  Neutron stars are one of the most mysterious wonders in the Universe. Their
extreme densities hint at new and exotic physics at work within. Gravitational
waves could be the key to unlocking their secrets. In particular, a first
detection of gravitational waves from rapidly-spinning, deformed neutron stars
could yield new insights into the physics of matter at extreme densities and
under strong gravity. Once a first detection is made, a critical challenge will
be to robustly extract physically interesting information from the detected
signals. In this essay, we describe initial research towards answering this
challenge, and thereby unleashing the full power of gravitational waves as an
engine for the discovery of new physics.
