Physical Foundations of Cosmology

6.3 Instability in an expanding universe 271

the pressure due to the energy density inhomogeneities is exactly compensated
by the corresponding contribution from the entropy perturbations, so thatδpk=
c^2 sδεk+σδSkvanishes.
Entropy perturbations can occur only in multi-component fluids. For example, in
a fluid consisting of baryons and radiation, the baryons can be distributed inhomo-
geneously on a homogeneous background of radiation. In such a case, the entropy,
which is equal to the number of photons per baryon, varies from place to place.
Thus we have found the complete set of modes−two adiabatic modes, two
vector modes and one entropy mode−describing perturbations in a gravitating
homogeneous non-expanding medium. The most interesting is the exponentially
growing adiabatic mode which is responsible for the origin of structure in the
universe.

6.3 Instability in an expanding universe

BackgroundIn an expanding homogeneous and isotropic universe, the background
energy density is a function of time, and the background velocities obey the Hubble
law:

ε=ε 0 (t), V=V 0 =H(t)·x. (6.32)

Substituting these expressions into (6.3), we obtain the familiar equation

ε ̇ 0 + 3 Hε 0 = 0 , (6.33)

which states that the total mass of nonrelativistic matter is conserved. The diver-
gence of the Euler equations (6.8) together with the Poisson equation (6.10) leads
to the Friedmann equation:

H ̇ +H^2 =−^4 πG

3

ε 0. (6.34)

PerturbationsIgnoring entropy perturbations and substituting the expressions

ε=ε 0 +δε(x,t), V=V 0 +δv,φ=φ 0 +δφ,

p=p 0 +δp=p 0 +c^2 sδε,

(6.35)

into (6.3), (6.8), (6.10), we derive the following set of linearized equations for small
perturbations:

∂δε

∂t

+ε 0 ∇δv+∇(δε·V 0 )= 0 , (6.36)

∂δv

∂t

+(V 0 ·∇)δv+(δv·∇)V 0 +

c^2 s

ε 0

∇δε+∇δφ= 0 , (6.37)

δφ= 4 πGδε. (6.38)