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

We build upon recent work by Antonelli et al. [Phys. Rev. Lett. 125 (2020) 1, 011103] to obtain, within the effective-one-body (EOB) 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 ($\rm DJS$) gauge, where the dependence on the angular momentum of the gyro-gravitomagnetic functions $(G_S,G_{S_*})$ is removed and (ii) to an alternative gauge (called anti-$\rm DJS$ gauge, $\overline{\rm DJS}$) 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 $(G_S,G_{S_*})$. We use these results to extend by one perturbative order, in PN sense, the analytical knowledge of the periastron advance. After performing a suitable factorization and resummation of $(G_S,G_{S_*})$, the $\rm DJS$ and $\overline{\rm DJS}$ performances are compared via various gauge-invariant quantities at the EOB last stable circular orbit. We eventually find some indications that the $\overline{\rm DJS}$ gauge might be advantageous in the description of the inspiral dynamics of circularized binaries.