Friedmann equations: Difference between revisions

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-disabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Friedmann equations: Difference between revisions - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-disabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy" ,"wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"7684ccd9-bc9d-4ea8-9e55-43433775f6a8","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Friedmann_equations","wgTitle":"Friedmann equations","wgCurRevisionId":1259169248,"wgRevisionId":1259169248,"wgArticleId":1147994,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 German-language sources (de)","Wikipedia articles needing clarification from September 2015","Articles with short description","Short description is different from Wikidata","Pages using sidebar with the child parameter","Articles needing additional references from September 2024","All articles needing additional references","Eponymous equations of physics","General relativity"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext", "wgRelevantPageName":"Friedmann_equations","wgRelevantArticleId":1147994,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"extRevisionSliderTimeOffset":0,"wgDiffOldId":1147470939,"wgDiffNewId":1259169248,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId": "Q467736","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.visualEditor.diffPage.init.styles":"ready","oojs-ui.styles.icons-accessibility":"ready","oojs-ui.styles.icons-editing-advanced":"ready","oojs-ui-core.styles":"ready","oojs-ui.styles.indicators":"ready","mediawiki.widgets.styles":"ready","oojs-ui-core.icons":"ready","ext.RevisionSlider.lazyCss":"ready","mediawiki.interface.helpers.styles":"ready","mediawiki.diff.styles":"ready","ext.cite.styles":"ready","ext.math.styles":"ready","mediawiki.helplink":"ready","skins.vector.search.codex.styles":"ready", "skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.visualEditor.diffPage.init","ext.RevisionSlider.lazyJs","mediawiki.diff","ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022", "ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.RevisionSlider.lazyCss%7Cext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.visualEditor.diffPage.init.styles%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cmediawiki.diff.styles%7Cmediawiki.helplink%7Cmediawiki.interface.helpers.styles%7Cmediawiki.widgets.styles%7Coojs-ui-core.icons%2Cstyles%7Coojs-ui.styles.icons-accessibility%2Cicons-editing-advanced%2Cindicators%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="noindex,nofollow,max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/1200px-WMAP_2012.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="600"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/800px-WMAP_2012.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="400"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/WMAP_2012.png/640px-WMAP_2012.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="320"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Friedmann equations: Difference between revisions - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Friedmann_equations"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Friedmann_equations&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Friedmann_equations"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="mw-article-diff skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Friedmann_equations rootpage-Friedmann_equations skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Friedmann+equations&returntoquery=diff%3D1259169248%26oldid%3D1147470939" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Friedmann+equations&returntoquery=diff%3D1259169248%26oldid%3D1147470939" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Friedmann+equations&returntoquery=diff%3D1259169248%26oldid%3D1147470939" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Friedmann+equations&returntoquery=diff%3D1259169248%26oldid%3D1147470939" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Assumptions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Assumptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Assumptions</span> </div> </a> <ul id="toc-Assumptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Equations</span> </div> </a> <ul id="toc-Equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Density_parameter" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Density_parameter"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Density parameter</span> </div> </a> <ul id="toc-Density_parameter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Useful_solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Useful_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Useful solutions</span> </div> </a> <button aria-controls="toc-Useful_solutions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Useful solutions 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 class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Friedmann equations: Difference between revisions</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q467736#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Friedmann_equations" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Friedmann_equations" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Friedmann_equations"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Friedmann_equations&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Friedmann_equations&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Friedmann_equations"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Friedmann_equations&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Friedmann_equations&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Friedmann_equations" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Friedmann_equations" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Friedmann_equations&oldid=1259169248" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Friedmann_equations&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Friedmann_equations&id=1259169248&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DFriedmann_equations%26diff%3D1259169248%26oldid%3D1147470939"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DFriedmann_equations%26diff%3D1259169248%26oldid%3D1147470939"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Friedmann_equations&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Friedmann_equations&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q467736" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-mw-helplink" class="mw-indicator"><a href="https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Diff" target="_blank" class="mw-helplink"><span class="mw-helplink-icon"></span>Help</a></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-revslider-container"><span style='width: 100%; text-align: center;' id='ooui-php-1' class='mw-revslider-toggle-button oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-indicatorElement oo-ui-labelElement oo-ui-buttonWidget' data-ooui='{"_":"OO.ui.ButtonWidget","rel":["nofollow"],"framed":false,"indicator":"down","label":"Browse history interactively","title":"Open revision slider","classes":["mw-revslider-toggle-button"]}'><a role='button' title='Open revision slider' tabindex='0' rel='nofollow' class='oo-ui-buttonElement-button'><span class='oo-ui-iconElement-icon oo-ui-iconElement-noIcon'></span><span class='oo-ui-labelElement-label'>Browse history interactively</span><span class='oo-ui-indicatorElement-indicator oo-ui-indicator-down'></span></a></span><div class="mw-revslider-slider-wrapper" style="display: none;"><div class="mw-revslider-placeholder"><div class="mw-revslider-spinner"><div class="mw-revslider-bounce"></div></div></div></div></div><div class="mw-diff-revision-history-links"><a href="/w/index.php?title=Friedmann_equations&diff=prev&oldid=1147470939" class="mw-diff-revision-history-link-previous" title="Friedmann equations">← Previous edit</a></div><div class="mw-diff-table-prefix" dir="ltr" lang="en"><div class="mw-diff-inline-legend oo-ui-element-hidden"><span class="mw-diff-inline-legend-del">Content deleted</span> <span class="mw-diff-inline-legend-ins">Content added</span></div><div class="ve-init-mw-diffPage-diffMode"><span class='oo-ui-widget oo-ui-widget-enabled oo-ui-buttonGroupWidget'><span aria-disabled='true' class='oo-ui-widget oo-ui-widget-disabled oo-ui-buttonElement oo-ui-buttonElement-framed oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget'><a role='button' tabindex='-1' aria-disabled='true' rel='nofollow' class='oo-ui-buttonElement-button'><span class='oo-ui-iconElement-icon oo-ui-icon-eye oo-ui-image-invert'></span><span class='oo-ui-labelElement-label'>Visual</span><span class='oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator oo-ui-image-invert'></span></a></span><span class='oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-framed oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget oo-ui-buttonElement-active'><a role='button' tabindex='0' rel='nofollow' class='oo-ui-buttonElement-button'><span class='oo-ui-iconElement-icon oo-ui-icon-wikiText oo-ui-image-invert'></span><span class='oo-ui-labelElement-label'>Wikitext</span><span class='oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator oo-ui-image-invert'></span></a></span></span></div><div class="mw-diffPage-inlineToggle-container"><div id='mw-diffPage-inline-toggle-switch-layout' class='oo-ui-layout oo-ui-labelElement oo-ui-fieldLayout oo-ui-fieldLayout-align-left' data-ooui='{"_":"OO.ui.FieldLayout","fieldWidget":{"tag":"mw-diffPage-inline-toggle-switch"},"$overlay":true,"label":"Inline","title":"Switches