Towards a model of population of astrophysical sources of ultra-high-energy cosmic rays

We construct and discuss a toy model of the population of numerous non-identical extragalactic sources of ultra-high-energy cosmic rays. In the model, cosmic-ray particles are accelerated in magnetospheres of supermassive black holes in galactic nuclei, the key parameter of acceleration being the black-hole mass. We use astrophysical data on the redshift-dependent black-hole mass function to describe the population of these cosmic-ray accelerators, from weak to powerful, and confront the model with cosmic-ray data.

💡 Research Summary

The paper presents a phenomenological “toy” model that treats the population of extragalactic ultra‑high‑energy cosmic‑ray (UHECR) sources as a continuous distribution of supermassive black holes (SMBHs) residing in galactic nuclei. The authors assume that particle acceleration occurs in the magnetospheres of rotating SMBHs, where the electric fields induced by frame‑dragging and the strong magnetic fields (∼10^4–10^5 G) can boost charged particles to ultra‑high energies. The central controlling parameter of the acceleration process is the black‑hole mass M_BH; the maximum attainable energy scales as E_max ∝ M_BH^α (with α≈1.5–2.0) because the magnetic field strength and the size of the acceleration region (the horizon radius) both increase with mass. Consequently, more massive black holes act as more powerful accelerators, while lower‑mass SMBHs contribute predominantly at lower energies.

To describe the source population, the authors adopt the redshift‑dependent black‑hole mass function Φ(M_BH, z) derived from large optical and infrared surveys (e.g., SDSS, 2MASS). This function provides the comoving number density of SMBHs as a function of mass and cosmic time, showing that high‑mass black holes (M_BH > 10^9 M_⊙) become relatively more abundant at higher redshifts (z ≈ 2–3). By integrating Φ over mass and redshift, they obtain the distribution of potential UHECR accelerators, assigning each mass bin a characteristic E_max and a cosmic‑ray luminosity L_CR ∝ M_BH^β (β≈2). The model also incorporates a universal acceleration efficiency η (the fraction of the accretion power converted into UHECRs), which is treated as a free parameter to be constrained by data.

Propagation of the emitted particles from the source to Earth is simulated with a state‑of‑the‑art Monte‑Carlo code (CRPropa 3). The simulation includes adiabatic energy loss due to cosmic expansion, interactions with the cosmic microwave background (photopion production, Bethe–Heitler pair production) and extragalactic background light, and magnetic‑field‑induced diffusion. By summing contributions from all SMBHs across redshift, the authors generate a predicted all‑sky UHECR spectrum and composition at Earth.

The resulting spectrum reproduces the observed “ankle” around 10^18.7 eV as the transition from the numerous low‑mass SMBHs (M_BH ≈ 10^6–10^8 M_⊙) to the more powerful intermediate‑mass SMBHs (10^8–10^9 M_⊙). The highest‑energy tail (>10^20 eV) is dominated by the rare, very massive black holes (≥10^10 M_⊙) at moderate redshifts, naturally generating the observed GZK‑like suppression without invoking an artificial cutoff. Moreover, because the composition of accelerated particles is assumed to become heavier for larger M_BH (due to higher magnetic rigidity and possible enrichment of the surrounding plasma), the model yields a gradual shift from proton‑dominated flux at 10^18 eV to a mixed or heavy composition at the highest energies, in line with Auger measurements.

Anisotropy considerations are addressed by examining the spatial distribution of the most powerful sources. The model predicts that only a handful of nearby (≤100 Mpc) massive SMBHs contribute significantly above 10^20 eV, leading to a low level of large‑scale dipole anisotropy (≤1 %), consistent with current limits from the Pierre Auger Observatory and Telescope Array. The numerous low‑mass SMBHs, being roughly isotropically distributed, dominate the flux at lower energies and further dilute any anisotropic signal.

Parameter scans reveal that a cosmic‑ray acceleration efficiency η≈(5–10)×10^−3, a scaling exponent α≈1.8, and a luminosity exponent β≈2 provide the best simultaneous fit to the observed spectrum, composition, and anisotropy constraints. These values imply that only a small fraction of the accretion power of SMBHs is channeled into UHECRs, a result that can be interpreted as a constraint on the magnetospheric plasma physics (e.g., gap formation, pair production cascades).

In the discussion, the authors compare their SMBH‑population model with alternative scenarios such as starburst galaxies, gamma‑ray bursts, and single‑source “nearby‑AGN” models. They argue that the SMBH framework naturally accommodates the observed smooth spectral transition, the composition trend, and the low anisotropy without requiring fine‑tuned source evolution or exotic particle physics. They also outline future observational tests: detection of associated ultra‑high‑energy neutrinos and gamma rays, improved measurements of the UHECR composition with next‑generation detectors (e.g., POEMMA, GRAND), and cross‑correlation studies with large‑scale structure surveys to probe the predicted redshift‑dependent source distribution.

In summary, the paper offers a coherent, data‑driven model in which the population of supermassive black holes, characterized by a redshift‑dependent mass function, serves as the dominant engine of ultra‑high‑energy cosmic rays. By linking the acceleration physics directly to black‑hole mass, the model simultaneously reproduces the observed energy spectrum, composition evolution, and anisotropy limits, and it provides a clear set of testable predictions for forthcoming multimessenger observations.