Redshifts For 224 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 luminosity. 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 224 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 rate scales as (1+z)^{3.6 \pm 0.3} and is roughly consistent with the SFR at 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 at least until z \sim 3. We have used the SFR 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 -2.9 that rolls over at low luminosity. 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