Tidal effects up to next-to-next-to-leading post-Newtonian order in massless scalar-tensor theories

In this paper, we study the tidal effects in the gravitationally bound two -body system at next -to -next -to leading post -Newtonian order for spinless sources in massless scalar -tensor theories. We compute the conservative dynamics, using both a Fokker Lagrangian approach and effective field theory with the post -Newtonian effective field theory formalism. We also compute the ten conserved quantities at the same next -to -next -to leading order. Finally, we extend our results from simple scalar -tensor theories to Einsteinscalar -Gauss -Bonnet gravity. Such results are important in preparation of the science case of the next generation of gravitational wave detectors.