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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Comparison of dynamical dark energy with ΛCDM in light of DESI DR2</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/2503.17342v1/"/></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/2503.17342v1#S1" title="In Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S2" title="In Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Data</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S3" title="In Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Methods</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S4" title="In Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Results</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S5" title="In Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Conclusions</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_fleqn"> <h1 class="ltx_title ltx_title_document">Comparison of dynamical dark energy with <math alttext="\Lambda" class="ltx_Math" display="inline" id="m1.1"><semantics id="m1.1a"><mi id="m1.1.1" mathvariant="normal" xref="m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="m1.1b"><ci id="m1.1.1.cmml" xref="m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="m1.1d">roman_Λ</annotation></semantics></math>CDM in light of DESI DR2</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> A.N. Ormondroyd,<sup class="ltx_sup" id="16.7.1"><span class="ltx_text ltx_font_italic" id="16.7.1.1">1,2</span></sup> W.J. Handley,<sup class="ltx_sup" id="17.8.2"><span class="ltx_text ltx_font_italic" id="17.8.2.1">1,2</span></sup> M.P. Hobson<sup class="ltx_sup" id="18.9.3"><span class="ltx_text ltx_font_italic" id="18.9.3.1">1</span></sup> and A.N. Lasenby<sup class="ltx_sup" id="19.10.4"><span class="ltx_text ltx_font_italic" id="19.10.4.1">1,2</span></sup> <br class="ltx_break"/><sup class="ltx_sup" id="20.11.5"><span class="ltx_text ltx_font_italic" id="20.11.5.1">1</span></sup>Astrophysics Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge, CB3 0HE, UK <br class="ltx_break"/><sup class="ltx_sup" id="21.12.6"><span class="ltx_text ltx_font_italic" id="21.12.6.1">2</span></sup>Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK <br class="ltx_break"/> </span><span class="ltx_author_notes">E-mail: ano23@cam.ac.uk</span></span> </div> <div class="ltx_dates">(Accepted XXX. Received YYY; in original form ZZZ)</div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id15.8">We present an updated reconstruction of the dark energy equation of state, <math alttext="w(a)" class="ltx_Math" display="inline" id="id8.1.m1.1"><semantics id="id8.1.m1.1a"><mrow id="id8.1.m1.1.2" xref="id8.1.m1.1.2.cmml"><mi id="id8.1.m1.1.2.2" xref="id8.1.m1.1.2.2.cmml">w</mi><mo id="id8.1.m1.1.2.1" xref="id8.1.m1.1.2.1.cmml">⁢</mo><mrow id="id8.1.m1.1.2.3.2" xref="id8.1.m1.1.2.cmml"><mo id="id8.1.m1.1.2.3.2.1" stretchy="false" xref="id8.1.m1.1.2.cmml">(</mo><mi id="id8.1.m1.1.1" xref="id8.1.m1.1.1.cmml">a</mi><mo id="id8.1.m1.1.2.3.2.2" stretchy="false" xref="id8.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id8.1.m1.1b"><apply id="id8.1.m1.1.2.cmml" xref="id8.1.m1.1.2"><times id="id8.1.m1.1.2.1.cmml" xref="id8.1.m1.1.2.1"></times><ci id="id8.1.m1.1.2.2.cmml" xref="id8.1.m1.1.2.2">𝑤</ci><ci id="id8.1.m1.1.1.cmml" xref="id8.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.1.m1.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="id8.1.m1.1d">italic_w ( italic_a )</annotation></semantics></math>, using the newly released DESI DR2 Baryon Acoustic Oscillation (BAO) data in combination with Pantheon+ and DES5Y Type Ia supernovae measurements, respectively. Building on our previous analysis in <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite>, which employed a nonparametric flexknot reconstruction approach, we examine whether the evidence for dynamical dark energy persists with the improved precision of the DESI DR2 dataset. We find that while the overall qualitative structure of <math alttext="w(a)" class="ltx_Math" display="inline" id="id9.2.m2.1"><semantics id="id9.2.m2.1a"><mrow id="id9.2.m2.1.2" xref="id9.2.m2.1.2.cmml"><mi id="id9.2.m2.1.2.2" xref="id9.2.m2.1.2.2.cmml">w</mi><mo id="id9.2.m2.1.2.1" xref="id9.2.m2.1.2.1.cmml">⁢</mo><mrow id="id9.2.m2.1.2.3.2" xref="id9.2.m2.1.2.cmml"><mo id="id9.2.m2.1.2.3.2.1" stretchy="false" xref="id9.2.m2.1.2.cmml">(</mo><mi id="id9.2.m2.1.1" xref="id9.2.m2.1.1.cmml">a</mi><mo id="id9.2.m2.1.2.3.2.2" stretchy="false" xref="id9.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id9.2.m2.1b"><apply id="id9.2.m2.1.2.cmml" xref="id9.2.m2.1.2"><times id="id9.2.m2.1.2.1.cmml" xref="id9.2.m2.1.2.1"></times><ci id="id9.2.m2.1.2.2.cmml" xref="id9.2.m2.1.2.2">𝑤</ci><ci id="id9.2.m2.1.1.cmml" xref="id9.2.m2.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id9.2.m2.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="id9.2.m2.1d">italic_w ( italic_a )</annotation></semantics></math> remains consistent with our earlier findings, the statistical support for dynamical dark energy is reduced when considering DESI DR2 data alone, particularly for more complex flexknot models with higher numbers of knots. However, the evidence for simpler dynamical models, such as <math alttext="w" class="ltx_Math" display="inline" id="id10.3.m3.1"><semantics id="id10.3.m3.1a"><mi id="id10.3.m3.1.1" xref="id10.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="id10.3.m3.1b"><ci id="id10.3.m3.1.1.cmml" xref="id10.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="id10.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="id10.3.m3.1d">italic_w</annotation></semantics></math>CDM and CPL (which correspond to <math alttext="n=1" class="ltx_Math" display="inline" id="id11.4.m4.1"><semantics id="id11.4.m4.1a"><mrow id="id11.4.m4.1.1" xref="id11.4.m4.1.1.cmml"><mi id="id11.4.m4.1.1.2" xref="id11.4.m4.1.1.2.cmml">n</mi><mo id="id11.4.m4.1.1.1" xref="id11.4.m4.1.1.1.cmml">=</mo><mn id="id11.4.m4.1.1.3" xref="id11.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="id11.4.m4.1b"><apply id="id11.4.m4.1.1.cmml" xref="id11.4.m4.1.1"><eq id="id11.4.m4.1.1.1.cmml" xref="id11.4.m4.1.1.1"></eq><ci id="id11.4.m4.1.1.2.cmml" xref="id11.4.m4.1.1.2">𝑛</ci><cn id="id11.4.m4.1.1.3.cmml" type="integer" xref="id11.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id11.4.m4.1c">n=1</annotation><annotation encoding="application/x-llamapun" id="id11.4.m4.1d">italic_n = 1</annotation></semantics></math> and <math alttext="n=2" class="ltx_Math" display="inline" id="id12.5.m5.1"><semantics id="id12.5.m5.1a"><mrow id="id12.5.m5.1.1" xref="id12.5.m5.1.1.cmml"><mi id="id12.5.m5.1.1.2" xref="id12.5.m5.1.1.2.cmml">n</mi><mo id="id12.5.m5.1.1.1" xref="id12.5.m5.1.1.1.cmml">=</mo><mn id="id12.5.m5.1.1.3" xref="id12.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="id12.5.m5.1b"><apply id="id12.5.m5.1.1.cmml" xref="id12.5.m5.1.1"><eq id="id12.5.m5.1.1.1.cmml" xref="id12.5.m5.1.1.1"></eq><ci id="id12.5.m5.1.1.2.cmml" xref="id12.5.m5.1.1.2">𝑛</ci><cn id="id12.5.m5.1.1.3.cmml" type="integer" xref="id12.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id12.5.m5.1c">n=2</annotation><annotation encoding="application/x-llamapun" id="id12.5.m5.1d">italic_n = 2</annotation></semantics></math> knots respectively), increases relative to <math alttext="\Lambda" class="ltx_Math" display="inline" id="id13.6.m6.1"><semantics id="id13.6.m6.1a"><mi id="id13.6.m6.1.1" mathvariant="normal" xref="id13.6.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="id13.6.m6.1b"><ci id="id13.6.m6.1.1.cmml" xref="id13.6.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="id13.6.m6.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="id13.6.m6.1d">roman_Λ</annotation></semantics></math>CDM with DESI DR2 alone, consistent with previous DESI analyses. When combined with Pantheon+ data, the conclusions remain broadly consistent with our earlier work, but the inclusion of DES5Y supernovae data leads to an increase of preference for flexknot models with more than two knots, placing <math alttext="w" class="ltx_Math" display="inline" id="id14.7.m7.1"><semantics id="id14.7.m7.1a"><mi id="id14.7.m7.1.1" xref="id14.7.m7.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="id14.7.m7.1b"><ci id="id14.7.m7.1.1.cmml" xref="id14.7.m7.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="id14.7.m7.1c">w</annotation><annotation encoding="application/x-llamapun" id="id14.7.m7.1d">italic_w</annotation></semantics></math>CDM and CPL on par with <math alttext="\Lambda" class="ltx_Math" display="inline" id="id15.8.m8.1"><semantics id="id15.8.m8.1a"><mi id="id15.8.m8.1.1" mathvariant="normal" xref="id15.8.m8.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="id15.8.m8.1b"><ci id="id15.8.m8.1.1.cmml" xref="id15.8.m8.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="id15.8.m8.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="id15.8.m8.1d">roman_Λ</annotation></semantics></math>CDM.</p> </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">keywords: </h6> methods: statistical – cosmology: dark energy, cosmological parameters </div> <span class="ltx_note ltx_note_frontmatter ltx_role_pubyear" id="22"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">pubyear: </span>2025</span></span></span><span class="ltx_note ltx_note_frontmatter ltx_role_pagerange" id="23"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">pagerange: </span><span class="ltx_ref ltx_ref_self"><span class="ltx_text ltx_ref_title">Comparison of dynamical dark energy with <math alttext="\Lambda" class="ltx_Math" display="inline"><semantics><mi mathvariant="normal">Λ</mi><annotation-xml encoding="MathML-Content"><ci>Λ</ci></annotation-xml><annotation encoding="application/x-tex">\Lambda</annotation><annotation encoding="application/x-llamapun">roman_Λ</annotation></semantics></math>CDM in light of DESI DR2</span></span>–<span class="ltx_ref ltx_ref_self"><span class="ltx_text ltx_ref_title">Comparison of dynamical dark energy with <math alttext="\Lambda" class="ltx_Math" display="inline"><semantics><mi mathvariant="normal">Λ</mi><annotation-xml encoding="MathML-Content"><ci>Λ</ci></annotation-xml><annotation encoding="application/x-tex">\Lambda</annotation><annotation encoding="application/x-llamapun">roman_Λ</annotation></semantics></math>CDM in light of DESI DR2</span></span></span></span></span> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">The standard model of cosmology, <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, has been remarkably successful in explaining a wide range of cosmological observations.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.5">Recent work has reinforced the importance of understanding the nature of dark energy through increasingly precise cosmological measurements. In our previous study <cite class="ltx_cite ltx_citemacro_citep">(Ormondroyd et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite>, we employed a nonparametric flexknot reconstruction <cite class="ltx_cite ltx_citemacro_cite">(originally termed ‘nodal reconstruction’ Vázquez et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib40" title="">2012a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib41" title="">b</a>)</cite> of the dark energy equation-of-state parameter, <math alttext="w(a)" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.2" xref="S1.p2.1.m1.1.2.cmml"><mi id="S1.p2.1.m1.1.2.2" xref="S1.p2.1.m1.1.2.2.cmml">w</mi><mo id="S1.p2.1.m1.1.2.1" xref="S1.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.p2.1.m1.1.2.3.2" xref="S1.p2.1.m1.1.2.cmml"><mo id="S1.