Data-driven extraction, phenomenology, and modeling of eccentric harmonics in binary black hole merger waveforms

Newtonian and post-Newtonian (PN) calculations suggest that each spherical harmonic mode of the gravitational waveforms (radiation) emitted by eccentric binaries can be further decomposed into several eccentricity-induced modes (indexed by $j=1$ to $j=\ensuremath{\infty}$), referred to as eccentric harmonics. These harmonics exhibit monotonically time-varying amplitudes and instantaneous frequencies, unlike the full eccentric spherical harmonic modes. However, computing or extracting these harmonics is not straightforward in current numerical relativity simulations and eccentric waveform models. To address this, Patterson et al. [Phys. Rev. D 111, 044073 (2025)] have developed a framework to extract the eccentric harmonics directly from effective-one-body formalism waveforms. In this paper, we build on the ideas presented in Patterson et al. and propose a data-driven framework, utilizing singular-value decomposition, that incorporates additional features based on PN intuition to ensure monotonicity in the extracted harmonics. We further demonstrate that the phase (frequency) of these harmonics is simply $j{\ensuremath{\phi}}_{\ensuremath{\lambda}}+{\ensuremath{\phi}}_{\mathrm{ecc}}$ ($j{f}_{\ensuremath{\lambda}}+{f}_{\mathrm{ecc}}$), where ${\ensuremath{\phi}}_{\ensuremath{\lambda}}$ (${f}_{\ensuremath{\lambda}}$) is related to the secular orbital phase (frequency) and ${\ensuremath{\phi}}_{\mathrm{ecc}}$ (${f}_{\mathrm{ecc}}$) is an additional phase (frequency) that only depends on the eccentricity. We also provide simple analytical fits to obtain the harmonics as a function of the mean anomaly. These relations may prove useful in constructing faithful models (such as [gwharmone: First data-driven surrogate for eccentric harmonics in binary black hole merger waveforms (to be published).]) that can be employed in cheap and efficient searches and parameter estimation of eccentric mergers. Our framework is modular and can be extended for any other eccentric waveform models or simulation frameworks. The framework is available through the gwminer package.