Introduction to Cosmology

Friedmann–Lemaitre Cosmologies 91

R(t) de Sitter

H constant

k = – 1

k = 0

future

0

• 1

+ 1

tmax

t

tH

to

past

k =

(^23) tH
Figure 5.1Time dependence of the cosmic scale푅(푡)=푎(푡)in various scenarios, all of which
correspond to the same constant slope퐻=퐻 0 at the present time푡 0 .푘=+1: a closed universe
with a total lifetime 2푡max. It started more recently than a flat universe would have.푘=0: a flat
universe which started^23 푡Hago.푘=−1: an open universe which started at a time^23 푡H<푡<푡H
before the present time. de Sitter: an exponential (inflationary) scenario corresponding to a
large cosmological constant. This is also called the Lemaitre cosmology.
Obviously,푅vanishes in a flat universe, and it is only meaningful when it is non-
negative, as in a closed universe. It is conventional to define a ‘radius of curvature’
that is also valid for open universes:

푟U≡

√

6

푅

=^1

퐻

√

|훺− 1 |

. (5.54)

For a closed universe,푟Uhas the physical meaning of the radius of a sphere.

Late Friedmann–Lemaitre Evolution. When휆>0, the recent past and the future
take an entirely different course (we do not consider the case휆<0, which is of mathe-
matical interest only). Since휌휆and훺휆are then scale-independent constants, they will
start to dominate over the matter term and the radiation term when the expansion has
reached a given scale. Friedmann’s Equation (5.18) can then be written

2 푎̈

푎

= 3 퐻 02 훺휆.

From this one sees that the expansion will accelerate regardless of the value of푘.In
particular, a closed universe with푘=+1 will ultimately not contract, but expand at
an accelerating pace.