Friedmann equations

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subsection</span> </button> <ul id="toc-Useful_solutions-sublist" class="vector-toc-list"> <li id="toc-Mixtures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mixtures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Mixtures</span> </div> </a> <ul id="toc-Mixtures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Detailed_derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Detailed_derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Detailed derivation</span> </div> </a> <ul id="toc-Detailed_derivation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In popular culture</span> </div> </a> <ul id="toc-In_popular_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span 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Available in 30 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-30" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">30 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%81%D8%B1%D9%8A%D8%AF%D9%85%D8%A7%D9%86" title="معادلات فريدمان – Arabic" lang="ar" hreflang="ar" data-title="معادلات فريدمان" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kainat%C4%B1n_Fridman_modeli" title="Kainatın Fridman modeli – Azerbaijani" lang="az" hreflang="az" data-title="Kainatın Fridman modeli" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%A6%E0%A6%AE%E0%A6%BE%E0%A6%A8_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="ফ্রিদমান সমীকরণ – Bangla" lang="bn" hreflang="bn" data-title="ফ্রিদমান সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equacions_de_Friedmann" title="Equacions de Friedmann – Catalan" lang="ca" hreflang="ca" data-title="Equacions de Friedmann" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Friedmann-Gleichungen" title="Friedmann-Gleichungen – German" lang="de" hreflang="de" data-title="Friedmann-Gleichungen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Friedmanni_v%C3%B5rrandid" title="Friedmanni võrrandid – Estonian" lang="et" hreflang="et" data-title="Friedmanni võrrandid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaciones_de_Friedmann" title="Ecuaciones de Friedmann – Spanish" lang="es" hreflang="es" data-title="Ecuaciones de Friedmann" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%81%D8%B1%DB%8C%D8%AF%D9%85%D8%A7%D9%86" title="معادلات فریدمان – Persian" lang="fa" hreflang="fa" data-title="معادلات فریدمان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quations_de_Friedmann" title="Équations de Friedmann – French" lang="fr" hreflang="fr" data-title="Équations de Friedmann" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%84%EB%A6%AC%EB%93%9C%EB%A7%8C_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="프리드만 방정식 – Korean" lang="ko" hreflang="ko" data-title="프리드만 방정식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D6%80%D5%AB%D5%A4%D5%B4%D5%A1%D5%B6%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Ֆրիդմանի հավասարումներ – Armenian" lang="hy" hreflang="hy" data-title="Ֆրիդմանի հավասարումներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_Friedmann" title="Persamaan Friedmann – Indonesian" lang="id" hreflang="id" data-title="Persamaan Friedmann" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazioni_di_Fridman" title="Equazioni di Fridman – Italian" lang="it" hreflang="it" data-title="Equazioni di Fridman" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%A4%D7%A8%D7%99%D7%93%D7%9E%D7%9F" title="משוואות פרידמן – Hebrew" lang="he" hreflang="he" data-title="משוואות פרידמן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%A0%E1%83%98%E1%83%93%E1%83%9B%E1%83%90%E1%83%9C%E1%83%98%E1%83%A1_%E1%83%92%E1%83%90%E1%83%9C%E1%83%A2%E1%83%9D%E1%83%9A%E1%83%94%E1%83%91%E1%83%94%E1%83%91%E1%83%98" title="ფრიდმანის განტოლებები – Georgian" lang="ka" hreflang="ka" data-title="ფრიდმანის განტოლებები" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Friedmannvergelijking" title="Friedmannvergelijking – Dutch" lang="nl" hreflang="nl" data-title="Friedmannvergelijking" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%83%AA%E3%83%BC%E3%83%89%E3%83%9E%E3%83%B3%E6%96%B9%E7%A8%8B%E5%BC%8F" title="フリードマン方程式 – Japanese" lang="ja" hreflang="ja" data-title="フリードマン方程式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnania_Friedmana" title="Równania Friedmana – Polish" lang="pl" hreflang="pl" data-title="Równania Friedmana" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%B5es_de_Friedmann" title="Equações de Friedmann – Portuguese" lang="pt" hreflang="pt" data-title="Equações de Friedmann" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%A4%D1%80%D0%B8%D0%B4%D0%BC%D0%B0%D0%BD%D0%B0" title="Уравнение Фридмана – Russian" lang="ru" hreflang="ru" data-title="Уравнение Фридмана" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Critical_density" title="Critical density – Simple English" lang="en-simple" hreflang="en-simple" data-title="Critical density" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fridmanovi_ena%C4%8Dbi" title="Fridmanovi enačbi – Slovenian" lang="sl" hreflang="sl" data-title="Fridmanovi enačbi" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fridmanin_yht%C3%A4l%C3%B6t" title="Fridmanin yhtälöt – Finnish" lang="fi" hreflang="fi" data-title="Fridmanin yhtälöt" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fridmans_ekvationer" title="Fridmans ekvationer – Swedish" lang="sv" hreflang="sv" data-title="Fridmans ekvationer" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9F%E0%B8%A3%E0%B8%B5%E0%B8%94%E0%B9%81%E0%B8%A1%E0%B8%99" title="สมการฟรีดแมน – Thai" lang="th" hreflang="th" data-title="สมการฟรีดแมน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Friedmann_denklemleri" title="Friedmann denklemleri – Turkish" lang="tr" hreflang="tr" data-title="Friedmann denklemleri" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%A4%D1%80%D1%96%D0%B4%D0%BC%D0%B0%D0%BD%D0%B0" title="Рівняння Фрідмана – Ukrainian" lang="uk" hreflang="uk" data-title="Рівняння Фрідмана" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%81%D8%B1%DB%8C%DA%88%D9%85%D8%A7%D9%86_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA" title="فریڈمان مساوات – Urdu" lang="ur" hreflang="ur" data-title="فریڈمان مساوات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_Friedmann" title="Phương trình Friedmann – Vietnamese" lang="vi" hreflang="vi" data-title="Phương trình Friedmann" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BC%97%E9%87%8C%E5%BE%B7%E6%9B%BC%E6%96%B9%E7%A8%8B" title="弗里德曼方程 – Chinese" lang="zh" hreflang="zh" data-title="弗里德曼方程" data-language-autonym="中文" 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href="/wiki/Physical_cosmology" title="Physical cosmology">Physical cosmology</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:WMAP_2012.png" class="mw-file-description"><img alt="Full-sky image derived from nine years' WMAP data" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/220px-WMAP_2012.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/330px-WMAP_2012.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/440px-WMAP_2012.png 2x" data-file-width="4096" data-file-height="2048" /></a></span></td></tr><tr><td class="sidebar-above" style="border:0;font-weight:normal; display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a> <b>·</b> <a href="/wiki/Universe" title="Universe">Universe</a></li> <li><a href="/wiki/Age_of_the_universe" title="Age of the universe">Age of the universe</a></li> <li><a href="/wiki/Chronology_of_the_universe" title="Chronology of the universe">Chronology of the universe</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddd8e7;text-align:center;;color: var(--color-base)">Early universe</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cosmic_inflation" title="Cosmic inflation">Inflation</a> <b>·</b> <a href="/wiki/Big_Bang_nucleosynthesis" title="Big Bang nucleosynthesis">Nucleosynthesis</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Backgrounds</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Gravitational_wave_background" title="Gravitational wave background">Gravitational wave (GWB)</a></li> <li><a href="/wiki/Cosmic_microwave_background" title="Cosmic microwave background">Microwave (CMB)</a> <b>·</b> <a href="/wiki/Cosmic_neutrino_background" title="Cosmic neutrino background">Neutrino (CNB)</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddd8e7;text-align:center;;color: var(--color-base)">Expansion <b>·</b> Future</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Hubble%27s_law" title="Hubble's law">Hubble's law</a> <b>·</b> <a href="/wiki/Redshift" title="Redshift">Redshift</a></li> <li><a href="/wiki/Expansion_of_the_universe" title="Expansion of the universe">Expansion of the universe</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">FLRW metric</a> <b>·</b> <a class="mw-selflink selflink">Friedmann equations</a></li> <li><a href="/wiki/Inhomogeneous_cosmology" title="Inhomogeneous cosmology">Inhomogeneous cosmology</a></li> <li><a href="/wiki/Future_of_an_expanding_universe" title="Future of an expanding universe">Future of an expanding universe</a></li> <li><a href="/wiki/Ultimate_fate_of_the_universe" title="Ultimate fate of the universe">Ultimate fate of the universe</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddd8e7;text-align:center;;color: var(--color-base)">Components <b>·</b> Structure</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Components</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Lambda-CDM_model" title="Lambda-CDM model">Lambda-CDM model</a></li> <li><a href="/wiki/Dark_energy" title="Dark energy">Dark energy</a> <b>·</b> <a href="/wiki/Dark_matter" title="Dark matter">Dark matter</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Structure</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Shape_of_the_universe" title="Shape of the universe">Shape of the universe</a></li> <li><a href="/wiki/Galaxy_filament" title="Galaxy filament">Galaxy filament</a> <b>·</b> <a href="/wiki/Galaxy_formation_and_evolution" title="Galaxy formation and evolution">Galaxy formation</a></li> <li><a href="/wiki/Large_quasar_group" title="Large quasar group">Large quasar group</a></li> <li><a