NRSur7dq4: Accurate Precessing BBH Waveforms

• NRSur7dq4 is a numerical-relativity surrogate model that approximates gravitational waveforms for quasicircular, precessing, unequal-mass black hole mergers with high accuracy.
• It extends previous models by covering mass ratios up to 4 and incorporating all spin-weighted spherical harmonics with ℓ ≤ 4, enhancing waveform fidelity.
• Trained on 1528 simulations using SVD and empirical interpolation, it serves as a robust benchmark for waveform systematics in gravitational-wave data analysis.

NRSur7dq4 is a numerical-relativity surrogate waveform model for quasicircular, precessing, unequal-mass binary black hole mergers. It was developed to provide a fast, highly accurate approximation to full numerical-relativity waveforms for gravitational-wave data analysis while extending earlier precessing surrogate models to higher mass ratios. The model predicts the gravitational waveform beginning about 20 orbits before merger, includes all spin-weighted spherical-harmonic modes with $\ell \le 4$, and also models the precession-frame dynamics and spin evolution of the black holes. In its training parameter range, it was shown to be more accurate than existing models by at least an order of magnitude, with errors comparable to the estimated errors in the numerical-relativity simulations (Varma et al., 2019).

1. Origins, target systems, and conceptual role

NRSur7dq4 was introduced in the context of a basic limitation of gravitational-wave inference: only numerical relativity can capture the full complexities of binary black hole mergers, but direct use of numerical-relativity simulations in parameter estimation is prohibitively expensive. The surrogate strategy is to learn the numerical-relativity solution space and then evaluate it rapidly enough for practical data analysis (Varma et al., 2019).

The model targets quasicircular, precessing, unequal-mass binary black hole systems. Its intrinsic parameter space is the standard seven-dimensional precessing binary-black-hole space,

$$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$

where $\vec{\chi}_{1,2}$ are the dimensionless spin vectors of the two black holes. The total mass scales out of the waveform model and is therefore not part of the intrinsic training space (Varma et al., 2019).

NRSur7dq4 extends earlier surrogate work, specifically NRSur7dq2 and NRSur7dq2Remnant, which were limited to mass ratio $q \le 2$ and spin magnitudes $\chi_1,\chi_2 \le 0.8$. NRSur7dq4 extends the domain to mass ratio $q \le 4$, spin magnitudes $\chi_1,\chi_2 \le 0.8$, and generic spin directions. It was released together with a companion remnant model, NRSur7dq4Remnant, for the final mass, spin, and recoil kick velocity of the merger remnant (Varma et al., 2019).

Later work established a second, equally important role for NRSur7dq4: it became a high-fidelity benchmark for waveform-systematics studies. In GWTC-3 reanalyses and waveform-comparison surveys, it is repeatedly used as the most numerical-relativity-informed reference model in the parameter regime where merger and ringdown dominate the observed signal (Islam et al., 2023, Uilliam et al., 2024).

2. Training set, parameter-space coverage, and usable domain

NRSur7dq4 was trained on 1528 numerical-relativity simulations. The training set comprises 890 precessing simulations from the earlier surrogate training set covering $q \le 2$, 64 aligned-spin simulations with $q \le 4$, and 574 new simulations with $2 < q \le 4$, generic spin directions, and $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$0. The new simulations were designed to fill the $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$1 region using sparse-grid and greedy farthest-point selection methods. The numerical-relativity runs were produced with the SpEC code of the SXS collaboration (Varma et al., 2019).

These simulations are long by numerical-relativity standards, starting about

$$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$2

before the peak of the waveform amplitude, with small eccentricity. After post-processing and truncating junk radiation, the surrogate data begin at

$$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$3

relative to the peak of the total waveform amplitude, corresponding to about 20 orbits before merger (Varma et al., 2019).

The original trained domain is therefore explicit and finite: mass ratio $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$4, spin magnitudes $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$5, and generic spin directions. The same paper also tested extrapolation to $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$6 and found that the waveform mismatch increases noticeably, as expected, but NRSur7dq4 remains better than SEOBNRv3 (Varma et al., 2019). A later application paper described the model as rigorously trained over $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$7 and $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$8, with safe extrapolation reported down to $$q = \frac{m_1}{m_2} \ge 1, \qquad \vec{\chi}_1, \vec{\chi}_2,$$9 and up to $\vec{\chi}_{1,2}$0 (Islam et al., 2020).

