Comment on the first-order Fermi acceleration at ultra-relativistic shocks

The first-order Fermi acceleration process at an ultra-relativistic shock wave is expected to create a particle spectrum with the unique asymptotic spectral index $\sigma_{\gamma \,\gg\, 1} \approx 2.2$. Below, we discuss this result and differences in its various derivations, which – explicitly or implicitly – always require highly turbulent conditions downstream of the shock. In the presence of medium amplitude turbulence the generated particle spectrum can be much steeper than the above asymptotic one. We also note problems with application of the pitch angle diffusion model for particle transport near the ultra-relativistic shocks