An αstable approach to the study of the P(D) distribution of unresolved point sources in CMB sky maps
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 αstable distributions. We show that the nonGaussian distribution of the intensity fluctuations produced by a generic point source population  whose number counts follow a simple power law  belongs to the αstable family of distributions. Even if source counts do not follow a simple power law, we show that the αstable model is still useful in many astrophysical scenarios. With the αstable model it is possible to totally describe the nonGaussian 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 P(D) 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 P(D). The formalism and methods here presented can be very useful also for experiments in other frequency ranges, e.g. Xray or radio Astronomy.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 September 2004
 DOI:
 10.1051/00046361:20035858
 arXiv:
 arXiv:astroph/0307114
 Bibcode:
 2004A&A...424.1081H
 Keywords:

 methods: statistical;
 galaxies: statistics;
 cosmology: cosmic microwave background;
 Astrophysics
 EPrint:
 16 pages, 6 figures, final version to appear in A&