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S1.p2.1.m1.1.2.cmml">(</mo><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">a</mi><mo id="S1.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.2.cmml" xref="S1.p2.1.m1.1.2"><times id="S1.p2.1.m1.1.2.1.cmml" xref="S1.p2.1.m1.1.2.1"></times><ci id="S1.p2.1.m1.1.2.2.cmml" xref="S1.p2.1.m1.1.2.2">𝑤</ci><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_w ( italic_a )</annotation></semantics></math>, to explore the possibility of dynamical dark energy. Using a flexible linear-spline approach with free-moving nodes, our analysis of DESI Baryon Acoustic Oscillation (BAO) combined with either Pantheon+ or DES5Y Type Ia supernovae data unexpectedly revealed a W-shaped structure in <math alttext="w(a)" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mrow id="S1.p2.2.m2.1.2" xref="S1.p2.2.m2.1.2.cmml"><mi id="S1.p2.2.m2.1.2.2" xref="S1.p2.2.m2.1.2.2.cmml">w</mi><mo id="S1.p2.2.m2.1.2.1" xref="S1.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.p2.2.m2.1.2.3.2" xref="S1.p2.2.m2.1.2.cmml"><mo id="S1.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S1.p2.2.m2.1.2.cmml">(</mo><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">a</mi><mo id="S1.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.2.cmml" xref="S1.p2.2.m2.1.2"><times id="S1.p2.2.m2.1.2.1.cmml" xref="S1.p2.2.m2.1.2.1"></times><ci id="S1.p2.2.m2.1.2.2.cmml" xref="S1.p2.2.m2.1.2.2">𝑤</ci><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_w ( italic_a )</annotation></semantics></math>. This structure, which deviates from the conventional constant-<math alttext="w" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mi id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><ci id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_w</annotation></semantics></math> (<math alttext="\Lambda" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" mathvariant="normal" xref="S1.p2.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">roman_Λ</annotation></semantics></math>CDM) picture, raised questions about whether standard parameterisations such as <math alttext="w" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">w</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_w</annotation></semantics></math>CDM or CPL might be too restrictive to capture the true dynamical behavior of dark energy. This is acknowledged in the DESI DR2 release <cite class="ltx_cite ltx_citemacro_citep">(DESI Collaboration et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib11" title="">2025d</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib9" title="">b</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib8" title="">a</a>)</cite>, which includes an entire paper dedicated to an extended dark energy analysis <cite class="ltx_cite ltx_citemacro_citep">(DESI Collaboration et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib10" title="">2025c</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">In this update, we investigate how the conclusions of <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite> change in light of DESI DR2.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Data</h2> <figure class="ltx_figure" id="S2.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="397" id="S2.F1.g1" src="x1.png" width="398"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span> Flexknot reconstruction of <math alttext="w(a)" class="ltx_Math" display="inline" id="S2.F1.12.m1.1"><semantics id="S2.F1.12.m1.1b"><mrow id="S2.F1.12.m1.1.2" xref="S2.F1.12.m1.1.2.cmml"><mi id="S2.F1.12.m1.1.2.2" xref="S2.F1.12.m1.1.2.2.cmml">w</mi><mo id="S2.F1.12.m1.1.2.1" xref="S2.F1.12.m1.1.2.1.cmml">⁢</mo><mrow id="S2.F1.12.m1.1.2.3.2" xref="S2.F1.12.m1.1.2.cmml"><mo id="S2.F1.12.m1.1.2.3.2.1" stretchy="false" xref="S2.F1.12.m1.1.2.cmml">(</mo><mi id="S2.F1.12.m1.1.1" xref="S2.F1.12.m1.1.1.cmml">a</mi><mo id="S2.F1.12.m1.1.2.3.2.2" stretchy="false" xref="S2.F1.12.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.12.m1.1c"><apply id="S2.F1.12.m1.1.2.cmml" xref="S2.F1.12.m1.1.2"><times id="S2.F1.12.m1.1.2.1.cmml" xref="S2.F1.12.m1.1.2.1"></times><ci id="S2.F1.12.m1.1.2.2.cmml" xref="S2.F1.12.m1.1.2.2">𝑤</ci><ci id="S2.F1.12.m1.1.1.cmml" xref="S2.F1.12.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.12.m1.1d">w(a)</annotation><annotation encoding="application/x-llamapun" id="S2.F1.12.m1.1e">italic_w ( italic_a )</annotation></semantics></math> using DESI DR2 BAO data using flexknots, compared to DR1. Upper left panel: the reconstructed <math alttext="w(a)" class="ltx_Math" display="inline" id="S2.F1.13.m2.1"><semantics id="S2.F1.13.m2.1b"><mrow id="S2.F1.13.m2.1.2" xref="S2.F1.13.m2.1.2.cmml"><mi id="S2.F1.13.m2.1.2.2" xref="S2.F1.13.m2.1.2.2.cmml">w</mi><mo id="S2.F1.13.m2.1.2.1" xref="S2.F1.13.m2.1.2.1.cmml">⁢</mo><mrow id="S2.F1.13.m2.1.2.3.2" xref="S2.F1.13.m2.1.2.cmml"><mo id="S2.F1.13.m2.1.2.3.2.1" stretchy="false" xref="S2.F1.13.m2.1.2.cmml">(</mo><mi id="S2.F1.13.m2.1.1" xref="S2.F1.13.m2.1.1.cmml">a</mi><mo id="S2.F1.13.m2.1.2.3.2.2" stretchy="false" xref="S2.F1.13.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.13.m2.1c"><apply id="S2.F1.13.m2.1.2.cmml" xref="S2.F1.13.m2.1.2"><times id="S2.F1.13.m2.1.2.1.cmml" xref="S2.F1.13.m2.1.2.1"></times><ci id="S2.F1.13.m2.1.2.2.cmml" xref="S2.F1.13.m2.1.2.2">𝑤</ci><ci id="S2.F1.13.m2.1.1.cmml" xref="S2.F1.13.m2.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.13.m2.1d">w(a)</annotation><annotation encoding="application/x-llamapun" id="S2.F1.13.m2.1e">italic_w ( italic_a )</annotation></semantics></math>. The dashed line is the mean of the posterior, and the shaded region is the <math alttext="1\sigma" class="ltx_Math" display="inline" id="S2.F1.14.m3.1"><semantics id="S2.F1.14.m3.1b"><mrow id="S2.F1.14.m3.1.1" xref="S2.F1.14.m3.1.1.cmml"><mn id="S2.F1.14.m3.1.1.2" xref="S2.F1.14.m3.1.1.2.cmml">1</mn><mo id="S2.F1.14.m3.1.1.1" xref="S2.F1.14.m3.1.1.1.cmml">⁢</mo><mi id="S2.F1.14.m3.1.1.3" xref="S2.F1.14.m3.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.14.m3.1c"><apply id="S2.F1.14.m3.1.1.cmml" xref="S2.F1.14.m3.1.1"><times id="S2.F1.14.m3.1.1.1.cmml" xref="S2.F1.14.m3.1.1.1"></times><cn id="S2.F1.14.m3.1.1.2.cmml" type="integer" xref="S2.F1.14.m3.1.1.2">1</cn><ci id="S2.F1.14.m3.1.1.3.cmml" xref="S2.F1.14.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.14.m3.1d">1\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.F1.14.m3.1e">1 italic_σ</annotation></semantics></math> contour. The overall shape is similar to DR1, but the transition around <math alttext="a=0.6-0.7" class="ltx_Math" display="inline" id="S2.F1.15.m4.1"><semantics id="S2.F1.15.m4.1b"><mrow id="S2.F1.15.m4.1.1" xref="S2.F1.15.m4.1.1.cmml"><mi id="S2.F1.15.m4.1.1.2" xref="S2.F1.15.m4.1.1.2.cmml">a</mi><mo id="S2.F1.15.m4.1.1.1" xref="S2.F1.15.m4.1.1.1.cmml">=</mo><mrow id="S2.F1.15.m4.1.1.3" xref="S2.F1.15.m4.1.1.3.cmml"><mn id="S2.F1.15.m4.1.1.3.2" xref="S2.F1.15.m4.1.1.3.2.cmml">0.6</mn><mo id="S2.F1.15.m4.1.1.3.1" xref="S2.F1.15.m4.1.1.3.1.cmml">−</mo><mn id="S2.F1.15.m4.1.1.3.3" xref="S2.F1.15.m4.1.1.3.3.cmml">0.7</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.15.m4.1c"><apply id="S2.F1.15.m4.1.1.cmml" xref="S2.F1.15.m4.1.1"><eq id="S2.F1.15.m4.1.1.1.cmml" xref="S2.F1.15.m4.1.1.1"></eq><ci id="S2.F1.15.m4.1.1.2.cmml" xref="S2.F1.15.m4.1.1.2">𝑎</ci><apply id="S2.F1.15.m4.1.1.3.cmml" xref="S2.F1.15.m4.1.1.3"><minus id="S2.F1.15.m4.1.1.3.1.cmml" xref="S2.F1.15.m4.1.1.3.1"></minus><cn id="S2.F1.15.m4.1.1.3.2.cmml" type="float" xref="S2.F1.15.m4.1.1.3.2">0.6</cn><cn id="S2.F1.15.m4.1.1.3.3.cmml" type="float" xref="S2.F1.15.m4.1.1.3.3">0.7</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.15.m4.1d">a=0.6-0.7</annotation><annotation encoding="application/x-llamapun" id="S2.F1.15.m4.1e">italic_a = 0.6 - 0.7</annotation></semantics></math> is less pronounced. Upper right panel: the evidence for each number of knots <math alttext="n" class="ltx_Math" display="inline" id="S2.F1.16.m5.1"><semantics id="S2.F1.16.m5.1b"><mi id="S2.F1.16.m5.1.1" xref="S2.F1.16.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.F1.16.m5.1c"><ci id="S2.F1.16.m5.1.1.cmml" xref="S2.F1.16.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.16.m5.1d">n</annotation><annotation encoding="application/x-llamapun" id="S2.F1.16.m5.1e">italic_n</annotation></semantics></math>, relative to <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.F1.17.m6.1"><semantics id="S2.F1.17.m6.1b"><mi id="S2.F1.17.m6.1.1" mathvariant="normal" xref="S2.F1.17.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.F1.17.m6.1c"><ci id="S2.F1.17.m6.1.1.cmml" xref="S2.F1.17.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.17.m6.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.F1.17.m6.1e">roman_Λ</annotation></semantics></math>CDM. The biggest change is the new data admits a constant <math alttext="w" class="ltx_Math" display="inline" id="S2.F1.18.m7.1"><semantics id="S2.F1.18.m7.1b"><mi id="S2.F1.18.m7.1.1" xref="S2.F1.18.m7.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.F1.18.m7.1c"><ci id="S2.F1.18.m7.1.1.cmml" xref="S2.F1.18.m7.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.18.m7.1d">w</annotation><annotation encoding="application/x-llamapun" id="S2.F1.18.m7.1e">italic_w</annotation></semantics></math> model (<math alttext="n=1" class="ltx_Math" display="inline" id="S2.F1.19.m8.1"><semantics id="S2.F1.19.m8.1b"><mrow id="S2.F1.19.m8.1.1" xref="S2.F1.19.m8.1.1.cmml"><mi id="S2.F1.19.m8.1.1.2" xref="S2.F1.19.m8.1.1.2.cmml">n</mi><mo id="S2.F1.19.m8.1.1.1" xref="S2.F1.19.m8.1.1.1.cmml">=</mo><mn id="S2.F1.19.m8.1.1.3" xref="S2.F1.19.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.19.m8.1c"><apply id="S2.F1.19.m8.1.1.cmml" xref="S2.F1.19.m8.1.1"><eq id="S2.F1.19.m8.1.1.1.cmml" xref="S2.F1.19.m8.1.1.1"></eq><ci id="S2.F1.19.m8.1.1.2.cmml" xref="S2.F1.19.m8.1.1.2">𝑛</ci><cn id="S2.F1.19.m8.1.1.3.cmml" type="integer" xref="S2.F1.19.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.19.m8.1d">n=1</annotation><annotation encoding="application/x-llamapun" id="S2.F1.19.m8.1e">italic_n = 1</annotation></semantics></math>) as a viable model. Evidence for models with more than four knots is reduced compared to DR1. Lower left panel: posterior distributions of <math alttext="\Omega_{\mathrm{m}}" class="ltx_Math" display="inline" id="S2.F1.20.m9.1"><semantics id="S2.F1.20.m9.1b"><msub id="S2.F1.20.m9.1.1" xref="S2.F1.20.m9.1.1.cmml"><mi id="S2.F1.20.m9.1.1.2" mathvariant="normal" xref="S2.F1.20.m9.1.1.2.cmml">Ω</mi><mi id="S2.F1.20.m9.1.1.3" mathvariant="normal" xref="S2.F1.20.m9.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S2.F1.20.m9.1c"><apply id="S2.F1.20.m9.1.1.cmml" xref="S2.F1.20.m9.1.1"><csymbol cd="ambiguous" id="S2.F1.20.m9.1.1.1.cmml" xref="S2.F1.20.m9.1.1">subscript</csymbol><ci id="S2.F1.20.m9.1.1.2.cmml" xref="S2.F1.20.m9.1.1.2">Ω</ci><ci id="S2.F1.20.m9.1.1.3.cmml" xref="S2.F1.20.m9.1.1.3">m</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.20.m9.1d">\Omega_{\mathrm{m}}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.20.m9.1e">roman_Ω start_POSTSUBSCRIPT roman_m end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H_{0}r_{\mathrm{d}}" class="ltx_Math" display="inline" id="S2.F1.21.m10.1"><semantics id="S2.F1.21.m10.1b"><mrow id="S2.F1.21.m10.1.1" xref="S2.F1.21.m10.1.1.cmml"><msub id="S2.F1.21.m10.1.1.2" xref="S2.F1.21.m10.1.1.2.cmml"><mi id="S2.F1.21.m10.1.1.2.2" xref="S2.F1.21.m10.1.1.2.2.cmml">H</mi><mn id="S2.F1.21.m10.1.1.2.3" xref="S2.F1.21.m10.1.1.2.3.cmml">0</mn></msub><mo id="S2.F1.21.m10.1.1.1" xref="S2.F1.21.m10.1.1.1.cmml">⁢</mo><msub id="S2.F1.21.m10.1.1.3" xref="S2.F1.21.m10.1.1.3.cmml"><mi id="S2.F1.21.m10.1.1.3.2" xref="S2.F1.21.m10.1.1.3.2.cmml">r</mi><mi id="S2.F1.21.m10.1.1.3.3" mathvariant="normal" xref="S2.F1.21.m10.1.1.