Gravitational wave triggered searches for high-energy neutrinos from binary neutron star mergers: prospects for next generation detectors

The next generation gravitational wave (GW) detectors -- Einstein Telescope (ET) and Cosmic Explorer (CE) will have distance horizons up to $\mathcal{O}(10)$ Gpc for detecting binary neutron star (BNS) mergers. This will make them ideal for triggering high-energy neutrino searches from BNS mergers at the next generation neutrino detectors, such as IceCube-Gen2. We calculate the distance limits as a function of the time window of neutrino analysis, up to which meaningful triggers from the GW detectors can be used to minimize backgrounds and collect a good sample of high-energy neutrino events at the neutrino detectors, using the sky localization capabilities of the GW detectors. We then discuss the prospects of the next generation detectors to work in synergy to facilitate coincident neutrino detections or to constrain the parameter space in the case of non-detection of neutrinos. We show that good localization of GW events, which can be achieved by multiple third generation GW detectors, is necessary to detect a GW-associated neutrino event or put a meaningful constraint ($\sim 3\sigma$ confidence level) on neutrino emission models. Such an analysis can also help constrain physical models and hence provide insights into neutrino production mechanisms in binary neutron star mergers.