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

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 $V_{\mathrm{max}}$ 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 $[\,0.7 < \rho_{0} < 0.8\,]\,\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$. 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 $-2.59 \pm0.93$ respectively, separated by a break luminosity $0.80 \pm0.43 \times 10^{52}$\,erg\,s$^{-1}$.