With current developments in experimental CMB physics, we will now be in a position to analyse very large data sets, with information about large patches of the sky measured with very high resolution and sensitivity. This means that we are in a position to seriously test whether the sky is Gaussian. If the sky is Gaussian indeed, then current techniques for estimating the power spectrum will become watertight methods. Should deviations from non-Gaussianity be detected then the power spectrum is not the end of the story in the quest for a statistical characterization of the fluctuations.

In the past this task has been tackled in a variety of ways
subject to very different philosophies. One approach has been to choose
a statistic which is easy to describe for a Gaussian random field and then
try to quantify, by means of this statistic, what are the chances
that a given data set comes from an underlying Gaussian ensemble. The
well known examples are peaks' statistics (Bond
& Efstathiou, Vittorio & Juskiewicz),
topological tests (Coles, Gott *et al*)
the 3-point correlation function (Kogut
*et al* ) and
skewness and kurtosis (Scaramela & Vittorio). Another approach has been
to devise statistics which are good discriminators between Gaussian
skies and specific non-Gaussian rivals. Such is the case in much
of the techniques involved in looking for topological defects, such
as strings and textures or even foregrounds, such as point sources.
These approaches have their merits. Non-Gaussian tests are very easy
to implement even in the context of very large data sets. Also
reducing the whole issue
to a single statistic allows one to concentrate on devising the
statistic ideally suited for detecting a given, pre-known, type
of non-Gaussianity. This is somewhat reminiscent of pattern recognition:
if we already know what we are looking for, we may improve our
chance of detecting an existing predefined pattern inside a noisy data-set.

One can, however, take a more humble approach, which is to admit that
we have little idea of what the underlying probability distribution
function of the CMB is. It then becomes necessary to devise as
complete a framework as possible, without prejudices with regards
to testing rival models, or ease of computation with regards to testing
non-Gaussianity. This is an alternative approach, which supports as
its underlying philosophy the quest for ruling out or detecting generic
non-Gaussianity.
The most well established formalism following this alternative
philosophy
is the n-point formalism (Peebles).
In spite of all its success it is argued in
Ferreira & Magueijo
that this framework is not systematic and is plagued by redundancy.
In principle, one can calculate an infinite
number of n-point functions, and there is no criteria where to
truncate such an evaluation. If one has a finite data set, then
many of these quantities will be algebraically dependent on each other.
There is a practical additional problem: to estimate the m-point correlation
function, one needs * O*(N_{pix}^{m})
operations, clearly a large number for the expected large data-sets.

We are developing tools that will allow us to tackle the problem of non-Gaussianity:

- High resolution simulations for non-Gaussian theories ( cosmic strings and texture)
- Non-Gaussian spectra for both small fields (such as interferometers) and large scales
- Cumulative techniques with the discrete wavelet transform

Pedro G. Ferreira

pgf@physics.berkeley.edu