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I have read that the chirp frequency in the first LIGO data is related to the period of the orbit of merging black holes. I have also read that that the frequency is related to time-dependent deformations of the final merged black hole. Which of these is correct (if indeed either of them are)?

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Both are correct. During the inspiral, the frequency of wave is simply twice the frequency at which the holes revolve around their center of mass. It increases as the holes get closer and closer and revolve faster and faster. After they merge, there is a ringdown phase while the merged hole oscillates in a damped (by gravitational wave emission) way into a stable Kerr metric. The dominant ringdown frequency is essentially the same as the frequency at merger, and it doesn’t appear to change much as the oscillations die away over just a few cycles.

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