Constraining the Luminosity Function and Delay-Time Distribution of Short Gamma-Ray Bursts for Multimessenger Gravitational-Wave Detection Rate Estimation

In this work, we analyze the most recent short gamma-ray burst (sGRB) sample detected by the \emph{Fermi} satellite to reassess the sGRB luminosity function and formation rate. Using the empirical redshift-luminosity correlation, we first determine the pseudo redshifts of 478 sGRBs. Then, we use the maximum likelihood method to constrain the luminosity function and formation rate of sGRBs under various delay-time distribution models, finding the Gaussian delay model statistically preferred over the power-law and lognormal delay models based on the Deviance Information Criterion. The local formation rate of sGRBs is $R_{\mathrm{sGRB}}(0)=1.37_{-0.27}^{+0.30}$ $\mathrm{Gpc^{-3}\,yr^{-1}}$, largely independent of the adopted delay-time distribution model. Additionally, we investigate the potential for joint detection of sGRBs and their gravitational wave (GW) counterparts from binary neutron star mergers using both current and future GRB and GW facilities. For sGRB detection, we consider three existing satellites: \emph{Fermi}, the Space-based multi-band astronomical Variable Objects Monitor (\emph{SVOM}), and the Einstein Probe (\emph{EP}). For GW detection, we examine two International GW Networks (IGWN): a four-detector network consisting of LIGO Hanford, Livingston, Virgo, and KAGRA (IGWN4) and an upcoming five-detector network that includes these four detectors plus LIGO India (IGWN5). Incorporating the angular dependence of sGRB jet emission energy, our results show that for different delay-time distribution models, the joint sGRB and GW detection rates for \emph{Fermi}, \emph{SVOM}, and \emph{EP} with IGWN4 (IGWN5) lie within 0.19--0.27 $\mathrm{yr^{-1}}$ (0.93--1.35 $\mathrm{yr^{-1}}$), 0.07--0.10 $\mathrm{yr^{-1}}$ (0.51--0.79 $\mathrm{yr^{-1}}$), and 0.01--0.03 $\mathrm{yr^{-1}}$ (0.15--0.27 $\mathrm{yr^{-1}}$), respectively.