Fourth post-Newtonian effective one-body dynamics
Thibault Damour, Piotr Jaranowski, Gerhard Schäfer
TL;DR
The authors translate the nonlocal-in-time 4PN two-body dynamics into the EOB formalism by splitting the Hamiltonian into local and nonlocal parts, applying an infinite-order reduction to a local action-angle framework, and matching to EOB potentials. The local part yields explicit 4PN inputs for $A(r)$, $\bar{D}(r)$, and $Q(r,p_r)$, while the nonlocal part is encoded as an infinite series in $p_r$ and represented by corresponding $A^{\rm II}$, $\bar{D}^{\rm II}$, and $Q^{\rm II}$ terms. The work further derives logarithmic (4PN/5PN) and half-integral (5.5PN) conservative corrections via an effective-action approach and tail calculations, and uses SF data to constrain linear-in-$\nu$ pieces of $\bar{D}(u)$. The resulting 4PN EOB dynamics align with known results and provide improved insights into strong-field behavior, with implications for gravitational-wave modeling and waveform generation across inspiral-merger phases.
Abstract
The conservative dynamics of gravitationally interacting two-point-mass systems has been recently determined at the fourth post-Newtonian (4PN) approximation [T.Damour, P.Jaranowski, and G.Schäfer, Phys. Rev. D 89, 064058 (2014)], and found to be nonlocal in time. We show how to transcribe this dynamics within the effective one-body (EOB) formalism. To achieve this EOB transcription, we develop a new strategy involving the (infinite-)order-reduction of a nonlocal dynamics to an ordinary action-angle Hamiltonian. Our final, equivalent EOB dynamics comprises two (local) radial potentials, $A(r)$ and $\bar{D}(r)$, and a nongeodesic mass-shell contribution $Q(r,p_r)$ given by an infinite series of even powers of the radial momentum $p_r$. Using an effective action technique, we complete our 4PN-level results by deriving two different, higher-order conservative contributions linked to tail-transported hereditary effects: the 5PN-level EOB logarithmic terms, as well as the 5.5PN-level, half-integral terms. We compare our improved analytical knowledge to previous, numerical gravitational-self-force computation of precession effects.