  Multimessenger signals from binary neutron star (BNS) mergers are promising
tools to infer the largely unknown properties of nuclear matter at densities
that are presently inaccessible to laboratory experiments. The gravitational
waves (GWs) emitted by BNS merger remnants, in particular, have the potential
of setting tight constraints on the neutron-star equation of state (EOS) that
would complement those coming from the late inspiral, direct mass-radius
measurements, or ab-initio dense-matter calculations. To explore this
possibility, we perform a representative series of general-relativistic
simulations of BNS systems with EOSs carefully constructed so as to cover
comprehensively the high-density regime of the EOS space. From these
simulations, we identify a novel and tight correlation between the ratio of the
energy and angular-momentum losses in the late-time portion of the post-merger
signal, i.e., the "long ringdown", and the properties of the EOS at the highest
pressures and densities in neutron-star cores. When applying this correlation
to post-merger GW signals, we find a significant reduction of the EOS
uncertainty at densities several times the nuclear saturation density, where no
direct constraints are currently available. Hence, the long ringdown has the
potential of providing new and stringent constraints on the state of matter in
neutron stars in general and, in particular, in their cores.
