MODERN COSMOLOGY

Dynamics of structure formation 51

For total radiation domination,p =ρc^2 /3, and it is easy to linearize these
equations as before. The main differences come from factors of 2 and 4/3 due to
the non-negligible contribution of the pressure. The result is a continuity equation
∇·u=−( 3 / 4 )δ ̇, and the evolution equation forδ:

δ ̈+ 2 a ̇

a

δ ̇=^32 π

3

Gρ 0 δ,

so the net result of all the relativistic corrections is a driving term on the right-hand
side that is a factor 8/3 higher than in the matter-dominated case.
In both matter- and radiation-dominated universes with
=1, we have
ρ 0 ∝ 1 /t^2 :

matter domination(a∝t^2 /^3 ):4πGρ 0 =

2

3 t^2

radiation domination(a∝t^1 /^2 ):32πGρ 0 / 3 =

1

t^2

.

Every term in the equation forδis thus the product of derivatives ofδand powers
oft, and a power-law solution is obviously possible. If we tryδ ∝tn,then
the result isn = 2 /3or−1 for matter domination; for radiation domination,
this becomesn=±1. For the growing mode, these can be combined rather
conveniently using theconformal timeη≡

∫

dt/a:

δ∝η^2.

Recall thatηis proportional to the comoving size of the horizon.
It is also interesting to think about the growth of matter perturbations in
universes with non-zero vacuum energy, or even possibly some other exotic
background with a peculiar equation of state. The differential equation forδis
as before, buta(t)is altered. The way to deal with this is to treat a spherical
perturbation as a small universe. Consider the Friedmann equation in the form

(a ̇)^2 =
tot 0 H 02 a^2 +K,

whereK =−kc^2 /R^20 ; this emphasizes thatKis a constant of integration. A
second constant of integration arises in the expression for time:

t=

∫a

0

a ̇−^1 da+C.

This lets us argue as before in the case of decaying modes: if a solution to the
Friedmann equation isa(t,K,C), then valid density perturbations are

δ∝

(

∂lna

∂K

)

t

or

(

∂lna

∂C

)

t

.