  We consider Quarkyonic Matter to naturally explain the observed properties of
neutron stars. We argue that such matter might exist at densities close to that
of nuclear matter and at the onset, the pressure and the sound velocity in
Quarkyonic matter increase rapidly. In the limit of large number of quark
colors $N_c$, this transition is characterized by a discontinuous change in
pressure as a function of baryon number density. We make a simple model of
Quarkyonic matter and show that generically the sound velocity is a
non-monotonic function of density -- it reaches a maximum at relatively low
density, decreases, and then increases again to its asymptotic value of
$1/\sqrt{3}$.
