MODERN COSMOLOGY

Cosmic background fluctuations 93

where

DLS=

2 c


^1 m/^2 H 0

( 1 +zLS)−^1 /^2 = 184 (
h^2 )−^1 /^2 Mpc

is the comoving horizon size at last scattering (a result that is independent of
whether there is a cosmological constant).
We can see immediately from these expressions the relative importance
of the various effects on different scales. The Sachs–Wolfe effect dominates
for wavelengths& 1 h−^1 Gpc; Doppler effects then take over but are almost
immediately dominated by adiabatic effects on the smallest scales.
These expressions apply to perturbations for which only gravity has been
important up until last scattering, i.e. those larger than the horizon atzeq.For
smaller wavelengths, a variety of additional physical processes act on the radiation
perturbations, generally reducing the predicted anisotropies. An accurate
treatment of these effects is not really possible without a more complicated
analysis, as is easily seen by considering the thickness of the last-scattering
shell,σr = 7 (
h^2 )−^1 /^2 Mpc. This clearly has to be of the same order of
magnitude as the photon mean free path at this time; on any smaller scales, a
fluid approximation for the radiation is inadequate and a proper solution of the
Boltzmann equation is needed. Nevertheless, some qualitative insight into the
small-scale processes is possible. The radiation fluctuations will be damped
relative to the baryon fluid by photon diffusion, characterized by the Silk-
damping scale,λS = 2. 7 (

Bh^6 )−^1 /^4 Mpc. Below the horizon scale atzeq,
16 (
h^2 )−^1 Mpc, there is also the possibility that dark-matter perturbations can
grow while the baryon fluid is still held back by radiation pressure, which results
in adiabatic radiation fluctuations that are less than would be predicted from the
dark-matter spectrum alone. In principle, this suggests a suppression factor of
( 1 +zeq)/( 1 +zLS)or roughly a factor 10. In detail, the effect is an oscillating
function of scale, since we have seen that baryonic perturbations oscillate as sound
waves when they come inside the horizon:

δb∝( 3 cS)^1 /^4 exp

(

±i

∫

kcSdτ

)

;

here,τstands for conformal time. There is thus an oscillating signal in the CMB,
depending on the exact phase of these waves at the time of last scattering. These
oscillations in the fluid of baryons plus radiation cause a set of acoustic peaks in
the small-scale power spectrum of the CMB fluctuations (see later).

2.8.4 Large-scale fluctuations and CMB power spectrum

The flat-space formalism becomes inadequate for very large angles; the proper
basis functions to use are the spherical harmonics:

δT

T

(qˆ)=

∑

am"Y"m(qˆ),