On the Injection Energy Distribution of Ultra--High-Energy Cosmic Rays

We investigate the injection spectrum of ultra-high-energy (>1015 eV) cosmic rays under the hypotheses that (1) these cosmic rays are protons and (2) the sources of these cosmic rays are extragalactic and are distributed homogeneously in space, although they may have had a different strength in the past; furthermore, we assume that we are not unusually close to any individual source(s). The most puzzling aspect of the observed ultra-high-energy cosmic-ray spectrum is the apparent nonexistence of a "Greisen cutoff" at about 1019.8 eV. Such a cutoff would be expected owing to rapid energy loss from photopion production caused by interactions with the microwave background. We show that this fact could be explained naturally if most (or all) of the cosmic rays presently observed above about 1019.6 eV were initially injected with energy above the Greisen cutoff. However, we find that the injection of cosmic rays above the Greisen cutoff cannot account for the observed flux below about 1019.6 eV unless the injection rate of these particles was enormously higher in the past, as would be the case if the injection resulted from the decay of an ultramassive particle with lifetime of order 10 yr. Even with such a rapid source evolution, the observed cosmic-ray spectrum below about 1018.5 eV cannot be explained by injection of particles above the Greisen cutoff in the distant past. However, we show that a 1/E^{3 }injection spectrum can account for the observed spectrum below 1018.5 eV, with the steepening observed by the Fly's Eye group between 1017.6 and 1018.5 eV being very naturally explained by e +e production effects. This latter fact lends support to the hypothesis that the cosmic rays in this energy regime are protons of extragalactic origin. However, owing to e + e - production effects, a 1/E^{3 }injection spectrum cannot account for the observed flux above about 1018.5 eV.