**2.4. Gaussian vs. non-Gaussian perturbations**

The two characteristics of the density perturbations generated in the simplest inflationary models that could, in principle, be more easily searched for in large-scale structure data are their initial Gaussian probability distribution and the near scale-invariance of their power spectrum.

In the simplest models of inflation, due to the nature of quantum
fluctuations, the phases of the
Fourier modes associated with the perturbations in the value of the
scalar field are
independent, drawn at random from a uniform distribution in the interval
[0, 2]. Therefore, the density
perturbations resulting from these perturbations
will also be a superposition of Fourier modes with independent random
phases. From the central limit theorem of statistics it then follows that the
density probability distribution at any point in space is *Gaussian*.
As previously mentioned, this result is extremely important, since only
one function, the
power spectrum at horizon re-entry, is then required to specify
all of the statistical properties of the initial density distribution.

Present large-scale structure data cannot be used to say whether the the initial density perturbations followed a Gaussian distribution or not. One problem with the detection of Gaussian initial conditions is that as the density perturbations grow under gravity, their distribution increasingly deviates from the initial Gaussian shape. If the density contrast had a perfect Gaussian distribution, with dispersion , there would always be a non-zero probability of having < - 1 in some region of space, which is clearly unphysical. Therefore, the real distribution of density perturbations will always be at least slightly non-Gaussian, with a cut-off at = - 1, i.e. positively skewed. Further, once the (initialy very rare) densest regions of the Universe start to turn-around and collapse, due to their own self-gravity, their density will increase much faster than that associated with most other regions at the same scale, which are still expanding with the Universe, i.e. evolving linearly. This leads to the development by the density distribution of a positive tail associated with high values for , thus further increasing the skewness. These deviations do not matter as long as << 1, for then the probability that < - 1 or the existence of regions in the process of turn-around is extremely small. Thus, in the early stages of the gravitational evolution of non-correlated random density perturbations it is a very reasonable assumption to take their distribution as perfectly Gaussian. However, as the value of approaches 1, the probability that < - 1 or of high values for cannot be neglected any longer, and assuming the density distribution to be Gaussian will induce significant errors in calculations.

In summary, gravitational evolution of density perturbations induces increasingly larger deviations from an initial Gaussian distribution, leading to a progressively more positively skewed distribution. Higher moments than the skewness, equally zero for a Gaussian distribution, are also generated in the process, though at increasingly later times for the same amplitude. This means that among these only the kurtosis (which compares the size of the side tails of some distribution against those of a Gaussian distribution) has developed significantly by today on scales larger than about 1 Mpc (on smaller scales astrophysical processes irremediably mess up the calculations).

In order to determine whether the initial probability distribution was
Gaussian one then
needs either to go to scales *R* which are still evolving linearly,
i.e. (*R*) << 1,
or one needs to quantify, using gravitational perturbation theory [see
e.g.
(9)],
the amount of skewness and kurtosis a Gaussian density field develops
under gravitational instability for the values of
(*R*) observed.

If the values for the skewness and kurtosis were found to be well in
excess of those
expected, either the initial density distribution was non-Gaussian or
structure did not form through gravitational instability. The problem is
that we presently do not
have direct access to the density field, though this will change in the
near future by its
reconstruction using data from large areas of the sky searched for
gravitational lensed galaxies [see e.g.
(81,
62)].
Traditionally, the galaxy distribution
has been used as a tracer of the density distribution, being assumed
that the skewness and
kurtosis of the density field should be the same as that of the galaxy
field. However, it was soon realised
that the galaxies could have a biased distribution with relation to the
matter, i.e. _{gal}
= *b* where *b* is
the *bias parameter*. And a biased mass distribution with respect
to galaxies closely resembles one which is unbiased, but which has a
stronger degree of gravitational
evolution. Dropping the assumption of linear bias, by considering the
possibility that the galaxy
distribution might depend on higher order terms of the density field,
further complicates. Observationally
the situation is also not very clear, with often incompatible values for
both the skewness and kurtosis derived from the same galaxy surveys
(41).

Another means of having access to the density field is by reconstructing
it using the
galaxy velocity field, under the assumption that galaxies move solely
under the action of
gravity. In fact, one can use directly the velocity field to constrain
the initial distribution of
the density perturbations, through its own skewness and kurtosis. The
scaled skewness and kurtosis of the divergence of the velocity field
have the advantage of
not depending on a possible bias between the mass and galaxy distributions. But
they depend on the value of
_{0}, in such a way
that the velocity
field of a low-density Universe is similar to that of a high-density Universe
whose density field has evolved further under gravity. Nevertheless, by
combining
measurements of both the scaled skewness and kurtosis it should in
principle be possible to
determine if the initial density field was Gaussian
(9).
However, only the line-of-sight velocity component is measurable,
hence reconstruction methods, like POTENT
(23),
are used to recover the full 3D velocity
field based solely on it. But such methods have their weaknesses, like
for example the need for very good distance indicators, which have up to today
hindered the use of the velocity field to determine whether the initial density
perturbations followed a Gaussian distribution.

The *topology* of the galaxy distribution can also be used to
determine whether the initial density
perturbations had a Gaussian distribution or not, if it is once more
assumed that it reflects the
underlying properties of the density field. The topological measure most
widely used to distinguish between different underlying distributions is
the *genus*, which essentially gives the number of holes minus the
number of isolated
regions, defined by a surface, plus one (e.g. it is zero
for a sphere, one for a doughnut). Up to today all measures of the genus
of the
galaxy distribution seem compatible with it being Gaussian on the
largest scales,
above about 10 *h*^{-1} Mpc [see e.g.
(13)],
with the prospects of even tighter constraints coming from the big
galaxy surveys under way like the 2dF and the SDSS
(15).