Intrinsic Spectral Curvature from Finite-Cycle Transport at Relativistic Shocks

Abstract

Power-law spectra are a central prediction of shock acceleration and are commonly associated with asymptotic scale invariance under diffusive transport. In finite relativistic shocks, strong anisotropy and limited residence times may restrict the number of effective shock crossings before the many-cycle diffusive limit is established. This work develops a reduced finite-cycle framework in which particle energization is described by discrete shock-crossing mappings, while downstream transport is encoded through an energy-dependent return probability. In this formulation, the local spectrum is controlled by the competition between the mean energy gain per cycle and the probability of surviving to the next cycle. A systematic decrease of the return probability with energy then produces intrinsic spectral curvature as a consequence of transport-limited cycle survival. The energy dependence of the return probability is estimated from the competition between magnetic deflection, downstream advection, and finite shock lifetime, yielding a characteristic steepening scale determined by macroscopic source parameters. For fiducial parameters relevant to compact blazar emission regions, the steepening scale lies below the ultimate acceleration cutoff, so that curvature can appear before the terminal maximum energy is reached. These results point to a pre-asymptotic finite-cycle limit of relativistic shock transport in which non-power-law spectra can arise from the limited survival of repeated shock-crossing cycles.