Gravitational self-force corrections to two-body tidal interactions and the effective one-body formalism

TL;DR

This work advances the theoretical modeling of tidal interactions in compact binaries by computing several tidal invariants to 7.5PN order in the linear self-force (1SF) regime and embedding these results into the effective-one-body (EOB) tidal framework. By combining high-precision analytic self-force calculations (via Regge-Wheeler–Zerilli techniques with Mano–Suzuki–Tagasugi expansions) and state-of-the-art numerical SF data, the authors derive explicit expressions for quadrupolar-electric, quadrupolar-magnetic, and octupolar invariants and translate them into dynamical EOB tidal factors. A key finding is a sign change in the X1-linear piece of the quadrupolar-electric tidal factor in the strong-field, accompanied by a proposed compensatory growth of nonlinear-in-mass-ratio terms, with consistent LR behavior explained analytically. The results yield simple, accurate global analytic representations of the strong-field tidal factors, validated against Dolan et al.’s tidal data, and provide a robust framework for improving tidal descriptions in late inspiral and neutron-star merger modeling. These insights enhance gravitational-wave modeling precision and guide future work on higher-order SF effects and cross-validation with numerical relativity.

Abstract

Tidal interactions have a significant influence on the late dynamics of compact binary systems, which constitute the prime targets of the upcoming network of gravitational-wave detectors. We refine the theoretical description of tidal interactions (hitherto known only to the second post-Newtonian level) by extending our recently developed analytic self-force formalism, for extreme mass-ratio binary systems, to the computation of several tidal invariants. Specifically, we compute, to linear order in the mass ratio and to the 7.5$^{\rm th}$ post-Newtonian order, the following tidal invariants: the square and the cube of the gravitoelectric quadrupolar tidal tensor, the square of the gravitomagnetic quadrupolar tidal tensor, and the square of the gravitoelectric octupolar tidal tensor. Our high-accuracy analytic results are compared to recent numerical self-force tidal data by Dolan et al. \cite{Dolan:2014pja}, and, notably, provide an analytic understanding of the light ring asymptotic behavior found by them. We transcribe our kinematical tidal-invariant results in the more dynamically significant effective one-body description of the tidal interaction energy. By combining, in a synergetic manner, analytical and numerical results, we provide simple, accurate analytic representations of the global, strong-field behavior of the gravitoelectric quadrupolar tidal factor. A striking finding is that the linear-in-mass-ratio piece in the latter tidal factor changes sign in the strong-field domain, to become negative (while its previously known second post-Newtonian approximant was always positive). We, however, argue that this will be more than compensated by a probable fast growth, in the strong-field domain, of the nonlinear-in-mass-ratio contributions in the tidal factor.

Figures (5)

• Figure 1: The base-10 logarithm of the Newtonian-rescaled numerical-minus-analytical difference $[\widehat{\lambda}_1^{\rm (E) 1SF}]^{\rm num}-[\widehat{\lambda}_1^{\rm (E) 1SF}]^{\rm 7.5PN}$ versus the base-10 logarithm of $y$. The slanting (red online) solid line indicates the analytical error estimate while the dashed horizontal line (located at $-11$) indicates the (rough) numerical error level as in Ref. Dolan:2014pja.
• Figure 2: The successive PN approximants to the $X_1-$linear piece $\hat{A}_{1}^{(2^+)1\rm SF}(u)$ in the quadrupolar-electric tidal factor are plotted as functions of $u$, starting from the 1PN approximant (straight line) up to the 7.5 PN approximant. The 2PN approximant is a parabola (with upward concavity) close to the 1PN straight line. The higher approximants can be identified by looking at the position of the zero (close to $u=0.12$) as given in Table I below. The boxes indicate the numerical SF data obtained by combining the results of Dolan:2014pja and Akcay:2012ea.
• Figure 3: In panel (a) we compare the plots of three different estimates of the light-ring-rescaled $X_1-$linear piece $\widetilde{A}_1^{(2^+)1\rm SF}(u)$ in the quadrupolar-electric tidal factor, Eq. (\ref{['eq:7.20']}): i) the numerical relativity data points Dolan:2014pja, Akcay:2012ea (boxes); ii) the fitting function $f_{23}(u)$, Eqs. (\ref{['eq:7.24']})-(\ref{['eq:7.27']}) (solid line); and iii) the product of the 7.5PN series for $\hat{A}_1^{(2^+)1\rm SF}(u)$ by $(1-3u)^{7/2}$ (dashed curve). Panel (b) shows (now up to $u=\frac{1}{3}$) the same estimates as in panel (a), except that the fitting function $f_{23}(u)$ is replaced by $f_{24}(u)$. The vertical dashed line indicates the position of the light ring $u=\frac{1}{3}$.
• Figure 4: The full quadrupolar-electric tidal factor $\hat{A}_{1}^{(2^+)}(u; X_1)$ is plotted as a function of the EOB variable $u$, Eq. (\ref{['eq:7.30']}) for the choice of parameters $p=4$ and $X_1=\frac{1}{2}$.
• Figure 5: The 7.5PN-accurate analytic expression for the $X_1-$linear piece $\hat{A}_{1}^{(2^-)1\rm SF}(u)$ in the quadrupolar-magnetic tidal factor is plotted as a function of the EOB variable $u$ (solid line). The numerical data (boxes) are obtained by combining the $1$SF results of Dolan:2014pja for $\Delta U$ and $\lambda^{\rm (B)1SF}$, and model #14 from Akcay:2012ea for the EOB function $a_{1\rm SF}(u)$.