between inline format and two-column format"}'><div class='oo-ui-fieldLayout-body'><span class='oo-ui-fieldLayout-header'><label title='Switches between inline format and two-column format' id='ooui-php-2' class='oo-ui-labelElement-label'>Inline</label></span><div class='oo-ui-fieldLayout-field'><div id='mw-diffPage-inline-toggle-switch' title='Switches between inline format and two-column format' aria-checked='false' tabindex='0' role='switch' aria-labelledby='ooui-php-2' class='oo-ui-widget oo-ui-widget-enabled oo-ui-toggleWidget oo-ui-toggleWidget-off oo-ui-toggleSwitchWidget' data-ooui='{"_":"OO.ui.ToggleSwitchWidget","href":"\/w\/index.php?title=Friedmann_equations&diff=1259169248&oldid=1147470939&diff-type=inline","value":false,"title":"Switches between inline format and two-column format"}'><span class='oo-ui-toggleSwitchWidget-glow'></span><a href='/w/index.php?title=Friedmann_equations&diff=1259169248&oldid=1147470939&diff-type=inline'><span class='oo-ui-toggleSwitchWidget-grip'></span></a></div></div></div></div></div></div><table class="diff diff-type-table diff-contentalign-left diff-editfont-monospace" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" class="diff-otitle diff-side-deleted"><div id="mw-diff-otitle1"><strong><a href="/w/index.php?title=Friedmann_equations&oldid=1147470939" title="Friedmann equations">Revision as of 05:09, 31 March 2023</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Friedmann_equations&action=edit&oldid=1147470939" title="Friedmann equations">edit</a></span><span class="mw-diff-timestamp" data-timestamp="2023-03-31T05:09:36Z"></span></strong></div><div id="mw-diff-otitle2"><a href="/w/index.php?title=User:CodeTalker&action=edit&redlink=1" class="new mw-userlink" title="User:CodeTalker (page does not exist)" data-mw-revid="1147470939"><bdi>CodeTalker</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:CodeTalker" class="mw-usertoollinks-talk" title="User talk:CodeTalker">talk</a> | <a href="/wiki/Special:Contributions/CodeTalker" class="mw-usertoollinks-contribs" title="Special:Contributions/CodeTalker">contribs</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"><a href="/wiki/Wikipedia:Extended_confirmed_editors" class="mw-redirect" title="Wikipedia:Extended confirmed editors">Extended confirmed users</a>, <a href="/wiki/Wikipedia:Rollback" title="Wikipedia:Rollback">Rollbackers</a></div><div class="mw-diff-usereditcount"><span>34,532</span> edits</div></div></div><div id="mw-diff-otitle3"> <span class="comment comment--without-parentheses">Reverted 1 edit by <a href="/wiki/Special:Contributions/118.149.73.176" title="Special:Contributions/118.149.73.176">118.149.73.176</a> (<a href="/wiki/User_talk:118.149.73.176" title="User talk:118.149.73.176">talk</a>) to last revision by Brienanni</span></div><div id="mw-diff-otitle5"><span class="mw-tag-markers"><a href="/wiki/Special:Tags" title="Special:Tags">Tags</a>: <span class="mw-tag-marker mw-tag-marker-twinkle"><a href="/wiki/Wikipedia:Twinkle" title="Wikipedia:Twinkle">Twinkle</a></span> <span class="mw-tag-marker mw-tag-marker-mw-undo"><a href="/wiki/Wikipedia:Undo" class="mw-redirect" title="Wikipedia:Undo">Undo</a></span></span></div><div id="mw-diff-otitle4"><a href="/w/index.php?title=Friedmann_equations&diff=prev&oldid=1147470939" title="Friedmann equations" id="differences-prevlink">← Previous edit</a></div></td> <td colspan="2" class="diff-ntitle diff-side-added"><div id="mw-diff-ntitle1"><strong><a href="/w/index.php?title=Friedmann_equations&oldid=1259169248" title="Friedmann equations">Latest revision as of 19:20, 23 November 2024</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Friedmann_equations&action=edit" title="Friedmann equations">edit</a></span><span class="mw-diff-timestamp" data-timestamp="2024-11-23T19:20:19Z"></span> <span class="mw-diff-undo"><a href="/w/index.php?title=Friedmann_equations&action=edit&undoafter=1147470939&undo=1259169248" title=""Undo" reverts this edit and opens the edit form in preview mode. It allows adding a reason in the summary.">undo</a></span></strong></div><div id="mw-diff-ntitle2"><a href="/wiki/User:Aseyhe" class="mw-userlink" title="User:Aseyhe" data-mw-revid="1259169248"><bdi>Aseyhe</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Aseyhe" class="mw-usertoollinks-talk" title="User talk:Aseyhe">talk</a> | <a href="/wiki/Special:Contributions/Aseyhe" class="mw-usertoollinks-contribs" title="Special:Contributions/Aseyhe">contribs</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"></div><div class="mw-diff-usereditcount"><span>184</span> edits</div></div></div><div id="mw-diff-ntitle3"><abbr class="minoredit" title="This is a minor edit">m</abbr> <span class="comment comment--without-parentheses">use better term</span></div><div id="mw-diff-ntitle5"></div><div id="mw-diff-ntitle4"> </div></td> </tr><tr><td colspan="4" class="diff-multi" lang="en">(21 intermediate revisions by 18 users not shown)</td></tr><tr> <td colspan="2" class="diff-lineno">Line 3:</td> <td colspan="2" class="diff-lineno">Line 3:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>[[File:Aleksandr Fridman.png|thumb|236px|[[Alexander Friedmann]]]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>[[File:Aleksandr Fridman.png|thumb|236px|[[Alexander Friedmann]]]]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>The '''Friedmann equations''' are a set of [[equation]]s in [[physical cosmology]] that govern<del class="diffchange diffchange-inline"> the</del> [[<del class="diffchange diffchange-inline">Metric expansion</del> of <del class="diffchange diffchange-inline">space</del>|<del class="diffchange diffchange-inline">expansion</del> <del class="diffchange diffchange-inline">of space</del>]] in [[Homogeneity (physics)|homogeneous]] and [[Isotropy|isotropic]] models of the universe within the context of [[general relativity]]. They were first derived by [[<del class="diffchange diffchange-inline">Alexander Alexandrovich Friedmann|</del>Alexander Friedmann]] in 1922 from [[Einstein field equations|Einstein's field equations]] of [[gravitation]] for the [[Friedmann–Lemaître–Robertson–Walker metric]] and a [[perfect fluid]] with a given [[Density|mass density]] {{mvar|[[Rho (letter)|ρ]]}} and [[pressure]] {{mvar|p}}.<ref name="af1922">{{cite journal |first=A |last=Friedman |author-link=Alexander Alexandrovich Friedman |title=Über die Krümmung des Raumes |journal=Z. Phys. |volume=10 |year=1922 |issue=1 |pages=377–386 |doi=10.1007/BF01332580 |bibcode = 1922ZPhy...10..377F|s2cid=125190902 |language=de}} (English translation: {{cite journal |first=A |last=Friedman |title=On the Curvature of Space |journal=General Relativity and Gravitation |volume=31 |issue=12 |year=1999 |pages= 1991–2000 |bibcode=1999GReGr..31.1991F |doi=10.1023/A:1026751225741|s2cid=122950995 }}). The original Russian manuscript of this paper is preserved in the [http://ilorentz.org/history/Friedmann_archive Ehrenfest archive].</ref> The equations for negative spatial curvature were given by Friedmann in 1924.<ref name="af1924">{{cite journal |first=A |last=Friedmann |author-link=Alexander Alexandrovich Friedman |title=Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes |journal=Z. Phys. |volume=21 |year=1924 |issue=1 |pages=326–332 |doi=10.1007/BF01328280 |bibcode=1924ZPhy...21..326F|s2cid=120551579 |language=de}} (English translation: {{cite journal |first=A |last=Friedmann |title=On the Possibility of a World with Constant Negative Curvature of Space |journal=General Relativity and Gravitation |volume=31 |issue=12 |year=1999 |pages=2001–2008 |bibcode=1999GReGr..31.2001F |doi=10.1023/A:1026755309811|s2cid=123512351 }})</ref></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>The '''Friedmann equations'''<ins class="diffchange diffchange-inline">, also known as the '''Friedmann–Lemaître''' ('''FL''') '''equations''',</ins> are a set of [[equation]]s in [[physical cosmology]] that govern [[<ins class="diffchange diffchange-inline">Expansion</ins> of <ins class="diffchange diffchange-inline">the universe</ins>|<ins class="diffchange diffchange-inline">cosmic</ins> <ins class="diffchange diffchange-inline">expansion</ins>]] in [[Homogeneity (physics)|homogeneous]] and [[Isotropy|isotropic]] models of the universe within the context of [[general relativity]]. They were first derived by [[Alexander Friedmann]] in 1922 from [[Einstein field equations|Einstein's field equations]] of [[gravitation]] for the [[Friedmann–Lemaître–Robertson–Walker metric]] and a [[perfect fluid]] with a given [[Density|mass density]] {{mvar|[[Rho (letter)|ρ]]}} and [[pressure]] {{mvar|p}}.<ref name="af1922">{{cite journal |first=A |last=Friedman |author-link=Alexander Alexandrovich Friedman |title=Über die Krümmung des Raumes |journal=Z. Phys. |volume=10 |year=1922 |issue=1 |pages=377–386 |doi=10.1007/BF01332580 |bibcode = 1922ZPhy...10..377F|s2cid=125190902 |language=de}} (English translation: {{cite journal |first=A |last=Friedman |title=On the Curvature of Space |journal=General Relativity and Gravitation |volume=31 |issue=12 |year=1999 |pages= 1991–2000 |bibcode=1999GReGr..31.1991F |doi=10.1023/A:1026751225741|s2cid=122950995 }}). The original Russian manuscript of this paper is preserved in the [http://ilorentz.org/history/Friedmann_archive Ehrenfest archive].</ref> The equations for negative spatial curvature were given by Friedmann in 1924.