Physical Foundations of Cosmology

9.2 Sachs–Wolfe effect 361

whereλis an affine parameter along the geodesic. Since the photons have zero
mass, the first integral of these equations ispαpα= 0 .Using this relation, we can
express the time component of the photon 4-momentum in terms of the spatial
components. Up to first order in the metric perturbations, one obtains

p^0 =

1

a^2

(

#p^2 i

) 1 / 2

≡

p

a^2

p 0 =( 1 + 2 )p. (9.14)

Then, from the first equation in (9.13) we can see that

dxi

dη

=

pi

p^0

=

−

1

a^2

( 1 + 2 )pi

p^0

=li( 1 + 2 ), (9.15)

whereli≡−pi/p.Expressingp^0 andpiin terms ofpiand substituting the metric
(9.12) into the second equation (9.13), we obtain

dpα

dη

=

1

2

∂gγδ

∂xα

pγpδ

p^0

= 2 p

∂

∂xα

. (9.16)

Equation for the temperature fluctuationsUsing the geodesic equations (9.15) and
(9.16), the Boltzmann equation (9.7) takes the form

∂f

∂η

+li( 1 + 2 )

∂f

∂xi

+ 2 p

∂

∂xj

∂f

∂pj

= 0. (9.17)

Sincefis the function of the single variable

y≡

ω

T

=

p 0

T

√

g 00



p

T 0 a

(

1 + −

δT

T 0

)

, (9.18)

the Boltzmann equation to zeroth order in the perturbations reduces to

(T 0 a)′= 0 , (9.19)

and to linear order becomes
(
∂
∂η

+li

∂

∂xi

)(

δT

T

+

)

= 2

∂

∂η

. (9.20)

SolutionsThe zeroth order equation (9.19) informs us that the temperature of the
background radiation in a homogeneous universe is inversely proportional to the
scale factor, while (9.20) determines the temperature fluctuations of the microwave
background. In the case of practical interest, the universe is matter-dominated after
recombination and the main mode in is constant. Therefore, the right hand side
of (9.20) vanishes. The operator on the left hand side is just a total time derivative