Hybridization of second-order gravitational self-force and numerical relativity waveforms for quasi-circular and non-spinning black hole binaries

TL;DR

This paper develops 2GSF-NR hybrid waveforms for non-spinning, quasi-circular black-hole binaries by stitching a 1PAT1 inspiral from second-order gravitational self-force theory to numerical-relativity merger data. The approach uses 68 NR waveforms from the SXS catalog and includes seven modes up to $(5,5)$ to construct inspiral-merger-ringdown hybrids, while analyzing hybridization errors and the impact of subdominant modes. Comparisons with NR surrogate models show mismatches at the level of a few times $10^{-3}$ for higher mass ratios and up to $\sim 10^{-2}$ for equal masses due to 1PAT1 limitations, illustrating both the potential and the current limits of the method. The results indicate that high-mass-ratio NR simulations can be initiated closer to merger, reducing computational cost, and they provide guidelines for optimal matching-window placement as a function of mass ratio, paving the way for efficient IMR modeling across broad binary parameter space.

Abstract

In the past few decades, the waveform community has made advances in producing waveforms that span the inspiral-merger-ringdown of comparable-mass-ratio black hole binaries using advances in post-Newtonian and numerical relativity (NR) theory along with state-of-the-art gravitational wave models. Current methods in NR have shown progress towards producing stable simulations reaching mass ratios of 1:100; however, the computational cost becomes prohibitively expensive as the mass ratio and the length of the simulation increases. Meanwhile, the gravitational self-force (GSF) community has developed waveform models that not only generate extreme mass ratio inspiral waveforms, but also generate near-equal-mass-ratio waveforms with high fidelity. To assess the limits of both the GSF and NR waveforms and alleviate the computational costs of NR, we present hybridized GSF-NR waveforms for non-spinning binary black hole systems in which GSF provides the inspiral, and NR the merger and ringdown. The hybrid waveforms are generated from a set of 68 non-spinning NR waveforms from the SXS catalogue with mass ratios spanning 1:1 to 1:20 and include the (2,2), (2,1), (3,3), (3,2), (4,4), (4,3), and (5,5) spin-weighted spherical harmonic modes. In this paper, we will highlight a selection of these hybrid waveforms and examine the error in the hybridization procedure. We will investigate the impact of subdominant modes on the accuracy of the hybrid waveforms by performing mismatch comparisons with surrogate models. To address the feasibility of hybridizing GSF inspirals with short, high-mass ratio NR waveforms, thereby alleviating computational costs, we will discuss the relationship between mass ratio and the placement of the matching window, which can be used to predict the necessary and optimal number of NR cycles that contribute to the hybrid waveform.

Figures (11)

• Figure 1: Comparison of NR and 1PAT1 (Sec. \ref{['subsec:1PAT1_model']}) waveforms for an example simulation. The real part of the $(2,2)$ and $(2,1)$ modes are shown in the top and bottom panels. The binary parameter and SXS identifier of the NR waveform are shown at the top of the plot. The waveforms are aligned in time and phase at the reference time of the NR simulation, which has been shifted to $t=0$. The inset shows a close-up of a region near the merger.
• Figure 2: Top: Waveform amplitudes for 1PAT1 (dashed) and NR (solid) for a set of spherical harmonic modes. The binary parameter, $q$, and SXS identifier of the NR waveform are shown at the top of the plot. The waveforms are aligned in time and phase at the reference time of the NR simulation, which has been shifted to $t=0$. We show the modes that have been excluded from the hybridization procedure. Bottom: Relative error in the waveform amplitudes for the excluded modes.
• Figure 3: An example hybrid waveform for a non-spinning $q=20$ simulation. We show the $(2,2)$, $(2,1)$, and $(3,3)$ modes in the top, middle, and bottom rows of the plot. The SXS identifier of the NR waveform is shown at the top of the plot. The vertical dashed red lines represent the matching window. Portions of the waveforms that appear gray are where the 1PAT1 model and NR become indistinguishable. We note that the plot shows the 1PAT1 and NR waveforms after having been transformed into the hybridization frame, and that the time has been shifted such that the peak of the total amplitude occurs at $t=0$.
• Figure 4: An example hybrid waveform for a non-spinning $q=1$ simulation. We show the $(2,2)$, $(3,2)$, and $(4,4)$ modes in the top, middle, and bottom rows of the plot. The SXS identifier of the NR waveform is shown at the top of the plot. The vertical dashed red lines represent the matching window. Portions of the waveforms that appear gray are where the 1PAT1 model and NR become indistinguishable. We note that the plot shows the 1PAT1 and NR waveforms after having been transformed into the hybridization frame, and that the time has been shifted such that the peak of the total amplitude occurs at $t=0$. Moreover, because the waveform is long, the plot for each mode has been divided into three sections: early inspiral, matching region, and merger.
• Figure 5: Estimates of the hybridization errors. The measure $\mathcal{E}[h_{\text{NR}}, h_{\text{hyb}}]$, using Eq. (\ref{['eq:L2_norm_error']}), computes the error between the NR and hybrid waveforms in the matching window. The measure $\mathcal{E}[h_{\text{NR}}^{\text{inspiral}}, h_{\text{hyb}}]$ computes the error between the NR and hybrid waveforms in a time segment with an end time of $t_2$, the end of the matching window, and an initial time of $t_1=t_2-3000\,M$. In addition, we include $\mathcal{E}[h_{\text{NR}}, h_{\text{NR}}^{\text{lower-res}}]$, the resolution error comparing the highest and second-highest resolution NR waveforms within the matching window. While $\mathcal{E}[h_{\text{NR}}, h_{\text{hyb}}]$ was computed for each of the 68 waveforms, $\mathcal{E}[h_{\text{NR}}^{\text{inspiral}}, h_{\text{hyb}}]$ and $\mathcal{E}[h_{\text{NR}}, h_{\text{NR}}^{\text{lower-res}}]$ were computed for the 46 waveforms for which longer and lower-resolution simulations were available; we note that there was a negligble difference in the blue histogram if the additional 22 waveforms were omitted. The histograms are normalized such that the area under each curve when integrated over $\text{log}_{10}(\mathcal{E})$ is 1. The dashed vertical lines represent the median values.