Accurate waveforms for eccentric, aligned-spin binary black holes: The multipolar effective-one-body model SEOBNRv5EHM

Aldo Gamboa; Alessandra Buonanno; Antoni Ramos-Buades; Hannes R. R\"uter; Harald P. Pfeiffer; H\'ector Estell\'es; Lawrence E. Kidder; Lorenzo Pompili; Mark A. Scheel; Michael Boyle

REVIEW 3 major objections 4 minor 19 cited by

Accurate waveforms for eccentric, aligned-spin binary black holes: The multipolar effective-one-body model SEOBNRv5EHM

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A new waveform model for eccentric, aligned-spin binary black holes achieves sub-percent (2,2)-mode accuracy for initial GW eccentricities below 0.5, about an order of magnitude better than previous models.

desk verdict First eccentric aligned-spin EOB model with 3PN eccentricity corrections in both the radiation reaction and waveform modes; the order-of-magnitude accuracy improvement over prior eccentric models is credible, though the headline 0.02% median is a best-fit over initial eccentricity and starting frequency, not a fixed-parameter predictive number. read the letter →

arxiv 2412.12823 v2 pith:3MSMAXPU submitted 2024-12-17 gr-qc

Aldo Gamboa , Alessandra Buonanno , Raffi Enficiaud , Mohammed Khalil , Antoni Ramos-Buades , Lorenzo Pompili , Héctor Estellés , Michael Boyle

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Lawrence E. Kidder Harald P. Pfeiffer Hannes R. Rüter Mark A. Scheel

This is my paper · ORCID

classification gr-qc

keywords effective-one-bodyeccentricbinaryblackholesalignedspinsgravitational-wavewaveformmodel3PNeccentricitycorrectionsinspiral-merger-ringdownnumericalrelativityvalidationSEOBNRv5EHM

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

SEOBNRv5EHM is a time-domain, effective-one-body waveform model that aims to give accurate gravitational-wave templates for black-hole binaries on eccentric orbits with spins aligned with the orbital angular momentum. It includes, for the first time in an eccentric EOB model, third-post-Newtonian-order eccentricity corrections to both the radiation-reaction force and the waveform modes, and it is built on a highly accurate circular-orbit model. The authors claim that for numerical-relativity waveforms with initial gravitational-wave eccentricity below 0.5, the worst (2,2)-mode mismatch over total masses 20–200 solar masses stays below or near 1%, with a median near 0.02%, roughly an order of magnitude better than the previous generation. If true, this makes eccentric templates accurate enough for unbiased parameter estimation just as detectors are expected to see a subpopulation of eccentric mergers.

What carries the argument

The load-bearing construction is the factorized eccentric EOB waveform: each mode is written as $h^\mathrm{F}_{\ell m} = h^{\mathrm{F},\mathrm{qc}}_{\ell m}(x)\, h^{\mathrm{ecc}}_{\ell m}(x,e,\zeta)$, where $h^{\mathrm{F},\mathrm{qc}}_{\ell m}$ is the SEOBNRv5HM quasi-circular mode and $h^{\mathrm{ecc}}_{\ell m}$ carries nonspinning eccentricity corrections through 3PN order. The orbital dynamics is a coupled system of Hamilton's equations plus evolution equations for the Keplerian radial phase $\zeta$, eccentricity $e$, and frequency parameter $x=\langle M\Omega\rangle^{2/3}$; a background quasi-circular evolution with the same masses and spins supplies the nonquasicircular corrections and the matching time, and the merger-ringdown is the same Kerr quasinormal-mode ansatz as in the circular model. The Keplerian parametrization $r = M/[u_p(1+e\cos\zeta)]$ and the 3PN radiation-reaction corrections are what carry the improved phasing at each periastron passage.

