Hidden structures in the UHE sky: unveiling potential cosmic-ray sources via spectral signatures

Abstract

The search for the origin of ultrahigh-energy cosmic rays (UHECRs) typically relies on identifying directional excesses in the particle flux. While this approach is well established, it cannot detect localized spectral features, which could provide constraints on the sources of UHECRs if measured. In this work, we propose extending directional searches through two complementary energy-weighted metrics: the accumulated energy $U$ and the local average energy per particle ${\mu }_E$. We demonstrate that $U$ inherently amplifies the statistical significance of sources with high-energy tails, while ${\mu }_E$ provides a distinct probe for direction-dependent spectra. To assess the statistical significance of these metrics, we derive a generalization of the Li-Ma significance formula for continuous energy variables. Using a maximum-likelihood analysis within an exploratory Monte-Carlo model, we show that these variables can enhance discovery potential by amplifying localized excesses and identifying hard-spectrum regions that are not apparent in standard particle-flux maps, thereby providing a comprehensive probe of potential UHECR sources. Furthermore, ${\mu }_E$ variations associated with localized excesses are smoothed into minor perturbations when the spectrum is integrated over right-ascension or declination bands. Consequently, the presence of small- or intermediate-scale regions with distinct spectral shapes remains consistent with the absence of declination-band spectral variations recently reported by the Pierre Auger Collaboration.