MODERN COSMOLOGY

Physics of temperature fluctuations 227

guarantees the distribution cannot have any azimuthal directional dependence.
(The azimuthal dependence for vector and tensor perturbations can also be
included in a similar decomposition). The moments of the distribution are defined
as

'(k,μ,t)=

∑∞

l= 0

(−i)l'l(k,t)Pl(μ); (7.1)

sometimes other normalizations are used. Tight coupling implies that'l=0for
l>1. Physically, thel=0 moment corresponds to the photon energy density
perturbation, whilel=1 corresponds to the bulk velocity. During tight coupling,
these two moments must match the baryon density and velocity perturbations.
Any higher moments rapidly decay due to the isotropizing effect of Thomson
scattering; this follows immediately from the photon Boltzmann equation.

7.2.4 Free-streaming

In the other regime, for temperatures significantly lower than 0.5 eV,tsH−^1
and photons on average never scatter again until the present time. This is known
as the ‘free-streaming’ epoch. Since the radiation is no longer tightly coupled
to the electrons, all higher moments in the radiation field develop as the photons
propagate. In a flat background spacetime, the exact solution is simple to derive.
After scattering ceases, the photons evolve according to the Liouville equation

'′+ikμ'= 0 (7.2)

with the trivial solution

'(k,μ,η)=e−ikμ(η−η∗)'(k,μ,η∗), (7.3)

where we have converted to conformal time defined by dη=dt/a(t)andη∗
corresponds to the time at which free-streaming begins. Taking moments of both
sides results in

'l(k,η)=( 2 l+ 1 )[' 0 (k,η∗)jl(kη−kη∗)+' 1 (k,η∗)jl′(kη−kη∗)] (7.4)

withjla spherical Bessel function. The process of free-streaming essentially
maps spatial variations in the photon distribution at the last-scattering surface
(wavenumberk) into angular variations on the sky today (momentl).

7.2.5 Diffusion damping

In the intermediate regime during recombination,tsH−^1. Photons propagate
a characteristic distance LDduring this time. Since some scattering is still
occurring, baryons experience a drag from the photons as long as the ionization
fraction is appreciable. A second-order perturbation analysis shows that the result