What would settle it

Take a numerical-relativity simulation not in the validation set with initial GW eccentricity around 0.45 at 20 Hz, mass ratio about 18, and strongly antialigned spins; compute the (2,2)-mode mismatch of SEOBNRv5EHM over total masses 20–200 solar masses with the same windowing and optimization used in the paper. If the maximum mismatch clearly exceeds 1%, the claim that eccentricities below 0.5 stay below or close to 1% fails.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that SEOBNRv5EHM is currently the most accurate public inspiral-merger-ringdown model for eccentric, aligned-spin black-hole binaries. It achieves this by generalizing the EOB dynamics with a Keplerian parametrization of the orbit and by folding in 3PN eccentricity corrections in the radiation reaction and in the factorized waveform modes, while taking the late-inspiral, merger, and ringdown from the circular-orbit model SEOBNRv5HM via a background circular dynamics. Validated against 99 eccentric numerical-relativity waveforms, the model keeps maximum (2,2)-mode unfaithfulness below or close to 1% for initial GW eccentricities up to 0.5 over total masses 20–200 solar masses, with a median of about 0.02%; it degrades to tens of percent only for eccentricities above about 0.7. In the circular limit it reproduces SEOBNRv5HM, and in two real-event analyses it finds no eccentricity signatures in GW150914 or GW190521.

Load-bearing premise

The model assumes that an eccentric binary has circularized enough by late inspiral that a circular-orbit evolution with the same masses and spins can stand in for the real dynamics when attaching the merger and ringdown; if a binary still has significant eccentricity at merger, this part of the waveform is wrong.

Editorial extensions

If this is right

For binaries with initial GW eccentricity below 0.5, the (2,2) mode has maximum unfaithfulness below or near 1% for total masses 20–200 solar masses, with a median near 0.02%.
This is roughly an order-of-magnitude accuracy gain over the previous-generation SEOBNRv4EHM and over the TEOBResumS-Dalí model on the same 99 simulations.
The model's circular limit agrees with SEOBNRv5HM, so eccentric templates can be used on quasi-circular events without introducing an eccentricity bias.
Zero-noise injections of numerical-relativity signals are recovered within 90% credible intervals, and analyses of GW150914 and GW190521 find no eccentricity signatures.
For initial GW eccentricities above about 0.7, mismatches rise to roughly 20%, so statements about highly eccentric waveforms should be made with caution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the circularization premise holds for most detectable binaries, the model makes it practical to search for moderate eccentricity with current detection pipelines without losing sensitivity to circular mergers, since the circular limit is already calibrated.
The strong dependence of accuracy on the 3PN corrections, compared with a 2PN variant, suggests that extending the eccentricity corrections to 4PN or adding spin-dependent eccentric terms could push the accuracy envelope further into the high-eccentricity regime.
The seven-equation overdetermined EOB-plus-Keplerian system that can desynchronize suggests a future reformulation with a single consistent orbital parametrization would remove a fragile part of the model.
If upcoming observing runs deliver eccentric events, this model should provide the first clean measurements of eccentricity at formation, directly testing dynamical-capture formation channels.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line up front: this paper delivers what it claims. SEOBNRv5EHM is the first eccentric, aligned-spin, multipolar EOB model with 3PN eccentricity corrections in both the radiation-reaction force and the waveform modes, and the validation evidence supports the main accuracy claim: roughly an order of magnitude better unfaithfulness than SEOBNRv4EHM and TEOBResumS-Dali against the same 99 eccentric NR waveforms. The model is public, fast enough for Bilby runs, and its QC limit reproduces SEOBNRv5HM, which matters for avoiding false eccentricity claims in parameter estimation.

What is genuinely new: the 3PN eccentricity corrections themselves (from the companion paper), the background-QC prescription for NQC corrections and matching time, and the post-adiabatic initial-condition scheme with backward secular evolution. The validation is extensive: 441 QC and 99 eccentric NR waveforms, a careful check of the QC limit, and a clean demonstration that the 3PN corrections matter for eccentricities above about 0.4. The NR injection recoveries and the GW150914/GW190521 analyses are sensible sanity checks rather than headline results, and the paper is appropriately cautious about them.

