.10" style="background-color:#EFEFEF;">Y</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.6.6.6.11" style="background-color:#EFEFEF;">J</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.6.6.6.12">H</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T2.6.6.6.13" style="background-color:#EFEFEF;">Reference</td> </tr> <tr class="ltx_tr" id="S3.T2.7.7.7"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.7.7.7.2"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.3">Optical Survey Wide</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.7.7.7.1.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.7.7.7.1.m1.1a"><mo id="S3.T2.7.7.7.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.7.7.7.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.7.7.7.1.m1.1b"><csymbol cd="latexml" id="S3.T2.7.7.7.1.m1.1.1.cmml" xref="S3.T2.7.7.7.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.7.7.7.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.7.7.7.1.m1.1d">∼</annotation></semantics></math> 17,500 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.4" style="background-color:#DAE8FC;">25.40</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.5" style="background-color:#DAE8FC;">25.40</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.6" style="background-color:#DAE8FC;">26.30</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.7" style="background-color:#DAE8FC;">26.00</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.8" style="background-color:#DAE8FC;"> 25.90</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.9" style="background-color:#DAE8FC;">25.20</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.7.7.7.10" style="background-color:#DAE8FC;">24.40</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.7.7.7.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.7.7.7.12"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.7.7.7.13"></td> </tr> <tr class="ltx_tr" id="S3.T2.8.8.8" style="background-color:#DAE8FC;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S3.T2.8.8.8.2" style="background-color:#DAE8FC;">CSST</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.3">Optical Survey Deep</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.8.8.8.1.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.8.8.8.1.m1.1a"><mo id="S3.T2.8.8.8.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.8.8.8.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.8.8.8.1.m1.1b"><csymbol cd="latexml" id="S3.T2.8.8.8.1.m1.1.1.cmml" xref="S3.T2.8.8.8.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.8.8.8.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.8.8.8.1.m1.1d">∼</annotation></semantics></math> 400 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.4" style="background-color:#DAE8FC;">26.70</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.5" style="background-color:#DAE8FC;">26.70</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.6" style="background-color:#DAE8FC;">27.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.7" style="background-color:#DAE8FC;">27.20</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.8" style="background-color:#DAE8FC;">27.00</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.9" style="background-color:#DAE8FC;">26.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.8.8.8.10" style="background-color:#DAE8FC;">25.70</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.8.8.8.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.8.8.8.12"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T2.8.8.8.13" style="background-color:#DAE8FC;"> Zhan et al. 2010; Cao et al. 2018; Zhan et al. 2021 </td> </tr> <tr class="ltx_tr" id="S3.T2.10.10.10" style="background-color:#FFF3F3;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.10.10.10.3">Euclid</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.10.10.10.4">Wide Survey</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.9.9.9.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.9.9.9.1.m1.1" style="background-color:#FFF3F3;"><semantics id="S3.T2.9.9.9.1.m1.1a"><mo id="S3.T2.9.9.9.1.m1.1.1" mathbackground="#FFF3F3" xref="S3.T2.9.9.9.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.9.9.9.1.m1.1b"><csymbol cd="latexml" id="S3.T2.9.9.9.1.m1.1.1.cmml" xref="S3.T2.9.9.9.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.9.9.9.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.9.9.9.1.m1.1d">∼</annotation></semantics></math> 14,500 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.10.10.10.5"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.10.10.10.6"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.10.10.10.7"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="3" id="S3.T2.10.10.10.2" style="background-color:#FFF3F3;"><math alttext="I_{\rm E}" class="ltx_Math" display="inline" id="S3.T2.10.10.10.2.1.m1.1" style="background-color:#FFF3F3;"><semantics id="S3.T2.10.10.10.2.1.m1.1a"><msub id="S3.T2.10.10.10.2.1.m1.1.1" xref="S3.T2.10.10.10.2.1.m1.1.1.cmml"><mi id="S3.T2.10.10.10.2.1.m1.1.1.2" mathbackground="#FFF3F3" mathcolor="#F56B00" xref="S3.T2.10.10.10.2.1.m1.1.1.2.cmml">I</mi><mi id="S3.T2.10.10.10.2.1.m1.1.1.3" mathbackground="#FFF3F3" mathcolor="#F56B00" mathvariant="normal" xref="S3.T2.10.10.10.2.1.m1.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T2.10.10.10.2.1.m1.1b"><apply id="S3.T2.10.10.10.2.1.m1.1.1.cmml" xref="S3.T2.10.10.10.2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T2.10.10.10.2.1.m1.1.1.1.cmml" xref="S3.T2.10.10.10.2.1.m1.1.1">subscript</csymbol><ci id="S3.T2.10.10.10.2.1.m1.1.1.2.cmml" xref="S3.T2.10.10.10.2.1.m1.1.1.2">𝐼</ci><ci id="S3.T2.10.10.10.2.1.m1.1.1.3.cmml" xref="S3.T2.10.10.10.2.1.m1.1.1.3">E</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.10.10.10.2.1.m1.1c">I_{\rm E}</annotation><annotation encoding="application/x-llamapun" id="S3.T2.10.10.10.2.1.m1.1d">italic_I start_POSTSUBSCRIPT roman_E end_POSTSUBSCRIPT</annotation></semantics></math> =26.20</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.10.10.10.8" style="background-color:#FFF3F3;">24.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.10.10.10.9" style="background-color:#FFF3F3;">24.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.10.10.10.10">24.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.10.10.10.11">Scarmamella et al. 2022</td> </tr> <tr class="ltx_tr" id="S3.T2.11.11.11"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.11.11.11.2"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.3">Wide Fast Deep (WFD)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.11.11.11.1.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.11.11.11.1.m1.1a"><mo id="S3.T2.11.11.11.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.11.11.11.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.11.11.11.1.m1.1b"><csymbol cd="latexml" id="S3.T2.11.11.11.1.m1.1.1.cmml" xref="S3.T2.11.11.11.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.11.11.11.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.11.11.11.1.m1.1d">∼</annotation></semantics></math> 14,500 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.11.11.11.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.5" style="background-color:#DAE8FC;">25.30</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.6" style="background-color:#DAE8FC;">26.84</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.7" style="background-color:#DAE8FC;">27.04</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.8" style="background-color:#DAE8FC;">26.35</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.9" style="background-color:#DAE8FC;">25.22</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.10" style="background-color:#DAE8FC;">24.47</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.11.11.11.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.11.11.11.12"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.11.11.11.13">DESC et al. 2018</td> </tr> <tr class="ltx_tr" id="S3.T2.13.13.13" style="background-color:#DAE8FC;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S3.T2.13.13.13.3" style="background-color:#DAE8FC;">LSST</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.13.13.13.4">North Ecliptic Spur (NES)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.12.12.12.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.12.12.12.1.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.12.12.12.1.m1.1a"><mo id="S3.T2.12.12.12.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.12.12.12.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.12.12.12.1.m1.1b"><csymbol cd="latexml" id="S3.T2.12.12.12.1.m1.1.1.cmml" xref="S3.T2.12.12.12.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.12.12.12.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.12.12.12.1.m1.1d">∼</annotation></semantics></math>4,160 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.13.13.13.5"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.13.13.13.6"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.13.13.13.7" style="background-color:#DAE8FC;"> 25.64</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.13.13.13.8" style="background-color:#DAE8FC;">25.84</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.13.13.13.9" style="background-color:#DAE8FC;">25.15</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.13.13.13.10" style="background-color:#DAE8FC;">24.02</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.13.13.13.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.13.13.13.12"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.13.13.13.13"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.13.13.13.2">Assuming <math alttext="1/3" class="ltx_Math" display="inline" id="S3.T2.13.13.13.2.1.m1.1"><semantics id="S3.T2.13.13.13.2.1.m1.1a"><mrow id="S3.T2.13.13.13.2.1.m1.1.1" xref="S3.T2.13.13.13.2.1.m1.1.1.cmml"><mn id="S3.T2.13.13.13.2.1.m1.1.1.2" mathbackground="#DAE8FC" xref="S3.T2.13.13.13.2.1.m1.1.1.2.cmml">1</mn><mo id="S3.T2.13.13.13.2.1.m1.1.1.1" mathbackground="#DAE8FC" xref="S3.T2.13.13.13.2.1.m1.1.1.1.cmml">/</mo><mn id="S3.T2.13.13.13.2.1.m1.1.1.3" mathbackground="#DAE8FC" xref="S3.T2.13.13.13.2.1.m1.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T2.13.13.13.2.1.m1.1b"><apply id="S3.T2.13.13.13.2.1.m1.1.1.cmml" xref="S3.T2.13.13.13.2.1.m1.1.1"><divide id="S3.T2.13.13.13.2.1.m1.1.1.1.cmml" xref="S3.T2.13.13.13.2.1.m1.1.1.1"></divide><cn id="S3.T2.13.13.13.2.1.m1.1.1.2.cmml" type="integer" xref="S3.T2.13.13.13.2.1.m1.1.1.2">1</cn><cn id="S3.T2.13.13.13.2.1.m1.1.1.3.cmml" type="integer" xref="S3.T2.13.13.13.2.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.13.13.13.2.1.m1.1c">1/3</annotation><annotation encoding="application/x-llamapun" id="S3.T2.13.13.13.2.1.m1.1d">1 / 3</annotation></semantics></math> visits of WFD</td> </tr> <tr class="ltx_tr" id="S3.T2.14.14.14"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.14.14.14.2"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.14.14.14.3">UNIONS/CFIS</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.14.14.14.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.14.14.14.1.m1.1" style="background-color:#FFF3F3;"><semantics id="S3.T2.14.14.14.1.m1.1a"><mo id="S3.T2.14.14.14.1.m1.1.1" mathbackground="#FFF3F3" xref="S3.T2.14.14.14.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.14.14.14.1.m1.1b"><csymbol cd="latexml" id="S3.T2.14.14.14.1.m1.1.1.cmml" xref="S3.T2.14.14.14.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.14.14.14.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.14.14.14.1.m1.1d">∼</annotation></semantics></math> 4,861 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.14.14.14.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.14.14.14.5" style="background-color:#FFF3F3;">24.30</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.14.14.14.6" style="background-color:#FFF3F3;">25.20</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.14.14.14.7" style="background-color:#FFF3F3;">24.90</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.14.14.14.8" style="background-color:#FFF3F3;">24.30</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.14.14.14.9" style="background-color:#FFF3F3;">24.10</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.14.14.14.10"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.14.14.14.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.14.14.14.12"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.14.14.14.13"></td> </tr> <tr class="ltx_tr" id="S3.T2.16.16.16" style="background-color:#FFF3F3;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S3.T2.16.16.16.3" style="background-color:#FFF3F3;">UNIONS</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.15.15.15.1">Extended UNIONS <math alttext="u" class="ltx_Math" display="inline" id="S3.T2.15.15.15.1.1.m1.1" style="background-color:#FFF3F3;"><semantics id="S3.T2.15.15.15.1.1.m1.1a"><mi id="S3.T2.15.15.15.1.1.m1.1.1" mathbackground="#FFF3F3" xref="S3.T2.15.15.15.1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S3.T2.15.15.15.1.1.m1.1b"><ci id="S3.T2.15.15.15.1.1.m1.1.1.cmml" xref="S3.T2.15.15.15.1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.15.15.15.1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S3.T2.15.15.15.1.1.m1.1d">italic_u</annotation></semantics></math>-band</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.16.16.16.2"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.16.16.16.2.m1.1" style="background-color:#FFF3F3;"><semantics id="S3.T2.16.16.16.2.m1.1a"><mo id="S3.T2.16.16.16.2.m1.1.1" mathbackground="#FFF3F3" xref="S3.T2.16.16.16.2.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.16.16.16.2.m1.1b"><csymbol cd="latexml" id="S3.T2.16.16.16.2.m1.1.1.cmml" xref="S3.T2.16.16.16.2.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.16.16.16.2.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.16.16.16.2.m1.1d">∼</annotation></semantics></math> 8,988 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.16.16.16.5" style="background-color:#FFF3F3;">24.30</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.6"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.7"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.8"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.9"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.10"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.16.16.16.12"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T2.16.16.16.13" style="background-color:#FFF3F3;">Ibata et al. 2017</td> </tr> <tr class="ltx_tr" id="S3.T2.17.17.17"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.17.17.17.2"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.17.17.17.3">DECaLS</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.17.17.17.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.17.17.17.1.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.17.17.17.1.m1.1a"><mo id="S3.T2.17.17.17.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.17.17.17.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.17.17.17.1.m1.1b"><csymbol cd="latexml" id="S3.T2.17.17.17.1.m1.1.1.cmml" xref="S3.T2.17.17.17.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.17.17.17.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.17.17.17.1.m1.1d">∼</annotation></semantics></math> 9,000 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.17.17.17.4"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.17.17.17.5"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.17.17.17.6" style="background-color:#DAE8FC;">24.65</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.17.17.17.7" style="background-color:#DAE8FC;">23.61</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.17.17.17.8" style="background-color:#DAE8FC;">22.84</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.17.17.17.9"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.17.17.17.10"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.17.17.17.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.17.17.17.12"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.17.17.17.13"></td> </tr> <tr class="ltx_tr" id="S3.T2.19.19.19" style="background-color:#DAE8FC;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S3.T2.19.19.19.3" style="background-color:#DAE8FC;">Legacy Survey</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.18.18.18.1">BASS <math alttext="+" class="ltx_Math" display="inline" id="S3.T2.18.18.18.1.1.m1.1"><semantics id="S3.T2.18.18.18.1.1.m1.1a"><mo id="S3.T2.18.18.18.1.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.18.18.18.1.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T2.18.18.18.1.1.m1.1b"><plus id="S3.T2.18.18.18.1.1.m1.1.1.cmml" xref="S3.T2.18.18.18.1.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.18.18.18.1.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T2.18.18.18.1.1.m1.1d">+</annotation></semantics></math> MzLS</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.19.19.19.2"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.19.19.19.2.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.19.19.19.2.m1.1a"><mo id="S3.T2.19.19.19.2.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.19.19.19.2.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.19.19.19.2.m1.1b"><csymbol cd="latexml" id="S3.T2.19.19.19.2.m1.1.1.cmml" xref="S3.T2.19.19.19.2.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.19.19.19.2.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.19.19.19.2.m1.1d">∼</annotation></semantics></math> 5,000 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.19.19.19.4"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.19.19.19.5"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.19.19.19.6" style="background-color:#DAE8FC;">24.30</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.19.19.19.7" style="background-color:#DAE8FC;">23.70</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.19.19.19.8" style="background-color:#DAE8FC;">23.04</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.19.19.19.9"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.19.19.19.10"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.19.19.19.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.19.19.19.12"></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T2.19.19.19.13" style="background-color:#DAE8FC;">Dey et al. 2019</td> </tr> <tr class="ltx_tr" id="S3.T2.20.20.20" style="background-color:#FEE8E7;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.20.20.20.2">Mozi</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.3">Wide Field Survey</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.20.20.20.1.m1.1" style="background-color:#FEE8E7;"><semantics id="S3.T2.20.20.20.1.m1.1a"><mo id="S3.T2.20.20.20.1.m1.1.1" mathbackground="#FEE8E7" xref="S3.T2.20.20.20.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.20.20.20.1.m1.1b"><csymbol cd="latexml" id="S3.T2.20.20.20.1.m1.1.1.cmml" xref="S3.T2.20.20.20.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.20.20.20.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.20.20.20.1.m1.1d">∼</annotation></semantics></math>8,000 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.20.20.20.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.5" style="background-color:#FEE8E7;">24.82</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.6" style="background-color:#FEE8E7;">25.85</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.7" style="background-color:#FEE8E7;">25.36</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.8" style="background-color:#FEE8E7;">24.83</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.9" style="background-color:#FEE8E7;"> 23.90</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.20.20.20.10"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.20.20.20.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.20.20.20.12"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.20.20.20.13">Wang et al. 2023</td> </tr> <tr class="ltx_tr" id="S3.T2.21.21.21"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S3.T2.21.21.21.2"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.21.21.21.3">SSP Wide</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.21.21.21.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.21.21.21.1.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.21.21.21.1.m1.1a"><mo id="S3.T2.21.21.21.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.21.21.21.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.21.21.21.1.m1.1b"><csymbol cd="latexml" id="S3.T2.21.21.21.1.m1.1.1.cmml" xref="S3.T2.21.21.21.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.21.21.21.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.21.21.21.1.m1.1d">∼</annotation></semantics></math> 1,400 </td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.21.21.21.4"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.21.21.21.5"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.21.21.21.6" style="background-color:#DAE8FC;">26.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.21.21.21.7" style="background-color:#DAE8FC;">26.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.21.21.21.8" style="background-color:#DAE8FC;">26.20</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.21.21.21.9" style="background-color:#DAE8FC;">25.20</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T2.21.21.21.10" style="background-color:#DAE8FC;">24.40</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.21.21.21.11"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.21.21.21.12"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S3.T2.21.21.21.13"></td> </tr> <tr class="ltx_tr" id="S3.T2.23.23.23" style="background-color:#DAE8FC;"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r" id="S3.T2.23.23.23.3" style="background-color:#DAE8FC;">HSC SSP</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.22.22.22.1">SSP Deep<math alttext="+" class="ltx_Math" display="inline" id="S3.T2.22.22.22.1.1.m1.1"><semantics id="S3.T2.22.22.22.1.1.m1.1a"><mo id="S3.T2.22.22.22.1.1.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.22.22.22.1.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S3.T2.22.22.22.1.1.m1.1b"><plus id="S3.T2.22.22.22.1.1.m1.1.1.cmml" xref="S3.T2.22.22.22.1.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.22.22.22.1.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S3.T2.22.22.22.1.1.m1.1d">+</annotation></semantics></math>UltraDeep</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.2"> <math alttext="\sim" class="ltx_Math" display="inline" id="S3.T2.23.23.23.2.m1.1" style="background-color:#DAE8FC;"><semantics id="S3.T2.23.23.23.2.m1.1a"><mo id="S3.T2.23.23.23.2.m1.1.1" mathbackground="#DAE8FC" xref="S3.T2.23.23.23.2.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S3.T2.23.23.23.2.m1.1b"><csymbol cd="latexml" id="S3.T2.23.23.23.2.m1.1.1.cmml" xref="S3.T2.23.23.23.2.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.23.23.23.2.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S3.T2.23.23.23.2.m1.1d">∼</annotation></semantics></math> 37 </td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.4"></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.5"></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.6" style="background-color:#DAE8FC;"> 27.40</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.7" style="background-color:#DAE8FC;">27.10</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.8" style="background-color:#DAE8FC;">26.90</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.9" style="background-color:#DAE8FC;">26.30</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.10" style="background-color:#DAE8FC;">25.30</td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.11"></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="S3.T2.23.23.23.12"></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S3.T2.23.23.23.13" style="background-color:#DAE8FC;">Aihara et al. 2022</td> </tr> </tbody> </table> </div> <figcaption class="ltx_caption ltx_centering">Table 2: Summary of available, ongoing, and planned imaging surveys that overlap with the potential footprint of MUST and can contribute to the target selection for LSS tracers of MUST. Please see the reference for details. The imaging depths represent the 5-<math alttext="\sigma" class="ltx_Math" display="inline" id="S3.T2.25.m1.1"><semantics id="S3.T2.25.m1.1b"><mi id="S3.T2.25.m1.1.1" xref="S3.T2.25.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.T2.25.m1.1c"><ci id="S3.T2.25.m1.1.1.cmml" xref="S3.T2.25.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T2.25.m1.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.T2.25.m1.1e">italic_σ</annotation></semantics></math> point source detection limits. However, we ignore the minor differences in aperture choices and filter differences between surveys. Please see Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.F2" title="Figure 2 ‣ 2.4 Survey Capability and Overall Scientific Potential ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2</a> to visualize their footprints. </figcaption> </figure> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="388" id="S3.F4.g1" src="extracted/5997959/figures/fig5.png" width="598"/> <figcaption class="ltx_caption ltx_centering">Figure 4: Here, we show the preliminary target selection strategy for the LSS spectroscopic surveys of MUST. The bottom panel shows the redshift distributions and comoving densities of the primary LSS tracers of MUST for low–intermediate redshift (<math alttext="z<1.5" class="ltx_Math" display="inline" id="S3.F4.13.m1.1"><semantics id="S3.F4.13.m1.1b"><mrow id="S3.F4.13.m1.1.1" xref="S3.F4.13.m1.1.1.cmml"><mi id="S3.F4.13.m1.1.1.2" xref="S3.F4.13.m1.1.1.2.cmml">z</mi><mo id="S3.F4.13.m1.1.1.1" xref="S3.F4.13.m1.1.1.1.cmml"><</mo><mn id="S3.F4.13.m1.1.1.3" xref="S3.F4.13.m1.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.13.m1.1c"><apply id="S3.F4.13.m1.1.1.cmml" xref="S3.F4.13.m1.1.1"><lt id="S3.F4.13.m1.1.1.1.cmml" xref="S3.F4.13.m1.1.1.1"></lt><ci id="S3.F4.13.m1.1.1.2.cmml" xref="S3.F4.13.m1.1.1.2">𝑧</ci><cn id="S3.F4.13.m1.1.1.3.cmml" type="float" xref="S3.F4.13.m1.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.13.m1.1d">z<1.5</annotation><annotation encoding="application/x-llamapun" id="S3.F4.13.m1.1e">italic_z < 1.5</annotation></semantics></math>) and high-redshift universe. At <math alttext="z<1.5" class="ltx_Math" display="inline" id="S3.F4.14.m2.1"><semantics id="S3.F4.14.m2.1b"><mrow id="S3.F4.14.m2.1.1" xref="S3.F4.14.m2.1.1.cmml"><mi id="S3.F4.14.m2.1.1.2" xref="S3.F4.14.m2.1.1.2.cmml">z</mi><mo id="S3.F4.14.m2.1.1.1" xref="S3.F4.14.m2.1.1.1.cmml"><</mo><mn id="S3.F4.14.m2.1.1.3" xref="S3.F4.14.m2.1.1.3.cmml">1.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.14.m2.1c"><apply id="S3.F4.14.m2.1.1.cmml" xref="S3.F4.14.m2.1.1"><lt id="S3.F4.14.m2.1.1.1.cmml" xref="S3.F4.14.m2.1.1.1"></lt><ci id="S3.F4.14.m2.1.1.2.cmml" xref="S3.F4.14.m2.1.1.2">𝑧</ci><cn id="S3.F4.14.m2.1.1.3.cmml" type="float" xref="S3.F4.14.m2.1.1.3">1.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.14.m2.1d">z<1.5</annotation><annotation encoding="application/x-llamapun" id="S3.F4.14.m2.1e">italic_z < 1.5</annotation></semantics></math>, MUST will densely sample the LSS with the help of low-redshift bright galaxies (BG), luminous red galaxies (LRG), and emission line galaxies (ELG) using their [O II] doublets along with other optical emission lines ([O III], Hβ, Hα, etc.) and significant absorption features. Using horizontal bars with distinct colors, we highlight the spectrograph wavelength coverage of different emission lines across the redshift space for MUST. 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The cosmological forecast presented here is based on these models. </figcaption> </figure> </section> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> 4 Target Selection</h2> <div class="ltx_para" id="S4.p1"> We must pay special attention to selecting targets from imaging surveys to achieve the scientific goals outlined in the previous section. There are two main requirements for cosmological target selection: 1) there should be easily identifiable spectral features, such as strong emission lines, absorption features, or continuum breaks, to enable redshift determination with a minimal exposure time; 2) the target density must be optimized to balance the cosmic variance and shot noise to maximize the precision of cosmological measurements. This section explores target selection criteria suitable for MUST and addresses these concerns. </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> 4.1 Challenges of Target Selection for Stage-V Spectroscopic Surveys</h3> <div class="ltx_para" id="S4.SS1.p1"> Unlike spectroscopic surveys using a prism (e.g., PRIMUS, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib198" title="">198</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib199" title="">199</a>]</cite>), grism (e.g., 3D-HST, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib200" title="">200</a>]</cite>; Euclid <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib18" title="">18</a>]</cite>), or IFS (e.g., HETDEX <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib201" title="">201</a>]</cite>), modern surveys using multi-slit instruments or robotic fiber positioners require careful and sophisticated target selection based on multi-band imaging data. Starting from the Main Galaxy Sample (MGS) in SDSS, primarily a flux-limited sample, the following surveys have gradually increased the number of different samples. As a Stage-IV survey, the main survey of DESI has already included low-redshift bright galaxies (BGS, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib202" title="">202</a>]</cite>), luminous red galaxies (LRG, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib203" title="">203</a>]</cite>), emission-line galaxies (ELG, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib204" title="">204</a>]</cite>), and QSOs (<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib205" title="">205</a>]</cite>). Selecting these samples now involves more complex selection criteria, including multiple color cuts, cross-matching with multi-wavelength datasets, and the application of machine learning-based methods (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib206" title="">206</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib207" title="">207</a>]</cite>). </div> <div class="ltx_para" id="S4.SS1.p2"> These changes reflect the increasingly demanding scientific and operational requirements of modern surveys. A survey must design flexible programs for bright & dark nights and different observing conditions while maximizing the fiber efficiency and the scientific output of the project. More importantly, for LSS redshift surveys, target selection design needs to ensure the volume densities, redshift distributions, and average halo biases of different LSS tracers meet the requirements for constraining the cosmological model. At the same time, the target selection should intentionally reduce the systematics inherited from the imaging surveys, such as the target density fluctuations induced by the variation of imaging depth, observing conditions, Galactic extinction, and data reduction issues. As the number of redshifts rapidly increases, the Stage-IV spectroscopic surveys officially entered the low shot-noise regime, where systematic issues from target selection are becoming critical for the cosmological potential of the survey. Looking forward to the Stage-V era for MUST, we will not only anticipate the continued development of these trends but also face new challenges. </div> <div class="ltx_para" id="S4.SS1.p3"> In particular, the LSS survey of MUST faces two outstanding challenges. 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The best candidates are Lyman Break Galaxies (LBG) selected using the Lyman-break “dropout” technique and the Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mi id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><ci id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">italic_α</annotation></semantics></math> Emitters (LAEs) traditionally identified in narrow-band imaging surveys. 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However, while the high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mi id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><ci id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">italic_z</annotation></semantics></math> galaxy community has been studying these populations extensively for two decades now (e.g., Hu & McMahon 1996 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib209" title="">209</a>]</cite>; Cowie & Hu 1998 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib210" title="">210</a>]</cite>), we still have not acquired the demanding broad- and narrow-band imaging capabilities to select them for a Stage-V survey and also have not fully understood their cosmological potential as existing deep field studies are still limited by cosmic variance among other systematics. As the LSS survey community has started to pay attention and organize pilot programs (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib211" title="">211</a>]</cite>), we expect to better understand these new LSS tracers before the first light of MUST. The proposed DESI-II project is exciting as it could start the first LSS survey using LBG and LAEs in <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.1"><semantics id="S4.SS1.p3.6.m6.1a"><mo id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.1b"><csymbol cd="latexml" id="S4.SS1.p3.6.m6.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.1d">∼</annotation></semantics></math>2029, gaining valuable insights before the operation of MUST. </div> <div class="ltx_para" id="S4.SS1.p4"> Secondly, given the site selection of MUST in the northern hemisphere, we do not enjoy the deep & uniform multi-band imaging coverage of the Legacy Survey of Space and Time (LSST) of the Vera C. Rubin Observatory in the south. Based on the landscape of imaging surveys in the north, no similar survey will be available before the 2030s. The panel (a) of Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S2.F2" title="Figure 2 ‣ 2.4 Survey Capability and Overall Scientific Potential ‣ 2 MUltiplexed Survey Telescope ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2</a> visualizes the footprints of major imaging surveys that overlap with the potential footprint of MUST and can contribute to the target selection of MUST. We also summarize their survey areas and imaging depths in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.T2" title="Table 2 ‣ 3.6.4 Gravitational waves and Fast Radio Bursts ‣ 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2</a>. Assuming MUST will observe targets with airmass better than 1.5, at the Lenghu site, MUST can cover the <math alttext="\delta\geq 10^{\circ}" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mrow id="S4.SS1.p4.1.m1.1.1" xref="S4.SS1.p4.1.m1.1.1.cmml"><mi id="S4.SS1.p4.1.m1.1.1.2" xref="S4.SS1.p4.1.m1.1.1.2.cmml">δ</mi><mo id="S4.SS1.p4.1.m1.1.1.1" xref="S4.SS1.p4.1.m1.1.1.1.cmml">≥</mo><msup id="S4.SS1.p4.1.m1.1.1.3" xref="S4.SS1.p4.1.m1.1.1.3.cmml"><mn id="S4.SS1.p4.1.m1.1.1.3.2" xref="S4.SS1.p4.1.m1.1.1.3.2.cmml">10</mn><mo id="S4.SS1.p4.1.m1.1.1.3.3" xref="S4.SS1.p4.1.m1.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><apply id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1"><geq id="S4.SS1.p4.1.m1.1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1.1"></geq><ci id="S4.SS1.p4.1.m1.1.1.2.cmml" xref="S4.SS1.p4.1.m1.1.1.2">𝛿</ci><apply id="S4.SS1.p4.1.m1.1.1.3.cmml" xref="S4.SS1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.1.1.3.1.cmml" xref="S4.SS1.p4.1.m1.1.1.3">superscript</csymbol><cn id="S4.SS1.p4.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.p4.1.m1.1.1.3.2">10</cn><compose id="S4.SS1.p4.1.m1.1.1.3.3.cmml" xref="S4.SS1.p4.1.m1.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">\delta\geq 10^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">italic_δ ≥ 10 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> footprint. Within this area, no imaging survey can cover the high Galactic latitude region suitable for an LSS survey. In the ideal scenario, the Chinese Space Station Telescope’s optical survey (CSST-OS) will provide the best multi-band support from the NUV to the <math alttext="Y" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mi id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><ci id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">italic_Y</annotation></semantics></math>-band. 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Meanwhile, the Euclid mission will provide valuable near-infrared (NIR) coverage and deep observations in the broad <math alttext="I_{\rm E}" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m5.1"><semantics id="S4.SS1.p4.5.m5.1a"><msub id="S4.SS1.p4.5.m5.1.1" xref="S4.SS1.p4.5.m5.1.1.cmml"><mi id="S4.SS1.p4.5.m5.1.1.2" xref="S4.SS1.p4.5.m5.1.1.2.cmml">I</mi><mi id="S4.SS1.p4.5.m5.1.1.3" mathvariant="normal" xref="S4.SS1.p4.5.m5.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.5.m5.1b"><apply id="S4.SS1.p4.5.m5.1.1.cmml" xref="S4.SS1.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m5.1.1.1.cmml" xref="S4.SS1.p4.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.p4.5.m5.1.1.2.cmml" xref="S4.SS1.p4.5.m5.1.1.2">𝐼</ci><ci id="S4.SS1.p4.5.m5.1.1.3.cmml" xref="S4.SS1.p4.5.m5.1.1.3">E</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m5.1c">I_{\rm E}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m5.1d">italic_I start_POSTSUBSCRIPT roman_E end_POSTSUBSCRIPT</annotation></semantics></math> optical band. While the exact strategy is yet to be investigated, the synergy between optical and NIR images from space should provide unique advantages and new angles for target selection. However, based on the current survey plan, both CSST and Euclid will avoid the region near the Ecliptic plane. For this region, the LSST survey should provide partial coverage, but most of it was covered by the Northern Ecliptic Spur (NES) mini-survey999<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://survey-strategy.lsst.io/baseline/minis.html" title="">https://survey-strategy.lsst.io/baseline/minis.html</a> of (<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib212" title="">212</a>]</cite>), which will only observe in <math alttext="g" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m6.1"><semantics id="S4.SS1.p4.6.m6.1a"><mi id="S4.SS1.p4.6.m6.1.1" xref="S4.SS1.p4.6.m6.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.6.m6.1b"><ci id="S4.SS1.p4.6.m6.1.1.cmml" xref="S4.SS1.p4.6.m6.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.6.m6.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.6.m6.1d">italic_g</annotation></semantics></math>, <math alttext="r" class="ltx_Math" display="inline" id="S4.SS1.p4.7.m7.1"><semantics id="S4.SS1.p4.7.m7.1a"><mi id="S4.SS1.p4.7.m7.1.1" xref="S4.SS1.p4.7.m7.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.7.m7.1b"><ci id="S4.SS1.p4.7.m7.1.1.cmml" xref="S4.SS1.p4.7.m7.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.7.m7.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.7.m7.1d">italic_r</annotation></semantics></math>, <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p4.8.m8.1"><semantics id="S4.SS1.p4.8.m8.1a"><mi id="S4.SS1.p4.8.m8.1.1" xref="S4.SS1.p4.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.8.m8.1b"><ci id="S4.SS1.p4.8.m8.1.1.cmml" xref="S4.SS1.p4.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.8.m8.1d">italic_i</annotation></semantics></math>, & <math alttext="z" class="ltx_Math" display="inline" id="S4.SS1.p4.9.m9.1"><semantics id="S4.SS1.p4.9.m9.1a"><mi id="S4.SS1.p4.9.m9.1.1" xref="S4.SS1.p4.9.m9.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.9.m9.1b"><ci id="S4.SS1.p4.9.m9.1.1.cmml" xref="S4.SS1.p4.9.m9.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.9.m9.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.9.m9.1d">italic_z</annotation></semantics></math>-filters with <math alttext="\sim 1/3" class="ltx_Math" display="inline" id="S4.SS1.p4.10.m10.1"><semantics id="S4.SS1.p4.10.m10.1a"><mrow id="S4.SS1.p4.10.m10.1.1" xref="S4.SS1.p4.10.m10.1.1.cmml"><mi id="S4.SS1.p4.10.m10.1.1.2" xref="S4.SS1.p4.10.m10.1.1.2.cmml"></mi><mo id="S4.SS1.p4.10.m10.1.1.1" xref="S4.SS1.p4.10.m10.1.1.1.cmml">∼</mo><mrow id="S4.SS1.p4.10.m10.1.1.3" xref="S4.SS1.p4.10.m10.1.1.3.cmml"><mn id="S4.SS1.p4.10.m10.1.1.3.2" xref="S4.SS1.p4.10.m10.1.1.3.2.cmml">1</mn><mo id="S4.SS1.p4.10.m10.1.1.3.1" xref="S4.SS1.p4.10.m10.1.1.3.1.cmml">/</mo><mn id="S4.SS1.p4.10.m10.1.1.3.3" xref="S4.SS1.p4.10.m10.1.1.3.3.cmml">3</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.10.m10.1b"><apply id="S4.SS1.p4.10.m10.1.1.cmml" xref="S4.SS1.p4.10.m10.1.1"><csymbol cd="latexml" id="S4.SS1.p4.10.m10.1.1.1.cmml" xref="S4.SS1.p4.10.m10.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS1.p4.10.m10.1.1.2.cmml" xref="S4.SS1.p4.10.m10.1.1.2">absent</csymbol><apply id="S4.SS1.p4.10.m10.1.1.3.cmml" xref="S4.SS1.p4.10.m10.1.1.3"><divide id="S4.SS1.p4.10.m10.1.1.3.1.cmml" xref="S4.SS1.p4.10.m10.1.1.3.1"></divide><cn id="S4.SS1.p4.10.m10.1.1.3.2.cmml" type="integer" xref="S4.SS1.p4.10.m10.1.1.3.2">1</cn><cn id="S4.SS1.p4.10.m10.1.1.3.3.cmml" type="integer" xref="S4.SS1.p4.10.m10.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.10.m10.1c">\sim 1/3</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.10.m10.1d">∼ 1 / 3</annotation></semantics></math> of the visits of the main Wide-Fast-Deep (WFD) survey. While the coadding depth is still competitive, the lack of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.p4.11.m11.1"><semantics id="S4.SS1.p4.11.m11.1a"><mi id="S4.SS1.p4.11.m11.1.1" xref="S4.SS1.p4.11.m11.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.11.m11.1b"><ci id="S4.SS1.p4.11.m11.1.1.cmml" xref="S4.SS1.p4.11.m11.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.11.m11.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.11.m11.1d">italic_u</annotation></semantics></math>-band is unfortunate for the LBG selection. The ongoing Ultraviolet Near Infrared Optical Northern Survey (UNIONS) campaign using the 3.6 m CFHT, 2<math alttext="\times" class="ltx_Math" display="inline" id="S4.SS1.p4.12.m12.1"><semantics id="S4.SS1.p4.12.m12.1a"><mo id="S4.SS1.p4.12.m12.1.1" xref="S4.SS1.p4.12.m12.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.12.m12.1b"><times id="S4.SS1.p4.12.m12.1.1.cmml" xref="S4.SS1.p4.12.m12.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.12.m12.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.12.m12.1d">×</annotation></semantics></math> 1.8 m Pan-STARRS, and 8.2 m Subaru telescopes is another major candidate for multi-band target selection. In an ideal situation, the southern extension of UNIONS in the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.p4.13.m13.1"><semantics id="S4.SS1.p4.13.m13.1a"><mi id="S4.SS1.p4.13.m13.1.1" xref="S4.SS1.p4.13.m13.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.13.m13.1b"><ci id="S4.SS1.p4.13.m13.1.1.cmml" xref="S4.SS1.p4.13.m13.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.13.m13.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.13.m13.1d">italic_u</annotation></semantics></math>-band to the northern limit (<math alttext="\delta=+12^{\circ}" class="ltx_Math" display="inline" id="S4.SS1.p4.14.m14.1"><semantics id="S4.SS1.p4.14.m14.1a"><mrow id="S4.SS1.p4.14.m14.1.1" xref="S4.SS1.p4.14.m14.1.1.cmml"><mi id="S4.SS1.p4.14.m14.1.1.2" xref="S4.SS1.p4.14.m14.1.1.2.cmml">δ</mi><mo id="S4.SS1.p4.14.m14.1.1.1" xref="S4.SS1.p4.14.m14.1.1.1.cmml">=</mo><mrow id="S4.SS1.p4.14.m14.1.1.3" xref="S4.SS1.p4.14.m14.1.1.3.cmml"><mo id="S4.SS1.p4.14.m14.1.1.3a" xref="S4.SS1.p4.14.m14.1.1.3.cmml">+</mo><msup id="S4.SS1.p4.14.m14.1.1.3.2" xref="S4.SS1.p4.14.m14.1.1.3.2.cmml"><mn id="S4.SS1.p4.14.m14.1.1.3.2.2" xref="S4.SS1.p4.14.m14.1.1.3.2.2.cmml">12</mn><mo id="S4.SS1.p4.14.m14.1.1.3.2.3" xref="S4.SS1.p4.14.m14.1.1.3.2.3.cmml">∘</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.14.m14.1b"><apply id="S4.SS1.p4.14.m14.1.1.cmml" xref="S4.SS1.p4.14.m14.1.1"><eq id="S4.SS1.p4.14.m14.1.1.1.cmml" xref="S4.SS1.p4.14.m14.1.1.1"></eq><ci id="S4.SS1.p4.14.m14.1.1.2.cmml" xref="S4.SS1.p4.14.m14.1.1.2">𝛿</ci><apply id="S4.SS1.p4.14.m14.1.1.3.cmml" xref="S4.SS1.p4.14.m14.1.1.3"><plus id="S4.SS1.p4.14.m14.1.1.3.1.cmml" xref="S4.SS1.p4.14.m14.1.1.3"></plus><apply id="S4.SS1.p4.14.m14.1.1.3.2.cmml" xref="S4.SS1.p4.14.m14.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p4.14.m14.1.1.3.2.1.cmml" xref="S4.SS1.p4.14.m14.1.1.3.2">superscript</csymbol><cn id="S4.SS1.p4.14.m14.1.1.3.2.2.cmml" type="integer" xref="S4.SS1.p4.14.m14.1.1.3.2.2">12</cn><compose id="S4.SS1.p4.14.m14.1.1.3.2.3.cmml" xref="S4.SS1.p4.14.m14.1.1.3.2.3"></compose></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.14.m14.1c">\delta=+12^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.14.m14.1d">italic_δ = + 12 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>) of the main survey of LSST will be especially valuable. </div> <div class="ltx_para" id="S4.SS1.p5"> In addition to these ongoing or planned imaging surveys, we expect the available multi-band data from the Legacy Survey (in <math alttext="g" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.1"><semantics id="S4.SS1.p5.1.m1.1a"><mi id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.1b"><ci id="S4.SS1.p5.1.m1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.1d">italic_g</annotation></semantics></math>, <math alttext="r" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m2.1"><semantics id="S4.SS1.p5.2.m2.1a"><mi id="S4.SS1.p5.2.m2.1.1" xref="S4.SS1.p5.2.m2.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.2.m2.1b"><ci id="S4.SS1.p5.2.m2.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.2.m2.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.2.m2.1d">italic_r</annotation></semantics></math>, and <math alttext="z" class="ltx_Math" display="inline" id="S4.SS1.p5.3.m3.1"><semantics id="S4.SS1.p5.3.m3.1a"><mi id="S4.SS1.p5.3.m3.1.1" xref="S4.SS1.p5.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.3.m3.1b"><ci id="S4.SS1.p5.3.m3.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.3.m3.1d">italic_z</annotation></semantics></math>-bands; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib213" title="">213</a>]</cite>) and the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib153" title="">153</a>]</cite>) will be essential for the development of target selection strategy for MUST: Legacy Survey satisfies the survey footprint requirements for a northern Stage-V project, while HSC-SSP, along with the UNIONS <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.p5.4.m4.1"><semantics id="S4.SS1.p5.4.m4.1a"><mi id="S4.SS1.p5.4.m4.1.1" xref="S4.SS1.p5.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.4.m4.1b"><ci id="S4.SS1.p5.4.m4.1.1.cmml" xref="S4.SS1.p5.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.4.m4.1d">italic_u</annotation></semantics></math>-band data should meet or surpass the imaging depth requirements for MUST. Also, the 2.5 m Mozi wide-field survey telescope at Peak C of the Lenghu site has started its operation. As a dedicated time-domain survey telescope that will continue its operation into the 2030s, its accumulated imaging depths will also be helpful to MUST. Moreover, the potential of imaging the northern sky using customized narrow- or medium-band filters to facilitate the selection of LAEs is an intriguing opportunity to be explored. We should also mention that CFHT, Subaru, and the 4.0 m Blanco telescope still have wide-field imaging capability that could support imaging campaigns to provide targeting selection support for any Stage-V spectroscopic surveys. In particular, the narrow- and medium-band surveys using the Dark Energy Camera (DECam; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib214" title="">214</a>]</cite>), such as the One-hundred-deg2 DECam Imaging in Narrowbands (ODIN; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib215" title="">215</a>]</cite>), the Merian survey (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib216" title="">216</a>]</cite>), and J-PAS (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib156" title="">156</a>]</cite>) could demonstrate an interesting new approach for target selection for Stage-V survey. </div> <div class="ltx_para" id="S4.SS1.p6"> Put all together, we can see that MUST is facing the highly challenging task of developing a consistent target selection strategy that meets the strict requirements for a Stage-V survey. As this work focuses on the theoretical prediction of the potential of LSS cosmology, we will not dive deep into the details of target selection as many of the imaging datasets mentioned above have yet to be available. Instead, in the following subsections, we will briefly introduce the concepts and assumptions of target selection for the LSS survey of MUST. </div> <div class="ltx_para" id="S4.SS1.p7"> For the cosmological prediction, we will assume two scenarios for the survey footprints: the ideal scenario for the grey (dark) time survey will have a 13,000 deg2 (11,000 deg2) footprint and a more conservative 11,000 deg2 (8,000 deg2) footprint. </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> 4.2 Low-Redshift Tracers</h3> <div class="ltx_para" id="S4.SS2.p1"> Similar to the ongoing DESI survey, MUST can measure the redshift of <math alttext="z<1.57" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mrow id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS2.p1.1.m1.1.1.1" xref="S4.SS2.p1.1.m1.1.1.1.cmml"><</mo><mn id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">1.57</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><lt id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1.1"></lt><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝑧</ci><cn id="S4.SS2.p1.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.p1.1.m1.1.1.3">1.57</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">z<1.57</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_z < 1.57</annotation></semantics></math> galaxies using the [O II] doublet emission lines at 3727 & 3729 <math alttext="\AA" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><mi id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml">Å</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><ci id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">italic-Å</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">\AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_Å</annotation></semantics></math>, along with the other significant emission and absorption features in galaxies’ spectra. While this “low-redshift” component is not the most critical one for the cosmological goals of MUST, it will be a valuable dataset that enables many other complementary cosmological probes, such as the multi-tracer clustering to explore the non-linear regime (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib217" title="">217</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib218" title="">218</a>]</cite>), the cross-correlations with weak lensing surveys like CSST & Euclid and other multi-wavelength datasets (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib219" title="">219</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib220" title="">220</a>]</cite>), and the spectroscopic survey of galaxy clusters (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib221" title="">221</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib222" title="">222</a>]</cite>). </div> <div class="ltx_para" id="S4.SS2.p2"> For now, we assume that MUST will follow the recipes of DESI to select BGS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib202" title="">202</a>]</cite>, LRG <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib203" title="">203</a>]</cite>, and ELG <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib204" title="">204</a>]</cite> samples as <math alttext="z<1.6" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mrow id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS2.p2.1.m1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.cmml"><</mo><mn id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><lt id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1"></lt><ci id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2">𝑧</ci><cn id="S4.SS2.p2.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.p2.1.m1.1.1.3">1.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">z<1.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_z < 1.6</annotation></semantics></math> LSS tracers. Conceptually, we plan to observe BGS and LRG samples as the bright night targets while observing ELG during the grey and dark nights. </div> <div class="ltx_para" id="S4.SS2.p3"> By 2030, DESI and its extension should have finished collecting for more than 40 million redshifts in this range. Therefore, we will only focus on targets fainter than those of DESI. Although the methods are similar, the BGS, LRG, and ELG samples for MUST were selected using photometric data from Legacy Survey DR10.1 instead of DR9. As Legacy Survey DR11 is scheduled to be released in 2025, it would be interesting to check if the samples can be improved with this new release. </div> <section class="ltx_subsubsection" id="S4.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> 4.2.1 Bright Galaxy Sample (BGS)</h4> <div class="ltx_para" id="S4.SS2.SSS1.p1"> For the high-density BGS sample, as DESI will have already observed BGS to <math alttext="r<20.175" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.1.m1.1"><semantics id="S4.SS2.SSS1.p1.1.m1.1a"><mrow id="S4.SS2.SSS1.p1.1.m1.1.1" xref="S4.SS2.SSS1.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p1.1.m1.1.1.2" xref="S4.SS2.SSS1.p1.1.m1.1.1.2.cmml">r</mi><mo id="S4.SS2.SSS1.p1.1.m1.1.1.1" xref="S4.SS2.SSS1.p1.1.m1.1.1.1.cmml"><</mo><mn id="S4.SS2.SSS1.p1.1.m1.1.1.3" xref="S4.SS2.SSS1.p1.1.m1.1.1.3.cmml">20.175</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.1.m1.1b"><apply id="S4.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p1.1.m1.1.1"><lt id="S4.SS2.SSS1.p1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.1.m1.1.1.1"></lt><ci id="S4.SS2.SSS1.p1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p1.1.m1.1.1.2">𝑟</ci><cn id="S4.SS2.SSS1.p1.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.SSS1.p1.1.m1.1.1.3">20.175</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.1.m1.1c">r<20.175</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.1.m1.1d">italic_r < 20.175</annotation></semantics></math> mag (including the FAINT sub-sample), MUST will observe fainter samples. Assuming that MUST can assign 2,000 fibers per deg2 to observe BGS during a bright night, based on the target selection results of DESI, MUST will focus on the magnitude range of <math alttext="20.175<r<21" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.3.m3.1"><semantics id="S4.SS2.SSS1.p1.3.m3.1a"><mrow id="S4.SS2.SSS1.p1.3.m3.1.1" xref="S4.SS2.SSS1.p1.3.m3.1.1.cmml"><mn id="S4.SS2.SSS1.p1.3.m3.1.1.2" xref="S4.SS2.SSS1.p1.3.m3.1.1.2.cmml">20.175</mn><mo id="S4.SS2.SSS1.p1.3.m3.1.1.3" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS1.p1.3.m3.1.1.4" xref="S4.SS2.SSS1.p1.3.m3.1.1.4.cmml">r</mi><mo id="S4.SS2.SSS1.p1.3.m3.1.1.5" xref="S4.SS2.SSS1.p1.3.m3.1.1.5.cmml"><</mo><mn id="S4.SS2.SSS1.p1.3.m3.1.1.6" xref="S4.SS2.SSS1.p1.3.m3.1.1.6.cmml">21</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.3.m3.1b"><apply id="S4.SS2.SSS1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1"><and id="S4.SS2.SSS1.p1.3.m3.1.1a.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1"></and><apply id="S4.SS2.SSS1.p1.3.m3.1.1b.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1"><lt id="S4.SS2.SSS1.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.3"></lt><cn id="S4.SS2.SSS1.p1.3.m3.1.1.2.cmml" type="float" xref="S4.SS2.SSS1.p1.3.m3.1.1.2">20.175</cn><ci id="S4.SS2.SSS1.p1.3.m3.1.1.4.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.4">𝑟</ci></apply><apply id="S4.SS2.SSS1.p1.3.m3.1.1c.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1"><lt id="S4.SS2.SSS1.p1.3.m3.1.1.5.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS1.p1.3.m3.1.1.4.cmml" id="S4.SS2.SSS1.p1.3.m3.1.1d.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1"></share><cn id="S4.SS2.SSS1.p1.3.m3.1.1.6.cmml" type="integer" xref="S4.SS2.SSS1.p1.3.m3.1.1.6">21</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.3.m3.1c">20.175<r<21</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.3.m3.1d">20.175 < italic_r < 21</annotation></semantics></math> mag in the Legacy Survey data after making similar quality cleaning cuts in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib202" title="">202</a>]</cite>. Firstly, we exclude stellar objects following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib202" title="">202</a>]</cite> using the Gaia catalog to perform an additional magnitude cut: </div> <div class="ltx_para" id="S4.SS2.SSS1.p2"> <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="G_{\textrm{Gaia}}-r_{\textrm{raw}}>0.6." class="ltx_Math" display="block" id="S4.E4.m1.1"><semantics id="S4.E4.m1.1a"><mrow id="S4.E4.m1.1.1.1" xref="S4.E4.m1.1.1.1.1.cmml"><mrow id="S4.E4.m1.1.1.1.1" xref="S4.E4.m1.1.1.1.1.cmml"><mrow id="S4.E4.m1.1.1.1.1.2" xref="S4.E4.m1.1.1.1.1.2.cmml"><msub id="S4.E4.m1.1.1.1.1.2.2" xref="S4.E4.m1.1.1.1.1.2.2.cmml"><mi id="S4.E4.m1.1.1.1.1.2.2.2" xref="S4.E4.m1.1.1.1.1.2.2.2.cmml">G</mi><mtext id="S4.E4.m1.1.1.1.1.2.2.3" xref="S4.E4.m1.1.1.1.1.2.2.3a.cmml">Gaia</mtext></msub><mo id="S4.E4.m1.1.1.1.1.2.1" xref="S4.E4.m1.1.1.1.1.2.1.cmml">−</mo><msub id="S4.E4.m1.1.1.1.1.2.3" xref="S4.E4.m1.1.1.1.1.2.3.cmml"><mi id="S4.E4.m1.1.1.1.1.2.3.2" xref="S4.E4.m1.1.1.1.1.2.3.2.cmml">r</mi><mtext id="S4.E4.m1.1.1.1.1.2.3.3" xref="S4.E4.m1.1.1.1.1.2.3.3a.cmml">raw</mtext></msub></mrow><mo id="S4.E4.m1.1.1.1.1.1" xref="S4.E4.m1.1.1.1.1.1.cmml">></mo><mn id="S4.E4.m1.1.1.1.1.3" xref="S4.E4.m1.1.1.1.1.3.cmml">0.6</mn></mrow><mo id="S4.E4.m1.1.1.1.2" lspace="0em" xref="S4.E4.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E4.m1.1b"><apply id="S4.E4.m1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1"><gt id="S4.E4.m1.1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1.1.1"></gt><apply id="S4.E4.m1.1.1.1.1.2.cmml" xref="S4.E4.m1.1.1.1.1.2"><minus id="S4.E4.m1.1.1.1.1.2.1.cmml" xref="S4.E4.m1.1.1.1.1.2.1"></minus><apply id="S4.E4.m1.1.1.1.1.2.2.cmml" xref="S4.E4.m1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.1.1.2.2.1.cmml" xref="S4.E4.m1.1.1.1.1.2.2">subscript</csymbol><ci id="S4.E4.m1.1.1.1.1.2.2.2.cmml" xref="S4.E4.m1.1.1.1.1.2.2.2">𝐺</ci><ci id="S4.E4.m1.1.1.1.1.2.2.3a.cmml" xref="S4.E4.m1.1.1.1.1.2.2.3"><mtext id="S4.E4.m1.1.1.1.1.2.2.3.cmml" mathsize="70%" xref="S4.E4.m1.1.1.1.1.2.2.3">Gaia</mtext></ci></apply><apply id="S4.E4.m1.1.1.1.1.2.3.cmml" xref="S4.E4.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.1.1.2.3.1.cmml" xref="S4.E4.m1.1.1.1.1.2.3">subscript</csymbol><ci id="S4.E4.m1.1.1.1.1.2.3.2.cmml" xref="S4.E4.m1.1.1.1.1.2.3.2">𝑟</ci><ci id="S4.E4.m1.1.1.1.1.2.3.3a.cmml" xref="S4.E4.m1.1.1.1.1.2.3.3"><mtext id="S4.E4.m1.1.1.1.1.2.3.3.cmml" mathsize="70%" xref="S4.E4.m1.1.1.1.1.2.3.3">raw</mtext></ci></apply></apply><cn id="S4.E4.m1.1.1.1.1.3.cmml" type="float" xref="S4.E4.m1.1.1.1.1.3">0.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.1c">G_{\textrm{Gaia}}-r_{\textrm{raw}}>0.6.</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.1d">italic_G start_POSTSUBSCRIPT Gaia end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT raw end_POSTSUBSCRIPT > 0.6 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(4)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.p3"> Then, we use the spatial mask bits provided by Legacy Survey to avoid targets polluted by bright stars in the vicinity. Those are defined as BITMASK 1, 12 and 13. Finally, to clean the sample from spurious objects and to ensure that the selection is based on high-quality observations, we perform the following final cut: </div> <div class="ltx_para" id="S4.SS2.SSS1.p4"> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E5"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E5X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-1<(g-r)<4" class="ltx_Math" display="inline" id="S4.E5X.2.1.1.m1.1"><semantics id="S4.E5X.2.1.1.m1.1a"><mrow id="S4.E5X.2.1.1.m1.1.1" xref="S4.E5X.2.1.1.m1.1.1.cmml"><mrow id="S4.E5X.2.1.1.m1.1.1.3" xref="S4.E5X.2.1.1.m1.1.1.3.cmml"><mo id="S4.E5X.2.1.1.m1.1.1.3a" xref="S4.E5X.2.1.1.m1.1.1.3.cmml">−</mo><mn id="S4.E5X.2.1.1.m1.1.1.3.2" xref="S4.E5X.2.1.1.m1.1.1.3.2.cmml">1</mn></mrow><mo id="S4.E5X.2.1.1.m1.1.1.4" xref="S4.E5X.2.1.1.m1.1.1.4.cmml"><</mo><mrow id="S4.E5X.2.1.1.m1.1.1.1.1" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.cmml"><mo id="S4.E5X.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E5X.2.1.1.m1.1.1.1.1.1" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S4.E5X.2.1.1.m1.1.1.1.1.1.2" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.2.cmml">g</mi><mo id="S4.E5X.2.1.1.m1.1.1.1.1.1.1" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E5X.2.1.1.m1.1.1.1.1.1.3" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S4.E5X.2.1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E5X.2.1.1.m1.1.1.5" xref="S4.E5X.2.1.1.m1.1.1.5.cmml"><</mo><mn id="S4.E5X.2.1.1.m1.1.1.6" xref="S4.E5X.2.1.1.m1.1.1.6.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.E5X.2.1.1.m1.1b"><apply id="S4.E5X.2.1.1.m1.1.1.cmml" xref="S4.E5X.2.1.1.m1.1.1"><and id="S4.E5X.2.1.1.m1.1.1a.cmml" xref="S4.E5X.2.1.1.m1.1.1"></and><apply id="S4.E5X.2.1.1.m1.1.1b.cmml" xref="S4.E5X.2.1.1.m1.1.1"><lt id="S4.E5X.2.1.1.m1.1.1.4.cmml" xref="S4.E5X.2.1.1.m1.1.1.4"></lt><apply id="S4.E5X.2.1.1.m1.1.1.3.cmml" xref="S4.E5X.2.1.1.m1.1.1.3"><minus id="S4.E5X.2.1.1.m1.1.1.3.1.cmml" xref="S4.E5X.2.1.1.m1.1.1.3"></minus><cn id="S4.E5X.2.1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.E5X.2.1.1.m1.1.1.3.2">1</cn></apply><apply id="S4.E5X.2.1.1.m1.1.1.1.1.1.cmml" xref="S4.E5X.2.1.1.m1.1.1.1.1"><minus id="S4.E5X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.1"></minus><ci id="S4.E5X.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.2">𝑔</ci><ci id="S4.E5X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S4.E5X.2.1.1.m1.1.1.1.1.1.3">𝑟</ci></apply></apply><apply id="S4.E5X.2.1.1.m1.1.1c.cmml" xref="S4.E5X.2.1.1.m1.1.1"><lt id="S4.E5X.2.1.1.m1.1.1.5.cmml" xref="S4.E5X.2.1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.E5X.2.1.1.m1.1.1.1.cmml" id="S4.E5X.2.1.1.m1.1.1d.cmml" xref="S4.E5X.2.1.1.m1.1.1"></share><cn id="S4.E5X.2.1.1.m1.1.1.6.cmml" type="integer" xref="S4.E5X.2.1.1.m1.1.1.6">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5X.2.1.1.m1.1c">\displaystyle-1<(g-r)<4</annotation><annotation encoding="application/x-llamapun" id="S4.E5X.2.1.1.m1.1d">- 1 < ( italic_g - italic_r ) < 4</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2">(5)</td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E5Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-1<(r-z)<4," class="ltx_Math" display="inline" id="S4.E5Xa.2.1.1.m1.1"><semantics id="S4.E5Xa.2.1.1.m1.1a"><mrow id="S4.E5Xa.2.1.1.m1.1.1.1" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E5Xa.2.1.1.m1.1.1.1.1" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E5Xa.2.1.1.m1.1.1.1.1.3" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.3.cmml"><mo id="S4.E5Xa.2.1.1.m1.1.1.1.1.3a" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.3.cmml">−</mo><mn id="S4.E5Xa.2.1.1.m1.1.1.1.1.3.2" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.3.2.cmml">1</mn></mrow><mo id="S4.E5Xa.2.1.1.m1.1.1.1.1.4" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.4.cmml"><</mo><mrow id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.2" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.2.cmml">r</mi><mo id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.1" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.3" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E5Xa.2.1.1.m1.1.1.1.1.5" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.5.cmml"><</mo><mn id="S4.E5Xa.2.1.1.m1.1.1.1.1.6" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.6.cmml">4</mn></mrow><mo id="S4.E5Xa.2.1.1.m1.1.1.1.2" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E5Xa.2.1.1.m1.1b"><apply id="S4.E5Xa.2.1.1.m1.1.1.1.1.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1"><and id="S4.E5Xa.2.1.1.m1.1.1.1.1a.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1"></and><apply id="S4.E5Xa.2.1.1.m1.1.1.1.1b.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1"><lt id="S4.E5Xa.2.1.1.m1.1.1.1.1.4.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.4"></lt><apply id="S4.E5Xa.2.1.1.m1.1.1.1.1.3.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.3"><minus id="S4.E5Xa.2.1.1.m1.1.1.1.1.3.1.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.3"></minus><cn id="S4.E5Xa.2.1.1.m1.1.1.1.1.3.2.cmml" type="integer" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.3.2">1</cn></apply><apply id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1"><minus id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.2">𝑟</ci><ci id="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.1.1.1.3">𝑧</ci></apply></apply><apply id="S4.E5Xa.2.1.1.m1.1.1.1.1c.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1"><lt id="S4.E5Xa.2.1.1.m1.1.1.1.1.5.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.E5Xa.2.1.1.m1.1.1.1.1.1.cmml" id="S4.E5Xa.2.1.1.m1.1.1.1.1d.cmml" xref="S4.E5Xa.2.1.1.m1.1.1.1"></share><cn id="S4.E5Xa.2.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S4.E5Xa.2.1.1.m1.1.1.1.1.6">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5Xa.2.1.1.m1.1c">\displaystyle-1<(r-z)<4,</annotation><annotation encoding="application/x-llamapun" id="S4.E5Xa.2.1.1.m1.1d">- 1 < ( italic_r - italic_z ) < 4 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.p5"> and we impose that the object has been observed at least once in the <math alttext="g,r" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.1.m1.2"><semantics id="S4.SS2.SSS1.p5.1.m1.2a"><mrow id="S4.SS2.SSS1.p5.1.m1.2.3.2" xref="S4.SS2.SSS1.p5.1.m1.2.3.1.cmml"><mi id="S4.SS2.SSS1.p5.1.m1.1.1" xref="S4.SS2.SSS1.p5.1.m1.1.1.cmml">g</mi><mo id="S4.SS2.SSS1.p5.1.m1.2.3.2.1" xref="S4.SS2.SSS1.p5.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS1.p5.1.m1.2.2" xref="S4.SS2.SSS1.p5.1.m1.2.2.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.1.m1.2b"><list id="S4.SS2.SSS1.p5.1.m1.2.3.1.cmml" xref="S4.SS2.SSS1.p5.1.m1.2.3.2"><ci id="S4.SS2.SSS1.p5.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p5.1.m1.1.1">𝑔</ci><ci id="S4.SS2.SSS1.p5.1.m1.2.2.cmml" xref="S4.SS2.SSS1.p5.1.m1.2.2">𝑟</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.1.m1.2c">g,r</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.1.m1.2d">italic_g , italic_r</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.2.m2.1"><semantics id="S4.SS2.SSS1.p5.2.m2.1a"><mi id="S4.SS2.SSS1.p5.2.m2.1.1" xref="S4.SS2.SSS1.p5.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.2.m2.1b"><ci id="S4.SS2.SSS1.p5.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.2.m2.1d">italic_z</annotation></semantics></math> bands. We have not made any cuts based on fiber magnitude as MUST will adopt a larger fiber size than DESI (1.3 v.s. 1.5 arcsec). We only require the target density and redshift distribution estimate for the cosmological forecast. For BGS, the detailed fiber magnitude or color cut should make little difference at this stage. We will apply them later for more accurate specifications. </div> <div class="ltx_para" id="S4.SS2.SSS1.p6"> In Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.F4" title="Figure 4 ‣ 3.6.4 Gravitational waves and Fast Radio Bursts ‣ 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4</a>, we show that the redshift distribution of the BGS sample ranges between <math alttext="0.1<z<0.8" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p6.1.m1.1"><semantics id="S4.SS2.SSS1.p6.1.m1.1a"><mrow id="S4.SS2.SSS1.p6.1.m1.1.1" xref="S4.SS2.SSS1.p6.1.m1.1.1.cmml"><mn id="S4.SS2.SSS1.p6.1.m1.1.1.2" xref="S4.SS2.SSS1.p6.1.m1.1.1.2.cmml">0.1</mn><mo id="S4.SS2.SSS1.p6.1.m1.1.1.3" xref="S4.SS2.SSS1.p6.1.m1.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS1.p6.1.m1.1.1.4" xref="S4.SS2.SSS1.p6.1.m1.1.1.4.cmml">z</mi><mo id="S4.SS2.SSS1.p6.1.m1.1.1.5" xref="S4.SS2.SSS1.p6.1.m1.1.1.5.cmml"><</mo><mn id="S4.SS2.SSS1.p6.1.m1.1.1.6" xref="S4.SS2.SSS1.p6.1.m1.1.1.6.cmml">0.8</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p6.1.m1.1b"><apply id="S4.SS2.SSS1.p6.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1"><and id="S4.SS2.SSS1.p6.1.m1.1.1a.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1"></and><apply id="S4.SS2.SSS1.p6.1.m1.1.1b.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1"><lt id="S4.SS2.SSS1.p6.1.m1.1.1.3.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1.3"></lt><cn id="S4.SS2.SSS1.p6.1.m1.1.1.2.cmml" type="float" xref="S4.SS2.SSS1.p6.1.m1.1.1.2">0.1</cn><ci id="S4.SS2.SSS1.p6.1.m1.1.1.4.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.SS2.SSS1.p6.1.m1.1.1c.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1"><lt id="S4.SS2.SSS1.p6.1.m1.1.1.5.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS1.p6.1.m1.1.1.4.cmml" id="S4.SS2.SSS1.p6.1.m1.1.1d.cmml" xref="S4.SS2.SSS1.p6.1.m1.1.1"></share><cn id="S4.SS2.SSS1.p6.1.m1.1.1.6.cmml" type="float" xref="S4.SS2.SSS1.p6.1.m1.1.1.6">0.8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p6.1.m1.1c">0.1<z<0.8</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p6.1.m1.1d">0.1 < italic_z < 0.8</annotation></semantics></math> with a broad peak between <math alttext="z=0.2" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p6.2.m2.1"><semantics id="S4.SS2.SSS1.p6.2.m2.1a"><mrow id="S4.SS2.SSS1.p6.2.m2.1.1" xref="S4.SS2.SSS1.p6.2.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p6.2.m2.1.1.2" xref="S4.SS2.SSS1.p6.2.m2.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS1.p6.2.m2.1.1.1" xref="S4.SS2.SSS1.p6.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p6.2.m2.1.1.3" xref="S4.SS2.SSS1.p6.2.m2.1.1.3.cmml">0.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p6.2.m2.1b"><apply id="S4.SS2.SSS1.p6.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p6.2.m2.1.1"><eq id="S4.SS2.SSS1.p6.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p6.2.m2.1.1.1"></eq><ci id="S4.SS2.SSS1.p6.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p6.2.m2.1.1.2">𝑧</ci><cn id="S4.SS2.SSS1.p6.2.m2.1.1.3.cmml" type="float" xref="S4.SS2.SSS1.p6.2.m2.1.1.3">0.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p6.2.m2.1c">z=0.2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p6.2.m2.1d">italic_z = 0.2</annotation></semantics></math> to 0.4. For the halo bias assumption for the BGS, as this work intends to compare with the forecast of DESI, we adopt the same bias evolution model used in the scientific requirement document of DESI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib223" title="">223</a>]</cite>: </div> <div class="ltx_para" id="S4.SS2.SSS1.p7"> <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="b(z)=1.34/D(z)" class="ltx_Math" display="block" id="S4.E6.m1.2"><semantics id="S4.E6.m1.2a"><mrow id="S4.E6.m1.2.3" xref="S4.E6.m1.2.3.cmml"><mrow id="S4.E6.m1.2.3.2" xref="S4.E6.m1.2.3.2.cmml"><mi id="S4.E6.m1.2.3.2.2" xref="S4.E6.m1.2.3.2.2.cmml">b</mi><mo id="S4.E6.m1.2.3.2.1" xref="S4.E6.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.E6.m1.2.3.2.3.2" xref="S4.E6.m1.2.3.2.cmml"><mo id="S4.E6.m1.2.3.2.3.2.1" stretchy="false" xref="S4.E6.m1.2.3.2.cmml">(</mo><mi id="S4.E6.m1.1.1" xref="S4.E6.m1.1.1.cmml">z</mi><mo id="S4.E6.m1.2.3.2.3.2.2" stretchy="false" xref="S4.E6.