Contribution of GRB Emission to the GeV Extragalactic Diffuse Gamma-Ray Flux

TeV gamma rays emitted by gamma-ray bursts (GRBs) are converted into electron-positron pairs via interactions with the extragalactic infrared radiation fields. In turn, the pairs produced, whose trajectories are randomized by magnetic fields, will inverse Compton scatter off the cosmic microwave background photons. The beamed TeV gamma-ray flux from GRBs is thus transformed into a GeV isotropic gamma-ray flux, which contributes to the total extragalactic gamma-ray background emission. Assuming a model for the extragalactic radiation fields, the GRB redshift distribution, and the GRB luminosity function, we evaluate the contribution of the GRB prompt and scattered emissions to the measured extragalactic gamma-ray flux. To estimate this contribution we optimistically require that the energy flux at TeV energies is about 10 times stronger than the energy flux at MeV energies. The resulting gamma-ray diffuse background is only a small fraction of what is observed, allowing blazars and other sources to give the dominant contribution.