Friedmann Equations - Assumptions

Assumptions

The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the Cosmological Principle; empirically, this is justified on scales larger than ~100 Mpc. The Cosmological Principle implies that the metric of the universe must be of the form:

where 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. The parameter discussed below takes the value 0, 1, -1 in these three cases respectively. It is this fact that allows us to sensibly speak of a "scale factor", .

Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. The resulting equations are described below.

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