Constraining deviations from the Kerr metric via a bumpy parametrization and particle swarm optimization in extreme mass-ratio inspirals

The measurement of deviations in the Kerr metric using gravitational-wave observations will provide a clear signal of new physics. Previous studies have developed multiple parametrizations (e.g., ``bumpy'' spacetime) to characterize such deviations in extreme mass-ratio inspirals (EMRIs). These approaches often rely on the Fisher information matrix (FIM) formalism to quantify the constraining power of future space-borne gravitational-wave detectors, such as LISA and Tianqin. For instance, using the analytical kludge waveform model under varying source configurations, such methods have achieved constraint sensitivity levels ranging from ${10}^{\ensuremath{-}4}$ to ${10}^{\ensuremath{-}2}$ on the dimensionless bumpy parameter $\ensuremath{\delta}\stackrel{\texttildelow{}}{Q}$ for LISA. In this paper, we advance prior analyses by integrating particle swarm optimization (PSO) with matched filtering under a restricted parameter search range to enforce a high probability of convergence for PSO. Our results reveal a significant number of degenerate peaks in the likelihood function over the signal parameter space with values that exceed the injected one. This extreme level of degeneracy arises from the involvement of the additional bumpy parameter $\ensuremath{\delta}\stackrel{\texttildelow{}}{Q}$ in the parameter space and introduces systematic errors in parameter estimation. We show that these systematic errors can be mitigated using information contained in the ensemble of degenerate peaks, thereby showing a promising potential method for improving local parameter estimation if the other immense challenges of a global search are first solved. This study highlights the critical importance of accounting for such degeneracies, which are absent in FIM-based analyses, and points out future directions for improving EMRI data analysis.