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.21.m10.1c"><apply id="S2.F1.21.m10.1.1.cmml" xref="S2.F1.21.m10.1.1"><times id="S2.F1.21.m10.1.1.1.cmml" xref="S2.F1.21.m10.1.1.1"></times><apply id="S2.F1.21.m10.1.1.2.cmml" xref="S2.F1.21.m10.1.1.2"><csymbol cd="ambiguous" id="S2.F1.21.m10.1.1.2.1.cmml" xref="S2.F1.21.m10.1.1.2">subscript</csymbol><ci id="S2.F1.21.m10.1.1.2.2.cmml" xref="S2.F1.21.m10.1.1.2.2">𝐻</ci><cn id="S2.F1.21.m10.1.1.2.3.cmml" type="integer" xref="S2.F1.21.m10.1.1.2.3">0</cn></apply><apply id="S2.F1.21.m10.1.1.3.cmml" xref="S2.F1.21.m10.1.1.3"><csymbol cd="ambiguous" id="S2.F1.21.m10.1.1.3.1.cmml" xref="S2.F1.21.m10.1.1.3">subscript</csymbol><ci id="S2.F1.21.m10.1.1.3.2.cmml" xref="S2.F1.21.m10.1.1.3.2">𝑟</ci><ci id="S2.F1.21.m10.1.1.3.3.cmml" xref="S2.F1.21.m10.1.1.3.3">d</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.21.m10.1d">H_{0}r_{\mathrm{d}}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.21.m10.1e">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math>. Lower right panel: the same reconstruction as the upper left panel, but transformed to <math alttext="w(z)" class="ltx_Math" display="inline" id="S2.F1.22.m11.1"><semantics id="S2.F1.22.m11.1b"><mrow id="S2.F1.22.m11.1.2" xref="S2.F1.22.m11.1.2.cmml"><mi id="S2.F1.22.m11.1.2.2" xref="S2.F1.22.m11.1.2.2.cmml">w</mi><mo id="S2.F1.22.m11.1.2.1" xref="S2.F1.22.m11.1.2.1.cmml">⁢</mo><mrow id="S2.F1.22.m11.1.2.3.2" xref="S2.F1.22.m11.1.2.cmml"><mo id="S2.F1.22.m11.1.2.3.2.1" stretchy="false" xref="S2.F1.22.m11.1.2.cmml">(</mo><mi id="S2.F1.22.m11.1.1" xref="S2.F1.22.m11.1.1.cmml">z</mi><mo id="S2.F1.22.m11.1.2.3.2.2" stretchy="false" xref="S2.F1.22.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.22.m11.1c"><apply id="S2.F1.22.m11.1.2.cmml" xref="S2.F1.22.m11.1.2"><times id="S2.F1.22.m11.1.2.1.cmml" xref="S2.F1.22.m11.1.2.1"></times><ci id="S2.F1.22.m11.1.2.2.cmml" xref="S2.F1.22.m11.1.2.2">𝑤</ci><ci id="S2.F1.22.m11.1.1.cmml" xref="S2.F1.22.m11.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.22.m11.1d">w(z)</annotation><annotation encoding="application/x-llamapun" id="S2.F1.22.m11.1e">italic_w ( italic_z )</annotation></semantics></math>. </figcaption> </figure> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="558" id="S2.F2.1.g1" src="x2.png" width="399"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span> BAO distances reconstructed from DESI BAO data. The left column is DESI DR1, the right column is DESI DR2. The best-fit <math alttext="\Lambda" class="ltx_Math" display="inline" id="S2.F2.6.m1.1"><semantics id="S2.F2.6.m1.1b"><mi id="S2.F2.6.m1.1.1" mathvariant="normal" xref="S2.F2.6.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S2.F2.6.m1.1c"><ci id="S2.F2.6.m1.1.1.cmml" xref="S2.F2.6.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.6.m1.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S2.F2.6.m1.1e">roman_Λ</annotation></semantics></math>CDM has been subtracted from each set of distances, and the <math alttext="w(a)" class="ltx_Math" display="inline" id="S2.F2.7.m2.1"><semantics id="S2.F2.7.m2.1b"><mrow id="S2.F2.7.m2.1.2" xref="S2.F2.7.m2.1.2.cmml"><mi id="S2.F2.7.m2.1.2.2" xref="S2.F2.7.m2.1.2.2.cmml">w</mi><mo id="S2.F2.7.m2.1.2.1" xref="S2.F2.7.m2.1.2.1.cmml">⁢</mo><mrow id="S2.F2.7.m2.1.2.3.2" xref="S2.F2.7.m2.1.2.cmml"><mo id="S2.F2.7.m2.1.2.3.2.1" stretchy="false" xref="S2.F2.7.m2.1.2.cmml">(</mo><mi id="S2.F2.7.m2.1.1" xref="S2.F2.7.m2.1.1.cmml">a</mi><mo id="S2.F2.7.m2.1.2.3.2.2" stretchy="false" xref="S2.F2.7.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.7.m2.1c"><apply id="S2.F2.7.m2.1.2.cmml" xref="S2.F2.7.m2.1.2"><times id="S2.F2.7.m2.1.2.1.cmml" xref="S2.F2.7.m2.1.2.1"></times><ci id="S2.F2.7.m2.1.2.2.cmml" xref="S2.F2.7.m2.1.2.2">𝑤</ci><ci id="S2.F2.7.m2.1.1.cmml" xref="S2.F2.7.m2.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.7.m2.1d">w(a)</annotation><annotation encoding="application/x-llamapun" id="S2.F2.7.m2.1e">italic_w ( italic_a )</annotation></semantics></math> posteriors are repeated in the bottom two panels. <math alttext="1\sigma" class="ltx_Math" display="inline" id="S2.F2.8.m3.1"><semantics id="S2.F2.8.m3.1b"><mrow id="S2.F2.8.m3.1.1" xref="S2.F2.8.m3.1.1.cmml"><mn id="S2.F2.8.m3.1.1.2" xref="S2.F2.8.m3.1.1.2.cmml">1</mn><mo id="S2.F2.8.m3.1.1.1" xref="S2.F2.8.m3.1.1.1.cmml">⁢</mo><mi id="S2.F2.8.m3.1.1.3" xref="S2.F2.8.m3.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.8.m3.1c"><apply id="S2.F2.8.m3.1.1.cmml" xref="S2.F2.8.m3.1.1"><times id="S2.F2.8.m3.1.1.1.cmml" xref="S2.F2.8.m3.1.1.1"></times><cn id="S2.F2.8.m3.1.1.2.cmml" type="integer" xref="S2.F2.8.m3.1.1.2">1</cn><ci id="S2.F2.8.m3.1.1.3.cmml" xref="S2.F2.8.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.8.m3.1d">1\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.F2.8.m3.1e">1 italic_σ</annotation></semantics></math> contours are also shown. It can been seen how the smaller error bars in DR2 have produced a narrower <math alttext="1\sigma" class="ltx_Math" display="inline" id="S2.F2.9.m4.1"><semantics id="S2.F2.9.m4.1b"><mrow id="S2.F2.9.m4.1.1" xref="S2.F2.9.m4.1.1.cmml"><mn id="S2.F2.9.m4.1.1.2" xref="S2.F2.9.m4.1.1.2.cmml">1</mn><mo id="S2.F2.9.m4.1.1.1" xref="S2.F2.9.m4.1.1.1.cmml">⁢</mo><mi id="S2.F2.9.m4.1.1.3" xref="S2.F2.9.m4.1.1.3.cmml">σ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.9.m4.1c"><apply id="S2.F2.9.m4.1.1.cmml" xref="S2.F2.9.m4.1.1"><times id="S2.F2.9.m4.1.1.1.cmml" xref="S2.F2.9.m4.1.1.1"></times><cn id="S2.F2.9.m4.1.1.2.cmml" type="integer" xref="S2.F2.9.m4.1.1.2">1</cn><ci id="S2.F2.9.m4.1.1.3.cmml" xref="S2.F2.9.m4.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.9.m4.1d">1\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.F2.9.m4.1e">1 italic_σ</annotation></semantics></math> contour for the reconstructions. </figcaption> </figure> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">We combine DESI DR2 BAO data with Pantheon+ <cite class="ltx_cite ltx_citemacro_citep">(Brout et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib3" title="">2022</a>)</cite> and DES5Y supernovae <cite class="ltx_cite ltx_citemacro_citep">(DES Collaboration et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib5" title="">2024</a>)</cite>, respectively. DESI DR2 cosmological distances are used as they appear in Table IV of <cite class="ltx_cite ltx_citemacro_cite">DESI Collaboration et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib8" title="">2025a</a>)</cite>. DESI DR1, Pantheon+, and DES5Y data are used in precisely the same manner as in <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.3">The DESI DR2 BAO improves constraints on cosmic expansion with a larger dataset of galaxies and quasars than DR1. In addition to the tightening of the error bars compared to the previous release, the Quasar Sample (QSO) now has sufficient signal-to-noise ratio that separate measurements of <math alttext="D_{\mathrm{M}}(z)" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mrow id="S2.p2.1.m1.1.2" xref="S2.p2.1.m1.1.2.cmml"><msub id="S2.p2.1.m1.1.2.2" xref="S2.p2.1.m1.1.2.2.cmml"><mi id="S2.p2.1.m1.1.2.2.2" xref="S2.p2.1.m1.1.2.2.2.cmml">D</mi><mi id="S2.p2.1.m1.1.2.2.3" mathvariant="normal" xref="S2.p2.1.m1.1.2.2.3.cmml">M</mi></msub><mo id="S2.p2.1.m1.1.2.1" xref="S2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p2.1.m1.1.2.3.2" xref="S2.p2.1.m1.1.2.cmml"><mo id="S2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S2.p2.1.m1.1.2.cmml">(</mo><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">z</mi><mo id="S2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.2.cmml" xref="S2.p2.1.m1.1.2"><times id="S2.p2.1.m1.1.2.1.cmml" xref="S2.p2.1.m1.1.2.1"></times><apply id="S2.p2.1.m1.1.2.2.cmml" xref="S2.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.p2.1.m1.1.2.2.1.cmml" xref="S2.p2.1.m1.1.2.2">subscript</csymbol><ci id="S2.p2.1.m1.1.2.2.2.cmml" xref="S2.p2.1.m1.1.2.2.2">𝐷</ci><ci id="S2.p2.1.m1.1.2.2.3.cmml" xref="S2.p2.1.m1.1.2.2.3">M</ci></apply><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">D_{\mathrm{M}}(z)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_D start_POSTSUBSCRIPT roman_M end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> and <math alttext="D_{\mathrm{H}}(z)" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mrow id="S2.p2.2.m2.1.2" xref="S2.p2.2.m2.1.2.cmml"><msub id="S2.p2.2.m2.1.2.2" xref="S2.p2.2.m2.1.2.2.cmml"><mi id="S2.p2.2.m2.1.2.2.2" xref="S2.p2.2.m2.1.2.2.2.cmml">D</mi><mi id="S2.p2.2.m2.1.2.2.3" mathvariant="normal" xref="S2.p2.2.m2.1.2.2.3.cmml">H</mi></msub><mo id="S2.p2.2.m2.1.2.1" xref="S2.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.p2.2.m2.1.2.3.2" xref="S2.p2.2.m2.1.2.cmml"><mo id="S2.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S2.p2.2.m2.1.2.cmml">(</mo><mi id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml">z</mi><mo id="S2.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S2.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><apply id="S2.p2.2.m2.1.2.cmml" xref="S2.p2.2.m2.1.2"><times id="S2.p2.2.m2.1.2.1.cmml" xref="S2.p2.2.m2.1.2.1"></times><apply id="S2.p2.2.m2.1.2.2.cmml" xref="S2.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.p2.2.m2.1.2.2.1.cmml" xref="S2.p2.2.m2.1.2.2">subscript</csymbol><ci id="S2.p2.2.m2.1.2.2.2.cmml" xref="S2.p2.2.m2.1.2.2.2">𝐷</ci><ci id="S2.p2.2.m2.1.2.2.3.cmml" xref="S2.p2.2.m2.1.2.2.3">H</ci></apply><ci id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">D_{\mathrm{H}}(z)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">italic_D start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> are reported in DR2, whereas in DR1 only a volume-averaged <math alttext="D_{\mathrm{V}}(z)" class="ltx_Math" display="inline" id="S2.p2.3.m3.1"><semantics id="S2.p2.3.m3.1a"><mrow id="S2.p2.3.m3.1.2" xref="S2.p2.3.m3.1.2.cmml"><msub id="S2.p2.3.m3.1.2.2" xref="S2.p2.3.m3.1.2.2.cmml"><mi id="S2.p2.3.m3.1.2.2.2" xref="S2.p2.3.m3.1.2.2.2.cmml">D</mi><mi id="S2.p2.3.m3.1.2.2.3" mathvariant="normal" xref="S2.p2.3.m3.1.2.2.3.cmml">V</mi></msub><mo id="S2.p2.3.m3.1.2.1" xref="S2.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.p2.3.m3.1.2.3.2" xref="S2.p2.3.m3.1.2.cmml"><mo id="S2.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S2.p2.3.m3.1.2.cmml">(</mo><mi id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml">z</mi><mo id="S2.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S2.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.1b"><apply id="S2.p2.3.m3.1.2.cmml" xref="S2.p2.3.m3.1.2"><times id="S2.p2.3.m3.1.2.1.cmml" xref="S2.p2.3.m3.1.2.1"></times><apply id="S2.p2.3.m3.1.2.2.cmml" xref="S2.p2.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.p2.3.m3.1.2.2.1.cmml" xref="S2.p2.3.m3.1.2.2">subscript</csymbol><ci id="S2.p2.3.m3.1.2.2.2.cmml" xref="S2.p2.3.m3.1.2.2.2">𝐷</ci><ci id="S2.p2.3.m3.1.2.2.3.cmml" xref="S2.p2.3.m3.1.2.2.3">V</ci></apply><ci id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.1c">D_{\mathrm{V}}(z)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.1d">italic_D start_POSTSUBSCRIPT roman_V end_POSTSUBSCRIPT ( italic_z )</annotation></semantics></math> value was reported <cite class="ltx_cite ltx_citemacro_cite">DESI Collaboration et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib8" title="">2025a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib6" title="">2024a</a>)</cite>.