Complete waveform comparison of post-Newtonian and numerical relativity in eccentric orbits

This study presents a thorough comparative analysis between post-Newtonian (PN) and numerically relativistic (NR) waveforms in eccentric orbits, covering nonspinning and spin-aligned configurations. The comparison examines frequency, amplitude, and phase characteristics of various harmonic modes, including $(\ensuremath{\ell},m)=(2,2),(2,1),(3,3),(3,2),(4,4),(5,5)$ modes. The study utilizes eccentric PN waveforms based on 3PN quasi-Keplerian parametrization with 3PN radiative reaction, surpassing Newtonian quadrupole moment with higher-order moments. NR waveforms from RIT and SXS catalogs span mass ratios from $1/4$ to 1, eccentricities up to 0.45, and durations exceeding $17000M$ across nonspinning and spin-aligned configurations. Focusing on the (2, 2) mode, frequency comparisons between quadrupole and higher-order moments of ${\mathrm{\ensuremath{\Psi}}}_{4}^{22}$ and ${h}^{22}$ were conducted. Amplitude comparisons revealed superior accuracy in quadrupole moments of ${\mathrm{\ensuremath{\Psi}}}_{4}^{22}$. Analysis of total 180 sets of eccentric waveforms showed increasing fitting residuals with rising eccentricity, correlating with smaller mass ratios. Comparisons of initial eccentricity from PN fitting, 3PN quasi-Keplerian parametrization, and RIT/SXS catalogs revealed alignment discrepancies. Frequency, phase, and amplitude comparisons of (2, 2) modes show consistent inspiral behavior between PN and NR, with divergences near merger for nonspinning PN and pre-$200M$ for spin-aligned PN. Average errors of frequency, phase, and amplitude up to $200M$ premerger amplify with increasing eccentricity. Average errors for eccentricities of 0--0.2 are below 3% for frequency, 0.2 for phase, and 6% for amplitude. For eccentricities of 0.2--0.4, errors increase. The higher-order modes demonstrate consistent trends for frequency and phase, and with increased amplitude errors, underscoring the self-consistency of the PN fitting process. Fittings on three RIT eccentric waveforms with low mass ratios highlight deviation between PN and NR for such scenarios. Refinements in PN and NR accuracy, especially at higher orders, and small mass ratios are essential for precise gravitational wave templates in eccentric orbits, reducing systematic errors in parameter estimation and advancing gravitational wave detection.