TEOBResumS-DALI: EOB Model for Eccentric Binaries

• TEOBResumS-DALI is an effective-one-body gravitational-wave model that generates inspiral–merger–ringdown waveforms for compact binaries with eccentric and generic orbits.
• It employs resummed metric potentials and a multipolar, factorized waveform structure to accurately capture non-circular, hyperbolic, and eccentric dynamics.
• The model supports direct parameter inference using LIGO-Virgo-KAGRA data and is validated against numerical relativity across diverse binary configurations.

TEOBResumS-DALI is an effective-one-body (EOB) gravitational-wave model for compact binaries with non-circular motion. In the cited literature it appears in two closely related senses: as the eccentric, aligned-spin extension of TEOBResumS used for nonprecessing binaries in present data-analysis applications, and as a more recent unified framework for black holes, neutron stars, and mixed binaries on arbitrary orbits, including quasi-circular, eccentric, non-planar, and hyperbolic trajectories, with inspiral–merger–ringdown waveforms, scattering observables, higher harmonics, spin couplings, and tidal interactions (Albanesi et al., 18 Mar 2025, Divyajyoti et al., 5 Oct 2025). It has become a central model for eccentric waveform generation, for direct parameter inference on LIGO-Virgo-KAGRA data, and for controlled studies of how eccentricity interacts with higher modes, spins, and waveform systematics (Kacanja et al., 31 Jul 2025, Malagon et al., 12 May 2026).

1. Lineage and domain of applicability

TEOBResumS-DALI belongs to the TEOBResumS family and is consistently described as its eccentric or generic-orbits branch. In the 2024 aligned-spin planar update, it is presented as an EOB model for non-circularized, planar, spin-aligned black-hole binaries, covering both bound eccentric inspirals and unbound hyperbolic scatterings while remaining compatible with the quasi-circular limit (Nagar et al., 2024). In later work it is recast as the first unified EOB model for generic compact binaries on arbitrary orbits, including black-hole binaries, binary neutron stars, and black-hole–neutron-star systems, with generic spins, tides, multipolar radiation reaction, and matter effects in a single code base (Albanesi et al., 18 Mar 2025).

The model name is written in the literature as TEOBResumS-DALI, TEOBResumS-Dali, and sometimes “Dalì.” The cited papers do not define the expansion of “Dali,” and some studies explicitly note that TEOBResumS-Dali and TEOBResumS-DALI refer to the same eccentric EOB implementation (Malagon et al., 12 May 2026, Bhaumik et al., 2024).

A persistent distinction in the literature is between the model’s formal scope and the submanifolds actually used in analyses. Several inference studies employ TEOBResumS-DALI as an eccentric, aligned-spin, nonprecessing model with spins restricted to aligned or anti-aligned components and with precession absent from the waveform family (Kacanja et al., 31 Jul 2025, Divyajyoti et al., 5 Oct 2025). By contrast, the unified arbitrary-orbit formulation includes non-planar motion, generic spins, and neutron-star matter sectors, together with detector-ready inspiral–merger–ringdown waveforms for black-hole binaries and black-hole–neutron-star binaries, inspiral-to-postmerger waveforms for binary neutron stars, and gauge-invariant strong-field observables such as the scattering angle (Albanesi et al., 18 Mar 2025).

2. Dynamical formulation and waveform structure

At its core, TEOBResumS-DALI uses the standard EOB mapping from the two-body problem to an effective one-body Hamiltonian with resummed metric potentials and radiation reaction. In the generic-orbit formulation, the Hamiltonian is written as

$$H(q,p,S_a) = M c^2 \sqrt{1 + 2 \nu \left[ \hat{H}_{\rm eff}(q,p,S_a) - 1 \right]},$$

with canonical variables $q=(r,\theta,\phi)$, $p=(p_r,p_\theta,p_\phi)$, inverse radius $u=GM/(c^2 r)$, symmetric mass ratio $\nu$, and spin couplings encoded through resummed gyro-gravitomagnetic functions and the centrifugal radius construction (Albanesi et al., 18 Mar 2025). In the aligned-spin planar implementation, the effective Hamiltonian is built from the potentials $A(u)$, $D(u)$, and $Q(u,p_r^*)$, with spin-orbit and spin-spin terms added to the orbital dynamics (Nagar et al., 2024).

