Prospects for optical detections from binary neutron star mergers with the next-generation multi-messenger observatories

Next-generation gravitational wave (GW) observatories, such as the Einstein Telescope (ET) and Cosmic Explorer, will observe binary neutron star (BNS) mergers across cosmic history, providing precise parameter estimates for the closest ones. Innovative wide-field observatories, like the Vera Rubin Observatory, will quickly cover large portions of the sky with unprecedented sensitivity to detect faint transients. This study aims to assess the prospects for detecting optical emissions from BNS mergers with next-generation detectors, considering how uncertainties in neutron star (NS) population properties and microphysics may affect detection rates. Starting from BNS merger populations exploiting different NS mass distributions and equations of state (EOSs), we model the GW and kilonova (KN) signals based on source properties. We model KNe ejecta through numerical-relativity informed fits, considering the effect of prompt collapse of the remnant to black hole and new fitting formulas appropriate for more massive BNS systems, like GW190425. We include optical afterglow emission from relativistic jets consistent with observed short gamma-ray bursts. We evaluate the detected mergers and the source parameter estimations for different geometries of ET, operating alone or in a network of current or next-generation GW detectors. Finally, we estimate the number of detected optical signals simulating realistic observational strategies by the Rubin Observatory. ET as a single observatory will enable the detection of about ten to a hundred KNe per year by the Rubin Observatory. This improves by a factor of $\sim 10$ already when operating in the network with current GW detectors. Detection rate uncertainties are dominated by the poorly constrained local BNS merger rate, and depend to a lesser extent on the NS mass distribution and EOS.