Modeling gravitational wave modes from the inspiral of binaries with arbitrary eccentricity

Eccentric binaries are key targets for current and future gravitational wave (GW) detectors, offering unique insights into the formation and environments of compact binaries. However, accurately and efficiently modeling eccentric waveforms remains challenging, in part due to their complex harmonic structure. In this work, we develop a post-Newtonian (PN) framework to compute the Fourier amplitudes of GWs from the inspiral of eccentric binaries, deriving simple expressions at 1PN order for all relevant (l, m) multipoles, valid for arbitrary eccentricities. We then characterize the GW emission by analyzing the contribution of each (l, m) mode to the strain, its mean frequency, frequency spread, and asymptotic behavior at high frequencies. Additionally, we introduce a method to determine the minimal set of Fourier modes needed to reconstruct the waveform to a given accuracy. Finally, we discuss how our framework can be extended to higher PN orders, obtaining closed-form expressions for the leading-order tail and spin contributions and outlining the steps required to include higher-order corrections. Our results provide both a deeper theoretical understanding of eccentric GW emission and practical tools for developing more accurate and efficient waveform models.