<ref name="af1924">{{cite journal |first=A |last=Friedmann |author-link=Alexander Alexandrovich Friedman |title=Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes |journal=Z. Phys. |volume=21 |year=1924 |issue=1 |pages=326–332 |doi=10.1007/BF01328280 |bibcode=1924ZPhy...21..326F|s2cid=120551579 |language=de}} (English translation: {{cite journal |first=A |last=Friedmann |title=On the Possibility of a World with Constant Negative Curvature of Space |journal=General Relativity and Gravitation |volume=31 |issue=12 |year=1999 |pages=2001–2008 |bibcode=1999GReGr..31.2001F |doi=10.1023/A:1026755309811|s2cid=123512351 }})</ref></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Assumptions ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Assumptions ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>{{Unreferenced section|date=September 2024}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{main|Friedmann–Lemaître–Robertson–Walker metric}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{main|Friedmann–Lemaître–Robertson–Walker metric}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and [[Isotropic manifold|isotropic]], that is, the [[cosmological principle]]; empirically, this is justified on scales larger than the order of 100 [[Parsec|Mpc]]. The cosmological principle implies that the metric of the universe must be of the form</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and [[Isotropic manifold|isotropic]], that is, the [[cosmological principle]]; empirically, this is justified on scales larger than the order of 100 [[Parsec|Mpc]]. The cosmological principle implies that the metric of the universe must be of the form</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math> -\mathrm{d}s^2 = a(t)^2 \, {\mathrm{d}s_3}^2 - c^2 \, \mathrm{d}t^2 </math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>> -\mathrm{d}s^2 = a(t)^2 \, {\mathrm{d}s_3}^2 - c^2 \, \mathrm{d}t^2 </math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>where {{math|d''s''<sub>3</sub><sup>2</sup>}} is a three-dimensional metric that must be one of '''(a)''' flat space, '''(b)''' a sphere of constant positive curvature or '''(c)''' a hyperbolic space with constant negative curvature. This metric is called Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The parameter {{mvar|k}} discussed below takes the value 0, 1, −1, or the [[Gaussian curvature]], in these three cases respectively. It is this fact that allows us to sensibly speak of a "[[Scale factor (Universe)|scale factor]]" {{math|''a''(''t'')}}.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>where {{math|d''s''<sub>3</sub><sup>2</sup>}} is a three-dimensional metric that must be one of '''(a)''' flat space, '''(b)''' a sphere of constant positive curvature or '''(c)''' a hyperbolic space with constant negative curvature. This metric is called<ins class="diffchange diffchange-inline"> the</ins> Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The parameter {{mvar|k}} discussed below takes the value 0, 1, −1, or the [[Gaussian curvature]], in these three cases respectively. It is this fact that allows us to sensibly speak of a "[[Scale factor (Universe)|scale factor]]" {{math|''a''(''t'')}}.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>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 [[Christoffel symbols]], then the [[Ricci tensor]]. With the [[stress–energy tensor]] for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>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 [[Christoffel symbols]], then the [[Ricci tensor]]. With the [[stress–energy tensor]] for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Equations ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Equations ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>{{Unreferenced section|date=September 2024}}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{General relativity sidebar |equations}} </div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>{{General relativity sidebar |equations}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block"> \frac{\dot{a}^2 + kc^2}{a^2} = \frac{8 \pi G \rho + \Lambda c^2}{3} ,</math></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>which is derived from the 00 component of the [[Einstein field equations]]. The second is:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}</math></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>which is derived from the first together with the [[Trace (linear algebra)|trace]] of Einstein's field equations (the dimension of the two equations is time<sup>&minus;2</sup>).</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>{{mvar|a}} is the [[scale factor (universe)|scale factor]], {{mvar|G}}, {{math|Λ}}, and {{mvar|c}} are universal constants ({{mvar|G}} is the [[Newtonian constant of gravitation]], {{math|Λ}} is the [[cosmological constant]] with dimension length<sup>&minus;2</sup>, and {{mvar|c}} is the [[speed of light|speed of light in vacuum]]). {{mvar|ρ}}&nbsp;and {{mvar|p}} are the volumetric mass density (and not the volumetric energy density) and the pressure, respectively. {{mvar|k}} is constant throughout a particular solution, but may vary from one solution to another.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math> \frac{\dot{a}^2 + kc^2}{a^2} = \frac{8 \pi G \rho + \Lambda c^2}{3} </math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>which is derived from the 00 component of [[Einstein field equations|Einstein's field equations]]. The second is:</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>which is derived from the first together with the [[Trace (linear algebra)|trace]] of Einstein's field equations (the dimension of the two equations is time<sup>&minus;2</sup>). </div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{mvar|a}} is the [[scale factor (universe)|scale factor]], {{mvar|G}}, {{mvar|Λ}}, and {{mvar|c}} are universal constants ({{mvar|G}} is Newton's [[gravitational constant]], {{mvar|Λ}} is the [[cosmological constant]] with dimension length<sup>&minus;2</sup>, and {{mvar|c}} is the [[speed of light|speed of light in vacuum]]). {{mvar|ρ}} and {{mvar|p}} are the volumetric mass density (and not the volumetric energy density) and the pressure, respectively. {{mvar|k}} is constant throughout a particular solution, but may vary from one solution to another.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>In previous equations, {{mvar|a}}, {{mvar|ρ}}, and {{mvar|p}} are functions of time. {{math|{{sfrac|''k''|''a''<sup>2</sup>}}}} is the [[curvature|spatial curvature]] in any time-slice of the universe; it is equal to one-sixth of the spatial [[scalar curvature|Ricci curvature scalar {{mvar|R}}]] since </div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>R = \frac{6}{c^2 a^2}(\ddot{a} a + \dot{a}^2 + kc^2)</math> </div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>In previous equations, {{mvar|a}}, {{mvar|ρ}}, and {{mvar|p}} are functions of time. {{math|{{sfrac|''k''|''a''<sup>2</sup>}}}} is the [[curvature|spatial curvature]] in any time-slice of the universe; it is equal to one-sixth of the spatial [[scalar curvature|Ricci curvature scalar {{mvar|R}}]] since</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">R = \frac{6}{c^2 a^2}(\ddot{a} a + \dot{a}^2 + kc^2)</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>in the Friedmann model. {{math|''H'' ≡ {{sfrac|''ȧ''|''a''}}}} is the [[Hubble parameter]].</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>in the Friedmann model. {{math|''H'' ≡ {{sfrac|''ȧ''|''a''}}}} is the [[Hubble parameter]].</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>We see that in the Friedmann equations, {{math|''a''(''t'')}} does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for {{mvar|a}} and {{mvar|k}} which describe the same physics:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>We see that in the Friedmann equations, {{math|''a''(''t'')}} does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for {{mvar|a}} and {{mvar|k}} which describe the same physics:</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>*{{math|''k'' {{=}} +1, 0}} or {{math|−1}} depending on whether the [[shape of the universe]] is a closed [[3-sphere]], flat ([[Euclidean space]]) or an open 3-[[hyperboloid]], respectively.<ref><del class="diffchange diffchange-inline">Ray</del> <del class="diffchange diffchange-inline">A</del> <del class="diffchange diffchange-inline">d</del>'Inverno<del class="diffchange diffchange-inline">,</del> <del class="diffchange diffchange-inline">''</del>Introducing Einstein's <del class="diffchange diffchange-inline">Relativity'',</del> <del class="diffchange diffchange-inline">{{ISBN</del>|0-19-859686-<del class="diffchange diffchange-inline">3</del>}}<del class="diffchange diffchange-inline">.</del></ref> If {{math|''k'' {{=}} +1}}, then {{mvar|a}} is the [[radius of curvature]] of the universe. If {{math|''k'' {{=}} 0}}, then {{mvar|a}} may be fixed to any arbitrary positive number at one particular time. If {{math|''k'' {{=}} −1}}, then (loosely speaking) one can say that {{math|[[Imaginary unit|''i'']] · ''a''}} is the radius of curvature of the universe.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>*<ins class="diffchange diffchange-inline"> </ins>{{math|''k'' {{=}} +1, 0}} or {{math|−1}} depending on whether the [[shape of the universe]] is a closed [[3-sphere]], flat ([[Euclidean space]]) or an open 3-[[hyperboloid]], respectively.<ref><ins class="diffchange diffchange-inline">{{Cite</ins> <ins class="diffchange diffchange-inline">book</ins> <ins class="diffchange diffchange-inline">|last=D</ins>'Inverno <ins class="diffchange diffchange-inline">|first=Ray |title=</ins>Introducing Einstein's <ins class="diffchange diffchange-inline">relativity</ins> |<ins class="diffchange diffchange-inline">date=2008 |publisher=Clarendon Press |isbn=978-</ins>0-19-859686-<ins class="diffchange diffchange-inline">8 |edition=Repr |location=Oxford</ins>}}</ref> If {{math|''k'' {{=}} +1}}, then {{mvar|a}} is the [[radius of curvature]] of the universe. If {{math|''k'' {{=}} 0}}, then {{mvar|a}} may be fixed to any arbitrary positive number at one particular time. If {{math|''k'' {{=}} −1}}, then (loosely speaking) one can say that {{math|[[Imaginary unit|''i'']] · ''a''}} is the radius of curvature of the universe.