Introduction to Cosmology

Friedmann–Lemaitre Cosmologies 87

of light, but freely propagating light did not exist at times when푧≳1080, so푧is then
no longer a true observable. Secondly, it is possible to describe the future in terms of
푎>1, but redshift is then not meaningful.
The value of the proportionality factor푤in Equations (5.28) and (5.29) follows
from the adiabaticity condition. Leaving the derivation of푤for a later discussion, we
shall anticipate here its value in three special cases of great importance.

Case I. Amatter-dominateduniverse filled with nonrelativistic cold matter in the

form of pressureless nonradiating dust for which푝=0. From Equation 5.28

then, this corresponds to푤=0, and the density evolves according to

휌m(푎)∝푎−^3 =( 1 +푧)^3. (5.30)

It follows that the evolution of the density parameter훺mis

훺m(푎)=훺m

퐻 02

퐻^2

푎−^3.

Solving for퐻^2 푎^2 훺and inserting it into Equation (5.13), one finds the evo-

lution of the Hubble parameter:

퐻(푎)=퐻 0 푎−^1

√

1 −훺m+훺m푎−^1 =퐻 0 ( 1 +푧)

√

1 +훺m푧. (5.31)

Case II. Aradiation-dominateduniverse filled with an ultra-relativistic hot gas com-

posed of elastically scattering particles of energy density 휀. Statistical

mechanics then tells us that the equation of state is

푝r=

1

3

휀=^1

3

휌r푐^2. (5.32)

This evidently corresponds to푤=^13 , so that the radiation density evolves

according to

휌r(푎)∝푎−^4 =( 1 +푧)^4. (5.33)

Case III. Thevacuum-energystate corresponds to a flat, static universe (푎̈=0,푎̇=
0) without dust or radiation, but with a cosmological term. From Equa-
tions (5.17) and (5.18) we then obtain

푝휆=−휌휆푐^2 ,푤=− 1. (5.34)

Thus the pressure of the vacuum energy is negative, in agreement with

the definition in Equation (5.19) of the vacuum-energy density as a nega-

tive quantity. In the equation of state [Equation (5.28)],휌휆and푝휆are then

scale-independent constants.

Early Time Dependence. It follows from the above scale dependences that the
curvature term in Equation (5.17) obeys the following inequality in the limit of
small푎:

푘푐^2

푎^2

≪

8 휋퐺

3

휌+

휆

3

. (5.35)