In the absence of a unique, gauge-independent definition of eccentricity in general relativity, there have been efforts to standardize the definition for gravitational-wave astronomy. Recently, Shaikh et al. proposed a model-independent measurement of eccentricity $e_{\mathrm{gw}}$ from the phase evolution of the dominant mode. Many works use loss functions (LFs) to assign eccentricity to a reference waveform, for instance by fitting a post-Newtonian expression to assign eccentricity to numerical relativity (NR) simulations. Therefore, we ask whether minimizing common LFs on gauge-dependent model parameters, such as the mismatch $\mathcal{M}$ or the $L_2$-norm of the dominant mode ${\mathcal{h}}_{22}$ residuals, for nonprecessing binaries, ensures a sufficient $e_{\mathrm{gw}}$ agreement. We use 10 eccentric NR simulations and the eccentric waveform teobresums-dalí as the parametric model to fit on eccentricity $e_0$ and reference frequency $f_0$. We first show that a minimized mismatch, the $\mathcal{M}\sim {10}^{-3}–{10}^{-2}$ results in better $e_{\mathrm{gw}}$ fractional differences ($\sim 1\%$) than with the minimized ${\mathcal{h}}_{22}$ residuals. Nonetheless, for small eccentricity NR simulations $(e_{\mathrm{gw}}\lesssim {10}^{-2})$, the mismatch can favor quasicircular ($e_0=0$) best-fit models. Thus, with sufficiently long NR simulations, we can include $e_{\mathrm{gw}}$ in the LF. We explain why solely fitting with $e_{\mathrm{gw}}$ constitutes a degenerate problem. To circumvent these limitations, we propose to minimize a convex sum of $\mathcal{M}$ and the $e_{\mathrm{gw}}$ difference to both assign nonzero eccentric values to NR strains and to control the mismatch threshold.

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• Classical black holes
• Gravitational waves
• Numerical relativity