Adopting a simple cosmological model for Gamma Ray Bursts ( GRBs ) , following Mao & Paczyński ( 1992 ) , we generate number vs. peak flux distributions for a range of values of \Omega _ { 0 } ( ratio of the density of the universe to the critical density ) and z _ { max } ( the redshift at which the faintest GRB in the present sample is located ) , and compare these distributions to one from BATSE GRBs in the 2B catalog .
The observed BATSE distribution is consistent with the faintest GRBs in our sample originating from a redshift of z _ { max } \sim 0.8-3.0 ( 90 % ) , with the most likely values in the range of 1.0-2.2 , and is largely insensitive to \Omega _ { 0 } for models with no evolution .

To constrain the model parameter \Omega _ { 0 } to the range 0.1-1.0 using only log N – log F _ { peak } distributions , more than 4000 GRBs , with a most likely value of \sim 9 , 000 GRBs , above the 1024 msec averaged peak flux of 0.3 phot cm ^ { -2 } sec ^ { -1 } would be needed .
This requires a live integration time of > 6 years with BATSE .
Detectors sensitive to much lower limits ( i.e . standard candle bursts out to z _ { max } =10 , \sim \times 70-400 in sensitivity ) require \sim 200 GRBs , with < 0.03 year 4 \pi ster coverage .

We place limits on the amount of frequency density and , separately , peak luminosity evolution in the sample of GRBs .
We find that frequency density evolution models can place the faintest GRBs at z _ { max } \sim 10-200 , without conflicting with the observations of relative time dilation of \sim 2 reported by Norris et al .
( 1994a ) and Wijers & Paczyński ( 1994 ) ( however , see ( ( Band 1994 ) ) ) , although this would require vastly different comoving burst rates in GRBs of different spectra .