Intermediate States in Chaotic Triple Evolution and Applications to Black Hole Merger Statistics

The gist Three-body interactions exhibit phases of strong chaotic evolution as well as hierarchical motion where one body separates from a binary and follows a hyperbolic or elliptic trajectory around it, and this study re-derives the elliptic outcome distribution using equilibrium statistical mechanics to quantify observably eccentric merger (OEM) GWs in dense star systems

Intermediate States in Chaotic Triple Evolution

The paper investigates the intermediate states of chaotic three-body evolution, which are phases where an inner binary evolves subject to perturbations from a bound but temporarily distant tertiary The derivation of the elliptic outcome distribution involves integrating over a specific phase space volume defined by energy and angular momentum conservation The resulting 3d phase volume for intermediate states in the chaotic three-body problem is given by an integral expression that accounts for energy and angular momentum conservation

Constraints on Excursions

The analysis explores constraints on excursions by examining regions where a real-valued phase volume is obtained, which requires that the tertiary’s pericenter q s should be inside the chaotic region This leads to constraints on angular momentum and energy, such as L 2 s < - G 2 M squared m 3 / 2 E s Furthermore, the constraint on the binary semi-major axis is imposed by demanding that a binary to be clearly separable from the tertiary and for a quasi-regular excursion to form, we demand that BaB ≤ as (16)

Features of the Elliptic Outcome Distribution

The marginal distribution of binary eccentricity d sigma / d eB is analyzed, revealing novel features such as a second-order discontinuity for intermediate angular momentum values> 1, L0 affects the location of a cutoff in the distribution; Stone & Leigh 2019> A second-order discontinuity in the d sigma / d eB distribution is identified and its analytic origin in the (Eb, eB, Cb = 1) phase plane is obtained in Eq. 21

Sampling from Cluster Distributions

A Monte Carlo algorithm is developed to sample initial conditions of binaries and incoming singles assuming a two-component cluster model with a black hole subsystem The sampling process involves drawing the impact parameter b from a linear distribution f b (b) ∝ b allowed between 0 and bmmax (v s), where bmmax is defined as min (√ 2 R G M v s, Ls+, m v s) This allows for the construction of the total angular momentum of the triple L 0, and also the cosine-inclination of the angle between the total angular momentum direction and the cluster axis C 0

Gravitational Waves from Binary-Single Scatterings

Numerical Monte Carlo results are presented for BH binaries embedded in NSCs, showing that there exists a critical total cluster mass M ej above which black hole binaries that would otherwise be destined for ejection merge first, in the cluster, and no ejected binaries escape The ratio of intermediate state observably eccentric mergers to all intermediate state mergers is about (40 ± 2)% − (57 ± 13)% for equal masses and (32 ± 3)% − (62 ± 17)% for unequal masses

Conclusions

The study concludes that the best agreement with numerical scattering experiments is achieved when B = alpha, the free parameter previously used to define the chaotic region in outcome space The fraction of eccentric mergers to total mergers lies within the 0.01-0.10 range for fmin ∈ [4, 40]Hz and emin = 0.1

Comparison to Past Investigations

The OEM fraction found in this work agrees well with the OEM fraction and ejected merger fraction of Dall’Amico et al. (2023) when comparing comparable cluster parameters

Code Availability

The open source code probab3 used to generate the dynamical binary sequence (DBS) evolution statistics is available as a Python package, a command-line tool and in raw repository form at https://github.com/DinaMeylakh/probab3

References

The paper cites numerous works, including Murray & Dermott 1999 and Poincaré's breakthrough on the three-body problem The paper also references work on chaotic dynamics and N-body simulations The paper includes comparisons to past theoretical methods like Samsing (2018) and direct N-body simulations The paper also references work on post-Newtonian (PN) dynamics and Henon-style Monte Carlo methods and direct N-body simulations The paper also references work on binary evolution sequences like Stone & Leigh (2019) and Heggie’s law<ref:2606.

1. Improve black hole merger population synthesis by incorporating intermediate state physics to predict prompt GW inspirals, which can make up 5-10% of all mergers produced by iterated binary-single scatterings.

2. Enhance gravitational wave data interpretation by refining the predicted fraction of observably eccentric mergers (OEM), finding that for current LIGO/Virgo sensitivity, this fraction is in the (1.6 − 3.1)% range for unequal mass binary black holes across explored cluster masses.

3. Develop a Monte Carlo algorithm to model repeated binary-single scatterings within dense star systems to estimate observably eccentric merger (OEM) GWs, specifically quantifying that the OEM fraction is (2.6−4.4)% for star cluster masses of [10 5 − 10 7]M⊙.

4. Create a statistical framework to determine the critical total cluster mass, Mej, above which black hole binaries that would otherwise be destined for ejection merge first, in the cluster, and no ejected binaries escape, which is found in the range of [10 6.67 − 10 7.0]M⊙.

5. Improve dynamical binary sequence evolution modeling by using analytic distributions to sample elliptic and hyperbolic outcomes from analytic distributions using the same Monte Carlo sampling technique to evolve binaries within clusters of different parameters, specifically noting that the elliptic and hyperbolic dsigma/dEB distributions share a common boundary when EB → E0.