The waveform is multipolar and factorized. In the generic-orbit description, the modes are written as

$$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$

where $h_{\ell m}^{\rm N}$ is the Newtonian prefactor, $q=(r,\theta,\phi)$0 the effective source, $q=(r,\theta,\phi)$1 the tail factor, $q=(r,\theta,\phi)$2 the residual phase correction, $q=(r,\theta,\phi)$3 the resummed amplitude residual, and $q=(r,\theta,\phi)$4 the non-quasi-circular factor (Albanesi et al., 18 Mar 2025). Radiation reaction is built from resummed multipolar fluxes and generalized beyond circular motion through non-circular Newtonian prefactors and iterative time derivatives, which are specifically designed to remain robust for eccentric and hyperbolic dynamics (Nagar et al., 2024, Placidi et al., 2023).

The model’s higher-mode content is broad but version-dependent. TEOBResumS-DALI supports higher-order modes up to multipole order $q=(r,\theta,\phi)$5; however, some implementations provide full inspiral-to-merger waveforms up to $q=(r,\theta,\phi)$6 while full inspiral–merger–ringdown completion is available only for selected modes such as $q=(r,\theta,\phi)$7, $q=(r,\theta,\phi)$8, and $q=(r,\theta,\phi)$9 in the 2024 aligned-spin planar release (Kacanja et al., 31 Jul 2025, Nagar et al., 2024). In data-analysis studies, higher modes are often truncated further; for example, eccentricity inference for low-mass sources used modes only up to $p=(p_r,p_\theta,p_\phi)$0 (Kacanja et al., 31 Jul 2025).

The dissipative sector has received sustained analytic refinement. The 2024 update incorporated nonspinning 4PN information into the quadrupolar waveform and flux through a Padé-resummed $p=(p_r,p_\theta,p_\phi)$1 and a 3.5PN Padé-resummed $p=(p_r,p_\theta,p_\phi)$2, and introduced both a fully analytic 4PN radiation-reaction model and a 4PN-NRTuned variant with an effective NR-informed term in $p=(p_r,p_\theta,p_\phi)$3 (Nagar et al., 2024). Complementary analytic work derived 2.5PN-accurate waveform information for generic planar orbits in EOB variables, including oscillatory memory terms that are purely non-circular and first appear at 1.5PN, together with ready-to-use non-circular factors explicitly designed for TEOBResumS-DALI (Placidi et al., 2023).

3. Eccentricity parameterization and interfaces to inference

In current data-analysis use, TEOBResumS-DALI parameterizes eccentricity by $p=(p_r,p_\theta,p_\phi)$4 at a chosen reference gravitational-wave frequency together with a mean anomaly describing the orbital phase along the eccentric trajectory. Several studies use $p=(p_r,p_\theta,p_\phi)$5 Hz, with TEOBResumS-DALI employing the mean anomaly $p=(p_r,p_\theta,p_\phi)$6 or $p=(p_r,p_\theta,p_\phi)$7, whereas the comparison model SEOBNRv5EHM uses a relativistic anomaly (Kacanja et al., 31 Jul 2025, Divyajyoti et al., 5 Oct 2025). In the large O3/O4a catalog study, the starting and reference frequencies are explicitly set equal on an event-by-event basis; for O4a analyses the standard choice is $p=(p_r,p_\theta,p_\phi)$8 Hz, while earlier runs often use $p=(p_r,p_\theta,p_\phi)$9 Hz (Malagon et al., 12 May 2026).

Because TEOBResumS-DALI is a time-domain model in the low-mass inference analyses, waveforms are converted to the frequency domain with an FFT and appropriate tapering before evaluation in frequency-domain likelihood pipelines (Kacanja et al., 31 Jul 2025). Inference has been carried out with both PyCBC Inference plus Dynesty and with the two-stage RIFT engine. In the PyCBC setup the posterior is written as

$u=GM/(c^2 r)$0

with eccentricity, aligned spin components, distance, sky position, and anomaly sampled under specified priors; time of coalescence, sky location, distance, and polarization can be marginalized to reduce dimensionality, while coalescence phase marginalization is unavailable once higher-order modes are included (Kacanja et al., 31 Jul 2025). In RIFT, the marginal likelihood over extrinsic parameters is computed first and then combined with intrinsic priors to obtain the posterior over masses, spins, eccentricity, and anomaly (Malagon et al., 12 May 2026).

Bayesian model selection for the eccentric versus circular hypotheses is often performed through Savage–Dickey density ratios. For the low-mass study, the relevant expression is reported as

$u=GM/(c^2 r)$1

with values greater than $u=GM/(c^2 r)$2 interpreted as favoring the eccentric model (Kacanja et al., 31 Jul 2025). In the larger catalog analysis, a uniform eccentricity prior on $u=GM/(c^2 r)$3 leads to

$u=GM/(c^2 r)$4

while direct evidence ratios are used when the posterior essentially excludes $u=GM/(c^2 r)$5 (Malagon et al., 12 May 2026).

