Data-driven extraction and phenomenology of eccentric harmonics in eccentric spinning binary black hole mergers

Newtonian and post-Newtonian (PN) calculations indicate that the phenomenology of eccentric binary black hole (BBH) merger waveforms is significantly more complex than that of their quasi-circular counterparts. Each spherical harmonic mode of the radiation can be further decomposed into several eccentricity-induced components, referred to as eccentric harmonics. Unlike the (cumulative) spherical harmonic modes, these constituent eccentric harmonics exhibit monotonically time-varying amplitudes and frequencies. However, these eccentric harmonics are not directly accessible in numerical relativity (NR) simulations or current eccentric waveform models. Using the recently developed data-driven framework gwMiner, which combines singular value decomposition, input from post-Newtonian theory, and signal processing techniques, we extract eccentric harmonics from eccentric, aligned-spin waveforms for six different spherical harmonic modes: (2,1), (2,2), (3,2), (3,3), (4,3), (4,4). We demonstrate that the phase (frequency) of each eccentric harmonic takes the form $j\,\phi_{\ell,m,\lambda} + \phi_{\ell,m,\rm ecc}$ ($j\,f_{\ell,m,\lambda} + f_{\ell,m,\rm ecc}$), where $\phi_{\ell,m,\lambda}$ ($f_{\ell,m,\lambda}$) corresponds to the secular orbital phase (frequency), and $\phi_{\ell,m,\rm ecc}$ ($f_{\ell,m,\rm ecc}$) is an additional contribution that depends solely on the eccentricity. We further find that $\phi_{\ell,m,\lambda}$ is the same across different spherical harmonic modes $(\ell, m)$, whereas the eccentric correction term $\phi_{\ell,m,\rm ecc}$ scales with $\ell$. Using effective-one-body dynamics, we further show that $\phi_{\ell,m,\lambda}$ is nothing but the relativistic anomaly and $\phi_{\ell,m,\rm ecc}$ is related to the precession advances.