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Methods</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.5">With these improved DESI DR2 measurements now available, we recap the flexknot-based methodology that enables us to explore dynamical dark energy in a nonparametric way. In this approach, <math alttext="w(a)" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mrow id="S3.p1.1.m1.1.2" xref="S3.p1.1.m1.1.2.cmml"><mi id="S3.p1.1.m1.1.2.2" xref="S3.p1.1.m1.1.2.2.cmml">w</mi><mo id="S3.p1.1.m1.1.2.1" xref="S3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p1.1.m1.1.2.3.2" xref="S3.p1.1.m1.1.2.cmml"><mo id="S3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S3.p1.1.m1.1.2.cmml">(</mo><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">a</mi><mo id="S3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S3.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.2.cmml" xref="S3.p1.1.m1.1.2"><times id="S3.p1.1.m1.1.2.1.cmml" xref="S3.p1.1.m1.1.2.1"></times><ci id="S3.p1.1.m1.1.2.2.cmml" xref="S3.p1.1.m1.1.2.2">𝑤</ci><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_w ( italic_a )</annotation></semantics></math> is modelled using a flexible linear spline between free-moving nodes. This technique is well-established in multiple fields within cosmology: it has been used to reconstruct history of the dark energy equation of state from CMB data <cite class="ltx_cite ltx_citemacro_citep">(Hee et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib22" title="">2016</a>; Vázquez et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib41" title="">2012b</a>)</cite>, the primordial power spectrum <cite class="ltx_cite ltx_citemacro_citep">(Handley et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib20" title="">2019</a>; Vázquez et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib40" title="">2012a</a>; Aslanyan et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib1" title="">2014</a>; Finelli et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib13" title="">2018</a>; Planck Collaboration et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib35" title="">2014</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib36" title="">2016</a>)</cite>, the cosmic reionisation history <cite class="ltx_cite ltx_citemacro_citep">(Millea & Bouchet, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib30" title="">2018</a>; Heimersheim et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib23" title="">2022</a>)</cite>, galaxy cluster profiles <cite class="ltx_cite ltx_citemacro_citep">(Olamaie et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib31" title="">2018</a>)</cite>, and the <math alttext="21\text{\,}\mathrm{cm}" class="ltx_markedasmath" display="inline" id="S3.p1.2.m2.1.1.m1.3"><semantics id="S3.p1.2.m2.1.1.m1.3a"><mrow id="S3.p1.2.m2.1.1.m1.3.3" xref="S3.p1.2.m2.1.1.m1.3.3.cmml"><mn id="S3.p1.2.m2.1.1.m1.1.1.1.1.1.1" xref="S3.p1.2.m2.1.1.m1.1.1.1.1.1.1.cmml">21</mn><mtext id="S3.p1.2.m2.1.1.m1.2.2.2.2.2.2" xref="S3.p1.2.m2.1.1.m1.2.2.2.2.2.2.cmml"> </mtext><mi class="ltx_unit" id="S3.p1.2.m2.1.1.m1.3.3.3.3.3.3" xref="S3.p1.2.m2.1.1.m1.3.3.3.3.3.3.cmml">cm</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1.1.m1.3b"><apply id="S3.p1.2.m2.1.1.m1.3.3.cmml" xref="S3.p1.2.m2.1.1.m1.3.3"><csymbol cd="latexml" id="S3.p1.2.m2.1.1.m1.2.2.2.2.2.2.cmml" xref="S3.p1.2.m2.1.1.m1.2.2.2.2.2.2">times</csymbol><cn id="S3.p1.2.m2.1.1.m1.1.1.1.1.1.1.cmml" type="integer" xref="S3.p1.2.m2.1.1.m1.1.1.1.1.1.1">21</cn><csymbol cd="latexml" id="S3.p1.2.m2.1.1.m1.3.3.3.3.3.3.cmml" xref="S3.p1.2.m2.1.1.m1.3.3.3.3.3.3">centimeter</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1.1.m1.3c">21\text{\,}\mathrm{cm}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1.1.m1.3d">start_ARG 21 end_ARG start_ARG times end_ARG start_ARG roman_cm end_ARG</annotation></semantics></math> signal <cite class="ltx_cite ltx_citemacro_citep">(Heimersheim et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib24" title="">2024</a>; Shen et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib37" title="">2024</a>)</cite>. Unlike dark energy reconstructions such as Gaussian processes <cite class="ltx_cite ltx_citemacro_cite">Dinda & Maartens (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib12" title="">2025</a>); Yang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib43" title="">2025</a>); Johnson & Jassal (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib27" title="">2025</a>); Gao et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib14" title="">2025</a>)</cite> or cubic splines <cite class="ltx_cite ltx_citemacro_citep">(Berti et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib2" title="">2025</a>)</cite>, flexknots can reconstruct arbitrarily sharp features, and have extremely weak functional correlation structure. Flexknots also have the advantage that the <math alttext="w" 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" xref="S3.p1.3.m3.1.1.cmml">w</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">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">italic_w</annotation></semantics></math>CDM and CPL models correspond to the special cases of <math alttext="n=1" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mrow id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mi id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml">n</mi><mo id="S3.p1.4.m4.1.1.1" xref="S3.p1.4.m4.1.1.1.cmml">=</mo><mn id="S3.p1.4.m4.1.1.3" xref="S3.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><apply 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id="S3.p1.5.m5.1.1.2.cmml" xref="S3.p1.5.m5.1.1.2">𝑛</ci><cn id="S3.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.p1.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">n=2</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">italic_n = 2</annotation></semantics></math> knots, respectively.</p> </div> <figure class="ltx_table" id="S3.T1"> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S3.T1.16"> <tr class="ltx_tr" id="S3.T1.16.17"> <td class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t" id="S3.T1.16.17.1">Parameter</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.16.17.2">Prior</td> </tr> <tr class="ltx_tr" id="S3.T1.2.2"> <td class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t" id="S3.T1.1.1.1"><math alttext="n" class="ltx_Math" display="inline" id="S3.T1.1.1.1.m1.1"><semantics id="S3.T1.1.1.1.m1.1a"><mi id="S3.T1.1.1.1.m1.1.1" xref="S3.T1.1.1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.T1.1.1.1.m1.1b"><ci id="S3.T1.1.1.1.m1.1.1.cmml" xref="S3.T1.1.1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.1.1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.T1.1.1.1.m1.1d">italic_n</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.2.2.2"><math alttext="[1,20]" class="ltx_Math" display="inline" id="S3.T1.2.2.2.m1.2"><semantics id="S3.T1.2.2.2.m1.2a"><mrow id="S3.T1.2.2.2.m1.2.3.2" xref="S3.T1.2.2.2.m1.2.3.1.cmml"><mo id="S3.T1.2.2.2.m1.2.3.2.1" stretchy="false" xref="S3.T1.2.2.2.m1.2.3.1.cmml">[</mo><mn id="S3.T1.2.2.2.m1.1.1" xref="S3.T1.2.2.2.m1.1.1.cmml">1</mn><mo id="S3.T1.2.2.2.m1.2.3.2.2" xref="S3.T1.2.2.2.m1.2.3.1.cmml">,</mo><mn id="S3.T1.2.2.2.m1.2.2" xref="S3.T1.2.2.2.m1.2.2.cmml">20</mn><mo id="S3.T1.2.2.2.m1.2.3.2.3" stretchy="false" xref="S3.T1.2.2.2.m1.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.2.2.2.m1.2b"><interval closure="closed" id="S3.T1.2.2.2.m1.2.3.1.cmml" xref="S3.T1.2.2.2.m1.2.3.2"><cn id="S3.T1.2.2.2.m1.1.1.cmml" type="integer" xref="S3.T1.2.2.2.m1.1.1">1</cn><cn id="S3.T1.2.2.2.m1.2.2.cmml" type="integer" xref="S3.T1.2.2.2.m1.2.2">20</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.2.2.2.m1.2c">[1,20]</annotation><annotation encoding="application/x-llamapun" id="S3.T1.2.2.2.m1.2d">[ 1 , 20 ]</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T1.4.4"> <td class="ltx_td ltx_align_left ltx_border_l ltx_border_r" id="S3.T1.3.3.1"><math alttext="a_{n-1}" class="ltx_Math" display="inline" id="S3.T1.3.3.1.m1.1"><semantics id="S3.T1.3.3.1.m1.1a"><msub id="S3.T1.3.3.1.m1.1.1" xref="S3.T1.3.3.1.m1.1.1.cmml"><mi id="S3.T1.3.3.1.m1.1.1.2" xref="S3.T1.3.3.1.m1.1.1.2.cmml">a</mi><mrow id="S3.T1.3.3.1.m1.1.1.3" xref="S3.T1.3.3.1.m1.1.1.3.cmml"><mi id="S3.T1.3.3.1.m1.1.1.3.2" xref="S3.T1.3.3.1.m1.1.1.3.2.cmml">n</mi><mo id="S3.T1.3.3.1.m1.1.1.3.1" xref="S3.T1.3.3.1.m1.1.1.3.1.cmml">−</mo><mn id="S3.T1.3.3.1.m1.1.1.3.3" xref="S3.T1.3.3.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.T1.3.3.1.m1.1b"><apply id="S3.T1.3.3.1.m1.1.1.cmml" xref="S3.T1.3.3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.3.3.1.m1.1.1.1.cmml" xref="S3.T1.3.3.1.m1.1.1">subscript</csymbol><ci id="S3.T1.3.3.1.m1.1.1.2.cmml" xref="S3.T1.3.3.1.m1.1.1.2">𝑎</ci><apply id="S3.T1.3.3.1.m1.1.1.3.cmml" xref="S3.T1.3.3.1.m1.1.1.3"><minus id="S3.T1.3.3.1.m1.1.1.3.1.cmml" xref="S3.T1.3.3.1.m1.1.1.3.1"></minus><ci id="S3.T1.3.3.1.m1.1.1.3.2.cmml" xref="S3.T1.3.3.1.m1.1.1.3.2">𝑛</ci><cn id="S3.T1.3.3.1.m1.1.1.3.3.cmml" type="integer" xref="S3.T1.3.3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.3.3.1.m1.1c">a_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.3.3.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.4.2"><math alttext="0" class="ltx_Math" display="inline" id="S3.T1.4.4.2.m1.1"><semantics id="S3.T1.4.4.2.m1.1a"><mn id="S3.T1.4.4.2.m1.1.1" xref="S3.T1.4.4.2.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S3.T1.4.4.2.m1.1b"><cn id="S3.T1.4.4.2.m1.1.1.cmml" type="integer" xref="S3.T1.4.4.2.m1.1.1">0</cn></annotation-xml></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T1.6.6"> <td class="ltx_td ltx_align_left ltx_border_l ltx_border_r" id="S3.T1.5.5.1"><math alttext="a_{n-2},\dots,a_{1}" class="ltx_Math" display="inline" id="S3.T1.5.5.1.m1.3"><semantics id="S3.T1.5.5.1.m1.3a"><mrow id="S3.T1.5.5.1.m1.3.3.2" xref="S3.T1.5.5.1.m1.3.3.3.cmml"><msub id="S3.T1.5.5.1.m1.2.2.1.1" xref="S3.T1.5.5.1.m1.2.2.1.1.cmml"><mi id="S3.T1.5.5.1.m1.2.2.1.1.2" xref="S3.T1.5.5.1.m1.2.2.1.1.2.cmml">a</mi><mrow id="S3.T1.5.5.1.m1.2.2.1.1.3" xref="S3.T1.5.5.1.m1.2.2.1.1.3.cmml"><mi id="S3.T1.5.5.1.m1.2.2.1.1.3.2" xref="S3.T1.5.5.1.m1.2.2.1.1.3.2.cmml">n</mi><mo id="S3.T1.5.5.1.m1.2.2.1.1.3.1" xref="S3.T1.5.5.1.m1.2.2.1.1.3.1.cmml">−</mo><mn id="S3.T1.5.5.1.m1.2.2.1.1.3.3" xref="S3.T1.5.5.1.m1.2.2.1.1.3.3.cmml">2</mn></mrow></msub><mo id="S3.T1.5.5.1.m1.3.3.2.3" xref="S3.T1.5.5.1.m1.3.3.3.cmml">,</mo><mi id="S3.T1.5.5.1.m1.1.1" mathvariant="normal" xref="S3.T1.5.5.1.m1.1.1.cmml">…</mi><mo id="S3.T1.5.5.1.m1.3.3.2.4" xref="S3.T1.5.5.1.m1.3.3.3.cmml">,</mo><msub id="S3.T1.5.5.1.m1.3.3.2.2" xref="S3.T1.5.5.1.m1.3.3.2.2.cmml"><mi id="S3.T1.5.5.1.m1.3.3.2.2.2" xref="S3.T1.5.5.1.m1.3.3.2.2.2.cmml">a</mi><mn id="S3.T1.5.5.1.m1.3.3.2.2.3" xref="S3.T1.5.5.1.m1.3.3.2.2.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.5.5.1.m1.3b"><list id="S3.T1.5.5.1.m1.3.3.3.cmml" xref="S3.T1.5.5.1.m1.3.3.2"><apply 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xref="S3.T1.5.5.1.m1.3.3.2.2.3">1</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.5.5.1.m1.3c">a_{n-2},\dots,a_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.5.5.1.m1.3d">italic_a start_POSTSUBSCRIPT italic_n - 2 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.6.6.2">sorted(<math alttext="[a_{n-1},a_{0}])" class="ltx_math_unparsed" display="inline" id="S3.T1.6.6.2.m1.1"><semantics id="S3.T1.6.6.2.m1.1a"><mrow id="S3.T1.6.6.2.m1.1b"><mrow id="S3.T1.6.6.2.m1.1.1"><mo id="S3.T1.6.6.2.m1.1.1.1" stretchy="false">[</mo><msub id="S3.T1.6.6.2.m1.1.1.2"><mi id="S3.T1.6.6.2.m1.1.1.2.2">a</mi><mrow id="S3.T1.6.6.2.m1.1.1.2.3"><mi id="S3.T1.6.6.2.m1.1.1.2.3.2">n</mi><mo id="S3.T1.6.6.2.m1.1.1.2.3.1">−</mo><mn id="S3.T1.6.6.2.m1.1.1.2.3.3">1</mn></mrow></msub><mo id="S3.T1.6.6.2.m1.1.1.3">,</mo><msub 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ltx_border_r" id="S3.T1.15.15.1"> <math alttext="H_{0}" class="ltx_Math" display="inline" id="S3.T1.15.15.1.m1.1"><semantics id="S3.T1.15.15.1.m1.1a"><msub id="S3.T1.15.15.1.m1.1.1" xref="S3.T1.15.15.1.m1.1.1.cmml"><mi id="S3.T1.15.15.1.m1.1.1.2" xref="S3.T1.15.15.1.m1.1.1.2.cmml">H</mi><mn id="S3.T1.15.15.1.m1.1.1.3" xref="S3.T1.15.15.