Computationally efficient models for the dominant and subdominant harmonic modes of precessing binary black holes

We present imrphenomxphm, a phenomenological frequency-domain model for the gravitational-wave signal emitted by quasicircular precessing binary black holes, which incorporates multipoles beyond the dominant quadrupole in the precessing frame. The model is a precessing extension of imrphenomxhm, [C. García-Quirós et al., Phys. Rev. D 102, 064002 (2020)] based on approximate maps between aligned-spin waveform modes in the coprecessing frame and precessing waveform modes in the inertial frame, which is commonly referred to as “twisting up” the nonprecessing waveforms. imrphenomxphm includes imrphenomxp as a special case, the restriction to the dominant quadrupole contribution in the coprecessing frame. We implement two alternative mappings, one based on a single-spin post-Newtonian approximation, as used in imrphenompv2 [M. Hannam et al., Phys. Rev. Lett. 113, 151101 (2014).], and one based on the double-spin multiple scale analysis approach of [K. Chatziioannou et al., Phys. Rev. D 95, 104004 (2017).]. We include a detailed discussion of conventions used in the description of precessing binaries and of all choices made in constructing the model. The computational cost of imrphenomxphm is further reduced by extending the interpolation technique of [C. García-Quirós et al., Classical Quant. Grav. 38, 015006 (2021).] to the Euler angles. The accuracy, speed, robustness, and modularity of the imrphenomx family will make these models productive tools for gravitational wave astronomy in the current era of greatly increased number and diversity of detected events.

Figure 1

The Euler angles $\alpha$ and $\gamma$ obtained by solving the MSA system of [24] and the NNLO PN equations for a binary of mass $M=20M_{\odot }$, $q=10$, ${\chi }_1=(0.4,0,0.4)$, and ${\chi }_2=(0.3,0,-0.3)$, with a starting frequency of 10 Hz. The vertical dashed lines correspond to the minimum energy circular orbit [51], innermost stable circular orbit, and ringdown frequency in increasing order. The multiple shaded lines denote the Euler angles evaluated using the NNLO angles and different orders for the MSA correction used in the MSA system where the default order is $\mathcal{O}(5)$.

Figure 2

Comparison of the NNLO Euler angle $\cos \beta$ against the MSA version [$\mathcal{O}(5)$], for the same binary as shown in Fig. 1. We show the impact of different PN orders for the orbital angular momentum $\mathit{L}$ on the determination of the opening angle for the NNLO prescription.

Figure 3

Comparison of Euler angles with NR. Left: SXS:BBH:0094 [$q=1.5$, ${\mathit{\chi }}_1=(0.5,0,0)$, ${\mathit{\chi }}_2=(0,0,0)$]. Right: SXS:BBH:0053 [$q=3$, ${\mathit{\chi }}_1=(0.5,0,0)$, ${\mathit{\chi }}_2=(-0.5,0,0)$]. Solid blue line: Quadrupole-aligned Euler angles extracted from NR simulation. Dashed orange line: NNLO implementation (version 102). Dashed green line: MSA implementation (version 223).

Figure 4

We compare the relative performance of NNLO and MSA angles against nrsur7dq4, by plotting the plus polarization returned by the different models for a double-spin configuration with high in-plane spins. Each panel refers to the same spin configuration, but we allow the mass ratio to vary from 1 to 4, which is the upper limit of the calibration region of nrsur7dq4. Overall NNLO angles perform well, although we do observe some disagreement with respect to nrsur7dq4 as we increase the mass ratio (lower panels).

Figure 5

We compare the behavior of the two twisting-up methods implemented in imrphenomxphm for the same spin configuration chosen in Fig. 4, varying the mass ratio between 4 and 16. For $q=4$ we observe good agreement between the two angle descriptions, especially during the inspiral. However, as the mass ratio increases the agreement degrades and NNLO angles tend to produce nonsmooth features in the waveform.

