and Cosmology

7.2 Gravitational Instability
i.e., the spatial and temporal dependences factorize in
these solutions. Here, ˜δ(x) is an arbitrary function of
the spatial coordinate, and D(t) satisfies the equation
¨D + 2ȧ a
Ḋ − 4πGρ(t) D = 0 . (7.13)
The Growth Factor. The differential equation (7.13)
has two linearly independent solutions. One can show
that one of them increases with time, whereas the other
decreases. If, at some early time, both functional dependences
were present, the increasing solution will
dominate at later times, whereas the solution decreasing
with t will become irrelevant. Therefore, we will
consider only the increasing solution, which is denoted
by D + (t), and normalize it such that D + (t 0 ) = 1. Then,
the density contrast becomes
δ(x, t) = D + (t)δ 0 (x). (7.14)
This mathematical consideration allows us to draw immediately
a number of conclusions. First, the solution
(7.14) indicates that in linear perturbation theory the
spatial shape of the density fluctuations is frozen in comoving
coordinates, only their amplitude increases. The
growth factor D + (t) of the amplitude follows a simple
differential equation that is easily solvable for any cosmological
model. In fact, one can show that for arbitrary
values of the density parameter in matter and vacuum
energy, the growth factor has the form
D + (a)
∝ H(a)
H 0
∫a
0
da ′
[
Ωm /a ′ + Ω Λ a ′2 − (Ω m + Ω Λ − 1) ] 3/2 ,
where the factor of proportionality is determined from
the condition D + (t 0 ) = 1.
In accordance with D + (t 0 ) = 1, δ 0 (x) would be the
distribution of density fluctuations today if the evolution
was indeed linear until the present epoch. Therefore,
δ 0 (x) is denoted as the linearly extrapolated density fluctuation
field. However, the linear approximation breaks
down if |δ| is no longer ≪ 1. In this case, the terms
that have been neglected in the above derivations are no
longer small and have to be included. The problem then
becomes considerably more difficult and defies analytical
treatment. Instead one needs, in general, to rely on
numerical procedures for analyzing the growth of density
perturbations. Furthermore, it shall be noted once
again that, for large density perturbations, the fluid approximation
is no longer valid, and that up to now we
have assumed the Universe to be matter dominated. At
early times, i.e., for z z eq (see Eq. 4.54), this assumption
becomes invalid, so that the above equations need
to be modified for these early epochs.
Example: Einstein–de Sitter Model. In the special
case of a universe with Ω m = 1, Ω Λ = 0, (7.13) can be
solved explicitly. In this case, a(t) = (t/t 0 ) 2/3 , so that
(ȧ
a
)
= 2 3t , and ρ(t) = a−3 ρ cr = 3H2 0
8πG
( t
t 0
) −2
;
furthermore, in this model t 0 H 0 = 2/3, so that (7.13)
reduces to
¨D + 4 3t Ḋ − 2
3t 2 D = 0 . (7.15)
This equation is easily solved by making the ansatz
D ∝ t q ; this ansatz is suggested because (7.15) is
equidimensional in t, i.e., each term has the dimension
D/(time) 2 . Inserting into (7.15) yields a quadratic
equation for q,
q(q − 1) + 4 3 q − 2 3 = 0 ,
with solutions q = 2/3 andq =−1. The latter corresponds
to fluctuations decreasing with time and will be
disregarded in the following. So, for the Einstein–de
Sitter model, the increasing solution
( ) t 2/3
D + (t) = = a(t), (7.16)
t 0
is found, i.e., in this case the growth factor equals the
scale factor. For different cosmological parameters this
is not the case, but the qualitative behavior is quite similar,
which is demonstrated in Fig. 7.3 for three models.
In particular, fluctuations were able to grow by a factor
∼ 1000 from the epoch of recombination at z ∼ 1000,
from which the CMB photons originate, to the present
day.
Evidence for Dark Matter on Cosmic Scales. At the
present epoch, δ ≫ 1 certainly on scales of clusters of
galaxies (∼ 2 Mpc), and δ ∼ 1 on scales of superclusters
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