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.E6.m1.2.3.1" xref="S4.E6.m1.2.3.1.cmml">=</mo><mrow id="S4.E6.m1.2.3.3" xref="S4.E6.m1.2.3.3.cmml"><mrow id="S4.E6.m1.2.3.3.2" xref="S4.E6.m1.2.3.3.2.cmml"><mn id="S4.E6.m1.2.3.3.2.2" xref="S4.E6.m1.2.3.3.2.2.cmml">1.34</mn><mo id="S4.E6.m1.2.3.3.2.1" xref="S4.E6.m1.2.3.3.2.1.cmml">/</mo><mi id="S4.E6.m1.2.3.3.2.3" xref="S4.E6.m1.2.3.3.2.3.cmml">D</mi></mrow><mo id="S4.E6.m1.2.3.3.1" xref="S4.E6.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.E6.m1.2.3.3.3.2" xref="S4.E6.m1.2.3.3.cmml"><mo id="S4.E6.m1.2.3.3.3.2.1" stretchy="false" xref="S4.E6.m1.2.3.3.cmml">(</mo><mi id="S4.E6.m1.2.2" xref="S4.E6.m1.2.2.cmml">z</mi><mo id="S4.E6.m1.2.3.3.3.2.2" stretchy="false" xref="S4.E6.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.2b"><apply id="S4.E6.m1.2.3.cmml" xref="S4.E6.m1.2.3"><eq id="S4.E6.m1.2.3.1.cmml" xref="S4.E6.m1.2.3.1"></eq><apply id="S4.E6.m1.2.3.2.cmml" xref="S4.E6.m1.2.3.2"><times id="S4.E6.m1.2.3.2.1.cmml" xref="S4.E6.m1.2.3.2.1"></times><ci id="S4.E6.m1.2.3.2.2.cmml" xref="S4.E6.m1.2.3.2.2">𝑏</ci><ci id="S4.E6.m1.1.1.cmml" xref="S4.E6.m1.1.1">𝑧</ci></apply><apply id="S4.E6.m1.2.3.3.cmml" xref="S4.E6.m1.2.3.3"><times id="S4.E6.m1.2.3.3.1.cmml" xref="S4.E6.m1.2.3.3.1"></times><apply id="S4.E6.m1.2.3.3.2.cmml" xref="S4.E6.m1.2.3.3.2"><divide id="S4.E6.m1.2.3.3.2.1.cmml" xref="S4.E6.m1.2.3.3.2.1"></divide><cn id="S4.E6.m1.2.3.3.2.2.cmml" type="float" xref="S4.E6.m1.2.3.3.2.2">1.34</cn><ci id="S4.E6.m1.2.3.3.2.3.cmml" xref="S4.E6.m1.2.3.3.2.3">𝐷</ci></apply><ci id="S4.E6.m1.2.2.cmml" xref="S4.E6.m1.2.2">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.2c">b(z)=1.34/D(z)</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.2d">italic_b ( italic_z ) = 1.34 / italic_D ( italic_z )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(6)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.p8"> where <math alttext="D(z)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.1.m1.1"><semantics id="S4.SS2.SSS1.p8.1.m1.1a"><mrow id="S4.SS2.SSS1.p8.1.m1.1.2" xref="S4.SS2.SSS1.p8.1.m1.1.2.cmml"><mi id="S4.SS2.SSS1.p8.1.m1.1.2.2" xref="S4.SS2.SSS1.p8.1.m1.1.2.2.cmml">D</mi><mo id="S4.SS2.SSS1.p8.1.m1.1.2.1" xref="S4.SS2.SSS1.p8.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p8.1.m1.1.2.3.2" xref="S4.SS2.SSS1.p8.1.m1.1.2.cmml"><mo id="S4.SS2.SSS1.p8.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS1.p8.1.m1.1.2.cmml">(</mo><mi id="S4.SS2.SSS1.p8.1.m1.1.1" xref="S4.SS2.SSS1.p8.1.m1.1.1.cmml">z</mi><mo id="S4.SS2.SSS1.p8.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS1.p8.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p8.1.m1.1b"><apply id="S4.SS2.SSS1.p8.1.m1.1.2.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.2"><times id="S4.SS2.SSS1.p8.1.m1.1.2.1.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.2.1"></times><ci id="S4.SS2.SSS1.p8.1.m1.1.2.2.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.2.2">𝐷</ci><ci id="S4.SS2.SSS1.p8.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.1.m1.1c">D(z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.1.m1.1d">italic_D ( italic_z )</annotation></semantics></math> is the growth factor that depends on the adopted cosmology <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib224" title="">224</a>]</cite>. </div> <div class="ltx_para" id="S4.SS2.SSS1.p9"> In addition, by reaching <math alttext="r<21.5" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p9.1.m1.1"><semantics id="S4.SS2.SSS1.p9.1.m1.1a"><mrow id="S4.SS2.SSS1.p9.1.m1.1.1" xref="S4.SS2.SSS1.p9.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p9.1.m1.1.1.2" xref="S4.SS2.SSS1.p9.1.m1.1.1.2.cmml">r</mi><mo id="S4.SS2.SSS1.p9.1.m1.1.1.1" xref="S4.SS2.SSS1.p9.1.m1.1.1.1.cmml"><</mo><mn id="S4.SS2.SSS1.p9.1.m1.1.1.3" xref="S4.SS2.SSS1.p9.1.m1.1.1.3.cmml">21.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p9.1.m1.1b"><apply id="S4.SS2.SSS1.p9.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p9.1.m1.1.1"><lt id="S4.SS2.SSS1.p9.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p9.1.m1.1.1.1"></lt><ci id="S4.SS2.SSS1.p9.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p9.1.m1.1.1.2">𝑟</ci><cn id="S4.SS2.SSS1.p9.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.SSS1.p9.1.m1.1.1.3">21.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p9.1.m1.1c">r<21.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p9.1.m1.1d">italic_r < 21.5</annotation></semantics></math> mag, MUST can increase the target density for BGS by <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p9.2.m2.1"><semantics id="S4.SS2.SSS1.p9.2.m2.1a"><mo id="S4.SS2.SSS1.p9.2.m2.1.1" xref="S4.SS2.SSS1.p9.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p9.2.m2.1b"><csymbol cd="latexml" id="S4.SS2.SSS1.p9.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p9.2.m2.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p9.2.m2.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p9.2.m2.1d">∼</annotation></semantics></math>2,500 deg-2. Even considering the fiber assignment efficiency and the redshift success rate, which should be higher than 95% given the fiducial exposure time, MUST can easily provide a dense sample of low-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p9.4.m4.1"><semantics id="S4.SS2.SSS1.p9.4.m4.1a"><mi id="S4.SS2.SSS1.p9.4.m4.1.1" xref="S4.SS2.SSS1.p9.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p9.4.m4.1b"><ci id="S4.SS2.SSS1.p9.4.m4.1.1.cmml" xref="S4.SS2.SSS1.p9.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p9.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p9.4.m4.1d">italic_z</annotation></semantics></math> galaxies for various cosmological probes. </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> 4.2.2 Luminous Red Galaxies (LRG)</h4> <div class="ltx_para" id="S4.SS2.SSS2.p1"> As their name suggests, LRG represents the massive & (mostly) quiescent galaxies at low-redshift (<math alttext="z<0.8" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.1.m1.1"><semantics id="S4.SS2.SSS2.p1.1.m1.1a"><mrow id="S4.SS2.SSS2.p1.1.m1.1.1" xref="S4.SS2.SSS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.p1.1.m1.1.1.2" xref="S4.SS2.SSS2.p1.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS2.p1.1.m1.1.1.1" xref="S4.SS2.SSS2.p1.1.m1.1.1.1.cmml"><</mo><mn id="S4.SS2.SSS2.p1.1.m1.1.1.3" xref="S4.SS2.SSS2.p1.1.m1.1.1.3.cmml">0.8</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.1.m1.1b"><apply id="S4.SS2.SSS2.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p1.1.m1.1.1"><lt id="S4.SS2.SSS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS2.p1.1.m1.1.1.1"></lt><ci id="S4.SS2.SSS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS2.p1.1.m1.1.1.2">𝑧</ci><cn id="S4.SS2.SSS2.p1.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.SSS2.p1.1.m1.1.1.3">0.8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.1.m1.1c">z<0.8</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.1.m1.1d">italic_z < 0.8</annotation></semantics></math>). LRG typically lives in relatively massive dark matter halos with high LSS bias, making them a great candidate to trace the matter distribution and provide high-quality BAO measurement (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib225" title="">225</a>]</cite>). As done for the BGS sample, we follow the LRG target selection recipe of DESI but avoid observing the same targets as the latter by reaching a fainter magnitude limit. DESI adopted the <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.2.m2.1"><semantics id="S4.SS2.SSS2.p1.2.m2.1a"><mi id="S4.SS2.SSS2.p1.2.m2.1.1" xref="S4.SS2.SSS2.p1.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.2.m2.1b"><ci id="S4.SS2.SSS2.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.2.m2.1d">italic_z</annotation></semantics></math>-band fiber magnitude <math alttext="z_{\mathrm{fiber}}<21.6" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.3.m3.1"><semantics id="S4.SS2.SSS2.p1.3.m3.1a"><mrow id="S4.SS2.SSS2.p1.3.m3.1.1" xref="S4.SS2.SSS2.p1.3.m3.1.1.cmml"><msub id="S4.SS2.SSS2.p1.3.m3.1.1.2" xref="S4.SS2.SSS2.p1.3.m3.1.1.2.cmml"><mi id="S4.SS2.SSS2.p1.3.m3.1.1.2.2" xref="S4.SS2.SSS2.p1.3.m3.1.1.2.2.cmml">z</mi><mi id="S4.SS2.SSS2.p1.3.m3.1.1.2.3" xref="S4.SS2.SSS2.p1.3.m3.1.1.2.3.cmml">fiber</mi></msub><mo id="S4.SS2.SSS2.p1.3.m3.1.1.1" xref="S4.SS2.SSS2.p1.3.m3.1.1.1.cmml"><</mo><mn id="S4.SS2.SSS2.p1.3.m3.1.1.3" xref="S4.SS2.SSS2.p1.3.m3.1.1.3.cmml">21.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.3.m3.1b"><apply id="S4.SS2.SSS2.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS2.p1.3.m3.1.1"><lt id="S4.SS2.SSS2.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS2.p1.3.m3.1.1.1"></lt><apply id="S4.SS2.SSS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.3.m3.1.1.2.1.cmml" xref="S4.SS2.SSS2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS2.p1.3.m3.1.1.2.2.cmml" xref="S4.SS2.SSS2.p1.3.m3.1.1.2.2">𝑧</ci><ci id="S4.SS2.SSS2.p1.3.m3.1.1.2.3.cmml" xref="S4.SS2.SSS2.p1.3.m3.1.1.2.3">fiber</ci></apply><cn id="S4.SS2.SSS2.p1.3.m3.1.1.3.cmml" type="float" xref="S4.SS2.SSS2.p1.3.m3.1.1.3">21.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.3.m3.1c">z_{\mathrm{fiber}}<21.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.3.m3.1d">italic_z start_POSTSUBSCRIPT roman_fiber end_POSTSUBSCRIPT < 21.6</annotation></semantics></math> cut as the primary flux-limit cut for LRG. Although MUST will use fiber with a different core size, we use the same Legacy Survey <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.4.m4.1"><semantics id="S4.SS2.SSS2.p1.4.m4.1a"><mi id="S4.SS2.SSS2.p1.4.m4.1.1" xref="S4.SS2.SSS2.p1.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.4.m4.1b"><ci id="S4.SS2.SSS2.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS2.p1.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.4.m4.1d">italic_z</annotation></semantics></math>-band fiber magnitude definition to estimate the sample density and compare it with DESI. </div> <div class="ltx_para" id="S4.SS2.SSS2.p2"> Similar to the BGS, we assume a 2,000 per deg2 fiber density for LRG as a bright or grey night target. 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xref="S4.SS2.SSS2.p2.3.m3.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.p2.3.m3.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.p2.3.m3.1.1.1.1.2">𝑟</ci><ci id="S4.SS2.SSS2.p2.3.m3.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.p2.3.m3.1.1.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.3.m3.1c">(r-z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.3.m3.1d">( italic_r - italic_z )</annotation></semantics></math> vs <math alttext="(z-W1)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.4.m4.1"><semantics id="S4.SS2.SSS2.p2.4.m4.1a"><mrow id="S4.SS2.SSS2.p2.4.m4.1.1.1" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.p2.4.m4.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.p2.4.m4.1.1.1.1" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.2" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.1" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.1.cmml">−</mo><mrow id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.2" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.2.cmml">W</mi><mo id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.1" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.3" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.SS2.SSS2.p2.4.m4.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.4.m4.1b"><apply id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.1"><minus id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.2">𝑧</ci><apply id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3"><times id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.1"></times><ci id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.2">𝑊</ci><cn id="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.4.m4.1c">(z-W1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.4.m4.1d">( italic_z - italic_W 1 )</annotation></semantics></math> plane: </div> <div class="ltx_para" id="S4.SS2.SSS2.p3"> <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(z-W1)>0.8\,\times\,(r-z)-0.6." class="ltx_Math" display="block" id="S4.E7.m1.1"><semantics id="S4.E7.m1.1a"><mrow id="S4.E7.m1.1.1.1" xref="S4.E7.m1.1.1.1.1.cmml"><mrow id="S4.E7.m1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.cmml"><mrow id="S4.E7.m1.1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.E7.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E7.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E7.m1.1.1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.E7.m1.1.1.1.1.1.1.1.2" xref="S4.E7.m1.1.1.1.1.1.1.1.2.cmml">z</mi><mo id="S4.E7.m1.1.1.1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E7.m1.1.1.1.1.1.1.1.3" xref="S4.E7.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E7.m1.1.1.1.1.1.1.1.3.2" xref="S4.E7.m1.1.1.1.1.1.1.1.3.2.cmml">W</mi><mo id="S4.E7.m1.1.1.1.1.1.1.1.3.1" xref="S4.E7.m1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.E7.m1.1.1.1.1.1.1.1.3.3" xref="S4.E7.m1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.E7.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E7.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E7.m1.1.1.1.1.3" xref="S4.E7.m1.1.1.1.1.3.cmml">></mo><mrow id="S4.E7.m1.1.1.1.1.2" xref="S4.E7.m1.1.1.1.1.2.cmml"><mrow id="S4.E7.m1.1.1.1.1.2.1" xref="S4.E7.m1.1.1.1.1.2.1.cmml"><mn id="S4.E7.m1.1.1.1.1.2.1.3" xref="S4.E7.m1.1.1.1.1.2.1.3.cmml">0.8</mn><mo id="S4.E7.m1.1.1.1.1.2.1.2" lspace="0.392em" rspace="0.392em" xref="S4.E7.m1.1.1.1.1.2.1.2.cmml">×</mo><mrow id="S4.E7.m1.1.1.1.1.2.1.1.1" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.cmml"><mo id="S4.E7.m1.1.1.1.1.2.1.1.1.2" stretchy="false" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S4.E7.m1.1.1.1.1.2.1.1.1.1" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.cmml"><mi id="S4.E7.m1.1.1.1.1.2.1.1.1.1.2" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.2.cmml">r</mi><mo id="S4.E7.m1.1.1.1.1.2.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.1.cmml">−</mo><mi id="S4.E7.m1.1.1.1.1.2.1.1.1.1.3" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.3.cmml">z</mi></mrow><mo id="S4.E7.m1.1.1.1.1.2.1.1.1.3" stretchy="false" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E7.m1.1.1.1.1.2.2" xref="S4.E7.m1.1.1.1.1.2.2.cmml">−</mo><mn id="S4.E7.m1.1.1.1.1.2.3" xref="S4.E7.m1.1.1.1.1.2.3.cmml">0.6</mn></mrow></mrow><mo id="S4.E7.m1.1.1.1.2" lspace="0em" xref="S4.E7.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E7.m1.1b"><apply id="S4.E7.m1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1"><gt id="S4.E7.m1.1.1.1.1.3.cmml" xref="S4.E7.m1.1.1.1.1.3"></gt><apply id="S4.E7.m1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.1.1"><minus id="S4.E7.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.E7.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.1.1.1.1.1.1.1.2">𝑧</ci><apply id="S4.E7.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E7.m1.1.1.1.1.1.1.1.3"><times id="S4.E7.m1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E7.m1.1.1.1.1.1.1.1.3.1"></times><ci id="S4.E7.m1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E7.m1.1.1.1.1.1.1.1.3.2">𝑊</ci><cn id="S4.E7.m1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.E7.m1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.E7.m1.1.1.1.1.2.cmml" xref="S4.E7.m1.1.1.1.1.2"><minus id="S4.E7.m1.1.1.1.1.2.2.cmml" xref="S4.E7.m1.1.1.1.1.2.2"></minus><apply id="S4.E7.m1.1.1.1.1.2.1.cmml" xref="S4.E7.m1.1.1.1.1.2.1"><times id="S4.E7.m1.1.1.1.1.2.1.2.cmml" xref="S4.E7.m1.1.1.1.1.2.1.2"></times><cn id="S4.E7.m1.1.1.1.1.2.1.3.cmml" type="float" xref="S4.E7.m1.1.1.1.1.2.1.3">0.8</cn><apply id="S4.E7.m1.1.1.1.1.2.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.2.1.1.1"><minus id="S4.E7.m1.1.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.1"></minus><ci id="S4.E7.m1.1.1.1.1.2.1.1.1.1.2.cmml" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.2">𝑟</ci><ci id="S4.E7.m1.1.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E7.m1.1.1.1.1.2.1.1.1.1.3">𝑧</ci></apply></apply><cn id="S4.E7.m1.1.1.1.1.2.3.cmml" type="float" xref="S4.E7.m1.1.1.1.1.2.3">0.6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.1c">(z-W1)>0.8\,\times\,(r-z)-0.6.</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.1d">( italic_z - italic_W 1 ) > 0.8 × ( italic_r - italic_z ) - 0.6 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(7)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS2.p4"> Then, to obtain targets at <math alttext="z>0.4" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p4.1.m1.1"><semantics id="S4.SS2.SSS2.p4.1.m1.1a"><mrow id="S4.SS2.SSS2.p4.1.m1.1.1" xref="S4.SS2.SSS2.p4.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.p4.1.m1.1.1.2" xref="S4.SS2.SSS2.p4.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS2.p4.1.m1.1.1.1" xref="S4.SS2.SSS2.p4.1.m1.1.1.1.cmml">></mo><mn id="S4.SS2.SSS2.p4.1.m1.1.1.3" xref="S4.SS2.SSS2.p4.1.m1.1.1.3.cmml">0.4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p4.1.m1.1b"><apply id="S4.SS2.SSS2.p4.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p4.1.m1.1.1"><gt id="S4.SS2.SSS2.p4.1.m1.1.1.1.cmml" xref="S4.SS2.SSS2.p4.1.m1.1.1.1"></gt><ci id="S4.SS2.SSS2.p4.1.m1.1.1.2.cmml" xref="S4.SS2.SSS2.p4.1.m1.1.1.2">𝑧</ci><cn id="S4.SS2.SSS2.p4.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.SSS2.p4.1.m1.1.1.3">0.4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p4.1.m1.1c">z>0.4</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p4.1.m1.1d">italic_z > 0.4</annotation></semantics></math>, we apply a <math alttext="(g-r)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p4.2.m2.1"><semantics id="S4.SS2.SSS2.p4.2.m2.1a"><mrow id="S4.SS2.SSS2.p4.2.m2.1.1.1" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.p4.2.m2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.p4.2.m2.1.1.1.1" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.p4.2.m2.1.1.1.1.2" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.2.cmml">g</mi><mo id="S4.SS2.SSS2.p4.2.m2.1.1.1.1.1" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS2.p4.2.m2.1.1.1.1.3" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.3.cmml">r</mi></mrow><mo id="S4.SS2.SSS2.p4.2.m2.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p4.2.m2.1b"><apply id="S4.SS2.SSS2.p4.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS2.p4.2.m2.1.1.1"><minus id="S4.SS2.SSS2.p4.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.p4.2.m2.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.2">𝑔</ci><ci id="S4.SS2.SSS2.p4.2.m2.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.p4.2.m2.1.1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p4.2.m2.1c">(g-r)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p4.2.m2.1d">( italic_g - italic_r )</annotation></semantics></math> vs <math alttext="(r-W1)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p4.3.m3.1"><semantics id="S4.SS2.SSS2.p4.3.m3.1a"><mrow id="S4.SS2.SSS2.p4.3.m3.1.1.1" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.p4.3.m3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.p4.3.m3.1.1.1.1" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.2" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.2.cmml">r</mi><mo id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.1" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.1.cmml">−</mo><mrow id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.2" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.2.cmml">W</mi><mo id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.1" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.3" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.SS2.SSS2.p4.3.m3.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p4.3.m3.1b"><apply id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.cmml" xref="S4.SS2.SSS2.p4.3.m3.1.1.1"><minus id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.2">𝑟</ci><apply id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3"><times id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.1"></times><ci id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.2">𝑊</ci><cn id="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.p4.3.m3.1.1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p4.3.m3.1c">(r-W1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p4.3.m3.1d">( italic_r - italic_W 1 )</annotation></semantics></math> color cut: </div> <div class="ltx_para" id="S4.SS2.SSS2.p5"> <table class="ltx_equation ltx_eqn_table" id="S4.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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id="S4.E8.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E8.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E8.m1.1.1.1.1.4" xref="S4.E8.m1.1.1.1.1.4.cmml">></mo><mrow id="S4.E8.m1.1.1.1.1.2" xref="S4.E8.m1.1.1.1.1.2.cmml"><mn id="S4.E8.m1.1.1.1.1.2.3" xref="S4.E8.m1.1.1.1.1.2.3.cmml">2.97</mn><mo id="S4.E8.m1.1.1.1.1.2.2" lspace="0.330em" xref="S4.E8.m1.1.1.1.1.2.2.cmml">⁢</mo><mtext id="S4.E8.m1.1.1.1.1.2.4" xref="S4.E8.m1.1.1.1.1.2.4a.cmml">OR</mtext><mo id="S4.E8.m1.1.1.1.1.2.2a" lspace="0.330em" xref="S4.E8.m1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.E8.m1.1.1.1.1.2.1.1" xref="S4.E8.m1.1.1.1.1.2.1.1.1.cmml"><mo id="S4.E8.m1.1.1.1.1.2.1.1.2" stretchy="false" xref="S4.E8.m1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.E8.m1.1.1.1.1.2.1.1.1" xref="S4.E8.m1.1.1.1.1.2.1.1.1.cmml"><mi id="S4.E8.m1.1.1.1.1.2.1.1.1.2" xref="S4.E8.m1.1.1.1.1.2.1.1.1.2.cmml">r</mi><mo id="S4.E8.m1.1.1.1.1.2.1.1.1.1" xref="S4.E8.m1.1.1.1.1.2.1.1.1.1.cmml">−</mo><mrow id="S4.E8.m1.1.1.1.1.2.1.1.1.3" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3.cmml"><mi id="S4.E8.m1.1.1.1.1.2.1.1.1.3.2" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3.2.cmml">W</mi><mo id="S4.E8.m1.1.1.1.1.2.1.1.1.3.1" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3.1.cmml">⁢</mo><mn id="S4.E8.m1.1.1.1.1.2.1.1.1.3.3" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.E8.m1.1.1.1.1.2.1.1.3" stretchy="false" xref="S4.E8.m1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E8.m1.1.1.1.1.5" xref="S4.E8.m1.1.1.1.1.5.cmml">></mo><mn id="S4.E8.m1.1.1.1.1.6" xref="S4.E8.m1.1.1.1.1.6.cmml">1.8</mn></mrow><mo id="S4.E8.m1.1.1.1.2" lspace="0em" xref="S4.E8.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E8.m1.1b"><apply id="S4.E8.m1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1"><and id="S4.E8.m1.1.1.1.1a.cmml" xref="S4.E8.m1.1.1.1"></and><apply id="S4.E8.m1.1.1.1.1b.cmml" xref="S4.E8.m1.1.1.1"><gt id="S4.E8.m1.1.1.1.1.4.cmml" xref="S4.E8.m1.1.1.1.1.4"></gt><apply id="S4.E8.m1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1"><minus id="S4.E8.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.E8.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.1.1.1.2">𝑔</ci><apply id="S4.E8.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E8.m1.1.1.1.1.1.1.1.3"><times id="S4.E8.m1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1.1.3.1"></times><ci id="S4.E8.m1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.E8.m1.1.1.1.1.1.1.1.3.2">𝑊</ci><cn id="S4.E8.m1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.E8.m1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.E8.m1.1.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.2"><times id="S4.E8.m1.1.1.1.1.2.2.cmml" xref="S4.E8.m1.1.1.1.1.2.2"></times><cn id="S4.E8.m1.1.1.1.1.2.3.cmml" type="float" xref="S4.E8.m1.1.1.1.1.2.3">2.97</cn><ci id="S4.E8.m1.1.1.1.1.2.4a.cmml" xref="S4.E8.m1.1.1.1.1.2.4"><mtext id="S4.E8.m1.1.1.1.1.2.4.cmml" xref="S4.E8.m1.1.1.1.1.2.4">OR</mtext></ci><apply id="S4.E8.m1.1.1.1.1.2.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.2.1.1"><minus id="S4.E8.m1.1.1.1.1.2.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.2.1.1.1.1"></minus><ci id="S4.E8.m1.1.1.1.1.2.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.2.1.1.1.2">𝑟</ci><apply id="S4.E8.m1.1.1.1.1.2.1.1.1.3.cmml" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3"><times id="S4.E8.m1.1.1.1.1.2.1.1.1.3.1.cmml" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3.1"></times><ci id="S4.E8.m1.1.1.1.1.2.1.1.1.3.2.cmml" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3.2">𝑊</ci><cn id="S4.E8.m1.1.1.1.1.2.1.1.1.3.3.cmml" type="integer" xref="S4.E8.m1.1.1.1.1.2.1.1.1.3.3">1</cn></apply></apply></apply></apply><apply id="S4.E8.m1.1.1.1.1c.cmml" xref="S4.E8.m1.1.1.1"><gt id="S4.E8.m1.1.1.1.1.5.cmml" xref="S4.E8.m1.1.1.1.1.5"></gt><share href="https://arxiv.org/html/2411.07970v2#S4.E8.m1.1.1.1.1.2.cmml" id="S4.E8.m1.1.1.1.1d.cmml" xref="S4.E8.m1.1.1.1"></share><cn id="S4.E8.m1.1.1.1.1.6.cmml" type="float" xref="S4.E8.m1.1.1.1.1.6">1.8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.1c">(g-W1)>2.97\,\,\text{OR}\,\,(r-W1)>1.8.</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.1d">( italic_g - italic_W 1 ) > 2.97 OR ( italic_r - italic_W 1 ) > 1.8 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(8)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS2.p6"> Lastly, we make a color cut on the <math alttext="(r-W1)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p6.1.m1.1"><semantics id="S4.SS2.SSS2.p6.1.m1.1a"><mrow id="S4.SS2.SSS2.p6.1.m1.1.1.1" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.p6.1.m1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.p6.1.m1.1.1.1.1" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.2" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.2.cmml">r</mi><mo id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.1" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.1.cmml">−</mo><mrow id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.2" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.2.cmml">W</mi><mo id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.1" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.3" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.SS2.SSS2.p6.1.m1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p6.1.m1.1b"><apply id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.cmml" xref="S4.SS2.SSS2.p6.1.m1.1.1.1"><minus id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.2">𝑟</ci><apply id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3"><times id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.1"></times><ci id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.2">𝑊</ci><cn id="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.p6.1.m1.1.1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p6.1.m1.1c">(r-W1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p6.1.m1.1d">( italic_r - italic_W 1 )</annotation></semantics></math> vs <math alttext="(W1)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p6.2.m2.1"><semantics id="S4.SS2.SSS2.p6.2.m2.1a"><mrow id="S4.SS2.SSS2.p6.2.m2.1.1.1" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.p6.2.m2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.p6.2.m2.1.1.1.1" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.p6.2.m2.1.1.1.1.2" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.2.cmml">W</mi><mo id="S4.SS2.SSS2.p6.2.m2.1.1.1.1.1" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS2.p6.2.m2.1.1.1.1.3" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.p6.2.m2.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p6.2.m2.1b"><apply id="S4.SS2.SSS2.p6.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS2.p6.2.m2.1.1.1"><times id="S4.SS2.SSS2.p6.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.1"></times><ci id="S4.SS2.SSS2.p6.2.m2.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.2">𝑊</ci><cn id="S4.SS2.SSS2.p6.2.m2.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p6.2.m2.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p6.2.m2.1c">(W1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p6.2.m2.1d">( italic_W 1 )</annotation></semantics></math> plane to isolate the most luminous galaxies (in <math alttext="W1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p6.3.m3.1"><semantics id="S4.SS2.SSS2.p6.3.m3.1a"><mrow id="S4.SS2.SSS2.p6.3.m3.1.1" xref="S4.SS2.SSS2.p6.3.m3.1.1.cmml"><mi id="S4.SS2.SSS2.p6.3.m3.1.1.2" xref="S4.SS2.SSS2.p6.3.m3.1.1.2.cmml">W</mi><mo id="S4.SS2.SSS2.p6.3.m3.1.1.1" xref="S4.SS2.SSS2.p6.3.m3.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS2.p6.3.m3.1.1.3" xref="S4.SS2.SSS2.p6.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p6.3.m3.1b"><apply id="S4.SS2.SSS2.p6.3.m3.1.1.cmml" xref="S4.SS2.SSS2.p6.3.m3.1.1"><times id="S4.SS2.SSS2.p6.3.m3.1.1.1.cmml" xref="S4.SS2.SSS2.p6.3.m3.1.1.1"></times><ci id="S4.SS2.SSS2.p6.3.m3.1.1.2.cmml" xref="S4.SS2.SSS2.p6.3.m3.1.1.2">𝑊</ci><cn id="S4.SS2.SSS2.p6.3.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p6.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p6.3.m3.1c">W1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p6.3.m3.1d">italic_W 1</annotation></semantics></math> band) while keeping the same redshift distribution: </div> <div class="ltx_para" id="S4.SS2.SSS2.p7"> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E9"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E9X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(r-W1)>1.83\,\times\,(W1-17.13)" class="ltx_Math" display="inline" id="S4.E9X.2.1.1.m1.2"><semantics id="S4.E9X.2.1.1.m1.2a"><mrow id="S4.E9X.2.1.1.m1.2.2" xref="S4.E9X.2.1.1.m1.2.2.cmml"><mrow id="S4.E9X.2.1.1.m1.1.1.1.1" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.cmml"><mo id="S4.E9X.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E9X.2.1.1.m1.1.1.1.1.1" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S4.E9X.2.1.1.m1.1.1.1.1.1.2" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.2.cmml">r</mi><mo id="S4.E9X.2.1.1.m1.1.1.1.1.1.1" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E9X.2.1.1.m1.1.1.1.1.1.3" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3.cmml"><mi id="S4.E9X.2.1.1.m1.1.1.1.1.1.3.2" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3.2.cmml">W</mi><mo id="S4.E9X.2.1.1.m1.1.1.1.1.1.3.1" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.E9X.2.1.1.m1.1.1.1.1.1.3.3" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.E9X.2.1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E9X.2.1.1.m1.2.2.3" xref="S4.E9X.2.1.1.m1.2.2.3.cmml">></mo><mrow id="S4.E9X.2.1.1.m1.2.2.2" xref="S4.E9X.2.1.1.m1.2.2.2.cmml"><mn id="S4.E9X.2.1.1.m1.2.2.2.3" xref="S4.E9X.2.1.1.m1.2.2.2.3.cmml">1.83</mn><mo id="S4.E9X.2.1.1.m1.2.2.2.2" lspace="0.392em" rspace="0.392em" xref="S4.E9X.2.1.1.m1.2.2.2.2.cmml">×</mo><mrow id="S4.E9X.2.1.1.m1.2.2.2.1.1" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.cmml"><mo id="S4.E9X.2.1.1.m1.2.2.2.1.1.2" stretchy="false" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.E9X.2.1.1.m1.2.2.2.1.1.1" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.cmml"><mrow id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.cmml"><mi id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.2" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.2.cmml">W</mi><mo id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.1" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.1.cmml">⁢</mo><mn id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.3" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.3.cmml">1</mn></mrow><mo id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.1" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.1.cmml">−</mo><mn id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.3" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.3.cmml">17.13</mn></mrow><mo id="S4.E9X.2.1.1.m1.2.2.2.1.1.3" stretchy="false" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E9X.2.1.1.m1.2b"><apply id="S4.E9X.2.1.1.m1.2.2.cmml" xref="S4.E9X.2.1.1.m1.2.2"><gt id="S4.E9X.2.1.1.m1.2.2.3.cmml" xref="S4.E9X.2.1.1.m1.2.2.3"></gt><apply id="S4.E9X.2.1.1.m1.1.1.1.1.1.cmml" xref="S4.E9X.2.1.1.m1.1.1.1.1"><minus id="S4.E9X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.1"></minus><ci id="S4.E9X.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.2">𝑟</ci><apply id="S4.E9X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3"><times id="S4.E9X.2.1.1.m1.1.1.1.1.1.3.1.cmml" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3.1"></times><ci id="S4.E9X.2.1.1.m1.1.1.1.1.1.3.2.cmml" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3.2">𝑊</ci><cn id="S4.E9X.2.1.1.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.E9X.2.1.1.m1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.E9X.2.1.1.m1.2.2.2.cmml" xref="S4.E9X.2.1.1.m1.2.2.2"><times id="S4.E9X.2.1.1.m1.2.2.2.2.cmml" xref="S4.E9X.2.1.1.m1.2.2.2.2"></times><cn id="S4.E9X.2.1.1.m1.2.2.2.3.cmml" type="float" xref="S4.E9X.2.1.1.m1.2.2.2.3">1.83</cn><apply id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.cmml" xref="S4.E9X.2.1.1.m1.2.2.2.1.1"><minus id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.1.cmml" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.1"></minus><apply id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.cmml" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2"><times id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.1.cmml" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.1"></times><ci id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.2.cmml" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.2">𝑊</ci><cn id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.2.3">1</cn></apply><cn id="S4.E9X.2.1.1.m1.2.2.2.1.1.1.3.cmml" type="float" xref="S4.E9X.2.1.1.m1.2.2.2.1.1.1.3">17.13</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9X.2.1.1.m1.2c">\displaystyle(r-W1)>1.83\,\times\,(W1-17.13)</annotation><annotation encoding="application/x-llamapun" id="S4.E9X.2.1.1.m1.2d">( italic_r - italic_W 1 ) > 1.83 × ( italic_W 1 - 17.13 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2">(9)</td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E9Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(r-W1)>W1-16.31\,\,\text{OR}\,\,(r-W1)>3.4." class="ltx_Math" display="inline" id="S4.E9Xa.2.1.1.m1.1"><semantics id="S4.E9Xa.2.1.1.m1.1a"><mrow id="S4.E9Xa.2.1.1.m1.1.1.1" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E9Xa.2.1.1.m1.1.1.1.1" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.2" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.2.cmml">r</mi><mo id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.1" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3.2" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3.2.cmml">W</mi><mo id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3.1" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mn id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3.3" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E9Xa.2.1.1.m1.1.1.1.1.4" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.4.cmml">></mo><mrow id="S4.E9Xa.2.1.1.m1.1.1.1.1.2" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.2.cmml"><mrow id="S4.E9Xa.2.1.1.m1.1.1.1.1.2.3" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.2.3.cmml"><mi id="S4.E9Xa.2.1.1.m1.1.1.1.1.2.3.2" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.2.3.2.cmml">W</mi><mo id="S4.E9Xa.2.1.1.m1.1.1.1.1.2.3.1" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mn 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id="S4.E9Xa.2.1.1.m1.1.1.1.1.5.cmml" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.5"></gt><share href="https://arxiv.org/html/2411.07970v2#S4.E9Xa.2.1.1.m1.1.1.1.1.2.cmml" id="S4.E9Xa.2.1.1.m1.1.1.1.1d.cmml" xref="S4.E9Xa.2.1.1.m1.1.1.1"></share><cn id="S4.E9Xa.2.1.1.m1.1.1.1.1.6.cmml" type="float" xref="S4.E9Xa.2.1.1.m1.1.1.1.1.6">3.4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9Xa.2.1.1.m1.1c">\displaystyle(r-W1)>W1-16.31\,\,\text{OR}\,\,(r-W1)>3.4.</annotation><annotation encoding="application/x-llamapun" id="S4.E9Xa.2.1.1.m1.1d">( italic_r - italic_W 1 ) > italic_W 1 - 16.31 OR ( italic_r - italic_W 1 ) > 3.4 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS2.p8"> In all the above cuts, <math alttext="W1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p8.1.m1.1"><semantics id="S4.SS2.SSS2.p8.1.m1.1a"><mrow id="S4.SS2.SSS2.p8.1.m1.1.1" xref="S4.SS2.SSS2.p8.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.p8.1.m1.1.1.2" xref="S4.SS2.SSS2.p8.1.m1.1.1.2.cmml">W</mi><mo id="S4.SS2.SSS2.p8.1.m1.1.1.1" xref="S4.SS2.SSS2.p8.1.m1.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS2.p8.1.m1.1.1.3" xref="S4.SS2.SSS2.p8.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p8.1.m1.1b"><apply id="S4.SS2.SSS2.p8.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p8.1.m1.1.1"><times id="S4.SS2.SSS2.p8.1.m1.1.1.1.cmml" xref="S4.SS2.SSS2.p8.1.m1.1.1.1"></times><ci id="S4.SS2.SSS2.p8.1.m1.1.1.2.cmml" xref="S4.SS2.SSS2.p8.1.m1.1.1.2">𝑊</ci><cn id="S4.SS2.SSS2.p8.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p8.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p8.1.m1.1c">W1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p8.1.m1.1d">italic_W 1</annotation></semantics></math> is the 3.4 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p8.2.m2.1"><semantics id="S4.SS2.SSS2.p8.2.m2.1a"><mi id="S4.SS2.SSS2.p8.2.m2.1.1" xref="S4.SS2.SSS2.p8.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p8.2.m2.1b"><ci id="S4.SS2.SSS2.p8.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p8.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p8.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p8.2.m2.1d">italic_μ</annotation></semantics></math>m infrared band from the NEOWISE (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib226" title="">226</a>]</cite>) data. Additional cuts to fine-tune the star-galaxy separation are performed by removing non-null proper motion or parallax having a signal-to-noise (SNR) ratio higher than 3 in the Gaia EDR3 catalogs. Most Gaia detected objects have a magnitude higher than 18 magnitude, making them irrelevant to the LRG selection of MUST. Finally, as for the BGS, we applied the spatial masking BITMASK 1, 12, and 13 and require the object to have a minimum of one valid observation in <math alttext="g" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p8.3.m3.1"><semantics id="S4.SS2.SSS2.p8.3.m3.1a"><mi id="S4.SS2.SSS2.p8.3.m3.1.1" xref="S4.SS2.SSS2.p8.3.m3.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p8.3.m3.1b"><ci id="S4.SS2.SSS2.p8.3.m3.1.1.cmml" xref="S4.SS2.SSS2.p8.3.m3.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p8.3.m3.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p8.3.m3.1d">italic_g</annotation></semantics></math>, <math alttext="r" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p8.4.m4.1"><semantics id="S4.SS2.SSS2.p8.4.m4.1a"><mi id="S4.SS2.SSS2.p8.4.m4.1.1" xref="S4.SS2.SSS2.p8.4.m4.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p8.4.m4.1b"><ci 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xref="S4.SS2.SSS2.p8.6.m6.1.1.cmml"><mi id="S4.SS2.SSS2.p8.6.m6.1.1.2" xref="S4.SS2.SSS2.p8.6.m6.1.1.2.cmml">W</mi><mo id="S4.SS2.SSS2.p8.6.m6.1.1.1" xref="S4.SS2.SSS2.p8.6.m6.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS2.p8.6.m6.1.1.3" xref="S4.SS2.SSS2.p8.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p8.6.m6.1b"><apply id="S4.SS2.SSS2.p8.6.m6.1.1.cmml" xref="S4.SS2.SSS2.p8.6.m6.1.1"><times id="S4.SS2.SSS2.p8.6.m6.1.1.1.cmml" xref="S4.SS2.SSS2.p8.6.m6.1.1.1"></times><ci id="S4.SS2.SSS2.p8.6.m6.1.1.2.cmml" xref="S4.SS2.SSS2.p8.6.m6.1.1.2">𝑊</ci><cn id="S4.SS2.SSS2.p8.6.m6.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p8.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p8.6.m6.1c">W1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p8.6.m6.1d">italic_W 1</annotation></semantics></math> band. </div> <div class="ltx_para" id="S4.SS2.SSS2.p9"> Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.F4" title="Figure 4 ‣ 3.6.4 Gravitational waves and Fast Radio Bursts ‣ 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4</a> visualizes the target density and redshift distribution of our LRG selection, which ranges between <math alttext="0.4<z<1.4" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p9.1.m1.1"><semantics id="S4.SS2.SSS2.p9.1.m1.1a"><mrow id="S4.SS2.SSS2.p9.1.m1.1.1" xref="S4.SS2.SSS2.p9.1.m1.1.1.cmml"><mn id="S4.SS2.SSS2.p9.1.m1.1.1.2" xref="S4.SS2.SSS2.p9.1.m1.1.1.2.cmml">0.4</mn><mo id="S4.SS2.SSS2.p9.1.m1.1.1.3" xref="S4.SS2.SSS2.p9.1.m1.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS2.p9.1.m1.1.1.4" xref="S4.SS2.SSS2.p9.1.m1.1.1.4.cmml">z</mi><mo id="S4.SS2.SSS2.p9.1.m1.1.1.5" xref="S4.SS2.SSS2.p9.1.m1.1.1.5.cmml"><</mo><mn id="S4.SS2.SSS2.p9.1.m1.1.1.6" xref="S4.SS2.SSS2.p9.1.m1.1.1.6.cmml">1.4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p9.1.m1.1b"><apply id="S4.SS2.SSS2.p9.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1"><and id="S4.SS2.SSS2.p9.1.m1.1.1a.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1"></and><apply id="S4.SS2.SSS2.p9.1.m1.1.1b.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1"><lt id="S4.SS2.SSS2.p9.1.m1.1.1.3.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1.3"></lt><cn id="S4.SS2.SSS2.p9.1.m1.1.1.2.cmml" type="float" xref="S4.SS2.SSS2.p9.1.m1.1.1.2">0.4</cn><ci id="S4.SS2.SSS2.p9.1.m1.1.1.4.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.SS2.SSS2.p9.1.m1.1.1c.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1"><lt id="S4.SS2.SSS2.p9.1.m1.1.1.5.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS2.p9.1.m1.1.1.4.cmml" id="S4.SS2.SSS2.p9.1.m1.1.1d.cmml" xref="S4.SS2.SSS2.p9.1.m1.1.1"></share><cn id="S4.SS2.SSS2.p9.1.m1.1.1.6.cmml" type="float" xref="S4.SS2.SSS2.p9.1.m1.1.1.6">1.4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p9.1.m1.1c">0.4<z<1.4</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p9.1.m1.1d">0.4 < italic_z < 1.4</annotation></semantics></math> and peaks at <math alttext="z\sim 1.0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p9.2.m2.1"><semantics id="S4.SS2.SSS2.p9.2.m2.1a"><mrow id="S4.SS2.SSS2.p9.2.m2.1.1" xref="S4.SS2.SSS2.p9.2.m2.1.1.cmml"><mi id="S4.SS2.SSS2.p9.2.m2.1.1.2" xref="S4.SS2.SSS2.p9.2.m2.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS2.p9.2.m2.1.1.1" xref="S4.SS2.SSS2.p9.2.m2.1.1.1.cmml">∼</mo><mn id="S4.SS2.SSS2.p9.2.m2.1.1.3" xref="S4.SS2.SSS2.p9.2.m2.1.1.3.cmml">1.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p9.2.m2.1b"><apply id="S4.SS2.SSS2.p9.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p9.2.m2.1.1"><csymbol cd="latexml" id="S4.SS2.SSS2.p9.2.m2.1.1.1.cmml" xref="S4.SS2.SSS2.p9.2.m2.1.1.1">similar-to</csymbol><ci id="S4.SS2.SSS2.p9.2.m2.1.1.2.cmml" xref="S4.SS2.SSS2.p9.2.m2.1.1.2">𝑧</ci><cn id="S4.SS2.SSS2.p9.2.m2.1.1.3.cmml" type="float" xref="S4.SS2.SSS2.p9.2.m2.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p9.2.m2.1c">z\sim 1.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p9.2.m2.1d">italic_z ∼ 1.0</annotation></semantics></math>. For the bias evolution model of the LRG, we adopt the DESI one: <math alttext="b(z)=1.7/D(z)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p9.3.m3.2"><semantics id="S4.SS2.SSS2.p9.3.m3.2a"><mrow id="S4.SS2.SSS2.p9.3.m3.2.3" xref="S4.SS2.SSS2.p9.3.m3.2.3.cmml"><mrow id="S4.SS2.SSS2.p9.3.m3.2.3.2" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.cmml"><mi id="S4.SS2.SSS2.p9.3.m3.2.3.2.2" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.2.cmml">b</mi><mo id="S4.SS2.SSS2.p9.3.m3.2.3.2.1" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p9.3.m3.2.3.2.3.2" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.cmml"><mo id="S4.SS2.SSS2.p9.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.cmml">(</mo><mi id="S4.SS2.SSS2.p9.3.m3.1.1" xref="S4.SS2.SSS2.p9.3.m3.1.1.cmml">z</mi><mo id="S4.SS2.SSS2.p9.3.m3.2.3.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.p9.3.m3.2.3.1" xref="S4.SS2.SSS2.p9.3.m3.2.3.1.cmml">=</mo><mrow id="S4.SS2.SSS2.p9.3.m3.2.3.3" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.cmml"><mrow id="S4.SS2.SSS2.p9.3.m3.2.3.3.2" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2.cmml"><mn id="S4.SS2.SSS2.p9.3.m3.2.3.3.2.2" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2.2.cmml">1.7</mn><mo id="S4.SS2.SSS2.p9.3.m3.2.3.3.2.1" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2.1.cmml">/</mo><mi id="S4.SS2.SSS2.p9.3.m3.2.3.3.2.3" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2.3.cmml">D</mi></mrow><mo id="S4.SS2.SSS2.p9.3.m3.2.3.3.1" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p9.3.m3.2.3.3.3.2" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.cmml"><mo id="S4.SS2.SSS2.p9.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.cmml">(</mo><mi id="S4.SS2.SSS2.p9.3.m3.2.2" xref="S4.SS2.SSS2.p9.3.m3.2.2.cmml">z</mi><mo id="S4.SS2.SSS2.p9.3.m3.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p9.3.m3.2b"><apply id="S4.SS2.SSS2.p9.3.m3.2.3.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3"><eq id="S4.SS2.SSS2.p9.3.m3.2.3.1.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.1"></eq><apply id="S4.SS2.SSS2.p9.3.m3.2.3.2.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.2"><times id="S4.SS2.SSS2.p9.3.m3.2.3.2.1.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.1"></times><ci id="S4.SS2.SSS2.p9.3.m3.2.3.2.2.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.2.2">𝑏</ci><ci id="S4.SS2.SSS2.p9.3.m3.1.1.cmml" xref="S4.SS2.SSS2.p9.3.m3.1.1">𝑧</ci></apply><apply id="S4.SS2.SSS2.p9.3.m3.2.3.3.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.3"><times id="S4.SS2.SSS2.p9.3.m3.2.3.3.1.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.1"></times><apply id="S4.SS2.SSS2.p9.3.m3.2.3.3.2.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2"><divide id="S4.SS2.SSS2.p9.3.m3.2.3.3.2.1.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2.1"></divide><cn id="S4.SS2.SSS2.p9.3.m3.2.3.3.2.2.cmml" type="float" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2.2">1.7</cn><ci id="S4.SS2.SSS2.p9.3.m3.2.3.3.2.3.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.3.3.2.3">𝐷</ci></apply><ci id="S4.SS2.SSS2.p9.3.m3.2.2.cmml" xref="S4.SS2.SSS2.p9.3.m3.2.2">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p9.3.m3.2c">b(z)=1.7/D(z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p9.3.m3.2d">italic_b ( italic_z ) = 1.7 / italic_D ( italic_z )</annotation></semantics></math>. </div> <div class="ltx_para" id="S4.SS2.SSS2.p10"> For LRG, our current <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p10.1.m1.1"><semantics id="S4.SS2.SSS2.p10.1.m1.1a"><mi id="S4.SS2.SSS2.p10.1.m1.1.1" xref="S4.SS2.SSS2.p10.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p10.1.m1.1b"><ci id="S4.SS2.SSS2.p10.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p10.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p10.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p10.1.m1.1d">italic_z</annotation></semantics></math>-band fiber magnitude limit approaches the detection limits of Legacy Survey. But, at <math alttext="z_{\mathrm{fiber}}<23.0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p10.2.m2.1"><semantics id="S4.SS2.SSS2.p10.2.m2.1a"><mrow id="S4.SS2.SSS2.p10.2.m2.1.1" xref="S4.SS2.SSS2.p10.2.m2.1.1.cmml"><msub id="S4.SS2.SSS2.p10.2.m2.1.1.2" xref="S4.SS2.SSS2.p10.2.m2.1.1.2.cmml"><mi id="S4.SS2.SSS2.p10.2.m2.1.1.2.2" xref="S4.SS2.SSS2.p10.2.m2.1.1.2.2.cmml">z</mi><mi id="S4.SS2.SSS2.p10.2.m2.1.1.2.3" xref="S4.SS2.SSS2.p10.2.m2.1.1.2.3.cmml">fiber</mi></msub><mo id="S4.SS2.SSS2.p10.2.m2.1.1.1" xref="S4.SS2.SSS2.p10.2.m2.1.1.1.cmml"><</mo><mn id="S4.SS2.SSS2.p10.2.m2.1.1.3" xref="S4.SS2.SSS2.p10.2.m2.1.1.3.cmml">23.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p10.2.m2.1b"><apply id="S4.SS2.SSS2.p10.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p10.2.m2.1.1"><lt id="S4.SS2.SSS2.p10.2.m2.1.1.1.cmml" xref="S4.SS2.SSS2.p10.2.m2.1.1.1"></lt><apply id="S4.SS2.SSS2.p10.2.m2.1.1.2.cmml" xref="S4.SS2.SSS2.p10.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p10.2.m2.1.1.2.1.cmml" xref="S4.SS2.SSS2.p10.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS2.p10.2.m2.1.1.2.2.cmml" xref="S4.SS2.SSS2.p10.2.m2.1.1.2.2">𝑧</ci><ci id="S4.SS2.SSS2.p10.2.m2.1.1.2.3.cmml" xref="S4.SS2.SSS2.p10.2.m2.1.1.2.3">fiber</ci></apply><cn id="S4.SS2.SSS2.p10.2.m2.1.1.3.cmml" type="float" xref="S4.SS2.SSS2.p10.2.m2.1.1.3">23.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p10.2.m2.1c">z_{\mathrm{fiber}}<23.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p10.2.m2.1d">italic_z start_POSTSUBSCRIPT roman_fiber end_POSTSUBSCRIPT < 23.0</annotation></semantics></math>, MUST can already select a few hundred more LRG per deg2 to compensate the fiber assignment and redshift measurement to loss to ensure the delivery of a 2,000 deg-2 density LRG sample for cosmology. As we reach to fainter magnitude limit, the redshift distribution of the LRG sample for MUST shifts significantly toward a higher average redshift than DESI. As the current selection is still using the LegacySurvey data, it is not clear whether this would be the best option for MUST, which should enjoy deeper imaging data from CSST, LSST, and Euclid. We will explore new approaches to select LRG using imaging data that are better suited to MUST in the upcoming work. </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> 4.2.3 Emission Line Galaxies (ELG)</h4> <div class="ltx_para" id="S4.SS2.SSS3.p1"> Emission-line galaxies represent the abundant, active star-forming galaxies at the peak of the cosmic star formation history (<math alttext="1<z<2" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p1.1.m1.1"><semantics id="S4.SS2.SSS3.p1.1.m1.1a"><mrow id="S4.SS2.SSS3.p1.1.m1.1.1" xref="S4.SS2.SSS3.p1.1.m1.1.1.cmml"><mn id="S4.SS2.SSS3.p1.1.m1.1.1.2" xref="S4.SS2.SSS3.p1.1.m1.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS3.p1.1.m1.1.1.3" xref="S4.SS2.SSS3.p1.1.m1.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS3.p1.1.m1.1.1.4" xref="S4.SS2.SSS3.p1.1.m1.1.1.4.cmml">z</mi><mo id="S4.SS2.SSS3.p1.1.m1.1.1.5" xref="S4.SS2.SSS3.p1.1.m1.1.1.5.cmml"><</mo><mn id="S4.SS2.SSS3.p1.1.m1.1.1.6" xref="S4.SS2.SSS3.p1.1.m1.1.1.6.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p1.1.m1.1b"><apply id="S4.SS2.SSS3.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1"><and id="S4.SS2.SSS3.p1.1.m1.1.1a.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1"></and><apply id="S4.SS2.SSS3.p1.1.m1.1.1b.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1"><lt id="S4.SS2.SSS3.p1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1.3"></lt><cn id="S4.SS2.SSS3.p1.1.m1.1.1.2.cmml" type="integer" xref="S4.SS2.SSS3.p1.1.m1.1.1.2">1</cn><ci id="S4.SS2.SSS3.p1.1.m1.1.1.4.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.SS2.SSS3.p1.1.m1.1.1c.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1"><lt id="S4.SS2.SSS3.p1.1.m1.1.1.5.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS3.p1.1.m1.1.1.4.cmml" id="S4.SS2.SSS3.p1.1.m1.1.1d.cmml" xref="S4.SS2.SSS3.p1.1.m1.1.1"></share><cn id="S4.SS2.SSS3.p1.1.m1.1.1.6.cmml" type="integer" xref="S4.SS2.SSS3.p1.1.m1.1.1.6">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p1.1.m1.1c">1<z<2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p1.1.m1.1d">1 < italic_z < 2</annotation></semantics></math>). Their high 3-D density and the relative ease to redshift confirmation vial the doublet signature of the [O II] emission lines at rest-frame 3727,3729 <math alttext="\AA" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p1.2.m2.1"><semantics id="S4.SS2.SSS3.p1.2.m2.1a"><mi id="S4.SS2.SSS3.p1.2.m2.1.1" xref="S4.SS2.SSS3.p1.2.m2.1.1.cmml">Å</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p1.2.m2.1b"><ci id="S4.SS2.SSS3.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p1.2.m2.1.1">italic-Å</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p1.2.m2.1c">\AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p1.2.m2.1d">italic_Å</annotation></semantics></math> make them appealing <math alttext="z>1" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p1.3.m3.1"><semantics id="S4.SS2.SSS3.p1.3.m3.1a"><mrow id="S4.SS2.SSS3.p1.3.m3.1.1" xref="S4.SS2.SSS3.p1.3.m3.1.1.cmml"><mi id="S4.SS2.SSS3.p1.3.m3.1.1.2" xref="S4.SS2.SSS3.p1.3.m3.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS3.p1.3.m3.1.1.1" xref="S4.SS2.SSS3.p1.3.m3.1.1.1.cmml">></mo><mn id="S4.SS2.SSS3.p1.3.m3.1.1.3" xref="S4.SS2.SSS3.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p1.3.m3.1b"><apply id="S4.SS2.SSS3.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p1.3.m3.1.1"><gt id="S4.SS2.SSS3.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.p1.3.m3.1.1.1"></gt><ci id="S4.SS2.SSS3.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.p1.3.m3.1.1.2">𝑧</ci><cn id="S4.SS2.SSS3.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p1.3.m3.1c">z>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p1.3.m3.1d">italic_z > 1</annotation></semantics></math> LSS tracers in the Stage-III and IV surveys (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib227" title="">227</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib204" title="">204</a>]</cite>). Within the same redshift range, ELG represents the galaxy population that is less massive than the LRG. Therefore, the average halo mass and bias of ELG are systematically lower than LRG. There are also questions about their galaxy-halo connection model (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib228" title="">228</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib229" title="">229</a>]</cite>), which could become a systematic for LSS analysis. </div> <div class="ltx_para" id="S4.SS2.SSS3.p2"> Stage-V surveys like MUST can reach a significantly fainter magnitude limit to observe an even larger population of ELG and explore their cosmological potential. Similar to the BGS and LRG samples, we will again follow the ELG selection of DESI in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib204" title="">204</a>]</cite> for our forecast. We assume that ELG are grey-time targets with an optimistic 3,600 per deg2 target density and a conservative 2,500 deg-2 density. Compared to the ELG_LOP sample of DESI with a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.3.m3.1"><semantics id="S4.SS2.SSS3.p2.3.m3.1a"><mo id="S4.SS2.SSS3.p2.3.m3.1.1" xref="S4.SS2.SSS3.p2.3.m3.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.3.m3.1b"><csymbol cd="latexml" id="S4.SS2.SSS3.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p2.3.m3.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.3.m3.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.3.m3.1d">∼</annotation></semantics></math>1,940 deg2 density at <math alttext="g" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.5.m5.1"><semantics id="S4.SS2.SSS3.p2.5.m5.1a"><mi id="S4.SS2.SSS3.p2.5.m5.1.1" xref="S4.SS2.SSS3.p2.5.m5.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.5.m5.1b"><ci id="S4.SS2.SSS3.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS3.p2.5.m5.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.5.m5.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.5.m5.1d">italic_g</annotation></semantics></math>-band fiber magnitude brighter than 24.1 mag, we can achieve the desired target density by selecting ELG within the <math alttext="24.1<g_{\rm fiber}<24.6" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.6.m6.1"><semantics id="S4.SS2.SSS3.p2.6.m6.1a"><mrow id="S4.SS2.SSS3.p2.6.m6.1.1" xref="S4.SS2.SSS3.p2.6.m6.1.1.cmml"><mn id="S4.SS2.SSS3.p2.6.m6.1.1.2" xref="S4.SS2.SSS3.p2.6.m6.1.1.2.cmml">24.1</mn><mo id="S4.SS2.SSS3.p2.6.m6.1.1.3" xref="S4.SS2.SSS3.p2.6.m6.1.1.3.cmml"><</mo><msub id="S4.SS2.SSS3.p2.6.m6.1.1.4" xref="S4.SS2.SSS3.p2.6.m6.1.1.4.cmml"><mi id="S4.SS2.SSS3.p2.6.m6.1.1.4.2" xref="S4.SS2.SSS3.p2.6.m6.1.1.4.2.cmml">g</mi><mi id="S4.SS2.SSS3.p2.6.m6.1.1.4.3" xref="S4.SS2.SSS3.p2.6.m6.1.1.4.3.cmml">fiber</mi></msub><mo id="S4.SS2.SSS3.p2.6.m6.1.1.5" xref="S4.SS2.SSS3.p2.6.m6.1.1.5.cmml"><</mo><mn id="S4.SS2.SSS3.p2.6.m6.1.1.6" xref="S4.SS2.SSS3.p2.6.m6.1.1.6.cmml">24.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.6.m6.1b"><apply id="S4.SS2.SSS3.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1"><and id="S4.SS2.SSS3.p2.6.m6.1.1a.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1"></and><apply id="S4.SS2.SSS3.p2.6.m6.1.1b.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1"><lt id="S4.SS2.SSS3.p2.6.m6.1.1.3.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1.3"></lt><cn id="S4.SS2.SSS3.p2.6.m6.1.1.2.cmml" type="float" xref="S4.SS2.SSS3.p2.6.m6.1.1.2">24.1</cn><apply id="S4.SS2.SSS3.p2.6.m6.1.1.4.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p2.6.m6.1.1.4.1.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1.4">subscript</csymbol><ci id="S4.SS2.SSS3.p2.6.m6.1.1.4.2.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1.4.2">𝑔</ci><ci id="S4.SS2.SSS3.p2.6.m6.1.1.4.3.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1.4.3">fiber</ci></apply></apply><apply id="S4.SS2.SSS3.p2.6.m6.1.1c.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1"><lt id="S4.SS2.SSS3.p2.6.m6.1.1.5.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS3.p2.6.m6.1.1.4.cmml" id="S4.SS2.SSS3.p2.6.m6.1.1d.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1"></share><cn id="S4.SS2.SSS3.p2.6.m6.1.1.6.cmml" type="float" xref="S4.SS2.SSS3.p2.6.m6.1.1.6">24.6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.6.m6.1c">24.1<g_{\rm fiber}<24.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.6.m6.1d">24.1 < italic_g start_POSTSUBSCRIPT roman_fiber end_POSTSUBSCRIPT < 24.6</annotation></semantics></math> mag range and applying the same color and quality cuts used in DESI. Again, we ignore the difference in fiber core size between MUST and DESI here. </div> <div class="ltx_para" id="S4.SS2.SSS3.p3"> Firstly, DESI designed two <math alttext="(g-r)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.1.m1.1"><semantics id="S4.SS2.SSS3.p3.1.m1.1a"><mrow id="S4.SS2.SSS3.p3.1.m1.1.1.1" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.p3.1.m1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.p3.1.m1.1.1.1.1" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p3.1.m1.1.1.1.1.2" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.2.cmml">g</mi><mo id="S4.SS2.SSS3.p3.1.m1.1.1.1.1.1" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS3.p3.1.m1.1.1.1.1.3" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S4.SS2.SSS3.p3.1.m1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.1.m1.1b"><apply id="S4.SS2.SSS3.p3.1.m1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.1"><minus id="S4.SS2.SSS3.p3.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS3.p3.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.2">𝑔</ci><ci id="S4.SS2.SSS3.p3.1.m1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.1.m1.1c">(g-r)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.1.m1.1d">( italic_g - italic_r )</annotation></semantics></math> v.s. <math alttext="(r-z)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.2.m2.1"><semantics id="S4.SS2.SSS3.p3.2.m2.1a"><mrow id="S4.SS2.SSS3.p3.2.m2.1.1.1" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.p3.2.m2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.p3.2.m2.1.1.1.1" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p3.2.m2.1.1.1.1.2" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.2.cmml">r</mi><mo id="S4.SS2.SSS3.p3.2.m2.1.1.1.1.1" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS3.p3.2.m2.1.1.1.1.3" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.3.cmml">z</mi></mrow><mo id="S4.SS2.SSS3.p3.2.m2.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.2.m2.1b"><apply id="S4.SS2.SSS3.p3.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS3.p3.2.m2.1.1.1"><minus id="S4.SS2.SSS3.p3.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.1"></minus><ci id="S4.SS2.SSS3.p3.2.m2.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.2">𝑟</ci><ci id="S4.SS2.SSS3.p3.2.m2.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.2.m2.1c">(r-z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.2.m2.1d">( italic_r - italic_z )</annotation></semantics></math> color cuts to select ELG within the <math alttext="0.6<z<1.6" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.3.m3.1"><semantics id="S4.SS2.SSS3.p3.3.m3.1a"><mrow id="S4.SS2.SSS3.p3.3.m3.1.1" xref="S4.SS2.SSS3.p3.3.m3.1.1.cmml"><mn id="S4.SS2.SSS3.p3.3.m3.1.1.2" xref="S4.SS2.SSS3.p3.3.m3.1.1.2.cmml">0.6</mn><mo id="S4.SS2.SSS3.p3.3.m3.1.1.3" xref="S4.SS2.SSS3.p3.3.m3.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS3.p3.3.m3.1.1.4" xref="S4.SS2.SSS3.p3.3.m3.1.1.4.cmml">z</mi><mo id="S4.SS2.SSS3.p3.3.m3.1.1.5" xref="S4.SS2.SSS3.p3.3.m3.1.1.5.cmml"><</mo><mn id="S4.SS2.SSS3.p3.3.m3.1.1.6" xref="S4.SS2.SSS3.p3.3.m3.1.1.6.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.3.m3.1b"><apply id="S4.SS2.SSS3.p3.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1"><and id="S4.SS2.SSS3.p3.3.m3.1.1a.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1"></and><apply id="S4.SS2.SSS3.p3.3.m3.1.1b.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1"><lt id="S4.SS2.SSS3.p3.3.m3.1.1.3.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1.3"></lt><cn id="S4.SS2.SSS3.p3.3.m3.1.1.2.cmml" type="float" xref="S4.SS2.SSS3.p3.3.m3.1.1.2">0.6</cn><ci id="S4.SS2.SSS3.p3.3.m3.1.1.4.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1.4">𝑧</ci></apply><apply id="S4.SS2.SSS3.p3.3.m3.1.1c.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1"><lt id="S4.SS2.SSS3.p3.3.m3.1.1.5.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS3.p3.3.m3.1.1.4.cmml" id="S4.SS2.SSS3.p3.3.m3.1.1d.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1"></share><cn id="S4.SS2.SSS3.p3.3.m3.1.1.6.cmml" type="float" xref="S4.SS2.SSS3.p3.3.m3.1.1.6">1.6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.3.m3.1c">0.6<z<1.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.3.m3.1d">0.6 < italic_z < 1.6</annotation></semantics></math> redshift range. These cuts also ensure the exclusion of stars while favoring the selection of strong [O II] emitters: </div> <div class="ltx_para" id="S4.SS2.SSS3.p4"> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E10"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E10X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(g-r)<0.5\,\times\,(r-z)+0.1" class="ltx_Math" display="inline" id="S4.E10X.2.1.1.m1.2"><semantics id="S4.E10X.2.1.1.m1.2a"><mrow id="S4.E10X.2.1.1.m1.2.2" xref="S4.E10X.2.1.1.m1.2.2.cmml"><mrow id="S4.E10X.2.1.1.m1.1.1.1.1" xref="S4.E10X.2.1.1.m1.1.1.1.1.1.cmml"><mo id="S4.E10X.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.E10X.2.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E10X.2.1.1.m1.1.1.1.1.1" xref="S4.E10X.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S4.E10X.2.1.1.m1.1.1.1.1.1.2" xref="S4.E10X.2.1.1.m1.1.1.1.1.1.2.cmml">g</mi><mo 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xref="S4.E10X.2.1.1.m1.1.1.1.1.1.2">𝑔</ci><ci id="S4.E10X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S4.E10X.2.1.1.m1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.E10X.2.1.1.m1.2.2.2.cmml" xref="S4.E10X.2.1.1.m1.2.2.2"><plus id="S4.E10X.2.1.1.m1.2.2.2.2.cmml" xref="S4.E10X.2.1.1.m1.2.2.2.2"></plus><apply id="S4.E10X.2.1.1.m1.2.2.2.1.cmml" xref="S4.E10X.2.1.1.m1.2.2.2.1"><times id="S4.E10X.2.1.1.m1.2.2.2.1.2.cmml" xref="S4.E10X.2.1.1.m1.2.2.2.1.2"></times><cn id="S4.E10X.2.1.1.m1.2.2.2.1.3.cmml" type="float" xref="S4.E10X.2.1.1.m1.2.2.2.1.3">0.5</cn><apply id="S4.E10X.2.1.1.m1.2.2.2.1.1.1.1.cmml" xref="S4.E10X.2.1.1.m1.2.2.2.1.1.1"><minus id="S4.E10X.2.1.1.m1.2.2.2.1.1.1.1.1.cmml" xref="S4.E10X.2.1.1.m1.2.2.2.1.1.1.1.1"></minus><ci id="S4.E10X.2.1.1.m1.2.2.2.1.1.1.1.2.cmml" xref="S4.E10X.2.1.1.m1.2.2.2.1.1.1.1.2">𝑟</ci><ci id="S4.E10X.2.1.1.m1.2.2.2.1.1.1.1.3.cmml" xref="S4.E10X.2.1.1.m1.2.2.2.1.1.1.1.3">𝑧</ci></apply></apply><cn id="S4.E10X.2.1.1.m1.2.2.2.3.cmml" type="float" xref="S4.E10X.2.1.1.m1.2.2.2.3">0.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10X.2.1.1.m1.2c">\displaystyle(g-r)<0.5\,\times\,(r-z)+0.1</annotation><annotation encoding="application/x-llamapun" id="S4.E10X.2.1.1.m1.2d">( italic_g - italic_r ) < 0.5 × ( italic_r - italic_z ) + 0.1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2">(10)</td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E10Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(g-r)<-1.2\,\times\,(r-z)+1.3," class="ltx_Math" display="inline" id="S4.E10Xa.2.1.1.m1.1"><semantics id="S4.E10Xa.2.1.1.m1.1a"><mrow id="S4.E10Xa.2.1.1.m1.1.1.1" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E10Xa.2.1.1.m1.1.1.1.1" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.2" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.2.cmml">g</mi><mo id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.1" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.3" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E10Xa.2.1.1.m1.1.1.1.1.3" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.3.cmml"><</mo><mrow id="S4.E10Xa.2.1.1.m1.1.1.1.1.2" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.cmml"><mrow id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.cmml"><mo 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xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.3" stretchy="false" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.2" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.2.cmml">+</mo><mn id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.3" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.3.cmml">1.3</mn></mrow></mrow><mo id="S4.E10Xa.2.1.1.m1.1.1.1.2" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E10Xa.2.1.1.m1.1b"><apply id="S4.E10Xa.2.1.1.m1.1.1.1.1.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1"><lt id="S4.E10Xa.2.1.1.m1.1.1.1.1.3.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.3"></lt><apply id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1"><minus id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.2">𝑔</ci><ci id="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2"><plus id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.2.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.2"></plus><apply id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1"><minus id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.2.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1"></minus><apply id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1"><times id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.2.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.2"></times><cn id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.3.cmml" type="float" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.3">1.2</cn><apply id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1"><minus id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.1.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.1"></minus><ci id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.2.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.2">𝑟</ci><ci id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.3.cmml" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.3">𝑧</ci></apply></apply></apply><cn id="S4.E10Xa.2.1.1.m1.1.1.1.1.2.3.cmml" type="float" xref="S4.E10Xa.2.1.1.m1.1.1.1.1.2.3">1.3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10Xa.2.1.1.m1.1c">\displaystyle(g-r)<-1.2\,\times\,(r-z)+1.3,</annotation><annotation encoding="application/x-llamapun" id="S4.E10Xa.2.1.1.m1.1d">( italic_g - italic_r ) < - 1.2 × ( italic_r - italic_z ) + 1.3 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p5"> At the same time, a <math alttext="(r-z)>0.15" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p5.1.m1.1"><semantics id="S4.SS2.SSS3.p5.1.m1.1a"><mrow id="S4.SS2.SSS3.p5.1.m1.1.1" xref="S4.SS2.SSS3.p5.1.m1.1.1.cmml"><mrow id="S4.SS2.SSS3.p5.1.m1.1.1.1.1" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.2" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.2.cmml">r</mi><mo id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.1" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.3" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS3.p5.1.m1.1.1.2" xref="S4.SS2.SSS3.p5.1.m1.1.1.2.cmml">></mo><mn id="S4.SS2.SSS3.p5.1.m1.1.1.3" xref="S4.SS2.SSS3.p5.1.m1.1.1.3.cmml">0.15</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p5.1.m1.1b"><apply id="S4.SS2.SSS3.p5.1.m1.1.1.cmml" xref="S4.SS2.SSS3.p5.1.m1.1.1"><gt id="S4.SS2.SSS3.p5.1.m1.1.1.2.cmml" xref="S4.SS2.SSS3.p5.1.m1.1.1.2"></gt><apply id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1"><minus id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.2">𝑟</ci><ci id="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p5.1.m1.1.1.1.1.1.3">𝑧</ci></apply><cn id="S4.SS2.SSS3.p5.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.SSS3.p5.1.m1.1.1.3">0.15</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p5.1.m1.1c">(r-z)>0.15</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p5.1.m1.1d">( italic_r - italic_z ) > 0.15</annotation></semantics></math> color cut is used to exclude <math alttext="z>1.6" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p5.2.m2.1"><semantics id="S4.SS2.SSS3.p5.2.m2.1a"><mrow id="S4.SS2.SSS3.p5.2.m2.1.1" xref="S4.SS2.SSS3.p5.2.m2.1.1.cmml"><mi id="S4.SS2.SSS3.p5.2.m2.1.1.2" xref="S4.SS2.SSS3.p5.2.m2.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS3.p5.2.m2.1.1.1" xref="S4.SS2.SSS3.p5.2.m2.1.1.1.cmml">></mo><mn id="S4.SS2.SSS3.p5.2.m2.1.1.3" xref="S4.SS2.SSS3.p5.2.m2.1.1.3.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p5.2.m2.1b"><apply id="S4.SS2.SSS3.p5.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p5.2.m2.1.1"><gt id="S4.SS2.SSS3.p5.2.m2.1.1.1.cmml" xref="S4.SS2.SSS3.p5.2.m2.1.1.1"></gt><ci id="S4.SS2.SSS3.p5.2.m2.1.1.2.cmml" xref="S4.SS2.SSS3.p5.2.m2.1.1.2">𝑧</ci><cn id="S4.SS2.SSS3.p5.2.m2.1.1.3.cmml" type="float" xref="S4.SS2.SSS3.p5.2.m2.1.1.3">1.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p5.2.m2.1c">z>1.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p5.2.m2.1d">italic_z > 1.6</annotation></semantics></math> galaxies when their [O II] lines redshift out of the wavelength coverage of the spectrograph of DESI, which is very similar to that of MUST. </div> <div class="ltx_para" id="S4.SS2.SSS3.p6"> Similarly, we require the targets to have clean photometry in <math alttext="g" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.1.m1.1"><semantics id="S4.SS2.SSS3.p6.1.m1.1a"><mi id="S4.SS2.SSS3.p6.1.m1.1.1" xref="S4.SS2.SSS3.p6.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p6.1.m1.1b"><ci id="S4.SS2.SSS3.p6.1.m1.1.1.cmml" xref="S4.SS2.SSS3.p6.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.1.m1.1d">italic_g</annotation></semantics></math>, <math alttext="r" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.2.m2.1"><semantics id="S4.SS2.SSS3.p6.2.m2.1a"><mi id="S4.SS2.SSS3.p6.2.m2.1.1" xref="S4.SS2.SSS3.p6.2.m2.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p6.2.m2.1b"><ci id="S4.SS2.SSS3.p6.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p6.2.m2.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.2.m2.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.2.m2.1d">italic_r</annotation></semantics></math>, and <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.3.m3.1"><semantics id="S4.SS2.SSS3.p6.3.m3.1a"><mi id="S4.SS2.SSS3.p6.3.m3.1.1" xref="S4.SS2.SSS3.p6.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p6.3.m3.1b"><ci id="S4.SS2.SSS3.p6.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p6.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.3.m3.1d">italic_z</annotation></semantics></math> bands and are not close to a bright star or galaxy using the spatial masking BITMASK 1, 12, and 13. </div> <div class="ltx_para" id="S4.SS2.SSS3.p7"> We show the redshift distribution of ELG and volume density in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.F4" title="Figure 4 ‣ 3.6.4 Gravitational waves and Fast Radio Bursts ‣ 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4</a>. Since we copied the ELG selection recipe of DESI, it is unsurprising that this sample occupies the identical <math alttext="0.6<z<1.6" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p7.1.m1.1"><semantics id="S4.SS2.SSS3.p7.1.m1.1a"><mrow id="S4.SS2.SSS3.p7.1.m1.1.1" xref="S4.SS2.SSS3.p7.1.m1.1.1.cmml"><mn id="S4.SS2.SSS3.p7.1.m1.1.1.2" xref="S4.SS2.SSS3.p7.1.m1.1.1.2.cmml">0.6</mn><mo id="S4.SS2.SSS3.p7.1.m1.1.1.3" xref="S4.SS2.SSS3.p7.1.m1.