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.T1.15.15.1.m1.1b"><apply id="S3.T1.15.15.1.m1.1.1.cmml" xref="S3.T1.15.15.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.15.15.1.m1.1.1.1.cmml" xref="S3.T1.15.15.1.m1.1.1">subscript</csymbol><ci id="S3.T1.15.15.1.m1.1.1.2.cmml" xref="S3.T1.15.15.1.m1.1.1.2">𝐻</ci><cn id="S3.T1.15.15.1.m1.1.1.3.cmml" type="integer" xref="S3.T1.15.15.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.15.15.1.m1.1c">H_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.15.15.1.m1.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> (Ia)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S3.T1.16.16.2"><math alttext="[20,100]" class="ltx_Math" display="inline" id="S3.T1.16.16.2.m1.2"><semantics id="S3.T1.16.16.2.m1.2a"><mrow id="S3.T1.16.16.2.m1.2.3.2" xref="S3.T1.16.16.2.m1.2.3.1.cmml"><mo id="S3.T1.16.16.2.m1.2.3.2.1" stretchy="false" xref="S3.T1.16.16.2.m1.2.3.1.cmml">[</mo><mn id="S3.T1.16.16.2.m1.1.1" xref="S3.T1.16.16.2.m1.1.1.cmml">20</mn><mo id="S3.T1.16.16.2.m1.2.3.2.2" xref="S3.T1.16.16.2.m1.2.3.1.cmml">,</mo><mn id="S3.T1.16.16.2.m1.2.2" xref="S3.T1.16.16.2.m1.2.2.cmml">100</mn><mo id="S3.T1.16.16.2.m1.2.3.2.3" stretchy="false" xref="S3.T1.16.16.2.m1.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.16.16.2.m1.2b"><interval closure="closed" id="S3.T1.16.16.2.m1.2.3.1.cmml" xref="S3.T1.16.16.2.m1.2.3.2"><cn id="S3.T1.16.16.2.m1.1.1.cmml" type="integer" xref="S3.T1.16.16.2.m1.1.1">20</cn><cn id="S3.T1.16.16.2.m1.2.2.cmml" type="integer" xref="S3.T1.16.16.2.m1.2.2">100</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.16.16.2.m1.2c">[20,100]</annotation><annotation encoding="application/x-llamapun" id="S3.T1.16.16.2.m1.2d">[ 20 , 100 ]</annotation></semantics></math></td> </tr> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 1: </span> Cosmological priors used in this work. Fixed values are indicated by a single number, while uniform priors are denoted by brackets. As BAO only depend on the product <math alttext="H_{0}r_{\mathrm{d}}" class="ltx_Math" display="inline" id="S3.T1.20.m1.1"><semantics id="S3.T1.20.m1.1b"><mrow id="S3.T1.20.m1.1.1" xref="S3.T1.20.m1.1.1.cmml"><msub id="S3.T1.20.m1.1.1.2" xref="S3.T1.20.m1.1.1.2.cmml"><mi id="S3.T1.20.m1.1.1.2.2" xref="S3.T1.20.m1.1.1.2.2.cmml">H</mi><mn id="S3.T1.20.m1.1.1.2.3" xref="S3.T1.20.m1.1.1.2.3.cmml">0</mn></msub><mo id="S3.T1.20.m1.1.1.1" xref="S3.T1.20.m1.1.1.1.cmml">⁢</mo><msub id="S3.T1.20.m1.1.1.3" xref="S3.T1.20.m1.1.1.3.cmml"><mi id="S3.T1.20.m1.1.1.3.2" xref="S3.T1.20.m1.1.1.3.2.cmml">r</mi><mi id="S3.T1.20.m1.1.1.3.3" mathvariant="normal" xref="S3.T1.20.m1.1.1.3.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.20.m1.1c"><apply id="S3.T1.20.m1.1.1.cmml" xref="S3.T1.20.m1.1.1"><times id="S3.T1.20.m1.1.1.1.cmml" xref="S3.T1.20.m1.1.1.1"></times><apply id="S3.T1.20.m1.1.1.2.cmml" xref="S3.T1.20.m1.1.1.2"><csymbol cd="ambiguous" id="S3.T1.20.m1.1.1.2.1.cmml" xref="S3.T1.20.m1.1.1.2">subscript</csymbol><ci id="S3.T1.20.m1.1.1.2.2.cmml" xref="S3.T1.20.m1.1.1.2.2">𝐻</ci><cn id="S3.T1.20.m1.1.1.2.3.cmml" type="integer" xref="S3.T1.20.m1.1.1.2.3">0</cn></apply><apply id="S3.T1.20.m1.1.1.3.cmml" xref="S3.T1.20.m1.1.1.3"><csymbol cd="ambiguous" id="S3.T1.20.m1.1.1.3.1.cmml" xref="S3.T1.20.m1.1.1.3">subscript</csymbol><ci id="S3.T1.20.m1.1.1.3.2.cmml" xref="S3.T1.20.m1.1.1.3.2">𝑟</ci><ci id="S3.T1.20.m1.1.1.3.3.cmml" xref="S3.T1.20.m1.1.1.3.3">d</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.20.m1.1d">H_{0}r_{\mathrm{d}}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.20.m1.1e">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT</annotation></semantics></math>, and supernovae depend on <math alttext="H_{0}" class="ltx_Math" display="inline" id="S3.T1.21.m2.1"><semantics id="S3.T1.21.m2.1b"><msub id="S3.T1.21.m2.1.1" xref="S3.T1.21.m2.1.1.cmml"><mi id="S3.T1.21.m2.1.1.2" xref="S3.T1.21.m2.1.1.2.cmml">H</mi><mn id="S3.T1.21.m2.1.1.3" xref="S3.T1.21.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.T1.21.m2.1c"><apply id="S3.T1.21.m2.1.1.cmml" xref="S3.T1.21.m2.1.1"><csymbol cd="ambiguous" id="S3.T1.21.m2.1.1.1.cmml" xref="S3.T1.21.m2.1.1">subscript</csymbol><ci id="S3.T1.21.m2.1.1.2.cmml" xref="S3.T1.21.m2.1.1.2">𝐻</ci><cn id="S3.T1.21.m2.1.1.3.cmml" type="integer" xref="S3.T1.21.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.21.m2.1d">H_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.21.m2.1e">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, those parameters are only included as necessary. Whilst the dynamical dark energy priors are broadly consistent with those in <cite class="ltx_cite ltx_citemacro_cite">DESI Collaboration et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib8" title="">2025a</a>)</cite>, these inevitably differ from CPL priors which instead put a uniform prior on the gradient <math alttext="w_{a}" class="ltx_Math" display="inline" id="S3.T1.22.m3.1"><semantics id="S3.T1.22.m3.1b"><msub id="S3.T1.22.m3.1.1" xref="S3.T1.22.m3.1.1.cmml"><mi id="S3.T1.22.m3.1.1.2" xref="S3.T1.22.m3.1.1.2.cmml">w</mi><mi id="S3.T1.22.m3.1.1.3" xref="S3.T1.22.m3.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.T1.22.m3.1c"><apply id="S3.T1.22.m3.1.1.cmml" xref="S3.T1.22.m3.1.1"><csymbol cd="ambiguous" id="S3.T1.22.m3.1.1.1.cmml" xref="S3.T1.22.m3.1.1">subscript</csymbol><ci id="S3.T1.22.m3.1.1.2.cmml" xref="S3.T1.22.m3.1.1.2">𝑤</ci><ci id="S3.T1.22.m3.1.1.3.cmml" xref="S3.T1.22.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.22.m3.1d">w_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.22.m3.1e">italic_w start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. </figcaption> </figure> <figure class="ltx_figure" id="S3.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_square" height="397" id="S3.F3.g1" src="x3.png" width="398"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_square" height="397" id="S3.F3.g2" src="x4.png" width="398"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span> Similar to Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S2.F1" title="Figure 1 ‣ 2 Data ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">1</span></a>, but comparing DESI DR2 BAO data with Pantheon+ supernovae (left) and DES5Y supernovae (right). The most significant difference between the DR1 and DR2 reconstructions is the same region of <math alttext="a" class="ltx_Math" display="inline" id="S3.F3.3.m1.1"><semantics id="S3.F3.3.m1.1b"><mi id="S3.F3.3.m1.1.1" xref="S3.F3.3.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S3.F3.3.m1.1c"><ci id="S3.F3.3.m1.1.1.cmml" xref="S3.F3.3.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.3.m1.1d">a</annotation><annotation encoding="application/x-llamapun" id="S3.F3.3.m1.1e">italic_a</annotation></semantics></math> as in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S2.F1" title="Figure 1 ‣ 2 Data ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">1</span></a>. Left four panels: DESI DR2 + Pantheon+, compared to DR1 + Pantheon+. The evidence for <math alttext="w" class="ltx_Math" display="inline" id="S3.F3.4.m2.1"><semantics id="S3.F3.4.m2.1b"><mi id="S3.F3.4.m2.1.1" xref="S3.F3.4.m2.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.F3.4.m2.1c"><ci id="S3.F3.4.m2.1.1.cmml" xref="S3.F3.4.m2.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.4.m2.1d">w</annotation><annotation encoding="application/x-llamapun" id="S3.F3.4.m2.1e">italic_w</annotation></semantics></math>CDM is very similar between DR1 and DR2, but the remaining flexknots are less favoured in DR2. Right four panels: DESI DR2 + DES5Y, compared to DR1 + DES5Y. The change is very much the opposite as it was with Pantheon+, with the evidence for all numbers of knots greater than or equal to two have increased significantly, with CPL remaining the preferred model. </figcaption> </figure> <figure class="ltx_figure" id="S3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="193" id="S3.F4.1.g1" src="x5.png" width="398"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 4: </span> Tension quantifications between combinations of datasets. For each knot reconstruction <math alttext="N" class="ltx_Math" display="inline" id="S3.F4.11.m1.1"><semantics id="S3.F4.11.m1.1b"><mi id="S3.F4.11.m1.1.1" xref="S3.F4.11.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.F4.11.m1.1c"><ci id="S3.F4.11.m1.1.1.cmml" xref="S3.F4.11.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.11.m1.1d">N</annotation><annotation encoding="application/x-llamapun" id="S3.F4.11.m1.1e">italic_N</annotation></semantics></math> we compare the tension between DESI (DR1 or DR2) and supernovae (Pantheon+ [left panel] or DES5Y [right panel]). In light of the update from DR1 to DR2, the tension has increased between DESI and Pantheon+ (<math alttext="\log R" class="ltx_Math" display="inline" id="S3.F4.12.m2.1"><semantics id="S3.F4.12.m2.1b"><mrow id="S3.F4.12.m2.1.1" xref="S3.F4.12.m2.1.1.cmml"><mi id="S3.F4.12.m2.1.1.1" xref="S3.F4.12.m2.1.1.1.cmml">log</mi><mo id="S3.F4.12.m2.1.1b" lspace="0.167em" xref="S3.F4.12.m2.1.1.cmml">⁡</mo><mi id="S3.F4.12.m2.1.1.2" xref="S3.F4.12.m2.1.1.2.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.12.m2.1c"><apply id="S3.F4.12.m2.1.1.cmml" xref="S3.F4.12.m2.1.1"><log id="S3.F4.12.m2.1.1.1.cmml" xref="S3.F4.12.m2.1.1.1"></log><ci id="S3.F4.12.m2.1.1.2.cmml" xref="S3.F4.12.m2.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.12.m2.1d">\log R</annotation><annotation encoding="application/x-llamapun" id="S3.F4.12.m2.1e">roman_log italic_R</annotation></semantics></math> lower), but remains consistent (<math alttext="\log R>0" class="ltx_Math" display="inline" id="S3.F4.13.m3.1"><semantics id="S3.F4.13.m3.1b"><mrow id="S3.F4.13.m3.1.1" xref="S3.F4.13.m3.1.1.cmml"><mrow id="S3.F4.13.m3.1.1.2" xref="S3.F4.13.m3.1.1.2.cmml"><mi id="S3.F4.13.m3.1.1.2.1" xref="S3.F4.13.m3.1.1.2.1.cmml">log</mi><mo id="S3.F4.13.m3.1.1.2b" lspace="0.167em" xref="S3.F4.13.m3.1.1.2.cmml">⁡</mo><mi id="S3.F4.13.m3.1.1.2.2" xref="S3.F4.13.m3.1.1.2.2.cmml">R</mi></mrow><mo id="S3.F4.13.m3.1.1.1" xref="S3.F4.13.m3.1.1.1.cmml">></mo><mn id="S3.F4.13.m3.1.1.3" xref="S3.F4.13.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.13.m3.1c"><apply id="S3.F4.13.m3.1.1.cmml" xref="S3.F4.13.m3.1.1"><gt id="S3.F4.13.m3.1.1.1.cmml" xref="S3.F4.13.m3.1.1.1"></gt><apply id="S3.F4.13.m3.1.1.2.cmml" xref="S3.F4.13.m3.1.1.2"><log id="S3.F4.13.m3.1.1.2.1.cmml" xref="S3.F4.13.m3.1.1.2.1"></log><ci id="S3.F4.13.m3.1.1.2.2.cmml" xref="S3.F4.13.m3.1.1.2.2">𝑅</ci></apply><cn id="S3.F4.13.m3.1.1.3.cmml" type="integer" xref="S3.F4.13.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.13.m3.1d">\log R>0</annotation><annotation encoding="application/x-llamapun" id="S3.F4.13.m3.1e">roman_log italic_R > 0</annotation></semantics></math>). For DESI and DES5Y in light of the update from DR1 to DR2 the tension has decreased (<math alttext="\log R" class="ltx_Math" display="inline" id="S3.F4.14.m4.1"><semantics id="S3.F4.14.m4.1b"><mrow id="S3.F4.14.m4.1.1" xref="S3.F4.14.m4.1.1.cmml"><mi id="S3.F4.14.m4.1.1.1" xref="S3.F4.14.m4.1.1.1.cmml">log</mi><mo id="S3.F4.14.m4.1.1b" lspace="0.167em" xref="S3.F4.14.m4.1.1.cmml">⁡</mo><mi id="S3.F4.14.m4.1.1.2" xref="S3.F4.14.m4.1.1.2.