On the universality of the spectrum of cosmic rays accelerated at highly relativistic shocks

I consider analytically particle acceleration at relativistic shocks, in the limit of pitch angle diffusion and large shock Lorentz factors. The derived energy spectrum index (k = (1 + √ 13)/2 ≈ 2.30), and particle pitch angle distribution at the shock are successfully compared with the results of numerical solutions. This totally analytic derivation is completely independent of the detailed dependence of the diffusion coefficient D(µ, p) on either parameter, and it is argued on a physical basis that the orientation of the magnetic field is also irrelevant, making k a universal index.