Nonlinear evolution of cosmic-ray driven instabilities

Extreme astrophysical objects such as relativistic jets and supernova remnants exhibit nonthermal radiative spectra highlighting the presence and the generation of suprathermal particle distributions -- $i.e.$ cosmic rays. The maximum energy reached through diffusive shock acceleration in such environments is directly conditioned by the energy injected~by the cosmic rays~in the self-generated turbulent electromagnetic field at~large scales comparable to~the Larmor radius of the most energetic particles.~Capturing the self-consistent, multiscale feedback of the turbulence on the nonthermal distribution, taking into account kinetic effects, has been a significant challenge. We will present the results of large-scale particle-in-cell kinetic simulations that capture~the generation and the nonlinear evolution of current- and pressure-driven instabilities relevant to the context of relativistic and sub-relativistic shocks.~~We will discuss how the coupling between different instabilities affects the nonlinear evolution of the system and shapes~the magnetic field energy injected at the~largest~scales.