IMRPhenomXPHM-SpinTaylor Waveform

• IMRPhenomXPHM-SpinTaylor is a frequency-domain phenomenological model that extends aligned-spin models by incorporating PN-derived precession dynamics through a twisting-up procedure.
• It uses the SpinTaylorT4 prescription to compute evolving Euler angles via coupled ODEs, enabling efficient parameter estimation for quasi-circular, precessing binary black hole signals.
• The model features multipolar content up to ℓ=5 and reliably recovers mass and effective spin for moderate mass ratios, though it may underestimate χp in cases of strong precession.

IMRPhenomXPHM-SpinTaylor is a frequency-domain phenomenological gravitational waveform model designed for the analysis of quasi-circular, precessing binary black hole (BBH) systems. It extends the aligned-spin, multipolar baseline IMRPhenomXHM by incorporating precession dynamics through a "twisting up" procedure, using approximate rotation mappings derived from post-Newtonian (PN) expansions. When equipped with the SpinTaylorT4 prescription for its precessional Euler angles, the model provides a computationally efficient approach to modeling precessing BBH signals with subdominant harmonic content for applications including gravitational-wave parameter estimation and tests of general relativity (Pratten et al., 2020, Akçay et al., 24 Jun 2025).

1. Mathematical Structure and Twisting-Up Procedure

IMRPhenomXPHM generates the frequency-domain waveform in the inertial (precessing) frame via the active rotation of a non-precessing multipolar signal. The core expression is:

$$h(f, \theta_s, \phi_s, ι, ψ, λ_\mathrm{intr}) = e^{-2iψ} \sum_{ℓ=2}^5 \sum_{m=-ℓ}^{ℓ} D^ℓ_{–2,m}(α(f), β(f), γ(f))\, h^co_{ℓm}(f; λ_\mathrm{intr})\, Y_{ℓm}(ι, \phi_s)$$

where:

• $h^co_{ℓm}(f)$: non-precessing (co-precessing frame) spherical harmonic modes, constructed from the IMRPhenomXHM model as a combination of PN inspiral, NR-calibrated phenomenological bridge, and quasinormal-mode ringdown.
• $D^ℓ_{–2,m}(α, β, γ)$: Wigner D-matrices parametrized by frequency-dependent Euler angles $(α, β, γ)$ determined by the precession formalism.
• $Y_{ℓm}(ι, \phi_s)$: spin-weighted spherical harmonics evaluated at the instantaneous inclination and azimuth.

The "twisting up" encapsulates the mapping from aligned-spin co-precessing modes to precessing frame modes via the application of PN-derived Euler-angle rotations (Pratten et al., 2020).

2. SpinTaylor Precession: Formalism and Implementation

In the SpinTaylorT4 ("SpinTaylor") prescription, the precession Euler angles are generated by integrating a set of coupled ordinary differential equations (ODEs):

$$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$

with precession-velocity vectors,

$$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$

The solution is propagated from an initial gravitational-wave frequency $f_0 = 20\,\mathrm{Hz}$ to merger, and the Euler angles $(α, β, γ)$ are computed using the next-to-next-to-leading-order (NNLO) co-precessing-frame expressions (Akçay et al., 24 Jun 2025).

The single-spin SpinTaylor mapping—originally developed for IMRPhenomPv2—assumes all in-plane spin resides on the larger black hole ($\mathbf{S}_2=0$). The approach is strictly valid for moderate mass ratios and spin magnitudes and approximately “simple precession” (constant $h^co_{ℓm}(f)$0).

3. Multipolar Content, PN Orders, and Phenomenological Calibration

IMRPhenomXPHM includes all spherical harmonic modes with $h^co_{ℓm}(f)$1:

• $h^co_{ℓm}(f)$2

The inspiral amplitude $h^co_{ℓm}(f)$3 incorporates spin–orbit terms to 3.5PN and spin–spin terms to 2PN (mode-dependent leading orders). The phase $h^co_{ℓm}(f)$4 is modeled to 4PN non-spinning, 3.5PN spin–orbit, and 2PN quadratic-in-spin, following Arun et al. (2008) and Bohé et al. (2013).

