Physical Foundations of Cosmology

364 Cosmic microwave background anisotropies

wherey≡ω/T andp 0 andphave been expressed in terms ofωusing (9.14)
and (9.18). The integral overycan be calculated explicitly and simply gives a
numerical factor that, combined with 4πT 04 ,represents the energy density of the
unperturbed radiation. This expression, describing the gas of photons immedi-
ately after recombination, should continuously match the 0−0 component of the
hydrodynamic energy–momentum tensor which characterizes radiation before re-
combination:T 00 =εγ

(

1 +δγ

)

. The matching condition implies

δγ= 4

∫

δT

T

d^2 l

4 π

. (9.28)

Similarly, one can derive from (9.26) that the other components of the kinetic
energy–momentum tensor are:

T 0 i 4 εγ

∫

δT

T

li

d^2 l

4 π

. (9.29)

Taking the divergence of this expression and comparing it to the divergence of
the hydrodynamical energy–momentum tensor for radiation before recombination,
which is given by (7.117), we get the second matching condition

δ′γ=− 4

∫

li∇i

(

δT

T

)

d^2 l

4 π

, (9.30)

where we have neglected the radiation contribution to the gravitational potential
and therefore set ′(ηr)=0.
It is straightforward to show that to satisfy both (9.28) and (9.30) the spatial
Fourier component of the temperature fluctuations should be related to the energy
density inhomogeneities in radiation as
(
δT
T

)

k

(l,ηr)=

1

4

(

δk+

3 i

k^2

(

kmlm

)

δ′k

)

. (9.31)

Here and throughout this chapter we drop the subscriptγ,keeping in mind that
the notationδis always used for the fractional energy fluctuations in the radiation
component itself. Substituting (9.31) in the Fourier expansion of (9.25), we obtain
the final expression for the temperature fluctuations in the directionl≡(l^1 ,l^2 ,l^3 )
as observed at locationx 0 ≡(x^1 ,x^2 ,x^3 ):

δT

T

(η 0 ,x 0 ,l)=

∫ [(

+

δ

4

)

k

−

3 δ′k

4 k^2

∂

∂η 0

]

ηr

eik·(x^0 +l(ηr−η^0 ))

d^3 k

( 2 π)^3 /^2

, (9.32)

wherek≡|k|,k·l≡kmlmandk·x 0 ≡knx 0 n.Becauseηr/η 0 is less than 1/30,
we can neglectηrin favor ofη 0 in this expression. The first term in square brackets
represents the combined result from the initial inhomogeneities in the radiation