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

TL;DR

This work delivers the first phenomenological waveform model that includes the complete $\ell=2$ mode content for precessing BBHs, incorporating the displacement memory in the $$(\ell=2,m=0)$$ mode in a co-precessing frame along with the oscillatory component. It develops memory via the Bondi-Metzner-Sachs balance laws and provides two coherent pathways: twisting-up the co-precessing modes with the memory integrated in the inertial frame, or computing the memory directly in the inertial frame; both approaches are shown to be consistent with NR. The model is implemented in the IMRPhenomTPHM/phenomxpy framework and validated against NR catalogs, showing improved matches and only modest computational overhead. A zero-noise Bayesian analysis demonstrates small biases and a likelihood improvement when including the $$(2,0)$$ mode and memory, indicating practical benefits for parameter estimation in precessing binaries, especially at lower masses where memory is more significant. This work therefore enhances GW data-analysis capabilities by enabling more accurate inference for precessing BBHs where memory and higher-$\ell$-mode contributions matter.

Abstract

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.

Figures (10)

• Figure 1: Comparison of our results for the oscillatory contribution of the $(2,0)$ mode in the inertial $\mathbf{J}$-frame with the NR simulation SXS:BBH:1520 with parameters: $q=3.03, \bm{\chi}_1^{\text{ref}}=\{0.540,-0.137,-0.435\}, \bm{\chi}_2^{\text{ref}}=\{0.056,0.258,0.129\}, \chi_{\text{p}}=0.557$ at $Mf_{\text{ref}}=5.15\times10^{-3}$. In light blue, we show the SXS waveform. In solid blue we present the mode computed with IMRPhenomTPHM, i.e., using the twisting-up approximation on the co-precessing modes from this model. In dashed yellow, we show the twisting-up of the SXS co-precessing modes. The solid red and dashed green curves show the contribution from the oscillatory component of the $(2,0)$ co-precessing mode for the IMRPhenomTPHM model and the SXS simulation, respectively. The top row corresponds to the real part of the mode, and the bottom row to the imaginary part. The left column shows the full evolution of the waveform, and the right column zooms in on the times around the merger.
• Figure 2: Top panel: comparison between the two methods to compute the memory contribution in the inertial frame. The solid blue curve is the correct result, which consists of twisting up the modes and then performing the time integral, whereas the yellow dashed curve is the incorrect result, where we first compute the memory and then twist up the result. Bottom panel: time evolution of the Euler angle $\beta$, corresponding to the rotation from the co-precessing frame to the inertial $\mathbf{J}$-frame, computed using Eq. (\ref{['beta_angle']}). This corresponds to a system with parameters: $q=3.03, \bm{\chi}_1^{\text{ref}}=\{0.540,-0.137,-0.435\}, \bm{\chi}_2^{\text{ref}}=\{0.056,0.258,0.129\}, \chi_{\text{p}}=0.557$ at $Mf_{\text{ref}}=5.15\times10^{-3}$.
• Figure 3: Comparison of our results for the memory contribution the inertial $\mathbf{J}$-frame with the NR simulation SXS:BBH:1520 with parameters: $q=3.03, \bm{\chi}_1^{\text{ref}}=\{0.540,-0.137,-0.435\}, \bm{\chi}_2^{\text{ref}}=\{0.056,0.258,0.129\}, \chi_{\text{p}}=0.557$ at $Mf_{\text{ref}}=5.15\times10^{-3}$. In light blue, we show the complete waveform with the memory correction added from the BMS balance laws (red dash-dotted line). In dashed yellow, we present the memory contribution computed by twisting up the co-precessing modes and then calculating the memory integral. In solid blue it is shown the memory contribution computed using the memory integration of the modes in the inertial $\mathbf{J}$-frame. The left column corresponds to the real part of the $\ell=2$ modes and the right column to the imaginary part.
• Figure 4: Comparison of the full $\ell=2, m\geq0$ spherical harmonic modes from the NR simulation SXS:BBH_ExtCCE:0008 (solid blue) with the IMRPhenomTPHM model (dashed yellow) in the inertial $\mathbf{J}$-frame. The first row shows the real part of the modes, while the third row shows the imaginary part. The insets show a zoom in on the times of the merger-ringdown stage of the evolution. The second and fourth rows, in red, stand for the waveform difference between NR and the model, in the modes on top of them. The mismatch between NR and our model for this waveform, computed as in Eq. (\ref{['mismatch']}), is $6.1\times10^{-2}$.
• Figure 5: Same as Fig. \ref{['fig:comp_CCE_8']}, but for SXS:BBH_ExtCCE:0013.The mismatch between NR and our model for this waveform, computed as in Eq. (\ref{['mismatch']}), is $1.2\times10^{-3}$.