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

¹ Istituto di Scienza e Tecnologie dell'Informazione, CNR, Area della Ricerca di Pisa, via G. Moruzzi 1, 56124 Pisa, Italy e-mail: [diego.herranz;kuruoglu]@isti.cnr.it
² Departamento de Física, Universidad de Oviedo, c/ Calvo Sotelo s/n, 33007 Oviedo, Spain

Received: 12 December 2003
Accepted: 8 April 2004

Abstract

We present a new approach to the statistical study and modelling of number counts of faint point sources in astronomical images, i.e. counts of sources whose flux falls below the detection limit of a survey. The approach is based on the theory of α-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 α-stable family of distributions. Even if source counts do not follow a simple power law, we show that the α-stable model is still useful in many astrophysical scenarios. With the α-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. We demonstrate that the model proves valid when applied to realistic point source number counts at microwave frequencies. In the case of point extragalactic sources observed at CMB frecuencies, our technique is able to successfully fit the distribution of deflections and to precisely determine the main parameters which describe the number counts. In the case of the Planck mission, the relative errors on these parameters are small either at low and at high frequencies. We provide a way to deal with the presence of Gaussian noise in the data using the empirical characteristic function of the . The formalism and methods here presented can be very useful also for experiments in other frequency ranges, e.g. X-ray or radio Astronomy.