Analytical derivation of long-term dephasing caused by phase transitions in the context of Kerr black holes

Extreme Mass Ratio Inspirals (EMRIs) constitute a prime target for future space-based gravitational-wave observatories such as LISA. In this paper, we analytically investigate the long-term phase shift (dephasing) in the gravitational wave signal induced by a first-order quantum chromodynamics (QCD) phase transition within a neutron star orbiting a supermassive Kerr black hole. By modeling the transition from a hadronic phase to a quark core phase, we quantify the sudden change in the tidal deformability ($Λ$) of the secondary object. Utilizing the Teukolsky formalism and Post-Newtonian expansions, we derive a strict analytical scaling law for the accumulated dephasing. We demonstrate that the Kerr spin parameter $a$ and the critical phase transition orbital velocity $v_c$ significantly amplify the dephasing effect. Our analytical framework provides a robust tool for probing the non-perturbative QCD equation of state at high baryon densities using gravitational wave astronomy.