High-accuracy numerical simulation of black-hole binaries: Computation of the gravitational-wave energy flux and comparisons with post-Newtonian approximants

TL;DR

This work benchmarks post‑Newtonian waveform models against a long, high‑accuracy numerical relativity simulation of an equal‑mass, non‑spinning black‑hole binary. It computes the gravitational‑wave energy flux $F$, GW frequency derivative, and, via energy balance, the derivative of the center‑of‑mass energy $dE/d\varpi$, then compares these NR results with adiabatic Taylor and Padé PN and non‑adiabatic EOB models. The Padé flux improves closeness to NR but does not accelerate convergence; non‑adiabatic EOB models, especially with non‑Keplerian flux, reproduce the NR flux, $\dot{\varpi}$, and $dE/d\varpi$ more faithfully, and their phase agreement can be dramatically improved by tuning a small set of PN‑order coefficients (e.g., $a_5$, $v_{\rm pole}$, and higher flux terms), reducing phase differences to well below the NR error. These results validate NR as a benchmark for PN template construction and demonstrate that tuned EOB or NR‑informed PN models can yield highly accurate analytic templates for gravitational‑wave data analysis of BBH inspirals.

Abstract

Expressions for the gravitational wave (GW) energy flux and center-of-mass energy of a compact binary are integral building blocks of post-Newtonian (PN) waveforms. In this paper, we compute the GW energy flux and GW frequency derivative from a highly accurate numerical simulation of an equal-mass, non-spinning black hole binary. We also estimate the (derivative of the) center-of-mass energy from the simulation by assuming energy balance. We compare these quantities with the predictions of various PN approximants (adiabatic Taylor and Pade models; non-adiabatic effective-one-body (EOB) models). We find that Pade summation of the energy flux does not accelerate the convergence of the flux series; nevertheless, the Pade flux is markedly closer to the numerical result for the whole range of the simulation (about 30 GW cycles). Taylor and Pade models overestimate the increase in flux and frequency derivative close to merger, whereas EOB models reproduce more faithfully the shape of and are closer to the numerical flux, frequency derivative and derivative of energy. We also compare the GW phase of the numerical simulation with Pade and EOB models. Matching numerical and untuned 3.5 PN order waveforms, we find that the phase difference accumulated until $M ω= 0.1$ is -0.12 radians for Pade approximants, and 0.50 (0.45) radians for an EOB approximant with Keplerian (non-Keplerian) flux. We fit free parameters within the EOB models to minimize the phase difference, and confirm degeneracies among these parameters. By tuning pseudo 4PN order coefficients in the radial potential or in the flux, or, if present, the location of the pole in the flux, we find that the accumulated phase difference can be reduced - if desired - to much less than the estimated numerical phase error (0.02 radians).

Figures (21)

• Figure 1: Some aspects of the numerical simulation. From top panel to bottom: the leading mode $\dot{h}_{22}$; the two next largest modes, $\dot h_{44}$ and $\dot h_{32}$ (smallest); the frequency of $\dot{h}_{22}$ [see Eq. (\ref{['eq:Def-hDotQuants']})].
• Figure 2: Lower panel: Relative difference between flux $F(\varpi)$ computed with 99 different intervals $[t_1,t_2]$ and the average of these. Upper panel: Relative change in the flux $F(\varpi)$ under various changes to the numerical simulation. The grey area in the upper panel indicates the uncertainty due to the choice of integration constants, which is always dominated by numerical error. The dashed line in the upper panel is our final error estimate, which we plot in later figures.
• Figure 3: Contributions of various $(l,m)$-modes to the total numerical gravitational wave flux. Upper panel: plotted as a function of time. Lower panel: Plotted as a function of frequency $M\varpi$. The lower panel also contains the error estimate derived in Fig. \ref{['fig:NumericalFlux']}.
• Figure 4: Lower panel: Difference between frequency derivative $\dot{\varpi}$ computed with 99 different intervals $[t_1, t_2]$ and the average of these. Upper panel: Change in the frequency derivative $\dot{\varpi}$ under various changes to the numerical simulation. The grey area in the upper panel indicates the uncertainty due to choice of integration constants, which dominates the overall uncertainty for low frequencies. The dashed line in the upper panel is our final error estimate, which we plot in later figures.
• Figure 5: Ratio of GW frequencies $\omega$ and $\varpi$ to orbital frequency, as a function of (twice) the orbital frequency, for different PN models. The GW frequencies $\omega$ and $\varpi$ are defined in Eqs. (\ref{['eq:Def-PsiFourQuants']}) and (\ref{['eq:Def-hDotQuants']}). Solid lines correspond to 3.5PN, dashed and dotted lines to 3PN and 2.5PN, respectively.