Post-Newtonian factorized multipolar waveforms for spinning, non-precessing black-hole binaries

TL;DR

This work generalizes the factorized, resummed, post-Newtonian (PN) multipolar waveforms to spinning, non-precessing black-hole binaries, validating the approach against high-precision Teukolsky-based numerics in the test-particle limit. The authors implement a $\rho$-resummation, $\rho_{lm}=f_{lm}^{1/l}$, for Kerr equatorial orbits and demonstrate substantially improved amplitude and energy-flux accuracy relative to standard Taylor-expanded PN results, especially for higher-order modes where PN information is limited. They explicitly derive Taylor-expanded waveforms, source and tail terms, and the residuals, then extend the framework to generic mass ratios, introducing spin-dependent terms and two prescriptions for mapping the spin to an effective Kerr-like parameter. Comparisons show that the $\rho^f$-resummed amplitudes and fluxes align with numerical results within a few percent for key modes over a broad parameter range, suggesting significant improvements for waveform templates used in LIGO/Virgo and future space-based detectors. The results offer a versatile pathway to accurate, spin-aware gravitational-wave templates and highlight potential calibrations with adjustable parameters to further tighten NR agreement.

Abstract

We generalize the factorized resummation of multipolar waveforms introduced by Damour, Iyer and Nagar to spinning black holes. For a nonspinning test-particle spiraling a Kerr black hole in the equatorial plane, we find that factorized multipolar amplitudes which replace the residual relativistic amplitude f_{l m} with its l-th root, ρ_{l m} = f_{l m}^{1/l}, agree quite well with the numerical amplitudes up to the Kerr-spin value q \leq 0.95 for orbital velocities v \leq 0.4. The numerical amplitudes are computed solving the Teukolsky equation with a spectral code. The agreement for prograde orbits and large spin values of the Kerr black hole can be further improved at high velocities by properly factoring out the lower-order post-Newtonian contributions in ρ_{l m}. The resummation procedure results in a better and systematic agreement between numerical and analytical amplitudes (and energy fluxes) than standard Taylor-expanded post-Newtonian approximants. This is particularly true for higher-order modes, such as (2,1), (3,3), (3,2), and (4,4) for which less spin post-Newtonian terms are known. We also extend the factorized resummation of multipolar amplitudes to generic mass-ratio, non-precessing, spinning black holes. Lastly, in our study we employ new, recently computed, higher-order post-Newtonian terms in several subdominant modes, and compute explicit expressions for the half and one-and-half post-Newtonian contributions to the odd-parity (current) and even-parity (odd) multipoles, respectively. Those results can be used to build more accurate templates for ground-based and space-based gravitational-wave detectors.

Figures (11)

• Figure 1: Hierarchy of the numerically-computed modes $h_{\ell m}$ relative to that of the dominant $h_{22}$ mode. The spin values in the three panels from left to right are $q=0.95$, $0$, $-0.95$, respectively. The $x$-axis ranges between $0$ and $v_{\rm LSO}(q)$.
• Figure 2: We plot the $\rho_{\ell m}$'s extracted from the numerical data as function of $x \equiv v^2$. The upper panels (blue colors) refer to $q = 0.95$, the lower panels (red colors) to $q =-0.95$. The variable $x$ ranges between $0 < x < x_{\rm LSO}(a)$.
• Figure 3: We plot the $\rho_{\ell m}$'s extracted from the numerical data as function of $x \equiv v^2$. The upper panels (blue colors) refer to $q = 0.95$, the lower panels (red colors) to $q =-0.95$. The variable $x$ ranges between $0 < x < x_{\rm LSO}(a)$.
• Figure 4: Numerical and analytical $\rho_{22}$'s as functions of $x=v^2$. The three panels are for spin values $q=0.95, 0$ and $-0.95$. The notation of the analytical $\rho_{22}$ models follows the definition in Sec. \ref{['sec:comparison']}. The $T_{10}[\rho_{22}]$ and $\rho^f_{22}$ lines overlap with each other and in the $q=0$ case they also overlap with the numerical $\rho_{22}$.
• Figure 5: Numerical and analytical $\rho_{33}$'s as functions of $x=v^2$. The three panels are for spin values $q=0.95, 0$ and $-0.95$. The notation of the analytical $\rho_{33}$ models follows the definition in Sec. \ref{['sec:comparison']}. In the antialigned $q=-0.95$ case, the numerical, $T_8[\rho_{33}]$ and $\rho^f_{33}$ lines overlap, while in the nonspinning $q=0$ case, they also overlap with $P^4_4[\rho_{33}]$.