Coprecessing Frame Transformation

• Coprecessing Frame Transformation is a method that applies time-dependent spatial rotations to gravitational-wave multipoles to remove orbital precession effects.
• It employs minimal-rotation gauges and Wigner-D matrices to align the waveform with the instantaneous dominant radiation axis in eccentric, precessing binaries.
• The technique leads to improved surrogate-model compression and reduced mismatches with aligned-spin baselines, though physical asymmetries remain.

Coprecessing frame transformation is a time-dependent spatial rotation applied to gravitational-wave multipoles in order to track the instantaneous dominant radiation axis and remove the leading kinematic effects of orbital-plane precession from the waveform. In the context of eccentric, precessing compact binaries, it is used to separate the effect of the precession of the orbital plane from waveform structure that remains intrinsically dynamical, notably periastron–apastron modulations associated with orbital eccentricity. The analysis in "Revisiting the Coprecessing Frame in the Presence of Orbital Eccentricity" shows that this transformation remains effective but incomplete in the eccentric regime: it suppresses most precession-induced amplitude and phase modulations and improves surrogate-model compression, yet residual mismatches against an eccentric, spin-aligned baseline remain at levels incompatible with high-fidelity waveform modeling in challenging configurations, especially at large inclinations and large effective precession spin (Thomas et al., 31 Mar 2026).

1. Formal definition and mode transformation

A gravitational waveform is written in spin-weighted spherical harmonics as

$$h(t,\theta,\phi) = \sum_{\ell,m} h_{\ell m}(t)\;{}_{-2}Y_{\ell m}(\theta,\phi)\,.$$

For nonprecessing, quasicircular binaries, the $(\ell=2,m=\pm 2)$ modes dominate, and the modes satisfy the equatorial-reflection symmetry

$h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$

so positive and negative $m$ modes are not independent (Thomas et al., 31 Mar 2026).

The coprecessing frame is specified by a rotation $R(t)\in SO(3)$ chosen to follow the instantaneous dominant radiation axis. If inertial-frame modes are denoted by $h^{\rm I}_{\ell m}(t)$ and coprecessing-frame modes by $h^{\rm CP}_{\ell m}(t)$, the relation between the two is given by a Wigner-$D$ rotation parameterized by Euler angles $\alpha(t)$, $\beta(t)$, and $(\ell=2,m=\pm 2)$0:

$(\ell=2,m=\pm 2)$1

or equivalently

$(\ell=2,m=\pm 2)$2

The Wigner matrix factorizes as

$(\ell=2,m=\pm 2)$3

with $(\ell=2,m=\pm 2)$4 the Wigner small-$(\ell=2,m=\pm 2)$5 matrix (Thomas et al., 31 Mar 2026).

A residual gauge freedom remains as a rotation about the coprecessing-frame $(\ell=2,m=\pm 2)$6-axis. The standard choice is the minimal-rotation condition,

$(\ell=2,m=\pm 2)$7

which ensures that the frame does not rotate about its own $(\ell=2,m=\pm 2)$8-axis more than is needed to track the radiation axis. In this minimally rotating coprecessing frame, precession-induced amplitude and phase modulations are strongly suppressed, and the transformed modes vary primarily on the radiation-reaction timescale (Thomas et al., 31 Mar 2026).

2. Construction from numerical-relativity data

The study computes the time-dependent rotation directly from numerical-relativity multipoles using the scri package, following Boyle et al. (2014) (Thomas et al., 31 Mar 2026). The preferred radiation axis is identified from the waveform, using the $(\ell=2,m=\pm 2)$9 modes, as the principal direction of gravitational-wave emission, equivalently the instantaneous radiation axis that maximizes the quadrupolar power. This axis defines the $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$0-axis of the coprecessing triad.

The spin about that axis is then fixed by imposing the minimal-rotation gauge, $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$1, after which the rotation is returned either as Euler angles $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$2 or as a unit quaternion $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$3. The resulting time series is applied to all modes through the Wigner-$h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$4 action to obtain $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$5 (Thomas et al., 31 Mar 2026).

