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Prompt GRB spectra: detailed calculations and the effect of pair production

We present detailed calculations of the prompt spectrum of gamma-ray bursts (GRBs) predicted within the fireball model framework, where emission is due to internal shocks in an expanding relativistic wind. Our time dependent numerical model describes cyclo-synchrotron emission and absorption, inverse and direct Compton scattering, and e^\pm pair production and annihilation (including the evolution of high energy electro-magnetic cascades). It allows, in particular, a self-consistent calculation of the energy distribution of e^\pm pairs produced by photon annihilation, and hence a calculation of the spectra resulting when the scattering optical depth due to pairs, τ_\pm, is high. We show that emission peaks at ~1MeV for moderate to large τ_\pm, reaching τ_\pm ~ 10^2. In this regime of large compactness we find that (i) A large fraction of shock energy can escape as radiation even for large τ_\pm; (ii) The spectrum depends only weakly on the magnetic field energy fraction; (iii) The spectrum is hard, epsilon^2 dN/depsilon \propto epsilon^αwith 0.5<α<1, between the self absorption (epsilon_{ssa}= 10^{0.5\pm0.5} keV) and peak (epsilon_{peak}= 10^{0.5\pm0.5} MeV) photon energy, (iv) and shows a sharp cutoff at ~10 MeV; (v) Thermal Comptonization leads to emission peaking at epsilon_peak>=30 MeV, and can not therefore account for observed GRB spectra. For small compactness, spectra extend to >10 GeV with flux detectable by GLAST, and the spectrum at low energy depends on the magnetic field energy fraction. Comparison of the flux at ~1 GeV and ~ 100 keV may therefore allow to determine the magnetic field strength. For both small and large compactness, the spectra depend only weakly on the spectral index of the energy distribution of accelerated electrons.