Laying the foundation of the effective-one-body waveform models SEOBNRv5: Improved accuracy and efficiency for spinning nonprecessing binary black holes

We present SEOBNRv5HM, a more accurate and faster inspiral-merger-ringdown gravitational waveform model for quasicircular, spinning, nonprecessing binary black holes within the effective-one-body (EOB) formalism. Compared to its predecessor, SEOBNRv4HM, the waveform model (i) incorporates recent high-order post-Newtonian results in the inspiral, with improved resummations, (ii) includes the gravitational modes $(\ell ,|m|)=(3,2),(4,3)$, in addition to the (2,2), (3,3), (2,1), (4,4), (5,5) modes already implemented in SEOBNRv4HM, (iii) is calibrated to larger mass ratios and spins using a catalog of 442 numerical-relativity (NR) simulations and 13 additional waveforms from black-hole perturbation theory, and (iv) incorporates information from second-order gravitational self-force in the nonspinning modes and radiation-reaction force. Computing the unfaithfulness against NR simulations, we find that for the dominant (2,2) mode the maximum unfaithfulness in the total mass range $10–300M_{\odot }$ is below ${10}^{-3}$ for 90% of the cases (38% for SEOBNRv4HM). When including all modes up to $\ell =5$ we find 98% (49%) of the cases with unfaithfulness below ${10}^{-2}$ (${10}^{-3}$), while these numbers reduce to 88% (5%) when using SEOBNRv4HM. Furthermore, the model shows improved agreement with NR in other dynamical quantities (e.g., the angular momentum flux and binding energy), providing a powerful check of its physical robustness. We implemented the waveform model in a high-performance python package (pyseobnr), which leads to evaluation times faster than SEOBNRv4HM by a factor of 10 to 50, depending on the configuration, and provides the flexibility to easily include spin-precession and eccentric effects, thus making it the starting point for a new generation of EOBNR waveform models (SEOBNRv5) to be employed for upcoming observing runs of the LIGO-Virgo-KAGRA detectors.

Figure 1

Mode mixing in the NR simulation SXS:BBH:2138 ($q=3.0,{\chi }_1=-0.6,{\chi }_2=0.4$). Upper panel: amplitude of the modes $|h_{\ell m}|$ and of $|{{}^{S}h}_{\ell m0}|$, after the mode-mixing removal [Eqs. (60a) and (60b)]. We denote with $t=0$ the time of the peak of the (2,2)-mode amplitude. Lower panel: frequencies of the modes $h_{\ell m}$ and of ${{}^{S}h}_{\ell m0}$. The ringdown frequencies of the ${{}^{S}h}_{320}$ and ${{}^{S}h}_{430}$ modes are well approximated by the (3,2,0) and (4,3,0) QNM frequencies (dashed horizontal lines) after the mode-mixing removal.

Figure 2

NR and BH-perturbation-theory waveforms used to calibrate SEOBNRv5HM, projected on the binary’s parameters $\nu$ and ${\chi }_{\mathrm{eff}}=({\chi }_1{m}_1+{\chi }_2{m}_2)/M$. We highlight four regions as explained in the text, and use different markers to distinguish between 327 simulations from the public SXS catalog [131], 114 private SXS waveforms, one einstein toolkit simulation, and 13 Teukolsky-code waveforms. We refer to private waveforms as all those which cannot be downloaded from the SXS website [130] at the time of this publication.

Figure 3

Posterior for the calibration parameters $\{d_{\mathrm{SO}},\mathrm{\Delta }{t}_{\mathrm{ISCO},\mathrm{S}}^{22}\}$, obtained by comparing to the NR simulation SXS:BBH:2420 ($q=1.0,{\chi }_1=0.2,{\chi }_2=0.2$). The blue posterior is the result of the nested-sampling analysis described in Sec. 5b, and shows values mostly clustered around two distinct regions or modes. The green posterior is what we obtain after removing one of the two modes and keeping only the points with ${\mathcal{M}}_{\max }<{10}^{-3}$ and $\delta t_{\text{merger}}<5M$. We use these processed posteriors to obtain fits for the calibration parameters across parameter space.

