TEOBResumS-Dal: Unified EOB Waveform for Generic Binaries

• TEOBResumS-Dal is an EOB modeling approach that extends waveform simulations to accommodate large mass ratios, eccentric orbits, and noncircular dynamics.
• It integrates PN, NR, and GSF information into a unified framework with calibrated Hamiltonians, radiation reaction forces, and multipolar waveform generation.
• The model is validated against extensive simulations and supports diverse compact binaries including BBHs, BNSs, and BHNSs with matter effects and spin precession.

TEOBResumS-Dal is a public branch of the TEOBResumS effective-one-body waveform framework that extends the EOB description of compact-binary dynamics beyond quasi-circular, comparable-mass black-hole binaries to regimes including large mass ratios, aligned spins, eccentricity, dynamical capture, scattering, and, in later formulations, generic spins, tides, and non-planar motion. Within the EOB picture, the binary is mapped to an effective particle moving in a deformed spacetime, with a conservative Hamiltonian, radiation reaction, and waveform generation. Across the literature, TEOBResumS-Dal and the spelling variant TEOBResumS-Dalí are used for implementations that target EMRI/IMRI-style inspirals, eccentric and unbound black-hole encounters, and eventually a unified model for binary black holes, binary neutron stars, and black-hole–neutron-star systems on arbitrary orbits through merger and including scattering (Albertini et al., 2023, Albanesi et al., 18 Mar 2025).

1. Definition, naming, and scope

TEOBResumS is an EOB waveform model in which the two-body problem is represented by an effective particle moving in an effective metric, and the waveform is built mode by mode from a factorized, resummed quasicircular baseline. In this framework, the key elements are a Hamiltonian and a radiation reaction force, which incorporate PN, NR, GSF, and test-mass information in a resummed way (Riemenschneider et al., 2021, Albanesi et al., 18 Mar 2025).

Within that family, the “Dal” branch is identified as the public implementation used for EMRI/IMRI-style inspirals, with large-mass-ratio capability, aligned spins, eccentricity, and later improvements in the spin-orbit sector. In subsequent work, TEOBResumS-Dalí is presented as an eccentric-orbit iteration of the TEOB family, as a model for dynamical captures and scatterings, and finally as a unified EOB model for generic compact binaries with arbitrary orbits (Albertini et al., 2024, Grilli et al., 2024, Albanesi et al., 2024, Albanesi et al., 18 Mar 2025).

The scope attributed to TEOBResumS-Dal broadens substantially across the cited works. One line of development focuses on inspiralling, quasi-circular large-mass-ratio binaries with aligned spins and eccentricity, benchmarked against second-order gravitational self-force information (Albertini et al., 2023, Albertini et al., 2024). Another line emphasizes noncircular and unbound black-hole dynamics, where the model is used for scatterings, dynamical captures, and waveform phenomenology close to the transition between bound and unbound motion (Albanesi et al., 2024). A later synthesis describes TEOBResumS-Dalí as a single framework for black holes, neutron stars, and black-hole–neutron-star systems evolving along quasi-circular, eccentric, hyperbolic/scattering, and non-planar orbits through merger, ringdown, or post-merger emission as appropriate (Albanesi et al., 18 Mar 2025).

This suggests that “TEOBResumS-Dal” is best understood not as a single frozen approximant, but as a branch of the TEOBResumS infrastructure whose defining feature is the extension of EOB modeling to generic-orbit and large-mass-ratio problems.

2. Dynamical and waveform structure

The EOB structure underlying TEOBResumS-Dal combines a conservative Hamiltonian, dissipative radiation reaction, and multipolar waveform generation. For BBHs, the unified formulation uses a 5PN Hamiltonian with two parameters calibrated to NR quasi-circular simulations. Spin effects include spin-orbit couplings, spin-spin couplings, and spin-induced precession; for spin-spin interactions, the model uses a centrifugal radius $r_c$, with PN information through NLO order. For BNSs, the Hamiltonian includes gravitoelectric terms, gravitomagnetic terms, and multipolar tidal couplings up to $\ell=8$, with the reduced tidal deformability written as

$$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$

The default tidal choice in the unified model is the GSF-resummed potential (Albanesi et al., 18 Mar 2025).

The waveform sector is multipolar. In the TEOBResumS formulation assessed in the NQC and post-adiabatic study, the modes are written as

$$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$

with next-to-quasicircular corrections parameterized by

$$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$

A central result of that work is the consistent use of NR-informed NQC information in both the waveform and the radiation reaction, replacing iterative determination of $(a_1^{(2,2)},a_2^{(2,2)})$ with direct fits across parameter space (Riemenschneider et al., 2021).