</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>*{{mvar|a}} is the [[Scale factor (Universe)|scale factor]] which is taken to be 1 at the present time. {{mvar|k}} is the current [[curvature|spatial curvature]] (when {{math|''a'' {{=}} 1}}). If the [[shape of the universe]] is [[Shape of the universe#Spherical universe|hyperspherical]] and {{math|''R<sub>t</sub>''}} is the radius of curvature ({{math|''R''<sub>0</sub>}} at the present), then {{math|''a'' {{=}} {{sfrac|''R<sub>t</sub>''|''R''<sub>0</sub>}}}}. If {{mvar|k}} is positive, then the universe is hyperspherical. If {{math|''k'' {{=}} 0}}, then the universe is [[Shape of the universe#Flat universe|flat]]. If {{mvar|k}} is negative, then the universe is [[Shape of the universe#Hyperbolic universe|hyperbolic]].</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>*<ins class="diffchange diffchange-inline"> </ins>{{mvar|a}} is the [[Scale factor (Universe)|scale factor]] which is taken to be 1 at the present time. {{mvar|k}} is the current [[curvature|spatial curvature]] (when {{math|''a'' {{=}} 1}}). If the [[shape of the universe]] is [[Shape of the universe#Spherical universe|hyperspherical]] and {{math|''R<sub>t</sub>''}} is the radius of curvature ({{math|''R''<sub>0</sub>}} at the present), then {{math|''a'' {{=}} {{sfrac|''R<sub>t</sub>''|''R''<sub>0</sub>}}}}. If {{mvar|k}} is positive, then the universe is hyperspherical. If {{math|''k'' {{=}} 0}}, then the universe is [[Shape of the universe#Flat universe|flat]]. If {{mvar|k}} is negative, then the universe is [[Shape of the universe#Hyperbolic universe|hyperbolic]].</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Using the first equation, the second equation can be re-expressed as<del class="diffchange diffchange-inline"> </del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Using the first equation, the second equation can be re-expressed as</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\dot{\rho} = -3 H \left(\rho + \frac{p}{c^2}\right),</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>which eliminates {{math|Λ}} and expresses the conservation of [[mass–energy]]:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\dot{\rho} = -3 H \left(\rho + \frac{p}{c^2}\right),</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block"> T^{\alpha\beta}{}_{;\beta}= 0.</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>which eliminates {{mvar|Λ}} and expresses the conservation of [[mass–energy]]:</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math> T^{\alpha\beta}{}_{;\beta}= 0.</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>These equations are sometimes simplified by replacing</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>These equations are sometimes simplified by replacing</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\begin{align}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>\rho &\to \rho - \frac{\Lambda c^2}{8 \pi G} &</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\begin{align}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>\rho &\to \rho - \frac{\Lambda c^2}{8 \pi G} \\</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>p &\to p + \frac{\Lambda c^4}{8 \pi G}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>p &\to p + \frac{\Lambda c^4}{8 \pi G}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>\end{align}</math></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>\end{align}</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>to give:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>to give:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\begin{align}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\begin{align}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>H^2 = \left(\frac{\dot{a}}{a}\right)^2 &= \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2} \\</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>H^2 = \left(\frac{\dot{a}}{a}\right)^2 &= \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2} \\</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>\dot{H} + H^2 = \frac{\ddot{a}}{a} &= - \frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right).</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>\dot{H} + H^2 = \frac{\ddot{a}}{a} &= - \frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right).</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Line 72:</td> <td colspan="2" class="diff-lineno">Line 62:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>The '''density parameter''' {{mvar|Ω}} is defined as the ratio of the actual (or observed) density {{mvar|ρ}} to the critical density {{math|''ρ''<sub>c</sub>}} 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 [[cosmological constant]] term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>The '''density parameter''' {{mvar|Ω}} is defined as the ratio of the actual (or observed) density {{mvar|ρ}} to the critical density {{math|''ρ''<sub>c</sub>}} 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 [[cosmological constant]] term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>To date, the critical density is estimated to be approximately five atoms (of [[monatomic]] [[hydrogen]]) per cubic metre, whereas the average density of [[Baryons#Baryonic matter|ordinary matter]] in the Universe is believed to be 0.2–0.25 atoms per cubic metre.<ref>Rees<del class="diffchange diffchange-inline">,</del> <del class="diffchange diffchange-inline">M.,</del> Just <del class="diffchange diffchange-inline">Six</del> <del class="diffchange diffchange-inline">Numbers,</del> <del class="diffchange diffchange-inline">(2000)</del> <del class="diffchange diffchange-inline">Orion</del> Books<del class="diffchange diffchange-inline">,</del> <del class="diffchange diffchange-inline">London,</del> <del class="diffchange diffchange-inline">p</del>. <del class="diffchange diffchange-inline">81,</del> <del class="diffchange diffchange-inline">p.</del> <del class="diffchange diffchange-inline">82</del>{{<del class="diffchange diffchange-inline"> </del>clarify<del class="diffchange diffchange-inline"> </del>|<del class="diffchange diffchange-inline"> </del>date<del class="diffchange diffchange-inline"> </del>=<del class="diffchange diffchange-inline"> </del>September 2015<del class="diffchange diffchange-inline"> </del>|<del class="diffchange diffchange-inline"> </del>reason<del class="diffchange diffchange-inline"> </del>=What kind of atoms?<del class="diffchange diffchange-inline"> </del>}}</ref><ref>{{cite web | publisher=[[NASA]] | title=Universe 101 | url=http://map.gsfc.nasa.gov/universe/uni_matter.html | access-date=September 9, 2015 | quote=The actual density of atoms is equivalent to roughly 1 proton per 4 cubic meters.}}</ref><del class="diffchange diffchange-inline"> </del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>To date, the critical density is estimated to be approximately five atoms (of [[monatomic]] [[hydrogen]]) per cubic metre, whereas the average density of [[Baryons#Baryonic matter|ordinary matter]] in the Universe is believed to be 0.2–0.25 atoms per cubic metre.<ref><ins class="diffchange diffchange-inline">{{Cite book |last=</ins>Rees <ins class="diffchange diffchange-inline">|first=Martin</ins> <ins class="diffchange diffchange-inline">|title=</ins>Just <ins class="diffchange diffchange-inline">six</ins> <ins class="diffchange diffchange-inline">numbers:</ins> <ins class="diffchange diffchange-inline">the</ins> <ins class="diffchange diffchange-inline">deep forces that shape the universe |date=2001 |publisher=Basic</ins> Books <ins class="diffchange diffchange-inline">|isbn=978-0-465-03673-8</ins> <ins class="diffchange diffchange-inline">|edition=Repr</ins>. <ins class="diffchange diffchange-inline">|series=Astronomy/science</ins> <ins class="diffchange diffchange-inline">|location=New</ins> <ins class="diffchange diffchange-inline">York, NY}}</ins>{{clarify|date=September 2015|reason=What kind of atoms?}}</ref><ref>{{cite web | publisher=[[NASA]] | title=Universe 101 | url=http://map.gsfc.nasa.gov/universe/uni_matter.html | access-date=September 9, 2015 | quote=The actual density of atoms is equivalent to roughly 1 proton per 4 cubic meters.}}</ref></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>[[File:UniverseComposition.svg|thumb|right|375px|Estimated relative distribution for components of the energy density of the universe. [[Dark energy]] dominates the total energy (74%) while [[dark matter]] (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 [[Planck (spacecraft)|Planck cosmology probe]] released new data refining these values to 4.9% ordinary matter, 25.9% dark matter and 69.1% dark energy.]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>[[File:UniverseComposition.svg|thumb|right|375px|Estimated relative distribution for components of the energy density of the universe. [[Dark energy]] dominates the total energy (74%) while [[dark matter]] (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 [[Planck (spacecraft)|Planck cosmology probe]] released new data refining these values to 4.9% ordinary matter, 25.9% dark matter and 69.1% dark energy.]]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>A much greater density comes from the unidentified [[dark matter]]<del class="diffchange diffchange-inline">;</del> both ordinary and dark matter contribute in favour of contraction of the universe. However, the largest part comes from so-called [[dark energy]], which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error),<del class="diffchange diffchange-inline"> the</del> dark energy does not lead to contraction of the universe but rather may accelerate its expansion.