Figure 4

Fits for the nonspinning calibration parameters ${\mathit{\theta }}_{\mathrm{noS}}=\{a_6,\mathrm{\Delta }{t}_{\mathrm{ISCO},\mathrm{noS}}^{22}\}$. The parameters are obtained by least-square fits of the maximum likelihood points (blue dots) of the calibration posteriors (shaded violins), for a set of NR simulations with different mass ratios $\nu$, together with estimates of the test-mass limit values (green dots). We rescale $\mathrm{\Delta }{t}_{\mathrm{ISCO},\mathrm{noS}}^{22}$ by ${\nu }^{1/5}$ to improve its extrapolation in the $\nu \to 0$ limit. No processing is needed for the nonspinning calibration posteriors, as the maximum likelihood point lies in the same mode for all configurations.

Figure 5

(2,2)-mode mismatch over a range of total masses between 10 and $300M_{\odot }$, between different aligned-spin approximants and the 442 NR simulations used in this work. The colored lines highlight cases with the worst maximum mismatch for each model. Note that SEOBNRv5 has no outliers beyond 0.3% and many more cases at lower unfaithfulness, especially compared to SEOBNRv4 and TEOBResumS-GIOTTO.

Figure 6

Top panel: histogram of the maximum (2,2)-mode mismatch over a range of total masses between 10 and $300M_{\odot }$, between different aligned-spin approximants and the 442 NR simulations used in this work. The NR error is estimated by computing the mismatch between NR simulations with the highest and second-highest resolutions. The vertical dashed lines show the medians. Bottom panel: distribution of the maximum (blue), median (orange), and minimum (green) mismatch over the same range of total masses for the different models.

Figure 7

The sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$, over a range of total masses between 20 and $300M_{\odot }$ between different aligned-spin multipolar approximants and the 441 SXS NR simulations used in this work. The colored lines highlight cases with the worst maximum mismatch for each model.

Figure 8

The sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$, over a range of total masses between 20 and $300M_{\odot }$, of NR waveforms with the same modes as SEOBNRv5HM $(\ell ,m)=(2,2),(3,3),(2,1),(4,4),(5,5),(3,2),(4,3)$ against NR waveforms with all ($\ell \le 5$) modes.

Figure 9

Top panel: histogram of the maximum sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$, over a range of total masses between 20 and $300M_{\odot }$, between different aligned-spin multipolar approximants and the 441 SXS NR simulations used in this work. The NR error is estimated by computing the mismatch between NR simulations with the highest and second-highest resolutions. The vertical dashed lines show the medians. Bottom panel: distribution of the maximum (blue), median (orange), and minimum (green) mismatch over the same range of total masses for the different models.

Figure 10

Mode-by-mode mismatches between SEOBNRv4HM, SEOBNRv5HM, and NRHybSur3dq8, for 5000 random configurations with $q\in [1,8],|{\chi }_i|\le 0.9$. For each mode we show the maximum mismatch over a range of total masses between 10 and $300M_{\odot }$. The horizontal lines show the medians.

Figure 11

Mode-by-mode mismatches between SEOBNRv4HM, SEOBNRv5HM, and NRHybSur2dq15, for 5000 random configurations with $q\in [1,15],|{\chi }_1|\le 0.6,{\chi }_2=0$. For each mode we show the maximum mismatch over a range of total masses between 10 and $300M_{\odot }$. The horizontal lines show the medians.

Figure 12

Sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$, between SEOBNRv5HM and NRHybSur3dq8, for 2000 random configurations with $q\in [1,8],|{\chi }_i|\le 0.8$. We show maximum mismatch over a range of total masses between 20 and $300M_{\odot }$ as a function of the mass-ratio $q$ and the primary spin ${\chi }_1$.

