  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.

Astrophysics