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The existence of merging black hole-neutron star (BHNS) binaries has beenascertained through the observation of their gravitational wave (GW) signals.However, to date, no definitive electromagnetic (EM) emission has beenconfidently associated with these mergers. Such an association could helpunravel crucial information on these systems, for example, their BH spindistribution, the equation of state (EoS) of NS and the rate of heavy elementproduction. We model the multi-messenger (MM) emission from BHNS mergersdetectable during the fourth (O4) and fifth (O5) observing runs of theLIGO-Virgo-KAGRA GW detector network, in order to provide detailed predictionsthat can help enhance the effectiveness of observational efforts and extractthe highest possible scientific information from such remarkable events. Ourmethodology is based on a population synthesis-approach, which includes themodelling of the signal-to-noise ratio of the GW signal in the detectors, theGW-inferred sky localization of the source, the kilonova (KN) optical andnear-infrared light curves, the relativistic jet gamma-ray burst (GRB) promptemission peak photon flux, and the GRB afterglow light curves in the radio,optical and X-ray bands. The resulting prospects for BHNS MM detections duringO4 are not promising, with a GW detection rate of $15.0^{+15.4}_{-8.8}$yr$^{-1}$, but joint MM rates of $\sim 10^{-1}$ yr$^{-1}$ for the KN and $\sim10^{-2}$ yr$^{-1}$ for the jet-related emission. In O5 we find an overallincrease in expected detection rates by around an order of magnitude, owing toboth the enhanced sensitivity of the GW detector network, and the coming onlineof future EM facilities. Finally, we discuss direct searches for the GRB radioafterglow with large-field-of-view instruments as a new possible follow-upstrategy in the context of ever-dimming prospects for KN detection.

Multi-messenger prospects for black hole - neutron star mergers in the O4 and O5 runs