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.

We propose semi-analytic models for the electron momentum distribution in weak shocks that accounts for both in situ acceleration and re-acceleration through diffusive shock acceleration (DSA). In the former case, a small fraction of incoming electrons is assumed to be reflected at the shock ramp and pre-accelerated to the so-called injection momentum, pinj, above which particles can diffuse across the shock transition and participate in the DSA process. This leads to the DSA power-law distribution extending from the smallest momentum of reflected electrons, pref, all the way to the cutoff momentum, peq, constrained by radiative cooling. In the latter case, fossil electrons, specified by a power-law spectrum with a cutoff, are assumed to be re-accelerated from pref up to peq via DSA. We show that, in the in situ acceleration model, the amplitude of radio synchrotron emission depends strongly on the shock Mach number, whereas it varies rather weakly in the re-acceleration model. Considering the rather turbulent nature of shocks in the intracluster medium, such extreme dependence for the in situ acceleration might not be compatible with the relatively smooth surface brightness of observed radio relics.

Galaxy clusters are known to host many active galaxies (AGNs) with radio jets, which could expand to form radio bubbles with relativistic electrons in the intracluster medium (ICM). It has been suggested that fossil relativistic electrons contained in remnant bubbles from extinct radio galaxies can be re-accelerated to radio-emitting energies by merger-driven shocks via diffusive shock acceleration (DSA), leading to the birth of radio relics detected in clusters. In this study we assume that such bubble consist primarily of thermal gas entrained from the surrounding medium and dynamically-insignificant amounts of relativistic electrons. We also consider several realistic models for magnetic fields in the cluster outskirts, including the ICM field that scales with the gas density as BICM ∝ n0.5 ICM. Then we perform time-dependent DSA simulations of a spherical shock that runs into a lower-density but higher-temperature bubble with the ratio nb/nICM ≈ TICM/Tb ≈ 0.5. We find that inside the bubble the shock speed increases by about 20 %, but the Mach number decreases by about 15% in the case under consideration. In this re-acceleration model, the observed properties of a radio relic such as radio flux, spectral index, and integrated spectrum would be governed mainly by the presence of seed relativistic electrons and the magnetic field profile as well as shock dynamics. Thus it is crucial to understand how fossil electrons are deposited by AGNs in the ICM and how the downstream magnetic field evolves behind the shock in detailed modeling of radio relics.

We explore the shock acceleration model for giant radio relics, in which relativistic electrons are accelerated via diffusive shock acceleration (DSA) by merger-driven shocks in the outskirts of galaxy clusters. In addition to DSA, turbulent acceleration by compressive MHD modes downstream of the shock are included as well as energy losses of postshock electrons due to Coulomb scattering, synchrotron emission, and inverse Compton scattering off the cosmic background radiation. Considering that only a small fraction of merging clusters host radio relics, we favor a reacceleration scenario in which radio relics are generated preferentially by shocks encountering the regions containing low-energy ( e . 300) cosmic ray electrons (CRe). We perform time-dependent DSA simulations of spherically expanding shocks with physical parameters relevant for the Sausage radio relic, and calculate the radio synchrotron emission from the accelerated CRe. We find that significant level of postshock turbulent acceleration is required in order to reproduce broad profiles of the observed radio flux densities of the Sausage relic. Moreover, the spectral curvature in the observed integrated radio spectrum can be explained, if the putative shock should have swept up and exited out of the preshock region of fossil CRe about 10~Myr ago.

The Sausage radio relic is the arc-like radio structure in the cluster CIZA J2242.8+5301, whose observed properties can be best understood by synchrotron emission from relativistic electrons accelerated at a merger-driven shock. However, there remain a few puzzles that cannot be explained by the shock acceleration model with only in-situ injection. In particular, the Mach number inferred from the observed radio spectral index, Mradio  4.6, while the Mach number estimated from X-ray observations, MX−ray  2.7. In an attempt to resolve such a discrepancy, here we consider the re-acceleration model in which a shock of Ms  3 sweeps through the intracluster gas with a pre-existing population of relativistic electrons. We find that observed brightness profiles at multi frequencies provide strong constraints on the spectral shape of pre-existing electrons. The models with a power-law momentum spectrum with the slope, s  4.1, and the cutoff Lorentz factor, e,c  3−5×104, can reproduce reasonably well the observed spatial profiles of radio fluxes and integrated radio spectrum of the Sausage relic. The possible origins of such relativistic electrons in the intracluster medium remain to be investigated further.

