Introduction to Cosmology

Friedmann–Lemaitre Cosmologies 89

and (5.18) that the Universe has always decelerated. It then follows that푎must have
been zero at some time in the past. Whether Friedmann’s equations can in fact be
trusted to that limit is another story which we shall come back to later. The time푡= 0
was sarcastically called theBig Bangby Fred Hoyle, who did not like the idea of an
expanding Universe starting from a singularity, but the name has stuck. Since about
1988 the steady state theory has been abandoned because of the discovery of early
quasars.

Late Einstein–de Sitter Evolution. The conclusions we derived from Equation
(5.35) were true for past times in the limit of small푎. However, the recent evolution
and the future depend on the value of푘and on the value of휆.For푘=0and푘=− 1
the expansion always continues, following Equation (5.38), and a positive value of휆
boosts the expansion further.
In a matter-dominated Einstein–de Sitter universe which is flat and has훺휆=0,
Friedmann’s Equation (5.4) can be integrated to give

푡(푧)=^2

3 퐻 0

( 1 +푧)−^3 ∕^2 , (5.43)

and the present age of the Universe at푧=0 would be

푡 0 =

2

3 퐻 0

. (5.44)

In that case the size of the Universe would be푐푡 0 = 2 ℎ−^1 Gpc. Inserting the value of퐻 0
used in Equation (1.21),퐻 0 = 0 .696kms−^1 Mpc−^1 , one finds

푡 0 = 9 .27 Gyr. (5.45)

This is in obvious conflict with푡 0 as determined from the ages of the oldest known
star in the Galaxy in Equation (1.24), 13. 5 ± 2 .9Gyr. Thus the flat-universe model with
훺휆=0 is in trouble.

Evolution of a Closed Universe. In a closed matter-dominated universe with푘=
+1and휆=0, the curvature term푘푐^2 ∕푎^2 drops with the second power of푎,while,
according to Equation (5.30), the density drops with the third power, so the inequality
[Equation (5.35)] is finally violated. This happens at a scale푎maxsuch that

푎−max^2 =

8 휋퐺휌m

3 푐^2

, (5.46)

and the expansion halts because푎̇=0 in Equation (5.4). Let us call this theturnover
time푡max. At later times the expansion turns into contraction, and the Universe returns
to zero size at time 2푡max. That time is usually called theBig Crunch.For푘=+1 Fried-
mann’s Equation (5.4) then takes the form

d푎

d푡

=

√

8 휋

3

퐺휌m(푎)푎^2 −푐^2.