374 Cosmic microwave background anisotropies
scale of radiation inhomogeneities one has to use (9.62) and the finite thickness
leads to the appearance of the overall factor
exp[
− 2 (σkηr)^2]
in the integrand of (9.38) for the multipole momentsCl.
9.7 Anisotropies on small angular scales
For largelor small angular scales, the main contribution toClcomes from pertur-
bations with angular sizeθ∼π/lon today’s sky. The multipole momentl∼ 200
corresponds to the sound horizon scale at recombination. Hence, the perturbations
responsible for the fluctuations withl>200 have wavenumbersk>ηr−^1 , that is,
they entered the horizon before recombination. These perturbations undergo evo-
lution, causing a significant modification of the primordial spectrum.
In Section 7.4.2, we found the transfer function relating the initial spectrum of
gravitational potential fluctuations (^0) kto the spectra of andδγat recombination
in two limiting cases: namely, for the perturbations that entered the horizon well
before equality, (7.143), and perturbations that enter well after equality, (7.134).
However, for realistic values of the cosmological parameters, neither of these limits
applies directly to the most interesting band of multipole moments corresponding
to the first few acoustic peaks. The approximation (7.143) is valid only for modes
which undergo at least one oscillation before equality (kηeq> 2
√
3 π∼10),and
approximation (7.134) is legitimate for modes which enter the horizon after the
radiation density has become negligible compared to the matter density. If (^) mh^275
0. 3 ,then it follows from (9.61) thatzeq/zr4, and the radiation still constitutes
about 20% of the energy density at recombination. Hence, the asymptotics (7.134)
and (7.143) are poor approximations for perturbations which enter the horizonnear
equality. These are precisely the modes responsible for the fluctuations in the region
of first few acoustic peaks. Since the precise positions and shapes of these peaks
provide valuable information about cosmological models, it is worth improving the
approximation for the source function( (^) k+δk/ 4 )rin this region.
9.7.1 Transfer functions
If the speed of sound changes slowly one can use after matter–radiation equality
the WKB solution (7.127), derived in Section 7.4, for the subhorizon modes. Since
the gravitational potential no longer changes significantly at this time, we find that
for a given amplitude of the gravitational potential (^0) kof the primordial spectrum