Universal phenomenological relations between spherical harmonic modes in non-precessing eccentric binary black hole merger waveforms and associated eccentricity estimator

Using publicly available numerical relativity (NR) simulations for non-precessing eccentric binary black hole (BBH) mergers, we demonstrate that the eccentricity-induced modulations in the amplitudes and frequencies of different spherical harmonic modes are mutually consistent and can be modeled using a single time series modulation. Based on these phenomenological relations, we provide a framework named gwNRXHME to compute multi-modal eccentric non-precessing waveforms using two inputs: quadrupolar eccentric waveforms, and the corresponding multi-modal quasi-circular non-precessing waveforms. Additionally, we present a simple analytical framework, rooted in post-Newtonian approximations, to characterize eccentricity directly from the modulation parameter. Our method (i) provides smooth time evolution of the eccentricity and (ii) yields meaningful estimates of eccentricity even near merger. Finally, we provide a roadmap for incorporating these modulation parameters and eccentricity estimates into eccentric waveform modeling. Our frameworks are publicly available through the gwModels package (https://github.com/tousifislam/gwModels).