File:Friedmann universes.svg

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Summary

DescriptionFriedmann universes.svg	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:

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

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