CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">     <html xmlns="http://www.w3.org/1999/xhtml" lang="ca" xml:lang="ca" xmlns:og="http://ogp.me/ns#" xmlns:fb="http://ogp.me/ns/fb#"> <head> <title>Bounds on the bubble wall velocity - CERN Document Server</title> <link href='https://framework.web.cern.ch/framework/2.0/fonts/PTSansWeb/PTSansWeb.css' rel='stylesheet' type='text/css' /> <link rel="stylesheet" href="https://cds.cern.ch/img/invenio.css?v=20141127" type="text/css" /> <link rel="stylesheet" href="https://cds.cern.ch/img/cern_theme/css/cern_theme.css?v=20141127" type="text/css" /> <link rel="stylesheet"href="/css/font-awesome.min.css"> <meta http-equiv="X-UA-Compatible" content="IE=Edge"/> <link rel="stylesheet" href="https://cds.cern.ch/img/cern_toolbar/css/toolbar.css" type="text/css" />    <link rel="canonical" href="https://cds.cern.ch/record/2917836/export/hm" /> <link rel="alternate" hreflang="el" href="https://cds.cern.ch/record/2917836/export/hm?ln=el" /> <link rel="alternate" hreflang="fr" href="https://cds.cern.ch/record/2917836/export/hm?ln=fr" /> <link rel="alternate" hreflang="bg" href="https://cds.cern.ch/record/2917836/export/hm?ln=bg" /> <link rel="alternate" hreflang="zh-TW" href="https://cds.cern.ch/record/2917836/export/hm?ln=zh_TW" /> <link rel="alternate" hreflang="pt" href="https://cds.cern.ch/record/2917836/export/hm?ln=pt" /> <link rel="alternate" hreflang="no" href="https://cds.cern.ch/record/2917836/export/hm?ln=no" /> <link rel="alternate" hreflang="hr" href="https://cds.cern.ch/record/2917836/export/hm?ln=hr" /> <link rel="alternate" hreflang="ca" href="https://cds.cern.ch/record/2917836/export/hm?ln=ca" /> <link rel="alternate" hreflang="de" href="https://cds.cern.ch/record/2917836/export/hm?ln=de" /> <link rel="alternate" hreflang="it" href="https://cds.cern.ch/record/2917836/export/hm?ln=it" /> <link rel="alternate" hreflang="zh-CN" href="https://cds.cern.ch/record/2917836/export/hm?ln=zh_CN" /> <link rel="alternate" hreflang="sv" href="https://cds.cern.ch/record/2917836/export/hm?ln=sv" /> <link rel="alternate" hreflang="sk" href="https://cds.cern.ch/record/2917836/export/hm?ln=sk" /> <link rel="alternate" hreflang="en" href="https://cds.cern.ch/record/2917836/export/hm?ln=en" /> <link rel="alternate" hreflang="pl" href="https://cds.cern.ch/record/2917836/export/hm?ln=pl" /> <link rel="alternate" hreflang="ru" href="https://cds.cern.ch/record/2917836/export/hm?ln=ru" /> <link rel="alternate" hreflang="ka" href="https://cds.cern.ch/record/2917836/export/hm?ln=ka" /> <link rel="alternate" hreflang="ja" href="https://cds.cern.ch/record/2917836/export/hm?ln=ja" /> <link rel="alternate" hreflang="es" href="https://cds.cern.ch/record/2917836/export/hm?ln=es" /> <link rel="alternate" type="application/rss+xml" title="CERN Document Server RSS" href="/rss?ln=ca" /> <link rel="search" type="application/opensearchdescription+xml" href="https://cds.cern.ch/opensearchdescription" title="CERN Document Server" /> <link rel="unapi-server" type="application/xml" title="unAPI" href="https://cds.cern.ch/unapi" /> <link rel="apple-touch-icon" href="/apple-touch-icon.png"/> <link rel="apple-touch-icon-precomposed" href="/apple-touch-icon-precomposed.png"/> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <meta http-equiv="Content-Language" content="ca" /> <meta name="description" content="Determining the bubble wall velocity in first-order phase transitions is a challenging task, requiring the solution of (coupled) equations of motion for the scalar field and Boltzmann equations for the particles in the plasma. The collision terms appearing in the Boltzmann equation present a prominent source of uncertainty as they are often known only at leading log accuracy. In this paper, we derive upper and lower bounds on the wall velocity, corresponding to the local thermal equilibrium and ballistic limits. These bounds