# File:Friedmann universes.svg

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## Summary

 Description Intended as a replacement for File:Universe.svg and File:Universos.gif. Improvements: better dash patterns, more accurate curves (actual solutions of the Friedmann equations, not hand-drawn). Date 23 September 2009 Source Own work Author BenRG Permission(Reusing this file) Public domain

## Formulas

This diagram uses the following exact solutions to the Friedmann equations:

${\displaystyle {\begin{cases}a(t)=H_{0}t&\Omega _{M}=\Omega _{\Lambda }=0\\{\begin{cases}a(q)={\tfrac {\Omega _{M}}{2(1-\Omega _{M})}}(\cosh q-1)\\t(q)={\tfrac {\Omega _{M}}{2H_{0}(1-\Omega _{M})^{3/2}}}(\sinh q-q)\end{cases}}&0<\Omega _{M}<1,\ \Omega _{\Lambda }=0\\a(t)=\left({\tfrac {3}{2}}H_{0}t\right)^{2/3}&\Omega _{M}=1,\ \Omega _{\Lambda }=0\\{\begin{cases}a(q)={\tfrac {\Omega _{M}}{2(\Omega _{M}-1)}}(1-\cos q)\\t(q)={\tfrac {\Omega _{M}}{2H_{0}(\Omega _{M}-1)^{3/2}}}(q-\sin q)\end{cases}}&\Omega _{M}>1,\ \Omega _{\Lambda }=0\\a(t)=\left({\tfrac {\Omega _{M}}{\Omega _{\Lambda }}}\sinh ^{2}\left({\tfrac {3}{2}}{\sqrt {\Omega _{\Lambda }}}H_{0}t\right)\right)^{1/3}&0<\Omega _{M}<1,\ \Omega _{\Lambda }=1-\Omega _{M}\end{cases}}}$

Some of the shown models are implemented as an animation at Cosmos-animation.

## Licensing

 I, the copyright holder of this work, release this work into the public domain. This applies worldwide.In some countries this may not be legally possible; if so:I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

## File history

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Date/TimeThumbnailDimensionsUserComment
current22:09, 23 September 2009620 × 500 (3 KB)BenRG (talk | contribs)Nimbus Roman doesn't have Greek letters; switch to DejaVu Serif
22:02, 23 September 2009620 × 500 (3 KB)BenRG (talk | contribs){{Information |Description=Intended as a replacement for File:Universe.svg and File:Universos.gif. Improvements: better dash patterns, more accurate curves (actual solutions of the Friedmann equations, not hand-drawn). |Source=Own work by upload
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