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Concept
Friedmann equations
Summary
The Friedmann equations are a set of equations in physical 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 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p. The equations for negative spatial curvature were given by Friedmann in 1924.
Friedmann–Lemaître–Robertson–Walker metric
The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, that is, the cosmological principle; empirically, this is justified on scales larger than the order of 100 Mpc. The cosmological principle implies that the metric of the universe must be of the form
where ds32 is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. This metric is called Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The parameter k discussed below takes the value 0, 1, −1, or the Gaussian curvature, in these three cases respectively. It is this fact that allows us to sensibly speak of a "scale factor" a(t).
Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute Christoffel symbols, then the Ricci tensor. With the stress–energy tensor for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below.
There are two independent Friedmann equations for modelling a homogeneous, isotropic universe. The first is:
which is derived from the 00 component of the Einstein field equations. The second is:
which is derived from the first together with the trace of Einstein's field equations (the dimension of the two equations is time−2).
Official source
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