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.