<del class="diffchange diffchange-inline"> </del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>A much greater density comes from the unidentified [[dark matter]]<ins class="diffchange diffchange-inline">, although</ins> both ordinary and dark matter contribute in favour of contraction of the universe. However, the largest part comes from so-called [[dark energy]], 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.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>An expression for the critical density is found by assuming {{mvar|Λ}} to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, {{mvar|k}}, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>An expression for the critical density is found by assuming {{mvar|Λ}} to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, {{mvar|k}}, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math>\rho_\mathrm{c} = \frac{3 H^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} ,</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>>\rho_\mathrm{c} = \frac{3 H^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} ,</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del>(where {{math|''h'' {{=}} ''H''<sub>0</sub>/(100 km/s/Mpc)}}. For {{math|''H<sub>o</sub>'' {{=}} 67.4 km/s/Mpc}}, i.e. {{math|''h'' {{=}} 0.674}}, {{math|''ρ''<sub>c</sub> {{=}} {{val|8.5e-27|u=kg/m<sup>3</sup>}}}}).</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><ins class="diffchange diffchange-inline">{{block indent | em = 1.5 | text = </ins>(where {{math|''h'' {{=}} ''H''<sub>0</sub>/(100 km/s/Mpc)}}. For {{math|''H<sub>o</sub>'' {{=}} 67.4 km/s/Mpc}}, i.e. {{math|''h'' {{=}} 0.674}}, {{math|''ρ''<sub>c</sub> {{=}} {{val|8.5e-27|u=kg/m<sup>3</sup>}}}}).<ins class="diffchange diffchange-inline">}}</ins></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div> </div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>The density parameter (useful for comparing different cosmological models) is then defined as:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>The density parameter (useful for comparing different cosmological models) is then defined as:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math>\Omega <del class="diffchange diffchange-inline">\equiv</del> \frac{\rho}{\rho_c} = \frac{8 \pi G\rho}{3 H^2}.</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>>\Omega <ins class="diffchange diffchange-inline">:=</ins> \frac{\rho}{\rho_c} = \frac{8 \pi G\rho}{3 H^2}.</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>This term originally was used as a means to determine the [[shape of the universe|spatial geometry]] of the universe, where {{math|''ρ''<sub>c</sub>}} is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if {{mvar|Ω}} is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If {{mvar|Ω}} 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 {{mvar|Ω}} 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 [[Lambda-CDM model|ΛCDM model]], there are important components of {{mvar|Ω}} due to [[baryon]]s, [[cold dark matter]] and [[dark energy]]. The spatial geometry of the [[universe]] has been measured by the [[Wilkinson Microwave Anisotropy Probe|WMAP]] spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter {{mvar|k}} 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.<del class="diffchange diffchange-inline"> </del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>This term originally was used as a means to determine the [[shape of the universe|spatial geometry]] of the universe, where {{math|''ρ''<sub>c</sub>}} is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if {{mvar|Ω}} is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If {{mvar|Ω}} 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 {{mvar|Ω}} 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 [[Lambda-CDM model|ΛCDM model]], there are important components of {{mvar|Ω}} due to [[baryon]]s, [[cold dark matter]] and [[dark energy]]. The spatial geometry of the [[universe]] has been measured by the [[Wilkinson Microwave Anisotropy Probe|WMAP]] spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter {{mvar|k}} 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.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>The first Friedmann equation is often seen in terms of the present values of the density parameters, that is<ref>{{cite journal | last=Nemiroff | first=Robert J. | author-link=Robert J. Nemiroff | author2=Patla, Bijunath |arxiv = astro-ph/0703739| doi = 10.1119/1.2830536 | volume=76 | title=Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations | journal=American Journal of Physics | year=2008 | issue=3 | pages=265–276 | bibcode = 2008AmJPh..76..265N| s2cid=51782808 }}</ref></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>The first Friedmann equation is often seen in terms of the present values of the density parameters, that is<ref>{{cite journal | last=Nemiroff | first=Robert J. | author-link=Robert J. Nemiroff | author2=Patla, Bijunath |arxiv = astro-ph/0703739| doi = 10.1119/1.2830536 | volume=76 | title=Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations | journal=American Journal of Physics | year=2008 | issue=3 | pages=265–276 | bibcode = 2008AmJPh..76..265N| s2cid=51782808 }}</ref></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math>\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}.</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>>\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}.</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Here {{math|Ω<sub>0,R</sub>}} is the radiation density today (when {{math|''a'' {{=}} 1}}), {{math|Ω<sub>0,M</sub>}} is the matter ([[dark matter|dark]] plus [[baryon]]ic) density today, {{math|Ω<sub>0,''k''</sub> {{=}} 1 − Ω<sub>0</sub>}} is the "spatial curvature density" today, and {{math|<del class="diffchange diffchange-inline">''</del>Ω<del class="diffchange diffchange-inline">''</del><sub>0,Λ</sub>}} is the cosmological constant or vacuum density today.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Here {{math|Ω<sub>0,R</sub>}} is the radiation density today (when {{math|''a'' {{=}} 1}}), {{math|Ω<sub>0,M</sub>}} is the matter ([[dark matter|dark]] plus [[baryon]]ic) density today, {{math|Ω<sub>0,''k''</sub> {{=}} 1 − Ω<sub>0</sub>}} is the "spatial curvature density" today, and {{math|Ω<sub>0,Λ</sub>}} is the cosmological constant or vacuum density today.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Useful solutions ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Useful solutions ==</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>The Friedmann equations can be solved exactly in presence of a [[perfect fluid]] with equation of state</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>The Friedmann equations can be solved exactly in presence of a [[perfect fluid]] with equation of state</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">p=w\rho c^2,</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>p=w\rho c^2,</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>where {{mvar|p}} is the [[pressure]], {{mvar|ρ}} is the mass density of the fluid in the comoving frame and {{mvar|w}} is some constant.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>where {{mvar|p}} is the [[pressure]], {{mvar|ρ}} is the mass density of the fluid in the comoving frame and {{mvar|w}} is some constant.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>In spatially flat case ({{math|''k'' {{=}} 0}}), the solution for the scale factor is</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>In spatially flat case ({{math|''k'' {{=}} 0}}), the solution for the scale factor is</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block"> a(t)=a_0\,t^{\frac{2}{3(w+1)}} </math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math> a(t)=a_0\,t^{\frac{2}{3(w+1)}} </math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>where {{math|''a''<sub>0</sub>}} is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by {{mvar|w}} is extremely important for cosmology. For example, {{math|''w'' {{=}} 0}} describes a [[matter-dominated era|matter-dominated]] 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</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>where {{math|''a''<sub>0</sub>}} is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by {{mvar|w}} is extremely important for cosmology. For example, {{math|''w'' {{=}} 0}} describes a [[matter-dominated era|matter-dominated]] 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</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">a(t) \propto t^{2/3}</math> matter-dominated</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>a(t)\propto t^\frac23</math> matter-dominated</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Another important example is the case of a [[radiation-dominated era|radiation-dominated]] universe, namely when {{math|''w'' {{=}} {{sfrac|1|3}}}}. This leads to</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Another important example is the case of a [[radiation-dominated era|radiation-dominated]] universe, namely when {{math|''w'' {{=}} {{sfrac|1|3}}}}. This leads to</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">a(t) \propto t^{1/2}</math> radiation-dominated</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>a(t)\propto t^\frac12</math> radiation-dominated</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Note that this solution is not valid for domination of the cosmological constant, which corresponds to an {{math|''w'' {{=}} −1}}. In this case the energy density is constant and the scale factor grows exponentially.