Redshifts For 220 BATSE Gamma-Ray Bursts Determined by Variability and   the Cosmological Consequences
  We show that the time variability of a gamma-ray burst (GRB) appears to be correlated with the absolute luminosity of the burst: smooth bursts are intrinsically less luminous. This Cepheid-like relationship can be used to determine the redshift of a GRB from parameters measured solely at gamma-ray energies. The relationship is based on only seven events at present and needs to be further confirmed with more events. We present the details of converting GRB observables to luminosities and redshifts for 220 bright, long GRBs from the Burst and Transient Source Experiment (BATSE) and explore the cosmological consequences. In particular, we derive the GRB rate as a function of $z$ without assuming either a luminosity function or that the GRB rate follows the star formation rate (SFR). We find that the GRB formation rate scales as (1+z)^{3.3 \pm 0.3}. The observations used to derive the SFR can be strongly affected by dust for z > 2 whereas GRB observations are not. If GRBs trace star formation, then our results indicate that the SFR does not peak at z\sim 2 but instead continues to increase until z \sim 10. We have used the burst formation rate to correct the observed GRB luminosity function for the incompleteness due to the detection threshold, resulting in a luminosity function with a power law index of \sim -2.3 that slightly rolls over at low luminosities. The reality of our variability--luminosity relationship requires confirmation but, if valid, will provide a powerful tool for studying both GRBs and the early universe. 
E. E. Fenimore (Los Alamos), E. Ramirez-Ruiz (Los Alamos, Cambridge   University)