Transitions from inspiral to merger–ringdown are governed by phenomenological coefficients (typically $h^co_{ℓm}(f)$5 per mode), fitted by least-squares to large banks of NR simulations. The ringdown segment relies on NR-calibrated quasinormal mode complexes (frequencies and amplitudes), referencing the formulae of Jiménez-Forteza et al. (2016) (Pratten et al., 2020).

4. Computational Aspects: Multibanding and Interpolation

The model accelerates waveform generation via "multibanding" interpolation. Coarse, uneven frequency grids $h^co_{ℓm}(f)$6 are constructed, so that the linear interpolation error for each phase or Euler angle does not exceed a user-defined threshold $h^co_{ℓm}(f)$7. For each function $h^co_{ℓm}(f)$8, the grid spacing satisfies $h^co_{ℓm}(f)$9, enabling much coarser grids for angular functions like $D^ℓ_{–2,m}(α, β, γ)$0 compared to the GW phase $D^ℓ_{–2,m}(α, β, γ)$1. The user controls accuracy through parameters such as “PrecThresholdMband” (default $D^ℓ_{–2,m}(α, β, γ)$2 rad for phases), balancing speed and fidelity (Pratten et al., 2020).

5. Domain of Validity and Theoretical Approximations

SpinTaylor-based IMRPhenomXPHM assumes:

• Single-spin: typically $D^ℓ_{–2,m}(α, β, γ)$3.
• Adiabatic precession: orbit-averaged PN treatment, neglecting spin–spin effects in the mapping.
• Simple precession: $D^ℓ_{–2,m}(α, β, γ)$4 approximately constant; transitions or "transitional precession" where $D^ℓ_{–2,m}(α, β, γ)$5 changes direction are not modeled.
• Valid up to moderate mass ratios $D^ℓ_{–2,m}(α, β, γ)$6 and spins $D^ℓ_{–2,m}(α, β, γ)$7; NNLO Euler angles can behave pathologically at higher $D^ℓ_{–2,m}(α, β, γ)$8 or high spin misalignment.

The stationary-phase approximation is used to connect the time- and frequency-domain representations, but may deteriorate near merger. The model does not capture asymmetries associated with large black hole recoils (Pratten et al., 2020).

6. Performance in Injection–Recovery and Parameter Estimation

A zero-noise injection–recovery study encompassing 35 strongly-precessing NR waveforms (10 each for $D^ℓ_{–2,m}(α, β, γ)$9; 5 single-spin $(α, β, γ)$0) in a two-detector Advanced LIGO O4 network (total SNR=40, precession SNR=10) found:

• For $(α, β, γ)$1 injections:
  • Mean recovery scores: $(α, β, γ)$2, $(α, β, γ)$3, $(α, β, γ)$4, $(α, β, γ)$5.
  • The 90% CIs for $(α, β, γ)$6 are biased low by $(α, β, γ)$72–3$(α, β, γ)$8 in 6/10 cases; $(α, β, γ)$9 is biased high by $Y_{ℓm}(ι, \phi_s)$0 in 6/10; $Y_{ℓm}(ι, \phi_s)$1 is well-recovered; $Y_{ℓm}(ι, \phi_s)$2 is substantially underestimated ($Y_{ℓm}(ι, \phi_s)$3 to $Y_{ℓm}(ι, \phi_s)$4 in 8/10).
• For $Y_{ℓm}(ι, \phi_s)$5 (single-spin):
  • Success rates for recovery within $Y_{ℓm}(ι, \phi_s)$6: $Y_{ℓm}(ι, \phi_s)$7: 2/5, $Y_{ℓm}(ι, \phi_s)$8: 2/5, $Y_{ℓm}(ι, \phi_s)$9: 5/5, $$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$0: 2/5.
  • For all $$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$1 cases, accurate inference of $$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$2 is unreliable with this and other phenomenological models.

