  We present a proof-of-concept study, based on numerical-relativity
simulations, of how gravitational waves (GWs) from neutron star merger remnants
can probe the nature of matter at extreme densities. Phase transitions and
extra degrees of freedom can emerge at densities beyond those reached during
the inspiral, and typically result in a softening of the equation of state
(EOS). We show that such physical effects change the qualitative dynamics of
the remnant evolution, but they are not identifiable as a signature in the GW
frequency, with the exception of possible black-hole formation effects. The EOS
softening is, instead, encoded in the GW luminosity and phase and is in
principle detectable up to distances of the order of several Mpcs with advanced
detectors and up to hundreds of Mpcs with third generation detectors. Probing
extreme-density matter will require going beyond the current paradigm and
developing a more holistic strategy for modeling and analyzing postmerger GW
signals.
