File:Friedmann universes.svg

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This file supersedes the file Universe.svg. It is recommended to use this file rather than the other one. Reason why to use this file over the other file: "Improvements: better dash patterns, more accurate curves (actual solutions of the Friedmann equations, not hand-drawn)." Bahasa Indonesia ∙ davvisámegiella ∙ Deutsch ∙ English ∙ español ∙ français ∙ italiano ∙ magyar ∙ Nederlands ∙ polski ∙ svenska ∙ македонски ∙ മലയാളം ∙ português do Brasil ∙ русский ∙ slovenščina ∙ 日本語 ∙ 中文（简体） ∙ 中文（繁體） ∙ farsi ∙ +/−

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

)=\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}}}"> { a ( t ) = H 0 t Ω M = Ω Λ = 0 { a ( q ) = Ω M 2 ( 1 − Ω M ) ( cosh ⁡ q − 1 ) t ( q ) = Ω M 2 H 0 ( 1 − Ω M ) 3 / 2 ( sinh ⁡ q − q ) 0 < Ω M < 1 ,   Ω Λ = 0 a ( t ) = ( 3 2 H 0 t ) 2 / 3 Ω M = 1 ,   Ω Λ = 0 { a ( q ) = Ω M 2 ( Ω M − 1 ) ( 1 − cos ⁡ q ) t ( q ) = Ω M 2 H 0 ( Ω M − 1 ) 3 / 2 ( q − sin ⁡ q ) Ω M > 1 ,   Ω Λ = 0 a ( t ) = ( Ω M Ω Λ sinh 2 ⁡ ( 3 2 Ω Λ H 0 t ) ) 1 / 3 0 < Ω M < 1 ,   Ω Λ = 1 − Ω M {\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}}} 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.

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File history

	Date/Time	Thumbnail	Dimensions	User	Comment
current	22:09, 23 September 2009		620 × 500 (3 KB)	BenRG	Nimbus Roman doesn't have Greek letters; switch to DejaVu Serif
	22:02, 23 September 2009		620 × 500 (3 KB)	BenRG	{{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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