Figure 6

The SNR-weighted mismatch $1-{\mathcal{M}}_w$ of imrphenomxphm against 99 SXS precessing waveforms of the lvcnr catalog, in order of ascending inclination. A dashed line and a dotted line mark the 5% and 10% mismatch thresholds, respectively.

Figure 7

In the upper panel we compare the nonprecessing models imrphenomd, imrphenomxas, imrphenomhm, imrphenomxphm, seobnrv4hm_rom and the precessing imrphenomxphm and nrsur7dq4 models with the nonprecessing nrhybsur3dq8 model as discussed in the main text. In the lower panel we compare two versions of our imrphenomxphm model (NNLO-based version 102 with final spin version 0 and MSA-based version 223 with final spin version 3; see Table 3), and imrphenompv3hm with nrsur7dq4, as discussed in the main text. For comparison we show also the mismatch between nrhybsur3dq8 and seobnrv4hm_rom as displayed in the upper panel. The number of samples for each comparison is indicated in the legend.

Figure 8

Histograms of a mismatch calculation for $h_{+}$ between versions of imrphenomxphm without using multibanding for the Euler angles, and with four different levels of multibanding threshold: ${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$. The top panel shows the results for the MSA prescription (version 223) and the bottom panel shows the result for NNLO angles (version 102). The ${10}^6$ waveforms were generated across a parameter space as described in the main text. The threshold ${10}^{-3}$ (MB3 in the plot) has been chosen as the default value in the lalsuite implementation.

Figure 9

Injection recovery results for SXS:BBH:0143 for imrphenomxphm without multibanding in the Euler angles and with four different thresholds (${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$) for multibanding in the angles. No appreciable differences arise in the posteriors, meaning that a more relaxed threshold than the default of ${10}^{-3}$ could be used in PE studies, reducing even more the computational cost of the runs.

Figure 10

Mean evaluation time for different precessing models as a function of the total mass (left panel) and as a function of the spacing of the frequency grid (right panel). The nrsur7dq4 model cannot be evaluated for such low frequencies as the phenom models; hence it only appears in the left panel for high masses, but not in the right panel, where we are using a total mass of $50M_{\odot }$.

Figure 11

Bayesian inference results for GW150914: one-dimensional posterior probability distributions for the effective precessing spin parameter ${\chi }_p$ and the mass ratio $q$. (Here $q$ is the inverse of our definition in Sec. 1, following the lalinference convention.) We show results for imrphenomxphm with and without higher modes, using NNLO and MSA angles, respectively, as discussed in Sec. 4. For comparison, we also show results for imrphenompv3 and imrphenompv3hm. The labels of the different versions of imrphenomxphm are explained in Appendix pp6.

Figure 12

Bayesian inference results for GW150914: posterior probability distributions for the effective spin parameters ${\chi }_{\mathrm{eff}}$ and ${\chi }_p$ with two-dimensional 90% credible intervals. Here we show results for imrphenomxphm with and without higher modes, using NNLO and MSA angles, respectively, as discussed in Sec. 4. For comparison, we also show results for imrphenompv3 and imrphenompv3hm.

Figure 13

Bayesian inference results for GW170729: Posterior probability distributions for the effective spin parameters ${\chi }_{\mathrm{eff}}$ and ${\chi }_p$, the component masses $m_1$ and $m_2$, distance $D_L$, and the angle ${\theta }_{JN}$ between angular momentum and line of sight. The 90% credible intervals are represented by vertical (contour) lines in the 1D (2D) plots.

Figure 14

Injection recovery results for SXS:BBH:0143. Top row: From left to right, posterior probability distributions for the mass ratio ($q=m_2/m_1$), total mass in the detector frame, and ${\chi }_{\mathrm{eff}}$. Bottom row: From left to right, posterior probability distributions for ${\chi }_p$, luminosity distance, and the angle ${\theta }_{\mathrm{JN}}$ between the total angular momentum and line of sight. The dashed vertical lines represent 90% credible intervals, while the thick black lines represent the injected value. The notation for the different versions of imrphenomxphm is described in Appendix pp6.