## Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 7, Number 3 (2013), 1249-1285.

### Multilevel Bayesian framework for modeling the production, propagation and detection of ultra-high energy cosmic rays

Kunlaya Soiaporn, David Chernoff, Thomas Loredo, David Ruppert, and Ira Wasserman

#### Abstract

Ultra-high energy cosmic rays (UHECRs) are atomic nuclei with energies over ten million times energies accessible to human-made particle accelerators. Evidence suggests that they originate from relatively nearby extragalactic sources, but the nature of the sources is unknown. We develop a multilevel Bayesian framework for assessing association of UHECRs and candidate source populations, and Markov chain Monte Carlo algorithms for estimating model parameters and comparing models by computing, via Chib’s method, marginal likelihoods and Bayes factors. We demonstrate the framework by analyzing measurements of 69 UHECRs observed by the Pierre Auger Observatory (PAO) from 2004–2009, using a volume-complete catalog of 17 local active galactic nuclei (AGN) out to 15 megaparsecs as candidate sources. An early portion of the data (“period 1,” with 14 events) was used by PAO to set an energy cut maximizing the anisotropy in period 1; the 69 measurements include this “tuned” subset, and subsequent “untuned” events with energies above the same cutoff. Also, measurement errors are approximately summarized. These factors are problematic for independent analyses of PAO data. Within the context of “standard candle” source models (i.e., with a common isotropic emission rate), and considering only the 55 untuned events, there is no significant evidence favoring association of UHECRs with local AGN vs. an isotropic background. The highest-probability associations are with the two nearest, adjacent AGN, Centaurus A and NGC 4945. If the association model is adopted, the fraction of UHECRs that may be associated is likely nonzero but is well below 50%. Our framework enables estimation of the angular scale for deflection of cosmic rays by cosmic magnetic fields; relatively modest scales of $\approx\!3^{\circ}$ to $30^{\circ}$ are favored. Models that assign a large fraction of UHECRs to a single nearby source (e.g., Centaurus A) are ruled out unless very large deflection scales are specified a priori, and even then they are disfavored. However, including the period 1 data alters the conclusions significantly, and a simulation study supports the idea that the period 1 data are anomalous, presumably due to the tuning. Accurate and optimal analysis of future data will likely require more complete disclosure of the data.

#### Article information

**Source**

Ann. Appl. Stat., Volume 7, Number 3 (2013), 1249-1285.

**Dates**

First available in Project Euclid: 3 October 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1380804795

**Digital Object Identifier**

doi:10.1214/13-AOAS654

**Mathematical Reviews number (MathSciNet)**

MR3127947

**Zentralblatt MATH identifier**

06237176

**Keywords**

Multilevel modeling hierarchical Bayes astrostatistics cosmic rays active galactic nuclei directional data coincidence assessment Bayes factors Chib’s method Fisher distribution

#### Citation

Soiaporn, Kunlaya; Chernoff, David; Loredo, Thomas; Ruppert, David; Wasserman, Ira. Multilevel Bayesian framework for modeling the production, propagation and detection of ultra-high energy cosmic rays. Ann. Appl. Stat. 7 (2013), no. 3, 1249--1285. doi:10.1214/13-AOAS654. https://projecteuclid.org/euclid.aoas/1380804795

#### Supplemental materials

- Supplementary material: Technical appendices. The online supplement contains six technical appendices with detailed material on the following topics:
A. Auger observatory exposure; B. Propagation effects on cosmic ray energies; C. Algorithm for Markov chain Monte Carlo; D. Cen A single-source model; E. Comparison with prior Bayesian work; F. Model checking.
Digital Object Identifier: doi:10.1214/13-AOAS654SUPP