Intergalactic Propagation of UHE Cosmic Rays

We discuss the intergalactic propagation of ultra-high-energy cosmic rays (UHECRs) with energies E \geq 10^{18.5} eV. We consider the propagation of UHECRs under the influence of the energy-dependent deflection by a weak random magnetic field in the intergalactic medium and energy losses by photo-pion and pair production. We calculate arrival spectra taking full account of the kinematics of photo-pion production and the Poisson statistics of the photo-pion interaction rate. We give estimates for the deflection of UHECRs from the line of sight to the source, time delays with respect to photons from the same source, arrival spectra and source statistics. These estimates are confirmed by numerical simulations of the propagation in energy evolution of UHECRs. These simulations demonstrate that the often-used continuous approximation in the treatment of energy losses due to photo-pion production on the cosmic microwave background (CMWB) cannot be justified for UHECRs. We discuss the implications of these results for the observed flux of particles above the Greisen-Zatsepin-Kuz'min cut-off in two of the scenarios that have been proposed for the production of these particles: continuous production in the large shock waves associated with powerful radio galaxies, or possibly large-scale structure formation, and the impulsive production at relativistic blast waves associated with cosmological gamma-ray bursts.