Surrogate model of hybridized numerical relativity binary black hole waveforms

Numerical relativity (NR) simulations provide the most accurate binary black hole gravitational waveforms, but are prohibitively expensive for applications such as parameter estimation. Surrogate models of NR waveforms have been shown to be both fast and accurate. However, NR-based surrogate models are limited by the training waveforms’ length, which is typically about 20 orbits before merger. We remedy this by hybridizing the NR waveforms using both post-Newtonian and effective one-body waveforms for the early inspiral. We present NRHybSur3dq8, a surrogate model for hybridized nonprecessing numerical relativity waveforms, that is valid for the entire LIGO band (starting at 20 Hz) for stellar mass binaries with total masses as low as $2.25M_{\odot }$. We include the $\ell \le 4$ and (5, 5) spin-weighted spherical harmonic modes but not the (4, 1) or (4, 0) modes. This model has been trained against hybridized waveforms based on 104 NR waveforms with mass ratios $q\le 8$, and $|{\chi }_{1z}|,|{\chi }_{2z}|\le 0.8$, where ${\chi }_{1z}$ (${\chi }_{2z}$) is the spin of the heavier (lighter) black hole in the direction of orbital angular momentum. The surrogate reproduces the hybrid waveforms accurately, with mismatches $\lesssim 3\times {10}^{-4}$ over the mass range $2.25M_{\odot }\le M\le 300M_{\odot }$. At high masses ($M\gtrsim 40M_{\odot }$), where the merger and ringdown are more prominent, we show roughly 2 orders of magnitude improvement over existing waveform models. We also show that the surrogate works well even when extrapolated outside its training parameter space range, including at spins as large as 0.998. Finally, we show that this model accurately reproduces the spheroidal-spherical mode mixing present in the NR ringdown signal.

Figure 1

Largest mismatch of the surrogate (over the entire validation set) as a function of number of greedy parameters used to train the PN surrogate. The PN surrogate is seen to converge to the validation waveforms as the size of the training dataset increases.

Figure 2

The parameter space covered by the 104 NR waveforms (circle markers) used in the construction of the surrogate model in Sec. 6. We also show the nine long NR waveforms (square markers) used to test hybridization in Sec. 7b and the eight NR waveforms (triangle markers) used to test extrapolation in Sec. 7c. The axes show the mass ratio and the spin on the heavier BH, while the colors indicate the spin on the lighter BH. The black rectangle indicates the bounds of the training region: $1\le q\le 8,-0.8\le {\chi }_{1z},{\chi }_{2z}\le 0.8$.

Figure 3

NR, PN (Sec. 4a), and EOB-corrected PN (Sec. 4b) waveforms for an example case. We show the (2, 2) and (2, 1) modes. The binary parameters are shown at the top of the plot. The EOB-corrected PN waveform [35, 40] stays faithful to the NR waveform until much later times, compared to the pure PN waveform.

Figure 4

(Left) The real part (top) and frequency (bottom) of the (3, 2) mode using the inertial frame stitching described in Sec. 5e1. The binary parameters are shown on the top of the plot. The vertical red dashed lines indicate the matching region. Note that this plot shows the inspiral and NR waveforms after the time and frame shifts are performed. (Right) Same, but using the amplitude-frequency stitching described in Sec. 5e2. Now we see that the frequency of the hybrid waveform agrees much better with the NR and inspiral data.

Figure 5

An example hybrid waveform used in this work. We show the $\ell =2$ modes of the inspiral, NR, and hybrid waveforms. The binary parameters are shown on the top of the plot. The vertical red dashed lines indicate the matching region. Note that this plot shows the inspiral and NR waveforms after the time and frame shifts are done.

Figure 6

Errors in NRHybSur3dq8 and SEOBNRv4HM when compared against hybrid waveforms. For NRHybSur3dq8, we show out-of-sample errors. Mismatches are computed at several points in the sky of the source frame using all available modes in each waveform: For the hybrid waveforms and NRHybSur3dq8, that is $\ell \le 4$ and (5, 5), but not (4, 1) or (4, 0). For SEOBNRv4HM that is (2, 2),(2, 1),(3, 3),(4, 4), and (5, 5). (Left) Mismatches computed using a flat noise curve, but including only the late inspiral part of the waveforms, starting at $-3500M$ before the peak. Therefore, we are essentially comparing only to the NR part of the hybrid waveforms. For comparison, we also show the NR resolution error, obtained by comparing the two highest available resolutions. The histograms are normalized such that the area under each curve is 1 when integrated over ${\log }_{10}(\mathrm{Mismatch})$. (Right) Mismatches as a function of total mass, computed using the advanced LIGO design sensitivity noise curve. Here we compare against the full hybrid waveforms. The solid (dashed) lines show the 95th percentile (median) mismatch values over points on the sky as well as different hybrid waveforms.

