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Eccentricity, an important parameter of gravitational waves, has been paidmore and more attention because it can reflect the dynamics of compact objectmergers. Obtaining an accurate and fast gravitational waveform template is ofgreat significance for the estimation of gravitational wave parameters. Thispaper aims to do an extended study of the phenomenological fitting modelproposed by Setyawati and Ohme for adding eccentricity to quasi-circularorbital waveforms. It can be applied to higher eccentricity up to e = 0.4. Butthe higher the eccentricity, the less the accuracy. For e in [0, 0.1], it givesan overlap of more than 99.99%. For e in [0.1, 0.2], it gives an overlap ofmore than 99.9%. For e in [0.2, 0.3], it gives an overlap of more than 99%. Fore in [0.3, 0.4], it gives an overlap of more than 90%. The reason for thesephenomena is that the larger the eccentricity, the larger the deviation of theeccentricity estimator from the cosine function due to the large change in themorphology of the eccentric waveform, and the worse the fitting effect of themodel. It can be applied to higher-order modes and gives the same overlapbehavior. After adding a shift parameter, it can be applied to spin-aligned orspin-antialigned waveforms. After obtaining spin-precessing effect, it can beapplied to the spin-precessing case. In summary, non-spining, spin-aligned,spin-antialigned or spin-precessing waveforms with eccentricity can beconstructed from quasi-circular non-spining waveforms by the phenomenologicalmodel, which is not only helpful for us to quickly construct phenomenologicalgravitational wave templates, but also reveals a phenomenological and universalrelationship between eccentric waveform and quasi-circular orbital waveform.

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