<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>String cosmology</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/String_cosmology"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-String_cosmology rootpage-String_cosmology skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">String cosmology</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1129693374">
/* start https://en.wikipedia.org/ */
.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}
/* end https://en.wikipedia.org/ */
</style><style data-mw-deduplicate="TemplateStyles:r1246091330">
/* start https://en.wikipedia.org/ */
.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}
/* end https://en.wikipedia.org/ */
</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><th class="sidebar-title"><a href="String_theory" title="String theory">String theory</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><th class="sidebar-heading" style="background: #ddf">
Fundamental objects</th></tr><tr><td class="sidebar-content">
<ul><li><a href="String_(physics)" title="String (physics)">String</a></li>
<li><a href="Cosmic_string" title="Cosmic string">Cosmic string</a></li>
<li><a href="Brane" title="Brane">Brane</a></li>
<li><a href="D-brane" title="D-brane">D-brane</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background: #ddf">
Perturbative theory</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li>
<li><a href="Superstring_theory" title="Superstring theory">Superstring</a> (<a href="Type_I_string_theory" title="Type I string theory">Type I</a>, <a href="Type_II_string_theory" title="Type II string theory">Type II</a>, <a href="Heterotic_string_theory" title="Heterotic string theory">Heterotic</a>)</li></ul></td>
</tr><tr><th class="sidebar-heading" style="background: #ddf">
Non-perturbative results</th></tr><tr><td class="sidebar-content">
<ul><li><a href="S-duality" title="S-duality">S-duality</a></li>
<li><a href="T-duality" title="T-duality">T-duality</a></li>
<li><a href="U-duality" title="U-duality">U-duality</a></li>
<li><a href="M-theory" title="M-theory">M-theory</a></li>
<li><a href="F-theory" title="F-theory">F-theory</a></li>
<li><a href="AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background: #ddf">
Phenomenology</th></tr><tr><td class="sidebar-content">
<ul><li><a href="String_phenomenology" title="String phenomenology">Phenomenology</a></li>
<li><a href="String_theory_landscape" title="String theory landscape">Landscape</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background: #ddf">
Mathematics</th></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;">
<ul><li><a href="Geometric_Langlands_correspondence" title="Geometric Langlands correspondence">Geometric Langlands correspondence</a></li>
<li><a href="Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">Mirror symmetry</a></li>
<li><a href="Monstrous_moonshine" title="Monstrous moonshine">Monstrous moonshine</a></li>
<li><a href="Vertex_algebra" class="mw-redirect" title="Vertex algebra">Vertex algebra</a></li>
<li><a href="K-theory_(physics)" title="K-theory (physics)">K-theory</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background: #ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Related concepts</div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">
<ul><li><a href="Theory_of_everything" title="Theory of everything">Theory of everything</a></li>
<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li>
<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li>
<li><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a></li>
<li><a href="Supergravity" title="Supergravity">Supergravity</a></li>
<li><a href="Twistor_string_theory" title="Twistor string theory">Twistor string theory</a></li>
<li><a href="N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory"><i>N</i> = 4 supersymmetric Yang–Mills theory</a></li>
<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>
<li><a href="Multiverse" title="Multiverse">Multiverse</a></li>
<li><a href="Holographic_principle" title="Holographic principle">Holographic principle</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background: #ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Theorists</div></div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;">
<ul><li><a href="Mina_Aganagi%C4%87" title="Mina Aganagić">Aganagić</a></li>
<li><a href="Nima_Arkani-Hamed" title="Nima Arkani-Hamed">Arkani-Hamed</a></li>
<li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>
<li><a href="Tom_Banks_(physicist)" title="Tom Banks (physicist)">Banks</a></li>
<li><a href="David_Berenstein" title="David Berenstein">Berenstein</a></li>
<li><a href="Raphael_Bousso" title="Raphael Bousso">Bousso</a></li>
<li><a href="Kevin_Costello" title="Kevin Costello">Costello</a></li>
<li><a href="Thomas_Curtright" title="Thomas Curtright">Curtright</a></li>
