Evolution of precessing binary black holes on eccentric orbits using orbit-averaged evolution equations

TL;DR

This work presents a publicly available code to evolve eccentric, precessing binary black holes (BBHs) using orbit-averaged post-Newtonian equations, enabling backward and forward tracking of spin orientations and eccentricity across large separations. The dynamics combine a 3PN nonspinning orbital evolution with leading spin-orbit and spin-spin corrections, a 2PN spin-precession framework, and spin-induced quadrupole effects specialized to black holes, supplemented by high-accuracy eccentricity enhancement functions. The authors systematically study how eccentricity influences spin morphologies and orbital dynamics, showing nonmonotonic transitions and population-dependent morphology fractions that depend on instantaneous eccentricity. They also quantify the sensitivity of results to the choice of enhancement functions and demonstrate the code's utility for waveform development, formation-channel inferences, and population studies of eccentric, precessing BBHs.

Abstract

The most general bound binary black hole (BBH) system has an eccentric orbit and precessing spins. The detection of such a system with significant eccentricity close to the merger would be a clear signature of dynamical formation. In order to study such systems, it is important to be able to evolve their spins and eccentricity from the larger separations at which the binary formed to the smaller separations at which it is detected, or vice versa. Knowledge of the precessional evolution of the binary's orbital angular momentum can also be used to twist up aligned-spin eccentric waveform models to create a spin-precessing eccentric waveform model. In this paper, we present a new publicly available code to evolve eccentric, precessing BBHs using orbit-averaged post-Newtonian (PN) equations from the literature. The spin-precession dynamics is 2PN accurate, i.e., with the leading spin-orbit and spin-spin corrections. The evolution of orbital parameters (orbital frequency, eccentricity, and periastron precession), which follow the quasi-Keplerian parametrization, is 3PN accurate in the point particle terms and includes the leading order spin-orbit and spin-spin effects. All the spin-spin terms include the quadrupole-monopole interaction. The eccentricity enhancement functions in the fluxes use the high-accuracy hyperasymptotic expansions from Loutrel and Yunes [Classical Quantum Gravity {\bf 34} 044003 (2017)]. We discuss various features of the code and study the evolution of the orbital and spin-precession parameters of eccentric, precessing BBHs. In particular, we study the dependence of the spin morphologies on eccentricity, where we find that the transition point from one spin morphology to another can depend nonmonotonically on eccentricity, and the fraction of binaries in a given morphology at a given point in the evolution of a population depends on the instantaneous eccentricity.

Figures (6)

• Figure 1: This figure depicts various vector quantities and the associated angles used in this study. The inertial frame is given by $(x, y, z)$ and the invariant plane lies on the $xy$ plane of the inertial frame. The coprecessing frame $(x', y', z')$ is defined such that its $x'y'$ plane represents the instantaneous orbital plane of the binary with $z'$-axis aligned with the orbital angular momentum vector $\mathbf{L}$. The angles $\theta_{1,2}^{}$ are the tilt angles of $\mathbf{S}^{}_{1,2}$, while $\phi^{}_{1,2}$ is the angle between the component of $\mathbf{S}^{}_{1,2}$ in the orbital plane and the $x'$-axis. The difference between the in-plane angles is denoted by $\phi^{}_{12}:= \phi^{}_2 - \phi^{}_1 = \psi^{}_{2} - \psi^{}_{1}$ and $\psi^{}_{1,2}$ is the angle between $\boldsymbol{\mathcal{P}}$ and the spin vector $\mathbf{S}^{}_{1,2}$. We do not show $\phi_2$ and $\psi_1$ in the figure to avoid crowding. The longitude of the line of nodes in the invariant plane is denoted by $\Omega$ and the angle $\varpi$ measures the orientation of the line of periastron relative to the $x'$-axis, which coincides with the $x$-axis initially and then evolves due to the spin-induced precession of the orbital plane.
• Figure 2: The figure shows the time evolution of orbital ($\omega$, $k$) and spin ($\cos\theta^{}_{1}$, $\cos\theta^{}_{2}$, $\cos\phi^{}_{12}$, $\cos \psi^{}_{1}$) parameters of binaries with $q=1/1.2$ using the hyperasymptotic enhancement function from Loutrel and Yunes Loutrel:2016cdw. Different colors depict different values of the initial eccentricity $e^{}_0$ at $f^{}_{\rm start}=10$ Hz.
• Figure 3: The figure shows the histogram of absolute differences in spin angles ($\cos\theta^{}_{1}$, $\cos\theta^{}_{2}$, $\cos\phi^{}_{12}$, and $\cos \psi^{}_{1}$) and orbital parameters ($k$ and $e^{}_{t}$) for binaries evolved with different enhancement functions. The top panel shows the differences between evolutions using the hyperasymptotic enhancement functions from Loutrel and Yunes Loutrel:2016cdw, and the $O(e_t^4)$ enhancement functions from Arun et al.Arun:2009mc with their respective highest eccentricity order. The bottom panel compares the hyperasymptotic enhancement functions with the superasymptotic enhancement functions. Different colors depict different values of the initial eccentricity $e^{}_0$ at $f^{}_{\rm start}=10$ Hz. It is evident that higher eccentricities correspond to larger differences in the top panel since the hyperasymptotic enhancement functions are more accurate for large eccentricities than the ones from Arun et al. In the bottom panel, we observe that the differences, especially in $e^{}_{t}$, are larger at smaller initial eccentricities. This is expected, as the hyperasymptotic corrections to the superasymptotic enhancement functions are more important for lower eccentricities.
• Figure 4: This figure illustrates the diversity in evolution of eccentricities in a population of precessing, eccentric binaries from an initial semimajor axis of $1000 M$ to a final semimajor axis of $10 M$. The panels show the eccentricity evolution of different binaries in the population, divided into three different possible scenarios: from top to bottom, monotonically increasing, monotonically decreasing, and nonmonotonic evolution, where the first and third scenarios are only possible due to the $2$PN spin-spin contributions to $\dot{e_t^2}$. The dotted line in each panel denotes the median value of eccentricity at a given separation in that subpopulation of binaries.
• Figure 5: Fraction of binaries in different morphologies as a function of instantaneous eccentricity $e_t^{}$ at a given semimajor axis. The eccentricities plotted are the upper edges of each eccentricity bin in which the fraction of binaries in different morphologies are computed. The green, blue, and red points denote binaries in the C, L$0$, and L$\pi$ morphologies, respectively. The range of $e^{}_t$ values plotted on the horizontal axis decreases dramatically as the semimajor axis decreases. The fractions of binaries in the three morphologies are computed in $50$ eccentricity bins with $\sim 1000$ binaries per bin. The gray shaded regions indicate where the estimated fractions fall below the sampling error of $\sim 0.03$.