The model’s computational cost is a recurring practical issue. No reduced-order model is available for the eccentric higher-mode parameter spaces used in the low-mass inference study, so that work relied on marginalization, explicit parallelization over eccentricity bins, and other efficiency measures (Kacanja et al., 31 Jul 2025). The O3/O4a catalog analysis instead used RIFT’s two-stage architecture together with sampling rates of at least $u=GM/(c^2 r)$6 Hz, and often $u=GM/(c^2 r)$7 Hz for binary neutron-star and neutron-star–black-hole systems, to maintain accurate discrete-time likelihood evaluations (Malagon et al., 12 May 2026).

4. Validation against numerical relativity and related models

TEOBResumS-DALI has been benchmarked against numerical relativity across quasi-circular, eccentric, scattering, capture, and matter sectors. The validation record is heterogeneous because different studies probe different limits of the model, but the common pattern is that the model is strongest in the mildly to moderately eccentric, NR-calibrated regime and becomes more delicate for very high eccentricity, very high mass ratio, or strongly precessing configurations.

Study	Regime	Representative findings
(Nagar et al., 2024)	Planar, spin-aligned BBH	Median quasi-circular unfaithfulness $u=GM/(c^2 r)$8; $u=GM/(c^2 r)$9 for 4PN-NRTuned
(Andrade et al., 2023)	Dynamical-capture BBH	Baseline matches about $\nu$0; NR-informed version about $\nu$1, with one direct-plunge outlier
(Bhaumik et al., 2024)	Nonspinning search sensitivity	Agreement with NR up to $\nu$2 at $\nu$3 Hz for $\nu$4 binaries
(Albanesi et al., 18 Mar 2025)	Unified arbitrary-orbit BBH/BNS/BHNS	Benchmarked against $\nu$5 NR simulations; $\nu$6 below $\nu$7 mismatch, $\nu$8 below $\nu$9, $A(u)$0 below $A(u)$1

The 2024 aligned-spin planar update is the most detailed accuracy study for the quasi-circular and mildly eccentric sector. With Advanced LIGO noise and total masses $A(u)$2–$A(u)$3, the dominant $A(u)$4 mode achieves $A(u)$5 across all $A(u)$6 spin-aligned SXS configurations, with median $A(u)$7, and similar figures are reported for $A(u)$8 public eccentric SXS simulations. When the 4PN radiation reaction is additionally NR-tuned, the median quasi-circular unfaithfulness drops to $A(u)$9, albeit with a tail up to $D(u)$0 for large positive spins (Nagar et al., 2024).

The dynamical-capture calibration study shows how TEOBResumS-DALI can be informed by explicitly non-quasi-circular numerical relativity data. Fourteen new NR simulations covering single- and double-encounter mergers were used to calibrate the non-quasi-circular correction and merger–ringdown sector. The uninformed model matches NR at about $D(u)$1, improving to about $D(u)$2 after inserting NR information into the NQC and ringdown sectors, and the maximum EOB–NR phase difference at merger decreases from $D(u)$3 rad to $D(u)$4 rad, with one outlier corresponding to a direct plunge (Andrade et al., 2023).

The model’s validity at high eccentricity has been examined in a search-sensitivity context using O3a data and comparisons to SXS and RIT numerical relativity waveforms. In that study TEOBResumS-DALI, run in a zero-spin configuration, agrees well with numerical relativity up to $D(u)$5 for source-frame total masses $D(u)$6–$D(u)$7 and mass ratios $D(u)$8. For $D(u)$9, however, the model’s quasi-circular merger–ringdown prescription causes the sensitive distance to saturate rather than continue the NR decline, so sensitivity estimates become optimistic relative to numerical relativity (Bhaumik et al., 2024).

The large unified arbitrary-orbit paper generalizes these validations substantially. It reports benchmarks against $Q(u,p_r^*)$0 black-hole-binary simulations spanning quasi-circular, quasi-spherical, eccentric, and dynamical-capture/scattering configurations, $Q(u,p_r^*)$1 binary-neutron-star simulations, and $Q(u,p_r^*)$2 black-hole–neutron-star simulations, totaling $Q(u,p_r^*)$3 NR waveforms. For the black-hole-binary sector at total mass $Q(u,p_r^*)$4 in the $Q(u,p_r^*)$5–$Q(u,p_r^*)$6 Hz band, $Q(u,p_r^*)$7 of configurations have mismatch below $Q(u,p_r^*)$8, $Q(u,p_r^*)$9 below $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$0, and $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$1 below $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$2 (Albanesi et al., 18 Mar 2025).

