We discuss several constraints on the properties of gamma-ray bursts (GRB) at cosmological distances. First we use the requirement that burst sources must be optically thin to pair production by photon-photon collisions in order to produce the observed nonthermal spectra. We derive probability distributions for the minimum Lorentz expansion factor, the radiation energy, the maximum baryonic mass, and the maximum surrounding gas density in the events, based on 254 events from the second BATSE catalog. Secondly, the time variabilities of the bursts in the BATSE catalog are used to place an upper bound on the curvature radius of the emitting surface in the events. This is based on the requirement that the emitting region seen by the observer must be sufficiently small to produce the observed variability without violating causality. Finally, we discuss the association of cosmological GRB with galaxies. We consider eight bright and well-localized bursts detected by the Pioneer Venus Orbiter, whose positional error boxes contain no bright galaxies. Using the local luminosity function of galaxies, we calculate the probability for not seeing the GRB host galaxy. This probability tends to increase as the width of the GRB luminosity function increases. However, the allowed width of the GRB luminosity function is restricted by the number-count statistics of the bursts.
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