The Toothbrush radio relic associated with the merging cluster 1RXS J060303.3 is presumed to be produced by relativistic electrons accelerated at merger-driven shocks. Since the shock Mach number inferred from the observed radio spectral index, Mradio  2:8, is larger than that estimated from X-ray observations, MX . 1:5, we consider the re-acceleration model in which a weak shock of Ms  1:2 - 1:5 sweeps through the intracluster plasma with a preshock population of relativistic electrons. We nd the models with a power-law momentum spectrum with the slope, s  4:6, and the cutoff Lorentz factor, e;c  7-8104 can reproduce reasonably well the observed proles of radio uxes and integrated radio spectrum of the head portion of the Toothbrush relic. This study conrms the strong connection between the ubiquitous presence of fossil relativistic plasma originated from AGNs and the shock-acceleration model of radio relics in the intracluster medium.

Most high energy cosmic rays (CRs) are thought to be produced by diffusive shock acceleration (DSA) in supernova remnants (SNRs) within the Galaxy. Plasma and MHD simulations have shown that the self-excitation of MHD waves and amplication of magnetic fields via plasma instabilities are an integral part of DSA for strong collisionless shocks. In this study we explore how plasma processes such as plasma instabilities and wave-particle interactions can affect the energy spectra of CR protons and electrons, using time-dependent DSA simulations of SNR shocks. We demonstrate that the time-dependent evolution of the shock dynamics, the self-amplified magnetic fields and Alfvenic drift govern the highest energy end of the CR energy spectra. As a result, the spectral cutoffs in nonthermal X-ray and γ-ray radiation spectra are regulated by the evolution of the highest energy particles, which are injected at the early phase of SNRs. We also nd that the maximum energy of CR protons can be boosted significantly only if the scale height of the magnetic field precursor is long enough to contain the diffusion lengths of the particles of interests. Thus, detailed understandings of nonlinear wave-particle interactions and time-dependent DSA simulations are crucial for understanding the nonthermal radiation from CR acceleration sources.

In Kang (2015) we calculated the acceleration of cosmic-ray electrons at weak spherical shocks that are expected to form in the cluster outskirts, and estimated the diffuse synchrotron radiation emitted by those electrons. There we demonstrated that, at decelerating spherical shocks, the volume integrated spectra of both electrons and radiation deviate significantly from the test-particle power-laws predicted for constant planar shocks, because the shock compression ratio and the flux of inject electrons decrease in time. In this study, we consider spherical blast waves propagating through a constant density core surrounded by an isothermal halo with  ∝ r−n in order to explore how the deceleration of the shock affects the radio emission from accelerated electrons. The surface brightness profile and the volumeintegrated radio spectrum of the model shocks are calculated by assuming a ribbon-like shock surface on a spherical shell and the associated downstream region of relativistic electrons. If the postshock magnetic field strength is about 0.7 or 7 μG, at the shock age of ∼ 50 Myr, the volume-integrated radio spectrum steepens gradually with the spectral index from inj to inj + 0.5 over 0.1–10 GHz, where inj is the injection index at the shock position expected from the diffusive shock acceleration theory. Such gradual steepening could explain the curved radio spectrum of the radio relic in cluster A2266, which was interpreted as a broken power-law by Trasatti et al. (2015), if the relic shock is young enough so that the break frequency is around 1 GHz.

