Post-merger chirps from binary black holes as probes of the final black-hole horizon

The paper addresses how gravitational-wave (GW) signals from binary black hole (BBH) mergers encode information about the dynamics and geometry of the highly distorted final black hole’s horizon, especially during merger and ringdown. With current detectors (LIGO/Virgo) having inaugurated GW astronomy and validated general relativity (GR) in the strong-field regime, increasing sensitivity and next-generation detectors (LISA, Einstein Telescope) motivate understanding beyond the dominant quadrupole chirp. For asymmetric (unequal-mass) binaries and edge-on orientations, higher-order modes significantly affect waveform morphology during merger and ringdown, potentially revealing new post-merger features. Prior work has correlated near-horizon dynamics with far-field quantities and developed frameworks to visualize spacetime curvature, but a direct link to an observable feature in the measured GW strain has been lacking. This study proposes and tests the hypothesis that observer-dependent, multiple post-merger frequency peaks (post-merger chirps) in the strain are directly linked to rotating, localized regions of extremal curvature on the dynamical apparent horizon, especially a cusp-like defect and associated “trident” structure, thus providing a probe of horizon geometry via GW observations.

The authors summarize existing GW detections and their implications for testing GR and astrophysics. They note that standard face-on, near-equal-mass signals are well described by a simple chirp dominated by the (l,m)=(2,±2) quadrupole. For asymmetric systems and edge-on views, sub-dominant higher-order modes become important and can produce complex morphologies. Two main prior approaches related horizon dynamics to GW emission: (1) empirical correlations between near-horizon fields/geometry and far-field GW flux, including phenomena like anti-kick (e.g., Rezzolla et al.; Jaramillo et al.); (2) analytical frameworks visualizing spacetime curvature (tendex/vortex fields) and linking geometrodynamics to GW generation (Owen et al., Nichols et al.). González et al. previously showed non-trivial post-merger emission visible in the time-domain via Ψ₄ for high mass ratios, but without analyzing the frequency-domain structure or connecting it to explicit strain features. No prior study directly connected a concrete, observable strain feature to specific geometrical structures on the final horizon; this work fills that gap.

- Numerical relativity: Simulations performed with the MAYA code, a branch of the EINSTEINTOOLKIT, evolving the BSSN formulation with the moving puncture gauge. Infrastructure components include CACTUS, KRANC for code generation, and CARPET for mesh refinement. Gravitational radiation extracted via the Newman–Penrose scalar Ψ₄ using WEYLSCAL4. Apparent horizons located with AHFINDERDIRECT, modified to output the induced 2-metric γ_{αβ} for curvature computations. - Systems and configuration: Nonspinning unequal-mass BBHs with mass ratios q from 1.1 to 10. Geometrized units with total mass M and c=1. Observers positioned at various azimuths in the orbital plane and face-on. A representative extraction radius r_ext ≈ 75M; analysis also considers far-field emission snapshots after waves propagate away from the source. - Mode content: Strain reconstructions include higher-order modes beyond the quadrupole: (l,m) = (2,1), (2,2), (3,±2), (3,3), (4,±3), (4,±4). Signals analyzed in time-domain and time-frequency domain. - Time–frequency analysis: Continuous wavelet transform (CWT) with a Morlet mother wavelet (center frequency f0=0.4), scales a ∈ [1,128], implemented via pyCWT, to resolve chirping substructure. Frequency–time maps derived from the relation between scale and frequency. The time of common apparent horizon formation marked for reference. - Near- and far-field linkage: Visualize Ψ₄ amplitude snapshots in the orbital plane at specific post-merger times, identifying distinct wavefront trains and their angular dependence. Compare spatial separations of wavefronts (~ΔR ≈ O(20M)) with temporal separations between observed frequency peaks (Δt ≈ O(20M)), accounting for travel time to the observer. - Horizon geometry diagnostics: On the intersection of the final horizon with the orbital plane (equator), compute mean curvature H (extrinsic; H=∇_μ n^μ) and Gaussian curvature K (intrinsic, from γ_{αβ} and its derivatives). Evaluate gradients along the equator via the arc-length parameter s, relating ds/dφ to γ components. Correlate azimuthal profiles of |dH/ds| and K with Ψ₄ maxima (the “arms”). Identify a cusp-like defect where |dH/ds| attains a maximum, aligned with the strongest Ψ₄ arm within a three-armed “trident” clustered on one side of the horizon and a fourth arm on the opposite side. - Retarded-time alignment: Use retarded time t_frame − r_ext to connect horizon-frame events (arm or back-side passages through the line-of-sight) to features in the recorded strain/Ψ₄ time-frequency maps and time series. - Robustness checks: Demonstrate correlation across multiple mass ratios (q=1.1–10) and verify in an alternative gauge choice (details in Supplementary). Assess temporal persistence of correlations as the horizon relaxes and the emission fades.

