Physical Foundations of Cosmology

9.9 Gravitational waves 393

Introducing the new variablex≡k(η 0 −η)instead ofη,and noting that

(l 1 k)

k

=−

∂

i∂x

e−i

kl 1

kx, (l^2 k)

k

=

∂

i∂x ̃

ei

kl 2

k ̃x,

after integration over the angular part ofkwe can rewrite (9.123) as

CT(θ)=

1

4

∫

∂hk

∂x

∂h∗k

∂x ̃

[

Fˆ·sin(|l^2 x ̃−l^1 x|)

|l 2 x ̃−l 1 x|

]

dxdx ̃

k^2 dk

2 π^2

, (9.125)

where

Fˆ= 2

(

cosθ−

∂

∂x

∂

∂x ̃

) 2

−

(

1 +

∂^2

∂x^2

)(

1 +

∂^2

∂x ̃^2

)

. (9.126)

Now we can use formula (9.36) to expandCT(θ)as a discrete sum over multipole
momenta (see (9.37)). After a lengthy but straightforward calculation, the result for
ClTcan be written in a rather simple form.

Problem 9.15Substitute (9.36) into (9.125) and use the recurrence relations for
zP(z), the Bessel functions equation and the recurrence relations for the spherical
Bessel functions to expressjl′′,jl− 2 ,jl′− 1 etc. throughjl,jl′,to verify that

ClT=

(l−1)l(l+1)(l+2)

2 π

∫∞

0

∣

∣

∣∣

∣∣

k(η∫ 0 −ηr)

0

∂hk

∂x

jl(x)

x^2

dx

∣

∣

∣∣

∣∣

2

k^2 dk. (9.127)

The derivative of the metric perturbations takes its maximal value atkη∼O( 1 )
and drops very fast after that.Hence, for those gravitational waves which entered
the horizon, the main contribution to the integral overxin (9.127) comes from the
relatively narrow region:kη 0 >x>kη 0 −O( 1 ).Forl1 andkη 0  1 ,the func-
tionjl(x)/x^2 does not change significantly within this interval and can therefore
be approximated by its value atx 0 =kη 0 .As a result, (9.127) simplifies forl 1
to

ClT

(l−1)l(l+1)(l+2)

2 π

∫∞

0

∣∣

h^2 k(ηr)k^3

∣∣jl^2 (x 0 )

x 05

dx 0 , (9.128)

where

∣

∣h^2 k(ηr)k^3

∣

∣should be expressed as a function ofx 0 =kη 0 .The gravity waves

generated during inflation, which enter the horizon after recombination but well
before the present time, have a nearly flat spectrum atη=ηr, so that
∣∣
h^2 k(ηr)k^3

∣∣

=Bgw≈const (9.129)

forη−r^1 >k>η− 01 .Taking into account that forl1 the main contribution to
(9.128) comes from the perturbations withk∼l/η 0 , and substituting (9.129) into