A Hierarchical Shock Model of Ultra-high-energy Cosmic Rays

Abstract We propose that a hierarchical shock model—including supernova remnant shocks, galactic wind termination shocks, and accretion shocks around cosmic filaments and galaxy clusters—can naturally explain the cosmic-ray spectrum from ∼1 GeV up to ∼ 200 EeV. While this framework applies to the entire cosmic-ray spectrum, in this work we focus on its implications for ultra-high-energy cosmic rays (UHECRs). We perform a hydrodynamic cosmological simulation to investigate the power processed at shocks around clusters and filaments. The downstream flux from nearby shocks around the local filament accounts for the softer, lower-energy extragalactic component around the ankle, and the upstream escaping flux from nearby clusters accounts for the transition to a hard spectral component at the highest energies. This interpretation is in agreement with UHECR observations. We suggest that a combination of early Universe galactic outflows, cosmic-ray streaming instabilities, and a small-scale turbulent dynamo can increase magnetic fields enough to attain the required rigidities. Our simulation suggests that the available volume-averaged power density of accretion shocks exceeds the required UHECR luminosity density by 3 orders of magnitude. We show that microgauss magnetic fields at these shocks could explain both the origin of UHECRs and potentially contribute to the diffuse radio synchrotron background below 10 GHz. The shock-accelerated electrons produce a hard radio background without overproducing diffuse inverse Compton emission. These results motivate further observational tests with upcoming facilities to help distinguish accretion shocks from other UHECR sources.