Luminosity Function of Gamma‐Ray Bursts Derived without Benefit of Redshifts

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Abstract
We show that the Euclidean value of for gamma-ray bursts (GRBs) selected on a timescale of 1024 ms is correlated with spectral hardness. The value of ranges from ~0.42 for soft bursts to ~0.26 for the hardest bursts. Given that the Euclidean value of for cosmological objects in a well-defined sample is a distance indicator, the hard bursts must reside at larger redshifts and therefore be more luminous than the soft bursts. The resulting luminosity-hardness correlation cannot be shown explicitly as a result of the small number of observed GRB redshifts at the present time. Based on the -hardness correlation, we derive the luminosity function of GRBs without using any redshifts, but we have to make an assumption how the comoving GRB rate varies with redshift. We present luminosity functions for three models of the GRB rate as a function of redshift, based on star formation rates. The peak luminosity functions are approximately broken power laws with an isotropic-equivalent break luminosity of ~1051.5 ergs s-1 in the 50-300 keV range and total local rate densities of ~0.5 Gpc-3 yr-1. Predicted GRB counts as a function of flux and redshift are presented. Based on the GRB luminosity function, we carry out a simulation to produce the luminosity-hardness correlation, which shows that the hardest GRBs are ~20 times more luminous than the softest ones.
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