On electromagnetic instabilities at ultra-relativistic shock waves

Recent work on Fermi acceleration at ultra-relativistic shock waves has demonstrated the need for strong amplification of the background magnetic field on very short scales. Amplification of the magnetic field by several orders of magnitude has also been suggested by observations of gamma-ray bursts afterglows, both in downstream and upstream plasmas. This paper addresses this issue of magnetic field generation in a relativistic shock precursor through micro-instabilities. In a generic superluminal configuration, the level of magnetization of the upstream plasma turns out to be a crucial parameter, notably because the length scale of the shock precursor is limited by the Larmor rotation of the accelerated particles in the background magnetic field and by the speed of the shock wave. We discuss in detail and calculate the growth rates of the following beam plasma instabilities seeded by the accelerated and reflected particle populations: for an unmagnetized shock, the Weibel and filamentation instabilities, as well as the Čerenkov resonant instabilities with electrostatic modes; for a magnetized shock, the Weibel instability and the resonant Čerenkov instabilities with the longitudinal electrostatic modes, as well as the Alfvén, Whisler and extraordinary modes. All these instabilities are generated upstream, then they are transmitted downstream. The modes excited by Čerenkov resonant instabilities take on particular importance with respect to the magnetisation of the downstream medium since, being plasma eigenmodes, they have a longer lifetime than the Weibel modes. We discuss the main limitation of the wave growth associated with the length of precursor and the magnetisation of the upstream medium for both oblique and parallel relativistic shock waves. We also characterize the proper conditions to obtain Fermi acceleration at ultra-relativistic shock waves: for superluminal shock waves, the Fermi process works for values of the magnetization parameter below some critical value, and there is an intrinsic limitation of the achievable cosmic ray energy depending on the ratio of the magnetization to its critical value. We recover results of most recent particle-in-cell simulations and conclude with some applications to astrophysical cases of interest. In particular, Fermi acceleration in pulsar winds is found to be unlikely whereas its development appears to hinge on the level of upstream magnetization in the case of ultra-relativistic gamma-ray burst external shock waves.