Beyond the Local Void: A data-driven search for the origins of the Amaterasu particle

TL;DR

The paper tackles the challenge of identifying the origins of individual ultra-high-energy cosmic rays by introducing a data-driven, simulation-based inference framework that combines 3D propagation modeling with Approximate Bayesian Computation. By using CRPropa 3 to simulate realistic energy losses and magnetic deflections, and by jointly constraining the observed energy and arrival direction, the method yields posterior distributions over a 3D source volume and key propagation parameters for Amaterasu. The study reveals that Amaterasu’s origins are consistent with regions outside the Local Void, with nearby galaxies such as M82, NGC 6946, and NGC 2403 emerging as plausible candidates depending on energy and composition assumptions, thereby providing a richer, interpretable view of potential sources. This framework offers a foundation for future simulation-based analyses of individual UHECR events and highlights the importance of energy, composition, and magnetic-field modeling in source identification.

Abstract

We introduce a simulation-based inference framework to constrain the origins of individual ultra-high-energy cosmic rays by combining realistic three-dimensional propagation modeling with Bayesian parameter estimation. Our method integrates CRPropa 3 simulations, including all relevant interactions and magnetic deflections in both Galactic and extra-Galactic fields, with Approximate Bayesian Computation to infer posterior distributions over key parameters such as source position, distance, energy, and magnetic field properties. This approach allows joint constraints from the observed energy and arrival direction to be applied simultaneously, naturally incorporating their correlations in addition to relevant modelling uncertainties. We demonstrate our method by applying it to the Amaterasu particle detected by the Telescope Array observatory, the second-highest-energy cosmic ray ever detected. The resulting posterior distributions quantify the regions of space consistent with its reconstructed properties under different energy and composition assumptions, revealing a broader set of nearby source candidates than found in previous analyses. This application highlights the framework's ability to translate individual UHECR observations into directly interpretable source constraints and provides a foundation for future simulation-based analyses of cosmic rays at the highest energies.

Figures (11)

• Figure 1: Sky maps of the prior on the source position for $E_{\text{nom}}$ (upper panel) and $E_{\text{low}}$ (lower panel). The shaded contours show the regions of highest prior density, for 10, 30, 70, 90 and 99%. As detailed in the text, our prior assumes that Amaterasu arrives as an iron nucleus when considering the Galactic deflections. For comparison, we also show the mean values and 90% contours that result from calculating our source direction prior assuming lighter nuclei for the Galactic backtracking. In particular protons, nitrogen and silicon are shown by the dark, mid and light green dashed lines, respectively.
• Figure 2: Sky maps resulting from the nominal energy run (case 1) showing the possible source positions of Amaterasu in Galactic coordinates. The measured arrival direction of Amaterasu and the outline of the Local Void are shown for reference. The markers show galaxies within the accepted $D_\mathrm{src}$ range, with stars indicating SBGs and AGN, and circles showing quiescent galaxies. In the upper panel, all accepted parameter sets are considered and the green contours outline the labelled regions of highest posterior density. The lower panel shows the arrival mass-dependent posterior distribution, conditional on the particle arriving with a mass number, $A$, in one of three groups. The contour levels are as in the upper panel.
• Figure 3: Sky maps of the low energy run (case 2). The layout is as in Fig. \ref{['fig:Enom_skymap']}, with the upper panel showing the total posterior distribution and the lower panel showing the composition-dependent distributions.
• Figure 4: The marginal posterior distribution of $D_\mathrm{src}$ is shown for the nominal and low energy cases in the upper and lower panels, respectively. The results are given as a stacked histogram to highlight the relative contributions of $D_\mathrm{src}$ values that lead to accepted events in different arrival mass groups. Orange, pink, and blue sections indicate the fraction of each bar that has arrival particle mass numbers of ${A \leq 4}$, ${4 < A \leq 28}$, and $A > 28$, respectively. The height of the bar gives the total density summed over all mass numbers and the histogram is normalised.
• Figure 5: The fractions of accepted iterations of our ABC method are shown for the selections on the arrival direction and the arrival energy. Note the different scales on the left and right y-axes.