Fast and Accurate Prediction of Numerical Relativity Waveforms from Binary Black Hole Coalescences Using Surrogate Models

Simulating a binary black hole coalescence by solving Einstein’s equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from nonspinning binary black hole coalescences with mass ratios in [1, 10] and durations corresponding to about 15 orbits before merger. We assess the model’s uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate’s training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic ${}_{-2}{Y}_{\ell m}$ waveform modes resolved by the NR code up to $\ell =8$. We compare our surrogate model to effective one body waveforms from $50M_{\odot }$ to $300M_{\odot }$ for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).

Figure 1

Top: The $+$ polarization (2,2) mode prediction for $q=2$, the surrogate model’s worst prediction over $q$ from a “leave-one-out” surrogate that was not trained with this waveform (see below). Our full surrogate, trained on the entire data set, is more accurate. Bottom: Phase $\delta {\phi }^{2,2}$ and waveform differences between the surrogate and highest resolution (Lev4) spec waveforms. Also shown is the spec numerical truncation error found by comparing the two highest resolution (Lev4 and Lev3) waveforms.

Figure 2

The relative error, $|h_i^{22}-h_{i+1}^{22}|/|h_{i+1}^{22}|$, successive resolutions of spec Lev$i$ for the (2,2) mode of simulation 198 in Table 1. Top: Waveform output as directly given by spec (“Unaligned”). Bottom: “Aligned,” which involves a multimode peak alignment scheme described by Eq. (2) followed by a rotation of the binary around the $z$ axis to align the waveform phases at $t_i=-2750M$. Our surrogate is built from NR waveform data after alignment, and so this measurement of truncation error is the most relevant for surrogate model building.

Figure 3

Numerical truncation errors (black) dominate all other sources of error for the (2,2) mode, except for simulation 180 ($q=1$), where the truncation errors are already very small. For some weaker modes, systematic amplitude oscillations primarily due to eccentricity may become more relevant.

Figure 4

Waveform differences between the two highest spec resolutions (black circles), surrogates built from the two highest spec resolutions (cyan line), the full surrogate and spec (red squares), and leave-one-out trial surrogates and spec (blue triangles). The largest surrogate error is for $q=2$, for which the (2,2) mode is shown in Fig. 1.

Figure 5

Unfaithfulness, from Eq. (4), comparing spec with our surrogate, eobnrv2, and seobnrv2 models using all available $m\ne 0$ modes. Dashed lines show the unfaithfulness for (2,2) modes only. All waveforms are Planck tapered [53] for $t\in [-2750,-2500]M$ and $t\in [50,90]M$. For the full multimodal waveforms, we maximize the unfaithfulness over $\theta$ and $\phi$ for the worst-case scenario. We use the “$+$” polarization, which is nonzero for all $(\theta ,\phi )$. Left: The shaded regions contain all 22 mass ratios, while the dashed lines maximize over mass ratio. The vertical grey line is the minimum total mass ($\approx 115M_{\odot }$) ensuring all (2,2) modes start with $\le 15\mathrm{Hz}$ at the end of the first tapering window. Right: Unfaithfulness for a $115M_{\odot }$ binary.