2.2 From the Einstein equation to the Friedmann equation

The next task is to write down the Einstein equation,
1 R μν − -Rg μν + Λgμν = 8πGT μν, (3 ) 2
using Equations (1View Equation) and (2View Equation). 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 (4View Equation) with respect to t yields

¨a 4πG ( a ) Λ --= ----- ρ˙--+ 2ρ + --. (6 ) a 3 ˙a 3
Then eliminating ¨a with Equation (5View Equation), one obtains
˙ρ = − 3a˙(ρ + p ). (7 ) a
This can be easily interpreted as the first law of thermodynamics, dQ = dU − pdV = d(ρa3) − pd(a3) = 0, in the present context. Equations (4View Equation) and (7View Equation) are often used as the two independent basic equations for a(t), instead of Equations (4View Equation) and (5View Equation).

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:

p = w ρ. (8 )
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 (7View Equation) immediately yields
ρ ∝ a−3(1+w). (9 )
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 (9View Equation) 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

∑N ρ = --ρi,0--, (10 ) i=1 a3(1+wi)
where the present value of the scale factor, a0, is set to be unity without loss of generality. Thus, Equation (4View Equation) becomes
( 1da )2 8πG ∑N ρ K Λ ---- = ----- ----i,0--− ---+ --. (11 ) adt 3 i=1 a3(1+wi) a2 3
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 (11View Equation) at the present epoch, one finds

∑N H2 = 8πG-- ρi,0 − K + Λ-, (12 ) 0 3 i=1 3
where H0 is the Hubble constant at the present epoch. The above equation is usually rewritten as follows:
( ) ∑N K ≡ H20ΩK = H20 Ωi,0 + ΩΛ − 1 , (13 ) i=1
where the density parameter for the i-th component is defined as
Ωi,0 ≡ 8πG--ρi,0, (14 ) 3H20
and similarly the dimensionless cosmological constant is
-Λ--- ΩΛ ≡ 3H2 . (15 ) 0
Incidentally Equation (13View Equation) 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:
N ∑ Ωi,0 + Ω Λ = 1. (16 ) i=1

Finally the cosmic expansion is described by

( )2 ( ∑N ∑N ) da- = H2 -Ωi,0--+ 1 − Ωi,0 − Ω Λ + Ω Λa2 . (17 ) dt 0 i=1 a1+3wi i=1
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 (17View Equation) as
( ) ( ) da 2 2 Ωm 2 dt- = H 0 -a--+ 1 − Ωm − Ω Λ + Ω Λa (18 )
unless otherwise stated.
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