Next generation of accurate and efficient multipolar precessing-spin effective-one-body waveforms for binary black holes

Spin precession is one of the key physical effects that coul unveil the origin of the compact binaries detected by ground- and space-based gravitational-wave (GW) detectors, and shed light on their possible formation channels. Efficiently and accurately modeling the GW signals emitted by these systems is crucial to extract their properties. Here, we present SEOBNRv5PHM, a multipolar precessing-spin waveform model within the effective-one-body formalism for the full signal (i.e. inspiral, merger and ringdown) of binary black holes (BBHs). In the nonprecessing limit, the model reduces to SEOBNRv5HM, which is calibrated to 442 numerical-relativity (NR) simulations, 13 waveforms from BH perturbation theory, and nonspinning energy flux from second-order gravitational self-force theory. We remark that SEOBNRv5PHM is not calibrated to precessing-spin NR waveforms from the Simulating eXtreme Spacetimes Collaboration. We validate SEOBNRv5PHM by computing the unfaithfulness against 1543 precessing-spin NR waveforms, and find that for 99.8% (84.4%) of the cases, the maximum value, in the total mass range $20–300M_{\odot }$, is below 3% (1%). These numbers reduce to 95.3% (60.8%) when using the previous version of the SEOBNR family, SEOBNRv4PHM, and to 78.2% (38.3%) when using the state-of-the-art frequency-domain multipolar precessing-spin phenomenological IMRPhenomXPHM model. Due to much better computational efficiency of SEOBNRv5PHM compared to SEOBNRv4PHM, we are also able to perform extensive Bayesian parameter estimation on synthetic signals and GW events observed by LIGO-Virgo detectors. We show that SEOBNRv5PHM can be used as a standard tool for inference analyses to extract astrophysical and cosmological information of large catalogs of BBHs.

Figure 1

Frames used in the construction of the SEOBNRv5PHM model. The coprecessing frame (red) is constructed such that its $z$ axis is instantaneously aligned with the Newtonian angular momentum ${\mathit{l}}_N(t)$ and can be described by the Euler angles $(\alpha ,\beta ,\gamma )$ with respect to ${\mathit{J}}_{\mathrm{f}}$ frame (blue), while the source frame (purple) corresponds to the inertial frame defined by the initial Newtonian angular momentum ${\mathit{l}}_N(t_{\mathrm{ref}})$ and unit separation vector $\widehat{\mathit{n}}(t_{\mathrm{ref}})$. For the SEOBNRv5PHM model we adopt the convention that at $t_{\mathrm{ref}}$, the source and coprecessing frames coincide.

Figure 2

Left panel: sky-and-polarization averaged, SNR-weighted unfaithfulness in the total mass range between $[20–300]M_{\odot }$ for an inclination $\iota =\pi /3$, of SEOBNRv5PHM with different prescriptions for the dynamics against the 118 highly precessing-spin NR simulations from Ref. [98]. The different prescriptions for the dynamics correspond to using the SEOBNRv5HM Hamiltonian, $H_{\mathrm{EOB}}^{\text{align}}$, with the spins projected onto ${\mathit{l}}_N$ (purple), using $H_{\mathrm{EOB}}^{\text{align}}$ with the spins projected onto ${\mathit{l}}_N$ and $\mathit{l}$ (yellow), and using the partially precessing Hamiltonian $H_{\mathrm{EOB}}^{\text{pprec}}$ of SEOBNRv5PHM (green), with the spins projected onto ${\mathit{l}}_N$ and $\mathit{l}$; see the main text for details. The dashed horizontal vertical lines correspond to the ${10}^{-3}$, 0.01 and 0.03 unfaithfulness values. Right panel: distribution of the maximum unfaithfulness over the total mass range for each NR simulation considered in the left plot. The vertical dashed lines indicate the median values of the distribution.

Figure 3

Parameter space coverage in $q-{\chi }_{\mathrm{eff}}-{\chi }_{\mathrm{p}}$ space for NR simulations used to build and validate the SEOBNRv5 models. For the nonprecessing runs used in the construction of the SEOBNRv5HM model the color is fixed to blue (see Ref. [149] for details about these simulations). The precessing-spin runs follow a color map depending on the value of the effective spin parameter ${\chi }_{\mathrm{p}}$ at the reference time of the simulation. The set of precessing NR waveforms is built upon simulations from the public SXS catalog [151], as well as the 118 simulations from Ref. [98], which are highlighted with red diamonds to ease their visualization.