href="/wiki/Observable_universe#Large-scale_structure" title="Observable universe">Large-scale structure</a></li> <li><a href="/wiki/Reionization" title="Reionization">Reionization</a> <b>·</b> <a href="/wiki/Structure_formation" title="Structure formation">Structure formation</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddd8e7;text-align:center;;color: var(--color-base)"><a href="/wiki/Observational_cosmology" title="Observational cosmology">Experiments</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Black_Hole_Initiative" title="Black Hole Initiative">Black Hole Initiative (BHI)</a></li> <li><a href="/wiki/BOOMERanG_experiment" title="BOOMERanG experiment">BOOMERanG</a></li> <li><a href="/wiki/Cosmic_Background_Explorer" title="Cosmic Background Explorer">Cosmic Background Explorer (COBE)</a></li> <li><a href="/wiki/Dark_Energy_Survey" title="Dark Energy Survey">Dark Energy Survey</a></li> <li><a href="/wiki/Planck_(spacecraft)" title="Planck (spacecraft)">Planck space observatory</a></li> <li><a href="/wiki/Sloan_Digital_Sky_Survey" title="Sloan Digital Sky Survey">Sloan Digital Sky Survey (SDSS)</a></li> <li><a href="/wiki/2dF_Galaxy_Redshift_Survey" title="2dF Galaxy Redshift Survey">2dF Galaxy Redshift Survey ("2dF")</a></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Wilkinson_Microwave_Anisotropy_Probe" title="Wilkinson Microwave Anisotropy Probe">Wilkinson Microwave Anisotropy<br />Probe (WMAP)</a></div></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:#ddd8e7;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist" style="padding:0 0.9em;"> <ul><li><a href="/wiki/Marc_Aaronson" title="Marc Aaronson">Aaronson</a></li> <li><a href="/wiki/Hannes_Alfv%C3%A9n" title="Hannes Alfvén">Alfvén</a></li> <li><a href="/wiki/Ralph_Alpher" title="Ralph Alpher">Alpher</a></li> <li><a href="/wiki/Nicolaus_Copernicus" title="Nicolaus Copernicus">Copernicus</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Robert_H._Dicke" title="Robert H. Dicke">Dicke</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/George_F._R._Ellis" title="George F. R. Ellis">Ellis</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/George_Gamow" title="George Gamow">Gamow</a></li> <li><a href="/wiki/Alan_Guth" title="Alan Guth">Guth</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Edwin_Hubble" title="Edwin Hubble">Hubble</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/John_C._Mather" title="John C. Mather">Mather</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Arno_Allan_Penzias" title="Arno Allan Penzias">Penzias</a></li> <li><a href="/wiki/Vera_Rubin" title="Vera Rubin">Rubin</a></li> <li><a href="/wiki/Brian_Schmidt" title="Brian Schmidt">Schmidt</a></li> <li><a href="/wiki/George_Smoot" title="George Smoot">Smoot</a></li> <li><a href="/wiki/Nicholas_B._Suntzeff" title="Nicholas B. Suntzeff">Suntzeff</a></li> <li><a href="/wiki/Rashid_Sunyaev" title="Rashid Sunyaev">Sunyaev</a></li> <li><a href="/wiki/Richard_C._Tolman" title="Richard C. 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line-height:1.6em;padding-bottom:0.5em;"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Physical_cosmology" title="Category:Physical cosmology">Category</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/16px-Crab_Nebula.jpg" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/24px-Crab_Nebula.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/32px-Crab_Nebula.jpg 2x" data-file-width="3864" data-file-height="3864" /></span></span> </span><a href="/wiki/Portal:Astronomy" title="Portal:Astronomy">Astronomy portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.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}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Physical_cosmology" title="Template:Physical cosmology"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Physical_cosmology" title="Template talk:Physical cosmology"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Physical_cosmology" title="Special:EditPage/Template:Physical cosmology"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Aleksandr_Fridman.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Aleksandr_Fridman.png/236px-Aleksandr_Fridman.png" decoding="async" width="236" height="352" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Aleksandr_Fridman.png/354px-Aleksandr_Fridman.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/62/Aleksandr_Fridman.png 2x" data-file-width="400" data-file-height="597" /></a><figcaption><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Alexander Friedmann</a></figcaption></figure> <p>The <b>Friedmann equations</b>, also known as the <b>Friedmann–Lemaître</b> (<b>FL</b>) <b>equations</b>, are a set of <a href="/wiki/Equation" title="Equation">equations</a> in <a href="/wiki/Physical_cosmology" title="Physical cosmology">physical cosmology</a> that govern <a href="/wiki/Expansion_of_the_universe" title="Expansion of the universe">cosmic expansion</a> in <a href="/wiki/Homogeneity_(physics)" title="Homogeneity (physics)">homogeneous</a> and <a href="/wiki/Isotropy" title="Isotropy">isotropic</a> models of the universe within the context of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. They were first derived by <a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Alexander Friedmann</a> in 1922 from <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein's field equations</a> of <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a> for the <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker metric</a> and a <a href="/wiki/Perfect_fluid" title="Perfect fluid">perfect fluid</a> with a given <a href="/wiki/Density" title="Density">mass density</a> <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Rho_(letter)" class="mw-redirect" title="Rho (letter)">ρ</a></span> and <a href="/wiki/Pressure" title="Pressure">pressure</a> <span class="texhtml mvar" style="font-style:italic;">p</span>.<sup id="cite_ref-af1922_1-0" class="reference"><a href="#cite_note-af1922-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The equations for negative spatial curvature were given by Friedmann in 1924.<sup id="cite_ref-af1924_2-0" class="reference"><a href="#cite_note-af1924-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Assumptions">Assumptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=1" title="Edit section: Assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Friedmann_equations" title="Special:EditPage/Friedmann equations">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker metric</a></div> <p>The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and <a href="/wiki/Isotropic_manifold" class="mw-redirect" title="Isotropic manifold">isotropic</a>, that is, the <a href="/wiki/Cosmological_principle" title="Cosmological principle">cosmological principle</a>; empirically, this is justified on scales larger than the order of 100 <a href="/wiki/Parsec" title="Parsec">Mpc</a>. The cosmological principle implies that the metric of the universe must be of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathrm {d} s^{2}=a(t)^{2}\,{\mathrm {d} s_{3}}^{2}-c^{2}\,\mathrm {d} t^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathrm {d} s^{2}=a(t)^{2}\,{\mathrm {d} s_{3}}^{2}-c^{2}\,\mathrm {d} t^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/515645f3080ffda1b8b0697d3966770b95c20122" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.63ex; height:3.176ex;" alt="{\displaystyle -\mathrm {d} s^{2}=a(t)^{2}\,{\mathrm {d} s_{3}}^{2}-c^{2}\,\mathrm {d} t^{2}}"></span> where <span class="texhtml">d<i>s</i><sub>3</sub><sup>2</sup></span> is a three-dimensional metric that must be one of <b>(a)</b> flat space, <b>(b)</b> a sphere of constant positive curvature or <b>(c)</b> a hyperbolic space with constant negative curvature. This metric is called the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The parameter <span class="texhtml mvar" style="font-style:italic;">k</span> discussed below takes the value 0, 1, −1, or the <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>, in these three cases respectively. It is this fact that allows us to sensibly speak of a "<a href="/wiki/Scale_factor_(Universe)" class="mw-redirect" title="Scale factor (Universe)">scale factor</a>" <span class="texhtml"><i>a</i>(<i>t</i>)</span>. </p><p>Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute <a href="/wiki/Christoffel_symbols" title="Christoffel symbols">Christoffel symbols</a>, then the <a href="/wiki/Ricci_tensor" class="mw-redirect" title="Ricci tensor">Ricci tensor</a>. With the <a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a> for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below. </p> <div class="mw-heading mw-heading2"><h2 id="Equations">Equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=2" title="Edit section: Equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Friedmann_equations" title="Special:EditPage/Friedmann equations">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/General_relativity" title="General relativity">General relativity</a></th></tr><tr><td class="sidebar-image"><span class="notpageimage" typeof="mw:File"><a href="/wiki/File:Spacetime_lattice_analogy.svg" class="mw-file-description" title="Spacetime curvature schematic"><img alt="Spacetime curvature schematic" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/220px-Spacetime_lattice_analogy.svg.png" decoding="async" width="220" height="82" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/330px-Spacetime_lattice_analogy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Spacetime_lattice_analogy.svg/440px-Spacetime_lattice_analogy.svg.png 2x" data-file-width="1260" data-file-height="469" /></a></span><div class="sidebar-caption" style="padding:0.5em 0.2em 0.6em;border-bottom:1px solid #aaa; display:block;margin-bottom:0.1em;"><span class="mwe-math-element"><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 }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"></span></div></td></tr><tr><td class="sidebar-content" style="padding-bottom:0.75em;"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><div class="hlist"><ul><li><a href="/wiki/History_of_general_relativity" title="History of