Complete conservative dynamics for inspiralling compact binaries with spins at the fourth post-Newtonian order

TL;DR

This work delivers the complete spin-dependent conservative dynamics of inspiralling compact binaries at the $4$PN order, deriving the NNLO spin-squared potential and the full spin-containing Hamiltonians. Equations of motion for position and spin are obtained from the action, and the Hamiltonian is constructed in the center-of-mass frame with a rigorous reduction of higher-time derivatives. The authors perform a thorough consistency check by constructing conserved integrals that realize the Poincaré algebra up to $4$PN, and they provide gauge-invariant relations among binding energy, angular momentum, and orbital frequency for circular, aligned-spin binaries to this order. The results have direct impact on high-precision gravitational-wave modeling and parameter estimation, especially within the EOB framework, and they align with cross-checks from self-force and scattering-amplitude approaches. Public resources and future extensions to precessing spins are discussed, marking a comprehensive step forward in PN spin dynamics.

Abstract

In this work we complete the spin-dependent conservative dynamics of inspiralling compact binaries at the fourth post-Newtonian order, and in particular the derivation of the next-to-next-to-leading order spin-squared interaction potential. We derive the physical equations of motion of the position and the spin from a direct variation of the action. Further, we derive the quadratic-in-spin Hamiltonians, as well as their expressions in the center-of-mass frame. We construct the conserved integrals of motion, which form the Poincaré algebra. This construction provided a consistency check for the validity of our result, which is crucial in particular in the current absence of another independent derivation of the next-to-next-to-leading order spin-squared interaction. Finally, we provide here the complete gauge-invariant relations among the binding energy, angular momentum, and orbital frequency of an inspiralling binary with generic compact spinning components to the fourth post-Newtonian order. These high post-Newtonian orders, in particular taking into account the spins of the binary constituents, will enable to gain more accurate information on the constituents from even more sensitive gravitational-wave detections to come.