Spin-dependent two-body interactions from gravitational self-force computations

TL;DR

This work develops a tight link between gravitational self-force corrections to Detweiler's redshift and spin-dependent two-body interactions in the spinning EOB framework. By combining a high-order PN expansion of the GSF redshift around a Kerr black hole with the spinning binary first law, the authors obtain analytic corrections to the EOB potentials $A$ and $G_S$ up to 8.5PN order and $\hat{a}^4$, and they use numerical GSF data to extract strong-field estimates of these corrections, validating the analytic results up to $u \approx 0.2$. A central result is the simple relation $\frac{1}{2} \delta z_1(y,\hat{a}) = \left[ \frac{\delta A(u,\hat{a})}{2 z_1} + p_\phi \hat{a} \delta G_S(u,\hat{a}) \right]_{u = y'(y,\hat{a})} + \nu \mathcal{K}(u,\hat{a})$, which enables translating redshift data into spin-dependent couplings within the EOB model. The study finds that the rescaled spin-orbit correction $\delta G_S^{\rm resc}$ grows from unity at large separations to order a few in the strong-field regime, generally acting to diminish the total spin-orbit coupling, and provides calibrated inputs for refining EOB waveform models in spinning binaries. Overall, the paper significantly advances analytic and numerical understanding of spin-dependent two-body dynamics in general relativity and informs future improvements of gravitational-wave templates.

Abstract

We analytically compute, through the eight-and-a-half post-Newtonian order and the fourth-order in spin, the gravitational self-force correction to Detweiler's gauge invariant redshift function for a small mass in circular orbit around a Kerr black hole. Using the first law of mechanics for black hole binaries with spin [L.~Blanchet, A.~Buonanno and A.~Le Tiec, Phys.\ Rev.\ D {\bf 87}, 024030 (2013)] we transcribe our results into a knowledge of various spin-dependent couplings, as encoded within the spinning effective-one-body model of T.~Damour and A.~Nagar [Phys.\ Rev.\ D {\bf 90}, 044018 (2014)]. We also compare our analytical results to the (corrected) numerical self-force results of A.~G.~Shah, J.~L.~Friedman and T.~S.~Keidl [Phys.\ Rev.\ D {\bf 86}, 084059 (2012)], from which we show how to directly extract physically relevant spin-dependent couplings.

Figures (5)

• Figure 1: Panel (a): The data points and the theoretical predictions for $\delta G_S^{(0)\rm resc}(u)$. Panel (b): The data points and the theoretical predictions for $\delta G_S^{(2)\rm resc}(u)$.
• Figure 2: Panel (a): The data points and the theoretical predictions for $f_A^{(0)\rm resc}(u)$. Panel (b): The data points and the theoretical predictions for $f_A^{(2)\rm resc}(u)$.
• Figure 3: The data points for $f_A^{(2)\rm resc}$, the theoretical PN prediction (dashed curve) and the fit $f_A^{(2)\rm resc, fit}(u)=f_A^{(2)\rm resc, PN}(u)+(c_1+ c_2\ln (u))u^6$ where $c_1=-24303.04$ and $c_2=-11754.74$.[The accuracy of the fit is found to be $2.30 \times 10^{-4}$.]
• Figure 4: The data points for the rescaled quantity $\frac{1}{u^3}\delta G_S^{(4)\rm resc}(u)$ for which no theoretical prediction is available. The solid curve superposed to the points corresponds to a (quadratic) fit by the function $262.1643 u^2+0.2878 u -3.1256$.
• Figure 5: The quantity $\hat{G}_S (u,\nu ,0,0)$ used in Damour-Nagar (dashed curve) Damour:2014sva as well as the present determination (solid curve) are compared in the case $\nu=0.25$.