Physical Foundations of Cosmology

344 Inflation II: origin of the primordial inhomogeneities

the following expansion for the operator ˆ:

ˆ(η,x)=^4 π(ε+p)

1 / 2

√

2

∫ [

u∗k(η)eikxaˆk−+uk(η)e−ikxaˆk+

] d^3 k

( 2 π)^3 /^2

, (8.94)

where the mode functionsuk(η)obey (8.59) and are related to the mode functions
vk(η)via (8.57). For the initial vacuum state| 0 〉the correlation function atη>ηi
is

〈 0 | ˆ(η,x) ˆ(η,y)| 0 〉=

∫

4 (ε+p)|uk|^2 k^3

sinkr

kr

dk

k

, (8.95)

wherer≡|x−y|. According to the definition of the power spectrum in (8.8) and
(8.9), we have

δ^2 (k,η)= 4 (ε+p)|uk(η)|^2 k^3. (8.96)

Givenvk(ηi)andvk′(ηi), the initial conditions forukcan be inferred from (8.57).
Let us consider a short-wavelength perturbation withc^2 sk^2 

(

z′′/z

)

i for which
ωk(ηi)csk. In this case the initial conditions (8.92) can be rewritten in terms of
ukas

uk(ηi)−

i

√

csk^3 /^2

, u′k(ηi)

√

cs

k^1 /^2

, (8.97)

where we have neglected higher-order terms, which are suppressed by powers
of(cskηi)−^1 1. The corresponding short-wavelength WKB solution, valid for
c^2 sk^2 

∣

∣θ′′/θ

∣

∣,is

uk(η)−

i

√

csk^3 /^2

exp

⎛

⎝ik

∫η

ηi

csdη ̃

⎞

⎠. (8.98)

During inflation the ratio|θ′′/θ|can be estimated roughly asη−^2 |H ̇/H^2 |. Because
|H ̇/H^2 |1, (8.98) is still applicable within the short time interval

1

csk

>|η|>

1

k

∣

∣H ̇/H^2

∣

∣^1 /^2 (8.99)

after the sound horizon crossing. At this time the argument in the exponent is almost
constant andukfreezes out. After a perturbation enters the long-wavelength regime
the time evolution of the gravitational potential is described by (8.67), and hence

uk(η)≡

4 π(ε+p)^1 /^2

=

Ak

4 π(ε+p)^1 /^2

(

1 −

H

a

∫

adt

)

. (8.100)