We study the evolution of the energy spectrum of cosmic-ray electrons accelerated at spherically expanding shocks with low Mach numbers and the ensuing spectral signatures imprinted in radio synchrotron emission. Time-dependent simulations of diffusive shock acceleration (DSA) of electrons in the test-particle limit have been performed for spherical shocks with parameters relevant for typical shocks in the intracluster medium. The electron and radiation spectra at the shock location can be described properly by the test-particle DSA predictions with instantaneous shock parameters. However, the volume integrated spectra of both electrons and radiation deviate significantly from the test-particle power-laws, because the shock compression ratio and the flux of injected electrons at the shock gradually decrease as the shock slows down in time. So one needs to be cautious about interpreting observed radio spectra of evolving shocks based on simple DSA models in the test-particle regime.

Nonthermal radiation from supernova remnants (SNRs) provides observational evidence and constraints on the diffusive shock acceleration (DSA) hypothesis for the origins of Galactic cosmic rays (CRs). Recently it has been recognized that a variety of plasma wave-particle interactions operate at astrophysical shocks and the detailed outcomes of DSA are governed by their complex and nonlinear interrelationships. Here we calculate the energy spectra of CR protons and electrons accelerated at Type Ia SNRs, using time-dependent, DSA simulations with phenomenological models for magnetic field amplification due to CR streaming instabilities, Alfv´enic drift, and free escape boundary. We show that, if scattering centers drift with the Alfv´en speed in the amplified magnetic fields, the CR energy spectrum is steepened and the acceleration efficiency is significantly reduced at strong CR modified SNR shocks. Even with fast Afv´enic drift, DSA can still be efficient enough to develop a substantial shock precursor due to CR pressure feedback and convert about 20-30% of the SN explosion energy into CRs. Since the high energy end of the CR proton spectrum is composed of the particles that are injected in the early stages, in order to predict nonthermal emissions, especially in X-ray and -ray bands, it is important to follow the time dependent evolution of the shock dynamics, CR injection process, magnetic field amplification, and particle escape. Thus it is crucial to understand the details of these plasma interactions associated with collisionless shocks in successful modeling of nonlinear DSA.

We calculate the energy spectra of cosmic ray (CR) protons and electrons at a plane shock with quasi-parallel magnetic fields, using time-dependent, diffusive shock acceleration (DSA) simulations, including energy losses via synchrotron emission and Inverse Compton (IC) scattering. A thermal leakage injection model and a Bohm type diffusion coefficient are adopted. The electron spectrum at the shock becomes steady after the DSA energy gains balance the synchrotron/IC losses, and it cuts off at the equilibrium momentum peq. In the postshock region the cutoff momentum of the electron spectrum decreases with the distance from the shock due to the energy losses and the thickness of the spatial distribution of electrons scales as p-1. Thus the slope of the downstream integrated spectrum steepens by one power of p for pb < p < peq, where the break momentum decreases with the shock age as pbr ∝ t-1. In a CR modified shock, both the proton and electron spectrum exhibit a concave curvature and deviate from the canonical test-particle power-law, and the upstream integrated electron spectrum could dominate over the downstream integrated spectrum near the cutoff momentum. Thus the spectral shape near the cutoff of X-ray synchrotron emission could reveal a signature of nonlinear DSA.

Shocks are ubiquitous in astrophysical environments and cosmic-rays (CRs) are known to be accelerated at collisionless shocks via diffusive shock acceleration. It is believed that the CR pressure is important in the evolution of the interstellar medium of our galaxy and most of galactic CRs with energies up to ~ 1015 eV are accelerated by supernova remnant shocks. In this contribution we have studied the CR acceleration at shocks through numerical simulation of 1D, quasi-parallel shocks for a wide range of shock Mach numbers and shock speeds. We show that CR modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies, and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We find that 10-4 - 10-3 of the particles passed through the shock are accelerated to form the CR population, and the injection rate is higher for shocks with higher Mach number. The time asymptotic value for the CR acceleration efficiency is controlled mainly by shock Mach number, and high Mach number shocks all evolve towards efficiencies ~ 50%, regardless of the injection rate and upstream CR pressure. We conclude that the injection rates in strong quasi-parallel shocks are sufficient to lead to significant nonlinear modifications to the shock structures, implying the importance of the CR acceleration at astrophysical shocks.