From the Einstein equation to the Friedmann equation

2.2 From the Einstein equation to the Friedmann equation

The next task is to write down the Einstein equation,

using Equations (1

) and (2

). In this case one is left with the following two independent equations:

( )2 1-da- = 8πG--ρ − K--+ Λ-, (4 ) a dt 3 a2 3 2 1d--a = − 4πG-(ρ + 3p) + Λ- (5 ) a dt2 3 3

for the three independent functions $a(t)$ , $ρ(t)$ , and $p(t)$ .

Differentiation of Equation (4 ) with respect to $t$ yields

Then eliminating $¨a$ with Equation (5

), one obtains

This can be easily interpreted as the first law of thermodynamics, $dQ = dU − pdV = d(ρa3) − pd(a3) = 0$ , in the present context. Equations (4

) and (7

) are often used as the two independent basic equations for $a(t)$ , instead of Equations (4

) and (5

In either case, however, one needs another independent equation to solve for $a(t)$ . This is usually given by an equation of state of the form $p = p(ρ)$ . In cosmology, the following simple relation is assumed:

While the value of $w$ may in principle change with redshift, it is often assumed that $w$ is independent of time just for simplicity. Then substituting this equation of state into Equation (7

) immediately yields

The non-relativistic matter (or dust), ultra-relativistic matter (or radiation), and the cosmological constant correspond to $w = 0$ , $1∕3$ , and $− 1$ , respectively.

If the Universe consists of different fluid species with $wi$ ( $i = 1,...,N$ ), Equation (9 ) still holds independently as long as the species do not interact with each other. If one denotes the present energy density of the $i$ -th component by $ρi,0$ , then the total energy density of the Universe at the epoch corresponding to the scale factor of $a(t)$ is given by

where the present value of the scale factor, $a0$ , is set to be unity without loss of generality. Thus, Equation (4

) becomes

Note that those components with $wi = − 1$ may be equivalent to the conventional cosmological constant $Λ$ at this level, although they may exhibit spatial variation unlike $Λ$ .

Evaluating Equation (11 ) at the present epoch, one finds

where $H0$ is the Hubble constant at the present epoch. The above equation is usually rewritten as follows:

where the density parameter for the $i$ -th component is defined as

and similarly the dimensionless cosmological constant is

Incidentally Equation (13

) clearly illustrates the Mach principle in the sense that the space curvature is simply determined by the amount of matter components in the Universe. In particular, the flat Universe $(K = 0)$ implies that the sum of the density parameters is unity:

Finally the cosmic expansion is described by

As will be shown below, the present Universe is supposed to be dominated by non-relativistic matter (baryons and collisionless dark matter) and the cosmological constant. So in the present review, we approximate Equation (17

) as

unless otherwise stated.