are completely independent of the collision terms. For the ballistic approximation, we argue that the inhomogeneous plasma temperature and velocity distributions across the bubble wall should be taken into account. This way, the hydrodynamic obstruction previously observed in local thermal equilibrium is also present for the ballistic approximation. This is essential for the ballistic approximation to provide a lower bound on the wall velocity. We use a model-independent approach to study the behaviour of the limiting wall velocities as a function of a few generic parameters, and we test our developments in the singlet extended Standard Model. Ai, Wen-Yuan; Laurent, Benoit; van de Vis, Jorinde" /> <meta name="keywords" content="CERN Document Server, WebSearch, CERN Document Server" /> <script type="text/javascript" src="https://cds.cern.ch/js/jquery.min.js"></script>  <link rel="stylesheet" href="https://cds.cern.ch/img/webnews.css" type="text/css" />  <script type="text/javascript" src="https://cds.cern.ch/js/webnews.js?v=20131009"></script> <meta property="fb:app_id" content="137353533001720"/>  <meta content="arXiv : Bounds on the bubble wall velocity" name="citation_title" /> <meta content="Ai, Wen-Yuan" name="citation_author" /> <meta content="Laurent, Benoit" name="citation_author" /> <meta content="van de Vis, Jorinde" name="citation_author" /> <meta content="2024/11/20" name="citation_publication_date" /> <meta name="citation_online_date" content="2024/11/23"> <meta content="arXiv:2411.13641" name="citation_technical_report_number" /> <meta content="KCL-PH-TH/2024-57" name="citation_technical_report_number" /> <meta content="CERN-TH-2024-198" name="citation_technical_report_number" /> <meta name="citation_pdf_url" content="https://cds.cern.ch/record/2917836/files/2411.13641.pdf" />  <meta content="arXiv" property="og:title" /> <meta content="Bounds on the bubble wall velocity" property="og:title" /> <meta content="website" property="og:type" /> <meta content="https://cds.cern.ch/record/2917836" property="og:url" /> <meta name="twitter:image" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_sigma.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_sigma.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_sigma.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/xSMPressureTn100.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/xSMPressureTn100.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/xSMPressureTn90.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/xSMPressureTn90.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwLTE-vwBall.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwLTE-vwBall.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/LMLDetonation_vwBall.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/LMLDetonation_vwBall.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwLTE.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwLTE.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/alphaVsB_detonation_m1=0_m2=1_sigma.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/alphaVsB_detonation_m1=0_m2=1_sigma.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwLTE.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwLTE.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_detonation_alphaP=0.01_b=0.1_vwBall.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_detonation_alphaP=0.01_b=0.1_vwBall.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/alphaVsB_detonation_m1=0_m2=1_vwBall.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/alphaVsB_detonation_m1=0_m2=1_vwBall.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_detonation_alphaP=0.01_b=0.1_sigma.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_detonation_alphaP=0.01_b=0.1_sigma.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwLTE-vwBall.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwLTE-vwBall.