# File:Friedmann universes.svg

## 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:

$\begin{cases} a(t) = H_0 t & \Omega_M = \Omega_\Lambda = 0 \\ \begin{cases} a(q) = \tfrac{\Omega_M}{2(1-\Omega_M)} (\cosh q - 1) \\ t(q) = \tfrac{\Omega_M}{2H_0(1-\Omega_M)^{3/2}} (\sinh q - q) \end{cases} & 0 < \Omega_M < 1,\ \Omega_\Lambda = 0 \\ a(t) = \left( \tfrac32 H_0 t \right)^{2/3} & \Omega_M = 1,\ \Omega_\Lambda = 0 \\ \begin{cases} a(q) = \tfrac{\Omega_M}{2(\Omega_M-1)} (1-\cos q) \\ t(q) = \tfrac{\Omega_M}{2H_0(\Omega_M-1)^{3/2}} (q - \sin q) \end{cases} & \Omega_M > 1,\ \Omega_\Lambda = 0 \\ a(t) = \left( \tfrac{\Omega_M}{\Omega_\Lambda} \sinh^2 \left( \tfrac32 \sqrt{\Omega_\Lambda} H_0 t \right) \right)^{1/3} & 0 < \Omega_M < 1,\ \Omega_\Lambda = 1 - \Omega_M \end{cases}$

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.

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