Analytical study of diffusive relativistic shock acceleration

Particle acceleration in relativistic shocks is studied analytically in the test-particle, small-angle scattering limit, for an arbitrary velocity-angle diffusion function D. The particle spectral index s is found to be sensitive to D, particularly downstream and at certain angles. The analysis, confirmed numerically, justifies and generalizes previous results for isotropic diffusion. It can be used to test collisionless shock models and to observationally constrain D. For example, strongly forward- or backward-enhanced diffusion downstream is ruled out by gamma-ray burst afterglow observations.