An alpha-stable approach to the study of the P(D) distribution of unresolved point sources in CMB sky maps

We present a new approach to the statistical study and modelling of point source counts in astronomical images. 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 point source population -- whose number counts follow a simple power law -- belongs to the alpha-stable family of distributions. Even if source counts do not follow a simple power law, we show that the alpha-stable model is still useful in many astrophysical scenarios. With the alpha-stable model it is possible to totally describe the non-Gaussian distribution with a few parameters which are closely related to the parameters describing the source counts, 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 apply the method to Cosmic Microwave Background (CMB) observations where point sources appear as superimposed to the cosmological signal as well as the instrumental noise, and propose a method to statistically disentangle these contributions. In the case of the Planck mission, our technique 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 experiments in other frequency ranges such as X-rays or radio Astronomy.