Prospects for multi-messenger detection of binary neutron star mergers in the fourth LIGO-Virgo-KAGRA observing run

The joint detection of GW170817 and GRB 170817A opened the era of multi-messenger astronomy with gravitational waves (GWs) and provided the first direct probe that at least some binary neutron star (BNS) mergers are progenitors of short gamma-ray bursts (S-GRBs). In the next years, we expect to have more multi-messenger detections of BNS mergers, thanks to the increasing sensitivity of GW detectors. Here, we present a comprehensive study on the prospects for joint GW and electromagnetic observations of merging BNSs in the fourth LIGO--Virgo--KAGRA observing run with \emph{Fermi}, \emph{Swift}, INTEGRAL and SVOM. This work combines accurate population synthesis models with simulations of the expected GW signals and the associated S-GRBs, considering different assumptions about the GRB jet structure. We show that the expected rate of joint GW and electromagnetic detections could be up to $\sim$ 6 yr$^{-1}$ when \emph{Fermi}/GBM is considered. Future joint observations will help us to better constrain the association between BNS mergers and S-GRBs, as well as the geometry of the GRB jets.