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.14.m4.1c"><apply id="S3.F4.14.m4.1.1.cmml" xref="S3.F4.14.m4.1.1"><log id="S3.F4.14.m4.1.1.1.cmml" xref="S3.F4.14.m4.1.1.1"></log><ci id="S3.F4.14.m4.1.1.2.cmml" xref="S3.F4.14.m4.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.14.m4.1d">\log R</annotation><annotation encoding="application/x-llamapun" id="S3.F4.14.m4.1e">roman_log italic_R</annotation></semantics></math> higher) and is now consistent <math alttext="\log R>0" class="ltx_Math" display="inline" id="S3.F4.15.m5.1"><semantics id="S3.F4.15.m5.1b"><mrow id="S3.F4.15.m5.1.1" xref="S3.F4.15.m5.1.1.cmml"><mrow id="S3.F4.15.m5.1.1.2" xref="S3.F4.15.m5.1.1.2.cmml"><mi id="S3.F4.15.m5.1.1.2.1" xref="S3.F4.15.m5.1.1.2.1.cmml">log</mi><mo id="S3.F4.15.m5.1.1.2b" lspace="0.167em" xref="S3.F4.15.m5.1.1.2.cmml">⁡</mo><mi id="S3.F4.15.m5.1.1.2.2" xref="S3.F4.15.m5.1.1.2.2.cmml">R</mi></mrow><mo id="S3.F4.15.m5.1.1.1" xref="S3.F4.15.m5.1.1.1.cmml">></mo><mn id="S3.F4.15.m5.1.1.3" xref="S3.F4.15.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.15.m5.1c"><apply id="S3.F4.15.m5.1.1.cmml" xref="S3.F4.15.m5.1.1"><gt id="S3.F4.15.m5.1.1.1.cmml" xref="S3.F4.15.m5.1.1.1"></gt><apply id="S3.F4.15.m5.1.1.2.cmml" xref="S3.F4.15.m5.1.1.2"><log id="S3.F4.15.m5.1.1.2.1.cmml" xref="S3.F4.15.m5.1.1.2.1"></log><ci id="S3.F4.15.m5.1.1.2.2.cmml" xref="S3.F4.15.m5.1.1.2.2">𝑅</ci></apply><cn id="S3.F4.15.m5.1.1.3.cmml" type="integer" xref="S3.F4.15.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.15.m5.1d">\log R>0</annotation><annotation encoding="application/x-llamapun" id="S3.F4.15.m5.1e">roman_log italic_R > 0</annotation></semantics></math> for all but the <math alttext="\Lambda" class="ltx_Math" display="inline" id="S3.F4.16.m6.1"><semantics id="S3.F4.16.m6.1b"><mi id="S3.F4.16.m6.1.1" mathvariant="normal" xref="S3.F4.16.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S3.F4.16.m6.1c"><ci id="S3.F4.16.m6.1.1.cmml" xref="S3.F4.16.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.16.m6.1d">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S3.F4.16.m6.1e">roman_Λ</annotation></semantics></math>CDM (<math alttext="N=0" class="ltx_Math" display="inline" id="S3.F4.17.m7.1"><semantics id="S3.F4.17.m7.1b"><mrow id="S3.F4.17.m7.1.1" xref="S3.F4.17.m7.1.1.cmml"><mi id="S3.F4.17.m7.1.1.2" xref="S3.F4.17.m7.1.1.2.cmml">N</mi><mo id="S3.F4.17.m7.1.1.1" xref="S3.F4.17.m7.1.1.1.cmml">=</mo><mn id="S3.F4.17.m7.1.1.3" xref="S3.F4.17.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.17.m7.1c"><apply id="S3.F4.17.m7.1.1.cmml" xref="S3.F4.17.m7.1.1"><eq id="S3.F4.17.m7.1.1.1.cmml" xref="S3.F4.17.m7.1.1.1"></eq><ci id="S3.F4.17.m7.1.1.2.cmml" xref="S3.F4.17.m7.1.1.2">𝑁</ci><cn id="S3.F4.17.m7.1.1.3.cmml" type="integer" xref="S3.F4.17.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.17.m7.1d">N=0</annotation><annotation encoding="application/x-llamapun" id="S3.F4.17.m7.1e">italic_N = 0</annotation></semantics></math>) and <math alttext="w" class="ltx_Math" display="inline" id="S3.F4.18.m8.1"><semantics id="S3.F4.18.m8.1b"><mi id="S3.F4.18.m8.1.1" xref="S3.F4.18.m8.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.F4.18.m8.1c"><ci id="S3.F4.18.m8.1.1.cmml" xref="S3.F4.18.m8.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.18.m8.1d">w</annotation><annotation encoding="application/x-llamapun" id="S3.F4.18.m8.1e">italic_w</annotation></semantics></math>CDM (<math alttext="N=1" class="ltx_Math" display="inline" id="S3.F4.19.m9.1"><semantics id="S3.F4.19.m9.1b"><mrow id="S3.F4.19.m9.1.1" xref="S3.F4.19.m9.1.1.cmml"><mi id="S3.F4.19.m9.1.1.2" xref="S3.F4.19.m9.1.1.2.cmml">N</mi><mo id="S3.F4.19.m9.1.1.1" xref="S3.F4.19.m9.1.1.1.cmml">=</mo><mn id="S3.F4.19.m9.1.1.3" xref="S3.F4.19.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.19.m9.1c"><apply id="S3.F4.19.m9.1.1.cmml" xref="S3.F4.19.m9.1.1"><eq id="S3.F4.19.m9.1.1.1.cmml" xref="S3.F4.19.m9.1.1.1"></eq><ci id="S3.F4.19.m9.1.1.2.cmml" xref="S3.F4.19.m9.1.1.2">𝑁</ci><cn id="S3.F4.19.m9.1.1.3.cmml" type="integer" xref="S3.F4.19.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.19.m9.1d">N=1</annotation><annotation encoding="application/x-llamapun" id="S3.F4.19.m9.1e">italic_N = 1</annotation></semantics></math>) cases . </figcaption> </figure> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">Posterior samples and evidences were obtained using the nested sampling algorithm <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.1">PolyChord</span> <cite class="ltx_cite ltx_citemacro_citep">(Skilling, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib38" title="">2004</a>; Handley et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib18" title="">2015a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib19" title="">b</a>)</cite>. A branch of <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.2">fgivenx</span> was used to produce the functional posterior plots <cite class="ltx_cite ltx_citemacro_citep">(Handley, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib15" title="">2019a</a>)</cite>, and <span class="ltx_text ltx_font_typewriter" id="S3.p2.1.3">anesthetic</span> was used to process the nested sampling chains <cite class="ltx_cite ltx_citemacro_citep">(Handley, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib16" title="">2019b</a>)</cite>.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.5">As in <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite>, <math alttext="H_{0}" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><msub id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml"><mi id="S3.p3.1.m1.1.1.2" xref="S3.p3.1.m1.1.1.2.cmml">H</mi><mn id="S3.p3.1.m1.1.1.3" xref="S3.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.p3.1.m1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.2">𝐻</ci><cn id="S3.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">H_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="M_{B}" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><msub id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml"><mi id="S3.p3.2.m2.1.1.2" xref="S3.p3.2.m2.1.1.2.cmml">M</mi><mi id="S3.p3.2.m2.1.1.3" xref="S3.p3.2.m2.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p3.2.m2.1.1.1.cmml" xref="S3.p3.2.m2.1.1">subscript</csymbol><ci id="S3.p3.2.m2.1.1.2.cmml" xref="S3.p3.2.m2.1.1.2">𝑀</ci><ci id="S3.p3.2.m2.1.1.3.cmml" xref="S3.p3.2.m2.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">M_{B}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_M start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> are marginalised over, and the prior on <math alttext="M_{B}" class="ltx_Math" display="inline" id="S3.p3.3.m3.1"><semantics id="S3.p3.3.m3.1a"><msub id="S3.p3.3.m3.1.1" xref="S3.p3.3.m3.1.1.cmml"><mi id="S3.p3.3.m3.1.1.2" xref="S3.p3.3.m3.1.1.2.cmml">M</mi><mi id="S3.p3.3.m3.1.1.3" xref="S3.p3.3.m3.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.1b"><apply id="S3.p3.3.m3.1.1.cmml" xref="S3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p3.3.m3.1.1.1.cmml" xref="S3.p3.3.m3.1.1">subscript</csymbol><ci id="S3.p3.3.m3.1.1.2.cmml" xref="S3.p3.3.m3.1.1.2">𝑀</ci><ci id="S3.p3.3.m3.1.1.3.cmml" xref="S3.p3.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.1c">M_{B}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.1d">italic_M start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> is taken to be uniform and sufficiently wide to contain the entire posterior. In the DESI analyses <cite class="ltx_cite ltx_citemacro_cite">DESI Collaboration et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib8" title="">2025a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib7" title="">2024b</a>)</cite>, it is enforced that <math alttext="w_{0}+w_{a}<0" class="ltx_Math" display="inline" id="S3.p3.4.m4.1"><semantics id="S3.p3.4.m4.1a"><mrow id="S3.p3.4.m4.1.1" xref="S3.p3.4.m4.1.1.cmml"><mrow id="S3.p3.4.m4.1.1.2" xref="S3.p3.4.m4.1.1.2.cmml"><msub id="S3.p3.4.m4.1.1.2.2" xref="S3.p3.4.m4.1.1.2.2.cmml"><mi id="S3.p3.4.m4.1.1.2.2.2" xref="S3.p3.4.m4.1.1.2.2.2.cmml">w</mi><mn id="S3.p3.4.m4.1.1.2.2.3" xref="S3.p3.4.m4.1.1.2.2.3.cmml">0</mn></msub><mo id="S3.p3.4.m4.1.1.2.1" xref="S3.p3.4.m4.1.1.2.1.cmml">+</mo><msub id="S3.p3.4.m4.1.1.2.3" xref="S3.p3.4.m4.1.1.2.3.cmml"><mi id="S3.p3.4.m4.1.1.2.3.2" xref="S3.p3.4.m4.1.1.2.3.2.cmml">w</mi><mi id="S3.p3.4.m4.1.1.2.3.3" xref="S3.p3.4.m4.1.1.2.3.3.cmml">a</mi></msub></mrow><mo id="S3.p3.4.m4.1.1.1" xref="S3.p3.4.m4.1.1.1.cmml"><</mo><mn id="S3.p3.4.m4.1.1.3" xref="S3.p3.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.4.m4.1b"><apply id="S3.p3.4.m4.1.1.cmml" xref="S3.p3.4.m4.1.1"><lt id="S3.p3.4.m4.1.1.1.cmml" xref="S3.p3.4.m4.1.1.1"></lt><apply id="S3.p3.4.m4.1.1.2.cmml" xref="S3.p3.4.m4.1.1.2"><plus id="S3.p3.4.m4.1.1.2.1.cmml" xref="S3.p3.4.m4.1.1.2.1"></plus><apply id="S3.p3.4.m4.1.1.2.2.cmml" xref="S3.p3.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S3.p3.4.m4.1.1.2.2.1.cmml" xref="S3.p3.4.m4.1.1.2.2">subscript</csymbol><ci id="S3.p3.4.m4.1.1.2.2.2.cmml" xref="S3.p3.4.m4.1.1.2.2.2">𝑤</ci><cn id="S3.p3.4.m4.1.1.2.2.3.cmml" type="integer" xref="S3.p3.4.m4.1.1.2.2.3">0</cn></apply><apply id="S3.p3.4.m4.1.1.2.3.cmml" xref="S3.p3.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S3.p3.4.m4.1.1.2.3.1.cmml" xref="S3.p3.4.m4.1.1.2.3">subscript</csymbol><ci id="S3.p3.4.m4.1.1.2.3.2.cmml" xref="S3.p3.4.m4.1.1.2.3.2">𝑤</ci><ci id="S3.p3.4.m4.1.1.2.3.3.cmml" xref="S3.p3.4.m4.1.1.2.3.3">𝑎</ci></apply></apply><cn id="S3.p3.4.m4.1.1.3.cmml" type="integer" xref="S3.p3.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.1c">w_{0}+w_{a}<0</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_w start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT < 0</annotation></semantics></math> to ensure that there is a period of matter domination at high redshifts. In this work, we take <math alttext="w_{i}<0" class="ltx_Math" display="inline" id="S3.p3.5.m5.1"><semantics id="S3.p3.5.m5.1a"><mrow id="S3.p3.5.m5.1.1" xref="S3.p3.5.m5.1.1.cmml"><msub id="S3.p3.5.m5.1.1.2" xref="S3.p3.5.m5.1.1.2.cmml"><mi id="S3.p3.5.m5.1.1.2.2" xref="S3.p3.5.m5.1.1.2.2.cmml">w</mi><mi id="S3.p3.5.m5.1.1.2.3" xref="S3.p3.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S3.p3.5.m5.1.1.1" xref="S3.p3.5.m5.1.1.1.cmml"><</mo><mn id="S3.p3.5.m5.1.1.3" xref="S3.p3.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.5.m5.1b"><apply id="S3.p3.5.m5.1.1.cmml" xref="S3.p3.5.m5.1.1"><lt id="S3.p3.5.m5.1.1.1.cmml" xref="S3.p3.5.m5.1.1.1"></lt><apply id="S3.p3.5.m5.1.1.2.cmml" xref="S3.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.p3.5.m5.1.1.2.1.cmml" xref="S3.p3.5.m5.1.1.2">subscript</csymbol><ci id="S3.p3.5.m5.1.1.2.2.cmml" xref="S3.p3.5.m5.1.1.2.2">𝑤</ci><ci id="S3.p3.5.m5.1.1.2.3.cmml" xref="S3.p3.5.m5.1.1.2.3">𝑖</ci></apply><cn id="S3.p3.5.m5.1.1.3.cmml" type="integer" xref="S3.p3.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m5.1c">w_{i}<0</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m5.1d">italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < 0</annotation></semantics></math> to achieve the same effect.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.1">Table <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S3.T1" title="Table 1 ‣ 3 Methods ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">1</span></a> lists the cosmological priors used in this work, the same as <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite>, which themselves were chosen to remain consistent with those used in <cite class="ltx_cite ltx_citemacro_cite">Calderon et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib4" title="">2024</a>)</cite>.