Towards a model of population of astrophysical sources of   ultra-high-energy cosmic rays
  We construct and discuss a toy model of the population of numerous non-identical extragalactic sources of ultra-high-energy cosmic rays. In the model, cosmic-ray particles are accelerated in magnetospheres of supermassive black holes in galactic nuclei, the key parameter of acceleration being the black-hole mass. We use astrophysical data on the redshift-dependent black-hole mass function to describe the population of these cosmic-ray accelerators, from weak to powerful, and confront the model with cosmic-ray data. 
O. E. Kalashev, K. V. Ptitsyna and S. V. Troitsky
astro-ph.HE