Physical Foundations of Cosmology

308 Gravitational instability in General Relativity

mode should describe inhomogeneities in the baryon distribution on a nearly ho-
mogeneous radiation background , we define the entropy mode to have the initial
condition

→0asη→ 0. (7.87)

Obviously, the particular solution (7.86) does not satisfy this condition. The solution
which does is obtained by adding to (7.86) a general solution (7.71) and choosing
the integration constants so that (7.87) is satisfied. The result is

=

1

5

ξ

ξ^2 + 6 ξ+ 10

(ξ+ 2 )^3

δS

S

. (7.88)

Problem 7.14Calculateδε/εandδεb/εb.

In Figure 7.3 we plot the time dependence of, δε/εandδεb/εbfor the entropy
mode. The amplitudes of andδε/εincrease linearly untilηeq, whereas they are
constant for adiabatic perturbations. The fluctuations in the cold-matter density
δεb/εbare somewhat frozen beforeηeqand decrease to 2/5 of their initial values

η^2

δεb

εb 1

2

5

δε

ε

∼

⏐Φ⏐

ηeq ∼k −^1 η

1

5

Fig. 7.3.