Gravitational Lensing Limits on the Average Redshift of Gamma-Ray Bursts

The lack of bright host galaxies in several recently examined gamma-ray burst (GRB) error boxes suggests that the redshifts of cosmological GRBs may be significantly higher than previously believed. On the other hand, the non-detection of multiple images in the BATSE 4B catalog implies an upper limit on the average redshift <z> of GRBs. Here, we calculate an upper limit to <z>, independent of the physical model for GRBs, using a new statistical lensing method that removes distance ambiguities, and thus permits accurate computation of the lensing rate at high z. The upper limit on <z> depends directly on the cosmological parameters Omega and Lambda. If there are no multiple images among the brightest 80% of the first 1802 bursts in the BATSE 4B catalog, then, at the 95% confidence level, <z> is less than 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 limit to the average redshift is comparable to or less than the median redshift of GRBs in scenarios in which the GRB rate is proportional to the rate of star formation, for any cosmology. The uncertainty in the lensing rate --- arising from uncertainties in the cosmological parameters and in the number density and average velocity dispersion of galaxies --- will be reduced significantly in the next few years by a new generation of experiments and surveys. Moreover, the continued increase in the number of GRBs observed by BATSE will greatly constrain their redshift distribution.