TROYE: Modeling Dynamic Phase Transitions in Gravitational Waves from Neutron Star-Black Hole Mergers

The Equation of State (EoS) of dense nuclear matter remains one of the most compelling open questions in high-energy astrophysics. While static EoS models are increasingly well-constrained by observations of binary neutron star (BNS) inspirals, the possibility of a dynamic phase transition occurring during the coalescence has been thus far deferred from standard gravitational-wave (GW) analyses. In this work, we investigate the detectability of such a phase transition, manifesting as a macroscopic shift in the tidal deformability parameter $Λ$, using GWs from Neutron Star-Black Hole (NSBH) coalescences. We argue that NSBH systems serve as a cleaner laboratory for this phenomenology than BNS systems due to the absence of the $\tildeΛ(Λ_1,Λ_2)$ degeneracy, allowing for the isolation of single-body tidal evolution. We introduce a phenomenological waveform model, TROYE (Transitional Representation Of varYing Equation-of-state), which stitches together two waveform approximants in the time domain to simulate a smooth but rapid transition between two equations of state during the late inspiral. We perform a comprehensive Bayesian injection and recovery campaign on 100 simulated events using the bilby inference library. Our results demonstrate that a phase transition corresponding to a tidal shift of $|ΔΛ| \gtrsim 400$ is detectable with Advanced LIGO design sensitivity, yielding decisive statistical evidence ($\ln B > 5$). We further identify a "V-shape" asymmetry in detectability, where "softening" transitions (decreasing $Λ$) are systematically easier to detect than "stiffening" ones due to the specific phase evolution of the tidal sector. Finally, we present "stress tests" showing that the transition remains recoverable even when marginalized over uncertainties in the stitching time and binary mass ratio.