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS3.p7.1.m1.1.1.4" xref="S4.SS2.SSS3.p7.1.m1.1.1.4.cmml">z</mi><mo id="S4.SS2.SSS3.p7.1.m1.1.1.5" xref="S4.SS2.SSS3.p7.1.m1.1.1.5.cmml"><</mo><mn id="S4.SS2.SSS3.p7.1.m1.1.1.6" xref="S4.SS2.SSS3.p7.1.m1.1.1.6.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p7.1.m1.1b"><apply id="S4.SS2.SSS3.p7.1.m1.1.1.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1"><and id="S4.SS2.SSS3.p7.1.m1.1.1a.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1"></and><apply id="S4.SS2.SSS3.p7.1.m1.1.1b.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1"><lt id="S4.SS2.SSS3.p7.1.m1.1.1.3.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1.3"></lt><cn id="S4.SS2.SSS3.p7.1.m1.1.1.2.cmml" type="float" xref="S4.SS2.SSS3.p7.1.m1.1.1.2">0.6</cn><ci id="S4.SS2.SSS3.p7.1.m1.1.1.4.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.SS2.SSS3.p7.1.m1.1.1c.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1"><lt id="S4.SS2.SSS3.p7.1.m1.1.1.5.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS2.SSS3.p7.1.m1.1.1.4.cmml" id="S4.SS2.SSS3.p7.1.m1.1.1d.cmml" xref="S4.SS2.SSS3.p7.1.m1.1.1"></share><cn id="S4.SS2.SSS3.p7.1.m1.1.1.6.cmml" type="float" xref="S4.SS2.SSS3.p7.1.m1.1.1.6">1.6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p7.1.m1.1c">0.6<z<1.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p7.1.m1.1d">0.6 < italic_z < 1.6</annotation></semantics></math> redshift range and peaks at <math alttext="z\sim 1.2" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p7.2.m2.1"><semantics id="S4.SS2.SSS3.p7.2.m2.1a"><mrow id="S4.SS2.SSS3.p7.2.m2.1.1" xref="S4.SS2.SSS3.p7.2.m2.1.1.cmml"><mi id="S4.SS2.SSS3.p7.2.m2.1.1.2" xref="S4.SS2.SSS3.p7.2.m2.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS3.p7.2.m2.1.1.1" xref="S4.SS2.SSS3.p7.2.m2.1.1.1.cmml">∼</mo><mn id="S4.SS2.SSS3.p7.2.m2.1.1.3" xref="S4.SS2.SSS3.p7.2.m2.1.1.3.cmml">1.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p7.2.m2.1b"><apply id="S4.SS2.SSS3.p7.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p7.2.m2.1.1"><csymbol cd="latexml" id="S4.SS2.SSS3.p7.2.m2.1.1.1.cmml" xref="S4.SS2.SSS3.p7.2.m2.1.1.1">similar-to</csymbol><ci id="S4.SS2.SSS3.p7.2.m2.1.1.2.cmml" xref="S4.SS2.SSS3.p7.2.m2.1.1.2">𝑧</ci><cn id="S4.SS2.SSS3.p7.2.m2.1.1.3.cmml" type="float" xref="S4.SS2.SSS3.p7.2.m2.1.1.3">1.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p7.2.m2.1c">z\sim 1.2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p7.2.m2.1d">italic_z ∼ 1.2</annotation></semantics></math>. We adopt the <math alttext="b(z)=0.84/D(z)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p7.3.m3.2"><semantics id="S4.SS2.SSS3.p7.3.m3.2a"><mrow id="S4.SS2.SSS3.p7.3.m3.2.3" xref="S4.SS2.SSS3.p7.3.m3.2.3.cmml"><mrow id="S4.SS2.SSS3.p7.3.m3.2.3.2" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.cmml"><mi id="S4.SS2.SSS3.p7.3.m3.2.3.2.2" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.p7.3.m3.2.3.2.1" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p7.3.m3.2.3.2.3.2" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.cmml"><mo id="S4.SS2.SSS3.p7.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.cmml">(</mo><mi id="S4.SS2.SSS3.p7.3.m3.1.1" xref="S4.SS2.SSS3.p7.3.m3.1.1.cmml">z</mi><mo id="S4.SS2.SSS3.p7.3.m3.2.3.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.p7.3.m3.2.3.1" xref="S4.SS2.SSS3.p7.3.m3.2.3.1.cmml">=</mo><mrow id="S4.SS2.SSS3.p7.3.m3.2.3.3" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.cmml"><mrow id="S4.SS2.SSS3.p7.3.m3.2.3.3.2" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2.cmml"><mn id="S4.SS2.SSS3.p7.3.m3.2.3.3.2.2" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2.2.cmml">0.84</mn><mo id="S4.SS2.SSS3.p7.3.m3.2.3.3.2.1" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2.1.cmml">/</mo><mi id="S4.SS2.SSS3.p7.3.m3.2.3.3.2.3" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2.3.cmml">D</mi></mrow><mo id="S4.SS2.SSS3.p7.3.m3.2.3.3.1" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p7.3.m3.2.3.3.3.2" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.cmml"><mo id="S4.SS2.SSS3.p7.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.cmml">(</mo><mi id="S4.SS2.SSS3.p7.3.m3.2.2" xref="S4.SS2.SSS3.p7.3.m3.2.2.cmml">z</mi><mo id="S4.SS2.SSS3.p7.3.m3.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p7.3.m3.2b"><apply id="S4.SS2.SSS3.p7.3.m3.2.3.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3"><eq id="S4.SS2.SSS3.p7.3.m3.2.3.1.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.1"></eq><apply id="S4.SS2.SSS3.p7.3.m3.2.3.2.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.2"><times id="S4.SS2.SSS3.p7.3.m3.2.3.2.1.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.1"></times><ci id="S4.SS2.SSS3.p7.3.m3.2.3.2.2.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.2.2">𝑏</ci><ci id="S4.SS2.SSS3.p7.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p7.3.m3.1.1">𝑧</ci></apply><apply id="S4.SS2.SSS3.p7.3.m3.2.3.3.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.3"><times id="S4.SS2.SSS3.p7.3.m3.2.3.3.1.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.1"></times><apply id="S4.SS2.SSS3.p7.3.m3.2.3.3.2.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2"><divide id="S4.SS2.SSS3.p7.3.m3.2.3.3.2.1.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2.1"></divide><cn id="S4.SS2.SSS3.p7.3.m3.2.3.3.2.2.cmml" type="float" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2.2">0.84</cn><ci id="S4.SS2.SSS3.p7.3.m3.2.3.3.2.3.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.3.3.2.3">𝐷</ci></apply><ci id="S4.SS2.SSS3.p7.3.m3.2.2.cmml" xref="S4.SS2.SSS3.p7.3.m3.2.2">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p7.3.m3.2c">b(z)=0.84/D(z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p7.3.m3.2d">italic_b ( italic_z ) = 0.84 / italic_D ( italic_z )</annotation></semantics></math> halo bias model for the ELG sample. It is worth noting that while the density of ELG is significantly higher than that of DESI, our current target selection is still based on the relatively shallow LegacySurveys data designed for Stage-IV surveys. Such selection is not optimized to isolate the targets at higher redshift, which should be the main focus of MUST. Deeper imaging data will improve that. Also, given the anticipated decrease of spectrograph throughput at <math alttext="\lambda<9000\ \AA" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p7.4.m4.1"><semantics id="S4.SS2.SSS3.p7.4.m4.1a"><mrow id="S4.SS2.SSS3.p7.4.m4.1.1" xref="S4.SS2.SSS3.p7.4.m4.1.1.cmml"><mi id="S4.SS2.SSS3.p7.4.m4.1.1.2" xref="S4.SS2.SSS3.p7.4.m4.1.1.2.cmml">λ</mi><mo id="S4.SS2.SSS3.p7.4.m4.1.1.1" xref="S4.SS2.SSS3.p7.4.m4.1.1.1.cmml"><</mo><mrow id="S4.SS2.SSS3.p7.4.m4.1.1.3" xref="S4.SS2.SSS3.p7.4.m4.1.1.3.cmml"><mn id="S4.SS2.SSS3.p7.4.m4.1.1.3.2" xref="S4.SS2.SSS3.p7.4.m4.1.1.3.2.cmml">9000</mn><mo id="S4.SS2.SSS3.p7.4.m4.1.1.3.1" lspace="0.500em" xref="S4.SS2.SSS3.p7.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.p7.4.m4.1.1.3.3" xref="S4.SS2.SSS3.p7.4.m4.1.1.3.3.cmml">Å</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p7.4.m4.1b"><apply id="S4.SS2.SSS3.p7.4.m4.1.1.cmml" xref="S4.SS2.SSS3.p7.4.m4.1.1"><lt id="S4.SS2.SSS3.p7.4.m4.1.1.1.cmml" xref="S4.SS2.SSS3.p7.4.m4.1.1.1"></lt><ci id="S4.SS2.SSS3.p7.4.m4.1.1.2.cmml" xref="S4.SS2.SSS3.p7.4.m4.1.1.2">𝜆</ci><apply id="S4.SS2.SSS3.p7.4.m4.1.1.3.cmml" xref="S4.SS2.SSS3.p7.4.m4.1.1.3"><times id="S4.SS2.SSS3.p7.4.m4.1.1.3.1.cmml" xref="S4.SS2.SSS3.p7.4.m4.1.1.3.1"></times><cn id="S4.SS2.SSS3.p7.4.m4.1.1.3.2.cmml" type="integer" xref="S4.SS2.SSS3.p7.4.m4.1.1.3.2">9000</cn><ci id="S4.SS2.SSS3.p7.4.m4.1.1.3.3.cmml" xref="S4.SS2.SSS3.p7.4.m4.1.1.3.3">italic-Å</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p7.4.m4.1c">\lambda<9000\ \AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p7.4.m4.1d">italic_λ < 9000 italic_Å</annotation></semantics></math>, the redshift success rate for faint ELG at the higher end of their redshift distribution strongly depends on the EW of the [O II] emission lines. We will search for improved color selections to isolate the strong-[O II] emitters at <math alttext="z>1" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p7.5.m5.1"><semantics id="S4.SS2.SSS3.p7.5.m5.1a"><mrow id="S4.SS2.SSS3.p7.5.m5.1.1" xref="S4.SS2.SSS3.p7.5.m5.1.1.cmml"><mi id="S4.SS2.SSS3.p7.5.m5.1.1.2" xref="S4.SS2.SSS3.p7.5.m5.1.1.2.cmml">z</mi><mo id="S4.SS2.SSS3.p7.5.m5.1.1.1" xref="S4.SS2.SSS3.p7.5.m5.1.1.1.cmml">></mo><mn id="S4.SS2.SSS3.p7.5.m5.1.1.3" xref="S4.SS2.SSS3.p7.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p7.5.m5.1b"><apply id="S4.SS2.SSS3.p7.5.m5.1.1.cmml" xref="S4.SS2.SSS3.p7.5.m5.1.1"><gt id="S4.SS2.SSS3.p7.5.m5.1.1.1.cmml" xref="S4.SS2.SSS3.p7.5.m5.1.1.1"></gt><ci id="S4.SS2.SSS3.p7.5.m5.1.1.2.cmml" xref="S4.SS2.SSS3.p7.5.m5.1.1.2">𝑧</ci><cn id="S4.SS2.SSS3.p7.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.p7.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p7.5.m5.1c">z>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p7.5.m5.1d">italic_z > 1</annotation></semantics></math>. </div> </section> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> 4.3 High-Redshift Tracers</h3> <div class="ltx_para" id="S4.SS3.p1"> Starting from <math alttext="z=2.1" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><mrow id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS3.p1.1.m1.1.1.1" xref="S4.SS3.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.1.m1.1.1.3" xref="S4.SS3.p1.1.m1.1.1.3.cmml">2.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><apply id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1"><eq id="S4.SS3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1.1"></eq><ci id="S4.SS3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2">𝑧</ci><cn id="S4.SS3.p1.1.m1.1.1.3.cmml" type="float" xref="S4.SS3.p1.1.m1.1.1.3">2.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">z=2.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_z = 2.1</annotation></semantics></math>, the Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mi id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><ci id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">italic_α</annotation></semantics></math> emission line at rest-frame <math alttext="1216\AA" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><mrow id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml"><mn id="S4.SS3.p1.3.m3.1.1.2" xref="S4.SS3.p1.3.m3.1.1.2.cmml">1216</mn><mo id="S4.SS3.p1.3.m3.1.1.1" xref="S4.SS3.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.3.m3.1.1.3" xref="S4.SS3.p1.3.m3.1.1.3.cmml">Å</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><apply id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1"><times id="S4.SS3.p1.3.m3.1.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1.1"></times><cn id="S4.SS3.p1.3.m3.1.1.2.cmml" type="integer" xref="S4.SS3.p1.3.m3.1.1.2">1216</cn><ci id="S4.SS3.p1.3.m3.1.1.3.cmml" xref="S4.SS3.p1.3.m3.1.1.3">italic-Å</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">1216\AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">1216 italic_Å</annotation></semantics></math> redshifts into the wavelength coverage at <math alttext="\sim 370" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><mrow id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml"><mi id="S4.SS3.p1.4.m4.1.1.2" xref="S4.SS3.p1.4.m4.1.1.2.cmml"></mi><mo id="S4.SS3.p1.4.m4.1.1.1" xref="S4.SS3.p1.4.m4.1.1.1.cmml">∼</mo><mn id="S4.SS3.p1.4.m4.1.1.3" xref="S4.SS3.p1.4.m4.1.1.3.cmml">370</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><apply id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1"><csymbol cd="latexml" id="S4.SS3.p1.4.m4.1.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.p1.4.m4.1.1.2.cmml" xref="S4.SS3.p1.4.m4.1.1.2">absent</csymbol><cn id="S4.SS3.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.SS3.p1.4.m4.1.1.3">370</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">\sim 370</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">∼ 370</annotation></semantics></math> nm for the spectrograph of MUST and enables our high-redshift target selection for the Stage-V cosmological survey. Along with the Lyman break feature at rest-frame <math alttext="912\AA" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m5.1"><semantics id="S4.SS3.p1.5.m5.1a"><mrow id="S4.SS3.p1.5.m5.1.1" xref="S4.SS3.p1.5.m5.1.1.cmml"><mn id="S4.SS3.p1.5.m5.1.1.2" xref="S4.SS3.p1.5.m5.1.1.2.cmml">912</mn><mo id="S4.SS3.p1.5.m5.1.1.1" xref="S4.SS3.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.5.m5.1.1.3" xref="S4.SS3.p1.5.m5.1.1.3.cmml">Å</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m5.1b"><apply id="S4.SS3.p1.5.m5.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1"><times id="S4.SS3.p1.5.m5.1.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1.1"></times><cn id="S4.SS3.p1.5.m5.1.1.2.cmml" type="integer" xref="S4.SS3.p1.5.m5.1.1.2">912</cn><ci id="S4.SS3.p1.5.m5.1.1.3.cmml" xref="S4.SS3.p1.5.m5.1.1.3">italic-Å</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m5.1c">912\AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m5.1d">912 italic_Å</annotation></semantics></math> and a series of interstellar absorption lines, such as C II 1335 <math alttext="\AA" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m6.1"><semantics id="S4.SS3.p1.6.m6.1a"><mi id="S4.SS3.p1.6.m6.1.1" xref="S4.SS3.p1.6.m6.1.1.cmml">Å</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.6.m6.1b"><ci id="S4.SS3.p1.6.m6.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1">italic-Å</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m6.1c">\AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m6.1d">italic_Å</annotation></semantics></math>, C IV 1548 <math alttext="\AA" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m7.1"><semantics id="S4.SS3.p1.7.m7.1a"><mi id="S4.SS3.p1.7.m7.1.1" xref="S4.SS3.p1.7.m7.1.1.cmml">Å</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m7.1b"><ci id="S4.SS3.p1.7.m7.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1">italic-Å</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.7.m7.1c">\AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.7.m7.1d">italic_Å</annotation></semantics></math>, O I 1302 <math alttext="\AA+" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m8.1"><semantics id="S4.SS3.p1.8.m8.1a"><mrow id="S4.SS3.p1.8.m8.1.1" xref="S4.SS3.p1.8.m8.1.1.cmml"><mi id="S4.SS3.p1.8.m8.1.1.2" xref="S4.SS3.p1.8.m8.1.1.2.cmml">Å</mi><mo id="S4.SS3.p1.8.m8.1.1.3" xref="S4.SS3.p1.8.m8.1.1.3.cmml">+</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m8.1b"><apply id="S4.SS3.p1.8.m8.1.1.cmml" xref="S4.SS3.p1.8.m8.1.1"><csymbol cd="latexml" id="S4.SS3.p1.8.m8.1.1.1.cmml" xref="S4.SS3.p1.8.m8.1.1">limit-from</csymbol><ci id="S4.SS3.p1.8.m8.1.1.2.cmml" xref="S4.SS3.p1.8.m8.1.1.2">italic-Å</ci><plus id="S4.SS3.p1.8.m8.1.1.3.cmml" xref="S4.SS3.p1.8.m8.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m8.1c">\AA+</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m8.1d">italic_Å +</annotation></semantics></math> Si II 1304 <math alttext="\AA" class="ltx_Math" display="inline" id="S4.SS3.p1.9.m9.1"><semantics id="S4.SS3.p1.9.m9.1a"><mi id="S4.SS3.p1.9.m9.1.1" xref="S4.SS3.p1.9.m9.1.1.cmml">Å</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.9.m9.1b"><ci id="S4.SS3.p1.9.m9.1.1.cmml" xref="S4.SS3.p1.9.m9.1.1">italic-Å</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.9.m9.1c">\AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.9.m9.1d">italic_Å</annotation></semantics></math>, and more (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib230" title="">230</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib21" title="">21</a>]</cite>), support the redshift measurements at <math alttext="2.1<z<5.0" class="ltx_Math" display="inline" id="S4.SS3.p1.10.m10.1"><semantics id="S4.SS3.p1.10.m10.1a"><mrow id="S4.SS3.p1.10.m10.1.1" xref="S4.SS3.p1.10.m10.1.1.cmml"><mn id="S4.SS3.p1.10.m10.1.1.2" xref="S4.SS3.p1.10.m10.1.1.2.cmml">2.1</mn><mo id="S4.SS3.p1.10.m10.1.1.3" xref="S4.SS3.p1.10.m10.1.1.3.cmml"><</mo><mi id="S4.SS3.p1.10.m10.1.1.4" xref="S4.SS3.p1.10.m10.1.1.4.cmml">z</mi><mo id="S4.SS3.p1.10.m10.1.1.5" xref="S4.SS3.p1.10.m10.1.1.5.cmml"><</mo><mn id="S4.SS3.p1.10.m10.1.1.6" xref="S4.SS3.p1.10.m10.1.1.6.cmml">5.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.10.m10.1b"><apply id="S4.SS3.p1.10.m10.1.1.cmml" xref="S4.SS3.p1.10.m10.1.1"><and id="S4.SS3.p1.10.m10.1.1a.cmml" xref="S4.SS3.p1.10.m10.1.1"></and><apply id="S4.SS3.p1.10.m10.1.1b.cmml" xref="S4.SS3.p1.10.m10.1.1"><lt id="S4.SS3.p1.10.m10.1.1.3.cmml" xref="S4.SS3.p1.10.m10.1.1.3"></lt><cn id="S4.SS3.p1.10.m10.1.1.2.cmml" type="float" xref="S4.SS3.p1.10.m10.1.1.2">2.1</cn><ci id="S4.SS3.p1.10.m10.1.1.4.cmml" xref="S4.SS3.p1.10.m10.1.1.4">𝑧</ci></apply><apply id="S4.SS3.p1.10.m10.1.1c.cmml" xref="S4.SS3.p1.10.m10.1.1"><lt id="S4.SS3.p1.10.m10.1.1.5.cmml" xref="S4.SS3.p1.10.m10.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.p1.10.m10.1.1.4.cmml" id="S4.SS3.p1.10.m10.1.1d.cmml" xref="S4.SS3.p1.10.m10.1.1"></share><cn id="S4.SS3.p1.10.m10.1.1.6.cmml" type="float" xref="S4.SS3.p1.10.m10.1.1.6">5.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.10.m10.1c">2.1<z<5.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.10.m10.1d">2.1 < italic_z < 5.0</annotation></semantics></math> of MUST. High-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.p1.11.m11.1"><semantics id="S4.SS3.p1.11.m11.1a"><mi id="S4.SS3.p1.11.m11.1.1" xref="S4.SS3.p1.11.m11.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.11.m11.1b"><ci id="S4.SS3.p1.11.m11.1.1.cmml" xref="S4.SS3.p1.11.m11.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.11.m11.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.11.m11.1d">italic_z</annotation></semantics></math> LSS tracers are the bread and butter of a Stage-V cosmological survey like MUST. Until the Stage-IV surveys such as DESI, high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.p1.12.m12.1"><semantics id="S4.SS3.p1.12.m12.1a"><mi id="S4.SS3.p1.12.m12.1.1" xref="S4.SS3.p1.12.m12.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.12.m12.1b"><ci id="S4.SS3.p1.12.m12.1.1.cmml" xref="S4.SS3.p1.12.m12.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.12.m12.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.12.m12.1d">italic_z</annotation></semantics></math> QSOs are the primary LSS tracers at <math alttext="z>2" class="ltx_Math" display="inline" id="S4.SS3.p1.13.m13.1"><semantics id="S4.SS3.p1.13.m13.1a"><mrow id="S4.SS3.p1.13.m13.1.1" xref="S4.SS3.p1.13.m13.1.1.cmml"><mi id="S4.SS3.p1.13.m13.1.1.2" xref="S4.SS3.p1.13.m13.1.1.2.cmml">z</mi><mo id="S4.SS3.p1.13.m13.1.1.1" xref="S4.SS3.p1.13.m13.1.1.1.cmml">></mo><mn id="S4.SS3.p1.13.m13.1.1.3" xref="S4.SS3.p1.13.m13.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.13.m13.1b"><apply id="S4.SS3.p1.13.m13.1.1.cmml" xref="S4.SS3.p1.13.m13.1.1"><gt id="S4.SS3.p1.13.m13.1.1.1.cmml" xref="S4.SS3.p1.13.m13.1.1.1"></gt><ci id="S4.SS3.p1.13.m13.1.1.2.cmml" xref="S4.SS3.p1.13.m13.1.1.2">𝑧</ci><cn id="S4.SS3.p1.13.m13.1.1.3.cmml" type="integer" xref="S4.SS3.p1.13.m13.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.13.m13.1c">z>2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.13.m13.1d">italic_z > 2</annotation></semantics></math> (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib231" title="">231</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib232" title="">232</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib233" title="">233</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib234" title="">234</a>]</cite>). While these accreting super-massive black holes (SMBHs) present us the unique opportunity to explore the inter-galactic medium (IGM) to constrain the nature of dark matter (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib147" title="">147</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib235" title="">235</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib236" title="">236</a>]</cite>), their volume density is typically too low to become the principal LSS tracer in the Stage-V era. </div> <div class="ltx_para" id="S4.SS3.p2"> Right now, the most promising candidates of <math alttext="z>2" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mrow id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml"><mi id="S4.SS3.p2.1.m1.1.1.2" xref="S4.SS3.p2.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS3.p2.1.m1.1.1.1" xref="S4.SS3.p2.1.m1.1.1.1.cmml">></mo><mn id="S4.SS3.p2.1.m1.1.1.3" xref="S4.SS3.p2.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><apply id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1"><gt id="S4.SS3.p2.1.m1.1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1.1"></gt><ci id="S4.SS3.p2.1.m1.1.1.2.cmml" xref="S4.SS3.p2.1.m1.1.1.2">𝑧</ci><cn id="S4.SS3.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.p2.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">z>2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">italic_z > 2</annotation></semantics></math> LSS tracers are the LBG and LAEs (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib237" title="">237</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib211" title="">211</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib208" title="">208</a>]</cite>). Compared to LAEs, selecting LBG based on the broad-band color criteria makes it easier to populate the <math alttext="2<z<5" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><mrow id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml"><mn id="S4.SS3.p2.2.m2.1.1.2" xref="S4.SS3.p2.2.m2.1.1.2.cmml">2</mn><mo id="S4.SS3.p2.2.m2.1.1.3" xref="S4.SS3.p2.2.m2.1.1.3.cmml"><</mo><mi id="S4.SS3.p2.2.m2.1.1.4" xref="S4.SS3.p2.2.m2.1.1.4.cmml">z</mi><mo id="S4.SS3.p2.2.m2.1.1.5" xref="S4.SS3.p2.2.m2.1.1.5.cmml"><</mo><mn id="S4.SS3.p2.2.m2.1.1.6" xref="S4.SS3.p2.2.m2.1.1.6.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><apply id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1"><and id="S4.SS3.p2.2.m2.1.1a.cmml" xref="S4.SS3.p2.2.m2.1.1"></and><apply id="S4.SS3.p2.2.m2.1.1b.cmml" xref="S4.SS3.p2.2.m2.1.1"><lt id="S4.SS3.p2.2.m2.1.1.3.cmml" xref="S4.SS3.p2.2.m2.1.1.3"></lt><cn id="S4.SS3.p2.2.m2.1.1.2.cmml" type="integer" xref="S4.SS3.p2.2.m2.1.1.2">2</cn><ci id="S4.SS3.p2.2.m2.1.1.4.cmml" xref="S4.SS3.p2.2.m2.1.1.4">𝑧</ci></apply><apply id="S4.SS3.p2.2.m2.1.1c.cmml" xref="S4.SS3.p2.2.m2.1.1"><lt id="S4.SS3.p2.2.m2.1.1.5.cmml" xref="S4.SS3.p2.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.p2.2.m2.1.1.4.cmml" id="S4.SS3.p2.2.m2.1.1d.cmml" xref="S4.SS3.p2.2.m2.1.1"></share><cn id="S4.SS3.p2.2.m2.1.1.6.cmml" type="integer" xref="S4.SS3.p2.2.m2.1.1.6">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">2<z<5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">2 < italic_z < 5</annotation></semantics></math> redshift space continuously with sufficient density. Given the available data and previous works, we focus on the LBG populations as the primary LSS tracers for our cosmological forecast in this work. While not included in the Fisher forecast, we briefly discuss the potential for LAEs and QSOs. </div> <div class="ltx_para" id="S4.SS3.p3"> We should note that LBG and LAEs are defined photometrically, not physically, with overlaps between these two populations. In fact, the success rate of redshift measurement for LBG strongly depends on the presence of a prominent Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.1"><semantics id="S4.SS3.p3.1.m1.1a"><mi id="S4.SS3.p3.1.m1.1.1" xref="S4.SS3.p3.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.1b"><ci id="S4.SS3.p3.1.m1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.1d">italic_α</annotation></semantics></math> emission line that often makes them LAEs. </div> <section class="ltx_subsubsection" id="S4.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> 4.3.1 Lyman-Break Galaxies (LBG)</h4> <div class="ltx_para" id="S4.SS3.SSS1.p1"> LBG are galaxies with significant flux decrement, or a “break”, at the rest-frame wavelength shorter than the Lyman limit (<math alttext="911.3\ \AA" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p1.1.m1.1"><semantics id="S4.SS3.SSS1.p1.1.m1.1a"><mrow id="S4.SS3.SSS1.p1.1.m1.1.1" xref="S4.SS3.SSS1.p1.1.m1.1.1.cmml"><mn id="S4.SS3.SSS1.p1.1.m1.1.1.2" xref="S4.SS3.SSS1.p1.1.m1.1.1.2.cmml">911.3</mn><mo id="S4.SS3.SSS1.p1.1.m1.1.1.1" lspace="0.500em" xref="S4.SS3.SSS1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p1.1.m1.1.1.3" xref="S4.SS3.SSS1.p1.1.m1.1.1.3.cmml">Å</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p1.1.m1.1b"><apply id="S4.SS3.SSS1.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p1.1.m1.1.1"><times id="S4.SS3.SSS1.p1.1.m1.1.1.1.cmml" xref="S4.SS3.SSS1.p1.1.m1.1.1.1"></times><cn id="S4.SS3.SSS1.p1.1.m1.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p1.1.m1.1.1.2">911.3</cn><ci id="S4.SS3.SSS1.p1.1.m1.1.1.3.cmml" xref="S4.SS3.SSS1.p1.1.m1.1.1.3">italic-Å</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p1.1.m1.1c">911.3\ \AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p1.1.m1.1d">911.3 italic_Å</annotation></semantics></math>) due to strong internal absorption by neutral Hydrogen. They represent the “normal”, young, star-forming high redshift galaxies (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib238" title="">238</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib239" title="">239</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib240" title="">240</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib241" title="">241</a>]</cite>) with a large enough density to support LSS surveys. In observation, this break often translates into the non-detection in filters bluer than the observed frame of the Lyman limit (or a “dropout”) or a high color value using filters that put the break between them. Starting from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib238" title="">238</a>]</cite>, this method and its further development have helped select millions of LBG as candidates of high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p1.2.m2.1"><semantics id="S4.SS3.SSS1.p1.2.m2.1a"><mi id="S4.SS3.SSS1.p1.2.m2.1.1" xref="S4.SS3.SSS1.p1.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p1.2.m2.1b"><ci id="S4.SS3.SSS1.p1.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p1.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p1.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p1.2.m2.1d">italic_z</annotation></semantics></math> galaxies in <math alttext="2<z<7" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p1.3.m3.1"><semantics id="S4.SS3.SSS1.p1.3.m3.1a"><mrow id="S4.SS3.SSS1.p1.3.m3.1.1" xref="S4.SS3.SSS1.p1.3.m3.1.1.cmml"><mn id="S4.SS3.SSS1.p1.3.m3.1.1.2" xref="S4.SS3.SSS1.p1.3.m3.1.1.2.cmml">2</mn><mo id="S4.SS3.SSS1.p1.3.m3.1.1.3" xref="S4.SS3.SSS1.p1.3.m3.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p1.3.m3.1.1.4" xref="S4.SS3.SSS1.p1.3.m3.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p1.3.m3.1.1.5" xref="S4.SS3.SSS1.p1.3.m3.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p1.3.m3.1.1.6" xref="S4.SS3.SSS1.p1.3.m3.1.1.6.cmml">7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p1.3.m3.1b"><apply id="S4.SS3.SSS1.p1.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1"><and id="S4.SS3.SSS1.p1.3.m3.1.1a.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1"></and><apply id="S4.SS3.SSS1.p1.3.m3.1.1b.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1"><lt id="S4.SS3.SSS1.p1.3.m3.1.1.3.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1.3"></lt><cn id="S4.SS3.SSS1.p1.3.m3.1.1.2.cmml" type="integer" xref="S4.SS3.SSS1.p1.3.m3.1.1.2">2</cn><ci id="S4.SS3.SSS1.p1.3.m3.1.1.4.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p1.3.m3.1.1c.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1"><lt id="S4.SS3.SSS1.p1.3.m3.1.1.5.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p1.3.m3.1.1.4.cmml" id="S4.SS3.SSS1.p1.3.m3.1.1d.cmml" xref="S4.SS3.SSS1.p1.3.m3.1.1"></share><cn id="S4.SS3.SSS1.p1.3.m3.1.1.6.cmml" type="integer" xref="S4.SS3.SSS1.p1.3.m3.1.1.6">7</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p1.3.m3.1c">2<z<7</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p1.3.m3.1d">2 < italic_z < 7</annotation></semantics></math> (e.g., <math alttext="\sim 4.1" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p1.4.m4.1"><semantics id="S4.SS3.SSS1.p1.4.m4.1a"><mrow id="S4.SS3.SSS1.p1.4.m4.1.1" xref="S4.SS3.SSS1.p1.4.m4.1.1.cmml"><mi id="S4.SS3.SSS1.p1.4.m4.1.1.2" xref="S4.SS3.SSS1.p1.4.m4.1.1.2.cmml"></mi><mo id="S4.SS3.SSS1.p1.4.m4.1.1.1" xref="S4.SS3.SSS1.p1.4.m4.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p1.4.m4.1.1.3" xref="S4.SS3.SSS1.p1.4.m4.1.1.3.cmml">4.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p1.4.m4.1b"><apply id="S4.SS3.SSS1.p1.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p1.4.m4.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p1.4.m4.1.1.1.cmml" xref="S4.SS3.SSS1.p1.4.m4.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.SSS1.p1.4.m4.1.1.2.cmml" xref="S4.SS3.SSS1.p1.4.m4.1.1.2">absent</csymbol><cn id="S4.SS3.SSS1.p1.4.m4.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p1.4.m4.1.1.3">4.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p1.4.m4.1c">\sim 4.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p1.4.m4.1d">∼ 4.1</annotation></semantics></math> million from the Great Optically Luminous Dropout Research Using Subaru HSC, or GOLDRUSH project <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib242" title="">242</a>]</cite>). This mature approach is the foundation of high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p1.5.m5.1"><semantics id="S4.SS3.SSS1.p1.5.m5.1a"><mi id="S4.SS3.SSS1.p1.5.m5.1.1" xref="S4.SS3.SSS1.p1.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p1.5.m5.1b"><ci id="S4.SS3.SSS1.p1.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p1.5.m5.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p1.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p1.5.m5.1d">italic_z</annotation></semantics></math> LSS tracer selection using deep broad-band imaging surveys for Stage-IV and Stage-V surveys. </div> <div class="ltx_para" id="S4.SS3.SSS1.p2"> Motivated by the target selection for the proposed DESI-II, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib21" title="">21</a>]</cite> validated the LBG selections based on a series of dropout criteria from the HSC SSP and the CLAUDS data using dedicated DESI campaigns and confirmed a <math alttext="\sim 620" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.1.m1.1"><semantics id="S4.SS3.SSS1.p2.1.m1.1a"><mrow id="S4.SS3.SSS1.p2.1.m1.1.1" xref="S4.SS3.SSS1.p2.1.m1.1.1.cmml"><mi id="S4.SS3.SSS1.p2.1.m1.1.1.2" xref="S4.SS3.SSS1.p2.1.m1.1.1.2.cmml"></mi><mo id="S4.SS3.SSS1.p2.1.m1.1.1.1" xref="S4.SS3.SSS1.p2.1.m1.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p2.1.m1.1.1.3" xref="S4.SS3.SSS1.p2.1.m1.1.1.3.cmml">620</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.1.m1.1b"><apply id="S4.SS3.SSS1.p2.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p2.1.m1.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p2.1.m1.1.1.1.cmml" xref="S4.SS3.SSS1.p2.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.SSS1.p2.1.m1.1.1.2.cmml" xref="S4.SS3.SSS1.p2.1.m1.1.1.2">absent</csymbol><cn id="S4.SS3.SSS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p2.1.m1.1.1.3">620</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.1.m1.1c">\sim 620</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.1.m1.1d">∼ 620</annotation></semantics></math> deg-2 <math alttext="r<24.2" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.3.m3.1"><semantics id="S4.SS3.SSS1.p2.3.m3.1a"><mrow id="S4.SS3.SSS1.p2.3.m3.1.1" xref="S4.SS3.SSS1.p2.3.m3.1.1.cmml"><mi id="S4.SS3.SSS1.p2.3.m3.1.1.2" xref="S4.SS3.SSS1.p2.3.m3.1.1.2.cmml">r</mi><mo id="S4.SS3.SSS1.p2.3.m3.1.1.1" xref="S4.SS3.SSS1.p2.3.m3.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p2.3.m3.1.1.3" xref="S4.SS3.SSS1.p2.3.m3.1.1.3.cmml">24.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.3.m3.1b"><apply id="S4.SS3.SSS1.p2.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p2.3.m3.1.1"><lt id="S4.SS3.SSS1.p2.3.m3.1.1.1.cmml" xref="S4.SS3.SSS1.p2.3.m3.1.1.1"></lt><ci id="S4.SS3.SSS1.p2.3.m3.1.1.2.cmml" xref="S4.SS3.SSS1.p2.3.m3.1.1.2">𝑟</ci><cn id="S4.SS3.SSS1.p2.3.m3.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p2.3.m3.1.1.3">24.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.3.m3.1c">r<24.2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.3.m3.1d">italic_r < 24.2</annotation></semantics></math> mag LBG density at <math alttext="2.3<z<3.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.4.m4.1"><semantics id="S4.SS3.SSS1.p2.4.m4.1a"><mrow id="S4.SS3.SSS1.p2.4.m4.1.1" xref="S4.SS3.SSS1.p2.4.m4.1.1.cmml"><mn id="S4.SS3.SSS1.p2.4.m4.1.1.2" xref="S4.SS3.SSS1.p2.4.m4.1.1.2.cmml">2.3</mn><mo id="S4.SS3.SSS1.p2.4.m4.1.1.3" xref="S4.SS3.SSS1.p2.4.m4.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p2.4.m4.1.1.4" xref="S4.SS3.SSS1.p2.4.m4.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p2.4.m4.1.1.5" xref="S4.SS3.SSS1.p2.4.m4.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p2.4.m4.1.1.6" xref="S4.SS3.SSS1.p2.4.m4.1.1.6.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.4.m4.1b"><apply id="S4.SS3.SSS1.p2.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1"><and id="S4.SS3.SSS1.p2.4.m4.1.1a.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1"></and><apply id="S4.SS3.SSS1.p2.4.m4.1.1b.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1"><lt id="S4.SS3.SSS1.p2.4.m4.1.1.3.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1.3"></lt><cn id="S4.SS3.SSS1.p2.4.m4.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p2.4.m4.1.1.2">2.3</cn><ci id="S4.SS3.SSS1.p2.4.m4.1.1.4.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p2.4.m4.1.1c.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1"><lt id="S4.SS3.SSS1.p2.4.m4.1.1.5.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p2.4.m4.1.1.4.cmml" id="S4.SS3.SSS1.p2.4.m4.1.1d.cmml" xref="S4.SS3.SSS1.p2.4.m4.1.1"></share><cn id="S4.SS3.SSS1.p2.4.m4.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p2.4.m4.1.1.6">3.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.4.m4.1c">2.3<z<3.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.4.m4.1d">2.3 < italic_z < 3.5</annotation></semantics></math> or a <math alttext="\sim 470" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.5.m5.1"><semantics id="S4.SS3.SSS1.p2.5.m5.1a"><mrow id="S4.SS3.SSS1.p2.5.m5.1.1" xref="S4.SS3.SSS1.p2.5.m5.1.1.cmml"><mi id="S4.SS3.SSS1.p2.5.m5.1.1.2" xref="S4.SS3.SSS1.p2.5.m5.1.1.2.cmml"></mi><mo id="S4.SS3.SSS1.p2.5.m5.1.1.1" xref="S4.SS3.SSS1.p2.5.m5.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p2.5.m5.1.1.3" xref="S4.SS3.SSS1.p2.5.m5.1.1.3.cmml">470</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.5.m5.1b"><apply id="S4.SS3.SSS1.p2.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p2.5.m5.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p2.5.m5.1.1.1.cmml" xref="S4.SS3.SSS1.p2.5.m5.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.SSS1.p2.5.m5.1.1.2.cmml" xref="S4.SS3.SSS1.p2.5.m5.1.1.2">absent</csymbol><cn id="S4.SS3.SSS1.p2.5.m5.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p2.5.m5.1.1.3">470</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.5.m5.1c">\sim 470</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.5.m5.1d">∼ 470</annotation></semantics></math> deg-2 <math alttext="r<24.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.7.m7.1"><semantics id="S4.SS3.SSS1.p2.7.m7.1a"><mrow id="S4.SS3.SSS1.p2.7.m7.1.1" xref="S4.SS3.SSS1.p2.7.m7.1.1.cmml"><mi id="S4.SS3.SSS1.p2.7.m7.1.1.2" xref="S4.SS3.SSS1.p2.7.m7.1.1.2.cmml">r</mi><mo id="S4.SS3.SSS1.p2.7.m7.1.1.1" xref="S4.SS3.SSS1.p2.7.m7.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p2.7.m7.1.1.3" xref="S4.SS3.SSS1.p2.7.m7.1.1.3.cmml">24.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.7.m7.1b"><apply id="S4.SS3.SSS1.p2.7.m7.1.1.cmml" xref="S4.SS3.SSS1.p2.7.m7.1.1"><lt id="S4.SS3.SSS1.p2.7.m7.1.1.1.cmml" xref="S4.SS3.SSS1.p2.7.m7.1.1.1"></lt><ci id="S4.SS3.SSS1.p2.7.m7.1.1.2.cmml" xref="S4.SS3.SSS1.p2.7.m7.1.1.2">𝑟</ci><cn id="S4.SS3.SSS1.p2.7.m7.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p2.7.m7.1.1.3">24.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.7.m7.1c">r<24.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.7.m7.1d">italic_r < 24.5</annotation></semantics></math> mag LBG density at <math alttext="2.8<z<3.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.8.m8.1"><semantics id="S4.SS3.SSS1.p2.8.m8.1a"><mrow id="S4.SS3.SSS1.p2.8.m8.1.1" xref="S4.SS3.SSS1.p2.8.m8.1.1.cmml"><mn id="S4.SS3.SSS1.p2.8.m8.1.1.2" xref="S4.SS3.SSS1.p2.8.m8.1.1.2.cmml">2.8</mn><mo id="S4.SS3.SSS1.p2.8.m8.1.1.3" xref="S4.SS3.SSS1.p2.8.m8.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p2.8.m8.1.1.4" xref="S4.SS3.SSS1.p2.8.m8.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p2.8.m8.1.1.5" xref="S4.SS3.SSS1.p2.8.m8.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p2.8.m8.1.1.6" xref="S4.SS3.SSS1.p2.8.m8.1.1.6.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.8.m8.1b"><apply id="S4.SS3.SSS1.p2.8.m8.1.1.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1"><and id="S4.SS3.SSS1.p2.8.m8.1.1a.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1"></and><apply id="S4.SS3.SSS1.p2.8.m8.1.1b.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1"><lt id="S4.SS3.SSS1.p2.8.m8.1.1.3.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1.3"></lt><cn id="S4.SS3.SSS1.p2.8.m8.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p2.8.m8.1.1.2">2.8</cn><ci id="S4.SS3.SSS1.p2.8.m8.1.1.4.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p2.8.m8.1.1c.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1"><lt id="S4.SS3.SSS1.p2.8.m8.1.1.5.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p2.8.m8.1.1.4.cmml" id="S4.SS3.SSS1.p2.8.m8.1.1d.cmml" xref="S4.SS3.SSS1.p2.8.m8.1.1"></share><cn id="S4.SS3.SSS1.p2.8.m8.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p2.8.m8.1.1.6">3.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.8.m8.1c">2.8<z<3.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.8.m8.1d">2.8 < italic_z < 3.5</annotation></semantics></math>. Also aimed at the fixed 1,100 deg-2 target density for DESI-II, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib22" title="">22</a>]</cite> applies a Random Forest selection to the HSC<math alttext="+" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.10.m10.1"><semantics id="S4.SS3.SSS1.p2.10.m10.1a"><mo id="S4.SS3.SSS1.p2.10.m10.1.1" xref="S4.SS3.SSS1.p2.10.m10.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.10.m10.1b"><plus id="S4.SS3.SSS1.p2.10.m10.1.1.cmml" xref="S4.SS3.SSS1.p2.10.m10.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.10.m10.1c">+</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.10.m10.1d">+</annotation></semantics></math>CLAUDS data and spectroscopically confirmed a 493 deg-2 <math alttext="z>2.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p2.12.m12.1"><semantics id="S4.SS3.SSS1.p2.12.m12.1a"><mrow id="S4.SS3.SSS1.p2.12.m12.1.1" xref="S4.SS3.SSS1.p2.12.m12.1.1.cmml"><mi id="S4.SS3.SSS1.p2.12.m12.1.1.2" xref="S4.SS3.SSS1.p2.12.m12.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p2.12.m12.1.1.1" xref="S4.SS3.SSS1.p2.12.m12.1.1.1.cmml">></mo><mn id="S4.SS3.SSS1.p2.12.m12.1.1.3" xref="S4.SS3.SSS1.p2.12.m12.1.1.3.cmml">2.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p2.12.m12.1b"><apply id="S4.SS3.SSS1.p2.12.m12.1.1.cmml" xref="S4.SS3.SSS1.p2.12.m12.1.1"><gt id="S4.SS3.SSS1.p2.12.m12.1.1.1.cmml" xref="S4.SS3.SSS1.p2.12.m12.1.1.1"></gt><ci id="S4.SS3.SSS1.p2.12.m12.1.1.2.cmml" xref="S4.SS3.SSS1.p2.12.m12.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p2.12.m12.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p2.12.m12.1.1.3">2.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p2.12.m12.1c">z>2.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p2.12.m12.1d">italic_z > 2.5</annotation></semantics></math> under the imaging depth afforded by a UNIONS-like survey. </div> <div class="ltx_para" id="S4.SS3.SSS1.p3"> In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib237" title="">237</a>]</cite>, the authors considered the BX color selection in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib243" title="">243</a>]</cite> centered at <math alttext="z\sim 2.20" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.1.m1.1"><semantics id="S4.SS3.SSS1.p3.1.m1.1a"><mrow id="S4.SS3.SSS1.p3.1.m1.1.1" xref="S4.SS3.SSS1.p3.1.m1.1.1.cmml"><mi id="S4.SS3.SSS1.p3.1.m1.1.1.2" xref="S4.SS3.SSS1.p3.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p3.1.m1.1.1.1" xref="S4.SS3.SSS1.p3.1.m1.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p3.1.m1.1.1.3" xref="S4.SS3.SSS1.p3.1.m1.1.1.3.cmml">2.20</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.1.m1.1b"><apply id="S4.SS3.SSS1.p3.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p3.1.m1.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p3.1.m1.1.1.1.cmml" xref="S4.SS3.SSS1.p3.1.m1.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS1.p3.1.m1.1.1.2.cmml" xref="S4.SS3.SSS1.p3.1.m1.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p3.1.m1.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p3.1.m1.1.1.3">2.20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.1.m1.1c">z\sim 2.20</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.1.m1.1d">italic_z ∼ 2.20</annotation></semantics></math>, the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.2.m2.1"><semantics id="S4.SS3.SSS1.p3.2.m2.1a"><mi id="S4.SS3.SSS1.p3.2.m2.1.1" xref="S4.SS3.SSS1.p3.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.2.m2.1b"><ci id="S4.SS3.SSS1.p3.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p3.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.2.m2.1d">italic_u</annotation></semantics></math>-dropout selection based on the CFHTLS-Archive-Research-Survey (CARS; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib244" title="">244</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib245" title="">245</a>]</cite>) centered at <math alttext="z\sim 2.96" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.3.m3.1"><semantics id="S4.SS3.SSS1.p3.3.m3.1a"><mrow id="S4.SS3.SSS1.p3.3.m3.1.1" xref="S4.SS3.SSS1.p3.3.m3.1.1.cmml"><mi id="S4.SS3.SSS1.p3.3.m3.1.1.2" xref="S4.SS3.SSS1.p3.3.m3.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p3.3.m3.1.1.1" xref="S4.SS3.SSS1.p3.3.m3.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p3.3.m3.1.1.3" xref="S4.SS3.SSS1.p3.3.m3.1.1.3.cmml">2.96</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.3.m3.1b"><apply id="S4.SS3.SSS1.p3.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p3.3.m3.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p3.3.m3.1.1.1.cmml" xref="S4.SS3.SSS1.p3.3.m3.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS1.p3.3.m3.1.1.2.cmml" xref="S4.SS3.SSS1.p3.3.m3.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p3.3.m3.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p3.3.m3.1.1.3">2.96</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.3.m3.1c">z\sim 2.96</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.3.m3.1d">italic_z ∼ 2.96</annotation></semantics></math>, and the <math alttext="g" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.4.m4.1"><semantics id="S4.SS3.SSS1.p3.4.m4.1a"><mi id="S4.SS3.SSS1.p3.4.m4.1.1" xref="S4.SS3.SSS1.p3.4.m4.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.4.m4.1b"><ci id="S4.SS3.SSS1.p3.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p3.4.m4.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.4.m4.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.4.m4.1d">italic_g</annotation></semantics></math>- & <math alttext="r" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.5.m5.1"><semantics id="S4.SS3.SSS1.p3.5.m5.1a"><mi id="S4.SS3.SSS1.p3.5.m5.1.1" xref="S4.SS3.SSS1.p3.5.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.5.m5.1b"><ci id="S4.SS3.SSS1.p3.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p3.5.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.5.m5.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.5.m5.1d">italic_r</annotation></semantics></math>-dropout selection based on GOLDRUSH centered at <math alttext="z\sim 3.8" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.6.m6.1"><semantics id="S4.SS3.SSS1.p3.6.m6.1a"><mrow id="S4.SS3.SSS1.p3.6.m6.1.1" xref="S4.SS3.SSS1.p3.6.m6.1.1.cmml"><mi id="S4.SS3.SSS1.p3.6.m6.1.1.2" xref="S4.SS3.SSS1.p3.6.m6.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p3.6.m6.1.1.1" xref="S4.SS3.SSS1.p3.6.m6.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p3.6.m6.1.1.3" xref="S4.SS3.SSS1.p3.6.m6.1.1.3.cmml">3.8</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.6.m6.1b"><apply id="S4.SS3.SSS1.p3.6.m6.1.1.cmml" xref="S4.SS3.SSS1.p3.6.m6.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p3.6.m6.1.1.1.cmml" xref="S4.SS3.SSS1.p3.6.m6.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS1.p3.6.m6.1.1.2.cmml" xref="S4.SS3.SSS1.p3.6.m6.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p3.6.m6.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p3.6.m6.1.1.3">3.8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.6.m6.1c">z\sim 3.8</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.6.m6.1d">italic_z ∼ 3.8</annotation></semantics></math> & 4.9. Based on LSST-Y10’s imaging depths, the authors concluded that it is practical to spectroscopically confirm 2,000 <math alttext="R<24.0" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.7.m7.1"><semantics id="S4.SS3.SSS1.p3.7.m7.1a"><mrow id="S4.SS3.SSS1.p3.7.m7.1.1" xref="S4.SS3.SSS1.p3.7.m7.1.1.cmml"><mi id="S4.SS3.SSS1.p3.7.m7.1.1.2" xref="S4.SS3.SSS1.p3.7.m7.1.1.2.cmml">R</mi><mo id="S4.SS3.SSS1.p3.7.m7.1.1.1" xref="S4.SS3.SSS1.p3.7.m7.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p3.7.m7.1.1.3" xref="S4.SS3.SSS1.p3.7.m7.1.1.3.cmml">24.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.7.m7.1b"><apply id="S4.SS3.SSS1.p3.7.m7.1.1.cmml" xref="S4.SS3.SSS1.p3.7.m7.1.1"><lt id="S4.SS3.SSS1.p3.7.m7.1.1.1.cmml" xref="S4.SS3.SSS1.p3.7.m7.1.1.1"></lt><ci id="S4.SS3.SSS1.p3.7.m7.1.1.2.cmml" xref="S4.SS3.SSS1.p3.7.m7.1.1.2">𝑅</ci><cn id="S4.SS3.SSS1.p3.7.m7.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p3.7.m7.1.1.3">24.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.7.m7.1c">R<24.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.7.m7.1d">italic_R < 24.0</annotation></semantics></math> mag BX galaxies, 500 <math alttext="i<24.0" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.8.m8.1"><semantics id="S4.SS3.SSS1.p3.8.m8.1a"><mrow id="S4.SS3.SSS1.p3.8.m8.1.1" xref="S4.SS3.SSS1.p3.8.m8.1.1.cmml"><mi id="S4.SS3.SSS1.p3.8.m8.1.1.2" xref="S4.SS3.SSS1.p3.8.m8.1.1.2.cmml">i</mi><mo id="S4.SS3.SSS1.p3.8.m8.1.1.1" xref="S4.SS3.SSS1.p3.8.m8.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p3.8.m8.1.1.3" xref="S4.SS3.SSS1.p3.8.m8.1.1.3.cmml">24.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.8.m8.1b"><apply id="S4.SS3.SSS1.p3.8.m8.1.1.cmml" xref="S4.SS3.SSS1.p3.8.m8.1.1"><lt id="S4.SS3.SSS1.p3.8.m8.1.1.1.cmml" xref="S4.SS3.SSS1.p3.8.m8.1.1.1"></lt><ci id="S4.SS3.SSS1.p3.8.m8.1.1.2.cmml" xref="S4.SS3.SSS1.p3.8.m8.1.1.2">𝑖</ci><cn id="S4.SS3.SSS1.p3.8.m8.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p3.8.m8.1.1.3">24.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.8.m8.1c">i<24.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.8.m8.1d">italic_i < 24.0</annotation></semantics></math> mag <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.9.m9.1"><semantics id="S4.SS3.SSS1.p3.9.m9.1a"><mi id="S4.SS3.SSS1.p3.9.m9.1.1" xref="S4.SS3.SSS1.p3.9.m9.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.9.m9.1b"><ci id="S4.SS3.SSS1.p3.9.m9.1.1.cmml" xref="S4.SS3.SSS1.p3.9.m9.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.9.m9.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.9.m9.1d">italic_u</annotation></semantics></math>-dropouts, 330 <math alttext="i<25.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.10.m10.1"><semantics id="S4.SS3.SSS1.p3.10.m10.1a"><mrow id="S4.SS3.SSS1.p3.10.m10.1.1" xref="S4.SS3.SSS1.p3.10.m10.1.1.cmml"><mi id="S4.SS3.SSS1.p3.10.m10.1.1.2" xref="S4.SS3.SSS1.p3.10.m10.1.1.2.cmml">i</mi><mo id="S4.SS3.SSS1.p3.10.m10.1.1.1" xref="S4.SS3.SSS1.p3.10.m10.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p3.10.m10.1.1.3" xref="S4.SS3.SSS1.p3.10.m10.1.1.3.cmml">25.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.10.m10.1b"><apply id="S4.SS3.SSS1.p3.10.m10.1.1.cmml" xref="S4.SS3.SSS1.p3.10.m10.1.1"><lt id="S4.SS3.SSS1.p3.10.m10.1.1.1.cmml" xref="S4.SS3.SSS1.p3.10.m10.1.1.1"></lt><ci id="S4.SS3.SSS1.p3.10.m10.1.1.2.cmml" xref="S4.SS3.SSS1.p3.10.m10.1.1.2">𝑖</ci><cn id="S4.SS3.SSS1.p3.10.m10.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p3.10.m10.1.1.3">25.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.10.m10.1c">i<25.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.10.m10.1d">italic_i < 25.5</annotation></semantics></math> mag <math alttext="g" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.11.m11.1"><semantics id="S4.SS3.SSS1.p3.11.m11.1a"><mi id="S4.SS3.SSS1.p3.11.m11.1.1" xref="S4.SS3.SSS1.p3.11.m11.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.11.m11.1b"><ci id="S4.SS3.SSS1.p3.11.m11.1.1.cmml" xref="S4.SS3.SSS1.p3.11.m11.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.11.m11.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.11.m11.1d">italic_g</annotation></semantics></math>-dropouts, and 100 <math alttext="z<25.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.12.m12.1"><semantics id="S4.SS3.SSS1.p3.12.m12.1a"><mrow id="S4.SS3.SSS1.p3.12.m12.1.1" xref="S4.SS3.SSS1.p3.12.m12.1.1.cmml"><mi id="S4.SS3.SSS1.p3.12.m12.1.1.2" xref="S4.SS3.SSS1.p3.12.m12.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p3.12.m12.1.1.1" xref="S4.SS3.SSS1.p3.12.m12.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p3.12.m12.1.1.3" xref="S4.SS3.SSS1.p3.12.m12.1.1.3.cmml">25.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.12.m12.1b"><apply id="S4.SS3.SSS1.p3.12.m12.1.1.cmml" xref="S4.SS3.SSS1.p3.12.m12.1.1"><lt id="S4.SS3.SSS1.p3.12.m12.1.1.1.cmml" xref="S4.SS3.SSS1.p3.12.m12.1.1.1"></lt><ci id="S4.SS3.SSS1.p3.12.m12.1.1.2.cmml" xref="S4.SS3.SSS1.p3.12.m12.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p3.12.m12.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p3.12.m12.1.1.3">25.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.12.m12.1c">z<25.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.12.m12.1d">italic_z < 25.5</annotation></semantics></math> mag <math alttext="r" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.13.m13.1"><semantics id="S4.SS3.SSS1.p3.13.m13.1a"><mi id="S4.SS3.SSS1.p3.13.m13.1.1" xref="S4.SS3.SSS1.p3.13.m13.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.13.m13.1b"><ci id="S4.SS3.SSS1.p3.13.m13.1.1.cmml" xref="S4.SS3.SSS1.p3.13.m13.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.13.m13.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.13.m13.1d">italic_r</annotation></semantics></math>-dropouts per square degree. Altogether, this supports a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.14.m14.1"><semantics id="S4.SS3.SSS1.p3.14.m14.1a"><mo id="S4.SS3.SSS1.p3.14.m14.1.1" xref="S4.SS3.SSS1.p3.14.m14.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.14.m14.1b"><csymbol cd="latexml" id="S4.SS3.SSS1.p3.14.m14.1.1.cmml" xref="S4.SS3.SSS1.p3.14.m14.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.14.m14.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.14.m14.1d">∼</annotation></semantics></math> 3,000 deg-2 confirmed high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.16.m16.1"><semantics id="S4.SS3.SSS1.p3.16.m16.1a"><mi id="S4.SS3.SSS1.p3.16.m16.1.1" xref="S4.SS3.SSS1.p3.16.m16.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.16.m16.1b"><ci id="S4.SS3.SSS1.p3.16.m16.1.1.cmml" xref="S4.SS3.SSS1.p3.16.m16.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.16.m16.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.16.m16.1d">italic_z</annotation></semantics></math> tracers that match the multiplex capability of MUST nicely. Interestingly, the authors point out that the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.17.m17.1"><semantics id="S4.SS3.SSS1.p3.17.m17.1a"><mi id="S4.SS3.SSS1.p3.17.m17.1.1" xref="S4.SS3.SSS1.p3.17.m17.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.17.m17.1b"><ci id="S4.SS3.SSS1.p3.17.m17.1.1.cmml" xref="S4.SS3.SSS1.p3.17.m17.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.17.m17.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.17.m17.1d">italic_u</annotation></semantics></math>-band imaging depth limits the photometric selection of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.18.m18.1"><semantics id="S4.SS3.SSS1.p3.18.m18.1a"><mi id="S4.SS3.SSS1.p3.18.m18.1.1" xref="S4.SS3.SSS1.p3.18.m18.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.18.m18.1b"><ci id="S4.SS3.SSS1.p3.18.m18.1.1.cmml" xref="S4.SS3.SSS1.p3.18.m18.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.18.m18.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.18.m18.1d">italic_u</annotation></semantics></math>-dropout. This assumes that one needs an actual detection in the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.19.m19.1"><semantics id="S4.SS3.SSS1.p3.19.m19.1a"><mi id="S4.SS3.SSS1.p3.19.m19.1.1" xref="S4.SS3.SSS1.p3.19.m19.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.19.m19.1b"><ci id="S4.SS3.SSS1.p3.19.m19.1.1.cmml" xref="S4.SS3.SSS1.p3.19.m19.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.19.m19.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.19.m19.1d">italic_u</annotation></semantics></math>-band (or the band on the bluer side of the observed frame Lyman limit) to estimate the <math alttext="u-g" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.20.m20.1"><semantics id="S4.SS3.SSS1.p3.20.m20.1a"><mrow id="S4.SS3.SSS1.p3.20.m20.1.1" xref="S4.SS3.SSS1.p3.20.m20.1.1.cmml"><mi id="S4.SS3.SSS1.p3.20.m20.1.1.2" xref="S4.SS3.SSS1.p3.20.m20.1.1.2.cmml">u</mi><mo id="S4.SS3.SSS1.p3.20.m20.1.1.1" xref="S4.SS3.SSS1.p3.20.m20.1.1.1.cmml">−</mo><mi id="S4.SS3.SSS1.p3.20.m20.1.1.3" xref="S4.SS3.SSS1.p3.20.m20.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.20.m20.1b"><apply id="S4.SS3.SSS1.p3.20.m20.1.1.cmml" xref="S4.SS3.SSS1.p3.20.m20.1.1"><minus id="S4.SS3.SSS1.p3.20.m20.1.1.1.cmml" xref="S4.SS3.SSS1.p3.20.m20.1.1.1"></minus><ci id="S4.SS3.SSS1.p3.20.m20.1.1.2.cmml" xref="S4.SS3.SSS1.p3.20.m20.1.1.2">𝑢</ci><ci id="S4.SS3.SSS1.p3.20.m20.1.1.3.cmml" xref="S4.SS3.SSS1.p3.20.m20.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.20.m20.1c">u-g</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.20.m20.1d">italic_u - italic_g</annotation></semantics></math> color to identify the break reliably, which is different from the dropout selections used to study the LBG populations (e.g., their luminosity function). It is worth investigating whether this is required to select LSS tracers with straightforward systematics. This is particularly important for MUST as heterogeneous imaging datasets will contribute to our target selection. In this work, we still assume that detection is optional in the band <math alttext="X" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.21.m21.1"><semantics id="S4.SS3.SSS1.p3.21.m21.1a"><mi id="S4.SS3.SSS1.p3.21.m21.1.1" xref="S4.SS3.SSS1.p3.21.m21.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.21.m21.1b"><ci id="S4.SS3.SSS1.p3.21.m21.1.1.cmml" xref="S4.SS3.SSS1.p3.21.m21.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.21.m21.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.21.m21.1d">italic_X</annotation></semantics></math> to select the <math alttext="X" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p3.22.m22.1"><semantics id="S4.SS3.SSS1.p3.22.m22.1a"><mi id="S4.SS3.SSS1.p3.22.m22.1.1" xref="S4.SS3.SSS1.p3.22.m22.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p3.22.m22.1b"><ci id="S4.SS3.SSS1.p3.22.m22.1.1.cmml" xref="S4.SS3.SSS1.p3.22.m22.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p3.22.m22.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p3.22.m22.1d">italic_X</annotation></semantics></math>-dropout populations. </div> <div class="ltx_para" id="S4.SS3.SSS1.p4"> In the WST Science White Paper <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib42" title="">42</a>]</cite>, the authors estimated the relations between the surface densities of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.1.m1.1"><semantics id="S4.SS3.SSS1.p4.1.m1.1a"><mi id="S4.SS3.SSS1.p4.1.m1.1.1" xref="S4.SS3.SSS1.p4.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.1.m1.1b"><ci id="S4.SS3.SSS1.p4.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p4.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.1.m1.1d">italic_u</annotation></semantics></math>-, <math alttext="g" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.2.m2.1"><semantics id="S4.SS3.SSS1.p4.2.m2.1a"><mi id="S4.SS3.SSS1.p4.2.m2.1.1" xref="S4.SS3.SSS1.p4.2.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.2.m2.1b"><ci id="S4.SS3.SSS1.p4.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p4.2.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.2.m2.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.2.m2.1d">italic_g</annotation></semantics></math>-, and <math alttext="r" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.3.m3.1"><semantics id="S4.SS3.SSS1.p4.3.m3.1a"><mi id="S4.SS3.SSS1.p4.3.m3.1.1" xref="S4.SS3.SSS1.p4.3.m3.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.3.m3.1b"><ci id="S4.SS3.SSS1.p4.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p4.3.m3.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.3.m3.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.3.m3.1d">italic_r</annotation></semantics></math>-dropouts down to different magnitude limits (see Figure 65): at <math alttext="r_{\rm lim}<24.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.4.m4.1"><semantics id="S4.SS3.SSS1.p4.4.m4.1a"><mrow id="S4.SS3.SSS1.p4.4.m4.1.1" xref="S4.SS3.SSS1.p4.4.m4.1.1.cmml"><msub id="S4.SS3.SSS1.p4.4.m4.1.1.2" xref="S4.SS3.SSS1.p4.4.m4.1.1.2.cmml"><mi id="S4.SS3.SSS1.p4.4.m4.1.1.2.2" xref="S4.SS3.SSS1.p4.4.m4.1.1.2.2.cmml">r</mi><mi id="S4.SS3.SSS1.p4.4.m4.1.1.2.3" xref="S4.SS3.SSS1.p4.4.m4.1.1.2.3.cmml">lim</mi></msub><mo id="S4.SS3.SSS1.p4.4.m4.1.1.1" xref="S4.SS3.SSS1.p4.4.m4.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p4.4.m4.1.1.3" xref="S4.SS3.SSS1.p4.4.m4.1.1.3.cmml">24.5</mn></mrow><annotation-xml 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<math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.5.m5.1"><semantics id="S4.SS3.SSS1.p4.5.m5.1a"><mi id="S4.SS3.SSS1.p4.5.m5.1.1" xref="S4.SS3.SSS1.p4.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.5.m5.1b"><ci id="S4.SS3.SSS1.p4.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p4.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.5.m5.1d">italic_u</annotation></semantics></math>-dropout should be within 1,500 to 2,000 deg-2 for MUST. And, at <math alttext="i_{\rm lim}<24.6" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.7.m7.1"><semantics id="S4.SS3.SSS1.p4.7.m7.1a"><mrow id="S4.SS3.SSS1.p4.7.m7.1.1" xref="S4.SS3.SSS1.p4.7.m7.1.1.cmml"><msub id="S4.SS3.SSS1.p4.7.m7.1.1.2" xref="S4.SS3.SSS1.p4.7.m7.1.1.2.cmml"><mi id="S4.SS3.SSS1.p4.7.m7.1.1.2.2" xref="S4.SS3.SSS1.p4.7.m7.1.1.2.2.cmml">i</mi><mi id="S4.SS3.SSS1.p4.7.m7.1.1.2.3" xref="S4.SS3.SSS1.p4.7.m7.1.1.2.3.cmml">lim</mi></msub><mo id="S4.SS3.SSS1.p4.7.m7.1.1.1" xref="S4.SS3.SSS1.p4.7.m7.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p4.7.m7.1.1.3" xref="S4.SS3.SSS1.p4.7.m7.1.1.3.cmml">24.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.7.m7.1b"><apply id="S4.SS3.SSS1.p4.7.m7.1.1.cmml" xref="S4.SS3.SSS1.p4.7.m7.1.1"><lt id="S4.SS3.SSS1.p4.7.m7.1.1.1.cmml" xref="S4.SS3.SSS1.p4.7.m7.1.1.1"></lt><apply id="S4.SS3.SSS1.p4.7.m7.1.1.2.cmml" xref="S4.SS3.SSS1.p4.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS1.p4.7.m7.1.1.2.1.cmml" xref="S4.SS3.SSS1.p4.7.m7.1.1.2">subscript</csymbol><ci id="S4.SS3.SSS1.p4.7.m7.1.1.2.2.cmml" xref="S4.SS3.SSS1.p4.7.m7.1.1.2.2">𝑖</ci><ci id="S4.SS3.SSS1.p4.7.m7.1.1.2.3.cmml" xref="S4.SS3.SSS1.p4.7.m7.1.1.2.3">lim</ci></apply><cn id="S4.SS3.SSS1.p4.7.m7.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p4.7.m7.1.1.3">24.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.7.m7.1c">i_{\rm lim}<24.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.7.m7.1d">italic_i start_POSTSUBSCRIPT roman_lim end_POSTSUBSCRIPT < 24.6</annotation></semantics></math>, there are <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.8.m8.1"><semantics id="S4.SS3.SSS1.p4.8.m8.1a"><mo id="S4.SS3.SSS1.p4.8.m8.1.1" xref="S4.SS3.SSS1.p4.8.m8.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.8.m8.1b"><csymbol cd="latexml" id="S4.SS3.SSS1.p4.8.m8.1.1.cmml" xref="S4.SS3.SSS1.p4.8.m8.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.8.m8.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.8.m8.1d">∼</annotation></semantics></math>1,000 deg-2 <math alttext="g" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.10.m10.1"><semantics id="S4.SS3.SSS1.p4.10.m10.1a"><mi id="S4.SS3.SSS1.p4.10.m10.1.1" xref="S4.SS3.SSS1.p4.10.m10.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.10.m10.1b"><ci id="S4.SS3.SSS1.p4.10.m10.1.1.cmml" xref="S4.SS3.SSS1.p4.10.m10.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.10.m10.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.10.m10.1d">italic_g</annotation></semantics></math>-dropout for MUST to target. For <math alttext="r" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.11.m11.1"><semantics id="S4.SS3.SSS1.p4.11.m11.1a"><mi id="S4.SS3.SSS1.p4.11.m11.1.1" xref="S4.SS3.SSS1.p4.11.m11.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.11.m11.1b"><ci id="S4.SS3.SSS1.p4.11.m11.1.1.cmml" xref="S4.SS3.SSS1.p4.11.m11.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.11.m11.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.11.m11.1d">italic_r</annotation></semantics></math>-dropout, the expected candidate density is <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.12.m12.1"><semantics id="S4.SS3.SSS1.p4.12.m12.1a"><mo id="S4.SS3.SSS1.p4.12.m12.1.1" xref="S4.SS3.SSS1.p4.12.m12.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.12.m12.1b"><csymbol cd="latexml" id="S4.SS3.SSS1.p4.12.m12.1.1.cmml" xref="S4.SS3.SSS1.p4.12.m12.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.12.m12.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.12.m12.1d">∼</annotation></semantics></math> 300 (600) deg-2 for a <math alttext="z_{\rm lim}<24.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.14.m14.1"><semantics id="S4.SS3.SSS1.p4.14.m14.1a"><mrow id="S4.SS3.SSS1.p4.14.m14.1.1" xref="S4.SS3.SSS1.p4.14.m14.1.1.cmml"><msub id="S4.SS3.SSS1.p4.14.m14.1.1.2" xref="S4.SS3.SSS1.p4.14.m14.1.1.2.cmml"><mi id="S4.SS3.SSS1.p4.14.m14.1.1.2.2" xref="S4.SS3.SSS1.p4.14.m14.1.1.2.2.cmml">z</mi><mi id="S4.SS3.SSS1.p4.14.m14.1.1.2.3" xref="S4.SS3.SSS1.p4.14.m14.1.1.2.3.cmml">lim</mi></msub><mo id="S4.SS3.SSS1.p4.14.m14.1.1.1" xref="S4.SS3.SSS1.p4.14.m14.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p4.14.m14.1.1.3" xref="S4.SS3.SSS1.p4.14.m14.1.1.3.cmml">24.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.14.m14.1b"><apply id="S4.SS3.SSS1.p4.14.m14.1.1.cmml" xref="S4.SS3.SSS1.p4.14.m14.1.1"><lt id="S4.SS3.SSS1.p4.14.m14.1.1.1.cmml" xref="S4.SS3.SSS1.p4.14.m14.1.1.1"></lt><apply id="S4.SS3.SSS1.p4.14.m14.1.1.2.cmml" xref="S4.SS3.SSS1.p4.14.m14.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS1.p4.14.m14.1.1.2.1.cmml" xref="S4.SS3.SSS1.p4.14.m14.1.1.2">subscript</csymbol><ci id="S4.SS3.SSS1.p4.14.m14.1.1.2.2.cmml" xref="S4.SS3.SSS1.p4.14.m14.1.1.2.2">𝑧</ci><ci id="S4.SS3.SSS1.p4.14.m14.1.1.2.3.cmml" xref="S4.SS3.SSS1.p4.14.m14.1.1.2.3">lim</ci></apply><cn id="S4.SS3.SSS1.p4.14.m14.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p4.14.m14.1.1.3">24.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.14.m14.1c">z_{\rm lim}<24.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.14.m14.1d">italic_z start_POSTSUBSCRIPT roman_lim end_POSTSUBSCRIPT < 24.5</annotation></semantics></math> (25.0) mag sample. These values represent the most optimistic estimations for MUST. In reality, the photometric selection will include contamination from galaxies outside of the desired redshift range. MUST will not be able to measure the redshifts of all candidates as the success rate should strongly depend on the presence and the strength of the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p4.15.m15.1"><semantics id="S4.SS3.SSS1.p4.15.m15.1a"><mi id="S4.SS3.SSS1.p4.15.m15.1.1" xref="S4.SS3.SSS1.p4.15.m15.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p4.15.m15.1b"><ci id="S4.SS3.SSS1.p4.15.m15.1.1.cmml" xref="S4.SS3.SSS1.p4.15.m15.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p4.15.m15.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p4.15.m15.1d">italic_α</annotation></semantics></math> emission line. </div> <div class="ltx_para" id="S4.SS3.SSS1.p5"> For the cosmological forecast here, we will not go into the details of the target selection as we do not yet have the multi-band images and spectroscopic validation data required for a definitive Stage-V selection strategy. Instead, based on the previous work, we provide optimistic and conservative number density estimates for each population. For <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.1.m1.1"><semantics id="S4.SS3.SSS1.p5.1.m1.1a"><mi id="S4.SS3.SSS1.p5.1.m1.1.1" xref="S4.SS3.SSS1.p5.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.1.m1.1b"><ci id="S4.SS3.SSS1.p5.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p5.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.1.m1.1d">italic_u</annotation></semantics></math>-dropout in <math alttext="2.1<z<3.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.2.m2.1"><semantics id="S4.SS3.SSS1.p5.2.m2.1a"><mrow id="S4.SS3.SSS1.p5.2.m2.1.1" xref="S4.SS3.SSS1.p5.2.m2.1.1.cmml"><mn id="S4.SS3.SSS1.p5.2.m2.1.1.2" xref="S4.SS3.SSS1.p5.2.m2.1.1.2.cmml">2.1</mn><mo id="S4.SS3.SSS1.p5.2.m2.1.1.3" xref="S4.SS3.SSS1.p5.2.m2.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p5.2.m2.1.1.4" xref="S4.SS3.SSS1.p5.2.m2.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p5.2.m2.1.1.5" xref="S4.SS3.SSS1.p5.2.m2.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p5.2.m2.1.1.6" xref="S4.SS3.SSS1.p5.2.m2.1.1.6.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.2.m2.1b"><apply id="S4.SS3.SSS1.p5.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1"><and id="S4.SS3.SSS1.p5.2.m2.1.1a.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1"></and><apply id="S4.SS3.SSS1.p5.2.m2.1.1b.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1"><lt id="S4.SS3.SSS1.p5.2.m2.1.1.3.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1.3"></lt><cn id="S4.SS3.SSS1.p5.2.m2.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p5.2.m2.1.1.2">2.1</cn><ci id="S4.SS3.SSS1.p5.2.m2.1.1.4.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p5.2.m2.1.1c.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1"><lt id="S4.SS3.SSS1.p5.2.m2.1.1.5.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p5.2.m2.1.1.4.cmml" id="S4.SS3.SSS1.p5.2.m2.1.1d.cmml" xref="S4.SS3.SSS1.p5.2.m2.1.1"></share><cn id="S4.SS3.SSS1.p5.2.m2.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p5.2.m2.1.1.6">3.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.2.m2.1c">2.1<z<3.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.2.m2.1d">2.1 < italic_z < 3.5</annotation></semantics></math>, we assume that a <math alttext="r<24.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.3.m3.1"><semantics id="S4.SS3.SSS1.p5.3.m3.1a"><mrow id="S4.SS3.SSS1.p5.3.m3.1.1" xref="S4.SS3.SSS1.p5.3.m3.1.1.cmml"><mi id="S4.SS3.SSS1.p5.3.m3.1.1.2" xref="S4.SS3.SSS1.p5.3.m3.1.1.2.cmml">r</mi><mo id="S4.SS3.SSS1.p5.3.m3.1.1.1" xref="S4.SS3.SSS1.p5.3.m3.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p5.3.m3.1.1.3" xref="S4.SS3.SSS1.p5.3.m3.1.1.3.cmml">24.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.3.m3.1b"><apply id="S4.SS3.SSS1.p5.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p5.3.m3.1.1"><lt id="S4.SS3.SSS1.p5.3.m3.1.1.1.cmml" xref="S4.SS3.SSS1.p5.3.m3.1.1.1"></lt><ci id="S4.SS3.SSS1.p5.3.m3.1.1.2.cmml" xref="S4.SS3.SSS1.p5.3.m3.1.1.2">𝑟</ci><cn id="S4.SS3.SSS1.p5.3.m3.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p5.3.m3.1.1.3">24.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.3.m3.1c">r<24.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.3.m3.1d">italic_r < 24.5</annotation></semantics></math> selection could result in a 1,200 deg-2 sample in an optimistic and a 600 deg-2 sample in a conservative situation. For <math alttext="g" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.6.m6.1"><semantics id="S4.SS3.SSS1.p5.6.m6.1a"><mi id="S4.SS3.SSS1.p5.6.m6.1.1" xref="S4.SS3.SSS1.p5.6.m6.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.6.m6.1b"><ci id="S4.SS3.SSS1.p5.6.m6.1.1.cmml" xref="S4.SS3.SSS1.p5.6.m6.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.6.m6.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.6.m6.1d">italic_g</annotation></semantics></math>-dropout in <math alttext="3.3<z<4.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.7.m7.1"><semantics id="S4.SS3.SSS1.p5.7.m7.1a"><mrow id="S4.SS3.SSS1.p5.7.m7.1.1" xref="S4.SS3.SSS1.p5.7.m7.1.1.cmml"><mn id="S4.SS3.SSS1.p5.7.m7.1.1.2" xref="S4.SS3.SSS1.p5.7.m7.1.1.2.cmml">3.3</mn><mo id="S4.SS3.SSS1.p5.7.m7.1.1.3" xref="S4.SS3.SSS1.p5.7.m7.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p5.7.m7.1.1.4" xref="S4.SS3.SSS1.p5.7.m7.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p5.7.m7.1.1.5" xref="S4.SS3.SSS1.p5.7.m7.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p5.7.m7.1.1.6" xref="S4.SS3.SSS1.p5.7.m7.1.1.6.cmml">4.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.7.m7.1b"><apply id="S4.SS3.SSS1.p5.7.m7.1.1.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1"><and id="S4.SS3.SSS1.p5.7.m7.1.1a.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1"></and><apply id="S4.SS3.SSS1.p5.7.m7.1.1b.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1"><lt id="S4.SS3.SSS1.p5.7.m7.1.1.3.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1.3"></lt><cn id="S4.SS3.SSS1.p5.7.m7.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p5.7.m7.1.1.2">3.3</cn><ci id="S4.SS3.SSS1.p5.7.m7.1.1.4.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p5.7.m7.1.1c.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1"><lt id="S4.SS3.SSS1.p5.7.m7.1.1.5.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p5.7.m7.1.1.4.cmml" id="S4.SS3.SSS1.p5.7.m7.1.1d.cmml" xref="S4.SS3.SSS1.p5.7.m7.1.1"></share><cn id="S4.SS3.SSS1.p5.7.m7.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p5.7.m7.1.1.6">4.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.7.m7.1c">3.3<z<4.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.7.m7.1d">3.3 < italic_z < 4.5</annotation></semantics></math>, we expect a 800 deg-2 and 300 deg-2 sample for the optimistic and conservative scenarios. As for the <math alttext="r" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.10.m10.1"><semantics id="S4.SS3.SSS1.p5.10.m10.1a"><mi id="S4.SS3.SSS1.p5.10.m10.1.1" xref="S4.SS3.SSS1.p5.10.m10.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.10.m10.1b"><ci id="S4.SS3.SSS1.p5.10.m10.1.1.cmml" xref="S4.SS3.SSS1.p5.10.m10.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.10.m10.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.10.m10.1d">italic_r</annotation></semantics></math>-dropout in <math alttext="4.5<z<5.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.11.m11.1"><semantics id="S4.SS3.SSS1.p5.11.m11.1a"><mrow id="S4.SS3.SSS1.p5.11.m11.1.1" xref="S4.SS3.SSS1.p5.11.m11.1.1.cmml"><mn id="S4.SS3.SSS1.p5.11.m11.1.1.2" xref="S4.SS3.SSS1.p5.11.m11.1.1.2.cmml">4.5</mn><mo id="S4.SS3.SSS1.p5.11.m11.1.1.3" xref="S4.SS3.SSS1.p5.11.m11.