On astrophysical solution to ultrahigh energy cosmic rays

We argue that an astrophysical solution to the ultrahigh energy cosmic ray (UHECR) problem is viable. The detailed study of UHECR energy spectra is performed. The spectral features of extragalactic protons interacting with the cosmic microwave background (CMB) are calculated in a model-independent way. Using the power-law generation spectrum $\propto E^{-{\gamma }_g}$ as the only assumption, we analyze four features of the proton spectrum: the GZK cutoff, dip, bump, and the second dip. We found the dip, induced by electron-positron production on the CMB, to be the most robust feature, existing in energy range $1\times {10}^{18}–4\times {10}^{19}\mathrm{eV}$. Its shape is stable relative to various phenomena included in calculations: discreteness of the source distribution, different modes of UHE proton propagation (from rectilinear to diffusive), local overdensity or deficit of the sources, large-scale inhomogeneities in the universe, and interaction fluctuations. The dip is well confirmed by observations of the AGASA, HiRes, Fly’s Eye, and Yakutsk detectors. With two free parameters (${\gamma }_g$ and flux normalization constant) the dip describes about 20 energy bins with ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.\approx 1$ for each experiment. The best fit is reached at ${\gamma }_g=2.7$, with the allowed range 2.55–2.75. The dip is used for energy calibration of the detectors. For each detector independently, the energy is shifted by factor $\lambda$ to reach the minimum ${\chi }^2$. We found ${\lambda }_{\mathrm{Ag}}=0.9$, ${\lambda }_{\mathrm{Hi}}=1.2$, and ${\lambda }_{\mathrm{Ya}}=0.75$ for the AGASA, HiRes, and Yakutsk detectors, respectively. Remarkably, after this energy shift the fluxes and spectra of all three detectors agree perfectly, with discrepancy between AGASA and HiRes at $E>1\times {10}^{20}\mathrm{eV}$ being not statistically significant. The excellent agreement of the dip with observations should be considered as confirmation of UHE proton interaction with the CMB. The dip has two flattenings. The high energy flattening at $E\approx 1\times {10}^{19}\mathrm{eV}$ automatically explains ankle, the feature observed in all experiments starting from the 1980s. The low-energy flattening at $E\approx 1\times {10}^{18}\mathrm{eV}$ reproduces the transition to galactic cosmic rays. This transition is studied quantitatively in this work. Inclusion of primary nuclei with a fraction of more than 20% upsets the agreement of the dip with observations, which we interpret as an indication of the acceleration mechanism. We study in detail the formal problems of spectra calculations: energy losses (the new detailed calculations are presented), the analytic method of spectrum calculations, and the study of fluctuations with the help of a kinetic equation. The UHECR sources, AGN and GRBs, are studied in a model-dependent way, and acceleration is discussed. Based on the agreement of the dip with existing data, we make the robust prediction for the spectrum at $1\times {10}^{18}–1\times {10}^{20}\mathrm{eV}$ to be measured in the nearest future by the Auger detector. We also predict the spectral signature of nearby sources, if they are observed by Auger. This paper is long and contains many technical details. For those who are interested only in physical content we recommend the Introduction and Conclusions, which are written as autonomous parts of the paper.