general relativity">History</a></li><li><a href="/wiki/Timeline_of_gravitational_physics_and_relativity" title="Timeline of gravitational physics and relativity">Timeline</a></li><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</a></li></ul></div></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</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">Fundamental concepts</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-Riemannian manifold</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">Phenomena</div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Kepler problem</a></li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">Gravitational lensing</a></li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">Gravitational redshift</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">Frame-dragging</a></li> <li><a href="/wiki/Geodetic_effect" title="Geodetic effect">Geodetic effect</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ececff; font-style:italic;font-weight:normal;"> <a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Spacetime_diagram" title="Spacetime diagram">Spacetime diagrams</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski spacetime</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Einstein–Rosen bridge</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Equations</li><li>Formalisms</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;padding-bottom:0;margin-bottom:0;"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Equations</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a class="mw-selflink selflink">Friedmann</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Formalisms</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;padding-bottom:0;"> Advanced theory</th></tr><tr><td class="sidebar-content" style="padding-top:0;"> <ul><li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul></td> </tr></tbody></table></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"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li> <li><a href="/wiki/Einstein%E2%80%93Rosen_metric" title="Einstein–Rosen metric">Einstein–Rosen waves</a></li> <li><a href="/wiki/Wormhole" title="Wormhole">Wormhole</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a></li> <li><a href="/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric" title="Kerr–Newman–de–Sitter metric">Kerr–Newman–de Sitter</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li> <li><a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Robertson–Walker</a></li> <li><a href="/wiki/Oppenheimer%E2%80%93Snyder_model" title="Oppenheimer–Snyder model">Oppenheimer–Snyder</a></li> <li><a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Weyl%E2%80%93Lewis%E2%80%93Papapetrou_coordinates" title="Weyl–Lewis–Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Hartle%E2%80%93Thorne_metric" title="Hartle–Thorne metric">Hartle–Thorne</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">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hans_Reissner" title="Hans Reissner">Reissner</a></li> <li><a href="/wiki/Gunnar_Nordstr%C3%B6m" title="Gunnar Nordström">Nordström</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">Oppenheimer</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/James_M._Bardeen" title="James M. Bardeen">Bardeen</a></li> <li><a href="/wiki/Arthur_Geoffrey_Walker" title="Arthur Geoffrey Walker">Walker</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Amal_Kumar_Raychaudhuri" title="Amal Kumar Raychaudhuri">Raychaudhuri</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Willem_Jacob_van_Stockum" title="Willem Jacob van Stockum">van Stockum</a></li> <li><a href="/wiki/Abraham_H._Taub" title="Abraham H. Taub">Taub</a></li> <li><a href="/wiki/Ezra_T._Newman" title="Ezra T. Newman">Newman</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:General_relativity" title="Category:General relativity">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_relativity_sidebar" title="Template:General relativity sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_relativity_sidebar" title="Template talk:General relativity sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_relativity_sidebar" title="Special:EditPage/Template:General relativity sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {8\pi G\rho +\Lambda c^{2}}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>k</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {8\pi G\rho +\Lambda c^{2}}{3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1325ba86e9983ded3ed181ee1f125cbdf25a0729" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.852ex; height:6.009ex;" alt="{\displaystyle {\frac {{\dot {a}}^{2}+kc^{2}}{a^{2}}}={\frac {8\pi G\rho +\Lambda c^{2}}{3}},}"></span> which is derived from the 00 component of the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>. The second is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>¨</mo> </mover> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>p</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Λ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab89105802b4d7dda583eb3e0053077dbd07ffde" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.499ex; height:6.343ex;" alt="{\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}"></span> which is derived from the first together with the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of Einstein's field equations (the dimension of the two equations is time<sup>−2</sup>). </p><p><span class="texhtml mvar" style="font-style:italic;">a</span> is the <a href="/wiki/Scale_factor_(universe)" class="mw-redirect" title="Scale factor (universe)">scale factor</a>, <span class="texhtml mvar" style="font-style:italic;">G</span>, <span class="texhtml">Λ</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> are universal constants (<span class="texhtml mvar" style="font-style:italic;">G</span> is the <a href="/wiki/Newtonian_constant_of_gravitation" class="mw-redirect" title="Newtonian constant of gravitation">Newtonian constant of gravitation</a>, <span class="texhtml">Λ</span> is the <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a> with dimension length<sup>−2</sup>, and <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light in vacuum</a>). <span class="texhtml mvar" style="font-style:italic;">ρ</span> and <span class="texhtml mvar" style="font-style:italic;">p</span> are the volumetric mass density (and not the volumetric energy density) and the pressure, respectively. <span class="texhtml mvar" style="font-style:italic;">k</span> is constant throughout a particular solution, but may vary from one solution to another. </p><p>In previous equations, <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">ρ</span>, and <span class="texhtml mvar" style="font-style:italic;">p</span> are functions of time. <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>k</i></span><span class="sr-only">/</span><span class="den"><i>a</i><sup>2</sup></span></span>⁠</span></span> is the <a href="/wiki/Curvature" title="Curvature">spatial curvature</a> in any time-slice of the universe; it is equal to one-sixth of the spatial <a href="/wiki/Scalar_curvature" title="Scalar curvature">Ricci curvature scalar <span class="texhtml mvar" style="font-style:italic;">R</span></a> since <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {6}{c^{2}a^{2}}}({\ddot {a}}a+{\dot {a}}^{2}+kc^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi>a</mi> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>k</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {6}{c^{2}a^{2}}}({\ddot {a}}a+{\dot {a}}^{2}+kc^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b44f39df832e6887fd706cb1d626f3483aa69dad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.55ex; height:5.509ex;" alt="{\displaystyle R={\frac {6}{c^{2}a^{2}}}({\ddot {a}}a+{\dot {a}}^{2}+kc^{2})}"></span> in the Friedmann model. <span class="texhtml"><i>H</i> ≡ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ȧ</i></span><span class="sr-only">/</span><span class="den"><i>a</i></span></span>⁠</span></span> is the <a href="/wiki/Hubble_parameter" class="mw-redirect" title="Hubble parameter">Hubble parameter</a>. </p><p>We see that in the Friedmann equations, <span class="texhtml"><i>a</i>(<i>t</i>)</span> does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">k</span> which describe the same physics: </p> <ul><li><span class="texhtml"><i>k</i> = +1, 0</span> or <span class="texhtml">−1</span> depending on whether the <a href="/wiki/Shape_of_the_universe" title="Shape of the universe">shape of the universe</a> is a closed <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a>, flat (<a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>) or an open 3-<a href="/wiki/Hyperboloid" title="Hyperboloid">hyperboloid</a>, respectively.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> If <span class="texhtml"><i>k</i> = +1</span>, then <span class="texhtml mvar" style="font-style:italic;">a</span> is the <a href="/wiki/Radius_of_curvature" title="Radius of curvature">radius of curvature</a> of the universe. If <span class="texhtml"><i>k</i> = 0</span>, then <span class="texhtml mvar" style="font-style:italic;">a</span> may be fixed to any arbitrary positive number at one particular time. If <span class="texhtml"><i>k</i> = −1</span>, then (loosely speaking) one can say that <span class="texhtml"><a href="/wiki/Imaginary_unit" title="Imaginary unit"><i>i</i></a> · <i>a</i></span> is the radius of curvature of the universe.