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Relativistic collisionless shocks are believed to be efficient particleaccelerators. Nonlinear outcome of the interaction of accelerated particlesthat run ahead of the shock, the so-called "precursor", with the unperturbedplasma of the shock upstream, is thought to facilitate additional accelerationof these particles and to possibly modify the hydrodynamic structure of theshock. We explore here the linear growth of kinetic modes appearing in theprecursor-upstream interaction in relativistic shocks propagating in non andweakly magnetized plasmas: electrostatic two-stream parallel mode andelectrostatic oblique modes. These modes are of particular interest becausethey are the fastest growing modes known in this type of system. Using asimplified distribution function for a dilute ultra-relativistic beam that isrelativistically hot in its own rest frame, yet has momenta that are narrowlycollimated in the frame of the cold upstream plasma into which it propagates,we identify the fastest growing mode in the full $k$-space and calculate itsgrowth rate. We consider all types of plasma (pairs and ions-electrons) andbeam (charged and charge-neutral). We find that unstable electrostatic modesare present in any type of plasma and for any shock parameters. We further findthat two modes, one parallel ($k_\perp=0$) and the other one oblique ($k_\perp\sim k_\|$), are competing for dominance and that either one may dominate thegrowth rate in different regions of the phase space. The dominant mode isdetermined mostly by the perpendicular spread of the accelerated particlemomenta in the upstream frame, which reflects the shock Lorentz factor. Theparallel mode becomes more dominant in shocks with lower Lorentz factors (i.e.,with larger momentum spreads). We briefly discuss possible implications of ourresults for external shocks in gamma-ray burst sources.

Two-stream-like instability in dilute hot relativistic beams and astrophysical relativistic shocks