A practical limitation follows directly from the waveform length. Because the surrogate begins only about 20 orbits before merger, it is naturally suited to heavy binaries. In the GW190412 analysis this forced the use of $\vec{\chi}_{1,2}$1 Hz rather than 20 Hz (Islam et al., 2020). In the GWTC-3 reanalysis with NRSur7dq4, the event selection was restricted to 47 binary-black-hole events with mass ratios greater than $\vec{\chi}_{1,2}$2 and total masses greater than $\vec{\chi}_{1,2}$3 (Islam et al., 2023). In a separate hypermodel comparison study, NRSur7dq4 was described as valid for mass ratio $\vec{\chi}_{1,2}$4 and total mass $\vec{\chi}_{1,2}$5, with the precise region depending on mass ratio (Puecher et al., 2023).

3. Waveform representation and surrogate architecture

The complex strain is written as

$\vec{\chi}_{1,2}$6

and decomposed into spin-weighted spherical harmonics,

$\vec{\chi}_{1,2}$7

NRSur7dq4 models all modes with $\vec{\chi}_{1,2}$8, including the full set of $\vec{\chi}_{1,2}$9 and asymmetric modes available in that truncation (Varma et al., 2019). Later application papers emphasize this richer harmonic content explicitly: compared with phenomenological models, NRSur7dq4 includes many modes absent from models such as IMRPhenomXPHM, including $q \le 2$0 (Islam et al., 2020).

A central design feature is the use of three frames. The inertial frame is the original waveform extraction frame. The coprecessing frame is a time-dependent frame whose $q \le 2$1-axis tracks the instantaneous orbital angular momentum $q \le 2$2. The coorbital frame is obtained by an additional rotation about the coprecessing $q \le 2$3-axis by the orbital phase so that the black holes lie on the $q \le 2$4-axis, with the heavier black hole on $q \le 2$5 (Varma et al., 2019).

The orbital phase is defined from the coprecessing-frame $q \le 2$6 modes:

$q \le 2$7

and the coorbital-frame transformation for each mode is

$q \le 2$8

In the coorbital frame, the model reorganizes the $q \le 2$9 and $\chi_1,\chi_2 \le 0.8$0 content through

$\chi_1,\chi_2 \le 0.8$1

For each $\chi_1,\chi_2 \le 0.8$2 and each $\chi_1,\chi_2 \le 0.8$3, the surrogate models the real and imaginary parts of $\chi_1,\chi_2 \le 0.8$4; for $\chi_1,\chi_2 \le 0.8$5, it models the real and imaginary parts of $\chi_1,\chi_2 \le 0.8$6 directly (Varma et al., 2019).

The surrogate itself has two coupled parts. The coorbital-frame waveform surrogate downsamples each waveform data piece to about 2000 time points spaced approximately uniformly in orbital phase, builds a linear basis via SVD, chooses an empirical interpolation set of the same size, and fits the coefficients at each empirical node as functions of the binary parameters. The SVD basis uses an RMS tolerance of

$\chi_1,\chi_2 \le 0.8$7

The dynamics surrogate models the time evolution of the orbital phase $\chi_1,\chi_2 \le 0.8$8, the coprecessing-frame quaternion $\chi_1,\chi_2 \le 0.8$9, and the coorbital-frame spin evolution $q \le 4$0. It fits the time derivatives of these quantities at 238 dynamical time nodes, roughly 10 nodes per orbit, and integrates them using a 4th-order Adams–Bashforth ODE scheme (Varma et al., 2019).

The modeled dynamical quantities are $q \le 4$1, the coprecessing-frame angular-velocity components $q \le 4$2 and $q \le 4$3, and the spin derivatives $q \le 4$4, with $q \le 4$5. The fit basis consists of tensor products of one-dimensional monomials in the parameters after affine rescaling to $q \le 4$6, with up to cubic powers in $q \le 4$7 and up to quadratic powers in spin variables. Fit selection uses a forward-stepwise greedy algorithm with 10 trial fits, 10% held out each time for validation, and a maximum of 100 basis functions (Varma et al., 2019).

4. Validation, accuracy, and known limitations

Validation in the original construction used 20-fold cross validation: the 1528 simulations were split into 20 groups of 76, the model was trained on 1452 simulations and validated on the held-out 76, and the process was repeated for all groups (Varma et al., 2019). Waveform accuracy was quantified with the standard mismatch,

$q \le 4$8

using the noise-weighted inner product

$q \le 4$9

The evaluation was carried out with both a flat noise curve and the Advanced LIGO design sensitivity curve, over many sky locations, optimizing over time shift, phase, and polarization angle (Varma et al., 2019).

