Large-Eccentricity Asymptotics and Fast Analytic Approximation for Fourier modes of Post-Newtonian Eccentric Waveforms

In this work, we developed analytic asymptotic methods for computing the Fourier modes of gravitational waves from post-Newtonian binary systems in the quasi-Keplerian parametrization in the high eccentricity regime. We have also derived the large-eccentricity asymptotic expansion of the eccentricity enhancement function appearing in the tail contributions to the radiation. Furthermore, based on these results, we constructed an endpoint-constrained analytic approximation that significantly accelerate the computation of the Fourier modes at large eccentricity.The overall error of this analytic approximation is controlled within $10^{-3}$, and it remains valid for Fourier modes with $p\le200$. This approach provides an analytic building blocks for modeling frequency-domain gravitational wave from highly eccentric binaries.