A systematic study of binary neutron star merger rate density history using simulated gravitational wave and short gamma-ray burst observations

Measuring the merger rate density history of binary neutron stars (BNS) can greatly aid in understanding the history of heavy element formation in the Universe. Currently, second-generation Gravitational Wave (GW) detectors can only measure the BNS merger rate density history at low redshifts ($z$ $\sim$ 0.1). Short gamma-ray bursts (sGRBs) may trace the BNS merger to higher redshifts ($z$ $\sim$ 3). However, not all BNS mergers result in sGRBs, and it is not certain that all sGRBs originate from BNS mergers. In this study, we simultaneously utilize simulated BNS merger GW signals detected by the advanced LIGO design and sGRB signals detected by {\it Fermi}/GBM to constrain the BNS merger rate density history up to $z$ $\sim$ 3. The results indicate that with $\sim$ 8 GWs and 571 sGRBs, the BNS merger rate density can be measured with an accuracy of about 50\% through $z=0$ to $z=1$. The ratio of the jet opening angle-corrected sGRB event rate density to the BNS merger rate density, denoted as $\eta$, can be constrained to a relative uncertainty of 45\%. With $\sim$ 21 GWs and 761 sGRBs, the BNS merger rate density can be measured to approximately 35\% and 40\% at $z=0$ and $z=1$, respectively. Meanwhile, $\eta$ can be constrained to a relative uncertainty of 28\%. Additionally, in our parameterized simulation, we find that at least approximately $\sim$550 sGRBs are needed to constrain the characteristic delay time in the star formation rate model, given a relative error of 50\% in the estimated redshift.