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 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. We present luminosity function fits to the BATSE log N-log P relation for different redshift distributions of the bursts. Our results 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 H_0=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/cm^2 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). 
Cristiano Porciani and Piero Madau