**5.3. Structure formation and constraints on dark energy**

Combining the initial power spectrum,
*P*(*k*) = *Ak*^{n},
*n* 1,
with the transfer function in (74) we find that
the final spectrum has the approximate form

(89) |

with 2
*k*_{eq}^{-1}
*d*_{H}(*z*_{eq})
13(_{NR}
*h*^{2})^{-1} Mpc =
13(
*h*)^{-1}*h*^{-1} Mpc [see equation (68)]
where
_{NR} *h*
is the shape parameter (see equation (74); we have assumed
_{B}
0 for
simplicity.) From equation (89), it is clear that *P*(*k*)
changes from an increasing function to a decreasing function at
*k*_{eq} , the numerical value of which is decided by the
shape parameter
. Smaller values of
_{NR} and
will lead
to more power at longer wavelengths.

One of the earliest investigations which used power spectrum to
determine
_{} was
based on the APM galaxy survey
[206].
This work showed that the existence of large scale power requires a non
zero cosmological constant.
This result was confirmed when the COBE observations fixed the amplitude
of the power spectrum unequivocally (see
section 6). It was pointed out in
[207,
208]
that the COBE normalization led to a wrong shape for the power spectrum
if we take
_{NR} = 1,
_{} = 0,
with more power at small scales than observed. This problem could be
solved by reducing
_{NR} and
changing the shape of the power spectrum. Current observations favour
0.25.
In fact, an analysis of a host of observational data, including those
mentioned above suggested
[209]
that _{}
0 even before the SN data
came up.

Another useful constraint on the models for structure formation can be
obtained from the abundance of rich clusters of galaxies with masses
*M*
10^{15}
*M*_{}.
This mass scale corresponds to a length scale of about
8*h*^{-1} Mpc and hence
the abundance of rich clusters is sensitive to the root-mean-square
fluctuation in the density contrast
at 8*h*^{-1} Mpc. It is conventional to denote this
quantity < (
/
)^{2}
> ^{1/2},
evaluated at 8*h*^{-1} Mpc, by
_{8}. To be
consistent with the observed abundance of rich clusters, equation (88)
requires _{8}
0.5_{NR}^{-1/2}. This is consistent
with COBE normalization for
_{NR}
0.3,
_{}
0.7. [Unfortunately,
there is still some uncertainty about the
_{8} -
_{NR}
relation. There is a claim
[210] that
recent analysis of SDSS data gives
_{8}
0.33 ±
0.03_{NR}^{-0.6}.]

The effect of dark energy component on the growth of
linear perturbations changes the value of
_{8}.
The results of section 5.1 translate into the
fitting function
[189]

(90) |

where
= (*n* - 1) +
(*h* - 0.65) and
(,
) = 0.21 -
0.22*w* +
0.33 +
0.25.
For constant *w* models with *w* = - 1, - 2/3 and -1/3, this
gives
_{8} = 0.96,
0.80 and 0.46 respectively.
Because of this effect, the abundance of clusters can be used to put
stronger constraints on cosmology when the data for high redshift
clusters improves. As mentioned before, linear perturbations grow more
slowly in a universe with cosmological constant compared to the
_{NR} = 1
universe. This means that clusters will be comparatively rare at high
redshifts in a
_{NR} = 1
universe compared to models with cosmological constant. Only less than
10 per cent of massive clusters form at *z* > 0.5 in a
_{NR} = 1
universe whereas almost all massive clusters would have formed by
*z* 0.5 in a
universe with cosmological constant
[211,
212,
213,
214,
75].
(A simple way of understanding this effect is by noting that if the
clusters are not in place by
*z* 0.5, say,
they could not have formed by today
in models with cosmological constant since there is very little growth
of fluctuation between these two epochs.)
Hence the evolution of cluster population as a function of redshift can be
used to discriminate between these models.

An indirect way of measuring this abundance is through the lensing
effect of a cluster of galaxy on extended background
sources. Typically, the foreground clusters shears the light
distribution of the background object and leads to giant arcs. Numerical
simulations suggest
[215]
that a model with
_{NR} = 0.3,
_{} =
0.7 will produce about 280 arcs which is nearly an order of magnitude
larger than the number of arcs produced in a
_{NR} = 1,
_{} = 0
model. (In fact, an open model with
_{NR} = 0.3,
_{} = 0
will produce about 2400 arcs.) To use this effect, one needs a well defined
data base of arcs and a controlled sample. At present it is not clear
which model is preferred though this is one test which seems to prefer
open model rather than a
-CDM model.

Given the solution to (64) in the presence of dark energy, we can repeat
the above analysis and obtain the abundance of different
kinds of structures in the universe in the presence of dark energy. In
particular this formalism can be used to study the abundance of weak
gravitational lenses and virialized x-ray clusters which could act as
gravitational lenses. The calculations again show
[204]
that the result is highly degenerate in *w* and
_{NR}.
If _{NR} is
known, then the number count of weak lenses will be about a
factor 2 smaller for *w* = - 2/3 compared to the
CDM model with a
cosmological constant. However, if
_{NR} and
*w* are allowed to vary in such a way that the matter power
spectrum matches with both COBE results and abundance of x-ray clusters,
then the predicted abundance of lenses is less than 25 per cent for
-1 *w*
- 0.4. It may be possible
to constrain the dark energy better by comparing relative abundance of
virialized lensing clusters with the abundance of x-ray under luminous
lensing halos. For example, a survey covering about 50 square degrees of
sky may be able to differentiate a
CDM model from
*w* = - 0.6 model at a
3 level.

Constraints on cosmological models can also arise from the modeling of
damped Lyman- systems
[75,
108,
109,
216,
217,
218]
when the observational situation improves. At present these observations
are consistent with
_{NR} = 0.3,
_{} =
0.7 model but do not exclude other models at a high significance level.

Finally, we comment on a direct relation between
(*a*) and
*H*(*a*). Expressing equation (65) in terms of
*H*(*a*) will lead to the form

(91) |

This can be used to determine *H*^{2}(*a*) from
(*a*) since this
equation is linear and first order in
*Q*(*a*)
*H*^{2}(*a*) (though it is second order in
).
Rewriting it in the form

(92) |

where

(93) |

We can integrate it to give the solution

(94) |

This shows that, given the non linear growth of perturbations
(*a*) as a function of
redshift and the approximate validity of spherical model,
one can determine *H*(*a*) and thus *w*(*a*) *even
during the nonlinear phases of the evolution*.
[A similar analysis with the linear equation (66) was done in
[219],
leading to the result which can be obtained by expanding (94) to linear
order in .]
Unfortunately, this is an impractical method from observational point of
view at present.