The implications of direct redshift measurement of gamma-ray bursts

The recent discoveries of X-ray and optical counterparts for gamma-ray bursts (GRBs), and the possible discovery of a host galaxy, imply that direct measurement of the redshift of some GRB host galaxies is imminent. We discuss the implications of such measurements. These measurements could enable us to determine the GRB luminosity distribution and the variation of the rate of GRBs with cosmic time and even, under favorable circumstances, to estimate Omega. Using GRB 970508 alone, assuming standard candles, and assuming the GRB source to be at the redshift of the absorption line observed in the optical transient spectrum, we constrain the intrinsic GRE evolution to rho(z) = (1+z)(-0.5+/-0.7).