**2.5. Scale-invariant perturbations ?**

The nearly scale-invariant nature of the initial perturbations, as expected from inflation, can only be probed using large-scale structure data insofar as one assumes a certain matter content in the Universe. This unfortunate situation results from the fact that after horizon re-entry the subsequent evolution of the density perturbations is greatly affected by both the type of matter present in the Universe and the cosmic expansion rate, which is in turn a function of the total quantity of matter present in the Universe.

The changes in the initial
power spectrum of density perturbations, which in our notation is given by
_{H}^{2}(*k*),
are encapsulated in the transfer function,
*T*(*k*, *t*). Thus, unless one assumes a certain
transfer function, we have
no hope of recovering the shape of
_{H}^{2}(*k*)
from large-scale structure data. The
evidence for
_{H}^{2}(*k*)
*k*^{n - 1}, as
expected in the simplest inflationary models
(in particular *n* 1), is
therefore dependent on assumptions regarding the parameters that
affect the transfer function. However, the choice of certain values for
some of these
parameters may come at a price, and be in conflict with observations not
directly
related with the power spectrum. For example, the observational data
regarding light element
abundances implies that the present cosmic baryon density,
_{b}, is about
0.02 *h*^{-2}, with an error at 95 per cent confidence of
less than about 20 per cent, if
homogeneous standard nucleosynthesis is assumed [see e.g.
(77)].
Heavy tinkering with the value of
_{b} is tantamount to
throwing away the standard
nucleosynthesis calculation, thus requiring a viable alternative to be
put in its place. It is
also not possible to have any type of dark matter one might want to
consider. If all the dark
matter particles had high intrinsic velocities, i.e. if they were hot,
like neutrinos, then the
effect of free-streaming would completely erase all the perturbations on
small scales, e.g.
in the case of standard neutrinos on all scales smaller than about
4 x 10^{14}(_{0} /
_{})
(_{} *h*^{2})^{-2}
*M*_{}. Not
only it would then be impossible to reproduce the abundance of
high-redshift objects, like proto-galaxies, quasars, or damped
Lyman- systems, but the actual
present-day
abundance and distribution of galaxies would be radically different from
that observed. It
seems very improbable that neutrinos presently contribute with more than
about 30 per cent
of the total matter density in the Universe [see e.g.
(78,
18,
66)].

What is needed is to reformulate slightly the original question and ask instead
whether for the simplest, best observationally supported assumptions one
can make, the
present-day power spectrum of density perturbations is compatible with
an initial
power-law shape, and in particular with one which is nearly
scale-invariant. These
assumptions are: the total energy density in the Universe is equal
to or less than the critical density, i.e.
1, and results
from a matter component, plus a possible classical cosmological constant;
the Hubble constant, in the form of *h*, has a value between 0.4
and 0.9; the baryon abundance, in the form of
_{b}*h*^{2},
is within 0.015 to 0.025; the (non-baryonic) dark matter
is essentially cold, with the possibility of any of the 3 known standard
neutrinos having a
cosmologically significant mass. Among all these assumptions, the one
which could be more easily
changed without conflicting with non-large scale structure data is the
nature of the (non-baryonic)
dark matter. There could be warm, decaying, or even self-interacting,
dark matter, though usually
the existence of some contribution by cold dark matter is found to be
required to fit all the available large-scale structure data.

In order to constrain the above free parameters in this *simplest*
model, one needs at least
the same number of independent observational constraints. The most
widely used are:
the slope of the galaxy/cluster power spectrum; the present number
density of rich galaxy clusters;
the high-redshift abundance of proto-galaxies, quasars and damped
Lyman- systems;
the amplitude of velocity bulk flows; the amplitude of CMBR temperature
anisotropies, both on
large-angular scales, as measured by *COBE*, and on
intermediate-angular scales, as presently
measured by balloon experiments. While the first constraint directly
limits the slope of the density power spectrum, the
next four constraints do such only indirectly, by imposing possible
intervals for the amplitude of
the density power spectrum at specific scales.

Other constraint that has started to be used recently, is the slope and
normalisation of the density
power spectrum on Mpc scales as inferred from the abundance and
distribution of Lyman-
forest absorption features in the spectra of distant (*z* ~ 2.5) quasars
(19).
However, these calculations depend on assuming
a relatively simple physical picture for the formation of such features,
being presently still
unclear if such a picture provides a good approximation to reality.

The comparison of the above defined *simplest* structure formation
models with
CMBR anisotropy data is also presently not as clean as one would like.
The two most important culprits are: the possibility of a gravity wave
contribution at the
*COBE* scale, thus allowing one to arbitrarily decrease the
amplitude imposed by the
*COBE* result on the density power spectrum; the possibility of
re-ionization,
which allows models with too much intermediate-scale power in the CMBR
anisotropy angular
power spectrum to evade the observational limits on such scales.

Again taking refuge in the simplicity assumption, the *simplest* models
would be those with a negligible contribution of gravity waves to the
large-angle CMBR anisotropy signal, together with no significant
re-ionization. In any case, it should be mentioned that relaxing these
two further assumptions
does not open up a large region of parameter space
(78).

Unfortunately, the comparison of these *simplest* structure
formation models with the
observational constraints just described does not tell us much about the
shape of the initial
density power spectrum. The assumption of a power-law shape is perfectly
compatible with the
data, with the value of the spectral index being loosely constrained to
be between 0.7 and
1.4 (78,
66).
However, by imposing further restrictions on the type of structure
formation model considered, stronger constraints can be obtained. For example,
if the Universe was Einstein-de Sitter then the value of the Hubble
constant would have to be
smaller than about 0.55, in order for the Universe to be more than 12
Gyr old. This does not matter
much, because high values for *h* increase the amount of
small-scale power relative to large-scale
power and at the same time suppress power at intermediate angular scales
on the CMBR, and
for *h* > 0.55 these two effects join together to exclude almost
all viable Einstein-de
Sitter structure formation models in the context of the *simplest*
assumptions laid down before
(78).
Restricting ourselves to
0.5 < *h* < 0.55,
then yields a preferred value for the spectral index *n* roughly
between 0.9 and 1.1, with a total neutrino density of
_{} ~ 0.15 [see e.g.
(78,
31,
66).
Another example, is
the case of the
_{0} = 0.3 flat model,
that preferred by the high-redshift type Ia
supernova data of
(68), for which
values for the spectral index in excess of 1 tend
to be preferred
(78,
66).
One should note however
that both models are only marginally compatible with the observational
data if *n* = 1.0.

Finally, it should be mentioned that, even within the simplified structure formation scenario we have been assuming, there is enough room for initial density power spectra with deviations from a power-law shape to be viable, as the survival of the broken scale-invariance model testifies (53). This only goes to show the still scarceness of good quality large-scale structure data at present.

In the near future the Sloan Digital Sky Survey
(SDSS) will allow a much better constraint to be imposed on the value of
*n*, by extending the measure of the slope of the galaxy power spectrum
to larger scales, thus probing a region of the power spectrum which has
not been in principle
too much affected by the dark matter properties, retaining therefore more
information about the shape of the initial density power spectrum
(58).