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 time-scale 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 time-scales 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.