Physical Foundations of Cosmology

276 Gravitational instability in Newtonian theory

In this particular case the perturbation preserves its initial spatial shape as it devel-
ops. Such a solution is said to be self-similar.
Generically the shape of inhomogeneity changes. However, at late times (tt 0 )
when the growing mode dominates, we can omit the second term in (6.59) and the
linearperturbation grows in a self-similar way.

6.3.4 Cold matter in the presence of radiation or dark energy

There is convincing evidence that along with the cold matter in the universe there
exists a smooth dark energy component. This dark energy changes the expansion
rate and, as a result, influences the growth of inhomogeneities in the cold matter.
To study the gravitational instability in the presence of relativistic matter, in prin-
ciple we need the full relativistic theory. However, on scales smaller than the Jeans
length for relativistic matter, which is comparable to the horizon scale, the inhomo-
geneities in the cold matter distribution do not disturb the relativistic component
and it remains practically homogeneous. As a result we can still apply modified
Newtonian theory tothe perturbations in the cold matter itselfon scales smaller
than the horizon. In the following we consider the growth of the perturbations in the
presence of a homogeneous relativistic energy component. This can be radiation,
with equation of statew= 1 /3, or dark matter withw<− 1 / 3.
It is easy to verify that the equation for the perturbation in the cold component
alone,δ≡δεd/εd, coincides with (6.46), but the Hubble constant is now determined
by thetotalenergy density

εtot=

εeq

2

((

aeq

a

) 3

+

(aeq

a

) 3 ( 1 +w))

, (6.61)

via the usual relation, which for aflat universeis

H^2 =

8 πG

3

εtot. (6.62)

Hereaeqis the scale factor at “equality” when the energy densities of both compo-
nents are equal. To find the explicit solutions of (6.46), it is convenient to rewrite it
using as a time variable the normalized scale factorx≡a/aeqinstead oft. Taking
into account thatε 0 entering (6.46) is the cold matter density alone, equal to

εd=

εeq

2

(aeq

a

) 3

, (6.63)

and using (6.62) to express the Hubble parameter in terms ofx, (6.46) becomes

x^2

(

1 +x−^3 w

)d^2 δ

dx^2

+

3

2

x

(

1 +( 1 −w)x−^3 w

)dδ

dx

−

3

2

δ= 0. (6.64)