Physical Foundations of Cosmology

8.3 Quantum cosmological perturbations 345

We can use (8.69) to simplify this expression during inflation:

uk(η)−

Ak

4 π(ε+p)^1 /^2

( ̇

H

H^2

)

=Ak

(ε+p)^1 /^2

H^2

. (8.101)

Taking into account that within the time interval (8.99) the ratio

(ε+p)^1 /^2

H^2

is almost constant and comparing (8.98) and (8.101), we obtain

Ak−

i

k^3 /^2

(

H^2

√

cs(ε+p)^1 /^2

)

cskHa

. (8.102)

Substituting (8.98) into (8.96) gives the scale-independent power spectrum

δ 2 (k,t)

4 (ε+p)

cs

(8.103)

for short-wavelength perturbations withk>Ha(t)/cs. Using (8.100) withAkas
given in (8.102), we obtain

δ^2 (k,t)

16

9

(

ε

cs( 1 +p/ε)

)

cskHa

(

1 −

H

a

∫

adt

) 2

(8.104)

for long-wavelength perturbations withHa(t)/cs>k>Hai/cs, whereai≡a(ti).

Problem 8.12Verify that for a massive scalar field of massm the spectrum
δ

(

λph,t

)

as a function of the physical scaleλph∼a(t)/kis given by

δ 

m

√

3 π

⎧

⎪⎨

⎪⎩

(^1 ,λph<H−^1 ,

1 +

ln

(

λphH

)

ln

(

af/a(t)

)

)

, H−^1

a(t)

ai

>λph>H−^1 ,

(8.105)

foraf >a(t)>ai, whereaiandafare the values of the scale factor at the beginning
and at the end of inflation respectively. The evolution of the spectrum (8.105) is
sketched in Figure 8.4.

It follows from (8.104) that in the post-inflationary, radiation-dominated epoch
the resulting power spectrum is

δ^2 

64

81

(

ε

cs( 1 +p/ε)

)

cskHa

. (8.106)

This formula is applicable only on scales corresponding to

(

c−s^1 Ha

)

f>
k>

(

cs−^1 Ha

)

i. This range surely encompasses the observable universe. The