Soft spots, in proportion: the stress-test concern is real but not fatal. Equation (34) maximizes over the template's initial eccentricity and starting frequency, so the reported 0.02% median is a best-fit under two extra degrees of freedom, not a fixed-parameter predictive mismatch. Since the same optimization is applied to all three models, the relative order-of-magnitude comparison stands; the absolute claim should be read as in-sample accuracy after tuning initial conditions. Also, 71 of the 99 eccentric NR waveforms are private, so independent recomputation of the headline numbers is not currently possible, and the 3PN formulas are in the companion paper rather than checkable from this text. The overdetermined EOB-plus-Keplerian system is acknowledged to desynchronize outside a visually defined conservative region, and the QC merger-ringdown assumption degrades at high eccentricity, with mismatches around 20% for egw above about 0.7. These are disclosed limitations, and none of them undercuts the central claim for moderate eccentricities below roughly 0.5.

My verdict: conditional accept. The central physics and engineering look sound; the caveats are about external dependencies and the mismatch protocol, not about a load-bearing error. This paper deserves serious peer review and will be cited by anyone doing eccentric waveform work. I would bring it to a reading group, mainly because the mismatch-optimization discussion is a useful case study in how validation metrics can flatter a model.

Referee Report

3 major / 4 minor

Summary. The paper presents SEOBNRv5EHM, a time-domain effective-one-body waveform model for eccentric, aligned-spin binary black holes, built on the quasicircular SEOBNRv5HM model. It incorporates nonspinning eccentricity corrections up to 3PN order in the radiation-reaction force and waveform modes, taken from the companion paper [241], and models six (ell, |m|) modes. The validation compares the model with 441 quasicircular and 99 eccentric SXS NR waveforms, using an unfaithfulness measure that optimizes over the template's initial eccentricity and starting frequency. The authors report a median (2,2)-mode mismatch of about 0.02% and an order-of-magnitude improvement over SEOBNRv4EHM and TEOBResumS-Dali, demonstrate the benefit of the 3PN terms, characterize speed and a conservative robustness region, and apply the model to NR-injection recovery and to GW150914 and GW190521.

Significance. The model, if validated as claimed, would be the most accurate public eccentric IMR model in its parameter space and a practical tool for Bayesian inference. The paper's strengths are substantial: the eccentric content is parameter-free analytical input from a dedicated companion paper, the accuracy is tested against 99 eccentric NR waveforms that were not used to calibrate the eccentric sector, the 3PN-versus-2PN improvement is demonstrated directly (Figs. 7 and 18), the QC limit is shown to match SEOBNRv5HM closely, and the implementation is public through pySEOBNR. The main caveat is that the headline mismatch is obtained under optimization over template initial conditions; this affects the absolute interpretation of the accuracy claim rather than undermining the relative comparison.

major comments (3)

