Time-dependent treatment of cosmic-ray spectral steepening due to turbulence driving

TL;DR

This paper investigates how turbulence driven by cosmic rays at supernova remnant shocks affects their energy spectrum, finding that the spectral steepening is generally negligible under typical conditions, with only extreme cases showing noticeable effects.

Contribution

It introduces a finite-time model for turbulence driving, refining previous steady-state estimates of cosmic-ray spectral steepening at shocks.

Findings

Spectral index change is at most η divided by Alfvénic Mach number.

For Mach numbers below 50, spectral steepening is minimal.

High shock speeds and efficient acceleration can cause a spectral change up to 0.1.

Abstract

Cosmic-ray acceleration at non-relativistic shocks relies on scattering by turbulence that the cosmic rays drive upstream of the shock. We explore the rate of energy transfer from cosmic rays to non-resonant Bell modes and the spectral softening it implies. Accounting for the finite time available for turbulence driving at supernova-remnant shocks yields a smaller spectral impact than found earlier with steady-state considerations. Generally, for diffusion scaling with the Bohm rate by a factor $\eta$, the change in spectral index is at most $\eta$ divided by the Alfv\'enic Mach number of the thermal sub-shock. For $M_\mathrm{A}\lesssim 50$ it is well below this limit. Only for very fast shocks and very efficient cosmic-ray acceleration the change in spectral index may reach $0.1$. For standard SNR parameters it is negligible. Independent confirmation is derived by considering the synchrotron energy losses of electrons: if intense nonthermal multi-keV emission is produced, the energy loss, and hence the spectral steepening, is very small for hadronic cosmic rays that produce TeV-band gamma-ray emission.