Diffusion of cosmic rays in the expanding universe. II. Energy spectra of ultra-high energy cosmic rays

We consider the astrophysical implications of the diffusion equation solution for ultra-high energy cosmic rays (UHECRs) in the expanding universe, which we obtained in Paper I of this series. The UHECR spectra are calculated in a model with sources located in vertices of a cubic grid with a linear constant ( source separation) d. The calculations are performed for various magnetic field configurations (B-c, l(c)), where l(c) is the basic scale of the turbulence and Bc is the coherent magnetic field on this scale. The main purpose of these calculations is to demonstrate the validity of the solution obtained in Paper I and to compare this solution with the Syrovatsky solution used in previous works. The Syrovatsky solution must be necessarily embedded in the static cosmological model. The formal comparison of the two solutions with all parameters being fixed identically reveals the appreciable discrepancies between two spectra. These discrepancies are less if different sets of best-fit parameters are used in these models.