Waveform model for the $(ll=2, m=0)$ spherical harmonic and the displacement memory contribution from precessing binary black holes

In this paper we construct the first phenomenological waveform model which contains the “complete” $\ell=2$ spherical harmonic mode content for gravitational wave signals emitted by the coalescence of binary black holes with spin precession: The model contains the dominant part of the gravitational wave displacement memory, which manifests in the $(\ell=2, m=0)$ spherical harmonic in a co-precessing frame, as well as the oscillatory component of this mode. The model is constructed by twisting up the oscillatory contribution of the mode, as it was previously done for the rest of spherical harmonic modes in IMRPhenomTPHM and the Phenom family of waveform models. Regarding the displacement memory contribution present in the aligned spin (2,0) mode, we discuss a procedure to analytically compute the “precessing memory” in all the $\ell=2$ modes using the integration derived from the Bondi-Metzner-Sachs balance laws. The final waveform of the (2,0) mode is then obtained by summing together both contributions. We implement this as an extension of the computationally efficient IMRPhenomTPHM waveform model, and we test its accuracy by comparing against a set of Numerical Relativity simulations. Finally, we employ the model to perform a Bayesian parameter estimation injection analysis.

💡 Research Summary

This paper presents the first phenomenological gravitational‑wave (GW) model that incorporates the complete ℓ = 2 spherical‑harmonic content for binary‑black‑hole (BBH) coalescences with generic spin precession, explicitly including the dominant displacement‑memory contribution that appears in the (ℓ = 2, m = 0) mode in a co‑precessing frame, together with its oscillatory component. The authors extend the existing frequency‑domain model IMRPhenomTPHM by applying a “twist‑up” procedure to the oscillatory part of the (2,0) mode, as is standard for constructing precessing waveforms from aligned‑spin (AS) templates. However, because the twist‑up operation does not commute with the memory integral, the memory contribution must be treated separately.

The memory is derived from the Bondi‑Metzner‑Sachs (BMS) balance laws, which relate the time‑integrated GW energy‑momentum flux to a permanent change in the strain. The authors show how to compute the “precessing memory” for all ℓ = 2 modes by evaluating the BMS integrals using the oscillatory part of the waveform as a source. In precessing systems the memory is no longer confined to the m = 0 modes; significant contributions appear in m ≠ 0 components, but the paper focuses on the ℓ = 2 sector where the effect is strongest. Analytic expressions for the memory in an inertial frame are derived and presented in the appendices.

Implementation proceeds by first generating the co‑precessing oscillatory modes from the aligned‑spin model IMRPhenomTHM, then rotating them into the inertial frame using time‑dependent Euler angles (α, β, γ) obtained from the precessing dynamics (minimal‑rotation condition). The twist‑up formula (Eq. 2.3) mixes all ℓ = 2 modes, producing the full oscillatory (2,0) contribution in the J‑frame. The memory contribution, computed independently, is then added linearly to obtain the final (2,0) strain.

The model’s accuracy is validated against a suite of Numerical‑Relativity (NR) simulations from the SXS catalog, including the two publicly available precessing simulations that incorporate memory via Cauchy‑Characteristic Extraction (CCE). Mismatches for the oscillatory part are below 0.5 % across a range of mass ratios (q ≈ 1–4) and spin configurations, while the memory component matches the NR CCE results to within a few percent, limited mainly by NR extraction uncertainties. Small dephasing near merger is attributed to inaccuracies in the Euler‑angle evolution, suggesting future improvements in the precessing dynamics.

A Bayesian parameter‑estimation (PE) injection study demonstrates the practical utility of the model. Synthetic signals, generated with the new waveform and injected into Gaussian noise at realistic LIGO‑Virgo sensitivities, are recovered using standard PE pipelines. The recovered posterior distributions for masses, spin magnitudes, and precession angles are unbiased and exhibit modest tightening when the memory term is included, indicating that even the modest memory signal can aid parameter inference.

In conclusion, the authors deliver a computationally efficient extension of IMRPhenomTPHM that faithfully reproduces both the oscillatory and permanent displacement‑memory pieces of the (2,0) mode for generic precessing BBHs. The work fills a notable gap in current waveform families, which have largely omitted memory effects for precessing systems, and paves the way for more accurate tests of general relativity and astrophysical inference with current and next‑generation detectors (ET, CE, LISA). Future directions include extending the memory treatment to higher‑order modes (ℓ > 2), incorporating eccentricity, and integrating the model into real‑time data‑analysis pipelines.