IMRPhenomXHM: Multi-Mode Gravitational Waveforms

• IMRPhenomXHM is a frequency-domain, phenomenological gravitational waveform model that extends previous models by including higher spherical harmonic modes such as (2,1), (3,3), (3,2), and (4,4).
• It employs a modular structure dividing the waveform into inspiral, merger, and ringdown phases with calibration against hybrid waveforms from NR, EOB, PN, and Teukolsky data.
• Its efficient implementation in the LSC Algorithm Library Suite facilitates fast waveform generation, improved parameter estimation, and robust general relativity tests in gravitational-wave astronomy.

The IMRPhenomXHM waveform model is a frequency-domain, phenomenological multi-mode gravitational waveform model for non-precessing, quasi-circular binary black hole mergers. It extends the IMRPhenomXAS model—an advanced (2,2)-mode baseline—by including higher spherical harmonic modes, specifically the (2,1), (3,3), (3,2), and (4,4) components, and models important mode-mixing effects, such as those in the (3,2) mode. The model is calibrated to hybrid waveforms, which combine effective-one-body (EOB) and post-Newtonian (PN) descriptions in the inspiral with numerical relativity (NR) and Teukolsky-equation results in the merger and ringdown. IMRPhenomXHM is implemented efficiently in the LSC Algorithm Library Suite, allowing for accurate and fast waveform generation necessary for gravitational-wave detection and parameter estimation.

1. Model Framework and Motivation

IMRPhenomXHM is formulated as a modular, frequency-domain waveform model designed to address shortcomings in previous models that neglected higher-multipole contributions. For non-precessing binaries, the gravitational-wave signal detected can be decomposed into spin-weighted spherical harmonic multipoles:

$$h_{+}(t) + i h_{\times}(t) = \sum_{\ell \geq 2} \sum_{m = -\ell}^{\ell} {}_{-2}Y_{\ell m}(\iota, \varphi) h_{\ell m}(t)$$

where $h_{\ell m}(t)$ are complex mode amplitudes, and ${}_{-2}Y_{\ell m}$ are spin-weighted spherical harmonics.

The motivation for creating IMRPhenomXHM derives from the increasing importance of higher modes for:

• Sources with large mass ratios $(q \gg 1)$
• Signals observed at high inclination (edge-on rather than face-on)
• Breaking parameter degeneracies, leading to less biased and more precise recovery of intrinsic (mass ratio, spins) and extrinsic (distance, orientation) parameters

2. Mathematical Structure and Mode Construction

IMRPhenomXHM divides the full waveform modeling into three primary frequency regimes:

1. Inspiral: Dominated by analytical post-Newtonian expansions. For higher modes, amplitudes and phases use PN expansions and frequency re-mapping. For most $(\ell,m)$ modes (excluding mixed or mode-mixed cases), the phase is related to that of the (2,2) mode via:

$$\Phi_{\ell m}(f) \approx (m/2) \Phi_{22}\bigg(\frac{2f}{m}\bigg)$$

PN amplitudes are similarly rescaled.

1. Intermediate (Merger): An explicitly phenomenological model bridges the inspiral and ringdown, with smooth $C^1$ continuity. Amplitudes and phases at a set of collocation points are directly calibrated to hybrid data, and interpolated as functions of frequency and intrinsic parameters.
2. Ringdown: Ringdown is modeled via a sum over quasi-normal modes (QNMs), with frequencies and damping times calibrated to the remnant black hole properties (as predicted by fits to NR results). Notably, for the (3,2) mode, the model incorporates mode-mixing by first representing the ringdown in the natural spheroidal-harmonic basis, then transforming to the spherical basis.

The mapping between (2,2) and $(\ell,m)$ modes is given schematically by:

$$\tilde{h}_{\ell m}(f) \approx |\beta_{\ell m}(f)| \tilde{h}_{22}(f_{22}(f)) \exp\{i[\kappa \tilde{\phi}_{22}(f_{22}(f)) + \Delta_{\ell m}]\}$$

where:

• $f_{22}(f)$ specifies frequency remapping; in the inspiral, $f_{22}(f) = 2f/m$, and in the ringdown, $$f_{22}(f) = f - (f_{\mathrm{RD},\ell m} - f_{\mathrm{RD},22})$$
• $\beta_{\ell m}(f)$ rescales the amplitude using PN results
• $\kappa = 1/f_{22}'(f)$ is a stretching factor
• $\Delta_{\ell m}$ is a phase offset, ensuring continuity and correct mode parity

3. Calibration and Data Sources

IMRPhenomXHM is calibrated against an extensive set of hybrid waveforms, including:

• Comparable-mass regime: Hundreds of NR simulations from codes such as SXS, BAM, and Einstein Toolkit, covering a wide range of mass ratios and spins.
• High mass ratio regime: Numerical solutions to the Teukolsky equation, anchoring the model up to $q \sim 1000$.
• Hybridization: The inspiral is primarily EOB- and PN-based and is smoothly combined with NR/Teukolsky data in the late inspiral and post-merger.

Calibration proceeds in two stages:

1. Direct fitting: For each frequency and mode, phenomenological ansätze for amplitude and phase are fit to individual hybrid waveforms.
2. Parameter-space fitting: The coefficients of these expressions are parameterized as functions over the physical binary parameters (mass ratio, component spins), producing smooth mappings over the three-dimensional space.

