Geodesics and scalar perturbations of Schwarzschild black holes embedded in a Dehnen-type dark matter halo with quintessence

Abstract We present a comprehensive analysis of a Schwarzschild black hole embedded in a Dehnen-type dark-matter halo and surrounded by a quintessence field. We construct the composite spacetime metric and analyze its geometric properties, including horizon structure and curvature invariants. We find that increasing both the dark-matter core density ₛ and the quintessence normalization parameter c enlarges the event horizon while reducing the cosmological horizon. We analyze timelike and null geodesics to determine the innermost stable circular orbits, photon-sphere radii, and shadow properties. Using the Gauss--Bonnet theorem, we compute weak-deflection angles and demonstrate that gravitational lensing intensifies as ₛ and rₛ increase. Scalar perturbations are examined using the sixth-order WKB method and Padé approximants, supported by time-domain checks. We observe that the quasinormal modes are suppressed as the dark-matter density increases. Greybody factors and the sparsity of Hawking radiation are also computed, revealing that increased halo density enhances transmission and modifies their deviation from blackbody emission. Our results highlight the observable signatures of dark matter and quintessence in black-hole phenomenology, with testable implications for Event Horizon Telescope imaging and gravitational-wave spectroscopy.