During the formation of large-scale structures in the universe, weak internal shocks are induced within the hot intracluster medium (ICM), while strong accretion shocks arise in the warm-hot intergalactic medium (WHIM) within filaments, and the warm-cold gas in voids surrounding galaxy clusters. These cosmological shocks are thought to accelerate cosmic ray (CR) protons and electrons via diffusive shock acceleration (DSA). Recent advances in particle-in-cell and hybrid simulations have provided deeper insights into the kinetic plasma processes that govern microinstabilities and particle acceleration in collisionless shocks in weakly magnetized astrophysical plasma. In this study, we adopt a thermal-leakage type injection model and DSA power-law distribution functions in the test-particle regime. The CR proton spectrum directly connects to the Maxwellian distribution of protons at the injection momentum pinj = Qppth,p. On the other hand, the CR electron spectrum extends down to pmin = Qepth,e and is linked to the Maxwellian distribution of electrons. Here, pth,p and pth,e, are the proton and electron thermal momenta, respectively. Moreover, we propose that the postshock gas temperature and the injection parameters, Qp and Qe are self-regulated to maintain the test-particle condition, as the thermal energy is gradually transferred to the CR energy. Under these constraints, we estimate the self-regulated values of the temperature reduction factor, RT , and the proton injection parameter, Qp, along with the resulting CR efficiencies, ηp and ηe. We then provide analytical fitting functions for these parameters as functions of the shock Mach number, Ms. These fitting formulas may serve as valuable tools for quantitatively assessing the impact of CR protons and electrons, as well as the resulting nonthermal emissions in galaxy clusters and cosmic filaments.

This study investigates the impact of magnetic turbulence on cosmic ray (CR) electrons through Fermi-II acceleration behind merger-driven shocks in the intracluster medium and examines how the ensuing synchrotron radio emission is influenced by the decay of magnetic energy through dissipation in the postshock region. We adopt simplified models for the momentum diffusion coefficient, specifically considering transit-time-damping resonance with fast-mode waves and gyroresonance with Alfvén waves. Utilizing analytic solutions derived from diffusive shock acceleration theory, at the shock location, we introduce a CR spectrum that is either shock-injected or shock-reaccelerated. We then track its temporal evolution along the Lagrangian fluid element in the time domain. The resulting CR spectra are mapped onto a spherical shell configuration to estimate the surface brightness profile of the model radio relics. Turbulent acceleration proves to be a significant factor in delaying the aging of postshock CR electrons, while decaying magnetic fields have marginal impacts due to the dominance of inverse Compton cooling over synchrotron cooling. However, the decay of magnetic fields substantially reduces synchrotron radiation. Consequently, the spatial distribution of the postshock magnetic fields affects the volume-integrated radio spectrum and its spectral index. We demonstrate that the Mach numbers estimated from the integrated spectral index tend to be higher than the actual shock Mach numbers, highlighting the necessity for accurate modeling of postshock magnetic turbulence in interpreting observations of radio relics.

The intracluster medium (ICM) is expected to experience on average about three passages of weak shocks with low sonic Mach numbers, M . 3, during the formation of galaxy clusters. Both protons and electrons could be accelerated to become high energy cosmic rays (CRs) at such ICM shocks via diusive shock acceleration (DSA). We examine the eects of DSA by multiple shocks on the spectrum of accelerated CRs by including in situ injection/acceleration at each shock, followed by repeated re- acceleration at successive shocks in the test-particle regime. For simplicity, the accelerated particles are assumed to undergo adiabatic decompression without energy loss and escape from the system, before they encounter subsequent shocks. We show that in general the CR spectrum is attened by multiple shock passages, compared to a single episode of DSA, and that the acceleration eciency increases with successive shock passages. However, the decompression due to the expansion of shocks into the cluster outskirts may reduce the amplication and attening of the CR spectrum by multiple shock passages. The nal CR spectrum behind the last shock is determined by the accumulated eects of repeated re-acceleration by all previous shocks, but it is relatively insensitive to the ordering of the shock Mach numbers. Thus multiple passages of shocks may cause the slope of the CR spectrum to deviate from the canonical DSA power-law slope of the current shock.