Friedmann equations

The Friedman equations relate various cosmological parameters within the context of general relativity. They were derived by Alexander Friedmann in 1922 as a solution to the Friedman-Lemaître-Robertson-Walker metric for a fluid with a given density and pressure. The equations are:

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

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

where $ρ$ and $p$ are the density and pressure of the fluid, $Λ$ is the vacuum energy, $G$ is the gravitational constant, $k$ gives the shape of the universe, and $a$ is the scale factor. The Hubble constant $H$ is the rate of expansion of the universe. Applied to a fluid with a given equation of state, the Friedman 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

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

which follows from the Friedman equations determines the geometry where $ρ c$ is the critical density for which the geometry is flat. If $Ω$ is larger than unity, the geometry is closed. If $Ω$ is less than unity, it is open.

Categories: General relativity | Equations

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01-04-2007 01:21:04

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