Coverage statistics (fraction of 90% CIs containing the true value):

Parameter	$$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$3 (out of 30)	$$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$4 (out of 5)
$$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$5	47%	40%
$$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$6	50%	40%
$$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$7	93%	100%
$$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$8	37%	40%

For moderate $$\begin{align*} \frac{dω}{dt} &= \frac{96}{5}\, ν\, ω^{11/3}\, [1 + \text{PN corrections up to 3.5PN incl. spins}] \ \frac{d\mathbf{S}_i}{dt} &= \mathbf{Ω}_i \times \mathbf{S}_i, \quad i=1,2 \ \frac{d\hat{\mathbf{L}}}{dt} &= -\frac{1}{|\mathbf{L}|}\left(\frac{d\mathbf{S}_1}{dt} + \frac{d\mathbf{S}_2}{dt}\right) \end{align*}$$9, IMRPhenomXPHM is appropriate for recovering $$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$0 and masses, but $$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$1 inference should be cross-checked with models such as IMRPhenomTPHM or SEOBNRv5PHM (Akçay et al., 24 Jun 2025).

7. IMR Consistency Testing and Limitations

Inspiral-merger-ringdown (IMR) consistency tests evaluated GR consistency by comparing low-frequency (inspiral) and high-frequency (ringdown) estimates of final mass and spin. Using two distinct ISCO-frequency splits (Schwarzschild and Kerr), IMRPhenomXPHM showed:

• False GR violation rates: 27% (8/30) at Schwarzschild ISCO; 7% (2/30) at Kerr ISCO, mainly driven by $$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$2.
• By contrast, SEOBNRv5PHM and IMRPhenomTPHM exhibited no false violations at either cutoff.
• For high mass ratio ($$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$3) and spin-perpendicular-to-orbital-angular-momentum regions ($$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$4), precessional-rotation pathologies and frame-twisting artifacts degrade model accuracy, biasing both parameter recovery and consistency tests.

A recommended procedure is that, whenever IMRPhenomXPHM indicates an apparent GR deviation at the Schwarzschild ISCO split, results should be cross-checked at the Kerr ISCO split and with alternate waveform models. Persistent inconsistencies almost always reflect systematic model error rather than genuine beyond-GR physics (Akçay et al., 24 Jun 2025).

8. Implications, Comparisons, and Recommendations

IMRPhenomXPHM-SpinTaylor provides competitive computational efficiency and coverage for the majority of BBH signals within its design domain, notably for moderate mass ratios and moderate precession. However, for $$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$5 or strong in-plane spin ($$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$6), neither this nor alternative current phenomenological models can fully resolve all source parameters robustly; averaging approaches that weight models by local NR mismatch can reduce parameter bias, including up to 30% reduction for $$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$7 using "NR-informed" posterior mixing (Akçay et al., 24 Jun 2025).

Researchers are advised to cross-check $$\mathbf{Ω}_1 = \frac{1}{r^3}\left[\frac{4 + 3 m_2/m_1}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_2\right], \quad \mathbf{Ω}_2 = \frac{1}{r^3}\left[\frac{4 + 3 m_1/m_2}{2} \mathbf{L} + \frac{3}{2}\mathbf{S}_1\right]$$8 inferencing across models, employ multiple IMR split frequencies for consistency testing, and incorporate model accuracy as an explicit variable in advanced Bayesian pipelines. IMRPhenomXPHM remains a leading tool for rapid, flexible parameter estimation and hypothesis testing for precessing BBH systems, but is not universally reliable in regimes of extreme mass ratio or precession.

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Key references: IMRPhenomXPHM and the SpinTaylor prescription are detailed in Pratten et al. (Pratten et al., 2020); injection–recovery performance and comparison with other models is discussed in "Waging a Campaign" (Akçay et al., 24 Jun 2025).