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Results</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.7">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S2.F1" title="Figure 1 ‣ 2 Data ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">1</span></a> shows reconstructions of <math alttext="w(a)" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mrow id="S4.p1.1.m1.1.2" xref="S4.p1.1.m1.1.2.cmml"><mi id="S4.p1.1.m1.1.2.2" xref="S4.p1.1.m1.1.2.2.cmml">w</mi><mo id="S4.p1.1.m1.1.2.1" xref="S4.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.p1.1.m1.1.2.3.2" xref="S4.p1.1.m1.1.2.cmml"><mo id="S4.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.p1.1.m1.1.2.cmml">(</mo><mi id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">a</mi><mo id="S4.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><apply id="S4.p1.1.m1.1.2.cmml" xref="S4.p1.1.m1.1.2"><times id="S4.p1.1.m1.1.2.1.cmml" xref="S4.p1.1.m1.1.2.1"></times><ci id="S4.p1.1.m1.1.2.2.cmml" xref="S4.p1.1.m1.1.2.2">𝑤</ci><ci id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">italic_w ( italic_a )</annotation></semantics></math> from DESI alone, and compares DR1 and DR2. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S2.F2" title="Figure 2 ‣ 2 Data ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">2</span></a> shows the corresponding BAO distances reconstructed from DESI BAO data, compared to the best-fit <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><mi id="S4.p1.2.m2.1.1" mathvariant="normal" xref="S4.p1.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><ci id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">roman_Λ</annotation></semantics></math>CDM model. The most significant change is not in the shape of the <math alttext="w(a)" class="ltx_Math" display="inline" id="S4.p1.3.m3.1"><semantics id="S4.p1.3.m3.1a"><mrow id="S4.p1.3.m3.1.2" xref="S4.p1.3.m3.1.2.cmml"><mi id="S4.p1.3.m3.1.2.2" xref="S4.p1.3.m3.1.2.2.cmml">w</mi><mo id="S4.p1.3.m3.1.2.1" xref="S4.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.p1.3.m3.1.2.3.2" xref="S4.p1.3.m3.1.2.cmml"><mo id="S4.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S4.p1.3.m3.1.2.cmml">(</mo><mi id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml">a</mi><mo id="S4.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S4.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.1b"><apply id="S4.p1.3.m3.1.2.cmml" xref="S4.p1.3.m3.1.2"><times id="S4.p1.3.m3.1.2.1.cmml" xref="S4.p1.3.m3.1.2.1"></times><ci id="S4.p1.3.m3.1.2.2.cmml" xref="S4.p1.3.m3.1.2.2">𝑤</ci><ci id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.1d">italic_w ( italic_a )</annotation></semantics></math> posterior, but the evidences for each number of knots <math alttext="n" class="ltx_Math" display="inline" id="S4.p1.4.m4.1"><semantics id="S4.p1.4.m4.1a"><mi id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.1b"><ci id="S4.p1.4.m4.1.1.cmml" xref="S4.p1.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.1d">italic_n</annotation></semantics></math> in the reconstruction. The greatest contrast is in the single knot case (<math alttext="w" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><mi id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><ci id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_w</annotation></semantics></math>CDM), for which there is now a very slight preference relative to <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.p1.6.m6.1"><semantics id="S4.p1.6.m6.1a"><mi id="S4.p1.6.m6.1.1" mathvariant="normal" xref="S4.p1.6.m6.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.p1.6.m6.1b"><ci id="S4.p1.6.m6.1.1.cmml" xref="S4.p1.6.m6.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.m6.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m6.1d">roman_Λ</annotation></semantics></math>CDM in DR2, whereas <math alttext="w" class="ltx_Math" display="inline" id="S4.p1.7.m7.1"><semantics id="S4.p1.7.m7.1a"><mi id="S4.p1.7.m7.1.1" xref="S4.p1.7.m7.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.p1.7.m7.1b"><ci id="S4.p1.7.m7.1.1.cmml" xref="S4.p1.7.m7.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m7.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m7.1d">italic_w</annotation></semantics></math>CDM was reasonably disfavoured in DR1. As the number of knots increases, the evidences tend to a lower value for DR2 than for DR1. This means that DR2 is less supportive than DR1 of the flexknot model.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.2">The most significant change to the reconstructed <math alttext="w(a)" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mrow id="S4.p2.1.m1.1.2" xref="S4.p2.1.m1.1.2.cmml"><mi id="S4.p2.1.m1.1.2.2" xref="S4.p2.1.m1.1.2.2.cmml">w</mi><mo id="S4.p2.1.m1.1.2.1" xref="S4.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.p2.1.m1.1.2.3.2" xref="S4.p2.1.m1.1.2.cmml"><mo id="S4.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S4.p2.1.m1.1.2.cmml">(</mo><mi id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">a</mi><mo id="S4.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S4.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.2.cmml" xref="S4.p2.1.m1.1.2"><times id="S4.p2.1.m1.1.2.1.cmml" xref="S4.p2.1.m1.1.2.1"></times><ci id="S4.p2.1.m1.1.2.2.cmml" xref="S4.p2.1.m1.1.2.2">𝑤</ci><ci id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_w ( italic_a )</annotation></semantics></math> is the transition around <math alttext="a=0.6-0.7" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><mrow id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml"><mi id="S4.p2.2.m2.1.1.2" xref="S4.p2.2.m2.1.1.2.cmml">a</mi><mo id="S4.p2.2.m2.1.1.1" xref="S4.p2.2.m2.1.1.1.cmml">=</mo><mrow id="S4.p2.2.m2.1.1.3" xref="S4.p2.2.m2.1.1.3.cmml"><mn id="S4.p2.2.m2.1.1.3.2" xref="S4.p2.2.m2.1.1.3.2.cmml">0.6</mn><mo id="S4.p2.2.m2.1.1.3.1" xref="S4.p2.2.m2.1.1.3.1.cmml">−</mo><mn id="S4.p2.2.m2.1.1.3.3" xref="S4.p2.2.m2.1.1.3.3.cmml">0.7</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><apply id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1"><eq id="S4.p2.2.m2.1.1.1.cmml" xref="S4.p2.2.m2.1.1.1"></eq><ci id="S4.p2.2.m2.1.1.2.cmml" xref="S4.p2.2.m2.1.1.2">𝑎</ci><apply id="S4.p2.2.m2.1.1.3.cmml" xref="S4.p2.2.m2.1.1.3"><minus id="S4.p2.2.m2.1.1.3.1.cmml" xref="S4.p2.2.m2.1.1.3.1"></minus><cn id="S4.p2.2.m2.1.1.3.2.cmml" type="float" xref="S4.p2.2.m2.1.1.3.2">0.6</cn><cn id="S4.p2.2.m2.1.1.3.3.cmml" type="float" xref="S4.p2.2.m2.1.1.3.3">0.7</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">a=0.6-0.7</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_a = 0.6 - 0.7</annotation></semantics></math>, which is the region containing the two LRG points whose affect was investigated in detail in <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite>. This has moved to slightly higher redshifts, and is tighter than in DR1. The DR2 reconstructions have a more pronounced transition at slightly higher redshifts between phantom and quintessence than DR1, which continues to be the case when supernovae are included.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.3">Looking specifically, however, at <math alttext="n=2" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mrow id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml"><mi id="S4.p3.1.m1.1.1.2" xref="S4.p3.1.m1.1.1.2.cmml">n</mi><mo id="S4.p3.1.m1.1.1.1" xref="S4.p3.1.m1.1.1.1.cmml">=</mo><mn id="S4.p3.1.m1.1.1.3" xref="S4.p3.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><apply id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"><eq id="S4.p3.1.m1.1.1.1.cmml" xref="S4.p3.1.m1.1.1.1"></eq><ci id="S4.p3.1.m1.1.1.2.cmml" xref="S4.p3.1.m1.1.1.2">𝑛</ci><cn id="S4.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.p3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">n=2</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_n = 2</annotation></semantics></math> (i.e. the CPL parameterisation), we note that the evidence slightly increases relative to <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mi id="S4.p3.2.m2.1.1" mathvariant="normal" xref="S4.p3.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><ci id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">roman_Λ</annotation></semantics></math>CDM with DR2 compared to DR1. This matches the conclusion of <cite class="ltx_cite ltx_citemacro_cite">DESI Collaboration et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib8" title="">2025a</a>)</cite> that DR2 more strongly supports CPL dark energy over <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.p3.3.m3.1"><semantics id="S4.p3.3.m3.1a"><mi id="S4.p3.3.m3.1.1" mathvariant="normal" xref="S4.p3.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.p3.3.m3.1b"><ci id="S4.p3.3.m3.1.1.cmml" xref="S4.p3.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m3.1d">roman_Λ</annotation></semantics></math>CDM than DR1, although the overall evidence for both flexknots, and CPL, is marginal.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.2">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S3.F3" title="Figure 3 ‣ 3 Methods ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">3</span></a> shows a comparison of results from the DESI DR1/DR2 BAO data when combined with Pantheon+ and DES5Y supernovae, respectively. The evidences for <math alttext="w" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><mi id="S4.p4.1.m1.1.1" xref="S4.p4.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.p4.1.m1.1b"><ci id="S4.p4.1.m1.1.1.cmml" xref="S4.p4.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">italic_w</annotation></semantics></math>CDM in DR2 are very similar to those with DR1 and, as when using DESI alone, the evidences tend to a lower value for DR2 than for DR1 as the number of knots <math alttext="n" class="ltx_Math" display="inline" id="S4.p4.2.m2.1"><semantics id="S4.p4.2.m2.1a"><mi id="S4.p4.2.m2.1.1" xref="S4.p4.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.p4.2.m2.1b"><ci id="S4.p4.2.m2.1.1.cmml" xref="S4.p4.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.p4.2.m2.1d">italic_n</annotation></semantics></math> increases. Most notably, there is now a preference for models with large numbers of knots with DESI DR2 + DES5Y, which was the only combination in <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite> which had any evidence in favour of dynamical dark energy, and the CPL model is still favoured.</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.3">When combining data it is prudent to check that the datasets are consistent with each other. Here we follow <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite> in using the tension analysis developed and deployed in <cite class="ltx_cite ltx_citemacro_cite">Handley & Lemos (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib17" title="">2019</a>); Hergt et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib25" title="">2021</a>); Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib33" title="">2023</a>)</cite> via the <math alttext="\log R" class="ltx_Math" display="inline" id="S4.p5.1.m1.1"><semantics id="S4.p5.1.m1.1a"><mrow id="S4.p5.1.m1.1.1" xref="S4.p5.1.m1.1.1.cmml"><mi id="S4.p5.1.m1.1.1.1" xref="S4.p5.1.m1.1.1.1.cmml">log</mi><mo id="S4.p5.1.m1.1.1a" lspace="0.167em" xref="S4.p5.1.m1.1.1.cmml">⁡</mo><mi id="S4.p5.1.m1.1.1.2" xref="S4.p5.1.m1.1.1.2.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.1.m1.1b"><apply id="S4.p5.1.m1.1.1.cmml" xref="S4.p5.1.m1.1.1"><log id="S4.p5.1.m1.1.1.1.cmml" xref="S4.p5.1.m1.1.1.1"></log><ci id="S4.p5.1.m1.1.1.2.cmml" xref="S4.p5.1.m1.1.1.