[Sec. IV A 3, Eq. (34), and footnote 14] The headline accuracy claim is obtained by maximizing the faithfulness over the template's initial eccentricity et and starting frequency <M Omega>_t (with zeta_t fixed to pi), and the template is then trimmed to the NR time to merger. Because the optimizer can choose a lower starting frequency and adjust et, any systematic dephasing in the early inspiral can be absorbed by adding unpenalized cycles that are later trimmed away. Thus the abstract statement that 'the maximum (2,2)-mode unfaithfulness ... is consistently below or close to 1%, with a median value of ~0.02%' is a best-fit statement over a two-dimensional family of initial conditions, not a fixed-parameter predictive accuracy. The comparison among SEOBNRv4EHM, SEOBNRv5EHM, and TEOBResumS-Dali is fairer since all three receive the same optimization, but I ask the authors to add a fixed-parameter mismatch at the NR's own (egw, <M Omega>) values and to report the distribution of the optimized (e_opt, <M Omega>_opt) offsets from the NR values, so readers can separate intrinsic model accuracy from optimizer flexibility.
[Sec. III A 2, Eq. (22), and Sec. III C] The late-inspiral and merger-ringdown content rests on the assumption that there exists t* after which the eccentric separation coincides with the background QC separation, and the merger-ringdown is taken from the QC SEOBNRv5HM ansatz. The paper documents that this premise fails at high eccentricity (mismatches ~20% for egw >~ 0.7), but the claim for egw < 0.5 also depends on it. Since the optimization in Eq. (34) can mask dephasing, indirect evidence from low mismatches is not enough; please provide direct evidence, e.g., compare (r, pr*, Omega) of the eccentric and background QC dynamics near merger for representative cases in the claimed accuracy range, and report mismatches computed with and without the merger-ringdown segment. This would pin down the eccentricity at which the QC-merger assumption starts to degrade.
[Sec. IV E and Appendix E] The recommended validity region (q in [1,20], chi_1,2 in [-0.999,0.999], e in [0,0.45], zeta in [0,2pi], M >= 10 M_sun at <f_start> = 20 Hz) is fixed by visual inspection of waveform envelopes, and Appendix E shows that the overdetermined system Eqs. (4a)-(4g) can desynchronize outside it without an automated diagnostic. The abstract's accuracy claim, however, is stated for the NR suite with egw < 0.5, and the optimized model eccentricities in Table V can exceed 0.45 in that suite (e.g., SXS:BBH:2549 and SXS:BBH:3966). I recommend a quantitative, reproducible definition of the boundary, and a statement of which validation cases fall inside it, so that the region in which accuracy, robustness, and speed are claimed is not a fuzzy envelope.

minor comments (4)

[Figure 4 and Figure 18 captions] The word 'Botttom' appears in the captions of Figs. 4 and 18; it should read 'Bottom'.
[Sec. IV A 3 / Table V] The abstract's phrase 'below or close to 1%' is somewhat loose: Table V lists a maximum (2,2)-mode mismatch of 1.131% for SXS:BBH:3826 within the egw < 0.5 subset. Please state the actual maximum value and the threshold used in the abstract.
[Sec. IV B / footnote 7] The replacement of v_phi by v_Omega in Eq. (18) is justified only by the QC-limit comparison; since v_Omega changes the eccentric waveform phasing, a brief discussion of the systematic effect for e > 0 would be useful.
[Table IV, GW150914 rows] The log10 Bayes factors from serial and parallel Bilby for GW150914 differ by 0.23 (-0.34 versus -0.57), which appears larger than the quoted statistical errors; the text attributes this to sampler implementation, but it would be helpful to state explicitly whether this reflects sampling noise or a systematic difference in sampler settings.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the eccentric content is parameter-free analytical PN input tested against external NR data; QC-limit agreement is inherited by construction but not presented as an independent prediction.

full rationale

The SEOBNRv5EHM derivation chain is self-contained with respect to circularity. The eccentric dynamics (Eqs. 4-6) are constructed from 3PN analytical results developed in the companion paper [241], which are not fitted to the eccentric NR data used in validation. The QC ingredients inherited from SEOBNRv5HM (NQC coefficients, merger-ringdown ansatz, Delta t22_ISCO) are calibrated to quasicircular NR simulations, not to the eccentric waveforms whose accuracy is being claimed. The eccentric validation (Sec. IV C) compares against 99 SXS NR simulations, an external dataset, and the model contains no eccentric calibration parameters. The QC-limit agreement is by construction (the eccentric corrections reduce to unity and the evolution reduces to SEOBNRv5HM), but the paper states this explicitly rather than presenting it as an independent prediction. The mismatch metric (Eq. 34) optimizes over template initial eccentricity and starting frequency to handle gauge-dependent eccentricity definitions; this is a methodological choice applied identically to all compared models, so it does not make the relative accuracy claim circular. No self-citation is load-bearing in a way that forbids alternatives: the companion paper's PN results are independently validated against NR waveforms in this work. Overall, no step in the claimed derivation reduces to its own inputs.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The eccentric sector of SEOBNRv5EHM is parameter-free: the 3PN eccentricity corrections are analytical, so the model's eccentric accuracy claim is not fitted to eccentric data. What the central claim does rest on is: (i) a companion-paper derivation this text does not reproduce; (ii) inherited QC calibrations (NQC and ringdown coefficients, matching-time offset) whose extrapolation to eccentric binaries is an assumption; (iii) domain assumptions about circularization by merger and synchronization of an overdetermined system of equations; and (iv) several hand-chosen thresholds (10 M backward-evolution cutoff, 10% A_inv peak threshold, e at most 0.45 recommended bound). The validation adds no free parameters, but it does optimize over template eccentricity and starting frequency per waveform, which is a form of fitting in the comparison metric.

