MODERN COSMOLOGY

52 An introduction to the physics of cosmology

Since∂(a ̇^2 )/∂K=1, this gives the growing and decaying modes as

δ∝

{

(a ̇/a)

∫a

0 (a ̇)

− (^3) da (growing mode)
(a ̇/a) (decaying mode).
(Heath 1977, see also section 10 of Peebles 1980).
The equation for the growing mode requires numerical integration in general,
witha ̇(a)given by the Friedmann equation. A very good approximation to the
answer is given by Carrollet al(1992):
δ(z= 0 ,
)
δ(z= 0 ,
= 1 )



5

2

m

[

^4 m/^7 −
v+

(

1 +

1

2

m

)(

1 +

1

70

v

)]− 1

.

This fitting formula for the growth suppression in low-density universes is an
invaluable practical tool. For flat models with
m+
v=1, it says that the growth
suppression is less marked than for an open universe—approximately
^0.^23 as
against
^0.^65 if=0. This reflects the more rapid variation of
vwith redshift;
if the cosmological constant is important dynamically, this only became so very
recently, and the universe spent more of its history in a nearly Einstein–de Sitter
state by comparison with an open universe of the same
m.
What about the case of collisionless matter in a radiation background? The
fluid treatment is not appropriate here, since the two species of particles can
interpenetrate. A particularly interesting limit is for perturbations well inside the
horizon: the radiation can then be treated as a smooth, unclustered background
that affects only the overall expansion rate. This is analogous to the effect of,
but an analytical solution does exist in this case. The perturbation equation is as
before

δ ̈+ 2 a ̇

a

δ ̇= 4 πGρmδ,

but nowH^2 = 8 πG(ρm+ρr)/3. If we change variable toy≡ρm/ρr=a/aeq,
and use the Friedmann equation, then the growth equation becomes

δ′′+

2 + 3 y

2 y( 1 +y)

δ′−

3

2 y( 1 +y)

δ= 0

(fork=0, as appropriate for early times). It may be seen by inspection that a
growing solution exists withδ′′=0:

δ∝y+ 2 / 3.

It is also possible to derive the decaying mode. This is simple in the radiation-
dominated case (y1):δ∝−lnyis easily seen to be an approximate solution
in this limit.
What this says is that, at early times, the dominant energy of radiation drives
the universe to expand so fast that the matter has no time to respond, andδis
frozen at a constant value. At late times, the radiation becomes negligible, and the