Gravitational spin-orbit coupling through the third-subleading post-Newtonian order: Exploring spin-gauge flexibility

We build upon recent work by Antonelli et al. [Gravitational spin-orbit coupling through third-subleading post-Newtonian order: From first-order self-force to arbitrary mass ratios, Phys. Rev. Lett. 125, 011103 (2020).] to obtain, within the effective-one-body formalism, and for an arbitrary choice of gauge, the third-subleading post-Newtonian (4.5PN) corrections to the spin-orbit conservative dynamics of spin-aligned binaries. This is then specialized to (i) the well-known Damour-Jaranowski-Sch\"afer (DJS) gauge, where the dependence on the angular momentum of the gyrogravitomagnetic functions (GS,G_S*) is removed and (ii) to an alternative gauge (called anti-DJS gauge, DJS̄) that is chosen so as to precisely reproduce the Hamiltonian of a spinning test particle at linear order in the particle spin and keep the full dependence on the radial and angular momentum in (GS,G_S*). We use these results to extend by one perturbative order, in post-Newtonian sense, the analytical knowledge of the periastron advance. After performing a suitable factorization and resummation of (GS,G_S*), the DJS and DJS̄ performances are compared via various gauge-invariant quantities at the effective-one-body last stable circular orbit. We eventually find some indications that the DJS̄ gauge might be advantageous in the description of the inspiral dynamics of circularized binaries.