Friedmann equations

Alexander Friedman

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 Einstein's field equations of gravitation for the Friedmann-Lemaître-Robertson-Walker metric and a fluid with a given mass density ρ and pressure $p$. The equations are:

$H^{2}\equiv \left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}+{\frac {\Lambda c^{2}}{3}}$

${\dot {H}}+H^{2}\equiv {\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}},$

where $\Lambda$ is the cosmological constant, possibly caused by vacuum energy, $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, $a$ is the scale factor, and $k$ is the normalised spatial curvature when $a=1$ (i.e. today). If the shape of the universe is hyperspherical and $R$ is the radius of curvature ($R_{0}$ in the present-day), then $a=R/R_{0}$. Generally, $k \over a^{2}$ is the spatial curvature. If $k$ is positive, then the universe is hyperspherical. If $k$ is zero, then the universe is flat. If $k$ is negative, then the universe is hyperbolic. Note that $\rho$ and $p$ are in general functions of $a$. The Hubble parameter, $H$, is the rate of expansion of the universe.

These equations are sometimes simplified by redefining

$\rho \rightarrow \rho -{\frac {\Lambda c^{2}}{8\pi G}}$

$p\rightarrow p+{\frac {\Lambda }{8\pi G}}$

to give:

$H^{2}\equiv \left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}$

${\dot {H}}+H^{2}\equiv {\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right).$

The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's 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 Friedmann acceleration equation and reserve the term Friedmann equation for only the first equation.

The density parameter

The density parameter, $\Omega$, is defined as the ratio of the actual (or observed) density $\rho$ to the critical density $\rho _{c}$ of the Friedmann universe. An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:

$\rho _{c}={\frac {3H^{2}}{8\pi G}}.$

The density parameter (useful for comparing different cosmological models) is then defined as:

$\Omega \equiv {\frac {\rho }{\rho _{c}}}={\frac {8\pi G\rho }{3H^{2}}}.$

This term originally was used as a means to determine the spatial geometry of the universe, where $\rho _{c}$ is the critical density for which the spatial geometry is flat (or Euclidian). Assuming a zero vacuum energy density, if $\Omega$ is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If $\Omega$ is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for $\Omega$ in which case this density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the ΛCDM model, there are important components of $\Omega$ due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP satellite to be nearly flat, meaning that the spatial curvature parameter $k$ is zero.

The first Friedmann equation is often seen in a form with density parameters.

${\frac {H^{2}}{H_{0}^{2}}}=\Omega _{R}a^{-4}+\Omega _{M}a^{-3}+\Omega _{k}a^{-2}+\Omega _{\Lambda }.$

Here $\Omega _{R}$ is the radiation density today, $\Omega _{M}$ is the matter (dark plus baryonic) density today, $\Omega _{k}$ is the spatial curvature density today, and $\Omega _{\Lambda }$ is the cosmological constant or vacuum density today.

Useful solutions

The Friedmann equations can be easily solved in presence of a perfect fluid with equation of state (ideal gas law)

$p=w\rho c^{2},$

where $p$ is the pressure, $\rho$ is the mass density of the fluid in the comoving frame and $w$ is some constant. The solution for the scale factor is

$a(t)=a_{0}\,t^{\frac {2}{3(w+1)}}$

where $a_{0}$ is some integration constant to be fixed by the choice of initial conditions. This family of solutions labelled by $w$ is extremely important for cosmology. E.g. $w=0$ describes a matter-dominated universe, where the pressure is negligible with respect to the mass density. From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as

$a(t)\propto t^{2/3}$ matter-dominated

Another important example is the case of a radiation-dominated universe, i.e., when $w=1/3$. This leads to

$a(t)\propto t^{1/2}$ radiation dominated

Rescaled Friedmann equation

Set a=ãa₀, ρ_c=3H₀²/8πG, ρ=ρ_cΩ, $t={\tilde {t}}/H_{0}$, Ω_c=-kc^2/H₀²a₀² where a₀ and H₀ are separately the scale factor and the Hubble parameter today. Then we can have

${\frac {1}{2}}\left({\frac {d{\tilde {a}}}{d{\tilde {t}}}}\right)^{2}+U_{\rm {eff}}({\tilde {a}})={\frac {1}{2}}\Omega _{c}$

where U_eff(ã)=Ωã²/2. For any form of the effective potential U_eff(ã), there is an equation of state p=p(ρ) that will produce it.