Characterizing the effect of eccentricity on the dynamics of binary black hole mergers in numerical relativity

Many articles have partially studied the configuration of eccentric orbital binary black hole (BBH) mergers. However, there is a scarcity of systematic and comprehensive research on the effect of eccentricity on BBH dynamics. Thanks to the rich and numerous numerical relativistic simulations of eccentric orbital BBH mergers from Rochester Institute of Technology catalog, this paper aims to investigate the impact of initial eccentricity $e_0$ on various dynamic quantities such as merger time $T_{\text{merger}}$, peak luminosity $L_{\mathrm{peak}}$ of gravitational waves, recoil velocity $V_f$, mass $M_f$, and spin ${\alpha }_f$ of merger remnants. We cover configurations of no spin, spin alignment, and spin precession, as well as a broad parameter space of mass ratio ranging from $1/32$ to 1 and initial eccentricity from 0 to 1. For nonspinning BBH with an initial coordinate separation of $11.3M$ ($M$ is the total mass of BBH), we make the first discovery of a ubiquitous oscillation in the relationship between dynamic quantities $L_{\mathrm{peak}}$, $V_f$, $M_f$, ${\alpha }_f$, and initial eccentricity $e_0$. Additionally, at $24.6M$, we observe the same oscillation phenomenon in the case of mass ratio $q=1$ but do not see it in other mass ratios, suggesting that this oscillation will be evident in numerical simulations with sufficiently dense initial eccentricity. By associating the integer numbers of the orbital cycle of $N_{\text{orbits}}$ with the peaks and valleys observed in the curves depicting the relationship between the dynamic quantities and the initial eccentricity, we reveal the significant oscillatory behavior attributed to orbital transitions. This discovery sheds light on the presence of additional orbital transitions in eccentric BBH mergers, extending beyond the widely recognized transition from inspiral to plunge. We perform an analysis to understand the different behaviors exhibited by the dynamic quantities and attribute them to variations in the calculation formulas. Furthermore, we demonstrate that finely adjusting the initial eccentricity can lead to the remnant black hole becoming a Schwarzschild black hole in the case of spin alignment. In a comprehensive analysis that surpasses previous studies by encompassing cases of no spin, spin alignment, and spin precession, we reveal consistent variations in the correlation between dynamic quantities and initial eccentricity, regardless of the presence of spin. This discovery underscores the universality of the impact of eccentricity on BBH dynamics and carries profound implications for astrophysical research.

Figure 1

The parameters used in our study include three configurations, no spin, spin alignment, and spin precession, in two initial coordinate separations $11.3M$ and $24.6M$, which cover the parameter space mass ratio $q$ from $1/32$ to 1, and the initial eccentricity $e_0$ from 0 to 1. The left panel uses effective spin ${\chi }_{\mathrm{eff}}$ to label nonspinning and spin-aligned configurations. The right panel marks the spin precession configuration using the effective precession spin parameter ${\chi }_p$.

Figure 2

Variations of dynamical quantities of the merger time $T_{\text{merger}}$ (a), peak luminosity $L_{\mathrm{peak}}$ (c), recoil velocity $V_f$ (b), mass $M_f$ (d), and spin ${\alpha }_f$ (e) of the merger remnants as a function of the initial eccentricity $e_0$ at the initial coordinate separation of $11.3M$ for nonspinning configuration with different mass ratio. Dynamic quantities of BBH mergers for various circular orbits are denoted by black marks “x.” These circular orbits correspond to RIT:BBH:0001 ($q=1$), RIT:BBH:0112 ($q=1$), RIT:BBH:0198 ($q=1$), RIT:BBH:0114 ($q=3/4$), RIT:BBH:0117 ($q=1/2$), and RIT:BBH:0119 ($q=1/4$) in the RIT catalog, respectively. Their initial orbital separations are 9.53, 20.0, 11.0, 11.0, 11.0, and 11.0, respectively.

Figure 3

Variations of dynamical quantities of the merger time $T_{\text{merger}}$ (a), peak luminosity $L_{\mathrm{peak}}$ (c), recoil velocity $V_f$ (b), mass $M_f$ (d), and spin ${\alpha }_f$ (e) of the merger remnants as a function of the initial eccentricity $e_0$ at the initial coordinate separation of $24.6M$ for nonspinning configuration with different mass ratio.

Figure 4

Relationship between the integer orbital cycle number $N_{\text{orbits}}$ and various quantities such as the peak luminosity $L_{\mathrm{peak}}$, recoil velocity $V_f$, the mass $M_f$ and the spin ${\alpha }_f$ of the remnant at initial coordinate separation of $11.3M$ and $24.6M$ for nonspinning configuration with different mass ratios. These points, denoted by red “x” markers, correspond to either an integer or are in close proximity to an integer of the orbital cycle number. Moving from right to left, each red “x” corresponds to successive orbital cycles, starting from cycle 1 and continuing indefinitely. In the case of a mass ratio of $11.3M$ being $q=1$, we illustrate the integer orbital cycle numbers in the figure using numerical values $1,2,3\dots$ (with the exception of the recoil velocity, which is represented by $q=1/4$). The upper four panels correspond to the initial coordinate separation of $11.3M$, and the lower four panels correspond to the initial coordinate separation of $24.6M$.