Figure 13

Sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$, between SEOBNRv5HM and IMRPhenomXHM, for 2000 random configurations with $q\in [1,20],|{\chi }_i|\le 0.99$. We show maximum mismatch over a range of total masses between 20 and $300M_{\odot }$ as a function of the mass ratio $q$ and the effective spin ${\chi }_{\mathrm{eff}}$.

Figure 14

Comparison of the Newtonian-normalized angular-momentum flux between SEOBNRv4, SEOBNRv5, and the NR simulation SXS:BBH:0156. The triangle, square, and diamond correspond, respectively, to 3, 1, and 0 GW cycles before merger, which is taken as the peak of the (2,2)-mode amplitude for each model.

Figure 15

Fractional difference between the Newtonian-normalized angular-momentum flux $\dot{\widehat{J}}$ of SEOBNRv4 and SEOBNRv5, and the one obtained from the NR simulations used in this work, evaluated 2 cycles before merger.

Figure 16

Fractional difference between the EOB and NR nonspinning binding energy as a function of $v$, for SEOBNRv5 and SEOBNRv4. The gray region represents an estimate of the NR error. Notice the improvement in the agreement of SEOBNRv5 compared to SEOBNRv4, especially between 3 and 1 cycles before the merger.

Figure 17

Spin-orbit contribution to the binding energy as a function of $v$ for SEOBNRv4 (blue), SEOBNRv5 (green), and NR (gray). The uncalibrated models are obtained by setting to zero the calibration parameters entering the Hamiltonian. The dashed vertical line represents the merger of the NR configuration in Eq. (94a) that merges at the lowest frequency, and the numbers of cycles also refer to the same simulation, while the EOB curves terminate at the EOB merger. The shaded regions represent the NR error. SEOBNRv5 has a better agreement with NR compared to SEOBNRv4, and remains within the NR error almost until the merger.

Figure 18

Same as Fig 17, but for spin-spin contributions (left panel) and for cubic-in-spin contributions (right panel).

Figure 19

Fractional difference between the EOB and NR binding energy, for SEOBNRv5 and SEOBNRv4, at $v=\sqrt{0.2}\simeq 0.45$.

Figure 20

Walltimes for SEOBNRv5HM and SEOBNRv4HM, with PA approximation (solid lines) and without (dashed lines), TEOBResumS-GIOTTO and IMRPhenomTHM, starting from $f_{\text{start}}=10\mathrm{Hz}$, as a function of the total mass $M$. SEOBNRv5HM outperforms SEOBNRv4HM, particularly for low total mass systems, both with and without the PA approximation, and shows walltimes close to TEOBResumS-GIOTTO. IMRPhenomTHM is the fastest model for low total masses due to its use of closed-form expressions, with the gap narrowing for lower total masses. The analytic PA approximation and several optimizations, such as the use of analytic derivatives of the Hamiltonian, play a crucial role in the SEOBNRv5HM performance.

Figure 21

2D and 1D posterior distributions for some relevant parameters measured from the synthetic BBH signal with mass ratio $q=15$, total source-frame mass of $162M_{\odot }$, dimensionless spins ${\chi }_{1z}=0.5$, and ${\chi }_{2z}=0.0$. The inclination with respect to the line of sight of the binary is $\iota =\pi /3\mathrm{rad}$. The other parameters are specified in the text and in Table 3. The injected signal is the SXS NR waveform SXS:BBH:2464. In the 2D posteriors the solid contours represent the 90% 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 SEOBNRv5HM model (green) and the IMRPhenomXHM model (orange).

Figure 22

1D and 2D posterior distributions for several parameters for the GW events GW150914, GW170729, and GW190412.

Figure 23

Network matched-filter SNR recovered by IMRPhenomXHM and SEOBNRv5HM for the three analyzed GW events.

Figure 24

Mismatch of SEOBNRv5_ROM against SEOBNRv5 for different values of the total mass, for ${10}^5$ random configurations. The dashed vertical lines show the medians.

