Friedmann equations

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Alexander Friedmann

The Friedmann equations are a set of equations in physical 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 $\!\rho$ and pressure $\!p$. The equations for negative spatial curvature were given by Friedmann in 1924.^[2]

Assumptions

Main article: Friedmann-Lemaître-Robertson-Walker metric

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

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

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:

$\frac{\dot{a}^2 + kc^2}{a^2} = \frac{8 \pi G \rho + \Lambda c^2}{3}$

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

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

which is derived from the trace of Einstein's field equations. $H \equiv \frac{\dot{a}}{a}$ is the Hubble parameter, G, Λ, and c are universal constants (G is Newton's gravitational constant, Λ is the cosmological constant, c is the speed of light in vacuum). k is constant throughout a particular solution, but may vary from one solution to another. a, H, ρ, and p are functions of time. $k \over a^2$ 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 Friedmann 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 (R₀ in the present-day), then a = R_t / R₀. 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 replacing

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

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

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.

Density parameter

The density parameter, Ω, is defined as the ratio of the actual (or observed) density ρ to the critical density ρ_c of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe. In earlier models, which did not include a cosmological constant term, critical density was regarded also as the watershed between an expanding and a contracting Universe.

To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2 atoms per cubic metre.^[4] A much greater density comes from the unidentified dark matter; both ordinary and dark matter contribute in favor of contraction of the universe. However, the largest part comes from so-called dark energy, which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), the dark energy does not lead to contraction of the universe but rather accelerates its expansion. Therefore, the universe will expand forever.

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 ρ_c is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if Ω is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If Ω 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 Ω 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 Ω due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter k is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see. (Similarly, the fact that Earth is approximately flat at the scale of a region does not imply that the Earth is flat: it only implies that it is much larger than this region.)

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 Ω_R is the radiation density today (i.e. when a = 1), Ω_M is the matter (dark plus baryonic) density today, Ω_k = 1 − Ω is the "spatial curvature density" today, and Ω_Λ is the cosmological constant or vacuum density today.

Useful solutions

The Friedmann equations can be solved exactly in presence of a perfect fluid with equation of state

$p=w\rho c^2,\!$

where $\!p$ is the pressure, $\!\rho$ is the mass density of the fluid in the comoving frame and $\!w$ is some constant.

In spatially flat case (k = 0), 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

Note that this solution is not valid for domination of the cosmological constant, which corresponds to an $\!w=-1$. In this case the energy density is constant and the scale factor grows exponentially.

Solutions for other values of k can be found at Tersic, Balsa. "Lecture Notes on Astrophysics". http://nicadd.niu.edu/~bterzic/PHYS652/PHYS652_notes.pdf. Retrieved 20 July 2011. .

Mixtures

If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then

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

holds separately for each such fluid f. In each case,

$\dot{\rho}_{f} = -3 H \left( \rho_{f} + w_{f} \rho_{f} \right) \,$

from which we get

${\rho}_{f} \propto a^{-3 (1 + w_{f})} \,.$

For example, one can form a linear combination of such terms

$\rho = A a^{-3} + B a^{-4} + C a^{0} \,$

where: A is the density of "dust" (ordinary matter, w=0) when a=1; B is the density of radiation (w=1/3) when a=1; and C is the density of "dark energy" (w=−1). One then substitutes this into

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

and solves for a as a function of time.

Rescaled Friedmann equation

Set $\tilde{a}=\frac{a}{a_0}, \;\rho_c=\frac{3H_0^2}{8\pi G},\; \Omega=\frac{\rho}{\rho_c},\; t=\frac{\tilde{t}}{H_0},\; \Omega_c=-\frac{kc^2}{H_0^2 a_0^2}\;$, 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 $U_\textrm{eff}(\tilde{a})=\frac{\Omega\tilde{a}^2}{2}\;$. For any form of the effective potential $U_\textrm{eff}(\tilde{a})\;$, there is an equation of state p = p(ρ) that will produce it.

See also

• Universe
• Mathematics of general relativity
• Solutions of Einstein's field equations

Notes

1. ^ Friedman, A (1922). "Über die Krümmung des Raumes". Z. Phys. 10 (1): 377–386. Bibcode 1922ZPhy...10..377F. doi:10.1007/BF01332580.  (German) (English translation in: Friedman, A (1999). "On the Curvature of Space". General Relativity and Gravitation 31 (12): 1991–2000. Bibcode 1999GReGr..31.1991F. doi:10.1023/A:1026751225741. )
2. ^ Friedmann, A (1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes". Z. Phys. 21 (1): 326–332. Bibcode 1924ZPhy...21..326F. 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 (12): 2001–2008. Bibcode 1999GReGr..31.2001F. doi:10.1023/A:1026755309811. )
3. ^ Ray A d'Inverno, Introducing Einstein's Relativity, ISBN 0-19-859686-3.
4. ^ Rees, M., Just Six Numbers, (2000) Orion Books, London, p. 81, p. 82