A distinct line of validation concerns the model’s large-mass-ratio behavior. In a comparison among numerical relativity, second-order self-force, and TEOBResumS for quasi-circular nonspinning inspirals, TEOBResumS is largely indistinguishable from numerical relativity for $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$3, remains within $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$4 rad of the 1PAT1 self-force model for $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$5, and can be improved further at $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$6 by inserting the test-particle 6PN correction into the dominant $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$7 flux contribution (Albertini et al., 2022). A dedicated GSF-informed eccentric large-mass-ratio variant later replaced the comparable-mass NR-informed potentials by $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$8-linear 8.5PN potentials fitted to numerical self-force data, explicitly targeting EMRI and IMRI inspirals (Nagar et al., 2022).

5. Use in gravitational-wave measurements

TEOBResumS-DALI has been used directly in eccentricity measurements for compact-binary events observed by the LIGO and Virgo detectors. In the low-mass analysis of six previously observed BNS and NSBH events, it served as one of two eccentric IMR waveform models and was run both with and without higher-order modes up to $$h_{\ell m} = h_{\ell m}^{\rm N} S_{\rm eff}^{(\epsilon)} T_{\ell m} e^{i\delta_{\ell m}} \left( \rho_{\ell m} \right)^{\ell} \mathcal{N}_{\ell m}^{\rm NQC},$$9. The study found non-negligible eccentricity only for GW200105, with $h_{\ell m}^{\rm N}$0 at $h_{\ell m}^{\rm N}$1 Hz under a uniform prior $h_{\ell m}^{\rm N}$2, together with a Bayes factor of approximately $h_{\ell m}^{\rm N}$3–$h_{\ell m}^{\rm N}$4 in favor of the eccentric hypothesis; under a log-uniform prior on $h_{\ell m}^{\rm N}$5, the Bayes factor falls to $h_{\ell m}^{\rm N}$6. The remaining sources were consistent with low eccentricity, with TEOBResumS-DALI upper limits of $h_{\ell m}^{\rm N}$7 for GW170817, $h_{\ell m}^{\rm N}$8 for GW190425, $h_{\ell m}^{\rm N}$9 for GW190814, $q=(r,\theta,\phi)$00 for GW200115, and $q=(r,\theta,\phi)$01 for GW230529, and the study explicitly reported no support for non-negligible eccentricity in GW190814 (Kacanja et al., 31 Jul 2025).

A broader O3/O4a analysis of $q=(r,\theta,\phi)$02 confident sources used TEOBResumS-Dali and SEOBNRv5EHM in parallel within RIFT. The overall conclusion was that both waveform families generally disfavor the eccentric hypothesis, but TEOBResumS-Dali identified three notable cases. For GW200129 it found strong support for small but nonzero eccentricity, with $q=(r,\theta,\phi)$03 and $q=(r,\theta,\phi)$04. For GW231001 it yielded modest positive support, $q=(r,\theta,\phi)$05, with $q=(r,\theta,\phi)$06, while for GW231123 it returned very strong support, $q=(r,\theta,\phi)$07, and $q=(r,\theta,\phi)$08, but with a posterior that rails at the upper prior $q=(r,\theta,\phi)$09. The same study emphasizes that the latter two events are ambiguous because the degree of support differs strongly between waveform models, and in the case of GW231123 other analyses favor precession over eccentricity (Malagon et al., 12 May 2026).

The model has also been used as an injection generator in controlled bias studies. For aligned-spin eccentric injections with total mass $q=(r,\theta,\phi)$10, mass ratios $q=(r,\theta,\phi)$11, spins $q=(r,\theta,\phi)$12, and eccentricities $q=(r,\theta,\phi)$13, recovery with the quasi-circular precessing model IMRPhenomXP produced increasing chirp-mass bias and increasingly artificial support for nonzero $q=(r,\theta,\phi)$14, whereas recovery with an eccentric aligned-spin model recovered the eccentricity and favored the eccentric hypothesis more strongly as $q=(r,\theta,\phi)$15 increased (Divyajyoti et al., 5 Oct 2025). An analogous study with equal-mass, nonspinning TEOBResumS-DALI injections over $q=(r,\theta,\phi)$16–$q=(r,\theta,\phi)$17 and $q=(r,\theta,\phi)$18 found that circular precessing templates remain reliable only for very small residual eccentricity and develop significant biases above $q=(r,\theta,\phi)$19, including artificial precession, inclination shifts toward edge-on, and severe mass-ratio distortions at high mass and high eccentricity (RoyChowdhury et al., 2 Mar 2026).