</li> <li><span class="texhtml mvar" style="font-style:italic;">a</span> is the <a href="/wiki/Scale_factor_(Universe)" class="mw-redirect" title="Scale factor (Universe)">scale factor</a> which is taken to be 1 at the present time. <span class="texhtml mvar" style="font-style:italic;">k</span> is the current <a href="/wiki/Curvature" title="Curvature">spatial curvature</a> (when <span class="texhtml"><i>a</i> = 1</span>). If the <a href="/wiki/Shape_of_the_universe" title="Shape of the universe">shape of the universe</a> is <a href="/wiki/Shape_of_the_universe#Spherical_universe" title="Shape of the universe">hyperspherical</a> and <span class="texhtml"><i>R<sub>t</sub></i></span> is the radius of curvature (<span class="texhtml"><i>R</i><sub>0</sub></span> at the present), then <span class="texhtml"><i>a</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>R<sub>t</sub></i></span><span class="sr-only">/</span><span class="den"><i>R</i><sub>0</sub></span></span>⁠</span></span>. If <span class="texhtml mvar" style="font-style:italic;">k</span> is positive, then the universe is hyperspherical. If <span class="texhtml"><i>k</i> = 0</span>, then the universe is <a href="/wiki/Shape_of_the_universe#Flat_universe" title="Shape of the universe">flat</a>. If <span class="texhtml mvar" style="font-style:italic;">k</span> is negative, then the universe is <a href="/wiki/Shape_of_the_universe#Hyperbolic_universe" title="Shape of the universe">hyperbolic</a>.</li></ul> <p>Using the first equation, the second equation can be re-expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\rho }}=-3H\left(\rho +{\frac {p}{c^{2}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\rho }}=-3H\left(\rho +{\frac {p}{c^{2}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ca4e41f6965bc7a2ac390a6c3d1464c1e4ccd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.29ex; height:6.176ex;" alt="{\displaystyle {\dot {\rho }}=-3H\left(\rho +{\frac {p}{c^{2}}}\right),}"></span> which eliminates <span class="texhtml">Λ</span> and expresses the conservation of <a href="/wiki/Mass%E2%80%93energy" class="mw-redirect" title="Mass–energy">mass–energy</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{\alpha \beta }{}_{;\beta }=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>;</mo> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{\alpha \beta }{}_{;\beta }=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c93b2114c4ad826aa8ebb6f24ef5d6db8f30d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.485ex; height:3.343ex;" alt="{\displaystyle T^{\alpha \beta }{}_{;\beta }=0.}"></span> </p><p>These equations are sometimes simplified by replacing <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\rho &\to \rho -{\frac {\Lambda c^{2}}{8\pi G}}&p&\to p+{\frac {\Lambda c^{4}}{8\pi G}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ρ</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>ρ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Λ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Λ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\rho &\to \rho -{\frac {\Lambda c^{2}}{8\pi G}}&p&\to p+{\frac {\Lambda c^{4}}{8\pi G}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3de150b7f9e61238393f7c456cdfd6e2871bc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.363ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\rho &\to \rho -{\frac {\Lambda c^{2}}{8\pi G}}&p&\to p+{\frac {\Lambda c^{4}}{8\pi G}}\end{aligned}}}"></span> to give: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}&={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\\{\dot {H}}+H^{2}={\frac {\ddot {a}}{a}}&=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mi>ρ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>¨</mo> </mover> </mrow> <mi>a</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>p</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}&={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\\{\dot {H}}+H^{2}={\frac {\ddot {a}}{a}}&=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6f4a59c5ecc96112b7357559768b90a81648d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:36.094ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}&={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\\{\dot {H}}+H^{2}={\frac {\ddot {a}}{a}}&=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right).\end{aligned}}}"></span> </p><p>The simplified form of the second equation is invariant under this transformation. </p><p>The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of <a href="/wiki/Hubble%27s_law" title="Hubble's law">Hubble's law</a>. Applied to a fluid with a given <a href="/wiki/Equation_of_state_(cosmology)" title="Equation of state (cosmology)">equation of state</a>, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density. </p><p>Some cosmologists call the second of these two equations the <b>Friedmann acceleration equation</b> and reserve the term <i>Friedmann equation</i> for only the first equation. </p> <div class="mw-heading mw-heading2"><h2 id="Density_parameter">Density parameter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=3" title="Edit section: Density parameter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>density parameter</b> <span class="texhtml mvar" style="font-style:italic;">Ω</span> is defined as the ratio of the actual (or observed) density <span class="texhtml mvar" style="font-style:italic;">ρ</span> to the critical density <span class="texhtml"><i>ρ</i><sub>c</sub></span> of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclidean). In earlier models, which did not include a <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a> term, critical density was initially defined as the watershed point between an expanding and a contracting Universe. </p><p>To date, the critical density is estimated to be approximately five atoms (of <a href="/wiki/Monatomic" class="mw-redirect" title="Monatomic">monatomic</a> <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a>) per cubic metre, whereas the average density of <a href="/wiki/Baryons#Baryonic_matter" class="mw-redirect" title="Baryons">ordinary matter</a> in the Universe is believed to be 0.2–0.25 atoms per cubic metre.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:UniverseComposition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/UniverseComposition.svg/375px-UniverseComposition.svg.png" decoding="async" width="375" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/UniverseComposition.svg/563px-UniverseComposition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/UniverseComposition.svg/750px-UniverseComposition.svg.png 2x" data-file-width="392" data-file-height="270" /></a><figcaption>Estimated relative distribution for components of the energy density of the universe. <a href="/wiki/Dark_energy" title="Dark energy">Dark energy</a> dominates the total energy (74%) while <a href="/wiki/Dark_matter" title="Dark matter">dark matter</a> (22%) constitutes most of the mass. Of the remaining baryonic matter (4%), only one tenth is compact. In February 2015, the European-led research team behind the <a href="/wiki/Planck_(spacecraft)" title="Planck (spacecraft)">Planck cosmology probe</a> released new data refining these values to 4.9% ordinary matter, 25.9% dark matter and 69.1% dark energy.</figcaption></figure> <p>A much greater density comes from the unidentified <a href="/wiki/Dark_matter" title="Dark matter">dark matter</a>, although both ordinary and dark matter contribute in favour of contraction of the universe. However, the largest part comes from so-called <a href="/wiki/Dark_energy" title="Dark energy">dark energy</a>, which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), dark energy does not lead to contraction of the universe but rather may accelerate its expansion. </p><p>An expression for the critical density is found by assuming <span class="texhtml mvar" style="font-style:italic;">Λ</span> to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, <span class="texhtml mvar" style="font-style:italic;">k</span>, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {c} }={\frac {3H^{2}}{8\pi G}}=1.8788\times 10^{-26}h^{2}{\rm {kg}}\,{\rm {m}}^{-3}=2.7754\times 10^{11}h^{2}M_{\odot }\,{\rm {Mpc}}^{-3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.8788</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>26</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>2.7754</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊙</mo> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {c} }={\frac {3H^{2}}{8\pi G}}=1.8788\times 10^{-26}h^{2}{\rm {kg}}\,{\rm {m}}^{-3}=2.7754\times 10^{11}h^{2}M_{\odot }\,{\rm {Mpc}}^{-3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86575f1a72a49c5b9de77ae09ace78c1899ced9a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:68.316ex; height:5.843ex;" alt="{\displaystyle \rho _{\mathrm {c} }={\frac {3H^{2}}{8\pi G}}=1.8788\times 10^{-26}h^{2}{\rm {kg}}\,{\rm {m}}^{-3}=2.7754\times 10^{11}h^{2}M_{\odot }\,{\rm {Mpc}}^{-3},}"></span> </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;">(where <span class="texhtml"><i>h</i> = <i>H</i><sub>0</sub>/(100 km/s/Mpc)</span>. For <span class="texhtml"><i>H<sub>o</sub></i> = 67.4 km/s/Mpc</span>, i.e. <span class="texhtml"><i>h</i> = 0.674</span>, <span class="texhtml"><i>ρ</i><sub>c</sub> = <span class="nowrap"><span data-sort-value="6973850000000000000♠"></span>8.5<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−27</sup> kg/m<sup>3</sup></span></span>).</div> <p>The density parameter (useful for comparing different cosmological models) is then defined as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega :={\frac {\rho }{\rho _{c}}}={\frac {8\pi G\rho }{3H^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> <mi>ρ</mi> </mrow> <mrow> <mn>3</mn> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega :={\frac {\rho }{\rho _{c}}}={\frac {8\pi G\rho }{3H^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92dd43938e99ab61b977367c1778dc9ced549411" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.51ex; height:6.009ex;" alt="{\displaystyle \Omega :={\frac {\rho }{\rho _{c}}}={\frac {8\pi G\rho }{3H^{2}}}.