Physical Foundations of Cosmology

360 Cosmic microwave background anisotropies

with the relationpαpα=0weget

ω ̃=ω

(

1 +

∂ξi

∂η

li

)

, (9.10)

where we have kept only the first order terms inξαand in the metric perturbations
around the homogeneous isotropic universe. Since the distribution function (9.8) is
a scalar,

ω/T(xα)=ω/ ̃ T ̃( ̃xα),

and hence the temperature fluctuations measured by two observers are related as

δ ̃T=δT−T 0 ′ξ^0 +T 0 ∂ξ

i

∂η

li. (9.11)

We can see from this expression that the monopole (li-independent) and dipole
(proportional toli) components of the temperature fluctuations depend on the par-
ticular coordinate system in which an observer is at rest. If we can only observe the
radiation from one vantage point, then the monopole term can always be removed
by redefinition of the background temperature. The dipole component depends on
the motion of the observer with respect to the “preferred frame” determined by
the background radiation. For these reasons, neither the monopole nor the dipole
components are very informative about the initial fluctuations. We have to look to
the quadrupole and higher-order multipoles instead, which do not depend on the
motion of the particular observer and coordinate system we use to calculate them.

9.2 Sachs–Wolfe effect

In this section we will solve the Boltzmann equation for freely propagating radiation
in a flat universe using the conformal Newtonian coordinate system where the metric
takes the form

ds^2 =a^2

{

( 1 + 2 )dη^2 −( 1 − 2 )δikdxidxk

}

. (9.12)

Here 1 is the gravitational potential of the scalar metric perturbations. We
discount gravitational waves for the moment and consider them later.

GeodesicsThe geodesic equations describing the propagation of the radiation in
an arbitrary curved spacetime can be rewritten as (see Problem 2.13)

dxα

dλ

=pα,

dpα

dλ

=

1

2

∂gγδ

∂xα

pγpδ, (9.13)