Scale factor dependence of the energy density
7
where m p is the Planck mass. and
Mp:::: 2.44 x 10 18 GeV
mp :::: 1.22 x 10 19 GeV.
(1.40)
Since the Hubble parameter varies with time. so does pc. The density parameter
n is defined as
n= £..
( 1.41)
Pc
and measures the density as a fraction of the 'critical' density Pc. The current
value of n, denoted by no. has a value [I]
no = 1.02 ± 0.02.
(1.42)
1.4 Scale factor dependence of the energy density
There is also conservation of the energy-momentum tensor to take into account:
DIITILII = 0
(1.43)
where
D).. VI' = a).. VI' + rr p v P
(1.44)
is the action of the covariant derivative D).. on a contravariant index. The J.L = 0
component of (1.43) yields (exercise 3)
.
R
P + 3(p + p)"R = o.
(1.45)
It is easy to see that this is just the first law of thermodynamics
dE+pdV=O
(1.46)
for a comoving volume V ex R 3 (t).
The energy density p may be related to the scale factor R(t) once we have
the equation of state. If this is of the form
p=wp
(1.47)
then ( 1.45) leads to
p ex R-3(l+w).
(1.48)
In particular. for w = ~. corresponding to radiation (massless matter)
p ex R- 4
_1
radiation
P - 'JP·
(1.49)
For w = O. corresponding to massive matter,
p ex R- 3
matter
p=O.
(1.50)