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The observational follow-up campaign of the gravitational wave (GW)multi-messenger event GW170817/GRB170817A has shown that the prompt$\gamma$-rays are consistent with a relativistic structured jet observed from awide viewing angle $\gtrsim 20$\deg. We perform Bayesian inference using thedata from early and late EM observations to determine the jet profile ofGRB170817A assuming a structured jet model. We use the geometric dependence onthe burst luminosity to produce a short duration gamma-ray burst (sGRB)efficiency function with redshift, which folded in with binary neutron stardetection rate, allows us to estimate the future joint GW/sGRB detection ratesfor LIGO and Virgo detectors. We show that, if the jet structured profile ofGRB170817A is a relatively common feature of sGRBs, then there is a realisticprobability of another off-axis coincident detection during the thirdaLIGO/Virgo observing run (O3). We also find that up to 4 yr$^{-1}$ jointevents may be observed during the advanced LIGO run at design sensitivity andup to 10 yr$^{-1}$ by the upgraded advanced LIGO configuration A+. We show thatthe detection efficiencies for wide-angled sGRB emissions will be limited byGRB satellites as the GW detection range increases through proposed upgrades.Therefore, although the number of coincident detections will increase with GWdetector sensitivity, the relative proportion of detected binary neutron starswith $\gamma$-ray counterparts will decrease; 11\% for O3 down to 2\% duringA+.

Joint gravitational wave -- gamma-ray burst detection rates in the aftermath of GW170817