2025A&A...704A.260C

The Einstein Telescope (ET), a proposed next-generation gravitational wave (GW) observatory, will expand the reach of GW astronomy of stellar-mass compact object binaries to unprecedented distances, enhancing opportunities for multi-messenger observations. Here we investigate multi-messenger emission properties of binary neutron star (NSNS) and black hole-neutron star (BHNS) mergers detectable by ET and provide projections to optimise observational strategies and maximise scientific insights from these sources. Using a synthetic population of compact binary mergers, we modelled each source's GW signal-to-noise ratio, sky localization uncertainty, kilonova light curves (in optical and near-infrared bands), and the fluence of the relativistic jet gamma-ray burst prompt emission and afterglow light curves across radio, optical, X-ray, and very high energy wavelengths. We analysed multi-messenger detectability prospects for ET as a standalone observatory with two different configurations and within a network of next-generation GW detectors. ET will detect over 10⁴ NSNS mergers annually, enabling the potential observation of tens to hundreds of electromagnetic (EM) counterparts. In contrast, BHNS mergers have more limited multi-messenger prospects, but joint GW-EM rates will increase by an order of magnitude compared to current-generation instruments. We quantified the uncertainties due to the NS equation of state (EoS) and mass distribution of NSNSs as well as the NS EoS and BH spin for BHNSs. While a single ET will achieve an impressive GW detection rate, the fraction of well-localised events (< 100 deg²) is orders of magnitude lower than in a network with additional detectors. This significantly limits efficient EM follow-up and science cases requiring well-characterized counterparts or early observations. The challenge is even greater for BHNS mergers due to their low EM rate. Thus, multi-messenger astronomy in the next decade will critically depend on a network of at least two detectors.