Physical Foundations of Cosmology

7.4 Baryon–radiation plasma and cold dark matter 315

the amplitude of the gravitational potential and the radiation fluctuationsδγat

recombination for a given initial (^0) kor, equivalently, to find thetransfer functions
relating the initial spectrum of perturbations with the resulting one at recombi-
nation. We will see in Chapter 9 that andδγdetermine the anisotropies in the
background radiation.
Long-wavelength perturbations (kηr< 1 )We first consider the long-wavelength
perturbations which are supercurvature modes at recombination. They are described
by solution (7.71) for adiabatic perturbations in a two-component medium consist-
ing of cold matter and radiation. Although the dark matter particles are not tightly
coupled to the radiation, the entropy per cold dark matter particle is nevertheless
conserved on supercurvature scales because, as can be seen intuitively, there is
insufficient time to move matter distances greater than the Hubble scale. One can
formally arrive at this conclusion by noting that, for long-wavelength perturbations,
theui,i-terms in (7.107) and (7.110) are negligible and, consequently, the steps which
lead to the entropy conservation law per baryon (see (7.111)) can be repeated here.
Knowing the gravitational potential, which is given by (7.71), we can easily find
δγ. Skipping the velocity term in (7.110), which is negligible for the long-wave-
length perturbations, and integrating, we obtain
δγ− 4 =C, (7.118)
whereCis a constant of integration. To determineCwe note that, during the radi-
ation-dominated epoch, the gravitational potential is mainly due to the fluctuations
in the radiation component and stays constant on supercurvature scales. At these
early times,
δγ− 2

(

ηηeq

)

≡− 20 ; (7.119)

henceC=− 60 .After equality, when the cold matter overtakes the radiation, the
gravitational potential changes its value by a factor of 9/10 and remains constant
afterwards, that is,
(
ηηeq

)

 1090. (7.120)

Therefore, assuming that cold dark matter dominates at recombination, we obtain
from (7.118)

δγ(ηr)=− 60 + 4 (ηr)=−^83 (ηr)=−^83

( 9

10

0 ). (7.121)

One arrives at the same conclusion by noting that, for adiabatic perturbations,
δγ= 4 δd/3 andδd− 2 (ηr)at recombination.
Standard inflation predicts adiabatic fluctuations. In principle, one can imag-
ine alternative possibilities for the initial inhomogeneities, such as entropy