Consistent eccentricities for gravitational wave astronomy: Resolving discrepancies between astrophysical simulations and waveform models

Detecting imprints of orbital eccentricity in GW signals promises to shed light on the formation mechanisms of binary black holes. To constrain formation mechanisms, distributions of eccentricity derived from numerical simulations of astrophysical formation channels are compared to the estimates of eccentricity inferred from GW signals. We report that the definition of eccentricity typically used in astrophysical simulations is inconsistent with the one used while modeling GW signals, with the differences mainly arising due to the choice of reference frequency used in both cases. We also posit a prescription for calculating eccentricity from astrophysical simulations by evolving ordinary differential equations obtained from post-Newtonian theory, and using the dominant ($\ell = m =2$) mode's frequency as the reference frequency; this ensures consistency in the definitions. On comparing the existing eccentricities of binaries present in the Cluster Monte Carlo catalog of globular cluster simulations with the eccentricities calculated using the prescription presented here, we find a significant discrepancy at $e \gtrsim 0.2$; this discrepancy becomes worse with increasing eccentricity. We note the implications this discrepancy has for existing studies, and recommend that care be taken when comparing data-driven constraints on eccentricity to expectations from astrophysical formation channels.