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

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.

💡 Research Summary

The paper presents a novel simulation‑based inference framework that combines three‑dimensional UHECR propagation simulations (CRPropa 3) with Approximate Bayesian Computation (ABC) to infer the origin of individual ultra‑high‑energy cosmic rays. By explicitly modeling all relevant interactions (photo‑pion production, photo‑disintegration, pair production, nuclear decay, adiabatic losses) and magnetic deflections in both the Galactic magnetic field (GMF) and extragalactic magnetic field (EGMF), the authors construct a forward model that maps source parameters (sky position, distance, source energy, EGMF strength and coherence length) to observable quantities (arrival direction and energy).

Two energy scenarios for the Amaterasu particle— a nominal 244 EeV and a lower systematic bound of 168 EeV— are explored. The source composition is fixed to iron, motivated by acceleration arguments and consistency with previous heavy‑composition studies. Priors are chosen to be as uninformative as possible while respecting physical constraints: log‑uniform priors for (B_{\rm rms}) (0.1–10 nG) and (L_{\rm c}) (60–1000 kpc), a power‑law (E^{-1}) prior for the source energy ranging from three sigma below the detected energy up to 3 ZeV, and a uniform‑in‑volume prior for the source distance limited to 12 Mpc (15 Mpc for the low‑energy case). The source direction prior is a von Mises‑Fisher distribution centered on the mean deflection obtained by back‑tracking an iron nucleus through the chosen GMF model, with a width that accommodates the larger of the GMF and estimated EGMF deflections (capped at 87° to retain computational efficiency).

The ABC algorithm samples from these priors, runs a CRPropa simulation with (10^{6}) isotropically emitted particles for each parameter set, and accepts those that reproduce the observed arrival direction and energy within three standard deviations. Accepted samples are weighted by Gaussian likelihood factors for each observable, yielding a posterior distribution over the six free parameters.

Results show that, contrary to earlier back‑tracking studies that placed the source deep inside the Local Void, the posterior probability density is concentrated in a broad region of the nearby universe (5–12 Mpc). For modest EGMF strengths ((B_{\rm rms}\lesssim0.1) nG) the deflections are small, and the most probable sources lie in known nearby galaxy groups such as the M81 group, the Centaurus A region, and the NGC 253 starburst galaxy. Higher EGMF values allow more distant sources to be compatible, but the log‑uniform prior heavily weights low field strengths, keeping the posterior probability for distant origins low. The analysis also demonstrates the importance of jointly fitting energy and direction: larger source distances naturally imply larger magnetic deflections and longer interaction paths, which in turn affect the expected energy loss, a correlation fully captured by the Bayesian framework.

The authors discuss several limitations. Only a single GMF model (UF23 base + JF12 turbulence) is used, which may introduce systematic bias; the distance prior is truncated at 12–15 Mpc due to computational constraints, potentially missing more distant candidates; and the ABC tolerance of 3σ is relatively generous, leading to broader posteriors. Future work will incorporate multiple GMF realizations, extend the distance prior, and explore more efficient ABC schemes (e.g., sequential Monte Carlo) to tighten tolerances.

In summary, this study introduces a powerful, fully probabilistic method for translating individual UHECR observations into source constraints, applies it to the record‑breaking Amaterasu event, and finds that a wider set of nearby astrophysical objects—not the Local Void—are viable origins. The framework sets the stage for systematic, data‑driven investigations of the highest‑energy cosmic rays, enabling the community to assess source hypotheses with quantified uncertainties and to integrate forthcoming multi‑messenger data.