Physical Foundations of Cosmology

350 Inflation II: origin of the primordial inhomogeneities

π

λph

ph

HΛ−^1

− 1

HΛ

HΛ−^1

ηi

η 1

η=η 1 η=ηf

η=ηi

HΛ−^1

ηi

ηf

8

∝λ

δh

Fig. 8.5.

integrationC 1 andC 2 and the solution becomes

vk(η)=

1

√

k

(

1 +

i

kη

)

exp(ik(η−ηi)). (8.122)

Substituting this into (8.119), we obtain

δ^2 h=

8 H^2

π

[

1 +(kη)^2

]

=

8 H^2

π

[

1 +

(

kph

H

) 2 ]

, (8.123)

wherekph≡k/a is the physical wavelength. This formula is applicable only
forkphH(η/ηi). The amplitudeδhas a function of the physical wavelength
λph∼k−ph^1 is sketched in Figure 8.5. Long-wavelength gravitational waves with
H−^1 (ηi/η)>λph>H−^1 have a flat spectrum with amplitude proportional toH.
The above consideration refers to a pure de Sitter universe whereHis exactly
constant. In realistic inflationary models the Hubble constant slowly changes with
time. Recalling that the nondecaying mode of a gravitational wave is frozen on
supercurvature scales (see Section 7.3.2), we obtain

δ^2 h

8 Hk^2 Ha

π

=

64

3

εkHa. (8.124)

The tensor spectral index is then equal to

nT≡

dlnδ^2 h

dlnk

− 3

(

1 +

p

ε

)

kHa

, (8.125)