<li><a href="Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf</a></li>
<li><a href="Jacques_Distler" title="Jacques Distler">Distler</a></li>
<li><a href="Michael_R._Douglas" title="Michael R. Douglas">Douglas</a></li>
<li><a href="Michael_Duff_(physicist)" title="Michael Duff (physicist)">Duff</a></li>
<li><a href="Gia_Dvali" class="mw-redirect" title="Gia Dvali">Dvali</a></li>
<li><a href="Sergio_Ferrara" title="Sergio Ferrara">Ferrara</a></li>
<li><a href="Willy_Fischler" title="Willy Fischler">Fischler</a></li>
<li><a href="Daniel_Friedan" title="Daniel Friedan">Friedan</a></li>
<li><a href="Sylvester_James_Gates" title="Sylvester James Gates">Gates</a></li>
<li><a href="Ferdinando_Gliozzi" title="Ferdinando Gliozzi">Gliozzi</a></li>
<li><a href="Rajesh_Gopakumar" title="Rajesh Gopakumar">Gopakumar</a></li>
<li><a href="Michael_Green_(physicist)" title="Michael Green (physicist)">Green</a></li>
<li><a href="Brian_Greene" title="Brian Greene">Greene</a></li>
<li><a href="David_Gross" title="David Gross">Gross</a></li>
<li><a href="Steven_Gubser" title="Steven Gubser">Gubser</a></li>
<li><a href="Sergei_Gukov" title="Sergei Gukov">Gukov</a></li>
<li><a href="Alan_Guth" title="Alan Guth">Guth</a></li>
<li><a href="Andrew_J._Hanson" title="Andrew J. Hanson">Hanson</a></li>
<li><a href="Jeffrey_A._Harvey" title="Jeffrey A. Harvey">Harvey</a></li>
<li><a href="Petr_Ho%C5%99ava_(theorist)" class="mw-redirect" title="Petr Hořava (theorist)">Hořava</a></li>
<li><a href="Gary_Horowitz" title="Gary Horowitz">Horowitz</a></li>
<li><a href="Gary_Gibbons" title="Gary Gibbons">Gibbons</a></li>
<li><a href="Shamit_Kachru" title="Shamit Kachru">Kachru</a></li>
<li><a href="Michio_Kaku" title="Michio Kaku">Kaku</a></li>
<li><a href="Renata_Kallosh" title="Renata Kallosh">Kallosh</a></li>
<li><a href="Theodor_Kaluza" title="Theodor Kaluza">Kaluza</a></li>
<li><a href="Anton_Kapustin" title="Anton Kapustin">Kapustin</a></li>
<li><a href="Igor_Klebanov" title="Igor Klebanov">Klebanov</a></li>
<li><a href="Vadim_Knizhnik" title="Vadim Knizhnik">Knizhnik</a></li>
<li><a href="Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li>
<li><a href="Oskar_Klein" title="Oskar Klein">Klein</a></li>
<li><a href="Andrei_Linde" title="Andrei Linde">Linde</a></li>
<li><a href="Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li>
<li><a href="Stanley_Mandelstam" title="Stanley Mandelstam">Mandelstam</a></li>
<li><a href="Donald_Marolf" title="Donald Marolf">Marolf</a></li>
<li><a href="Emil_Martinec" title="Emil Martinec">Martinec</a></li>
<li><a href="Shiraz_Minwalla" title="Shiraz Minwalla">Minwalla</a></li>
<li><a href="Greg_Moore_(physicist)" title="Greg Moore (physicist)">Moore</a></li>
<li><a href="Lubo%C5%A1_Motl" title="Luboš Motl">Motl</a></li>
<li><a href="Sunil_Mukhi" title="Sunil Mukhi">Mukhi</a></li>
<li><a href="Robert_Myers_(physicist)" title="Robert Myers (physicist)">Myers</a></li>
<li><a href="Dimitri_Nanopoulos" title="Dimitri Nanopoulos">Nanopoulos</a></li>
<li><a href="Hora%C8%9Biu_N%C4%83stase" title="Horațiu Năstase">Năstase</a></li>
<li><a href="Nikita_Nekrasov" title="Nikita Nekrasov">Nekrasov</a></li>
<li><a href="Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li>
<li><a href="Holger_Bech_Nielsen" title="Holger Bech Nielsen">Nielsen</a></li>
<li><a href="Peter_van_Nieuwenhuizen" title="Peter van Nieuwenhuizen">van Nieuwenhuizen</a></li>
<li><a href="Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov</a></li>
<li><a href="David_Olive" title="David Olive">Olive</a></li>
<li><a href="Hirosi_Ooguri" title="Hirosi Ooguri">Ooguri</a></li>
<li><a href="Burt_Ovrut" title="Burt Ovrut">Ovrut</a></li>
<li><a href="Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li>
<li><a href="Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li>
<li><a href="Arvind_Rajaraman" title="Arvind Rajaraman">Rajaraman</a></li>
<li><a href="Pierre_Ramond" title="Pierre Ramond">Ramond</a></li>
<li><a href="Lisa_Randall" title="Lisa Randall">Randall</a></li>
<li><a href="Seifallah_Randjbar-Daemi" title="Seifallah Randjbar-Daemi">Randjbar-Daemi</a></li>
<li><a href="Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li>
<li><a href="Ryan_Rohm" title="Ryan Rohm">Rohm</a></li>
<li><a href="Augusto_Sagnotti" title="Augusto Sagnotti">Sagnotti</a></li>
<li><a href="Jo%C3%ABl_Scherk" title="Joël Scherk">Scherk</a></li>
<li><a href="John_Henry_Schwarz" title="John Henry Schwarz">Schwarz</a></li>
<li><a href="Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li>
<li><a href="Ashoke_Sen" title="Ashoke Sen">Sen</a></li>
<li><a href="Stephen_Shenker" title="Stephen Shenker">Shenker</a></li>
<li><a href="Warren_Siegel" title="Warren Siegel">Siegel</a></li>
<li><a href="Eva_Silverstein" title="Eva Silverstein">Silverstein</a></li>