Figure 5

Increment percentages of $V_f$, $L_{\mathrm{peak}}$, $M_f$, and ${\alpha }_f$ relative to the corresponding circular orbit at initial coordinate separations of $11.3M$ and $24.6M$ for nonspinning configuration with different mass ratios. The upper four panels correspond to the initial coordinate separation of $11.3M$, and the lower four panels correspond to the initial coordinate separation of $24.6M$.

Figure 6

Variations of dynamical quantities of the merger time $T_{\text{merger}}$ (a), peak luminosity $L_{\mathrm{peak}}$ (c), recoil velocity $V_f$ (b), mass $M_f$ (d), and spin ${\alpha }_f$ (e) of the merger remnants as a function of the initial eccentricity $e_0$ at the initial coordinate separation of $24.6M$ for spin aligned configuration with different mass ratios. We mark the position where $N_{\text{orbits}}$ is approximately equal to one orbit with a red “x.” The dashed line in (e) represents the Schwarzschild black hole whose corresponding spin is 0.

Figure 7

Variations of dynamical quantities of the merger time $T_{\text{merger}}$ (a), peak luminosity $L_{\mathrm{peak}}$ (c), recoil velocity $V_f$ (b), mass $M_f$ (d), and spin ${\alpha }_f$ (e) of the merger remnants as a function of the initial eccentricity $e_0$ at the initial coordinate separation of $24.6M$ for spin precession configuration with mass ratio $q=1$.

Figure 8

Recoil velocity $V_f$ (a), peak luminosity $L_{\mathrm{peak}}$ (b), mass $M_f$ (c), and spin ${\alpha }_f$ (d) with error bars based on the typical error estimate provided by RIT with 5% for recoil velocity and peak luminosity and 0.1% for mass and spin. For spin ${\alpha }_f$, an enlarged view is shown in (e) due to its smaller error bar.

Figure 9

Errors of the nearest continuous integer orbital cycle $N_{\text{orbits}}$ compared to the corresponding integer value for the $11.3M$ nonspinning case.

Figure 10

Variations of dynamical quantities of the merger time $T_{\text{merger}}$ (a), peak luminosity $L_{\mathrm{peak}}$ (c), recoil velocity $V_f$ (b), mass $M_f$ (d), and spin ${\alpha }_f$ (e) of the merger remnants as a function of the initial eccentricity $e_{\mathrm{ADM},0}$ at the initial coordinate separation of $11.3M$ for nonspinning configuration. Initial eccentricities are measured under ADM coordinate of 3PN.

Figure 11

Variations of dynamical quantities of the merger time $T_{\text{merger}}$ (a), peak luminosity $L_{\mathrm{peak}}$ (c), recoil velocity $V_f$ (b), mass $M_f$ (d), and spin ${\alpha }_f$ (e) of the merger remnants as a function of the initial eccentricity $e_{\mathrm{harmonic},0}$ at the initial coordinate separation of $11.3M$ for nonspinning configuration. Initial eccentricities are measured under harmonic coordinate of 3PN.

Figure 12

Variations of dynamical quantities of the merger time $T_{\text{merger}}$ (a), peak luminosity $L_{\mathrm{peak}}$ (c), recoil velocity $V_f$ (b), mass $M_f$ (d), and spin ${\alpha }_f$ (e) of the merger remnants as a function of the initial eccentricity at the initial coordinate separation of $11.3M$ for nonspinning configuration of mass ratio $q=1/4$. Initial eccentricities are measured through three methods $e_{\mathrm{ADM},0}$, $e_{\mathrm{harmonic},0}$, $e_{\mathrm{RIT},0}$.

Figure 13

Comparisons of dynamic quantities merger time $T_{\text{merger}}$ (a), recoil velocity $V_f$ (b), peak luminosity $L_{\mathrm{peak}}$ (c), mass $M_f$ (d), and spin ${\alpha }_f$ (e) under initial coordinate separations of $11.3M$ and $24.6M$ (only for mass ratio $q=1$) for the nonspinning configuration under eccentricity measurement methods in ADM coordinates.

Figure 14

Comparisons of dynamic quantities merger time $T_{\text{merger}}$ (a), recoil velocity $V_f$ (b), peak luminosity $L_{\mathrm{peak}}$ (c), mass $M_f$ (d), and spin ${\alpha }_f$ (e) under initial coordinate separations of $11.3M$ and $24.6M$ (only for mass ratio $q=1$) for the nonspinning configuration under eccentricity measurement methods in harmonic coordinates.

Figure 15

Comparisons of dynamic quantities merger time $T_{\text{merger}}$ (a), recoil velocity $V_f$ (b), peak luminosity $L_{\mathrm{peak}}$ (c), mass $M_f$ (d), and spin ${\alpha }_f$ (e) under initial coordinate separations of $11.3M$ and $24.6M$ (only for mass ratio $q=1$) for the nonspinning configuration under eccentricity measurement methods from RIT.