The same paper emphasizes the post-adiabatic approximation as a computational strategy for accelerating long inspirals. In that implementation, the PA evolution is typically carried out to 8th order on a radial grid with step $dr=0.1$, from the starting radius $r_0$ down to $r=14$, after which the full ODEs are used. Comparing to 611 NR simulations for total mass $10M_\odot\leq M \leq 200M_\odot$, the model has EOB/NR unfaithfulness well below $\ell=8$0, with $\ell=8$1 of the cases below $\ell=8$2 (Riemenschneider et al., 2021).

For generic-orbit applications, the unified TEOBResumS-Dal formulation uses multipolar waveform amplitudes and phases, multipolar radiation reaction, non-circular corrections for eccentric and hyperbolic motion, Padé resummations for strong-field robustness, and NR-informed waveform completion near merger. For BBHs, the model uses a $\ell=8$3PN description of all multipoles up to $\ell=8$4, with the $\ell=8$5 mode treated at full 4PN order, and includes spin-dependent horizon absorption effects to LO (Albanesi et al., 18 Mar 2025).

3. Large-mass-ratio and GSF-informed developments

A defining application of TEOBResumS-Dal is the large-mass-ratio regime. The EMRI-oriented implementation incorporates aligned spins on both bodies and orbital eccentricity, while recasting conservative first-order gravitational self-force information into resummed EOB potentials and using a $\ell=8$6PN-accurate flux at infinity together with an improved horizon-flux prescription (Albertini et al., 2023).

In that formulation, the conservative dynamics is encoded in EOB potentials such as $\ell=8$7, $\ell=8$8, and $\ell=8$9, and the phasing is analyzed through

$$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$0

with expansion

$$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$1

The infinity flux is computed as

$$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$2

and the practical improvement highlighted in the paper is the inclusion of the $$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$3 and $$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$4 modes. For a standard EMRI with $$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$5 and $$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$6, the accumulated EOB/2GSF dephasing changes from $$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$7 to $$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$8 over $$\tilde\Lambda=16/13\, (m_1+12m_2)m_1^4\Lambda_1/M + (1\leftrightarrow2).$$9, and from $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$0 to $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$1 over $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$2. Over about 1.2382 years of evolution, corresponding to roughly $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$3 cycles, the dephasing is $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$4, which the paper states is consistent with standard EMRI waveform accuracy requirements (Albertini et al., 2023).

The 2024 spin-sector upgrade focuses on inspiralling, quasi-circular black-hole binaries with a non-spinning primary and a spinning secondary. The main change is a new gauge choice for the gyro-gravitomagnetic functions

$$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$5

which enter the spin-orbit Hamiltonian contribution

$$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$6

The earlier TEOBResumS formulation uses the DJS gauge, in which $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$7 and $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$8 depend only on inverse radius and radial momentum. The updated TEOBResumS-Dal implementation adopts the $$h_{\ell m} = h_{\ell m}^{(N,\epsilon)} \hat{h}_{\ell m}\,\hat{h}^{\rm NQC}_{\ell m},$$9 gauge, allowing dependence on

$$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$0

so that one can factor out the complete Kerr spinning-particle expression $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$1 as

$$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$2

The paper concludes that the new $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$3 gauge improves agreement with GSF, especially in the linear-in-spin 1PA contribution: the extracted spin piece $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$4 is closer to the GSF result, and the integrated spin-driven dephasing is roughly halved compared with the standard DJS choice over the comparison interval $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$5. These improved functions are implemented in the public TEOBResumS-Dal code (Albertini et al., 2024).

The broader GSF motivation was already established in the earlier comparison between TEOBResumS, 2GSF/1PAT1, and NR. That work found that for $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$6, TEOBResumS is largely indistinguishable from NR while 1PAT1 has significant dephasing; for $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$7, TEOBResumS and 1PAT1 agree well; and for $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$8, 1PAT1 becomes more accurate than TEOBResumS, with EOB–GSF dephasing reaching about $$\hat{h}^{\rm NQC}_{\ell m} = \left(1+a_1^{(\ell,m)}n^{(\ell,m)}_1+a_2^{(\ell,m)}n^{(\ell,m)}_2\right) \exp\!\left[i\left(b_1^{(\ell,m)}n^{(\ell,m)}_3+b_2^{(\ell,m)}n^{(\ell,m)}_4\right)\right].$$9 at $(a_1^{(2,2)},a_2^{(2,2)})$0. The paper identifies a path toward GSF-informed EOB modeling through improved conservative potentials and dissipative fluxes, which is the direction later realized in the TEOBResumS-Dal large-mass-ratio program (Albertini et al., 2022).

