Ultra-high energy cosmic rays from a finite number of point sources

We have calculated the probability that the clustering of arrival directions of ultra-high energy cosmic rays (UHECRs) is consistent with a finite number of uniformly distributed proton sources. The case of a continuous source distribution is reached only for an unrealisticly high source density, $n_s\gg 10^{-2}/Mpc^3$. Even for densities as large as $n_s=10^{-3}/Mpc^3$, less than half of the observed cluster are on average by chance. For the best-fit value $n_s=(1-3)\times 10^{-5}/Mpc^3$ derived from the AGASA data, the probability that at least one observed cluster is from a true point source is larger than 99.97%, while on average almost all observed clusters are true. The best-fit value found is comparable to the density of AGNs and consistent with the recent HiRes stereo data. In this scenario, the Pierre Auger Observatory will not only establish the clustering of UHECRs but also determine the density of UHECR sources within a factor of a few after one year of data taking.