  The characteristic photon energy for Gamma Ray Bursts, E_peak, has a
remarkably narrow distribution for bursts of similar peak flux, with values
between 150 and 600 keV for most faint bursts. This result is surprising within
the framework of internal shock models, since spectral shifts associated with
the jet's blue shift (by a Lorentz factor of Gamma) and the cosmological red
shift (by a factor of 1+z) should cause substantial smearing in the
distribution of the spectral peak in the jet's co-moving frame, E_rest. For the
general case where the luminosity (L) varies as Gamma^N and E_rest varies as
Gamma^M, then the observed E_peak will vary as L^{(M+1)/N}(1+z)^{-1}. For two
independent set of 20 and 84 bursts, E_peak(1+z) varies as a power law of the
luminosity with an index of (M+1)/N=0.36+-0.03. With this measured value, the
above functional dependence of E_peak on L and z results in E_peak being
roughly constant for bursts of similar peak flux, P_256. Thus, the kinematic
smearing will be small, hence allowing the E_peak distribution to be narrow.
This model also predicts that bright bursts will have high E_peak values
because they all have some combination of high luminosity (and hence a large
blue shift Gamma) and a nearby distance (and hence a small cosmological red
shift). Quantitatively, E_peak should vary roughly as P_256^0.36, and this
model prediction is strikingly confirmed with BATSE data by Mallozzi et al. A
prediction of this model is that GRBs at very high red shift z~10 should all
appear with E_peak at ~200 keV. A further prediction of this model is that
normal bursts with P_256 below the BATSE trigger threshold will appear as x-ray
flashes with E_peak~70 keV; just as is reported by Kippen et al. and Heise et
al.