free parameters (6)

Matching-time offset Delta t22_ISCO = Calibrated to 441 QC NR simulations; inherited from SEOBNRv5HM
Sec. III B: the inspiral/merger-ringdown attachment time for eccentric modes is taken from the QC calibration as a first approximation; the paper acknowledges it as a source of systematic error for high-spin, high-eccentricity systems.
NQC correction coefficients a_lm, b_lm = Fixed by QC NR input values at the matching time
Eq. (21): five constants per mode are matched to QC NR amplitude and frequency values using the background QC dynamics; inherited calibration that shapes the late-inspiral waveform.
Merger-ringdown ansatz coefficients c_lm, d_lm = Polynomial fits in nu and chi to QC NR and test-mass-limit waveforms
Eqs. (26)-(27): ringdown amplitude and phase functions use coefficients fit to QC NR and QC test-mass-limit waveforms, plus QNM frequencies from the qnm package; applied to eccentric binaries under the circularization assumption.
Backward-evolution threshold r0 = 10 M = 10 M (hand-chosen)
Sec. II B and Appendix A: if the starting separation is below 10 M, the code evolves (x, e, zeta) backward in time; Appendix A shows backward and non-backward waveforms differ by up to ~80% mismatch in ~13% of tested cases, so the threshold materially shapes the model.
A_inv peak-selection threshold = 10% of the largest A_inv peak
Sec. III (Eq. 14): the threshold defining t = 0 through the last peak of A_inv is a convention that matters for high-eccentricity waveforms with several comparable amplitude peaks.
Recommended eccentricity bound e <= 0.45 = 0.45 (by visual inspection)
Sec. IV E and Appendix E: the boundary of the safe region is fixed by inspecting waveform-envelope monotonicity for desynchronization; a soft criterion on which the robustness claim rests.

assumptions (8)

