Stochastic analysis of ultra-high energy cosmic ray interactions

Abstract

Photonuclear interactions between ultra-high-energy cosmic ray (UHECR) nuclei and surrounding photon fields are key to understanding the connection between the compositions observed at Earth and those emitted from the sources. These interactions can completely disintegrate a nucleus of iron over trajectory lengths of a few and up to hundreds of megaparsecs, depending on the energy of the UHECR. The stochastic nature of these interactions means that it is not possible to describe them deterministically for a single cosmic ray, and an exact formulation of the probability distributions is not yet available. Current approaches describe these interactions using either Monte Carlo simulations or solving ordinary differential equations that neglect stochasticity. Because of the limitations of these approaches, only partial capture of the process is achieved. This paper presents an analytic probabilistic description of UHECR interactions and the resulting nuclear cascades, establishing their connection to Markov jump processes. The fundamental properties of these cascades are presented, as is the computation of the usual quantities of interest, such as the horizon, spectrum, and composition. The benefits of this description are outlined using astrophysical examples related to extragalactic propagation and UHECR sources.

Figures (8)

• Figure 1: Estimation of the deviation from regularity of photonuclear cross-sections. Top: Dependence of the energy-weighted photodisintegration cross-section divided by the nuclear mass as a function of photon energy. The lines show the average over all nuclear species in the respective model. The shaded bands represent the standard deviation at each energy and are centered on the mean. Bottom: The coefficient of variation (standard deviation divided by the mean) at each energy.
• Figure 2: Left: Probability of finding an injected nucleus or composition of nuclei fully disintegrated after propagating a specified distance ($\gamma=7\cdot10^9$). The green and purple solid lines in the extremes correspond to $^4$He and $^{56}$Fe injection, respectively, representing the lightest and heaviest initial compositions. The black solid line uses a similar composition as obtained in fits of the UHECR spectrum Abdul_Halim_2024. The specific nuclei and their approximate fractions are given in the legend. The dashed black line represents the case where all species share the same fraction and the black dot-dashed black line a composition reflecting solar abundances. Right: Occupation probabilities for species in the nuclear cascade for $^{56}$Fe injection ($\gamma=7\cdot10^9$). The values are given for two propagation distances in the range where the full disintegration probability is negligible: 2 Mpc (purple) and 10 Mpc (green).
• Figure 3: Density functions of distance until reaching different values of nuclear mass, the variation for the boost $\gamma \in [4\cdot10^8, 3\cdot10^{10}]$ is represented by the shaded bands. The distributions are standardized and centered at the expected value, as they span different scales at different boosts.
• Figure 4: Cosmic ray horizons of $^{14}$N (left) and $^{56}$Fe (right) in the background photon fields. The widely used energy loss length (dashed red) overestimates the effect of interactions. The $L_{\rm FD}$ horizons correspond to the distance where the full disintegration probability is 99 %: in dot-dashed green the values for the homogeneous case, in solid purple the values assuming coherent inhomogeneities, and in dotted orange the values obtained numerically treating the redshift dependence of the rates and adiabatic losses.
• Figure 5: Expected distance until full disintegration in units of the inverse of the mean interaction rate per nucleon $1/\lambda_1$. The black line corresponds to the canonical cascade (RSeC), the dashed lines represent ISeCs, and the scattered points show values for CoCs, where multiple points appear for each mass corresponding the multiple isobars. The boost is indicated by the color, as listed in the legend.