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_sigma.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_sigma.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwBall.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwBall.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/pressures_m1=0_m2=3_alpha=0.3_b=1.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/pressures_m1=0_m2=3_alpha=0.3_b=1.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwBall.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwBall.png" /> <meta property="og:image" content="https://cds.cern.ch/record/2917836/files/bubble_wall_dynamics.png" /> <meta property="og:image:secure_url" content="https://cds.cern.ch/record/2917836/files/bubble_wall_dynamics.png" /> <meta content="CERN Document Server" property="og:site_name" /> <meta content="Determining the bubble wall velocity in first-order phase transitions is a challenging task, requiring the solution of (coupled) equations of motion for the scalar field and Boltzmann equations for the particles in the plasma. The collision terms appearing in the Boltzmann equation present a prominent source of uncertainty as they are often known only at leading log accuracy. In this paper, we derive upper and lower bounds on the wall velocity, corresponding to the local thermal equilibrium and ballistic limits. These bounds are completely independent of the collision terms. For the ballistic approximation, we argue that the inhomogeneous plasma temperature and velocity distributions across the bubble wall should be taken into account. This way, the hydrodynamic obstruction previously observed in local thermal equilibrium is also present for the ballistic approximation. This is essential for the ballistic approximation to provide a lower bound on the wall velocity. We use a model-independent approach to study the behaviour of the limiting wall velocities as a function of a few generic parameters, and we test our developments in the singlet extended Standard Model." property="og:description" /> <meta content="arXiv" property="og:description" />  <meta content="summary" name="twitter:card" /> <style></style> </head> <body class="CERN32Document32Server search" lang="ca">  <div id="cern-toolbar"> <h1><a href="http://cern.ch" title="CERN">CERN <span>Accelerating science</span></a></h1> <ul> <li class="cern-accountlinks"><a class="cern-account" href="https://cds.cern.ch/youraccount/login?ln=ca&referer=https%3A//cds.cern.ch/record/2917836/export/hm%3Fln%3Dca" title="Sign in to your CERN account">Sign in</a></li> <li><a class="cern-directory" href="http://cern.ch/directory" title="Search CERN resources and browse the directory">Directory</a></li> </ul> </div>   <div role="banner" class="clearfix" id="header"> <div class="header-inner inner"> <hgroup class="clearfix"> <h2 id="site-name"> <a rel="home" title="Home" href="/"><span>CERN Document Server</span></a> </h2> <h3 id="site-slogan">Access articles, reports and multimedia content in HEP</h3> </hgroup> <div role="navigation" id="main-navigation" class="cdsmenu"> <h2 class="element-invisible">Main menu</h2><ul class="links inline clearfix"> <li class="menu-386 first active-trail"><a class="active-trail" href="https://cds.cern.ch/?ln=ca">Cerca</a></li> <li class="menu-444 "><a class="" title="" href="https://cds.cern.ch/submit?ln=ca">Lliura</a></li> <li class="menu-426 "><a class="" href="https://cds.cern.ch/help/?ln=ca">Ajuda</a></li> <li class="leaf hassubcdsmenu"> <a hreflang="en" class="header" href="https://cds.cern.ch/youraccount/display?ln=ca">Personalitza</a> <ul class="subsubcdsmenu"><li><a href="https://cds.cern.ch/youralerts/list?ln=ca">Les vostres alertes</a></li><li><a href="https://cds.cern.ch/yourbaskets/display?ln=ca">Els vostres cistells</a></li><li><a href="https://cds.cern.ch/yourcomments?ln=ca">Your comments</a></li><li><a href="https://cds.cern.ch/youralerts/display?ln=ca">Les