Reconciling gamma-ray burst time-dilation measurements to the brightness distribution in the context of standard cosmology

If gamma-ray bursts are located at cosmological distances, it is natural to expect that measurements of cosmological time dilation in the burst profiles or cosmological energy shifting in burst spectral forms must agree with other cosmological signatures such as the brightness distribution's deviation from a -3/2 logarithmic slope. Recently, several works have offered evidence for time dilation and energy stretching in gamma-ray bursts observed with the Burst and Transient Source Experiment (BATSE). We therefore present a means to assess the consistency of time-dilation and energy-shifting measurements with both the brightness distribution of BATSE and various Friedmann cosmological models. This methodology can be utilized with energy-width corrections of various sizes that are inherent to time-dilation analyses, yet absent in energy-shifting measurements. Following the presentation of the general technique, we utilize the methodology to assess the compatibility of a recent claim by Norris et al. of a time dilation factor of 2.25 from bursts separated by a factor of 21 in brightness. We show that these results are not compatible with the brightness distribution of BATSE assuming a nonevolving population of standard-candle bursts embedded in an Einstein-de Sitter universe. Recent revisions of this value downward to 1.75 published by Norris in 1995 help to alleviate the discrepancy; however, they are still somewhat marginal in light of the energy-width corrections that need to be applied.