Pablo F. Muguruza, Carlos F. Sopuerta

Extreme-Mass-Ratio Inspirals (EMRIs) are one of the main sources of gravitational waves expected in the low-frequency band, where space-based detectors like Laser Interferometer Space Antenna (LISA) will operate. The large number of gravitational-wave cycles accumulated in the EMRI signal in the strong-field regime makes them precise probes of the local spacetime geometry, highly sensitive to deviations from the Kerr black hole paradigm. In this work, we investigate EMRIs around generic, non-Kerr compact objects characterized by a rich multipolar structure. At leading post-Newtonian and linear mass-ratio orders, we incorporate in the waveform model both axisymmetric and non-axisymmetric components of the mass quadrupole and octupole moments, parameterizing the breaking of two fundamental symmetries of the Kerr metric. We study the impact of these modifications on the waveform following the philosophy of EMRI \emph{Analytic Kludge} models. Then, using Fisher-matrix analysis, we assess LISA's capability to constrain deviations of the multipole moments from their Kerr values and the detection of symmetry-breaking effects. We analyze how effectively LISA will probe models beyond General Relativity that predict horizon-scale modifications, such as the fuzzball model proposed in string theory. Our results demonstrate that future LISA observations of EMRIs will provide powerful tests of black hole structure and the underlying theory of gravity. In particular, with one year of LISA data from the inspiral of a $10 M_{\odot}$ compact object into a rotating supermassive black hole of $10^{6} M_{\odot}$ and signal-to-noise ratio of 30, it will be possible to place tight bounds on deviations from the two fundamental symmetries of the Kerr metric, constraining equatorial symmetry breaking to the $10^{-2}$ level and axial symmetry breaking to the $10^{-3}$ level.