Figure 4

Left panel: sky-and-polarization averaged, SNR-weighted unfaithfulness in the total mass range between $[20–300]M_{\odot }$ for an inclination $\iota =\pi /3$, between SEOBNRv4PHM (blue), IMRPhenomXPHM (orange), TEOBResumS-GIOTTO (pink) and SEOBNRv5PHM (green) against NR for the 118 highly precessing-spin BBH simulations from Ref. [98]. The dashed horizontal vertical lines correspond to the ${10}^{-3}$, 0.01 and 0.03 unfaithfulness values. Right panel: distribution of the maximum unfaithfulness over the total mass range for each NR simulation considered in the left plot. The vertical dashed lines indicate the median values of the distribution.

Figure 5

Time-domain comparison of the SEOBNRv5PHM and SEOBNRv4PHM models to the NR waveform PrecBBH000001 from Ref. [98] with mass ratio 1.25, black-hole spin magnitudes 0.8 and total mass $M=60M_{\odot }$. The source parameters are ${\iota }_s=\pi /3$, ${\varphi }_s=\pi$, ${\kappa }_s=0$. The NR waveform includes all the multipoles with $l\le 5$. Both waveform models resemble accurately the features of the NR waveform at the inspiral, merger and ringdown, with a more faithful agreement of SEOBNRv5PHM which translates into an unfaithfulness of 0.69%, while for SEOBNRv4PHM it increases to 1.1%.

Figure 6

Sky-averaged SNR weighted unfaithfulness as a function of the total mass of the system $[20,300]M_{\odot }$, of the SEOBNRv4PHM model (top left panel), the SEOBNRv5PHM model (top right panel), the IMRPhenomXPHM model (left bottom panel) and TEOBResumS-GIOTTO (right bottom panel), against 1543 precessing-spin SXS simulations. The simulations with the highest unfaithfulness for each model are highlighted in each panel. For SEOBNRv5PHM and SEOBNRv4PHM the case with highest unfaithfulness coincides, and thus the highlighted curves overlap.

Figure 7

Distribution of the sky-and-polarization averaged, SNR-weighted unfaithfulness as a function of the binary’s total mass for inclination $\iota =\pi /3$, between SEOBNRv4PHM (blue), SEOBNRv5PHM (green), IMRPhenomXPHM (orange) and TEOBResumS-GIOTTO (pink) against NR for 1543 quasicircular precessing-spin BBH simulations. For each total mass considered the distributions of SEOBNRv4PHM and SEOBNRv5PHM, and IMRPhenomXPHM and TEOBResumS-GIOTTO have been shifted in the x axis by $4M_{\odot }$ from the total mass value used to compute the unfaithfulness to ease the visualization of the results. The bracket indicates the total mass value used for the unfaithfulness calculation. In each distribution the median value is highlighted with thicker lines.

Figure 8

Distribution of maximum (blue), median (orange) and minimum (green) sky-and-polarization averaged, SNR-weighted unfaithfulness over the binary’s total mass range $[20–30]M_{\odot }$ for inclination $\iota =\pi /3$, between the different waveform families (SEOBNRv4, IMRPhenomX, TEOBResumS-GIOTTO and SEOBNRv5) against NR for aligned spins (Aligned), precessing spins (Precessing) and combining the two previous distributions (Combined). The nonprecessing NR simulations correspond to the 441 cases presented in Ref. [149], while the precessing NR simulations correspond to the 1543 cases used in Fig. 6. In the violin plots the median values of the distributions are highlighted with thicker lines.

Figure 9

Sky-and-polarization-averaged, SNR-weighted unfaithfulness as a function of the total mass of the binary for inclination ${\iota }_s=\pi /3$, among the NRSur7dq4 model and the SEOBNRv4PHM (blue), IMRPhenomXPHM (orange), TEOBResumS-GIOTTO (pink) and SEOBNRv5PHM (green) models for 5000 randomly distributed precessing-spin configurations. Left: the solid (dashed) lines show the median (95th percentile) as a function of the total mass. Right: distribution of maximum unfaithfulness over all the total masses considered. The vertical dashed lines indicated the median values of the distributions.

Figure 10

Maximum sky-and-polarization-averaged unfaithfulness weighted by the SNR over the total mass range $[20–30]M_{\odot }$ between SEOBNRv5PHM and IMRPhenomXPHM for 5000 random configurations with inclination ${\iota }_s=\pi /3$. The unfaithfulness grows with increasing mass ratio and spin magnitude values, and it can reach very large values for mass ratios $q\sim 20$ and ${\chi }_{\mathrm{p}}\sim 1$.

