Hot Dense Matter in Neutron Star Merger Remnants: Finite-Temperature Equation of State, Neutrino Transport, Bulk Viscosity, Convective Instabilities, and the Helium Spectroscopic Clock — A Unified Framework for Reading the QCD Phase Diagram from Transient Observations

This paper constitutes the work on dense nuclear matter, advancing beyond the cold-matter equation of state (EoS), the hyperon puzzle, and the multimessenger aftermath of binary neutron star (BNS) mergers to address the most poorly understood regime of the QCD phase diagram: hot, dense, out-of-equilibrium nuclear matter at temperatures T ≈ 10–100 MeV and baryon densities (1–5)ρ₀ as realised in hypermassive neutron star (HMNS) remnants. We synthesise four interconnected theoretical frontiers that have each seen transformative advances in 2024–2025. First, we examine the three-dimensional finite-temperature EoS problem — the challenge of constructing fully tabulated P(ρ, T, Ye) tables consistent with nuclear theory, heavy-ion constraints, and astrophysical observations — and evaluate the accuracy of the M*-parametric framework for thermal extensions versus fully microscopic finite-temperature EoS tables (SFHo, DD2, BLh), demonstrating that finite-temperature EoS choices produce distinct postmerger gravitational-wave frequency evolutions at late times (t > 50 ms) that deviate significantly from quasi-universal relations. Second, we present a comprehensive treatment of neutrino transport physics in the merger remnant: the emergence and saturation of bulk viscosity driven by weak-interaction Urca processes, the role of trapped neutrinos in modifying flavor equilibration timescales, and the magnetic-field-induced modification of Urca rates through the Nucleon Width Approximation framework. Third, we analyse the Tayler–Spruit (TS) dynamo operating in the differentially-rotating HMNS as the dominant mechanism for field amplification beyond the Kelvin–Helmholtz phase, generating toroidal fields B ∼ 10¹⁶ G on timescales of tens of milliseconds and driving thermal instabilities and convective patterns that excite sub-dominant inertial modes with frequencies f < 2 kHz, potentially detectable by third-generation GW detectors. Fourth, and most originally, we develop the helium spectroscopic clock as a quantitative EoS constraint: the absence of a prominent He I λ1083.3 nm feature in AT2017gfo at 4.4 days post-merger limits the helium mass fraction in polar ejecta to X_He < 0.05, which, combined with neutrino-hydrodynamic simulations, constrains the remnant lifetime to τ_rem ≤ 20–30 ms. This implies M_thres ≤ 2.93 M☉ and, combined with causality arguments, limits the maximum NS mass to M_max ≤ 2.3 M☉ and the radius of 1.6 M☉ NS to approximately 11–13 km. We place all four threads within a unified observational framework, demonstrating that neutrino emission, convective mode excitation, bulk viscous dissipation, and kilonova spectroscopic features constitute a quartet of orthogonal, complementary constraints on the finite-temperature EoS, each sensitive to a distinct region of the (ρ, T) plane inaccessible to cold-matter probes.