The age of the universe, in terms of the Hubble time, is then
t 0 =
2
3(1 + w)
H −1
0 . (5.44)
In a spatially flat universe, if w > − 1/3, the universe is younger than the
Hubble time. If w < − 1/3, the universe is older than the Hubble time.
As a function of scale factor, the energy density of a component with
equation-of-state parameter w is
ε(a) = ε 0 a −3(1+w) , (5.45)
so in a spatially flat universe with only a single component, the energy density
as a function of time is (combining equations 5.41 and 5.45)
ε(t) = ε 0 (t/t 0 ) −2 ,