Gravitational lensing limits on the average redshift of gamma-ray bursts

The lack of bright host galaxies in several recently examined gamma-ray bur st (GRB) error boxes suggests that the redshifts of cosmological GRBs may b e significantly higher than previously hypothesized. On the other hand, the nondetection of multiple images in the Fourth BATSE Gamma-Ray Burst Catalo g thereafter 4B catalog) implies an upper limit: to the average redshift [z ] of CRBs. Were we calculate an upper limit to [z], independent of the phys ical model for CRBs, using a new statistical lensing method that removes di stance ambiguities and thus permits accurate computation of the lensing rat e at high z. The upper limit on [z] depends directly on the cosmological pa rameters Omega and Lambda. If there are no multiple images among the bright est 80% of the first 1802 bursts in the 4B catalog, then, at the 95% confid ence level, [z] < 2.2 2.8, 4.3, or 5.3 for (Omega, Lambda) values of (0.3, 0.7), (0.5, 0.5), (0.5, 0.0), or (1.0, 0.0), respectively. The 68% upper li mit to the average redshift is comparable to or less than the median redshi ft of CRBs in scenarios in which the GRB rate is proportional to the rate o f star formation, for any cosmology. The uncertainty in the lensing rate-ar ising from uncertainties in the cosmological parameters and in the number d ensity and average velocity dispersion of galaxies-will be reduced signific antly in the next few years by a new generation of experiments and surveys. Moreover, the continued increase in the number of CRBs observed by BATSE w ill greatly constrain their redshift distribution.