Figure 7

The plus polarization of the waveforms for the cases that result in the largest mismatch for NRHybSur3dq8 (top) and SEOBNRv4HM (bottom) in the left panel of Fig. 6. We also show the corresponding hybrid waveforms (labeled as NR because only the late part is shown). Each waveform is projected using all available modes for that model, along the direction which results in the largest mismatch for NRHybSur3dq8 (SEOBNRv4HM) in the top (bottom) panel. Note that NRHybSur3dq8 is evaluated using trial surrogates that are not trained using these cases. The binary parameters and the direction in the source frame are indicated in the inset text. All waveforms are time shifted such that the peak of the total waveform amplitude occurs at $t=0$ [using all available modes, according to Eq. (38)]. Then the waveform modes are rotated about the $z$ axis such that the orbital phase is zero at $t=-3500M$.

Figure 8

Comparisons between the NRHybSur3dq8 surrogate model and a few NR waveforms of $\sim {10}^{5}M$ in duration. We also show the NR resolution error. The 95th percentile mismatches (over points in the sky) are shown as a function of total mass. (Inset) Indicates the mass ratio and component spins. Mismatches are computed using the advanced LIGO design sensitivity noise curve. To best assess the error introduced by the hybridization procedure we use the same set of modes for the NR waveforms as the surrogate. At low masses, the hybridization errors (red circles) become less reliable measures of accuracy due to the large NR resolution error (black circles) itself. Figure 9 describes a refined comparison to improve the assessment at low masses.

Figure 9

Errors in the NRHybSur3dq8 surrogate model against long NR waveforms, but only looking at segments of length $\mathrm{\Delta }t=5\times {10}^{3}M$ individually. Each point represents one segment that ends at a specified number of orbits before the waveform peak, as plotted on the horizontal axis. Therefore, going from left to right in the figure, we plot segments that start earlier in the inspiral. We also show the NR resolution error in the same segments. (Inset) Indicates the mass ratio and component spins. We show 95th percentile mismatches (over points in the sky), computed using a flat noise curve. We use the same set of modes for the NR waveforms as the surrogate. We find that, in general, the surrogate error is lower than or comparable to the NR resolution error throughout the inspiral.

Figure 10

Errors in NRHybSur3dq8 when evaluated outside its training range. The 95th percentile mismatches (over points in the sky) are shown as a function of total mass for different extrapolated cases. These are computed using the advanced LIGO design sensitivity noise curve. To best assess the error introduced by the extrapolation, we use the same set of modes for the NR waveforms as the surrogate. The labels indicate the mass ratio and component spins ($q$, ${\chi }_{1z}$, ${\chi }_{2z}$). For comparison we reproduce the 95th percentile mismatches for NRHybSur3dq8 within its training range from the right panel of Fig. 6.

Figure 11

Mode mixing between spherical harmonic modes is clearly seen in the ringdown signal of the NR waveform and is accurately reproduced by the surrogate. The absolute values of the Fourier transform of different spherical harmonic modes are shown as solid (dashed) curves for the surrogate (NR). The dotted vertical lines indicate the frequencies of the fundamental QNM overtone of these modes. The component parameters as well as the remnant mass and spin are shown in the text above the figure.

Figure 12

Evaluation cost for NRHybSur3dq8 including the cost of a FFT. We show the cost for evaluating all 11 modes modeled by NRHybSur3dq8, as well as the cost per mode. The FFT cost is included in both of the above but also shown separately. We also show the evaluation cost of SEOBNRv4_ROM, which includes only the (2, 2) mode. The evaluation cost is computed by averaging over 64 points uniformly distributed in the parameter space, $q\le 8$ and $|{\chi }_{1z}|,|{\chi }_{2z}|\le 0.8$.