Numerical relativity surrogate model with memory effects and post-Newtonian hybridization

Numerical relativity simulations provide the most precise templates for the gravitational waves produced by binary black hole mergers. However, many of these simulations use an incomplete waveform extraction technique—extrapolation—that fails to capture important physics, such as gravitational memory effects. Cauchy-characteristic evolution (CCE), by contrast, is a much more physically accurate extraction procedure that fully evolves Einstein’s equations to future null infinity and accurately captures the expected physics. In this work, we present a new surrogate model, NRHybSur3dq8_CCE, built from CCE waveforms that have been mapped to the post-Newtonian (PN) BMS frame and then hybridized with PN and effective one-body (EOB) waveforms. This model is trained on 102 waveforms with mass ratios $q\le 8$ and aligned spins ${\chi }_{1z},{\chi }_{2z}\in [-0.8,0.8]$. The model spans the entire LIGO-Virgo-KAGRA (LVK) frequency band (with $f_{\mathrm{low}}=20\mathrm{Hz}$) for total masses $M\gtrsim 2.25M_{\odot }$ and includes the $\ell \le 4$ and $(\ell ,m)=(5,5)$ spin-weight $-2$ spherical harmonic modes, but not the (3, 1), (4, 2) or (4, 1) modes. We find that NRHybSur3dq8_CCE can accurately reproduce the training waveforms with mismatches $\lesssim 2\times {10}^{-4}$ for total masses $2.25M_{\odot }\le M\le 300M_{\odot }$ and can, for a modest degree of extrapolation, capably model outside of its training region. Most importantly, unlike previous waveform models, the new surrogate model successfully captures memory effects.

Figure 1

Plus polarization of the strain for a GW150914-like event computed using the previous surrogate NRHybSur3dq8 (left) and the new surrogate NRHybSur3dq8_CCE (right). The exact parameters that are used to calculate these two waveforms are $m_1=36M_{\odot }$, $m_2=29M_{\odot }$, ${\chi }_{1z}=0.32$, ${\chi }_{2z}=-0.44$, $D_L=410\mathrm{Mpc}$, $\iota =\pi /2$, and $\phi =0$. Because we wish to highlight the main difference between these surrogates, we used an inclination angle of $\iota =\pi /2$, i.e., the “edge on” orientation, for which the memory (red) (computed using Eq. (17b) of Ref. [66]) is maximized. Note that the memory roughly scales as ${\sin }^2(\iota )$.

Figure 2

Training set parameters used in the construction of the new surrogate model NRHybSur3dq8_CCE. There are a total of 102 training data points used for NRHybSur3dq8_CCE, which exactly match those of NRHybSur3dq8 minus two points, for which the initial world tube data for CCE was not available. The boundary of the training region is represented with the black rectangle: $1\le q\le 8$ and $-0.8\le {\chi }_{1z}$, ${\chi }_{2z}\le 0.8$.

Figure 3

NR (black), PN (red), and EOB-corrected PN (blue) waveforms for an example simulation with binary parameters $q=8.0$, ${\chi }_{1z}=-0.8$, and ${\chi }_{2z}=-0.8$. The real part of both the (2, 2) and (2, 1) modes are shown in the top and bottom panels. Notice that the EOB-corrected PN waveform is more faithful to the NR waveform than the PN waveform.

