7.4 Baryon–radiation plasma and cold dark matter 311the anisotropy of the cosmic microwave background and to determine the transfer
function relating the primordial spectrum of density inhomogeneities created during
inflation to the spectrum after matter–radiation equality. As a prelude to analysis of
the microwave background anisotropies, we consider in this section the calculation
of the gravitational potential and the radiation fluctuations at recombination.
Before recombination, baryons are strongly coupled to radiation and the baryon–
radiation component can be treated in the hydrodynamical approximation as a
singleimperfectfluid. The other component consists of heavy cold particles, which
we assume interact only gravitationally with the baryon–photon plasma and are
otherwise free to move with respect to it.
We assume that the number of photons per cold dark matter particle is initially
spatially uniform on supercurvature scales (wavelengths greater thanH−^1 ) but that
the matter and radiation densities vary in space. In other words, we consideradia-
baticperturbations. As the universe expands and the inhomogeneity scale becomes
smaller than the curvature scale, the components move with respect to one another
and the entropy (number of photons) percold dark matter particlevaries spatially.
In contrast, the entropy perbaryonremains spatially uniform on all scales until the
baryons decouple from the radiation.
7.4.1 Equations
Because the baryon–radiation and cold dark matter components interact only gravi-
tationally, their energy–momentum tensors satisfy the conservation laws,Tβα;α= 0 ,
separately. Thecoldparticles have negligible relative velocities and hence the cold
dark matter can be described as a dust-like perfect fluid with zero pressure. In the
baryon–radiation plasma, the photons can efficiently transfer energy from one re-
gion of the fluid to another over distances determined by the mean free path of the
photons (e.g. through diffusion). Shear viscosity and heat conduction play an im-
portant role in this case and lead to the dissipation of perturbations on small scales
(Silk damping). In the limit of low baryon density, corresponding to current obser-
vations, heat conduction is not as important as shear viscosity, and so we will only
consider the latter. The derivation of the energy–momentum tensor for an imperfect
fluid is given in many books and we will not repeat it here. The energy–momentum
tensor is found to be
Tβα=(ε+p)uαuβ−pδαβ−η(
Pγαuβ;γ+Pβγuα;γ−^23 Pβαuγ;γ)
, (7.101)
whereηis the shear viscosity coefficient and
Pβα≡δαβ−uαuβ