Post-Newtonian observables for aligned-spin binaries to sixth order in spin from gravitational self-force and Compton amplitudes

Accurate modeling of compact binaries is essential for gravitational-wave detection and parameter estimation, with spin being an important effect to include in waveform models. In this paper, we derive new post-Newtonian (PN) results for the conservative aligned-spin dynamics at next-to-next-to-leading order for the spin$^3$ and spin$^4$ contributions, in addition to the next-to-leading order (NLO) spin$^5$ and spin$^6$ contributions. One approach we follow is the Tutti Frutti method, which relates PN and gravitational self-force results through the redshift and spin-precession invariants, by making use of the simple dependence of the scattering angle on the symmetric mass ratio. However, an ambiguity arises at the NLO spin$^5$ contribution, due to transcendental functions of the Kerr spin in the redshift; this is also the order at which Compton amplitudes calculations are affected by spurious poles. Therefore, we follow an additional approach to determine the NLO spin$^5$ and spin$^6$ dynamics: using on-shell Compton amplitudes obtained from black hole perturbation theory. The Compton amplitude used in this work is composed of the unambiguous tree-level far-zone part reported in [Phys. Rev. D 109, 084071 (2024)], as well as the full, non-interfering with the far-zone, $\ell=2$ partial wave contributions from the near zone, which are responsible for capturing Kerr finite-size effects. Other results in this paper include deriving the scattering angle of a spinning test body in a Kerr background from a parametrized worldline action, and computing the redshift and spin-precession invariants for eccentric orbits without an eccentricity expansion.