Figure 4

Two unique estimates of the hybridization error. $\mathcal{E}[{\mathcal{h}}_{\mathrm{NR}},{\mathcal{h}}_{\mathrm{hyb}}]$, using Eq. (12), computes the error between the hybrid and the NR waveforms in the hybridization window. Meanwhile, $\mathcal{E}[{\mathcal{h}}_{\mathrm{long}-\mathrm{NR}},{\mathcal{h}}_{\mathrm{hyb}}]$ computes the error between the hybrid and the long NR waveforms from an initial time of $t_1=-6000M$ to the end of hybridization window. We also include $\mathcal{E}[{\mathcal{h}}_{\mathrm{NR}}^{\mathrm{higher}-\mathrm{res}},{\mathcal{h}}_{\mathrm{NR}}]$ as a resolution error between the two highest resolution NR waveforms, computed within the hybridization window. $\mathcal{E}[{\mathcal{h}}_{\mathrm{NR}},{\mathcal{h}}_{\mathrm{hyb}}]$ was computed for every one of the 102 training waveforms, while $\mathcal{E}[{\mathcal{h}}_{\mathrm{long}-\mathrm{NR}},{\mathcal{h}}_{\mathrm{hyb}}]$ and $\mathcal{E}[{\mathcal{h}}_{\mathrm{NR}}^{\mathrm{higher}-\mathrm{res}},{\mathcal{h}}_{\mathrm{NR}}]$ were computed for the 37 waveforms for which longer and higher resolution simulations were available. The dashed lines represent the median values.

Figure 5

Frequency-domain mismatches for the NRHybSur3dq8 and NRHybSur3dq8_CCE surrogates when compared to their respective training hybridized waveforms (extrapolated and CCE, respectively). The results for NRHybSur3dq8 are taken from Fig. 6 of Ref. [22] and are plotted for comparison with NRHybSur3dq8_CCE. For both of these surrogate models, we compute leave-five-out errors at several points in the sky of the source frame using all available modes in each model: $\ell \le 4$ and (5, 5), but excluding (3, 1), (4, 2), and (4, 1) for NRHybSur3dq8_CCE and (4, 1) and (4, 0) for NRHybSur3dq8. Left: mismatches computed using a flat noise curve for only the late inspiral part (NR part) of the hybrid waveforms. The dashed vertical line represents the median mismatch value. Right: mismatches computed using the Advanced-LIGO noise curve as a function of the system’s total mass. The solid (dashed) line represents 95th percentile (median) mismatch values.

Figure 6

Errors for the displacement (blue) and spin (orange) memory effects computed by comparing the training waveforms to NRHybSur3dq8_CCE ’s output using Eq. (12). For this result, we show the leave-five-out cross-validation errors (solid line). As a reference, we also include the errors computed between the two highest resolution waveforms (dashed line).

Figure 7

Amplitudes of the most dominant modes of the displacement and spin memory effects when computed from the NR waveform and the surrogate evaluation for the case with the largest errors in Fig. 6. The top and bottom panels show the (2, 0) mode of the displacement null memory and the (3, 0) mode of the spin null memory.

Figure 8

Noise-weighted frequency-domain mismatches between NRHybSur3dq8_CCE and NR simulations that are outside the model’s training region. The numbers in the legend correspond to the NR simulation’s mass ratio $q$, primary spin ${\chi }_{1z}$, and secondary spin ${\chi }_{2z}$. The mismatches that are shown are computed at several points in the sky of the source frame using the Advanced-LIGO noise curve. Each of the solid lines represents the 95th percentile mismatch values.

Figure 9

Noise-weighted frequency-domain mismatches between extrapolated and CCE hybrid waveforms in red. These are the hybrid waveforms that were used to train NRHybSur3dq8 and NRHybSur3dq8_CCE. Apart from this, we also include the mismatches between the memory-corrected extrapolated and CCE waveforms in green. The mismatches that are shown are computed at several points in the sky of the source frame using the Advanced-LIGO noise curve. The solid (dashed) line represents the 95th percentile (median) mismatch values.

Figure 10

An example of the unphysical oscillations that are seen in the amplitude of the (3, 1) mode for one of the CCE training waveforms (blue). As a reference, we also show the extrapolated waveform (orange) for the same simulation and the higher-resolution CCE waveform (green).

Figure 11

Phase and amplitude of the (3, 1) mode for a CCE (blue) and an extrapolated (orange) waveform. We also include the memory contribution (red) to highlight the fact that the CCE waveform does not decay to zero as $t\to \infty$.