Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism

Using a recently presented numerical code for calculating the Lorenz-gauge gravitational self-force (GSF), we compute the $O(m)$ conservative correction to the precession rate of the small-eccentricity orbits of a particle of mass $m$ moving around a Schwarzschild black hole of mass $\mathsf{M}\gg m$. Specifically, we study the gauge-invariant function $\rho (x)$, where $\rho$ is defined as the $O(m)$ part of the dimensionless ratio $({\widehat{\Omega }}_r/{\widehat{\Omega }}_{\phi }{)}^2$ between the squares of the radial and azimuthal frequencies of the orbit, and where $x=[G{c}^{-3}(\mathsf{M}+m){\widehat{\Omega }}_{\phi }{]}^{2/3}$ is a gauge-invariant measure of the dimensionless gravitational potential (mass over radius) associated with the mean circular orbit. Our GSF computation of the function $\rho (x)$ in the interval $0<x\le 1/6$ determines, for the first time, the strong-field behavior of a combination of two of the basic functions entering the effective one-body (EOB) description of the conservative dynamics of binary systems. We show that our results agree well in the weak-field regime (small $x$) with the 3rd post-Newtonian (PN) expansion of the EOB results, and that this agreement is improved when taking into account the analytic values of some of the logarithmic-running terms occurring at higher PN orders. Furthermore, we demonstrate that GSF data give access to higher-order PN terms of $\rho (x)$ and can be used to set useful new constraints on the values of yet-undetermined EOB parameters. Most significantly, we observe that an excellent global representation of $\rho (x)$ can be obtained using a simple “2-point” Padé approximant which combines 3PN knowledge at $x=0$ with GSF information at a single strong-field point (say, $x=1/6$).

Figure 1

Numerical GSF data for $\rho (x)$ compared with various EOB/PN approximations. For clarity, we show $\rho (x)/x^2$ rather than $\rho (x)$ itself, recalling the small-$x$ asymptotic behavior $\rho (x)\propto x^2$. Dots represent the GSF data points shown in Table , and the blue (darkest) line is a simple cubic-spline interpolation of these data. Numerical error bars are too small to show on this scale. Other lines display various analytic PN models ${\rho }^{\mathrm{PN}}$ constructed from Eq. (34); the plots labeled “$n\mathrm{PN}$” show ${\rho }^{\mathrm{PN}}$ through $O(x^n)$, where at 4PN and 5PN only the known, logarithmic terms are included. The inset shows an expansion of the small-$x$ portion of the plot. Recall $x$ is the (dimensionless) gravitational potential, so $x\to 0$ corresponds to $r_0\to \infty$. The $x$ domain here extends to the ISCO location at $x=1/6$.

Figure 2

The fractional difference $|{\rho }^{\mathrm{PN}}-\rho |/\rho$ between the various PN models shown in Fig. 1 and the numerical GSF data, as a function of $x$. (As in Fig. 1, the 4PN and 5PN models include the logarithmic terms only.) Note the logarithmic scale of the vertical axis. The magnitude of the “kink” at the far left end of the 5PN plot is well within the numerical error for the left-most data point (at $x=1/80$).

Figure 3

Numerical GSF data for the 4PN residual quantities ${\Delta }_4(x)$, ${\Delta }_4^{+}(x)$, and ${\Delta }_4^{++}(x)$ [see Eqs. (44) for definitions]. Curves are simple cubic-spline interpolations of the numerical data, and the inset displays the ${\Delta }_4^{+}$ data at a modified aspect ratio for better clarity. Error bars are computed from the estimated numerical errors in $\rho (x)$ indicated in Table . As discussed in the text, these graphs illustrate the consistency of the numerical GSF data with the EOB/PN prediction through 3PN order. Furthermore, they suggest that the GSF shows the correct (analytically predicted) logarithmic running at 4PN and 5PN.

Figure 4

“Signal” amplitudes from various PN contributions to $\rho (x)$ (thin lines), compared with the “noise” amplitude from numerical error (thick red line). For this plot, all unknown PN coefficients through 8PN were crudely estimated by fitting the numerical GSF data to a tentative 8PN model. Each of the signal lines displays the amplitude of the total contribution from a particular PN order, which at 4PN through 6PN also includes a logarithmic term of the form ${\rho }_n{x}^n\ln x$ (at 7PN and 8PN we crudely absorb all possible logarithmic running in an effective term of the form ${\rho }_n{x}^n$). Note the logarithmic scale of the vertical axis. The total 5PN contribution changes its sign around $x=0.0535$, where the (negative) ${\rho }_5^c{x}^5$ term conspires to cancel out the (positive) ${\rho }_5^{\log }{x}^5\ln x$ term. (Note that in this figure the labels $n\mathrm{PN}$ indicate the contribution of an individual PN order, unlike elsewhere in the text where $n\mathrm{PN}$ stands for a partial PN sum.)

Figure 5

Global fit for $\rho (x)$ based on the simple 2-point Padé model (56) [with Eq. (57)]. Thick (blue) points are the numerical GSF data, while the solid (magenta) line shows the 2-point Padé model, which is based solely on four pieces of information: two at $x=0$ (the 2PN and 3PN coefficients) and two at the ISCO [$\rho (1/6)$ and ${\rho }^{\prime }(1/6)$]. For comparison, “broken” curves show various analytic PN approximations: 2PN (dotted), 3PN (dashed), 3PN including the 4PN logarithmic term (dash-dotted line, below the numerical data points; light green online) and 3PN including both 4PN and 5PN logarithmic terms (dash-dotted, top curve; dark blue online). Evidently, a mere knowledge of the GSF (and its derivative) at $x=1/6$ already improves significantly our ability to construct a faithful model of $\rho$ in the strong-field regime.