# Binary Black-Hole Spin Precession: Dynamical Evolution and Gravitational-Wave Parameter Estimation

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## Abstract

Einstein’s general relativity predicts the existence of black holes. A binary black hole is a system of two black holes orbiting each other. During the inspiral of a binary black hole, the separation between the two black holes shrinks slowly and gravitational waves are emitted until the merger, at which time the gravitational waves peak. We use analytic solutions for generic spin precession at second post-Newtonian(2PN) order to derive Fourier series for the total and orbital angular momenta. On the precession timescale, the angle between the total and orbital angular momenta oscillates with nutation period ⌧ , during which the orbital angular momentum L precesses about the total angular momentum J by an angle ↵. As black holes inspiral, they can pass through nutational resonances (⌦ = n!), at which the total angular momentum tilts by an angle ✓tilt, where ⌦ ⌘ ↵/⌧ is the precession frequency and ! ⌘ 2⇡/⌧ is the nutation frequency. We are the first to identify and study the nutational resonances. We derive an approximate expression for this tilt angle and show the angle is usually less than 103 radians at PN regime. We also show that both precession and nutation can modulate the frequency and amplitude of the gravitational waveforms. In the absence of radiation reaction, the amplitudes and frequencies of precession and nutation can be treated as constant parameters. Using two different approximations to calculate the gravitational-waveform modulation (frequency modulation and amplitude modulation), we calculate the Fisher matrix for waveform models that include these parameters and use this to determine how accurately these parameters can be measured.