What can cosmic-ray knees reveal about source populations?

¹ University of Crete, Department of Physics & Institute of Theoretical & Computational Physics 70013 Herakleio, Greece
² Institute of Astrophysics, Foundation for Research and Technology-Hellas 71110 Heraklion Crete, Greece

^★ Corresponding authors: This email address is being protected from spambots. You need JavaScript enabled to view it. ; This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 23 April 2024
Accepted: 26 October 2025

Abstract

Context. Breaks in the cosmic-ray (CR) flux spectrum encode information on the properties of CR accelerator populations producing the observed flux. Spectral steepenings, known as knees, are generally accompanied by a transition to a higher-mass composition.

Aims. We seek generic features of CR source populations that are robustly enough imprinted on knee observables to be discernible even in the presence of significant uncertainties in CR data. We explore how the diversity among population members imprints on the knee phenomenology under the assumption that a knee arises from a fixed-rigidity cutoff in the source spectrum. Our scope is explicitly exclusionary: We did not fit specific datasets, but determined which observed spectral features are incompatible with a single-population fixed-rigidity cutoff picture, which would indicate additional physics.

Methods. We used a simple theoretical model for a population of CR accelerators. Each member of the population stochastically accelerated CR to a power-law spectrum up to a cutoff rigidity that resulted from source-confinement requirements. We allowed variance among the members in the cutoff rigidity and in the power-law slope.

Results. We found that (a) the slope step of the flux spectrum is ∼0.5 and decreased weakly with increasing spread in either property, (b) composition always broke first, and (c) the difference between the break energies in composition and flux increased with increasing diversity. These trends are robust under our assumptions. Deviations from them in observed data would indicate more complex physics than encoded in our simple model.

Conclusions. From comparing these trends with observed CR knees, we conclude that (i) the primary knee at ∼4 × 10¹⁵ eV is consistent with a constant-rigidity cutoff according to KASCADE-Grande data processed with post-LHC hadronic models, but not according to other datasets, (ii) the second knee at ∼5 × 10¹⁷ eV conclusively requires more complexity than the cutoff of a single CR source population, (iii) the constant-rigidity source cutoff interpretation of the spectral feature identified by Auger at ∼10¹⁹ eV cannot be rejected when the cutoff rigidity and slope in the parent source population are substantial. Interestingly, a significant spread in slope would also result in the spectral curvature before the break, which in turn might contribute to the ankle feature.