Third post-Newtonian effective-one-body Hamiltonian in scalar-tensor and Einstein-scalar-Gauss-Bonnet gravity

We build an effective-one-body (EOB) Hamiltonian at third post-Newtonian(3PN) order in scalar-tensor (ST) and Einstein-scalar-Gauss-Bonnet (ESGB)theories of gravity. The latter is an extension of general relativity thatpredicts scalar hair for black holes. We start from the known two-bodyLagrangian at 3PN order, and use order-reduction methods to construct itsordinary Hamiltonian counterpart. We then reduce the conservative two-bodydynamics to the (non-geodesic) motion of a test particle in an effective metricby means of canonical transformations. The resulting EOB Hamiltonian is amodification of the general relativistic Hamiltonian, and already at 3PN order,it must account for nonlocal-in-time tail contributions. We include the latterbeyond circular orbits and up to sixth order in the binary's orbitaleccentricity. We finally calculate the orbital frequency at the innermoststable circular orbit (ISCO) of binary black holes in the shift-symmetric ESGBmodel. Our work extends F.L. Juli\'e and N. Deruelle [Phys. Rev. D95, 124054(2017)], and it is an essential step towards the accurate modeling ofgravitational waveforms beyond general relativity.