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:
for the three independent functions , , and .
Differentiation of Equation (4) with respect to yields
Then eliminating with Equation (5), one obtains
This can be easily interpreted as the first law of thermodynamics, ,
in the present context. Equations (4) and (7) are often used as the two independent basic equations for
, instead of Equations (4) and (5).
In either case, however, one needs another independent equation to solve for . This is usually given
by an equation of state of the form . In cosmology, the following simple relation is assumed:
While the value of may in principle change with redshift, it is often assumed that 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 , , and , respectively.
If the Universe consists of different fluid species with (), 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 -th component by , then the total energy density of the Universe at the epoch
corresponding to the scale factor of is given by
where the present value of the scale factor, , is set to be unity without loss of generality. Thus,
Equation (4) becomes
Note that those components with may be equivalent to the conventional cosmological constant
at this level, although they may exhibit spatial variation unlike .
Evaluating Equation (11) at the present epoch, one finds
where is the Hubble constant at the present epoch. The above equation is usually rewritten as follows:
where the density parameter for the -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
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.