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<title>Friedmann-Modell</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Friedmann-Modell</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Unter einem <b>Friedmann-Modell</b> oder <b>Friedmann-Lemaître-Modell</b> (benannt nach dem russischen Mathematiker und Meteorologen <a href="Alexander_Alexandrowitsch_Friedmann" title="Alexander Alexandrowitsch Friedmann">Alexander Friedmann</a> und dem belgischen Astrophysiker <a href="Georges_Lema%C3%AEtre" title="Georges Lemaître">Georges Lemaître</a>)<sup id="cite_ref-Goenner1999_1-0" class="reference"><a href="#cite_note-Goenner1999-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> versteht man in der <a href="Kosmologie" title="Kosmologie">Kosmologie</a> Lösungen der <a href="Friedmann-Gleichung" class="mw-redirect" title="Friedmann-Gleichung">Friedmann-Gleichung</a>, d. h. eine Lösung der <a href="Einsteinsche_Feldgleichungen" title="Einsteinsche Feldgleichungen">Einsteinschen Feldgleichungen</a> mit konstanter <a href="Kr%C3%BCmmung" title="Krümmung">Krümmung</a>, die um jeden Punkt räumlich <a href="Isotrop" class="mw-redirect" title="Isotrop">isotrop</a> ist.

</p><p>Friedmann-Modelle unterscheiden sich durch den Parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> aus der <a href="Robertson-Walker-Metrik" class="mw-redirect" title="Robertson-Walker-Metrik">Robertson-Walker-Metrik</a>

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<annotation encoding="application/x-tex">{\displaystyle k=+1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c94932b3d937c93f7a546fef65639101deafb492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.28ex; height:2.343ex;" alt="{\displaystyle k=+1}" loading="lazy"></span>: positive Krümmung</li>

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<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy"></span>: keine Krümmung, flacher Raum</li>

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<annotation encoding="application/x-tex">{\displaystyle k=-1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/654e2ed2859d2ddf7f69949359f20f28977f829d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.28ex; height:2.343ex;" alt="{\displaystyle k=-1}" loading="lazy"></span>: negative Krümmung</li></ul>

<p>und den Wert der <a href="Kosmologische_Konstante" title="Kosmologische Konstante">kosmologischen Konstante</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Sonderfälle_der_Friedmann-Modelle"><span id="Sonderf.C3.A4lle_der_Friedmann-Modelle"></span>Sonderfälle der Friedmann-Modelle</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Einstein-Kosmos">Einstein-Kosmos</h3></div>

<p>Es handelt sich um ein

nicht expandierendes oder kontrahierendes, statisches (gegenüber kleinen Änderungen instabiles) Universum mit

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<annotation encoding="application/x-tex">{\displaystyle k=+1,\quad \Lambda =\Lambda _{c}\ ,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7856f5301a363274c69d52692e84d93a81c600d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.133ex; height:2.509ex;" alt="{\displaystyle k=+1,\quad \Lambda =\Lambda _{c}\ ,}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{c}=4/(\kappa M)^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda _{c}=4/(\kappa M)^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1903af37adcf01a495bc90150330b76c811939b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.625ex; height:3.176ex;" alt="{\displaystyle \Lambda _{c}=4/(\kappa M)^{2}}" loading="lazy"></span> ist.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2.1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Lemaître-Universum"><span id="Lema.C3.AEtre-Universum"></span>Lemaître-Universum</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle k=+1,\quad \Lambda =\Lambda _{c}(1+\epsilon )\ ,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbe50efad1b0fd437ca5e6864d752dac2e49b166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.889ex; height:2.843ex;" alt="{\displaystyle k=+1,\quad \Lambda =\Lambda _{c}(1+\epsilon )\ ,}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"></span> ein sehr kleiner Parameter ist. Durch die Wahl eines geeigneten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"></span> ist die Zeitskala der <a href="Expansion_des_Universums" title="Expansion des Universums">Expansion des Universums</a> so gedehnt, dass zwischen zwei expandierenden Zeitphasen ein fast statisches Universum besteht.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>2.2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="De-Sitter-Modell">De-Sitter-Modell</h3></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="De-Sitter-Modell" title="De-Sitter-Modell">De-Sitter-Modell</a></i></div>

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<annotation encoding="application/x-tex">{\displaystyle \rho =0,\quad \Lambda >0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd9c90162986b8343bb86810477d32f453ad9f42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.693ex; height:2.676ex;" alt="{\displaystyle \rho =0,\quad \Lambda >0}" loading="lazy"></span></dd></dl>

<p>Die drei verschiedenen Werte für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> ergeben drei mögliche Modelle, die aber nur verschiedene Schnitte derselben <a href="Raumzeit" title="Raumzeit">Raumzeit</a> sind.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>2.3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Einstein-de-Sitter-Modell">Einstein-de-Sitter-Modell</h3></div>

<p>Das Einstein-de-Sitter-Universum ergibt sich mit

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k=0,\quad \Lambda =0\ .}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9270bdbf2923a1d437f51192071a37902a4fa45e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.93ex; height:2.509ex;" alt="{\displaystyle k=0,\quad \Lambda =0\ .}" loading="lazy"></span></dd></dl>

<p>Für dieses flache, unendlich ausgedehnte Universum entwickelt sich der Parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> der Robertson-Walker-Metrik gerade mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\sim t^{2/3}}">

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<annotation encoding="application/x-tex">{\displaystyle R\sim t^{2/3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a20bccb800505160c668d3934f6b4fd2a8fdac09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.4ex; height:2.843ex;" alt="{\displaystyle R\sim t^{2/3}}" loading="lazy"></span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>2.4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-Goenner1999-1"><span class="mw-cite-backlink"><a href="#cite_ref-Goenner1999_1-0">↑</a></span> <span class="reference-text"><span class="book">Hubert Goenner: <cite style="font-style:italic">Einsteins Relativitätstheorien: Raum, Zeit, Masse, Gravitation</cite>. C.H.Beck, 1999, ISBN 978-3-406-45669-5, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>96</span> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=5XcsOGhE0j0C&pg=PA96&hl=de">google.de</a> [abgerufen am 9. April 2012]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Friedmann-Modell&rft.au=Hubert+Goenner&rft.btitle=Einsteins+Relativit%C3%A4tstheorien%3A+Raum%2C+Zeit%2C+Masse%2C+Gravitation&rft.date=1999&rft.genre=book&rft.isbn=9783406456695&rft.pages=96&rft.pub=C.H.Beck" style="display:none"> </span></span></span>

</li>

<li><span class="mw-cite-backlink">↑ </span> <span class="reference-text">R. Sexl, H. Urbantke: <cite style="font-style:italic">Gravitation und Kosmologie</cite>. 3., korrigierte Auflage. BI-Wissenschaftsverlag, Mannheim 1987, ISBN 3-411-03177-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Friedmann-Modell&rft.au=R.+Sexl%2C+H.+Urbantke&rft.btitle=Gravitation+und+Kosmologie&rft.date=1987&rft.edition=3.%2C+korrigierte&rft.genre=book&rft.isbn=3411031778&rft.place=Mannheim&rft.pub=BI-Wissenschaftsverlag" style="display:none"> </span></span>

</li>

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