Following merger, a neutron star (NS) binary can produce roughly one of three different outcomes: (1) a stable NS, (2) a black hole (BH), or (3) a supra-massive, rotationally-supported NS, which then collapses to a BH following angular momentum losses. Which of these fates occur and in what proportion has important implications for the electromagnetic transient associated with the mergers and the expected gravitational wave signatures, which in turn depend on the high density equation of state (EOS). Here we combine relativistic calculations of NS masses using realistic EOSs with Monte Carlo population synthesis based on the mass distribution of NS binaries in our Galaxy to predict the distribution of fates expected. For many EOSs, a significant fraction of the remnants are NSs or supra-massive NSs. This lends support to scenarios where a quickly spinning, highly magnetized NS may be powering an electromagnetic transient. This also indicates that it will be important for future gravitational wave (GW) observatories to focus on high frequencies to study the post merger GW emission. Even in cases where individual GW events are too low in signal to noise to study the post merger signature in detail, the statistics of how many mergers produce NSs versus BHs can be compared with our work to constrain the EOS. To match short gamma-ray burst (SGRB) X-ray afterglow statistics, we find that the stiffest EOSs are ruled out. Furthermore, many popular EOSs require ~60-70% of SGRBs to be from NS-BH mergers rather than just binary NSs.
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