domain assumption The 3PN eccentricity corrections to the RR force and waveform modes (F_ecc_phi, F_ecc_r, h_ecc_lm) are correct as derived in the companion paper [241].
Invoked in Sec. II A (Eqs. 4-6) and Sec. III A 1 (Eq. 16); the model's eccentric dynamics and phasing rest entirely on these analytic results, not derived in this text.
domain assumption Eccentric binaries circularize sufficiently by merger so that r is approximately r_QC after some time t* (Eq. 22).
Sec. III A 2: the background-QC NQC corrections and matching time, and the QC merger-ringdown of Sec. III C, all assume late-time circularization; the paper documents ~20% mismatches when this fails at egw above about 0.7.
domain assumption The overdetermined EOB-plus-Keplerian system (Eqs. 4a-4g, 7 equations for 4 degrees of freedom) stays synchronized in the recommended parameter region.
Appendix E: desynchronization produces unphysical amplitude envelopes in challenging regions; the stated validity region is exactly where the authors detected no desynchronization by visual inspection.
domain assumption The post-adiabatic approximation and the 3PN Keplerian formulas map (langle M Omega rangle, e, zeta) to correct EOB initial conditions.
Sec. II B and Appendix A: the PA initial-condition prescription plus backward secular evolution is assumed accurate; Appendix A shows backward and non-backward waveforms differ by up to ~80% mismatch for ~13% of sampled cases.
domain assumption Orbit-averaged frequency x = langle M Omega rangle^(2/3) via Eq. (B7) of the companion paper is a faithful reparametrization of the semilatus rectum for eccentric orbits.
Footnote 5: the model replaces u_p by x through a PN transformation; the e to 0 limit is verified, but the finite-eccentricity mapping inherits PN truncation error.
domain assumption The Kerr-geodesic ISCO radius, computed with QC-fitted remnant mass and spin, is the right reference for eccentric matching times.
Sec. III B: r_ISCO from QC remnant fits anchors the matching time for eccentric binaries by Eq. (23); the paper calls this a source of systematic errors for high-eccentricity and high-spin systems.
standard math QNM frequencies of the remnant Kerr BH computed with the qnm package describe the merger-ringdown.
Sec. III C (Eq. 26): the ringdown ansatz uses least-damped QNM frequencies from BH perturbation theory; standard background, not fitted here.
standard math The -2 spin-weighted spherical-harmonic decomposition (Eq. 12) with modes up to l = 4 is sufficient for the stated accuracy.
Sec. III: the multipolar decomposition is exact in principle; truncation at l = 4 is a domain choice, and the mismatch growth with total mass shows the higher-mode modeling remains the weakest part.

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Pith. "Pith review of Accurate waveforms for eccentric, aligned-spin binary black holes: The multipolar effective-one-body model SEOBNRv5EHM." pith.science (2026). https://pith.science/paper/3MSMAXPU

@misc{pith2026241212823,
  author       = {Pith},
  title        = {Pith review of: Accurate waveforms for eccentric, aligned-spin binary black holes: The multipolar effective-one-body model SEOBNRv5EHM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MSMAXPU}},
  note         = {Machine review of arXiv:2412.12823}
}

abstract

The measurement of orbital eccentricity in gravitational-wave (GW) signals will provide unique insights into the astrophysical origin of binary systems, while ignoring eccentricity in waveform models could introduce significant biases in parameter estimation and tests of General Relativity. Upcoming LIGO-Virgo-KAGRA observing runs are expected to detect a subpopulation of eccentric signals, making it vital to develop accurate waveform models for eccentric orbits. Here, employing recent analytical results through the third post-Newtonian order, we develop SEOBNRv5EHM: a new time-domain, effective-one-body, multipolar waveform model for eccentric binary black holes with spins aligned (or antialigned) with the orbital angular momentum. Besides the dominant (2,2) mode, the model includes the (2,1), (3,3), (3,2), (4,4) and (4,3) modes. We validate the model's accuracy by computing its unfaithfulness against 99 (28 public and 71 private) eccentric numerical-relativity (NR) simulations, produced by the Simulating eXtreme Spacetimes Collaboration. Importantly, for NR waveforms with initial GW eccentricities below 0.5, the maximum (2,2)-mode unfaithfulness across the total mass range 20-200 $M_\odot$ is consistently below or close to $1 \%$, with a median value of $ \sim 0.02 \% $, reflecting an accuracy improvement of approximately an order of magnitude compared to the previous-generation SEOBNRv4EHM and the state-of-the-art TEOBResumS-Dal\'i eccentric model. In the quasi-circular-orbit limit, SEOBNRv5EHM is in excellent agreement with the highly accurate SEOBNRv5HM model. The accuracy, robustness, and speed of SEOBNRv5EHM make it suitable for data analysis and astrophysical studies. We demonstrate this by performing a set of recovery studies of synthetic NR-signal injections, and parameter-estimation analyses of the events GW150914 and GW190521, which we find to have no eccentricity signatures.

Figures

Figures reproduced from arXiv: 2412.12823 by the authors.