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Note that this solution is not valid for domination of the cosmological constant, which corresponds to an {{math|''w'' {{=}} −1}}. In this case the energy density is constant and the scale factor grows exponentially.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Solutions for other values of {{mvar|k}} can be found at {{cite web|last=Tersic|first=Balsa|title=Lecture Notes on Astrophysics|url=https://www.academia.edu/5025956|access-date=24 February 2022}}</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Solutions for other values of {{mvar|k}} can be found at {{cite web<ins class="diffchange diffchange-inline"> </ins>|<ins class="diffchange diffchange-inline"> </ins>last=Tersic<ins class="diffchange diffchange-inline"> </ins>|<ins class="diffchange diffchange-inline"> </ins>first=Balsa<ins class="diffchange diffchange-inline"> </ins>|<ins class="diffchange diffchange-inline"> </ins>title=Lecture Notes on Astrophysics<ins class="diffchange diffchange-inline"> </ins>|<ins class="diffchange diffchange-inline"> </ins>url=https://www.academia.edu/5025956|access-date=24 February 2022}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>===Mixtures===</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>===Mixtures===</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math>\dot{\rho}_{f} = -3 H \left( \rho_{f} + \frac{p_{f}}{c^2} \right) \,</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>>\dot{\rho}_{f} = -3 H \left( \rho_{f} + \frac{p_{f}}{c^2} \right) \,</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>holds separately for each such fluid {{mvar|f}}. In each case,</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>holds separately for each such fluid {{mvar|f}}. In each case,</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math>\dot{\rho}_{f} = -3 H \left( \rho_{f} + w_{f} \rho_{f} \right) \,</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>>\dot{\rho}_{f} = -3 H \left( \rho_{f} + w_{f} \rho_{f} \right) \,</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>from which we get</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>from which we get</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math>{\rho}_{f} \propto a^{-3 \left(1 + w_{f}\right)} \,.</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>>{\rho}_{f} \propto a^{-3 \left(1 + w_{f}\right)} \,.</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>For example, one can form a linear combination of such terms</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>For example, one can form a linear combination of such terms</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div><del class="diffchange diffchange-inline">:</del><math>\rho = A a^{-3} + B a^{-4} + C a^{0} \,</math></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math<ins class="diffchange diffchange-inline"> display="block"</ins>>\rho = A a^{-3} + B a^{-4} + C a^{0} \,</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>where {{mvar|A}} is the density of "dust" (ordinary matter, {{math|''w'' {{=}} 0}}) when {{math|''a'' {{=}} 1}}; {{mvar|B}} is the density of radiation ({{math|''w'' {{=}} {{sfrac|1|3}}}}) when {{math|''a'' {{=}} 1}}; and {{mvar|C}} is the density of "dark energy" ({{math|''w'' {{=}} &minus;1}}). One then substitutes this into</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>where {{mvar|A}} is the density of "dust" (ordinary matter, {{math|''w'' {{=}} 0}}) when {{math|''a'' {{=}} 1}}; {{mvar|B}} is the density of radiation ({{math|''w'' {{=}} {{sfrac|1|3}}}}) when {{math|''a'' {{=}} 1}}; and {{mvar|C}} is the density of "dark energy" ({{math|''w'' {{=}} &minus;1}}). One then substitutes this into</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2} \,</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2} \,</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>and solves for {{mvar|a}} as a function of time.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>and solves for {{mvar|a}} as a function of time.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>===Detailed derivation===</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>===Detailed derivation===</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>To make the solutions more explicit, we can derive the full relationships from the first <del class="diffchange diffchange-inline">Friedman</del> equation:</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>To make the solutions more explicit, we can derive the full relationships from the first <ins class="diffchange diffchange-inline">Friedmann</ins> equation:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\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}</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\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}</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>with</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>with</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\begin{align}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\begin{align}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>H &= \frac{\dot{a}}{a} \\[6px]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>H &= \frac{\dot{a}}{a} \\[6px]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>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]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>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]</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Line 149:</td> <td colspan="2" class="diff-lineno">Line 124:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Rearranging and changing to use variables {{math|''a''′}} and {{math|''t''′}} for the integration</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Rearranging and changing to use variables {{math|''a''′}} and {{math|''t''′}} for the integration</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">t H_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}}</math></div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>t H_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}}</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>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 {{math|''Ω''<sub>0,''k''</sub> ≈ 0}}, which is the same as assuming that the dominating source of energy density is approximately 1.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>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 {{math|''Ω''<sub>0,''k''</sub> ≈ 0}}, which is the same as assuming that the dominating source of energy density is approximately 1.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>For matter-dominated universes, where {{math|''Ω''<sub>0,M</sub> ≫ ''Ω''<sub>0,R</sub>}} and {{math|''Ω''<sub>0,''Λ''</sub>}}, as well as {{math|''Ω''<sub>0,M</sub> ≈ 1}}:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>For matter-dominated universes, where {{math|''Ω''<sub>0,M</sub> ≫ ''Ω''<sub>0,R</sub>}} and {{math|''Ω''<sub>0,''Λ''</sub>}}, as well as {{math|''Ω''<sub>0,M</sub> ≈ 1}}:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\begin{align}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\begin{align}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{\Omega_{0,\mathrm M} a'^{-1}}} \\[6px]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{\Omega_{0,\mathrm M} a'^{-1}}} \\[6px]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>t H_0 \sqrt{\Omega_{0,\mathrm M}} &= \left.\left( \tfrac23 a'^<del class="diffchange diffchange-inline">\frac32</del> \right) \,\right|^a_0 \\[6px]</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>t H_0 \sqrt{\Omega_{0,\mathrm M}} &= \left.\left( \tfrac23 <ins class="diffchange diffchange-inline">{</ins>a'<ins class="diffchange diffchange-inline">}</ins>^<ins class="diffchange diffchange-inline">{3/2}</ins> \right) \,\right|^a_0 \\[6px]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>\left( \tfrac32 t H_0 \sqrt{\Omega_{0,\mathrm M}}\right)^<del class="diffchange diffchange-inline">\frac23</del> &= a(t)<del class="diffchange diffchange-inline"> </del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>\left( \tfrac32 t H_0 \sqrt{\Omega_{0,\mathrm M}}\right)^<ins class="diffchange diffchange-inline">{2/3}</ins> &= a(t)</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>\end{align}</math></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>\end{align}</math></div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>which recovers the aforementioned {{math|''a'' ∝ ''t''<sup>2/3</sup>}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>For radiation-dominated universes, where {{math|Ω<sub>0,R</sub> ≫ Ω<sub>0,M</sub>}} and {{math|Ω<sub>0,Λ</sub>}}, as well as {{math|Ω<sub>0,R</sub> ≈ 1}}:</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>which recovers the aforementioned {{math|''a'' ∝ ''t''<sup>{{sfrac|2|3}}</sup>}}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\begin{align}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>For radiation-dominated universes, where {{math|''Ω''<sub>0,R</sub> ≫ ''Ω''<sub>0,M</sub>}} and {{math|''Ω''<sub>0,''Λ''</sub>}}, as well as {{math|''Ω''<sub>0,R</sub> ≈ 1}}:</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\begin{align}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{\Omega_{0,\mathrm R} a'^{-2}}} \\[6px]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>t H_0 &= \int_{0}^{a} \frac{\mathrm{d}a'}{\sqrt{\Omega_{0,\mathrm R} a'^{-2}}} \\[6px]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>t H_0 \sqrt{\Omega_{0,\mathrm R}} &= \left.