Figure 11

Wall times of the SEOBNRv4PHM, IMRPhenomTPHM, TEOBResumS-GIOTTO, and SEOBNRv5PHM models for a configuration with dimensionless spins ${\mathit{\chi }}_1=[0.5,0,0.8]$, ${\mathit{\chi }}_2=[0,0.5,0.3]$, total mass range $M\in [10,100]M_{\odot }$, starting frequency $f_{\text{start}}=10\mathrm{Hz}$ and three different mass ratios: 1 (top panel), 3 (mid panel) and 10 (bottom panel).

Figure 12

2D and 1D posterior distributions for some relevant parameters measured from the synthetic BBH signal with mass ratio $q=6$, total source-frame mass of $95M_{\odot }$, dimensionless spins of the BHs ${\mathit{\chi }}_1=[-0.06,0.78.-0.4]$ and ${\mathit{\chi }}_2=[0.08,-0.17,-0.23]$ defined at 20 Hz. The inclination with respect to the line of sight of the binary is $\iota =0.69\mathrm{rad}$. The other parameters are specified in the text and in Table 2. The signal waveform is a NR waveform from the public SXS catalog SXS:BBH:0165. In the 2D posteriors the solid contours represent the 95% credible intervals, and black dots show the values of the parameters of the injected signal. In the 1D posteriors they are represented by dashed and solid vertical lines, respectively. The parameter estimation is performed with the SEOBNRv5PHM model (green) and the IMRPhenomXPHM model (orange). Left: component masses in the detector frame. Right: effective spin parameters, ${\chi }_{\mathrm{eff}}$ and ${\chi }_{\mathrm{p}}$.

Figure 13

Component masses in the source frame inferred for the real GW events reanalyzed with SEOBNRv5PHM. Comparisons are presented with SEOBNRv4PHM (when available) and IMRPhenomXPHM from GWTC-2.1 [7] and GWTC-3 [8] catalogs, except for GW190412 for which we present the SEOBNRv4PHM from the discovery paper [216], since the convergence of the posteriors is larger than in the GWTC-2.1 catalog. For GW190521 and GW200129 we include the posterior samples of the NRSur7dq4 model from [217] and [218], respectively.

Figure 14

Effective-spin parameters ${\chi }_{\mathrm{eff}}$ and ${\chi }_{\mathrm{p}}$ inferred for the GW events reanalyzed with SEOBNRv5PHM. Comparisons are presented with SEOBNRv4PHM (when available) and IMRPhenomXPHM from GWTC-2.1 [7] and GWTC-3 [8] catalogs, except for GW190412 for which we present the SEOBNRv4PHM from the discovery paper [216] as in Fig. 13. For GW190521 and GW200129 we include the posterior samples of the NRSur7dq4 model from [217] and [218], respectively.

Figure 15

Comparison of mass-ratio and effective spin parameter inferred for GW190521 between SEOBNRv5PHM, the phenomenological models IMRPhenomTPHM from Ref. [223] and IMRPhenomXPHM from GWTC-2.1 [7] and the samples of the NRSur7dq4 model from [217].

Figure 16

Distribution of network matched-filter SNR inferred for some of the GW events reanalyzed with SEOBNRv5PHM. Comparisons are presented with IMRPhenomXPHM results obtained with the same settings and data as the SEOBNRv5PHM results.

Figure 17

Left panel: sky-averaged SNR weighted unfaithfulness as a function of the total mass of the system $[20,300]M_{\odot }$, of IMRPhenomXPHM (orange), IMRPhenomTPHM (brown) and SEOBNRv5PHM (green), against the 118 highly precessing simulations from Ref. [98]. Right panel: distribution of the maximum unfaithfulness over the total mass range for each NR simulation considered in the left plot. The vertical dashed lines indicate the median values of the distribution.

Figure 18

Sky-averaged SNR-weighted unfaithfulness as a function of the total mass of the system $[20,300]M_{\odot }$, of the SEOBNRv5PHM model (left panel), and the state-of-the-art phenomenological models, IMRPhenomXPHM (midpanel) and IMRPhenomTPHM (right panel), against 1543 precessing-spin SXS NR waveforms.

Figure 19

Maximum sky-and-polarization-averaged unfaithfulness weighted by the SNR over the total mass range $[20–30]M_{\odot }$ between SEOBNRv5PHM and IMRPhenomTPHM for 5000 random configurations with inclination ${\iota }_s=\pi /3$. The unfaithfulness grows with increasing mass ratio and spin magnitude values, and it can reach very large values for mass ratios $q\sim 20$ and ${\chi }_{\mathrm{p}}\sim 1$.