Conservative self-force correction to the innermost stable circular orbit: Comparison with multiple post-Newtonian-based methods

Barack and Sago [Phys. Rev. Lett. 102, 191101 (2009)] have recently computed the shift of the innermost stable circular orbit (ISCO) of the Schwarzschild spacetime due to the conservative self-force that arises from the finite-mass of an orbiting test-particle. This calculation of the ISCO shift is one of the first concrete results of the self-force program, and provides an exact (fully relativistic) point of comparison with approximate post-Newtonian (PN) computations of the ISCO. Here this exact ISCO shift is compared with nearly all known PN-based methods. These include both “nonresummed” and “resummed” approaches (the latter reproduce the test-particle limit by construction). The best agreement with the exact (Barack-Sago) result is found when the pseudo-4PN coefficient of the effective-one-body (EOB) metric is fit to numerical relativity simulations. However, if one considers uncalibrated methods based only on the currently known 3PN-order conservative dynamics, the best agreement is found from the gauge-invariant ISCO condition of Blanchet and Iyer [Classical Quantum Gravity 20, 755 (2003)], which relies only on the (nonresummed) 3PN equations of motion. This method reproduces the exact test-particle limit without any resummation. A comparison of PN methods with the ISCO in the equal-mass case (computed via sequences of numerical relativity initial-data sets) is also performed. Here a (different) nonresummed method also performs very well (as was previously shown). These results suggest that the EOB approach—while exactly incorporating the conservative test-particle dynamics and having several other important advantages—does not (in the absence of calibration) incorporate conservative self-force effects more accurately than standard PN methods. I also consider how the conservative self-force ISCO shift, combined in some cases with numerical relativity computations of the ISCO, can be used to constrain our knowledge of (1) the EOB effective metric, (2) phenomenological inspiral-merger-ringdown templates, and (3) 4PN- and 5PN-order terms in the PN orbital energy. These constraints could help in constructing better gravitational-wave templates. Lastly, I suggest a new method to calibrate unknown PN terms in inspiral templates using numerical-relativity calculations.

Figure 1

The difference between various PN methods and the renormalized Barack-Sago conservative correction to the ISCO frequency $c_{\Omega }^{\mathrm{ren}}$. We show only the methods from Table  that have reasonable agreement (within $\sim 50\%$) with the exact value for $c_{\Omega }^{\mathrm{ren}}$ [Eq. (2.20)].

Figure 2

ISCO orbital angular frequency as a function of the reduced mass ratio $\eta$. The solid black line in each plot is the renormalized Barack-Sago result (extended to large $\eta$). The abbreviations for each of the methods used are explained in the text. All EOB methods that use Padé approximants for $A(r)$ are shown in the upper-left, and those that use Taylor expansions for $A(r)$ (as well as the logarithmic approach of 94) are in the upper-right. The lower-left plot shows the ISCO computed from different PN orders of the gauge-invariant ISCO condition in Sec. 3b1. The lower-right plot shows results for the $e$-method, $j$-method, and the 3PN hybrid energy function. The arrow near the point (0.25, 0.12) indicates the equal-mass ISCO from quasicircular initial data calculations in 86.