ECCentric: Decomposing Eccentric Waveforms

• ECCentric is a data-driven framework that decomposes gravitational wave signals from eccentric binary black holes into monotonic eccentric harmonics.
• It utilizes singular value decomposition and advanced signal-processing techniques to extract and rank harmonics by instantaneous frequency and amplitude trends.
• The universal phase-frequency law, derived from effective-one-body dynamics, provides a robust basis for accurately reconstructing inspiral-merger signals.

Searching arXiv for the focal paper and closely related work on eccentric waveform decomposition, waveform-based eccentricity definitions, and eccentric EOB/IMR modeling. ECCentric denotes a data-driven reorganization of eccentric binary black hole merger waveforms in which each spin-weighted-spherical-harmonic mode $h_{\ell m}(t)$ is decomposed into eccentricity-induced subcomponents, or eccentric harmonics, $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$. In aligned-spin eccentric binary black holes, these constituent harmonics exhibit monotonically time-varying amplitudes and frequencies even when the cumulative mode $h_{\ell m}(t)$ shows strong amplitude and frequency modulations and multiple overlapping time-frequency tracks. The central phenomenology is that the phase and frequency of each eccentric harmonic separate into a secular orbital part and a purely eccentric correction, with the secular phase identified with the relativistic anomaly and the eccentric correction related to precession advances (Islam et al., 24 Sep 2025).

1. Definition and basic phenomenology

For an eccentric binary black hole, the standard radiation mode $h_{\ell m}(t)$ does not arise from a single, slowly evolving carrier as in the quasi-circular case. Newtonian and post-Newtonian theory instead predict that each $h_{\ell m}(t)$ is a superposition of several eccentricity-induced components, referred to as eccentric harmonics, so that

$$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$

where $\lambda$ denotes the intrinsic parameters and $j$ is the eccentric-harmonic index (Islam et al., 24 Sep 2025).

The circular limit is structurally simple. As $e\to 0$, a single harmonic survives; for the modes studied, the dominant $j$ equals $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$0, while all other harmonics vanish smoothly with eccentricity. Away from that limit, the full mode $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$1 acquires beatings, amplitude-frequency modulations, and multiple time-frequency tracks. By contrast, each extracted $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$2 has monotonic amplitude and frequency evolution over most of the inspiral. This distinction is the defining feature of ECCentric: it isolates the complicated eccentric structure of the observable mode into a small number of monotonic constituents (Islam et al., 24 Sep 2025).

The study analyzes six spherical harmonic modes, $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$3, $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$4, $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$5, $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$6, $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$7, and $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$8. For each $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$9, the leading eccentric harmonic is $h_{\ell m}(t)$0, and significant power begins at $h_{\ell m}(t)$1. In the $h_{\ell m}(t)$2 mode, the analysis uses $h_{\ell m}(t)$3 (Islam et al., 24 Sep 2025).

2. Data-driven extraction with gwMiner

ECCentric is extracted with the data-driven framework gwMiner, which combines singular value decomposition, input from post-Newtonian theory, and signal-processing techniques. The method extends earlier gwMiner work on nonspinning quadrupole radiation to aligned-spin systems and higher modes (Islam et al., 24 Sep 2025).

The pipeline is organized around ensembles at fixed eccentricity and uniformly sampled relativistic anomaly. Waveforms are generated with the aligned-spin eccentric SEOBNRv5EHM model, with mass ratios $h_{\ell m}(t)$4, spins $h_{\ell m}(t)$5, and $h_{\ell m}(t)$6 up to $h_{\ell m}(t)$7, together with a very high-eccentricity example $h_{\ell m}(t)$8 used to visualize track crowding. The reference anomaly $h_{\ell m}(t)$9 is sampled uniformly over $h_{\ell m}(t)$0 with 50 snapshots per configuration, and the waveforms are long, of order $h_{\ell m}(t)$1, with merger at $h_{\ell m}(t)$2 (Islam et al., 24 Sep 2025).

Step	Operation	Purpose
1	Ensemble generation at fixed $h_{\ell m}(t)$3	Sample $h_{\ell m}(t)$4
2	Preprocessing and STFT spectrograms	Verify multiple monotonic tracks
3	SVD across anomaly snapshots	Build eccentric-harmonic bases $h_{\ell m}(t)$5
4	Indexing and smoothing	Rank by instantaneous frequency and enforce monotonicity
5	Diagnostics and validation	Reconstruct modes and test phase relations

For each $h_{\ell m}(t)$6, the SVD is performed on a data matrix whose rows are times and whose columns correspond to different $h_{\ell m}(t)$7. The left-singular vectors define normalized eccentric-harmonic bases $h_{\ell m}(t)$8, and the right-singular vectors provide complex coefficients $h_{\ell m}(t)$9. Harmonics are indexed by instantaneous-frequency ranking. Amplitudes $h_{\ell m}(t)$0 are then smoothed with post-Newtonian-inspired fits to enforce monotonicity; for higher modes, only the early inspiral, approximately the first $h_{\ell m}(t)$1, is fit to prevent power leakage among overlapping harmonics (Islam et al., 24 Sep 2025).

3. Universal phase-frequency law

The central ECCentric result is a universal phase-frequency structure. For the quadrupole, the phase and instantaneous frequency take the form

$h_{\ell m}(t)$2

where the secular quantities $h_{\ell m}(t)$3 survive in the circular limit and scale linearly with $h_{\ell m}(t)$4, while the eccentric contributions $h_{\ell m}(t)$5 are independent of $h_{\ell m}(t)$6 and vanish as $h_{\ell m}(t)$7. A $h_{\ell m}(t)$8 phase offset is observed for the $h_{\ell m}(t)$9 harmonic in the $$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$0 mode (Islam et al., 24 Sep 2025).

