Two-body gravitational spin-orbit interaction at linear order in the mass ratio

TL;DR

This paper computes the first-order (linear in mass ratio) spin-orbit interaction for a small spinning body in a circular orbit around a large non-spinning mass, extending analytical reach to eight-and-a-half post-Newtonian order by exploiting gravitational self-force theory and Detweiler’s gauge-invariant first-derivative redshift $|\nabla k|(\Omega)$. The authors derive a gauge-invariant 1SF correction $\delta^{\rm R}(y)$ to $|\nabla k|$, express it in terms of RWZ perturbations, and validate it against strong-field numerical data, predicting a pole-like divergence near the light ring for the associated spin-precession measure $\delta\psi(y)$. They then translate these kinematical results into refined, high-order analytic knowledge of the gauge-invariant gyro-gravitomagnetic ratio $g_{S^*}^{\rm eff}$ entering the EOB spin-orbit sector, and provide accurate analytic fits, including a robust 1SF fit $g_{S^*}^{1\rm SF,fit}(u)$ with a zero near $u\approx0.2833$. Overall, the work strengthens the analytic modeling of spinning binaries in the strong-field regime and offers concrete tools to improve gravitational-wave templates through the EOB framework.

Abstract

We analytically compute, to linear order in the mass-ratio, the "geodetic" spin precession frequency of a small spinning body orbiting a large (non-spinning) body to the eight-and-a-half post-Newtonian order, thereby extending previous analytical knowledge which was limited to the third post-Newtonian level. These results are obtained applying analytical gravitational self-force theory to the first-derivative level generalization of Detweiler's gauge-invariant redshift variable. We compare our analytic results with strong-field numerical data recently obtained by S.~R.~Dolan et al. [Phys.\ Rev.\ D {\bf 89}, 064011 (2014)]. Our new, high-post-Newtonian-order results capture the strong-field features exhibited by the numerical data. We argue that the spin-precession will diverge as $\approx -0.14/(1-3y)$ as the light-ring is approached. We transcribe our kinematical spin-precession results into a corresponding improved analytic knowledge of one of the two (gauge-invariant) effective gyro-gravitomagnetic ratios characterizing spin-orbit couplings within the effective-one-body formalism. We provide simple, accurate analytic fits both for spin-precession and the effective gyro-gravitomagnetic ratio. The latter fit predicts that the linear-in-mass-ratio correction to the gyro-gravitomagnetic ratio changes sign before reaching the light-ring. This strong-field prediction might be important for improving the analytic modeling of coalescing spinning binaries.

Figures (3)

• Figure 1: The quantity $\delta \psi$ either from numerical relativity or from PN theory is plotted as a function of $y\in [0,1/3[$. The boxes are the points taken from the numerical results of Dolan:2013roa while the dotted curve corresponds to the fit (\ref{['5.11']}). The other curves are the PN approximants of order $2\le n \le 8.5$. The curves are distinguishable from their values when $y\ge 0.2$ where they are ordered from top to bottom as follows: 2PN, 3PN, 4PN, 6PN, 5PN, 6.5PN, 7PN, 7.5PN, 8.5PN, 8PN.
• Figure 2: The quantity $g_{S_\ast}^{1 \rm SF}$ either exact from numerical relativity or approximated by using the data fits is plotted as a function of $u\in [0,1/3]$. The used data fits are obtained from our model (\ref{['5.11']}) for $\delta \psi$, and from model $\# 14$ of Ref. Akcay:2012ea for $a_{1\rm SF}(y)$.
• Figure 3: The full function $g_{S_\ast}^{\rm eff,circ}(u; \nu)$ is plotted as a function of $u$, for selected values of $\nu=[0,0.05,0.1,0.15,0.2,0.25]$.