FERMI constraints on the high energy, ~1 GeV, emission of long GRBs

We investigate the constraints imposed on the luminosity function (LF) of long duration Gamma Ray Bursts (LGRBs) by the flux distribution of bursts detected by the GBM at ~1 MeV, and the implications of the non detection of the vast majority, ~95%, of the LGRBs at higher energy, ~1 GeV, by the LAT detector. We find a LF that is consistent with those determined by BATSE and Swift. The non detections by LAT set upper limits on the ratio R of the prompt fluence at ~1 GeV to that at ~1 MeV. The upper limits are more stringent for brighter bursts, with R<{0.1,0.3,1} for {5,30,60}% of the bursts. This implies that for most bursts the prompt ~1 GeV emission may be comparable to the ~1 MeV emission, but can not dominate it. The value of R is not universal, with a spread of (at least) an order of magnitude around R~10^(-1). For several bright bursts with reliable determination of the photon spectral index at ~1 MeV, the LAT non detection implies an upper limit to the ~100 MeV flux which is <0.1 of the flux obtained by extrapolating the ~1 MeV flux to high energy. For the widely accepted models, in which the ~1 MeV power-law photon spectrum reflects the power-law energy distribution of fast cooling electrons, this suggests that either the electron energy distribution does not follow a power-law over a wide energy range, or that the high energy photons are absorbed. Requiring an order unity pair production optical depth at ~100 MeV sets an upper limit for the Lorentz factor, Gamma<=10^(2.5).