Defining eccentricity for gravitational wave astronomy

Eccentric compact binary mergers are significant scientific targets for current and future gravitational wave observatories. To detect and analyze eccentric signals, there is an increasing effort to develop waveform models, numerical relativity simulations, and parameter estimation frameworks for eccentric binaries. Unfortunately, current models and simulations use different internal parametrizations of eccentricity in the absence of a unique natural definition of eccentricity in general relativity, which can result in incompatible eccentricity measurements. In this paper, we adopt a standardized definition of eccentricity and mean anomaly based solely on waveform quantities and make our implementation publicly available through an easy-to-use Python package, gw_eccentricity. This definition is free of gauge ambiguities, has the correct Newtonian limit, and can be applied as a postprocessing step when comparing eccentricity measurements from different models. This standardization puts all models and simulations on the same footing and enables direct comparisons between eccentricity estimates from gravitational wave observations and astrophysical predictions. We demonstrate the applicability of this definition and the robustness of our implementation for waveforms of different origins, including post-Newtonian theory, effective-one-body, extreme mass ratio inspirals, and numerical relativity simulations. We focus on binaries without spin precession in this work, but possible generalizations to spin-precessing binaries are discussed.