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Recent evidence appears to link gamma-ray bursts (GRBs) to star-forming regions in galaxies at cosmological distances. If short-lived massive stars are the progenitors of GRBs, the rate of events per unit cosmological volume should be an unbiased tracer (i. e. unaffected by dust obscuration and surface brightness limits) of the cosmic history of star formation. Here we use realistic estimates for the evolution of the stellar birthrate in galaxies to model the number counts, redshift distribution, and time-delay factors of GRBs, and indentify some of the remaining uncertainties. We find that present-day data cannot fully discriminate between different plausible star formation histories. The best-fitting parameters to the BATSE log N- log P relation imply about 1-2 GRBs every one million Type II supernovae, and a characteristic `isotropic-equivalent' burst luminosity in the range 3-20 x 1e51 ergs/s (for H₀=65 km/s/Mpc). We compute the rate of multiple imaging of background GRBs due to foreground mass condensations in a Lambda-dominated cold dark matter cosmology, assuming that dark halos approximate singular isothermal spheres on galaxy scales and Navarro-Frenk-White profiles on group/cluster scales, and are distributed in mass according to the Press-Schechter model. We show that the expected sensivity increase of Swift relative to BATSE could result in a few strongly lensed individual bursts detected down to a photon flux of 0. 1 phot/s/cm² in a 3-year survey. Because of the partial sky coverage, however, it is unlikely that the \ Swift satellite will observe recurrent events (lensed pairs).
Porciani et al. (Tue,) studied this question.