Friedmann equations

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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^[1] 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 for negative spatial curvature were given by Friedmann in 1924.^[2]

1 Assumptions
2 The equations
3 The density parameter
4 Useful solutions
5 Rescaled Friedmann equation
6 See also
7 References

Assumptions

Main article: Friedmann-Lemaître-Robertson-Walker metric

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

$ds^2 = {a(t)}^2 ds_3^2 - dt^2$

where $ds_3^2$ 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 $k$ 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", $a(t)$ .

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.

The equations

There are two independent Friedmann equations for modeling a homogeneous, isotropic universe. They are:

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

which is derived from the 00 component of Einstein's field equations, and

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

which is derived from the trace of Einstein's field equations. G, Λ, and c are universal constants. k is constant throughout a particular solution, but may vary from one solution to another. a, H, ρ, and p are functions of time. Where $H \equiv \frac{\dot{a}}{a}$ , the Hubble parameter, is the rate of expansion of the universe. $\Lambda$ is the cosmological constant. $G$ is Newton's gravitational constant. $c$ is the speed of light in vacuum. $k \over a^2$ is the spatial curvature in any time-slice of the universe; it is equal to one-sixth of the spatial Ricci curvature scalar R since $R = \frac{6}{a^2}(\ddot{a} a + \dot{a}^2 + kc^2)$ in the Friedman model. There are two commonly used choices for $a$ and $k$ which describe the same physics:

$k$ = +1, 0 or -1 depending on whether the shape of the universe is a closed 3-sphere, flat (i.e. Euclidean space) or an open 3-hyperboloid, respectively.^[3] If k = +1, then $a$ is the radius of curvature of the universe. If k = 0, then a may be fixed to any arbitrary positive number at one particular time. If k = -1, then (loosely speaking) one can say that i·a is the radius of curvature of the universe.
$a$ is the scale factor which is taken to be 1 at the present time. $k$ is the spatial curvature when $a = 1$ (i.e. today). If the shape of the universe is hyperspherical and $R_t$ is the radius of curvature ( $R_0$ in the present-day), then $a = R_t/R_0$ . 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.

Using the first equation, the second equation can be re-expressed as

$\dot{\rho} = -3 H \left(\rho + \frac{p}{c^2}\right),$

which eliminates $\Lambda \!$ and expresses the conservation of mass-energy.

These equations are sometimes simplified by redefining

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

$p \rightarrow p - \frac{\Lambda c^4}{8 \pi G}$

to give:

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

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

And the simplified form of the second equation is invariant under this transformation.

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:

$\rho_c = \frac{3 H^2}{8 \pi G}.$

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

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

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 (i.e. when $a=1$ ), $\Omega_M$ is the matter (dark plus baryonic) density today, $\Omega_k = 1 - \Omega$ 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=-kc²/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.

References

↑ Friedman, A (1922). "Über die Krümmung des Raumes". Z. Phys. 10: 377–386. doi:10.1007/BF01332580. (German) (English translation in: Friedman, A (1999). "On the Curvature of Space". General Relativity and Gravitation 31: 1991–2000. doi:10.1023/A:1026751225741. http://adsabs.harvard.edu/abs/1999GReGr..31.1991F. )
↑ Friedmann, A (1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes". Z. Phys. 21: 326–332. doi:10.1007/BF01328280. (German) (English translation in: Friedmann, A (1999). "On the Possibility of a World with Constant Negative Curvature of Space". General Relativity and Gravitation 31: 2001–2008. doi:10.1023/A:1026755309811. http://adsabs.harvard.edu/abs/1999GReGr..31.2001F. )
↑ Ray A d'Inverno, Introducing Einstein's Relativity, ISBN 0198596863.