Physical Foundations of Cosmology

268 Gravitational instability in Newtonian theory

Slightly disturbing the matter distribution, we have:

ε(x,t)=ε 0 +δε(x,t), V(x,t)=V 0 +δv=δv(x,t),

φ(x,t)=φ 0 +δφ(x,t), S(x,t)=S 0 +δS(x,t),

(6.12)

whereδεε 0 , etc. The pressure is equal to

p(x,t)=p(ε 0 +δε,S 0 +δS)=p 0 +δp(x,t), (6.13)

and in linear approximation its perturbationδpcan be expressed in terms of the
energy density and entropy perturbations as

δp=c^2 sδε+σδS. (6.14)

Herec^2 s≡(∂p/∂ε)Sis the square of the speed of sound andσ≡(∂p/∂S)ε.For
nonrelativistic matter (pε), the speed of sound as well as the velocitiesδvare
much less than the speed of light.
Substituting (6.12) and (6.14) into (6.3), (6.8)–(6.10) and keeping only the terms
which are linear in the perturbations, we obtain:

∂δε

∂t

+ε 0 ∇(δv)= 0 , (6.15)

∂δv

∂t

+

cs^2

ε 0

∇δε+

σ

ε 0

∇δS+∇δφ= 0 , (6.16)

∂δS

∂t

= 0 , (6.17)

δφ= 4 πGδε. (6.18)

Equation (6.17) has a simple general solution

δS(x,t)=δS(x), (6.19)

which states that the entropy is an arbitrary time-independent function of the spatial
coordinates.
Taking the divergence of (6.16) and using the continuity and Poisson equations
to express∇δvandδφin terms ofδε, we obtain

∂^2 δε

∂t^2

−c^2 sδε− 4 πGε 0 δε=σδS(x). (6.20)

This is a closed, linear equation forδε, where the entropy perturbation serves as a
given source.