Operationally, the procedure on numerical-relativity data begins after removing junk radiation at the simulation’s metadata “relaxation” reference time. The rotation $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$6 is then computed from the $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$7 radiation in the minimal-rotation gauge, and the entire harmonic expansion is rotated to the coprecessing frame. This is the stated procedure used throughout the work to generate coprecessing-frame waveforms (Thomas et al., 31 Mar 2026).

The waveform alignment used for subsequent analysis is also part of the construction. The merger time $h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$8 is defined as the peak of the frame-invariant amplitude

$h_{\ell m}(t) = (-1)^{\ell}\, h^*_{\ell,-m}(t)\,,$9

the time coordinate is shifted so that $m$0, and post-merger data are removed so that the analysis focuses on inspiral (Thomas et al., 31 Mar 2026).

3. Separation of precession and eccentricity

The central question is whether the coprecessing frame can disentangle orbital-plane precession from waveform structure when orbital eccentricity is non-negligible. The study distinguishes two modulation scales. Eccentricity introduces periastron–apastron modulations of both amplitude and frequency on the orbital timescale, whereas spin precession modulates the waveform on the longer precession timescale and causes mode mixing among modes with the same $m$1 in the inertial frame (Thomas et al., 31 Mar 2026).

For quasicircular precessing binaries, the coprecessing rotation largely removes the kinematic imprint of precession, restores the usual mode hierarchy, and leaves slowly varying amplitudes and phases. In eccentric, precessing binaries, the same qualitative simplification persists but is not complete. The coprecessing rotation still removes most precession-induced modulations and mode mixing. An explicit example given in the study is that the inertial-frame $m$2 mode can be strongly contaminated by $m$3 content, whereas in the coprecessing frame its frequency becomes proportional to $m$4 and the spurious twice-orbital-frequency oscillations are removed (Thomas et al., 31 Mar 2026).

Residual modulations remain after the transformation, but these are described as genuinely dynamical rather than kinematic. They include periastron/apastron-driven amplitude oscillations and instantaneous-frequency oscillations from eccentricity. These structures are present even in nonprecessing eccentric systems and are therefore not expected to disappear under a rotation that only tracks the dominant emission direction (Thomas et al., 31 Mar 2026).

A fundamental limitation follows from symmetry. Precession breaks the relation

$m$5

and no time-dependent rotation can restore it. Consequently, mode asymmetries persist even in the coprecessing frame. This result is significant because it identifies a source of residual disagreement with spin-aligned baselines that is physical rather than gauge-related. A plausible implication is that eccentric coprecessing-frame models cannot achieve high fidelity by rotation alone; they must also represent asymmetry structure explicitly (Thomas et al., 31 Mar 2026).

4. Numerical-relativity dataset and waveform-based eccentricity

The numerical study selected 22 eccentric and precessing simulations from the SXS catalog, requiring metadata eccentricity $m$6, at least one in-plane spin $m$7, and at least 8 inspiral orbits. For quantitative mismatch calculations, 2 simulations were excluded because their waveform-based initial eccentricity fell below $m$8, leaving 20 simulations in the mismatch analyses (Thomas et al., 31 Mar 2026).

Across the 22 selected simulations, the mass ratio range is $m$9. The spins span a wide variety of magnitudes and directions, with effective inspiral spin $R(t)\in SO(3)$0 and effective precession spin $R(t)\in SO(3)$1 at the simulation reference time. For the 20 simulations used in mismatch studies, waveform-based initial eccentricities $R(t)\in SO(3)$2 span approximately $R(t)\in SO(3)$3 to $R(t)\in SO(3)$4, while the two excluded cases have $R(t)\in SO(3)$5. The number of orbits from reference time to common-horizon formation ranges from approximately $R(t)\in SO(3)$6 to $R(t)\in SO(3)$7 (Thomas et al., 31 Mar 2026).

Detector-frame strains are formed at fixed inclinations $R(t)\in SO(3)$8 (Thomas et al., 31 Mar 2026). These three viewing geometries are used throughout the strain-comparison analysis and are important because the paper finds markedly different residual behavior between face-on and edge-on observations.