}"></span> </p><p>This term originally was used as a means to determine the <a href="/wiki/Shape_of_the_universe" title="Shape of the universe">spatial geometry</a> of the universe, where <span class="texhtml"><i>ρ</i><sub>c</sub></span> is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if <span class="texhtml mvar" style="font-style:italic;">Ω</span> is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If <span class="texhtml mvar" style="font-style:italic;">Ω</span> is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for <span class="texhtml mvar" style="font-style:italic;">Ω</span> in which case this density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the <a href="/wiki/Lambda-CDM_model" title="Lambda-CDM model">ΛCDM model</a>, there are important components of <span class="texhtml mvar" style="font-style:italic;">Ω</span> due to <a href="/wiki/Baryon" title="Baryon">baryons</a>, <a href="/wiki/Cold_dark_matter" title="Cold dark matter">cold dark matter</a> and <a href="/wiki/Dark_energy" title="Dark energy">dark energy</a>. The spatial geometry of the <a href="/wiki/Universe" title="Universe">universe</a> has been measured by the <a href="/wiki/Wilkinson_Microwave_Anisotropy_Probe" title="Wilkinson Microwave Anisotropy Probe">WMAP</a> spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter <span class="texhtml mvar" style="font-style:italic;">k</span> is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see. </p><p>The first Friedmann equation is often seen in terms of the present values of the density parameters, that is<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {H^{2}}{H_{0}^{2}}}=\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H^{2}}{H_{0}^{2}}}=\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008e5b14492bb3f8ace995f0a87c41407d6ed504" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.42ex; height:6.843ex;" alt="{\displaystyle {\frac {H^{2}}{H_{0}^{2}}}=\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }.}"></span> Here <span class="texhtml">Ω<sub>0,R</sub></span> is the radiation density today (when <span class="texhtml"><i>a</i> = 1</span>), <span class="texhtml">Ω<sub>0,M</sub></span> is the matter (<a href="/wiki/Dark_matter" title="Dark matter">dark</a> plus <a href="/wiki/Baryon" title="Baryon">baryonic</a>) density today, <span class="texhtml">Ω<sub>0,<i>k</i></sub> = 1 − Ω<sub>0</sub></span> is the "spatial curvature density" today, and <span class="texhtml">Ω<sub>0,Λ</sub></span> is the cosmological constant or vacuum density today. </p> <div class="mw-heading mw-heading2"><h2 id="Useful_solutions">Useful solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=4" title="Edit section: Useful solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Friedmann equations can be solved exactly in presence of a <a href="/wiki/Perfect_fluid" title="Perfect fluid">perfect fluid</a> with equation of state <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=w\rho c^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>w</mi> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=w\rho c^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1a2048fb4ef752b0b80ea6f7c16e71a15d1064" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.931ex; height:3.176ex;" alt="{\displaystyle p=w\rho c^{2},}"></span> where <span class="texhtml mvar" style="font-style:italic;">p</span> is the <a href="/wiki/Pressure" title="Pressure">pressure</a>, <span class="texhtml mvar" style="font-style:italic;">ρ</span> is the mass density of the fluid in the comoving frame and <span class="texhtml mvar" style="font-style:italic;">w</span> is some constant. </p><p>In spatially flat case (<span class="texhtml"><i>k</i> = 0</span>), the solution for the scale factor is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(t)=a_{0}\,t^{\frac {2}{3(w+1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>w</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(t)=a_{0}\,t^{\frac {2}{3(w+1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab721149ea500bb467e22d3aa887e7120cc2491" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.923ex; height:4.509ex;" alt="{\displaystyle a(t)=a_{0}\,t^{\frac {2}{3(w+1)}}}"></span> where <span class="texhtml"><i>a</i><sub>0</sub></span> is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by <span class="texhtml mvar" style="font-style:italic;">w</span> is extremely important for cosmology. For example, <span class="texhtml"><i>w</i> = 0</span> describes a <a href="/wiki/Matter-dominated_era" class="mw-redirect" title="Matter-dominated era">matter-dominated</a> universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(t)\propto t^{2/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(t)\propto t^{2/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c47f17d2db28023149a7d02cde580374e244ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.515ex; height:3.343ex;" alt="{\displaystyle a(t)\propto t^{2/3}}"></span> matter-dominated Another important example is the case of a <a href="/wiki/Radiation-dominated_era" class="mw-redirect" title="Radiation-dominated era">radiation-dominated</a> universe, namely when <span class="texhtml"><i>w</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span></span>. This leads to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(t)\propto t^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(t)\propto t^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5c2427ec6c9e3477573806e246aaca10375ab3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.515ex; height:3.343ex;" alt="{\displaystyle a(t)\propto t^{1/2}}"></span> radiation-dominated </p><p>Note that this solution is not valid for domination of the cosmological constant, which corresponds to an <span class="texhtml"><i>w</i> = −1</span>. In this case the energy density is constant and the scale factor grows exponentially. </p><p>Solutions for other values of <span class="texhtml mvar" style="font-style:italic;">k</span> can be found at <style data-mw-deduplicate="TemplateStyles:r1238218222">.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("//upload.wikimedia.org/wikipedia/commons/6/65/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("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/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}}</style><cite id="CITEREFTersic" class="citation web cs1">Tersic, Balsa. <a rel="nofollow" class="external text" href="https://www.academia.edu/5025956">"Lecture Notes on Astrophysics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">24 February</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lecture+Notes+on+Astrophysics&rft.aulast=Tersic&rft.aufirst=Balsa&rft_id=https%3A%2F%2Fwww.academia.edu%2F5025956&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Mixtures">Mixtures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=5" title="Edit section: Mixtures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+{\frac {p_{f}}{c^{2}}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+{\frac {p_{f}}{c^{2}}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b6f7cad45e65c70adf90b6d167fabf95232abb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.161ex; height:6.176ex;" alt="{\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+{\frac {p_{f}}{c^{2}}}\right)\,}"></span> holds separately for each such fluid <span class="texhtml mvar" style="font-style:italic;">f</span>. In each case, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+w_{f}\rho _{f}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+w_{f}\rho _{f}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8adcab227461baf79e9dd9f66187cb48ef757cd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.546ex; height:3.176ex;" alt="{\displaystyle {\dot {\rho }}_{f}=-3H\left(\rho _{f}+w_{f}\rho _{f}\right)\,}"></span> from which we get <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rho }_{f}\propto a^{-3\left(1+w_{f}\right)}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>∝</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rho }_{f}\propto a^{-3\left(1+w_{f}\right)}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8e7fdf2148af6c4c60be3d63389ebdadd117287" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.715ex; height:3.843ex;" alt="{\displaystyle {\rho }_{f}\propto a^{-3\left(1+w_{f}\right)}\,.}"></span> </p><p>For example, one can form a linear combination of such terms <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =Aa^{-3}+Ba^{-4}+Ca^{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>A</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>C</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =Aa^{-3}+Ba^{-4}+Ca^{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32b77c1858948327665905e37677713967768d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.051ex; height:3.176ex;" alt="{\displaystyle \rho =Aa^{-3}+Ba^{-4}+Ca^{0}\,}"></span> where <span class="texhtml mvar" style="font-style:italic;">A</span> is the density of "dust" (ordinary matter, <span class="texhtml"><i>w</i> = 0</span>) when <span class="texhtml"><i>a</i> = 1</span>; <span class="texhtml mvar" style="font-style:italic;">B</span> is the density of radiation (<span class="texhtml"><i>w</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span></span>) when <span class="texhtml"><i>a</i> = 1</span>; and <span class="texhtml mvar" style="font-style:italic;">C</span> is the density of "dark energy" (<span class="texhtml"><i>w</i> = −1</span>). One then substitutes this into <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>π</mi> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mi>ρ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fd414834ca113227432384766e8b1724ca32fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.335ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\,}"></span> and solves for <span class="texhtml mvar" style="font-style:italic;">a</span> as a function of time. </p> <div class="mw-heading mw-heading3"><h3 id="Detailed_derivation">Detailed derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=6" title="Edit section: Detailed derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To make the solutions more explicit, we can derive the full relationships from the first Friedmann equation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {H^{2}}{H_{0}^{2}}}=\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H^{2}}{H_{0}^{2}}}=\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd3babff4c54bbe60e17853d0cce40d0d704efc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.774ex; height:6.843ex;" alt="{\displaystyle {\frac {H^{2}}{H_{0}^{2}}}=\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}H&={\frac {\dot {a}}{a}}\\[6px]H^{2}&=H_{0}^{2}\left(\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }\right)\\[6pt]H&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}}\\[6pt]{\frac {\dot {a}}{a}}&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}}\\[6pt]{\frac {\mathrm {d} a}{\mathrm {d} t}}&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-2}+\Omega _{0,\mathrm {M} }a^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a^{2}}}\\[6pt]\mathrm {d} a&=\mathrm {d} tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-2}+\Omega _{0,\mathrm {M} }a^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a^{2}}}\\[6pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.9em 0.9em 0.9em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>H</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> <mi>a</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˙</mo> </mover> </mrow> <mi>a</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>a</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H&={\frac {\dot {a}}{a}}\\[6px]H^{2}&=H_{0}^{2}\left(\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }\right)\\[6pt]H&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}}\\[6pt]{\frac {\dot {a}}{a}}&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}}\\[6pt]{\frac {\mathrm {d} a}{\mathrm {d} t}}&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-2}+\Omega _{0,\mathrm {M} }a^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a^{2}}}\\[6pt]\mathrm {d} a&=\mathrm {d} tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-2}+\Omega _{0,\mathrm {M} }a^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a^{2}}}\\[6pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ac295f458a52350717883d08eec15b8b05ffd5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.838ex; width:50.052ex; height:36.843ex;" alt="{\displaystyle {\begin{aligned}H&={\frac {\dot {a}}{a}}\\[6px]H^{2}&=H_{0}^{2}\left(\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }\right)\\[6pt]H&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}}\\[6pt]{\frac {\dot {a}}{a}}&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-4}+\Omega _{0,\mathrm {M} }a^{-3}+\Omega _{0,k}a^{-2}+\Omega _{0,\Lambda }}}\\[6pt]{\frac {\mathrm {d} a}{\mathrm {d} t}}&=H_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-2}+\Omega _{0,\mathrm {M} }a^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a^{2}}}\\[6pt]\mathrm {d} a&=\mathrm {d} tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }a^{-2}+\Omega _{0,\mathrm {M} }a^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a^{2}}}\\[6pt]\end{aligned}}}"></span> </p><p>Rearranging and changing to use variables <span class="texhtml"><i>a</i>′</span> and <span class="texhtml"><i>t</i>′</span> for the integration <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tH_{0}=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {R} }a'^{-2}+\Omega _{0,\mathrm {M} }a'^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a'^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tH_{0}=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {R} }a'^{-2}+\Omega _{0,\mathrm {M} }a'^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a'^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010ab94f15ed6bf38443ed5348eb8bf220cdddbd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:50.797ex; height:8.343ex;" alt="{\displaystyle tH_{0}=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {R} }a'^{-2}+\Omega _{0,\mathrm {M} }a'^{-1}+\Omega _{0,k}+\Omega _{0,\Lambda }a'^{2}}}}}"></span> </p><p>Solutions for the dependence of the scale factor with respect to time for universes dominated by each component can be found. In each we also have assumed that <span class="texhtml"><i>Ω</i><sub>0,<i>k</i></sub> ≈ 0</span>, which is the same as assuming that the dominating source of energy density is approximately 1. </p><p>For matter-dominated universes, where <span class="texhtml"><i>Ω</i><sub>0,M</sub> ≫ <i>Ω</i><sub>0,R</sub></span> and <span class="texhtml"><i>Ω</i><sub>0,<i>Λ</i></sub></span>, as well as <span class="texhtml"><i>Ω</i><sub>0,M</sub> ≈ 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}tH_{0}&=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {M} }a'^{-1}}}}\\[6px]tH_{0}{\sqrt {\Omega _{0,\mathrm {M} }}}&=\left.\left({\tfrac {2}{3}}{a'}^{3/2}\right)\,\right|_{0}^{a}\\[6px]\left({\tfrac {3}{2}}tH_{0}{\sqrt {\Omega _{0,\mathrm {M} }}}\right)^{2/3}&=a(t)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}tH_{0}&=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {M} }a'^{-1}}}}\\[6px]tH_{0}{\sqrt {\Omega _{0,\mathrm {M} }}}&=\left.\left({\tfrac {2}{3}}{a'}^{3/2}\right)\,\right|_{0}^{a}\\[6px]\left({\tfrac {3}{2}}tH_{0}{\sqrt {\Omega _{0,\mathrm {M} }}}\right)^{2/3}&=a(t)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e58758d5ef654eb31d1ee67c044cdd44f825b06d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:37.652ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}tH_{0}&=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {M} }a'^{-1}}}}\\[6px]tH_{0}{\sqrt {\Omega _{0,\mathrm {M} }}}&=\left.\left({\tfrac {2}{3}}{a'}^{3/2}\right)\,\right|_{0}^{a}\\[6px]\left({\tfrac {3}{2}}tH_{0}{\sqrt {\Omega _{0,\mathrm {M} }}}\right)^{2/3}&=a(t)\end{aligned}}}"></span> which recovers the aforementioned <span class="texhtml"><i>a</i> ∝ <i>t</i><sup>2/3</sup></span> </p><p>For radiation-dominated universes, where <span class="texhtml">Ω<sub>0,R</sub> ≫ Ω<sub>0,M</sub></span> and <span class="texhtml">Ω<sub>0,Λ</sub></span>, as well as <span class="texhtml">Ω<sub>0,R</sub> ≈ 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}tH_{0}&=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {R} }a'^{-2}}}}\\[6px]tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }}}&=\left.{\frac {a'^{2}}{2}}\,\right|_{0}^{a}\\[6px]\left(2tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }}}\right)^{1/2}&=a(t)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <msup> <mi>a</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}tH_{0}&=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {R} }a'^{-2}}}}\\[6px]tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }}}&=\left.{\frac {a'^{2}}{2}}\,\right|_{0}^{a}\\[6px]\left(2tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }}}\right)^{1/2}&=a(t)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5f31e287530f47ec22fc3bad5a3ca4b6d11103" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.171ex; width:36.561ex; height:23.509ex;" alt="{\displaystyle {\begin{aligned}tH_{0}&=\int _{0}^{a}{\frac {\mathrm {d} a'}{\sqrt {\Omega _{0,\mathrm {R} }a'^{-2}}}}\\[6px]tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }}}&=\left.{\frac {a'^{2}}{2}}\,\right|_{0}^{a}\\[6px]\left(2tH_{0}{\sqrt {\Omega _{0,\mathrm {R} }}}\right)^{1/2}&=a(t)\end{aligned}}}"></span> </p><p>For <span class="texhtml mvar" style="font-style:italic;">Λ</span>-dominated universes, where <span class="texhtml"><i>Ω</i><sub>0,<i>Λ</i></sub> ≫ <i>Ω</i><sub>0,R</sub></span> and <span class="texhtml"><i>Ω</i><sub>0,M</sub></span>, as well as <span class="texhtml"><i>Ω</i><sub>0,<i>Λ</i></sub> ≈ 1</span>, and where we now will change our bounds of integration from <span class="texhtml"><i>t<sub>i</sub></i></span> to <span class="texhtml mvar" style="font-style:italic;">t</span> and likewise <span class="texhtml"><i>a<sub>i</sub></i></span> to <span class="texhtml mvar" style="font-style:italic;">a</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(t-t_{i}\right)H_{0}&=\int _{a_{i}}^{a}{\frac {\mathrm {d} a'}{\sqrt {(\Omega _{0,\Lambda }a'^{2})}}}\\[6px]\left(t-t_{i}\right)H_{0}{\sqrt {\Omega _{0,\Lambda }}}&={\bigl .}\ln |a'|\,{\bigr |}_{a_{i}}^{a}\\[6px]a_{i}\exp \left((t-t_{i})H_{0}{\sqrt {\Omega _{0,\Lambda }}}\right)&=a(t)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> <msqrt> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em"></mo> </mrow> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(t-t_{i}\right)H_{0}&=\int _{a_{i}}^{a}{\frac {\mathrm {d} a'}{\sqrt {(\Omega _{0,\Lambda }a'^{2})}}}\\[6px]\left(t-t_{i}\right)H_{0}{\sqrt {\Omega _{0,\Lambda }}}&={\bigl .}\ln |a'|\,{\bigr |}_{a_{i}}^{a}\\[6px]a_{i}\exp \left((t-t_{i})H_{0}{\sqrt {\Omega _{0,\Lambda }}}\right)&=a(t)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f90402286787d1facaea641bc3a7b8c7b08687" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:45.352ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}\left(t-t_{i}\right)H_{0}&=\int _{a_{i}}^{a}{\frac {\mathrm {d} a'}{\sqrt {(\Omega _{0,\Lambda }a'^{2})}}}\\[6px]\left(t-t_{i}\right)H_{0}{\sqrt {\Omega _{0,\Lambda }}}&={\bigl .}\ln |a'|\,{\bigr |}_{a_{i}}^{a}\\[6px]a_{i}\exp \left((t-t_{i})H_{0}{\sqrt {\Omega _{0,\Lambda }}}\right)&=a(t)\end{aligned}}}"></span> </p><p>The <span class="texhtml mvar" style="font-style:italic;">Λ</span>-dominated universe solution is of particular interest because the second derivative with respect to time is positive, non-zero; in other words implying an accelerating expansion of the universe, making <span class="texhtml"><i>ρ<sub>Λ</sub></i></span> a candidate for <a href="/wiki/Dark_energy" title="Dark energy">dark energy</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a(t)&=a_{i}\exp \left((t-t_{i})H_{0}\textstyle {\sqrt {\Omega _{0,\Lambda }}}\right)\\[6px]{\frac {\mathrm {d} ^{2}a(t)}{\mathrm {d} t^{2}}}&=a_{i}{H_{0}}^{2}\,\Omega _{0,\Lambda }\exp \left((t-t_{i})H_{0}\textstyle {\sqrt {\Omega _{0,\Lambda }}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a(t)&=a_{i}\exp \left((t-t_{i})H_{0}\textstyle {\sqrt {\Omega _{0,\Lambda }}}\right)\\[6px]{\frac {\mathrm {d} ^{2}a(t)}{\mathrm {d} t^{2}}}&=a_{i}{H_{0}}^{2}\,\Omega _{0,\Lambda }\exp \left((t-t_{i})H_{0}\textstyle {\sqrt {\Omega _{0,\Lambda }}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8296ac17625a996bf2b9af1c035b337aae0fd41e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:44.536ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}a(t)&=a_{i}\exp \left((t-t_{i})H_{0}\textstyle {\sqrt {\Omega _{0,\Lambda }}}\right)\\[6px]{\frac {\mathrm {d} ^{2}a(t)}{\mathrm {d} t^{2}}}&=a_{i}{H_{0}}^{2}\,\Omega _{0,\Lambda }\exp \left((t-t_{i})H_{0}\textstyle {\sqrt {\Omega _{0,\Lambda }}}\right)\end{aligned}}}"></span> </p><p>Where by construction <span class="texhtml"><i>a<sub>i</sub></i> > 0</span>, our assumptions were <span class="texhtml"><i>Ω</i><sub>0,<i>Λ</i></sub> ≈ 1</span>, and <span class="texhtml"><i>H</i><sub>0</sub></span> has been measured to be positive, forcing the acceleration to be greater than zero. