Stochastic analysis of ultrahigh-energy cosmic ray interactions

Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany

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Received: 25 September 2025
Accepted: 13 February 2026

Abstract

Context. Photonuclear interactions between ultrahigh-energy cosmic ray (UHECR) nuclei and photon fields are key to connecting the compositions observed at Earth to those emitted from the sources as well as understanding their evolution during propagation. The stochastic nature of these interactions implies that any deterministic description is an approximation, but an exact probabilistic formulation is not yet available. Currently, Monte Carlo simulations are the only means to probe the probability space of these events.

Aims. We aim to obtain a rigorous probabilistic description of UHECR interactions based on the theory of stochastic processes and to derive closed-form expressions for the probability distributions of the distance, with the inclusion of all energy losses and cosmological effects. Additionally, we aim to demonstrate the application of this description in astrophysical scenarios where UHECR-photon interactions are relevant, such as radiation models of UHECR sources and extragalactic propagation of UHECRs.

Methods. We applied the theory of Markov jump processes to the nuclear cascades produced by photonuclear interactions of UHECRs, providing definitions to characterize the cascades within this new framework. By comparing different choices of input originating from nuclear physics experiments and electromagnetic observations, we improved our understanding of the underlying physics.

Results. We developed the sought-after probabilistic description and derived analytical expressions for various types of cascades, which are defined in terms of the number of nuclei and disintegration channels involved. We discuss the fundamental properties of these cascades and their relation to physical parameters, such as the UHECR composition, the Lorentz boost, and the target photon thickness. We discuss the role of continuous energy losses in this description and show their impact in the propagation horizons of UHECRs. We computed several quantities of interest, such as the UHECR disintegration distance as a function of the nuclear mass, the composition evolution in a gamma-ray burst (GRB) source scenario, and the angular diffusion of disintegration products produced during extragalactic propagation, while accounting for interactions.