The Signature of a Correlation between $>10^{19}{\rm eV}$ Cosmic Ray Sources and Large Scale Structure

We analyze the anisotropy signature expected if the high energy (above $10^{19}$eV) cosmic ray (CR) sources are extra-Galactic and trace the distribution of luminous matter on large scales. We investigate the dependence of the anisotropy on both the relative bias between the CR sources and the galaxy distribution and on the (unknown) intrinsic CR source density. We find that the expected anisotropy associated with the large scale structure (LSS) should be detected once the number of CR events observed above $10^{19}{\rm eV}$ is increased by a factor of $\sim10$. This would require $\sim30$ observation-years with existing experiments, but less then $1$ year with the proposed $\sim5000\ {\rm km}^2$ Auger detectors. We find that the recently reported concentration of the Haverah Park CR events towards the super-galactic plane is not consistent with the known LSS. If real, the Haverah Park result suggests that the CR sources are much more concentrated towards the super-galactic plane than the known LSS. Our results are not sensitive to the number density of CR sources. We show that once the number of detected events is increased by a factor of $\sim10$, the number density would be strongly constrained by considering the probability for having repeating sources.