4. Eccentricity, noncircular motion, memory, and unbound dynamics

TEOBResumS-Dal is explicitly described as already equipped with eccentricity in the large-mass-ratio branch, and eccentricity remains a central ingredient in later generic-binary formulations (Albertini et al., 2024, Albanesi et al., 18 Mar 2025). In the direct-current memory work, the model becomes the first EOB model to include DC memory contributions. The DC memory is defined as a non-oscillatory, hereditary component of the gravitational-wave signal, with permanent strain offset

$(a_1^{(2,2)},a_2^{(2,2)})$1

The paper focuses on the purely non-oscillatory memory contribution in the $(a_1^{(2,2)},a_2^{(2,2)})$2 spherical-harmonic modes and transforms prior harmonic-coordinate, quasi-Keplerian results into EOB phase-space variables

$(a_1^{(2,2)},a_2^{(2,2)})$3

using known 2PN coordinate transformations. The implementation has overall 2.5PN accuracy and preserves the small-eccentricity expansion up to $(a_1^{(2,2)},a_2^{(2,2)})$4 (Grilli et al., 2024).

For $(a_1^{(2,2)},a_2^{(2,2)})$5 modes, the model uses

$(a_1^{(2,2)},a_2^{(2,2)})$6

with

$(a_1^{(2,2)},a_2^{(2,2)})$7

and the full $(a_1^{(2,2)},a_2^{(2,2)})$8 waveform assembled in a non-factorized form. A notable implementation detail is the use of $(a_1^{(2,2)},a_2^{(2,2)})$9 and the auxiliary combination

$dr=0.1$0

which makes the eccentricity counting cleaner and helps remove spurious higher-order terms beyond the sixth eccentricity order. The paper reports that the DC term produces a clear, cumulative offset in the plus polarization and depends sensitively on the formation eccentricity $dr=0.1$1, with $dr=0.1$2 argued to be the most physically motivated default (Grilli et al., 2024).

For highly noncircular and unbound BBH motion, TEOBResumS-Dal is validated against NR near the transition from scattering to dynamical capture. In that setting, the model uses a waveform factorization

$dr=0.1$3

and for the dominant mode the generic Newtonian noncircular factor is

$dr=0.1$4

The paper emphasizes that factorizing this generic Newtonian term is crucial for good agreement at apastron and in dynamical capture. In the low-energy regime $dr=0.1$5, mismatches weighted with the zero-detuned, high-power noise spectral density of Advanced LIGO are typically below or around the $dr=0.1$6 level, with only a few spinning binaries slightly above the $dr=0.1$7 threshold (Albanesi et al., 2024).

The unified generic-orbit paper systematizes these capabilities by treating scattering angle as a gauge-invariant diagnostic and by defining

$dr=0.1$8

for interpreting turning points and transitions between scattering and plunge. This places eccentric inspiral, hyperbolic scattering, and dynamical capture within a single EOB dynamical picture (Albanesi et al., 18 Mar 2025).

5. Matter effects, precession, and generic compact binaries

A major expansion of TEOBResumS-Dal occurs in BHNS and generic compact-binary modeling. The BHNS extension described in 2025 uses 52 new NR simulations to inform TEOBResumS-Dalí as a multipolar EOB model including precession and eccentricity. The model is designed to handle generic orbits, spin precession, and eccentricity, while incorporating BHNS-specific tidal-disruption physics in the merger and ringdown regime (Gonzalez et al., 30 Jun 2025).

That work identifies a BHNS waveform morphology not reducible to BBH behavior with tides added. The merger phenomenology is classified into Type I, Type II, and Type III according to the ratio $dr=0.1$9, with thresholds $r_0$0, $r_0$1, and the intermediate range, respectively. The ringdown prescription depends on that classification, and the NQC quantities are extracted at

$r_0$2

except for $r_0$3, $r_0$4, and $r_0$5, where they are evaluated at $r_0$6. The paper finds that $r_0$7 works better for BHNS than the BBH-inspired value 1, and reports that the new $r_0$8 NQC prescriptions improve the merger amplitude by about an order of magnitude relative to the older model (Gonzalez et al., 30 Jun 2025).

The same BHNS study highlights the special role of the $r_0$9 and $r=14$0 modes. It states that the $r=14$1 amplitude can be of the same order as $r=14$2 for $r=14$3, and that the $r=14$4 can approach the $r=14$5 contribution in the same region. The authors note that current TEOBResumS-Dalí does not yet include $r=14$6 modes there, identifying them as an important future direction (Gonzalez et al., 30 Jun 2025). This is consistent with the separate DC-memory implementation for $r=14$7 modes in the eccentric-orbit branch (Grilli et al., 2024).

