The Multi-Messenger Matrix: the Future of Neutron Star Merger Constraints on the Nuclear Equation of State

The electromagnetic (EM) signal of a binary neutron star (BNS) merger depends sensitively on the total binary mass, $M_{\rm tot}$, relative to various threshold masses set by the neutron star (NS) equation-of-state (EOS), parameterized through the NS maximum mass, $M_{\rm TOV}$, and characteristic radius, $R_{1.6}$. EM observations of a BNS merger detected through its gravitational wave (GW) emission, which are of sufficient quality to ascertain the identity of the merger remnant, can therefore constrain the values of $M_{\rm TOV}$ and $R_{1.6}$, given the tight connection between $M_{\rm tot}$ and the well-measured chirp mass. We elucidate the present and future landscape of EOS constraints from BNS mergers, introducing the "Multi-Messenger Matrix", a mapping between GW and EM measurables that defines the ranges of event chirp masses which provide the most leverage on constraining the EOS. By simulating a population of BNS mergers drawn from the Galactic double NS mass distribution we show that $\sim 10$ joint detections can constrain $M_{\rm TOV}$ and $R_{1.6}$ to several percent level where systematic uncertainties may become significant. Current EOS constraints imply that most mergers will produce supramassive or hypermassive remnants, a smaller minority (possibly zero) will undergo prompt-collapse, while at most only a few percent of events will form indefinitely stable NSs. In support of the envisioned program, we advocate in favor of LIGO/Virgo releasing chirp mass estimates as early as possible to the scientific community, enabling observational resources to be allocated in the most efficient way to maximize the scientific gain from multi-messenger discoveries.