Analytic constraints on gamma-ray burst luminosity functions

The brightness distribution of detected gamma-ray bursts enables constraints to be placed on their luminosities; specifically, analyses by previous authors have demonstrated that the range of luminosity from which approximately 90% of the observed bursts are drawn is very likely less than an order of magnitude. It has also been demonstrated that when power-law functional forms with infinite ranges of luminosity (a) from 0 to L(max) or (b) from L(min) to infinity are employed, it is necessary (but perhaps not sufficient) to restrict the power-law index beta < 1.8 and beta > 2.5, respectively, in order to obtain consistency with the observed brightness distribution. Similarly, these previous works have demonstrated that when 1.8 < beta < 2.5, the range of the burst luminosity function must be restricted and cannot be arbitrarily large. We present here analytic derivations of these results valid for any spatial density function and for both Euclidean and cosmological scenarios. This analysis shows that the limiting values of beta are simply the absolute values of the maximum and minimum logarithmic slopes of the observed brightness distribution. We demonstrate also that this result is consistent with those requiring a narrow range in luminosity for a majority of the detected bursts.