SIMBAD references

We propose an observable for ultra-high-energy cosmic ray (UHECR) physics: the harmonic-space cross-correlation power spectrum between the arrival directions of UHECRs and the large-scale cosmic structure mapped by galaxies. This cross-correlation has not yet been considered in the literature, and it permits a direct theoretical modelling of the main astrophysical components. We describe the expected form of the cross-correlation and show how, if the distribution of UHECR sources traces the large-scale cosmic structure, it could be easier to detect with current data than the UHECR auto-correlation. Moreover, the cross-correlation is more sensitive to UHECR anisotropies on smaller angular scales, it is more robust to systematic uncertainties, and it could be used to determine the redshift distribution of UHECR sources, making it a valuable tool for determining their origins and properties.