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p5.11.m11.1.1.4" xref="S4.SS3.SSS1.p5.11.m11.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p5.11.m11.1.1.5" xref="S4.SS3.SSS1.p5.11.m11.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p5.11.m11.1.1.6" xref="S4.SS3.SSS1.p5.11.m11.1.1.6.cmml">5.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.11.m11.1b"><apply id="S4.SS3.SSS1.p5.11.m11.1.1.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1"><and id="S4.SS3.SSS1.p5.11.m11.1.1a.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1"></and><apply id="S4.SS3.SSS1.p5.11.m11.1.1b.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1"><lt id="S4.SS3.SSS1.p5.11.m11.1.1.3.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1.3"></lt><cn id="S4.SS3.SSS1.p5.11.m11.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p5.11.m11.1.1.2">4.5</cn><ci id="S4.SS3.SSS1.p5.11.m11.1.1.4.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p5.11.m11.1.1c.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1"><lt id="S4.SS3.SSS1.p5.11.m11.1.1.5.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p5.11.m11.1.1.4.cmml" id="S4.SS3.SSS1.p5.11.m11.1.1d.cmml" xref="S4.SS3.SSS1.p5.11.m11.1.1"></share><cn id="S4.SS3.SSS1.p5.11.m11.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p5.11.m11.1.1.6">5.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.11.m11.1c">4.5<z<5.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.11.m11.1d">4.5 < italic_z < 5.5</annotation></semantics></math>, we assume the optimistic sample has a 200 deg-2 density, while the conservative sample halves this value. We summarize these assumptions in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.T2" title="Table 2 ‣ 3.6.4 Gravitational waves and Fast Radio Bursts ‣ 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">2</a>. We do not include the BX selection sample in our forecast as it was defined using a different series of broad-band filters than the Sloan <math alttext="ugriz" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.13.m13.1"><semantics id="S4.SS3.SSS1.p5.13.m13.1a"><mrow id="S4.SS3.SSS1.p5.13.m13.1.1" xref="S4.SS3.SSS1.p5.13.m13.1.1.cmml"><mi id="S4.SS3.SSS1.p5.13.m13.1.1.2" xref="S4.SS3.SSS1.p5.13.m13.1.1.2.cmml">u</mi><mo id="S4.SS3.SSS1.p5.13.m13.1.1.1" xref="S4.SS3.SSS1.p5.13.m13.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p5.13.m13.1.1.3" xref="S4.SS3.SSS1.p5.13.m13.1.1.3.cmml">g</mi><mo id="S4.SS3.SSS1.p5.13.m13.1.1.1a" xref="S4.SS3.SSS1.p5.13.m13.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p5.13.m13.1.1.4" xref="S4.SS3.SSS1.p5.13.m13.1.1.4.cmml">r</mi><mo id="S4.SS3.SSS1.p5.13.m13.1.1.1b" xref="S4.SS3.SSS1.p5.13.m13.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p5.13.m13.1.1.5" xref="S4.SS3.SSS1.p5.13.m13.1.1.5.cmml">i</mi><mo id="S4.SS3.SSS1.p5.13.m13.1.1.1c" xref="S4.SS3.SSS1.p5.13.m13.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p5.13.m13.1.1.6" xref="S4.SS3.SSS1.p5.13.m13.1.1.6.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.13.m13.1b"><apply id="S4.SS3.SSS1.p5.13.m13.1.1.cmml" xref="S4.SS3.SSS1.p5.13.m13.1.1"><times id="S4.SS3.SSS1.p5.13.m13.1.1.1.cmml" xref="S4.SS3.SSS1.p5.13.m13.1.1.1"></times><ci id="S4.SS3.SSS1.p5.13.m13.1.1.2.cmml" xref="S4.SS3.SSS1.p5.13.m13.1.1.2">𝑢</ci><ci id="S4.SS3.SSS1.p5.13.m13.1.1.3.cmml" xref="S4.SS3.SSS1.p5.13.m13.1.1.3">𝑔</ci><ci id="S4.SS3.SSS1.p5.13.m13.1.1.4.cmml" xref="S4.SS3.SSS1.p5.13.m13.1.1.4">𝑟</ci><ci id="S4.SS3.SSS1.p5.13.m13.1.1.5.cmml" xref="S4.SS3.SSS1.p5.13.m13.1.1.5">𝑖</ci><ci id="S4.SS3.SSS1.p5.13.m13.1.1.6.cmml" xref="S4.SS3.SSS1.p5.13.m13.1.1.6">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.13.m13.1c">ugriz</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.13.m13.1d">italic_u italic_g italic_r italic_i italic_z</annotation></semantics></math> ones <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib246" title="">246</a>]</cite>. Hence, the BX criteria require filter conversions to work on the imaging data for MUST or need to be updated. Moreover, the BX-selected sample often centers at <math alttext="z\sim 2" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.14.m14.1"><semantics id="S4.SS3.SSS1.p5.14.m14.1a"><mrow id="S4.SS3.SSS1.p5.14.m14.1.1" xref="S4.SS3.SSS1.p5.14.m14.1.1.cmml"><mi id="S4.SS3.SSS1.p5.14.m14.1.1.2" xref="S4.SS3.SSS1.p5.14.m14.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p5.14.m14.1.1.1" xref="S4.SS3.SSS1.p5.14.m14.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p5.14.m14.1.1.3" xref="S4.SS3.SSS1.p5.14.m14.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.14.m14.1b"><apply id="S4.SS3.SSS1.p5.14.m14.1.1.cmml" xref="S4.SS3.SSS1.p5.14.m14.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p5.14.m14.1.1.1.cmml" xref="S4.SS3.SSS1.p5.14.m14.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS1.p5.14.m14.1.1.2.cmml" xref="S4.SS3.SSS1.p5.14.m14.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p5.14.m14.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p5.14.m14.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.14.m14.1c">z\sim 2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.14.m14.1d">italic_z ∼ 2</annotation></semantics></math>, leading to a significant fraction out of the redshift range covered by MUST. Still, the current estimation suggests that the potential LSS tracer density at <math alttext="z<2.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p5.15.m15.1"><semantics id="S4.SS3.SSS1.p5.15.m15.1a"><mrow id="S4.SS3.SSS1.p5.15.m15.1.1" xref="S4.SS3.SSS1.p5.15.m15.1.1.cmml"><mi id="S4.SS3.SSS1.p5.15.m15.1.1.2" xref="S4.SS3.SSS1.p5.15.m15.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p5.15.m15.1.1.1" xref="S4.SS3.SSS1.p5.15.m15.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p5.15.m15.1.1.3" xref="S4.SS3.SSS1.p5.15.m15.1.1.3.cmml">2.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p5.15.m15.1b"><apply id="S4.SS3.SSS1.p5.15.m15.1.1.cmml" xref="S4.SS3.SSS1.p5.15.m15.1.1"><lt id="S4.SS3.SSS1.p5.15.m15.1.1.1.cmml" xref="S4.SS3.SSS1.p5.15.m15.1.1.1"></lt><ci id="S4.SS3.SSS1.p5.15.m15.1.1.2.cmml" xref="S4.SS3.SSS1.p5.15.m15.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p5.15.m15.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p5.15.m15.1.1.3">2.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p5.15.m15.1c">z<2.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p5.15.m15.1d">italic_z < 2.5</annotation></semantics></math> can be much higher when better selection criteria are designed. </div> <div class="ltx_para" id="S4.SS3.SSS1.p6"> To provide the initial estimates of the redshift distributions for each population, we adopt a simple method based on the COSMOS2020101010<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://astroweaver.github.io/project/cosmos2020-galaxy-catalog/" title="">https://astroweaver.github.io/project/cosmos2020-galaxy-catalog/</a> photometric and photo-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p6.1.m1.1"><semantics id="S4.SS3.SSS1.p6.1.m1.1a"><mi id="S4.SS3.SSS1.p6.1.m1.1.1" xref="S4.SS3.SSS1.p6.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p6.1.m1.1b"><ci id="S4.SS3.SSS1.p6.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p6.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p6.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p6.1.m1.1d">italic_z</annotation></semantics></math> catalog <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib247" title="">247</a>]</cite>. Following the previous dropout-selection recipes, we designed the color cuts for the magnitude-limited samples for MUST to roughly reach the desired target density while ensuring their photometric redshift distribution is consistent with the design. We apply the bright object masks from the COSMOS field of the HSC-SSP Ultra-Deep survey. It results in a 1.5 deg2 effective footprint to estimate the target density. We confirm that the choice of photometric measurements and the photo-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p6.3.m3.1"><semantics id="S4.SS3.SSS1.p6.3.m3.1a"><mi id="S4.SS3.SSS1.p6.3.m3.1.1" xref="S4.SS3.SSS1.p6.3.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p6.3.m3.1b"><ci id="S4.SS3.SSS1.p6.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p6.3.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p6.3.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p6.3.m3.1d">italic_z</annotation></semantics></math> estimations do not affect the derived redshift distribution. </div> <div class="ltx_para" id="S4.SS3.SSS1.p7"> For the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p7.1.m1.1"><semantics id="S4.SS3.SSS1.p7.1.m1.1a"><mi id="S4.SS3.SSS1.p7.1.m1.1.1" xref="S4.SS3.SSS1.p7.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p7.1.m1.1b"><ci id="S4.SS3.SSS1.p7.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p7.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p7.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p7.1.m1.1d">italic_u</annotation></semantics></math>-dropouts, we recover a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p7.2.m2.1"><semantics id="S4.SS3.SSS1.p7.2.m2.1a"><mo id="S4.SS3.SSS1.p7.2.m2.1.1" xref="S4.SS3.SSS1.p7.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p7.2.m2.1b"><csymbol cd="latexml" id="S4.SS3.SSS1.p7.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p7.2.m2.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p7.2.m2.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p7.2.m2.1d">∼</annotation></semantics></math>1,500 deg-2 sample within <math alttext="22.4<r<24.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p7.4.m4.1"><semantics id="S4.SS3.SSS1.p7.4.m4.1a"><mrow id="S4.SS3.SSS1.p7.4.m4.1.1" xref="S4.SS3.SSS1.p7.4.m4.1.1.cmml"><mn id="S4.SS3.SSS1.p7.4.m4.1.1.2" xref="S4.SS3.SSS1.p7.4.m4.1.1.2.cmml">22.4</mn><mo id="S4.SS3.SSS1.p7.4.m4.1.1.3" xref="S4.SS3.SSS1.p7.4.m4.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p7.4.m4.1.1.4" xref="S4.SS3.SSS1.p7.4.m4.1.1.4.cmml">r</mi><mo id="S4.SS3.SSS1.p7.4.m4.1.1.5" xref="S4.SS3.SSS1.p7.4.m4.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p7.4.m4.1.1.6" xref="S4.SS3.SSS1.p7.4.m4.1.1.6.cmml">24.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p7.4.m4.1b"><apply id="S4.SS3.SSS1.p7.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1"><and id="S4.SS3.SSS1.p7.4.m4.1.1a.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1"></and><apply id="S4.SS3.SSS1.p7.4.m4.1.1b.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1"><lt id="S4.SS3.SSS1.p7.4.m4.1.1.3.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1.3"></lt><cn id="S4.SS3.SSS1.p7.4.m4.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p7.4.m4.1.1.2">22.4</cn><ci id="S4.SS3.SSS1.p7.4.m4.1.1.4.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1.4">𝑟</ci></apply><apply id="S4.SS3.SSS1.p7.4.m4.1.1c.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1"><lt id="S4.SS3.SSS1.p7.4.m4.1.1.5.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p7.4.m4.1.1.4.cmml" id="S4.SS3.SSS1.p7.4.m4.1.1d.cmml" xref="S4.SS3.SSS1.p7.4.m4.1.1"></share><cn id="S4.SS3.SSS1.p7.4.m4.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p7.4.m4.1.1.6">24.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p7.4.m4.1c">22.4<r<24.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p7.4.m4.1d">22.4 < italic_r < 24.5</annotation></semantics></math> mag using the following criteria: </div> <div class="ltx_para" id="S4.SS3.SSS1.p8"> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E11"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" 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xref="S4.E11X.2.1.1.m1.1.1.2.cmml">></mo><mn id="S4.E11X.2.1.1.m1.1.1.3" xref="S4.E11X.2.1.1.m1.1.1.3.cmml">0.95</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.E11X.2.1.1.m1.1b"><apply id="S4.E11X.2.1.1.m1.1.1.cmml" xref="S4.E11X.2.1.1.m1.1.1"><gt id="S4.E11X.2.1.1.m1.1.1.2.cmml" xref="S4.E11X.2.1.1.m1.1.1.2"></gt><apply id="S4.E11X.2.1.1.m1.1.1.1.1.1.cmml" xref="S4.E11X.2.1.1.m1.1.1.1.1"><minus id="S4.E11X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.E11X.2.1.1.m1.1.1.1.1.1.1"></minus><ci id="S4.E11X.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.E11X.2.1.1.m1.1.1.1.1.1.2">𝑢</ci><ci id="S4.E11X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S4.E11X.2.1.1.m1.1.1.1.1.1.3">𝑔</ci></apply><cn id="S4.E11X.2.1.1.m1.1.1.3.cmml" type="float" xref="S4.E11X.2.1.1.m1.1.1.3">0.95</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E11X.2.1.1.m1.1c">\displaystyle(u-g)>0.95</annotation><annotation encoding="application/x-llamapun" id="S4.E11X.2.1.1.m1.1d">( italic_u - italic_g ) > 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id="S4.E11Xa.2.1.1.m1.1.1b.cmml" xref="S4.E11Xa.2.1.1.m1.1.1"><lt id="S4.E11Xa.2.1.1.m1.1.1.4.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.4"></lt><apply id="S4.E11Xa.2.1.1.m1.1.1.3.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.3"><minus id="S4.E11Xa.2.1.1.m1.1.1.3.1.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.3"></minus><cn id="S4.E11Xa.2.1.1.m1.1.1.3.2.cmml" type="float" xref="S4.E11Xa.2.1.1.m1.1.1.3.2">0.5</cn></apply><apply id="S4.E11Xa.2.1.1.m1.1.1.1.1.1.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.1.1"><minus id="S4.E11Xa.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.1.1.1.1"></minus><ci id="S4.E11Xa.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.1.1.1.2">𝑔</ci><ci id="S4.E11Xa.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.1.1.1.3">𝑟</ci></apply></apply><apply id="S4.E11Xa.2.1.1.m1.1.1c.cmml" xref="S4.E11Xa.2.1.1.m1.1.1"><lt id="S4.E11Xa.2.1.1.m1.1.1.5.cmml" xref="S4.E11Xa.2.1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.E11Xa.2.1.1.m1.1.1.1.cmml" id="S4.E11Xa.2.1.1.m1.1.1d.cmml" xref="S4.E11Xa.2.1.1.m1.1.1"></share><cn id="S4.E11Xa.2.1.1.m1.1.1.6.cmml" type="float" xref="S4.E11Xa.2.1.1.m1.1.1.6">1.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E11Xa.2.1.1.m1.1c">\displaystyle-0.5<(g-r)<1.1</annotation><annotation encoding="application/x-llamapun" id="S4.E11Xa.2.1.1.m1.1d">- 0.5 < ( italic_g - italic_r ) < 1.1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E11Xb"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(u-g)>1.17\times(g-r)+0.71." class="ltx_Math" display="inline" id="S4.E11Xb.2.1.1.m1.1"><semantics id="S4.E11Xb.2.1.1.m1.1a"><mrow id="S4.E11Xb.2.1.1.m1.1.1.1" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E11Xb.2.1.1.m1.1.1.1.1" 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id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.2" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.2.cmml">+</mo><mn id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.3" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.3.cmml">0.71</mn></mrow></mrow><mo id="S4.E11Xb.2.1.1.m1.1.1.1.2" lspace="0em" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E11Xb.2.1.1.m1.1b"><apply id="S4.E11Xb.2.1.1.m1.1.1.1.1.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1"><gt id="S4.E11Xb.2.1.1.m1.1.1.1.1.3.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.3"></gt><apply id="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1"><minus id="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.1.1.1.3">𝑔</ci></apply><apply id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2"><plus id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.2.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.2"></plus><apply id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1"><times id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.2.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.2"></times><cn id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.3.cmml" type="float" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.3">1.17</cn><apply id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1"><minus id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.1"></minus><ci id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.2.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.2">𝑔</ci><ci id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.3">𝑟</ci></apply></apply><cn id="S4.E11Xb.2.1.1.m1.1.1.1.1.2.3.cmml" type="float" xref="S4.E11Xb.2.1.1.m1.1.1.1.1.2.3">0.71</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E11Xb.2.1.1.m1.1c">\displaystyle(u-g)>1.17\times(g-r)+0.71.</annotation><annotation encoding="application/x-llamapun" id="S4.E11Xb.2.1.1.m1.1d">( italic_u - italic_g ) > 1.17 × ( italic_g - italic_r ) + 0.71 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S4.SS3.SSS1.p9"> About 88% of the target has photo-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p9.1.m1.1"><semantics id="S4.SS3.SSS1.p9.1.m1.1a"><mi id="S4.SS3.SSS1.p9.1.m1.1.1" xref="S4.SS3.SSS1.p9.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p9.1.m1.1b"><ci id="S4.SS3.SSS1.p9.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p9.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p9.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p9.1.m1.1d">italic_z</annotation></semantics></math> within <math alttext="2.2<z<3.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p9.2.m2.1"><semantics id="S4.SS3.SSS1.p9.2.m2.1a"><mrow id="S4.SS3.SSS1.p9.2.m2.1.1" xref="S4.SS3.SSS1.p9.2.m2.1.1.cmml"><mn id="S4.SS3.SSS1.p9.2.m2.1.1.2" xref="S4.SS3.SSS1.p9.2.m2.1.1.2.cmml">2.2</mn><mo id="S4.SS3.SSS1.p9.2.m2.1.1.3" xref="S4.SS3.SSS1.p9.2.m2.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p9.2.m2.1.1.4" xref="S4.SS3.SSS1.p9.2.m2.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p9.2.m2.1.1.5" xref="S4.SS3.SSS1.p9.2.m2.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p9.2.m2.1.1.6" xref="S4.SS3.SSS1.p9.2.m2.1.1.6.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p9.2.m2.1b"><apply id="S4.SS3.SSS1.p9.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1"><and id="S4.SS3.SSS1.p9.2.m2.1.1a.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1"></and><apply id="S4.SS3.SSS1.p9.2.m2.1.1b.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1"><lt id="S4.SS3.SSS1.p9.2.m2.1.1.3.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1.3"></lt><cn id="S4.SS3.SSS1.p9.2.m2.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p9.2.m2.1.1.2">2.2</cn><ci id="S4.SS3.SSS1.p9.2.m2.1.1.4.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p9.2.m2.1.1c.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1"><lt id="S4.SS3.SSS1.p9.2.m2.1.1.5.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p9.2.m2.1.1.4.cmml" id="S4.SS3.SSS1.p9.2.m2.1.1d.cmml" xref="S4.SS3.SSS1.p9.2.m2.1.1"></share><cn id="S4.SS3.SSS1.p9.2.m2.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p9.2.m2.1.1.6">3.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p9.2.m2.1c">2.2<z<3.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p9.2.m2.1d">2.2 < italic_z < 3.5</annotation></semantics></math> and shows a clear peak at <math alttext="z=3" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p9.3.m3.1"><semantics id="S4.SS3.SSS1.p9.3.m3.1a"><mrow id="S4.SS3.SSS1.p9.3.m3.1.1" xref="S4.SS3.SSS1.p9.3.m3.1.1.cmml"><mi id="S4.SS3.SSS1.p9.3.m3.1.1.2" xref="S4.SS3.SSS1.p9.3.m3.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p9.3.m3.1.1.1" xref="S4.SS3.SSS1.p9.3.m3.1.1.1.cmml">=</mo><mn id="S4.SS3.SSS1.p9.3.m3.1.1.3" xref="S4.SS3.SSS1.p9.3.m3.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p9.3.m3.1b"><apply id="S4.SS3.SSS1.p9.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p9.3.m3.1.1"><eq id="S4.SS3.SSS1.p9.3.m3.1.1.1.cmml" xref="S4.SS3.SSS1.p9.3.m3.1.1.1"></eq><ci id="S4.SS3.SSS1.p9.3.m3.1.1.2.cmml" xref="S4.SS3.SSS1.p9.3.m3.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p9.3.m3.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p9.3.m3.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p9.3.m3.1c">z=3</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p9.3.m3.1d">italic_z = 3</annotation></semantics></math>. We confirm that, while using different choices of <math alttext="ugr" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p9.4.m4.1"><semantics id="S4.SS3.SSS1.p9.4.m4.1a"><mrow id="S4.SS3.SSS1.p9.4.m4.1.1" xref="S4.SS3.SSS1.p9.4.m4.1.1.cmml"><mi id="S4.SS3.SSS1.p9.4.m4.1.1.2" xref="S4.SS3.SSS1.p9.4.m4.1.1.2.cmml">u</mi><mo id="S4.SS3.SSS1.p9.4.m4.1.1.1" xref="S4.SS3.SSS1.p9.4.m4.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p9.4.m4.1.1.3" xref="S4.SS3.SSS1.p9.4.m4.1.1.3.cmml">g</mi><mo id="S4.SS3.SSS1.p9.4.m4.1.1.1a" xref="S4.SS3.SSS1.p9.4.m4.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p9.4.m4.1.1.4" xref="S4.SS3.SSS1.p9.4.m4.1.1.4.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p9.4.m4.1b"><apply id="S4.SS3.SSS1.p9.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p9.4.m4.1.1"><times id="S4.SS3.SSS1.p9.4.m4.1.1.1.cmml" xref="S4.SS3.SSS1.p9.4.m4.1.1.1"></times><ci id="S4.SS3.SSS1.p9.4.m4.1.1.2.cmml" xref="S4.SS3.SSS1.p9.4.m4.1.1.2">𝑢</ci><ci id="S4.SS3.SSS1.p9.4.m4.1.1.3.cmml" xref="S4.SS3.SSS1.p9.4.m4.1.1.3">𝑔</ci><ci id="S4.SS3.SSS1.p9.4.m4.1.1.4.cmml" xref="S4.SS3.SSS1.p9.4.m4.1.1.4">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p9.4.m4.1c">ugr</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p9.4.m4.1d">italic_u italic_g italic_r</annotation></semantics></math> or <math alttext="ugi" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p9.5.m5.1"><semantics id="S4.SS3.SSS1.p9.5.m5.1a"><mrow id="S4.SS3.SSS1.p9.5.m5.1.1" xref="S4.SS3.SSS1.p9.5.m5.1.1.cmml"><mi id="S4.SS3.SSS1.p9.5.m5.1.1.2" xref="S4.SS3.SSS1.p9.5.m5.1.1.2.cmml">u</mi><mo id="S4.SS3.SSS1.p9.5.m5.1.1.1" xref="S4.SS3.SSS1.p9.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p9.5.m5.1.1.3" xref="S4.SS3.SSS1.p9.5.m5.1.1.3.cmml">g</mi><mo id="S4.SS3.SSS1.p9.5.m5.1.1.1a" xref="S4.SS3.SSS1.p9.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p9.5.m5.1.1.4" xref="S4.SS3.SSS1.p9.5.m5.1.1.4.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p9.5.m5.1b"><apply id="S4.SS3.SSS1.p9.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p9.5.m5.1.1"><times id="S4.SS3.SSS1.p9.5.m5.1.1.1.cmml" xref="S4.SS3.SSS1.p9.5.m5.1.1.1"></times><ci id="S4.SS3.SSS1.p9.5.m5.1.1.2.cmml" xref="S4.SS3.SSS1.p9.5.m5.1.1.2">𝑢</ci><ci id="S4.SS3.SSS1.p9.5.m5.1.1.3.cmml" xref="S4.SS3.SSS1.p9.5.m5.1.1.3">𝑔</ci><ci id="S4.SS3.SSS1.p9.5.m5.1.1.4.cmml" xref="S4.SS3.SSS1.p9.5.m5.1.1.4">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p9.5.m5.1c">ugi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p9.5.m5.1d">italic_u italic_g italic_i</annotation></semantics></math> color cuts can change the target density from <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p9.6.m6.1"><semantics id="S4.SS3.SSS1.p9.6.m6.1a"><mo id="S4.SS3.SSS1.p9.6.m6.1.1" xref="S4.SS3.SSS1.p9.6.m6.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p9.6.m6.1b"><csymbol cd="latexml" id="S4.SS3.SSS1.p9.6.m6.1.1.cmml" xref="S4.SS3.SSS1.p9.6.m6.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p9.6.m6.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p9.6.m6.1d">∼</annotation></semantics></math>800 to <math alttext=">" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p9.7.m7.1"><semantics id="S4.SS3.SSS1.p9.7.m7.1a"><mo id="S4.SS3.SSS1.p9.7.m7.1.1" xref="S4.SS3.SSS1.p9.7.m7.1.1.cmml">></mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p9.7.m7.1b"><gt id="S4.SS3.SSS1.p9.7.m7.1.1.cmml" xref="S4.SS3.SSS1.p9.7.m7.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p9.7.m7.1c">></annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p9.7.m7.1d">></annotation></semantics></math>2,000 deg-2, the redshift distribution does not vary significantly. </div> <div class="ltx_para" id="S4.SS3.SSS1.p10"> For the <math alttext="g" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.1.m1.1"><semantics id="S4.SS3.SSS1.p10.1.m1.1a"><mi id="S4.SS3.SSS1.p10.1.m1.1.1" xref="S4.SS3.SSS1.p10.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.1.m1.1b"><ci id="S4.SS3.SSS1.p10.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p10.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.1.m1.1d">italic_g</annotation></semantics></math>-dropouts, we design the following <math alttext="grz" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.2.m2.1"><semantics id="S4.SS3.SSS1.p10.2.m2.1a"><mrow id="S4.SS3.SSS1.p10.2.m2.1.1" xref="S4.SS3.SSS1.p10.2.m2.1.1.cmml"><mi id="S4.SS3.SSS1.p10.2.m2.1.1.2" xref="S4.SS3.SSS1.p10.2.m2.1.1.2.cmml">g</mi><mo id="S4.SS3.SSS1.p10.2.m2.1.1.1" xref="S4.SS3.SSS1.p10.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p10.2.m2.1.1.3" xref="S4.SS3.SSS1.p10.2.m2.1.1.3.cmml">r</mi><mo id="S4.SS3.SSS1.p10.2.m2.1.1.1a" xref="S4.SS3.SSS1.p10.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p10.2.m2.1.1.4" xref="S4.SS3.SSS1.p10.2.m2.1.1.4.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.2.m2.1b"><apply id="S4.SS3.SSS1.p10.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p10.2.m2.1.1"><times id="S4.SS3.SSS1.p10.2.m2.1.1.1.cmml" xref="S4.SS3.SSS1.p10.2.m2.1.1.1"></times><ci id="S4.SS3.SSS1.p10.2.m2.1.1.2.cmml" xref="S4.SS3.SSS1.p10.2.m2.1.1.2">𝑔</ci><ci id="S4.SS3.SSS1.p10.2.m2.1.1.3.cmml" xref="S4.SS3.SSS1.p10.2.m2.1.1.3">𝑟</ci><ci id="S4.SS3.SSS1.p10.2.m2.1.1.4.cmml" xref="S4.SS3.SSS1.p10.2.m2.1.1.4">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.2.m2.1c">grz</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.2.m2.1d">italic_g italic_r italic_z</annotation></semantics></math> color cuts to select a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.3.m3.1"><semantics id="S4.SS3.SSS1.p10.3.m3.1a"><mo id="S4.SS3.SSS1.p10.3.m3.1.1" xref="S4.SS3.SSS1.p10.3.m3.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.3.m3.1b"><csymbol cd="latexml" id="S4.SS3.SSS1.p10.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p10.3.m3.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.3.m3.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.3.m3.1d">∼</annotation></semantics></math> 870 deg-2 sample peaked at <math alttext="z\sim 3.4" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.5.m5.1"><semantics id="S4.SS3.SSS1.p10.5.m5.1a"><mrow id="S4.SS3.SSS1.p10.5.m5.1.1" xref="S4.SS3.SSS1.p10.5.m5.1.1.cmml"><mi id="S4.SS3.SSS1.p10.5.m5.1.1.2" xref="S4.SS3.SSS1.p10.5.m5.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p10.5.m5.1.1.1" xref="S4.SS3.SSS1.p10.5.m5.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p10.5.m5.1.1.3" xref="S4.SS3.SSS1.p10.5.m5.1.1.3.cmml">3.4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.5.m5.1b"><apply id="S4.SS3.SSS1.p10.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p10.5.m5.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p10.5.m5.1.1.1.cmml" xref="S4.SS3.SSS1.p10.5.m5.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS1.p10.5.m5.1.1.2.cmml" xref="S4.SS3.SSS1.p10.5.m5.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p10.5.m5.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p10.5.m5.1.1.3">3.4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.5.m5.1c">z\sim 3.4</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.5.m5.1d">italic_z ∼ 3.4</annotation></semantics></math>. <math alttext="\sim 84" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.6.m6.1"><semantics id="S4.SS3.SSS1.p10.6.m6.1a"><mrow id="S4.SS3.SSS1.p10.6.m6.1.1" xref="S4.SS3.SSS1.p10.6.m6.1.1.cmml"><mi id="S4.SS3.SSS1.p10.6.m6.1.1.2" xref="S4.SS3.SSS1.p10.6.m6.1.1.2.cmml"></mi><mo id="S4.SS3.SSS1.p10.6.m6.1.1.1" xref="S4.SS3.SSS1.p10.6.m6.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p10.6.m6.1.1.3" xref="S4.SS3.SSS1.p10.6.m6.1.1.3.cmml">84</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.6.m6.1b"><apply id="S4.SS3.SSS1.p10.6.m6.1.1.cmml" xref="S4.SS3.SSS1.p10.6.m6.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p10.6.m6.1.1.1.cmml" xref="S4.SS3.SSS1.p10.6.m6.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.SSS1.p10.6.m6.1.1.2.cmml" xref="S4.SS3.SSS1.p10.6.m6.1.1.2">absent</csymbol><cn id="S4.SS3.SSS1.p10.6.m6.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p10.6.m6.1.1.3">84</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.6.m6.1c">\sim 84</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.6.m6.1d">∼ 84</annotation></semantics></math>% of the sample falls into <math alttext="3.2<z<4.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.7.m7.1"><semantics id="S4.SS3.SSS1.p10.7.m7.1a"><mrow id="S4.SS3.SSS1.p10.7.m7.1.1" xref="S4.SS3.SSS1.p10.7.m7.1.1.cmml"><mn id="S4.SS3.SSS1.p10.7.m7.1.1.2" xref="S4.SS3.SSS1.p10.7.m7.1.1.2.cmml">3.2</mn><mo id="S4.SS3.SSS1.p10.7.m7.1.1.3" xref="S4.SS3.SSS1.p10.7.m7.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p10.7.m7.1.1.4" xref="S4.SS3.SSS1.p10.7.m7.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p10.7.m7.1.1.5" xref="S4.SS3.SSS1.p10.7.m7.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p10.7.m7.1.1.6" xref="S4.SS3.SSS1.p10.7.m7.1.1.6.cmml">4.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.7.m7.1b"><apply id="S4.SS3.SSS1.p10.7.m7.1.1.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1"><and id="S4.SS3.SSS1.p10.7.m7.1.1a.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1"></and><apply id="S4.SS3.SSS1.p10.7.m7.1.1b.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1"><lt id="S4.SS3.SSS1.p10.7.m7.1.1.3.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1.3"></lt><cn id="S4.SS3.SSS1.p10.7.m7.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p10.7.m7.1.1.2">3.2</cn><ci id="S4.SS3.SSS1.p10.7.m7.1.1.4.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p10.7.m7.1.1c.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1"><lt id="S4.SS3.SSS1.p10.7.m7.1.1.5.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p10.7.m7.1.1.4.cmml" id="S4.SS3.SSS1.p10.7.m7.1.1d.cmml" xref="S4.SS3.SSS1.p10.7.m7.1.1"></share><cn id="S4.SS3.SSS1.p10.7.m7.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p10.7.m7.1.1.6">4.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.7.m7.1c">3.2<z<4.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.7.m7.1d">3.2 < italic_z < 4.5</annotation></semantics></math> based on photo-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.8.m8.1"><semantics id="S4.SS3.SSS1.p10.8.m8.1a"><mi id="S4.SS3.SSS1.p10.8.m8.1.1" xref="S4.SS3.SSS1.p10.8.m8.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.8.m8.1b"><ci id="S4.SS3.SSS1.p10.8.m8.1.1.cmml" xref="S4.SS3.SSS1.p10.8.m8.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.8.m8.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.8.m8.1d">italic_z</annotation></semantics></math>, resulting in a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.9.m9.1"><semantics id="S4.SS3.SSS1.p10.9.m9.1a"><mo id="S4.SS3.SSS1.p10.9.m9.1.1" xref="S4.SS3.SSS1.p10.9.m9.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.9.m9.1b"><csymbol cd="latexml" id="S4.SS3.SSS1.p10.9.m9.1.1.cmml" xref="S4.SS3.SSS1.p10.9.m9.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.9.m9.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.9.m9.1d">∼</annotation></semantics></math> 760 deg-2 sample within <math alttext="23.0<i<24.6" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p10.11.m11.1"><semantics id="S4.SS3.SSS1.p10.11.m11.1a"><mrow id="S4.SS3.SSS1.p10.11.m11.1.1" xref="S4.SS3.SSS1.p10.11.m11.1.1.cmml"><mn id="S4.SS3.SSS1.p10.11.m11.1.1.2" xref="S4.SS3.SSS1.p10.11.m11.1.1.2.cmml">23.0</mn><mo id="S4.SS3.SSS1.p10.11.m11.1.1.3" xref="S4.SS3.SSS1.p10.11.m11.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p10.11.m11.1.1.4" xref="S4.SS3.SSS1.p10.11.m11.1.1.4.cmml">i</mi><mo id="S4.SS3.SSS1.p10.11.m11.1.1.5" xref="S4.SS3.SSS1.p10.11.m11.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p10.11.m11.1.1.6" xref="S4.SS3.SSS1.p10.11.m11.1.1.6.cmml">24.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p10.11.m11.1b"><apply id="S4.SS3.SSS1.p10.11.m11.1.1.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1"><and id="S4.SS3.SSS1.p10.11.m11.1.1a.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1"></and><apply id="S4.SS3.SSS1.p10.11.m11.1.1b.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1"><lt id="S4.SS3.SSS1.p10.11.m11.1.1.3.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1.3"></lt><cn id="S4.SS3.SSS1.p10.11.m11.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p10.11.m11.1.1.2">23.0</cn><ci id="S4.SS3.SSS1.p10.11.m11.1.1.4.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1.4">𝑖</ci></apply><apply id="S4.SS3.SSS1.p10.11.m11.1.1c.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1"><lt id="S4.SS3.SSS1.p10.11.m11.1.1.5.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p10.11.m11.1.1.4.cmml" id="S4.SS3.SSS1.p10.11.m11.1.1d.cmml" xref="S4.SS3.SSS1.p10.11.m11.1.1"></share><cn id="S4.SS3.SSS1.p10.11.m11.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p10.11.m11.1.1.6">24.6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p10.11.m11.1c">23.0<i<24.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p10.11.m11.1d">23.0 < italic_i < 24.6</annotation></semantics></math> mag: </div> <div class="ltx_para" id="S4.SS3.SSS1.p11"> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E12"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E12X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(g-r)>1.0" class="ltx_Math" display="inline" id="S4.E12X.2.1.1.m1.1"><semantics id="S4.E12X.2.1.1.m1.1a"><mrow id="S4.E12X.2.1.1.m1.1.1" xref="S4.E12X.2.1.1.m1.1.1.cmml"><mrow id="S4.E12X.2.1.1.m1.1.1.1.1" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.cmml"><mo id="S4.E12X.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E12X.2.1.1.m1.1.1.1.1.1" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S4.E12X.2.1.1.m1.1.1.1.1.1.2" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.2.cmml">g</mi><mo id="S4.E12X.2.1.1.m1.1.1.1.1.1.1" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E12X.2.1.1.m1.1.1.1.1.1.3" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S4.E12X.2.1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E12X.2.1.1.m1.1.1.2" xref="S4.E12X.2.1.1.m1.1.1.2.cmml">></mo><mn id="S4.E12X.2.1.1.m1.1.1.3" xref="S4.E12X.2.1.1.m1.1.1.3.cmml">1.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.E12X.2.1.1.m1.1b"><apply id="S4.E12X.2.1.1.m1.1.1.cmml" xref="S4.E12X.2.1.1.m1.1.1"><gt id="S4.E12X.2.1.1.m1.1.1.2.cmml" xref="S4.E12X.2.1.1.m1.1.1.2"></gt><apply id="S4.E12X.2.1.1.m1.1.1.1.1.1.cmml" xref="S4.E12X.2.1.1.m1.1.1.1.1"><minus id="S4.E12X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.1"></minus><ci id="S4.E12X.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.2">𝑔</ci><ci id="S4.E12X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S4.E12X.2.1.1.m1.1.1.1.1.1.3">𝑟</ci></apply><cn id="S4.E12X.2.1.1.m1.1.1.3.cmml" type="float" xref="S4.E12X.2.1.1.m1.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E12X.2.1.1.m1.1c">\displaystyle(g-r)>1.0</annotation><annotation encoding="application/x-llamapun" id="S4.E12X.2.1.1.m1.1d">( italic_g - italic_r ) > 1.0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2">(12)</td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E12Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(g-r)>1.2\times(r-z)+0.65," class="ltx_Math" display="inline" id="S4.E12Xa.2.1.1.m1.1"><semantics id="S4.E12Xa.2.1.1.m1.1a"><mrow id="S4.E12Xa.2.1.1.m1.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E12Xa.2.1.1.m1.1.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.2" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.2.cmml">g</mi><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.3" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.3" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.3.cmml">></mo><mrow id="S4.E12Xa.2.1.1.m1.1.1.1.1.2" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.cmml"><mrow id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.cmml"><mn id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.3" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.3.cmml">1.2</mn><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.2" lspace="0.222em" rspace="0.222em" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.2.cmml">×</mo><mrow id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.cmml"><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.2" stretchy="false" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.cmml"><mi id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.2" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.2.cmml">r</mi><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.cmml">−</mo><mi id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.3" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.3.cmml">z</mi></mrow><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.3" stretchy="false" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.2" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.2.cmml">+</mo><mn id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.3" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.3.cmml">0.65</mn></mrow></mrow><mo id="S4.E12Xa.2.1.1.m1.1.1.1.2" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E12Xa.2.1.1.m1.1b"><apply id="S4.E12Xa.2.1.1.m1.1.1.1.1.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1"><gt id="S4.E12Xa.2.1.1.m1.1.1.1.1.3.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.3"></gt><apply id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1"><minus id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.2">𝑔</ci><ci id="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2"><plus id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.2.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.2"></plus><apply id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1"><times id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.2.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.2"></times><cn id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.3.cmml" type="float" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.3">1.2</cn><apply id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1"><minus id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.1"></minus><ci id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.2.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.2">𝑟</ci><ci id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.1.1.1.1.3">𝑧</ci></apply></apply><cn id="S4.E12Xa.2.1.1.m1.1.1.1.1.2.3.cmml" type="float" xref="S4.E12Xa.2.1.1.m1.1.1.1.1.2.3">0.65</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E12Xa.2.1.1.m1.1c">\displaystyle(g-r)>1.2\times(r-z)+0.65,</annotation><annotation encoding="application/x-llamapun" id="S4.E12Xa.2.1.1.m1.1d">( italic_g - italic_r ) > 1.2 × ( italic_r - italic_z ) + 0.65 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S4.SS3.SSS1.p12"> As for the <math alttext="r" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p12.1.m1.1"><semantics id="S4.SS3.SSS1.p12.1.m1.1a"><mi id="S4.SS3.SSS1.p12.1.m1.1.1" xref="S4.SS3.SSS1.p12.1.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p12.1.m1.1b"><ci id="S4.SS3.SSS1.p12.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p12.1.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p12.1.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p12.1.m1.1d">italic_r</annotation></semantics></math>-dropouts, we can define a <math alttext="23.0<z<25.0" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p12.2.m2.1"><semantics id="S4.SS3.SSS1.p12.2.m2.1a"><mrow id="S4.SS3.SSS1.p12.2.m2.1.1" xref="S4.SS3.SSS1.p12.2.m2.1.1.cmml"><mn id="S4.SS3.SSS1.p12.2.m2.1.1.2" xref="S4.SS3.SSS1.p12.2.m2.1.1.2.cmml">23.0</mn><mo id="S4.SS3.SSS1.p12.2.m2.1.1.3" xref="S4.SS3.SSS1.p12.2.m2.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p12.2.m2.1.1.4" xref="S4.SS3.SSS1.p12.2.m2.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p12.2.m2.1.1.5" xref="S4.SS3.SSS1.p12.2.m2.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p12.2.m2.1.1.6" xref="S4.SS3.SSS1.p12.2.m2.1.1.6.cmml">25.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p12.2.m2.1b"><apply id="S4.SS3.SSS1.p12.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1"><and id="S4.SS3.SSS1.p12.2.m2.1.1a.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1"></and><apply id="S4.SS3.SSS1.p12.2.m2.1.1b.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1"><lt id="S4.SS3.SSS1.p12.2.m2.1.1.3.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1.3"></lt><cn id="S4.SS3.SSS1.p12.2.m2.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p12.2.m2.1.1.2">23.0</cn><ci id="S4.SS3.SSS1.p12.2.m2.1.1.4.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p12.2.m2.1.1c.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1"><lt id="S4.SS3.SSS1.p12.2.m2.1.1.5.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p12.2.m2.1.1.4.cmml" id="S4.SS3.SSS1.p12.2.m2.1.1d.cmml" xref="S4.SS3.SSS1.p12.2.m2.1.1"></share><cn id="S4.SS3.SSS1.p12.2.m2.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p12.2.m2.1.1.6">25.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p12.2.m2.1c">23.0<z<25.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p12.2.m2.1d">23.0 < italic_z < 25.0</annotation></semantics></math> mag sample within <math alttext="4.0<z<5.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p12.3.m3.1"><semantics id="S4.SS3.SSS1.p12.3.m3.1a"><mrow id="S4.SS3.SSS1.p12.3.m3.1.1" xref="S4.SS3.SSS1.p12.3.m3.1.1.cmml"><mn id="S4.SS3.SSS1.p12.3.m3.1.1.2" xref="S4.SS3.SSS1.p12.3.m3.1.1.2.cmml">4.0</mn><mo id="S4.SS3.SSS1.p12.3.m3.1.1.3" xref="S4.SS3.SSS1.p12.3.m3.1.1.3.cmml"><</mo><mi id="S4.SS3.SSS1.p12.3.m3.1.1.4" xref="S4.SS3.SSS1.p12.3.m3.1.1.4.cmml">z</mi><mo id="S4.SS3.SSS1.p12.3.m3.1.1.5" xref="S4.SS3.SSS1.p12.3.m3.1.1.5.cmml"><</mo><mn id="S4.SS3.SSS1.p12.3.m3.1.1.6" xref="S4.SS3.SSS1.p12.3.m3.1.1.6.cmml">5.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p12.3.m3.1b"><apply id="S4.SS3.SSS1.p12.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1"><and id="S4.SS3.SSS1.p12.3.m3.1.1a.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1"></and><apply id="S4.SS3.SSS1.p12.3.m3.1.1b.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1"><lt id="S4.SS3.SSS1.p12.3.m3.1.1.3.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1.3"></lt><cn id="S4.SS3.SSS1.p12.3.m3.1.1.2.cmml" type="float" xref="S4.SS3.SSS1.p12.3.m3.1.1.2">4.0</cn><ci id="S4.SS3.SSS1.p12.3.m3.1.1.4.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1.4">𝑧</ci></apply><apply id="S4.SS3.SSS1.p12.3.m3.1.1c.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1"><lt id="S4.SS3.SSS1.p12.3.m3.1.1.5.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS3.SSS1.p12.3.m3.1.1.4.cmml" id="S4.SS3.SSS1.p12.3.m3.1.1d.cmml" xref="S4.SS3.SSS1.p12.3.m3.1.1"></share><cn id="S4.SS3.SSS1.p12.3.m3.1.1.6.cmml" type="float" xref="S4.SS3.SSS1.p12.3.m3.1.1.6">5.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p12.3.m3.1c">4.0<z<5.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p12.3.m3.1d">4.0 < italic_z < 5.5</annotation></semantics></math>, centered at <math alttext="z\sim 4.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p12.4.m4.1"><semantics id="S4.SS3.SSS1.p12.4.m4.1a"><mrow id="S4.SS3.SSS1.p12.4.m4.1.1" xref="S4.SS3.SSS1.p12.4.m4.1.1.cmml"><mi id="S4.SS3.SSS1.p12.4.m4.1.1.2" xref="S4.SS3.SSS1.p12.4.m4.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p12.4.m4.1.1.1" xref="S4.SS3.SSS1.p12.4.m4.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p12.4.m4.1.1.3" xref="S4.SS3.SSS1.p12.4.m4.1.1.3.cmml">4.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p12.4.m4.1b"><apply id="S4.SS3.SSS1.p12.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p12.4.m4.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p12.4.m4.1.1.1.cmml" xref="S4.SS3.SSS1.p12.4.m4.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS1.p12.4.m4.1.1.2.cmml" xref="S4.SS3.SSS1.p12.4.m4.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p12.4.m4.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p12.4.m4.1.1.3">4.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p12.4.m4.1c">z\sim 4.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p12.4.m4.1d">italic_z ∼ 4.5</annotation></semantics></math> using these <math alttext="riz" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p12.5.m5.1"><semantics id="S4.SS3.SSS1.p12.5.m5.1a"><mrow id="S4.SS3.SSS1.p12.5.m5.1.1" xref="S4.SS3.SSS1.p12.5.m5.1.1.cmml"><mi id="S4.SS3.SSS1.p12.5.m5.1.1.2" xref="S4.SS3.SSS1.p12.5.m5.1.1.2.cmml">r</mi><mo id="S4.SS3.SSS1.p12.5.m5.1.1.1" xref="S4.SS3.SSS1.p12.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p12.5.m5.1.1.3" xref="S4.SS3.SSS1.p12.5.m5.1.1.3.cmml">i</mi><mo id="S4.SS3.SSS1.p12.5.m5.1.1.1a" xref="S4.SS3.SSS1.p12.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p12.5.m5.1.1.4" xref="S4.SS3.SSS1.p12.5.m5.1.1.4.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p12.5.m5.1b"><apply id="S4.SS3.SSS1.p12.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p12.5.m5.1.1"><times id="S4.SS3.SSS1.p12.5.m5.1.1.1.cmml" xref="S4.SS3.SSS1.p12.5.m5.1.1.1"></times><ci id="S4.SS3.SSS1.p12.5.m5.1.1.2.cmml" xref="S4.SS3.SSS1.p12.5.m5.1.1.2">𝑟</ci><ci id="S4.SS3.SSS1.p12.5.m5.1.1.3.cmml" xref="S4.SS3.SSS1.p12.5.m5.1.1.3">𝑖</ci><ci id="S4.SS3.SSS1.p12.5.m5.1.1.4.cmml" xref="S4.SS3.SSS1.p12.5.m5.1.1.4">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p12.5.m5.1c">riz</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p12.5.m5.1d">italic_r italic_i italic_z</annotation></semantics></math> color cuts: </div> <div class="ltx_para" id="S4.SS3.SSS1.p13"> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E13"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E13X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(r-i)>0.65" class="ltx_Math" display="inline" id="S4.E13X.2.1.1.m1.1"><semantics id="S4.E13X.2.1.1.m1.1a"><mrow id="S4.E13X.2.1.1.m1.1.1" xref="S4.E13X.2.1.1.m1.1.1.cmml"><mrow id="S4.E13X.2.1.1.m1.1.1.1.1" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.cmml"><mo id="S4.E13X.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E13X.2.1.1.m1.1.1.1.1.1" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S4.E13X.2.1.1.m1.1.1.1.1.1.2" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.2.cmml">r</mi><mo id="S4.E13X.2.1.1.m1.1.1.1.1.1.1" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E13X.2.1.1.m1.1.1.1.1.1.3" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.3.cmml">i</mi></mrow><mo id="S4.E13X.2.1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E13X.2.1.1.m1.1.1.2" xref="S4.E13X.2.1.1.m1.1.1.2.cmml">></mo><mn id="S4.E13X.2.1.1.m1.1.1.3" xref="S4.E13X.2.1.1.m1.1.1.3.cmml">0.65</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.E13X.2.1.1.m1.1b"><apply id="S4.E13X.2.1.1.m1.1.1.cmml" xref="S4.E13X.2.1.1.m1.1.1"><gt id="S4.E13X.2.1.1.m1.1.1.2.cmml" xref="S4.E13X.2.1.1.m1.1.1.2"></gt><apply id="S4.E13X.2.1.1.m1.1.1.1.1.1.cmml" xref="S4.E13X.2.1.1.m1.1.1.1.1"><minus id="S4.E13X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.1"></minus><ci id="S4.E13X.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.2">𝑟</ci><ci id="S4.E13X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S4.E13X.2.1.1.m1.1.1.1.1.1.3">𝑖</ci></apply><cn id="S4.E13X.2.1.1.m1.1.1.3.cmml" type="float" xref="S4.E13X.2.1.1.m1.1.1.3">0.65</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13X.2.1.1.m1.1c">\displaystyle(r-i)>0.65</annotation><annotation encoding="application/x-llamapun" id="S4.E13X.2.1.1.m1.1d">( italic_r - italic_i ) > 0.65</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="3">(13)</td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E13Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle-0.5<(i-z)<0.9" class="ltx_Math" display="inline" id="S4.E13Xa.2.1.1.m1.1"><semantics id="S4.E13Xa.2.1.1.m1.1a"><mrow id="S4.E13Xa.2.1.1.m1.1.1" xref="S4.E13Xa.2.1.1.m1.1.1.cmml"><mrow id="S4.E13Xa.2.1.1.m1.1.1.3" xref="S4.E13Xa.2.1.1.m1.1.1.3.cmml"><mo id="S4.E13Xa.2.1.1.m1.1.1.3a" xref="S4.E13Xa.2.1.1.m1.1.1.3.cmml">−</mo><mn id="S4.E13Xa.2.1.1.m1.1.1.3.2" xref="S4.E13Xa.2.1.1.m1.1.1.3.2.cmml">0.5</mn></mrow><mo id="S4.E13Xa.2.1.1.m1.1.1.4" xref="S4.E13Xa.2.1.1.m1.1.1.4.cmml"><</mo><mrow id="S4.E13Xa.2.1.1.m1.1.1.1.1" xref="S4.E13Xa.2.1.1.m1.1.1.1.1.1.cmml"><mo id="S4.E13Xa.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.E13Xa.2.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E13Xa.2.1.1.m1.1.1.1.1.1" xref="S4.E13Xa.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S4.E13Xa.2.1.1.m1.1.1.1.1.1.2" xref="S4.E13Xa.2.1.1.m1.1.1.1.1.1.2.cmml">i</mi><mo id="S4.E13Xa.2.1.1.m1.1.1.1.1.1.1" xref="S4.E13Xa.2.1.1.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E13Xa.2.1.1.m1.1.1.1.1.1.3" xref="S4.E13Xa.2.1.1.m1.1.1.1.1.1.3.cmml">z</mi></mrow><mo 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xref="S4.E13Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.2">𝑖</ci><ci id="S4.E13Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.3.cmml" xref="S4.E13Xb.2.1.1.m1.1.1.1.1.2.1.1.1.1.3">𝑧</ci></apply></apply><cn id="S4.E13Xb.2.1.1.m1.1.1.1.1.2.3.cmml" type="float" xref="S4.E13Xb.2.1.1.m1.1.1.1.1.2.3">0.65</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13Xb.2.1.1.m1.1c">\displaystyle(r-i)>1.5\times(i-z)+0.65,</annotation><annotation encoding="application/x-llamapun" id="S4.E13Xb.2.1.1.m1.1d">( italic_r - italic_i ) > 1.5 × ( italic_i - italic_z ) + 0.65 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="S4.SS3.SSS1.p14"> Using photo-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p14.1.m1.1"><semantics id="S4.SS3.SSS1.p14.1.m1.1a"><mi id="S4.SS3.SSS1.p14.1.m1.1.1" xref="S4.SS3.SSS1.p14.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p14.1.m1.1b"><ci id="S4.SS3.SSS1.p14.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p14.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p14.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p14.1.m1.1d">italic_z</annotation></semantics></math>, this sample has a <math alttext="\sim 390" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p14.2.m2.1"><semantics id="S4.SS3.SSS1.p14.2.m2.1a"><mrow id="S4.SS3.SSS1.p14.2.m2.1.1" xref="S4.SS3.SSS1.p14.2.m2.1.1.cmml"><mi id="S4.SS3.SSS1.p14.2.m2.1.1.2" xref="S4.SS3.SSS1.p14.2.m2.1.1.2.cmml"></mi><mo id="S4.SS3.SSS1.p14.2.m2.1.1.1" xref="S4.SS3.SSS1.p14.2.m2.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p14.2.m2.1.1.3" xref="S4.SS3.SSS1.p14.2.m2.1.1.3.cmml">390</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p14.2.m2.1b"><apply id="S4.SS3.SSS1.p14.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p14.2.m2.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p14.2.m2.1.1.1.cmml" xref="S4.SS3.SSS1.p14.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.SSS1.p14.2.m2.1.1.2.cmml" xref="S4.SS3.SSS1.p14.2.m2.1.1.2">absent</csymbol><cn id="S4.SS3.SSS1.p14.2.m2.1.1.3.cmml" type="integer" xref="S4.SS3.SSS1.p14.2.m2.1.1.3">390</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p14.2.m2.1c">\sim 390</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p14.2.m2.1d">∼ 390</annotation></semantics></math> deg-2 density within the desired redshift range. Like the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p14.4.m4.1"><semantics id="S4.SS3.SSS1.p14.4.m4.1a"><mi id="S4.SS3.SSS1.p14.4.m4.1.1" xref="S4.SS3.SSS1.p14.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p14.4.m4.1b"><ci id="S4.SS3.SSS1.p14.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p14.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p14.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p14.4.m4.1d">italic_u</annotation></semantics></math>-dropouts, tweaking these preliminary color cuts can significantly change the target density but will not affect the redshift distributions. We want to emphasize that these color cuts are not designed for actual target selection, but only to support our assumptions for a more optimistic high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p14.5.m5.1"><semantics id="S4.SS3.SSS1.p14.5.m5.1a"><mi id="S4.SS3.SSS1.p14.5.m5.1.1" xref="S4.SS3.SSS1.p14.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p14.5.m5.1b"><ci id="S4.SS3.SSS1.p14.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p14.5.m5.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p14.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p14.5.m5.1d">italic_z</annotation></semantics></math> LSS tracer selection in the 2030s. </div> <div class="ltx_para" id="S4.SS3.SSS1.p15"> LBG will be the highest priority target for the dark time program of MUST. However, based on the empirical estimation of the required exposure time (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib237" title="">237</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib42" title="">42</a>]</cite>) for LBG and the results from recent DESI campaigns (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib22" title="">22</a>]</cite>), successfully recover redshifts down to <math alttext="r\sim 24.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p15.1.m1.1"><semantics id="S4.SS3.SSS1.p15.1.m1.1a"><mrow id="S4.SS3.SSS1.p15.1.m1.1.1" xref="S4.SS3.SSS1.p15.1.m1.1.1.cmml"><mi id="S4.SS3.SSS1.p15.1.m1.1.1.2" xref="S4.SS3.SSS1.p15.1.m1.1.1.2.cmml">r</mi><mo id="S4.SS3.SSS1.p15.1.m1.1.1.1" xref="S4.SS3.SSS1.p15.1.m1.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS1.p15.1.m1.1.1.3" xref="S4.SS3.SSS1.p15.1.m1.1.1.3.cmml">24.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p15.1.m1.1b"><apply id="S4.SS3.SSS1.p15.1.m1.1.1.cmml" xref="S4.SS3.SSS1.p15.1.m1.1.1"><csymbol cd="latexml" id="S4.SS3.SSS1.p15.1.m1.1.1.1.cmml" xref="S4.SS3.SSS1.p15.1.m1.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS1.p15.1.m1.1.1.2.cmml" xref="S4.SS3.SSS1.p15.1.m1.1.1.2">𝑟</ci><cn id="S4.SS3.SSS1.p15.1.m1.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p15.1.m1.1.1.3">24.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p15.1.m1.1c">r\sim 24.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p15.1.m1.1d">italic_r ∼ 24.5</annotation></semantics></math> mag or deeper is still a challenging task for a 6.5 m telescope. Not only does this challenge demand a careful scientific requirements rundown to guide the design of the scientific instruments, but it also motivates us to investigate an improved selection strategy that focuses on LBG with a clear Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p15.2.m2.1"><semantics id="S4.SS3.SSS1.p15.2.m2.1a"><mi id="S4.SS3.SSS1.p15.2.m2.1.1" xref="S4.SS3.SSS1.p15.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p15.2.m2.1b"><ci id="S4.SS3.SSS1.p15.2.m2.1.1.cmml" xref="S4.SS3.SSS1.p15.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p15.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p15.2.m2.1d">italic_α</annotation></semantics></math> emission line. As the throughput of the telescope, fiber, and detector all drop significantly toward the blue end at <math alttext="\lambda<4500\ \AA" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p15.3.m3.1"><semantics id="S4.SS3.SSS1.p15.3.m3.1a"><mrow id="S4.SS3.SSS1.p15.3.m3.1.1" xref="S4.SS3.SSS1.p15.3.m3.1.1.cmml"><mi id="S4.SS3.SSS1.p15.3.m3.1.1.2" xref="S4.SS3.SSS1.p15.3.m3.1.1.2.cmml">λ</mi><mo id="S4.SS3.SSS1.p15.3.m3.1.1.1" xref="S4.SS3.SSS1.p15.3.m3.1.1.1.cmml"><</mo><mrow id="S4.SS3.SSS1.p15.3.m3.1.1.3" xref="S4.SS3.SSS1.p15.3.m3.1.1.3.cmml"><mn id="S4.SS3.SSS1.p15.3.m3.1.1.3.2" xref="S4.SS3.SSS1.p15.3.m3.1.1.3.2.cmml">4500</mn><mo id="S4.SS3.SSS1.p15.3.m3.1.1.3.1" lspace="0.500em" xref="S4.SS3.SSS1.p15.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.SSS1.p15.3.m3.1.1.3.3" xref="S4.SS3.SSS1.p15.3.m3.1.1.3.3.cmml">Å</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p15.3.m3.1b"><apply id="S4.SS3.SSS1.p15.3.m3.1.1.cmml" xref="S4.SS3.SSS1.p15.3.m3.1.1"><lt id="S4.SS3.SSS1.p15.3.m3.1.1.1.cmml" xref="S4.SS3.SSS1.p15.3.m3.1.1.1"></lt><ci id="S4.SS3.SSS1.p15.3.m3.1.1.2.cmml" xref="S4.SS3.SSS1.p15.3.m3.1.1.2">𝜆</ci><apply id="S4.SS3.SSS1.p15.3.m3.1.1.3.cmml" xref="S4.SS3.SSS1.p15.3.m3.1.1.3"><times id="S4.SS3.SSS1.p15.3.m3.1.1.3.1.cmml" xref="S4.SS3.SSS1.p15.3.m3.1.1.3.1"></times><cn id="S4.SS3.SSS1.p15.3.m3.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS1.p15.3.m3.1.1.3.2">4500</cn><ci id="S4.SS3.SSS1.p15.3.m3.1.1.3.3.cmml" xref="S4.SS3.SSS1.p15.3.m3.1.1.3.3">italic-Å</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p15.3.m3.1c">\lambda<4500\ \AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p15.3.m3.1d">italic_λ < 4500 italic_Å</annotation></semantics></math>, we expect the redshift success rate for <math alttext="u" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p15.4.m4.1"><semantics id="S4.SS3.SSS1.p15.4.m4.1a"><mi id="S4.SS3.SSS1.p15.4.m4.1.1" xref="S4.SS3.SSS1.p15.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p15.4.m4.1b"><ci id="S4.SS3.SSS1.p15.4.m4.1.1.cmml" xref="S4.SS3.SSS1.p15.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p15.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p15.4.m4.1d">italic_u</annotation></semantics></math>-dropout also reflects this at <math alttext="z<2.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p15.5.m5.1"><semantics id="S4.SS3.SSS1.p15.5.m5.1a"><mrow id="S4.SS3.SSS1.p15.5.m5.1.1" xref="S4.SS3.SSS1.p15.5.m5.1.1.cmml"><mi id="S4.SS3.SSS1.p15.5.m5.1.1.2" xref="S4.SS3.SSS1.p15.5.m5.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p15.5.m5.1.1.1" xref="S4.SS3.SSS1.p15.5.m5.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p15.5.m5.1.1.3" xref="S4.SS3.SSS1.p15.5.m5.1.1.3.cmml">2.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p15.5.m5.1b"><apply id="S4.SS3.SSS1.p15.5.m5.1.1.cmml" xref="S4.SS3.SSS1.p15.5.m5.1.1"><lt id="S4.SS3.SSS1.p15.5.m5.1.1.1.cmml" xref="S4.SS3.SSS1.p15.5.m5.1.1.1"></lt><ci id="S4.SS3.SSS1.p15.5.m5.1.1.2.cmml" xref="S4.SS3.SSS1.p15.5.m5.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p15.5.m5.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p15.5.m5.1.1.3">2.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p15.5.m5.1c">z<2.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p15.5.m5.1d">italic_z < 2.5</annotation></semantics></math>. Given that we have not yet included the BX selection with a high target density, in principle, MUST should be able to guarantee the 3-D tracer density for cosmology at <math alttext="z<2.5" class="ltx_Math" display="inline" id="S4.SS3.SSS1.p15.6.m6.1"><semantics id="S4.SS3.SSS1.p15.6.m6.1a"><mrow id="S4.SS3.SSS1.p15.6.m6.1.1" xref="S4.SS3.SSS1.p15.6.m6.1.1.cmml"><mi id="S4.SS3.SSS1.p15.6.m6.1.1.2" xref="S4.SS3.SSS1.p15.6.m6.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS1.p15.6.m6.1.1.1" xref="S4.SS3.SSS1.p15.6.m6.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS1.p15.6.m6.1.1.3" xref="S4.SS3.SSS1.p15.6.m6.1.1.3.cmml">2.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS1.p15.6.m6.1b"><apply id="S4.SS3.SSS1.p15.6.m6.1.1.cmml" xref="S4.SS3.SSS1.p15.6.m6.1.1"><lt id="S4.SS3.SSS1.p15.6.m6.1.1.1.cmml" xref="S4.SS3.SSS1.p15.6.m6.1.1.1"></lt><ci id="S4.SS3.SSS1.p15.6.m6.1.1.2.cmml" xref="S4.SS3.SSS1.p15.6.m6.1.1.2">𝑧</ci><cn id="S4.SS3.SSS1.p15.6.m6.1.1.3.cmml" type="float" xref="S4.SS3.SSS1.p15.6.m6.1.1.3">2.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS1.p15.6.m6.1c">z<2.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS1.p15.6.m6.1d">italic_z < 2.5</annotation></semantics></math>. More importantly, despite the uncertainties around the fiber assignment and redshift efficiencies for LBG, the contract between our optimistic and conservative target density estimation should make our cosmological forecast more realistic for MUST. </div> </section> <section class="ltx_subsubsection" id="S4.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> 4.3.2 Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS2.1.m1.1"><semantics id="S4.SS3.SSS2.1.m1.1b"><mi id="S4.SS3.SSS2.1.m1.1.1" xref="S4.SS3.SSS2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.1.m1.1c"><ci id="S4.SS3.SSS2.1.m1.1.1.cmml" xref="S4.SS3.SSS2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.1.m1.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.1.m1.1e">italic_α</annotation></semantics></math> Emitters (LAE)</h4> <div class="ltx_para" id="S4.SS3.SSS2.p1"> LAEs are high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.1.m1.1"><semantics id="S4.SS3.SSS2.p1.1.m1.1a"><mi id="S4.SS3.SSS2.p1.1.m1.1.1" xref="S4.SS3.SSS2.p1.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.1.m1.1b"><ci id="S4.SS3.SSS2.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS2.p1.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.1.m1.1d">italic_z</annotation></semantics></math> galaxies with strong (<math alttext="{\rm EW}\geq 20\ \AA" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.2.m2.1"><semantics id="S4.SS3.SSS2.p1.2.m2.1a"><mrow id="S4.SS3.SSS2.p1.2.m2.1.1" xref="S4.SS3.SSS2.p1.2.m2.1.1.cmml"><mi id="S4.SS3.SSS2.p1.2.m2.1.1.2" xref="S4.SS3.SSS2.p1.2.m2.1.1.2.cmml">EW</mi><mo id="S4.SS3.SSS2.p1.2.m2.1.1.1" xref="S4.SS3.SSS2.p1.2.m2.1.1.1.cmml">≥</mo><mrow id="S4.SS3.SSS2.p1.2.m2.1.1.3" xref="S4.SS3.SSS2.p1.2.m2.1.1.3.cmml"><mn id="S4.SS3.SSS2.p1.2.m2.1.1.3.2" xref="S4.SS3.SSS2.p1.2.m2.1.1.3.2.cmml">20</mn><mo id="S4.SS3.SSS2.p1.2.m2.1.1.3.1" lspace="0.500em" xref="S4.SS3.SSS2.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.SSS2.p1.2.m2.1.1.3.3" xref="S4.SS3.SSS2.p1.2.m2.1.1.3.3.cmml">Å</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.2.m2.1b"><apply id="S4.SS3.SSS2.p1.2.m2.1.1.cmml" xref="S4.SS3.SSS2.p1.2.m2.1.1"><geq id="S4.SS3.SSS2.p1.2.m2.1.1.1.cmml" xref="S4.SS3.SSS2.p1.2.m2.1.1.1"></geq><ci id="S4.SS3.SSS2.p1.2.m2.1.1.2.cmml" xref="S4.SS3.SSS2.p1.2.m2.1.1.2">EW</ci><apply id="S4.SS3.SSS2.p1.2.m2.1.1.3.cmml" xref="S4.SS3.SSS2.p1.2.m2.1.1.3"><times id="S4.SS3.SSS2.p1.2.m2.1.1.3.1.cmml" xref="S4.SS3.SSS2.p1.2.m2.1.1.3.1"></times><cn id="S4.SS3.SSS2.p1.2.m2.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS2.p1.2.m2.1.1.3.2">20</cn><ci id="S4.SS3.SSS2.p1.2.m2.1.1.3.3.cmml" xref="S4.SS3.SSS2.p1.2.m2.1.1.3.3">italic-Å</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.2.m2.1c">{\rm EW}\geq 20\ \AA</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.2.m2.1d">roman_EW ≥ 20 italic_Å</annotation></semantics></math>) Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.3.m3.1"><semantics id="S4.SS3.SSS2.p1.3.m3.1a"><mi id="S4.SS3.SSS2.p1.3.m3.1.1" xref="S4.SS3.SSS2.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.3.m3.1b"><ci id="S4.SS3.SSS2.p1.3.m3.1.1.cmml" xref="S4.SS3.SSS2.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.3.m3.1d">italic_α</annotation></semantics></math> emission lines (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib248" title="">248</a>]</cite>), which are typically young, star-forming galaxies with low stellar and dust mass. As LAEs live in dark matter halos with lower average halo mass than LBG, their potential density should be significant enough to make them competitive high-redshift LSS tracers. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib211" title="">211</a>]</cite>, the authors spectroscopically confirm 822 (1,099) <math alttext="z=2.40\pm 0.03" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.4.m4.1"><semantics id="S4.SS3.SSS2.p1.4.m4.1a"><mrow id="S4.SS3.SSS2.p1.4.m4.1.1" xref="S4.SS3.SSS2.p1.4.m4.1.1.cmml"><mi id="S4.SS3.SSS2.p1.4.m4.1.1.2" xref="S4.SS3.SSS2.p1.4.m4.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS2.p1.4.m4.1.1.1" xref="S4.SS3.SSS2.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S4.SS3.SSS2.p1.4.m4.1.1.3" xref="S4.SS3.SSS2.p1.4.m4.1.1.3.cmml"><mn id="S4.SS3.SSS2.p1.4.m4.1.1.3.2" xref="S4.SS3.SSS2.p1.4.m4.1.1.3.2.cmml">2.40</mn><mo id="S4.SS3.SSS2.p1.4.m4.1.1.3.1" xref="S4.SS3.SSS2.p1.4.m4.1.1.3.1.cmml">±</mo><mn id="S4.SS3.SSS2.p1.4.m4.1.1.3.3" xref="S4.SS3.SSS2.p1.4.m4.1.1.3.3.cmml">0.03</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.4.m4.1b"><apply id="S4.SS3.SSS2.p1.4.m4.1.1.cmml" xref="S4.SS3.SSS2.p1.4.m4.1.1"><eq id="S4.SS3.SSS2.p1.4.m4.1.1.1.cmml" xref="S4.SS3.SSS2.p1.4.m4.1.1.1"></eq><ci id="S4.SS3.SSS2.p1.4.m4.1.1.2.cmml" xref="S4.SS3.SSS2.p1.4.m4.1.1.2">𝑧</ci><apply id="S4.SS3.SSS2.p1.4.m4.1.1.3.cmml" xref="S4.SS3.SSS2.p1.4.m4.1.1.3"><csymbol cd="latexml" id="S4.SS3.SSS2.p1.4.m4.1.1.3.1.cmml" xref="S4.SS3.SSS2.p1.4.m4.1.1.3.1">plus-or-minus</csymbol><cn id="S4.SS3.SSS2.p1.4.m4.1.1.3.2.cmml" type="float" xref="S4.SS3.SSS2.p1.4.m4.1.1.3.2">2.40</cn><cn id="S4.SS3.SSS2.p1.4.m4.1.1.3.3.cmml" type="float" xref="S4.SS3.SSS2.p1.4.m4.1.1.3.3">0.03</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.4.m4.1c">z=2.40\pm 0.03</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.4.m4.1d">italic_z = 2.40 ± 0.03</annotation></semantics></math> (<math alttext="3.10\pm 0.03" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.5.m5.1"><semantics id="S4.SS3.SSS2.p1.5.m5.1a"><mrow id="S4.SS3.SSS2.p1.5.m5.1.1" xref="S4.SS3.SSS2.p1.5.m5.1.1.cmml"><mn id="S4.SS3.SSS2.p1.5.m5.1.1.2" xref="S4.SS3.SSS2.p1.5.m5.1.1.2.cmml">3.10</mn><mo id="S4.SS3.SSS2.p1.5.m5.1.1.1" xref="S4.SS3.SSS2.p1.5.m5.1.1.1.cmml">±</mo><mn id="S4.SS3.SSS2.p1.5.m5.1.1.3" xref="S4.SS3.SSS2.p1.5.m5.1.1.3.cmml">0.03</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.5.m5.1b"><apply id="S4.SS3.SSS2.p1.5.m5.1.1.cmml" xref="S4.SS3.SSS2.p1.5.m5.1.1"><csymbol cd="latexml" id="S4.SS3.SSS2.p1.5.m5.1.1.1.cmml" xref="S4.SS3.SSS2.p1.5.m5.1.1.1">plus-or-minus</csymbol><cn id="S4.SS3.SSS2.p1.5.m5.1.1.2.cmml" type="float" xref="S4.SS3.SSS2.p1.5.m5.1.1.2">3.10</cn><cn id="S4.SS3.SSS2.p1.5.m5.1.1.3.cmml" type="float" xref="S4.SS3.SSS2.p1.5.m5.1.1.3">0.03</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.5.m5.1c">3.10\pm 0.03</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.5.m5.1d">3.10 ± 0.03</annotation></semantics></math>) LAEs in the 8.90 (9.34) deg2 ODIN fields using the N419 (N501) narrow-band filters down to 25.5 (25.7) 5-<math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.7.m7.1"><semantics id="S4.SS3.SSS2.p1.7.m7.1a"><mi id="S4.SS3.SSS2.p1.7.m7.1.1" xref="S4.SS3.SSS2.p1.7.m7.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.7.m7.1b"><ci id="S4.SS3.SSS2.p1.7.m7.1.1.cmml" xref="S4.SS3.SSS2.p1.7.m7.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.7.m7.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.7.m7.1d">italic_σ</annotation></semantics></math> detection limit. These samples correspond to a <math alttext="\sim 10^{-3}h^{-1}{\rm Mpc}^{-3}" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.8.m8.1"><semantics id="S4.SS3.SSS2.p1.8.m8.1a"><mrow id="S4.SS3.SSS2.p1.8.m8.1.1" xref="S4.SS3.SSS2.p1.8.m8.1.1.cmml"><mi id="S4.SS3.SSS2.p1.8.m8.1.1.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.2.cmml"></mi><mo id="S4.SS3.SSS2.p1.8.m8.1.1.1" xref="S4.SS3.SSS2.p1.8.m8.1.1.1.cmml">∼</mo><mrow id="S4.SS3.SSS2.p1.8.m8.1.1.3" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.cmml"><msup id="S4.SS3.SSS2.p1.8.m8.1.1.3.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.cmml"><mn id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.2.cmml">10</mn><mrow id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.cmml"><mo id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3a" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.cmml">−</mo><mn id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.2.cmml">3</mn></mrow></msup><mo id="S4.SS3.SSS2.p1.8.m8.1.1.3.1" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.1.cmml">⁢</mo><msup id="S4.SS3.SSS2.p1.8.m8.1.1.3.3" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.cmml"><mi id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.2.cmml">h</mi><mrow id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.cmml"><mo id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3a" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.cmml">−</mo><mn id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS3.SSS2.p1.8.m8.1.1.3.1a" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.1.cmml">⁢</mo><msup id="S4.SS3.SSS2.p1.8.m8.1.1.3.4" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.cmml"><mi id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.2.cmml">Mpc</mi><mrow id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.cmml"><mo id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3a" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.cmml">−</mo><mn id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.2" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.2.cmml">3</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.8.m8.1b"><apply id="S4.SS3.SSS2.p1.8.m8.1.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1"><csymbol cd="latexml" id="S4.SS3.SSS2.p1.8.m8.1.1.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.SSS2.p1.8.m8.1.1.2.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.2">absent</csymbol><apply id="S4.SS3.SSS2.p1.8.m8.1.1.3.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3"><times id="S4.SS3.SSS2.p1.8.m8.1.1.3.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.1"></times><apply id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2">superscript</csymbol><cn id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.2.cmml" type="integer" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.2">10</cn><apply id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3"><minus id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3"></minus><cn id="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.2.cmml" type="integer" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.2.3.2">3</cn></apply></apply><apply id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3">superscript</csymbol><ci id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.2.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.2">ℎ</ci><apply id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3"><minus id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3"></minus><cn id="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.2.cmml" type="integer" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.3.3.2">1</cn></apply></apply><apply id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4">superscript</csymbol><ci id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.2.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.2">Mpc</ci><apply id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3"><minus id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.1.cmml" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3"></minus><cn id="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.2.cmml" type="integer" xref="S4.SS3.SSS2.p1.8.m8.1.1.3.4.3.2">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.8.m8.1c">\sim 10^{-3}h^{-1}{\rm Mpc}^{-3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.8.m8.1d">∼ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_Mpc start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT</annotation></semantics></math> 3D density in two narrow redshift windows, higher than the optimistic prediction of LBG 3D density across the whole redshift range. If such performance can be extrapolated to a broader redshift range, LAEs can become an extremely interesting LSS tracer: not only do they dramatically improve the density of high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.9.m9.1"><semantics id="S4.SS3.SSS2.p1.9.m9.1a"><mi id="S4.SS3.SSS2.p1.9.m9.1.1" xref="S4.SS3.SSS2.p1.9.m9.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.9.m9.1b"><ci id="S4.SS3.SSS2.p1.9.m9.1.1.cmml" xref="S4.SS3.SSS2.p1.9.m9.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.9.m9.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.9.m9.1d">italic_z</annotation></semantics></math> LSS tracers, but their halo bias should also be significantly lower than the dropout-selected LBG (e.g., <math alttext="b\sim 1.7" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.10.m10.1"><semantics id="S4.SS3.SSS2.p1.10.m10.1a"><mrow id="S4.SS3.SSS2.p1.10.m10.1.1" xref="S4.SS3.SSS2.p1.10.m10.1.1.cmml"><mi id="S4.SS3.SSS2.p1.10.m10.1.1.2" xref="S4.SS3.SSS2.p1.10.m10.1.1.2.cmml">b</mi><mo id="S4.SS3.SSS2.p1.10.m10.1.1.1" xref="S4.SS3.SSS2.p1.10.m10.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS2.p1.10.m10.1.1.3" xref="S4.SS3.SSS2.p1.10.m10.1.1.3.cmml">1.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.10.m10.1b"><apply id="S4.SS3.SSS2.p1.10.m10.1.1.cmml" xref="S4.SS3.SSS2.p1.10.m10.1.1"><csymbol cd="latexml" id="S4.SS3.SSS2.p1.10.m10.1.1.1.cmml" xref="S4.SS3.SSS2.p1.10.m10.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS2.p1.10.m10.1.1.2.cmml" xref="S4.SS3.SSS2.p1.10.m10.1.1.2">𝑏</ci><cn id="S4.SS3.SSS2.p1.10.m10.1.1.3.cmml" type="float" xref="S4.SS3.SSS2.p1.10.m10.1.1.3">1.