Gravitational-wave imprints of compact and galactic-scale environments in extreme-mass-ratio binaries

Circumambient and galactic-scale environments are intermittently present around black holes that reside in active galactic nuclei. As supermassive black holes impart energy on their host galaxy, so the galactic environment affects the dynamics of solar-mass objects around black holes and the gravitational waves emitted from non-vacuum asymmetric binaries. Only recently an exact general-relativistic solution has been found that describes a Schwarzschild black hole immersed in a dark matter halo of the Hernquist type. We perform an extensive analysis of generic geodesics delving in such non-vacuum spacetimes and compare our results with those obtained in Schwarzschild, as well as calculate their gravitational-wave emission. Our findings indicate that the radial and polar oscillation frequency ratios descend deeper into the strong gravity region as the compactness of the halo increases. This translates to a redshift of non-vacuum geodesics and their resulting waveforms with respect to the vacuum ones. We calculate the overlap between waveforms resulting from Schwarzschild and non-vacuum geometries and find that it decreases as the halo compactness grows, meaning that dark matter environments should be distinguishable by space-borne detectors. For compact environments, we find that the apsidal precession is strongly affected due to the gravitational pull of dark matter; the orbit's axis can rotate in the opposite direction as that of the orbital motion, leading to a retrograde precession drift that depends on the halo mass, as opposed to the typical prograde precession transpiring in galactic-scale environments. Gravitational waves in retrograde-to-prograde alterations demonstrate transient frequency phenomena around critical non-precessing turning points, thus they may serve as `smoking guns' for the presence of compact dark matter environments around supermassive black holes.