The influence of the mean anomaly on the dynamical quantities of binary black hole mergers in eccentric orbits

TL;DR

This paper shows that the mean anomaly $l_0$ fundamentally influences the dynamics of eccentric binary black hole mergers, driving oscillations in radiated energy and angular momentum that propagate into remnant mass and spin, as well as recoil and peak luminosity. By combining NR data with 3PN eccentric waveforms, the authors demonstrate that these oscillations persist across inspiral, merger, and ringdown and originate from the initial orbital phase encoded in $l_0$. They introduce the concept of an envelope formed by varying $l_0$ within PN fits to NR waveforms, which bounds the oscillatory behavior and grows with increasing ${e_t}_0$; orbital averaging in PN fluxes removes these $l_0$ effects. The findings imply substantial amplitude deviations (up to tens of percent in some quantities) relative to circular-orbit predictions, highlighting the importance of incorporating mean-anomaly effects in gravitational-wave modeling for accurate parameter estimation and astrophysical inferences. Overall, the work provides a concrete mechanism—the initial mean anomaly—for the observed oscillary structure in eccentric BBH mergers and outlines a path toward more complete waveform templates that account for both eccentricity and mean anomaly.

Abstract

In studies of binary black hole (BBH) mergers in eccentric orbits, the mean anomaly, traditionally regarded as less significant than eccentricity, has been thought to encode only the orbital phase, leading to the assumption that it exerts minimal influence on the dynamics of eccentric mergers. In a previous investigation, we identified consistent oscillations in dynamical quantities peak luminosity $L_{\text{peak}}$, remnant mass $M_{\text{rem}}$, spin $α_{\text{rem}}$, and recoil velocity $V_{\text{rem}}$ in relation to the initial eccentricity $e_0$. These oscillations are associated with integer orbital cycles within a phenomenological framework. In this paper, we aim to explore the underlying physical nature of these oscillations through gravitational waveforms. Our examination of remnant mass and spin reveals that while the initial ADM mass $M_{\mathrm{ADM}}$ and orbital angular momentum $L_0$ exhibit gradual variations with $e_0$, the radiated energy $E_{\text{rad}}$ and angular momentum $L_{\text{rad}}$ display oscillatory patterns akin to those observed in $M_{\text{rem}}$ and $α_{\text{rem}}$. By decomposing the waveforms into three distinct phases inspiral, late inspiral to merger, and ringdown, we demonstrate that these oscillations persist across all phases, suggesting a common origin. Through a comparative analysis of $E_{\text{rad}}$ and $L_{\text{rad}}$ derived from numerical relativity (NR), post-Newtonian (PN) waveforms, and orbital-averaged PN fluxes during the inspiral phase, we identify the initial mean anomaly $l_0$ as the source of the observed oscillations. ...

Figures (13)

• Figure 1: Parameter space of all nonspinning eccentric ($e_0\neq0$) orbital and circular ($e_0=0$) orbital simulations from SXS and RIT, which contains 81 sets of SXS circular orbital simulations, 19 sets of RIT circular orbital simulations, and 316 sets of RIT eccentric orbit simulations. The mass ratio $q$ for the circular orbit simulations ranges from 1/10 to 1, while for the eccentric orbit simulations, it spans from 1/7 to 1. The initial eccentricity $e_0$ varies from 0 to 1 for $q = 1,3/4,1/2,1/4$ with initial distances $D_{\text{ini}}=11.3M$ and $q=1$ with $D_{\text{ini}}=24.6M$. For other mass ratios with $D_{\text{ini}}=24.6M$, we only include cases where the orbital cycle number exceeds 1, excluding those with an orbital cycle number less than 1 for the purposes of this study. The calculation of the orbital cycle number can be found in our previous work Wang:2023vka.
• Figure 2: Relationship between initial ADM mass $M_{\mathrm{ADM}}$ and angular momentum $L_0$ with initial eccentricity $e_0$ in the RIT simulation series $q=1, 3/4,1/2,1/4$ for $D_{\text{ini}}=11.3M$ and $q=1$ for $D_{\text{ini}}=24.6M$. Each point represents a NR simulation, and the lines of the same color are drawn just to highlight the trend of change.
• Figure 3: Variation of radiated energy $E_{\mathrm{rad}}$ and radiated angular momentum $L_{\mathrm{rad}}$ with the initial eccentricity $e_0$ for all oscillating waveforms, which are calculated by $h$ from the initial time $t_0$ to the end time $t_{\text{end}}$. The blue solid line in the figures indicates BBH orbital motion lasting less than one cycle, corresponding to the plunge phase, during which no oscillatory behavior is observed; consequently, this phase is not shown explicitly.
• Figure 4: Relationship between the radiative energy $E_{\mathrm{rad}}$ and angular momentum $L_{\mathrm{rad}}$ and the initial eccentricity $e_0$ in three phases $[t_0,-200M]$ ((a), (d)), $[-200M,0]$ ((b), (e)), $[0,t_{\text{end}}]$ ((c), (f)). Similar to the FIG. \ref{['FIG:3']}, the blue solid line in the figures indicates BBH orbital motion lasting less than one cycle, corresponding to the plunge phase, during which no oscillatory behavior is observed; consequently, this phase is not shown explicitly.
• Figure 5: Comparison of radiative energy $E_{\mathrm{rad}}$ and angular momentum $L_{\mathrm{rad}}$ between panels (a) and (d) in FIG. \ref{['FIG:4']} and FIG. \ref{['FIG:3']}. The former corresponds to the inspiral phase of the waveforms, while the latter covers the entire duration; nevertheless, both display the same oscillatory modes.