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The complete non-spinning effective-one-body metric at linear order in the mass ratio

Enrico Barausse, Alessandra Buonanno, Alexandre Le Tiec

TL;DR

The paper derives exact linear-in-$\nu$ corrections to the EOB effective metric for non-spinning binaries by exploiting gravitational self-force data for circular orbits. It provides a closed-form relation between the SF redshift $z_{\rm SF}$ and the EOB potential $A_{\rm SF}$, and uses SF data on the periastron advance to fix $\bar{D}_{\rm SF}$, thereby completing the linear-in-$\nu$ EOB metric. It further extends PN knowledge by determining 4PN–6PN corrections to $A(u)$ and 4PN–5PN corrections to $\bar{D}(u)$ from the 6PN binding energy and SF inputs, yielding explicit coefficients and validating consistency with SF fits and PN expansions. The results demonstrate the value of resumming PN dynamics about the test-particle limit for smoothly bridging the test-mass and equal-mass regimes and lay groundwork for future SF–NR comparisons across parameter space.

Abstract

Using the main result of a companion paper, in which the binding energy of a circular-orbit non-spinning compact binary system is computed at leading-order beyond the test-particle approximation, the exact expression of the effective-one-body (EOB) metric component g^eff_tt is obtained through first order in the mass ratio. Combining these results with the recent gravitational self-force calculation of the periastron advance for circular orbits in the Schwarzschild geometry, the EOB metric component g^eff_rr is also determined at linear order in the mass ratio. These results assume that the mapping between the real and effective Hamiltonians at the second and third post-Newtonian (PN) orders holds at all PN orders. Our findings also confirm the advantage of resumming the PN dynamics around the test-particle limit if the goal is to obtain a flexible model that can smoothly connect the test-mass and equal-mass limits.

The complete non-spinning effective-one-body metric at linear order in the mass ratio

TL;DR

The paper derives exact linear-in- corrections to the EOB effective metric for non-spinning binaries by exploiting gravitational self-force data for circular orbits. It provides a closed-form relation between the SF redshift and the EOB potential , and uses SF data on the periastron advance to fix , thereby completing the linear-in- EOB metric. It further extends PN knowledge by determining 4PN–6PN corrections to and 4PN–5PN corrections to from the 6PN binding energy and SF inputs, yielding explicit coefficients and validating consistency with SF fits and PN expansions. The results demonstrate the value of resumming PN dynamics about the test-particle limit for smoothly bridging the test-mass and equal-mass regimes and lay groundwork for future SF–NR comparisons across parameter space.

Abstract

Using the main result of a companion paper, in which the binding energy of a circular-orbit non-spinning compact binary system is computed at leading-order beyond the test-particle approximation, the exact expression of the effective-one-body (EOB) metric component g^eff_tt is obtained through first order in the mass ratio. Combining these results with the recent gravitational self-force calculation of the periastron advance for circular orbits in the Schwarzschild geometry, the EOB metric component g^eff_rr is also determined at linear order in the mass ratio. These results assume that the mapping between the real and effective Hamiltonians at the second and third post-Newtonian (PN) orders holds at all PN orders. Our findings also confirm the advantage of resumming the PN dynamics around the test-particle limit if the goal is to obtain a flexible model that can smoothly connect the test-mass and equal-mass limits.

Paper Structure

This paper contains 9 sections, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Self-force contributions to the EOB potentials
  3. Self-force contribution to the EOB effective metric potential $A$
  4. Self-force contribution to the EOB effective metric potential $\bar{D}$
  5. High-order post-Newtonian terms in the EOB potentials
  6. High-order post-Newtonian terms in the EOB metric potential $A$
  7. High-order post-Newtonian terms in the EOB metric potential $\bar{D}$
  8. Discussion and conclusions
  9. On the generic structure of the effective-one-body Hamiltonian

Figures (2)

  • Figure 1: The fractional difference between the GSF-accurate (specific) binding energy and the (specific) binding energy calculated at linear order in $\nu$ using standard PN approximants (upper plot), and EOB approximants (lower plot), for $\nu = 0.01$ (left panel) and $\nu = 0.1$ (right panel).
  • Figure 2: Same as in Fig. \ref{['fig:E001-01']}, for an equal-mass binary ($\nu=0.25$).