Waveform-based eccentricity is defined from coprecessing-frame quadrupole modes using gw_eccentricity. The orbit-averaged gravitational-wave frequency is

$R(t)\in SO(3)$9

Successive periastron and apastron passages are identified, smooth interpolants of $h^{\rm I}_{\ell m}(t)$0 through those extrema are constructed, and then

$h^{\rm I}_{\ell m}(t)$1

and

$h^{\rm I}_{\ell m}(t)$2

Here the superscripts $h^{\rm I}_{\ell m}(t)$3 and $h^{\rm I}_{\ell m}(t)$4 denote values on the periastron and apastron sequences, and $h^{\rm I}_{\ell m}(t)$5 are consecutive periastron times. Before extrema are selected, a secular amplitude trend is removed with the AmplitudeFits option of gw_eccentricity (Thomas et al., 31 Mar 2026).

For mismatch comparisons, the analysis start time is taken to be the first apastron after the beginning of the periastron–apastron interpolants, the initial $h^{\rm I}_{\ell m}(t)$6 is used, and the mean anomaly is set to $h^{\rm I}_{\ell m}(t)$7, corresponding to apastron (Thomas et al., 31 Mar 2026).

5. Mismatch methodology and quantitative performance

The mismatch is defined using the standard maximized normalized inner product with white noise, $h^{\rm I}_{\ell m}(t)$8:

$h^{\rm I}_{\ell m}(t)$9

with

$h^{\rm CP}_{\ell m}(t)$0

and normalized strain

$h^{\rm CP}_{\ell m}(t)$1

The study compares each numerical-relativity strain to a bank of SEOBNRv5EHM strains in both inertial and coprecessing frames, for $h^{\rm CP}_{\ell m}(t)$2 (Thomas et al., 31 Mar 2026).

The optimization holds fixed the numerical-relativity mass ratio $h^{\rm CP}_{\ell m}(t)$3 and aligned spin components $h^{\rm CP}_{\ell m}(t)$4 in SEOBNRv5EHM, adopts the same mean anomaly $h^{\rm CP}_{\ell m}(t)$5 at the start of the comparison window, and works with dimensionless strains, without distance or noise scaling. The model’s eccentric initial parameters, defined at apastron, are optimized over $h^{\rm CP}_{\ell m}(t)$6 on a grid centered on $h^{\rm CP}_{\ell m}(t)$7 with width $h^{\rm CP}_{\ell m}(t)$8, and over $h^{\rm CP}_{\ell m}(t)$9 by minimizing the time-to-merger difference relative to numerical relativity for each grid value. Time and phase shifts are also optimized, with $D$0 limited to $D$1 of the waveform length and $D$2 maximized analytically (Thomas et al., 31 Mar 2026).

The principal quantitative result is that, for virtually all simulations and inclinations, transforming the numerical-relativity waveform to the coprecessing frame lowers the mismatch against the eccentric, aligned-spin baseline SEOBNRv5EHM. This is interpreted as a quantitative demonstration that the coprecessing rotation removes most precession-driven mode mixing and amplitude/phase modulations, making the waveform resemble a spin-aligned eccentric signal more closely (Thomas et al., 31 Mar 2026).

The improvement is not sufficient for uniformly accurate waveform modeling. At edge-on inclination, $D$3, coprecessing-frame mismatches typically do not drop appreciably below $D$4, and residuals remain at or above the approximately $D$5 level. At face-on inclination, $D$6, low-$D$7 cases can already have low inertial-frame mismatches, while others exhibit order-of-magnitude improvements in the coprecessing frame; however, for large precessing spins of approximately $D$8, mismatches remain in the $D$9 to $\alpha(t)$0 range in both frames (Thomas et al., 31 Mar 2026).

One example highlighted in the study is SXS:BBH:3714 at $\alpha(t)$1, where the mismatch improves from approximately $\alpha(t)$2 in the inertial frame to approximately $\alpha(t)$3 in the coprecessing frame. The reported interpretation is that a large precession modulation near merger dominates the inertial-frame mismatch, whereas the coprecessing rotation removes it and leaves excellent agreement, including the eccentric amplitude modulation (Thomas et al., 31 Mar 2026).