vostres cerques</a></li></ul></li> </ul> </div> </div> </div>  <table class="navtrailbox"> <tr> <td class="navtrailboxbody"> <a href="/?ln=ca" class="navtrail">Pàgina inicial</a> > <a class="navtrail" href="/record/2917836">Bounds on the bubble wall velocity</a> > HTML MARC </td> </tr> </table> </div> <div class="pagebody"><div class="pagebodystripemiddle"> <pre style="margin: 1em 0px;">002917836 001__ 2917836 002917836 005__ 20241124041847.0 002917836 0248_ $$aoai:cds.cern.ch:2917836$$pcerncds:FULLTEXT$$pcerncds:CERN:FULLTEXT$$pcerncds:CERN 002917836 037__ $$9arXiv:reportnumber$$aCERN-TH-2024-198 002917836 037__ $$9arXiv:reportnumber$$aKCL-PH-TH/2024-57 002917836 035__ $$9arXiv$$aoai:arXiv.org:2411.13641 002917836 037__ $$9arXiv$$aarXiv:2411.13641$$chep-ph 002917836 035__ $$9Inspire$$aoai:inspirehep.net:2850555$$d2024-11-23T20:06:32Z$$h2024-11-24T03:00:22Z$$mmarcxml$$ttrue$$uhttps://inspirehep.net/api/oai2d 002917836 035__ $$9Inspire$$a2850555 002917836 041__ $$aeng 002917836 100__ $$aAi, Wen-Yuan$$mwenyuan.ai@kcl.ac.uk$$uKing's Coll. London$$vTheoretical Particle Physics and Cosmology, King's College London, Strand, London WC2R 2LS, United Kingdom 002917836 245__ $$9arXiv$$aBounds on the bubble wall velocity 002917836 269__ $$c2024-11-20 002917836 300__ $$a47 p 002917836 500__ $$9arXiv$$a47 pages, 8 figures 002917836 520__ $$9arXiv$$aDetermining the bubble wall velocity in first-order phase transitions is a challenging task, requiring the solution of (coupled) equations of motion for the scalar field and Boltzmann equations for the particles in the plasma. The collision terms appearing in the Boltzmann equation present a prominent source of uncertainty as they are often known only at leading log accuracy. In this paper, we derive upper and lower bounds on the wall velocity, corresponding to the local thermal equilibrium and ballistic limits. These bounds are completely independent of the collision terms. For the ballistic approximation, we argue that the inhomogeneous plasma temperature and velocity distributions across the bubble wall should be taken into account. This way, the hydrodynamic obstruction previously observed in local thermal equilibrium is also present for the ballistic approximation. This is essential for the ballistic approximation to provide a lower bound on the wall velocity. We use a model-independent approach to study the behaviour of the limiting wall velocities as a function of a few generic parameters, and we test our developments in the singlet extended Standard Model. 002917836 541__ $$aarXiv$$chepcrawl$$d2024-11-22T09:48:38.560668$$e8500281 002917836 540__ $$3preprint$$aCC BY 4.0$$uhttp://creativecommons.org/licenses/by/4.0/ 002917836 595__ $$aCERN-TH 002917836 65017 $$2arXiv$$aastro-ph.CO 002917836 65017 $$2SzGeCERN$$aAstrophysics and Astronomy 002917836 65017 $$2arXiv$$ahep-ph 002917836 65017 $$2SzGeCERN$$aParticle Physics - Phenomenology 002917836 690C_ $$aCERN 002917836 690C_ $$aPREPRINT 002917836 700__ $$aLaurent, Benoit$$mbenoit.laurent@mail.mcgill.ca$$uMcGill U., Montreal (main)$$vMcGill University, Department of Physics, 3600 University St., Montréal, QC H3A2T8 Canada 002917836 700__ $$avan de Vis, Jorinde$$mjorinde.van.de.vis@cern.ch$$uCERN$$vTheoretical Physics Department, CERN, 1 Esplanade des Particules, CH-1211 Geneva 23, Switzerland 002917836 8564_ $$82691833$$s13021$$uhttps://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_sigma.png$$y00009 \footnotesize{Contour plots of scans varying the parameters $m_+$ and $m_-$ with fixed $\alpha_p$ and $b$ for deflagration and hybrid solutions only. They show the same quantities as in Fig.\ \ref{fig:alphaBScan}. The left side is for a rather weak PT with $\alpha_p=0.01$ and $b=0.1$, while the right side is for a stronger PT with $\alpha_p=1$ and $b=1$.