Eccentricity-Modulated Phase Degeneracy and Distinguishability between Dark Matter and Accretion Disk Environmental Effects in EMRIs
Extreme mass-ratio inspirals (EMRIs) are sensitive probes of weak environmental effects around massive black holes, since such effects can accumulate into observable gravitational-wave phase shifts. In this work, we study the phase degeneracy between dark matter halos and accretion disks in eccentric EMRI waveforms. We model the dark matter (DM) environment with NFW and Beta halo profiles, and describe the disk using a thin $α$-disk model. Their distinguishability is quantified through eccentricity-dependent phase diagnostics and the residual signal-to-noise ratio (SNR) in the LISA band. Our results show that DM-induced dephasing depends only weakly on the initial eccentricity $e_0$, whereas disk-induced dephasing is strongly suppressed as $e_0$ increases. The distinguishability time exhibits a smooth and weakly non-monotonic dependence on the initial eccentricity, while being more strongly controlled by the environmental strength. For the benchmark systems considered here, the DM--disk waveform difference can be detectable by LISA, and $e_0$ can serve as an auxiliary diagnostic in addition to the observation duration.