On the association of gamma-ray bursts with massive stars: Implications for number counts and lensing statistics

Recent evidence appears to link gamma-ray bursts (GRBs) to star-forming reg ions in galaxies at cosmological distances. If short-lived massive stars ar e the progenitors of GRBs, the rate of events per unit cosmological volume should be an unbiased tracer (i.e., unaffected by dust obscuration and surf ace 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. We present luminosity function fits to the BATSE log N- log P rela tion for different redshift distributions of the bursts. Our results imply about 1-2 GRBs for every one million Type II supernovae, and a characterist ic "isotropic-equivalent" burst luminosity in the range 3-20 x 10(51) ergs s(-1) (for km s(-1) Mpc(-1)). We compute the rate of multiple imaging of ba ckground GRBs due to foreground mass condensations in a Lambda -dominated c old dark matter cosmology, assuming that dark halos approximate singular is othermal spheres on galaxy scales and Navarro-Frenk-White profiles on group /cluster scales, and are distributed in mass according to the Press-Schecht er model. We show that the expected sensitivity increase of Swift relative to BATSE could result in a few strongly lensed individual bursts detected d own to a photon flux of 0.1 cm(-2) s(-1) in a 3 yr survey. Because of the p artial sky coverage, however, it is unlikely that the Swift satellite will observe recurrent events (lensed pairs).