Constraint on environments with eccentric extreme-mass-ratio inspirals: Bayesian inference and Fisher-matrix for dark-matter spikes

Abstract

Extreme-mass-ratio inspirals (EMRIs) provide sensitive probes of the astrophysical environments surrounding massive black holes. We investigate the inference of dark-matter (DM) spikes with eccentric EMRI gravitational waves within the analytic-kludge waveform framework, including DM-induced dynamical friction and accretion in the secular orbital evolution. We perform Bayesian parameter estimation using Markov-chain Monte Carlo sampling, complemented by Fisher-matrix analyses. We find that steeper DM spikes produce stronger accumulated waveform modifications and substantially improve the measurement of the spike slope $α_{\rm DM}$. More importantly, the constraints exhibit a nonmonotonic dependence on orbital eccentricity. At moderate eccentricity, higher harmonics help break the degeneracy between $e_{\rm LSO}$ and $α_{\rm DM}$, whereas this correlation can persist or the parameter recovery can deteriorate for highly eccentric systems. These results demonstrate that the harmonic structure of eccentric EMRIs can provide valuable information for probing DM distributions around massive black holes with future LISA observations.