A Bayesian Approach to Comparing Cosmic Ray Energy Spectra

TL;DR

This paper introduces a Bayesian method to compare cosmic ray energy spectra from different sky regions, effectively handling uncertainties and small sample sizes without relying on theoretical models.

Contribution

It develops a novel Bayesian approach using Bayes factors to compare spectra, accounting for non-Gaussian uncertainties and small sample sizes in cosmic ray physics.

Findings

The method accurately compares spectra without model dependence.

It correctly accounts for non-Gaussian uncertainties.

The approach is effective at high energies with few events.

Abstract

A common problem in ultra-high energy cosmic ray physics is the comparison of energy spectra. The question is whether the spectra from two experiments or two regions of the sky agree within their statistical and systematic uncertainties. We develop a method to directly compare energy spectra for ultra-high energy cosmic rays from two different regions of the sky in the same experiment without reliance on agreement with a theoretical model of the energy spectra. The consistency between the two spectra is expressed in terms of a Bayes factor, defined here as the ratio of the likelihood of the two-parent source hypothesis to the likelihood of the one-parent source hypothesis. Unlike other methods, for example chi^2 tests, the Bayes factor allows for the calculation of the posterior odds ratio and correctly accounts for non-Gaussian uncertainties. The latter is particularly important at the highest energies, where the number of events is very small.