Small Scale Clustering in the Isotropic Arrival Distribution of   Ultra-High Energy Cosmic Rays and Implications for Their Source Candidates
  We present numerical simulations on the propagation of UHE protons with energies of $(10^{19.5}-10^{22})$ eV in extragalactic magnetic fields over 1 Gpc. We use the ORS galaxy sample, which allow us to accurately quantify the contribution of nearby sources to the energy spectrum and the arrival distribution, as a source model. We calculate three observable quantities, cosmic ray spectrum, harmonic amplitude, and two point correlation function from our data of numerical simulations. With these quantities, we compare the results of our numerical calculations with the observation. We show that the three observable quantities including the GZK cutoff of the energy spectrum can be reproduced in the case that the number fraction $\sim 10^{-1.7}$ of the ORS galaxies more luminous than -20.5 mag is selected as UHECR sources. In terms of the source number density, this constraint corresponds to $10^{-6}$ Mpc$^{-3}$. However, since mean number of sources within the GZK sphere is only $\sim 0.5$ in this case, the AGASA 8 events above $10^{20.0}$ eV, which do not constitute the clustered events with each other, can not be reproduced. On the other hand, if the cosmic ray flux measured by the HiRes, which is consistent with the GZK cutoff, is correct and observational features about the arrival distribution of UHECRs are same as the AGASA, our source model can explain both the arrival distribution and the flux at the same time. Thus, we conclude that large fraction of the AGASA 8 events above $10^{20}$ eV might originate in the topdown scenarios, or that the cosmic ray flux measured by the HiRes experiment might be better. We also discuss the origin of UHECRs below $10^{20.0}$ eV through comparisons between the number density of astrophysical source candidates and our result ($\sim 10^{-6}$ Mpc$^{-3}$). 
H. Yoshiguchi, S. Nagataki, S. Tsubaki, K. Sato