Age of the universe
9
approximation because. as can be seen from section 1.8. the universe was matterdominated for most of its history. First. rewrite the Friedmann equation (1.34) in
tenns of the value Po of the energy density p today. From (1.50).
~ = (:0)-3
(1.56)
Thus. the Friedmann equation may be written as
• 2
( ~) + ~ = 8TfGN Po Ro.
( 1.57)
Ro
R5
3
R
Next rewrite this in tenns of the present value 00 of the density parameter (1.41):
Po
- (3/S1rGN)H o
2'
( 1.58)
00 -
Then. at t = to. (1.57) gives
k
8TfGN
2
2
2 = - - P o - Ho = Ho(Oo-l)
(1.59)
Ro
3
where the last equality employs (1.58). Thus. the Friedmann equation may be
written as
( R)
• 2
- +
2
=
2 O
Ho (00 - I) OoHo-.
R
(1.60)
Ro
R
This may be rewritten in tenns of the variable
R
(1.61 )
x == Ro
as
x 2 + HJ(Oo - 1) = OoHJx- 1
(1.62)
with solution
1 r
dx'
10
(1.63)
t = Ho
JOo(x H - I) + 1
In particular. today. when R = Ro. x has the value I and the current age of the
universe is
If'
dx
(1.64)
to = Ho 10 JOo(x-1 - 1) + 1
We see that to '" HO-I with the precise value depending on the value of 00. For
example. for an exactly flat universe (which is not consistent with observations)
00 = I and to = jHOl. It is usual to write HOI in the fonn
HO-I::: h- 1 9.78 x 10 9 yr
(1.65)