Effective two-body approach to the hierarchical three-body problem : quadrupole to 1PN

Many binary systems of interest for gravitational-wave astronomy are orbited by a third distant body, which can considerably alter their relativistic dynamics. Precision computations are needed to understand the interplay between relativistic corrections and three-body interactions. We use an effective field theory approach to derive the effective action describing the long-timescale dynamics of hierarchical three-body systems up to 1PN quadrupole order. At this level of approximation, computations are complicated by the backreaction of small oscillations on orbital timescales as well as deviations from the adiabatic approximation. We address these difficulties by eliminating the fast modes through the method of near-identity transformations. This allows us to compute for the first time the complete expression of the 1PN quadrupole cross terms in generic configurations of three-body systems. We numerically integrate the resulting equations of motion and show that 1PN quadrupole terms can affect the long term dynamics of relativistic three-body systems.