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An advanced LIGO and Virgo’s third observing run brought another binary neutron star merger (BNS) and the first neutron-star black hole mergers. While no confirmed kilonovae were identified in conjunction with any of these events, continued improvements of analyses surrounding GW170817 allow us to project constraints on the Hubble Constant (H₀), the Galactic enrichment fromr-process nucleosynthesis, and ultra-dense matter possible from forthcoming events. Here, we describe the expected constraints based on the latest expected event rates from the international gravitational-wave network and analyses of GW170817. We show the expected detection rate of gravitational waves and their counterparts, as well as how sensitive potential constraints are to the observed numbers of counterparts. We intend this analysis as support for the community when creating scientifically driven electromagnetic follow-up proposals. During the next observing run O4, we predict an annual detection rate of electromagnetic counterparts from BNS of ${0.43}_{-0.26}^{+0.58}$( ${1.97}_{-1.2}^{+2.68}$) for the Zwicky Transient Facility (Rubin Observatory). 

Abstract The Einstein Telescope (ET), the European project for a third-generation gravitational-wave detector, has a reference configuration based on a triangular shape consisting of three nested detectors with 10 km arms, where each detector has a 'xylophone' configuration made of an interferometer tuned toward high frequencies, and an interferometer tuned toward low frequencies and working at cryogenic temperature. Here, we examine the scientific perspectives under possible variations of this reference design. We perform a detailed evaluation of the science case for a single triangular geometry observatory, and we compare it with the results obtained for a network of two L-shaped detectors (either parallel or misaligned) located in Europe, considering different choices of arm-length for both the triangle and the 2L geometries. We also study how the science output changes in the absence of the low-frequency instrument, both for the triangle and the 2L configurations. We examine a broad class of simple 'metrics' that quantify the science output, related to compact binary coalescences, multi-messenger astronomy and stochastic backgrounds, and we then examine the impact of different detector designs on a more specific set of scientific objectives.