Mismatch behavior in both frames increases with $\alpha(t)$4, and the amount of mismatch reduction produced by the coprecessing transformation correlates with $\alpha(t)$5. This suggests that most, but not all, precession effects are kinematic and removable by an appropriate rotation even when eccentricity is present (Thomas et al., 31 Mar 2026). The waveform-based eccentricity inferred directly from the waveforms also agrees well between numerical relativity and best-fit SEOBNRv5EHM and generally improves further in the coprecessing frame, although the study notes exceptions when improvements are concentrated late in the comparison window rather than in the first few cycles where $\alpha(t)$6 is measured (Thomas et al., 31 Mar 2026).

6. Residual asymmetries, surrogate modeling, and model-building implications

The dominant residual limitation identified in the work is the persistence of mode asymmetries. Because precession breaks equatorial symmetry and no time-dependent rotation can restore

$\alpha(t)$7

genuine physical asymmetries between $\alpha(t)$8 and $\alpha(t)$9 modes remain in the coprecessing frame. The study indicates that modeling these asymmetries, already incorporated in modern quasicircular precessing models, will likely be essential for high-fidelity eccentric, precessing models (Thomas et al., 31 Mar 2026).

The same coprecessing transformation is evaluated from a different perspective in surrogate modeling, where it is treated as a data-driven simplification tool rather than as a claim of physical equivalence to spin-aligned waveforms. Reduced bases are constructed across the available numerical-relativity set for the amplitude and phase of the $\beta(t)$0, $\beta(t)$1, and $\beta(t)$2 modes, after alignment to the peak of $\beta(t)$3 and removal of post-peak data. The reduced-basis projection errors decrease faster with basis size in the coprecessing frame for all modeled components, implying that for a fixed number of basis elements, coprecessing-frame components are represented more accurately than inertial-frame components (Thomas et al., 31 Mar 2026).

The largest improvements occur for the amplitude of $\beta(t)$4 and the phase of $\beta(t)$5. The first is described as being strongly cleaned by the removal of precession-induced mode mixing; the second is described as the most impactful element for overall faithfulness. These findings support the conclusion that the coprecessing frame substantially improves basis compression and accuracy for eccentric, precessing surrogates (Thomas et al., 31 Mar 2026).

The study also provides practical guidance for model builders. Recommended steps are to remove junk radiation at the provided “relaxation” reference time, define $\beta(t)$6 from the maximum of $\beta(t)$7, shift to $\beta(t)$8, remove post-merger, compute the radiation axis from the $\beta(t)$9 modes with scri, impose minimal rotation, and rotate all modes using Wigner $(\ell=2,m=\pm 2)$00 matrices to obtain $(\ell=2,m=\pm 2)$01. If eccentricity estimates are required, $(\ell=2,m=\pm 2)$02, $(\ell=2,m=\pm 2)$03, and $(\ell=2,m=\pm 2)$04 are then extracted in the coprecessing frame using periastron/apastron sequences and the AmplitudeFits option of gw_eccentricity. Comparisons should begin at the first clean apastron, with $(\ell=2,m=\pm 2)$05 when possible (Thomas et al., 31 Mar 2026).

Additional recommendations are more explicitly prospective. To reduce residuals, the study recommends including mode asymmetries in the coprecessing frame, refining radiation-axis tracking where needed, adding eccentricity-dependent corrections in the coprecessing frame, considering the modeling of precessing ringdown properties, and exploiting the coprecessing frame for data compression in surrogates. It also notes potential failure modes: edge-on and large-$(\ell=2,m=\pm 2)$06 systems retain mismatches $(\ell=2,m=\pm 2)$07 even in the coprecessing frame; aligned-spin initial-condition solvers may perform poorly if the comparison window begins too near periastron in the strong-field regime; and residual amplitude oscillations near merger, particularly in short signals, can dominate mismatches if they are not modeled (Thomas et al., 31 Mar 2026).

Taken together, these results establish a dual characterization of the coprecessing frame in the eccentric, precessing problem. It is a cornerstone decomposition because it robustly suppresses precession-induced modulations and improves representational efficiency, but it is not a complete solution because physically meaningful asymmetries and strong-field couplings survive any rotation. This suggests that future high-accuracy waveform models will need to combine coprecessing-frame simplification with explicit modeling of the residual structures that remain after the transformation (Thomas et al., 31 Mar 2026).