### 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.