Introduction to Cosmology

152 Cosmic Inflation

7.1 Paradoxes of the Expansion

Particle Horizons. Recall the definition of the particle horizon, Equation (2.48),
whichinaspatiallyflatmetricis

휒ph=휎ph=푐

∫

푡 0

푡min

d푡

푎(푡)

=푐∫

0

푎min

d푎

푎̇푎

. (7.1)

This was illustrated in Figure 2.1. In expanding Friedmann models, the particle hori-
zon is finite. Let us go back to the derivation of the time dependence of the scale factor
푎(푡)in Equations (5.39)–(5.41). At very early times, the mass density term in the Fried-
mann Equation (5.4) dominates over the curvature term (we have also called it the
vacuum-energy term),

푘푐^2

푎^2

≪

8 휋퐺

3

휌. (7.2)

This permits us to drop the curvature term and solve for the Hubble parameter,

푎̇

푎

=퐻(푡)=

(

8 휋퐺

3

휌

) 1 ∕ 2

. (7.3)

Substituting this relation into Equation (7.1) we obtain

휎ph=푐

∫

푎 0

푎min

d푎

푎^2 (푎̇∕푎)

=

(

3 푐^2

8 휋퐺

) 1 ∕ 2

∫

푎 0

푎min

d푎

푎^2

√

휌

. (7.4)

In a radiation-dominated Universe,휌scales like푎−^4 , so the integral on the right con-
verges in the lower limit푎min=0, and the result is that the particle horizon is finite:

휎ph∝

∫

푎 0

0

d푎

푎^2 푎−^2

=푎 0. (7.5)

Similarly, in a matter-dominated Universe,휌scales like푎−^3 , so the integral also con-
verges, now yielding

√

푎 0. Note that an observer living at a time푡 1 <푡 0 would see a

smaller particle horizon,푎 1 <푎 0 , in a radiation-dominated Universe or

√

푅 1 <

√

푅 0 in
a matter-dominated Universe.
Suppose however, that the curvature term or a cosmological constant dominates
the Friedmann equation at some epoch. Then the conditions in Equations (5.35)
and (5.36) are not fulfilled; on the contrary, we have a negative net pressure

푝<−^1

3

휌푐^2. (7.6)

Substituting this into the law of energy conservation [Equation (5.24)] we find

휌<̇ − 2

푅̇

푅

휌. (7.7)

This can easily be integrated to give the푅dependence of휌,

휌<푅−^2. (7.8)