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Improved resummation of post-Newtonian multipolar waveforms from circularized compact binaries

Thibault Damour, Bala R. Iyer, Alessandro Nagar

TL;DR

The paper develops a refined resummation of post-Newtonian multipolar gravitational waveforms for circular compact binaries by moving from an additive to a multiplicative decomposition and introducing ρ_{lm} = f_{lm}^{1/ℓ}. This approach factors the waveform into physically meaningful blocks (effective source, tail, phase, and a resummed modulus) and demonstrates that ρ_{lm} improves convergence, matching exact BH perturbation results in the ν→0 limit and remaining robust as ν grows toward the equal-mass case. Explicit 1PN corrections to odd-parity multipoles are computed, and extensive comparisons show that Padé-resummed ρ_{lm} produce highly accurate waveforms (notably for ρ_{22}) across a range of mass ratios. The method yields small sensitivity to the 4PN EOB parameter a5 and to ν, supporting its potential to generate reliable gravitational-wave templates up to merger for both extreme and comparable mass binaries.

Abstract

We improve and generalize a resummation method of post-Newtonian multipolar waveforms from circular compact binaries introduced in Refs. \cite{Damour:2007xr,Damour:2007yf}. One of the characteristic features of this resummation method is to replace the usual {\it additive} decomposition of the standard post-Newtonian approach by a {\it multiplicative} decomposition of the complex multipolar waveform $h_{\lm}$ into several (physically motivated) factors: (i) the "Newtonian" waveform, (ii) a relativistic correction coming from an "effective source", (iii) leading-order tail effects linked to propagation on a Schwarzschild background, (iv) a residual tail dephasing, and (v) residual relativistic amplitude corrections $f_{\lm}$. We explore here a new route for resumming $f_{\lm}$ based on replacing it by its $\ell$-th root: $ρ_{\lm}=f_{\lm}^{1/\ell}$. In the extreme-mass-ratio case, this resummation procedure results in a much better agreement between analytical and numerical waveforms than when using standard post-Newtonian approximants. We then show that our best approximants behave in a robust and continuous manner as we "deform" them by increasing the symmetric mass ratio $ν\equiv m_1 m_2/(m_1+m_2)^2$ from 0 (extreme-mass-ratio case) to 1/4 (equal-mass case). The present paper also completes our knowledge of the first post-Newtonian corrections to multipole moments by computing ready-to-use explicit expressions for the first post-Newtonian contributions to the odd-parity (current) multipoles.

Improved resummation of post-Newtonian multipolar waveforms from circularized compact binaries

TL;DR

The paper develops a refined resummation of post-Newtonian multipolar gravitational waveforms for circular compact binaries by moving from an additive to a multiplicative decomposition and introducing ρ_{lm} = f_{lm}^{1/ℓ}. This approach factors the waveform into physically meaningful blocks (effective source, tail, phase, and a resummed modulus) and demonstrates that ρ_{lm} improves convergence, matching exact BH perturbation results in the ν→0 limit and remaining robust as ν grows toward the equal-mass case. Explicit 1PN corrections to odd-parity multipoles are computed, and extensive comparisons show that Padé-resummed ρ_{lm} produce highly accurate waveforms (notably for ρ_{22}) across a range of mass ratios. The method yields small sensitivity to the 4PN EOB parameter a5 and to ν, supporting its potential to generate reliable gravitational-wave templates up to merger for both extreme and comparable mass binaries.

