On the shape of the UHE cosmic ray spectrum

We fit the ultra high energy cosmic ray spectra above 10$^{19}$ eV with different injection spectra at cosmic ray sources that are uniformly and homogeneously distributed in the Universe. We conclude that the current UHE spectra are consistent with power laws of index $\alpha$ between 2.4 and 2.7. There is a slow dependence of these indices on the cosmological evolution of the cosmic ray sources, which in this model determines the end of the galactic cosmic rays spectrum.