Special care is taken for $(3,2)$, implementing mode-mixing and ensuring appropriate behavior as mass ratio and spin vary.

4. Computational Implementation and Efficiency

IMRPhenomXHM is implemented in the LALSimulation module of LALSuite, with several features that ensure practical utility:

• Modular structure: Each frequency region and mode can be updated or recalibrated independently.
• Multibanding: Waveforms are computed on an adaptively spaced grid over frequency and interpolated, significantly reducing computational cost for Bayesian parameter estimation (critical for millions of evaluations).
• Interfaces: Both C and Python interfaces (via SWIG) are available.

Additionally, acceleration strategies such as evolutionary algorithms for frequency sampling can further speed up waveform generation, achieving over 30% reduction in waveform evaluation time with negligible loss in SNR reconstruction accuracy (Meijer et al., 12 Apr 2024).

5. Quantitative Accuracy and Performance

IMRPhenomXHM achieves improved accuracy in multiple metrics:

• Mode-by-mode mismatches: These are computed between the model and hybrid waveforms, with significantly reduced mismatches compared to earlier models (e.g., IMRPhenomHM), particularly for systems exhibiting strong higher-mode content.
• Multi-mode mismatches: By combining modes, the overall polarization mismatches for a range of inclinations remain low—typically $< 10^{-3}$ to $10^{-2}$—across parameter space.
• Parameter estimation: Case studies (e.g., GW170729) demonstrate that IMRPhenomXHM recovers parameters (mass ratio, spins, inclination, distance) that are consistent with independent models and show less systematic bias relative to models omitting higher modes.
• Systematic evaluation: In tests relevant for tests of general relativity (e.g., higher-mode consistency in GW190814 (Islam, 2021)), omission of higher modes can result in apparent deviations from the multipolar prediction of general relativity as detector sensitivity improves, emphasizing the model's importance for robust null-hypothesis testing.

6. Applications, Extensions, and Limitations

Applications

• Detection pipelines: IMRPhenomXHM’s efficiency and accuracy make it suitable for template-bank searches in non-precessing binary black hole coalescence.
• Parameter estimation: Inclusion of higher modes reduces parameter estimation biases, breaks inclination–distance degeneracies, and improves the constraint on source parameters—especially for high-mass-ratio, edge-on, or otherwise strongly asymmetric systems.
• Astrophysical inference and joint EM-GW analyses: Reduced parameter uncertainties propagate directly to astrophysical event classification, rate estimation, and cross-identification with electromagnetic signals (Kalaghatgi et al., 2019).
• Population and remnant studies: Consistent recoil (remnant velocity) estimates and improved modeling of the coalescence process.

Extensions

• Tidal effects: Modular corrections such as the NRTidalv3 HM extension facilitate accurate binary neutron star analyses by attaching multipolar tidal phase corrections with the prescription $$\psi_{\ell m}^T(f) \approx (|m|/2) \psi_T^{NRT3}(2f/|m|)$$ (Abac et al., 21 Jul 2025).
• Precessing binaries: The IMRPhenomXHM model forms the non-precessing backbone for precessing models such as IMRPhenomXPHM/IMRPhenomXODE, where precessional effects are included by “twisting up” with frequency-dependent Euler angles calibrated or numerically solved on the fly (Colleoni et al., 21 Dec 2024).

Limitations

• Some physical effects remain only approximately included:
  • Not all $(\ell, m)$ modes are present; important in rare cases where, e.g., $(3,1)$, $(4,2)$, $(4,1)$ may be significant.
  • Direct mode–mixing calibration is currently limited to the (3,2) mode.
  • The model as originally formulated does not include matter effects or prompt eccentricity, although tidal and eccentric corrections can now be modularly attached.
• The overlap decreases marginally for highly asymmetric binaries in extreme corners of parameter space—a topic addressed via continuous NR and Teukolsky calibration upgrades and the addition of systematic uncertainty quantification (Yu et al., 2023, Mezzasoma et al., 30 Mar 2025).

7. Future Developments

The IMRPhenomXHM model serves as a foundation for continued advancements in waveform modeling:

• Uncertainty quantification: Sampling and marginalizing over calibration coefficients to propagate NR and model uncertainties into parameter inferences, essential for high-SNR observations (Mezzasoma et al., 30 Mar 2025).
• Improved calibration: Simultaneous, automated calibration of all waveform segments using gradient-based optimization strategies reduces median model–NR mismatches by up to 50%, systematically lowering model error (Lam et al., 2023).
• Multipolar memory and (2,0) content: Recent extensions have constructed complete ℓ=2 mode content incorporating both oscillatory components and the nonlinear displacement (memory) contributions for precessing binaries, providing a more physically complete signal description (Rosselló-Sastre et al., 10 Jun 2025).
• Integration with PhenomXO4a: The inclusion of multipolar asymmetries and numerically tuned precession dynamics into the X-family framework (IMRPhenomXHM-CP) further enhances precessing modeling accuracy, enabling robust population studies and high-precision astrophysics (Thompson et al., 2023).

As detector sensitivity increases and detection rates grow, the modular, extensible structure of IMRPhenomXHM—anchored by a robust NR-calibrated backbone—enables continuous upgrades in accuracy, physical completeness, computational performance, and scientific impact for gravitational-wave astronomy.