Multmessenger Signatures of Neutron Star Mergers: Postmerger Gravitational Waves, Kilonova Emission, R-Process Nucleosynthesis, and the Nuclear Equation of State in the Era of Third-Generation Detectors

This paper examins dense nuclear matter from the nuclear saturation density through the supra-nuclear exotic matter regime. In our preceding papers we established: (i) multi messenger constraints on the nuclear symmetry energy and neutron skin thickness from PREX-II, CREX, NICER, and GW170817; and (ii) the hadron–quark phase transition, the hyperon puzzle, and the speed-of-sound diagnostic. Here we integrate these threads into a comprehensive treatment of the full observational and nucleosynthetic aftermath of binary neutron star (BNS) mergers. We examine, in detail, the postmerger gravitational-wave spectrum and its dependence on the equation of state (EoS) — including the dominant f₂ oscillation mode, the long-ringdown correlation with ultra-high-density EoS, and the role of magnetic fields in shifting postmerger frequencies. We analyze the kilonova electromagnetic transient AT2017gfo — the only spectroscopically observed electromagnetic counterpart of a BNS merger — and the physical pipeline connecting merger ejecta dynamics to r-process nucleosynthesis, radiative transfer, and observable light curves and spectra. The identification of strontium (Z=38) and emerging evidence for tellurium (Z=52), lanthanum (Z=57), and cerium (Z=58) in kilo nova spectra is reviewed within the current state of atomic data and radiative transfer modelling. The sensitivity of kilonova ejecta mass, electron fraction Ye, and lanthanide opacity to the underlying EoS is discussed as a new indirect nuclear probe. We further present a forward-looking analysis of the science enabled by the Einstein Telescope (ET) and Cosmic Explorer (CE) — third-generation gravitational-wave observatories expected to detect O(10⁵) BNS mergers per year — including postmerger frequency extraction, Hubble constant measurement, and the precision mapping of r-process enrichment across cosmic history. The paper concludes that the convergence of gravitational-wave, electromagnetic, and nuclear laboratory observations is transforming BNS mergers into the most powerful known laboratory for exploring nuclear matter under the most extreme conditions that exist in the observable universe.