Breaking the universal power law: Advanced models of diffusive shock acceleration in Galactic sources

Diffusive shock acceleration (DSA) is a promising acceleration process in a hierarchy of sources, reaching from stellar termination to Galaxy cluster shocks. Two of the relevant parameters defining the maximal energy reached at the accelerator are the turbulent magnetic field and the shock's lifetime. In this contribution, we show how time-dependent models in complex geometries and diffusion descriptions beyond a simple scaling in energy, taking structured turbulence and non-linear feedback of the CRs into account, lead to deviations from the universal $E^{-2}$ power law at a strong shock. By modeling shocks in the Galactic halo, we show how time-dependent transport gives rise to harder energy spectra and the collision of shocks can lead to an enhancement at high energies. The influence of energy-dependent escape including adiabatic cooling on the spectrum is discussed. Harder energy spectra can also be obtained when superdiffusion is considered, which might result from structured turbulence close to astrophysical sources. All modeling is done in a single framework, a modified version of CRPropa 3.2 based on the integration of stochastic differential equations.