90 Cosmological Models
Then푡maxis obtained by integrating푡from 0 to푡maxand푎from 0 tomax,푡max=^1
푐∫푎max0d푎(
8 휋퐺
3 푐^2
휌m(푎)푎^2 − 1)− 1 ∕ 2
. (5.47)
To solve the푎integral we need to know the energy density휌m(푎)in terms of the
scale factor, and we need to know푎max. Let us take the mass of the Universe to be푀.
We have already found in Equation (2.43) that the volume of a closed universe with
Robertson–Walker metric is
푉= 2 휋^2 푎^3.Since the energy density in a matter-dominated universe is mostly pressureless
dust,
휌m=푀
푉= 푀
2 휋^2 푎^3
. (5.48)
This agrees perfectly with the result [Equation (5.30)] that the density is inversely
proportional to푎^3. Obviously, the missing proportionality factor in Equation (5.30) is
then푀∕ 2 휋^2. Inserting the density [Equation (5.48)] with푎=푎maxinto Equation (5.46)
we obtain
푎max=4 MG
3 휋푐^2
. (5.49)
We can now complete the integral in Equation (5.47):
푡max= 휋
2 푐푎max=^2 MG
3 푐^3. (5.50)
Although we might not know whether we live in a closed universe, we certainly
know from the ongoing expansion that푡max>푡 0. Using the value for푡 0 from Equa-
tion (1.21) we find a lower limit to the mass of the Universe:
푀>3 푡 0 푐^3
2 퐺
= 1. 30 × 1023 푀⊙. (5.51)
Actually, the total mass inside the present horizon is estimated to be about 10^22 푀⊙.
The dependence of푡maxon훺mcan also be obtained:
푡max=휋훺m
2 퐻 0 (훺m− 1 )^3 ∕^2. (5.52)
The three cases푘=−1, 0,+1with휆=0 are illustrated qualitatively in Figure 5.1.
All models have to be consistent with the scale and rate of expansion today,푎=1and
푎̇ 0 , at time푡 0. Following the curves back in time one notices that they intersect the
time axis at different times. Thus what may be called time푡=0 is more recent in a
flat universe than in an open universe, and in a closed universe it is even more recent.
The Radius of the Universe. The spatial curvature is given by the Ricci scalar푅
introduced in Equation (3.18), and it can be expressed in terms of훺:
푅= 6 퐻^2 (훺− 1 ). (5.53)