6. Systematics, degeneracies, and ongoing extensions

Several limitations recur across the TEOBResumS-DALI literature. First, most deployed inference studies use aligned-spin or nonspinning sectors only, so precession is not modeled even though precession-like signatures can compete with eccentricity in the data (Kacanja et al., 31 Jul 2025, Malagon et al., 12 May 2026). Second, high-mass high-eccentricity configurations can produce artificial support near the upper prior boundary. The O3/O4a catalog study explicitly reports that TEOBResumS-Dali can rail at $q=(r,\theta,\phi)$20 in high total-mass systems and recommends capping the prior at $q=(r,\theta,\phi)$21 and interpreting such posteriors cautiously (Malagon et al., 12 May 2026). Third, the nonspinning search-sensitivity analysis shows that once eccentricity remains high enough at plunge, the model’s quasi-circular merger–ringdown prescription ceases to track numerical relativity reliably, which is why that study treats $q=(r,\theta,\phi)$22 at $q=(r,\theta,\phi)$23 Hz as a model-limited regime (Bhaumik et al., 2024).

The statistical structure of eccentric inference with TEOBResumS-DALI is itself nontrivial. The low-mass event analysis reports multi-modal posterior structure in eccentricity, anomaly, and mass parameters, and attributes these islands to waveform physics rather than to sampling artifacts; it also shows strong prior sensitivity, exemplified by the drop in GW200105’s Bayes factor from approximately $q=(r,\theta,\phi)$24–$q=(r,\theta,\phi)$25 under a uniform eccentricity prior to $q=(r,\theta,\phi)$26 under a log-uniform prior (Kacanja et al., 31 Jul 2025). Injection studies then add a separate degeneracy: quasi-circular precessing templates can absorb eccentric amplitude and phase modulations by pushing $q=(r,\theta,\phi)$27 to artificially large values. In the equal-mass nonspinning injections, this effect becomes pronounced for $q=(r,\theta,\phi)$28, while in the aligned-spin $q=(r,\theta,\phi)$29 injections it yields systematically biased chirp masses and increasing preference for the eccentric aligned-spin recovery over the quasi-circular precessing one (Divyajyoti et al., 5 Oct 2025, RoyChowdhury et al., 2 Mar 2026).

A further source of ambiguity is the definition of eccentricity itself. One methodological study used TEOBResumS-DALI as a parametric model to fit eccentric numerical-relativity strains and compared mismatch-based fitting to the standardized gravitational-wave eccentricity $q=(r,\theta,\phi)$30. It found that minimizing mismatch alone can favor quasi-circular fits with $q=(r,\theta,\phi)$31 for very small-eccentricity NR simulations, even when the standardized $q=(r,\theta,\phi)$32 is nonzero, whereas minimizing only the $q=(r,\theta,\phi)$33 discrepancy is degenerate and can yield mismatches up to about $q=(r,\theta,\phi)$34. The proposed remedy is a convex loss

$q=(r,\theta,\phi)$35

with $q=(r,\theta,\phi)$36 used as a default, which keeps $q=(r,\theta,\phi)$37 and avoids zero-eccentricity fits in the small-$q=(r,\theta,\phi)$38 cases (Chartier et al., 25 Mar 2025). This suggests that TEOBResumS-DALI-based eccentricity assignment is sensitive both to waveform agreement and to the eccentricity convention used to label the waveform.

Ongoing extensions attack these issues from several directions. The 2.5PN generic-planar waveform construction provides explicit non-circular factors, including oscillatory memory, for direct insertion into TEOBResumS-DALI’s factorized waveform (Placidi et al., 2023). The GSF-informed large-mass-ratio variant replaces comparable-mass NR-informed potentials by resummed $q=(r,\theta,\phi)$39-linear 8.5PN functions fitted to self-force data, thereby targeting eccentric EMRIs and IMRIs (Nagar et al., 2022). A separate modular development, gwNRHME, uses the nonspinning limit of TEOBResumS-DALI as a quasi-circular baseline whose higher modes are “eccentricized” by universal eccentric modulation functions; the resulting model, gwNRHME_TEOB_q4, achieves median mismatch $q=(r,\theta,\phi)$40 with standard deviation $q=(r,\theta,\phi)$41 against $q=(r,\theta,\phi)$42 eccentric SXS waveforms (Islam et al., 20 Apr 2026). Together, these lines of work indicate that TEOBResumS-DALI is not a single frozen approximant but an evolving EOB framework whose main open fronts are high-eccentricity merger physics, spin-precessing eccentric IMR completion, and the control of definition-dependent and prior-dependent eccentricity systematics.