<li><a href="%C4%90%C3%A0m_Thanh_S%C6%A1n" title="Đàm Thanh Sơn">Sơn</a></li>
<li><a href="Matthias_Staudacher" title="Matthias Staudacher">Staudacher</a></li>
<li><a href="Paul_Steinhardt" title="Paul Steinhardt">Steinhardt</a></li>
<li><a href="Andrew_Strominger" title="Andrew Strominger">Strominger</a></li>
<li><a href="Raman_Sundrum" title="Raman Sundrum">Sundrum</a></li>
<li><a href="Leonard_Susskind" title="Leonard Susskind">Susskind</a></li>
<li><a href="Gerard_'t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li>
<li><a href="Paul_Townsend" title="Paul Townsend">Townsend</a></li>
<li><a href="Sandip_Trivedi" title="Sandip Trivedi">Trivedi</a></li>
<li><a href="Neil_Turok" title="Neil Turok">Turok</a></li>
<li><a href="Cumrun_Vafa" title="Cumrun Vafa">Vafa</a></li>
<li><a href="Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li>
<li><a href="Erik_Verlinde" title="Erik Verlinde">Verlinde</a></li>
<li><a href="Herman_Verlinde" title="Herman Verlinde">Verlinde</a></li>
<li><a href="Julius_Wess" title="Julius Wess">Wess</a></li>
<li><a href="Edward_Witten" title="Edward Witten">Witten</a></li>
<li><a href="Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li>
<li><a href="Tamiaki_Yoneya" title="Tamiaki Yoneya">Yoneya</a></li>
<li><a href="Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li>
<li><a href="Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li>
<li><a href="Eric_Zaslow" title="Eric Zaslow">Zaslow</a></li>
<li><a href="Bruno_Zumino" title="Bruno Zumino">Zumino</a></li>
<li><a href="Barton_Zwiebach" title="Barton Zwiebach">Zwiebach</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-below hlist" style="display:block;margin-top:0.3em;">
<ul><li><a href="History_of_string_theory" title="History of string theory">History</a></li>
<li><a href="Glossary_of_string_theory" title="Glossary of string theory">Glossary</a></li></ul></td></tr><tr><td class="sidebar-navbar" style="padding-top:0.25em;"><style data-mw-deduplicate="TemplateStyles:r1239400231">
/* start https://en.wikipedia.org/ */
.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}
/* end https://en.wikipedia.org/ */
</style></td></tr></tbody></table>
<p><b>String cosmology</b> is a relatively new field that tries to apply equations of <a href="String_theory" title="String theory">string theory</a> to solve the questions of early <a href="Physical_cosmology" title="Physical cosmology">cosmology</a>. A related area of study is <a href="Brane_cosmology" title="Brane cosmology">brane cosmology</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>
<p>This approach can be dated back to a paper by <a href="Gabriele_Veneziano" title="Gabriele Veneziano">Gabriele Veneziano</a><sup id="cite_ref-Ven91_1-0" class="reference"><a href="#cite_note-Ven91-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> that shows how an inflationary cosmological model can be obtained from string theory, thus opening the door to a description of pre-<a href="Big_Bang" title="Big Bang">Big Bang</a> scenarios.
</p><p>The idea is related to a property of the <a href="Bosonic_string_theory" title="Bosonic string theory">bosonic string</a> in a curve background, better known as the <a href="Nonlinear_sigma_model" class="mw-redirect" title="Nonlinear sigma model">nonlinear sigma model</a>. First calculations from this model<sup id="cite_ref-Frie80_2-0" class="reference"><a href="#cite_note-Frie80-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> showed that the <a href="Beta_function" title="Beta function">beta function</a>, representing the running of the metric of the model as a function of an energy scale, is proportional to the <a href="Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensor</a> giving rise to a <a href="Ricci_flow" title="Ricci flow">Ricci flow</a>. As this model has <a href="Conformal_field_theory" title="Conformal field theory">conformal invariance</a> and this must be kept to have a sensible <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, the <a href="Beta_function" title="Beta function">beta function</a> must be zero producing immediately the <a href="Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>. While Einstein equations seem to appear somewhat out of place, nevertheless this result is surely striking showing as a background two-dimensional model could produce higher-dimensional physics. An interesting point here is that such a string theory can be formulated without a requirement of criticality at 26 dimensions for consistency as happens on a flat background. This is a serious hint that the underlying physics of Einstein equations could be described by an effective two-dimensional <a href="Conformal_field_theory" title="Conformal field theory">conformal field theory</a>. Indeed, the fact that we have evidence for an inflationary universe is an important support to string cosmology.