- Discovery of observer-dependent multiple post-merger frequency peaks (“post-merger chirps”) in the strain for unequal-mass BBHs, most prominent for observers near the orbital plane and approximately 55° from the final recoil (kick) direction measured in the direction of the original orbit. The double-chirp feature strengthens with increasing mass ratio (q=1.1–10). - Far-field wavefront trains exhibit radial separations ΔR ≈ O(20M), consistent with temporal separations between frequency peaks Δt ≈ O(20M) recorded by observers (accounting for propagation time), and with direction-dependent intensities (stronger secondary wavefronts toward kick-on vs. weaker toward kick-off). - Near-horizon emission forms an asymmetric pattern shortly after merger: three prominent Ψ₄ “arms” cluster on one side of the horizon (a trident), while a fourth arm lies on the opposite (“back”) side. The central, strongest arm aligns with a cusp-like defect on the horizon. - Direct timing correlation: After the cusp (trident arms) cross the line-of-sight, frequency peaks appear in the strain and Ψ₄ maps a time ≈ r_ext later; when the back side crosses, frequency minima occur. Time-domain instantaneous wavelength changes are consistent with the spacing of arriving wavefronts. - Geometric correlation: Azimuthal maxima of Ψ₄ on the horizon equator co-locate with local maxima of |dH/ds| (mean curvature gradient) and with locally extremal Gaussian curvature K. The cusp corresponds to the global maximum of |dH/ds| and the strongest Ψ₄ arm. These correlations persist across the post-merger evolution as the structure rotates and fades, with slight degradation only at very late times (~30M after common horizon formation) when the signal is weak and more susceptible to numerical noise. - Robustness across systems and gauges: The Ψ₄–curvature correlations hold for all analyzed mass ratios and persist under an alternative gauge choice (per Supplementary material). - Observational prospect: Simulations indicate that Advanced LIGO at design sensitivity could observe the post-merger chirp signature for a favorably oriented analog of BBH event GW170729, suggesting feasibility prior to next-generation detectors.

The study establishes a concrete, observable feature in the GW strain—the presence and timing of post-merger chirps—and links it directly to specific, localized geometric structures on the dynamical apparent horizon of the remnant black hole. This addresses the research goal of probing horizon dynamics via far-field GWs, moving beyond qualitative correlations or indirect measures. The frequency peaks arise when strongly emitting regions with large mean curvature gradients and extremal Gaussian curvature (the trident arms, particularly the cusp) cross the observer’s line-of-sight; corresponding minima occur when the smoother opposite region passes. The direction- and mass-ratio-dependent morphology is consistent with the excitation and interference of higher-order quasi-normal modes during merger and ringdown, providing an interpretable mapping from waveform substructure to horizon geometry. The results imply that with improving detector sensitivity, post-merger chirps can serve as probes of strong-field GR and tests related to black hole horizon properties, complementing ringdown spectroscopy and offering a novel window distinct from proposed exotic-object echo signatures. The correlation’s persistence over time and across gauges supports its physical robustness, though its visibility depends on orientation and asymmetry.

This work provides the first explicit connection between a concrete, observable feature in the GW strain—multiple post-merger frequency peaks—and geometric structures on the final black hole’s dynamical apparent horizon. The key contributions are: (1) identification of post-merger double-chirp features for unequal-mass, edge-on binaries; (2) demonstration that these features’ timing and intensity correspond to the passage of a rotating trident of emission tied to regions of extremal Gaussian curvature and maximal mean curvature gradient, with the strongest arm aligned with a cusp-like defect; and (3) validation across mass ratios and gauges, with clear observational implications. The findings suggest that current and near-future detectors could use post-merger chirps to probe horizon geometry and test GR in the strongest-field regime. Future work should extend to spinning and precessing binaries to map how spin-induced dynamics alter the orientation and evolution of the cusp/trident structure, refine data-analysis methods to detect and quantify post-merger chirps in real detector noise, and explore synergies with ringdown spectroscopy to jointly constrain horizon geometry and black hole parameters.

- Numerical limitations: At late post-merger times (~30M after common horizon formation), emission is weak and correlations between Ψ₄ and curvature exhibit slight degradation, likely due to numerical artifacts and low signal strength. - System scope: The presented results focus on nonspinning unequal-mass binaries; spin and precession, which can modulate emission patterns and viewing-angle dependence, are not explored here and may shift the direction of maximal chirp visibility. - Observational constraints: Detectability depends strongly on source orientation (near the orbital plane and ~55° from the kick direction) and mass asymmetry; current detector sensitivities may limit routine observation until higher SNR events or next-generation detectors. Horizon location is gauge-dependent, though correlations with curvature and Ψ₄ appear robust under gauge changes examined. - Mode content and reconstruction: While several dominant higher modes are included, incomplete mode content or extraction uncertainties could affect detailed morphology for certain configurations.