232 Inflation I: homogeneous limit
since the main contribution to the integral comes froma∼ai. At the end of inflation
rp
(
tf)
∼rh(
tf)
, that is, the size of the homogeneous region, originating from a
causal domain, is of order the particle horizon scale.
Thus, instead of considering a homogeneous universe in many causally discon-
nected regions, we can begin with a small homogeneous causal domain which
inflation blows up to a very large size, preserving the homogeneity irrespective of
the conditions outside this domain.Problem 5.2Why does the above consideration fail in a decelerating universe?The next question is whether we can relax the restriction of homogeneity on
the initial conditions. Namely, if we begin with a strongly inhomogeneous causal
domain, can inflation still produce a large homogeneous universe?
The answer to this question is positive. Let us assume that the initial energy
density inhomogeneity is of order unity on scales∼Hi−^1 , that is,
(
δε
ε)
ti∼
1
ε|∇ε|
ai
Hi−^1 =|∇ε|
ε1
a ̇i∼O( 1 ), (5.12)
where∇is the spatial derivative with respect to the comoving coordinates. At
tti, the contribution of this inhomogeneity to the variation of the energy density
within the Hubble scaleH(t)−^1 can be estimated as
(
δε
ε)
t∼
1
ε|∇ε|
a(t)H(t)−^1 ∼O( 1 )
a ̇i
a ̇(t), (5.13)
where we have assumed that|∇ε|/εdoes not change substantially during ex-
pansion. This assumption is supported by the analysis of the behavior of linear
perturbations on scales larger than the curvature scaleH−^1 (see Chapters 7 and
8). It follows from (5.13) that if the universe undergoes a stage of acceleration,
that is,a ̇(t)>a ̇ifort>ti, then the contribution of a large initial inhomogeneity
to the energy variation on the curvature scale disappears. A patch of sizeH−^1 be-
comes more and more homogeneous because the initial inhomogeneity is “kicked
out”: the physical size of the perturbation,∝a, grows faster than the curvature
scale,H−^1 =a/a ̇, while the perturbation amplitude does not change substantially.
Since inhomogeneities are “devalued” within the curvature scale, the name “infla-
tion” fairly captures the physical effect of accelerated expansion. The consideration
above is far from rigorous. However, it gives the flavor of the “no-hair” theorem
for an inflationary stage.
To sum up, inflation demolishes large initial inhomogeneities and produces a
homogeneous, isotropic domain. It follows from (5.13) that if we want to avoid
the situation of a large initial perturbation re-entering the present horizon,∼H 0 −^1 ,
and inducing a large inhomogeneity, we have to assume that the initial expansion