A Hierarchical Shock Model of Ultra-High-Energy Cosmic Rays

TL;DR

The paper introduces a hierarchical shock acceleration model in which cosmic rays are progressively energized by nonrelativistic shocks from supernova remnants to galactic wind termination shocks, filament accretion shocks, and cluster accretion shocks, yielding a spectrum that spans ~1 GeV to ~200 EeV. A high-resolution radiation-hydrodynamic AMR simulation demonstrates that large-scale structure shocks deliver sufficient power, and magnetic fields amplified by early outflows, CR instabilities, and a turbulent dynamo can reach the μG level required for high-rigidity acceleration. The model predicts a soft, downstream component from nearby filament shocks dominating around the ankle and a hard, upstream component from cluster shocks driving the highest-energy CRs, with heavy nuclei (e.g., iron) prevailing at the extreme end. Multi-messenger and radio synchrotron signatures are explored, including constraints from gamma rays, neutrinos, and the diffuse radio background, and upcoming facilities are identified as critical tests to distinguish accretion-shock CRs from alternative UHECR sources.

Abstract

We propose that a hierarchical shock model$\unicode{x2014}$including supernova remnant shocks, galactic wind termination shocks, and accretion shocks around cosmic filaments and galaxy clusters$\unicode{x2014}$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 three 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.

Figures (4)

• Figure 1: Left: virial radius for each halo in the simulation. Center: the radius for each halo that contains half the total power being dissipated by shocks that pass our cuts. Error bars represent one standard deviation above and below the mean. Right: the maximum (orange) and the 95th percentile (blue) upstream speed, $u_1$, taken from the histogram of shock velocities for each halo.
• Figure 2: Virgo analog halo, selected for its roughly spherical accretion shock with attached filaments and a virial mass similar to the Virgo Cluster. In this analysis, the virial radius was defined as $r_{200}$ and the virial mass as $M_{200}$. All shocks were selected with $M >5$ and preshock overdensity less than $10^3$. Top left: cumulative kinetic energy processed by shocks as a function of radius. Top center: cumulative surface area weighted by Mach number. Top right: histogram of shock surface area vs. shock speed ($u_1$). Shocks within $3 r_{\rm vir}$ of any halo larger than 1% of the central halo are labeled "Halo." All other shocks were labeled as "Filament." Bottom left: projection of logarithmic energy processed by shocks. Bottom center: projection of Mach number of selected shocks. Mach numbers appear lower than their true values due to volume-averaged projection effects. Bottom right: projection of gas temperature.
• Figure 3: Coma analog halo, the most massive halo in the simulation box, with a virial mass of $10^{15.11} M_{\odot}$ comparable to the Coma Cluster.
• Figure 4: Loss timescales (colored curves) as a function of particle rigidity for different species: protons, helium, oxygen, silicon, iron. Adiabatic (Hubble) expansion losses are labeled with $t_H$. The acceleration timescales as a function of rigidity following Equation (\ref{['eq:tau_accel-numeric']}) are plotted as black solid lines with different values of $u_{1}$ (400, 1000, 2000, and 4000 km s$^{-1}$). The upstream amplified magnetic field at the length scales resonant with the cosmic rays is assumed to be $B_{\rm tot} = 1\,\mu{\rm G}$, except for one line labeled with 4 $\mu$G. The shaded band indicates the diffusion timescale, $\tau_{\rm diff}$, for $L_{\rm esc}=0.5$–2 Mpc assuming uniform magnetic field. The dashed line is the shock lifetime, $t_{\rm sh}$. The crossing points of $\tau_{\rm acc}$ with $\tau_{\rm loss}$ marks the maximum rigidity allowed for that species with the specified shock parameters.