CMB Analysis: methods for quantifying non-Gaussianity

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(Npixm) operations, clearly a large number for the expected large data-sets.

Goals

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

Pedro G. Ferreira
pgf@physics.berkeley.edu

Last Updated: April 11 1997