\frac{a'^2}{2} \,\right|^a_0 \\[6px]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>t H_0 \sqrt{\Omega_{0,\mathrm R}} &= \left.\frac{a'^2}{2} \,\right|^a_0 \\[6px]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>\left(2 t H_0 \sqrt{\Omega_{0,\mathrm R}}\right)^<del class="diffchange diffchange-inline">\frac12</del> &= a(t)</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>\left(2 t H_0 \sqrt{\Omega_{0,\mathrm R}}\right)^<ins class="diffchange diffchange-inline">{1/2}</ins> &= a(t)</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>\end{align}</math></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>\end{align}</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>For {{mvar|Λ}}-dominated universes, where {{math|''Ω''<sub>0,''Λ''</sub> ≫ ''Ω''<sub>0,R</sub>}} and {{math|''Ω''<sub>0,M</sub>}}, as well as {{math|''Ω''<sub>0,''Λ''</sub> ≈ 1}}, and where we now will change our bounds of integration from {{math|''t<sub>i</sub>''}} to {{mvar|t}} and likewise {{math|''a<sub>i</sub>''}} to {{mvar|a}}:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>For {{mvar|Λ}}-dominated universes, where {{math|''Ω''<sub>0,''Λ''</sub> ≫ ''Ω''<sub>0,R</sub>}} and {{math|''Ω''<sub>0,M</sub>}}, as well as {{math|''Ω''<sub>0,''Λ''</sub> ≈ 1}}, and where we now will change our bounds of integration from {{math|''t<sub>i</sub>''}} to {{mvar|t}} and likewise {{math|''a<sub>i</sub>''}} to {{mvar|a}}:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\begin{align}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\begin{align}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>\left(t-t_i\right) H_0 &= \int_{a_i}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,\Lambda} a'^2)}} \\[6px]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>\left(t-t_i\right) H_0 &= \int_{a_i}^{a} \frac{\mathrm{d}a'}{\sqrt{(\Omega_{0,\Lambda} a'^2)}} \\[6px]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>\left(t - t_i\right) H_0 \sqrt{\Omega_{0,\Lambda}} &= \bigl. \ln|a'| \,\bigr|^a_{a_i} \\[6px]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>\left(t - t_i\right) H_0 \sqrt{\Omega_{0,\Lambda}} &= \bigl. \ln|a'| \,\bigr|^a_{a_i} \\[6px]</div></td> </tr> <tr> <td colspan="2" class="diff-lineno">Line 181:</td> <td colspan="2" class="diff-lineno">Line 151:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>The {{mvar|Λ}}-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 {{math|''ρ<sub>Λ</sub>''}} a candidate for [[dark energy]]:</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>The {{mvar|Λ}}-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 {{math|''ρ<sub>Λ</sub>''}} a candidate for [[dark energy]]:</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><math display="block">\begin{align}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>a(t) &= a_i \exp\left( (t - t_i) H_0 \textstyle\sqrt{\Omega_{0,\Lambda}}\right) \\[6px]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\begin{align}</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>a(t) &= a_i \exp\left( (t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}}\right)<del class="diffchange diffchange-inline"> \\[6px]</del></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><ins class="diffchange diffchange-inline">\frac{\mathrm{d}^2 </ins>a(t)<ins class="diffchange diffchange-inline">}{\mathrm{d}t^2}</ins> &= a_i<ins class="diffchange diffchange-inline"> {H_0}^2 \, \Omega_{0,\Lambda}</ins> \exp\left( (t - t_i) H_0 <ins class="diffchange diffchange-inline">\textstyle</ins>\sqrt{\Omega_{0,\Lambda}}\right)</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>\frac{\mathrm{d}^2 a(t)}{\mathrm{d}t^2} &= a_i \left(H_0\right)^2 \Omega_{0,\Lambda} \exp\left( (t - t_i) H_0 \sqrt{\Omega_{0,\Lambda}}\right)</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>\end{align}</math></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>\end{align}</math></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Where by construction {{math|''a<sub>i</sub>'' > 0}}, our assumptions were {{math|''Ω''<sub>0,''Λ''</sub> ≈ 1}}, and {{math|''H''<sub>0</sub>}} has been measured to be positive, forcing the acceleration to be greater than zero.</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Where by construction {{math|''a<sub>i</sub>'' > 0}}, our assumptions were {{math|''Ω''<sub>0,''Λ''</sub> ≈ 1}}, and {{math|''H''<sub>0</sub>}} has been measured to be positive, forcing the acceleration to be greater than zero.</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== Rescaled Friedmann equation ==</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Set</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\tilde{a}=\frac{a}{a_0}, \quad\rho_c=\frac{3H_0^2}{8\pi G},\quad</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>\Omega=\frac{\rho}{\rho_\mathrm{c}},\quad t=\frac{\tilde{t}}{H_0},\quad</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>\Omega_\mathrm{k}=-\frac{kc^2}{H_0^2 a_0^2},</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>where {{math|''a''<sub>0</sub>}} and {{math|''H''<sub>0</sub>}} are separately the [[Scale factor (Universe)|scale factor]] and the [[Hubble parameter]] today.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Then we can have</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>\frac12\left( \frac{d\tilde{a}}{d\tilde{t}}\right)^2 + U_\text{eff}(\tilde{a})=\frac12\Omega_\mathrm{k}</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>where</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>:<math>U_\text{eff}(\tilde{a})=\frac{-\Omega\tilde{a}^2}{2}.</math></div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><br /></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>For any form of the effective potential {{math|''U''<sub>eff</sub>(''ã'')}}, there is an equation of state {{math|''p'' {{=}} ''p''(''ρ'')}} that will produce it.</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== In popular culture ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== In popular culture ==</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>Several students at [[Tsinghua University]] ([[<del class="diffchange diffchange-inline">People's</del> <del class="diffchange diffchange-inline">Republic</del> of <del class="diffchange diffchange-inline">China</del>|<del class="diffchange diffchange-inline">PRC</del>]]<del class="diffchange diffchange-inline"> President</del> [[Xi Jinping]]'s [[alma mater]]) participating in the [[2022 COVID-19 protests in China]] 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.<del class="diffchange diffchange-inline"> </del><ref>{{Cite news|<del class="diffchange diffchange-inline">url</del>=<del class="diffchange diffchange-inline">https://www.bbc.com/news/world-asia-china-63778871</del>|title=China's protests: Blank paper becomes the symbol of rare demonstrations|<del class="diffchange diffchange-inline">work</del>=<del class="diffchange diffchange-inline">BBC News</del> |<del class="diffchange diffchange-inline">date</del>=<del class="diffchange diffchange-inline">November</del> <del class="diffchange diffchange-inline">28, 2022</del>}}</ref></div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Several students at [[Tsinghua University]] ([[<ins class="diffchange diffchange-inline">Chinese</ins> <ins class="diffchange diffchange-inline">Communist Party|CCP]] [[Leader</ins> of <ins class="diffchange diffchange-inline">the Chinese Communist Party</ins>|<ins class="diffchange diffchange-inline">leader</ins>]] [[Xi Jinping]]'s [[alma mater]]) participating in the [[2022 COVID-19 protests in China]] 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.<ref>{{Cite news<ins class="diffchange diffchange-inline"> </ins>|<ins class="diffchange diffchange-inline">last</ins>=<ins class="diffchange diffchange-inline">Murphy |first=Matt |date=November 28, 2022 </ins>|title=China's protests: Blank paper becomes the symbol of rare demonstrations<ins class="diffchange diffchange-inline"> </ins>|<ins class="diffchange diffchange-inline">url</ins>=<ins class="diffchange diffchange-inline">https://www.bbc.com/news/world-asia-china-63778871</ins> |<ins class="diffchange diffchange-inline">work</ins>=<ins class="diffchange diffchange-inline">[[BBC</ins> <ins class="diffchange diffchange-inline">News]]</ins>}}</ref></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== See also ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== See also ==</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* [[Mathematics of general relativity]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* [[Mathematics of general relativity]]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>* [[Solutions of the Einstein<del class="diffchange diffchange-inline"> field equations|Solutions of Einstein's</del> field equations]]</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>* [[Solutions of the Einstein field equations]]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* [[Warm inflation]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* [[Warm inflation]]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== <del class="diffchange diffchange-inline">Notes</del> ==</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== <ins class="diffchange diffchange-inline">Sources</ins> ==</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{reflist}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{reflist}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-lineno">Line 224:</td> <td colspan="2" class="diff-lineno">Line 178:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{DEFAULTSORT:Friedmann Equations}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{DEFAULTSORT:Friedmann Equations}}</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>[[Category:Eponymous