Ringdown frequencies in black holes formed from precessing black-hole binaries

We present a simple formula for the effective ringdown frequencies of the gravitational-wave signal of a precessing black-hole binary in the coprecessing frame. This formula requires only knowledge of the quasi -normal mode frequencies of the system and the value of the precession angle beta during ringdown. Such a formula will be useful in modeling precessing systems. We also provide a comprehensive description of the oscillations in the ringdown frequency in an inertial frame where the spin of the final black hole is in the z -direction. These oscillations arise due to the superposition of the prograde and retrograde frequencies. Our understanding of these oscillations can be used to extract the ratio of the amplitudes of the prograde and retrograde frequencies from numerical data. Alternatively, knowledge of this ratio of the amplitudes can be used to produce a simple model of the time-domain oscillations in the ringdown frequency.