</p><p>In the evolution of the universe, after the inflationary phase, the expansion observed today sets in that is well described by <a href="Friedmann_equations" title="Friedmann equations">Friedmann equations</a>. A smooth transition is expected between these two different phases. String cosmology appears to have difficulties in explaining this transition. This is known in the literature as the <b>graceful exit problem</b>.
</p><p>An <a href="Inflation_(cosmology)" class="mw-redirect" title="Inflation (cosmology)">inflationary cosmology</a> implies the presence of a scalar field that drives inflation. In string cosmology, this arises from the so-called <a href="Dilaton" title="Dilaton">dilaton</a> field. This is a scalar term entering into the description of the <a href="Bosonic_string_theory" title="Bosonic string theory">bosonic string</a> that produces a scalar field term into the effective theory at low energies. The corresponding equations resemble those of a <a href="Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a>.
</p><p>Analysis has been worked out from a critical number of dimension (26) down to four. In general, one gets <a href="Friedmann_equations" title="Friedmann equations">Friedmann equations</a> in an arbitrary number of dimensions. The other way round is to assume that a certain number of dimensions is <a href="Compactification_(physics)" title="Compactification (physics)">compactified</a> producing an effective four-dimensional theory to work with. Such a theory is a typical <a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a> with a set of scalar fields arising from <a href="Compactification_(physics)" title="Compactification (physics)">compactified</a> dimensions. Such fields are called <b>moduli</b>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Technical_details">Technical details</h2></div>
<p>This section presents some of the relevant equations entering into string cosmology. The starting point is the <a href="Polyakov_action" title="Polyakov action">Polyakov action</a>, which can be written as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}={\frac {1}{4\pi \alpha '}}\int d^{2}z{\sqrt {\gamma }}\left[\gamma ^{ab}G_{\mu \nu }(X)\partial _{a}X^{\mu }\partial _{b}X^{\nu }+\alpha '\ ^{(2)}R\Phi (X)\right],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>4</mn>
<mi>π<!-- π --></mi>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>γ<!-- γ --></mi>
</msqrt>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<msup>
<mi>γ<!-- γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
<mi>b</mi>
</mrow>
</msup>
<msub>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msup>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
<msup>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi>R</mi>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{2}={\frac {1}{4\pi \alpha '}}\int d^{2}z{\sqrt {\gamma }}\left[\gamma ^{ab}G_{\mu \nu }(X)\partial _{a}X^{\mu }\partial _{b}X^{\nu }+\alpha '\ ^{(2)}R\Phi (X)\right],}</annotation>
</semantics>
</math></span><img src="./30b41c2bbb31905c125c1fafcdcddc42972cc785.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.008ex; height:5.676ex;" alt="{\displaystyle S_{2}={\frac {1}{4\pi \alpha '}}\int d^{2}z{\sqrt {\gamma }}\left[\gamma ^{ab}G_{\mu \nu }(X)\partial _{a}X^{\mu }\partial _{b}X^{\nu }+\alpha '\ ^{(2)}R\Phi (X)\right],}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ ^{(2)}R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ ^{(2)}R}</annotation>
</semantics>
</math></span><img src="./3b7971c3cb77ef897400b7850a3ce416542a81fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.678ex; height:2.843ex;" alt="{\displaystyle \ ^{(2)}R}" loading="lazy"></span> is the <a href="Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci scalar</a> in two dimensions, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Φ<!-- Φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>
</semantics>
</math></span><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"></span> the <a href="Dilaton" title="Dilaton">dilaton</a> field, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha '}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha '}</annotation>
</semantics>
</math></span><img src="./6cb0468d39268c4405a9286d2cba77c2e4631fed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.172ex; height:2.509ex;" alt="{\displaystyle \alpha '}" loading="lazy"></span> the string constant. The indices <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a,b}</annotation>
</semantics>
</math></span><img src="./181523deba732fda302fd176275a0739121d3bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}" loading="lazy"></span> range over 1,2, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\nu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>ν<!-- ν --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ,\nu }</annotation>
</semantics>
</math></span><img src="./489a4ac1b5d76663fa5328e6f3c1d382655cb496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.668ex; height:2.176ex;" alt="{\displaystyle \mu ,\nu }" loading="lazy"></span> over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,\ldots ,D}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1,\ldots ,D}</annotation>
</semantics>
</math></span><img src="./d8371ae3f7b22777fb1af81286fd7c47519d28d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.265ex; height:2.509ex;" alt="{\displaystyle 1,\ldots ,D}" loading="lazy"></span>, where <i>D</i> the dimension of the target space. A further antisymmetric field could be added. This is generally considered when one wants this action generating a potential for inflation.<sup id="cite_ref-Wands96_3-0" class="reference"><a href="#cite_note-Wands96-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Otherwise, a generic potential is inserted by hand, as well as a cosmological constant.