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.10.m10.1c">b\sim 1.7</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.10.m10.1d">italic_b ∼ 1.7</annotation></semantics></math> and 2.0 for the two ODIN samples), making multi-tracer probe potentially possible at high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p1.11.m11.1"><semantics id="S4.SS3.SSS2.p1.11.m11.1a"><mi id="S4.SS3.SSS2.p1.11.m11.1.1" xref="S4.SS3.SSS2.p1.11.m11.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p1.11.m11.1b"><ci id="S4.SS3.SSS2.p1.11.m11.1.1.cmml" xref="S4.SS3.SSS2.p1.11.m11.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p1.11.m11.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p1.11.m11.1d">italic_z</annotation></semantics></math>. </div> <div class="ltx_para" id="S4.SS3.SSS2.p2"> However, due to their faint continuum emission and the limited number of spectral features (often just the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p2.1.m1.1"><semantics id="S4.SS3.SSS2.p2.1.m1.1a"><mi id="S4.SS3.SSS2.p2.1.m1.1.1" xref="S4.SS3.SSS2.p2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p2.1.m1.1b"><ci id="S4.SS3.SSS2.p2.1.m1.1.1.cmml" xref="S4.SS3.SSS2.p2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p2.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p2.1.m1.1d">italic_α</annotation></semantics></math> line), the selection of LAEs demands deep narrow-band imaging (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib215" title="">215</a>]</cite>) or IFS (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib249" title="">249</a>]</cite>) observations. Most ongoing LAE surveys either cover much smaller areas than the MUST footprint (e.g., ODIN; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib215" title="">215</a>]</cite>) or are not deep enough for MUST (e.g., JPAS; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib250" title="">250</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib251" title="">251</a>]</cite>). Furthermore, the bias of LAEs has only been measured on relatively small fields (<math alttext="\lesssim 100{\rm deg}^{2}" class="ltx_Math" display="inline" id="S4.SS3.SSS2.p2.2.m2.1"><semantics id="S4.SS3.SSS2.p2.2.m2.1a"><mrow id="S4.SS3.SSS2.p2.2.m2.1.1" xref="S4.SS3.SSS2.p2.2.m2.1.1.cmml"><mi id="S4.SS3.SSS2.p2.2.m2.1.1.2" xref="S4.SS3.SSS2.p2.2.m2.1.1.2.cmml"></mi><mo id="S4.SS3.SSS2.p2.2.m2.1.1.1" xref="S4.SS3.SSS2.p2.2.m2.1.1.1.cmml">≲</mo><mrow id="S4.SS3.SSS2.p2.2.m2.1.1.3" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.cmml"><mn id="S4.SS3.SSS2.p2.2.m2.1.1.3.2" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.2.cmml">100</mn><mo id="S4.SS3.SSS2.p2.2.m2.1.1.3.1" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.SSS2.p2.2.m2.1.1.3.3" mathvariant="normal" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.3.cmml">d</mi><mo id="S4.SS3.SSS2.p2.2.m2.1.1.3.1a" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS3.SSS2.p2.2.m2.1.1.3.4" mathvariant="normal" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.4.cmml">e</mi><mo id="S4.SS3.SSS2.p2.2.m2.1.1.3.1b" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><msup id="S4.SS3.SSS2.p2.2.m2.1.1.3.5" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.5.cmml"><mi id="S4.SS3.SSS2.p2.2.m2.1.1.3.5.2" mathvariant="normal" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.5.2.cmml">g</mi><mn id="S4.SS3.SSS2.p2.2.m2.1.1.3.5.3" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.5.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS2.p2.2.m2.1b"><apply id="S4.SS3.SSS2.p2.2.m2.1.1.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1"><csymbol cd="latexml" id="S4.SS3.SSS2.p2.2.m2.1.1.1.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.1">less-than-or-similar-to</csymbol><csymbol cd="latexml" id="S4.SS3.SSS2.p2.2.m2.1.1.2.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.2">absent</csymbol><apply id="S4.SS3.SSS2.p2.2.m2.1.1.3.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.3"><times id="S4.SS3.SSS2.p2.2.m2.1.1.3.1.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.1"></times><cn id="S4.SS3.SSS2.p2.2.m2.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.2">100</cn><ci id="S4.SS3.SSS2.p2.2.m2.1.1.3.3.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.3">d</ci><ci id="S4.SS3.SSS2.p2.2.m2.1.1.3.4.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.4">e</ci><apply id="S4.SS3.SSS2.p2.2.m2.1.1.3.5.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.5"><csymbol cd="ambiguous" id="S4.SS3.SSS2.p2.2.m2.1.1.3.5.1.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.5">superscript</csymbol><ci id="S4.SS3.SSS2.p2.2.m2.1.1.3.5.2.cmml" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.5.2">g</ci><cn id="S4.SS3.SSS2.p2.2.m2.1.1.3.5.3.cmml" type="integer" xref="S4.SS3.SSS2.p2.2.m2.1.1.3.5.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS2.p2.2.m2.1c">\lesssim 100{\rm deg}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS2.p2.2.m2.1d">≲ 100 roman_d roman_e roman_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib252" title="">252</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib211" title="">211</a>]</cite>). We do not include LAEs in our current cosmological forecast for these reasons. At the same time, this implies that the cosmological constraining power of MUST could be further enhanced when a wide & deep LAE sample becomes available in the future. We should note that the DECam on the 4 m Blanco telescope (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib214" title="">214</a>]</cite>) and the HSC on the 8.2 m Subaru (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib153" title="">153</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib154" title="">154</a>]</cite>) can carry out deep narrow- or medium-band surveys in the following years. It is worth noting that DESI has conducted pilot spectroscopic observations of LAEs, demonstrating their promise as LSS tracers of the high-redshift Universe. </div> </section> <section class="ltx_subsubsection" id="S4.SS3.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> 4.3.3 Quasi-Stellar Objects (QSO)</h4> <div class="ltx_para" id="S4.SS3.SSS3.p1"> Quasi-stellar objects (QSOs) or quasars111111We use the term QSO and quasar interchangeably in this work. are both direct tracers of the dark matter field and sources for Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p1.1.m1.1"><semantics id="S4.SS3.SSS3.p1.1.m1.1a"><mi id="S4.SS3.SSS3.p1.1.m1.1.1" xref="S4.SS3.SSS3.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p1.1.m1.1b"><ci id="S4.SS3.SSS3.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS3.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p1.1.m1.1d">italic_α</annotation></semantics></math> forest detections. The transition in their usage – from tracers to sources – is typically set around <math alttext="z\sim 2" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p1.2.m2.1"><semantics id="S4.SS3.SSS3.p1.2.m2.1a"><mrow id="S4.SS3.SSS3.p1.2.m2.1.1" xref="S4.SS3.SSS3.p1.2.m2.1.1.cmml"><mi id="S4.SS3.SSS3.p1.2.m2.1.1.2" xref="S4.SS3.SSS3.p1.2.m2.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS3.p1.2.m2.1.1.1" xref="S4.SS3.SSS3.p1.2.m2.1.1.1.cmml">∼</mo><mn id="S4.SS3.SSS3.p1.2.m2.1.1.3" xref="S4.SS3.SSS3.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p1.2.m2.1b"><apply id="S4.SS3.SSS3.p1.2.m2.1.1.cmml" xref="S4.SS3.SSS3.p1.2.m2.1.1"><csymbol cd="latexml" id="S4.SS3.SSS3.p1.2.m2.1.1.1.cmml" xref="S4.SS3.SSS3.p1.2.m2.1.1.1">similar-to</csymbol><ci id="S4.SS3.SSS3.p1.2.m2.1.1.2.cmml" xref="S4.SS3.SSS3.p1.2.m2.1.1.2">𝑧</ci><cn id="S4.SS3.SSS3.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS3.SSS3.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p1.2.m2.1c">z\sim 2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p1.2.m2.1d">italic_z ∼ 2</annotation></semantics></math>, depending on their comoving densities. QSOs have been an essential ingredient of major spectroscopic surveys as the primary targets for probing the <math alttext="z\gtrsim 2" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p1.3.m3.1"><semantics id="S4.SS3.SSS3.p1.3.m3.1a"><mrow id="S4.SS3.SSS3.p1.3.m3.1.1" xref="S4.SS3.SSS3.p1.3.m3.1.1.cmml"><mi id="S4.SS3.SSS3.p1.3.m3.1.1.2" xref="S4.SS3.SSS3.p1.3.m3.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS3.p1.3.m3.1.1.1" xref="S4.SS3.SSS3.p1.3.m3.1.1.1.cmml">≳</mo><mn id="S4.SS3.SSS3.p1.3.m3.1.1.3" xref="S4.SS3.SSS3.p1.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p1.3.m3.1b"><apply id="S4.SS3.SSS3.p1.3.m3.1.1.cmml" xref="S4.SS3.SSS3.p1.3.m3.1.1"><csymbol cd="latexml" id="S4.SS3.SSS3.p1.3.m3.1.1.1.cmml" xref="S4.SS3.SSS3.p1.3.m3.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S4.SS3.SSS3.p1.3.m3.1.1.2.cmml" xref="S4.SS3.SSS3.p1.3.m3.1.1.2">𝑧</ci><cn id="S4.SS3.SSS3.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.SS3.SSS3.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p1.3.m3.1c">z\gtrsim 2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p1.3.m3.1d">italic_z ≳ 2</annotation></semantics></math> Universe. The Stage-III and IV spectroscopic surveys, such as BOSS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib253" title="">253</a>]</cite>, eBOSS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib254" title="">254</a>]</cite>, and DESI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib255" title="">255</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib205" title="">205</a>]</cite>, define the modern standard of QSO target selection using multi-band deep imaging data and broad-band color cuts to isolate quasar candidates from stars and other galaxies (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib256" title="">256</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib257" title="">257</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib258" title="">258</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib259" title="">259</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib260" title="">260</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib261" title="">261</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib262" title="">262</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib263" title="">263</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib264" title="">264</a>]</cite>). At the same time, these surveys have explored QSO selection based on their flux variability (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib265" title="">265</a>]</cite>) and using machine learning algorithms (e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib266" title="">266</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib267" title="">267</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib205" title="">205</a>]</cite>). </div> <div class="ltx_para" id="S4.SS3.SSS3.p2"> While QSOs will certainly enable a wide range of interesting scientific topics in the age of Stage-V surveys, their low 3D density and color degeneracy with abundant Milky Way stars at <math alttext="z>2" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p2.1.m1.1"><semantics id="S4.SS3.SSS3.p2.1.m1.1a"><mrow id="S4.SS3.SSS3.p2.1.m1.1.1" xref="S4.SS3.SSS3.p2.1.m1.1.1.cmml"><mi id="S4.SS3.SSS3.p2.1.m1.1.1.2" xref="S4.SS3.SSS3.p2.1.m1.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS3.p2.1.m1.1.1.1" xref="S4.SS3.SSS3.p2.1.m1.1.1.1.cmml">></mo><mn id="S4.SS3.SSS3.p2.1.m1.1.1.3" xref="S4.SS3.SSS3.p2.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p2.1.m1.1b"><apply id="S4.SS3.SSS3.p2.1.m1.1.1.cmml" xref="S4.SS3.SSS3.p2.1.m1.1.1"><gt id="S4.SS3.SSS3.p2.1.m1.1.1.1.cmml" xref="S4.SS3.SSS3.p2.1.m1.1.1.1"></gt><ci id="S4.SS3.SSS3.p2.1.m1.1.1.2.cmml" xref="S4.SS3.SSS3.p2.1.m1.1.1.2">𝑧</ci><cn id="S4.SS3.SSS3.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS3.p2.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p2.1.m1.1c">z>2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p2.1.m1.1d">italic_z > 2</annotation></semantics></math> (e.g., see Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.F4" title="Figure 4 ‣ 3.6.4 Gravitational waves and Fast Radio Bursts ‣ 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4</a>) make them a less appealing LSS tracers. Therefore, we do not include QSO in the Fisher forecast to constrain cosmological parameters. In Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.F4" title="Figure 4 ‣ 3.6.4 Gravitational waves and Fast Radio Bursts ‣ 3.6 Synergy with Other Probes ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4</a>, we adopt the pure luminosity function evolution model in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib265" title="">265</a>]</cite> to estimate the 3D density of QSOs at <math alttext="r<23.5" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p2.2.m2.1"><semantics id="S4.SS3.SSS3.p2.2.m2.1a"><mrow id="S4.SS3.SSS3.p2.2.m2.1.1" xref="S4.SS3.SSS3.p2.2.m2.1.1.cmml"><mi id="S4.SS3.SSS3.p2.2.m2.1.1.2" xref="S4.SS3.SSS3.p2.2.m2.1.1.2.cmml">r</mi><mo id="S4.SS3.SSS3.p2.2.m2.1.1.1" xref="S4.SS3.SSS3.p2.2.m2.1.1.1.cmml"><</mo><mn id="S4.SS3.SSS3.p2.2.m2.1.1.3" xref="S4.SS3.SSS3.p2.2.m2.1.1.3.cmml">23.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p2.2.m2.1b"><apply id="S4.SS3.SSS3.p2.2.m2.1.1.cmml" xref="S4.SS3.SSS3.p2.2.m2.1.1"><lt id="S4.SS3.SSS3.p2.2.m2.1.1.1.cmml" xref="S4.SS3.SSS3.p2.2.m2.1.1.1"></lt><ci id="S4.SS3.SSS3.p2.2.m2.1.1.2.cmml" xref="S4.SS3.SSS3.p2.2.m2.1.1.2">𝑟</ci><cn id="S4.SS3.SSS3.p2.2.m2.1.1.3.cmml" type="float" xref="S4.SS3.SSS3.p2.2.m2.1.1.3">23.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p2.2.m2.1c">r<23.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p2.2.m2.1d">italic_r < 23.5</annotation></semantics></math>. Assuming 80% of the QSOs will be observed, MUST will reach a QSO density of <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p2.3.m3.1"><semantics id="S4.SS3.SSS3.p2.3.m3.1a"><mo id="S4.SS3.SSS3.p2.3.m3.1.1" xref="S4.SS3.SSS3.p2.3.m3.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p2.3.m3.1b"><csymbol cd="latexml" id="S4.SS3.SSS3.p2.3.m3.1.1.cmml" xref="S4.SS3.SSS3.p2.3.m3.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p2.3.m3.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p2.3.m3.1d">∼</annotation></semantics></math>310 per deg-2 with <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p2.5.m5.1"><semantics id="S4.SS3.SSS3.p2.5.m5.1a"><mo id="S4.SS3.SSS3.p2.5.m5.1.1" xref="S4.SS3.SSS3.p2.5.m5.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p2.5.m5.1b"><csymbol cd="latexml" id="S4.SS3.SSS3.p2.5.m5.1.1.cmml" xref="S4.SS3.SSS3.p2.5.m5.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p2.5.m5.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p2.5.m5.1d">∼</annotation></semantics></math>90 deg-2 at <math alttext="z>2.1" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p2.7.m7.1"><semantics id="S4.SS3.SSS3.p2.7.m7.1a"><mrow id="S4.SS3.SSS3.p2.7.m7.1.1" xref="S4.SS3.SSS3.p2.7.m7.1.1.cmml"><mi id="S4.SS3.SSS3.p2.7.m7.1.1.2" xref="S4.SS3.SSS3.p2.7.m7.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS3.p2.7.m7.1.1.1" xref="S4.SS3.SSS3.p2.7.m7.1.1.1.cmml">></mo><mn id="S4.SS3.SSS3.p2.7.m7.1.1.3" xref="S4.SS3.SSS3.p2.7.m7.1.1.3.cmml">2.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p2.7.m7.1b"><apply id="S4.SS3.SSS3.p2.7.m7.1.1.cmml" xref="S4.SS3.SSS3.p2.7.m7.1.1"><gt id="S4.SS3.SSS3.p2.7.m7.1.1.1.cmml" xref="S4.SS3.SSS3.p2.7.m7.1.1.1"></gt><ci id="S4.SS3.SSS3.p2.7.m7.1.1.2.cmml" xref="S4.SS3.SSS3.p2.7.m7.1.1.2">𝑧</ci><cn id="S4.SS3.SSS3.p2.7.m7.1.1.3.cmml" type="float" xref="S4.SS3.SSS3.p2.7.m7.1.1.3">2.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p2.7.m7.1c">z>2.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p2.7.m7.1d">italic_z > 2.1</annotation></semantics></math>, including the QSOs already observed by BOSS/eBOSS and DESI (<math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p2.8.m8.1"><semantics id="S4.SS3.SSS3.p2.8.m8.1a"><mo id="S4.SS3.SSS3.p2.8.m8.1.1" xref="S4.SS3.SSS3.p2.8.m8.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p2.8.m8.1b"><csymbol cd="latexml" id="S4.SS3.SSS3.p2.8.m8.1.1.cmml" xref="S4.SS3.SSS3.p2.8.m8.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p2.8.m8.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p2.8.m8.1d">∼</annotation></semantics></math>60 deg-2 at <math alttext="z>2.1" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p2.10.m10.1"><semantics id="S4.SS3.SSS3.p2.10.m10.1a"><mrow id="S4.SS3.SSS3.p2.10.m10.1.1" xref="S4.SS3.SSS3.p2.10.m10.1.1.cmml"><mi id="S4.SS3.SSS3.p2.10.m10.1.1.2" xref="S4.SS3.SSS3.p2.10.m10.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS3.p2.10.m10.1.1.1" xref="S4.SS3.SSS3.p2.10.m10.1.1.1.cmml">></mo><mn id="S4.SS3.SSS3.p2.10.m10.1.1.3" xref="S4.SS3.SSS3.p2.10.m10.1.1.3.cmml">2.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p2.10.m10.1b"><apply id="S4.SS3.SSS3.p2.10.m10.1.1.cmml" xref="S4.SS3.SSS3.p2.10.m10.1.1"><gt id="S4.SS3.SSS3.p2.10.m10.1.1.1.cmml" xref="S4.SS3.SSS3.p2.10.m10.1.1.1"></gt><ci id="S4.SS3.SSS3.p2.10.m10.1.1.2.cmml" xref="S4.SS3.SSS3.p2.10.m10.1.1.2">𝑧</ci><cn id="S4.SS3.SSS3.p2.10.m10.1.1.3.cmml" type="float" xref="S4.SS3.SSS3.p2.10.m10.1.1.3">2.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p2.10.m10.1c">z>2.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p2.10.m10.1d">italic_z > 2.1</annotation></semantics></math>). </div> <div class="ltx_para" id="S4.SS3.SSS3.p3"> It is worth noting that, while not being the most promising high-<math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p3.1.m1.1"><semantics id="S4.SS3.SSS3.p3.1.m1.1a"><mi id="S4.SS3.SSS3.p3.1.m1.1.1" xref="S4.SS3.SSS3.p3.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p3.1.m1.1b"><ci id="S4.SS3.SSS3.p3.1.m1.1.1.cmml" xref="S4.SS3.SSS3.p3.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p3.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p3.1.m1.1d">italic_z</annotation></semantics></math> LSS tracers, at <math alttext="z>2.1" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p3.2.m2.1"><semantics id="S4.SS3.SSS3.p3.2.m2.1a"><mrow id="S4.SS3.SSS3.p3.2.m2.1.1" xref="S4.SS3.SSS3.p3.2.m2.1.1.cmml"><mi id="S4.SS3.SSS3.p3.2.m2.1.1.2" xref="S4.SS3.SSS3.p3.2.m2.1.1.2.cmml">z</mi><mo id="S4.SS3.SSS3.p3.2.m2.1.1.1" xref="S4.SS3.SSS3.p3.2.m2.1.1.1.cmml">></mo><mn id="S4.SS3.SSS3.p3.2.m2.1.1.3" xref="S4.SS3.SSS3.p3.2.m2.1.1.3.cmml">2.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p3.2.m2.1b"><apply id="S4.SS3.SSS3.p3.2.m2.1.1.cmml" xref="S4.SS3.SSS3.p3.2.m2.1.1"><gt id="S4.SS3.SSS3.p3.2.m2.1.1.1.cmml" xref="S4.SS3.SSS3.p3.2.m2.1.1.1"></gt><ci id="S4.SS3.SSS3.p3.2.m2.1.1.2.cmml" xref="S4.SS3.SSS3.p3.2.m2.1.1.2">𝑧</ci><cn id="S4.SS3.SSS3.p3.2.m2.1.1.3.cmml" type="float" xref="S4.SS3.SSS3.p3.2.m2.1.1.3">2.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p3.2.m2.1c">z>2.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p3.2.m2.1d">italic_z > 2.1</annotation></semantics></math>, QSOs provide the vital capability for studying the IGM using the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p3.3.m3.1"><semantics id="S4.SS3.SSS3.p3.3.m3.1a"><mi id="S4.SS3.SSS3.p3.3.m3.1.1" xref="S4.SS3.SSS3.p3.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p3.3.m3.1b"><ci id="S4.SS3.SSS3.p3.3.m3.1.1.cmml" xref="S4.SS3.SSS3.p3.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p3.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p3.3.m3.1d">italic_α</annotation></semantics></math> forest, which contains intriguing cosmological potential such as constraining the nature of dark matter. We will forecast the potential of Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.SSS3.p3.4.m4.1"><semantics id="S4.SS3.SSS3.p3.4.m4.1a"><mi id="S4.SS3.SSS3.p3.4.m4.1.1" xref="S4.SS3.SSS3.p3.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS3.p3.4.m4.1b"><ci id="S4.SS3.SSS3.p3.4.m4.1.1.cmml" xref="S4.SS3.SSS3.p3.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS3.p3.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS3.p3.4.m4.1d">italic_α</annotation></semantics></math>-QSO and provide a more thorough discussion in the following work of this series. </div> <figure class="ltx_table" id="S4.T3"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T3.36" style="width:433.6pt;height:131.2pt;vertical-align:-0.0pt;"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.36.36"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T3.8.8.8" style="background-color:#ECF4FF;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.8.8.8.9"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.8.8.8.9.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.8.8.8.9.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.9.1.1.1">Sample</td> </tr> <tr class="ltx_tr" id="S4.T3.8.8.8.9.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.9.1.2.1">Name</td> </tr> </table> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.8.8.8.10"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.8.8.8.10.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.8.8.8.10.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.10.1.1.1">Magnitude Limit</td> </tr> <tr class="ltx_tr" id="S4.T3.8.8.8.10.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.10.1.2.1">(AB mag)</td> </tr> </table> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.8.8.8.11"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.8.8.8.11.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.8.8.8.11.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.11.1.1.1">Redshift</td> </tr> <tr class="ltx_tr" id="S4.T3.8.8.8.11.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.11.1.2.1">Distribution</td> </tr> </table> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.1.1.1.1"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.1.1.1.1.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.1.1.1.1.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.1.1.1.1.1.2.1">Angular Density</td> </tr> <tr class="ltx_tr" id="S4.T3.1.1.1.1.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.1.1.1.1.1.1.1"> (deg-2) </td> </tr> </table> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.8.8.8.12"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.8.8.8.12.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.8.8.8.12.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.12.1.1.1">Number of</td> </tr> <tr class="ltx_tr" id="S4.T3.8.8.8.12.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.12.1.2.1">Redshift Bins</td> </tr> </table> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.4.4.4.4"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.4.4.4.4.3" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.4.4.4.4.3.4"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.4.4.4.4.3.4.1">3D Density</td> </tr> <tr class="ltx_tr" id="S4.T3.4.4.4.4.3.3"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.4.4.4.4.3.3.3"> (<math alttext="10^{-3}" class="ltx_Math" display="inline" id="S4.T3.2.2.2.2.1.1.1.m1.1"><semantics id="S4.T3.2.2.2.2.1.1.1.m1.1a"><msup id="S4.T3.2.2.2.2.1.1.1.m1.1.1" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1.cmml"><mn id="S4.T3.2.2.2.2.1.1.1.m1.1.1.2" mathbackground="#ECF4FF" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1.2.cmml">10</mn><mrow id="S4.T3.2.2.2.2.1.1.1.m1.1.1.3" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1.3.cmml"><mo id="S4.T3.2.2.2.2.1.1.1.m1.1.1.3a" mathbackground="#ECF4FF" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1.3.cmml">−</mo><mn id="S4.T3.2.2.2.2.1.1.1.m1.1.1.3.2" mathbackground="#ECF4FF" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1.3.2.cmml">3</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.T3.2.2.2.2.1.1.1.m1.1b"><apply id="S4.T3.2.2.2.2.1.1.1.m1.1.1.cmml" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T3.2.2.2.2.1.1.1.m1.1.1.1.cmml" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1">superscript</csymbol><cn id="S4.T3.2.2.2.2.1.1.1.m1.1.1.2.cmml" type="integer" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1.2">10</cn><apply id="S4.T3.2.2.2.2.1.1.1.m1.1.1.3.cmml" xref="S4.T3.2.2.2.2.1.1.1.m1.1.1.3"><minus 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ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.5.5.5.5"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.5.5.5.5.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.5.5.5.5.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.5.5.5.5.1.2.1">Survey Area</td> </tr> <tr class="ltx_tr" id="S4.T3.5.5.5.5.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.5.5.5.5.1.1.1"> (deg2) </td> </tr> </table> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.6.6.6.6"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.6.6.6.6.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.6.6.6.6.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.6.6.6.6.1.1.1"><math alttext="T_{\rm Exp}" class="ltx_Math" display="inline" id="S4.T3.6.6.6.6.1.1.1.m1.1"><semantics id="S4.T3.6.6.6.6.1.1.1.m1.1a"><msub id="S4.T3.6.6.6.6.1.1.1.m1.1.1" xref="S4.T3.6.6.6.6.1.1.1.m1.1.1.cmml"><mi id="S4.T3.6.6.6.6.1.1.1.m1.1.1.2" mathbackground="#ECF4FF" xref="S4.T3.6.6.6.6.1.1.1.m1.1.1.2.cmml">T</mi><mi id="S4.T3.6.6.6.6.1.1.1.m1.1.1.3" mathbackground="#ECF4FF" xref="S4.T3.6.6.6.6.1.1.1.m1.1.1.3.cmml">Exp</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T3.6.6.6.6.1.1.1.m1.1b"><apply id="S4.T3.6.6.6.6.1.1.1.m1.1.1.cmml" xref="S4.T3.6.6.6.6.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T3.6.6.6.6.1.1.1.m1.1.1.1.cmml" xref="S4.T3.6.6.6.6.1.1.1.m1.1.1">subscript</csymbol><ci id="S4.T3.6.6.6.6.1.1.1.m1.1.1.2.cmml" xref="S4.T3.6.6.6.6.1.1.1.m1.1.1.2">𝑇</ci><ci id="S4.T3.6.6.6.6.1.1.1.m1.1.1.3.cmml" xref="S4.T3.6.6.6.6.1.1.1.m1.1.1.3">Exp</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.6.6.6.6.1.1.1.m1.1c">T_{\rm Exp}</annotation><annotation encoding="application/x-llamapun" id="S4.T3.6.6.6.6.1.1.1.m1.1d">italic_T start_POSTSUBSCRIPT roman_Exp end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S4.T3.6.6.6.6.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.6.6.6.6.1.2.1">(Hour)</td> </tr> </table> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.7.7.7.7"> <table class="ltx_tabular ltx_align_middle" id="S4.T3.7.7.7.7.1" style="background-color:#ECF4FF;"> <tr class="ltx_tr" id="S4.T3.7.7.7.7.1.2"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.7.7.7.7.1.2.1">Total Number</td> </tr> <tr class="ltx_tr" id="S4.T3.7.7.7.7.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.7.7.7.7.1.1.1"> of Redshift (<math alttext="10^{6}" class="ltx_Math" display="inline" id="S4.T3.7.7.7.7.1.1.1.m1.1"><semantics id="S4.T3.7.7.7.7.1.1.1.m1.1a"><msup id="S4.T3.7.7.7.7.1.1.1.m1.1.1" xref="S4.T3.7.7.7.7.1.1.1.m1.1.1.cmml"><mn id="S4.T3.7.7.7.7.1.1.1.m1.1.1.2" mathbackground="#ECF4FF" xref="S4.T3.7.7.7.7.1.1.1.m1.1.1.2.cmml">10</mn><mn id="S4.T3.7.7.7.7.1.1.1.m1.1.1.3" mathbackground="#ECF4FF" xref="S4.T3.7.7.7.7.1.1.1.m1.1.1.3.cmml">6</mn></msup><annotation-xml 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ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.8.1.2.1">Total Fiber</td> </tr> <tr class="ltx_tr" id="S4.T3.8.8.8.8.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S4.T3.8.8.8.8.1.1.1"> Hours (<math alttext="10^{6}" class="ltx_Math" display="inline" id="S4.T3.8.8.8.8.1.1.1.m1.1"><semantics id="S4.T3.8.8.8.8.1.1.1.m1.1a"><msup id="S4.T3.8.8.8.8.1.1.1.m1.1.1" xref="S4.T3.8.8.8.8.1.1.1.m1.1.1.cmml"><mn id="S4.T3.8.8.8.8.1.1.1.m1.1.1.2" mathbackground="#ECF4FF" xref="S4.T3.8.8.8.8.1.1.1.m1.1.1.2.cmml">10</mn><mn id="S4.T3.8.8.8.8.1.1.1.m1.1.1.3" mathbackground="#ECF4FF" xref="S4.T3.8.8.8.8.1.1.1.m1.1.1.3.cmml">6</mn></msup><annotation-xml encoding="MathML-Content" id="S4.T3.8.8.8.8.1.1.1.m1.1b"><apply id="S4.T3.8.8.8.8.1.1.1.m1.1.1.cmml" xref="S4.T3.8.8.8.8.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T3.8.8.8.8.1.1.1.m1.1.1.1.cmml" xref="S4.T3.8.8.8.8.1.1.1.m1.1.1">superscript</csymbol><cn id="S4.T3.8.8.8.8.1.1.1.m1.1.1.2.cmml" type="integer" 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xref="S4.T3.9.9.9.1.m1.1.1"><lt id="S4.T3.9.9.9.1.m1.1.1.5.cmml" xref="S4.T3.9.9.9.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.9.9.9.1.m1.1.1.4.cmml" id="S4.T3.9.9.9.1.m1.1.1d.cmml" xref="S4.T3.9.9.9.1.m1.1.1"></share><cn id="S4.T3.9.9.9.1.m1.1.1.6.cmml" type="float" xref="S4.T3.9.9.9.1.m1.1.1.6">21.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.9.9.9.1.m1.1c">20.18<r<21.0</annotation><annotation encoding="application/x-llamapun" id="S4.T3.9.9.9.1.m1.1d">20.18 < italic_r < 21.0</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.10.10.10.2"><math alttext="0.1<z<0.7" class="ltx_Math" display="inline" id="S4.T3.10.10.10.2.m1.1" style="background-color:#FFCB2F;"><semantics id="S4.T3.10.10.10.2.m1.1a"><mrow id="S4.T3.10.10.10.2.m1.1.1" xref="S4.T3.10.10.10.2.m1.1.1.cmml"><mn id="S4.T3.10.10.10.2.m1.1.1.2" mathbackground="#FFCB2F" xref="S4.T3.10.10.10.2.m1.1.1.2.cmml">0.1</mn><mo id="S4.T3.10.10.10.2.m1.1.1.3" mathbackground="#FFCB2F" xref="S4.T3.10.10.10.2.m1.1.1.3.cmml"><</mo><mi id="S4.T3.10.10.10.2.m1.1.1.4" mathbackground="#FFCB2F" xref="S4.T3.10.10.10.2.m1.1.1.4.cmml">z</mi><mo id="S4.T3.10.10.10.2.m1.1.1.5" mathbackground="#FFCB2F" xref="S4.T3.10.10.10.2.m1.1.1.5.cmml"><</mo><mn id="S4.T3.10.10.10.2.m1.1.1.6" mathbackground="#FFCB2F" xref="S4.T3.10.10.10.2.m1.1.1.6.cmml">0.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.10.10.10.2.m1.1b"><apply id="S4.T3.10.10.10.2.m1.1.1.cmml" xref="S4.T3.10.10.10.2.m1.1.1"><and id="S4.T3.10.10.10.2.m1.1.1a.cmml" xref="S4.T3.10.10.10.2.m1.1.1"></and><apply id="S4.T3.10.10.10.2.m1.1.1b.cmml" xref="S4.T3.10.10.10.2.m1.1.1"><lt id="S4.T3.10.10.10.2.m1.1.1.3.cmml" xref="S4.T3.10.10.10.2.m1.1.1.3"></lt><cn id="S4.T3.10.10.10.2.m1.1.1.2.cmml" type="float" xref="S4.T3.10.10.10.2.m1.1.1.2">0.1</cn><ci id="S4.T3.10.10.10.2.m1.1.1.4.cmml" xref="S4.T3.10.10.10.2.m1.1.1.4">𝑧</ci></apply><apply id="S4.T3.10.10.10.2.m1.1.1c.cmml" xref="S4.T3.10.10.10.2.m1.1.1"><lt id="S4.T3.10.10.10.2.m1.1.1.5.cmml" xref="S4.T3.10.10.10.2.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.10.10.10.2.m1.1.1.4.cmml" id="S4.T3.10.10.10.2.m1.1.1d.cmml" xref="S4.T3.10.10.10.2.m1.1.1"></share><cn id="S4.T3.10.10.10.2.m1.1.1.6.cmml" type="float" xref="S4.T3.10.10.10.2.m1.1.1.6">0.7</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.10.10.10.2.m1.1c">0.1<z<0.7</annotation><annotation encoding="application/x-llamapun" id="S4.T3.10.10.10.2.m1.1d">0.1 < italic_z < 0.7</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.5">2,000</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.6">2</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.7">5.7</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.8">1.6</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.3"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.11.11.11.3.m1.1" style="background-color:#FFCB2F;"><semantics id="S4.T3.11.11.11.3.m1.1a"><mo id="S4.T3.11.11.11.3.m1.1.1" mathbackground="#FFCB2F" xref="S4.T3.11.11.11.3.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.11.11.11.3.m1.1b"><csymbol cd="latexml" id="S4.T3.11.11.11.3.m1.1.1.cmml" xref="S4.T3.11.11.11.3.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.11.11.11.3.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.11.11.11.3.m1.1d">∼</annotation></semantics></math>13,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.9">0.12</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.10">26.0</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.11.11.11.11">3.12</td> </tr> <tr class="ltx_tr" id="S4.T3.14.14.14" style="background-color:#FFCCC9;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.14.14.14.4">LRG</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.12.12.12.1"><math alttext="21.6<z_{\rm fib}<22.8" class="ltx_Math" display="inline" id="S4.T3.12.12.12.1.m1.1" style="background-color:#FFCCC9;"><semantics id="S4.T3.12.12.12.1.m1.1a"><mrow id="S4.T3.12.12.12.1.m1.1.1" xref="S4.T3.12.12.12.1.m1.1.1.cmml"><mn id="S4.T3.12.12.12.1.m1.1.1.2" mathbackground="#FFCCC9" xref="S4.T3.12.12.12.1.m1.1.1.2.cmml">21.6</mn><mo id="S4.T3.12.12.12.1.m1.1.1.3" mathbackground="#FFCCC9" xref="S4.T3.12.12.12.1.m1.1.1.3.cmml"><</mo><msub id="S4.T3.12.12.12.1.m1.1.1.4" xref="S4.T3.12.12.12.1.m1.1.1.4.cmml"><mi id="S4.T3.12.12.12.1.m1.1.1.4.2" mathbackground="#FFCCC9" xref="S4.T3.12.12.12.1.m1.1.1.4.2.cmml">z</mi><mi id="S4.T3.12.12.12.1.m1.1.1.4.3" mathbackground="#FFCCC9" xref="S4.T3.12.12.12.1.m1.1.1.4.3.cmml">fib</mi></msub><mo id="S4.T3.12.12.12.1.m1.1.1.5" mathbackground="#FFCCC9" xref="S4.T3.12.12.12.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.12.12.12.1.m1.1.1.6" mathbackground="#FFCCC9" xref="S4.T3.12.12.12.1.m1.1.1.6.cmml">22.8</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.12.12.12.1.m1.1b"><apply id="S4.T3.12.12.12.1.m1.1.1.cmml" xref="S4.T3.12.12.12.1.m1.1.1"><and id="S4.T3.12.12.12.1.m1.1.1a.cmml" xref="S4.T3.12.12.12.1.m1.1.1"></and><apply id="S4.T3.12.12.12.1.m1.1.1b.cmml" xref="S4.T3.12.12.12.1.m1.1.1"><lt id="S4.T3.12.12.12.1.m1.1.1.3.cmml" xref="S4.T3.12.12.12.1.m1.1.1.3"></lt><cn id="S4.T3.12.12.12.1.m1.1.1.2.cmml" type="float" xref="S4.T3.12.12.12.1.m1.1.1.2">21.6</cn><apply id="S4.T3.12.12.12.1.m1.1.1.4.cmml" xref="S4.T3.12.12.12.1.m1.1.1.4"><csymbol cd="ambiguous" id="S4.T3.12.12.12.1.m1.1.1.4.1.cmml" xref="S4.T3.12.12.12.1.m1.1.1.4">subscript</csymbol><ci id="S4.T3.12.12.12.1.m1.1.1.4.2.cmml" xref="S4.T3.12.12.12.1.m1.1.1.4.2">𝑧</ci><ci id="S4.T3.12.12.12.1.m1.1.1.4.3.cmml" xref="S4.T3.12.12.12.1.m1.1.1.4.3">fib</ci></apply></apply><apply id="S4.T3.12.12.12.1.m1.1.1c.cmml" xref="S4.T3.12.12.12.1.m1.1.1"><lt id="S4.T3.12.12.12.1.m1.1.1.5.cmml" xref="S4.T3.12.12.12.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.12.12.12.1.m1.1.1.4.cmml" id="S4.T3.12.12.12.1.m1.1.1d.cmml" xref="S4.T3.12.12.12.1.m1.1.1"></share><cn id="S4.T3.12.12.12.1.m1.1.1.6.cmml" type="float" xref="S4.T3.12.12.12.1.m1.1.1.6">22.8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.12.12.12.1.m1.1c">21.6<z_{\rm fib}<22.8</annotation><annotation encoding="application/x-llamapun" id="S4.T3.12.12.12.1.m1.1d">21.6 < italic_z start_POSTSUBSCRIPT roman_fib end_POSTSUBSCRIPT < 22.8</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.13.13.13.2"><math alttext="0.8<z<1.3" class="ltx_Math" display="inline" id="S4.T3.13.13.13.2.m1.1" style="background-color:#FFCCC9;"><semantics id="S4.T3.13.13.13.2.m1.1a"><mrow id="S4.T3.13.13.13.2.m1.1.1" xref="S4.T3.13.13.13.2.m1.1.1.cmml"><mn id="S4.T3.13.13.13.2.m1.1.1.2" mathbackground="#FFCCC9" xref="S4.T3.13.13.13.2.m1.1.1.2.cmml">0.8</mn><mo id="S4.T3.13.13.13.2.m1.1.1.3" mathbackground="#FFCCC9" xref="S4.T3.13.13.13.2.m1.1.1.3.cmml"><</mo><mi id="S4.T3.13.13.13.2.m1.1.1.4" mathbackground="#FFCCC9" xref="S4.T3.13.13.13.2.m1.1.1.4.cmml">z</mi><mo id="S4.T3.13.13.13.2.m1.1.1.5" mathbackground="#FFCCC9" xref="S4.T3.13.13.13.2.m1.1.1.5.cmml"><</mo><mn id="S4.T3.13.13.13.2.m1.1.1.6" mathbackground="#FFCCC9" xref="S4.T3.13.13.13.2.m1.1.1.6.cmml">1.3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.13.13.13.2.m1.1b"><apply id="S4.T3.13.13.13.2.m1.1.1.cmml" xref="S4.T3.13.13.13.2.m1.1.1"><and id="S4.T3.13.13.13.2.m1.1.1a.cmml" xref="S4.T3.13.13.13.2.m1.1.1"></and><apply id="S4.T3.13.13.13.2.m1.1.1b.cmml" xref="S4.T3.13.13.13.2.m1.1.1"><lt id="S4.T3.13.13.13.2.m1.1.1.3.cmml" xref="S4.T3.13.13.13.2.m1.1.1.3"></lt><cn id="S4.T3.13.13.13.2.m1.1.1.2.cmml" type="float" xref="S4.T3.13.13.13.2.m1.1.1.2">0.8</cn><ci id="S4.T3.13.13.13.2.m1.1.1.4.cmml" xref="S4.T3.13.13.13.2.m1.1.1.4">𝑧</ci></apply><apply id="S4.T3.13.13.13.2.m1.1.1c.cmml" xref="S4.T3.13.13.13.2.m1.1.1"><lt id="S4.T3.13.13.13.2.m1.1.1.5.cmml" xref="S4.T3.13.13.13.2.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.13.13.13.2.m1.1.1.4.cmml" id="S4.T3.13.13.13.2.m1.1.1d.cmml" xref="S4.T3.13.13.13.2.m1.1.1"></share><cn id="S4.T3.13.13.13.2.m1.1.1.6.cmml" type="float" xref="S4.T3.13.13.13.2.m1.1.1.6">1.3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.13.13.13.2.m1.1c">0.8<z<1.3</annotation><annotation encoding="application/x-llamapun" id="S4.T3.13.13.13.2.m1.1d">0.8 < italic_z < 1.3</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.5">2,000</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.6">2</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.7">1.0</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.8">2.8</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.3"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.14.14.14.3.m1.1" style="background-color:#FFCCC9;"><semantics id="S4.T3.14.14.14.3.m1.1a"><mo id="S4.T3.14.14.14.3.m1.1.1" mathbackground="#FFCCC9" xref="S4.T3.14.14.14.3.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.14.14.14.3.m1.1b"><csymbol cd="latexml" id="S4.T3.14.14.14.3.m1.1.1.cmml" xref="S4.T3.14.14.14.3.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.14.14.14.3.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.14.14.14.3.m1.1d">∼</annotation></semantics></math>13,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.9">1.00</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.10">26.0</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.14.14.14.11">26.0</td> </tr> <tr class="ltx_tr" id="S4.T3.15.15.15"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.15.15.15.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.15.15.15.3"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.15.15.15.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.5">Opt:3,600</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.6">2</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.7">1.49</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.8">1.5</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.15.15.15.1.m1.1" style="background-color:#99DAED;"><semantics id="S4.T3.15.15.15.1.m1.1a"><mo id="S4.T3.15.15.15.1.m1.1.1" mathbackground="#99DAED" xref="S4.T3.15.15.15.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.15.15.15.1.m1.1b"><csymbol cd="latexml" id="S4.T3.15.15.15.1.m1.1.1.cmml" xref="S4.T3.15.15.15.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.15.15.15.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.15.15.15.1.m1.1d">∼</annotation></semantics></math>13,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.9">0.30</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.10">46.8</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.15.15.15.11">14.0</td> </tr> <tr class="ltx_tr" id="S4.T3.18.18.18" style="background-color:#99DAED;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S4.T3.18.18.18.4" style="background-color:#99DAED;">ELG</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.16.16.16.1" style="background-color:#99DAED;"><math alttext="24.1<g_{\rm fib}<24.6" class="ltx_Math" display="inline" id="S4.T3.16.16.16.1.1.m1.1"><semantics id="S4.T3.16.16.16.1.1.m1.1a"><mrow id="S4.T3.16.16.16.1.1.m1.1.1" xref="S4.T3.16.16.16.1.1.m1.1.1.cmml"><mn id="S4.T3.16.16.16.1.1.m1.1.1.2" mathbackground="#99DAED" xref="S4.T3.16.16.16.1.1.m1.1.1.2.cmml">24.1</mn><mo id="S4.T3.16.16.16.1.1.m1.1.1.3" mathbackground="#99DAED" xref="S4.T3.16.16.16.1.1.m1.1.1.3.cmml"><</mo><msub id="S4.T3.16.16.16.1.1.m1.1.1.4" xref="S4.T3.16.16.16.1.1.m1.1.1.4.cmml"><mi id="S4.T3.16.16.16.1.1.m1.1.1.4.2" mathbackground="#99DAED" xref="S4.T3.16.16.16.1.1.m1.1.1.4.2.cmml">g</mi><mi id="S4.T3.16.16.16.1.1.m1.1.1.4.3" mathbackground="#99DAED" xref="S4.T3.16.16.16.1.1.m1.1.1.4.3.cmml">fib</mi></msub><mo id="S4.T3.16.16.16.1.1.m1.1.1.5" mathbackground="#99DAED" xref="S4.T3.16.16.16.1.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.16.16.16.1.1.m1.1.1.6" mathbackground="#99DAED" xref="S4.T3.16.16.16.1.1.m1.1.1.6.cmml">24.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.16.16.16.1.1.m1.1b"><apply id="S4.T3.16.16.16.1.1.m1.1.1.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1"><and id="S4.T3.16.16.16.1.1.m1.1.1a.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1"></and><apply id="S4.T3.16.16.16.1.1.m1.1.1b.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1"><lt id="S4.T3.16.16.16.1.1.m1.1.1.3.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1.3"></lt><cn id="S4.T3.16.16.16.1.1.m1.1.1.2.cmml" type="float" xref="S4.T3.16.16.16.1.1.m1.1.1.2">24.1</cn><apply id="S4.T3.16.16.16.1.1.m1.1.1.4.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S4.T3.16.16.16.1.1.m1.1.1.4.1.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1.4">subscript</csymbol><ci id="S4.T3.16.16.16.1.1.m1.1.1.4.2.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1.4.2">𝑔</ci><ci id="S4.T3.16.16.16.1.1.m1.1.1.4.3.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1.4.3">fib</ci></apply></apply><apply id="S4.T3.16.16.16.1.1.m1.1.1c.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1"><lt id="S4.T3.16.16.16.1.1.m1.1.1.5.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.16.16.16.1.1.m1.1.1.4.cmml" id="S4.T3.16.16.16.1.1.m1.1.1d.cmml" xref="S4.T3.16.16.16.1.1.m1.1.1"></share><cn id="S4.T3.16.16.16.1.1.m1.1.1.6.cmml" type="float" xref="S4.T3.16.16.16.1.1.m1.1.1.6">24.6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.16.16.16.1.1.m1.1c">24.1<g_{\rm fib}<24.6</annotation><annotation encoding="application/x-llamapun" id="S4.T3.16.16.16.1.1.m1.1d">24.1 < italic_g start_POSTSUBSCRIPT roman_fib end_POSTSUBSCRIPT < 24.6</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.17.17.17.2" style="background-color:#99DAED;"><math alttext="0.8<z<1.4" class="ltx_Math" display="inline" id="S4.T3.17.17.17.2.1.m1.1"><semantics id="S4.T3.17.17.17.2.1.m1.1a"><mrow id="S4.T3.17.17.17.2.1.m1.1.1" xref="S4.T3.17.17.17.2.1.m1.1.1.cmml"><mn id="S4.T3.17.17.17.2.1.m1.1.1.2" mathbackground="#99DAED" xref="S4.T3.17.17.17.2.1.m1.1.1.2.cmml">0.8</mn><mo id="S4.T3.17.17.17.2.1.m1.1.1.3" mathbackground="#99DAED" xref="S4.T3.17.17.17.2.1.m1.1.1.3.cmml"><</mo><mi id="S4.T3.17.17.17.2.1.m1.1.1.4" mathbackground="#99DAED" xref="S4.T3.17.17.17.2.1.m1.1.1.4.cmml">z</mi><mo id="S4.T3.17.17.17.2.1.m1.1.1.5" mathbackground="#99DAED" xref="S4.T3.17.17.17.2.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.17.17.17.2.1.m1.1.1.6" mathbackground="#99DAED" xref="S4.T3.17.17.17.2.1.m1.1.1.6.cmml">1.4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.17.17.17.2.1.m1.1b"><apply id="S4.T3.17.17.17.2.1.m1.1.1.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1"><and id="S4.T3.17.17.17.2.1.m1.1.1a.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1"></and><apply id="S4.T3.17.17.17.2.1.m1.1.1b.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1"><lt id="S4.T3.17.17.17.2.1.m1.1.1.3.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1.3"></lt><cn id="S4.T3.17.17.17.2.1.m1.1.1.2.cmml" type="float" xref="S4.T3.17.17.17.2.1.m1.1.1.2">0.8</cn><ci id="S4.T3.17.17.17.2.1.m1.1.1.4.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.T3.17.17.17.2.1.m1.1.1c.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1"><lt id="S4.T3.17.17.17.2.1.m1.1.1.5.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.17.17.17.2.1.m1.1.1.4.cmml" id="S4.T3.17.17.17.2.1.m1.1.1d.cmml" xref="S4.T3.17.17.17.2.1.m1.1.1"></share><cn id="S4.T3.17.17.17.2.1.m1.1.1.6.cmml" type="float" xref="S4.T3.17.17.17.2.1.m1.1.1.6">1.4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.17.17.17.2.1.m1.1c">0.8<z<1.4</annotation><annotation encoding="application/x-llamapun" id="S4.T3.17.17.17.2.1.m1.1d">0.8 < italic_z < 1.4</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.5">Con:2,500</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.6">2</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.7">1.03</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.8">1.5</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.3"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.18.18.18.3.m1.1" style="background-color:#99DAED;"><semantics id="S4.T3.18.18.18.3.m1.1a"><mo id="S4.T3.18.18.18.3.m1.1.1" mathbackground="#99DAED" xref="S4.T3.18.18.18.3.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.18.18.18.3.m1.1b"><csymbol cd="latexml" id="S4.T3.18.18.18.3.m1.1.1.cmml" xref="S4.T3.18.18.18.3.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.18.18.18.3.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.18.18.18.3.m1.1d">∼</annotation></semantics></math>11,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.9">0.30</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.10">27.5</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.18.18.18.11">8.25</td> </tr> <tr class="ltx_tr" id="S4.T3.19.19.19"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.19.19.19.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.19.19.19.3"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.19.19.19.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.19.19.19.5">Opt:1,200</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.19.19.19.6">3</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.19.19.19.7">0.21</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.19.19.19.8"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.19.19.19.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.19.19.19.1.m1.1" style="background-color:#CBCEFB;"><semantics id="S4.T3.19.19.19.1.m1.1a"><mo id="S4.T3.19.19.19.1.m1.1.1" mathbackground="#CBCEFB" xref="S4.T3.19.19.19.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.19.19.19.1.m1.1b"><csymbol cd="latexml" id="S4.T3.19.19.19.1.m1.1.1.cmml" xref="S4.T3.19.19.19.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.19.19.19.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.19.19.19.1.m1.1d">∼</annotation></semantics></math>11,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.19.19.19.9">2.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.19.19.19.10">15.6</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.19.19.19.11">39.0</td> </tr> <tr class="ltx_tr" id="S4.T3.23.23.23" style="background-color:#CBCEFB;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S4.T3.20.20.20.1" style="background-color:#CBCEFB;"><math alttext="u" class="ltx_Math" display="inline" id="S4.T3.20.20.20.1.1.m1.1"><semantics id="S4.T3.20.20.20.1.1.m1.1a"><mi id="S4.T3.20.20.20.1.1.m1.1.1" mathbackground="#CBCEFB" xref="S4.T3.20.20.20.1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.T3.20.20.20.1.1.m1.1b"><ci id="S4.T3.20.20.20.1.1.m1.1.1.cmml" xref="S4.T3.20.20.20.1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.20.20.20.1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.T3.20.20.20.1.1.m1.1d">italic_u</annotation></semantics></math>-Dropout</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.21.21.21.2" style="background-color:#CBCEFB;"><math alttext="22.4<r<24.5" class="ltx_Math" display="inline" id="S4.T3.21.21.21.2.1.m1.1"><semantics id="S4.T3.21.21.21.2.1.m1.1a"><mrow id="S4.T3.21.21.21.2.1.m1.1.1" xref="S4.T3.21.21.21.2.1.m1.1.1.cmml"><mn id="S4.T3.21.21.21.2.1.m1.1.1.2" mathbackground="#CBCEFB" xref="S4.T3.21.21.21.2.1.m1.1.1.2.cmml">22.4</mn><mo id="S4.T3.21.21.21.2.1.m1.1.1.3" mathbackground="#CBCEFB" xref="S4.T3.21.21.21.2.1.m1.1.1.3.cmml"><</mo><mi id="S4.T3.21.21.21.2.1.m1.1.1.4" mathbackground="#CBCEFB" xref="S4.T3.21.21.21.2.1.m1.1.1.4.cmml">r</mi><mo id="S4.T3.21.21.21.2.1.m1.1.1.5" mathbackground="#CBCEFB" xref="S4.T3.21.21.21.2.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.21.21.21.2.1.m1.1.1.6" mathbackground="#CBCEFB" xref="S4.T3.21.21.21.2.1.m1.1.1.6.cmml">24.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.21.21.21.2.1.m1.1b"><apply id="S4.T3.21.21.21.2.1.m1.1.1.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1"><and id="S4.T3.21.21.21.2.1.m1.1.1a.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1"></and><apply id="S4.T3.21.21.21.2.1.m1.1.1b.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1"><lt id="S4.T3.21.21.21.2.1.m1.1.1.3.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1.3"></lt><cn id="S4.T3.21.21.21.2.1.m1.1.1.2.cmml" type="float" xref="S4.T3.21.21.21.2.1.m1.1.1.2">22.4</cn><ci id="S4.T3.21.21.21.2.1.m1.1.1.4.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1.4">𝑟</ci></apply><apply id="S4.T3.21.21.21.2.1.m1.1.1c.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1"><lt id="S4.T3.21.21.21.2.1.m1.1.1.5.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.21.21.21.2.1.m1.1.1.4.cmml" id="S4.T3.21.21.21.2.1.m1.1.1d.cmml" xref="S4.T3.21.21.21.2.1.m1.1.1"></share><cn id="S4.T3.21.21.21.2.1.m1.1.1.6.cmml" type="float" xref="S4.T3.21.21.21.2.1.m1.1.1.6">24.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.21.21.21.2.1.m1.1c">22.4<r<24.5</annotation><annotation encoding="application/x-llamapun" id="S4.T3.21.21.21.2.1.m1.1d">22.4 < italic_r < 24.5</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.22.22.22.3" style="background-color:#CBCEFB;"><math alttext="2.2<z<3.5" class="ltx_Math" display="inline" id="S4.T3.22.22.22.3.1.m1.1"><semantics id="S4.T3.22.22.22.3.1.m1.1a"><mrow id="S4.T3.22.22.22.3.1.m1.1.1" xref="S4.T3.22.22.22.3.1.m1.1.1.cmml"><mn id="S4.T3.22.22.22.3.1.m1.1.1.2" mathbackground="#CBCEFB" xref="S4.T3.22.22.22.3.1.m1.1.1.2.cmml">2.2</mn><mo id="S4.T3.22.22.22.3.1.m1.1.1.3" mathbackground="#CBCEFB" xref="S4.T3.22.22.22.3.1.m1.1.1.3.cmml"><</mo><mi id="S4.T3.22.22.22.3.1.m1.1.1.4" mathbackground="#CBCEFB" xref="S4.T3.22.22.22.3.1.m1.1.1.4.cmml">z</mi><mo id="S4.T3.22.22.22.3.1.m1.1.1.5" mathbackground="#CBCEFB" xref="S4.T3.22.22.22.3.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.22.22.22.3.1.m1.1.1.6" mathbackground="#CBCEFB" xref="S4.T3.22.22.22.3.1.m1.1.1.6.cmml">3.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.22.22.22.3.1.m1.1b"><apply id="S4.T3.22.22.22.3.1.m1.1.1.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1"><and id="S4.T3.22.22.22.3.1.m1.1.1a.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1"></and><apply id="S4.T3.22.22.22.3.1.m1.1.1b.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1"><lt id="S4.T3.22.22.22.3.1.m1.1.1.3.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1.3"></lt><cn id="S4.T3.22.22.22.3.1.m1.1.1.2.cmml" type="float" xref="S4.T3.22.22.22.3.1.m1.1.1.2">2.2</cn><ci id="S4.T3.22.22.22.3.1.m1.1.1.4.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.T3.22.22.22.3.1.m1.1.1c.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1"><lt id="S4.T3.22.22.22.3.1.m1.1.1.5.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.22.22.22.3.1.m1.1.1.4.cmml" id="S4.T3.22.22.22.3.1.m1.1.1d.cmml" xref="S4.T3.22.22.22.3.1.m1.1.1"></share><cn id="S4.T3.22.22.22.3.1.m1.1.1.6.cmml" type="float" xref="S4.T3.22.22.22.3.1.m1.1.1.6">3.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.22.22.22.3.1.m1.1c">2.2<z<3.5</annotation><annotation encoding="application/x-llamapun" id="S4.T3.22.22.22.3.1.m1.1d">2.2 < italic_z < 3.5</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.23.23.23.5">Con:600</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.23.23.23.6">3</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.23.23.23.7">0.11</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.23.23.23.8" style="background-color:#CBCEFB;">4.5</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.23.23.23.4"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.23.23.23.4.m1.1" style="background-color:#CBCEFB;"><semantics id="S4.T3.23.23.23.4.m1.1a"><mo id="S4.T3.23.23.23.4.m1.1.1" mathbackground="#CBCEFB" xref="S4.T3.23.23.23.4.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.23.23.23.4.m1.1b"><csymbol cd="latexml" id="S4.T3.23.23.23.4.m1.1.1.cmml" xref="S4.T3.23.23.23.4.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.23.23.23.4.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.23.23.23.4.m1.1d">∼</annotation></semantics></math>8,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.23.23.23.9">2.50</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.23.23.23.10">6.0</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.23.23.23.11">15.0</td> </tr> <tr class="ltx_tr" id="S4.T3.24.24.24"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.24.24.24.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.24.24.24.3"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.24.24.24.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.24.24.24.5">Opt:800</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.24.24.24.6">2</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.24.24.24.7">0.16</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.24.24.24.8"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.24.24.24.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.24.24.24.1.m1.1" style="background-color:#9AE881;"><semantics id="S4.T3.24.24.24.1.m1.1a"><mo id="S4.T3.24.24.24.1.m1.1.1" mathbackground="#9AE881" xref="S4.T3.24.24.24.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.24.24.24.1.m1.1b"><csymbol cd="latexml" id="S4.T3.24.24.24.1.m1.1.1.cmml" xref="S4.T3.24.24.24.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.24.24.24.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.24.24.24.1.m1.1d">∼</annotation></semantics></math>11,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.24.24.24.9">6.00</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.24.24.24.10">10.4</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.24.24.24.11">62.4</td> </tr> <tr class="ltx_tr" id="S4.T3.28.28.28" style="background-color:#9AE881;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S4.T3.25.25.25.1" style="background-color:#9AE881;"><math alttext="g" class="ltx_Math" display="inline" id="S4.T3.25.25.25.1.1.m1.1"><semantics id="S4.T3.25.25.25.1.1.m1.1a"><mi id="S4.T3.25.25.25.1.1.m1.1.1" mathbackground="#9AE881" xref="S4.T3.25.25.25.1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.T3.25.25.25.1.1.m1.1b"><ci id="S4.T3.25.25.25.1.1.m1.1.1.cmml" xref="S4.T3.25.25.25.1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.25.25.25.1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.T3.25.25.25.1.1.m1.1d">italic_g</annotation></semantics></math>-Dropout</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.26.26.26.2" style="background-color:#9AE881;"><math alttext="23.0<i<24.6" class="ltx_Math" display="inline" id="S4.T3.26.26.26.2.1.m1.1"><semantics id="S4.T3.26.26.26.2.1.m1.1a"><mrow id="S4.T3.26.26.26.2.1.m1.1.1" xref="S4.T3.26.26.26.2.1.m1.1.1.cmml"><mn id="S4.T3.26.26.26.2.1.m1.1.1.2" mathbackground="#9AE881" xref="S4.T3.26.26.26.2.1.m1.1.1.2.cmml">23.0</mn><mo id="S4.T3.26.26.26.2.1.m1.1.1.3" mathbackground="#9AE881" xref="S4.T3.26.26.26.2.1.m1.1.1.3.cmml"><</mo><mi id="S4.T3.26.26.26.2.1.m1.1.1.4" mathbackground="#9AE881" xref="S4.T3.26.26.26.2.1.m1.1.1.4.cmml">i</mi><mo id="S4.T3.26.26.26.2.1.m1.1.1.5" mathbackground="#9AE881" xref="S4.T3.26.26.26.2.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.26.26.26.2.1.m1.1.1.6" mathbackground="#9AE881" xref="S4.T3.26.26.26.2.1.m1.1.1.6.cmml">24.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.26.26.26.2.1.m1.1b"><apply id="S4.T3.26.26.26.2.1.m1.1.1.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1"><and id="S4.T3.26.26.26.2.1.m1.1.1a.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1"></and><apply id="S4.T3.26.26.26.2.1.m1.1.1b.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1"><lt id="S4.T3.26.26.26.2.1.m1.1.1.3.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1.3"></lt><cn id="S4.T3.26.26.26.2.1.m1.1.1.2.cmml" type="float" xref="S4.T3.26.26.26.2.1.m1.1.1.2">23.0</cn><ci id="S4.T3.26.26.26.2.1.m1.1.1.4.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1.4">𝑖</ci></apply><apply id="S4.T3.26.26.26.2.1.m1.1.1c.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1"><lt id="S4.T3.26.26.26.2.1.m1.1.1.5.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.26.26.26.2.1.m1.1.1.4.cmml" id="S4.T3.26.26.26.2.1.m1.1.1d.cmml" xref="S4.T3.26.26.26.2.1.m1.1.1"></share><cn id="S4.T3.26.26.26.2.1.m1.1.1.6.cmml" type="float" xref="S4.T3.26.26.26.2.1.m1.1.1.6">24.6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.26.26.26.2.1.m1.1c">23.0<i<24.6</annotation><annotation encoding="application/x-llamapun" id="S4.T3.26.26.26.2.1.m1.1d">23.0 < italic_i < 24.6</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.27.27.27.3" style="background-color:#9AE881;"><math alttext="3.2<z<4.5" class="ltx_Math" display="inline" id="S4.T3.27.27.27.3.1.m1.1"><semantics id="S4.T3.27.27.27.3.1.m1.1a"><mrow id="S4.T3.27.27.27.3.1.m1.1.1" xref="S4.T3.27.27.27.3.1.m1.1.1.cmml"><mn id="S4.T3.27.27.27.3.1.m1.1.1.2" mathbackground="#9AE881" xref="S4.T3.27.27.27.3.1.m1.1.1.2.cmml">3.2</mn><mo id="S4.T3.27.27.27.3.1.m1.1.1.3" mathbackground="#9AE881" xref="S4.T3.27.27.27.3.1.m1.1.1.3.cmml"><</mo><mi id="S4.T3.27.27.27.3.1.m1.1.1.4" mathbackground="#9AE881" xref="S4.T3.27.27.27.3.1.m1.1.1.4.cmml">z</mi><mo id="S4.T3.27.27.27.3.1.m1.1.1.5" mathbackground="#9AE881" xref="S4.T3.27.27.27.3.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.27.27.27.3.1.m1.1.1.6" mathbackground="#9AE881" xref="S4.T3.27.27.27.3.1.m1.1.1.6.cmml">4.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.27.27.27.3.1.m1.1b"><apply id="S4.T3.27.27.27.3.1.m1.1.1.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1"><and id="S4.T3.27.27.27.3.1.m1.1.1a.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1"></and><apply id="S4.T3.27.27.27.3.1.m1.1.1b.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1"><lt id="S4.T3.27.27.27.3.1.m1.1.1.3.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1.3"></lt><cn id="S4.T3.27.27.27.3.1.m1.1.1.2.cmml" type="float" xref="S4.T3.27.27.27.3.1.m1.1.1.2">3.2</cn><ci id="S4.T3.27.27.27.3.1.m1.1.1.4.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.T3.27.27.27.3.1.m1.1.1c.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1"><lt id="S4.T3.27.27.27.3.1.m1.1.1.5.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.27.27.27.3.1.m1.1.1.4.cmml" id="S4.T3.27.27.27.3.1.m1.1.1d.cmml" xref="S4.T3.27.27.27.3.1.m1.1.1"></share><cn id="S4.T3.27.27.27.3.1.m1.1.1.6.cmml" type="float" xref="S4.T3.27.27.27.3.1.m1.1.1.6">4.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.27.27.27.3.1.m1.1c">3.2<z<4.5</annotation><annotation encoding="application/x-llamapun" id="S4.T3.27.27.27.3.1.m1.1d">3.2 < italic_z < 4.5</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.28.28.28.5">Con:300</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.28.28.28.6">2</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.28.28.28.7">0.06</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.28.28.28.8" style="background-color:#9AE881;">5.3</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.28.28.28.4"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.28.28.28.4.m1.1" style="background-color:#9AE881;"><semantics id="S4.T3.28.28.28.4.m1.1a"><mo id="S4.T3.28.28.28.4.m1.1.1" mathbackground="#9AE881" xref="S4.T3.28.28.28.4.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.28.28.28.4.m1.1b"><csymbol cd="latexml" id="S4.T3.28.28.28.4.m1.1.1.cmml" xref="S4.T3.28.28.28.4.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.28.28.28.4.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.28.28.28.4.m1.1d">∼</annotation></semantics></math>8,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.28.28.28.9">6.00</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.28.28.28.10">3.0</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.28.28.28.11">18.0</td> </tr> <tr class="ltx_tr" id="S4.T3.29.29.29"> <td class="ltx_td ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.29.29.29.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.29.29.29.3"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.29.29.29.4"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.29.29.29.5">Opt:200</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.29.29.29.6">1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.29.29.29.7">0.06</td> <td class="ltx_td ltx_border_r ltx_border_t" id="S4.T3.29.29.29.8"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.29.29.29.1"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.29.29.29.1.m1.1" style="background-color:#FFACAC;"><semantics id="S4.T3.29.29.29.1.m1.1a"><mo id="S4.T3.29.29.29.1.m1.1.1" mathbackground="#FFACAC" xref="S4.T3.29.29.29.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.29.29.29.1.m1.1b"><csymbol cd="latexml" id="S4.T3.29.29.29.1.m1.1.1.cmml" xref="S4.T3.29.29.29.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.29.29.29.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.29.29.29.1.m1.1d">∼</annotation></semantics></math>11,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.29.29.29.9">8.00</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.29.29.29.10">2.6</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.29.29.29.11">20.8</td> </tr> <tr class="ltx_tr" id="S4.T3.33.33.33" style="background-color:#FFACAC;"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r" id="S4.T3.30.30.30.1" style="background-color:#FFACAC;"><math alttext="r" class="ltx_Math" display="inline" id="S4.T3.30.30.30.1.1.m1.1"><semantics id="S4.T3.30.30.30.1.1.m1.1a"><mi id="S4.T3.30.30.30.1.1.m1.1.1" mathbackground="#FFACAC" xref="S4.T3.30.30.30.1.1.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.T3.30.30.30.1.1.m1.1b"><ci id="S4.T3.30.30.30.1.1.m1.1.1.cmml" xref="S4.T3.30.30.30.1.1.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.30.30.30.1.1.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.T3.30.30.30.1.1.m1.1d">italic_r</annotation></semantics></math>-Dropout</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.31.31.31.2" style="background-color:#FFACAC;"><math alttext="23.0<z<25.0" class="ltx_Math" display="inline" id="S4.T3.31.31.31.2.1.m1.1"><semantics id="S4.T3.31.31.31.2.1.m1.1a"><mrow id="S4.T3.31.31.31.2.1.m1.1.1" xref="S4.T3.31.31.31.2.1.m1.1.1.cmml"><mn id="S4.T3.31.31.31.2.1.m1.1.1.2" mathbackground="#FFACAC" xref="S4.T3.31.31.31.2.1.m1.1.1.2.cmml">23.0</mn><mo id="S4.T3.31.31.31.2.1.m1.1.1.3" mathbackground="#FFACAC" xref="S4.T3.31.31.31.2.1.m1.1.1.3.cmml"><</mo><mi id="S4.T3.31.31.31.2.1.m1.1.1.4" mathbackground="#FFACAC" xref="S4.T3.31.31.31.2.1.m1.1.1.4.cmml">z</mi><mo id="S4.T3.31.31.31.2.1.m1.1.1.5" mathbackground="#FFACAC" xref="S4.T3.31.31.31.2.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.31.31.31.2.1.m1.1.1.6" mathbackground="#FFACAC" xref="S4.T3.31.31.31.2.1.m1.1.1.6.cmml">25.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.31.31.31.2.1.m1.1b"><apply id="S4.T3.31.31.31.2.1.m1.1.1.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1"><and id="S4.T3.31.31.31.2.1.m1.1.1a.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1"></and><apply id="S4.T3.31.31.31.2.1.m1.1.1b.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1"><lt id="S4.T3.31.31.31.2.1.m1.1.1.3.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1.3"></lt><cn id="S4.T3.31.31.31.2.1.m1.1.1.2.cmml" type="float" xref="S4.T3.31.31.31.2.1.m1.1.1.2">23.0</cn><ci id="S4.T3.31.31.31.2.1.m1.1.1.4.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.T3.31.31.31.2.1.m1.1.1c.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1"><lt id="S4.T3.31.31.31.2.1.m1.1.1.5.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.31.31.31.2.1.m1.1.1.4.cmml" id="S4.T3.31.31.31.2.1.m1.1.1d.cmml" xref="S4.T3.31.31.31.2.1.m1.1.1"></share><cn id="S4.T3.31.31.31.2.1.m1.1.1.6.cmml" type="float" xref="S4.T3.31.31.31.2.1.m1.1.1.6">25.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.31.31.31.2.1.m1.1c">23.0<z<25.0</annotation><annotation encoding="application/x-llamapun" id="S4.T3.31.31.31.2.1.m1.1d">23.0 < italic_z < 25.0</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.32.32.32.3" style="background-color:#FFACAC;"><math alttext="4.0<z<5.5" class="ltx_Math" display="inline" id="S4.T3.32.32.32.3.1.m1.1"><semantics id="S4.T3.32.32.32.3.1.m1.1a"><mrow id="S4.T3.32.32.32.3.1.m1.1.1" xref="S4.T3.32.32.32.3.1.m1.1.1.cmml"><mn id="S4.T3.32.32.32.3.1.m1.1.1.2" mathbackground="#FFACAC" xref="S4.T3.32.32.32.3.1.m1.1.1.2.cmml">4.0</mn><mo id="S4.T3.32.32.32.3.1.m1.1.1.3" mathbackground="#FFACAC" xref="S4.T3.32.32.32.3.1.m1.1.1.3.cmml"><</mo><mi id="S4.T3.32.32.32.3.1.m1.1.1.4" mathbackground="#FFACAC" xref="S4.T3.32.32.32.3.1.m1.1.1.4.cmml">z</mi><mo id="S4.T3.32.32.32.3.1.m1.1.1.5" mathbackground="#FFACAC" xref="S4.T3.32.32.32.3.1.m1.1.1.5.cmml"><</mo><mn id="S4.T3.32.32.32.3.1.m1.1.1.6" mathbackground="#FFACAC" xref="S4.T3.32.32.32.3.1.m1.1.1.6.cmml">5.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.32.32.32.3.1.m1.1b"><apply id="S4.T3.32.32.32.3.1.m1.1.1.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1"><and id="S4.T3.32.32.32.3.1.m1.1.1a.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1"></and><apply id="S4.T3.32.32.32.3.1.m1.1.1b.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1"><lt id="S4.T3.32.32.32.3.1.m1.1.1.3.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1.3"></lt><cn id="S4.T3.32.32.32.3.1.m1.1.1.2.cmml" type="float" xref="S4.T3.32.32.32.3.1.m1.1.1.2">4.0</cn><ci id="S4.T3.32.32.32.3.1.m1.1.1.4.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.T3.32.32.32.3.1.m1.1.1c.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1"><lt id="S4.T3.32.32.32.3.1.m1.1.1.5.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.T3.32.32.32.3.1.m1.1.1.4.cmml" id="S4.T3.32.32.32.3.1.m1.1.1d.cmml" xref="S4.T3.32.32.32.3.1.m1.1.1"></share><cn id="S4.T3.32.32.32.3.1.m1.1.1.6.cmml" type="float" xref="S4.T3.32.32.32.3.1.m1.1.1.6">5.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.32.32.32.3.1.m1.1c">4.0<z<5.5</annotation><annotation encoding="application/x-llamapun" id="S4.T3.32.32.32.3.1.m1.1d">4.0 < italic_z < 5.5</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.33.33.33.5">Con:100</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.33.33.33.6">1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.33.33.33.7">0.03</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S4.T3.33.33.33.8" style="background-color:#FFACAC;">6.4</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.33.33.33.4"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.33.33.33.4.m1.1" style="background-color:#FFACAC;"><semantics id="S4.T3.33.33.33.4.m1.1a"><mo id="S4.T3.33.33.33.4.m1.1.1" mathbackground="#FFACAC" xref="S4.T3.33.33.33.4.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.33.33.33.4.m1.1b"><csymbol cd="latexml" id="S4.T3.33.33.33.4.m1.1.1.cmml" xref="S4.T3.33.33.33.4.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.33.33.33.4.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.33.33.33.4.m1.1d">∼</annotation></semantics></math>8,000 </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.33.33.33.9">8.00</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.33.33.33.10">1.0</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T3.33.33.33.11">8.0</td> </tr> <tr class="ltx_tr" id="S4.T3.36.36.36" style="background-color:#DAE8FC;"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T3.36.36.36.4">QSO</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.34.34.34.1"><math alttext="r<23.5" class="ltx_Math" display="inline" id="S4.T3.34.34.34.1.m1.1" style="background-color:#DAE8FC;"><semantics id="S4.T3.34.34.34.1.m1.1a"><mrow id="S4.T3.34.34.34.1.m1.1.1" xref="S4.T3.34.34.34.1.m1.1.1.cmml"><mi id="S4.T3.34.34.34.1.m1.1.1.2" mathbackground="#DAE8FC" xref="S4.T3.34.34.34.1.m1.1.1.2.cmml">r</mi><mo id="S4.T3.34.34.34.1.m1.1.1.1" mathbackground="#DAE8FC" xref="S4.T3.34.34.34.1.m1.1.1.1.cmml"><</mo><mn id="S4.T3.34.34.34.1.m1.1.1.3" mathbackground="#DAE8FC" xref="S4.T3.34.34.34.1.m1.1.1.3.cmml">23.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.34.34.34.1.m1.1b"><apply id="S4.T3.34.34.34.1.m1.1.1.cmml" xref="S4.T3.34.34.34.1.m1.1.1"><lt id="S4.T3.34.34.34.1.m1.1.1.1.cmml" xref="S4.T3.34.34.34.1.m1.1.1.1"></lt><ci id="S4.T3.34.34.34.1.m1.1.1.2.cmml" xref="S4.T3.34.34.34.1.m1.1.1.2">𝑟</ci><cn id="S4.T3.34.34.34.1.m1.1.1.3.cmml" type="float" xref="S4.T3.34.34.34.1.m1.1.1.3">23.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.34.34.34.1.m1.1c">r<23.5</annotation><annotation encoding="application/x-llamapun" id="S4.T3.34.34.34.1.m1.1d">italic_r < 23.5</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.35.35.35.2"><math alttext="z>2.1" class="ltx_Math" display="inline" id="S4.T3.35.35.35.2.m1.1" style="background-color:#DAE8FC;"><semantics id="S4.T3.35.35.35.2.m1.1a"><mrow id="S4.T3.35.35.35.2.m1.1.1" xref="S4.T3.35.35.35.2.m1.1.1.cmml"><mi id="S4.T3.35.35.35.2.m1.1.1.2" mathbackground="#DAE8FC" xref="S4.T3.35.35.35.2.m1.1.1.2.cmml">z</mi><mo id="S4.T3.35.35.35.2.m1.1.1.1" mathbackground="#DAE8FC" xref="S4.T3.35.35.35.2.m1.1.1.1.cmml">></mo><mn id="S4.T3.35.35.35.2.m1.1.1.3" mathbackground="#DAE8FC" xref="S4.T3.35.35.35.2.m1.1.1.3.cmml">2.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T3.35.35.35.2.m1.1b"><apply id="S4.T3.35.35.35.2.m1.1.1.cmml" xref="S4.T3.35.35.35.2.m1.1.1"><gt id="S4.T3.35.35.35.2.m1.1.1.1.cmml" xref="S4.T3.35.35.35.2.m1.1.1.1"></gt><ci id="S4.T3.35.35.35.2.m1.1.1.2.cmml" xref="S4.T3.35.35.35.2.m1.1.1.2">𝑧</ci><cn id="S4.T3.35.35.35.2.m1.1.1.3.cmml" type="float" xref="S4.T3.35.35.35.2.m1.1.1.3">2.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.35.35.35.2.m1.1c">z>2.1</annotation><annotation encoding="application/x-llamapun" id="S4.T3.35.35.35.2.m1.1d">italic_z > 2.1</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.5">90</td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.6"></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.7"></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.8"></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.3"> <math alttext="\sim" class="ltx_Math" display="inline" id="S4.T3.36.36.36.3.m1.1" style="background-color:#DAE8FC;"><semantics id="S4.T3.36.36.36.3.m1.1a"><mo id="S4.T3.36.36.36.3.m1.1.1" mathbackground="#DAE8FC" xref="S4.T3.36.36.36.3.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.T3.36.36.36.3.m1.1b"><csymbol cd="latexml" id="S4.T3.36.36.36.3.m1.1.1.cmml" xref="S4.T3.36.36.36.3.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.36.36.36.3.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.T3.36.36.36.3.m1.1d">∼</annotation></semantics></math>13,000 </td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.9">0.12</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.10">1.2</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T3.36.36.36.11">0.14</td> </tr> </tbody> </table> </div> <figcaption class="ltx_caption ltx_centering">Table 3: Summary of the low- and high-redshift samples selected in this work. All but the QSO samples are considered in the cosmological forecast. For the ELG, <math alttext="u" class="ltx_Math" display="inline" id="S4.T3.40.m1.1"><semantics id="S4.T3.40.m1.1b"><mi id="S4.T3.40.m1.1.1" xref="S4.T3.40.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.T3.40.m1.1c"><ci id="S4.T3.40.m1.1.1.cmml" xref="S4.T3.40.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.40.m1.1d">u</annotation><annotation encoding="application/x-llamapun" id="S4.T3.40.m1.1e">italic_u</annotation></semantics></math>-, <math alttext="g" class="ltx_Math" display="inline" id="S4.T3.41.m2.1"><semantics id="S4.T3.41.m2.1b"><mi id="S4.T3.41.m2.1.1" xref="S4.T3.41.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.T3.41.m2.1c"><ci id="S4.T3.41.m2.1.1.cmml" xref="S4.T3.41.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.41.m2.1d">g</annotation><annotation encoding="application/x-llamapun" id="S4.T3.41.m2.1e">italic_g</annotation></semantics></math>, and <math alttext="r" class="ltx_Math" display="inline" id="S4.T3.42.m3.1"><semantics id="S4.T3.42.m3.1b"><mi id="S4.T3.42.m3.1.1" xref="S4.T3.42.m3.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.T3.42.m3.1c"><ci id="S4.T3.42.m3.1.1.cmml" xref="S4.T3.42.m3.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T3.42.m3.1d">r</annotation><annotation encoding="application/x-llamapun" id="S4.T3.42.m3.1e">italic_r</annotation></semantics></math>-dropout galaxies, we estimate an optimistic (“Opt”) and conservative (“Con”) surface density to design the survey.