Directional clustering in highest energy cosmic rays

An unexpected degree of small-scale clustering is observed in highest-energy cosmic ray events. Some directional clustering can be expected due to purely statistical fluctuations for sources distributed randomly in the sky. This creates a background for events originating in clustered sources. We derive analytic formulas to estimate the probability of random cluster configurations, and use these formulas to study the strong potential of the HiRes, Auger, Telescope Array and EUSO-OWL-AirWatch facilities for deciding whether any observed clustering is most likely due to nonrandom sources. For a detailed comparison to data, our analytical approach cannot compete with Monte Carlo simulations, including experimental systematics. However, our derived formulas do offer two advantages: (i) easy assessment of the significance of any observed clustering, and most importantly, (ii) an explicit dependence of cluster probabilities on the chosen angular bin size