Physical Foundations of Cosmology

352 Inflation II: origin of the primordial inhomogeneities

λph

δh

H 0 −^1 zeq H 0 −^1

HΛzeq

−1/2

2

ph

ph

∝λ

∝λ

∼ −^1

∼HΛ

Fig. 8.6.

8.5 Self-reproduction of the universe

The amplitude of scalar perturbations takes its maximum value on scales corre-
sponding tokHiai, that is, those which left the horizon at the very beginning of
inflation. For a massive scalar field at the end of inflation, this maximal amplitude
can be estimated from (8.37) and is equal to

δmax ∼mln

(

af/ai

)

∼mφi^2. (8.129)

If the initial valueφiis larger thanm−^1 /^2 , then inhomogeneities on scalesλph∼
Hi−^1 a/ai become very large(δ > 1 )before inflation ends. Therefore, for large
initial values of the scalar field, the initial homogeneity is completely spoiled by
amplified quantum fluctuations on scales exceedingλph

(

tf

)

∼H−^1 exp

(

m−^1

)

.For
realistic values ofmthese scales are enormous. For example, ifm∼ 10 −^6 , they
are larger thanH−^1 exp

(

106

)

and exceed the observable scales,∼H−^1 exp( 70 ),
by many orders of magnitude. On scales smaller thanH−^1 exp

(

m−^1

)

the universe
remains quasi-homogeneous. Thus, if inflation begins atm−^1 >φi>m−^1 /^2 , then,
on one hand, it produces a very homogeneous, isotropic piece of space which is
large enough to encompass the observed universe while, on the other hand, quantum
fluctuations induce a large inhomogeneity on scales much larger than the observable
scale.
Futhermore, ifφi>m−^1 /^2 in one causally connected region, then inflation never
ends but continues eternally somewhere in space. To see why this happens let us
consider a causal domain of sizeH−^1. In a typical Hubble time,tH∼H−^1 ,