It is known that the maximum energy of Cosmic Rays (CRs) reaches to around $10^{20}$ eV. One of the candidates of the acceleration mechanism of these highest-energy CRs is shock acceleration in a relativistic shock. However, it is also known that in a relativistic shock, particles cannot go back and forth around the shock many times, because there is almost no difference between the shock velocity and the particle velocity.
In this work, we consider density fluctuations in the shock upstream region. The interaction of density fluctuations and a relativistic shock drives the turbulence in the shock downstream region. This turbulence could change their direction of particle motion. We carried out the MagnetoHydroDynamic (MHD) simulations and test-particle simulations to check this mechanism can accelerate the CRs.
We found that particles can cross the shock front many times thanks to scattering by the downstream region. In addition, the amplitude of the density fluctuation is an important parameter to determine the acceleration efficiency. We revealed a condition where particles can cross the shock many times.