On the determination of the last stable orbit for circular general relativistic binaries at the third post-Newtonian approximation

TL;DR

This work analyzes the Last Stable Orbit (LSO) for circular binaries at the third post-Newtonian (3PN) level, focusing on how to extract nonperturbative LSO information from a slowly convergent PN expansion. It compares several resummation strategies based on gauge-invariant invariants, notably E(x), e(x), and j^2(x), and augments them with the effective-one-body (EOB) framework, including a generalized non-geodesic dynamics at 3PN. A key finding is that LSO predictions are highly sensitive to the 3PN regularization ambiguity parameter $\omega_{ ext{static}}$, but can be brought into agreement across methods if $\omega_{ ext{static}} \approx -9$, with the special value $\omega_{ ext{static}}^*=-9.3439...$ playing a prominent role. The authors show that Padé resummation and Shanks transformations improve convergence and stabilize LSO estimates, though the unresolved static ambiguity remains a crucial limitation. Overall, the results bolster the use of resummation (and EOB) approaches for analytic predictions near the LSO, while underscoring the need to resolve the 3PN static regularization to confirm the exact LSO location.

Abstract

We discuss the analytical determination of the location of the Last Stable Orbit (LSO) in circular general relativistic orbits of two point masses. We use several different ``resummation methods'' (including new ones) based on the consideration of gauge-invariant functions, and compare the results they give at the third post-Newtonian (3PN) approximation of general relativity. Our treatment is based on the 3PN Hamiltonian of Jaranowski and Schäfer. One of the new methods we introduce is based on the consideration of the (invariant) function linking the angular momentum and the angular frequency. We also generalize the ``effective one-body'' approach of Buonanno and Damour by introducing a non-minimal (i.e. ``non-geodesic'') effective dynamics at the 3PN level. We find that the location of the LSO sensitively depends on the (currently unknown) value of the dimensionless quantity $\oms$ which parametrizes a certain regularization ambiguity of the 3PN dynamics. We find, however, that all the analytical methods we use numerically agree between themselves if the value of this parameter is $\oms\simeq-9$. This suggests that the correct value of $\oms$ is near -9 (the precise value $\oms^*\equiv-{47/3}+{41/64}π^2=-9.3439...$ seems to play a special role). If this is the case, we then show how to further improve the analytical determination of various LSO quantities by using a ``Shanks'' transformation to accelerate the convergence of the successive (already resummed) PN estimates.

Figures (3)

• Figure 1: Binary-system LSO parameters obtained by means of the $j$-method as functions of the symmetric mass-ratio $\nu$. The reduced binding energy $E_{\rm LSO}$, Eq. (\ref{['eq2.2']}), and the reduced angular momentum $j_{\rm LSO}$, Eq. (\ref{['eq2.17']}), are divided by their "Schwarzschild values": $\left\vert{E_{\rm LSO}^{\rm Schw}}\right\vert=1-\frac{1}{3}\sqrt{8}$ and $j_{\rm LSO}^{\rm Schw}=\sqrt{12}$; the dimensionless orbital frequency $\widehat{\omega}_{\rm LSO}$ is defined in Eq. (\ref{['e08']}). The lines shown in the plots correspond to different PN orders of approximation: 1PN (solid), 2PN (dotted), and 3PN (dashed).
• Figure 2: Equal-mass ($\nu=\frac{1}{4}$) binary systems LSO parameters obtained by means of different methods as functions of the ambiguity parameter $\omega_{\text{static}}$. The reduced binding energy $E_{\rm LSO}$, Eq. (\ref{['eq2.2']}), and the reduced angular momentum $j_{\rm LSO}$, Eq. (\ref{['eq2.17']}), are divided by their "Schwarzschild values": $\left\vert{E_{\rm LSO}^{\rm Schw}}\right\vert=1-\frac{1}{3}\sqrt{8}$ and $j_{\rm LSO}^{\rm Schw}=\sqrt{12}$; the dimensionless orbital frequency $\widehat{\omega}_{\rm LSO}$ is defined in Eq. (\ref{['e08']}). The lines shown in the plots correspond to different methods: $j$-method (solid), effective one-body method (dotted), and $e$-method (dashed). The solid vertical lines correspond to $\omega_{\text{static}}=\omega_{\text{static}}^*$, Eq. (\ref{['omstar']}).
• Figure 3: Equal-mass ($\nu=1/4$) binary systems reduced binding energy $E_{\rm LSO}/\vert{E_{\rm LSO}^{\rm Schw}}\vert$ versus the dimensionless orbital frequency $\widehat{\omega}_{\rm LSO}$, for different methods discussed in our paper. For $j$- and effective one-body methods we have plotted the results at the 1PN level, 2PN level, and $S$-approximants; they are all exhibited in Table \ref{['tab:shanks']}. We have also shown the results obtained in Refs. [8] (labelled by KWW), and by applying the 2PN effective-one-body method to the "Wilson-Mathews" truncation of general relativity (labelled by WM, see Appendix B).