All Order Gravitational Wave Tails for Subleading Post-Newtonian Expansion

Abstract

The low-frequency gravitational waveform emitted in a classical scattering process contains an infinite tower of late- and early-time tail terms of the form $u^{-r}(\ln|u|)^{r-1}$, $r \geq 1$. In the Newtonian limit, the coefficients of this entire tower have recently been shown to be universal, i.e. fixed by the asymptotic data of the scattering objects alone. At the retained order in the velocity expansion, these leading-logarithmic tails are determined to all orders in Newton's constant $G$. We extend this result to the first subleading order in the post-Newtonian expansion of the asymptotic states. We consider scattering of an arbitrary number of incoming and outgoing nonrelativistic objects interacting only through gravity at long range, and we derive the complete tower of tail terms at this order. The correction arises from the first subleading term in the retardation expansion and is determined by the asymptotic Newtonian trajectories of the massive particles. Its coefficients are again determined entirely by the asymptotic trajectories and are independent of the details of the interaction. For $2\to2$ scattering, the leading-logarithmic series can be summed in closed form. Our results agree with a relativistic conjecture proposed in earlier work arXiv:2610.07134.