On the Evidence for Clustering in the Arrival Directions of AGASA's Ultrahigh Energy Cosmic Rays

Previous analyses of cosmic rays above 40 EeV observed by the AGASA experiment have suggested that their arrival directions may be clustered. However, estimates of the chance probability of this clustering signal vary from 10^{-2} to 10^{-6} and beyond. It is essential that the strength of this evidence be well understood in order to compare it with anisotropy studies in other cosmic ray experiments. We apply two methods for extracting a meaningful significance from this data set: one can scan for the cuts which optimize the clustering signal, using simulations to determine the appropriate statistical penalty for the scan. This analysis finds a chance probability of about 0.3%. Alternatively, one can optimize the cuts with a first set of data, and then apply them to the remaining data directly without statistical penalty. One can extend the statistical power of this test by considering cross-correlation between the initial data and the remaining data, as long as the initial clustering signal is not included. While the scan is more useful in general, in the present case only splitting the data set offers an unbiased test of the clustering hypothesis. Using this test we find that the AGASA data is consistent at the 8% level with the null hypothesis of isotropically distributed arrival directions.Comment: 14 pages, 3 figures. Unbiased test expanded to include cross-correlation between initial and later data sets for greater statistical power; minor revisions to discussion. Accepted by Astropart. Phy

A Bayesian Approach to Comparing Cosmic Ray Energy Spectra

A common problem in ultra-high energy cosmic ray physics is the comparison of energy spectra. The question is whether the spectra from two experiments or two regions of the sky agree within their statistical and systematic uncertainties. We develop a method to directly compare energy spectra for ultra-high energy cosmic rays from two different regions of the sky in the same experiment without reliance on agreement with a theoretical model of the energy spectra. The consistency between the two spectra is expressed in terms of a Bayes factor, defined here as the ratio of the likelihood of the two-parent source hypothesis to the likelihood of the one-parent source hypothesis. Unlike other methods, for example chi^2 tests, the Bayes factor allows for the calculation of the posterior odds ratio and correctly accounts for non-Gaussian uncertainties. The latter is particularly important at the highest energies, where the number of events is very small.Comment: 22 pages, 10 figures, accepted for publication in Ap

Generalized Smoluchowski equation with correlation between clusters

In this paper we compute new reaction rates of the Smoluchowski equation which takes into account correlations. The new rate K = KMF + KC is the sum of two terms. The first term is the known Smoluchowski rate with the mean-field approximation. The second takes into account a correlation between clusters. For this purpose we introduce the average path of a cluster. We relate the length of this path to the reaction rate of the Smoluchowski equation. We solve the implicit dependence between the average path and the density of clusters. We show that this correlation length is the same for all clusters. Our result depends strongly on the spatial dimension d. The mean-field term KMFi,j = (Di + Dj)(rj + ri)d-2, which vanishes for d = 1 and is valid up to logarithmic correction for d = 2, is the usual rate found with the Smoluchowski model without correlation (where ri is the radius and Di is the diffusion constant of the cluster). We compute a new rate: the correlation rate K_{i,j}^{C} (D_i+D_j)(r_j+r_i)^{d-1}M{\big(\frac{d-1}{d_f}}\big) is valid for d \leq 1(where M(\alpha) = \sum+\infty i=1i\alphaNi is the moment of the density of clusters and df is the fractal dimension of the cluster). The result is valid for a large class of diffusion processes and mass radius relations. This approach confirms some analytical solutions in d 1 found with other methods. We also show Monte Carlo simulations which illustrate some exact new solvable models