Friedmann equations

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Alexander Friedman

The Friedmann equations are a set of equations in cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922^[1] from the Einstein field equations for the Friedmann-Lemaître-Robertson-Walker metric and a fluid with a given energy density ρ and pressure $p$ . The equations are:

$H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G \rho + \Lambda}{3} - K\frac{c^2}{a^2}$

$3 \, \frac{\ddot{a}}{a} = \Lambda - 4 \pi G \left(\rho + \frac{3p}{c^2}\right)$

where $Λ$ is the cosmological constant possibly caused by vacuum energy, $G$ is the gravitational constant, $c$ is the speed of light, $a$ is the scale factor, and $K$ is the Gaussian curvature when $a = 1$ (i.e. today). If the shape of the universe is hyperspherical and $R$ is the radius of curvature ( $R 0$ in the present-day), then $a = R / R 0$ . Generally, $K \over a^2$ is the Gaussian curvature. If $K$ is positive, then the universe is hyperspherical. If $K$ is zero, then the universe is flat. If $K$ is negative, then the universe is hyperbolic. Note that $ρ$ and $p$ are in general functions of $a$ . The Hubble parameter, $H$ , is the rate of expansion of the universe.

These equations are sometimes simplified by redefining

$\rho \rightarrow \rho - \frac{\Lambda}{8 \pi G}$

$p \rightarrow p + \frac{\Lambda c^2}{8 \pi G}$

to give:

$H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - K\frac{c^2}{a^2}$

$3 \, \frac{\ddot{a}}{a} = - 4 \pi G \left(\rho + \frac{3p}{c^2}\right)$

The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the energy density, vacuum energy, and curvature). Evaluating the Hubble parameter at the present time yields the Hubble constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.

Some cosmologists call the second of these two equations the acceleration equation and reserve the term Friedmann equation for only the first equation.

The density parameter, $Ω$ , is defined as the ratio of actual (or observed) density $ρ$ to the critical density $ρ c$ of the Friedmann universe. An expression for critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the curvature, K, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:

$\rho_c = \frac{3 H^2}{8 \pi G}$

And the expression for the density parameter (useful for comparing different cosmological models) then follows:

$\Omega \equiv \frac{\rho}{\rho_c} = \frac{8 \pi G}{3 H^2}\rho$

This term originally was used as a means to determine the geometry of the field where $ρ c$ is the critical density for which the geometry is flat. Assuming a zero vacuum energy density, if $Ω$ is larger than unity, the geometry is closed; the universe will eventually stop expanding, then collapse. If $Ω$ is less than unity, it is open; and the universe expands forever. However, one can also subsume the curvature and vacuum energy terms into a more general expression for $Ω$ in which case this energy density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the Lambda-CDM model, there are important components of $Ω$ due to baryons, cold dark matter and dark energy. The geometry of spacetime has been measured by the WMAP probe to be nearly flat meaning that the curvature parameter κ is zero.

The first Friedmann Equation is often seen in a form with density parameters.

$\frac{H^2}{H_0^2} = \Omega_R a^{-4} + \Omega_M a^{-3} + \Omega_{\Lambda} - K c^2 a^{-2}$

Here $Ω R$ is the radiation density today, $Ω M$ is the matter (dark plus baryonic) density today, and $Ω Λ$ is the cosmological constant or vacuum density today.

Set a=ãa₀, ρ_c=3H₀²/8πG, ρ=ρ_cΩ, $t=\tilde{t}/H_0$ , Ω_c=-κ/H₀²a₀² where a₀ and H₀ are separately the scale factor and the Hubble parameter today. Then we can have

$\frac{1}{2}\left( \frac{d\tilde{a}}{d\tilde{t}}\right)^2 + U_{\rm eff}(\tilde{a})=\frac{1}{2}\Omega_c$

where U_eff(ã)=Ωã²/2. For any form of the effective potential U_eff(ã), there is an equation of state p=p(ρ) that will produce it.

^ Friedmann, A: Über die Krümmung des Raumes, Z. Phys. 10 (1922), 377-386. (English translation in: Gen. Rel. Grav. 31 (1999), 1991-2000.)

Please help improve this article or section by expanding it.
Further information might be found on the talk page or at requests for expansion. (January 2007)

Retrieved from "http://en.wikipedia.org/wiki/Friedmann_equations"

Categories: Articles to be expanded since January 2007 | All articles to be expanded | General relativity | Equations

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