The reported performance is unusually strong within the calibrated region. The model errors are comparable to numerical-relativity truncation error and at least an order of magnitude smaller than SEOBNRv3. For Advanced LIGO noise and total masses $\chi_1,\chi_2 \le 0.8$0–$\chi_1,\chi_2 \le 0.8$1, the 95th-percentile mismatch is always $\chi_1,\chi_2 \le 0.8$2 (Varma et al., 2019). In a GW190412-specific study, the mismatch with numerical relativity for GW190412-like masses was reported to have median mismatch around $\chi_1,\chi_2 \le 0.8$3 (Islam et al., 2020). A later survey of precessing waveform models described the numerical-relativity surrogate family as having NR faithfulness at the level of $\chi_1,\chi_2 \le 0.8$4 and therefore used NRSur7dq4 as the reference target for faithfulness tests of SEOBNRv5PHM, TEOBResumS, IMRPhenomTPHM, and IMRPhenomXPHM (Uilliam et al., 2024).

That survey also clarifies where model disagreement becomes most visible relative to NRSur7dq4. All four approximants become more faithful as the mass ratio approaches unity and when the merger-ringdown portion of the waveforms are excluded. The overall strain mismatch is strongly correlated with the mismatch of the co-precessing $\chi_1,\chi_2 \le 0.8$5 modes, while the $\chi_1,\chi_2 \le 0.8$6 modes matter more at high inclination; omitting the co-precessing $\chi_1,\chi_2 \le 0.8$7 modes can increase the strain mismatch by up to $\chi_1,\chi_2 \le 0.8$8 at $\chi_1,\chi_2 \le 0.8$9 and up to $q \le 2$0 at $q \le 2$1, depending on the case (Uilliam et al., 2024).

The main limitation remains waveform duration. Because NRSur7dq4 begins only about 20 orbits before merger, low-mass analyses require a higher low-frequency cutoff or are excluded outright. This is why the most extensive catalog-scale uses of the model concentrate on high-mass systems, where late inspiral, merger, and ringdown dominate the information content (Islam et al., 2020, Islam et al., 2023).

5. Use in parameter estimation and waveform-systematics studies

NRSur7dq4 has become a standard instrument for studying waveform systematics in Bayesian gravitational-wave inference. In the GWTC-3 reanalysis with RIFT and asimov, it is treated as one of the state-of-the-art waveform models against which contemporary approximants are compared, especially for higher-mass O3b events (Fernando et al., 2024).

In that framework, parameter inference is organized by separating intrinsic and extrinsic parameters. For a compact binary there are 15 parameters in total, with intrinsic parameters denoted $q \le 2$2 and extrinsic parameters denoted $q \le 2$3. RIFT first computes the marginalized likelihood over extrinsic parameters,

$q \le 2$4

and then turns that into the posterior over intrinsic parameters,

$q \le 2$5

RIFT works with marginalized-likelihood evaluations $q \le 2$6 and builds an interpolant for $q \le 2$7 before producing posterior samples. asimov stores event-specific settings, analysis choices, and workflow metadata in reproducible form and then executes analyses consistently at scale (Fernando et al., 2024).

The astrophysical reason this matters is that high-mass binaries place a large fraction of the signal-to-noise ratio in late inspiral, merger, and ringdown, where waveform differences are most consequential. In that regime, subtle modeling differences can shift inferred masses, mass ratios, spins, and precession. The RIFT/asimov reanalysis finds that for several events all models largely agree, but for a few exceptional events they disagree substantially on the nature of the merging binary. It further reports that every pair of waveform models has at least one event where the posterior shifts by more than the informal Jensen–Shannon divergence threshold of $q \le 2$8 (Fernando et al., 2024).

A complementary hypermodel study reached a more cautious conclusion about global model preference. Sampling directly over NRSur7dq4, SEOBNRv4PHM, IMRPhenomXPHM, and IMRPhenomTPHM for 13 heavy GWTC-3 events, it found a nominal joint odds ratio

$q \le 2$9

but this result was mainly determined by three events with possible data-quality issues. Without GW200129 the odds ratio becomes $q \le 4$0, and without GW190521a, GW191109, and GW200129 together it becomes $q \le 4$1, which the authors interpret as no statistically meaningful overall preference among waveform families (Puecher et al., 2023). This corrects a common overstatement: NRSur7dq4 is often preferred in difficult events, but not universally preferred across catalog populations.

