  Recent detections of GeV photons in a few GRBs by Fermi-LAT imply huge bulk
Lorentz factors to avoid a large gamma gamma optical depth at high energy.
Estimates can be as high as Gamma ~ 1000 in the most extreme cases. This puts
severe constraints on models of the central engine and the jet acceleration in
GRBs. These estimates are however obtained from a simplified single zone model.
We present here a more realistic calculation which takes into account the time,
space and direction dependent photon field existing in an outflow with several
relativistically moving emitting zones. The formalism is general and can be
applied to many models of the prompt GRB emission. We present results obtained
for a numerical implementation in the framework of the internal shock model. We
show that (i) the minimum Lorentz factor Gamma_min in bright LAT GRBs is
reduced by a factor ~ 2-3 compared to previous estimates if the GeV and MeV
emission are produced in the same region, and by an additional factor ~ 2-8 if
the GeV emission is produced at larger radii. We provide an improved
approximate formula for Gamma_min which is in good agreement with our numerical
results and can be directly applied to LAT GRB data; (ii) a delayed GeV onset
can be due to the time evolution of the opacity. As an illustration of these
two first results, we present a synthetic GRB that reproduces most features of
GRB 080916C with a mean Lorentz factor of ~ 340, an optically thin regime for
gamma gamma opacity at 3GeV in bin 'b', a variability timescale of 0.5s in the
MeV lightcurve and a delayed onset of ~ 5s of the GeV emission; (iii) the gamma
gamma opacity can smooth the short timescale variability in the GeV lightcurve.
This last result implies that the observed variability at high energy is not
necessarily a good test to distinguish between an internal and an external
origin for the GeV emission in GRBs. [abridged]