Figure 25

Walltime comparison between SEOBNRv5 and SEOBNRv5_ROM, using a starting frequency $f_{\text{start}}=10\mathrm{Hz}$, as a function of the total mass $M$. For the time domain SEOBNRv5 model this also includes the conversion in the Fourier domain.

Figure 26

Histogram of the maximum (2,2)-mode mismatch over a range of total masses between 10 and $300M_{\odot }$, between the 442 NR simulations used in this work and SEOBNRv4 (blue), SEOBNRv5 calibrated to 442 NR simulations (green), and SEOBNRv5 calibrated to 137 NR simulations (orange). The vertical dashed lines show the medians.

Figure 27

Histogram of the maximum (2,2)-mode mismatch over a range of total masses between 10 and $300M_{\odot }$, between the 442 NR simulations used in this work and SEOBNRv5, using a white noise curve and the PSDs of Advanced LIGO, Einstein Telescope and Cosmic Explorer. The vertical dashed lines show the medians. The (2,2) mode of SEOBNRv5 is calibrated using the Advanced LIGO PSD, but performs as well using different noise curves.

Figure 28

(2,2)-mode mode mismatch against the NR simulation SXS:BBH:1432 for the maximum likelihood points of the SEOBNRv5 calibration posteriors, with and without NQCs in the RR force. The unfaithfulness is very similar, as the calibration of the Hamiltonian reabsorbs any difference in the dissipative dynamics. We also show an estimate of the NR error obtained by comparing simulations of different resolutions.

Figure 29

Comparison of the Newtonian normalized angular-momentum flux between SEOBNRv5 with and without NQCs in the radiation reaction, and the NR simulation SXS:BBH:1432. The triangle, square, and diamond correspond respectively to 3, 1, and 0 cycles before the merger, which for each model is taken as the peak of the (2,2)-mode amplitude. SEOBNRv5 with NQCs matches NR at the merger as expected, but does not agree with NR as well as SEOBNRv5 without NQCs at low frequencies.

Figure 30

(2,2)-mode mismatch over a range of total masses between 10 and $300M_{\odot }$, between SEOBNRv5, IMRPhenomXAS, IMRPhenomT, and the 442 NR simulations used in this work. The colored lines highlight cases with the worst maximum mismatch for each model.

Figure 31

Histogram of the maximum (2,2)-mode mismatch over a range of total masses between 10 and $300M_{\odot }$, between SEOBNRv5, IMRPhenomXAS, IMRPhenomT, and the 442 NR simulations used in this work. The NR error is estimated by computing the mismatch between NR simulations with the highest and second-highest resolutions. The vertical dashed lines show the medians.

Figure 32

The sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$, over a range of total masses between 20 and $300M_{\odot }$ between SEOBNRv5HM, IMRPhenomXHM, IMRPhenomTHM and the 441 SXS NR simulations used in this work. The colored lines highlight cases with the worst maximum mismatch for each model. This comparison highlights the similarity in performance of the time-domain models SEOBNRv5HM and IMRPhenomTHM.

Figure 33

Histogram of the maximum sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$, over a range of total masses between 20 and $300M_{\odot }$, between SEOBNRv5HM, IMRPhenomXMH, IMRPhenomTHM and the 441 SXS NR simulations used in this work. The NR error is estimated by computing the mismatch between NR simulations with the highest and second-highest resolutions. The vertical dashed lines show the medians.

Figure 34

Sky-and-polarization averaged, SNR-weighted mismatch, for inclination $\iota =\pi /3$ as a function of the total mass between SEOBNRv5HM, IMRPhenomTHM and the NR simulation BFI:ExtremeAligned:0003, with $q=10.0,{\chi }_1=0.8,{\chi }_2=0.5$, using ${\ell }_{\max }=5$ and ${\ell }_{\max }=4$, in both the models and the NR waveform.