Abstract

We improve and generalize a resummation method of post-Newtonian multipolar waveforms from circular compact binaries introduced in Refs. \cite{Damour:2007xr,Damour:2007yf}. One of the characteristic features of this resummation method is to replace the usual {\it additive} decomposition of the standard post-Newtonian approach by a {\it multiplicative} decomposition of the complex multipolar waveform into several (physically motivated) factors: (i) the "Newtonian" waveform, (ii) a relativistic correction coming from an "effective source", (iii) leading-order tail effects linked to propagation on a Schwarzschild background, (iv) a residual tail dephasing, and (v) residual relativistic amplitude corrections . We explore here a new route for resumming based on replacing it by its -th root: . In the extreme-mass-ratio case, this resummation procedure results in a much better agreement between analytical and numerical waveforms than when using standard post-Newtonian approximants. We then show that our best approximants behave in a robust and continuous manner as we "deform" them by increasing the symmetric mass ratio from 0 (extreme-mass-ratio case) to 1/4 (equal-mass case). The present paper also completes our knowledge of the first post-Newtonian corrections to multipole moments by computing ready-to-use explicit expressions for the first post-Newtonian contributions to the odd-parity (current) multipoles.

Paper Structure

This paper contains 22 sections, 71 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Defining the building blocks of the resummation of the multipolar gravitational waveform
  3. The Newtonian multipolar waveform
  4. The first three factors in the multiplicative decomposition of the PN fractional correction $\hat{h}_{{\ell m}}^{(\epsilon)}$
  5. The fourth factor in the multiplicative decomposition of the PN fractional correction $\hat{h}_{{\ell m}}^{(\epsilon)}$
  6. Resumming the modulus factor $f_{{\ell m}}$
  7. Results for the extreme-mass-ratio case ($\nu=0$)
  8. Extracting the $\rho_{{\ell m}}^{\text{Exact}}$ multipoles from black-hole perturbation numerical data
  9. Finding structure in the $\rho_{{\ell m}}^{\text{Exact}}$ multipoles extracted from numerical data
  10. Comparing Taylor and Padé approximants of $\rho_{22}$
  11. Comparing resummed waveforms to Taylor-expanded and exact ones
  12. Results for the comparable mass case (notably the equal mass case, $\nu=0.25$)
  13. Mild dependence of $\rho_{{\ell m}}$ on $\nu$
  14. Mild sensitivity of $|\hat{h}_{{\ell m}}|$ to $\nu$
  15. Weak dependence of $|\hat{h}_{{\ell m}}(x)|$ on $a_5$.
  16. ...and 7 more sections

Figures (12)

  • Figure 1: Extreme-mass-ratio limit ($\nu=0$). Comparing various resummations of the (Newton-normalized) gravitational wave energy flux: (a) standard Taylor expansion; (b) Padé resummation as proposed in Ref. Damour:1997ub with $v_{\rm pole}=1/\sqrt{3}$; (c) Padé resummation flexing$v_{\rm pole}$ according to the discussion of Sec. II of Ref. Damour:2007yf; (d) new resummation technique based on the $\rho_{{\ell m}}$ functions discussed in this paper.
  • Figure 2: Extreme-mass-ratio limit ($\nu=0$). Complement to panel (d) of Fig. \ref{['fig:vpole']}. Difference between the resummed and exact energy flux, for different approaches to the resummation of the $\rho_{22}$ function. See text for explanations.
  • Figure 3: Extreme-mass-ratio limit ($\nu=0$). The "exact" functions $\rho_{{\ell m}}(x)$ for $0<x<1/6$ extracted from E. Berti's numerical fluxes. Multipoles up to $\ell=6$ are considered. Each panel corresponds to one value of $\ell$ and shows the even-parity partial amplitudes (black online) together with the J-normalized odd-parity ones (red online).
  • Figure 4: Extreme-mass-ratio limit ($\nu=0$). Comparison between the "exact" leading and subleading quadrupolar amplitudes $\rho_{22}$ and $\rho^J_{21}$ and the corresponding 1PN-accurate analytical ones.
  • Figure 5: Extreme-mass-ratio limit ($\nu=0$). Resummation of the function $\rho_{22}(x)$ on the interval $0\leq x\leq 1/3$: comparison between various Taylor and Padé approximants and the "exact" function obtained from (both frequency-domain and time-domain) numerical calculations. The time-domain data points (see Table \ref{['tab:table3']}) are indicated as empty circles.
  • ...and 7 more figures