Phenomenological relationship between eccentric and quasi-circular orbital binary black hole waveform

Eccentricity, an important parameter of gravitational waves, has been paid more and more attention because it can reflect the dynamics of compact object mergers. Obtaining an accurate and fast gravitational waveform template is of great significance for the estimation of gravitational wave parameters. This paper aims to do an extended study of the phenomenological fitting model proposed by Setyawati and Ohme for adding eccentricity to quasi-circular orbital waveforms. It can be applied to higher eccentricity up to e = 0.4. But the higher the eccentricity, the less the accuracy. For e in [0, 0.1], it gives an overlap of more than 99.99%. For e in [0.1, 0.2], it gives an overlap of more than 99.9%. For e in [0.2, 0.3], it gives an overlap of more than 99%. For e in [0.3, 0.4], it gives an overlap of more than 90%. The reason for these phenomena is that the larger the eccentricity, the larger the deviation of the eccentricity estimator from the cosine function due to the large change in the morphology of the eccentric waveform, and the worse the fitting effect of the model. It can be applied to higher-order modes and gives the same overlap behavior. After adding a shift parameter, it can be applied to spin-aligned or spin-antialigned waveforms. After obtaining spin-precessing effect, it can be applied to the spin-precessing case. In summary, non-spining, spin-aligned, spin-antialigned or spin-precessing waveforms with eccentricity can be constructed from quasi-circular non-spining waveforms by the phenomenological model, which is not only helpful for us to quickly construct phenomenological gravitational wave templates, but also reveals a phenomenological and universal relationship between eccentric waveform and quasi-circular orbital waveform.