Eccentricity Effects on Modeling Dynamic Quantities and Their Correlations in Binary Black Hole Mergers

In this study, we begin by revisiting the oscillatory behavior of radiative quantities-energy, angular momentum, and linear momentum-linked with initial eccentricities in binary black hole (BBH) mergers. By varying the mean anomaly $l_0$ across the parameter range $[0,2\pi]$ from a post-Newtonian perspective, we establish an envelope that encapsulates the oscillations of these radiative quantities. Our analysis reveals that while the oscillations are influenced by the specific initial condition $l_0$, the effect of eccentricity contributes to the formation of this envelope. Subsequently, we model dynamical quantities such as peak luminosity $L_{\text{peak}}$, remnant mass $M_{\text{rem}}$, spin $\alpha_{\text{rem}}$, and recoil velocity $V_{\text{rem}}$ in circular orbits. Through polynomial modeling, we explore their relationships with mass ratios and correlations. Our results demonstrate the effectiveness of these polynomials in capturing the intricate relationships and correlations among these quantities in circular orbits. Furthermore, we synthesize and analyze dynamical quantities for both circular and eccentric orbits, revealing continuous variations within specific ranges corresponding to distinct mass ratios. These variations are influenced by continuous changes in initial eccentricity and the associated envelope, which can be extrapolated to encompass other mass ratios. By interpolating the maximum and minimum values of these dynamical quantities, we unveil considerably broad domains relative to circular orbits in both orbital and non-orbital BBH mergers. These domains provide robust constraints on the relationships between dynamical quantities, mass ratios, and their correlations. Finally, we discuss the extension of this eccentricity effect to spin alignment and spin precession configurations of BBHs.