Constraining the luminosity function of GRBs from time dilation, brightness distribution and redshift data

We constrain the luminosity function of gamma ray bursts (GRBs) by testing several cosmological models versus a combination of (a) observed time dilation in the peak-to-peak time scales of GRB light curves, (b) the $LogN−LogP$ number counts relation, and (c) GRB luminosities directly determined from recent redshift measurements and apparent GRB brightness. A power law luminosity function $φ(L)=C⋅L−β$ $(Lmin<L<Lmax)$ is examined within three cosmological models, with cosmological parameters $(Ω0,ΩΛ)=(1.0,0.0),$ (0.3, 0.0), or (0.3, 0.7). It is found that (a) GRB models with constant comoving density rate do not fit the data while models with GRB density evolution tracing the observed star formation rate can accommodate all the observed data; (b) the width $K=Lmax/Lmin$ of the luminosity function is found to be $K>100$ at the 1 σ level; (c) the intrinsic average luminosity 〈L〉 is found to be $2∼6×1051 ergs⋅s−1$ at the 3σ level, this implies that bursts at the BATSE 100% efficiency threshold have typical red shifts of $〈z0.42〉=2∼3.$