Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring

We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass $m_1$ as it moves along an (unstable) circular geodesic orbit between the innermost stable orbit and the light ring of a Schwarzschild black hole of mass $m_2\gg m_1$. More precisely, we construct the function $h_{uu}^{R,L}(x)\equiv h_{\mu \nu }^{R,L}{u}^{\mu }{u}^{\nu }$ (related to Detweiler’s gauge-invariant “redshift” variable), where $h_{\mu \nu }^{R,L}(\propto m_1)$ is the regularized metric perturbation in the Lorenz gauge, $u^{\mu }$ is the four-velocity of $m_1$ in the background Schwarzschild metric of $m_2$, and $x\equiv [G{c}^{-3}(m_1+m_2)\Omega {]}^{2/3}$ is an invariant coordinate constructed from the orbital frequency $\Omega$. In particular, we explore the behavior of $h_{uu}^{R,L}$ just outside the “light ring” at $x=\frac{1}{3}$ (i.e., $r=3G{m}_2/c^2$), where the circular orbit becomes null. Using the recently discovered link between $h_{uu}^{R,L}$ and the piece $a(u)$, linear in the symmetric mass ratio $\nu \equiv m_1{m}_2/(m_1+m_2{)}^2$, of the main radial potential $A(u,\nu )=1-2u+\nu a(u)+O({\nu }^2)$ of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function $a(u)$ over the entire domain $0<u<\frac{1}{3}$ (thereby extending previous results limited to $u\le \frac{1}{5}$). We find that $a(u)$ diverges like $a(u)\approx 0.25(1-3u{)}^{-1/2}$ at the light-ring limit, $u\to (\frac{1}{3}{)}^{-}$, explain the physical origin of this divergent behavior, and discuss its consequences for the EOB formalism. We construct accurate global analytic fits for $a(u)$, valid on the entire domain $0<u<\frac{1}{3}$ (and possibly beyond), and give accurate numerical estimates of the values of $a(u)$ and its first three derivatives at the innermost stable circular orbit $u=\frac{1}{6}$, as well as the associated $O(\nu )$ shift in the frequency of that orbit. In previous work we used GSF data on slightly eccentric orbits to compute a certain linear combination of $a(u)$ and its first two derivatives, involving also the $O(\nu )$ piece of a second EOB radial potential $\overline{D}(u)=1+\nu \overline{d}(u)+O({\nu }^2)$. Combining these results with our present global analytic representation of $a(u)$, we numerically compute $\overline{d}(u)$ on the interval $0<u\le \frac{1}{6}$.

Figure 1

Broadening of the $l$-mode spectrum near the LR, $z=1-3m_2/r=0$. We plot here the (absolute values of the) regularized modes $h_{uu}^l-D_0-\frac{D_2}{L_2}-\frac{D_4}{L_4}$ [see Eq. (29)] for $0\le l\le 80$, for a range of radii on and below the ISCO. Solid curves refer to (top to bottom) $z=0.005,0.025,0.05,0.1,\mathrm{and}0.5$. The dashed lines are arbitrary $\propto l^{-6}$ references. Away from the LR, the regularized modes are expected to fall off at large $l$ with an $\sim l^{-6}$ tail, as is clearly manifest in the case $z=1/2$ (the ISCO, lower curve). As the radius gets closer to the LR, the onset of the $l^{-6}$ tail shifts to larger $l$-values, with the standard tail not developing until around $l\sim 1/z$ (the regularized mode contributions turn from negative to positive around that value of $l$). In the near-LR case $z=1/200$ (upper curve) no transition to a power law is evident below $l=80$. How these data are obtained is described in Sec. 2d.

Figure 2

Our raw numerical data for the Lorenz-gauge quantity $h_{uu}^{R,L}(x)$ (blue data points; the solid line is an interpolation). The numerical values are tabulated, with error bars, in the Appendix. Note in the main plot the orbital radius increases to the left; the locations of the geodesic ISCO ($x=1/6$) and LR ($x=1/3$) are marked with vertical dashed lines. The inset shows the same data (in absolute value) plotted against $z=1-3x$ on a double logarithmic scale (note here the orbital radius increases to the right, and the LR limit is $z\to 0$ asymptotically far to the left). The dashed (magenta) line is a simple power-law model $h_{uu}^{R,L}\sim -\frac{q}{2}{z}^{-3/2}$.

Figure 3

Numerical data for the doubly rescaled function ${\widehat{a}}_E(x)$ [see Eq. (50)]. The solid line is a cubic interpolation of the numerical data points (beads). The inset shows, on a semilogarithmic scale, the relative numerical error in the ${\widehat{a}}_E$ data, computed based on the estimated errors tabulated in the Appendix. Note that the relative error is between ${10}^{-8}$ and ${10}^{-10}$ over most of the domain, and it never exceeds ${10}^{-5}$ (except at a single point, closest to the LR, where it is $\sim 0.1\%$).

Figure 4

Faithfulness of the analytic best-fit model (52), with parameters as given in Table . The left panel shows, on a semilogarithmic scale, the magnitude of the absolute difference between the model and the data; we use here the variable $a_E(x)$ [rather than ${\widehat{a}}_E(x)$], which is the relevant one entering the EOB potential. The right panel shows (now on a linear scale) that same difference divided by the estimated numerical error for each data point. For most data points the model reproduces the data down to the level of our numerical noise.

Figure 5

Extension of our analytic ${\widehat{a}}_E(x)$ models below the LR. The thick (blue) curve shows the behavior of our selected model ($\#14$ in Tables ) over the entire domain $0<x<1$. Other curves, labelled by model numbers from Table , show the behavior of other models for comparison. Shown, from top to bottom at $x=0.8$, are models number 5, 8, 6, 4, 13, 14, 19 and 10.

Figure 6

Performance of the simpler analytic model (54), with the parameters given in Table . The plot shows, on a semilogarithmic scale, the magnitude of the absolute difference between the model and the numerical data for the function $a_E(x)$.

Figure 7

The GSF corrections to the binding energy $e(u(x))$ and angular momentum $j(u(x))$. We show here the rescaled functions ${\widehat{e}}_{SF}(x)$ (lower curve) and ${\widehat{j}}_{SF}(x)$ (upper curve) defined in Eqs. (83, 84). The curves shown represent the analytic functions obtained by inserting our global analytic fit for $a(x)$, Eq. (52), into Eqs. (69, 70).

Figure 8

The $O(\nu )$ EOB function $\overline{d}(x)$. We plot here the quasinumerical data from Table , normalized by the leading-order PN term $6x^2$ for convenience.