Physical Foundations of Cosmology

8.4 Gravitational waves from inflation 349

Rewritten in terms of the new variable

vk=

√

eijeij

32 π

ahk, (8.114)

the action becomes

S=

1

2

∫(

vk′v−′k−

(

k^2 −

a′′

a

)

vkv−k

)

dηd^3 k. (8.115)

It describes a real scalar field in terms of its Fourier components. The resulting
equations of motion are

v′′k+ω^2 k(η)vk= 0 ,ω^2 k(η)≡k^2 −a′′/a. (8.116)

There is no need to repeat the quantization procedure for this case. Taking into
account (8.114) and (8.112), we immediately find the correlation function

〈 0 |hij(η,x)hij(η,y)| 0 〉=

8

πa^2

∫

|vk|^2 k^3

sinkr

kr

dk

k

, (8.117)

wherevkis the solution of (8.116) with initial conditions

vk(ηi)=

1

√

ωk

,vk′(ηi)=i

√

ωk. (8.118)

These initial conditions make sense only ifωk>0, that is, for gravitational waves
withk^2 >

(

a′′/a

)

ηi. The power spectrum, characterizing the strength of a gravita-
tional wave with comoving wavenumberk, is correspondingly

δ^2 h(k,η)=

8 |vk|^2 k^3

πa^2

. (8.119)

InflationIn contrast to scalar perturbations, the deviation of the equation of state
from the vacuum equation of state is not so crucial to the evolution of gravitational
waves. Therefore, we first consider a pure de Sitter universe wherea=−(Hη)−^1.
In this case (8.116) simplifies to

v′′k+

(

k^2 −

2

η^2

)

vk= 0 (8.120)

and has the exact solution

vk(η)=

1

η

{C 1 [kηcos(kη)−sin(kη)]+C 2 [kηsin(kη)+cos(kη)]}. (8.121)

Let us consider gravitational waves withk|ηi|1 for whichωkk. Taking
into account the initial conditions in (8.118), we can determine the constants of