regardless of the value of w. Making the substitutions
ε 0 = ε c,0 =
3c 2
8πG
H 2
0 (5.47)
and
t 0 =
2
3(1 + w)
H −1
0 , (5.48)
equation (5.46) can be written in the form
ε(t) =
1
6π(1 + w) 2
c 2
G
t −2 . (5.49)
Expressed in terms of Planck units (introduced in Chapter 1), this relation
between energy density and cosmic time is
ε(t) =
1
6π(1 + w) 2
E p
` 3
p µ t
t P ¶
−2
.