Ultra high energy cosmic rays from extragalactic astrophysical sources: energy losses and spectra

The energy losses and spectra of Ultra High Energy Cosmic Rays (UHECR) are calculated for protons as primary particles. The attention is given to the energy losses due to electron-positron production in collisions with the microwave 2.73 K photons. The energy spectra are calculated for several models, which differ by production spectra and by source distribution, namely: (i) Uniform distribution of the sources with steep generation spectra with indices 2.4 - 2.7, with cosmological evolution and without it. In this case it is possible to fit the shape of the observed spectrum up to 8.10^{19} eV, but the required CR emissivity is too high and the GZK cutoff is present. (ii) Uniform distribution of the sources with flat generation spectrum dE/E^2 which is relevant to GRBs. The calculated spectrum is in disagreement with the observed one. The agreement at E\lesssim 8.10^{19} eV can be reached using a complex generation spectrum, but the required CR emissivity is three orders of magnitude higher than that of GRBs, and the predicted spectrum has the GZK cutoff. (iii) The case of local enhancement within region of size 10 - 30 Mpc with overdensity given by factor 3- 30. The overdensity larger than 10 is needed to eliminate the GZK cutoff.