Efficiency of cosmic ray reflections from an ultrarelativistic shock wave

The process of cosmic ray acceleration up to energies in excess of 1020 eV at relativistic shock waves with large Lorentz factors, Γ»1, requires ∼Γ2 particle energy gains at single reflections from the shock (cf. Gallant & Achterberg). In the present Letter, by applying numerical simulations we address an efficiency problem arising for such models. The actual efficiency of the acceleration process is expected to be substantially lower than the estimates of previous authors.