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

Abstract

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.

Figures (4)

• Figure 1: Schematic illustration of an EMRI in which the neutron-star secondary undergoes a microphysical transition during inspiral around a Kerr black hole. Outside the critical radius $r_c$, the companion remains in a hadronic configuration with tidal deformability $\Lambda_{\rm H}$ (blue segment). Once the orbit enters the critical regime, the star develops a denser quark core and the effective tidal response changes to $\Lambda_{\rm Q}<\Lambda_{\rm H}$ (red segment). The strong-field endpoint is set by $r_{\rm ISCO}(\hat{a})$, while the phase evolution is correspondingly modified by the transition-induced jump $\Delta\Lambda<0$.
• Figure 2: Dimensionless tidal deformability $\Lambda$ as a function of the normalized central pressure $p_c/p_{\rm trans}$. The dashed curve shows an idealized sharp first-order transition, while the solid curve shows a finite-width mixed-phase transition. The reduction from the hadronic branch to the quark-core branch defines the effective microphysical input $\delta\Lambda$ that enters the dephasing integral.
• Figure 3: Contour plot of the accumulated dephasing $\log_{10}|\Delta\Phi|$ in the $(v_c,\hat{a})$ plane, computed from the $v^9$-truncated analytical formula. The red curve denotes the boundary $v_c=v_{\rm ISCO}(\hat{a})$, beyond which the transition would occur only at or after plunge.
• Figure 4: Waveform-level manifestation of the phase-transition-induced dephasing. The main waveform panel compares the baseline signal with the phase-transition-modified signal during the late inspiral. The accompanying phase-accumulation panels show the growth of $\Delta\Phi$ with gravitational-wave frequency and the corresponding normalized cumulative profile, making explicit where in the inspiral the microphysical signal is accumulated.