The amplitudes and instantaneous frequencies of the spherical harmonic modes of eccentric binary black hole (BBH) merger waveforms show quasi-universal, mutually consistent modulations that can be modeled by a single time series, and the gwNRXHME framework exploits this to convert multi-modal quasi-circular waveforms into eccentric ones. These results have been established only for non-precessing (non-spinning and aligned/anti-aligned-spin) binaries. The open problem is to establish the domain of validity of these relations beyond that regime: whether they survive spin-induced precession, requiring dedicated precessing eccentric numerical relativity (NR) simulations and a suitable reformulation of the mode-modulation comparison; and whether the larger departures observed in RIT and MAYA NR data are numerical artifacts or physical breakdown, requiring targeted convergence and waveform-extraction studies.

Background

Gravitational waves from merging binary black holes (BBHs) are observed by the LIGO, Virgo, and KAGRA detectors, and their interpretation relies on waveform models calibrated against full numerical relativity (NR) solutions of the Einstein equations. The strain is conventionally decomposed into spin-weighted spherical harmonic modes hℓm(t)=Aℓm(t)eiϕℓm(t)h_{\ell m}(t)=A_{\ell m}(t)e^{i\phi_{\ell m}(t)}, where AℓmA_{\ell m} is a real amplitude and ϕℓm\phi_{\ell m} a phase; the dominant quadrupolar mode is (2,2)(2,2), and the subdominant higher-order modes are important for parameter estimation of asymmetric binaries. Most existing models assume quasi-circular orbits, in which eccentricity has been radiated away, but formation channels such as dynamical capture in dense stellar clusters can deliver binaries to the detector band with residual orbital eccentricity ee, conventionally quoted together with a mean anomaly (the orbital phase within the radial oscillation cycle) at a chosen reference time or frequency.

Eccentricity imprints characteristic, approximately periodic modulations on the amplitudes and instantaneous frequencies of all modes. Islam (arXiv:2403.15506) found, using non-spinning eccentric SXS simulations, that these modulations are quasi-universal across modes. Writing λ\lambda for the binary parameters and λ0\lambda_0 for the corresponding quasi-circular parameters, the fractional frequency and amplitude modulations are defined as ξℓmω(t)=[ωℓm(t;λ)−ωℓm(t;λ0)]/ωℓm(t;λ0)\xi^{\omega}_{\ell m}(t)=[\omega_{\ell m}(t;\lambda)-\omega_{\ell m}(t;\lambda_0)]/\omega_{\ell m}(t;\lambda_0) and ξℓmA(t)=2ℓ[Aℓm(t;λ)−Aℓm(t;λ0)]/Aℓm(t;λ0)\xi^{A}_{\ell m}(t)=\frac{2}{\ell}[A_{\ell m}(t;\lambda)-A_{\ell m}(t;\lambda_0)]/A_{\ell m}(t;\lambda_0), where ωℓm=dϕℓm/dt\omega_{\ell m}=d\phi_{\ell m}/dt. The empirical findings are that the ξℓmA\xi^{A}_{\ell m} extracted from different modes agree with one another, that the ξℓmω\xi^{\omega}_{\ell m} from different modes agree with one another, and that the two are proportional, ξℓmA=B ξℓmω\xi^{A}_{\ell m}=B\,\xi^{\omega}_{\ell m}, with a single fitted constant. Islam and Venumadhav (Phys. Rev. D 111, L081503 (2025)) extended these tests to all non-precessing binaries — spins aligned or anti-aligned with the orbital angular momentum — using 83 NR simulations from the SXS, RIT, and MAYA catalogs (simulation catalogs produced by three independent numerical-relativity groups with independent codes) with mass ratios from 1:11:1 to 1:41:4. They built the public gwNRXHME framework, which uses these relations to combine a quadrupolar eccentric waveform with a multi-modal quasi-circular non-precessing waveform and thereby obtain all higher-order eccentric modes, reducing the modeling of eccentricity to a single modulation time series. They also defined an overall degree of departure Eξ\mathcal{E}_{\xi}, a relative-L2L^2 combination of the differences between modulations extracted from different modes and of the amplitude-frequency proportionality, and found that SXS simulations adhere to the relations more strictly than RIT and MAYA simulations, with the origin of these larger departures undetermined.

The tested regime excludes precession: when a component spin is misaligned with the orbital angular momentum, the orbital plane and spins precess around the total angular momentum, producing additional amplitude and phase modulations, and the symmetry relation hℓ,−m=(−1)ℓhℓm∗h_{\ell,-m}=(-1)^{\ell}h^{*}_{\ell m} between positive- and negative-mm modes that holds for non-precessing binaries breaks down, so the mode structure itself changes. Independent waveform models cover adjacent territory — aligned-spin eccentric multi-modal effective-one-body (an analytical framework that replaces the relativistic two-body problem by a test particle moving in an effective, deformed black-hole spacetime, calibrated to numerical relativity) and phenomenological models (SEOBNRv5EHM, IMRPhenomTEHM), and effective-one-body models that include both eccentricity and precession — but none of them tests whether the quasi-universal eccentric mode-modulation relations themselves hold when precession is present, nor settles whether the RIT/MAYA departures are numerical or physical.

Problem Statement

Establish the domain of validity of the quasi-universal eccentric mode-modulation relations between spherical harmonic modes of binary black hole merger waveforms — and of the gwNRXHME/gwNRHME construction built on them — beyond the non-precessing regime in which they were verified (non-spinning and aligned/anti-aligned-spin systems with mass ratios 1:11:1 to 1:41:4). Two jointly required results serve this single objective. First, determine whether the single-time-series eccentric amplitude and frequency modulation ansatz, and the algebraic promotion rules derived from it, remain valid for precessing eccentric binary black hole mergers; since the symmetry hℓ,−m=(−1)ℓhℓm∗h_{\ell,-m}=(-1)^{\ell}h^{*}_{\ell m} between positive- and negative-mm modes breaks down under precession, and since precessing eccentric merger data spanning mass ratio, spin magnitudes and directions, eccentricity, and mean anomaly are currently essentially absent from public catalogs, this requires a mode-by-mode reformulation or refutation of the modulation comparison in the appropriate frame. Second, determine whether the larger departures Eξ\mathcal{E}_{\xi} observed in many RIT and MAYA simulations are caused by catalog-dependent numerical error (finite resolution, waveform extraction accuracy) or reflect physical parameter regimes in which the relations genuinely degrade.

References

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