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Abstract Diffusive shock acceleration at collisionless shocks remains the most likely process for accelerating particles in a variety of astrophysical sources. While the standard prediction for strong shocks is that the spectrum of accelerated particles is universal, f ( p ) ∝ p −4 , numerous phenomena affect this simple conclusion. In general, the nonlinear dynamical reaction of accelerated particles leads to a concave spectrum, steeper than p −4 at momenta below a few tens of GeV c −1 and harder than the standard prediction at high energies. However, the nonlinear effects become important in the presence of magnetic field amplification, which in turn leads to higher values of the maximum momentum p max . It was recently discovered that the self-generated perturbations that enhance particle scattering, when advected downstream, move in the same direction as the background plasma, so that the effective compression factor at the shock decreases and the spectrum becomes steeper. We investigate the implications of the excitation of the non-resonant streaming instability on these spectral deformations, the dependence of the spectral steepening on the shock velocity, and the role played by the injection momentum. 

The recent discoveries in the theory of diffusive shock acceleration (DSA) that stem from first-principle kinetic plasma simulations are discussed. When ion acceleration is efficient, the back-reaction of non-thermal particles and self-generated magnetic fields becomes prominent and leads to both enhanced shock compression and particle spectra significantly softer than the standard test-particle DSA theory. These results are discussed in the context of the non-thermal phenomenology of astrophysical shocks, with a special focus on the supernova remnant SN1006. 

Cosmic rays leave their sources mainly along the local magnetic field present in the region around the source and in doing so they excite both resonant and non-resonant modes through streaming instabilities. The excitation of these modes leads to enhanced particle scattering and in turn to a large pressure gradient that causes the formation of expanding bubbles of gas and self-generated magnetic fields. By means of hybrid particle-in-cell simulations, we demonstrate that, by exciting this instability, cosmic rays excavate a cavity around their source where the diffusivity is strongly suppressed. This phenomenon is general and is expected to occur around any sufficiently powerful cosmic ray source in the Galaxy. Our results are consistent with recent γ-ray observations where emission from the region around supernova remnants and stellar clusters have been used to infer that the diffusion coefficient around these sources is ∼10−100 times smaller than the typical Galactic one.