</figcaption> </figure> </section> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> 4.4 Summary of the Targets Selection</h3> <div class="ltx_para" id="S4.SS4.p1"> In Table <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.T3" title="Table 3 ‣ 4.3.3 Quasi-Stellar Objects (QSO) ‣ 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3</a>, we summarize the preliminary target selection discussed above and provide an initial estimate of the survey cost measured using the total fiber-hour. All but the QSO samples are considered in the cosmological forecast. </div> <div class="ltx_para" id="S4.SS4.p2"> The effective collecting area of MUST will be <math alttext="\sim 3.2\times" class="ltx_math_unparsed" display="inline" id="S4.SS4.p2.1.m1.1"><semantics id="S4.SS4.p2.1.m1.1a"><mrow id="S4.SS4.p2.1.m1.1b"><mo id="S4.SS4.p2.1.m1.1.1">∼</mo><mn id="S4.SS4.p2.1.m1.1.2">3.2</mn><mo id="S4.SS4.p2.1.m1.1.3" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="S4.SS4.p2.1.m1.1c">\sim 3.2\times</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.1.m1.1d">∼ 3.2 ×</annotation></semantics></math> than the one for DESI121212We assume the primary mirror of DESI is 3.8 m in diameter with a 1.8 m central obscuration. For MUST, we assume the central obscuration has a 2.5 m central obscuration. Considering the additional throughput loss caused by the secondary mirror, WFC, fiber route, and spectrograph, we assume that for the same target, MUST can reach the same S/N 2.5<math alttext="\times" class="ltx_Math" display="inline" id="S4.SS4.p2.2.m2.1"><semantics id="S4.SS4.p2.2.m2.1a"><mo id="S4.SS4.p2.2.m2.1.1" xref="S4.SS4.p2.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.2.m2.1b"><times id="S4.SS4.p2.2.m2.1.1.cmml" xref="S4.SS4.p2.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.2.m2.1d">×</annotation></semantics></math> faster than DESI. This ignores the difference caused by seeing (Lenghu site is likely to have better median seeing than Kitt Peak) and fiber core diameter (1.5 arcsec for DESI v.s. 1.3 arcsec for MUST). For the same target type, MUST will obverse fainter targets than DESI. We assume the exposure time will increase <math alttext="10^{2\delta{m}/2.5}" class="ltx_Math" display="inline" id="S4.SS4.p2.3.m3.1"><semantics id="S4.SS4.p2.3.m3.1a"><msup id="S4.SS4.p2.3.m3.1.1" xref="S4.SS4.p2.3.m3.1.1.cmml"><mn id="S4.SS4.p2.3.m3.1.1.2" xref="S4.SS4.p2.3.m3.1.1.2.cmml">10</mn><mrow id="S4.SS4.p2.3.m3.1.1.3" xref="S4.SS4.p2.3.m3.1.1.3.cmml"><mrow id="S4.SS4.p2.3.m3.1.1.3.2" xref="S4.SS4.p2.3.m3.1.1.3.2.cmml"><mn id="S4.SS4.p2.3.m3.1.1.3.2.2" xref="S4.SS4.p2.3.m3.1.1.3.2.2.cmml">2</mn><mo id="S4.SS4.p2.3.m3.1.1.3.2.1" xref="S4.SS4.p2.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="S4.SS4.p2.3.m3.1.1.3.2.3" xref="S4.SS4.p2.3.m3.1.1.3.2.3.cmml">δ</mi><mo id="S4.SS4.p2.3.m3.1.1.3.2.1a" xref="S4.SS4.p2.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="S4.SS4.p2.3.m3.1.1.3.2.4" xref="S4.SS4.p2.3.m3.1.1.3.2.4.cmml">m</mi></mrow><mo id="S4.SS4.p2.3.m3.1.1.3.1" xref="S4.SS4.p2.3.m3.1.1.3.1.cmml">/</mo><mn id="S4.SS4.p2.3.m3.1.1.3.3" xref="S4.SS4.p2.3.m3.1.1.3.3.cmml">2.5</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.3.m3.1b"><apply id="S4.SS4.p2.3.m3.1.1.cmml" xref="S4.SS4.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS4.p2.3.m3.1.1.1.cmml" xref="S4.SS4.p2.3.m3.1.1">superscript</csymbol><cn id="S4.SS4.p2.3.m3.1.1.2.cmml" type="integer" xref="S4.SS4.p2.3.m3.1.1.2">10</cn><apply id="S4.SS4.p2.3.m3.1.1.3.cmml" xref="S4.SS4.p2.3.m3.1.1.3"><divide id="S4.SS4.p2.3.m3.1.1.3.1.cmml" xref="S4.SS4.p2.3.m3.1.1.3.1"></divide><apply id="S4.SS4.p2.3.m3.1.1.3.2.cmml" xref="S4.SS4.p2.3.m3.1.1.3.2"><times id="S4.SS4.p2.3.m3.1.1.3.2.1.cmml" xref="S4.SS4.p2.3.m3.1.1.3.2.1"></times><cn id="S4.SS4.p2.3.m3.1.1.3.2.2.cmml" type="integer" xref="S4.SS4.p2.3.m3.1.1.3.2.2">2</cn><ci id="S4.SS4.p2.3.m3.1.1.3.2.3.cmml" xref="S4.SS4.p2.3.m3.1.1.3.2.3">𝛿</ci><ci id="S4.SS4.p2.3.m3.1.1.3.2.4.cmml" xref="S4.SS4.p2.3.m3.1.1.3.2.4">𝑚</ci></apply><cn id="S4.SS4.p2.3.m3.1.1.3.3.cmml" type="float" xref="S4.SS4.p2.3.m3.1.1.3.3">2.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.3.m3.1c">10^{2\delta{m}/2.5}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.3.m3.1d">10 start_POSTSUPERSCRIPT 2 italic_δ italic_m / 2.5 end_POSTSUPERSCRIPT</annotation></semantics></math> times for the fainter sample, where <math alttext="\delta{m}" class="ltx_Math" display="inline" id="S4.SS4.p2.4.m4.1"><semantics id="S4.SS4.p2.4.m4.1a"><mrow id="S4.SS4.p2.4.m4.1.1" xref="S4.SS4.p2.4.m4.1.1.cmml"><mi id="S4.SS4.p2.4.m4.1.1.2" xref="S4.SS4.p2.4.m4.1.1.2.cmml">δ</mi><mo id="S4.SS4.p2.4.m4.1.1.1" xref="S4.SS4.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S4.SS4.p2.4.m4.1.1.3" xref="S4.SS4.p2.4.m4.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.4.m4.1b"><apply id="S4.SS4.p2.4.m4.1.1.cmml" xref="S4.SS4.p2.4.m4.1.1"><times id="S4.SS4.p2.4.m4.1.1.1.cmml" xref="S4.SS4.p2.4.m4.1.1.1"></times><ci id="S4.SS4.p2.4.m4.1.1.2.cmml" xref="S4.SS4.p2.4.m4.1.1.2">𝛿</ci><ci id="S4.SS4.p2.4.m4.1.1.3.cmml" xref="S4.SS4.p2.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.4.m4.1c">\delta{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.4.m4.1d">italic_δ italic_m</annotation></semantics></math> is the magnitude difference. We assume DESI can achieve the designed redshift success rate using 200 seconds of exposure time for BGS and 1,000 seconds of exposure time for LRG, ELG, and QSO. Hence, we estimate that MUST can achieve the same 95% to 99% redshift success rate using 440, 3600, 729, and 440 seconds of exposure time for BGS, LRG, ELG, and QSO. </div> <div class="ltx_para" id="S4.SS4.p3"> Estimating the exposure time for the LBG is more difficult. Based on the recent LBG observation campaign by DESI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib21" title="">21</a>]</cite>, for a <math alttext="r<24.2" class="ltx_Math" display="inline" id="S4.SS4.p3.1.m1.1"><semantics id="S4.SS4.p3.1.m1.1a"><mrow id="S4.SS4.p3.1.m1.1.1" xref="S4.SS4.p3.1.m1.1.1.cmml"><mi id="S4.SS4.p3.1.m1.1.1.2" xref="S4.SS4.p3.1.m1.1.1.2.cmml">r</mi><mo id="S4.SS4.p3.1.m1.1.1.1" xref="S4.SS4.p3.1.m1.1.1.1.cmml"><</mo><mn id="S4.SS4.p3.1.m1.1.1.3" xref="S4.SS4.p3.1.m1.1.1.3.cmml">24.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.1.m1.1b"><apply id="S4.SS4.p3.1.m1.1.1.cmml" xref="S4.SS4.p3.1.m1.1.1"><lt id="S4.SS4.p3.1.m1.1.1.1.cmml" xref="S4.SS4.p3.1.m1.1.1.1"></lt><ci id="S4.SS4.p3.1.m1.1.1.2.cmml" xref="S4.SS4.p3.1.m1.1.1.2">𝑟</ci><cn id="S4.SS4.p3.1.m1.1.1.3.cmml" type="float" xref="S4.SS4.p3.1.m1.1.1.3">24.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.1.m1.1c">r<24.2</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.1.m1.1d">italic_r < 24.2</annotation></semantics></math> mag <math alttext="u" class="ltx_Math" display="inline" id="S4.SS4.p3.2.m2.1"><semantics id="S4.SS4.p3.2.m2.1a"><mi id="S4.SS4.p3.2.m2.1.1" xref="S4.SS4.p3.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.2.m2.1b"><ci id="S4.SS4.p3.2.m2.1.1.cmml" xref="S4.SS4.p3.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.2.m2.1d">italic_u</annotation></semantics></math>-dropout sample, DESI can already achieve a <math alttext="\sim 70" class="ltx_Math" display="inline" id="S4.SS4.p3.3.m3.1"><semantics id="S4.SS4.p3.3.m3.1a"><mrow id="S4.SS4.p3.3.m3.1.1" xref="S4.SS4.p3.3.m3.1.1.cmml"><mi id="S4.SS4.p3.3.m3.1.1.2" xref="S4.SS4.p3.3.m3.1.1.2.cmml"></mi><mo id="S4.SS4.p3.3.m3.1.1.1" xref="S4.SS4.p3.3.m3.1.1.1.cmml">∼</mo><mn id="S4.SS4.p3.3.m3.1.1.3" xref="S4.SS4.p3.3.m3.1.1.3.cmml">70</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.3.m3.1b"><apply id="S4.SS4.p3.3.m3.1.1.cmml" xref="S4.SS4.p3.3.m3.1.1"><csymbol cd="latexml" id="S4.SS4.p3.3.m3.1.1.1.cmml" xref="S4.SS4.p3.3.m3.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S4.SS4.p3.3.m3.1.1.2.cmml" xref="S4.SS4.p3.3.m3.1.1.2">absent</csymbol><cn id="S4.SS4.p3.3.m3.1.1.3.cmml" type="integer" xref="S4.SS4.p3.3.m3.1.1.3">70</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.3.m3.1c">\sim 70</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.3.m3.1d">∼ 70</annotation></semantics></math> redshift efficiency in 2.5-hour of exposure time. However, the efficiency seems to plateau after 2.5-hour hours as a 5-hour exposure still cannot improve it to <math alttext=">80" class="ltx_Math" display="inline" id="S4.SS4.p3.4.m4.1"><semantics id="S4.SS4.p3.4.m4.1a"><mrow id="S4.SS4.p3.4.m4.1.1" xref="S4.SS4.p3.4.m4.1.1.cmml"><mi id="S4.SS4.p3.4.m4.1.1.2" xref="S4.SS4.p3.4.m4.1.1.2.cmml"></mi><mo id="S4.SS4.p3.4.m4.1.1.1" xref="S4.SS4.p3.4.m4.1.1.1.cmml">></mo><mn id="S4.SS4.p3.4.m4.1.1.3" xref="S4.SS4.p3.4.m4.1.1.3.cmml">80</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.4.m4.1b"><apply id="S4.SS4.p3.4.m4.1.1.cmml" xref="S4.SS4.p3.4.m4.1.1"><gt id="S4.SS4.p3.4.m4.1.1.1.cmml" xref="S4.SS4.p3.4.m4.1.1.1"></gt><csymbol cd="latexml" id="S4.SS4.p3.4.m4.1.1.2.cmml" xref="S4.SS4.p3.4.m4.1.1.2">absent</csymbol><cn id="S4.SS4.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.SS4.p3.4.m4.1.1.3">80</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.4.m4.1c">>80</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.4.m4.1d">> 80</annotation></semantics></math>%. It is worth noting that there is certainly room to improve the LBG redshift estimation soon with the arrival of better spectral models or templates. So, for the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS4.p3.5.m5.1"><semantics id="S4.SS4.p3.5.m5.1a"><mi id="S4.SS4.p3.5.m5.1.1" xref="S4.SS4.p3.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.5.m5.1b"><ci id="S4.SS4.p3.5.m5.1.1.cmml" xref="S4.SS4.p3.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.5.m5.1d">italic_u</annotation></semantics></math>-dropout sample, we apply the simple empirical scaling relation and assume that MUST can achieve the same success rate for our <math alttext="r<24.5" class="ltx_Math" display="inline" id="S4.SS4.p3.6.m6.1"><semantics id="S4.SS4.p3.6.m6.1a"><mrow id="S4.SS4.p3.6.m6.1.1" xref="S4.SS4.p3.6.m6.1.1.cmml"><mi id="S4.SS4.p3.6.m6.1.1.2" xref="S4.SS4.p3.6.m6.1.1.2.cmml">r</mi><mo id="S4.SS4.p3.6.m6.1.1.1" xref="S4.SS4.p3.6.m6.1.1.1.cmml"><</mo><mn id="S4.SS4.p3.6.m6.1.1.3" xref="S4.SS4.p3.6.m6.1.1.3.cmml">24.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.6.m6.1b"><apply id="S4.SS4.p3.6.m6.1.1.cmml" xref="S4.SS4.p3.6.m6.1.1"><lt id="S4.SS4.p3.6.m6.1.1.1.cmml" xref="S4.SS4.p3.6.m6.1.1.1"></lt><ci id="S4.SS4.p3.6.m6.1.1.2.cmml" xref="S4.SS4.p3.6.m6.1.1.2">𝑟</ci><cn id="S4.SS4.p3.6.m6.1.1.3.cmml" type="float" xref="S4.SS4.p3.6.m6.1.1.3">24.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.6.m6.1c">r<24.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.6.m6.1d">italic_r < 24.5</annotation></semantics></math> mag sample with a 2.5-hour exposure. For the <math alttext="g" class="ltx_Math" display="inline" id="S4.SS4.p3.7.m7.1"><semantics id="S4.SS4.p3.7.m7.1a"><mi id="S4.SS4.p3.7.m7.1.1" xref="S4.SS4.p3.7.m7.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.7.m7.1b"><ci id="S4.SS4.p3.7.m7.1.1.cmml" xref="S4.SS4.p3.7.m7.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.7.m7.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.7.m7.1d">italic_g</annotation></semantics></math>- and <math alttext="r" class="ltx_Math" display="inline" id="S4.SS4.p3.8.m8.1"><semantics id="S4.SS4.p3.8.m8.1a"><mi id="S4.SS4.p3.8.m8.1.1" xref="S4.SS4.p3.8.m8.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.8.m8.1b"><ci id="S4.SS4.p3.8.m8.1.1.cmml" xref="S4.SS4.p3.8.m8.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.8.m8.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.8.m8.1d">italic_r</annotation></semantics></math>-dropouts, it is less straightforward to estimate the required exposure time as there is no statistically meaningful DESI sample to compare with. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib237" title="">237</a>]</cite>, the authors estimated an 80-minute exposure would be enough for a <math alttext="z=24" class="ltx_Math" display="inline" id="S4.SS4.p3.9.m9.1"><semantics id="S4.SS4.p3.9.m9.1a"><mrow id="S4.SS4.p3.9.m9.1.1" xref="S4.SS4.p3.9.m9.1.1.cmml"><mi id="S4.SS4.p3.9.m9.1.1.2" xref="S4.SS4.p3.9.m9.1.1.2.cmml">z</mi><mo id="S4.SS4.p3.9.m9.1.1.1" xref="S4.SS4.p3.9.m9.1.1.1.cmml">=</mo><mn id="S4.SS4.p3.9.m9.1.1.3" xref="S4.SS4.p3.9.m9.1.1.3.cmml">24</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.9.m9.1b"><apply id="S4.SS4.p3.9.m9.1.1.cmml" xref="S4.SS4.p3.9.m9.1.1"><eq id="S4.SS4.p3.9.m9.1.1.1.cmml" xref="S4.SS4.p3.9.m9.1.1.1"></eq><ci id="S4.SS4.p3.9.m9.1.1.2.cmml" xref="S4.SS4.p3.9.m9.1.1.2">𝑧</ci><cn id="S4.SS4.p3.9.m9.1.1.3.cmml" type="integer" xref="S4.SS4.p3.9.m9.1.1.3">24</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.9.m9.1c">z=24</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.9.m9.1d">italic_z = 24</annotation></semantics></math> mag galaxy using a 6.5 m telescope. Scaling this up to a <math alttext="z<24.50" class="ltx_Math" display="inline" id="S4.SS4.p3.10.m10.1"><semantics id="S4.SS4.p3.10.m10.1a"><mrow id="S4.SS4.p3.10.m10.1.1" xref="S4.SS4.p3.10.m10.1.1.cmml"><mi id="S4.SS4.p3.10.m10.1.1.2" xref="S4.SS4.p3.10.m10.1.1.2.cmml">z</mi><mo id="S4.SS4.p3.10.m10.1.1.1" xref="S4.SS4.p3.10.m10.1.1.1.cmml"><</mo><mn id="S4.SS4.p3.10.m10.1.1.3" xref="S4.SS4.p3.10.m10.1.1.3.cmml">24.50</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.10.m10.1b"><apply id="S4.SS4.p3.10.m10.1.1.cmml" xref="S4.SS4.p3.10.m10.1.1"><lt id="S4.SS4.p3.10.m10.1.1.1.cmml" xref="S4.SS4.p3.10.m10.1.1.1"></lt><ci id="S4.SS4.p3.10.m10.1.1.2.cmml" xref="S4.SS4.p3.10.m10.1.1.2">𝑧</ci><cn id="S4.SS4.p3.10.m10.1.1.3.cmml" type="float" xref="S4.SS4.p3.10.m10.1.1.3">24.50</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.10.m10.1c">z<24.50</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.10.m10.1d">italic_z < 24.50</annotation></semantics></math> mag sample would require a 3.4-hour exposure time. </div> </section> <section class="ltx_subsection" id="S4.SS5"> <h3 class="ltx_title ltx_title_subsection"> 4.5 Conceptual Survey Design</h3> <div class="ltx_para" id="S4.SS5.p1"> This section briefly describes the conceptual survey designs for the bright-grey and dark-time programs of MUST based on the previous description of the telescope & instrument design and the target selection under several oversimplified assumptions. In the upcoming work of this series, we will present a more proper survey design backed by detailed instrument specifications, an exposure time calculator (ETC) for MUST, and an improved target selection strategy. </div> <section class="ltx_subsubsection" id="S4.SS5.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> 4.5.1 Dark Time Survey</h4> <div class="ltx_para" id="S4.SS5.SSS1.p1"> We assume that the CSST photometry will support the selection of dark-time targets for MUST, which will mainly include the LBG at <math alttext="2.2<z<5.0" class="ltx_Math" display="inline" id="S4.SS5.SSS1.p1.1.m1.1"><semantics id="S4.SS5.SSS1.p1.1.m1.1a"><mrow id="S4.SS5.SSS1.p1.1.m1.1.1" xref="S4.SS5.SSS1.p1.1.m1.1.1.cmml"><mn id="S4.SS5.SSS1.p1.1.m1.1.1.2" xref="S4.SS5.SSS1.p1.1.m1.1.1.2.cmml">2.2</mn><mo id="S4.SS5.SSS1.p1.1.m1.1.1.3" xref="S4.SS5.SSS1.p1.1.m1.1.1.3.cmml"><</mo><mi id="S4.SS5.SSS1.p1.1.m1.1.1.4" xref="S4.SS5.SSS1.p1.1.m1.1.1.4.cmml">z</mi><mo id="S4.SS5.SSS1.p1.1.m1.1.1.5" xref="S4.SS5.SSS1.p1.1.m1.1.1.5.cmml"><</mo><mn id="S4.SS5.SSS1.p1.1.m1.1.1.6" xref="S4.SS5.SSS1.p1.1.m1.1.1.6.cmml">5.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS1.p1.1.m1.1b"><apply id="S4.SS5.SSS1.p1.1.m1.1.1.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1"><and id="S4.SS5.SSS1.p1.1.m1.1.1a.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1"></and><apply id="S4.SS5.SSS1.p1.1.m1.1.1b.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1"><lt id="S4.SS5.SSS1.p1.1.m1.1.1.3.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1.3"></lt><cn id="S4.SS5.SSS1.p1.1.m1.1.1.2.cmml" type="float" xref="S4.SS5.SSS1.p1.1.m1.1.1.2">2.2</cn><ci id="S4.SS5.SSS1.p1.1.m1.1.1.4.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1.4">𝑧</ci></apply><apply id="S4.SS5.SSS1.p1.1.m1.1.1c.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1"><lt id="S4.SS5.SSS1.p1.1.m1.1.1.5.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS5.SSS1.p1.1.m1.1.1.4.cmml" id="S4.SS5.SSS1.p1.1.m1.1.1d.cmml" xref="S4.SS5.SSS1.p1.1.m1.1.1"></share><cn id="S4.SS5.SSS1.p1.1.m1.1.1.6.cmml" type="float" xref="S4.SS5.SSS1.p1.1.m1.1.1.6">5.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS1.p1.1.m1.1c">2.2<z<5.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS1.p1.1.m1.1d">2.2 < italic_z < 5.0</annotation></semantics></math>. The right panel of Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.F5" title="Figure 5 ‣ 4.5.2 Grey Time Survey ‣ 4.5 Conceptual Survey Design ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5</a> shows a preliminary observing pattern. Each single pointing represents the 5 deg2 effective FoV of MUST. Given the latitude of the Lenghu site, MUST can not observe the whole footprint for CSST. Assuming a maximum of 1.5 airmass (<math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS1.p1.3.m3.1"><semantics id="S4.SS5.SSS1.p1.3.m3.1a"><mo id="S4.SS5.SSS1.p1.3.m3.1.1" xref="S4.SS5.SSS1.p1.3.m3.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS1.p1.3.m3.1b"><csymbol cd="latexml" id="S4.SS5.SSS1.p1.3.m3.1.1.cmml" xref="S4.SS5.SSS1.p1.3.m3.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS1.p1.3.m3.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS1.p1.3.m3.1d">∼</annotation></semantics></math>40∘ above the horizon) for observation, we cut the footprint at Dec<math alttext=">-10^{\circ}" class="ltx_Math" display="inline" id="S4.SS5.SSS1.p1.5.m5.1"><semantics id="S4.SS5.SSS1.p1.5.m5.1a"><mrow id="S4.SS5.SSS1.p1.5.m5.1.1" xref="S4.SS5.SSS1.p1.5.m5.1.1.cmml"><mi id="S4.SS5.SSS1.p1.5.m5.1.1.2" xref="S4.SS5.SSS1.p1.5.m5.1.1.2.cmml"></mi><mo id="S4.SS5.SSS1.p1.5.m5.1.1.1" xref="S4.SS5.SSS1.p1.5.m5.1.1.1.cmml">></mo><mrow id="S4.SS5.SSS1.p1.5.m5.1.1.3" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.cmml"><mo id="S4.SS5.SSS1.p1.5.m5.1.1.3a" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.cmml">−</mo><msup id="S4.SS5.SSS1.p1.5.m5.1.1.3.2" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.2.cmml"><mn id="S4.SS5.SSS1.p1.5.m5.1.1.3.2.2" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.2.2.cmml">10</mn><mo id="S4.SS5.SSS1.p1.5.m5.1.1.3.2.3" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.2.3.cmml">∘</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS1.p1.5.m5.1b"><apply id="S4.SS5.SSS1.p1.5.m5.1.1.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1"><gt id="S4.SS5.SSS1.p1.5.m5.1.1.1.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1.1"></gt><csymbol cd="latexml" id="S4.SS5.SSS1.p1.5.m5.1.1.2.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1.2">absent</csymbol><apply id="S4.SS5.SSS1.p1.5.m5.1.1.3.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1.3"><minus id="S4.SS5.SSS1.p1.5.m5.1.1.3.1.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1.3"></minus><apply id="S4.SS5.SSS1.p1.5.m5.1.1.3.2.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS5.SSS1.p1.5.m5.1.1.3.2.1.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.2">superscript</csymbol><cn id="S4.SS5.SSS1.p1.5.m5.1.1.3.2.2.cmml" type="integer" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.2.2">10</cn><compose id="S4.SS5.SSS1.p1.5.m5.1.1.3.2.3.cmml" xref="S4.SS5.SSS1.p1.5.m5.1.1.3.2.3"></compose></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS1.p1.5.m5.1c">>-10^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS1.p1.5.m5.1d">> - 10 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>. Observing 11 000 deg2 or 2600-pointing observations (with a minimal overlap for a continuous coverage) and with an average of 2.5 hours exposure per target, this represents a total of 650 effective nights of dark time with an average 10/nights. Assuming a 70% clear night fraction and 35% of Dark Time during the year and accounting for 80% survey efficiency, this translates into a 7.5-year observing program. Altogether, a total of about 30 million redshift measurements should be obtained during this dark-time survey, assuming an overall 70% redshift measurement efficiency. </div> </section> <section class="ltx_subsubsection" id="S4.SS5.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> 4.5.2 Grey Time Survey</h4> <div class="ltx_para" id="S4.SS5.SSS2.p1"> As the grey time program will primarily focus on BGS, LRG, and ELG, we use the Legacy Survey footprint to simulate the pointings needed for MUST as shown in the left panel of Fig.<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.F5" title="Figure 5 ‣ 4.5.2 Grey Time Survey ‣ 4.5 Conceptual Survey Design ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5</a>. With the limitation in declination, MUST can observe a total of <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p1.1.m1.1"><semantics id="S4.SS5.SSS2.p1.1.m1.1a"><mo id="S4.SS5.SSS2.p1.1.m1.1.1" xref="S4.SS5.SSS2.p1.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p1.1.m1.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p1.1.m1.1.1.cmml" xref="S4.SS5.SSS2.p1.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p1.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p1.1.m1.1d">∼</annotation></semantics></math> 13 000 deg2 of the Legacy Survey footprint, corresponding to <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p1.3.m3.1"><semantics id="S4.SS5.SSS2.p1.3.m3.1a"><mo id="S4.SS5.SSS2.p1.3.m3.1.1" xref="S4.SS5.SSS2.p1.3.m3.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p1.3.m3.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p1.3.m3.1.1.cmml" xref="S4.SS5.SSS2.p1.3.m3.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p1.3.m3.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p1.3.m3.1d">∼</annotation></semantics></math>3100 contiguous MUST paintings. </div> <div class="ltx_para" id="S4.SS5.SSS2.p2"> With an average of 1.0 hours exposure per target with a target density of <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.1.m1.1"><semantics id="S4.SS5.SSS2.p2.1.m1.1a"><mo id="S4.SS5.SSS2.p2.1.m1.1.1" xref="S4.SS5.SSS2.p2.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.1.m1.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.1.m1.1.1.cmml" xref="S4.SS5.SSS2.p2.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.1.m1.1d">∼</annotation></semantics></math>2 times the fiber density for MUST (<math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.2.m2.1"><semantics id="S4.SS5.SSS2.p2.2.m2.1a"><mo id="S4.SS5.SSS2.p2.2.m2.1.1" xref="S4.SS5.SSS2.p2.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.2.m2.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.2.m2.1.1.cmml" xref="S4.SS5.SSS2.p2.2.m2.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.2.m2.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.2.m2.1d">∼</annotation></semantics></math> 4,000 deg-2; this could include 2,000 deg-2 BGS, 2,000 deg-2 LRG, 3,600 deg-2 ELG, and 400 deg-2 of other targets), such survey represents a total of 620 effective nights of grey time with 10 hours per night. Assuming 40% of Grey Time during the year, accounting for 80% survey efficiency, and a 70% clear night fraction, this translates into a 6.0-year observing program. Altogether, the grey-time program of MUST can accumulate <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.8.m8.1"><semantics id="S4.SS5.SSS2.p2.8.m8.1a"><mo id="S4.SS5.SSS2.p2.8.m8.1.1" xref="S4.SS5.SSS2.p2.8.m8.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.8.m8.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.8.m8.1.1.cmml" xref="S4.SS5.SSS2.p2.8.m8.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.8.m8.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.8.m8.1d">∼</annotation></semantics></math>88 million redshift measurements during this time, assuming an overall 85% redshift measurement efficiency. Among these grey-time targets, the exposure time required for LRG (<math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.9.m9.1"><semantics id="S4.SS5.SSS2.p2.9.m9.1a"><mo id="S4.SS5.SSS2.p2.9.m9.1.1" xref="S4.SS5.SSS2.p2.9.m9.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.9.m9.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.9.m9.1.1.cmml" xref="S4.SS5.SSS2.p2.9.m9.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.9.m9.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.9.m9.1d">∼</annotation></semantics></math> 1 hour for a <math alttext="z_{\rm fiber}=22.8" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.10.m10.1"><semantics id="S4.SS5.SSS2.p2.10.m10.1a"><mrow id="S4.SS5.SSS2.p2.10.m10.1.1" xref="S4.SS5.SSS2.p2.10.m10.1.1.cmml"><msub id="S4.SS5.SSS2.p2.10.m10.1.1.2" xref="S4.SS5.SSS2.p2.10.m10.1.1.2.cmml"><mi id="S4.SS5.SSS2.p2.10.m10.1.1.2.2" xref="S4.SS5.SSS2.p2.10.m10.1.1.2.2.cmml">z</mi><mi id="S4.SS5.SSS2.p2.10.m10.1.1.2.3" xref="S4.SS5.SSS2.p2.10.m10.1.1.2.3.cmml">fiber</mi></msub><mo id="S4.SS5.SSS2.p2.10.m10.1.1.1" xref="S4.SS5.SSS2.p2.10.m10.1.1.1.cmml">=</mo><mn id="S4.SS5.SSS2.p2.10.m10.1.1.3" xref="S4.SS5.SSS2.p2.10.m10.1.1.3.cmml">22.8</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.10.m10.1b"><apply id="S4.SS5.SSS2.p2.10.m10.1.1.cmml" xref="S4.SS5.SSS2.p2.10.m10.1.1"><eq id="S4.SS5.SSS2.p2.10.m10.1.1.1.cmml" xref="S4.SS5.SSS2.p2.10.m10.1.1.1"></eq><apply id="S4.SS5.SSS2.p2.10.m10.1.1.2.cmml" xref="S4.SS5.SSS2.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS5.SSS2.p2.10.m10.1.1.2.1.cmml" xref="S4.SS5.SSS2.p2.10.m10.1.1.2">subscript</csymbol><ci id="S4.SS5.SSS2.p2.10.m10.1.1.2.2.cmml" xref="S4.SS5.SSS2.p2.10.m10.1.1.2.2">𝑧</ci><ci id="S4.SS5.SSS2.p2.10.m10.1.1.2.3.cmml" xref="S4.SS5.SSS2.p2.10.m10.1.1.2.3">fiber</ci></apply><cn id="S4.SS5.SSS2.p2.10.m10.1.1.3.cmml" type="float" xref="S4.SS5.SSS2.p2.10.m10.1.1.3">22.8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.10.m10.1c">z_{\rm fiber}=22.8</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.10.m10.1d">italic_z start_POSTSUBSCRIPT roman_fiber end_POSTSUBSCRIPT = 22.8</annotation></semantics></math> mag target) is significantly longer than the others. With a careful survey strategy, the average exposure time per grey-time target could be shortened to speed up the current program, leaving space for further increase of the grey-time target density. For example, within the current detection limits of the LegacySurveys, we could select BGS down to <math alttext="r<21.5" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.11.m11.1"><semantics id="S4.SS5.SSS2.p2.11.m11.1a"><mrow id="S4.SS5.SSS2.p2.11.m11.1.1" xref="S4.SS5.SSS2.p2.11.m11.1.1.cmml"><mi id="S4.SS5.SSS2.p2.11.m11.1.1.2" xref="S4.SS5.SSS2.p2.11.m11.1.1.2.cmml">r</mi><mo id="S4.SS5.SSS2.p2.11.m11.1.1.1" xref="S4.SS5.SSS2.p2.11.m11.1.1.1.cmml"><</mo><mn id="S4.SS5.SSS2.p2.11.m11.1.1.3" xref="S4.SS5.SSS2.p2.11.m11.1.1.3.cmml">21.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.11.m11.1b"><apply id="S4.SS5.SSS2.p2.11.m11.1.1.cmml" xref="S4.SS5.SSS2.p2.11.m11.1.1"><lt id="S4.SS5.SSS2.p2.11.m11.1.1.1.cmml" xref="S4.SS5.SSS2.p2.11.m11.1.1.1"></lt><ci id="S4.SS5.SSS2.p2.11.m11.1.1.2.cmml" xref="S4.SS5.SSS2.p2.11.m11.1.1.2">𝑟</ci><cn id="S4.SS5.SSS2.p2.11.m11.1.1.3.cmml" type="float" xref="S4.SS5.SSS2.p2.11.m11.1.1.3">21.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.11.m11.1c">r<21.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.11.m11.1d">italic_r < 21.5</annotation></semantics></math> mag with a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.12.m12.1"><semantics id="S4.SS5.SSS2.p2.12.m12.1a"><mo id="S4.SS5.SSS2.p2.12.m12.1.1" xref="S4.SS5.SSS2.p2.12.m12.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.12.m12.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.12.m12.1.1.cmml" xref="S4.SS5.SSS2.p2.12.m12.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.12.m12.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.12.m12.1d">∼</annotation></semantics></math>4,000 deg-2 density, LRG down to <math alttext="z_{\rm fiber}=23.0" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.14.m14.1"><semantics id="S4.SS5.SSS2.p2.14.m14.1a"><mrow id="S4.SS5.SSS2.p2.14.m14.1.1" xref="S4.SS5.SSS2.p2.14.m14.1.1.cmml"><msub id="S4.SS5.SSS2.p2.14.m14.1.1.2" xref="S4.SS5.SSS2.p2.14.m14.1.1.2.cmml"><mi id="S4.SS5.SSS2.p2.14.m14.1.1.2.2" xref="S4.SS5.SSS2.p2.14.m14.1.1.2.2.cmml">z</mi><mi id="S4.SS5.SSS2.p2.14.m14.1.1.2.3" xref="S4.SS5.SSS2.p2.14.m14.1.1.2.3.cmml">fiber</mi></msub><mo id="S4.SS5.SSS2.p2.14.m14.1.1.1" xref="S4.SS5.SSS2.p2.14.m14.1.1.1.cmml">=</mo><mn id="S4.SS5.SSS2.p2.14.m14.1.1.3" xref="S4.SS5.SSS2.p2.14.m14.1.1.3.cmml">23.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.14.m14.1b"><apply id="S4.SS5.SSS2.p2.14.m14.1.1.cmml" xref="S4.SS5.SSS2.p2.14.m14.1.1"><eq id="S4.SS5.SSS2.p2.14.m14.1.1.1.cmml" xref="S4.SS5.SSS2.p2.14.m14.1.1.1"></eq><apply id="S4.SS5.SSS2.p2.14.m14.1.1.2.cmml" xref="S4.SS5.SSS2.p2.14.m14.1.1.2"><csymbol cd="ambiguous" id="S4.SS5.SSS2.p2.14.m14.1.1.2.1.cmml" xref="S4.SS5.SSS2.p2.14.m14.1.1.2">subscript</csymbol><ci id="S4.SS5.SSS2.p2.14.m14.1.1.2.2.cmml" xref="S4.SS5.SSS2.p2.14.m14.1.1.2.2">𝑧</ci><ci id="S4.SS5.SSS2.p2.14.m14.1.1.2.3.cmml" xref="S4.SS5.SSS2.p2.14.m14.1.1.2.3">fiber</ci></apply><cn id="S4.SS5.SSS2.p2.14.m14.1.1.3.cmml" type="float" xref="S4.SS5.SSS2.p2.14.m14.1.1.3">23.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.14.m14.1c">z_{\rm fiber}=23.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.14.m14.1d">italic_z start_POSTSUBSCRIPT roman_fiber end_POSTSUBSCRIPT = 23.0</annotation></semantics></math> mag with a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.15.m15.1"><semantics id="S4.SS5.SSS2.p2.15.m15.1a"><mo id="S4.SS5.SSS2.p2.15.m15.1.1" xref="S4.SS5.SSS2.p2.15.m15.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.15.m15.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.15.m15.1.1.cmml" xref="S4.SS5.SSS2.p2.15.m15.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.15.m15.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.15.m15.1d">∼</annotation></semantics></math>3,000 deg-2 density. Combining them with the current ELG sample will result in a target density <math alttext="\sim 2.7\times" class="ltx_math_unparsed" display="inline" id="S4.SS5.SSS2.p2.17.m17.1"><semantics id="S4.SS5.SSS2.p2.17.m17.1a"><mrow id="S4.SS5.SSS2.p2.17.m17.1b"><mo id="S4.SS5.SSS2.p2.17.m17.1.1">∼</mo><mn id="S4.SS5.SSS2.p2.17.m17.1.2">2.7</mn><mo id="S4.SS5.SSS2.p2.17.m17.1.3" lspace="0.222em">×</mo></mrow><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.17.m17.1c">\sim 2.7\times</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.17.m17.1d">∼ 2.7 ×</annotation></semantics></math> the fiber density. Assuming a 1.0 hour average exposure time, this project will take <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.18.m18.1"><semantics id="S4.SS5.SSS2.p2.18.m18.1a"><mo id="S4.SS5.SSS2.p2.18.m18.1.1" xref="S4.SS5.SSS2.p2.18.m18.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.18.m18.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.18.m18.1.1.cmml" xref="S4.SS5.SSS2.p2.18.m18.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.18.m18.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.18.m18.1d">∼</annotation></semantics></math>8.3-year of grey-time to finish with a <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.19.m19.1"><semantics id="S4.SS5.SSS2.p2.19.m19.1a"><mo id="S4.SS5.SSS2.p2.19.m19.1.1" xref="S4.SS5.SSS2.p2.19.m19.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.19.m19.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p2.19.m19.1.1.cmml" xref="S4.SS5.SSS2.p2.19.m19.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.19.m19.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.19.m19.1d">∼</annotation></semantics></math>114 million redshifts at <math alttext="z<1.6" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p2.20.m20.1"><semantics id="S4.SS5.SSS2.p2.20.m20.1a"><mrow id="S4.SS5.SSS2.p2.20.m20.1.1" xref="S4.SS5.SSS2.p2.20.m20.1.1.cmml"><mi id="S4.SS5.SSS2.p2.20.m20.1.1.2" xref="S4.SS5.SSS2.p2.20.m20.1.1.2.cmml">z</mi><mo id="S4.SS5.SSS2.p2.20.m20.1.1.1" xref="S4.SS5.SSS2.p2.20.m20.1.1.1.cmml"><</mo><mn id="S4.SS5.SSS2.p2.20.m20.1.1.3" xref="S4.SS5.SSS2.p2.20.m20.1.1.3.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p2.20.m20.1b"><apply id="S4.SS5.SSS2.p2.20.m20.1.1.cmml" xref="S4.SS5.SSS2.p2.20.m20.1.1"><lt id="S4.SS5.SSS2.p2.20.m20.1.1.1.cmml" xref="S4.SS5.SSS2.p2.20.m20.1.1.1"></lt><ci id="S4.SS5.SSS2.p2.20.m20.1.1.2.cmml" xref="S4.SS5.SSS2.p2.20.m20.1.1.2">𝑧</ci><cn id="S4.SS5.SSS2.p2.20.m20.1.1.3.cmml" type="float" xref="S4.SS5.SSS2.p2.20.m20.1.1.3">1.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p2.20.m20.1c">z<1.6</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p2.20.m20.1d">italic_z < 1.6</annotation></semantics></math> with an 80% average redshift efficiency. Safely assuming a fraction of these grey-time targets can be observed during bright time, it is possible to finish this along with the 7.5-year dark-time survey. </div> <div class="ltx_para" id="S4.SS5.SSS2.p3"> Given the required length of the dark-time program, an extension of the grey-night program down to a fainter magnitude limit could reach a target density of 3 times the fiber density of MUST. Assuming a 75% overall redshift success rate, MUST can obtain a total of <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p3.1.m1.1"><semantics id="S4.SS5.SSS2.p3.1.m1.1a"><mo id="S4.SS5.SSS2.p3.1.m1.1.1" xref="S4.SS5.SSS2.p3.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p3.1.m1.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p3.1.m1.1.1.cmml" xref="S4.SS5.SSS2.p3.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p3.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p3.1.m1.1d">∼</annotation></semantics></math>93 million redshifts during a 7.0-year grey-time survey. </div> <div class="ltx_para" id="S4.SS5.SSS2.p4"> Putting the grey & dark programs together, MUST has the capability of collecting <math alttext="\sim" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p4.1.m1.1"><semantics id="S4.SS5.SSS2.p4.1.m1.1a"><mo id="S4.SS5.SSS2.p4.1.m1.1.1" xref="S4.SS5.SSS2.p4.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p4.1.m1.1b"><csymbol cd="latexml" id="S4.SS5.SSS2.p4.1.m1.1.1.cmml" xref="S4.SS5.SSS2.p4.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p4.1.m1.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p4.1.m1.1d">∼</annotation></semantics></math>120 million redshifts at <math alttext="0.1<z<5.0" class="ltx_Math" display="inline" id="S4.SS5.SSS2.p4.2.m2.1"><semantics id="S4.SS5.SSS2.p4.2.m2.1a"><mrow id="S4.SS5.SSS2.p4.2.m2.1.1" xref="S4.SS5.SSS2.p4.2.m2.1.1.cmml"><mn id="S4.SS5.SSS2.p4.2.m2.1.1.2" xref="S4.SS5.SSS2.p4.2.m2.1.1.2.cmml">0.1</mn><mo id="S4.SS5.SSS2.p4.2.m2.1.1.3" xref="S4.SS5.SSS2.p4.2.m2.1.1.3.cmml"><</mo><mi id="S4.SS5.SSS2.p4.2.m2.1.1.4" xref="S4.SS5.SSS2.p4.2.m2.1.1.4.cmml">z</mi><mo id="S4.SS5.SSS2.p4.2.m2.1.1.5" xref="S4.SS5.SSS2.p4.2.m2.1.1.5.cmml"><</mo><mn id="S4.SS5.SSS2.p4.2.m2.1.1.6" xref="S4.SS5.SSS2.p4.2.m2.1.1.6.cmml">5.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS5.SSS2.p4.2.m2.1b"><apply id="S4.SS5.SSS2.p4.2.m2.1.1.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1"><and id="S4.SS5.SSS2.p4.2.m2.1.1a.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1"></and><apply id="S4.SS5.SSS2.p4.2.m2.1.1b.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1"><lt id="S4.SS5.SSS2.p4.2.m2.1.1.3.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1.3"></lt><cn id="S4.SS5.SSS2.p4.2.m2.1.1.2.cmml" type="float" xref="S4.SS5.SSS2.p4.2.m2.1.1.2">0.1</cn><ci id="S4.SS5.SSS2.p4.2.m2.1.1.4.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1.4">𝑧</ci></apply><apply id="S4.SS5.SSS2.p4.2.m2.1.1c.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1"><lt id="S4.SS5.SSS2.p4.2.m2.1.1.5.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S4.SS5.SSS2.p4.2.m2.1.1.4.cmml" id="S4.SS5.SSS2.p4.2.m2.1.1d.cmml" xref="S4.SS5.SSS2.p4.2.m2.1.1"></share><cn id="S4.SS5.SSS2.p4.2.m2.1.1.6.cmml" type="float" xref="S4.SS5.SSS2.p4.2.m2.1.1.6">5.0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS5.SSS2.p4.2.m2.1c">0.1<z<5.0</annotation><annotation encoding="application/x-llamapun" id="S4.SS5.SSS2.p4.2.m2.1d">0.1 < italic_z < 5.0</annotation></semantics></math> before 2040. </div> <figure class="ltx_figure" id="S4.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="153" id="S4.F5.g1" src="extracted/5997959/figures/MUST_grey_time_footprint.png" width="293"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_figure_panel ltx_img_landscape" height="153" id="S4.F5.g2" src="extracted/5997959/figures/MUST_dark_time_footprint.png" width="293"/></div> </div> <figcaption class="ltx_caption">Figure 5: Example of an observation sequence. Each red circle represents the footprint of MUST (5 deg2). Left: grey time based on the DECaLS footprint modified by a cut on declination (Dec <math alttext=">-10^{\circ}" class="ltx_Math" display="inline" id="S4.F5.6.m2.1"><semantics id="S4.F5.6.m2.1b"><mrow id="S4.F5.6.m2.1.1" xref="S4.F5.6.m2.1.1.cmml"><mi id="S4.F5.6.m2.1.1.2" xref="S4.F5.6.m2.1.1.2.cmml"></mi><mo id="S4.F5.6.m2.1.1.1" xref="S4.F5.6.m2.1.1.1.cmml">></mo><mrow id="S4.F5.6.m2.1.1.3" xref="S4.F5.6.m2.1.1.3.cmml"><mo id="S4.F5.6.m2.1.1.3b" xref="S4.F5.6.m2.1.1.3.cmml">−</mo><msup id="S4.F5.6.m2.1.1.3.2" xref="S4.F5.6.m2.1.1.3.2.cmml"><mn id="S4.F5.6.m2.1.1.3.2.2" xref="S4.F5.6.m2.1.1.3.2.2.cmml">10</mn><mo id="S4.F5.6.m2.1.1.3.2.3" xref="S4.F5.6.m2.1.1.3.2.3.cmml">∘</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.6.m2.1c"><apply id="S4.F5.6.m2.1.1.cmml" xref="S4.F5.6.m2.1.1"><gt id="S4.F5.6.m2.1.1.1.cmml" xref="S4.F5.6.m2.1.1.1"></gt><csymbol cd="latexml" id="S4.F5.6.m2.1.1.2.cmml" xref="S4.F5.6.m2.1.1.2">absent</csymbol><apply id="S4.F5.6.m2.1.1.3.cmml" xref="S4.F5.6.m2.1.1.3"><minus id="S4.F5.6.m2.1.1.3.1.cmml" xref="S4.F5.6.m2.1.1.3"></minus><apply id="S4.F5.6.m2.1.1.3.2.cmml" xref="S4.F5.6.m2.1.1.3.2"><csymbol cd="ambiguous" id="S4.F5.6.m2.1.1.3.2.1.cmml" xref="S4.F5.6.m2.1.1.3.2">superscript</csymbol><cn id="S4.F5.6.m2.1.1.3.2.2.cmml" type="integer" xref="S4.F5.6.m2.1.1.3.2.2">10</cn><compose id="S4.F5.6.m2.1.1.3.2.3.cmml" xref="S4.F5.6.m2.1.1.3.2.3"></compose></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.6.m2.1d">>-10^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.6.m2.1e">> - 10 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>). Right: dark time based on the CSST footprint. We get a total of 3100 and 2600 pointings reaching <math alttext="\sim" class="ltx_Math" display="inline" id="S4.F5.7.m3.1"><semantics id="S4.F5.7.m3.1b"><mo id="S4.F5.7.m3.1.1" xref="S4.F5.7.m3.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.F5.7.m3.1c"><csymbol cd="latexml" id="S4.F5.7.m3.1.1.cmml" xref="S4.F5.7.m3.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.7.m3.1d">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.F5.7.m3.1e">∼</annotation></semantics></math>13 000 and 11 000 deg2, respectively. On both maps, the stellar density of the Milky Way is represented using a white-red color map, and the Ecliptic plane region is highlighted in black on the right panel.</figcaption> </figure> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> 5 Cosmological Forecasts</h2> <div class="ltx_para" id="S5.p1"> To quantitatively evaluate the scientific potential of MUST in addressing the fundamental cosmological questions outlined in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3" title="3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3</a>, we perform forecasts for the statistical errors associated with several key science cases based on the expected target densities and biases detailed in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4" title="4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">4</a>. To highlight potential improvements offered by MUST, we compare our forecasts with those for DESI, the current state-of-the-art Stage-IV survey, using the target specifications given by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib223" title="">223</a>]</cite>. In particular, we focus on DESI galaxy samples, including BGSs, LRG, and ELG. Furthermore, to comprehensively assess the scientific potential of next-generation cosmological data, we include external priors from CMB surveys, including Planck <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib268" title="">268</a>]</cite> and the Simons Observatory (SO; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib45" title="">45</a>]</cite>), following the strategy detailed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib16" title="">16</a>]</cite>. </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> 5.1 Methodology</h3> <div class="ltx_para" id="S5.SS1.p1"> The Fisher forecast method is widely used to estimate the statistical uncertainty of cosmological parameters inferred from the observed power spectrum. This method is based on the Cramér-Rao inequality <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib269" title="">269</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib270" title="">270</a>]</cite>, which establishes that no unbiased estimator can achieve a covariance matrix smaller than the inverse of the Fisher information matrix <math alttext="\mathbf{F}" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.1"><semantics id="S5.SS1.p1.1.m1.1a"><mi id="S5.SS1.p1.1.m1.1.1" xref="S5.SS1.p1.1.m1.1.1.cmml">𝐅</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.1.m1.1b"><ci id="S5.SS1.p1.1.m1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1">𝐅</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.1c">\mathbf{F}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.1d">bold_F</annotation></semantics></math>. 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id="S5.SS1.p1.4.m4.1.1.1.1.1.1.2" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml">F</mi><mrow id="S5.SS1.p1.4.m4.1.1.1.1.1.1.3" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml"><mo id="S5.SS1.p1.4.m4.1.1.1.1.1.1.3a" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml">−</mo><mn id="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.2" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S5.SS1.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.SS1.p1.4.m4.1.1.1.3" xref="S5.SS1.p1.4.m4.1.1.1.3.cmml"><mi id="S5.SS1.p1.4.m4.1.1.1.3.2" xref="S5.SS1.p1.4.m4.1.1.1.3.2.cmml">i</mi><mo id="S5.SS1.p1.4.m4.1.1.1.3.1" xref="S5.SS1.p1.4.m4.1.1.1.3.1.cmml">⁢</mo><mi id="S5.SS1.p1.4.m4.1.1.1.3.3" xref="S5.SS1.p1.4.m4.1.1.1.3.3.cmml">i</mi></mrow></msub></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.4.m4.2b"><apply id="S5.SS1.p1.4.m4.2.2.cmml" xref="S5.SS1.p1.4.m4.2.2"><geq id="S5.SS1.p1.4.m4.2.2.2.cmml" xref="S5.SS1.p1.4.m4.2.2.2"></geq><apply id="S5.SS1.p1.4.m4.2.2.1.cmml" xref="S5.SS1.p1.4.m4.2.2.1"><times id="S5.SS1.p1.4.m4.2.2.1.2.cmml" xref="S5.SS1.p1.4.m4.2.2.1.2"></times><ci id="S5.SS1.p1.4.m4.2.2.1.3.cmml" xref="S5.SS1.p1.4.m4.2.2.1.3">𝜎</ci><apply id="S5.SS1.p1.4.m4.2.2.1.1.1.1.cmml" xref="S5.SS1.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S5.SS1.p1.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S5.SS1.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S5.SS1.p1.4.m4.2.2.1.1.1.1.2">𝜃</ci><ci id="S5.SS1.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="S5.SS1.p1.4.m4.2.2.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S5.SS1.p1.4.m4.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1"><root id="S5.SS1.p1.4.m4.1.1a.cmml" xref="S5.SS1.p1.4.m4.1.1"></root><apply id="S5.SS1.p1.4.m4.1.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.4.m4.1.1.1.2.cmml" xref="S5.SS1.p1.4.m4.1.1.1">subscript</csymbol><apply id="S5.SS1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S5.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.2">𝐹</ci><apply id="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.3"><minus id="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.3"></minus><cn id="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S5.SS1.p1.4.m4.1.1.1.1.1.1.3.2">1</cn></apply></apply><apply id="S5.SS1.p1.4.m4.1.1.1.3.cmml" xref="S5.SS1.p1.4.m4.1.1.1.3"><times id="S5.SS1.p1.4.m4.1.1.1.3.1.cmml" xref="S5.SS1.p1.4.m4.1.1.1.3.1"></times><ci id="S5.SS1.p1.4.m4.1.1.1.3.2.cmml" xref="S5.SS1.p1.4.m4.1.1.1.3.2">𝑖</ci><ci id="S5.SS1.p1.4.m4.1.1.1.3.3.cmml" xref="S5.SS1.p1.4.m4.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.4.m4.2c">\sigma(\theta_{i})\geq\sqrt{(F^{-1})_{ii}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.4.m4.2d">italic_σ ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≥ square-root start_ARG ( italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>. Following the formulation in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib271" title="">271</a>]</cite>, the Fisher information matrix for a redshift survey can be approximated as <table class="ltx_equation ltx_eqn_table" id="S5.E14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F_{ij}\sim\frac{\partial\bm{P}^{\mathrm{T}}}{\partial\theta_{i}}\mathbf{C}^{-1% }\frac{\partial\bm{P}}{\partial\theta_{j}}," class="ltx_Math" display="block" id="S5.E14.m1.1"><semantics id="S5.E14.m1.1a"><mrow id="S5.E14.m1.1.1.1" xref="S5.E14.m1.1.1.1.1.cmml"><mrow id="S5.E14.m1.1.1.1.1" xref="S5.E14.m1.1.1.1.1.cmml"><msub id="S5.E14.m1.1.1.1.1.2" xref="S5.E14.m1.1.1.1.1.2.cmml"><mi id="S5.E14.m1.1.1.1.1.2.2" xref="S5.E14.m1.1.1.1.1.2.2.cmml">F</mi><mrow id="S5.E14.m1.1.1.1.1.2.3" xref="S5.E14.m1.1.1.1.1.2.3.cmml"><mi id="S5.E14.m1.1.1.1.1.2.3.2" xref="S5.E14.m1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S5.E14.m1.1.1.1.1.2.3.1" xref="S5.E14.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S5.E14.m1.1.1.1.1.2.3.3" xref="S5.E14.m1.1.1.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="S5.E14.m1.1.1.1.1.1" xref="S5.E14.m1.1.1.1.1.1.cmml">∼</mo><mrow id="S5.E14.m1.1.1.1.1.3" xref="S5.E14.m1.1.1.1.1.3.cmml"><mfrac id="S5.E14.m1.1.1.1.1.3.2" xref="S5.E14.m1.1.1.1.1.3.2.cmml"><mrow id="S5.E14.m1.1.1.1.1.3.2.2" xref="S5.E14.m1.1.1.1.1.3.2.2.cmml"><mo id="S5.E14.m1.1.1.1.1.3.2.2.1" rspace="0em" xref="S5.E14.m1.1.1.1.1.3.2.2.1.cmml">∂</mo><msup id="S5.E14.m1.1.1.1.1.3.2.2.2" xref="S5.E14.m1.1.1.1.1.3.2.2.2.cmml"><mi id="S5.E14.m1.1.1.1.1.3.2.2.2.2" xref="S5.E14.m1.1.1.1.1.3.2.2.2.2.cmml">𝑷</mi><mi id="S5.E14.m1.1.1.1.1.3.2.2.2.3" mathvariant="normal" xref="S5.E14.m1.1.1.1.1.3.2.2.2.3.cmml">T</mi></msup></mrow><mrow id="S5.E14.m1.1.1.1.1.3.2.3" xref="S5.E14.m1.1.1.1.1.3.2.3.cmml"><mo id="S5.E14.m1.1.1.1.1.3.2.3.1" rspace="0em" xref="S5.E14.m1.1.1.1.1.3.2.3.1.cmml">∂</mo><msub id="S5.E14.m1.1.1.1.1.3.2.3.2" xref="S5.E14.m1.1.1.1.1.3.2.3.2.cmml"><mi id="S5.E14.m1.1.1.1.1.3.2.3.2.2" xref="S5.E14.m1.1.1.1.1.3.2.3.2.2.cmml">θ</mi><mi id="S5.E14.m1.1.1.1.1.3.2.3.2.3" xref="S5.E14.m1.1.1.1.1.3.2.3.2.3.cmml">i</mi></msub></mrow></mfrac><mo id="S5.E14.m1.1.1.1.1.3.1" xref="S5.E14.m1.1.1.1.1.3.1.cmml">⁢</mo><msup id="S5.E14.m1.1.1.1.1.3.3" xref="S5.E14.m1.1.1.1.1.3.3.cmml"><mi id="S5.E14.m1.1.1.1.1.3.3.2" xref="S5.E14.m1.1.1.1.1.3.3.2.cmml">𝐂</mi><mrow id="S5.E14.m1.1.1.1.1.3.3.3" xref="S5.E14.m1.1.1.1.1.3.3.3.cmml"><mo id="S5.E14.m1.1.1.1.1.3.3.3a" xref="S5.E14.m1.1.1.1.1.3.3.3.cmml">−</mo><mn id="S5.E14.m1.1.1.1.1.3.3.3.2" xref="S5.E14.m1.1.1.1.1.3.3.3.2.cmml">1</mn></mrow></msup><mo id="S5.E14.m1.1.1.1.1.3.1a" xref="S5.E14.m1.1.1.1.1.3.1.cmml">⁢</mo><mfrac id="S5.E14.m1.1.1.1.1.3.4" xref="S5.E14.m1.1.1.1.1.3.4.cmml"><mrow id="S5.E14.m1.1.1.1.1.3.4.2" xref="S5.E14.m1.1.1.1.1.3.4.2.cmml"><mo id="S5.E14.m1.1.1.1.1.3.4.2.1" rspace="0em" xref="S5.E14.m1.1.1.1.1.3.4.2.1.cmml">∂</mo><mi id="S5.E14.m1.1.1.1.1.3.4.2.2" xref="S5.E14.m1.1.1.1.1.3.4.2.2.cmml">𝑷</mi></mrow><mrow id="S5.E14.m1.1.1.1.1.3.4.3" xref="S5.E14.m1.1.1.1.1.3.4.3.cmml"><mo id="S5.E14.m1.1.1.1.1.3.4.3.1" rspace="0em" xref="S5.E14.m1.1.1.1.1.3.4.3.1.cmml">∂</mo><msub id="S5.E14.m1.1.1.1.1.3.4.3.2" xref="S5.E14.m1.1.1.1.1.3.4.3.2.cmml"><mi id="S5.E14.m1.1.1.1.1.3.4.3.2.2" xref="S5.E14.m1.1.1.1.1.3.4.3.2.2.cmml">θ</mi><mi id="S5.E14.m1.1.1.1.1.3.4.3.2.3" xref="S5.E14.m1.1.1.1.1.3.4.3.2.3.cmml">j</mi></msub></mrow></mfrac></mrow></mrow><mo id="S5.E14.m1.1.1.1.2" xref="S5.E14.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E14.m1.1b"><apply id="S5.E14.m1.1.1.1.1.cmml" xref="S5.E14.m1.1.1.1"><csymbol cd="latexml" id="S5.E14.m1.1.1.1.1.1.cmml" xref="S5.E14.m1.1.1.1.1.1">similar-to</csymbol><apply id="S5.E14.m1.1.1.1.1.2.cmml" xref="S5.E14.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.E14.m1.1.1.1.1.2.1.cmml" xref="S5.E14.m1.1.1.1.1.2">subscript</csymbol><ci id="S5.E14.m1.1.1.1.1.2.2.cmml" xref="S5.E14.m1.1.1.1.1.2.2">𝐹</ci><apply id="S5.E14.m1.1.1.1.1.2.3.cmml" xref="S5.E14.m1.1.1.1.1.2.3"><times id="S5.E14.m1.1.1.1.1.2.3.1.cmml" xref="S5.E14.m1.1.1.1.1.2.3.1"></times><ci id="S5.E14.m1.1.1.1.1.2.3.2.cmml" xref="S5.E14.m1.1.1.1.1.2.3.2">𝑖</ci><ci id="S5.E14.m1.1.1.1.1.2.3.3.cmml" xref="S5.E14.m1.1.1.1.1.2.3.3">𝑗</ci></apply></apply><apply id="S5.E14.m1.1.1.1.1.3.cmml" xref="S5.E14.m1.1.1.1.1.3"><times id="S5.E14.m1.1.1.1.1.3.1.cmml" xref="S5.E14.m1.1.1.1.1.3.1"></times><apply id="S5.E14.m1.1.1.1.1.3.2.cmml" xref="S5.E14.m1.1.1.1.1.3.2"><divide id="S5.E14.m1.1.1.1.1.3.2.1.cmml" xref="S5.E14.m1.1.1.1.1.3.2"></divide><apply id="S5.E14.m1.1.1.1.1.3.2.2.cmml" xref="S5.E14.m1.1.1.1.1.3.2.2"><partialdiff id="S5.E14.m1.1.1.1.1.3.2.2.1.cmml" xref="S5.E14.m1.1.1.1.1.3.2.2.1"></partialdiff><apply id="S5.E14.m1.1.1.1.1.3.2.2.2.cmml" xref="S5.E14.m1.1.1.1.1.3.2.2.2"><csymbol cd="ambiguous" id="S5.E14.m1.1.1.1.1.3.2.2.2.1.cmml" xref="S5.E14.m1.1.1.1.1.3.2.2.2">superscript</csymbol><ci id="S5.E14.m1.1.1.1.1.3.2.2.2.2.cmml" xref="S5.E14.m1.1.1.1.1.3.2.2.2.2">𝑷</ci><ci id="S5.E14.m1.1.1.1.1.3.2.2.2.3.cmml" xref="S5.E14.m1.1.1.1.1.3.2.2.2.3">T</ci></apply></apply><apply id="S5.E14.m1.1.1.1.1.3.2.3.cmml" xref="S5.E14.m1.1.1.1.1.3.2.3"><partialdiff id="S5.E14.m1.1.1.1.1.3.2.3.1.cmml" xref="S5.E14.m1.1.1.1.1.3.2.3.1"></partialdiff><apply id="S5.E14.m1.1.1.1.1.3.2.3.2.cmml" xref="S5.E14.m1.1.1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S5.E14.m1.1.1.1.1.3.2.3.2.1.cmml" xref="S5.E14.m1.1.1.1.1.3.2.3.2">subscript</csymbol><ci id="S5.E14.m1.1.1.1.1.3.2.3.2.2.cmml" xref="S5.E14.m1.1.1.1.1.3.2.3.2.2">𝜃</ci><ci id="S5.E14.m1.1.1.1.1.3.2.3.2.3.cmml" xref="S5.E14.m1.1.1.1.1.3.2.3.2.3">𝑖</ci></apply></apply></apply><apply id="S5.E14.m1.1.1.1.1.3.3.cmml" xref="S5.E14.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S5.E14.m1.1.1.1.1.3.3.1.cmml" xref="S5.E14.m1.1.1.1.1.3.3">superscript</csymbol><ci id="S5.E14.m1.1.1.1.1.3.3.2.cmml" xref="S5.E14.m1.1.1.1.1.3.3.2">𝐂</ci><apply id="S5.E14.m1.1.1.1.1.3.3.3.cmml" xref="S5.E14.m1.1.1.1.1.3.3.3"><minus id="S5.E14.m1.1.1.1.1.3.3.3.1.cmml" xref="S5.E14.m1.1.1.1.1.3.3.3"></minus><cn id="S5.E14.m1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S5.E14.m1.1.1.1.1.3.3.3.2">1</cn></apply></apply><apply id="S5.E14.m1.1.1.1.1.3.4.cmml" xref="S5.E14.m1.1.1.1.1.3.4"><divide id="S5.E14.m1.1.1.1.1.3.4.1.cmml" xref="S5.E14.m1.1.1.1.1.3.4"></divide><apply id="S5.E14.m1.1.1.1.1.3.4.2.cmml" xref="S5.E14.m1.1.1.1.1.3.4.2"><partialdiff id="S5.E14.m1.1.1.1.1.3.4.2.1.cmml" xref="S5.E14.m1.1.1.1.1.3.4.2.1"></partialdiff><ci id="S5.E14.m1.1.1.1.1.3.4.2.2.cmml" xref="S5.E14.m1.1.1.1.1.3.4.2.2">𝑷</ci></apply><apply id="S5.E14.m1.1.1.1.1.3.4.3.cmml" xref="S5.E14.m1.1.1.1.1.3.4.3"><partialdiff id="S5.E14.m1.1.1.1.1.3.4.3.1.cmml" xref="S5.E14.m1.1.1.1.1.3.4.3.1"></partialdiff><apply id="S5.E14.m1.1.1.1.1.3.4.3.2.cmml" xref="S5.E14.m1.1.1.1.1.3.4.3.2"><csymbol cd="ambiguous" id="S5.E14.m1.1.1.1.1.3.4.3.2.1.cmml" xref="S5.E14.m1.1.1.1.1.3.4.3.2">subscript</csymbol><ci id="S5.E14.m1.1.1.1.1.3.4.3.2.2.cmml" xref="S5.E14.m1.1.1.1.1.3.4.3.2.2">𝜃</ci><ci id="S5.E14.m1.1.1.1.1.3.4.3.2.3.cmml" xref="S5.E14.m1.1.1.1.1.3.4.3.2.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E14.m1.1c">F_{ij}\sim\frac{\partial\bm{P}^{\mathrm{T}}}{\partial\theta_{i}}\mathbf{C}^{-1% }\frac{\partial\bm{P}}{\partial\theta_{j}},</annotation><annotation encoding="application/x-llamapun" id="S5.E14.m1.1d">italic_F start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∼ divide start_ARG ∂ bold_italic_P start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG bold_C start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT divide start_ARG ∂ bold_italic_P end_ARG start_ARG ∂ italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(14)</td> </tr></tbody> </table> where <math alttext="\bm{P}" class="ltx_Math" display="inline" id="S5.SS1.p1.5.m1.1"><semantics id="S5.SS1.p1.5.m1.1a"><mi id="S5.SS1.p1.5.m1.1.1" xref="S5.SS1.p1.5.m1.1.1.cmml">𝑷</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.5.m1.1b"><ci id="S5.SS1.p1.5.m1.1.1.cmml" xref="S5.SS1.p1.5.m1.1.1">𝑷</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.5.m1.1c">\bm{P}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.5.m1.1d">bold_italic_P</annotation></semantics></math> is the data vector with <math alttext="P_{i}=P(k_{i},\mu_{i})" class="ltx_Math" display="inline" id="S5.SS1.p1.6.m2.2"><semantics id="S5.SS1.p1.6.m2.2a"><mrow id="S5.SS1.p1.6.m2.2.2" xref="S5.SS1.p1.6.m2.2.2.cmml"><msub id="S5.SS1.p1.6.m2.2.2.4" xref="S5.SS1.p1.6.m2.2.2.4.cmml"><mi id="S5.SS1.p1.6.m2.2.2.4.2" xref="S5.SS1.p1.6.m2.2.2.4.2.cmml">P</mi><mi id="S5.SS1.p1.6.m2.2.2.4.3" xref="S5.SS1.p1.6.m2.2.2.4.3.cmml">i</mi></msub><mo id="S5.SS1.p1.6.m2.2.2.3" xref="S5.SS1.p1.6.m2.2.2.3.cmml">=</mo><mrow id="S5.SS1.p1.6.m2.2.2.2" xref="S5.SS1.p1.6.m2.2.2.2.cmml"><mi id="S5.SS1.p1.6.m2.2.2.2.4" xref="S5.SS1.p1.6.m2.2.2.2.4.cmml">P</mi><mo id="S5.SS1.p1.6.m2.2.2.2.3" xref="S5.SS1.p1.6.m2.2.2.2.3.cmml">⁢</mo><mrow id="S5.SS1.p1.6.m2.2.2.2.2.2" xref="S5.SS1.p1.6.m2.2.2.2.2.3.cmml"><mo id="S5.SS1.p1.6.m2.2.2.2.2.2.3" stretchy="false" xref="S5.SS1.p1.6.m2.2.2.2.2.3.cmml">(</mo><msub id="S5.SS1.p1.6.m2.1.1.1.1.1.1" xref="S5.SS1.p1.6.m2.1.1.1.1.1.1.cmml"><mi id="S5.SS1.p1.6.m2.1.1.1.1.1.1.2" xref="S5.SS1.p1.6.m2.1.1.1.1.1.1.2.cmml">k</mi><mi id="S5.SS1.p1.6.m2.1.1.1.1.1.1.3" xref="S5.SS1.p1.6.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS1.p1.6.m2.2.2.2.2.2.4" xref="S5.SS1.p1.6.m2.2.2.2.2.3.cmml">,</mo><msub id="S5.SS1.p1.6.m2.2.2.2.2.2.2" xref="S5.SS1.p1.6.m2.2.2.2.2.2.2.cmml"><mi id="S5.SS1.p1.6.m2.2.2.2.2.2.2.2" xref="S5.SS1.p1.6.m2.2.2.2.2.2.2.2.cmml">μ</mi><mi id="S5.SS1.p1.6.m2.2.2.2.2.2.2.3" xref="S5.SS1.p1.6.m2.2.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S5.SS1.p1.6.m2.2.2.2.2.2.5" stretchy="false" xref="S5.SS1.p1.6.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.6.m2.2b"><apply id="S5.SS1.p1.6.m2.2.2.cmml" xref="S5.SS1.p1.6.m2.2.2"><eq id="S5.SS1.p1.6.m2.2.2.3.cmml" xref="S5.SS1.p1.6.m2.2.2.3"></eq><apply id="S5.SS1.p1.6.m2.2.2.4.cmml" xref="S5.SS1.p1.6.m2.2.2.4"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m2.2.2.4.1.cmml" xref="S5.SS1.p1.6.m2.2.2.4">subscript</csymbol><ci id="S5.SS1.p1.6.m2.2.2.4.2.cmml" xref="S5.SS1.p1.6.m2.2.2.4.2">𝑃</ci><ci id="S5.SS1.p1.6.m2.2.2.4.3.cmml" xref="S5.SS1.p1.6.m2.2.2.4.3">𝑖</ci></apply><apply id="S5.SS1.p1.6.m2.2.2.2.cmml" xref="S5.SS1.p1.6.m2.2.2.2"><times id="S5.SS1.p1.6.m2.2.2.2.3.cmml" xref="S5.SS1.p1.6.m2.2.2.2.3"></times><ci id="S5.SS1.p1.6.m2.2.2.2.4.cmml" xref="S5.SS1.p1.6.m2.2.2.2.4">𝑃</ci><interval closure="open" id="S5.SS1.p1.6.m2.2.2.2.2.3.cmml" xref="S5.SS1.p1.6.m2.2.2.2.2.2"><apply id="S5.SS1.p1.6.m2.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.6.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m2.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p1.6.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS1.p1.6.m2.1.1.1.1.1.1.2.cmml" xref="S5.SS1.p1.6.m2.1.1.1.1.1.1.2">𝑘</ci><ci id="S5.SS1.p1.6.m2.1.1.1.1.1.1.3.cmml" xref="S5.SS1.p1.6.m2.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S5.SS1.p1.6.m2.2.2.2.2.2.2.cmml" xref="S5.SS1.p1.6.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.SS1.p1.6.m2.2.2.2.2.2.2.1.cmml" xref="S5.SS1.p1.6.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S5.SS1.p1.6.m2.2.2.2.2.2.2.2.cmml" xref="S5.SS1.p1.6.m2.2.2.2.2.2.2.2">𝜇</ci><ci id="S5.SS1.p1.6.m2.2.2.2.2.2.2.3.cmml" xref="S5.SS1.p1.6.m2.2.2.2.2.2.2.3">𝑖</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.6.m2.2c">P_{i}=P(k_{i},\mu_{i})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.6.m2.2d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_P ( italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> representing the power spectrum of the <math alttext="i" class="ltx_Math" display="inline" id="S5.SS1.p1.7.m3.1"><semantics id="S5.SS1.p1.7.m3.1a"><mi id="S5.SS1.p1.7.m3.1.1" xref="S5.SS1.p1.7.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.7.m3.1b"><ci id="S5.SS1.p1.7.m3.1.1.cmml" xref="S5.SS1.p1.7.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.7.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.7.m3.1d">italic_i</annotation></semantics></math>-th bin in <math alttext="\bm{k}" class="ltx_Math" display="inline" id="S5.SS1.p1.8.m4.1"><semantics id="S5.SS1.p1.8.m4.1a"><mi id="S5.SS1.p1.8.m4.1.1" xref="S5.SS1.p1.8.m4.1.1.cmml">𝒌</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.8.m4.1b"><ci id="S5.SS1.p1.8.m4.1.1.cmml" xref="S5.SS1.p1.8.m4.1.1">𝒌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.8.m4.1c">\bm{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.8.m4.1d">bold_italic_k</annotation></semantics></math>-space, and <math alttext="\mathbf{C}" class="ltx_Math" display="inline" id="S5.SS1.p1.9.m5.1"><semantics id="S5.SS1.p1.9.m5.1a"><mi id="S5.SS1.p1.9.m5.1.1" xref="S5.SS1.p1.9.m5.1.1.cmml">𝐂</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.9.m5.1b"><ci id="S5.SS1.p1.9.m5.1.1.cmml" xref="S5.SS1.p1.9.m5.1.1">𝐂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.9.m5.1c">\mathbf{C}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.9.m5.1d">bold_C</annotation></semantics></math> is the associated covariance matrix of the power spectrum. With the usual assumption that the overdensity field is approximately a Gaussian random field, the covariance matrix <math alttext="\mathbf{C}" class="ltx_Math" display="inline" id="S5.SS1.p1.10.m6.1"><semantics id="S5.SS1.p1.10.m6.1a"><mi id="S5.SS1.p1.10.m6.1.1" xref="S5.SS1.p1.10.m6.1.1.cmml">𝐂</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.10.m6.1b"><ci id="S5.SS1.p1.10.m6.1.1.cmml" xref="S5.SS1.p1.10.m6.1.1">𝐂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.10.m6.1c">\mathbf{C}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.10.m6.1d">bold_C</annotation></semantics></math> is given by <table class="ltx_equation ltx_eqn_table" id="S5.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C_{ij}\sim\delta_{ij}\frac{2P_{i}^{2}}{V_{i}V_{\rm eff}}," class="ltx_Math" display="block" id="S5.E15.m1.1"><semantics id="S5.E15.m1.1a"><mrow id="S5.E15.m1.1.1.1" xref="S5.E15.m1.1.1.1.1.cmml"><mrow id="S5.E15.m1.1.1.1.1" xref="S5.E15.m1.1.1.1.1.cmml"><msub id="S5.E15.m1.1.1.1.1.2" xref="S5.E15.m1.1.1.1.1.2.cmml"><mi id="S5.E15.m1.1.1.1.1.2.2" xref="S5.E15.m1.1.1.1.1.2.2.cmml">C</mi><mrow id="S5.E15.m1.1.1.1.1.2.3" xref="S5.E15.m1.1.1.1.1.2.3.cmml"><mi id="S5.E15.m1.1.1.1.1.2.3.2" xref="S5.E15.m1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S5.E15.m1.1.1.1.1.2.3.1" xref="S5.E15.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S5.E15.m1.1.1.1.1.2.3.3" xref="S5.E15.m1.1.1.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="S5.E15.m1.1.1.1.1.1" xref="S5.E15.m1.1.1.1.1.1.cmml">∼</mo><mrow id="S5.E15.m1.1.1.1.1.3" xref="S5.E15.m1.1.1.1.1.3.cmml"><msub id="S5.E15.m1.1.1.1.1.3.2" xref="S5.E15.m1.1.1.1.1.3.2.cmml"><mi id="S5.E15.m1.1.1.1.1.3.2.2" xref="S5.E15.m1.1.1.1.1.3.2.2.cmml">δ</mi><mrow id="S5.E15.m1.1.1.1.1.3.2.3" xref="S5.E15.m1.1.1.1.1.3.2.3.cmml"><mi id="S5.E15.m1.1.1.1.1.3.2.3.2" xref="S5.E15.m1.1.1.1.1.3.2.3.2.cmml">i</mi><mo id="S5.E15.m1.1.1.1.1.3.2.3.1" xref="S5.E15.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S5.E15.m1.1.1.1.1.3.2.3.3" xref="S5.E15.m1.1.1.1.1.3.2.3.3.cmml">j</mi></mrow></msub><mo id="S5.E15.m1.1.1.1.1.3.1" xref="S5.E15.m1.1.1.1.1.3.1.cmml">⁢</mo><mfrac id="S5.E15.m1.1.1.1.1.3.3" xref="S5.E15.m1.1.1.1.1.3.3.cmml"><mrow id="S5.E15.m1.1.1.1.1.3.3.2" xref="S5.E15.m1.1.1.1.1.3.3.2.cmml"><mn id="S5.E15.m1.1.1.1.1.3.3.2.2" xref="S5.E15.m1.1.1.1.1.3.3.2.2.cmml">2</mn><mo id="S5.E15.m1.1.1.1.1.3.3.2.1" xref="S5.E15.m1.1.1.1.1.3.3.2.1.cmml">⁢</mo><msubsup id="S5.E15.m1.1.1.1.1.3.3.2.3" xref="S5.E15.m1.1.1.1.1.3.3.2.3.cmml"><mi id="S5.E15.m1.1.1.1.1.3.3.2.3.2.2" xref="S5.E15.m1.1.1.1.1.3.3.2.3.2.2.cmml">P</mi><mi id="S5.E15.m1.1.1.1.1.3.3.2.3.2.3" xref="S5.E15.m1.1.1.1.1.3.3.2.3.2.3.cmml">i</mi><mn id="S5.E15.m1.1.1.1.1.3.3.2.3.3" xref="S5.E15.m1.1.1.1.1.3.3.2.3.3.cmml">2</mn></msubsup></mrow><mrow id="S5.E15.m1.1.1.1.1.3.3.3" xref="S5.E15.m1.1.1.1.1.3.3.3.cmml"><msub id="S5.E15.m1.1.1.1.1.3.3.3.2" xref="S5.E15.m1.1.1.1.1.3.3.3.2.cmml"><mi id="S5.E15.m1.1.1.1.1.3.3.3.2.2" xref="S5.E15.m1.1.1.1.1.3.3.3.2.2.cmml">V</mi><mi id="S5.E15.m1.1.1.1.1.3.3.3.2.3" xref="S5.E15.m1.1.1.1.1.3.3.3.2.3.cmml">i</mi></msub><mo id="S5.E15.m1.1.1.1.1.3.3.3.1" xref="S5.E15.m1.1.1.1.1.3.3.3.1.cmml">⁢</mo><msub id="S5.E15.m1.1.1.1.1.3.3.3.3" xref="S5.E15.m1.1.1.1.1.3.3.3.3.cmml"><mi id="S5.E15.m1.1.1.1.1.3.3.3.3.2" xref="S5.E15.m1.1.1.1.1.3.3.3.3.2.cmml">V</mi><mi id="S5.E15.m1.1.1.1.1.3.3.3.3.3" xref="S5.E15.m1.1.1.1.1.3.3.3.3.3.cmml">eff</mi></msub></mrow></mfrac></mrow></mrow><mo id="S5.E15.m1.1.1.1.2" xref="S5.E15.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E15.m1.1b"><apply id="S5.E15.m1.1.1.1.1.cmml" xref="S5.E15.m1.1.1.1"><csymbol cd="latexml" id="S5.E15.m1.1.1.1.1.1.cmml" xref="S5.E15.m1.1.1.1.1.1">similar-to</csymbol><apply id="S5.E15.m1.1.1.1.1.2.cmml" xref="S5.E15.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.E15.m1.1.1.1.1.2.1.cmml" xref="S5.E15.m1.1.1.1.1.2">subscript</csymbol><ci id="S5.E15.m1.1.1.1.1.2.2.cmml" xref="S5.E15.m1.1.1.1.1.2.2">𝐶</ci><apply id="S5.E15.m1.1.1.1.1.2.3.cmml" xref="S5.E15.m1.1.1.1.1.2.3"><times id="S5.E15.m1.1.1.1.1.2.3.1.cmml" xref="S5.E15.m1.1.1.1.1.2.3.1"></times><ci id="S5.E15.m1.1.1.1.1.2.3.2.cmml" xref="S5.E15.m1.1.1.1.1.2.3.2">𝑖</ci><ci id="S5.E15.m1.1.1.1.1.2.3.3.cmml" xref="S5.E15.m1.1.1.1.1.2.3.3">𝑗</ci></apply></apply><apply id="S5.E15.m1.1.1.1.1.3.cmml" xref="S5.E15.m1.1.1.1.1.3"><times id="S5.E15.m1.1.1.1.1.3.1.cmml" xref="S5.E15.m1.1.1.1.1.3.1"></times><apply id="S5.E15.m1.1.1.1.1.3.2.cmml" xref="S5.E15.m1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.E15.m1.1.1.1.1.3.2.1.cmml" xref="S5.E15.m1.1.1.1.1.3.2">subscript</csymbol><ci id="S5.E15.m1.1.1.1.1.3.2.2.cmml" xref="S5.E15.m1.1.1.1.1.3.2.2">𝛿</ci><apply id="S5.E15.m1.1.1.1.1.3.2.3.cmml" xref="S5.E15.m1.1.1.1.1.3.2.3"><times id="S5.E15.m1.1.1.1.1.3.2.3.1.cmml" xref="S5.E15.m1.1.1.1.1.3.2.3.1"></times><ci id="S5.E15.m1.1.1.1.1.3.2.3.2.cmml" xref="S5.E15.m1.1.1.1.1.3.2.3.2">𝑖</ci><ci id="S5.E15.m1.1.1.1.1.3.2.3.3.cmml" xref="S5.E15.m1.1.1.1.1.3.2.3.3">𝑗</ci></apply></apply><apply id="S5.E15.m1.1.1.1.1.3.3.cmml" xref="S5.E15.m1.1.1.1.1.3.3"><divide id="S5.E15.m1.1.1.1.1.3.3.1.cmml" xref="S5.E15.m1.1.1.1.1.3.3"></divide><apply id="S5.E15.m1.1.1.1.1.3.3.2.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2"><times id="S5.E15.m1.1.1.1.1.3.3.2.1.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2.1"></times><cn id="S5.E15.m1.1.1.1.1.3.3.2.2.cmml" type="integer" xref="S5.E15.m1.1.1.1.1.3.3.2.2">2</cn><apply id="S5.E15.m1.1.1.1.1.3.3.2.3.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S5.E15.m1.1.1.1.1.3.3.2.3.1.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2.3">superscript</csymbol><apply id="S5.E15.m1.1.1.1.1.3.3.2.3.2.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S5.E15.m1.1.1.1.1.3.3.2.3.2.1.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2.3">subscript</csymbol><ci id="S5.E15.m1.1.1.1.1.3.3.2.3.2.2.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2.3.2.2">𝑃</ci><ci id="S5.E15.m1.1.1.1.1.3.3.2.3.2.3.cmml" xref="S5.E15.m1.1.1.1.1.3.3.2.3.2.3">𝑖</ci></apply><cn id="S5.E15.m1.1.1.1.1.3.3.2.3.3.cmml" type="integer" xref="S5.E15.m1.1.1.1.1.3.3.2.3.3">2</cn></apply></apply><apply id="S5.E15.m1.1.1.1.1.3.3.3.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3"><times id="S5.E15.m1.1.1.1.1.3.3.3.1.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.1"></times><apply id="S5.E15.m1.1.1.1.1.3.3.3.2.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.2"><csymbol cd="ambiguous" id="S5.E15.m1.1.1.1.1.3.3.3.2.1.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.2">subscript</csymbol><ci id="S5.E15.m1.1.1.1.1.3.3.3.2.2.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.2.2">𝑉</ci><ci id="S5.E15.m1.1.1.1.1.3.3.3.2.3.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.2.3">𝑖</ci></apply><apply id="S5.E15.m1.1.1.1.1.3.3.3.3.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.3"><csymbol cd="ambiguous" id="S5.E15.m1.1.1.1.1.3.3.3.3.1.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.3">subscript</csymbol><ci id="S5.E15.m1.1.1.1.1.3.3.3.3.2.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.3.2">𝑉</ci><ci id="S5.E15.m1.1.1.1.1.3.3.3.3.3.cmml" xref="S5.E15.m1.1.1.1.1.3.3.3.3.3">eff</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E15.m1.1c">C_{ij}\sim\delta_{ij}\frac{2P_{i}^{2}}{V_{i}V_{\rm eff}},</annotation><annotation encoding="application/x-llamapun" id="S5.E15.m1.1d">italic_C start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∼ italic_δ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT divide start_ARG 2 italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(15)</td> </tr></tbody> </table> where <math alttext="V_{i}" class="ltx_Math" display="inline" id="S5.SS1.p1.11.m1.1"><semantics id="S5.SS1.p1.11.m1.1a"><msub id="S5.SS1.p1.11.m1.1.1" xref="S5.SS1.p1.11.m1.1.1.cmml"><mi id="S5.SS1.p1.11.m1.1.1.2" xref="S5.SS1.p1.11.m1.1.1.2.cmml">V</mi><mi id="S5.SS1.p1.11.m1.1.1.3" xref="S5.SS1.p1.11.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.11.m1.1b"><apply id="S5.SS1.p1.11.m1.1.1.cmml" xref="S5.SS1.p1.11.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.11.m1.1.1.1.cmml" xref="S5.SS1.p1.11.m1.1.1">subscript</csymbol><ci id="S5.SS1.p1.11.m1.1.1.2.cmml" xref="S5.SS1.p1.11.m1.1.1.2">𝑉</ci><ci id="S5.SS1.p1.11.m1.1.1.3.cmml" xref="S5.SS1.p1.11.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.11.m1.1c">V_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.11.m1.1d">italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the volume of the <math alttext="i" class="ltx_Math" display="inline" id="S5.SS1.p1.12.m2.1"><semantics id="S5.SS1.p1.12.m2.1a"><mi id="S5.SS1.p1.12.m2.1.1" xref="S5.SS1.p1.12.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.12.m2.1b"><ci id="S5.SS1.p1.12.m2.1.1.cmml" xref="S5.SS1.p1.12.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.12.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.12.m2.1d">italic_i</annotation></semantics></math>-th bin in <math alttext="\bm{k}" class="ltx_Math" display="inline" id="S5.SS1.p1.13.m3.1"><semantics id="S5.SS1.p1.13.m3.1a"><mi id="S5.SS1.p1.13.m3.1.1" xref="S5.SS1.p1.13.m3.1.1.cmml">𝒌</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.13.m3.1b"><ci id="S5.SS1.p1.13.m3.1.1.cmml" xref="S5.SS1.p1.13.m3.1.1">𝒌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.13.m3.1c">\bm{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.13.m3.1d">bold_italic_k</annotation></semantics></math>-space, and <math alttext="V_{\rm eff}" class="ltx_Math" display="inline" id="S5.SS1.p1.14.m4.1"><semantics id="S5.SS1.p1.14.m4.1a"><msub id="S5.SS1.p1.14.m4.1.1" xref="S5.SS1.p1.14.m4.1.1.cmml"><mi id="S5.SS1.p1.14.m4.1.1.2" xref="S5.SS1.p1.14.m4.1.1.2.cmml">V</mi><mi id="S5.SS1.p1.14.m4.1.1.3" xref="S5.SS1.p1.14.m4.1.1.3.cmml">eff</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.14.m4.1b"><apply id="S5.SS1.p1.14.m4.1.1.cmml" xref="S5.SS1.p1.14.m4.1.1"><csymbol cd="ambiguous" id="S5.SS1.p1.14.m4.1.1.1.cmml" xref="S5.SS1.p1.14.m4.1.1">subscript</csymbol><ci id="S5.SS1.p1.14.m4.1.1.2.cmml" xref="S5.SS1.p1.14.m4.1.1.2">𝑉</ci><ci id="S5.SS1.p1.14.m4.1.1.3.cmml" xref="S5.SS1.p1.14.m4.1.1.3">eff</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.14.m4.1c">V_{\rm eff}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.14.m4.1d">italic_V start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT</annotation></semantics></math> is the effective survey volume. </div> <div class="ltx_para" id="S5.SS1.p2"> Throughout this work, we perform Fisher forecasts using the tracer power spectrum <math alttext="P(\bm{k})" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.1"><semantics id="S5.SS1.p2.1.m1.1a"><mrow id="S5.SS1.p2.1.m1.1.2" xref="S5.SS1.p2.1.m1.1.2.cmml"><mi id="S5.SS1.p2.1.m1.1.2.2" xref="S5.SS1.p2.1.m1.1.2.2.cmml">P</mi><mo id="S5.SS1.p2.1.m1.1.2.1" xref="S5.SS1.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS1.p2.1.m1.1.2.3.2" xref="S5.SS1.p2.1.m1.1.2.cmml"><mo id="S5.SS1.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S5.SS1.p2.1.m1.1.2.cmml">(</mo><mi id="S5.SS1.p2.1.m1.1.1" xref="S5.SS1.p2.1.m1.1.1.cmml">𝒌</mi><mo id="S5.SS1.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S5.SS1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.1b"><apply id="S5.SS1.p2.1.m1.1.2.cmml" xref="S5.SS1.p2.1.m1.1.2"><times id="S5.SS1.p2.1.m1.1.2.1.cmml" xref="S5.SS1.p2.1.m1.1.2.1"></times><ci id="S5.SS1.p2.1.m1.1.2.2.cmml" xref="S5.SS1.p2.1.m1.1.2.2">𝑃</ci><ci id="S5.SS1.p2.1.m1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1">𝒌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.1c">P(\bm{k})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.1d">italic_P ( bold_italic_k )</annotation></semantics></math>, with <table class="ltx_equation ltx_eqn_table" id="S5.E16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="k_{\rm max}=0.4\,h\,{\rm Mpc}^{-1}," 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start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(16)</td> </tr></tbody> </table> which can be computed using either full-shape models or a template-based manner. However, we use Bayesian inference to estimate error for forecasts related to dark matter constraints. For the full-shape model, we make use of the forecast tool FishLSS131313<a class="ltx_ref ltx_url ltx_font_typewriter ltx_font_upright" href="https://github.com/NoahSailer/FishLSS" title="">https://github.com/NoahSailer/FishLSS</a>, which is described in detail in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib16" title="">16</a>]</cite>. FishLSS computes the nonlinear tracer power spectrum using Lagrangian perturbation theory with a linear tracer bias model, further refined by fitting a single-parameter counter-term to the HaloFit prescription. Shot noise and the finger-of-god effect are incorporated based on the density of the tracers. The template-based approach is only used for the <math alttext="f\sigma_{8}" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m1.1"><semantics id="S5.SS1.p2.2.m1.1a"><mrow id="S5.SS1.p2.2.m1.1.1" xref="S5.SS1.p2.2.m1.1.1.cmml"><mi id="S5.SS1.p2.2.m1.1.1.2" xref="S5.SS1.p2.2.m1.1.1.2.cmml">f</mi><mo id="S5.SS1.p2.2.m1.1.1.1" xref="S5.SS1.p2.2.m1.1.1.1.cmml">⁢</mo><msub id="S5.SS1.p2.2.m1.1.1.3" xref="S5.SS1.p2.2.m1.1.1.3.cmml"><mi id="S5.SS1.p2.2.m1.1.1.3.2" xref="S5.SS1.p2.2.m1.1.1.3.2.cmml">σ</mi><mn id="S5.SS1.p2.2.m1.1.1.3.3" xref="S5.SS1.p2.2.m1.1.1.3.3.cmml">8</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m1.1b"><apply id="S5.SS1.p2.2.m1.1.1.cmml" xref="S5.SS1.p2.2.m1.1.1"><times id="S5.SS1.p2.2.m1.1.1.1.cmml" xref="S5.SS1.p2.2.m1.1.1.1"></times><ci id="S5.SS1.p2.2.m1.1.1.2.cmml" xref="S5.SS1.p2.2.m1.1.1.2">𝑓</ci><apply id="S5.SS1.p2.2.m1.1.1.3.cmml" xref="S5.SS1.p2.