Standard cosmology and the BATSE number versus peak flux distribution

The observed 2B BATSE distribution is consistent with the faintest GRBs inour sample originating from a redshift of Zmax ~ 0.8-3.0 (90\%), with the mostlikely values in the range of 1.0-2.2, and is largely insensitive to Omega formodels with no evolution. To constrain the model parameter Omega to the range0.1-1.0 using only Log N -- Log P distributions, more than 4000 GRBs, with amost likely value of ~ 9,000 GRBs to BATSE sensitivity. This requires a liveintegration time of >6 years with BATSE. Detectors sensitive to much lowerlimits (~ 70-400 in sensitivity) require ~ 200 GRBs, with <0.03 year 4pi stercoverage. We place limits on the amount of frequency density and, separately,peak luminosity evolution in the sample of GRBs. We find that frequency densityevolution models can place the faintest GRBs at Zmax ~ 10-200, withoutconflicting with the observations of relative time dilation of ~ 2.