The Generalized Pareto Distribution: Its Applications to Cosmic Microwave Background

Abstract

In this paper the authors want to demonstrate new possibilities of the generalized Pareto distribution (GPD) that contains two or three power-law exponents. The eigen-coordinates method allows to detect these exponents that can contain complexconjugated terms also. This GPD is applied successfully to Cosmic Microwave Background (CMB) data from the ESA’s Planck space mission. The data analyzed in the paper are CMB temperature fluctuations around the 2.725 K mean, measured by Planck instruments at the frequencies of 100 GHz, 143 GHz, 217 GHz, and 353 GHz. Planck CMB temperature maps are provided by ESA in HEALPix nested format based on the division of the sky into 12 regions, called primary pixels. Because the data are really huge, we considered only one of the twelve primary pixels for each frequency mentioned above. For us it is important only to read these data quantitatively and prove that the detected distribution does not coincides with conventional one as the Gaussian. For this purpose, we used the GPD for description the temperature fluctuation envelopes. This distribution demonstrates its efficiency and can be used for more detailed description of many data.