2.7 Linear growth rate of the
density fluctuation
Most likely our Universe is dominated by collisionless dark matter,
and thus
is negligibly small. Thus, at most
scales of cosmological interest, Equation (54) is well approximated
as
For a given set of cosmological parameters, one can solve the above
equation by substituting the expansion law for
as described in Section 2.3. Since Equation (55) is the second-order
differential equation with respect to
, there are two
independent solutions; a decaying mode and a growing mode which
monotonically decreases and increases as
, respectively. The
former mode becomes negligibly small as the Universe expands, and
thus one is usually interested in the growing mode alone.
More specifically those solutions are explicitly
obtained as follows. First note that the l.h.s. of
Equation (18) is the Hubble
parameter at
,
:
The first and second differentiation of Equation (56) with respect to
yields
and
respectively. Thus the differential equation for
reduces to
This coincides with the linear perturbation equation for
, Equation (55). Since
is a decreasing function of
, this implies that
is the decaying
solution for Equation (55). Then the
corresponding growing solution
can be obtained
according to the standard procedure: Subtracting Equation (55) from
Equation (60) yields
and therefore the formal expression for the growing solution in
linear theory is
It is often more useful to rewrite
in terms of the
redshift
as follows:
where the proportional factor is chosen so as to reproduce
for
. Linear
growth rates for the models described in Section 2.3 are summarized below:
- Einstein-de Sitter model (
):
- Open model with vanishing cosmological constant
(
):
- Spatially-flat model with cosmological constant
(
):
For most purposes, the following fitting
formulae [67
] provide
sufficiently accurate approximations:
where
Note that
and
refer to the
present values of the density
parameter and the dimensionless cosmological constant,
respectively, which will be frequently used in the rest of the
review.
Figure 2 shows the comparison
of the numerically computed growth rate (thick lines) against the
above fitting formulae (thin lines), which are practically
indistinguishable.