Tidal effects in the equations of motion of compact binary systems to next-to-next-to-leading post-Newtonian order

As a first step in the computation of the orbital phase evolution of spinless compact binaries including tidal effects up to the next-to-next-to-leading (NNL) order, we obtain the equations of motion of those systems and the associated conserved integrals in harmonic coordinates. The internal structure and finite size effects of the compact objects are described by means of an effective Fokker-type action. Our results, complete to the NNL order, correspond to the second-post-Newtonian (2PN) approximation beyond the leading tidal effect itself, already occurring at the 5PN order. They are parametrized by three polarizability (or deformability) coefficients describing the mass quadrupolar, mass octupolar and current quadrupolar deformations of the objects through tidal interactions. Up to the next-to-leading (NL) order, we recover previous results in the literature; up to the NNL order for quasi-circular orbits, we confirm the known tidal effects in the (PN re-expansion of the) effective-one-body (EOB) Hamiltonian. In a future work, we shall derive the tidal contributions to the gravitational-wave flux up to the NNL order, which is the second step required to find the orbital phase evolution.