Nutational resonances, transitional precession, and precession-averaged evolution in binary black-hole systems

Abstract

In the post-Newtonian (PN) regime, the time scale on which the spins of binary black holes precess is much shorter than the radiation-reaction time scale on which the black holes inspiral to smaller separations. On the precession time scale, the angle between the total and orbital angular momenta oscillates with nutation period τ, during which the orbital angular momentum precesses about the total angular momentum by an angle α. This defines two distinct frequencies that vary on the radiation-reaction time scale: the nutation frequency ω ≡ 2π=τ and the precession frequency Ω ≡ α=τ. We use analytic solutions for generic spin precession at 2PN order to derive Fourier series for the total and orbital angular momenta in which each term is a sinusoid with frequency Ω − nω for integer n. As black holes inspiral, they can pass through nutational resonances (Ω ¼ nω) at which the total angular momentum tilts. We derive an approximate expression for this tilt angle and show that it is usually less than 10 −3 radians for nutational resonances at binary separations r > 10M. The large tilts occurring during transitional precession (near zero total angular momentum) are a consequence of such states being approximate n ¼ 0 nutational resonances. Our new Fourier series for the total and orbital angular momenta converge rapidly with n providing an intuitive and computationally efficient approach to understanding generic precession that may facilitate future calculations of gravitational waveforms in the PN regime.

Up-down instability of binary black holes in numerical relativity

“…The angle θ L completes ∼3-5 cycles compared to only ∼1 full period for θ 1;2 . This is in qualitative agreement with previous PN predictions 1 [41].…”

“…7 in Ref. [41] for binaries at r ∼ 10M. In their notation, the ratio between the number ofL cycles and spinnutation cycles is given by α=2π.…”

Spin orientations of merging black holes formed from the evolution of stellar binaries

“…Spin precession naturally separates populations that formed with different tilt angles θ 1 , θ 2 into different distributions for ∆Φ . This is a well-known PN effect, first discovered by Schnittman [104] and later explored in detail by [39,56,93,105,106] (see also [99,100,[107][108][109] for later investigations). Tidal interactions affect whether the tilt angle of the second formed BH is set to zero between the two explosions, thus strongly impacting its tilt angle at formation.…”

Endpoint of the up-down instability in precessing binary black holes

“…The inequality t orb ≪ t pre has been used to study precession in binary BHs by averaging the motion over the orbital period (e.g., [42,47]). Further, the inequality t pre ≪ t RR has been used to separate the precessional motion from the GW-driven inspiral [38,39,[48][49][50]. The 2PN orbit-averaged equations describing the evolutions of the BH spins and the orbital angular momentum read [47]…”

“…Excluding the case of transitional precession where J ∼ 0 [9], the directionĴ is conserved to very high accuracy also on the longer radiation-reaction timescale [49]. In a noninertial frame coprecessing withL, we define the relative orientations of the spin directions by the angles θ i betweenŜ i andL and the angle ΔΦ between the projections of the spins onto the orbital plane (see Fig.…”