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  1. Neutron stars were first posited in the early thirties and discovered as pulsars in late sixties; however, only recently are we beginning to understand the matter they contain. This talk describes the continuing development of a consistent picture of the liquid interiors of neutron stars, driven by four advances: observations of heavy neutron stars with masses in the range of two solar masses; inferences of masses and radii simultaneously for an increasing number of neutron stars in low mass X-ray binaries, and ongoing determinations via the NICER observatory; the observation of the binary neutron star merger, GW170817, through gravitational waves as well as across the electromagnetic spectrum; and an emerging understanding in QCD of how nuclear matter can turn into deconfined quark matter in the interior. We describe the modern quark-hadron crossover equation of state, QHC18 and now QHC19, and the corresponding neutron stars, which agree well with current observations.
  2. We present a much improved equation of state for neutron star matter, QHC19, with a smooth crossover from the hadronic regime at lower densities to the quark regime at higher densities. We now use the Togashi et al.equation of state, a generalization of the Akmal–Pandharipande–Ravenhall equation of state of uniform nuclear matter, in the entire hadronic regime; the Togashi equation of state consistently describes nonuniform as well as uniform matter, and matter at beta equilibrium without the need for an interpolation between pure neutron and symmetric nuclear matter. We describe the quark matter regime at higher densities with the Nambu–Jona–Lasinio model, now identifying tight constraints on the phenomenological universal vector repulsion between quarks and the pairing interaction between quarks arising from the requirements of thermodynamic stability and causal propagation of sound. The resultant neutron star properties agree very well with the inferences of the LIGO/Virgo collaboration, from GW170817, of the pressure versus baryon density, neutron star radii, and tidal deformabilities. The maximum neutron star mass allowed by QHC19 is 2.35 solar masses, consistent with all neutron star mass determinations.