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 from the
Einstein field equations for the
Friedmann-Lemaître-Robertson-Walker metric and a fluid with a given energy
density ρ and
pressure . The equations are: where is the
cosmological constant possibly caused by
vacuum energy, is the
gravitational constant, is the speed of light, is the
scale factor, and is the
Gaussian curvature when (i.e. today). If the
shape of the universe is hyperspherical and is the radius of curvature ( in the present-day), then . Generally, is the
Gaussian curvature. If is positive, then the universe is hyperspherical. If is zero, then the universe is
flat. If is negative, then the universe is hyperbolic. Note that and are in general functions of . The
Hubble parameter, , is the rate of expansion of the universe.
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The average density of matter required to halt the expansion of the universe. According to present estimates, the luminous matter in the universe only provides ten per cent of the required material.
Dark matter is theorised to make up the rest.
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