Unfolding the Energy Spectrum of Ultra-High-Energy Cosmic Rays Using Pierre Auger Open Data

TL;DR

The paper tackles reproducing the Pierre Auger unfolded UHECR energy spectrum using open data alone, where generator-level truth is unavailable. It constructs the absolute migration matrix $M_{ij}=R_{ij}N_{corr,j}$ from published quantities, then samples a correlated truth–reco pseudo-MC using this matrix to enable both classical and ML unfolding, including OmniFold, and validates the results against the published $N_{corr}$. The study demonstrates that the open data, when combined with the published response, suffice to recreate the unfolded spectrum and to benchmark ML-based unfolding in a low-statistics regime, highlighting both the promise and current limitations of Open Data-driven unfolding. This work provides a practical pipeline for detector-model reconstruction from public data and offers guidance for future open-data analyses in high-energy astroparticle physics.

Abstract

We reconstruct the energy spectrum of ultra-high-energy cosmic rays using the publicly released Pierre Auger Observatory data set. Since event-level Monte Carlo truth information is not included in the open data, we develop a consistent procedure to regenerate a pseudo-Monte Carlo sample directly from the published quantities: the registered event counts $N$, the unfolded spectrum $N_\mathrm{corr}$, and the detector response matrix $R_{ij}$ from the Auger 2020 spectrum data analysis. Using the row-normalized response matrix and the published unfolded spectrum as a truth prior, we construct an absolute-level migration matrix and generate the event-by-event truth and reconstructed-level pairs by drawing from a two-dimensional probability distribution function. The resulting sample statistically replicates the detector response properties of the Pierre Auger Surface Detector. This pseudo-MC sample allows for the application of classical unfolding techniques (bin-by-bin and iterative Bayesian unfolding via RooUnfold) as well as a machine-learning-based unfolding using OmniFold. We demonstrate that using such publicly available information this approach allows the full unfolding procedure.

Figures (7)

• Figure 1: Flow diagram of the preparatory work and unfolding methods used in this work. Measured counts $N_i$, unfolded reference counts $N_{\mathrm{corr},i}$, and the published response matrix $R_{ij}$ are combined to construct an absolute migration matrix $M_{ij}$. A pseudo-MC sample is generated by randomly sampling a two-dimensional histogram, enabling both classical and machine-learning unfolding approaches, results of which are then compared to the published unfolded spectrum.
• Figure 2: Reconstructed absolute migration matrix $M_{ij} = R_{ij}\,N_{\mathrm{corr},j}$ obtained by scaling the published Pierre Auger row-normalized response matrix $R_{ij}$ with the unfolded spectrum $N_\mathrm{corr}$ from Table VI of Ref. PierreAuger:2020qqz. This matrix represents the expected number of events migrating between true and reconstructed energy bins and forms the basis for generating a self-consistent pseudo-Monte Carlo dataset.
• Figure 3: Two-dimensional distribution of the (truth,reconstructed) energies for pseudo--Monte Carlo events generated from the original migration matrix $M_{ij}$ by randomly sampling a two-dimensional histogram. This sampling converts the expected migration rates into an event-level dataset with correlated truth and reconstructed energies. This pseudo--MC sample is used to train the OmniFold unfolding as well as to validate classical unfolding methods.
• Figure 4: Comparison of the EAS unfolded energy spectra obtained using several methods. Shown are the measured counts $N$, the published unfolded spectrum $N_\mathrm{corr}$, the RooUnfold Bayesian and bin-by-bin results using the reconstructed response matrix, and the OmniFold machine-learning unfolding of the 10% Pierre Auger Open Data sample. A second, doubly-binned, OmniFold curve is included to emphasize the possibility of finer binning choice. The lower panel displays the ratio of all unfolded spectra to $N_\mathrm{corr}$.
• Figure 5: The binned EAS energy spectrum expressed as $J(E)\,E^{3}$ in the measured counts $N$, the published unfolded spectrum $N_\mathrm{corr}$, the RooUnfold Bayesian and bin-by-bin results, and the OmniFold unfolding of the 10% open-data sample. Multiplication by $E^{3}$ reduces the dynamic range of the flux and highlights differences in the spectral shape. The lower panel shows the ratio to $N_\mathrm{corr}$.