### Gravitational-wave tail effects to quartic non-linear order

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Gravitational-wave tails are due to the backscattering of linear waves onto the space-time curvature generated by the total mass of the matter source. The dominant tails correspond to quadratic non-linear interactions and arise at the one-and-a-half post-Newtonian (1.5PN) order in the gravitational waveform. Also known are the "tails-of-tails", which are cubically non-linear effects appearing at the 3PN order in the waveform. Here we derive still higher non-linear tail effects, namely those associated with quartic non-linear interactions or "tails-of-tails-of-tails", which are shown to arise at the 4.5PN order. As an application we obtain at that order the complete coefficient in the total gravitational-wave energy flux of compact binary systems moving on circular orbits. Our result perfectly agrees with black-hole perturbation calculations in the limit of extreme mass ratio of the two compact objects.