6. Event-level results and astrophysical implications

The first major parameter-estimation application highlighted in the supplied literature is GW190412. Using NRSur7dq4, the source was inferred to have mass ratio

$q \le 4$2

in broad agreement with IMRPhenomXPHM and the LIGO-Virgo Collaboration’s SEOBNRv4PHM-based analysis, but in tension with IMRPhenomPv3HM. The NRSur7dq4 analysis also favored a more face-on system and found a broader posterior for the precession parameter. A notable result is that even though the $q \le 4$3 harmonic modes have negligible signal-to-noise ratio, their omission still influences posterior distributions for chirp mass, effective inspiral spin, luminosity distance, and inclination (Islam et al., 2020).

Catalog-scale reanalysis sharpened the picture. For 47 GWTC-3 binary-black-hole events within the usable regime of the surrogate, most NRSur7dq4 inferences matched the LIGO-Virgo-KAGRA analyses obtained with IMRPhenomXPHM and SEOBNRv4PHM, but more than 20\% of events showed noticeable differences relative to IMRPhenomXPHM and about 55\% showed significant differences relative to SEOBNRv4PHM using a Jensen–Shannon divergence threshold of $q \le 4$4 bits. Among the most consequential results were the 99.3\% credible-level constraint that $q \le 4$5 for GW191109_010717 and the strong-precession, large-kick interpretation of GW200129_065458, although the latter was repeatedly flagged as sensitive to glitch subtraction issues (Islam et al., 2023).

The later RIFT/asimov reanalysis refined those conclusions. For GW200129, NRSur7dq4 favored more extreme mass ratio, larger primary spin, and larger precession $q \le 4$6 than the SEOBNR models, but did not find strong support for very asymmetric mass ratio $q \le 4$7 or near-extremal spin; the marginal-likelihood points themselves did not support those extreme regions. For GW191109, NRSur7dq4, SEOBNRv4PHM, and SEOBNRv5PHM were described as “coarsely” consistent, with only modest shifts in total mass and spin-related parameters. For GW200216, GW200220_061928, and GW200220_124850, NRSur7dq4 preferred a wider mass-ratio distribution and slightly larger spins than the SEOBNR models (Fernando et al., 2024).

Subsequent work on systematic-error mitigation showed that some inter-model tension can be reduced without erasing the astrophysical content. In a framework that augments the reference waveform by smooth amplitude and phase corrections, the posteriors from IMRPhenomXPHM, IMRPhenomXO4a, and NRSur7dq4 became much more consistent for GW191109_010717 and GW200129_065458. For the latter event, the nonzero precession inference remained, with

$q \le 4$8

for IMRPhenomXPHM, IMRPhenomXO4a, and NRSur7dq4 respectively (Kumar et al., 23 Apr 2026). This suggests that some apparent disagreement between waveform families can arise from a combination of waveform systematics and data-quality artifacts rather than from a wholesale failure of the underlying astrophysical interpretation.

The most extreme later example is GW231123. There NRSur7dq4 was used as the main waveform model to test whether the unusual parameter estimates were intrinsic to the event or artifacts of waveform choice and Gaussian noise. Simulations built from the NRSur7dq4 maximum-likelihood waveform reproduced the qualitative inter-model disagreements seen in the real event, while analyses over 20 Gaussian-noise realizations consistently supported large masses, high spin magnitudes, and strong precession. For the real event, the combined-network NRSur medians were

$q \le 4$9

$2 < q \le 4$0

The paper’s conclusion is that the heavy masses and large spin magnitudes inferred with NRSur are robust against Gaussian-noise perturbations (Bini et al., 14 Jan 2026).

Taken together, these studies define the present status of NRSur7dq4. Within its validity region it functions both as a high-fidelity inference model and as a benchmark against which approximate precessing waveform families are judged. Its most visible impact appears in merger-dominated, high-mass, strongly precessing events, where waveform choice can materially alter inferred mass ratio, spin magnitudes, inclination, distance, remnant properties, and the astrophysical interpretation of formation channels (Islam et al., 2023, Fernando et al., 2024).