Curious case of the maximum rigidity distribution of cosmic-ray accelerators

Abstract

In many models, the sources of ultra-high-energy cosmic rays (UHECRs) are assumed to accelerate particles to the same maximum energy. Motivated by the fact that candidate astrophysical accelerators exhibit a vast diversity in terms of their relevant properties such as luminosity, Lorentz factor, and magnetic field strength, we study the compatibility of a population of sources with non-identical maximum cosmic-ray energies with the observed energy spectrum and composition of UHECRs at Earth. For this purpose, we compute the UHECR spectrum emerging from a population of sources with a power-law, or broken-power-law, distribution of maximum energies, applicable to a broad range of astrophysical scenarios. We find that the allowed source-to-source variance of the maximum energy must be small to describe the data if a power-law distribution is considered. Even in the most extreme scenario, with a very sharp cutoff of individual source spectra and negative redshift evolution of the accelerators, the maximum energies of 90\% of sources must be identical within a factor of three -- in contrast to the variance expected for astrophysical sources. Substantial variance of the maximum energy in the source population is only possible if the maximum energies follow a broken power-law distribution with a very steep spectrum above the break. However, in this scenario, the individual source energy spectra are required to be unusually hard with increasing energy output as a function of energy.