Physical Foundations of Cosmology

392 Cosmic microwave background anisotropies

and substituting (9.116) in the Boltzmann equation (9.7), we find that the temper-
ature fluctuations satisfy the equation
(
∂
∂η

+lj

∂

∂xj

)

δT

T

=−

1

2

∂hik

∂η

lilk, (9.118)

which has the obvious solution

δT(l)

T

=−

1

2

∫η^0

ηr

∂hij

∂η

liljdη. (9.119)

Note that until now we have not used the fact thathikis a traceless, divergence-
free tensor. Therefore, (9.119) is a general result, which can also be applied when
calculating the temperature fluctuations induced by the scalar metric perturbations
in the synchronous gauge. For tensor perturbations,hiksatisfies the extra conditions
hii=hik,i=0 (here we raise and lower the indices with the unit tensorδik), which
reduce the number of independent components ofhikto two, corresponding to
two independent polarizations of the gravitational waves. For random Gaussian
fluctuations, the tensor metric perturbations can be written as

hik(x,η)=

∫

hk(η)eik(k)eikx

d^3 k

( 2 π)^3 /^2

, (9.120)

whereeik(k)is the time-independent random polarization tensor. Because of the
conditionseii=eijki= 0 ,eik(k)should satisfy
〈
eik(k)ejl

(

k′

)〉

=

(

PijPkl+PilPkj−PikPjl

)

δ

(

k+k′

)

, (9.121)

where

Pij≡

(

δij−kikj/k^2

)

, (9.122)

is the projection operator. Substituting (9.120) into (9.119) and calculating the
tensor contribution to the correlation function of the temperature fluctuations (see
the definition in (9.33)), we obtain

CT(θ)=

1

4

∫

F(l 1 ,l 2 ,k)h′k(η)h∗′k(η ̃)eik[l^1 (η−η^0 )−l^2 ( ̃η−η^0 )]dηdη ̃

d^3 k

( 2 π)^3

, (9.123)

where cosθ=l 1 ·l 2 .In deriving (9.123) we have averaged over the random polar-
ization with the help of (9.121). The functionFentering this expression does not
depend on time and is equal to

F= 2

(

(l 1 l 2 )−

(l 1 k)(l 2 k)

k^2

) 2

−

(

1 −

(l 1 k)^2

k^2

)(

1 −

(l 2 k)^2

k^2

)

. (9.124)