The dynamics of precessing binary black holes using the post-Newtonian approximation

TL;DR

This paper studies the conservative dynamics of binary black holes using post-Newtonian Hamiltonian equations, finding chaos only in specific, less astrophysically relevant scenarios, and emphasizing the importance of higher-order corrections.

Contribution

It provides a detailed analysis of chaos in binary black hole orbits within the post-Newtonian framework, including spin effects and eccentricities, with implications for gravitational wave modeling.

Findings

No chaos in quasi-circular orbits at typical astrophysical separations.

Chaos appears only at small separations with significant spin-spin oscillations.

Chaotic solutions in eccentric orbits occur at very close pericenters, requiring higher-order corrections.

Abstract

We investigate the (conservative) dynamics of binary black holes using the Hamiltonian formulation of the post-Newtonian (PN) equations of motion. The Hamiltonian we use includes spin-orbit coupling, spin-spin coupling, and mass monopole/spin-induced quadrupole interaction terms. In the case of both quasi-circular and eccentric orbits, we search for the presence of chaos (using the method of Lyapunov exponents) for a large variety of initial conditions. For quasi-circular orbits, we find no chaotic behavior for black holes with total mass 10 - 40 solar masses when initially at a separation corresponding to a Newtonian gravitational-wave frequency less than 150 Hz. Only for rather small initial radial distances, for which spin-spin induced oscillations in the radial separation are rather important, do we find chaotic solutions, and even then they are rare. Moreover, these chaotic quasi-circular orbits are of questionable astrophysical significance, since they originate from direct parametrization of the equations of motion rather than from widely separated binaries evolving to small separations under gravitational radiation reaction. In the case of highly eccentric orbits, which for ground-based interferometers are not astrophysically favored, we again find chaotic solutions, but only at pericenters so small that higher order PN corrections, especially higher spin PN corrections, should also be taken into account.