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=7" title="Edit section: In popular culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several students at <a href="/wiki/Tsinghua_University" title="Tsinghua University">Tsinghua University</a> (<a href="/wiki/Chinese_Communist_Party" title="Chinese Communist Party">CCP</a> <a href="/wiki/Leader_of_the_Chinese_Communist_Party" title="Leader of the Chinese Communist Party">leader</a> <a href="/wiki/Xi_Jinping" title="Xi Jinping">Xi Jinping</a>'s <a href="/wiki/Alma_mater" title="Alma mater">alma mater</a>) participating in the <a href="/wiki/2022_COVID-19_protests_in_China" title="2022 COVID-19 protests in China">2022 COVID-19 protests in China</a> carried placards with Friedmann equations scrawled on them, interpreted by some as a play on the words "Free man". Others have interpreted the use of the equations as a call to “open up” China and stop its Zero Covid policy, as the Friedmann equations relate to the expansion, or “opening” of the universe.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics of general relativity</a></li> <li><a href="/wiki/Solutions_of_the_Einstein_field_equations" title="Solutions of the Einstein field equations">Solutions of the Einstein field equations</a></li> <li><a href="/wiki/Warm_inflation" title="Warm inflation">Warm inflation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=9" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-af1922-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-af1922_1-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedman1922" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Alexander_Alexandrovich_Friedman" class="mw-redirect" title="Alexander Alexandrovich Friedman">Friedman, A</a> (1922). "Über die Krümmung des Raumes". <i>Z. Phys.</i> (in German). <b>10</b> (1): 377–386. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1922ZPhy...10..377F">1922ZPhy...10..377F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01332580">10.1007/BF01332580</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125190902">125190902</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Z.+Phys.&rft.atitle=%C3%9Cber+die+Kr%C3%BCmmung+des+Raumes&rft.volume=10&rft.issue=1&rft.pages=377-386&rft.date=1922&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125190902%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01332580&rft_id=info%3Abibcode%2F1922ZPhy...10..377F&rft.aulast=Friedman&rft.aufirst=A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span> (English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedman1999" class="citation journal cs1">Friedman, A (1999). "On the Curvature of Space". <i>General Relativity and Gravitation</i>. <b>31</b> (12): 1991–2000. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999GReGr..31.1991F">1999GReGr..31.1991F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1026751225741">10.1023/A:1026751225741</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122950995">122950995</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=General+Relativity+and+Gravitation&rft.atitle=On+the+Curvature+of+Space&rft.volume=31&rft.issue=12&rft.pages=1991-2000&rft.date=1999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122950995%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FA%3A1026751225741&rft_id=info%3Abibcode%2F1999GReGr..31.1991F&rft.aulast=Friedman&rft.aufirst=A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span>). The original Russian manuscript of this paper is preserved in the <a rel="nofollow" class="external text" href="http://ilorentz.org/history/Friedmann_archive">Ehrenfest archive</a>.</span> </li> <li id="cite_note-af1924-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-af1924_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedmann1924" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Alexander_Alexandrovich_Friedman" class="mw-redirect" title="Alexander Alexandrovich Friedman">Friedmann, A</a> (1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes". <i>Z. Phys.</i> (in German). <b>21</b> (1): 326–332. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1924ZPhy...21..326F">1924ZPhy...21..326F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01328280">10.1007/BF01328280</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120551579">120551579</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Z.+Phys.&rft.atitle=%C3%9Cber+die+M%C3%B6glichkeit+einer+Welt+mit+konstanter+negativer+Kr%C3%BCmmung+des+Raumes&rft.volume=21&rft.issue=1&rft.pages=326-332&rft.date=1924&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120551579%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01328280&rft_id=info%3Abibcode%2F1924ZPhy...21..326F&rft.aulast=Friedmann&rft.aufirst=A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span> (English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedmann1999" class="citation journal cs1">Friedmann, A (1999). "On the Possibility of a World with Constant Negative Curvature of Space". <i>General Relativity and Gravitation</i>. <b>31</b> (12): 2001–2008. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999GReGr..31.2001F">1999GReGr..31.2001F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1026755309811">10.1023/A:1026755309811</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123512351">123512351</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=General+Relativity+and+Gravitation&rft.atitle=On+the+Possibility+of+a+World+with+Constant+Negative+Curvature+of+Space&rft.volume=31&rft.issue=12&rft.pages=2001-2008&rft.date=1999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123512351%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FA%3A1026755309811&rft_id=info%3Abibcode%2F1999GReGr..31.2001F&rft.aulast=Friedmann&rft.aufirst=A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span>)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD'Inverno2008" class="citation book cs1">D'Inverno, Ray (2008). <i>Introducing Einstein's relativity</i> (Repr ed.). Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-859686-8" title="Special:BookSources/978-0-19-859686-8"><bdi>978-0-19-859686-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introducing+Einstein%27s+relativity&rft.place=Oxford&rft.edition=Repr&rft.pub=Clarendon+Press&rft.date=2008&rft.isbn=978-0-19-859686-8&rft.aulast=D%27Inverno&rft.aufirst=Ray&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRees2001" class="citation book cs1">Rees, Martin (2001). <i>Just six numbers: the deep forces that shape the universe</i>. Astronomy/science (Repr. ed.). New York, NY: Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-03673-8" title="Special:BookSources/978-0-465-03673-8"><bdi>978-0-465-03673-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Just+six+numbers%3A+the+deep+forces+that+shape+the+universe&rft.place=New+York%2C+NY&rft.series=Astronomy%2Fscience&rft.edition=Repr.&rft.pub=Basic+Books&rft.date=2001&rft.isbn=978-0-465-03673-8&rft.aulast=Rees&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="What kind of atoms? (September 2015)">clarification needed</span></a></i>]</sup></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://map.gsfc.nasa.gov/universe/uni_matter.html">"Universe 101"</a>. <a href="/wiki/NASA" title="NASA">NASA</a><span class="reference-accessdate">. Retrieved <span class="nowrap">September 9,</span> 2015</span>. <q>The actual density of atoms is equivalent to roughly 1 proton per 4 cubic meters.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Universe+101&rft.pub=NASA&rft_id=http%3A%2F%2Fmap.gsfc.nasa.gov%2Funiverse%2Funi_matter.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNemiroffPatla,_Bijunath2008" class="citation journal cs1"><a href="/wiki/Robert_J._Nemiroff" title="Robert J. Nemiroff">Nemiroff, Robert J.</a>; Patla, Bijunath (2008). "Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations". <i>American Journal of Physics</i>. <b>76</b> (3): 265–276. <a href="/wiki/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/astro-ph/0703739">astro-ph/0703739</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008AmJPh..76..265N">2008AmJPh..76..265N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.2830536">10.1119/1.2830536</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51782808">51782808</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Adventures+in+Friedmann+cosmology%3A+A+detailed+expansion+of+the+cosmological+Friedmann+equations&rft.volume=76&rft.issue=3&rft.pages=265-276&rft.date=2008&rft_id=info%3Aarxiv%2Fastro-ph%2F0703739&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51782808%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.2830536&rft_id=info%3Abibcode%2F2008AmJPh..76..265N&rft.aulast=Nemiroff&rft.aufirst=Robert+J.&rft.au=Patla%2C+Bijunath&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurphy2022" class="citation news cs1">Murphy, Matt (November 28, 2022). <a rel="nofollow" class="external text" href="https://www.bbc.com/news/world-asia-china-63778871">"China's protests: Blank paper becomes the symbol of rare demonstrations"</a>. <i><a href="/wiki/BBC_News" title="BBC News">BBC News</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BBC+News&rft.atitle=China%27s+protests%3A+Blank+paper+becomes+the+symbol+of+rare+demonstrations&rft.date=2022-11-28&rft.aulast=Murphy&rft.aufirst=Matt&rft_id=https%3A%2F%2Fwww.bbc.com%2Fnews%2Fworld-asia-china-63778871&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Friedmann_equations&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiebscher2005" class="citation book cs1">Liebscher, Dierck-Ekkehard (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VK_rbBR61eUC&pg=PA53">"Expansion"</a>. <i>Cosmology</i>. Berlin: Springer. pp. 53–77. