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The gravitational-wave signal from inspiralling neutron-star--neutron-star(or black-hole--neutron-star) binaries will be influenced by tidal coupling inthe system. An important science goal in the gravitational-wave detection ofthese systems is to obtain information about the equation of state of neutronstar matter via the measurement of the tidal polarizability parameters ofneutron stars. To extract this piece of information will require to haveaccurate analytical descriptions of both the motion and the radiation oftidally interacting binaries. We improve the analytical description of the lateinspiral dynamics by computing the next-to-next-to-leading order relativisticcorrection to the tidal interaction energy. Our calculation is based on aneffective-action approach to tidal interactions, and on its transcriptionwithin the effective-one-body formalism. We find that second-order relativisticeffects (quadratic in the relativistic gravitational potential $u=G(m_1+m_2)/(c^2 r)$) significantly increase the effective tidal polarizability ofneutron stars by a distance-dependent amplification factor of the form $1 +\alpha_1 \, u + \alpha_2 \, u^2 +...$ where, say for an equal-mass binary,$\alpha_1=5/4=1.25$ (as previously known) and $\alpha_2=85/14\simeq6.07143$ (asdetermined here for the first time). We argue that higher-order relativisticeffects will lead to further amplification, and we suggest a Pad\'e-type way ofresumming them. We recommend to test our results by comparingresolution-extrapolated numerical simulations of inspiralling-binary neutronstars to their effective one body description.

Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description