</p><p>The above string action has a conformal invariance. This is a property of a two dimensional <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. At the quantum level, this property is lost due to anomalies and the theory itself is not consistent, having no <a href="Unitarity" class="mw-redirect" title="Unitarity">unitarity</a>. So it is necessary to require that <a href="Conformal_field_theory" title="Conformal field theory">conformal invariance</a> is kept at any order of <a href="Perturbation_theory" title="Perturbation theory">perturbation theory</a>. <a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a> is the only known approach to manage the <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. Indeed, the <a href="Beta_function" title="Beta function">beta functions</a> at two loops are
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{\mu \nu }^{G}=R_{\mu \nu }+2\alpha '\nabla _{\mu }\Phi \nabla _{\nu }\Phi +O(\alpha '^{2}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo>=</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow>
<mo class="MJX-variant">′</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{\mu \nu }^{G}=R_{\mu \nu }+2\alpha '\nabla _{\mu }\Phi \nabla _{\nu }\Phi +O(\alpha '^{2}),}</annotation>
</semantics>
</math></span><img src="./a665d80aa624cdc25ee5405ab2ca84425a1e2e4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.162ex; height:3.343ex;" alt="{\displaystyle \beta _{\mu \nu }^{G}=R_{\mu \nu }+2\alpha '\nabla _{\mu }\Phi \nabla _{\nu }\Phi +O(\alpha '^{2}),}" loading="lazy"></span></dd></dl>
<p>and
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{\Phi }={\frac {D-26}{6}}-{\frac {\alpha '}{2}}\nabla ^{2}\Phi +\alpha '\nabla _{\kappa }\Phi \nabla ^{\kappa }\Phi +O(\alpha '^{2}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Φ<!-- Φ --></mi>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>26</mn>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo>+</mo>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
<msub>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>κ<!-- κ --></mi>
</mrow>
</msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>κ<!-- κ --></mi>
</mrow>
</msup>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow>
<mo class="MJX-variant">′</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta ^{\Phi }={\frac {D-26}{6}}-{\frac {\alpha '}{2}}\nabla ^{2}\Phi +\alpha '\nabla _{\kappa }\Phi \nabla ^{\kappa }\Phi +O(\alpha '^{2}).}</annotation>
</semantics>
</math></span><img src="./6f1d267b417adfe2fa83e9c827ebf50c4d25b9c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.96ex; height:5.509ex;" alt="{\displaystyle \beta ^{\Phi }={\frac {D-26}{6}}-{\frac {\alpha '}{2}}\nabla ^{2}\Phi +\alpha '\nabla _{\kappa }\Phi \nabla ^{\kappa }\Phi +O(\alpha '^{2}).}" loading="lazy"></span></dd></dl>
<p>The assumption that <a href="Conformal_field_theory" title="Conformal field theory">conformal invariance</a> holds implies that
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{\mu \nu }^{G}=\beta ^{\Phi }=0,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msubsup>
<mo>=</mo>
<msup>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Φ<!-- Φ --></mi>
</mrow>
</msup>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{\mu \nu }^{G}=\beta ^{\Phi }=0,}</annotation>
</semantics>
</math></span><img src="./994a42a62a28f58a8b175ae047ac6f349e64869f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.172ex; height:3.343ex;" alt="{\displaystyle \beta _{\mu \nu }^{G}=\beta ^{\Phi }=0,}" loading="lazy"></span></dd></dl>
<p>producing the corresponding equations of motion of low-energy physics. These conditions can only be satisfied perturbatively, but this has to hold at any order of <a href="Perturbation_theory" title="Perturbation theory">perturbation theory</a>. The first term in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{\Phi }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Φ<!-- Φ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta ^{\Phi }}</annotation>
</semantics>
</math></span><img src="./b8dc995a89772dab010a234347d4a2d2d4f04152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.756ex; height:3.009ex;" alt="{\displaystyle \beta ^{\Phi }}" loading="lazy"></span> is just the anomaly of the <a href="Bosonic_string_theory" title="Bosonic string theory">bosonic string theory</a> in a flat spacetime. But here there are further terms that can grant compensation of the anomaly also when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\neq 26}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo>≠<!-- ≠ --></mo>
<mn>26</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D\neq 26}</annotation>
</semantics>
</math></span><img src="./1ed8ed39c10e891129a8721c54fb5815dbc26367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.348ex; height:2.676ex;" alt="{\displaystyle D\neq 26}" loading="lazy"></span>, and from this cosmological models of a pre-big bang, scenario can be constructed. Indeed, this low energy equations can be obtained from the following action:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {1}{2\kappa _{0}^{2}}}\int d^{D}x{\sqrt {-G}}e^{-2\Phi }\left[-{\frac {2(D-26)}{3\alpha '}}+R+4\partial _{\mu }\Phi \partial ^{\mu }\Phi +O(\alpha ')\right],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msubsup>
<mi>κ<!-- κ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo>−<!-- − --></mo>