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-23261-3" title="Special:BookSources/3-540-23261-3"><bdi>3-540-23261-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Expansion&rft.btitle=Cosmology&rft.place=Berlin&rft.pages=53-77&rft.pub=Springer&rft.date=2005&rft.isbn=3-540-23261-3&rft.aulast=Liebscher&rft.aufirst=Dierck-Ekkehard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVK_rbBR61eUC%26pg%3DPA53&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFriedmann+equations" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Relativity" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Relativity" title="Template:Relativity"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Relativity" title="Template talk:Relativity"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Relativity" title="Special:EditPage/Template:Relativity"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Relativity" style="font-size:114%;margin:0 4em"><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Relativity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Special_relativity" title="Special relativity">Special<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principle_of_relativity" title="Principle of relativity">Principle of relativity</a> (<a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean relativity</a></li> <li><a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>)</li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Speed_of_light" title="Speed of light">Speed of light</a></li> <li><a href="/wiki/Hyperbolic_orthogonality" title="Hyperbolic orthogonality">Hyperbolic orthogonality</a></li> <li><a href="/wiki/Rapidity" title="Rapidity">Rapidity</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_time" title="Proper time">Proper time</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Relativistic mass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/List_of_textbooks_on_relativity" title="List of textbooks on relativity">Textbooks</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">Mass–energy equivalence (E=mc<sup>2</sup>)</a></li> <li><a href="/wiki/Length_contraction" title="Length contraction">Length contraction</a></li> <li><a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">Relativity of simultaneity</a></li> <li><a href="/wiki/Relativistic_Doppler_effect" title="Relativistic Doppler effect">Relativistic Doppler effect</a></li> <li><a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a></li> <li><a href="/wiki/Ladder_paradox" title="Ladder paradox">Ladder paradox</a></li> <li><a href="/wiki/Twin_paradox" title="Twin paradox">Twin paradox</a></li> <li><a href="/wiki/Terrell_rotation" title="Terrell rotation">Terrell rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Light_cone" title="Light cone">Light cone</a></li> <li><a href="/wiki/World_line" title="World line">World line</a></li> <li><a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagram</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/General_relativity" title="General relativity">General<br />relativity</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Background</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</a></li> <li><a href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematical formulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Fundamental<br />concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li> <li><a href="/wiki/Penrose_diagram" title="Penrose diagram">Penrose diagram</a></li> <li><a href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics</a></li> <li><a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</a></li> <li><a href="/wiki/BSSN_formalism" title="BSSN formalism">BSSN formalism</a></li> <li><a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li> <li><a href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</a></li> <li><a href="/wiki/Raychaudhuri_equation" title="Raychaudhuri equation">Raychaudhuri equation</a></li> <li><a href="/wiki/Hamilton%E2%80%93Jacobi%E2%80%93Einstein_equation" title="Hamilton–Jacobi–Einstein equation">Hamilton–Jacobi–Einstein equation</a></li> <li><a href="/wiki/Ernst_equation" title="Ernst equation">Ernst equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_hole" title="Black hole">Black hole</a></li> <li><a href="/wiki/Event_horizon" title="Event horizon">Event horizon</a></li> <li><a href="/wiki/Gravitational_singularity" title="Gravitational singularity">Singularity</a></li> <li><a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem</a></li></ul> <ul><li><a href="/wiki/Gravitational_wave" title="Gravitational wave">Gravitational waves</a>: <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">astronomy</a></li> <li><a href="/wiki/Gravitational-wave_observatory" title="Gravitational-wave observatory">detectors</a> (<a href="/wiki/LIGO" title="LIGO">LIGO</a> and <a href="/wiki/LIGO_Scientific_Collaboration" title="LIGO Scientific Collaboration">collaboration</a></li> <li><a href="/wiki/Virgo_interferometer" title="Virgo interferometer">Virgo</a></li> <li><a href="/wiki/LISA_Pathfinder" title="LISA Pathfinder">LISA Pathfinder</a></li> <li><a href="/wiki/GEO600" title="GEO600">GEO</a>)</li> <li><a href="/wiki/Hulse%E2%80%93Taylor_binary" class="mw-redirect" title="Hulse–Taylor binary">Hulse–Taylor binary</a></li></ul> <ul><li><a href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Other tests</a>: <a href="/wiki/Apsidal_precession" title="Apsidal precession">precession</a> of Mercury</li> <li><a href="/wiki/Gravitational_lens" title="Gravitational lens">lensing</a> (together with <a href="/wiki/Einstein_cross" class="mw-redirect" title="Einstein cross">Einstein cross</a> and <a href="/wiki/Einstein_rings" class="mw-redirect" title="Einstein rings">Einstein rings</a>)</li> <li><a href="/wiki/Gravitational_redshift" title="Gravitational redshift">redshift</a></li> <li><a href="/wiki/Shapiro_time_delay" title="Shapiro time delay">Shapiro delay</a></li> <li><a href="/wiki/Frame-dragging" title="Frame-dragging">frame-dragging</a> / <a href="/wiki/Geodetic_effect" title="Geodetic effect">geodetic effect</a> (<a href="/wiki/Lense%E2%80%93Thirring_precession" title="Lense–Thirring precession">Lense–Thirring precession</a>)</li> <li><a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;">Advanced<br />theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;text-align:center;"><a href="/wiki/Exact_solutions_in_general_relativity" title="Exact solutions in general relativity">Solutions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Cosmological: <a href="/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric" title="Friedmann–Lemaître–Robertson–Walker metric">Friedmann–Lemaître–Robertson–Walker</a> (<a class="mw-selflink selflink">Friedmann equations</a>)</li> <li><a href="/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric" title="Lemaître–Tolman metric">Lemaître–Tolman</a></li> <li><a href="/wiki/Kasner_metric" title="Kasner metric">Kasner</a></li> <li><a href="/wiki/BKL_singularity" title="BKL singularity">BKL singularity</a></li> <li><a href="/wiki/G%C3%B6del_metric" title="Gödel metric">Gödel</a></li> <li><a href="/wiki/Milne_model" title="Milne model">Milne</a></li></ul> <ul><li>Spherical: <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild</a> (<a href="/wiki/Interior_Schwarzschild_metric" title="Interior Schwarzschild metric">interior</a></li> <li><a href="/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" title="Tolman–Oppenheimer–Volkoff equation">Tolman–Oppenheimer–Volkoff equation</a>)</li> <li><a href="/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric" title="Reissner–Nordström metric">Reissner–Nordström</a></li></ul> <ul><li>Axisymmetric: <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr</a> (<a href="/wiki/Kerr%E2%80%93Newman_metric" title="Kerr–Newman metric">Kerr–Newman</a>)</li> <li><a href="/wiki/Weyl%E2%88%92Lewis%E2%88%92Papapetrou_coordinates" class="mw-redirect" title="Weyl−Lewis−Papapetrou coordinates">Weyl−Lewis−Papapetrou</a></li> <li><a href="/wiki/Taub%E2%80%93NUT_space" title="Taub–NUT space">Taub–NUT</a></li> <li><a href="/wiki/Van_Stockum_dust" title="Van Stockum dust">van Stockum dust</a></li> <li><a href="/wiki/Relativistic_disk" title="Relativistic disk">discs</a></li></ul> <ul><li>Others: <a href="/wiki/Pp-wave_spacetime" title="Pp-wave spacetime">pp-wave</a></li> <li><a href="/wiki/Ozsv%C3%A1th%E2%80%93Sch%C3%BCcking_metric" title="Ozsváth–Schücking metric">Ozsváth–Schücking</a></li> <li><a href="/wiki/Alcubierre_drive" title="Alcubierre drive">Alcubierre</a></li></ul> <ul><li>In computational physics: <a href="/wiki/Numerical_relativity" title="Numerical relativity">Numerical relativity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Scientists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a></li> <li><a href="/wiki/Willem_de_Sitter" title="Willem de Sitter">de Sitter</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Arthur_Eddington" title="Arthur Eddington">Eddington</a></li> <li><a href="/wiki/Alexander_Friedmann" title="Alexander Friedmann">Friedmann</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a></li> <li><a href="/wiki/Howard_P._Robertson" title="Howard P. Robertson">Robertson</a></li> <li><a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">Chandrasekhar</a></li> <li><a href="/wiki/Fritz_Zwicky" title="Fritz Zwicky">Zwicky</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Choquet-Bruhat</a></li> <li><a href="/wiki/Roy_Kerr" title="Roy Kerr">Kerr</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zel'dovich</a></li> <li><a href="/wiki/Igor_Dmitriyevich_Novikov" title="Igor Dmitriyevich Novikov">Novikov</a></li> <li><a href="/wiki/J%C3%BCrgen_Ehlers" title="Jürgen Ehlers">Ehlers</a></li> <li><a href="/wiki/Robert_Geroch" title="Robert Geroch">Geroch</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Joseph_Hooton_Taylor_Jr." title="Joseph Hooton Taylor Jr.">Taylor</a></li> <li><a href="/wiki/Russell_Alan_Hulse" title="Russell Alan Hulse">Hulse</a></li> <li><a href="/wiki/Hermann_Bondi" title="Hermann Bondi">Bondi</a></li> <li><a href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne</a></li> <li><a href="/wiki/Rainer_Weiss" title="Rainer Weiss">Weiss</a></li> <li><a href="/wiki/List_of_contributors_to_general_relativity" title="List of contributors to general relativity"><i>others</i></a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align:center;"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" 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Friedmann equations - Wikipedia