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p2.2.m1.1.1.3.1.cmml" xref="S5.SS1.p2.2.m1.1.1.3">subscript</csymbol><ci id="S5.SS1.p2.2.m1.1.1.3.2.cmml" xref="S5.SS1.p2.2.m1.1.1.3.2">𝜎</ci><cn id="S5.SS1.p2.2.m1.1.1.3.3.cmml" type="integer" xref="S5.SS1.p2.2.m1.1.1.3.3">8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m1.1c">f\sigma_{8}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m1.1d">italic_f italic_σ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT</annotation></semantics></math> forecasts (Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.SS3" title="5.3 Structure Growth & Gravity ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">5.3</a>). In this approach, we adopt a code141414<a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/wdoumerg/Forecast_highz_spectroscopic_survey" title="">https://github.com/wdoumerg/Forecast_highz_spectroscopic_survey</a> developed in a recent forecast for redshift surveys <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib272" title="">272</a>]</cite>, which refers to the RSD template developed by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib273" title="">273</a>]</cite>. </div> <div class="ltx_para" id="S5.SS1.p3"> The number densities and biases of different tracers used for our forecasts are detailed in Table <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S4.T3" title="Table 3 ‣ 4.3.3 Quasi-Stellar Objects (QSO) ‣ 4.3 High-Redshift Tracers ‣ 4 Target Selection ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3</a>. We perform two sets of forecasts – optimistic and conservative – based on the different estimations of available imaging data to ensure a comprehensive evaluation of the scientific potential of MUST. </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> 5.2 Dark Energy</h3> <div class="ltx_para" id="S5.SS2.p1"> As a critical driver of cosmic expansion history, dark energy – particularly its dynamic behavior – can be unveiled by geometrical measurements, with BAO as the standard ruler (see Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS1" title="3.1 Nature & Evolution of Dark Energy ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.1</a>). 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z</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.3.m3.1d">roman_Δ italic_z</annotation></semantics></math> of tracer pairs that exhibit excess correlation in the transverse and line-of-sight directions, respectively: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx2"> <tbody id="S5.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle r_{d}=(1+z)D_{A}(z)\Delta\theta=\Delta\theta\int^{z}_{0}\frac{% cdz^{\prime}}{H(z^{\prime})}," class="ltx_Math" display="inline" id="S5.E17.m1.3"><semantics id="S5.E17.m1.3a"><mrow id="S5.E17.m1.3.3.1" xref="S5.E17.m1.3.3.1.1.cmml"><mrow id="S5.E17.m1.3.3.1.1" xref="S5.E17.m1.3.3.1.1.cmml"><msub id="S5.E17.m1.3.3.1.1.3" xref="S5.E17.m1.3.3.1.1.3.cmml"><mi id="S5.E17.m1.3.3.1.1.3.2" xref="S5.E17.m1.3.3.1.1.3.2.cmml">r</mi><mi id="S5.E17.m1.3.3.1.1.3.3" 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xref="S5.E17.m1.1.1.1.3">𝐻</ci><apply id="S5.E17.m1.1.1.1.1.1.1.cmml" xref="S5.E17.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E17.m1.1.1.1.1.1.1.1.cmml" xref="S5.E17.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.E17.m1.1.1.1.1.1.1.2.cmml" xref="S5.E17.m1.1.1.1.1.1.1.2">𝑧</ci><ci id="S5.E17.m1.1.1.1.1.1.1.3.cmml" xref="S5.E17.m1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E17.m1.3c">\displaystyle r_{d}=(1+z)D_{A}(z)\Delta\theta=\Delta\theta\int^{z}_{0}\frac{% cdz^{\prime}}{H(z^{\prime})},</annotation><annotation encoding="application/x-llamapun" id="S5.E17.m1.3d">italic_r start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = ( 1 + italic_z ) italic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_z ) roman_Δ italic_θ = roman_Δ italic_θ ∫ start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG italic_c italic_d italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_H ( italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(17)</td> </tr></tbody> <tbody id="S5.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle r_{d}\approx\frac{c\Delta z}{H(z)}," class="ltx_Math" display="inline" id="S5.E18.m1.2"><semantics id="S5.E18.m1.2a"><mrow id="S5.E18.m1.2.2.1" xref="S5.E18.m1.2.2.1.1.cmml"><mrow id="S5.E18.m1.2.2.1.1" xref="S5.E18.m1.2.2.1.1.cmml"><msub id="S5.E18.m1.2.2.1.1.2" xref="S5.E18.m1.2.2.1.1.2.cmml"><mi id="S5.E18.m1.2.2.1.1.2.2" xref="S5.E18.m1.2.2.1.1.2.2.cmml">r</mi><mi id="S5.E18.m1.2.2.1.1.2.3" xref="S5.E18.m1.2.2.1.1.2.3.cmml">d</mi></msub><mo id="S5.E18.m1.2.2.1.1.1" xref="S5.E18.m1.2.2.1.1.1.cmml">≈</mo><mstyle displaystyle="true" id="S5.E18.m1.1.1" xref="S5.E18.m1.1.1.cmml"><mfrac id="S5.E18.m1.1.1a" xref="S5.E18.m1.1.1.cmml"><mrow id="S5.E18.m1.1.1.3" xref="S5.E18.m1.1.1.3.cmml"><mi id="S5.E18.m1.1.1.3.2" xref="S5.E18.m1.1.1.3.2.cmml">c</mi><mo id="S5.E18.m1.1.1.3.1" xref="S5.E18.m1.1.1.3.1.cmml">⁢</mo><mi id="S5.E18.m1.1.1.3.3" mathvariant="normal" xref="S5.E18.m1.1.1.3.3.cmml">Δ</mi><mo id="S5.E18.m1.1.1.3.1a" xref="S5.E18.m1.1.1.3.1.cmml">⁢</mo><mi id="S5.E18.m1.1.1.3.4" xref="S5.E18.m1.1.1.3.4.cmml">z</mi></mrow><mrow id="S5.E18.m1.1.1.1" xref="S5.E18.m1.1.1.1.cmml"><mi id="S5.E18.m1.1.1.1.3" xref="S5.E18.m1.1.1.1.3.cmml">H</mi><mo id="S5.E18.m1.1.1.1.2" xref="S5.E18.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E18.m1.1.1.1.4.2" xref="S5.E18.m1.1.1.1.cmml"><mo id="S5.E18.m1.1.1.1.4.2.1" stretchy="false" xref="S5.E18.m1.1.1.1.cmml">(</mo><mi id="S5.E18.m1.1.1.1.1" xref="S5.E18.m1.1.1.1.1.cmml">z</mi><mo id="S5.E18.m1.1.1.1.4.2.2" stretchy="false" xref="S5.E18.m1.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo id="S5.E18.m1.2.2.1.2" xref="S5.E18.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E18.m1.2b"><apply id="S5.E18.m1.2.2.1.1.cmml" xref="S5.E18.m1.2.2.1"><approx id="S5.E18.m1.2.2.1.1.1.cmml" xref="S5.E18.m1.2.2.1.1.1"></approx><apply id="S5.E18.m1.2.2.1.1.2.cmml" xref="S5.E18.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="S5.E18.m1.2.2.1.1.2.1.cmml" xref="S5.E18.m1.2.2.1.1.2">subscript</csymbol><ci id="S5.E18.m1.2.2.1.1.2.2.cmml" xref="S5.E18.m1.2.2.1.1.2.2">𝑟</ci><ci id="S5.E18.m1.2.2.1.1.2.3.cmml" xref="S5.E18.m1.2.2.1.1.2.3">𝑑</ci></apply><apply id="S5.E18.m1.1.1.cmml" xref="S5.E18.m1.1.1"><divide id="S5.E18.m1.1.1.2.cmml" xref="S5.E18.m1.1.1"></divide><apply id="S5.E18.m1.1.1.3.cmml" xref="S5.E18.m1.1.1.3"><times id="S5.E18.m1.1.1.3.1.cmml" xref="S5.E18.m1.1.1.3.1"></times><ci id="S5.E18.m1.1.1.3.2.cmml" xref="S5.E18.m1.1.1.3.2">𝑐</ci><ci id="S5.E18.m1.1.1.3.3.cmml" xref="S5.E18.m1.1.1.3.3">Δ</ci><ci id="S5.E18.m1.1.1.3.4.cmml" xref="S5.E18.m1.1.1.3.4">𝑧</ci></apply><apply id="S5.E18.m1.1.1.1.cmml" xref="S5.E18.m1.1.1.1"><times id="S5.E18.m1.1.1.1.2.cmml" xref="S5.E18.m1.1.1.1.2"></times><ci id="S5.E18.m1.1.1.1.3.cmml" xref="S5.E18.m1.1.1.1.3">𝐻</ci><ci id="S5.E18.m1.1.1.1.1.cmml" xref="S5.E18.m1.1.1.1.1">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E18.m1.2c">\displaystyle r_{d}\approx\frac{c\Delta z}{H(z)},</annotation><annotation encoding="application/x-llamapun" id="S5.E18.m1.2d">italic_r start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ≈ divide start_ARG italic_c roman_Δ italic_z end_ARG start_ARG italic_H ( italic_z ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(18)</td> </tr></tbody> </table> where <math alttext="D_{A}(z)" 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xref="S5.SS2.p1.4.m1.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.p1.4.m1.1.2.2.1.cmml" xref="S5.SS2.p1.4.m1.1.2.2">subscript</csymbol><ci id="S5.SS2.p1.4.m1.1.2.2.2.cmml" xref="S5.SS2.p1.4.m1.1.2.2.2">𝐷</ci><ci id="S5.SS2.p1.4.m1.1.2.2.3.cmml" xref="S5.SS2.p1.4.m1.1.2.2.3">𝐴</ci></apply><ci id="S5.SS2.p1.4.m1.1.1.cmml" xref="S5.SS2.p1.4.m1.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.4.m1.1c">D_{A}(z)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.4.m1.1d">italic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> denotes the angular diameter distance and <math alttext="H(z)" class="ltx_Math" display="inline" id="S5.SS2.p1.5.m2.1"><semantics id="S5.SS2.p1.5.m2.1a"><mrow id="S5.SS2.p1.5.m2.1.2" xref="S5.SS2.p1.5.m2.1.2.cmml"><mi id="S5.SS2.p1.5.m2.1.2.2" xref="S5.SS2.p1.5.m2.1.2.2.cmml">H</mi><mo id="S5.SS2.p1.5.m2.1.2.1" xref="S5.SS2.p1.5.m2.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.p1.5.m2.1.2.3.2" xref="S5.SS2.p1.5.m2.1.2.cmml"><mo id="S5.SS2.p1.5.m2.1.2.3.2.1" stretchy="false" xref="S5.SS2.p1.5.m2.1.2.cmml">(</mo><mi id="S5.SS2.p1.5.m2.1.1" xref="S5.SS2.p1.5.m2.1.1.cmml">z</mi><mo id="S5.SS2.p1.5.m2.1.2.3.2.2" stretchy="false" xref="S5.SS2.p1.5.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.5.m2.1b"><apply id="S5.SS2.p1.5.m2.1.2.cmml" xref="S5.SS2.p1.5.m2.1.2"><times id="S5.SS2.p1.5.m2.1.2.1.cmml" xref="S5.SS2.p1.5.m2.1.2.1"></times><ci id="S5.SS2.p1.5.m2.1.2.2.cmml" xref="S5.SS2.p1.5.m2.1.2.2">𝐻</ci><ci id="S5.SS2.p1.5.m2.1.1.cmml" xref="S5.SS2.p1.5.m2.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.5.m2.1c">H(z)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.5.m2.1d">italic_H ( italic_z )</annotation></semantics></math> is the Hubble parameter. Thus, the BAO feature permits measurements of the combinations <math alttext="D_{A}(z)/r_{d}" class="ltx_Math" display="inline" id="S5.SS2.p1.6.m3.1"><semantics id="S5.SS2.p1.6.m3.1a"><mrow id="S5.SS2.p1.6.m3.1.2" xref="S5.SS2.p1.6.m3.1.2.cmml"><mrow id="S5.SS2.p1.6.m3.1.2.2" xref="S5.SS2.p1.6.m3.1.2.2.cmml"><msub id="S5.SS2.p1.6.m3.1.2.2.2" xref="S5.SS2.p1.6.m3.1.2.2.2.cmml"><mi id="S5.SS2.p1.6.m3.1.2.2.2.2" xref="S5.SS2.p1.6.m3.1.2.2.2.2.cmml">D</mi><mi id="S5.SS2.p1.6.m3.1.2.2.2.3" xref="S5.SS2.p1.6.m3.1.2.2.2.3.cmml">A</mi></msub><mo id="S5.SS2.p1.6.m3.1.2.2.1" xref="S5.SS2.p1.6.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.p1.6.m3.1.2.2.3.2" xref="S5.SS2.p1.6.m3.1.2.2.cmml"><mo id="S5.SS2.p1.6.m3.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.p1.6.m3.1.2.2.cmml">(</mo><mi id="S5.SS2.p1.6.m3.1.1" xref="S5.SS2.p1.6.m3.1.1.cmml">z</mi><mo id="S5.SS2.p1.6.m3.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.p1.6.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.p1.6.m3.1.2.1" xref="S5.SS2.p1.6.m3.1.2.1.cmml">/</mo><msub id="S5.SS2.p1.6.m3.1.2.3" xref="S5.SS2.p1.6.m3.1.2.3.cmml"><mi id="S5.SS2.p1.6.m3.1.2.3.2" xref="S5.SS2.p1.6.m3.1.2.3.2.cmml">r</mi><mi id="S5.SS2.p1.6.m3.1.2.3.3" xref="S5.SS2.p1.6.m3.1.2.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.6.m3.1b"><apply id="S5.SS2.p1.6.m3.1.2.cmml" xref="S5.SS2.p1.6.m3.1.2"><divide id="S5.SS2.p1.6.m3.1.2.1.cmml" xref="S5.SS2.p1.6.m3.1.2.1"></divide><apply id="S5.SS2.p1.6.m3.1.2.2.cmml" xref="S5.SS2.p1.6.m3.1.2.2"><times id="S5.SS2.p1.6.m3.1.2.2.1.cmml" xref="S5.SS2.p1.6.m3.1.2.2.1"></times><apply id="S5.SS2.p1.6.m3.1.2.2.2.cmml" xref="S5.SS2.p1.6.m3.1.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.p1.6.m3.1.2.2.2.1.cmml" xref="S5.SS2.p1.6.m3.1.2.2.2">subscript</csymbol><ci id="S5.SS2.p1.6.m3.1.2.2.2.2.cmml" xref="S5.SS2.p1.6.m3.1.2.2.2.2">𝐷</ci><ci id="S5.SS2.p1.6.m3.1.2.2.2.3.cmml" xref="S5.SS2.p1.6.m3.1.2.2.2.3">𝐴</ci></apply><ci id="S5.SS2.p1.6.m3.1.1.cmml" xref="S5.SS2.p1.6.m3.1.1">𝑧</ci></apply><apply id="S5.SS2.p1.6.m3.1.2.3.cmml" xref="S5.SS2.p1.6.m3.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.p1.6.m3.1.2.3.1.cmml" xref="S5.SS2.p1.6.m3.1.2.3">subscript</csymbol><ci id="S5.SS2.p1.6.m3.1.2.3.2.cmml" xref="S5.SS2.p1.6.m3.1.2.3.2">𝑟</ci><ci id="S5.SS2.p1.6.m3.1.2.3.3.cmml" xref="S5.SS2.p1.6.m3.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.6.m3.1c">D_{A}(z)/r_{d}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.6.m3.1d">italic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_z ) / italic_r start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H(z)r_{d}" class="ltx_Math" display="inline" id="S5.SS2.p1.7.m4.1"><semantics id="S5.SS2.p1.7.m4.1a"><mrow id="S5.SS2.p1.7.m4.1.2" xref="S5.SS2.p1.7.m4.1.2.cmml"><mi id="S5.SS2.p1.7.m4.1.2.2" xref="S5.SS2.p1.7.m4.1.2.2.cmml">H</mi><mo id="S5.SS2.p1.7.m4.1.2.1" xref="S5.SS2.p1.7.m4.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.p1.7.m4.1.2.3.2" xref="S5.SS2.p1.7.m4.1.2.cmml"><mo id="S5.SS2.p1.7.m4.1.2.3.2.1" stretchy="false" xref="S5.SS2.p1.7.m4.1.2.cmml">(</mo><mi id="S5.SS2.p1.7.m4.1.1" xref="S5.SS2.p1.7.m4.1.1.cmml">z</mi><mo id="S5.SS2.p1.7.m4.1.2.3.2.2" stretchy="false" xref="S5.SS2.p1.7.m4.1.2.cmml">)</mo></mrow><mo id="S5.SS2.p1.7.m4.1.2.1a" xref="S5.SS2.p1.7.m4.1.2.1.cmml">⁢</mo><msub id="S5.SS2.p1.7.m4.1.2.4" xref="S5.SS2.p1.7.m4.1.2.4.cmml"><mi id="S5.SS2.p1.7.m4.1.2.4.2" xref="S5.SS2.p1.7.m4.1.2.4.2.cmml">r</mi><mi id="S5.SS2.p1.7.m4.1.2.4.3" xref="S5.SS2.p1.7.m4.1.2.4.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.7.m4.1b"><apply id="S5.SS2.p1.7.m4.1.2.cmml" xref="S5.SS2.p1.7.m4.1.2"><times id="S5.SS2.p1.7.m4.1.2.1.cmml" xref="S5.SS2.p1.7.m4.1.2.1"></times><ci id="S5.SS2.p1.7.m4.1.2.2.cmml" xref="S5.SS2.p1.7.m4.1.2.2">𝐻</ci><ci id="S5.SS2.p1.7.m4.1.1.cmml" xref="S5.SS2.p1.7.m4.1.1">𝑧</ci><apply 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The top panels show Fisher forecast results for MUST tracers across different redshifts, where the upper and lower boundaries of the shaded regions correspond to results based on the conservative and optimistic target density estimations, respectively. The error bars in the bottom panels are all evaluated in the optimistic scenario, with the red lines representing predictions of the best-fit flat-<math alttext="w_{0}w_{a}" class="ltx_Math" display="inline" id="S5.SS2.p2.5.m5.1"><semantics id="S5.SS2.p2.5.m5.1a"><mrow id="S5.SS2.p2.5.m5.1.1" xref="S5.SS2.p2.5.m5.1.1.cmml"><msub id="S5.SS2.p2.5.m5.1.1.2" xref="S5.SS2.p2.5.m5.1.1.2.cmml"><mi id="S5.SS2.p2.5.m5.1.1.2.2" xref="S5.SS2.p2.5.m5.1.1.2.2.cmml">w</mi><mn id="S5.SS2.p2.5.m5.1.1.2.3" xref="S5.SS2.p2.5.m5.1.1.2.3.cmml">0</mn></msub><mo id="S5.SS2.p2.5.m5.1.1.1" xref="S5.SS2.p2.5.m5.1.1.1.cmml">⁢</mo><msub id="S5.SS2.p2.5.m5.1.1.3" xref="S5.SS2.p2.5.m5.1.1.3.cmml"><mi id="S5.SS2.p2.5.m5.1.1.3.2" xref="S5.SS2.p2.5.m5.1.1.3.2.cmml">w</mi><mi id="S5.SS2.p2.5.m5.1.1.3.3" xref="S5.SS2.p2.5.m5.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.5.m5.1b"><apply id="S5.SS2.p2.5.m5.1.1.cmml" xref="S5.SS2.p2.5.m5.1.1"><times id="S5.SS2.p2.5.m5.1.1.1.cmml" xref="S5.SS2.p2.5.m5.1.1.1"></times><apply id="S5.SS2.p2.5.m5.1.1.2.cmml" xref="S5.SS2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.p2.5.m5.1.1.2.1.cmml" xref="S5.SS2.p2.5.m5.1.1.2">subscript</csymbol><ci id="S5.SS2.p2.5.m5.1.1.2.2.cmml" xref="S5.SS2.p2.5.m5.1.1.2.2">𝑤</ci><cn id="S5.SS2.p2.5.m5.1.1.2.3.cmml" type="integer" xref="S5.SS2.p2.5.m5.1.1.2.3">0</cn></apply><apply id="S5.SS2.p2.5.m5.1.1.3.cmml" xref="S5.SS2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.p2.5.m5.1.1.3.1.cmml" xref="S5.SS2.p2.5.m5.1.1.3">subscript</csymbol><ci id="S5.SS2.p2.5.m5.1.1.3.2.cmml" xref="S5.SS2.p2.5.m5.1.1.3.2">𝑤</ci><ci id="S5.SS2.p2.5.m5.1.1.3.3.cmml" xref="S5.SS2.p2.5.m5.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.5.m5.1c">w_{0}w_{a}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.5.m5.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>CDM model based on the combination of DESI Y1, CMB, and DES Y5 data <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib73" title="">73</a>]</cite>, illustrating how distance measurements can differentiate between dark energy models. </div> <div class="ltx_para" id="S5.SS2.p3"> The precision in distance measurements of MUST is generally at the percent level with the assumption of a fiducial <math alttext="{\rm\Lambda}" class="ltx_Math" display="inline" id="S5.SS2.p3.1.m1.1"><semantics id="S5.SS2.p3.1.m1.1a"><mi id="S5.SS2.p3.1.m1.1.1" mathvariant="normal" xref="S5.SS2.p3.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.1.m1.1b"><ci id="S5.SS2.p3.1.m1.1.1.cmml" xref="S5.SS2.p3.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.1.m1.1c">{\rm\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.1.m1.1d">roman_Λ</annotation></semantics></math>CDM cosmology, demonstrating strong potential for constraining the equation of state. In the local Universe (<math alttext="z<0.7" class="ltx_Math" display="inline" id="S5.SS2.p3.2.m2.1"><semantics id="S5.SS2.p3.2.m2.1a"><mrow id="S5.SS2.p3.2.m2.1.1" xref="S5.SS2.p3.2.m2.1.1.cmml"><mi id="S5.SS2.p3.2.m2.1.1.2" xref="S5.SS2.p3.2.m2.1.1.2.cmml">z</mi><mo id="S5.SS2.p3.2.m2.1.1.1" xref="S5.SS2.p3.2.m2.1.1.1.cmml"><</mo><mn id="S5.SS2.p3.2.m2.1.1.3" xref="S5.SS2.p3.2.m2.1.1.3.cmml">0.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.2.m2.1b"><apply id="S5.SS2.p3.2.m2.1.1.cmml" xref="S5.SS2.p3.2.m2.1.1"><lt id="S5.SS2.p3.2.m2.1.1.1.cmml" xref="S5.SS2.p3.2.m2.1.1.1"></lt><ci id="S5.SS2.p3.2.m2.1.1.2.cmml" xref="S5.SS2.p3.2.m2.1.1.2">𝑧</ci><cn id="S5.SS2.p3.2.m2.1.1.3.cmml" type="float" xref="S5.SS2.p3.2.m2.1.1.3">0.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.2.m2.1c">z<0.7</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.2.m2.1d">italic_z < 0.7</annotation></semantics></math>), MUST provides constraints comparable to existing surveys due to the limitation of cosmic variance. In the intermediate redshift range, the ELG samples of MUST are expected to yield significantly tighter constraints. 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However, by combining tracers among all redshift, MUST is anticipated to achieve a much tighter constraint on the model of dark energy. </div> <figure class="ltx_figure" id="S5.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="349" id="S5.F6.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 6: Top panels: errors on <math alttext="D_{A}(z)/r_{d}" class="ltx_Math" display="inline" id="S5.F6.6.m1.1"><semantics id="S5.F6.6.m1.1b"><mrow id="S5.F6.6.m1.1.2" xref="S5.F6.6.m1.1.2.cmml"><mrow id="S5.F6.6.m1.1.2.2" xref="S5.F6.6.m1.1.2.2.cmml"><msub id="S5.F6.6.m1.1.2.2.2" xref="S5.F6.6.m1.1.2.2.2.cmml"><mi id="S5.F6.6.m1.1.2.2.2.2" xref="S5.F6.6.m1.1.2.2.2.2.cmml">D</mi><mi id="S5.F6.6.m1.1.2.2.2.3" xref="S5.F6.6.m1.1.2.2.2.3.cmml">A</mi></msub><mo id="S5.F6.6.m1.1.2.2.1" xref="S5.F6.6.m1.1.2.2.1.cmml">⁢</mo><mrow id="S5.F6.6.m1.1.2.2.3.2" xref="S5.F6.6.m1.1.2.2.cmml"><mo id="S5.F6.6.m1.1.2.2.3.2.1" stretchy="false" xref="S5.F6.6.m1.1.2.2.cmml">(</mo><mi id="S5.F6.6.m1.1.1" xref="S5.F6.6.m1.1.1.cmml">z</mi><mo id="S5.F6.6.m1.1.2.2.3.2.2" stretchy="false" xref="S5.F6.6.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.F6.6.m1.1.2.1" xref="S5.F6.6.m1.1.2.1.cmml">/</mo><msub id="S5.F6.6.m1.1.2.3" xref="S5.F6.6.m1.1.2.3.cmml"><mi id="S5.F6.6.m1.1.2.3.2" xref="S5.F6.6.m1.1.2.3.2.cmml">r</mi><mi id="S5.F6.6.m1.1.2.3.3" xref="S5.F6.6.m1.1.2.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.F6.6.m1.1c"><apply id="S5.F6.6.m1.1.2.cmml" xref="S5.F6.6.m1.1.2"><divide id="S5.F6.6.m1.1.2.1.cmml" xref="S5.F6.6.m1.1.2.1"></divide><apply id="S5.F6.6.m1.1.2.2.cmml" xref="S5.F6.6.m1.1.2.2"><times id="S5.F6.6.m1.1.2.2.1.cmml" xref="S5.F6.6.m1.1.2.2.1"></times><apply id="S5.F6.6.m1.1.2.2.2.cmml" xref="S5.F6.6.m1.1.2.2.2"><csymbol cd="ambiguous" id="S5.F6.6.m1.1.2.2.2.1.cmml" xref="S5.F6.6.m1.1.2.2.2">subscript</csymbol><ci id="S5.F6.6.m1.1.2.2.2.2.cmml" xref="S5.F6.6.m1.1.2.2.2.2">𝐷</ci><ci id="S5.F6.6.m1.1.2.2.2.3.cmml" xref="S5.F6.6.m1.1.2.2.2.3">𝐴</ci></apply><ci id="S5.F6.6.m1.1.1.cmml" xref="S5.F6.6.m1.1.1">𝑧</ci></apply><apply id="S5.F6.6.m1.1.2.3.cmml" xref="S5.F6.6.m1.1.2.3"><csymbol cd="ambiguous" id="S5.F6.6.m1.1.2.3.1.cmml" xref="S5.F6.6.m1.1.2.3">subscript</csymbol><ci id="S5.F6.6.m1.1.2.3.2.cmml" xref="S5.F6.6.m1.1.2.3.2">𝑟</ci><ci id="S5.F6.6.m1.1.2.3.3.cmml" xref="S5.F6.6.m1.1.2.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F6.6.m1.1d">D_{A}(z)/r_{d}</annotation><annotation encoding="application/x-llamapun" id="S5.F6.6.m1.1e">italic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_z ) / italic_r start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> (left) and <math alttext="H(z)r_{d}" class="ltx_Math" display="inline" id="S5.F6.7.m2.1"><semantics id="S5.F6.7.m2.1b"><mrow id="S5.F6.7.m2.1.2" xref="S5.F6.7.m2.1.2.cmml"><mi id="S5.F6.7.m2.1.2.2" xref="S5.F6.7.m2.1.2.2.cmml">H</mi><mo id="S5.F6.7.m2.1.2.1" xref="S5.F6.7.m2.1.2.1.cmml">⁢</mo><mrow id="S5.F6.7.m2.1.2.3.2" xref="S5.F6.7.m2.1.2.cmml"><mo id="S5.F6.7.m2.1.2.3.2.1" stretchy="false" xref="S5.F6.7.m2.1.2.cmml">(</mo><mi id="S5.F6.7.m2.1.1" xref="S5.F6.7.m2.1.1.cmml">z</mi><mo id="S5.F6.7.m2.1.2.3.2.2" stretchy="false" xref="S5.F6.7.m2.1.2.cmml">)</mo></mrow><mo id="S5.F6.7.m2.1.2.1b" xref="S5.F6.7.m2.1.2.1.cmml">⁢</mo><msub id="S5.F6.7.m2.1.2.4" xref="S5.F6.7.m2.1.2.4.cmml"><mi id="S5.F6.7.m2.1.2.4.2" xref="S5.F6.7.m2.1.2.4.2.cmml">r</mi><mi id="S5.F6.7.m2.1.2.4.3" xref="S5.F6.7.m2.1.2.4.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.F6.7.m2.1c"><apply id="S5.F6.7.m2.1.2.cmml" xref="S5.F6.7.m2.1.2"><times id="S5.F6.7.m2.1.2.1.cmml" xref="S5.F6.7.m2.1.2.1"></times><ci id="S5.F6.7.m2.1.2.2.cmml" xref="S5.F6.7.m2.1.2.2">𝐻</ci><ci id="S5.F6.7.m2.1.1.cmml" xref="S5.F6.7.m2.1.1">𝑧</ci><apply id="S5.F6.7.m2.1.2.4.cmml" xref="S5.F6.7.m2.1.2.4"><csymbol cd="ambiguous" id="S5.F6.7.m2.1.2.4.1.cmml" xref="S5.F6.7.m2.1.2.4">subscript</csymbol><ci id="S5.F6.7.m2.1.2.4.2.cmml" xref="S5.F6.7.m2.1.2.4.2">𝑟</ci><ci id="S5.F6.7.m2.1.2.4.3.cmml" xref="S5.F6.7.m2.1.2.4.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F6.7.m2.1d">H(z)r_{d}</annotation><annotation encoding="application/x-llamapun" id="S5.F6.7.m2.1e">italic_H ( italic_z ) italic_r start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> (right) from SDSS measurements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib10" title="">10</a>]</cite> and our forecasts for DESI and MUST galaxies. The lower and upper limits of the shaded regions show the estimation with optimistic and conservative MUST observational conditions, respectively. Bottom panels: the same errors are centred on the flat-<math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.F6.8.m3.1"><semantics id="S5.F6.8.m3.1b"><mi id="S5.F6.8.m3.1.1" mathvariant="normal" xref="S5.F6.8.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.F6.8.m3.1c"><ci id="S5.F6.8.m3.1.1.cmml" xref="S5.F6.8.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F6.8.m3.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.F6.8.m3.1e">roman_Λ</annotation></semantics></math>CDM cosmology, in comparison with deviations from <math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.F6.9.m4.1"><semantics id="S5.F6.9.m4.1b"><mi id="S5.F6.9.m4.1.1" mathvariant="normal" xref="S5.F6.9.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.F6.9.m4.1c"><ci id="S5.F6.9.m4.1.1.cmml" xref="S5.F6.9.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F6.9.m4.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.F6.9.m4.1e">roman_Λ</annotation></semantics></math>CDM for the best-fit flat-<math alttext="w_{0}w_{a}" class="ltx_Math" display="inline" id="S5.F6.10.m5.1"><semantics id="S5.F6.10.m5.1b"><mrow id="S5.F6.10.m5.1.1" xref="S5.F6.10.m5.1.1.cmml"><msub id="S5.F6.10.m5.1.1.2" xref="S5.F6.10.m5.1.1.2.cmml"><mi id="S5.F6.10.m5.1.1.2.2" xref="S5.F6.10.m5.1.1.2.2.cmml">w</mi><mn id="S5.F6.10.m5.1.1.2.3" xref="S5.F6.10.m5.1.1.2.3.cmml">0</mn></msub><mo id="S5.F6.10.m5.1.1.1" xref="S5.F6.10.m5.1.1.1.cmml">⁢</mo><msub id="S5.F6.10.m5.1.1.3" xref="S5.F6.10.m5.1.1.3.cmml"><mi id="S5.F6.10.m5.1.1.3.2" xref="S5.F6.10.m5.1.1.3.2.cmml">w</mi><mi id="S5.F6.10.m5.1.1.3.3" xref="S5.F6.10.m5.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.F6.10.m5.1c"><apply id="S5.F6.10.m5.1.1.cmml" xref="S5.F6.10.m5.1.1"><times id="S5.F6.10.m5.1.1.1.cmml" xref="S5.F6.10.m5.1.1.1"></times><apply 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ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib73" title="">73</a>]</cite>.</figcaption> </figure> </section> <section class="ltx_subsection" id="S5.SS3"> <h3 class="ltx_title ltx_title_subsection"> 5.3 Structure Growth & Gravity</h3> <div class="ltx_para" id="S5.SS3.p1"> The distinction between GR and alternative gravity models can be inferred through measurements of structure growth derived from the anisotropic galaxy clustering, often parameterized by <math alttext="f\sigma_{8}" class="ltx_Math" display="inline" id="S5.SS3.p1.1.m1.1"><semantics id="S5.SS3.p1.1.m1.1a"><mrow id="S5.SS3.p1.1.m1.1.1" xref="S5.SS3.p1.1.m1.1.1.cmml"><mi id="S5.SS3.p1.1.m1.1.1.2" xref="S5.SS3.p1.1.m1.1.1.2.cmml">f</mi><mo id="S5.SS3.p1.1.m1.1.1.1" xref="S5.SS3.p1.1.m1.1.1.1.cmml">⁢</mo><msub id="S5.SS3.p1.1.m1.1.1.3" xref="S5.SS3.p1.1.m1.1.1.3.cmml"><mi id="S5.SS3.p1.1.m1.1.1.3.2" xref="S5.SS3.p1.1.m1.1.1.3.2.cmml">σ</mi><mn id="S5.SS3.p1.1.m1.1.1.3.3" 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href="https://arxiv.org/html/2411.07970v2#S3.SS2" title="3.2 Growth of Structure & Nature of Gravity ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.2</a>). Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.F7" title="Figure 7 ‣ 5.3 Structure Growth & Gravity ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">7</a> shows Fisher forecasts for the statistical errors of <math alttext="f\sigma_{8}" class="ltx_Math" display="inline" id="S5.SS3.p1.2.m2.1"><semantics id="S5.SS3.p1.2.m2.1a"><mrow id="S5.SS3.p1.2.m2.1.1" xref="S5.SS3.p1.2.m2.1.1.cmml"><mi id="S5.SS3.p1.2.m2.1.1.2" xref="S5.SS3.p1.2.m2.1.1.2.cmml">f</mi><mo id="S5.SS3.p1.2.m2.1.1.1" xref="S5.SS3.p1.2.m2.1.1.1.cmml">⁢</mo><msub id="S5.SS3.p1.2.m2.1.1.3" xref="S5.SS3.p1.2.m2.1.1.3.cmml"><mi id="S5.SS3.p1.2.m2.1.1.3.2" xref="S5.SS3.p1.2.m2.1.1.3.2.cmml">σ</mi><mn id="S5.SS3.p1.2.m2.1.1.3.3" xref="S5.SS3.p1.2.m2.1.1.3.3.cmml">8</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.2.m2.1b"><apply id="S5.SS3.p1.2.m2.1.1.cmml" xref="S5.SS3.p1.2.m2.1.1"><times id="S5.SS3.p1.2.m2.1.1.1.cmml" xref="S5.SS3.p1.2.m2.1.1.1"></times><ci id="S5.SS3.p1.2.m2.1.1.2.cmml" xref="S5.SS3.p1.2.m2.1.1.2">𝑓</ci><apply id="S5.SS3.p1.2.m2.1.1.3.cmml" xref="S5.SS3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.2.m2.1.1.3.1.cmml" xref="S5.SS3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.SS3.p1.2.m2.1.1.3.2.cmml" xref="S5.SS3.p1.2.m2.1.1.3.2">𝜎</ci><cn id="S5.SS3.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S5.SS3.p1.2.m2.1.1.3.3">8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.2.m2.1c">f\sigma_{8}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.2.m2.1d">italic_f italic_σ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT</annotation></semantics></math> with MUST tracers, based on RSD measurements with template power spectra, compared with constraints from some existing surveys. Overall, MUST is expected to achieve more than a 4-fold improvement in precision at <math alttext="z<1.6" class="ltx_Math" display="inline" id="S5.SS3.p1.3.m3.1"><semantics id="S5.SS3.p1.3.m3.1a"><mrow id="S5.SS3.p1.3.m3.1.1" xref="S5.SS3.p1.3.m3.1.1.cmml"><mi id="S5.SS3.p1.3.m3.1.1.2" xref="S5.SS3.p1.3.m3.1.1.2.cmml">z</mi><mo id="S5.SS3.p1.3.m3.1.1.1" xref="S5.SS3.p1.3.m3.1.1.1.cmml"><</mo><mn id="S5.SS3.p1.3.m3.1.1.3" xref="S5.SS3.p1.3.m3.1.1.3.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.3.m3.1b"><apply id="S5.SS3.p1.3.m3.1.1.cmml" xref="S5.SS3.p1.3.m3.1.1"><lt id="S5.SS3.p1.3.m3.1.1.1.cmml" xref="S5.SS3.p1.3.m3.1.1.1"></lt><ci id="S5.SS3.p1.3.m3.1.1.2.cmml" xref="S5.SS3.p1.3.m3.1.1.2">𝑧</ci><cn id="S5.SS3.p1.3.m3.1.1.3.cmml" type="float" xref="S5.SS3.p1.3.m3.1.1.3">1.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.3.m3.1c">z<1.6</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.3.m3.1d">italic_z < 1.6</annotation></semantics></math>, thanks to its high galaxy number density. Furthermore, MUST LBG will enable measurements of <math alttext="f\sigma_{8}" class="ltx_Math" display="inline" id="S5.SS3.p1.4.m4.1"><semantics id="S5.SS3.p1.4.m4.1a"><mrow id="S5.SS3.p1.4.m4.1.1" xref="S5.SS3.p1.4.m4.1.1.cmml"><mi id="S5.SS3.p1.4.m4.1.1.2" xref="S5.SS3.p1.4.m4.1.1.2.cmml">f</mi><mo id="S5.SS3.p1.4.m4.1.1.1" xref="S5.SS3.p1.4.m4.1.1.1.cmml">⁢</mo><msub id="S5.SS3.p1.4.m4.1.1.3" xref="S5.SS3.p1.4.m4.1.1.3.cmml"><mi id="S5.SS3.p1.4.m4.1.1.3.2" xref="S5.SS3.p1.4.m4.1.1.3.2.cmml">σ</mi><mn id="S5.SS3.p1.4.m4.1.1.3.3" xref="S5.SS3.p1.4.m4.1.1.3.3.cmml">8</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.4.m4.1b"><apply id="S5.SS3.p1.4.m4.1.1.cmml" xref="S5.SS3.p1.4.m4.1.1"><times id="S5.SS3.p1.4.m4.1.1.1.cmml" xref="S5.SS3.p1.4.m4.1.1.1"></times><ci id="S5.SS3.p1.4.m4.1.1.2.cmml" xref="S5.SS3.p1.4.m4.1.1.2">𝑓</ci><apply id="S5.SS3.p1.4.m4.1.1.3.cmml" xref="S5.SS3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p1.4.m4.1.1.3.1.cmml" 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id="S5.SS3.p2.1.m1.1.1.1" xref="S5.SS3.p2.1.m1.1.1.1.cmml">∼</mo><mn id="S5.SS3.p2.1.m1.1.1.3" xref="S5.SS3.p2.1.m1.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.1.m1.1b"><apply id="S5.SS3.p2.1.m1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1"><csymbol cd="latexml" id="S5.SS3.p2.1.m1.1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1.1">similar-to</csymbol><ci id="S5.SS3.p2.1.m1.1.1.2.cmml" xref="S5.SS3.p2.1.m1.1.1.2">𝑧</ci><cn id="S5.SS3.p2.1.m1.1.1.3.cmml" type="integer" xref="S5.SS3.p2.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.1.m1.1c">z\sim 5</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.1.m1.1d">italic_z ∼ 5</annotation></semantics></math>, with the only exception of <math alttext="\sim" class="ltx_Math" display="inline" id="S5.SS3.p2.2.m2.1"><semantics id="S5.SS3.p2.2.m2.1a"><mo id="S5.SS3.p2.2.m2.1.1" xref="S5.SS3.p2.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" 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The conservative estimation mainly affects the constraints of high-<math alttext="z" class="ltx_Math" display="inline" id="S5.SS3.p2.4.m4.1"><semantics id="S5.SS3.p2.4.m4.1a"><mi id="S5.SS3.p2.4.m4.1.1" xref="S5.SS3.p2.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.4.m4.1b"><ci id="S5.SS3.p2.4.m4.1.1.cmml" xref="S5.SS3.p2.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.4.m4.1d">italic_z</annotation></semantics></math> LBGs, resulting in a <math alttext="\sim" class="ltx_Math" display="inline" id="S5.SS3.p2.5.m5.1"><semantics id="S5.SS3.p2.5.m5.1a"><mo id="S5.SS3.p2.5.m5.1.1" xref="S5.SS3.p2.5.m5.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.5.m5.1b"><csymbol cd="latexml" id="S5.SS3.p2.5.m5.1.1.cmml" xref="S5.SS3.p2.5.m5.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.5.m5.1c">\sim</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.5.m5.1d">∼</annotation></semantics></math>3–4% precision. In any case, the unprecedented precision of structure growth measurements enables a significant potential for testing gravity theories with MUST data. Assuming a fiducial GR-<math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.SS3.p2.6.m6.1"><semantics id="S5.SS3.p2.6.m6.1a"><mi id="S5.SS3.p2.6.m6.1.1" mathvariant="normal" xref="S5.SS3.p2.6.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.6.m6.1b"><ci id="S5.SS3.p2.6.m6.1.1.cmml" xref="S5.SS3.p2.6.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.6.m6.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.6.m6.1d">roman_Λ</annotation></semantics></math>CDM framework, several gravity theories currently consistent with the state-of-the-art spectroscopic data – with their best-fitting parameters – could potentially be ruled out by MUST. For instance, with notable contributions of constraints at <math alttext="z>2" class="ltx_Math" display="inline" id="S5.SS3.p2.7.m7.1"><semantics id="S5.SS3.p2.7.m7.1a"><mrow id="S5.SS3.p2.7.m7.1.1" xref="S5.SS3.p2.7.m7.1.1.cmml"><mi id="S5.SS3.p2.7.m7.1.1.2" xref="S5.SS3.p2.7.m7.1.1.2.cmml">z</mi><mo id="S5.SS3.p2.7.m7.1.1.1" xref="S5.SS3.p2.7.m7.1.1.1.cmml">></mo><mn id="S5.SS3.p2.7.m7.1.1.3" xref="S5.SS3.p2.7.m7.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.7.m7.1b"><apply id="S5.SS3.p2.7.m7.1.1.cmml" xref="S5.SS3.p2.7.m7.1.1"><gt id="S5.SS3.p2.7.m7.1.1.1.cmml" xref="S5.SS3.p2.7.m7.1.1.1"></gt><ci id="S5.SS3.p2.7.m7.1.1.2.cmml" xref="S5.SS3.p2.7.m7.1.1.2">𝑧</ci><cn id="S5.SS3.p2.7.m7.1.1.3.cmml" type="integer" xref="S5.SS3.p2.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.7.m7.1c">z>2</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.7.m7.1d">italic_z > 2</annotation></semantics></math>, MUST is expected to probe <math alttext="|f_{R0}|" class="ltx_Math" display="inline" id="S5.SS3.p2.8.m8.1"><semantics id="S5.SS3.p2.8.m8.1a"><mrow id="S5.SS3.p2.8.m8.1.1.1" xref="S5.SS3.p2.8.m8.1.1.2.cmml"><mo id="S5.SS3.p2.8.m8.1.1.1.2" stretchy="false" xref="S5.SS3.p2.8.m8.1.1.2.1.cmml">|</mo><msub id="S5.SS3.p2.8.m8.1.1.1.1" xref="S5.SS3.p2.8.m8.1.1.1.1.cmml"><mi id="S5.SS3.p2.8.m8.1.1.1.1.2" xref="S5.SS3.p2.8.m8.1.1.1.1.2.cmml">f</mi><mrow id="S5.SS3.p2.8.m8.1.1.1.1.3" xref="S5.SS3.p2.8.m8.1.1.1.1.3.cmml"><mi id="S5.SS3.p2.8.m8.1.1.1.1.3.2" xref="S5.SS3.p2.8.m8.1.1.1.1.3.2.cmml">R</mi><mo id="S5.SS3.p2.8.m8.1.1.1.1.3.1" xref="S5.SS3.p2.8.m8.1.1.1.1.3.1.cmml">⁢</mo><mn id="S5.SS3.p2.8.m8.1.1.1.1.3.3" xref="S5.SS3.p2.8.m8.1.1.1.1.3.3.cmml">0</mn></mrow></msub><mo id="S5.SS3.p2.8.m8.1.1.1.3" stretchy="false" xref="S5.SS3.p2.8.m8.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.8.m8.1b"><apply 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Bottom panel: Relative error of MUST measurements and the deviation of different gravity models to the fiducial GR-<math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.F7.10.m2.1"><semantics id="S5.F7.10.m2.1b"><mi id="S5.F7.10.m2.1.1" mathvariant="normal" xref="S5.F7.10.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.F7.10.m2.1c"><ci id="S5.F7.10.m2.1.1.cmml" xref="S5.F7.10.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F7.10.m2.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.F7.10.m2.1e">roman_Λ</annotation></semantics></math>CDM model. The shaded regions represent the optimistic scenario, while the lines indicate the more conservative forecast. Results from existing surveys are shown with open symbols on the top panel: SDSS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib274" title="">274</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib275" title="">275</a>]</cite>, 2dFGRS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib276" title="">276</a>]</cite>, VVDS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib277" title="">277</a>]</cite>, VIPERS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib278" title="">278</a>]</cite>, WiggleZ <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib279" title="">279</a>]</cite>, 6dFGRS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib280" title="">280</a>]</cite>, FastSound <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib281" title="">281</a>]</cite>. In both panels, MUST targets are indicated by filled stars color-coded based on the galaxy sample. The black solid line denotes the GR-<math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.F7.11.m3.1"><semantics id="S5.F7.11.m3.1b"><mi id="S5.F7.11.m3.1.1" mathvariant="normal" xref="S5.F7.11.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.F7.11.m3.1c"><ci id="S5.F7.11.m3.1.1.cmml" xref="S5.F7.11.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F7.11.m3.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.F7.11.m3.1e">roman_Λ</annotation></semantics></math>CDM model. In contrast, other lines represent several alternative gravity theories: the covariant Galileon gravity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib282" title="">282</a>]</cite>, the extended Galileon gravity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib283" title="">283</a>]</cite>, the varying gravitational constant model with <math alttext="|\dot{G}/G|=3.5\times 10^{-11}\rm{yr}^{-1}" class="ltx_Math" display="inline" id="S5.F7.12.m4.1"><semantics id="S5.F7.12.m4.1b"><mrow id="S5.F7.12.m4.1.1" xref="S5.F7.12.m4.1.1.cmml"><mrow id="S5.F7.12.m4.1.1.1.1" xref="S5.F7.12.m4.1.1.1.2.cmml"><mo id="S5.F7.12.m4.1.1.1.1.2" stretchy="false" xref="S5.F7.12.m4.1.1.1.2.1.cmml">|</mo><mrow id="S5.F7.12.m4.1.1.1.1.1" xref="S5.F7.12.m4.1.1.1.1.1.cmml"><mover accent="true" id="S5.F7.12.m4.1.1.1.1.1.2" xref="S5.F7.12.m4.1.1.1.1.1.2.cmml"><mi id="S5.F7.12.m4.1.1.1.1.1.2.2" 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xref="S5.F7.14.m6.1.1.3">superscript</csymbol><cn id="S5.F7.14.m6.1.1.3.2.cmml" type="integer" xref="S5.F7.14.m6.1.1.3.2">10</cn><apply id="S5.F7.14.m6.1.1.3.3.cmml" xref="S5.F7.14.m6.1.1.3.3"><minus id="S5.F7.14.m6.1.1.3.3.1.cmml" xref="S5.F7.14.m6.1.1.3.3"></minus><cn id="S5.F7.14.m6.1.1.3.3.2.cmml" type="integer" xref="S5.F7.14.m6.1.1.3.3.2">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F7.14.m6.1d">|f_{R0}|=10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S5.F7.14.m6.1e">| italic_f start_POSTSUBSCRIPT italic_R 0 end_POSTSUBSCRIPT | = 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="10^{-5}" class="ltx_Math" display="inline" id="S5.F7.15.m7.1"><semantics id="S5.F7.15.m7.1b"><msup id="S5.F7.15.m7.1.1" xref="S5.F7.15.m7.1.1.cmml"><mn id="S5.F7.15.m7.1.1.2" xref="S5.F7.15.m7.1.1.2.cmml">10</mn><mrow id="S5.F7.15.m7.1.1.3" xref="S5.F7.15.m7.1.1.3.cmml"><mo id="S5.F7.15.m7.1.1.3b" xref="S5.F7.15.m7.1.1.3.cmml">−</mo><mn id="S5.F7.15.m7.1.1.3.2" xref="S5.F7.15.m7.1.1.3.2.cmml">5</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.F7.15.m7.1c"><apply id="S5.F7.15.m7.1.1.cmml" xref="S5.F7.15.m7.1.1"><csymbol cd="ambiguous" id="S5.F7.15.m7.1.1.1.cmml" xref="S5.F7.15.m7.1.1">superscript</csymbol><cn id="S5.F7.15.m7.1.1.2.cmml" type="integer" xref="S5.F7.15.m7.1.1.2">10</cn><apply id="S5.F7.15.m7.1.1.3.cmml" xref="S5.F7.15.m7.1.1.3"><minus id="S5.F7.15.m7.1.1.3.1.cmml" xref="S5.F7.15.m7.1.1.3"></minus><cn id="S5.F7.15.m7.1.1.3.2.cmml" type="integer" xref="S5.F7.15.m7.1.1.3.2">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F7.15.m7.1d">10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S5.F7.15.m7.1e">10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="10^{-6}" class="ltx_Math" display="inline" id="S5.F7.16.m8.1"><semantics id="S5.F7.16.m8.1b"><msup id="S5.F7.16.m8.1.1" xref="S5.F7.16.m8.1.1.cmml"><mn id="S5.F7.16.m8.1.1.2" xref="S5.F7.16.m8.1.1.2.cmml">10</mn><mrow id="S5.F7.16.m8.1.1.3" xref="S5.F7.16.m8.1.1.3.cmml"><mo id="S5.F7.16.m8.1.1.3b" xref="S5.F7.16.m8.1.1.3.cmml">−</mo><mn id="S5.F7.16.m8.1.1.3.2" xref="S5.F7.16.m8.1.1.3.2.cmml">6</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.F7.16.m8.1c"><apply id="S5.F7.16.m8.1.1.cmml" xref="S5.F7.16.m8.1.1"><csymbol cd="ambiguous" id="S5.F7.16.m8.1.1.1.cmml" xref="S5.F7.16.m8.1.1">superscript</csymbol><cn id="S5.F7.16.m8.1.1.2.cmml" type="integer" xref="S5.F7.16.m8.1.1.2">10</cn><apply id="S5.F7.16.m8.1.1.3.cmml" xref="S5.F7.16.m8.1.1.3"><minus id="S5.F7.16.m8.1.1.3.1.cmml" xref="S5.F7.16.m8.1.1.3"></minus><cn id="S5.F7.16.m8.1.1.3.2.cmml" type="integer" xref="S5.F7.16.m8.1.1.3.2">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F7.16.m8.1d">10^{-6}</annotation><annotation encoding="application/x-llamapun" id="S5.F7.16.m8.1e">10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT</annotation></semantics></math>.</figcaption> </figure> </section> <section class="ltx_subsection" id="S5.SS4"> <h3 class="ltx_title ltx_title_subsection"> 5.4 Primordial Non-Gaussianity</h3> <div class="ltx_para" id="S5.SS4.p1"> The presence of PNG incorporates a form of dependence on the initial potential field, <math alttext="\phi" class="ltx_Math" display="inline" id="S5.SS4.p1.1.m1.1"><semantics id="S5.SS4.p1.1.m1.1a"><mi id="S5.SS4.p1.1.m1.1.1" xref="S5.SS4.p1.1.m1.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.1.m1.1b"><ci id="S5.SS4.p1.1.m1.1.1.cmml" xref="S5.SS4.p1.1.m1.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.1.m1.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.1.m1.1d">italic_ϕ</annotation></semantics></math>, into the galaxy bias model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib286" title="">286</a>]</cite>. At first order, this dependence adds a so-called ”scale-dependent bias” term to the galaxy power spectrum <table class="ltx_equation ltx_eqn_table" id="S5.E19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P_{\rm gg}(k,z)={\left(b_{1}+f_{\text{NL}}^{\text{local}}b_{\phi}\frac{3H_{0}^% {2}\Omega_{\rm m}}{2k^{2}T(k)D_{\rm md}(z)}\right)}^{2}P_{\rm mm}(k,z)+P_{\rm shot% }(z)," class="ltx_Math" display="block" id="S5.E19.m1.8"><semantics id="S5.E19.m1.8a"><mrow id="S5.E19.m1.8.8.1" xref="S5.E19.m1.8.8.1.1.cmml"><mrow id="S5.E19.m1.8.8.1.1" xref="S5.E19.m1.8.8.1.1.cmml"><mrow id="S5.E19.m1.8.8.1.1.3" xref="S5.E19.m1.8.8.1.1.3.cmml"><msub id="S5.E19.m1.8.8.1.1.3.2" xref="S5.E19.m1.8.8.1.1.3.2.cmml"><mi id="S5.E19.m1.8.8.1.1.3.2.2" xref="S5.E19.m1.8.8.1.1.3.2.2.cmml">P</mi><mi id="S5.E19.m1.8.8.1.1.3.2.3" xref="S5.E19.m1.8.8.1.1.3.2.3.cmml">gg</mi></msub><mo 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id="S5.E19.m1.2.2.4.3.2.cmml" xref="S5.E19.m1.2.2.4.3"><csymbol cd="ambiguous" id="S5.E19.m1.2.2.4.3.2.1.cmml" xref="S5.E19.m1.2.2.4.3">subscript</csymbol><ci id="S5.E19.m1.2.2.4.3.2.2.cmml" xref="S5.E19.m1.2.2.4.3.2.2">𝐻</ci><cn id="S5.E19.m1.2.2.4.3.2.3.cmml" type="integer" xref="S5.E19.m1.2.2.4.3.2.3">0</cn></apply><cn id="S5.E19.m1.2.2.4.3.3.cmml" type="integer" xref="S5.E19.m1.2.2.4.3.3">2</cn></apply><apply id="S5.E19.m1.2.2.4.4.cmml" xref="S5.E19.m1.2.2.4.4"><csymbol cd="ambiguous" id="S5.E19.m1.2.2.4.4.1.cmml" xref="S5.E19.m1.2.2.4.4">subscript</csymbol><ci id="S5.E19.m1.2.2.4.4.2.cmml" xref="S5.E19.m1.2.2.4.4.2">Ω</ci><ci id="S5.E19.m1.2.2.4.4.3.cmml" xref="S5.E19.m1.2.2.4.4.3">m</ci></apply></apply><apply id="S5.E19.m1.2.2.2.cmml" xref="S5.E19.m1.2.2.2"><times id="S5.E19.m1.2.2.2.3.cmml" xref="S5.E19.m1.2.2.2.3"></times><cn id="S5.E19.m1.2.2.2.4.cmml" type="integer" xref="S5.E19.m1.2.2.2.4">2</cn><apply id="S5.E19.m1.2.2.2.5.cmml" xref="S5.E19.m1.2.2.2.5"><csymbol cd="ambiguous" id="S5.E19.m1.2.2.2.5.1.cmml" xref="S5.E19.m1.2.2.2.5">superscript</csymbol><ci id="S5.E19.m1.2.2.2.5.2.cmml" xref="S5.E19.m1.2.2.2.5.2">𝑘</ci><cn id="S5.E19.m1.2.2.2.5.3.cmml" type="integer" xref="S5.E19.m1.2.2.2.5.3">2</cn></apply><ci id="S5.E19.m1.2.2.2.6.cmml" xref="S5.E19.m1.2.2.2.6">𝑇</ci><ci id="S5.E19.m1.1.1.1.1.cmml" xref="S5.E19.m1.1.1.1.1">𝑘</ci><apply id="S5.E19.m1.2.2.2.8.cmml" xref="S5.E19.m1.2.2.2.8"><csymbol cd="ambiguous" id="S5.E19.m1.2.2.2.8.1.cmml" xref="S5.E19.m1.2.2.2.8">subscript</csymbol><ci id="S5.E19.m1.2.2.2.8.2.cmml" xref="S5.E19.m1.2.2.2.8.2">𝐷</ci><ci id="S5.E19.m1.2.2.2.8.3.cmml" xref="S5.E19.m1.2.2.2.8.3">md</ci></apply><ci id="S5.E19.m1.2.2.2.2.cmml" xref="S5.E19.m1.2.2.2.2">𝑧</ci></apply></apply></apply></apply><cn id="S5.E19.m1.8.8.1.1.1.1.1.3.cmml" type="integer" xref="S5.E19.m1.8.8.1.1.1.1.1.3">2</cn></apply><apply id="S5.E19.m1.8.8.1.1.1.1.3.cmml" xref="S5.E19.m1.8.8.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E19.m1.8.8.1.1.1.1.3.1.cmml" xref="S5.E19.m1.8.8.1.1.1.1.3">subscript</csymbol><ci id="S5.E19.m1.8.8.1.1.1.1.3.2.cmml" xref="S5.E19.m1.8.8.1.1.1.1.3.2">𝑃</ci><ci id="S5.E19.m1.8.8.1.1.1.1.3.3.cmml" xref="S5.E19.m1.8.8.1.1.1.1.3.3">mm</ci></apply><interval closure="open" id="S5.E19.m1.8.8.1.1.1.1.4.1.cmml" xref="S5.E19.m1.8.8.1.1.1.1.4.2"><ci id="S5.E19.m1.5.5.cmml" xref="S5.E19.m1.5.5">𝑘</ci><ci id="S5.E19.m1.6.6.cmml" xref="S5.E19.m1.6.6">𝑧</ci></interval></apply><apply id="S5.E19.m1.8.8.1.1.1.3.cmml" xref="S5.E19.m1.8.8.1.1.1.3"><times id="S5.E19.m1.8.8.1.1.1.3.1.cmml" xref="S5.E19.m1.8.8.1.1.1.3.1"></times><apply id="S5.E19.m1.8.8.1.1.1.3.2.cmml" xref="S5.E19.m1.8.8.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.E19.m1.8.8.1.1.1.3.2.1.cmml" xref="S5.E19.m1.8.8.1.1.1.3.2">subscript</csymbol><ci id="S5.E19.m1.8.8.1.1.1.3.2.2.cmml" xref="S5.E19.m1.8.8.1.1.1.3.2.2">𝑃</ci><ci id="S5.E19.m1.8.8.1.1.1.3.2.3.cmml" xref="S5.E19.m1.8.8.1.1.1.3.2.3">shot</ci></apply><ci id="S5.E19.m1.7.7.cmml" xref="S5.E19.m1.7.7">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E19.m1.8c">P_{\rm gg}(k,z)={\left(b_{1}+f_{\text{NL}}^{\text{local}}b_{\phi}\frac{3H_{0}^% {2}\Omega_{\rm m}}{2k^{2}T(k)D_{\rm md}(z)}\right)}^{2}P_{\rm mm}(k,z)+P_{\rm shot% }(z),</annotation><annotation encoding="application/x-llamapun" id="S5.E19.m1.8d">italic_P start_POSTSUBSCRIPT roman_gg end_POSTSUBSCRIPT ( italic_k , italic_z ) = ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT divide start_ARG 3 italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ω start_POSTSUBSCRIPT roman_m end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_T ( italic_k ) italic_D start_POSTSUBSCRIPT roman_md end_POSTSUBSCRIPT ( italic_z ) end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT roman_mm end_POSTSUBSCRIPT ( italic_k , italic_z ) + italic_P start_POSTSUBSCRIPT roman_shot end_POSTSUBSCRIPT ( italic_z ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(19)</td> </tr></tbody> </table> where <math alttext="b_{1}(z)" class="ltx_Math" display="inline" id="S5.SS4.p1.2.m1.1"><semantics id="S5.SS4.p1.2.m1.1a"><mrow id="S5.SS4.p1.2.m1.1.2" xref="S5.SS4.p1.2.m1.1.2.cmml"><msub id="S5.SS4.p1.2.m1.1.2.2" xref="S5.SS4.p1.2.m1.1.2.2.cmml"><mi id="S5.SS4.p1.2.m1.1.2.2.2" xref="S5.SS4.p1.2.m1.1.2.2.2.cmml">b</mi><mn id="S5.SS4.p1.2.m1.1.2.2.3" xref="S5.SS4.p1.2.m1.1.2.2.3.cmml">1</mn></msub><mo id="S5.SS4.p1.2.m1.1.2.1" xref="S5.SS4.p1.2.m1.1.2.1.cmml">⁢</mo><mrow id="S5.SS4.p1.2.m1.1.2.3.2" xref="S5.SS4.p1.2.m1.1.2.cmml"><mo id="S5.SS4.p1.2.m1.1.2.3.2.1" stretchy="false" xref="S5.SS4.p1.2.m1.1.2.cmml">(</mo><mi id="S5.SS4.p1.2.m1.1.1" xref="S5.SS4.p1.2.m1.1.1.cmml">z</mi><mo id="S5.SS4.p1.2.m1.1.2.3.2.2" stretchy="false" xref="S5.SS4.p1.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.2.m1.1b"><apply id="S5.SS4.p1.2.m1.1.2.cmml" xref="S5.SS4.p1.2.m1.1.2"><times id="S5.SS4.p1.2.m1.1.2.1.cmml" xref="S5.SS4.p1.2.m1.1.2.1"></times><apply id="S5.SS4.p1.2.m1.1.2.2.cmml" xref="S5.SS4.p1.2.m1.1.2.2"><csymbol cd="ambiguous" id="S5.SS4.p1.2.m1.1.2.2.1.cmml" xref="S5.SS4.p1.2.m1.1.2.2">subscript</csymbol><ci id="S5.SS4.p1.2.m1.1.2.2.2.cmml" xref="S5.SS4.p1.2.m1.1.2.2.2">𝑏</ci><cn id="S5.SS4.p1.2.m1.1.2.2.3.cmml" type="integer" xref="S5.SS4.p1.2.m1.1.2.2.3">1</cn></apply><ci id="S5.SS4.p1.2.m1.1.1.cmml" xref="S5.SS4.p1.2.m1.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.2.m1.1c">b_{1}(z)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.2.m1.1d">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> denotes the linear galaxy bias, <math alttext="b_{\phi}(z)" class="ltx_Math" display="inline" id="S5.SS4.p1.3.m2.1"><semantics id="S5.SS4.p1.3.m2.1a"><mrow id="S5.SS4.p1.3.m2.1.2" xref="S5.SS4.p1.3.m2.1.2.cmml"><msub id="S5.SS4.p1.3.m2.1.2.2" xref="S5.SS4.p1.3.m2.1.2.2.cmml"><mi id="S5.SS4.p1.3.m2.1.2.2.2" xref="S5.SS4.p1.3.m2.1.2.2.2.cmml">b</mi><mi id="S5.SS4.p1.3.m2.1.2.2.3" xref="S5.SS4.p1.3.m2.1.2.2.3.cmml">ϕ</mi></msub><mo id="S5.SS4.p1.3.m2.1.2.1" xref="S5.SS4.p1.3.m2.1.2.1.cmml">⁢</mo><mrow id="S5.SS4.p1.3.m2.1.2.3.2" xref="S5.SS4.p1.3.m2.1.2.cmml"><mo id="S5.SS4.p1.3.m2.1.2.3.2.1" stretchy="false" xref="S5.SS4.p1.3.m2.1.2.cmml">(</mo><mi id="S5.SS4.p1.3.m2.1.1" xref="S5.SS4.p1.3.m2.1.1.cmml">z</mi><mo id="S5.SS4.p1.3.m2.1.2.3.2.2" stretchy="false" xref="S5.SS4.p1.3.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.3.m2.1b"><apply id="S5.SS4.p1.3.m2.1.2.cmml" xref="S5.SS4.p1.3.m2.1.2"><times id="S5.SS4.p1.3.m2.1.2.1.cmml" xref="S5.SS4.p1.3.m2.1.2.1"></times><apply id="S5.SS4.p1.3.m2.1.2.2.cmml" xref="S5.SS4.p1.3.m2.1.2.2"><csymbol cd="ambiguous" id="S5.SS4.p1.3.m2.1.2.2.1.cmml" xref="S5.SS4.p1.3.m2.1.2.2">subscript</csymbol><ci id="S5.SS4.p1.3.m2.1.2.2.2.cmml" xref="S5.SS4.p1.3.m2.1.2.2.2">𝑏</ci><ci id="S5.SS4.p1.3.m2.1.2.2.3.cmml" xref="S5.SS4.p1.3.m2.1.2.2.3">italic-ϕ</ci></apply><ci id="S5.SS4.p1.3.m2.1.1.cmml" xref="S5.SS4.p1.3.m2.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.3.m2.1c">b_{\phi}(z)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.3.m2.1d">italic_b start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> represents the galaxy bias on <math alttext="\phi" class="ltx_Math" display="inline" id="S5.SS4.p1.4.m3.1"><semantics id="S5.SS4.p1.4.m3.1a"><mi id="S5.SS4.p1.4.m3.1.1" xref="S5.SS4.p1.4.m3.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.4.m3.1b"><ci id="S5.SS4.p1.4.m3.1.1.cmml" xref="S5.SS4.p1.4.m3.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.4.m3.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.4.m3.1d">italic_ϕ</annotation></semantics></math>, <math alttext="T(k)" class="ltx_Math" display="inline" id="S5.SS4.p1.5.m4.1"><semantics id="S5.SS4.p1.5.m4.1a"><mrow id="S5.SS4.p1.5.m4.1.2" xref="S5.SS4.p1.5.m4.1.2.cmml"><mi id="S5.SS4.p1.5.m4.1.2.2" xref="S5.SS4.p1.5.m4.1.2.2.cmml">T</mi><mo id="S5.SS4.p1.5.m4.1.2.1" xref="S5.SS4.p1.5.m4.1.2.1.cmml">⁢</mo><mrow id="S5.SS4.p1.5.m4.1.2.3.2" xref="S5.SS4.p1.5.m4.1.2.cmml"><mo id="S5.SS4.p1.5.m4.1.2.3.2.1" stretchy="false" xref="S5.SS4.p1.5.m4.1.2.cmml">(</mo><mi id="S5.SS4.p1.5.m4.1.1" xref="S5.SS4.p1.5.m4.1.1.cmml">k</mi><mo id="S5.SS4.p1.5.m4.1.2.3.2.2" stretchy="false" xref="S5.SS4.p1.5.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.5.m4.1b"><apply id="S5.SS4.p1.5.m4.1.2.cmml" xref="S5.SS4.p1.5.m4.1.2"><times id="S5.SS4.p1.5.m4.1.2.1.cmml" xref="S5.SS4.p1.5.m4.1.2.1"></times><ci id="S5.SS4.p1.5.m4.1.2.2.cmml" xref="S5.SS4.p1.5.m4.1.2.2">𝑇</ci><ci id="S5.SS4.p1.5.m4.1.1.cmml" xref="S5.SS4.p1.5.m4.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.5.m4.1c">T(k)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.5.m4.1d">italic_T ( italic_k )</annotation></semantics></math> is the transfer function and <math alttext="D_{\rm md}(z)" class="ltx_Math" display="inline" id="S5.SS4.p1.6.m5.1"><semantics id="S5.SS4.p1.6.m5.1a"><mrow id="S5.SS4.p1.6.m5.1.2" xref="S5.SS4.p1.6.m5.1.2.cmml"><msub id="S5.SS4.p1.6.m5.1.2.2" xref="S5.SS4.p1.6.m5.1.2.2.cmml"><mi id="S5.SS4.p1.6.m5.1.2.2.2" xref="S5.SS4.p1.6.m5.1.2.2.2.cmml">D</mi><mi id="S5.SS4.p1.6.m5.1.2.2.3" xref="S5.SS4.p1.6.m5.1.2.2.3.cmml">md</mi></msub><mo id="S5.SS4.p1.6.m5.1.2.1" xref="S5.SS4.p1.6.m5.1.2.1.cmml">⁢</mo><mrow id="S5.SS4.p1.6.m5.1.2.3.2" xref="S5.SS4.p1.6.m5.1.2.cmml"><mo id="S5.SS4.p1.6.m5.1.2.3.2.1" stretchy="false" xref="S5.SS4.p1.6.m5.1.2.cmml">(</mo><mi id="S5.SS4.p1.6.m5.1.1" xref="S5.SS4.p1.6.m5.1.1.cmml">z</mi><mo id="S5.SS4.p1.6.m5.1.2.3.2.2" stretchy="false" xref="S5.SS4.p1.6.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.6.m5.1b"><apply id="S5.SS4.p1.6.m5.1.2.cmml" xref="S5.SS4.p1.6.m5.1.2"><times id="S5.SS4.p1.6.m5.1.2.1.cmml" xref="S5.SS4.p1.6.m5.1.2.1"></times><apply id="S5.SS4.p1.6.m5.1.2.2.cmml" xref="S5.SS4.p1.6.m5.1.2.2"><csymbol cd="ambiguous" id="S5.SS4.p1.6.m5.1.2.2.1.cmml" xref="S5.SS4.p1.6.m5.1.2.2">subscript</csymbol><ci id="S5.SS4.p1.6.m5.1.2.2.2.cmml" xref="S5.SS4.p1.6.m5.1.2.2.2">𝐷</ci><ci id="S5.SS4.p1.6.m5.1.2.2.3.cmml" xref="S5.SS4.p1.6.m5.1.2.2.3">md</ci></apply><ci id="S5.SS4.p1.6.m5.1.1.cmml" xref="S5.SS4.p1.6.m5.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.6.m5.1c">D_{\rm md}(z)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.6.m5.1d">italic_D start_POSTSUBSCRIPT roman_md end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> is the linear growth factor normalized to <math alttext="1/(1+z)" class="ltx_Math" display="inline" id="S5.SS4.p1.7.m6.1"><semantics id="S5.SS4.p1.7.m6.1a"><mrow id="S5.SS4.p1.7.m6.1.1" xref="S5.SS4.p1.7.m6.1.1.cmml"><mn id="S5.SS4.p1.7.m6.1.1.3" xref="S5.SS4.p1.7.m6.1.1.3.cmml">1</mn><mo id="S5.SS4.p1.7.m6.1.1.2" xref="S5.SS4.p1.7.m6.1.1.2.cmml">/</mo><mrow id="S5.SS4.p1.7.m6.1.1.1.1" xref="S5.SS4.p1.7.m6.1.1.1.1.1.cmml"><mo id="S5.SS4.p1.7.m6.1.1.1.1.2" stretchy="false" xref="S5.SS4.p1.7.m6.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS4.p1.7.m6.1.1.1.1.1" xref="S5.SS4.p1.7.m6.1.1.1.1.1.cmml"><mn id="S5.SS4.p1.7.m6.1.1.1.1.1.2" xref="S5.SS4.p1.7.m6.1.1.1.1.1.2.cmml">1</mn><mo id="S5.SS4.p1.7.m6.1.1.1.1.1.1" xref="S5.SS4.p1.7.m6.1.1.1.1.1.1.cmml">+</mo><mi id="S5.SS4.p1.7.m6.1.1.1.1.1.3" xref="S5.SS4.p1.7.m6.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="S5.SS4.p1.7.m6.1.1.1.1.3" stretchy="false" xref="S5.SS4.p1.7.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.7.m6.1b"><apply id="S5.SS4.p1.7.m6.1.1.cmml" xref="S5.SS4.p1.7.m6.1.1"><divide id="S5.SS4.p1.7.m6.1.1.2.cmml" xref="S5.SS4.p1.7.m6.1.1.2"></divide><cn id="S5.SS4.p1.7.m6.1.1.3.cmml" type="integer" xref="S5.SS4.p1.7.m6.1.1.3">1</cn><apply id="S5.SS4.p1.7.m6.1.1.1.1.1.cmml" xref="S5.SS4.p1.7.m6.1.1.1.1"><plus id="S5.SS4.p1.7.m6.1.1.1.1.1.1.cmml" xref="S5.SS4.p1.7.m6.1.1.1.1.1.1"></plus><cn id="S5.SS4.p1.7.m6.1.1.1.1.1.2.cmml" type="integer" xref="S5.SS4.p1.7.m6.1.1.1.1.1.2">1</cn><ci id="S5.SS4.p1.7.m6.1.1.1.1.1.3.cmml" xref="S5.SS4.p1.7.m6.1.1.1.1.1.3">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.7.m6.1c">1/(1+z)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.7.m6.1d">1 / ( 1 + italic_z )</annotation></semantics></math> at the matter-dominated epoch. <math alttext="T(k)" class="ltx_Math" display="inline" id="S5.SS4.p1.8.m7.1"><semantics id="S5.SS4.p1.8.m7.1a"><mrow id="S5.SS4.p1.8.m7.1.2" xref="S5.SS4.p1.8.m7.1.2.cmml"><mi id="S5.SS4.p1.8.m7.1.2.2" xref="S5.SS4.p1.8.m7.1.2.2.cmml">T</mi><mo id="S5.SS4.p1.8.m7.1.2.1" xref="S5.SS4.p1.8.m7.1.2.1.cmml">⁢</mo><mrow id="S5.SS4.p1.8.m7.1.2.3.2" xref="S5.SS4.p1.8.m7.1.2.cmml"><mo id="S5.SS4.p1.8.m7.1.2.3.2.1" stretchy="false" xref="S5.SS4.p1.8.m7.1.2.cmml">(</mo><mi id="S5.SS4.p1.8.m7.1.1" xref="S5.SS4.p1.8.m7.1.1.cmml">k</mi><mo id="S5.SS4.p1.8.m7.1.2.3.2.2" stretchy="false" xref="S5.SS4.p1.8.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.8.m7.1b"><apply id="S5.SS4.p1.8.m7.1.2.cmml" xref="S5.SS4.p1.8.m7.1.2"><times id="S5.SS4.p1.8.m7.1.2.1.cmml" xref="S5.SS4.p1.8.m7.1.2.1"></times><ci id="S5.SS4.p1.8.m7.1.2.2.cmml" xref="S5.SS4.p1.8.m7.1.2.2">𝑇</ci><ci id="S5.SS4.p1.8.m7.1.1.cmml" xref="S5.SS4.p1.8.m7.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.8.m7.1c">T(k)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.8.m7.1d">italic_T ( italic_k )</annotation></semantics></math> is normalized to <math alttext="1" class="ltx_Math" display="inline" id="S5.SS4.p1.9.m8.1"><semantics id="S5.SS4.p1.9.m8.1a"><mn id="S5.SS4.p1.9.m8.1.1" xref="S5.SS4.p1.9.m8.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.9.m8.1b"><cn id="S5.SS4.p1.9.m8.1.1.cmml" type="integer" xref="S5.SS4.p1.9.m8.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.9.m8.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.9.m8.1d">1</annotation></semantics></math> on large scales, where the PNG signal is the most prominent, so the “scale-dependent” effect typically follows a <math alttext="k^{-2}" class="ltx_Math" display="inline" id="S5.SS4.p1.10.m9.1"><semantics id="S5.SS4.p1.10.m9.1a"><msup id="S5.SS4.p1.10.m9.1.1" xref="S5.SS4.p1.10.m9.1.1.cmml"><mi id="S5.SS4.p1.10.m9.1.1.2" xref="S5.SS4.p1.10.m9.1.1.2.cmml">k</mi><mrow id="S5.SS4.p1.10.m9.1.1.3" xref="S5.SS4.p1.10.m9.1.1.3.cmml"><mo id="S5.SS4.p1.10.m9.1.1.3a" xref="S5.SS4.p1.10.m9.1.1.3.cmml">−</mo><mn id="S5.SS4.p1.10.m9.1.1.3.2" xref="S5.SS4.p1.10.m9.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.10.m9.1b"><apply id="S5.SS4.p1.10.m9.1.1.cmml" xref="S5.SS4.p1.10.m9.1.1"><csymbol cd="ambiguous" id="S5.SS4.p1.10.m9.1.1.1.cmml" xref="S5.SS4.p1.10.m9.1.1">superscript</csymbol><ci id="S5.SS4.p1.10.m9.1.1.2.cmml" xref="S5.SS4.p1.10.m9.1.1.2">𝑘</ci><apply id="S5.SS4.p1.10.m9.1.1.3.cmml" xref="S5.SS4.p1.10.m9.1.1.3"><minus id="S5.SS4.p1.10.m9.1.1.3.1.cmml" xref="S5.SS4.p1.10.m9.1.1.3"></minus><cn id="S5.SS4.p1.10.m9.1.1.3.2.cmml" type="integer" xref="S5.SS4.p1.10.m9.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.10.m9.1c">k^{-2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.10.m9.1d">italic_k start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> behavior. In our current forecasts, we focus exclusively on the local-shape configuration and apply a commonly used universality relation, <math alttext="b_{\phi}=2\delta_{\text{c}}(b_{1}-1)" class="ltx_Math" display="inline" id="S5.SS4.p1.11.m10.1"><semantics id="S5.SS4.p1.11.m10.1a"><mrow id="S5.SS4.p1.11.m10.1.1" xref="S5.SS4.p1.11.m10.1.1.cmml"><msub id="S5.SS4.p1.11.m10.1.1.3" xref="S5.SS4.p1.11.m10.1.1.3.cmml"><mi id="S5.SS4.p1.11.m10.1.1.3.2" xref="S5.SS4.p1.11.m10.1.1.3.2.cmml">b</mi><mi id="S5.SS4.p1.11.m10.1.1.3.3" xref="S5.SS4.p1.11.m10.1.1.3.3.cmml">ϕ</mi></msub><mo id="S5.SS4.p1.11.m10.1.1.2" xref="S5.SS4.p1.11.m10.1.1.2.cmml">=</mo><mrow id="S5.SS4.p1.11.m10.1.1.1" xref="S5.SS4.p1.11.m10.1.1.1.cmml"><mn id="S5.SS4.p1.11.m10.1.1.1.3" xref="S5.SS4.p1.11.m10.1.1.1.3.cmml">2</mn><mo id="S5.SS4.p1.11.m10.1.1.1.2" xref="S5.SS4.p1.11.m10.1.1.1.2.cmml">⁢</mo><msub id="S5.SS4.p1.11.m10.1.1.1.4" xref="S5.SS4.p1.11.m10.1.1.1.4.cmml"><mi id="S5.SS4.p1.11.m10.1.1.1.4.2" xref="S5.SS4.p1.11.m10.1.1.1.4.2.cmml">δ</mi><mtext id="S5.SS4.p1.11.m10.1.1.1.4.3" xref="S5.SS4.p1.11.m10.1.1.1.4.3a.cmml">c</mtext></msub><mo id="S5.SS4.p1.11.m10.1.1.1.2a" xref="S5.SS4.p1.11.m10.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS4.p1.11.m10.1.1.1.1.1" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.cmml"><mo id="S5.SS4.p1.11.m10.1.1.1.1.1.2" stretchy="false" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS4.p1.11.m10.1.1.1.1.1.1" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.cmml"><msub id="S5.SS4.p1.11.m10.1.1.1.1.1.1.2" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.cmml"><mi id="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.2" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.2.cmml">b</mi><mn id="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.3" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S5.SS4.p1.11.m10.1.1.1.1.1.1.1" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.1.cmml">−</mo><mn id="S5.SS4.p1.11.m10.1.1.1.1.1.1.3" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS4.p1.11.m10.1.1.1.1.1.3" stretchy="false" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.11.m10.1b"><apply id="S5.SS4.p1.11.m10.1.1.cmml" xref="S5.SS4.p1.11.m10.1.1"><eq id="S5.SS4.p1.11.m10.1.1.2.cmml" xref="S5.SS4.p1.11.m10.1.1.2"></eq><apply id="S5.SS4.p1.11.m10.1.1.3.cmml" xref="S5.SS4.p1.11.m10.1.1.3"><csymbol cd="ambiguous" id="S5.SS4.p1.11.m10.1.1.3.1.cmml" xref="S5.SS4.p1.11.m10.1.1.3">subscript</csymbol><ci id="S5.SS4.p1.11.m10.1.1.3.2.cmml" xref="S5.SS4.p1.11.m10.1.1.3.2">𝑏</ci><ci id="S5.SS4.p1.11.m10.1.1.3.3.cmml" xref="S5.SS4.p1.11.m10.1.1.3.3">italic-ϕ</ci></apply><apply id="S5.SS4.p1.11.m10.1.1.1.cmml" xref="S5.SS4.p1.11.m10.1.1.1"><times id="S5.SS4.p1.11.m10.1.1.1.2.cmml" xref="S5.SS4.p1.11.m10.1.1.1.2"></times><cn id="S5.SS4.p1.11.m10.1.1.1.3.cmml" type="integer" xref="S5.SS4.p1.11.m10.1.1.1.3">2</cn><apply id="S5.SS4.p1.11.m10.1.1.1.4.cmml" xref="S5.SS4.p1.11.m10.1.1.1.4"><csymbol cd="ambiguous" id="S5.SS4.p1.11.m10.1.1.1.4.1.cmml" xref="S5.SS4.p1.11.m10.1.1.1.4">subscript</csymbol><ci id="S5.SS4.p1.11.m10.1.1.1.4.2.cmml" xref="S5.SS4.p1.11.m10.1.1.1.4.2">𝛿</ci><ci id="S5.SS4.p1.11.m10.1.1.1.4.3a.cmml" xref="S5.SS4.p1.11.m10.1.1.1.4.3"><mtext id="S5.SS4.p1.11.m10.1.1.1.4.3.cmml" mathsize="70%" xref="S5.SS4.p1.11.m10.1.1.1.4.3">c</mtext></ci></apply><apply id="S5.SS4.p1.11.m10.1.1.1.1.1.1.cmml" xref="S5.SS4.p1.11.m10.1.1.1.1.1"><minus id="S5.SS4.p1.11.m10.1.1.1.1.1.1.1.cmml" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.1"></minus><apply id="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.cmml" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.1.cmml" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.2">subscript</csymbol><ci id="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.2.cmml" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.2">𝑏</ci><cn id="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.2.3">1</cn></apply><cn id="S5.SS4.p1.11.m10.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS4.p1.11.m10.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.11.m10.1c">b_{\phi}=2\delta_{\text{c}}(b_{1}-1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.11.m10.1d">italic_b start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT = 2 italic_δ start_POSTSUBSCRIPT c end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 )</annotation></semantics></math>, where <math alttext="\delta_{\text{c}}" class="ltx_Math" display="inline" id="S5.SS4.p1.12.m11.1"><semantics id="S5.SS4.p1.12.m11.1a"><msub id="S5.SS4.p1.12.m11.1.1" xref="S5.SS4.p1.12.m11.1.1.cmml"><mi id="S5.SS4.p1.12.m11.1.1.2" xref="S5.SS4.p1.12.m11.1.1.2.cmml">δ</mi><mtext id="S5.SS4.p1.12.m11.1.1.3" xref="S5.SS4.p1.12.m11.1.1.3a.cmml">c</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.12.m11.1b"><apply id="S5.SS4.p1.12.m11.1.1.cmml" xref="S5.SS4.p1.12.m11.1.1"><csymbol cd="ambiguous" id="S5.SS4.p1.12.m11.1.1.1.cmml" xref="S5.SS4.p1.12.m11.1.1">subscript</csymbol><ci id="S5.SS4.p1.12.m11.1.1.2.cmml" xref="S5.SS4.p1.12.m11.1.1.2">𝛿</ci><ci id="S5.SS4.p1.12.m11.1.1.3a.cmml" xref="S5.SS4.p1.12.m11.1.1.3"><mtext id="S5.SS4.p1.12.m11.1.1.3.cmml" mathsize="70%" xref="S5.SS4.p1.12.m11.1.1.3">c</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.12.m11.1c">\delta_{\text{c}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.12.m11.1d">italic_δ start_POSTSUBSCRIPT c end_POSTSUBSCRIPT</annotation></semantics></math> is the critical density in the spherical collapse model. Recent studies, however, suggest that this relation may vary depending on the specific galaxy sample <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib287" title="">287</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib288" title="">288</a>]</cite>, and we leave further investigations to future works. </div> <div class="ltx_para" id="S5.SS4.p2"> Our forecast results are presented in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.F8" title="Figure 8 ‣ 5.4 Primordial Non-Gaussianity ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">8</a>. For comparison, we perform similar forecasts for DESI and show the existing constraints from BOSS and Planck. Notably, an independent constraint from MUST is expected to limit <math alttext="f_{\text{NL}}^{\text{local}}" class="ltx_Math" display="inline" id="S5.SS4.p2.1.m1.1"><semantics id="S5.SS4.p2.1.m1.1a"><msubsup id="S5.SS4.p2.1.m1.1.1" xref="S5.SS4.p2.1.m1.1.1.cmml"><mi id="S5.SS4.p2.1.m1.1.1.2.2" xref="S5.SS4.p2.1.m1.1.1.2.2.cmml">f</mi><mtext id="S5.SS4.p2.1.m1.1.1.2.3" xref="S5.SS4.p2.1.m1.1.1.2.3a.cmml">NL</mtext><mtext id="S5.SS4.p2.1.m1.1.1.3" xref="S5.SS4.p2.1.m1.1.1.3a.cmml">local</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.1.m1.1b"><apply id="S5.SS4.p2.1.m1.1.1.cmml" xref="S5.SS4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.1.m1.1.1.1.cmml" xref="S5.SS4.p2.1.m1.1.1">superscript</csymbol><apply id="S5.SS4.p2.1.m1.1.1.2.cmml" xref="S5.SS4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.1.m1.1.1.2.1.cmml" xref="S5.SS4.p2.1.m1.1.1">subscript</csymbol><ci id="S5.SS4.p2.1.m1.1.1.2.2.cmml" xref="S5.SS4.p2.1.m1.1.1.2.2">𝑓</ci><ci 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id="S5.SS4.p2.2.m2.1d">italic_σ ( italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT )</annotation></semantics></math> ranging from 1.83 to 2.08, providing a five-fold improvement over DESI and outperforming the state-of-the-art CMB constraint of <math alttext="\sigma(f_{\text{NL}}^{\text{local}})=5.1" class="ltx_Math" display="inline" id="S5.SS4.p2.3.m3.1"><semantics id="S5.SS4.p2.3.m3.1a"><mrow id="S5.SS4.p2.3.m3.1.1" xref="S5.SS4.p2.3.m3.1.1.cmml"><mrow id="S5.SS4.p2.3.m3.1.1.1" xref="S5.SS4.p2.3.m3.1.1.1.cmml"><mi id="S5.SS4.p2.3.m3.1.1.1.3" xref="S5.SS4.p2.3.m3.1.1.1.3.cmml">σ</mi><mo id="S5.SS4.p2.3.m3.1.1.1.2" xref="S5.SS4.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS4.p2.3.m3.1.1.1.1.1" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.SS4.p2.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.SS4.p2.3.m3.1.1.1.1.1.1" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.cmml"><mi 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id="S5.SS4.p2.3.m3.1.1.1.1.1.1.cmml" xref="S5.SS4.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.SS4.p2.3.m3.1.1.1.1.1">superscript</csymbol><apply id="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.SS4.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.1.cmml" xref="S5.SS4.p2.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.2.cmml" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.2">𝑓</ci><ci id="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.3a.cmml" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.3"><mtext id="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.2.3">NL</mtext></ci></apply><ci id="S5.SS4.p2.3.m3.1.1.1.1.1.1.3a.cmml" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.3"><mtext id="S5.SS4.p2.3.m3.1.1.1.1.1.1.3.cmml" mathsize="70%" xref="S5.SS4.p2.3.m3.1.1.1.1.1.1.3">local</mtext></ci></apply></apply><cn id="S5.SS4.p2.3.m3.1.1.3.cmml" type="float" xref="S5.SS4.p2.3.m3.1.1.3">5.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.3.m3.1c">\sigma(f_{\text{NL}}^{\text{local}})=5.1</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.3.m3.1d">italic_σ ( italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT ) = 5.1</annotation></semantics></math> from Planck <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib97" title="">97</a>]</cite>. Moreover, the combined constraints from MUST and CMB will tighten the limit to <math alttext="1.48<\sigma(f_{\text{NL}}^{\text{local}})<1.68" class="ltx_Math" display="inline" id="S5.SS4.p2.4.m4.1"><semantics id="S5.SS4.p2.4.m4.1a"><mrow id="S5.SS4.p2.4.m4.1.1" xref="S5.SS4.p2.4.m4.1.1.cmml"><mn id="S5.SS4.p2.4.m4.1.1.3" xref="S5.SS4.p2.4.m4.1.1.3.cmml">1.48</mn><mo id="S5.SS4.p2.4.m4.1.1.4" xref="S5.SS4.p2.4.m4.1.1.4.cmml"><</mo><mrow id="S5.SS4.p2.4.m4.1.1.1" xref="S5.SS4.p2.4.m4.1.1.1.cmml"><mi id="S5.SS4.p2.4.m4.1.1.1.3" xref="S5.SS4.p2.4.m4.1.1.1.3.cmml">σ</mi><mo id="S5.SS4.p2.4.m4.1.1.1.2" xref="S5.SS4.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS4.p2.4.m4.1.1.1.1.1" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.SS4.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.SS4.p2.4.m4.1.1.1.1.1.1" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.2" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.2.cmml">f</mi><mtext id="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.3" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.3a.cmml">NL</mtext><mtext id="S5.SS4.p2.4.m4.1.1.1.1.1.1.3" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.3a.cmml">local</mtext></msubsup><mo id="S5.SS4.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS4.p2.4.m4.1.1.5" xref="S5.SS4.p2.4.m4.1.1.5.cmml"><</mo><mn id="S5.SS4.p2.4.m4.1.1.6" xref="S5.SS4.p2.4.m4.1.1.6.cmml">1.68</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.4.m4.1b"><apply id="S5.SS4.p2.4.m4.1.1.cmml" xref="S5.SS4.p2.4.m4.1.1"><and id="S5.SS4.p2.4.m4.1.1a.cmml" xref="S5.SS4.p2.4.m4.1.1"></and><apply id="S5.SS4.p2.4.m4.1.1b.cmml" xref="S5.SS4.p2.4.m4.1.1"><lt id="S5.SS4.p2.4.m4.1.1.4.cmml" xref="S5.SS4.p2.4.m4.1.1.4"></lt><cn id="S5.SS4.p2.4.m4.1.1.3.cmml" type="float" xref="S5.SS4.p2.4.m4.1.1.3">1.48</cn><apply id="S5.SS4.p2.4.m4.1.1.1.cmml" xref="S5.SS4.p2.4.m4.1.1.1"><times id="S5.SS4.p2.4.m4.1.1.1.2.cmml" xref="S5.SS4.p2.4.m4.1.1.1.2"></times><ci id="S5.SS4.p2.4.m4.1.1.1.3.cmml" xref="S5.SS4.p2.4.m4.1.1.1.3">𝜎</ci><apply id="S5.SS4.p2.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS4.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.SS4.p2.4.m4.1.1.1.1.1">superscript</csymbol><apply id="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.SS4.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.1.cmml" xref="S5.SS4.p2.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.2.cmml" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.2">𝑓</ci><ci id="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.3a.cmml" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.3"><mtext id="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.2.3">NL</mtext></ci></apply><ci id="S5.SS4.p2.4.m4.1.1.1.1.1.1.3a.cmml" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.3"><mtext id="S5.SS4.p2.4.m4.1.1.1.1.1.1.3.cmml" mathsize="70%" xref="S5.SS4.p2.4.m4.1.1.1.1.1.1.3">local</mtext></ci></apply></apply></apply><apply id="S5.SS4.p2.4.m4.1.1c.cmml" xref="S5.SS4.p2.4.m4.1.1"><lt id="S5.SS4.p2.4.m4.1.1.5.cmml" xref="S5.SS4.p2.4.m4.1.1.5"></lt><share href="https://arxiv.org/html/2411.07970v2#S5.SS4.p2.4.m4.1.1.1.cmml" id="S5.SS4.p2.4.m4.1.1d.cmml" xref="S5.SS4.p2.4.m4.1.1"></share><cn id="S5.SS4.p2.4.m4.1.1.6.cmml" type="float" xref="S5.SS4.p2.4.m4.1.1.6">1.68</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.4.m4.1c">1.48<\sigma(f_{\text{NL}}^{\text{local}})<1.68</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.4.m4.1d">1.48 < italic_σ ( italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT ) < 1.68</annotation></semantics></math>. Additional methodologies not included in this forecast, such as the multi-tracer approach <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib289" title="">289</a>]</cite> and bispectrum <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib290" title="">290</a>]</cite>, are anticipated to further enhance MUST’s capability to <math alttext="\sigma(f_{\text{NL}}^{\text{local}})<1.0" class="ltx_Math" display="inline" id="S5.SS4.p2.5.m5.1"><semantics id="S5.SS4.p2.5.m5.1a"><mrow id="S5.SS4.p2.5.m5.1.1" xref="S5.SS4.p2.5.m5.1.1.cmml"><mrow id="S5.SS4.p2.5.m5.1.1.1" xref="S5.SS4.p2.5.m5.1.1.1.cmml"><mi id="S5.SS4.p2.5.m5.1.1.1.3" xref="S5.SS4.p2.5.m5.1.1.1.3.cmml">σ</mi><mo id="S5.SS4.p2.5.m5.1.1.1.2" xref="S5.SS4.p2.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS4.p2.5.m5.1.1.1.1.1" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.cmml"><mo id="S5.SS4.p2.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.SS4.p2.5.m5.1.1.1.1.1.1" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.2" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.2.cmml">f</mi><mtext id="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.3" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.3a.cmml">NL</mtext><mtext id="S5.SS4.p2.5.m5.1.1.1.1.1.1.3" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.3a.cmml">local</mtext></msubsup><mo id="S5.SS4.p2.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS4.p2.5.m5.1.1.2" xref="S5.SS4.p2.5.m5.1.1.2.cmml"><</mo><mn id="S5.SS4.p2.5.m5.1.1.3" xref="S5.SS4.p2.5.m5.1.1.3.cmml">1.0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.5.m5.1b"><apply id="S5.SS4.p2.5.m5.1.1.cmml" xref="S5.SS4.p2.5.m5.1.1"><lt id="S5.SS4.p2.5.m5.1.1.2.cmml" xref="S5.SS4.p2.5.m5.1.1.2"></lt><apply id="S5.SS4.p2.5.m5.1.1.1.cmml" xref="S5.SS4.p2.5.m5.1.1.1"><times id="S5.SS4.p2.5.m5.1.1.1.2.cmml" xref="S5.SS4.p2.5.m5.1.1.1.2"></times><ci id="S5.SS4.p2.5.m5.1.1.1.3.cmml" xref="S5.SS4.p2.5.m5.1.1.1.3">𝜎</ci><apply id="S5.SS4.p2.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS4.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.SS4.p2.5.m5.1.1.1.1.1">superscript</csymbol><apply id="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.SS4.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S5.SS4.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.2">𝑓</ci><ci id="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.3a.cmml" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.3"><mtext id="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.2.3">NL</mtext></ci></apply><ci id="S5.SS4.p2.5.m5.1.1.1.1.1.1.3a.cmml" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.3"><mtext id="S5.SS4.p2.5.m5.1.1.1.1.1.1.3.cmml" mathsize="70%" xref="S5.SS4.p2.5.m5.1.1.1.1.1.1.3">local</mtext></ci></apply></apply><cn id="S5.SS4.p2.5.m5.1.1.3.cmml" type="float" xref="S5.SS4.p2.5.m5.1.1.3">1.0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.5.m5.1c">\sigma(f_{\text{NL}}^{\text{local}})<1.0</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.5.m5.1d">italic_σ ( italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT ) < 1.0</annotation></semantics></math>, which is crucial from a theoretical perspective, as a non-detection at this level would rule out many models involving the curvaton mechanism <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib95" title="">95</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib291" title="">291</a>]</cite>. </div> <figure class="ltx_figure" id="S5.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="529" id="S5.F8.g1" src="x4.png" width="831"/> <figcaption class="ltx_caption ltx_centering">Figure 8: The 1-<math alttext="\sigma" class="ltx_Math" display="inline" id="S5.F8.5.m1.1"><semantics id="S5.F8.5.m1.1b"><mi id="S5.F8.5.m1.1.1" xref="S5.F8.5.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.F8.5.m1.1c"><ci id="S5.F8.5.m1.1.1.cmml" xref="S5.F8.5.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F8.5.m1.