<mi>G</mi>
</msqrt>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
</mrow>
</msup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>26</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>3</mn>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mi>R</mi>
<mo>+</mo>
<mn>4</mn>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msup>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2\kappa _{0}^{2}}}\int d^{D}x{\sqrt {-G}}e^{-2\Phi }\left[-{\frac {2(D-26)}{3\alpha '}}+R+4\partial _{\mu }\Phi \partial ^{\mu }\Phi +O(\alpha ')\right],}</annotation>
</semantics>
</math></span><img src="./47c063f4cc57a882846e139f56a5b544827bb6c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:68.824ex; height:6.843ex;" alt="{\displaystyle S={\frac {1}{2\kappa _{0}^{2}}}\int d^{D}x{\sqrt {-G}}e^{-2\Phi }\left[-{\frac {2(D-26)}{3\alpha '}}+R+4\partial _{\mu }\Phi \partial ^{\mu }\Phi +O(\alpha ')\right],}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{0}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>κ<!-- κ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \kappa _{0}^{2}}</annotation>
</semantics>
</math></span><img src="./6c4803fd804972b7120d134370fdfb6c49711fb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.393ex; height:3.176ex;" alt="{\displaystyle \kappa _{0}^{2}}" loading="lazy"></span> is a constant that can always be changed by redefining the dilaton field. One can also rewrite this action in a more familiar form by redefining the fields (Einstein frame) as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,g_{\mu \nu }=e^{2\omega }G_{\mu \nu }\!,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>ω<!-- ω --></mi>
</mrow>
</msup>
<msub>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mspace width="negativethinmathspace"></mspace>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,g_{\mu \nu }=e^{2\omega }G_{\mu \nu }\!,}</annotation>
</semantics>
</math></span><img src="./8da7f97a47f4e5ca9751f2a38c234dfff04cd222.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.03ex; height:3.343ex;" alt="{\displaystyle \,g_{\mu \nu }=e^{2\omega }G_{\mu \nu }\!,}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\frac {2(\Phi _{0}-\Phi )}{D-2}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {2(\Phi _{0}-\Phi )}{D-2}},}</annotation>
</semantics>
</math></span><img src="./275a3f42efacafe6a0d6ff184cc105544e64f7f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.25ex; height:5.843ex;" alt="{\displaystyle \omega ={\frac {2(\Phi _{0}-\Phi )}{D-2}},}" loading="lazy"></span></dd></dl>
<p>and using <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\Phi }}=\Phi -\Phi _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo>−<!-- − --></mo>
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {\Phi }}=\Phi -\Phi _{0}}</annotation>
</semantics>
</math></span><img src="./5d01ca2e823af838cd225fad100290aa1aa3d876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.027ex; height:3.009ex;" alt="{\displaystyle {\tilde {\Phi }}=\Phi -\Phi _{0}}" loading="lazy"></span> one can write
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {1}{2\kappa ^{2}}}\int d^{D}x{\sqrt {-g}}\left[-{\frac {2(D-26)}{3\alpha '}}e^{\frac {4{\tilde {\Phi }}}{D-2}}+{\tilde {R}}-{\frac {4}{D-2}}\partial _{\mu }{\tilde {\Phi }}\partial ^{\mu }{\tilde {\Phi }}+O(\alpha ')\right],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msup>
<mi>κ<!-- κ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo>−<!-- − --></mo>
<mi>g</mi>
</msqrt>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>26</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>3</mn>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
</mrow>
</mfrac>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>4</mn>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
<mrow>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mfrac>
</mrow>
</msup>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>R</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>4</mn>
<mrow>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mfrac>
</mrow>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>α<!-- α --></mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2\kappa ^{2}}}\int d^{D}x{\sqrt {-g}}\left[-{\frac {2(D-26)}{3\alpha '}}e^{\frac {4{\tilde {\Phi }}}{D-2}}+{\tilde {R}}-{\frac {4}{D-2}}\partial _{\mu }{\tilde {\Phi }}\partial ^{\mu }{\tilde {\Phi }}+O(\alpha ')\right],}</annotation>
</semantics>
</math></span><img src="./6f1253d4ee32d42a41ac00fed0dc76786e6d11c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.073ex; height:6.343ex;" alt="{\displaystyle S={\frac {1}{2\kappa ^{2}}}\int d^{D}x{\sqrt {-g}}\left[-{\frac {2(D-26)}{3\alpha '}}e^{\frac {4{\tilde {\Phi }}}{D-2}}+{\tilde {R}}-{\frac {4}{D-2}}\partial _{\mu }{\tilde {\Phi }}\partial ^{\mu }{\tilde {\Phi }}+O(\alpha ')\right],}" loading="lazy"></span></dd></dl>
<p>where
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {R}}=e^{-2\omega }[R-(D-1)\nabla ^{2}\omega -(D-2)(D-1)\partial _{\mu }\omega \partial ^{\mu }\omega ].}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>R</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>ω<!-- ω --></mi>
</mrow>
</msup>
<mo stretchy="false">[</mo>
<mi>R</mi>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>ω<!-- ω --></mi>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>D</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msub>