The unified generic model extends beyond BHNS. It is presented as the first EOB model for the general relativistic dynamics and gravitational radiation of generic compact binaries, explicitly encompassing BBH, BNS, and BHNS systems with tidal interactions, generic spins, multipolar radiation reaction and waveform, and NR-informed merger, ringdown, or post-merger completion. For BNSs, TEOBResumS-Dal includes NRPM or NRPMw post-merger descriptions; for BHNSs, it uses NR-informed remnant black-hole mass and spin models that smoothly deform the BBH case (Albanesi et al., 18 Mar 2025).

Precession in the unified formulation is treated via a hybrid PN-EOB scheme based on the separation of timescales between orbital motion and spin-precession motion. The parameter

$r=14$8

is used in the generic-binary and BHNS descriptions (Albanesi et al., 18 Mar 2025, Gonzalez et al., 30 Jun 2025). The BHNS validation against the 12-orbit precessing simulation BAM:0223 reports phase differences below $r=14$9 rad throughout the inspiral and mismatches including all the modes at the one percent level for low inclinations (Gonzalez et al., 30 Jun 2025).

6. Validation, systematics, and known limitations

The broadest validation claim appears in the generic-binary formulation, which reports comparisons with 1395 high-accuracy NR simulations: 1183 BBHs, 27 BNSs, and 185 BHNSs. For BBHs, $10M_\odot\leq M \leq 200M_\odot$0 of configurations have mismatch below $10M_\odot\leq M \leq 200M_\odot$1, $10M_\odot\leq M \leq 200M_\odot$2 below $10M_\odot\leq M \leq 200M_\odot$3, and $10M_\odot\leq M \leq 200M_\odot$4 below $10M_\odot\leq M \leq 200M_\odot$5. For an eccentric BNS example with $10M_\odot\leq M \leq 200M_\odot$6, the mismatch is $10M_\odot\leq M \leq 200M_\odot$7. For a long BHNS simulation with $10M_\odot\leq M \leq 200M_\odot$8, $10M_\odot\leq M \leq 200M_\odot$9, and $\ell=8$00, the mismatch is about $\ell=8$01, while a precessing BHNS with $\ell=8$02 and $\ell=8$03 yields mismatch $\ell=8$04 (Albanesi et al., 18 Mar 2025).

The low-energy scattering study provides a complementary validation in a regime especially sensitive to noncircular structure. It finds good agreement for waveform phenomenologies, scattering angles, mismatches, and energetics when $\ell=8$05, but identifies deteriorating performance for higher energies, larger mass ratios, and some spinning direct captures. The paper attributes this to the PN nature of the Hamiltonian and radiation reaction and to the quasi-circular calibration of $\ell=8$06 and $\ell=8$07, while also noting that NQC corrections are turned off in that work (Albanesi et al., 2024).

For large-mass-ratio inspirals, the GSF-based benchmark papers supply a different notion of validation. The 2022 TEOBResumS-versus-1PAT1 comparison establishes that current EOB accuracy depends strongly on mass ratio, with a mutual-consistency region around $\ell=8$08–32 and growing dephasing at larger $\ell=8$09 unless the flux is improved (Albertini et al., 2022). The EMRI-oriented Dal implementation responds directly to this by using GSF-informed potentials and a $\ell=8$10PN flux with $\ell=8$11, obtaining sub-radian dephasing over a year-long standard EMRI (Albertini et al., 2023). The 2024 spin-orbit update then reduces the linear-in-spin 1PA discrepancy by changing the gauge choice for $\ell=8$12 and $\ell=8$13 (Albertini et al., 2024).

A distinct limitation emerges from parameter-estimation systematics. In the 2026 SEOBNRv6EHM study, TEOBResumS-Dal is used as a comparison model for eccentric aligned-spin inference. The paper states that TEOBResumS-Dal can yield biased estimates of eccentricity, masses, and spins in the most challenging high-eccentricity configurations. For SXS:BBH:2527, the optimized mismatch quoted against NR is $\ell=8$14, compared with $\ell=8$15 for SEOBNRv6EHM, and the model is not used in the main 26-event LVK analysis because of its higher computational cost for long-duration signals. The same paper also notes that TEOBResumS-Dal does not currently support the backward-in-time waveform integration option used in some SEOBNR analyses (Pompili et al., 27 May 2026).

These results indicate that TEOBResumS-Dal is highly developed and unusually broad in scope, but not uniformly optimal across all inference settings. A plausible implication is that its strongest role is as a physically unified EOB framework—especially for generic-orbit dynamics, GSF-informed large-mass-ratio inspirals, and matter-rich compact binaries—while continued calibration and algorithmic refinement remain necessary in the most demanding high-eccentricity and long-in-band parameter-estimation problems.