A comprehensive look into the accuracy of SpEC binary black hole waveforms

TL;DR

The paper introduces three precision metrics to assess SpEC binary black hole waveform accuracy beyond the conventional match: a generalized, frequency-weighted mismatch that emphasizes late-time merger dynamics, a standard mismatch under multiple alignment schemes, and asymmetric per-mode amplitude/phase differences to detect signed, mode-resolved errors. Across the SXS catalog, numerical errors accumulate over evolution, but merger timing is not intrinsically less accurate when properly aligned; mismatches grow with increasing in-plane spin $\chi_p$, highlighting the heightened complexity of precessing systems. The asymmetric amplitude/phase analyses show no global systemic bias, with phase differences typically small and amplitude differences smallest for the dominant $(\ell,m)=(2,2)$ mode, though subdominant modes are more sensitive to numerical noise. Collectively, these metrics offer a comprehensive framework for NR waveform validation applicable to SpEC and other NR codes, with clear guidance on how precession and waveform portion weighting influence accuracy and model building for GW data analysis.

Abstract

Numerical relativity simulations provide a full description of the dynamics of binary systems, including gravitational radiation. The waveforms produced by these simulations have a number of applications in gravitational-wave detection and inference. In this work, we revisit the accuracy of the waveforms produced by the Spectral Einstein Code. Motivated by the wide range of waveform applications, we propose and explore three accuracy metrics between simulation resolutions: (i) the generalized frequency-weighted mismatch, (ii) the relative amplitude difference, and (iii) the phase difference at different times. We find that numerical errors accumulate over the binary evolution, but the error is not intrinsically larger during the latest, more dynamical stages. Studying errors across the parameter space, we identify a positive correlation between both the mismatch and the phase difference with precessing spin, but little correlation with aligned spin or eccentricity. Lastly, amplitude and phases differences are symmetric upon exchanging resolutions across the catalog, suggesting that there is no systematic error.

Figures (10)

• Figure 1: Waveform conditioning demonstrated with the amplitude of the $\ell=m=2$ mode $A^{(2,2)}_i$ for SXS:BBH:1141, which is a quasicircular, aligned-spin simulation with parameters given in Table \ref{['tab:ex_sim_properties']}. The highest (second-highest) resolution is shown in the top (bottom) panel. Dashed vertical lines denote $T_{\rm ref}$, the approximate end of junk radiation for each resolution. The right-most plots show a zoom-in of the early portions of the waveform to highlight the junk radiation of each resolution (black boxes). Dotted vertical lines denote $T_{\rm peak}$, the time of largest amplitude for each resolution. Both times differ between resolutions. When comparing two resolutions, we pick the latest of the two $T_{\rm ref}$'s and the $T_{\rm peak}$ of the highest resolution.
• Figure 2: Demonstration of various time and phase alignment schemes used throughout this work between the highest (blue) and second-highest (orange) resolutions of SXS:BBH:0803. The left column shows the full waveform while the right column zooms in on the merger portion (black boxes). In the top row, we align waveforms by $(\delta t_0, \delta \phi_0)$, obtained by minimizing the normalized $L^2$ norm between resolutions, see Sec. \ref{['sec:mismatches']}. In the second row, we align the waveforms at the beginning of the usable part of the waveforms, $(T_{\rm ref},\phi_{\rm ref})$ (dashed vertical line). In the third row, we align them at the peak, $(T_{\rm peak},\phi_{\rm peak})$ (dashed vertical line). We refer to Sec. \ref{['sec:asymmetric']} for discussion about the alignments for the second and third rows. Each alignment choice trades the amount of (dis)agreement between different parts of the waveform.
• Figure 3: Comparison of alignment prescriptions for SXS:BBH:0803 for $n=3$. The highest resolution, $h_{n=3}^{\rm I}(t)$, is plotted in blue. We also plot the second highest resolution, $h_{n=3}^{\rm II}(t)$, shifted by $(\delta t_0,\delta \phi_0)$ (orange dotted) and by $(\delta t_3,\delta \phi_3)$ (orange dashed). When realigning after frequency-weighting the waveform, the two resolutions have better agreement (blue and orange dashed) than when with the one-time alignment (blue and orange dotted).
• Figure 4: Normalized weighted waveforms and mismatches for three example simulations: the quasicircular spin-aligned SXS:BBH:1141 (top row), the nonspinning eccentric SXS:BBH:1356 (middle row), and the quasicircular spin-precessing SXS:BBH:0803 (bottom). Parameters for each are given in Table \ref{['tab:ex_sim_properties']}. We plot the frequency-weighted strain up to $n=3$, Eq. \ref{['eqn:n_deg_strain']}, for the highest resolution in the left column. Strains are normalized by the peak amplitude, ${\rm max}|h_n(t)|$. Since the mismatch is scale-invariant, this normalization does not affect the mismatch and the visual comparison is faithful. The merger portion of each weighted waveform is shown in the middle column (black box). Generalized mismatches for two alignments, $\overline{ \mathcal{M}}_{n,0}$ (solid lines) and $\overline{ \mathcal{M}}_{n,n}$ (dashed lines), are shown in the right column. As $n$ increases, the inspiral is downweighted compared to the merger, so generalized mismatches for larger $n$ are increasingly targeting the later coalescence stages.
• Figure 5: Generalized mismatches $\overline{ \mathcal{M}}_{n,0}$ (left column) and $\overline{ \mathcal{M}}_{n,n}$ (right column) as a function of the degree of frequency weighting $n$ for all BBH simulations in the SXS Catalog. The top row shows distributions for all simulations with the solid lines connecting the means. The bottom three rows show results for individual simulations (lines) separated into three categories and colored by pertinent parameters: eccentric (second row; colored by $e$), aligned spin (third row; colored by $\chi_{\rm eff}$), and precessing (bottom row; colored by $\chi_p$) simulations. The accuracy of precessing simulations is anticorrelated with $\chi_p$.