Ultrahigh energy cosmic rays from compact sources

The clustering of ultrahigh energy (above ${10}^{20}$ eV) cosmic rays (UHECR) suggests that they might be emitted by compact sources. The statistical analysis of Dubovsky et al. [Phys. Rev. Lett. 85, 1154 (2000)] estimated the source density. We extend their analysis to give also the confidence intervals for the source density using (a) no assumptions on the relationship between clustered and unclustered events, (b) nontrivial distributions for the source luminosities and energies, and (c) the energy dependence of the propagation. We also determine the probability that a proton created at a distance r with energy E arrives at Earth above a threshold $E_c.$ Using this function one can determine the observed spectrum just by one numerical integration for any injection spectrum. The observed 14 UHECR events above ${10}^{20}$ eV with one doublet give, for the source densities, ${180}_{-165}^{+2730\mathrm{}}\times {10}^{-3}$ ${\mathrm{Mpc}}^{-3}$ (on the 68% confidence level). We present detailed results for future experiments with larger UHECR statistics.