A Rosetta Stone for eccentric gravitational waveform models

Orbital eccentricity is a key signature of dynamical binary black hole formation. The gravitational waves from a coalescing binary contain information about its orbital eccentricity, which may be measured if the binary retains sufficient eccentricity near merger. Dedicated waveforms are required to measure eccentricity. Several models have been put forward, and show good agreement with numerical relativity at the level of a few percent or better. However, there are multiple ways to define eccentricity for inspiralling systems, and different models internally use different definitions of eccentricity, making it difficult to directly compare eccentricity measurements. In this work, we systematically compare two eccentric waveform models, $\texttt{SEOBNRE}$ and $\texttt{TEOBResumS}$, by developing a framework to translate between different definitions of eccentricity. This mapping is constructed by minimizing the relative mismatch between the two models over eccentricity and reference frequency, before evolving the eccentricity of one model to the same reference frequency as the other model. We show that for a given value of eccentricity passed to $\texttt{SEOBNRE}$, one must input a $20$-$50\%$ smaller value of eccentricity to $\texttt{TEOBResumS}$ in order to obtain a waveform with the same empirical eccentricity. We verify this mapping by repeating our analysis for eccentric numerical relativity simulations, demonstrating that $\texttt{TEOBResumS}$ reports a correspondingly smaller value of eccentricity than $\texttt{SEOBNRE}$.