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, especially those residing in active galactic nuclei. As supermassive black holes impart energy on their host galaxy, so the galactic environment affects the geodesic dynamics of solar-mass objects around supermassive black holes and subsequently the gravitational waves emitted from such nonvacuum extreme-mass-ratio binaries. Only recently an exact general-relativistic solution has been found that describes a Schwarzschild black hole immersed in a dark matter halo profile of the Hernquist type. We perform an extensive analysis generic geodesics delving in such nonvacuum spacetimes and compare our results with those obtained in vacuum Schwarzschild spacetime, as well as calculate their dominant gravitational-wave emission. Our findings indicate that the radial and polar oscillation frequency ratios, which designate resonances, descend deeper into the extreme gravity regime as the compactness of the halo increases. This translates to a gravitational redshift of nonvacuum geodesics and their resulting waveforms with respect to the vacuum ones; a phenomenon that has also been observed for ringdown signals in these setups. We calculate the maximized overlap between waveforms resulting from orbital evolutions around Schwarzschild and nonvacuum geometries and find that it decreases as the halo compactness grows, meaning that dark matter environments should be distinguishable by space-borne gravitational-wave detectors. For compact environments, we find that the apsidal precession of orbits 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's mass, as opposed to the typical prograde precession transpiring in vacuum and galactic-scale environments. Gravitational waves in retrograde-to-prograde orbital alterations demonstrate transient frequency phenomena around a critical nonprecessing turning point, thus they may serve as a "smoking gun" for the presence of dense dark matter environments around supermassive black holes.