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

TL;DR

This work introduces TROYE, a time-domain waveform framework that stitches two tidal baselines to simulate a dynamical phase transition in a neutron star's EoS during NSBH mergers. Using 100 Bayesian injections with Advanced LIGO design sensitivity, the authors show transitions with |ΔΛ| ≳ 400 are detectable (ln B > 5) for d_L ≲ 100 Mpc, and reveal a robust softening vs. stiffening asymmetry in detectability. They demonstrate the method's resilience to uncertainties in transition timing and mass ratio through targeted stress tests, and discuss avenues for hierarchical analyses across multiple events. The study provides a practical, testable pathway to probe exotic states of dense matter in the late inspiral, with implications for constraining strong-field QCD and the high-density EoS using NSBH GW observations.

Abstract

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.

Figures (10)

• Figure 1: Detailed diagnostic of the troye stitching and phase alignment methodology. The panels display the evolution of the GW signal for a pre-transition model ($\Lambda=400$, solid light blue line), a post-transition model ($\Lambda=4000$, solid orange line), and the resulting stitched waveform (dashed black line). Top Panel: The strain amplitude $h(t)$. Middle Panel: The instantaneous frequency evolution $f(t)$. Bottom Panel: The unwrapped instantaneous phase (left axis) and the absolute phase difference between the aligned pre- and post-transition waveforms (red curve, right logarithmic axis). The vertical dashed line marks the stitching time $t_{stitch}$. Note that the dynamic phase alignment ensures the relative phase difference vanishes ($|\Delta\phi| \to 0$) precisely at $t_{stitch}$, guaranteeing a smooth "hand-off" between the models without discontinuities. A measurable phase discrepancy accumulates only as the system evolves away from $t_{stitch}$ in either direction, encoding the distinct tidal signatures of the two physical states. Note how the stitched signal perfectly tracks the pre-transition evolution (hiding the blue trace) prior to the window (light blue shaded region) and seamlessly transitions to follow the post-transition evolution (hiding the orange trace) thereafter, ensuring phase continuity and a physically smooth transition in the tidal parameter.
• Figure 2: The chosen mass prior distribution of the generated waveforms. On the top panel, a uniform distribution in the chirp mass ($\mathcal{M}$) - mass ratio ($q$) space. On the bottom, a mapping of that same prior to the $m_{BH}-m_{NS}$ space. The color brightness represents the density of the Jacobian for the transformation. The solid pink (magenta) lines show the inverse transformation of equal $m_{BH}$ ($m_{NS}$) lines, respectively, to the $\mathcal{M}-q$ plane. The dashed orange lines show the prior "bounding box" on both planes.
• Figure 3: Domain-validity map for the injection campaign in the $\mathcal{M}$--$q$ plane. Colors show the probability of a safe inspiral (marginalized over other intrinsic parameters), estimated from $N=1000$ random draws per pixel with $\chi_{BH}\sim U(0,0.99)$ and $\Lambda_{NS}\sim U(10,4900)$. A draw is labeled safe if it (i) avoids early plunge relative to the stitching time ($t_{\rm ISCO}<t_{\rm stitch}=40\,\mathrm{ms}$) and (ii) does not tidally disrupt ($M_{\rm rem}=0$ from the Foucart criterion foucart2012, with $C$ inferred from $\Lambda$ via the Yagi--Yunes universal relations YagiYunes2017). Solid white (cyan) contours denote constant $m_{NS}\in\{0.9,2.3\}\,M_\odot$ ($m_{BH}\in\{4,12\}\,M_\odot$). The dashed white curve is the conservative zero-spin plunge boundary ($\chi_{BH}=0$), above which early plunge is excluded for any spin. Magenta diamonds mark LVK events abbott190425abac2024gw230529abbott2021gwtc3abac2025gwtc; from left to right: GW190425, GW230529, GW200115, GW230518, GW200105, GW191219. The hatched box shows our adopted prior, $\mathcal{M}\in[2.0,3.4]\,M_\odot$ and $q\in[0.133,0.286]$, chosen to maximize the yield of valid signals while covering realistic NSBH systems.
• Figure 4: Representative posterior distributions for $\Lambda_{pre}$ (blue) and $\Lambda_{post}$ (orange) illustrating three distinct detection scenarios. Dashed vertical lines indicate the true injected values, while shaded regions denote 90% credible intervals (CIs). Top panel: A successful detection (SP $\checkmark$, PT $\checkmark$). The distributions are clearly distinct with non-overlapping CIs and strong Bayesian evidence ($\ln \mathcal{B} > 5.0$). Middle panel: A visually separated case with insufficient evidence (SP $\checkmark$, PT $\times$). While the CIs do not overlap, the broad uncertainty in the pre-transition posterior results in a Bayes factor below the threshold. Bottom panel: A clear non-detection (SP $\times$, PT $\times$) where the posteriors heavily overlap.
• Figure 5: Distribution of simulated events in the $\Lambda_{pre}$ vs. $\Lambda_{post}$ plane, color-coded by detection status. Blue circles indicate successful phase transition detections (PT?=True), while red crosses denote non-detections. Orange circles (crosses) indicate a successful (unsuccessful) detection that was classified as a TDE by the Foucart criterion foucart2012, with $C$ inferred from $\Lambda$ via the Yagi--Yunes universal relations YagiYunes2017. The dashed grey diagonal line represents the null hypothesis ($\Lambda_{pre} = \Lambda_{post}$). The shaded regions highlight zones of subtle transitions, with $|\Delta \Lambda| < 400$ (darker yellow) and $|\Delta \Lambda| < 800$ (lighter yellow). The clustering of non-detections within the narrow band around the diagonal demonstrates that a minimum shift magnitude is required for the sampler to resolve the transition against the detector noise.