Closed-form solutions of spinning, eccentric binary black holes at 1.5 post-Newtonian order

TL;DR

This paper derives two fully 1.5PN accurate closed-form solutions for spinning, eccentric binary black hole systems, combining previous methods and providing a Mathematica toolkit for modeling gravitational waves.

Contribution

It combines existing solution methods to produce two complete 1.5PN solutions for BBH dynamics, filling a gap in analytical modeling tools.

Findings

Solutions agree well with numerical treatments

Provides a Mathematica package for practical use

Verifies the five actions constructed in prior work

Abstract

The closed-form solution of the 1.5 post-Newtonian (PN) accurate binary black hole (BBH) Hamiltonian system has proven to be difficult to obtain for a long time since its introduction in 1966. Closed-form solutions of the PN BBH systems with arbitrary parameters (masses, spins, eccentricity) are required for modeling the gravitational waves (GWs) emitted by them. Accurate models of GWs are crucial for their detection by LIGO/Virgo and LISA. Only recently, two solution methods for solving the BBH dynamics were proposed in arXiv:1908.02927 (without using action-angle variables), and arXiv:2012.06586, arXiv:2110.15351 (action-angle based). This paper combines the ideas laid out in the above articles, fills the missing gaps and provides the two solutions which are fully 1.5PN accurate. We also present a public Mathematica package BBHpnToolkit which implements these two solutions and compares them with a fully numerical treatment. The level of agreement between these solutions provides a numerical verification for all the five actions constructed in arXiv:2012.06586, and arXiv:2110.15351. This paper hence serves as a stepping stone for pushing the action-angle-based solution to 2PN order via canonical perturbation theory.