equations of physics]]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>[[Category:General relativity]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>[[Category:General relativity]]</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>[[Category:Equations]]</div></td> <td colspan="2" class="diff-empty diff-side-added"></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Latest revision as of 19:20, 23 November 2024</h2> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Equations in physical cosmology</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><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:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a 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. Tolman">Tolman</a></li> <li><a href="/wiki/Robert_Woodrow_Wilson" title="Robert Woodrow Wilson">Wilson</a></li> <li><a href="/wiki/Yakov_Zeldovich" title="Yakov Zeldovich">Zeldovich</a></li></ul> </div> <ul><li><a href="/wiki/List_of_cosmologists" title="List of cosmologists">List of cosmologists</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:#ddd8e7;text-align:center;;color: var(--color-base)"><a href="/wiki/Physical_cosmology#Subject_history" title="Physical cosmology">Subject history</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Discovery_of_cosmic_microwave_background_radiation" title="Discovery of cosmic microwave background radiation">Discovery of cosmic microwave<br />background radiation</a></div></li> <li><a href="/wiki/History_of_the_Big_Bang_theory" title="History of the Big Bang theory">History of the Big Bang theory</a></li> <li><a href="/wiki/Timeline_of_cosmological_theories" title="Timeline of cosmological theories">Timeline of cosmological theories</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="display:block;margin-top:0.4em; 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 #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</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" /></span></span> <a href="/wiki/Category:Theory_of_relativity" title="Category:Theory of relativity">Category</a></div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;flex:0 1 auto;padding:0.15em 0;column-gap:1em;align-items:baseline;margin:0;list-style:none}.mw-parser-output .portal-bar-content-related{margin:0;list-style:none}.mw-parser-output .portal-bar-item{display:inline-block;margin:0.15em 0.2em;min-height:24px;line-height:24px}@media screen and (max-width:768px){.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;flex-flow:column wrap;align-items:baseline}.mw-parser-output .portal-bar-header{text-align:center;flex:0;padding-left:0.5em;margin:0 auto}.mw-parser-output .portal-bar-related{font-size:100%;align-items:flex-start}.mw-parser-output .portal-bar-content{display:flex;flex-flow:row wrap;align-items:center;flex:0;column-gap:1em;border-top:1px solid #a2a9b1;margin:0 auto;list-style:none}.mw-parser-output .portal-bar-content-related{border-top:none;margin:0;list-style:none}}.mw-parser-output .navbox+link+.portal-bar,.mw-parser-output .navbox+style+.portal-bar,.mw-parser-output .navbox+link+.portal-bar-bordered,.mw-parser-output .navbox+style+.portal-bar-bordered,.mw-parser-output .sister-bar+link+.portal-bar,.mw-parser-output .sister-bar+style+.portal-bar,.mw-parser-output .portal-bar+.navbox-styles+.navbox,.mw-parser-output .portal-bar+.navbox-styles+.sister-bar{margin-top:-1px}</style><div class="portal-bar noprint metadata noviewer portal-bar-bordered" role="navigation" aria-label="Portals"><span class="portal-bar-header"><a href="/wiki/Wikipedia:Contents/Portals" title="Wikipedia:Contents/Portals">Portals</a>:</span><ul class="portal-bar-content"><li class="portal-bar-item"><span class="nowrap"><span 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/17px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="17" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/26px-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/34px-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</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/19px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/29px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/38px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/19px-Crab_Nebula.jpg" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/29px-Crab_Nebula.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Crab_Nebula.jpg/38px-Crab_Nebula.jpg 2x" data-file-width="3864" data-file-height="3864" /></span></span> </span><a href="/wiki/Portal:Astronomy" title="Portal:Astronomy">Astronomy</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><a href="/wiki/File:He1523a.jpg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/16px-He1523a.jpg" decoding="async" width="16" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/25px-He1523a.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/He1523a.jpg/33px-He1523a.jpg 2x" data-file-width="180" data-file-height="207" /></a></span> </span><a href="/wiki/Portal:Stars" title="Portal:Stars">Stars</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/19px-RocketSunIcon.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/29px-RocketSunIcon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/38px-RocketSunIcon.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> </span><a href="/wiki/Portal:Spaceflight" title="Portal:Spaceflight">Spaceflight</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/21px-Earth-moon.jpg" decoding="async" width="21" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/32px-Earth-moon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/42px-Earth-moon.jpg 2x" data-file-width="3000" data-file-height="2400" /></span></span> </span><a href="/wiki/Portal:Outer_space" title="Portal:Outer space">Outer space</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Solar_system.jpg/15px-Solar_system.jpg" decoding="async" width="15" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Solar_system.jpg/23px-Solar_system.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Solar_system.jpg/30px-Solar_system.jpg 2x" data-file-width="4500" data-file-height="5600" /></span></span> </span><a href="/wiki/Portal:Solar_System" title="Portal:Solar System">Solar System</a></li><li class="portal-bar-item"><span class="nowrap"><span typeof="mw:File"><a href="/wiki/File:Nuvola_apps_kalzium.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Nuvola_apps_kalzium.svg/19px-Nuvola_apps_kalzium.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Nuvola_apps_kalzium.svg/29px-Nuvola_apps_kalzium.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Nuvola_apps_kalzium.svg/38px-Nuvola_apps_kalzium.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Science" title="Portal:Science">Science</a></li></ul></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Friedmann_equations&oldid=1259169248">https://en.wikipedia.org/w/index.php?title=Friedmann_equations&oldid=1259169248</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Eponymous_equations_of_physics" title="Category:Eponymous equations of physics">Eponymous equations of physics</a></li><li><a href="/wiki/Category:General_relativity" title="Category:General relativity">General relativity</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_September_2015" title="Category:Wikipedia articles needing clarification from September 2015">Wikipedia articles needing clarification from September 2015</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Pages_using_sidebar_with_the_child_parameter" title="Category:Pages using sidebar with the child parameter">Pages using sidebar with the child parameter</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_September_2024" title="Category:Articles needing additional references from September 2024">Articles needing additional references from September 2024</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 23 November 2024, at 19:20<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Friedmann_equations&diff=1259169248&oldid=1147470939&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-gm8xq","wgBackendResponseTime":294,"wgPageParseReport":{"limitreport":{"cputime":"0.815","walltime":"1.088","ppvisitednodes":{"value":5166,"limit":1000000},"postexpandincludesize":{"value":144576,"limit":2097152},"templateargumentsize":{"value":8127,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":12,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":113795,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 838.270 1 -total"," 19.71% 165.188 1 Template:Physical_cosmology"," 14.04% 117.698 59 Template:Math"," 12.55% 105.236 1 Template:Reflist"," 11.93% 100.018 1 Template:Short_description"," 11.60% 97.258 2 Template:Cite_web"," 8.50% 71.219 2 Template:Unreferenced_section"," 8.27% 69.288 2 Template:Pagetype"," 7.69% 64.426 2 Template:Unreferenced"," 7.16% 60.050 2 Template:Ambox"]},"scribunto":{"limitreport-timeusage":{"value":"0.436","limit":"10.000"},"limitreport-memusage":{"value":9665780,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\ntable#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.main-849f99967d-8m92z","timestamp":"20241123232106","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Friedmann equations","url":"https:\/\/en.wikipedia.org\/wiki\/Friedmann_equations","sameAs":"http:\/\/www.wikidata.org\/entity\/Q467736","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q467736","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-11-09T06:27:46Z","dateModified":"2024-11-23T19:20:19Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/ed\/WMAP_2012.png","headline":"equation"}</script> </body> </html>

CINXE.COM

Friedmann equations: Difference between revisions - Wikipedia