Effective Potentials and Morphological Transitions for Binary Black Hole Spin Precession

Abstract

We derive an effective potential for binary black hole (BBH) spin precession at second post-Newtonian order. This effective potential allows us to solve the orbit-averaged spin-precession equations analytically for arbitrary mass ratios and spins. These solutions are quasiperiodic functions of time: after a fixed period, the BBH spins return to their initial relative orientations and jointly precess about the total angular momentum by a fixed angle. Using these solutions, we classify BBH spin precession into three distinct morphologies between which BBHs can transition during their inspiral. We also derive a precessionaveraged evolution equation for the total angular momentum that can be integrated on the radiation-reaction time and identify a new class of spin-orbit resonances that can tilt the direction of the total angular momentum during the inspiral. Our new results will help efforts to model and interpret gravitational waves from generic BBH mergers and predict the distributions of final spins and gravitational recoils. Introduction.-The classic two-body problem was a major engine of historical progress in physics and astronomy. This problem can be solved analytically in Newtonian gravity; its solutions are the well-known Keplerian orbits. The analogs to Newtonian point masses in general relativity are binary black holes (BBHs). Astrophysical BBHs have spins S i [1] in addition to their masses m i [the masses determine the total mass M ≡ m 1 þ m 2 , mass ratio q ≡ m 2 =m 1 ≤ 1 and symmetric mass ratio η ≡ m 1 m 2 =M 2 ¼ q=ð1 þ qÞ 2 ]. Full solutions to the two-body problem in general relativity must, therefore, include spin evolution in addition to orbital motion. Einstein's equations must be solved numerically [2][3][4] when the binary separation r is comparable to the gravitational radius r g ≡ GM=c 2 , but post-Newtonian (PN) approximations may be used when r ≫ r g . BBH evolution in the PN limit occurs on three distinct time scales: the orbital time t orb ∼ ðr 3 =GMÞ 1=2 on which the binary separation r evolves, the precession time t pre ∼ c 2 r 5=2 = ½ηðGMÞ 3=2 ∼ ðt orb =ηÞðr=r g Þ on which the spin directions change, and the radiation-reaction time t RR ∼ E= jdE GW =dtj ∼ c 5 r 4 =½ηðGMÞ 3 ∼ ðt orb =ηÞðr=r g Þ 5=2 on which the energy E ¼ −GηM 2 =ð2rÞ and orbital angular momentum L ¼ ηðrGM 3 Þ 1=2 decrease.The hierarchy t orb ≪ t pre ≪ t RR implies that when considering evolution on one time scale, quantities evolving on a shorter (longer) time scale can be averaged (held constant). This has been used to derive orbit-averaged spin-precession equations _ S i ¼Ω i × S i [5][6][7][8], where the precession frequenciesΩ i depend on the orbital angular momentum L and spins S i but not on the instantaneous

Multi-timescale analysis of phase transitions in precessing black-hole binaries

“…Our new solutions for spin precession allow us to fully exploit the timescale hierarchy of Eq. (1), expanding and detailing the ideas put forward in our previous paper [28]. We showed that spin precession is quasiperiodic implying that the relative orientations of the three angular momenta are fully specified by a single parameter, the magnitude of the total spin, that oscillates on the precession time.…”

“…2. The simple dependence of ξðS; φ 0 Þ on φ 0 motivates us to define two "effective potentials" [28] corresponding to the extreme cases cos φ 0 ¼ ∓1 for which L, S 1 and S 2 are all coplanar:…”

“…Expanding on the analysis in our previous paper [28], we introduce a multi-timescale analysis of the dynamics of spinning, precessing BBHs. Multi-timescale analyses are commonly used in binary dynamics.…”

Cassini states for black hole binaries

“…However, in the range 10 < r/rg < 10 4 , the binary dynamics can be described using the spinning Taylor-expanded PN Hamiltonian. As in previous studies (Schnittman 2004;Kesden et al 2010a;Berti et al 2012;Gerosa et al 2013Gerosa et al , 2014Kesden et al 2015;Gerosa et al 2015a,b), we focus our analysis in this range. In the barycenter frame, the Hamiltonian depends on the canonical variables (r, p) and on the spins vectors.…”

“…One of the most difficult aspects of studying the spinning black hole binary system is the problem of visualizing and analyzing the orientation of the two spins and the angular momentum in an informative way. Previous studies always describe the libration around the coplanar configurations as spin-orbit resonances (Schnittman 2004;Kesden et al 2010a;Berti et al 2012;Gerosa et al 2013Gerosa et al , 2014Kesden et al 2015;Gerosa et al 2015a,b). However, this description is not consistent with the expectation that the two normal modes of this problem become resonant, since for black hole binaries one frequency is usually smaller than the other (Racine 2008).…”