Does the ringdown phase of a black hole merger ever stop? When binary black holes merge they emit gravitational waves in three stages, the inspiral in which the two black holes shed angular momentum through gravitational waves at a rate that becomes detectable just before they merge, the merger where the event horizons actually touch and become one and which produces the highest amplitude and frequency gravitational waves, and the ringdown where the new black hole settles into a spherical shape and sheds excess energy by emitting gravitational waves the amplitude of which quickly diminishes to the point where LIGO can't detect it.
My question is if the ringdown phase ever truly finishes, if there is a finite amount of time before it sheds all its energy and settles into an "ideal" Kerr black hole or if it's like blackbody radiation where the rate at which energy is emitted is proportional to the amount of energy that remains to be emitted, meaning that a black hole born of a black hole merger would continuing radiating exponentially weaker and weaker gravitational waves forever(or at least until it decays from Hawking radiation in the far, far future.)
 A: The decay of the ringdown modes is exponential, with a decay "half-time" determined by the mass and spin of the black hole (and the mode numbers of the mode). So (at least in the context of classical general relativity) this decay takes forever.
However, at some point the decaying field will be so weak that we need to start worrying about the quantization of the decaying field. (This would technically require understanding quantum gravity, which we don't. Although much can be inferred from pertubative quantum gravity which is relatively well understood.) It is possible that after some finite time that all the quanta in a mode actually evaporate and the mode will have reached its ground state.  The time scale on which one would expect this to happen is surprisingly short.
EDIT:
Here is a back of the envelope calculation:
Suppose a black hole of 100 solar mass has been formed of a black hole merger with a spin $\chi=0.7$ (as is typical). The dominant mode in the ringdown of this merger will have a frequency of about 1 kHz and a decay half-time of about 4 milliseconds.
Now suppose quantization of the gravitational field follows standard canonical quantization rules, meaning that the quanta of this mode have energy $h\omega \approx 0.4 \times 10^{-12} eV$.
Further suppose that about 10% of the rest mass of the binary has been deposited in this ringdown mode (an exaggeration). This would translate to about $10^{78} \approx 2^{260}$ quanta of this mode. This sound like a lot but, it will be reduced to about 1 quantum in only 260 decay half-times, which in this case translates to about 1 second.
We can thus be fairly certain that within seconds after a BH hole merger we are in a situation that there are no quanta in the ringdown modes, and the black hole has reached its ground state.