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.F8.5.m1.1e">italic_σ</annotation></semantics></math> constraints on <math alttext="f_{\text{NL}}^{\text{local}}" class="ltx_Math" display="inline" id="S5.F8.6.m2.1"><semantics id="S5.F8.6.m2.1b"><msubsup id="S5.F8.6.m2.1.1" xref="S5.F8.6.m2.1.1.cmml"><mi id="S5.F8.6.m2.1.1.2.2" xref="S5.F8.6.m2.1.1.2.2.cmml">f</mi><mtext id="S5.F8.6.m2.1.1.2.3" xref="S5.F8.6.m2.1.1.2.3a.cmml">NL</mtext><mtext id="S5.F8.6.m2.1.1.3" xref="S5.F8.6.m2.1.1.3a.cmml">local</mtext></msubsup><annotation-xml encoding="MathML-Content" id="S5.F8.6.m2.1c"><apply id="S5.F8.6.m2.1.1.cmml" xref="S5.F8.6.m2.1.1"><csymbol cd="ambiguous" id="S5.F8.6.m2.1.1.1.cmml" xref="S5.F8.6.m2.1.1">superscript</csymbol><apply id="S5.F8.6.m2.1.1.2.cmml" xref="S5.F8.6.m2.1.1"><csymbol cd="ambiguous" id="S5.F8.6.m2.1.1.2.1.cmml" xref="S5.F8.6.m2.1.1">subscript</csymbol><ci id="S5.F8.6.m2.1.1.2.2.cmml" xref="S5.F8.6.m2.1.1.2.2">𝑓</ci><ci id="S5.F8.6.m2.1.1.2.3a.cmml" xref="S5.F8.6.m2.1.1.2.3"><mtext id="S5.F8.6.m2.1.1.2.3.cmml" mathsize="70%" xref="S5.F8.6.m2.1.1.2.3">NL</mtext></ci></apply><ci id="S5.F8.6.m2.1.1.3a.cmml" xref="S5.F8.6.m2.1.1.3"><mtext id="S5.F8.6.m2.1.1.3.cmml" mathsize="70%" xref="S5.F8.6.m2.1.1.3">local</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F8.6.m2.1d">f_{\text{NL}}^{\text{local}}</annotation><annotation encoding="application/x-llamapun" id="S5.F8.6.m2.1e">italic_f start_POSTSUBSCRIPT NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT local end_POSTSUPERSCRIPT</annotation></semantics></math> from LSS surveys include BOSS (gray, see Table B1 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib292" title="">292</a>]</cite>) and our forecasts for DESI (green) and MUST (red, dark and light for the optimistic and conservative estimations respectively). The state-of-the-art constraint from Planck is shown as the blue dashed line for comparison. The slashed bars indicate the constraints combining the constraints from Planck <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib268" title="">268</a>]</cite> low-<math alttext="\ell" class="ltx_Math" display="inline" id="S5.F8.7.m3.1"><semantics id="S5.F8.7.m3.1b"><mi id="S5.F8.7.m3.1.1" mathvariant="normal" xref="S5.F8.7.m3.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S5.F8.7.m3.1c"><ci id="S5.F8.7.m3.1.1.cmml" xref="S5.F8.7.m3.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F8.7.m3.1d">\ell</annotation><annotation encoding="application/x-llamapun" id="S5.F8.7.m3.1e">roman_ℓ</annotation></semantics></math> and Simons Observatory (SO; <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib45" title="">45</a>]</cite>) high-<math alttext="\ell" class="ltx_Math" display="inline" id="S5.F8.8.m4.1"><semantics id="S5.F8.8.m4.1b"><mi id="S5.F8.8.m4.1.1" mathvariant="normal" xref="S5.F8.8.m4.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S5.F8.8.m4.1c"><ci id="S5.F8.8.m4.1.1.cmml" xref="S5.F8.8.m4.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F8.8.m4.1d">\ell</annotation><annotation encoding="application/x-llamapun" id="S5.F8.8.m4.1e">roman_ℓ</annotation></semantics></math> measurements.</figcaption> </figure> </section> <section class="ltx_subsection" id="S5.SS5"> <h3 class="ltx_title ltx_title_subsection"> 5.5 Neutrinos</h3> <div class="ltx_para" id="S5.SS5.p1"> The impacts of neutrinos on the expansion history of the Universe and the structure formation process permit constraints on the total neutrino mass <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S5.SS5.p1.1.m1.1"><semantics id="S5.SS5.p1.1.m1.1a"><msub id="S5.SS5.p1.1.m1.1.1" xref="S5.SS5.p1.1.m1.1.1.cmml"><mi id="S5.SS5.p1.1.m1.1.1.2" xref="S5.SS5.p1.1.m1.1.1.2.cmml">M</mi><mi id="S5.SS5.p1.1.m1.1.1.3" xref="S5.SS5.p1.1.m1.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS5.p1.1.m1.1b"><apply id="S5.SS5.p1.1.m1.1.1.cmml" xref="S5.SS5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS5.p1.1.m1.1.1.1.cmml" xref="S5.SS5.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS5.p1.1.m1.1.1.2.cmml" xref="S5.SS5.p1.1.m1.1.1.2">𝑀</ci><ci id="S5.SS5.p1.1.m1.1.1.3.cmml" xref="S5.SS5.p1.1.m1.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p1.1.m1.1c">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p1.1.m1.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> through galaxy clustering measurements (see Section <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S3.SS4" title="3.4 Neutrinos & Light Relics ‣ 3 Scientific Motivations of the Stage-V Cosmological Surveys ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">3.4</a>). As a Stage-V spectroscopic survey with high tracer number density and extensive survey volume, MUST is expected to place stringent constraints on neutrino mass and potentially distinguish between the normal and inverted mass hierarchies when combined with CMB measurements. </div> <div class="ltx_para" id="S5.SS5.p2"> We present a Fisher forecast for the error on <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S5.SS5.p2.1.m1.1"><semantics id="S5.SS5.p2.1.m1.1a"><msub id="S5.SS5.p2.1.m1.1.1" xref="S5.SS5.p2.1.m1.1.1.cmml"><mi id="S5.SS5.p2.1.m1.1.1.2" xref="S5.SS5.p2.1.m1.1.1.2.cmml">M</mi><mi id="S5.SS5.p2.1.m1.1.1.3" xref="S5.SS5.p2.1.m1.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.1.m1.1b"><apply id="S5.SS5.p2.1.m1.1.1.cmml" xref="S5.SS5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS5.p2.1.m1.1.1.1.cmml" xref="S5.SS5.p2.1.m1.1.1">subscript</csymbol><ci id="S5.SS5.p2.1.m1.1.1.2.cmml" xref="S5.SS5.p2.1.m1.1.1.2">𝑀</ci><ci id="S5.SS5.p2.1.m1.1.1.3.cmml" xref="S5.SS5.p2.1.m1.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p2.1.m1.1c">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p2.1.m1.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> derived from joint constraints of MUST and CMB measurements, assuming a fiducial value of <math alttext="M_{\nu}=0.06\,{\rm eV}" class="ltx_Math" display="inline" id="S5.SS5.p2.2.m2.1"><semantics id="S5.SS5.p2.2.m2.1a"><mrow id="S5.SS5.p2.2.m2.1.1" xref="S5.SS5.p2.2.m2.1.1.cmml"><msub id="S5.SS5.p2.2.m2.1.1.2" xref="S5.SS5.p2.2.m2.1.1.2.cmml"><mi id="S5.SS5.p2.2.m2.1.1.2.2" xref="S5.SS5.p2.2.m2.1.1.2.2.cmml">M</mi><mi id="S5.SS5.p2.2.m2.1.1.2.3" xref="S5.SS5.p2.2.m2.1.1.2.3.cmml">ν</mi></msub><mo id="S5.SS5.p2.2.m2.1.1.1" xref="S5.SS5.p2.2.m2.1.1.1.cmml">=</mo><mrow id="S5.SS5.p2.2.m2.1.1.3" xref="S5.SS5.p2.2.m2.1.1.3.cmml"><mn id="S5.SS5.p2.2.m2.1.1.3.2" xref="S5.SS5.p2.2.m2.1.1.3.2.cmml">0.06</mn><mo id="S5.SS5.p2.2.m2.1.1.3.1" lspace="0.170em" xref="S5.SS5.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S5.SS5.p2.2.m2.1.1.3.3" xref="S5.SS5.p2.2.m2.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.2.m2.1b"><apply id="S5.SS5.p2.2.m2.1.1.cmml" xref="S5.SS5.p2.2.m2.1.1"><eq id="S5.SS5.p2.2.m2.1.1.1.cmml" xref="S5.SS5.p2.2.m2.1.1.1"></eq><apply id="S5.SS5.p2.2.m2.1.1.2.cmml" xref="S5.SS5.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS5.p2.2.m2.1.1.2.1.cmml" xref="S5.SS5.p2.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS5.p2.2.m2.1.1.2.2.cmml" xref="S5.SS5.p2.2.m2.1.1.2.2">𝑀</ci><ci id="S5.SS5.p2.2.m2.1.1.2.3.cmml" xref="S5.SS5.p2.2.m2.1.1.2.3">𝜈</ci></apply><apply id="S5.SS5.p2.2.m2.1.1.3.cmml" xref="S5.SS5.p2.2.m2.1.1.3"><times id="S5.SS5.p2.2.m2.1.1.3.1.cmml" xref="S5.SS5.p2.2.m2.1.1.3.1"></times><cn id="S5.SS5.p2.2.m2.1.1.3.2.cmml" type="float" xref="S5.SS5.p2.2.m2.1.1.3.2">0.06</cn><ci id="S5.SS5.p2.2.m2.1.1.3.3.cmml" xref="S5.SS5.p2.2.m2.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p2.2.m2.1c">M_{\nu}=0.06\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p2.2.m2.1d">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 0.06 roman_eV</annotation></semantics></math> within the <math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.SS5.p2.3.m3.1"><semantics id="S5.SS5.p2.3.m3.1a"><mi id="S5.SS5.p2.3.m3.1.1" mathvariant="normal" xref="S5.SS5.p2.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.3.m3.1b"><ci id="S5.SS5.p2.3.m3.1.1.cmml" xref="S5.SS5.p2.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p2.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p2.3.m3.1d">roman_Λ</annotation></semantics></math>CDM framework, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.F9" title="Figure 9 ‣ 5.5 Neutrinos ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">9</a>. Under the assumption of a normal mass hierarchy, where neutrinos consist of a single massive eigenstate and two massless states, the statistical error, ranging from <math alttext="0.31\,{\rm eV}" class="ltx_Math" display="inline" id="S5.SS5.p2.4.m4.1"><semantics id="S5.SS5.p2.4.m4.1a"><mrow id="S5.SS5.p2.4.m4.1.1" xref="S5.SS5.p2.4.m4.1.1.cmml"><mn id="S5.SS5.p2.4.m4.1.1.2" xref="S5.SS5.p2.4.m4.1.1.2.cmml">0.31</mn><mo id="S5.SS5.p2.4.m4.1.1.1" lspace="0.170em" xref="S5.SS5.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S5.SS5.p2.4.m4.1.1.3" xref="S5.SS5.p2.4.m4.1.1.3.cmml">eV</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.4.m4.1b"><apply id="S5.SS5.p2.4.m4.1.1.cmml" xref="S5.SS5.p2.4.m4.1.1"><times id="S5.SS5.p2.4.m4.1.1.1.cmml" xref="S5.SS5.p2.4.m4.1.1.1"></times><cn id="S5.SS5.p2.4.m4.1.1.2.cmml" type="float" xref="S5.SS5.p2.4.m4.1.1.2">0.31</cn><ci id="S5.SS5.p2.4.m4.1.1.3.cmml" xref="S5.SS5.p2.4.m4.1.1.3">eV</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p2.4.m4.1c">0.31\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p2.4.m4.1d">0.31 roman_eV</annotation></semantics></math> in the optimistic estimation and <math alttext="0.40\,{\rm eV}" class="ltx_Math" display="inline" id="S5.SS5.p2.5.m5.1"><semantics id="S5.SS5.p2.5.m5.1a"><mrow id="S5.SS5.p2.5.m5.1.1" xref="S5.SS5.p2.5.m5.1.1.cmml"><mn id="S5.SS5.p2.5.m5.1.1.2" xref="S5.SS5.p2.5.m5.1.1.2.cmml">0.40</mn><mo id="S5.SS5.p2.5.m5.1.1.1" lspace="0.170em" xref="S5.SS5.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S5.SS5.p2.5.m5.1.1.3" xref="S5.SS5.p2.5.m5.1.1.3.cmml">eV</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.5.m5.1b"><apply id="S5.SS5.p2.5.m5.1.1.cmml" xref="S5.SS5.p2.5.m5.1.1"><times id="S5.SS5.p2.5.m5.1.1.1.cmml" xref="S5.SS5.p2.5.m5.1.1.1"></times><cn id="S5.SS5.p2.5.m5.1.1.2.cmml" type="float" xref="S5.SS5.p2.5.m5.1.1.2">0.40</cn><ci id="S5.SS5.p2.5.m5.1.1.3.cmml" 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xref="S5.SS5.p2.6.m6.1.1.3.3.cmml">σ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.6.m6.1b"><apply id="S5.SS5.p2.6.m6.1.1.cmml" xref="S5.SS5.p2.6.m6.1.1"><gt id="S5.SS5.p2.6.m6.1.1.1.cmml" xref="S5.SS5.p2.6.m6.1.1.1"></gt><csymbol cd="latexml" id="S5.SS5.p2.6.m6.1.1.2.cmml" xref="S5.SS5.p2.6.m6.1.1.2">absent</csymbol><apply id="S5.SS5.p2.6.m6.1.1.3.cmml" xref="S5.SS5.p2.6.m6.1.1.3"><times id="S5.SS5.p2.6.m6.1.1.3.1.cmml" xref="S5.SS5.p2.6.m6.1.1.3.1"></times><cn id="S5.SS5.p2.6.m6.1.1.3.2.cmml" type="float" xref="S5.SS5.p2.6.m6.1.1.3.2">1.5</cn><ci id="S5.SS5.p2.6.m6.1.1.3.3.cmml" xref="S5.SS5.p2.6.m6.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p2.6.m6.1c">>1.5\,\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p2.6.m6.1d">> 1.5 italic_σ</annotation></semantics></math> CL, which significantly enhances the best constraint from Planck with <math 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xref="S5.SS5.p2.7.m7.1.1.2.cmml">=</mo><mrow id="S5.SS5.p2.7.m7.1.1.3" xref="S5.SS5.p2.7.m7.1.1.3.cmml"><mn id="S5.SS5.p2.7.m7.1.1.3.2" xref="S5.SS5.p2.7.m7.1.1.3.2.cmml">0.11</mn><mo id="S5.SS5.p2.7.m7.1.1.3.1" lspace="0.170em" xref="S5.SS5.p2.7.m7.1.1.3.1.cmml">⁢</mo><mi id="S5.SS5.p2.7.m7.1.1.3.3" xref="S5.SS5.p2.7.m7.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.7.m7.1b"><apply id="S5.SS5.p2.7.m7.1.1.cmml" xref="S5.SS5.p2.7.m7.1.1"><eq id="S5.SS5.p2.7.m7.1.1.2.cmml" xref="S5.SS5.p2.7.m7.1.1.2"></eq><apply id="S5.SS5.p2.7.m7.1.1.1.cmml" xref="S5.SS5.p2.7.m7.1.1.1"><times id="S5.SS5.p2.7.m7.1.1.1.2.cmml" xref="S5.SS5.p2.7.m7.1.1.1.2"></times><ci id="S5.SS5.p2.7.m7.1.1.1.3.cmml" xref="S5.SS5.p2.7.m7.1.1.1.3">𝜎</ci><apply id="S5.SS5.p2.7.m7.1.1.1.1.1.1.cmml" xref="S5.SS5.p2.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS5.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="S5.SS5.p2.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S5.SS5.p2.7.m7.1.1.1.1.1.1.2.cmml" xref="S5.SS5.p2.7.m7.1.1.1.1.1.1.2">𝑀</ci><ci id="S5.SS5.p2.7.m7.1.1.1.1.1.1.3.cmml" xref="S5.SS5.p2.7.m7.1.1.1.1.1.1.3">𝜈</ci></apply></apply><apply id="S5.SS5.p2.7.m7.1.1.3.cmml" xref="S5.SS5.p2.7.m7.1.1.3"><times id="S5.SS5.p2.7.m7.1.1.3.1.cmml" xref="S5.SS5.p2.7.m7.1.1.3.1"></times><cn id="S5.SS5.p2.7.m7.1.1.3.2.cmml" type="float" xref="S5.SS5.p2.7.m7.1.1.3.2">0.11</cn><ci id="S5.SS5.p2.7.m7.1.1.3.3.cmml" xref="S5.SS5.p2.7.m7.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p2.7.m7.1c">\sigma(M_{\nu})=0.11\,\rm{eV}</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p2.7.m7.1d">italic_σ ( italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ) = 0.11 roman_eV</annotation></semantics></math>. The inverted mass hierarchy can also be tested with a significance of <math alttext=">1.0\,\sigma" class="ltx_Math" display="inline" id="S5.SS5.p2.8.m8.1"><semantics id="S5.SS5.p2.8.m8.1a"><mrow id="S5.SS5.p2.8.m8.1.1" xref="S5.SS5.p2.8.m8.1.1.cmml"><mi id="S5.SS5.p2.8.m8.1.1.2" xref="S5.SS5.p2.8.m8.1.1.2.cmml"></mi><mo id="S5.SS5.p2.8.m8.1.1.1" xref="S5.SS5.p2.8.m8.1.1.1.cmml">></mo><mrow id="S5.SS5.p2.8.m8.1.1.3" xref="S5.SS5.p2.8.m8.1.1.3.cmml"><mn id="S5.SS5.p2.8.m8.1.1.3.2" xref="S5.SS5.p2.8.m8.1.1.3.2.cmml">1.0</mn><mo id="S5.SS5.p2.8.m8.1.1.3.1" lspace="0.170em" xref="S5.SS5.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S5.SS5.p2.8.m8.1.1.3.3" xref="S5.SS5.p2.8.m8.1.1.3.3.cmml">σ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS5.p2.8.m8.1b"><apply id="S5.SS5.p2.8.m8.1.1.cmml" xref="S5.SS5.p2.8.m8.1.1"><gt id="S5.SS5.p2.8.m8.1.1.1.cmml" xref="S5.SS5.p2.8.m8.1.1.1"></gt><csymbol cd="latexml" id="S5.SS5.p2.8.m8.1.1.2.cmml" xref="S5.SS5.p2.8.m8.1.1.2">absent</csymbol><apply id="S5.SS5.p2.8.m8.1.1.3.cmml" xref="S5.SS5.p2.8.m8.1.1.3"><times id="S5.SS5.p2.8.m8.1.1.3.1.cmml" xref="S5.SS5.p2.8.m8.1.1.3.1"></times><cn id="S5.SS5.p2.8.m8.1.1.3.2.cmml" type="float" xref="S5.SS5.p2.8.m8.1.1.3.2">1.0</cn><ci id="S5.SS5.p2.8.m8.1.1.3.3.cmml" xref="S5.SS5.p2.8.m8.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS5.p2.8.m8.1c">>1.0\,\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.SS5.p2.8.m8.1d">> 1.0 italic_σ</annotation></semantics></math>. </div> <figure class="ltx_figure" id="S5.F9"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="580" id="S5.F9.g1" src="x5.png" width="748"/> <figcaption class="ltx_caption ltx_centering">Figure 9: Fisher forecast for the total neutrino mass <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S5.F9.7.m1.1"><semantics id="S5.F9.7.m1.1b"><msub id="S5.F9.7.m1.1.1" xref="S5.F9.7.m1.1.1.cmml"><mi id="S5.F9.7.m1.1.1.2" xref="S5.F9.7.m1.1.1.2.cmml">M</mi><mi id="S5.F9.7.m1.1.1.3" xref="S5.F9.7.m1.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S5.F9.7.m1.1c"><apply id="S5.F9.7.m1.1.1.cmml" xref="S5.F9.7.m1.1.1"><csymbol cd="ambiguous" id="S5.F9.7.m1.1.1.1.cmml" xref="S5.F9.7.m1.1.1">subscript</csymbol><ci id="S5.F9.7.m1.1.1.2.cmml" xref="S5.F9.7.m1.1.1.2">𝑀</ci><ci id="S5.F9.7.m1.1.1.3.cmml" xref="S5.F9.7.m1.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F9.7.m1.1d">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S5.F9.7.m1.1e">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> based on the galaxy power spectrum with <math alttext="k_{\rm max}=0.4\,h\,{\rm Mpc}^{-1}" class="ltx_Math" display="inline" id="S5.F9.8.m2.1"><semantics id="S5.F9.8.m2.1b"><mrow id="S5.F9.8.m2.1.1" xref="S5.F9.8.m2.1.1.cmml"><msub id="S5.F9.8.m2.1.1.2" xref="S5.F9.8.m2.1.1.2.cmml"><mi id="S5.F9.8.m2.1.1.2.2" xref="S5.F9.8.m2.1.1.2.2.cmml">k</mi><mi id="S5.F9.8.m2.1.1.2.3" xref="S5.F9.8.m2.1.1.2.3.cmml">max</mi></msub><mo id="S5.F9.8.m2.1.1.1" xref="S5.F9.8.m2.1.1.1.cmml">=</mo><mrow id="S5.F9.8.m2.1.1.3" xref="S5.F9.8.m2.1.1.3.cmml"><mn id="S5.F9.8.m2.1.1.3.2" xref="S5.F9.8.m2.1.1.3.2.cmml">0.4</mn><mo id="S5.F9.8.m2.1.1.3.1" lspace="0.170em" xref="S5.F9.8.m2.1.1.3.1.cmml">⁢</mo><mi id="S5.F9.8.m2.1.1.3.3" xref="S5.F9.8.m2.1.1.3.3.cmml">h</mi><mo id="S5.F9.8.m2.1.1.3.1b" lspace="0.170em" xref="S5.F9.8.m2.1.1.3.1.cmml">⁢</mo><msup id="S5.F9.8.m2.1.1.3.4" xref="S5.F9.8.m2.1.1.3.4.cmml"><mi id="S5.F9.8.m2.1.1.3.4.2" xref="S5.F9.8.m2.1.1.3.4.2.cmml">Mpc</mi><mrow id="S5.F9.8.m2.1.1.3.4.3" xref="S5.F9.8.m2.1.1.3.4.3.cmml"><mo id="S5.F9.8.m2.1.1.3.4.3b" xref="S5.F9.8.m2.1.1.3.4.3.cmml">−</mo><mn id="S5.F9.8.m2.1.1.3.4.3.2" xref="S5.F9.8.m2.1.1.3.4.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.F9.8.m2.1c"><apply id="S5.F9.8.m2.1.1.cmml" xref="S5.F9.8.m2.1.1"><eq id="S5.F9.8.m2.1.1.1.cmml" xref="S5.F9.8.m2.1.1.1"></eq><apply id="S5.F9.8.m2.1.1.2.cmml" xref="S5.F9.8.m2.1.1.2"><csymbol cd="ambiguous" id="S5.F9.8.m2.1.1.2.1.cmml" xref="S5.F9.8.m2.1.1.2">subscript</csymbol><ci id="S5.F9.8.m2.1.1.2.2.cmml" xref="S5.F9.8.m2.1.1.2.2">𝑘</ci><ci id="S5.F9.8.m2.1.1.2.3.cmml" xref="S5.F9.8.m2.1.1.2.3">max</ci></apply><apply id="S5.F9.8.m2.1.1.3.cmml" xref="S5.F9.8.m2.1.1.3"><times id="S5.F9.8.m2.1.1.3.1.cmml" xref="S5.F9.8.m2.1.1.3.1"></times><cn id="S5.F9.8.m2.1.1.3.2.cmml" type="float" xref="S5.F9.8.m2.1.1.3.2">0.4</cn><ci id="S5.F9.8.m2.1.1.3.3.cmml" xref="S5.F9.8.m2.1.1.3.3">ℎ</ci><apply id="S5.F9.8.m2.1.1.3.4.cmml" xref="S5.F9.8.m2.1.1.3.4"><csymbol cd="ambiguous" id="S5.F9.8.m2.1.1.3.4.1.cmml" xref="S5.F9.8.m2.1.1.3.4">superscript</csymbol><ci id="S5.F9.8.m2.1.1.3.4.2.cmml" xref="S5.F9.8.m2.1.1.3.4.2">Mpc</ci><apply id="S5.F9.8.m2.1.1.3.4.3.cmml" xref="S5.F9.8.m2.1.1.3.4.3"><minus id="S5.F9.8.m2.1.1.3.4.3.1.cmml" xref="S5.F9.8.m2.1.1.3.4.3"></minus><cn id="S5.F9.8.m2.1.1.3.4.3.2.cmml" type="integer" xref="S5.F9.8.m2.1.1.3.4.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F9.8.m2.1d">k_{\rm max}=0.4\,h\,{\rm Mpc}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.F9.8.m2.1e">italic_k start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT = 0.4 italic_h roman_Mpc start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. Solid lines represent the variation of <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S5.F9.9.m3.1"><semantics id="S5.F9.9.m3.1b"><msub id="S5.F9.9.m3.1.1" xref="S5.F9.9.m3.1.1.cmml"><mi id="S5.F9.9.m3.1.1.2" xref="S5.F9.9.m3.1.1.2.cmml">M</mi><mi id="S5.F9.9.m3.1.1.3" xref="S5.F9.9.m3.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S5.F9.9.m3.1c"><apply id="S5.F9.9.m3.1.1.cmml" xref="S5.F9.9.m3.1.1"><csymbol cd="ambiguous" id="S5.F9.9.m3.1.1.1.cmml" xref="S5.F9.9.m3.1.1">subscript</csymbol><ci id="S5.F9.9.m3.1.1.2.cmml" xref="S5.F9.9.m3.1.1.2">𝑀</ci><ci id="S5.F9.9.m3.1.1.3.cmml" xref="S5.F9.9.m3.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F9.9.m3.1d">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S5.F9.9.m3.1e">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>, with the mass of the lightest neutrino species in both mass hierarchies: inverted hierarchy (green) and normal hierarchy (blue). The red dashed line indicates the fiducial cosmology with <math alttext="M_{\nu}=0.06\,\rm{eV}" class="ltx_Math" display="inline" id="S5.F9.10.m4.1"><semantics id="S5.F9.10.m4.1b"><mrow id="S5.F9.10.m4.1.1" xref="S5.F9.10.m4.1.1.cmml"><msub id="S5.F9.10.m4.1.1.2" xref="S5.F9.10.m4.1.1.2.cmml"><mi id="S5.F9.10.m4.1.1.2.2" xref="S5.F9.10.m4.1.1.2.2.cmml">M</mi><mi id="S5.F9.10.m4.1.1.2.3" xref="S5.F9.10.m4.1.1.2.3.cmml">ν</mi></msub><mo id="S5.F9.10.m4.1.1.1" xref="S5.F9.10.m4.1.1.1.cmml">=</mo><mrow id="S5.F9.10.m4.1.1.3" xref="S5.F9.10.m4.1.1.3.cmml"><mn id="S5.F9.10.m4.1.1.3.2" xref="S5.F9.10.m4.1.1.3.2.cmml">0.06</mn><mo id="S5.F9.10.m4.1.1.3.1" lspace="0.170em" xref="S5.F9.10.m4.1.1.3.1.cmml">⁢</mo><mi id="S5.F9.10.m4.1.1.3.3" xref="S5.F9.10.m4.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.F9.10.m4.1c"><apply id="S5.F9.10.m4.1.1.cmml" xref="S5.F9.10.m4.1.1"><eq id="S5.F9.10.m4.1.1.1.cmml" xref="S5.F9.10.m4.1.1.1"></eq><apply id="S5.F9.10.m4.1.1.2.cmml" xref="S5.F9.10.m4.1.1.2"><csymbol cd="ambiguous" id="S5.F9.10.m4.1.1.2.1.cmml" xref="S5.F9.10.m4.1.1.2">subscript</csymbol><ci id="S5.F9.10.m4.1.1.2.2.cmml" xref="S5.F9.10.m4.1.1.2.2">𝑀</ci><ci id="S5.F9.10.m4.1.1.2.3.cmml" xref="S5.F9.10.m4.1.1.2.3">𝜈</ci></apply><apply id="S5.F9.10.m4.1.1.3.cmml" xref="S5.F9.10.m4.1.1.3"><times id="S5.F9.10.m4.1.1.3.1.cmml" xref="S5.F9.10.m4.1.1.3.1"></times><cn id="S5.F9.10.m4.1.1.3.2.cmml" type="float" xref="S5.F9.10.m4.1.1.3.2">0.06</cn><ci id="S5.F9.10.m4.1.1.3.3.cmml" xref="S5.F9.10.m4.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F9.10.m4.1d">M_{\nu}=0.06\,\rm{eV}</annotation><annotation encoding="application/x-llamapun" id="S5.F9.10.m4.1e">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 0.06 roman_eV</annotation></semantics></math>, while the dark (light) red-shaded region represents the optimistic (conservative) <math alttext="1\,\sigma" class="ltx_Math" display="inline" id="S5.F9.11.m5.1"><semantics id="S5.F9.11.m5.1b"><mrow id="S5.F9.11.m5.1.1" xref="S5.F9.11.m5.1.1.cmml"><mn id="S5.F9.11.m5.1.1.2" xref="S5.F9.11.m5.1.1.2.cmml">1</mn><mo id="S5.F9.11.m5.1.1.1" lspace="0.170em" xref="S5.F9.11.m5.1.1.1.cmml">⁢</mo><mi id="S5.F9.11.m5.1.1.3" xref="S5.F9.11.m5.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.F9.11.m5.1c"><apply id="S5.F9.11.m5.1.1.cmml" xref="S5.F9.11.m5.1.1"><times id="S5.F9.11.m5.1.1.1.cmml" xref="S5.F9.11.m5.1.1.1"></times><cn id="S5.F9.11.m5.1.1.2.cmml" type="integer" xref="S5.F9.11.m5.1.1.2">1</cn><ci id="S5.F9.11.m5.1.1.3.cmml" xref="S5.F9.11.m5.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F9.11.m5.1d">1\,\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.F9.11.m5.1e">1 italic_σ</annotation></semantics></math> uncertainty. The gray dashed line marks the upper limit of <math alttext="M_{\nu}" class="ltx_Math" display="inline" id="S5.F9.12.m6.1"><semantics id="S5.F9.12.m6.1b"><msub id="S5.F9.12.m6.1.1" xref="S5.F9.12.m6.1.1.cmml"><mi id="S5.F9.12.m6.1.1.2" xref="S5.F9.12.m6.1.1.2.cmml">M</mi><mi id="S5.F9.12.m6.1.1.3" xref="S5.F9.12.m6.1.1.3.cmml">ν</mi></msub><annotation-xml encoding="MathML-Content" id="S5.F9.12.m6.1c"><apply id="S5.F9.12.m6.1.1.cmml" xref="S5.F9.12.m6.1.1"><csymbol cd="ambiguous" id="S5.F9.12.m6.1.1.1.cmml" xref="S5.F9.12.m6.1.1">subscript</csymbol><ci id="S5.F9.12.m6.1.1.2.cmml" xref="S5.F9.12.m6.1.1.2">𝑀</ci><ci id="S5.F9.12.m6.1.1.3.cmml" xref="S5.F9.12.m6.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F9.12.m6.1d">M_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S5.F9.12.m6.1e">italic_M start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> by combining the CMB and BAO data from Planck <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib293" title="">293</a>]</cite>.</figcaption> </figure> </section> <section class="ltx_subsection" id="S5.SS6"> <h3 class="ltx_title ltx_title_subsection"> 5.6 Warm Dark Matter</h3> <div class="ltx_para" id="S5.SS6.p1"> The Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS6.p1.1.m1.1"><semantics id="S5.SS6.p1.1.m1.1a"><mi id="S5.SS6.p1.1.m1.1.1" xref="S5.SS6.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS6.p1.1.m1.1b"><ci id="S5.SS6.p1.1.m1.1.1.cmml" xref="S5.SS6.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p1.1.m1.1d">italic_α</annotation></semantics></math> forest is a powerful tool for probing small-scale structures and constraining dark matter models with small-scale cutoffs. When accounting for the repercussions of inhomogeneous H I reionization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib149" title="">149</a>]</cite>, the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS6.p1.2.m2.1"><semantics id="S5.SS6.p1.2.m2.1a"><mi id="S5.SS6.p1.2.m2.1.1" xref="S5.SS6.p1.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS6.p1.2.m2.1b"><ci id="S5.SS6.p1.2.m2.1.1.cmml" xref="S5.SS6.p1.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p1.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p1.2.m2.1d">italic_α</annotation></semantics></math> forest power spectrum can differentiate between various dark matter models even on large scales (<math alttext="k\lesssim 0.6\," class="ltx_Math" display="inline" id="S5.SS6.p1.3.m3.1"><semantics id="S5.SS6.p1.3.m3.1a"><mrow id="S5.SS6.p1.3.m3.1.1" xref="S5.SS6.p1.3.m3.1.1.cmml"><mi id="S5.SS6.p1.3.m3.1.1.2" xref="S5.SS6.p1.3.m3.1.1.2.cmml">k</mi><mo id="S5.SS6.p1.3.m3.1.1.1" xref="S5.SS6.p1.3.m3.1.1.1.cmml">≲</mo><mn id="S5.SS6.p1.3.m3.1.1.3" xref="S5.SS6.p1.3.m3.1.1.3.cmml">0.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p1.3.m3.1b"><apply id="S5.SS6.p1.3.m3.1.1.cmml" xref="S5.SS6.p1.3.m3.1.1"><csymbol cd="latexml" id="S5.SS6.p1.3.m3.1.1.1.cmml" xref="S5.SS6.p1.3.m3.1.1.1">less-than-or-similar-to</csymbol><ci id="S5.SS6.p1.3.m3.1.1.2.cmml" xref="S5.SS6.p1.3.m3.1.1.2">𝑘</ci><cn id="S5.SS6.p1.3.m3.1.1.3.cmml" type="float" xref="S5.SS6.p1.3.m3.1.1.3">0.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p1.3.m3.1c">k\lesssim 0.6\,</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p1.3.m3.1d">italic_k ≲ 0.6</annotation></semantics></math><math alttext="h{\rm Mpc^{-1}}" class="ltx_Math" display="inline" id="S5.SS6.p1.4.m4.1"><semantics id="S5.SS6.p1.4.m4.1a"><mrow id="S5.SS6.p1.4.m4.1.1" xref="S5.SS6.p1.4.m4.1.1.cmml"><mi id="S5.SS6.p1.4.m4.1.1.2" xref="S5.SS6.p1.4.m4.1.1.2.cmml">h</mi><mo id="S5.SS6.p1.4.m4.1.1.1" xref="S5.SS6.p1.4.m4.1.1.1.cmml">⁢</mo><msup id="S5.SS6.p1.4.m4.1.1.3" xref="S5.SS6.p1.4.m4.1.1.3.cmml"><mi id="S5.SS6.p1.4.m4.1.1.3.2" xref="S5.SS6.p1.4.m4.1.1.3.2.cmml">Mpc</mi><mrow id="S5.SS6.p1.4.m4.1.1.3.3" xref="S5.SS6.p1.4.m4.1.1.3.3.cmml"><mo id="S5.SS6.p1.4.m4.1.1.3.3a" xref="S5.SS6.p1.4.m4.1.1.3.3.cmml">−</mo><mn id="S5.SS6.p1.4.m4.1.1.3.3.2" xref="S5.SS6.p1.4.m4.1.1.3.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p1.4.m4.1b"><apply id="S5.SS6.p1.4.m4.1.1.cmml" xref="S5.SS6.p1.4.m4.1.1"><times id="S5.SS6.p1.4.m4.1.1.1.cmml" xref="S5.SS6.p1.4.m4.1.1.1"></times><ci id="S5.SS6.p1.4.m4.1.1.2.cmml" xref="S5.SS6.p1.4.m4.1.1.2">ℎ</ci><apply id="S5.SS6.p1.4.m4.1.1.3.cmml" xref="S5.SS6.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS6.p1.4.m4.1.1.3.1.cmml" xref="S5.SS6.p1.4.m4.1.1.3">superscript</csymbol><ci id="S5.SS6.p1.4.m4.1.1.3.2.cmml" xref="S5.SS6.p1.4.m4.1.1.3.2">Mpc</ci><apply id="S5.SS6.p1.4.m4.1.1.3.3.cmml" xref="S5.SS6.p1.4.m4.1.1.3.3"><minus id="S5.SS6.p1.4.m4.1.1.3.3.1.cmml" xref="S5.SS6.p1.4.m4.1.1.3.3"></minus><cn id="S5.SS6.p1.4.m4.1.1.3.3.2.cmml" type="integer" xref="S5.SS6.p1.4.m4.1.1.3.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p1.4.m4.1c">h{\rm Mpc^{-1}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p1.4.m4.1d">italic_h roman_Mpc start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>) (Zhang et al. in prep.). Here, we focus on warm dark matter models whose characteristic suppression of small-scale structure mirrors that of other dark matter models with similar small-scale cutoffs. We forecast the ability of MUST to place competitive constraints on warm dark matter particle mass by leveraging the reionization imprints in the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS6.p1.5.m5.1"><semantics id="S5.SS6.p1.5.m5.1a"><mi id="S5.SS6.p1.5.m5.1.1" xref="S5.SS6.p1.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS6.p1.5.m5.1b"><ci id="S5.SS6.p1.5.m5.1.1.cmml" xref="S5.SS6.p1.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p1.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p1.5.m5.1d">italic_α</annotation></semantics></math> forest power spectrum. Following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib149" title="">149</a>]</cite>, the flux power spectrum is given by </div> <div class="ltx_para" id="S5.SS6.p2"> <table class="ltx_equation ltx_eqn_table" id="S5.E20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P^{\rm 3D}_{\rm F}(\bm{k},z,m_{\rm X})=b^{2}_{\rm F}(1+\beta_{\rm F}\mu^{2})^{% 2}P_{m}+2b_{\rm F}b_{\Gamma}(1+\beta_{\rm F}\mu^{2})P_{m,\psi}," class="ltx_Math" display="block" id="S5.E20.m1.5"><semantics id="S5.E20.m1.5a"><mrow id="S5.E20.m1.5.5.1" xref="S5.E20.m1.5.5.1.1.cmml"><mrow id="S5.E20.m1.5.5.1.1" xref="S5.E20.m1.5.5.1.1.cmml"><mrow id="S5.E20.m1.5.5.1.1.1" xref="S5.E20.m1.5.5.1.1.1.cmml"><msubsup id="S5.E20.m1.5.5.1.1.1.3" xref="S5.E20.m1.5.5.1.1.1.3.cmml"><mi id="S5.E20.m1.5.5.1.1.1.3.2.2" 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xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.1"></times><apply id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2.cmml" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2.1.cmml" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2">subscript</csymbol><ci id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2.2.cmml" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2.2">𝛽</ci><ci id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2.3.cmml" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.2.3">F</ci></apply><apply id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3.cmml" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3"><csymbol cd="ambiguous" id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3.1.cmml" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3">superscript</csymbol><ci id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3.2.cmml" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3.2">𝜇</ci><cn id="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3.3.cmml" type="integer" xref="S5.E20.m1.5.5.1.1.3.2.1.1.1.3.3.3">2</cn></apply></apply></apply><apply id="S5.E20.m1.5.5.1.1.3.2.6.cmml" xref="S5.E20.m1.5.5.1.1.3.2.6"><csymbol cd="ambiguous" id="S5.E20.m1.5.5.1.1.3.2.6.1.cmml" xref="S5.E20.m1.5.5.1.1.3.2.6">subscript</csymbol><ci id="S5.E20.m1.5.5.1.1.3.2.6.2.cmml" xref="S5.E20.m1.5.5.1.1.3.2.6.2">𝑃</ci><list id="S5.E20.m1.2.2.2.3.cmml" xref="S5.E20.m1.2.2.2.4"><ci id="S5.E20.m1.1.1.1.1.cmml" xref="S5.E20.m1.1.1.1.1">𝑚</ci><ci id="S5.E20.m1.2.2.2.2.cmml" xref="S5.E20.m1.2.2.2.2">𝜓</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E20.m1.5c">P^{\rm 3D}_{\rm F}(\bm{k},z,m_{\rm X})=b^{2}_{\rm F}(1+\beta_{\rm F}\mu^{2})^{% 2}P_{m}+2b_{\rm F}b_{\Gamma}(1+\beta_{\rm F}\mu^{2})P_{m,\psi},</annotation><annotation encoding="application/x-llamapun" id="S5.E20.m1.5d">italic_P start_POSTSUPERSCRIPT 3 roman_D end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT ( bold_italic_k , italic_z , italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT ) = italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT ( 1 + italic_β start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + 2 italic_b start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT ( 1 + italic_β start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_P start_POSTSUBSCRIPT italic_m , italic_ψ end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(20)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.SS6.p3"> where <math alttext="b_{\rm F}" class="ltx_Math" display="inline" id="S5.SS6.p3.1.m1.1"><semantics id="S5.SS6.p3.1.m1.1a"><msub id="S5.SS6.p3.1.m1.1.1" xref="S5.SS6.p3.1.m1.1.1.cmml"><mi id="S5.SS6.p3.1.m1.1.1.2" xref="S5.SS6.p3.1.m1.1.1.2.cmml">b</mi><mi id="S5.SS6.p3.1.m1.1.1.3" mathvariant="normal" xref="S5.SS6.p3.1.m1.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p3.1.m1.1b"><apply id="S5.SS6.p3.1.m1.1.1.cmml" xref="S5.SS6.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS6.p3.1.m1.1.1.1.cmml" xref="S5.SS6.p3.1.m1.1.1">subscript</csymbol><ci id="S5.SS6.p3.1.m1.1.1.2.cmml" xref="S5.SS6.p3.1.m1.1.1.2">𝑏</ci><ci id="S5.SS6.p3.1.m1.1.1.3.cmml" xref="S5.SS6.p3.1.m1.1.1.3">F</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p3.1.m1.1c">b_{\rm F}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p3.1.m1.1d">italic_b start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\beta_{\rm F}" class="ltx_Math" display="inline" id="S5.SS6.p3.2.m2.1"><semantics id="S5.SS6.p3.2.m2.1a"><msub id="S5.SS6.p3.2.m2.1.1" xref="S5.SS6.p3.2.m2.1.1.cmml"><mi id="S5.SS6.p3.2.m2.1.1.2" xref="S5.SS6.p3.2.m2.1.1.2.cmml">β</mi><mi id="S5.SS6.p3.2.m2.1.1.3" mathvariant="normal" xref="S5.SS6.p3.2.m2.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p3.2.m2.1b"><apply id="S5.SS6.p3.2.m2.1.1.cmml" xref="S5.SS6.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS6.p3.2.m2.1.1.1.cmml" xref="S5.SS6.p3.2.m2.1.1">subscript</csymbol><ci id="S5.SS6.p3.2.m2.1.1.2.cmml" xref="S5.SS6.p3.2.m2.1.1.2">𝛽</ci><ci id="S5.SS6.p3.2.m2.1.1.3.cmml" xref="S5.SS6.p3.2.m2.1.1.3">F</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p3.2.m2.1c">\beta_{\rm F}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p3.2.m2.1d">italic_β start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT</annotation></semantics></math> are the bias coefficient and redshift-space distortion parameters, respectively. <math alttext="b_{\Gamma}" class="ltx_Math" display="inline" id="S5.SS6.p3.3.m3.1"><semantics id="S5.SS6.p3.3.m3.1a"><msub id="S5.SS6.p3.3.m3.1.1" xref="S5.SS6.p3.3.m3.1.1.cmml"><mi id="S5.SS6.p3.3.m3.1.1.2" xref="S5.SS6.p3.3.m3.1.1.2.cmml">b</mi><mi id="S5.SS6.p3.3.m3.1.1.3" mathvariant="normal" xref="S5.SS6.p3.3.m3.1.1.3.cmml">Γ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p3.3.m3.1b"><apply id="S5.SS6.p3.3.m3.1.1.cmml" xref="S5.SS6.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS6.p3.3.m3.1.1.1.cmml" xref="S5.SS6.p3.3.m3.1.1">subscript</csymbol><ci id="S5.SS6.p3.3.m3.1.1.2.cmml" xref="S5.SS6.p3.3.m3.1.1.2">𝑏</ci><ci id="S5.SS6.p3.3.m3.1.1.3.cmml" xref="S5.SS6.p3.3.m3.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p3.3.m3.1c">b_{\Gamma}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p3.3.m3.1d">italic_b start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT</annotation></semantics></math> is the radiation bias <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib294" title="">294</a>]</cite>, <math alttext="\mu\equiv k_{\parallel}/k" class="ltx_Math" display="inline" id="S5.SS6.p3.4.m4.1"><semantics id="S5.SS6.p3.4.m4.1a"><mrow id="S5.SS6.p3.4.m4.1.1" xref="S5.SS6.p3.4.m4.1.1.cmml"><mi id="S5.SS6.p3.4.m4.1.1.2" xref="S5.SS6.p3.4.m4.1.1.2.cmml">μ</mi><mo id="S5.SS6.p3.4.m4.1.1.1" xref="S5.SS6.p3.4.m4.1.1.1.cmml">≡</mo><mrow id="S5.SS6.p3.4.m4.1.1.3" xref="S5.SS6.p3.4.m4.1.1.3.cmml"><msub id="S5.SS6.p3.4.m4.1.1.3.2" xref="S5.SS6.p3.4.m4.1.1.3.2.cmml"><mi id="S5.SS6.p3.4.m4.1.1.3.2.2" xref="S5.SS6.p3.4.m4.1.1.3.2.2.cmml">k</mi><mo id="S5.SS6.p3.4.m4.1.1.3.2.3" xref="S5.SS6.p3.4.m4.1.1.3.2.3.cmml">∥</mo></msub><mo id="S5.SS6.p3.4.m4.1.1.3.1" xref="S5.SS6.p3.4.m4.1.1.3.1.cmml">/</mo><mi id="S5.SS6.p3.4.m4.1.1.3.3" xref="S5.SS6.p3.4.m4.1.1.3.3.cmml">k</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p3.4.m4.1b"><apply id="S5.SS6.p3.4.m4.1.1.cmml" xref="S5.SS6.p3.4.m4.1.1"><equivalent id="S5.SS6.p3.4.m4.1.1.1.cmml" xref="S5.SS6.p3.4.m4.1.1.1"></equivalent><ci id="S5.SS6.p3.4.m4.1.1.2.cmml" xref="S5.SS6.p3.4.m4.1.1.2">𝜇</ci><apply id="S5.SS6.p3.4.m4.1.1.3.cmml" xref="S5.SS6.p3.4.m4.1.1.3"><divide id="S5.SS6.p3.4.m4.1.1.3.1.cmml" xref="S5.SS6.p3.4.m4.1.1.3.1"></divide><apply id="S5.SS6.p3.4.m4.1.1.3.2.cmml" xref="S5.SS6.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS6.p3.4.m4.1.1.3.2.1.cmml" xref="S5.SS6.p3.4.m4.1.1.3.2">subscript</csymbol><ci id="S5.SS6.p3.4.m4.1.1.3.2.2.cmml" xref="S5.SS6.p3.4.m4.1.1.3.2.2">𝑘</ci><csymbol cd="latexml" id="S5.SS6.p3.4.m4.1.1.3.2.3.cmml" xref="S5.SS6.p3.4.m4.1.1.3.2.3">parallel-to</csymbol></apply><ci id="S5.SS6.p3.4.m4.1.1.3.3.cmml" xref="S5.SS6.p3.4.m4.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p3.4.m4.1c">\mu\equiv k_{\parallel}/k</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p3.4.m4.1d">italic_μ ≡ italic_k start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT / italic_k</annotation></semantics></math>, and <math alttext="P_{m,\psi}" class="ltx_Math" display="inline" id="S5.SS6.p3.5.m5.2"><semantics id="S5.SS6.p3.5.m5.2a"><msub id="S5.SS6.p3.5.m5.2.3" xref="S5.SS6.p3.5.m5.2.3.cmml"><mi id="S5.SS6.p3.5.m5.2.3.2" xref="S5.SS6.p3.5.m5.2.3.2.cmml">P</mi><mrow id="S5.SS6.p3.5.m5.2.2.2.4" xref="S5.SS6.p3.5.m5.2.2.2.3.cmml"><mi id="S5.SS6.p3.5.m5.1.1.1.1" xref="S5.SS6.p3.5.m5.1.1.1.1.cmml">m</mi><mo id="S5.SS6.p3.5.m5.2.2.2.4.1" xref="S5.SS6.p3.5.m5.2.2.2.3.cmml">,</mo><mi id="S5.SS6.p3.5.m5.2.2.2.2" xref="S5.SS6.p3.5.m5.2.2.2.2.cmml">ψ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p3.5.m5.2b"><apply id="S5.SS6.p3.5.m5.2.3.cmml" xref="S5.SS6.p3.5.m5.2.3"><csymbol cd="ambiguous" id="S5.SS6.p3.5.m5.2.3.1.cmml" xref="S5.SS6.p3.5.m5.2.3">subscript</csymbol><ci id="S5.SS6.p3.5.m5.2.3.2.cmml" xref="S5.SS6.p3.5.m5.2.3.2">𝑃</ci><list id="S5.SS6.p3.5.m5.2.2.2.3.cmml" xref="S5.SS6.p3.5.m5.2.2.2.4"><ci id="S5.SS6.p3.5.m5.1.1.1.1.cmml" xref="S5.SS6.p3.5.m5.1.1.1.1">𝑚</ci><ci id="S5.SS6.p3.5.m5.2.2.2.2.cmml" xref="S5.SS6.p3.5.m5.2.2.2.2">𝜓</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p3.5.m5.2c">P_{m,\psi}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p3.5.m5.2d">italic_P start_POSTSUBSCRIPT italic_m , italic_ψ end_POSTSUBSCRIPT</annotation></semantics></math> is the cross-power spectrum of matter and transparency of the IGM, which correspond to the response of the IGM to reionization. </div> <div class="ltx_para" id="S5.SS6.p4"> The second term in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.E20" title="Equation 20 ‣ 5.6 Warm Dark Matter ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">20</a>) enables us to relate the small-scale suppression caused by warm dark matter models during reionization to a distinct large-scale signature, i.e., the impact of inhomogeneous reionization on the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS6.p4.1.m1.1"><semantics id="S5.SS6.p4.1.m1.1a"><mi id="S5.SS6.p4.1.m1.1.1" xref="S5.SS6.p4.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS6.p4.1.m1.1b"><ci id="S5.SS6.p4.1.m1.1.1.cmml" xref="S5.SS6.p4.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p4.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p4.1.m1.1d">italic_α</annotation></semantics></math>. These reionization relics are coupled to the scale of the ionized bubbles, spanning a few Mpc to tens of Mpc. On such large scales, the dependence of the bias coefficients and the need for non-linear modeling concerning <math alttext="m_{\rm X}" class="ltx_Math" display="inline" id="S5.SS6.p4.2.m2.1"><semantics id="S5.SS6.p4.2.m2.1a"><msub id="S5.SS6.p4.2.m2.1.1" xref="S5.SS6.p4.2.m2.1.1.cmml"><mi id="S5.SS6.p4.2.m2.1.1.2" xref="S5.SS6.p4.2.m2.1.1.2.cmml">m</mi><mi id="S5.SS6.p4.2.m2.1.1.3" mathvariant="normal" xref="S5.SS6.p4.2.m2.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p4.2.m2.1b"><apply id="S5.SS6.p4.2.m2.1.1.cmml" xref="S5.SS6.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS6.p4.2.m2.1.1.1.cmml" xref="S5.SS6.p4.2.m2.1.1">subscript</csymbol><ci id="S5.SS6.p4.2.m2.1.1.2.cmml" xref="S5.SS6.p4.2.m2.1.1.2">𝑚</ci><ci id="S5.SS6.p4.2.m2.1.1.3.cmml" xref="S5.SS6.p4.2.m2.1.1.3">X</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p4.2.m2.1c">m_{\rm X}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p4.2.m2.1d">italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT</annotation></semantics></math> can be reasonably disregarded. </div> <div class="ltx_para" id="S5.SS6.p5"> We use Monte Carlo Markov Chain (MCMC) implemented with emcee <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib295" title="">295</a>]</cite> to forecast the constraints on warm dark matter mass <math alttext="m_{\rm X}" class="ltx_Math" display="inline" id="S5.SS6.p5.1.m1.1"><semantics id="S5.SS6.p5.1.m1.1a"><msub id="S5.SS6.p5.1.m1.1.1" xref="S5.SS6.p5.1.m1.1.1.cmml"><mi id="S5.SS6.p5.1.m1.1.1.2" xref="S5.SS6.p5.1.m1.1.1.2.cmml">m</mi><mi id="S5.SS6.p5.1.m1.1.1.3" mathvariant="normal" xref="S5.SS6.p5.1.m1.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.1.m1.1b"><apply id="S5.SS6.p5.1.m1.1.1.cmml" xref="S5.SS6.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS6.p5.1.m1.1.1.1.cmml" xref="S5.SS6.p5.1.m1.1.1">subscript</csymbol><ci id="S5.SS6.p5.1.m1.1.1.2.cmml" xref="S5.SS6.p5.1.m1.1.1.2">𝑚</ci><ci id="S5.SS6.p5.1.m1.1.1.3.cmml" xref="S5.SS6.p5.1.m1.1.1.3">X</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.1.m1.1c">m_{\rm X}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.1.m1.1d">italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT</annotation></semantics></math> by MUST. In addition to <math alttext="m_{\rm X}" class="ltx_Math" display="inline" id="S5.SS6.p5.2.m2.1"><semantics id="S5.SS6.p5.2.m2.1a"><msub id="S5.SS6.p5.2.m2.1.1" xref="S5.SS6.p5.2.m2.1.1.cmml"><mi id="S5.SS6.p5.2.m2.1.1.2" xref="S5.SS6.p5.2.m2.1.1.2.cmml">m</mi><mi id="S5.SS6.p5.2.m2.1.1.3" mathvariant="normal" xref="S5.SS6.p5.2.m2.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.2.m2.1b"><apply id="S5.SS6.p5.2.m2.1.1.cmml" xref="S5.SS6.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS6.p5.2.m2.1.1.1.cmml" xref="S5.SS6.p5.2.m2.1.1">subscript</csymbol><ci id="S5.SS6.p5.2.m2.1.1.2.cmml" xref="S5.SS6.p5.2.m2.1.1.2">𝑚</ci><ci id="S5.SS6.p5.2.m2.1.1.3.cmml" xref="S5.SS6.p5.2.m2.1.1.3">X</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.2.m2.1c">m_{\rm X}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.2.m2.1d">italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT</annotation></semantics></math>, we include <math alttext="\sigma_{8}" class="ltx_Math" display="inline" id="S5.SS6.p5.3.m3.1"><semantics id="S5.SS6.p5.3.m3.1a"><msub id="S5.SS6.p5.3.m3.1.1" xref="S5.SS6.p5.3.m3.1.1.cmml"><mi id="S5.SS6.p5.3.m3.1.1.2" xref="S5.SS6.p5.3.m3.1.1.2.cmml">σ</mi><mn id="S5.SS6.p5.3.m3.1.1.3" xref="S5.SS6.p5.3.m3.1.1.3.cmml">8</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.3.m3.1b"><apply id="S5.SS6.p5.3.m3.1.1.cmml" xref="S5.SS6.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS6.p5.3.m3.1.1.1.cmml" xref="S5.SS6.p5.3.m3.1.1">subscript</csymbol><ci id="S5.SS6.p5.3.m3.1.1.2.cmml" xref="S5.SS6.p5.3.m3.1.1.2">𝜎</ci><cn id="S5.SS6.p5.3.m3.1.1.3.cmml" type="integer" xref="S5.SS6.p5.3.m3.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.3.m3.1c">\sigma_{8}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.3.m3.1d">italic_σ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT</annotation></semantics></math> as a free parameter to account for potential degeneracies. We consider 12 redshift bins from 2.0 to 3.8 and 45 <math alttext="k" class="ltx_Math" display="inline" id="S5.SS6.p5.4.m4.1"><semantics id="S5.SS6.p5.4.m4.1a"><mi id="S5.SS6.p5.4.m4.1.1" xref="S5.SS6.p5.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.4.m4.1b"><ci id="S5.SS6.p5.4.m4.1.1.cmml" xref="S5.SS6.p5.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.4.m4.1d">italic_k</annotation></semantics></math>-bins spanning 0.09 to 0.67 <math alttext="h\,{\rm Mpc^{-1}}" class="ltx_Math" display="inline" id="S5.SS6.p5.5.m5.1"><semantics id="S5.SS6.p5.5.m5.1a"><mrow id="S5.SS6.p5.5.m5.1.1" xref="S5.SS6.p5.5.m5.1.1.cmml"><mi id="S5.SS6.p5.5.m5.1.1.2" xref="S5.SS6.p5.5.m5.1.1.2.cmml">h</mi><mo id="S5.SS6.p5.5.m5.1.1.1" lspace="0.170em" xref="S5.SS6.p5.5.m5.1.1.1.cmml">⁢</mo><msup id="S5.SS6.p5.5.m5.1.1.3" xref="S5.SS6.p5.5.m5.1.1.3.cmml"><mi id="S5.SS6.p5.5.m5.1.1.3.2" xref="S5.SS6.p5.5.m5.1.1.3.2.cmml">Mpc</mi><mrow id="S5.SS6.p5.5.m5.1.1.3.3" xref="S5.SS6.p5.5.m5.1.1.3.3.cmml"><mo id="S5.SS6.p5.5.m5.1.1.3.3a" xref="S5.SS6.p5.5.m5.1.1.3.3.cmml">−</mo><mn id="S5.SS6.p5.5.m5.1.1.3.3.2" xref="S5.SS6.p5.5.m5.1.1.3.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.5.m5.1b"><apply id="S5.SS6.p5.5.m5.1.1.cmml" xref="S5.SS6.p5.5.m5.1.1"><times id="S5.SS6.p5.5.m5.1.1.1.cmml" xref="S5.SS6.p5.5.m5.1.1.1"></times><ci id="S5.SS6.p5.5.m5.1.1.2.cmml" xref="S5.SS6.p5.5.m5.1.1.2">ℎ</ci><apply id="S5.SS6.p5.5.m5.1.1.3.cmml" xref="S5.SS6.p5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS6.p5.5.m5.1.1.3.1.cmml" xref="S5.SS6.p5.5.m5.1.1.3">superscript</csymbol><ci id="S5.SS6.p5.5.m5.1.1.3.2.cmml" xref="S5.SS6.p5.5.m5.1.1.3.2">Mpc</ci><apply id="S5.SS6.p5.5.m5.1.1.3.3.cmml" xref="S5.SS6.p5.5.m5.1.1.3.3"><minus id="S5.SS6.p5.5.m5.1.1.3.3.1.cmml" xref="S5.SS6.p5.5.m5.1.1.3.3"></minus><cn id="S5.SS6.p5.5.m5.1.1.3.3.2.cmml" type="integer" xref="S5.SS6.p5.5.m5.1.1.3.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.5.m5.1c">h\,{\rm Mpc^{-1}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.5.m5.1d">italic_h roman_Mpc start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. We use an estimated quasar luminosity function for MUST with <math alttext="r<23.5" class="ltx_Math" display="inline" id="S5.SS6.p5.6.m6.1"><semantics id="S5.SS6.p5.6.m6.1a"><mrow id="S5.SS6.p5.6.m6.1.1" xref="S5.SS6.p5.6.m6.1.1.cmml"><mi id="S5.SS6.p5.6.m6.1.1.2" xref="S5.SS6.p5.6.m6.1.1.2.cmml">r</mi><mo id="S5.SS6.p5.6.m6.1.1.1" xref="S5.SS6.p5.6.m6.1.1.1.cmml"><</mo><mn id="S5.SS6.p5.6.m6.1.1.3" xref="S5.SS6.p5.6.m6.1.1.3.cmml">23.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.6.m6.1b"><apply id="S5.SS6.p5.6.m6.1.1.cmml" xref="S5.SS6.p5.6.m6.1.1"><lt id="S5.SS6.p5.6.m6.1.1.1.cmml" xref="S5.SS6.p5.6.m6.1.1.1"></lt><ci id="S5.SS6.p5.6.m6.1.1.2.cmml" xref="S5.SS6.p5.6.m6.1.1.2">𝑟</ci><cn id="S5.SS6.p5.6.m6.1.1.3.cmml" type="float" xref="S5.SS6.p5.6.m6.1.1.3">23.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.6.m6.1c">r<23.5</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.6.m6.1d">italic_r < 23.5</annotation></semantics></math> and consider different survey areas. We follow <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib296" title="">296</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib297" title="">297</a>]</cite> to compute the corresponding covariance, which includes an aliasing term that accounts for the sparse sampling of quasars. Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#S5.F10" title="Figure 10 ‣ 5.6 Warm Dark Matter ‣ 5 Cosmological Forecasts ‣ MUltiplexed Survey Telescope: Perspectives for Large-Scale Structure Cosmology in the Era of Stage-V Spectroscopic Survey">10</a> showcases the constraining power of MUST across three survey areas (5,000, 10,000, and 14,000 deg2). An optimistic survey area of 14,000 deg2 would yield the most stringent constraint on warm dark matter mass to date: <math alttext="m_{\rm X}>10.5" class="ltx_Math" display="inline" id="S5.SS6.p5.9.m9.1"><semantics id="S5.SS6.p5.9.m9.1a"><mrow id="S5.SS6.p5.9.m9.1.1" xref="S5.SS6.p5.9.m9.1.1.cmml"><msub id="S5.SS6.p5.9.m9.1.1.2" xref="S5.SS6.p5.9.m9.1.1.2.cmml"><mi id="S5.SS6.p5.9.m9.1.1.2.2" xref="S5.SS6.p5.9.m9.1.1.2.2.cmml">m</mi><mi id="S5.SS6.p5.9.m9.1.1.2.3" mathvariant="normal" xref="S5.SS6.p5.9.m9.1.1.2.3.cmml">X</mi></msub><mo id="S5.SS6.p5.9.m9.1.1.1" xref="S5.SS6.p5.9.m9.1.1.1.cmml">></mo><mn id="S5.SS6.p5.9.m9.1.1.3" xref="S5.SS6.p5.9.m9.1.1.3.cmml">10.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.9.m9.1b"><apply id="S5.SS6.p5.9.m9.1.1.cmml" xref="S5.SS6.p5.9.m9.1.1"><gt id="S5.SS6.p5.9.m9.1.1.1.cmml" xref="S5.SS6.p5.9.m9.1.1.1"></gt><apply id="S5.SS6.p5.9.m9.1.1.2.cmml" xref="S5.SS6.p5.9.m9.1.1.2"><csymbol cd="ambiguous" id="S5.SS6.p5.9.m9.1.1.2.1.cmml" xref="S5.SS6.p5.9.m9.1.1.2">subscript</csymbol><ci id="S5.SS6.p5.9.m9.1.1.2.2.cmml" xref="S5.SS6.p5.9.m9.1.1.2.2">𝑚</ci><ci id="S5.SS6.p5.9.m9.1.1.2.3.cmml" xref="S5.SS6.p5.9.m9.1.1.2.3">X</ci></apply><cn id="S5.SS6.p5.9.m9.1.1.3.cmml" type="float" xref="S5.SS6.p5.9.m9.1.1.3">10.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.9.m9.1c">m_{\rm X}>10.5</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.9.m9.1d">italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT > 10.5</annotation></semantics></math> keV (at 95 % confidence level). Even a conservative 10,000 deg2 survey would yield <math alttext="m_{\rm X}>7.6" class="ltx_Math" display="inline" id="S5.SS6.p5.11.m11.1"><semantics id="S5.SS6.p5.11.m11.1a"><mrow id="S5.SS6.p5.11.m11.1.1" xref="S5.SS6.p5.11.m11.1.1.cmml"><msub id="S5.SS6.p5.11.m11.1.1.2" xref="S5.SS6.p5.11.m11.1.1.2.cmml"><mi id="S5.SS6.p5.11.m11.1.1.2.2" xref="S5.SS6.p5.11.m11.1.1.2.2.cmml">m</mi><mi id="S5.SS6.p5.11.m11.1.1.2.3" mathvariant="normal" xref="S5.SS6.p5.11.m11.1.1.2.3.cmml">X</mi></msub><mo id="S5.SS6.p5.11.m11.1.1.1" xref="S5.SS6.p5.11.m11.1.1.1.cmml">></mo><mn id="S5.SS6.p5.11.m11.1.1.3" xref="S5.SS6.p5.11.m11.1.1.3.cmml">7.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.11.m11.1b"><apply id="S5.SS6.p5.11.m11.1.1.cmml" xref="S5.SS6.p5.11.m11.1.1"><gt id="S5.SS6.p5.11.m11.1.1.1.cmml" xref="S5.SS6.p5.11.m11.1.1.1"></gt><apply id="S5.SS6.p5.11.m11.1.1.2.cmml" xref="S5.SS6.p5.11.m11.1.1.2"><csymbol cd="ambiguous" id="S5.SS6.p5.11.m11.1.1.2.1.cmml" xref="S5.SS6.p5.11.m11.1.1.2">subscript</csymbol><ci id="S5.SS6.p5.11.m11.1.1.2.2.cmml" xref="S5.SS6.p5.11.m11.1.1.2.2">𝑚</ci><ci id="S5.SS6.p5.11.m11.1.1.2.3.cmml" xref="S5.SS6.p5.11.m11.1.1.2.3">X</ci></apply><cn id="S5.SS6.p5.11.m11.1.1.3.cmml" type="float" xref="S5.SS6.p5.11.m11.1.1.3">7.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.11.m11.1c">m_{\rm X}>7.6</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.11.m11.1d">italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT > 7.6</annotation></semantics></math> keV, exceeding the current strongest constraint from the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS6.p5.12.m12.1"><semantics id="S5.SS6.p5.12.m12.1a"><mi id="S5.SS6.p5.12.m12.1.1" xref="S5.SS6.p5.12.m12.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.12.m12.1b"><ci id="S5.SS6.p5.12.m12.1.1.cmml" xref="S5.SS6.p5.12.m12.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.12.m12.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.12.m12.1d">italic_α</annotation></semantics></math> forest alone, <math alttext="m_{\rm X}>5.7" class="ltx_Math" display="inline" id="S5.SS6.p5.13.m13.1"><semantics id="S5.SS6.p5.13.m13.1a"><mrow id="S5.SS6.p5.13.m13.1.1" xref="S5.SS6.p5.13.m13.1.1.cmml"><msub id="S5.SS6.p5.13.m13.1.1.2" xref="S5.SS6.p5.13.m13.1.1.2.cmml"><mi id="S5.SS6.p5.13.m13.1.1.2.2" xref="S5.SS6.p5.13.m13.1.1.2.2.cmml">m</mi><mi id="S5.SS6.p5.13.m13.1.1.2.3" mathvariant="normal" xref="S5.SS6.p5.13.m13.1.1.2.3.cmml">X</mi></msub><mo id="S5.SS6.p5.13.m13.1.1.1" xref="S5.SS6.p5.13.m13.1.1.1.cmml">></mo><mn id="S5.SS6.p5.13.m13.1.1.3" xref="S5.SS6.p5.13.m13.1.1.3.cmml">5.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS6.p5.13.m13.1b"><apply id="S5.SS6.p5.13.m13.1.1.cmml" xref="S5.SS6.p5.13.m13.1.1"><gt id="S5.SS6.p5.13.m13.1.1.1.cmml" xref="S5.SS6.p5.13.m13.1.1.1"></gt><apply id="S5.SS6.p5.13.m13.1.1.2.cmml" xref="S5.SS6.p5.13.m13.1.1.2"><csymbol cd="ambiguous" id="S5.SS6.p5.13.m13.1.1.2.1.cmml" xref="S5.SS6.p5.13.m13.1.1.2">subscript</csymbol><ci id="S5.SS6.p5.13.m13.1.1.2.2.cmml" xref="S5.SS6.p5.13.m13.1.1.2.2">𝑚</ci><ci id="S5.SS6.p5.13.m13.1.1.2.3.cmml" xref="S5.SS6.p5.13.m13.1.1.2.3">X</ci></apply><cn id="S5.SS6.p5.13.m13.1.1.3.cmml" type="float" xref="S5.SS6.p5.13.m13.1.1.3">5.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS6.p5.13.m13.1c">m_{\rm X}>5.7</annotation><annotation encoding="application/x-llamapun" id="S5.SS6.p5.13.m13.1d">italic_m start_POSTSUBSCRIPT roman_X end_POSTSUBSCRIPT > 5.7</annotation></semantics></math> keV <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib146" title="">146</a>]</cite>. </div> <figure class="ltx_figure" id="S5.F10"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="479" id="S5.F10.g1" src="x6.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 10: Warm dark matter constraints. Arrows indicate the allowed parameter space (not the associated error), which extends towards infinity, representing CDM. Forecasted MUST constraints using the Ly<math alttext="\alpha" class="ltx_Math" display="inline" id="S5.F10.5.m1.1"><semantics id="S5.F10.5.m1.1b"><mi id="S5.F10.5.m1.1.1" xref="S5.F10.5.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.F10.5.m1.1c"><ci id="S5.F10.5.m1.1.1.cmml" xref="S5.F10.5.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F10.5.m1.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.F10.5.m1.1e">italic_α</annotation></semantics></math> forest for the warm dark matter mass are highlighted by stars for different survey areas. For context, we include representative constraints from MW stellar streams <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib298" title="">298</a>]</cite>, MW satellite counts <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib299" title="">299</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib300" title="">300</a>]</cite>, Lyman-<math alttext="\alpha" class="ltx_Math" display="inline" id="S5.F10.6.m2.1"><semantics id="S5.F10.6.m2.1b"><mi id="S5.F10.6.m2.1.1" xref="S5.F10.6.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.F10.6.m2.1c"><ci id="S5.F10.6.m2.1.1.cmml" xref="S5.F10.6.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F10.6.m2.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.F10.6.m2.1e">italic_α</annotation></semantics></math> forest measured by eBOSS <math alttext="+" class="ltx_Math" display="inline" id="S5.F10.7.m3.1"><semantics id="S5.F10.7.m3.1b"><mo id="S5.F10.7.m3.1.1" xref="S5.F10.7.m3.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S5.F10.7.m3.1c"><plus id="S5.F10.7.m3.1.1.cmml" xref="S5.F10.7.m3.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S5.F10.7.m3.1d">+</annotation><annotation encoding="application/x-llamapun" id="S5.F10.7.m3.1e">+</annotation></semantics></math> XQ-100 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib145" title="">145</a>]</cite> and by HIRES <math alttext="+" class="ltx_Math" display="inline" id="S5.F10.8.m4.1"><semantics id="S5.F10.8.m4.1b"><mo id="S5.F10.8.m4.1.1" xref="S5.F10.8.m4.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S5.F10.8.m4.1c"><plus id="S5.F10.8.m4.1.1.cmml" xref="S5.F10.8.m4.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S5.F10.8.m4.1d">+</annotation><annotation encoding="application/x-llamapun" id="S5.F10.8.m4.1e">+</annotation></semantics></math> UVES <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib146" title="">146</a>]</cite>, as well as strongly lensed quasars from JWST <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib301" title="">301</a>]</cite>.</figcaption> </figure> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> 6 Conclusions</h2> <div class="ltx_para" id="S6.p1"> The MUltiplexed Survey Telescope (MUST) is a 6.5-meter telescope dedicated to multi-object spectroscopic observations. It features a modularized focal plane with 336 triangular modules that accommodate over 20,000 fibers at the Cassegrain focus. The spectrographs are designed with three channels to cover a wavelength range of <math alttext="\sim 3700" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mrow id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mi id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2.cmml"></mi><mo id="S6.p1.1.m1.1.1.1" xref="S6.p1.1.m1.1.1.1.cmml">∼</mo><mn id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml">3700</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><csymbol cd="latexml" id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">absent</csymbol><cn id="S6.p1.1.m1.1.1.3.cmml" type="integer" xref="S6.p1.1.m1.1.1.3">3700</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">\sim 3700</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">∼ 3700</annotation></semantics></math>–<math alttext="9600\,\AA" class="ltx_Math" display="inline" id="S6.p1.2.m2.1"><semantics id="S6.p1.2.m2.1a"><mrow id="S6.p1.2.m2.1.1" xref="S6.p1.2.m2.1.1.cmml"><mn id="S6.p1.2.m2.1.1.2" xref="S6.p1.2.m2.1.1.2.cmml">9600</mn><mo id="S6.p1.2.m2.1.1.1" lspace="0.170em" xref="S6.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S6.p1.2.m2.1.1.3" xref="S6.p1.2.m2.1.1.3.cmml">Å</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.1b"><apply id="S6.p1.2.m2.1.1.cmml" xref="S6.p1.2.m2.1.1"><times id="S6.p1.2.m2.1.1.1.cmml" xref="S6.p1.2.m2.1.1.1"></times><cn id="S6.p1.2.m2.1.1.2.cmml" type="integer" xref="S6.p1.2.m2.1.1.2">9600</cn><ci id="S6.p1.2.m2.1.1.3.cmml" xref="S6.p1.2.m2.1.1.3">italic-Å</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.1c">9600\,\AA</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.1d">9600 italic_Å</annotation></semantics></math> at a resolution of <math alttext="R\sim 2000" class="ltx_Math" display="inline" id="S6.p1.3.m3.1"><semantics id="S6.p1.3.m3.1a"><mrow id="S6.p1.3.m3.1.1" xref="S6.p1.3.m3.1.1.cmml"><mi id="S6.p1.3.m3.1.1.2" xref="S6.p1.3.m3.1.1.2.cmml">R</mi><mo id="S6.p1.3.m3.1.1.1" xref="S6.p1.3.m3.1.1.1.cmml">∼</mo><mn id="S6.p1.3.m3.1.1.3" xref="S6.p1.3.m3.1.1.3.cmml">2000</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.3.m3.1b"><apply id="S6.p1.3.m3.1.1.cmml" xref="S6.p1.3.m3.1.1"><csymbol cd="latexml" id="S6.p1.3.m3.1.1.1.cmml" xref="S6.p1.3.m3.1.1.1">similar-to</csymbol><ci id="S6.p1.3.m3.1.1.2.cmml" xref="S6.p1.3.m3.1.1.2">𝑅</ci><cn id="S6.p1.3.m3.1.1.3.cmml" type="integer" xref="S6.p1.3.m3.1.1.3">2000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.3.m3.1c">R\sim 2000</annotation><annotation encoding="application/x-llamapun" id="S6.p1.3.m3.1d">italic_R ∼ 2000</annotation></semantics></math>–<math alttext="4000" class="ltx_Math" display="inline" id="S6.p1.4.m4.1"><semantics id="S6.p1.4.m4.1a"><mn id="S6.p1.4.m4.1.1" xref="S6.p1.4.m4.1.1.cmml">4000</mn><annotation-xml encoding="MathML-Content" id="S6.p1.4.m4.1b"><cn id="S6.p1.4.m4.1.1.cmml" type="integer" xref="S6.p1.4.m4.1.1">4000</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.4.m4.1c">4000</annotation><annotation encoding="application/x-llamapun" id="S6.p1.4.m4.1d">4000</annotation></semantics></math>. Initial monitoring of observing conditions indicates that the MUST candidate site, Peak A of the Saishiteng Mountain near Lenghu Town, could provide over 2,400 observing hours annually, making MUST at least 10 times more efficient than currently operating spectroscopic surveys. As a result, MUST is expected to collect over 100 million spectra in the 2030s. </div> <div class="ltx_para" id="S6.p2"> While designed as a flexible spectroscopic platform, the primary mission of MUST is to conduct the first Stage-V galaxy spectroscopic survey151515proposed by the Snowmass Cosmic Frontier report <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib75" title="">75</a>]</cite> as a continuation of the stages defined by the Dark Energy Task Force <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.07970v2#bib.bib15" title="">15</a>]</cite> to address fundamental questions in cosmology and physics. With its unprecedented photon collecting capability as a spectroscopic telescope, MUST will produce the first 3D map of galaxies spanning from the nearby Universe out to <math alttext="z\sim 5.5" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mrow id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml"><mi id="S6.p2.1.m1.1.1.2" xref="S6.p2.1.m1.1.1.2.cmml">z</mi><mo id="S6.p2.1.m1.1.1.1" xref="S6.p2.1.m1.1.1.1.cmml">∼</mo><mn id="S6.p2.1.m1.1.1.3" xref="S6.p2.1.m1.1.1.3.cmml">5.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><apply id="S6.p2.1.m1.1.1.cmml" xref="S6.p2.1.m1.1.1"><csymbol cd="latexml" id="S6.p2.1.m1.1.1.1.cmml" xref="S6.p2.1.m1.1.1.1">similar-to</csymbol><ci id="S6.p2.1.m1.1.1.2.cmml" xref="S6.p2.1.m1.1.1.2">𝑧</ci><cn id="S6.p2.1.m1.1.1.3.cmml" type="float" xref="S6.p2.1.m1.1.1.3">5.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">z\sim 5.5</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">italic_z ∼ 5.5</annotation></semantics></math>, which corresponds to <math alttext="\sim 1" class="ltx_Math" display="inline" id="S6.p2.2.m2.1"><semantics id="S6.p2.2.m2.1a"><mrow id="S6.p2.2.m2.1.1" xref="S6.p2.2.m2.1.1.cmml"><mi id="S6.p2.2.m2.1.1.2" xref="S6.p2.2.m2.1.1.2.cmml"></mi><mo id="S6.p2.2.m2.1.1.1" xref="S6.p2.2.m2.1.1.1.cmml">∼</mo><mn id="S6.p2.2.m2.1.1.3" xref="S6.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.2.m2.1b"><apply id="S6.p2.2.m2.1.1.cmml" xref="S6.p2.2.m2.1.1"><csymbol cd="latexml" id="S6.p2.2.m2.1.1.1.cmml" xref="S6.p2.2.m2.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S6.p2.2.m2.1.1.2.cmml" xref="S6.p2.2.m2.1.1.2">absent</csymbol><cn id="S6.p2.2.m2.1.1.3.cmml" type="integer" xref="S6.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.2.m2.1c">\sim 1</annotation><annotation encoding="application/x-llamapun" id="S6.p2.2.m2.1d">∼ 1</annotation></semantics></math> billion years after the Big Bang. This large survey volume enables in-depth studies of the dynamic evolution of the dark Universe and signals of fundamental physics in the primordial Universe. Thus, the primary scientific objectives of MUST include investigating the nature and evolution of dark energy, testing gravity theories, probing inflation physics through primordial non-Gaussianity, exploring properties of neutrinos and dark matter, etc. </div> <div class="ltx_para" id="S6.p3"> Different types of targets are required to cover the extensive redshift range with sufficient density to achieve these ambitious scientific goals. For the low-redshift Universe (<math alttext="z\lesssim 1.6" class="ltx_Math" display="inline" id="S6.p3.1.m1.1"><semantics id="S6.p3.1.m1.1a"><mrow id="S6.p3.1.m1.1.1" xref="S6.p3.1.m1.1.1.cmml"><mi id="S6.p3.1.m1.1.1.2" xref="S6.p3.1.m1.1.1.2.cmml">z</mi><mo id="S6.p3.1.m1.1.1.1" xref="S6.p3.1.m1.1.1.1.cmml">≲</mo><mn id="S6.p3.1.m1.1.1.3" xref="S6.p3.1.m1.1.1.3.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.1.m1.1b"><apply id="S6.p3.1.m1.1.1.cmml" xref="S6.p3.1.m1.1.1"><csymbol cd="latexml" id="S6.p3.1.m1.1.1.1.cmml" xref="S6.p3.1.m1.1.1.1">less-than-or-similar-to</csymbol><ci id="S6.p3.1.m1.1.1.2.cmml" xref="S6.p3.1.m1.1.1.2">𝑧</ci><cn id="S6.p3.1.m1.1.1.3.cmml" type="float" xref="S6.p3.1.m1.1.1.3">1.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.1.m1.1c">z\lesssim 1.6</annotation><annotation encoding="application/x-llamapun" id="S6.p3.1.m1.1d">italic_z ≲ 1.6</annotation></semantics></math>), MUST will use the same galaxy targets and similar selection criteria as DESI but will focus on fainter samplers to avoid redundant observations. Beyond <math alttext="z\sim 2" class="ltx_Math" display="inline" id="S6.p3.2.m2.1"><semantics id="S6.p3.2.m2.1a"><mrow id="S6.p3.2.m2.1.1" xref="S6.p3.2.m2.1.1.cmml"><mi id="S6.p3.2.m2.1.1.2" xref="S6.p3.2.m2.1.1.2.cmml">z</mi><mo id="S6.p3.2.m2.1.1.1" xref="S6.p3.2.m2.1.1.1.cmml">∼</mo><mn id="S6.p3.2.m2.1.1.3" xref="S6.p3.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.2.m2.1b"><apply id="S6.p3.2.m2.1.1.cmml" xref="S6.p3.2.m2.1.1"><csymbol cd="latexml" id="S6.p3.2.m2.1.1.1.cmml" xref="S6.p3.2.m2.1.1.1">similar-to</csymbol><ci id="S6.p3.2.m2.1.1.2.cmml" xref="S6.p3.2.m2.1.1.2">𝑧</ci><cn id="S6.p3.2.m2.1.1.3.cmml" type="integer" xref="S6.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.2.m2.1c">z\sim 2</annotation><annotation encoding="application/x-llamapun" id="S6.p3.2.m2.1d">italic_z ∼ 2</annotation></semantics></math>, MUST will mainly rely on LBG and LAEs to map large-scale structures. Although the actual target selection and redshift measurement efficiencies are not yet fully verified, current multi-band imaging and spectroscopic validation data in limited areas suggest that these tracers will be abundant enough for precise cosmological inferences. Meanwhile, QSOs will play a vital complementary role in probing the IGM, which is crucial for studying specific physical effects, such as the mass of dark matter particles. </div> <div class="ltx_para" id="S6.p4"> We have performed forecasts for some key scientific motivations of MUST. At low redshifts (<math alttext="z\lesssim 1.6" class="ltx_Math" display="inline" id="S6.p4.1.m1.1"><semantics id="S6.p4.1.m1.1a"><mrow id="S6.p4.1.m1.1.1" xref="S6.p4.1.m1.1.1.cmml"><mi id="S6.p4.1.m1.1.1.2" xref="S6.p4.1.m1.1.1.2.cmml">z</mi><mo id="S6.p4.1.m1.1.1.1" xref="S6.p4.1.m1.1.1.1.cmml">≲</mo><mn id="S6.p4.1.m1.1.1.3" xref="S6.p4.1.m1.1.1.3.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.1.m1.1b"><apply id="S6.p4.1.m1.1.1.cmml" xref="S6.p4.1.m1.1.1"><csymbol cd="latexml" id="S6.p4.1.m1.1.1.1.cmml" xref="S6.p4.1.m1.1.1.1">less-than-or-similar-to</csymbol><ci id="S6.p4.1.m1.1.1.2.cmml" xref="S6.p4.1.m1.1.1.2">𝑧</ci><cn id="S6.p4.1.m1.1.1.3.cmml" type="float" xref="S6.p4.1.m1.1.1.3">1.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.1.m1.1c">z\lesssim 1.6</annotation><annotation encoding="application/x-llamapun" id="S6.p4.1.m1.1d">italic_z ≲ 1.6</annotation></semantics></math>), the increased galaxy density provides stringent constraints on geometric measurements with percent-level precision for nearly all tracers, demonstrating the potential to distinguish between dark energy models. Meanwhile, the <math alttext="z\gtrsim 2" class="ltx_Math" display="inline" id="S6.p4.2.m2.1"><semantics id="S6.p4.2.m2.1a"><mrow id="S6.p4.2.m2.1.1" xref="S6.p4.2.m2.1.1.cmml"><mi id="S6.p4.2.m2.1.1.2" xref="S6.p4.2.m2.1.1.2.cmml">z</mi><mo id="S6.p4.2.m2.1.1.1" xref="S6.p4.2.m2.1.1.1.cmml">≳</mo><mn id="S6.p4.2.m2.1.1.3" xref="S6.p4.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.2.m2.1b"><apply id="S6.p4.2.m2.1.1.cmml" xref="S6.p4.2.m2.1.1"><csymbol cd="latexml" id="S6.p4.2.m2.1.1.1.cmml" xref="S6.p4.2.m2.1.1.1">greater-than-or-equivalent-to</csymbol><ci id="S6.p4.2.m2.1.1.2.cmml" xref="S6.p4.2.m2.1.1.2">𝑧</ci><cn id="S6.p4.2.m2.1.1.3.cmml" type="integer" xref="S6.p4.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.2.m2.1c">z\gtrsim 2</annotation><annotation encoding="application/x-llamapun" id="S6.p4.2.m2.1d">italic_z ≳ 2</annotation></semantics></math> galaxies significantly enhance the constraining power of structure growth measurements, reaching <math alttext="2\,\%" class="ltx_Math" display="inline" id="S6.p4.3.m3.1"><semantics id="S6.p4.3.m3.1a"><mrow id="S6.p4.3.m3.1.1" xref="S6.p4.3.m3.1.1.cmml"><mn id="S6.p4.3.m3.1.1.2" xref="S6.p4.3.m3.1.1.2.cmml">2</mn><mo id="S6.p4.3.m3.1.1.1" lspace="0.170em" xref="S6.p4.3.m3.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.3.m3.1b"><apply id="S6.p4.3.m3.1.1.cmml" xref="S6.p4.3.m3.1.1"><csymbol cd="latexml" id="S6.p4.3.m3.1.1.1.cmml" xref="S6.p4.3.m3.1.1.1">percent</csymbol><cn id="S6.p4.3.m3.1.1.2.cmml" type="integer" xref="S6.p4.3.m3.1.1.2">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.3.m3.1c">2\,\%</annotation><annotation encoding="application/x-llamapun" id="S6.p4.3.m3.1d">2 %</annotation></semantics></math>-level precision at most redshifts, thus enabling robust tests of gravity models. With the MUST data, we foresee that constraints on the primordial non-Gaussianity parameterized by <math alttext="f_{\rm NL}^{\rm local}" class="ltx_Math" display="inline" id="S6.p4.4.m4.1"><semantics id="S6.p4.4.m4.1a"><msubsup id="S6.p4.4.m4.1.1" xref="S6.p4.4.m4.1.1.cmml"><mi id="S6.p4.4.m4.1.1.2.2" xref="S6.p4.4.m4.1.1.2.2.cmml">f</mi><mi id="S6.p4.4.m4.1.1.2.3" xref="S6.p4.4.m4.1.1.2.3.cmml">NL</mi><mi id="S6.p4.4.m4.1.1.3" xref="S6.p4.4.m4.1.1.3.cmml">local</mi></msubsup><annotation-xml encoding="MathML-Content" id="S6.p4.4.m4.1b"><apply id="S6.p4.4.m4.1.1.cmml" xref="S6.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S6.p4.4.m4.1.1.1.cmml" xref="S6.p4.4.m4.1.1">superscript</csymbol><apply id="S6.p4.4.m4.1.1.2.cmml" xref="S6.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S6.p4.4.m4.1.1.2.1.cmml" xref="S6.p4.4.m4.1.1">subscript</csymbol><ci id="S6.p4.4.m4.1.1.2.2.cmml" xref="S6.p4.4.m4.1.1.2.2">𝑓</ci><ci id="S6.p4.4.m4.1.1.2.3.cmml" xref="S6.p4.4.m4.1.1.2.3">NL</ci></apply><ci id="S6.p4.4.m4.1.1.3.cmml" xref="S6.p4.4.m4.1.1.3">local</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.4.m4.1c">f_{\rm NL}^{\rm local}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.4.m4.1d">italic_f start_POSTSUBSCRIPT roman_NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_local end_POSTSUPERSCRIPT</annotation></semantics></math> from LSS data will surpass the precision of Planck measurements for the first time. Combined with CMB data from Planck and Simons Observatory, MUST could achieve <math alttext="f_{\rm NL}^{\rm local}" class="ltx_Math" display="inline" id="S6.p4.5.m5.1"><semantics id="S6.p4.5.m5.1a"><msubsup id="S6.p4.5.m5.1.1" xref="S6.p4.5.m5.1.1.cmml"><mi id="S6.p4.5.m5.1.1.2.2" xref="S6.p4.5.m5.1.1.2.2.cmml">f</mi><mi id="S6.p4.5.m5.1.1.2.3" xref="S6.p4.5.m5.1.1.2.3.cmml">NL</mi><mi id="S6.p4.5.m5.1.1.3" xref="S6.p4.5.m5.1.1.3.cmml">local</mi></msubsup><annotation-xml encoding="MathML-Content" id="S6.p4.5.m5.1b"><apply id="S6.p4.5.m5.1.1.cmml" xref="S6.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S6.p4.5.m5.1.1.1.cmml" xref="S6.p4.5.m5.1.1">superscript</csymbol><apply id="S6.p4.5.m5.1.1.2.cmml" xref="S6.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S6.p4.5.m5.1.1.2.1.cmml" xref="S6.p4.5.m5.1.1">subscript</csymbol><ci id="S6.p4.5.m5.1.1.2.2.cmml" xref="S6.p4.5.m5.1.1.2.2">𝑓</ci><ci id="S6.p4.5.m5.1.1.2.3.cmml" xref="S6.p4.5.m5.1.1.2.3">NL</ci></apply><ci id="S6.p4.5.m5.1.1.3.cmml" xref="S6.p4.5.m5.1.1.3">local</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.5.m5.1c">f_{\rm NL}^{\rm local}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.5.m5.1d">italic_f start_POSTSUBSCRIPT roman_NL end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_local end_POSTSUPERSCRIPT</annotation></semantics></math> measurements with precision on the order of unity, highlighting its potential in probing primordial physics. The combination of MUST and CMB data further gives a statistical error on the total neutrino mass of <math alttext="\sim 0.03\,{\rm eV}" class="ltx_Math" display="inline" id="S6.p4.6.m6.1"><semantics id="S6.p4.6.m6.1a"><mrow id="S6.p4.6.m6.1.1" xref="S6.p4.6.m6.1.1.cmml"><mi id="S6.p4.6.m6.1.1.2" xref="S6.p4.6.m6.1.1.2.cmml"></mi><mo id="S6.p4.6.m6.1.1.1" xref="S6.p4.6.m6.1.1.1.cmml">∼</mo><mrow id="S6.p4.6.m6.1.1.3" xref="S6.p4.6.m6.1.1.3.cmml"><mn id="S6.p4.6.m6.1.1.3.2" xref="S6.p4.6.m6.1.1.3.2.cmml">0.03</mn><mo id="S6.p4.6.m6.1.1.3.1" lspace="0.170em" xref="S6.p4.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S6.p4.6.m6.1.1.3.3" xref="S6.p4.6.m6.1.1.3.3.cmml">eV</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.6.m6.1b"><apply id="S6.p4.6.m6.1.1.cmml" xref="S6.p4.6.m6.1.1"><csymbol cd="latexml" id="S6.p4.6.m6.1.1.1.cmml" xref="S6.p4.6.m6.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S6.p4.6.m6.1.1.2.cmml" xref="S6.p4.6.m6.1.1.2">absent</csymbol><apply id="S6.p4.6.m6.1.1.3.cmml" xref="S6.p4.6.m6.1.1.3"><times id="S6.p4.6.m6.1.1.3.1.cmml" xref="S6.p4.6.m6.1.1.3.1"></times><cn id="S6.p4.6.m6.1.1.3.2.cmml" type="float" xref="S6.p4.6.m6.1.1.3.2">0.03</cn><ci id="S6.p4.6.m6.1.1.3.3.cmml" xref="S6.p4.6.m6.1.1.3.3">eV</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.6.m6.1c">\sim 0.03\,{\rm eV}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.6.m6.1d">∼ 0.03 roman_eV</annotation></semantics></math>, enabling evidence of non-zero neutrino mass confirmation. 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These forecasts demonstrate that, once completed, the MUST spectroscopic survey will contribute substantially to addressing fundamental cosmological and physical questions. </div> <div class="ltx_para" id="S6.p5"> \Acknowledgements The MUST project is supported by the Ministry of Science and Technology, China (Grant No. 2023YFA1605600) and the Ministry of Education, China. We want to express our sincere gratitude to Mr. Tian-Pei Chen for his generous donation and unwavering trust, which has significantly supported the MUST project. We also thank Mr. Dong-Hua Dong for his donation. The authors thanks Noah Sailer, Anand Raichoor, Ji Yao for useful discussions. Song Huang acknowledges the support from the National Natural Science Foundation of China (NSFC) Grant No. 12273015 & No. 12433003 and the China Crewed Space Program through its Space Application System. JPK, AR, DF, JY, AV, RG, and SH acknowledge the support from the SNF 200020_175751 and 200020_207379 “Cosmology with 3D Maps of the Universe” research grant. Si-Wei Zou acknowledges the support from the National Science Foundation of China (No. 12303011). Yu Liu acknowledges the support from the National Science Foundation of China (No. 12303005) and the Shuimu Tsinghua Scholar Program (No. 2022SM173). Pablo Renard acknowledges the support by the Tsinghua Shui Mu Scholarship, funding of the National Key R&D Program of China (grant no. 2023YFA1605600), the science research grants from the China Manned Space Project with No. CMS-CSST2021-A05, the Tsinghua University Initiative Scientific Research Program (No. 20223080023), and the National Science Foundation of China (grant no. 12350410365). </div> <div class="ltx_para" id="S6.p6"> \InterestConflict The authors declare that they have no conflict of interest. </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> [1] J. Huchra, M. Davis, D. Latham, and J. Tonry, ApJS52, 89 (1983). </li> <li class="ltx_bibitem" id="bib.bib2"> [2] V. de Lapparent, M. J. Geller, and J. P. Huchra, ApJ302, L1 (1986). </li> <li class="ltx_bibitem" id="bib.bib3"> [3] C. L. Bennett, D. Larson, J. L. 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