We have recently completed a search of 6 years of archival BATSE data for gamma-ray bursts (GRBs) that were too faint to activate the real-time burst detection system running onboard the spacecraft. These non-triggered bursts can be combined with the triggered bursts detected onboard to produce a GRB intensity distribution that reaches peak fluxes a factor of 2 lower than could be studied previously. The value of the V/Vmax statistic (in Euclidean space) for the bursts we detect is 0.177 +/- 0.006. This surprisingly low value is obtained because we detected very few bursts on the 4.096 s and 8.192 s time scales (where most bursts have their highest signal-to-noise ratio) that were not already detected on the 1.024 s time scale. If allowance is made for a power-law distribution of intrinsic peak luminosities, the extended peak flux distribution is consistent with models in which the redshift distribution of the gamma-ray burst rate approximately traces the star formation history of the Universe. We argue that this class of models is preferred over those in which the burst rate is independent of redshift. We use the peak flux distribution to derive a limit of 10% (99% confidence) on the fraction of the total burst rate that could be contributed by a spatially homogeneous (in Euclidean space) subpopulation of burst sources, such as type Ib/c supernovae. These results lend support to the conclusions of previous studies predicting that relatively few faint classical GRBs will be found below the BATSE onboard detection threshold.
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