Constraining the rate and luminosity function of Swift gamma-ray bursts

Abstract

We compute the intrinsic isotropic peak luminosity function (LF) and formation rate of long gamma-ray bursts (LGRBs) using a novel approach. We complement a standard log\,$N$\,--\,log\,$P$ brightness distribution and $V_{\max }$ estimations with two observation-time relations: a redshift--observation-time relation (log\,$z$\,--\,log\,$T$) and a new luminosity--observation-time relation (log\,$L$\,--\,log\,$T$). We show that this approach reduces degeneracies that exist between the rate and LF of a brightness distribution. To account for the complex triggering algorithm employed by \emph{Swift} we use recent results of \citet{Lien_2014ApJ} to produce a suite of efficiency functions. Using these functions with the above methods, we show that a log\,$L$\,--\,log\,$T$ method can provide good constraints on the form of the LF, particularly the high end. Using a sample of 175 peak luminosities determined from redshifts with well defined selection criteria our results suggest that LGRBs occur at a local rate (without beaming corrections) of $[0.7<\ddot{I}{}_0<0.8]{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$. Within this range, assuming a broken-power-law LF, we find best estimates for the low and high energy indices of $-0.95\pm 0.09$ and $-2.59\pm 0.93$ respectively, separated by a break luminosity $0.80\pm 0.43\times 10^{52}$\,erg\,s${}^{-1}$.

Accepted for publication in MNRAS