Physical Foundations of Cosmology

9.3 Initial conditions 363
Unlike Silk damping, free-streaming causes the spatial variation (the x-
dependence) of the photon distribution to decrease as a power law rather than
exponentially withk. Furthermore, free-streaming does not make the distribution
function isotropic. Although the spatial variation (thex-dependence) of the photon
distribution is damped, the initial angular anisotropy (thel-dependence) is pre-
served. Note that the perturbations on superhorizon scales are unaffected because
the photons have no chance to mix.

9.3 Initial conditions

As follows from (9.15), the photons arriving from directionliseen at the present
timeη 0 by an observer located atx 0 ipropagate along geodesics

xi(η)x 0 i+li(η−η 0 ). (9.24)

Therefore, from (9.21), we find thatδT/T in the directionlion the sky today is
equal to

δT

T

(η 0 ,x 0 i,li)=

δT

T

(ηr,xi(ηr),li)+ (ηr,xi(ηr))−

(

η 0 ,x 0 i

)

, (9.25)

whereηris the conformal time of recombination andxi(ηr) is given by (9.24). Since
we can observe from only one vantage point in the universe, we are only interested in
theli-dependence of the temperature fluctuations. Hence the last term, which only
contributes to the monopole component, can be ignored. The angular dependence
of(δT/T) 0 is set by two contributions: (a) the “initial” temperature fluctuations
on the last scattering surface; and (b) the value of the gravitational potential at
this same location. The first contribution,(δT/T)r, can be expressed in terms of
the gravitational potential and the fluctuations of the photon energy densityδγ≡
δεγ/εγon the last scattering surface. For this purpose, we use matching conditions
for thehydrodynamicenergy–momentum tensor, which describes the radiation
before decoupling, and thekineticenergy–momentum tensor, which characterizes
the gas of free photons after decoupling,

Tβα=

1

√

−g

∫

f

pαpβ

p^0

d^3 p. (9.26)

Substituting the metric (9.12) into (9.26) and assuming a Planckian distribution
(9.8), we get an expression to linear order for the 0−0 component of the kinetic
energy–momentum tensor

T 00 

1

a^4 ( 1 − 2 )

∫

̄f

(ω

T

)

p 0 d^3 p(T 0 )^4

∫(

1 + 4

δT

T 0

)

̄f(y)y^3 dyd^2 l,

(9.27)