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.1.m1.1c">\log R</annotation><annotation encoding="application/x-llamapun" id="S4.p5.1.m1.1d">roman_log italic_R</annotation></semantics></math> statistic. In Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S3.F4" title="Figure 4 ‣ 3 Methods ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">4</span></a> we show the tension quantifications between DESI and supernovae datasets. For DESI vs Pantheon+, there has been a slight increase in tension between DR1 and DR2, but they remain consistent. For DESI vs DES5Y the tension has decreased between DR1 and DR2, and is now consistent for all but the <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.p5.2.m2.1"><semantics id="S4.p5.2.m2.1a"><mi id="S4.p5.2.m2.1.1" mathvariant="normal" xref="S4.p5.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.p5.2.m2.1b"><ci id="S4.p5.2.m2.1.1.cmml" xref="S4.p5.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p5.2.m2.1d">roman_Λ</annotation></semantics></math>CDM and <math alttext="w" class="ltx_Math" display="inline" id="S4.p5.3.m3.1"><semantics id="S4.p5.3.m3.1a"><mi id="S4.p5.3.m3.1.1" xref="S4.p5.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.p5.3.m3.1b"><ci id="S4.p5.3.m3.1.1.cmml" xref="S4.p5.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.p5.3.m3.1d">italic_w</annotation></semantics></math>CDM cases. In this sense, dynamical dark energy models can be viewed as resolving a discrepancy between DESI and DES5Y data, in addition to providing a better fit to the data.</p> </div> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.1">For reference, we show the non-overlayed reconstructions of <math alttext="w(a)" class="ltx_Math" display="inline" id="S4.p6.1.m1.1"><semantics id="S4.p6.1.m1.1a"><mrow id="S4.p6.1.m1.1.2" xref="S4.p6.1.m1.1.2.cmml"><mi id="S4.p6.1.m1.1.2.2" xref="S4.p6.1.m1.1.2.2.cmml">w</mi><mo id="S4.p6.1.m1.1.2.1" xref="S4.p6.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.p6.1.m1.1.2.3.2" xref="S4.p6.1.m1.1.2.cmml"><mo id="S4.p6.1.m1.1.2.3.2.1" stretchy="false" xref="S4.p6.1.m1.1.2.cmml">(</mo><mi id="S4.p6.1.m1.1.1" xref="S4.p6.1.m1.1.1.cmml">a</mi><mo id="S4.p6.1.m1.1.2.3.2.2" stretchy="false" xref="S4.p6.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p6.1.m1.1b"><apply id="S4.p6.1.m1.1.2.cmml" xref="S4.p6.1.m1.1.2"><times id="S4.p6.1.m1.1.2.1.cmml" xref="S4.p6.1.m1.1.2.1"></times><ci id="S4.p6.1.m1.1.2.2.cmml" xref="S4.p6.1.m1.1.2.2">𝑤</ci><ci id="S4.p6.1.m1.1.1.cmml" xref="S4.p6.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.1.m1.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S4.p6.1.m1.1d">italic_w ( italic_a )</annotation></semantics></math> for the remaining combinations of datasets in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#Sx2.F5" title="Figure 5 ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">5</span></a>. DESI alone can be seen in the lower panels of Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#S2.F2" title="Figure 2 ‣ 2 Data ‣ Comparison of dynamical dark energy with ΛCDM in light of DESI DR2"><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Conclusions</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.2">We revisited the dynamical dark energy reconstructions presented in <cite class="ltx_cite ltx_citemacro_cite">Ormondroyd et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib34" title="">2025</a>)</cite> using the newly-released DESI DR2 BAO data in combination with Pantheon+ and DES5Y supernovae measurements, respectively. Our analysis employed a flexknot methodology to reconstruct the evolution of the dark energy equation of state <math alttext="w(a)" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mrow id="S5.p1.1.m1.1.2" xref="S5.p1.1.m1.1.2.cmml"><mi id="S5.p1.1.m1.1.2.2" xref="S5.p1.1.m1.1.2.2.cmml">w</mi><mo id="S5.p1.1.m1.1.2.1" xref="S5.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.p1.1.m1.1.2.3.2" xref="S5.p1.1.m1.1.2.cmml"><mo id="S5.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.p1.1.m1.1.2.cmml">(</mo><mi id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">a</mi><mo id="S5.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.2.cmml" xref="S5.p1.1.m1.1.2"><times id="S5.p1.1.m1.1.2.1.cmml" xref="S5.p1.1.m1.1.2.1"></times><ci id="S5.p1.1.m1.1.2.2.cmml" xref="S5.p1.1.m1.1.2.2">𝑤</ci><ci id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">italic_w ( italic_a )</annotation></semantics></math>. Overall, we find that while the qualitative shape of the reconstructed <math alttext="w(a)" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mrow id="S5.p1.2.m2.1.2" xref="S5.p1.2.m2.1.2.cmml"><mi id="S5.p1.2.m2.1.2.2" xref="S5.p1.2.m2.1.2.2.cmml">w</mi><mo id="S5.p1.2.m2.1.2.1" xref="S5.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S5.p1.2.m2.1.2.3.2" xref="S5.p1.2.m2.1.2.cmml"><mo id="S5.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S5.p1.2.m2.1.2.cmml">(</mo><mi id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml">a</mi><mo id="S5.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S5.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.2.cmml" xref="S5.p1.2.m2.1.2"><times id="S5.p1.2.m2.1.2.1.cmml" xref="S5.p1.2.m2.1.2.1"></times><ci id="S5.p1.2.m2.1.2.2.cmml" xref="S5.p1.2.m2.1.2.2">𝑤</ci><ci id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">w(a)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">italic_w ( italic_a )</annotation></semantics></math> remains consistent with our previous work, there is a marked change in the statistical evidence for dynamical dark energy.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.4">The DESI DR2 data alone lead to a reduction in the evidence as compared with DR1 for flexknot models with larger numbers of knots. However, the evidence for the <math alttext="w" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mi id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_w</annotation></semantics></math>CDM and CPL models, which correspond to <math alttext="n=1" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mrow id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml"><mi id="S5.p2.2.m2.1.1.2" xref="S5.p2.2.m2.1.1.2.cmml">n</mi><mo id="S5.p2.2.m2.1.1.1" xref="S5.p2.2.m2.1.1.1.cmml">=</mo><mn id="S5.p2.2.m2.1.1.3" xref="S5.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><apply id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1"><eq id="S5.p2.2.m2.1.1.1.cmml" xref="S5.p2.2.m2.1.1.1"></eq><ci id="S5.p2.2.m2.1.1.2.cmml" xref="S5.p2.2.m2.1.1.2">𝑛</ci><cn id="S5.p2.2.m2.1.1.3.cmml" type="integer" xref="S5.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">n=1</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_n = 1</annotation></semantics></math> and <math alttext="n=2" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><mrow id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml"><mi id="S5.p2.3.m3.1.1.2" xref="S5.p2.3.m3.1.1.2.cmml">n</mi><mo id="S5.p2.3.m3.1.1.1" xref="S5.p2.3.m3.1.1.1.cmml">=</mo><mn id="S5.p2.3.m3.1.1.3" xref="S5.p2.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.1b"><apply id="S5.p2.3.m3.1.1.cmml" xref="S5.p2.3.m3.1.1"><eq id="S5.p2.3.m3.1.1.1.cmml" xref="S5.p2.3.m3.1.1.1"></eq><ci id="S5.p2.3.m3.1.1.2.cmml" xref="S5.p2.3.m3.1.1.2">𝑛</ci><cn id="S5.p2.3.m3.1.1.3.cmml" type="integer" xref="S5.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">n=2</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">italic_n = 2</annotation></semantics></math> knots respectively, have increased compared to <math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><mi id="S5.p2.4.m4.1.1" mathvariant="normal" xref="S5.p2.4.m4.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><ci id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">roman_Λ</annotation></semantics></math>CDM with DESI alone, which aligns with the conclusions of <cite class="ltx_cite ltx_citemacro_cite">DESI Collaboration et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib8" title="">2025a</a>)</cite>. When the DESI DR2 BAO data are combined with Pantheon+ supernovae, the conclusions are similar to those in our original work.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.2">However, with DES5Y supernovae, there is now increased evidence for models with a larger number of knots, with evidence for CPL, which remains the preferred model, also increasing, and with <math alttext="w" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mi id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><ci id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">italic_w</annotation></semantics></math>CDM remaining on-par with <math alttext="\Lambda" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mi id="S5.p3.2.m2.1.1" mathvariant="normal" xref="S5.p3.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><ci id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">roman_Λ</annotation></semantics></math>CDM. In addition to providing a better fit, dynamical dark energy models serve to resolve a discrepancy between DESI and DES5Y data.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">This work was performed using the Cambridge Service for Data Driven Discovery (CSD3), part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility (<a class="ltx_ref ltx_url ltx_font_typewriter" href="www.dirac.ac.uk" title="">www.dirac.ac.uk</a>). The DiRAC component of CSD3 was funded by BEIS capital funding via STFC capital grants ST/P002307/1 and ST/R002452/1 and STFC operations grant ST/R00689X/1. DiRAC is part of the National e-Infrastructure.</p> </div> <div class="ltx_para" id="Sx1.p2"> <p class="ltx_p" id="Sx1.p2.1">The tension calculations in this work made use of <span class="ltx_text ltx_font_typewriter" id="Sx1.p2.1.1">NumPy</span> <cite class="ltx_cite ltx_citemacro_citep">(Harris et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib21" title="">2020</a>)</cite>, <span class="ltx_text ltx_font_typewriter" id="Sx1.p2.1.2">SciPy</span> <cite class="ltx_cite ltx_citemacro_citep">(Virtanen et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib42" title="">2020</a>)</cite>, and <span class="ltx_text ltx_font_typewriter" id="Sx1.p2.1.3">pandas</span> <cite class="ltx_cite ltx_citemacro_citep">(The pandas development team, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib39" title="">2023</a>; McKinney, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib29" title="">2010</a>)</cite>. The plots were produced in <span class="ltx_text ltx_font_typewriter" id="Sx1.p2.1.4">matplotlib</span> <cite class="ltx_cite ltx_citemacro_citep">(Hunter, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib26" title="">2007</a>)</cite>, using the <span class="ltx_text ltx_font_typewriter" id="Sx1.p2.1.5">smplotlib</span> template created by <cite class="ltx_cite ltx_citemacro_citet">Li (<a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib28" title="">2023</a>)</cite>.</p> </div> </section> <section class="ltx_section" id="Sx2"> <h2 class="ltx_title ltx_title_section">Data Availability</h2> <div class="ltx_para" id="Sx2.p1"> <p class="ltx_p" id="Sx2.p1.1">The pared-down Python pipeline and nested sampling chains used in this work can be obtained from Zenodo <cite class="ltx_cite ltx_citemacro_citep">(Ormondroyd, <a class="ltx_ref" href="https://arxiv.org/html/2503.17342v1#bib.bib32" title="">2025</a>)</cite>.</p> </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"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Aslanyan et al. (2014)</span> <span class="ltx_bibblock">