Friedmann equations
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The Friedmann equations relate various cosmological parameters within the context of general relativity. They were derived by Alexander Friedmann in 1922 from the Einstein field equations under some assumptions of symmetry appropriate for a cosmological model. From his equations, the Friedmann-Lemaître-Robertson-Walker metric was derived for a fluid with a given density and pressure. The equations are:
- [H^2 = \left(\frac}\right)^2 = \frac\rho + \frac - \frac]
- [\frac} = -\frac\left(\rho + 3p\right) + \frac]
where [\rho] and [p] are the density and pressure of the fluid, [\Lambda] is the
cosmological constant possibly caused by
vacuum energy, [G] is the
gravitational constant,
[k] gives the
shape of the universe, and [a] is the
scale factor. In these equations, [c], the speed of light is set to equal 1, and would appear in the last term otherwise. The
Hubble parameter [H] is the rate of
expansion of the universe, a value that 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
The first of the Friedmann equations defines a density parameter useful for comparing different cosmological models:
- [\Omega \equiv \frac = \frac\rho]
This term originally was used as a means to determine
the geometry of the field where [\rho_c] is the critical density for which the geometry is flat. Assuming a zero vacuum energy density, if [\Omega] is larger than unity, the geometry is closed. If [\Omega] is less than unity, it is open. However, one can also subsume the curvature and vacuum energy terms into a more general expression for [\Omega] 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 [\Omega] 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 = \Omega_R a^ + \Omega_M a^ + \Omega_ - \frac]
Here [\Omega_R] is the radiation density parameter, [\Omega_M] is the matter (
dark plus
baryonic) density parameter, and [\Omega_\lambda] is the
cosmological constant or vacuum density parameter.
Rescaled Friedmann equation
Set a=ãa
0, ρ
c=3H
02/8π,
ρ=ρ
cΩ, [t=\tilde/H_0],
Ω
c=-κ/H
02a
02
where a
0 and H
0 are separately the
scale factor and the
Hubble parameter today.
Then we can have
- [\frac\left( \frac}}\right)^2 + U_(\tilde)=\frac\Omega_c]
where U
eff(ã)=Ωã
2/2.
For any form of the effective potential U
eff(ã), there is an equation of state p=p(ρ) that will produce it.
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