<mi>ω<!-- ω --></mi>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
</mrow>
</msup>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">]</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {R}}=e^{-2\omega }[R-(D-1)\nabla ^{2}\omega -(D-2)(D-1)\partial _{\mu }\omega \partial ^{\mu }\omega ].}</annotation>
</semantics>
</math></span><img src="./2622a14c33d501411d9d375ac3d2b987e46659ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.248ex; height:3.343ex;" alt="{\displaystyle {\tilde {R}}=e^{-2\omega }[R-(D-1)\nabla ^{2}\omega -(D-2)(D-1)\partial _{\mu }\omega \partial ^{\mu }\omega ].}" loading="lazy"></span></dd></dl>
<p>This is the formula for the Einstein action describing a scalar field interacting with a gravitational field in D dimensions. Indeed, the following identity holds:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa =\kappa _{0}e^{2\Phi _{0}}=(8\pi G_{D})^{\frac {1}{2}}={\frac {\sqrt {8\pi }}{M_{p}}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>κ<!-- κ --></mi>
<mo>=</mo>
<msub>
<mi>κ<!-- κ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</msup>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mi>π<!-- π --></mi>
<msub>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msqrt>
<mn>8</mn>
<mi>π<!-- π --></mi>
</msqrt>
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \kappa =\kappa _{0}e^{2\Phi _{0}}=(8\pi G_{D})^{\frac {1}{2}}={\frac {\sqrt {8\pi }}{M_{p}}},}</annotation>
</semantics>
</math></span><img src="./9e67b777fe0d333c0cc4f60dc7bdad70c470fe73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.556ex; height:6.509ex;" alt="{\displaystyle \kappa =\kappa _{0}e^{2\Phi _{0}}=(8\pi G_{D})^{\frac {1}{2}}={\frac {\sqrt {8\pi }}{M_{p}}},}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{D}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>D</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G_{D}}</annotation>
</semantics>
</math></span><img src="./5f3f76e984daf24cd9a54cfe1d27f8e460efce18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.42ex; height:2.509ex;" alt="{\displaystyle G_{D}}" loading="lazy"></span> is the Newton constant in D dimensions and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M_{p}}</annotation>
</semantics>
</math></span><img src="./4a8004b0c032f7742153c5994fb87d923fdb4493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.313ex; height:2.843ex;" alt="{\displaystyle M_{p}}" loading="lazy"></span> the corresponding Planck mass. When setting <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=4}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo>=</mo>
<mn>4</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D=4}</annotation>
</semantics>
</math></span><img src="./7f4978172a19bddad4da20a7bfd0e91882aee24e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.185ex; height:2.176ex;" alt="{\displaystyle D=4}" loading="lazy"></span> in this action, the conditions for inflation are not fulfilled unless a potential or antisymmetric term is added to the string action,<sup id="cite_ref-Wands96_3-1" class="reference"><a href="#cite_note-Wands96-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> in which case power-law inflation is possible.
</p>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
/* start https://en.wikipedia.org/ */
.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}
/* end https://en.wikipedia.org/ */
</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">
<ol class="references">
<li id="cite_note-Ven91-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ven91_1-0">^</a></b></span> <span class="reference-text">
<style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}
/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFVeneziano1991" class="citation journal cs1"><a href="Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano, G.</a> (1991). "Scale factor duality for classical and quantum strings". <i><a href="Physics_Letters_B" class="mw-redirect" title="Physics Letters B">Physics Letters B</a></i>. <b>265</b> (<span class="nowrap">3–</span>4): <span class="nowrap">287–</span>294. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhLB..265..287V">1991PhLB..265..287V</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.8098">10.1.1.8.8098</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2891%2990055-U">10.1016/0370-2693(91)90055-U</a>.</cite></span>
</li>
<li id="cite_note-Frie80-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Frie80_2-0">^</a></b></span> <span class="reference-text">
<cite id="CITEREFFriedan1980" class="citation journal cs1"><a href="Daniel_Friedan" title="Daniel Friedan">Friedan, D.</a> (1980). <a rel="nofollow" class="external text" href="http://www.physics.rutgers.edu/~friedan/papers/PRL_45_1980_1057.pdf">"Nonlinear Models in 2+<i>ϵ</i> Dimensions"</a> <span class="cs1-format">(PDF)</span>. <i><a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>45</b> (13): <span class="nowrap">1057–</span>1060. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980PhRvL..45.1057F">1980PhRvL..45.1057F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.45.1057">10.1103/PhysRevLett.45.1057</a>.</cite></span>
</li>
<li id="cite_note-Wands96-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wands96_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wands96_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">
