Physical Foundations of Cosmology

7.4 Baryon–radiation plasma and cold dark matter 313

satisfies the equation

(δd− 3 )′+aui,i= 0. (7.107)

Solving forui,iin terms ofδdand and substituting into (7.106), the resulting
equation takes the form
(
a(δd− 3 )′

)′

−a= 0. (7.108)

Baryon–radiation plasmaThe baryons and radiation are tightly coupled before
recombination and, therefore, their energy and momentum are not conserved sep-
arately. Nevertheless, when the baryons are nonrelativistic, (7.105), in contrast to
(7.106), is valid for the baryon and radiation components separately, because the
energy conservation law for the baryons,T0;αα= 0 ,reduces to the conservation law
for total baryon number. (Specifically, ifT 0 α=mbnbuαu 0 , wherembis the baryon
mass, thenT0;αα=0 is equivalent to(nbuα);α=0 up to linear order in perturbations.)
Hence, the fractional baryon density fluctuation,δb≡δεb/εb,satisfies an equation
similar to (7.107):

(δb− 3 )′+aui,i= 0. (7.109)

The corresponding equation for the perturbations in the radiation component,δγ≡
δεγ/εγ,is, according to (7.105),
(
δγ− 4

)′

+^43 aui,i= 0. (7.110)

Since the photons and baryons are tightly coupled, they move together, and hence
the velocities entering both of these equations are the same. Multiplying (7.110) by
3 /4, subtracting (7.109), and integrating, we obtain

δS

S

≡

3

4

δγ−δb=const, (7.111)

whereδS/Sis the fractional entropy fluctuation in the baryon–radiation plasma
(see also (7.83)). Equation (7.111) states thatδS/Sis conservedon all scales.In
the case of adiabatic perturbations,δS=0 and therefore

δb=^34 δγ. (7.112)

Expressingui,iin terms ofδγand and substituting into (7.106), we obtain

(

δ′γ

cs^2

)′

−

3 η

εγa

δγ′−δγ=

4

3 c^2 s

+

(

4 ′

c^2 s

)′

−

12 η

εγa

′, (7.113)