The Cosmological Signatures and Flux Distribution of $\gamma$-Ray Bursts with a Broad Luminosity Distribution

The cosmological expansion cannot produce the reported correlations of the gamma-ray burst timescale and spectral energy with peak flux if the burst model reproduces the BATSE 3B peak-flux distribution for a non-evolving burst source density. Power law luminosity distribution models proportional to L^{-beta} produce acceptables fits to the data for most values of the spectral parameters when beta 3 for peak fluxes near the BATSE threshold, and with an average redshift of < 1 for all values of peak flux. The reported correlations of the burst timescale and the spectral energy with peak flux are systematically 1 standard deviation above the monoluminous model and 1.5 to 2 standard deviation above the power-law luminosity model. These results suggest that an intrinsic correlation of burst timescale and spectral energy with luminosity is present