} 002917836 8564_ $$82691834$$s14616$$uhttps://cds.cern.ch/record/2917836/files/xSMPressureTn100.png$$y00015 Pressure as a function of the wall velocity in four different approximations for the xSM. 002917836 8564_ $$82691835$$s13278$$uhttps://cds.cern.ch/record/2917836/files/xSMPressureTn90.png$$y00016 Pressure as a function of the wall velocity in four different approximations for the xSM. 002917836 8564_ $$82691836$$s17944$$uhttps://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwLTE-vwBall.png$$y00004 \footnotesize{Contour plots of scans varying the parameters $\alpha_p$ and $b$ with fixed $m_+$ and $m_-$ for deflagration and hybrid solutions only. They show (from top to bottom) the ballistic solution $v_w^{\rm b}$, the LTE solution $v_w^{\rm LTE}$, the difference between the ballistic and LTE solutions $v_w^{\rm LTE}-v_w^{\rm b}$ and the entropy fraction generated by the ballistic solution $\sigma$. The left side is with $m_+=0$ and $m_-=T_n$, and the right side with $m_+=0$ and $m_-=\infty$ (the large mass limit). } 002917836 8564_ $$82691837$$s9547$$uhttps://cds.cern.ch/record/2917836/files/LMLDetonation_vwBall.png$$y00014 Contour plot of $\gamma_w^{\rm det}(\alpha_p,\Psi)$ for the velocity given in Eq.~\eqref{eq:largeMassDet}. The region below the red line violates the condition~\eqref{eq:condition_Psidet}. If further $m_-/T_n$ is much larger than $\gamma_w^{\rm det}$, then there is no detonation solution for the parameter region below the red line. 002917836 8564_ $$82691838$$s13759$$uhttps://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwLTE.png$$y00003 \footnotesize{Contour plots of scans varying the parameters $\alpha_p$ and $b$ with fixed $m_+$ and $m_-$ for deflagration and hybrid solutions only. They show (from top to bottom) the ballistic solution $v_w^{\rm b}$, the LTE solution $v_w^{\rm LTE}$, the difference between the ballistic and LTE solutions $v_w^{\rm LTE}-v_w^{\rm b}$ and the entropy fraction generated by the ballistic solution $\sigma$. The left side is with $m_+=0$ and $m_-=T_n$, and the right side with $m_+=0$ and $m_-=\infty$ (the large mass limit). } 002917836 8564_ $$82691839$$s7644$$uhttps://cds.cern.ch/record/2917836/files/alphaVsB_detonation_m1=0_m2=1_sigma.png$$y00011 \footnotesize{Contour plots of scans varying the parameters $\alpha_p$ and $b$ with fixed $m_+$ and $m_-$ for detonation solutions only. They show the ballistic solution $v_w^{\rm b}$ and the entropy fraction generated $\sigma$. The left side is with $m_+=0$ and $m_-=T_n$, and the right side with $m_+=0$ and $m_-=5T_n$.} 002917836 8564_ $$82691840$$s12313$$uhttps://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwLTE.png$$y00007 \footnotesize{Contour plots of scans varying the parameters $m_+$ and $m_-$ with fixed $\alpha_p$ and $b$ for deflagration and hybrid solutions only. They show the same quantities as in Fig.\ \ref{fig:alphaBScan}. The left side is for a rather weak PT with $\alpha_p=0.01$ and $b=0.1$, while the right side is for a stronger PT with $\alpha_p=1$ and $b=1$.} 002917836 8564_ $$82691841$$s8765$$uhttps://cds.cern.ch/record/2917836/files/m1Vsm2_detonation_alphaP=0.01_b=0.1_vwBall.png$$y00012 \footnotesize{Same as in Fig.\ \ref{fig:alphaBScanDeton} but varying $m_\pm$ with $\alpha_p=0.01$ and $b=0.1$ (left) and $\alpha_p=b=1$ (right).} 002917836 8564_ $$82691842$$s10148$$uhttps://cds.cern.ch/record/2917836/files/alphaVsB_detonation_m1=0_m2=1_vwBall.png$$y00010 \footnotesize{Contour plots of scans varying the parameters $\alpha_p$ and $b$ with fixed $m_+$ and $m_-$ for detonation solutions only. They show the ballistic solution $v_w^{\rm b}$ and the entropy fraction generated $\sigma$. The left side is with $m_+=0$ and $m_-=T_n$, and the right side with $m_+=0$ and $m_-=5T_n$.} 002917836 8564_ $$82691843$$s3136441$$uhttps://cds.cern.ch/record/2917836/files/2411.13641.pdf$$yFulltext 002917836 8564_ $$82691844$$s9128$$uhttps://cds.cern.ch/record/2917836/files/m1Vsm2_detonation_alphaP=0.01_b=0.1_sigma.png$$y00013 \footnotesize{Same as in Fig.