  At supranuclear densities, explored in the core of neutron stars, a strong
phase transition from hadronic matter to more exotic forms of matter might be
present. To test this hypothesis, binary neutron-star mergers offer a unique
possibility to probe matter at densities that we can not create in any existing
terrestrial experiment. In this work, we show that, if present, strong phase
transitions can have a measurable imprint on the binary neutron-star
coalescence and the emitted gravitational-wave signal. We construct a new
parameterization of the supranuclear equation of state that allows us to test
for the existence of a strong phase transition and extract its characteristic
properties purely from the gravitational-wave signal of the inspiraling neutron
stars. We test our approach using a Bayesian inference study simulating 600
signals with three different equations of state and find that for current
gravitational-wave detector networks already twelve events might be sufficient
to verify the presence of a strong phase transition. Finally, we use our
methodology to analyze GW170817 and GW190425, but do not find any indication
that a strong phase transition is present at densities probed during the
inspiral.