<cite id="CITEREFEastherMaedaWands1996" class="citation journal cs1">Easther, R.; Maeda, Kei-ichi; <a href="David_Wands" title="David Wands">Wands, D.</a> (1996). "Tree-level string cosmology". <i><a href="Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>53</b> (8): <span class="nowrap">4247–</span>4256. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9509074">hep-th/9509074</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhRvD..53.4247E">1996PhRvD..53.4247E</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.53.4247">10.1103/PhysRevD.53.4247</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10020421">10020421</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8124718">8124718</a>.</cite></span>
</li>
</ol></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><cite id="CITEREFPolchinski1998a" class="citation book cs1"><a href="Joseph_Polchinski" title="Joseph Polchinski">Polchinski, Joseph</a> (1998a). <i>String Theory Vol. I: An Introduction to the Bosonic String</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-63303-1</bdi>.</cite></li>
<li><cite id="CITEREFPolchinski1998b" class="citation book cs1"><a href="Joseph_Polchinski" title="Joseph Polchinski">Polchinski, Joseph</a> (1998b). <i>String Theory Vol. II: Superstring Theory and Beyond</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-63304-8</bdi>.</cite></li>
<li><cite id="CITEREFLidseyWandsCopeland2000" class="citation journal cs1">Lidsey, James D.; <a href="David_Wands" title="David Wands">Wands, David</a>; <a href="Edmund_Copeland" title="Edmund Copeland">Copeland, E. J.</a> (2000). "Superstring Cosmology". <i>Physics Reports</i>. <b>337</b> (<span class="nowrap">4–</span>5): <span class="nowrap">343–</span>492. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9909061">hep-th/9909061</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000PhR...337..343L">2000PhR...337..343L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0370-1573%2800%2900064-8">10.1016/S0370-1573(00)00064-8</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119349072">119349072</a>.</cite></li>
<li><cite id="CITEREFCicoliConlonMaharanaParameswaran2024" class="citation journal cs1">Cicoli, Michele; Conlon, Joseph P; Maharana, Anshuman; Parameswaran, Susha; <a href="Fernando_Quevedo" title="Fernando Quevedo">Quevedo, Fernando</a>; Zavala, Ivonne (2024). "String Cosmology: from the Early Universe to Today". <i>Phys. Rep</i>. <b>1059</b>: <span class="nowrap">1–</span>155. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2303.04819">2303.04819</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2024PhR..1059....1C">2024PhR..1059....1C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physrep.2024.01.002">10.1016/j.physrep.2024.01.002</a>.</cite></li>
<li><cite id="CITEREFNăstase2019" class="citation book cs1"><a href="Hora%C8%9Biu_N%C4%83stase" title="Horațiu Năstase">Năstase, Horaţiu</a> (2019). <i>Cosmology and String Theory</i>. <a href="Springer_Publishing" title="Springer Publishing">Springer Publishing</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3030150761</bdi>.</cite></li>
<li><cite id="CITEREFBaumannMcAllister2015" class="citation book cs1">Baumann, Daniel; McAllister, Liam (2015). <i>Inflation and String Theory</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1107089693</bdi>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=15204">String cosmology on arxiv.org</a></li>
<li><a rel="nofollow" class="external text" href="http://www.ba.infn.it/~gasperin/">Maurizio Gasperini's homepage</a></li></ul>
<div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1236075235">
/* start https://en.wikipedia.org/ */
.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}
/* end https://en.wikipedia.org/ */
</style></div><div role="navigation" class="navbox" aria-labelledby="Major_subfields_of_astronomy32" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Major_subfields_of_astronomy32" style="font-size:114%;margin:0 4em">Major subfields of <a href="Astronomy" title="Astronomy">astronomy</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Astrobiology" title="Astrobiology">Astrobiology</a></li>
<li><a href="Astrochemistry" title="Astrochemistry">Astrochemistry</a></li>
<li><a href="Astrometry" title="Astrometry">Astrometry</a></li>
<li><a href="Astrophysics" title="Astrophysics">Astrophysics</a></li>
<li><a href="Cosmochemistry" title="Cosmochemistry">Cosmochemistry</a></li>
<li><a href="Cosmology" title="Cosmology">Cosmology</a></li>
<li><a href="Extragalactic_astronomy" title="Extragalactic astronomy">Extragalactic astronomy</a></li>
<li><a href="Galactic_astronomy" title="Galactic astronomy">Galactic astronomy</a></li>
<li><a href="Orbital_mechanics" title="Orbital mechanics">Orbital mechanics</a></li>
<li><a href="Physical_cosmology" title="Physical cosmology">Physical cosmology</a></li>
<li><a href="Planetary_geology" title="Planetary geology">Planetary geology</a></li>
<li><a href="Planetary_science" title="Planetary science">Planetary science</a></li>
<li><a href="Sun" title="Sun">Solar astronomy</a></li>
<li><a href="Astronomy#Stellar_astronomy" title="Astronomy">Stellar astronomy</a></li></ul>
</div></td></tr></tbody></table></div></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-08-03" href="https://en.wikipedia.org/wiki/?title=String_cosmology&oldid=1304096075">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
</body></html>