MODERN COSMOLOGY

92 An introduction to the physics of cosmology

Doppler source term.The effect here is just the Doppler effect from the
scattering of photons by moving plasma:
δT
T

=

δv·ˆr

c

.

Using the standard expression for the linear peculiar velocity, the

correspondingk-space result is

Tk=−i

√

( 1 +zLS)

(

H 0

c

)

δk(zLS)

k

kˆ·ˆr.

Adiabaticsourceterm.This is the simplest of the three effects mentioned
earlier:
Tk=

δk(zLS)

3

,

becauseδnγ/nγ=δρ/ρandnγ∝T^3. However, this simplicity conceals a

paradox. Last scattering occurs only when the universe recombines, which

occurs at roughly a fixed temperature: kT∼χ, the ionization potential

of hydrogen. Surely, then, we should just be looking back to a surface

of constant temperature? Hot and cold spots should normalize themselves

away, so that the last-scattering sphere appears uniform. The solution is that

a denser spot recombineslater: it is therefore less redshifted and appears

hotter. In algebraic terms, the observed temperature perturbation is

(

δT

T

)

obs

=−

δz

1 +z

=

δρ

ρ

,

where the last expression assumes linear growth,δ ∝( 1 +z)−^1. Thus,

even though a more correct picture for the temperature anisotropies seen

on the sky is of a crinkled surface at constant temperature, thinking of hot

and cold spots gives the right answer. Any observable cross-talk between

density perturbations and delayed recombination is confined to effects of

order higher than linear.

We now draw these results together to form the spatial power spectrum of
CMB fluctuations in terms of the power spectrum of mass fluctuations at last
scattering:
T3D^2 =[(fA+fSW)^2 (k)+fV^2 (k)μ^2 ]^2 k(zLS),

whereμ≡kˆ·ˆr. The dimensionless factors can be written most simply as

fSW=−

2

(kDLS)^2

fV=

2

kDLS

fA= 1 / 3 ,