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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. 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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="zh">  <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=zh_CN&referer=http%3A//cds.cern.ch/record/2917836%3Fln%3Dzh_CN" 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="http://cds.cern.ch/?ln=zh_CN">搜寻</a></li> <li class="menu-444 "><a class="" title="" href="http://cds.cern.ch/submit?ln=zh_CN">提交</a></li> <li class="menu-426 "><a class="" href="http://cds.cern.ch/help/?ln=zh_CN">帮助</a></li> <li class="leaf hassubcdsmenu"> <a hreflang="en" class="header" href="https://cds.cern.ch/youraccount/display?ln=zh_CN">个人化</a> <ul class="subsubcdsmenu"><li><a href="https://cds.cern.ch/youralerts/list?ln=zh_CN">Your alerts</a></li><li><a href="https://cds.cern.ch/yourbaskets/display?ln=zh_CN">Your baskets</a></li><li><a href="https://cds.cern.ch/yourcomments?ln=zh_CN">Your comments</a></li><li><a href="https://cds.cern.ch/youralerts/display?ln=zh_CN">Your searches</a></li></ul></li> </ul> </div> </div> </div>  <table class="navtrailbox"> <tr> <td class="navtrailboxbody"> <a href="/?ln=zh_CN" class="navtrail">主页</a> > Bounds on the bubble wall velocity </td> </tr> </table> </div> <div class="pagebody"><div class="pagebodystripemiddle"> <div class="detailedrecordbox"> <div class="detailedrecordtabs"> <div> <ul class="detailedrecordtabs"><li class="on first"><a href="/record/2917836/?ln=zh_CN">Information </a></li><li class=""><a href="/record/2917836/files?ln=zh_CN">Files </a></li></ul> <div id="tabsSpacer" style="clear:both;height:0px"> </div></div> </div> <div class="detailedrecordboxcontent"> <div class="top-left-folded"></div> <div class="top-right-folded"></div> <div class="inside">  <abbr class="unapi-id" title="2917836"></abbr>  <link href="/img/download_and_embed_buttons.css" rel="stylesheet" type="text/css" /> <style type="text/css">  .video_talk_wrapper{ max-width: 640px; min-height: 360px; } </style> <table class="formatRecordTableFullWidth"> <tr> <td class="formatRecordHeader" style="background-image: url('/img/journals.jpg');" colspan="2">  Preprint </td> </tr> <tr><td class="formatRecordLabel"> Report number </td><td style="padding-left:5px;">CERN-TH-2024-198 ; KCL-PH-TH/2024-57 ; <a href="http://arxiv.org/abs/arXiv:2411.13641">arXiv:2411.13641</a></td></tr> <tr><td class="formatRecordLabel"> Title </td><td style="padding-left:5px;"><b>Bounds on the bubble wall velocity</b></td></tr> <style> .video_control_container, #video_controlbutton { display: none; }#video_detailbox_download {display: none;}#video_detailbox_embed {display: none;}</style> <div class="video_control_container"> <div style="box-sizing: inherit;position: relative;min-height: 1em;margin: 1em 0;background: #f8f8f9;padding: 1em 1.5em;line-height: 1.4285em;transition: opacity .1s ease,color .1s ease,background .1s ease,box-shadow .1s ease,-webkit-box-shadow .1s ease;border-radius: .28571429rem;font-size: 1.1em;margin-top: 0;margin-bottom: 6px;background-color: #f8ffff;color: #276f86;box-shadow: 0 0 0 1px #a9d5de inset,0 0 0 0 transparent;"> <i class="fa fa-info"></i><span style="padding-left: 9px;">If you experience any problem watching the video, click the download button below</span> </div> <a id="video_controlbutton_download" rel="download_button" class="video_controlbutton" href="#">Download</a> <a id="video_controlbutton_embed" class="video_controlbutton" href="#">Embed</a> </div> <div id="video_detailbox_download" style="margin:10px"> </div> <div id="video_detailbox_embed" style="margin:10px"> </div> <tr><td class="formatRecordLabel"> <span style="white-space:nowrap;">Author(s)</span> </td><td style="padding-left:5px;"><a href="http://cds.cern.ch/search?f=author&p=Ai%2C%20Wen-Yuan&ln=zh_CN">Ai, Wen-Yuan</a> (King's Coll. London) ; <a href="http://cds.cern.ch/search?f=author&p=Laurent%2C%20Benoit&ln=zh_CN">Laurent, Benoit</a> (McGill U., Montreal (main)) ; <a href="http://cds.cern.ch/search?f=author&p=van%20de%20Vis%2C%20Jorinde&ln=zh_CN">van de Vis, Jorinde</a> (CERN)</td></tr> <tr><td class="formatRecordLabel"> Document contact </td><td style="padding-left:5px;"><span style="">Contact: </span>arXiv</td></tr>  <script type="text/javascript"> /* * Changes the Imprint title to Date - Duration as requested by RQF0820114 * This is a hack not to touch the general imprint format element */ $(document).ready(function() { var collections = 'PREPRINT'; if (collections.indexOf('E-LEARNING') > -1) { $('.cern-imprint-label').text('Date - Duration'); } }) </script> <tr><td class="formatRecordLabel cern-imprint-label"> Imprint </td><td style="padding-left:5px;">2024-11-20</td></tr> <tr><td class="formatRecordLabel"> Number of pages </td><td style="padding-left:5px;">47</td></tr>  <tr><td class="formatRecordLabel"> Note </td><td style="padding-left:5px;">47 pages, 8 figures</td></tr> <tr><td class="formatRecordLabel"> Subject category </td><td style="padding-left:5px;">astro-ph.CO ; Astrophysics and Astronomy ; hep-ph ; Particle Physics - Phenomenology</td></tr> <tr><td class="formatRecordLabel"> Abstract </td><td style="padding-left:5px;">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.</td></tr> <tr><td class="formatRecordLabel"> Other source </td><td style="padding-left:5px;"><a href="http://inspirehep.net/record/2850555">Inspire</a></td></tr> <tr><td class="formatRecordLabel"> Copyright/License </td><td style="padding-left:5px;">preprint: (License: <a href="http://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>)</td></tr> </table> <br /><div style="max-width:1024px;margin:auto"><div style="overflow-x:auto;display:inline;width:100%;"><a href="/record/2917836/plots#0"><img style="vertical-align:middle;" src="http://cds.cern.ch/record/2917836/files/bubble_wall_dynamics.png" title=" 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." width="200px"/></a> <a href="/record/2917836/plots#1"><img style="vertical-align:middle;" src="http://cds.cern.ch/record/2917836/files/pressures_m1=0_m2=3_alpha=0.3_b=1.png" title=" 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 $\mathcal{P}_{\rm friction}^{\rm b}/\mathcal{P}_{\rm driving}=1$." width="200px"/></a> <a href="/record/2917836/plots#2"><img style="vertical-align:middle;" src="http://cds.cern.ch/record/2917836/files/alphaVsB_m1=0_m2=1_vwBall.png" title=" \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). }" width="200px"/></a> <a href='/record/2917836/plots'>Show more plots</a></div></div><br /> <br/><div><div style="clear: both;"> </div></div> <script type="text/javascript"> // Initially hide: $(".longCaption").hide(); // Allow to toggle visibility: $(".toggleLongCaption").toggle(function(){ $(this).siblings(".longCaption").show('fast'); var thisElem = $(this); thisElem.text(thisElem.text() === "更多" ? "less" : "更多"); },function(){ $(this).siblings(".longCaption").hide('fast'); var thisElem = $(this); thisElem.text(thisElem.text() === "更多" ? 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