\ \ref{fig:alphaBScanDeton} but varying $m_\pm$ with $\alpha_p=0.01$ and $b=0.1$ (left) and $\alpha_p=b=1$ (right).} 002917836 8564_ $$82691845$$s12822$$uhttps://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwLTE-vwBall.png$$y00008 \footnotesize{Contour plots of scans varying the parameters $m_+$ and $m_-$ with fixed $\alpha_p$ and $b$ for deflagration and hybrid solutions only. They show the same quantities as in Fig.\ \ref{fig:alphaBScan}. The left side is for a rather weak PT with $\alpha_p=0.01$ and $b=0.1$, while the right side is for a stronger PT with $\alpha_p=1$ and $b=1$.} 002917836 8564_ $$82691846$$s7236$$uhttps://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_sigma.png$$y00005 \footnotesize{Contour plots of scans varying the parameters $\alpha_p$ and $b$ with fixed $m_+$ and $m_-$ for deflagration and hybrid solutions only. They show (from top to bottom) the ballistic solution $v_w^{\rm b}$, the LTE solution $v_w^{\rm LTE}$, the difference between the ballistic and LTE solutions $v_w^{\rm LTE}-v_w^{\rm b}$ and the entropy fraction generated by the ballistic solution $\sigma$. The left side is with $m_+=0$ and $m_-=T_n$, and the right side with $m_+=0$ and $m_-=\infty$ (the large mass limit). } 002917836 8564_ $$82691847$$s15654$$uhttps://cds.cern.ch/record/2917836/files/m1Vsm2_alphaP=0.01_b=0.1_vwBall.png$$y00006 \footnotesize{Contour plots of scans varying the parameters $m_+$ and $m_-$ with fixed $\alpha_p$ and $b$ for deflagration and hybrid solutions only. They show the same quantities as in Fig.\ \ref{fig:alphaBScan}. The left side is for a rather weak PT with $\alpha_p=0.01$ and $b=0.1$, while the right side is for a stronger PT with $\alpha_p=1$ and $b=1$.} 002917836 8564_ $$82691848$$s12454$$uhttps://cds.cern.ch/record/2917836/files/pressures_m1=0_m2=3_alpha=0.3_b=1.png$$y00001 Different contributions to the ballistic pressure for a strong (left) and a weak (right) PT in the model presented in Section \ref{sec:model-independent-ballistic}. In both cases, the $t^-$-modes (which correspond to $\delta f_+$) give the smallest contribution. The points show the solutions satisfying $\mathcal{P}_{\rm friction}^{\rm b}/\mathcal{P}_{\rm driving}=1$. 002917836 8564_ $$82691849$$s15793$$uhttps://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwBall.png$$y00002 \footnotesize{Contour plots of scans varying the parameters $\alpha_p$ and $b$ with fixed $m_+$ and $m_-$ for deflagration and hybrid solutions only. They show (from top to bottom) the ballistic solution $v_w^{\rm b}$, the LTE solution $v_w^{\rm LTE}$, the difference between the ballistic and LTE solutions $v_w^{\rm LTE}-v_w^{\rm b}$ and the entropy fraction generated by the ballistic solution $\sigma$. The left side is with $m_+=0$ and $m_-=T_n$, and the right side with $m_+=0$ and $m_-=\infty$ (the large mass limit). } 002917836 8564_ $$82691850$$s28160$$uhttps://cds.cern.ch/record/2917836/files/bubble_wall_dynamics.png$$y00000 Sketch of bubble wall dynamics using the deflagration mode as an example. The bubble wall and shock front are viewed as having zero size at the hydrodynamic scale. The hydrodynamic quantities $T$ and $v$ are discontinuous at these fronts. To study the matching conditions, one then needs to zoom in on these fronts. 