**Figure 1.** Waveform mismatches between different aligned-spin approximants and the 441 SXS QC NR simulations used in this work, [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** Parameter-space distribution of the 99 eccentric SXS NR [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

Figures from the paper (18 more)

**Figure 3.** Figure 3: Left column: Real part of the (2, 2) mode for different eccentric, nonspinning NR waveforms (black) with increasing eccentricity (one for each panel) along with the best-fitting waveforms for the state-of-the-art eccentric, aligned-spin models SEOBNRv5EHM (blue, dashed…

**Figure 4.** Figure 4: Top panel: Mismatches of a subset of 84 eccentric NR waveforms with initial GW eccentricities egw < 0.5 against different eccentric, aligned-spin waveform models: SEOBNRv4EHM (first column), SEOBNRv5EHM (second column), and TEOBResumS-Dal´ı (third column), calculated …

**Figure 5.** Figure 5: Top panels: Distributions of the maximum (purple), median (green), and minimum (yellow) (2, 2)-mode mismatches M22 (left) and sky-and-polarization-averaged, SNR-weighted mismatches MSNR (right) over a range of total masses from 20 and 200 M⊙, for different eccentric, a…

**Figure 6.** Figure 6: Evolution of the mass-scaled, orbit-averaged [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Summary of mismatches for the 99 eccentric [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: Real part of different waveform modes for the nonspinning [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Walltimes for the models SEOBNRv5HM (dashed), SEOBNRv5PHM (dash-dotted), and SEOBNRv5EHM (solid), as functions of the total mass M ∈ [10, 100]M⊙. The systems are characterized by a starting orbit-averaged frequency ⟨fstart⟩ = 10 Hz, dimensionless spin components χ1 = …

**Figure 10.** Figure 10: Uniform variation of eccentricity (left panels) and relativistic anomaly (right panels) for the different [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: Marginalized 1D and 2D posterior distributions obtained with the [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

**Figure 12.** Figure 12: Marginalized 1D and 2D posterior distributions corresponding to the real events GW150914 and GW190521 obtained with [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

**Figure 13.** Figure 13: Normalized histograms of the (2, 2)-mode mismatches between systems with and without backward secular evolution, for binary configurations which have a starting separation r0 ≤ 10M. The blue histogram corresponds to a test involving 105 waveform evaluations over the…

**Figure 14.** Figure 14: Amplitude and frequency of the (2, 2) mode, as well as the eccentricity evolution of systems which have the highest mismatch when comparing the waveforms affected and not affected by the backward secular evolution (see [PITH_FULL_IMAGE:figures/full_fig_p031_14.png]

**Figure 15.** Figure 15: Waveform mismatches of the 28 publicly-available SXS eccentric NR waveforms against different eccentric, aligned-spin wave [PITH_FULL_IMAGE:figures/full_fig_p032_15.png]

**Figure 16.** Figure 16: Amplitude of different waveform modes corresponding to two, publicly available SXS eccentric NR waveforms ( [PITH_FULL_IMAGE:figures/full_fig_p033_16.png]

**Figure 17.** Figure 17: Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p034_17.png]

**Figure 18.** Figure 18: Top panel: (2, 2)-mode mismatches between a subset of 75 eccentric NR waveforms with initial GW eccentricities egw < 0.4 and different eccentric, aligned-spin approximants, calculated over a range of total masses M ∈ [20, 200] M⊙. The first column corresponds to a ver…

**Figure 19.** Figure 19: (2, 2)-mode mismatches between 99 eccentric NR waveforms and different versions of the SEOBNRv5E RR force and waveform modes, calculated over a range of total masses M ∈ [20, 200] M⊙. The first panel corresponds to the default SEOBNRv5EHM model, the second panel corre…

**Figure 20.** Figure 20: Amplitude and frequency of the (2, 2) mode (top panels), and the trajectory in the (r, ζ) polar plane (bottom panels) are shown for a standard eccentric binary (left panels) and for a system affected by a desynchronization between the Keplerian and Hamiltonian evoluti…

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