Effective metric for bound state in an effective-one-body theory based on the third-post-Minkowskian approximation

Abstract The effective-one-body (EOB) framework, originally formulated within the post-Newtonian (PN) expansion, is central to modeling and interpreting gravitational-wave signals. Recent developments have incorporated the post-Minkowskian (PM) expansion into EOB theory. In this work, we focus on the bound state dynamics in the PM approximation up to the third order and construct the core ingredients of EOB: the effective metric. We examine two correspondence strategies, one based on the radial action variable and the other on the precession angle. We demonstrate that the radial action variable provides a consistent correspondence, which is further verified by the correspondence based on the precession angle. Building on these results, we adopt an isotropic gauge with a Schwarzschild-like parametrization to fix the remaining freedom. Within this parametrization, the effective metric coefficients are determined at 3PM order. Our results provide a consistent effective metric for the bound state dynamics.