Physical Foundations of Cosmology

9.1 Basics 359

Since phase volume is invariant under coordinate transformations,fis a spacetime
scalar. In the absence of particle interactions (scatterings), the particle number
within the conserved phase volume does not change. As a result, the distribution
function obeys the collisionless Boltzmann equation

Df

(

xi(η),pi(η),η

)

Dη

≡

∂f

∂η

+

dxi

dη

∂f

∂xi

+

dpi

dη

∂f

∂pi

= 0 , (9.7)

wheredxi/dηanddpi/dηare the derivatives calculated along the geodesics.

Temperature and its transformation propertiesLet us consider a nearly homoge-
neous isotropic universe filled by slightly perturbed thermal radiation. The fre-
quency (energy) of the photon measured by an observer is equal to the time com-
ponent of the photon’s 4-momentum in thecomoving local inertial frameof the
observer. Therefore in an arbitrary coordinate system where an observer has 4-
velocityuαand the photon 4-momentum ispα,this frequency can be expressed as
ω=pαuα.If the radiation coming to an observer from different directions

li≡−

pi

#p^2 i

has the Planckian spectra, then the distribution function is

f= ̄f

(ω

T

)

≡

2

exp(ω/T(xα,li))− 1

. (9.8)

The effective temperatureT(xα,li) depends not only on the directionlibut also
on the observer’s locationxiand on the moment of timeη. The factor of 2 in the
numerator accounts for the two possible polarizations of the photons.
In a nearly isotropic universe, this temperature can be written as

T(xα,li)=T 0 (η)+δT(xα,li), (9.9)

whereδTT 0 .To understand how the fluctuationsδTdepend on the coordinate
system, let us consider two observersOandO ̃, who are at rest with respect to two
different frames related by the coordinate trasformation ̃xα=xα+ξα.In the rest
frame of each observer, the zeroth component of 4-velocity can be expressed through
the metric by using the relationgαβuαuβ=g 00

(

u^0

) 2

= 1 .Thus we conclude that
the frequency of the same photon differs as measured by different observers. They
are equal to

ω=pαuα=p 0 /

√

g 00 and ̃ω=p ̃ 0 /

√

g ̃ 00

respectively, where the photon’s momentum and metric components in different
frames are related by coordinate transformation laws. Using these laws together