Comparing Effective-One-Body gravitational waveforms to accurate numerical data

TL;DR

The paper tackles the challenge of generating accurate analytical gravitational waveforms for inspiralling and coalescing binary black holes by advancing the Effective-One-Body (EOB) framework with a resummed 3PN-inspiral waveform and two tunable parameters, $a_5$ and $v_{ m pole}$. It first calibrates $v_{ m pole}$ in the small-mass-ratio limit using Padé-resummed flux comparisons to exact results, then extends the approach to the comparable-mass regime by adopting a resummed $h_{22}$ waveform that includes tail resummation and Padé-amplitude improvements, all built on the EOB relative dynamics with a Padé-resummed $A(u)$ potential containing $a_5$. By comparing to a high-accuracy NR simulation of 15 equal-mass orbits, the authors demonstrate phase agreement down to about $10^{-3}$ GW cycles over 30 GW cycles (with tuned $a_5\approx 40$ and $v_{ m pole}\approx 0.5074$) and superior amplitude fidelity relative to non-resummed PN. The results validate the EOB framework as a highly accurate analytic template for gravitational waves, with practical implications for GW detection and parameter estimation, and they illustrate how NR data can be leveraged to refine analytical models and inform template banks.

Abstract

We continue the program of constructing, within the Effective-One-Body (EOB) approach, high accuracy, faithful analytic waveforms describing the gravitational wave signal emitted by inspiralling and coalescing binary black holes (BHs). We present the comparable-mass version of a new, resummed 3PN-accurate EOB quadrupolar waveform recently introduced in the small-mass-ratio limit. We compare the phase and the amplitude of this waveform to the recently published results of a high-accuracy numerical relativity (NR) simulation of 15 orbits of an inspiralling equal-mass binary BHs system performed by the Caltech-Cornell group. We find a remarkable agreement, both in phase and in amplitude, between the new EOB waveform and the published numerical data. More precisely: (i) in the gravitational wave (GW) frequency domain $Mω<0.08$ where the phase of one of the non-resummed ``Taylor approximant'' (T4) waveform matches well with the numerical relativity one, we find that the EOB phase fares as well, while (ii) for higher GW frequencies, $0.08<Mω\lesssim 0.14$, where the TaylorT4 approximant starts to significantly diverge from the NR phase, we show that the EOB phase continues to match well the NR one. We further propose various methods of tuning the two inspiral flexibility parameters, $a_5$ and $v_{\rm pole}$, of the EOB waveform so as to ``best fit'' EOB predictions to numerical data. We find that the maximal dephasing between EOB and NR can then be reduced below $10^{-3}$ GW cycles over the entire span (30 GW cycles) of the simulation. Our resummed EOB amplitude agrees much better with the NR one than any of the previously considered non-resummed, post-Newtonian one.

Figures (7)

• Figure 1: Panel (a) compares the "exact" Newton-normalized flux function $\hat{F}(v)$gr-qc/9505030 to two different Padé resummed, $v^{11}$--accurate analytical flux functions: one using the standard value $v_{\rm pole}=1/\sqrt{3}=0.57735$ and the other one using an "optimized" flexed value $v_{\rm pole}=0.5398$. The bottom part of (a) plots the corresponding differences $\Delta = \hat{F}^{\rm Pade}-\hat{F}^{\rm Exact}$. Panel (b) plots the same quantities, except for the fact that it uses only $v^6$--accurate analytical flux functions.
• Figure 2: Reduced phase-acceleration curves as defined in Eq. (\ref{['eq:aomega']}) (with $M\equiv m_1+m_2=1$). The inset highlights how the "tuned" EOB curve nearly coincides (for $\omega\lesssim 0.08$) with the NR (and T4) curves, while the "non-tuned" EOB one lies slightly below.
• Figure 3: Correlation between $v_{\rm pole}$ and $a_5$ (top panel) obtained by imposing the constraint (\ref{['eq:ratio4']}). The numerical accuracy with which Eq. (\ref{['eq:ratio4']}) is satisfied is displayed in the left-bottom panel. The right-bottom panel displays the extent to which, as $a_5$ varies, the other ratios $\rho_{\omega_m}$, Eq. (\ref{['eq:ratio_rho']}), approximate unity.
• Figure 4: The $L_{\infty}$ norm of the phase difference between EOB (when $v_{\rm pole}$ is correlated to $a_5$ as in Fig. \ref{['fig:fig3']}) and numerical relativity, as defined by Eq. (\ref{['eq:linfinity']}).
• Figure 5: The upper panel compares various phase differences $\Delta^{\omega_m}\phi_{\rm T4X}$ versus time (with $M=1$), $\omega_m$ denoting a matching frequency and the label X being either EOB or NR. The lower panel exhibits the $\omega_4$--matched phase difference between EOB and NR. The flexibility parameters of EOB have been tuned here to $a_5=40$ and $v_{\rm pole}=0.5074$.