For aligned-spin eccentric binary black holes and higher modes, the general relation is

$$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$1

Two phenomenological facts emerge. First, $$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$2 is identical for all $$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$3, so one may write $$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$4. Second, the eccentric correction scales with $$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$5, giving the universal form

$$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$6

When divided by $$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$7, the eccentric correction collapses across modes onto a single curve (Islam et al., 24 Sep 2025).

Effective-one-body dynamics gives these terms a direct dynamical interpretation. Using SEOBNRv5EHM, the study finds

$$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$8

with $$h_{\ell m}(t;\lambda)=\sum_j h_{\ell m,j}(t;\lambda)=\sum_j A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)},$$9 the relativistic anomaly and $\lambda$0 the accumulated orbital phase. Thus $\lambda$1 is the relativistic anomaly, while $\lambda$2 is the precession advance, or periastron-precession-driven correction to the orbital phase. Residuals between the data-driven phases and the EOB quantities are small, of order $\lambda$3 rad, consistent with post-Newtonian-scale oscillations in EOB-averaged quantities close to merger (Islam et al., 24 Sep 2025).

4. Amplitudes, coefficients, and reconstruction

The amplitudes $\lambda$4 are monotonic and hierarchical, and they follow simple scaling with $\lambda$5. Representative fits for the fiducial example include

$\lambda$6

together with

$\lambda$7

Some subdominant harmonics scale as $\lambda$8 in the early inspiral (Islam et al., 24 Sep 2025).

The complex SVD coefficients $\lambda$9 also show simple structure. For fixed $j$0, the arguments of the dominant coefficients obey

$j$1

with a universal secondary oscillation

$j$2

The magnitude $j$3 of the leading harmonic is nearly constant with $j$4, varying by less than $j$5, whereas subleading magnitudes are periodic in $j$6 (Islam et al., 24 Sep 2025).

These findings imply an explicit phenomenological reconstruction,

$j$7

with

$j$8

Using the first four harmonics gives excellent reconstruction: relative $j$9 errors are approximately $e\to 0$0 for $e\to 0$1, $e\to 0$2 for $e\to 0$3, and $e\to 0$4 for $e\to 0$5 over the full signal, with further improvement when restricted to inspiral, $e\to 0$6. Solving for $e\to 0$7 and $e\to 0$8 from different harmonic pairs yields residuals of approximately $e\to 0$9 rad for $j$0 and $j$1 rad across modes (Islam et al., 24 Sep 2025).

5. Relation to eccentricity characterization and waveform modeling

ECCentric is complementary to waveform-based eccentricity standards. “Defining eccentricity for gravitational wave astronomy” defines $j$2 and mean anomaly $j$3 directly from the waveform at future null infinity by using pericenter and apocenter structure of the $j$4 mode and an orbit-averaged $j$5 to define $j$6 (Shaikh et al., 2023). “Post-Newtonian theory-inspired framework for characterizing eccentricity in gravitational waveforms” also exploits common eccentricity-induced modulations across spherical-harmonic modes, but its primary goal is a smooth, post-Newtonian-connected estimate of $j$7 rather than a $j$8-resolved decomposition into eccentric harmonics (Islam et al., 4 Feb 2025).

ECCentric also sits alongside existing eccentric inspiral-merger-ringdown models rather than replacing them. Related developments include the spin-aligned, moderately eccentric EOB model built on TEOBResumS (Chiaramello et al., 2020) and the ENIGMA family of non-spinning eccentric IMR models (Huerta et al., 2017, Chen et al., 2020). The distinctive point is that standard numerical relativity outputs and current eccentric IMR models do not directly provide a $j$9-decomposition through merger-ringdown, whereas ECCentric extracts a robust, data-driven basis $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$00 and coefficients $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$01 from those waveforms (Islam et al., 24 Sep 2025).

This suggests a useful division of labor. Waveform-only eccentricity definitions standardize what is meant by “eccentricity” across models, while ECCentric resolves how eccentric structure is distributed across monotonic constituents inside each spherical-harmonic mode.

6. Scope, limitations, and prospective use

The present framework is restricted to aligned-spin, non-precessing SEOBNRv5EHM waveforms. Robust extraction is demonstrated for moderate eccentricities $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$02. At higher eccentricity, harmonic tracks overlap more strongly and reconstruction degrades; the $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$03 example is included specifically to visualize track crowding. Near merger, overlap among tracks increases sharply, especially for higher modes, which complicates amplitude smoothing and raises reconstruction errors (Islam et al., 24 Sep 2025).

Reference conventions also matter. Small differences appear when choosing different prescriptions for $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$04 across the ensemble: the phases of $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$05 are robust, but subleading magnitudes can vary. Extension to fully precessing eccentric systems is presently limited by the lack of validated precessing-eccentric IMR models and numerical-relativity catalogs. The stated next steps are extension to precessing-eccentric binary black holes and construction of a fast surrogate of the eccentric harmonics themselves (Islam et al., 24 Sep 2025).

The practical significance follows directly from the decomposition. Because each harmonic is monotonic, these constituents are simpler to model, template, and filter against than the cumulative multi-track mode. The paper explicitly notes potential for rapid searches and parameter estimation of eccentric asymmetric binaries, and it notes that extraction per mode takes approximately $$h_{\ell m,j}(t)=A_{\ell m,j}(t)e^{i\phi_{\ell m,j}(t)}$$06 s, which motivates reduced-order or surrogate models of the eccentric harmonics (Islam et al., 24 Sep 2025).