002917836 960__ $$a11 002917836 980__ $$aPREPRINT</pre></div></div> <footer id="footer" class="pagefooter clearfix">  <div class="pagefooterstripeleft"> CERN Document Server :: <a class="footer" href="https://cds.cern.ch/?ln=ca">Cerca</a> :: <a class="footer" href="https://cds.cern.ch/submit?ln=ca">Lliura</a> :: <a class="footer" href="https://cds.cern.ch/youraccount/display?ln=ca">Personalitza</a> :: <a class="footer" href="https://cds.cern.ch/help/?ln=ca">Ajuda</a> :: <a class="footer" href="https://cern.service-now.com/service-portal?id=privacy_policy&se=CDS-Service" target="_blank">Privacy Notice</a> <br /> Powered by <a class="footer" href="http://invenio-software.org/">Invenio</a> <br /> Mantingut per <a class="footer" href="https://cern.service-now.com/service-portal?id=service_element&name=CDS-Service">CDS Service</a> - Need help? Contact <a href="https://cern.service-now.com/service-portal?id=service_element&name=CDS-Service">CDS Support</a>. <br /> </div> <div class="pagefooterstriperight"> <div class="cern-logo"> <a id="logo" href="http://cern.ch" title="CERN" rel="CERN" ><img src="https://cds.cern.ch/img/cern_theme/img/cern-logo-large.png" alt="CERN" /></a> </div> <div class="cern-languagebox"> Aquest lloc està també disponible en els següents idiomes:<br /><a href="/record/2917836/export/hm?ln=bg" class="langinfo">Български</a>  <span class="langinfo">Català</span>  <a href="/record/2917836/export/hm?ln=de" class="langinfo">Deutsch</a>  <a href="/record/2917836/export/hm?ln=el" class="langinfo">Ελληνικά</a>  <a href="/record/2917836/export/hm?ln=en" class="langinfo">English</a>  <a href="/record/2917836/export/hm?ln=es" class="langinfo">Español</a>  <a href="/record/2917836/export/hm?ln=fr" class="langinfo">Français</a>  <a href="/record/2917836/export/hm?ln=hr" class="langinfo">Hrvatski</a>  <a href="/record/2917836/export/hm?ln=it" class="langinfo">Italiano</a>  <a href="/record/2917836/export/hm?ln=ja" class="langinfo">日本語</a>  <a href="/record/2917836/export/hm?ln=ka" class="langinfo">ქართული</a>  <a href="/record/2917836/export/hm?ln=no" class="langinfo">Norsk/Bokmål</a>  <a href="/record/2917836/export/hm?ln=pl" class="langinfo">Polski</a>  <a href="/record/2917836/export/hm?ln=pt" class="langinfo">Português</a>  <a href="/record/2917836/export/hm?ln=ru" class="langinfo">Русский</a>  <a href="/record/2917836/export/hm?ln=sk" class="langinfo">Slovensky</a>  <a href="/record/2917836/export/hm?ln=sv" class="langinfo">Svenska</a>  <a href="/record/2917836/export/hm?ln=zh_CN" class="langinfo">中文(简)</a>  <a href="/record/2917836/export/hm?ln=zh_TW" class="langinfo">中文(繁)</a> </div> </div>  </footer> <script type="text/javascript"> var SyndeticsBookCovers = (function() { var SMALL_SIZE = "sc.gif", MEDIUM_SIZE = "mc.gif", RAW_URL = "https://secure.syndetics.com/index.aspx?isbn=THEISBN/THESIZE&client=cernlibrary"; replaceCover = function(imgElement, isbns, hdFormat) { var img = new Image(), size = hdFormat ? MEDIUM_SIZE : SMALL_SIZE; var _isbns = isbns.sort(function(a, b) { // sort from shortest to longest ISBN (more modern) return a.length > b.length ? 1 : -1; }); function next() { var isbn = _isbns.pop(); if (isbn) { var url = RAW_URL.replace("THEISBN", isbn).replace("THESIZE", size); img.src = url; } } function done() { imgElement.src = img.src; } img.onload = function() { if (this.width > 1) { done(); } else { next(); } }; next(); }; return { replaceCover: replaceCover }; })(); $(document).ready(function() { // get book covers $("img.book-cover").each(function() { var $this = $(this), strIsbns = $this.data("isbns") || "", isbnsArray = String(strIsbns).split(","), hdFormat = $this.hasClass("hd"); SyndeticsBookCovers.replaceCover(this, isbnsArray, hdFormat); }); // WebNews tooltips $.ajax({ url: "/news/tooltips", success: function(data) { create_tooltips(data); }, dataType: "json", cache: false }); }); </script>  <script src="//cds.cern.ch/js/feedback.js"></script>   <script> var _paq = window._paq = window._paq || []; /* tracker methods like "setCustomDimension" should be called before "trackPageView" */ _paq.push(['trackPageView']); _paq.push(['enableLinkTracking']); (function() { var u="https://webanalytics.web.cern.ch/"; _paq.push(['setTrackerUrl', u+'matomo.php']); _paq.push(['setSiteId', '756']); var d=document, g=d.createElement('script'), s=d.getElementsByTagName('script')[0]; g.async=true; g.src=u+'matomo.js'; s.parentNode.insertBefore(g,s); })(); </script>  </body> </html>