# A 3PN Fourier Domain Waveform for Non-Spinning Binaries with Moderate Eccentricity

## TL;DR

This paper develops a 3PN order Fourier domain gravitational waveform model for non-spinning eccentric binaries, improving accuracy and including effects like periastron precession, to aid future gravitational wave data analysis.

## Contribution

It extends previous eccentric waveform models to 3PN order, incorporating higher-order effects and validating the model against numerical simulations.

## Key findings

- Achieves 97-99% match with numerical models in key parameter regions.
- Identifies where eccentric effects significantly impact gravitational wave signals.
- Provides a validated, accurate model suitable for data analysis of eccentric binaries.

## Abstract

While current gravitational wave observations with ground based detectors have been consistent with compact binaries in quasi-circular orbits, eccentric binaries may be detectable by ground-based and space-based instruments in the near future. Eccentricity significantly complicates the gravitational wave signal, and we currently lack fast and accurate models that are valid in the moderate to large eccentricity range. In a previous paper, we built a Fourier domain, eccentric waveform model at leading order in the post-Newtonian approximation, i.e. as an expansion in small velocities and weak fields. Here we extend this model to 3rd post-Newtonian order, incorporating the effects of periastron precession and higher post-Newtonian order effects that qualitatively change the waveform behavior. Our 3PN model combines the stationary phase approximation, a truncated sum of harmonics of combinations of two orbital frequencies (an azimuthal one and a radial one), and a bivariate expansion in the orbital separation and the eccentricity. We validate the model through comparisons with a fully-numerical, time domain post-Newtonian model, and find good agreement (matches between $97\% - 99\%$) in much of the parameter space. We estimate in what regions of parameter space eccentric effects are important by exploring the signal to noise ratio of eccentric corrections. We also examine the effects of higher post-Newtonian order terms in the waveform amplitude, and the agreement between different PN-consistent, numerical, time-domain models. In an effort to guide future improvements to the model, we gauge the error in our 3PN model incurred by the different analytic approximations used to construct it. This model is useful for preliminary data analysis investigations and it could allow for a phenomenological hybrid that incorporates eccentricity into an inspiral-merger-ringdown model.