Friedmann equations relate various cosmological limits within the context of General relativity. Friedmann equations 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. Friedmann equations are:
where? andp are the density and pressure of the fluid,? is the cosmological constant possibly caused by Vacuum energy,G is the gravitational constant,K = 1,0, - 1 according to whether the Shape of the universe is hyperspherical, flat or hyperbolic respectively,a is the scale factor andc is the speed of light. Note that? andp are in general functions ofa. The Hubble parameter,H, is the rate of expansion of the universe.
These equations are sometimes simplified by redefining
The Hubble parameter 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:
This term originally was used as a means to determine the geometry of the field where?c is the critical density for which the geometry is flat. Assuming a zero vacuum energy density, ifO is larger than unity, the geometry is closed. IfO is less than unity, it is open. However, one can also subsume the curvature and vacuum energy terms into a more general expression forO 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 ofO 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 ? is zero.
The first Friedmann Equation is often seen in a form with density parameters.
HereOR is the radiation density today,OM is the matter (dark plus baryonic) density today, andO? is the cosmological constant or vacuum density today.
Rescaled Friedmann equation
Set a=ãa0, ?c=3H02/8p, ?=?cO, , Oc=-?/H02a02 where a0 and H0 are separately the scale factor and the Hubble parameter today. Then we can have
where Ueff(ã)=Oã2/2. For any form of the effective potential Ueff(ã), there is an equation of state p=p(?) that will produce it.