Using alpha-stable distributions to model the point source population in CMB sky maps

We present a new approach to the statistical study and modelling of point source populations in astronomical images obtained with pencil-beam detectors. The approach is based on the theory of alpha-stable distributions. We show that the non-Gaussian distribution of the intensity fluctuations produced by a generic power-law point source distribution belongs to the alpha_stable family of distributions. Even when the point source counts do not follow a pure power law, we show that the alpha-stable model is still valid in many cases of interest. With the alpha-stable model, it is possible to totally describe the non-Gaussian distribution with a few parameters, that are closely related to the physical parameters of the point source population, instead of an infinite number of moments. Using statistical tools available in the signal processing literature, we show how to estimate these parameters in an easy and fast way. Then we focus on the case of Cosmic Microwave Background observations where the point sources are superimposed to the cosmological signal as well as instrumental noise, and propose a method to statistically disentangle these contributions. In the case of the Planck mission, our thecnique is able to determine the parameters of the dominant point source populations with relative errors < 5% for the 30 GHz and 857 GHz channels. The formalism and methods presented here can be useful also for other experiments in other frequency ranges such as X-rays or radio Astronomy.