Tidal effects in the total flux and waveform in massless scalar-tensor theories to, respectively, relative 2PN and 1.5PN orders

Within scalar-tensor (ST) theories, neutron stars in binary systems experience tidal deformations caused by both their companion and the scalar field. These deformations are strongly correlated to the star's internal structure and composition. Accurately modeling their imprint on the emitted gravitational waves will be essential for interpreting the high-precision data expected from future detectors and for disentangling potential signatures of modified gravity from those arising due to the properties of neutron star matter. Using the post-Newtonian multipolar-post-Minkowskian formalism adapted to ST theories, and working within the adiabatic approximation, we compute the tidal corrections to the total energy flux, accounting for both gravitational and scalar radiation, and to the waveform phasing, at the next-to-next-to-leading order (NNLO). This corresponds to second post-Newtonian (2PN) order beyond the leading-order dipolar tidal contribution. At this accuracy, three independent types of tidal deformability (scalar, tensorial, and mixed scalar-tensorial) contribute to the signal. We also derive the full waveform amplitude modes, including gravitational and scalar modes, as well as the memory (m=0) ones, to the $\text{N}^{1.5}\text{LO}$ (i.e. to relative 1.5PN order).