  Gravitational waves (GWs) from primordial black holes (PBHs) inspiralling
inside neutron stars (NSs) within the Galaxy are detectable by ground-based
detectors and provide a unique insight into the interior structure of NSs. To
provide the most accurate templates for GW searches, we solve the Einstein
equations inside NSs, and calculate the orbital motion of the captured PBH by
taking the dynamical friction, accretion, and gravitational radiation into
account. Armed with the most accurate GW waveforms for PBHs inspiralling inside
NSs, we find that Einstein Telescope can distinguish different equations of the
state of NSs. As PBHs inspiral deeper inside NSs, the GW frequency increases
near the surface, then it decreases to a constant deeper inside NSs. The unique
characteristics of GW frequency serve as the smoking gun of GW signals emitted
by PBHs inspiralling inside NSs and can be used to probe the nuclear matter in
the crust and core of NSs.
