Dispersion and thermal effects on electromagnetic instabilities in the
  precursor of relativistic shocks
  Fermi acceleration can develop efficiently at relativistic collisionless
shock waves provided the upstream (unshocked) plasma is weakly magnetized. At
low magnetization, the large size of the shock precursor indeed provides enough
time for electromagnetic micro-instabilities to grow and such
micro-instabilities generate small scale turbulence that in turn provides the
scattering required. The present paper extends our previous analysis on the
development of these micro-instabilities to account for the finite angular
dispersion of the beam of reflected and accelerated particles and to account
for the expected heating of the upstream electrons in the shock precursor. We
show that the oblique two stream instability may operate down to values of the
shock Lorentz factor gamma_{sh}~10 as long as the electrons of the upstream
plasma remain cold, while the filamentation instability is strongly inhibited
in this limit; however, as electrons get heated to relativistic temperatures,
the situation becomes opposite and the two stream instability becomes inhibited
while the filamentation mode becomes efficient, even at moderate values of the
shock Lorentz factor. The peak wavelength of these instabilities migrates from
the inertial electron scale towards the proton inertial scale as the background
electrons get progressively heated during the crossing of the shock precursor.
We also discuss the role of current driven instabilities upstream of the shock.
In particular, we show that the returning/accelerated particles give rise to a
transverse current through their rotation in the background magnetic field. We
find that the compensating current in the background plasma can lead to a
Buneman instability which provides an efficient source of electron heating.
[Abridged]
