# Black hole ringdown beyond general relativity

Nicola Francchini (APC)

In this talk, I will sketch a procedure to compute quasi-normal modes of rotating black holes in alternative theories of gravity, when the modification from general relativity is determined by a small coupling constant that can be treated perturbatively. In general, one obtains a Teukolsky equation which is modified by terms proportional to the small coupling parameter. I will show a toy model involving only a scalar perturbation and comment on how one can generalize the procedure to the tensor case. In general, one would find a radial equation whose potential is modified with respect to general relativity. With a simplified non-rotating set up, I will show how these modifications can be constrained with high-precision gravitational-wave measurements of the black hole’s quasi-normal mode frequencies. By assuming the modifications to be proportional to the small coupling parameter, and further performing a Taylor expansion in M/r, one can compute the quasi-normal modes of the modified potential up to desired order in the perturbative parameter. Either through a principal component analysis or via Markov-chain Monte-Carlo methods, I will recover the Taylor coefficients in the M/r expansion. In both cases, even if the overall reconstruction is good, the bounds on the individual parameters are not robust. Since quasi-normal mode frequencies are related to the behaviour of the perturbation potential near the light ring, I will show a different strategy. By mapping the Taylor expansion to the value of the potential and its derivatives at the peak of the light-ring by using Wentzel–Kramers–Brillouin theory, I can show that the value of the potential and its second derivative at the light ring can be robustly constrained.