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This mostly has to do with the "ringing" or "ring down" immediately after the merger of two black holes.

Seems to me that after the merger, there shouldn't be anything detectable that would be reflective of any sloshing around of mass or whatever inside the new event horizon.

Do they have a model of what goes on inside the event horizon? If so, how does any "resonant sloshing" of mass (or whatever) inside of the new event horizon show up in this "ring down" tail of the gravity wave resulting from the merger?

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The event horizon is not a very useful concept in a highly dynamical numerical simulation. It is defined as the boundary of the region that cannot send signals to infinity, so by definition nothing comes out during merger (or at any other time). But this means that to find the event horizon you actually have to know the complete evolution of the spacetime (i.e. finish the numerical simulation). Sometimes this is done (numerically) so that pretty pictures can be drawn, but there isn't too much physics in it since the pictures depend on arbitrary embeddings (choices of coordinates) anyway.

There isn't much research (if any) on the interiors of merging black holes, mainly because the focus is on observations at infinity (e.g. LIGO), for which it doesn't matter. I am not a numericist myself, but as I understand it they basically just cut off the spacetime in some arbitrary way inside the event horizon. (You don't know exactly where the event horizon is during the numerical evolution, but you can guess and make sure you leave plenty of room for error.) It might be interesting to find out more details of what is going on inside the horizon, but by definition it would not affect ringdown or any other property of the gravitational-wave signal.

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