Black holes with dark matter halos and a positive cosmological constant: Energy conditions and signatures

We study static, spherically symmetric black holes surrounded by dark matter halos in the presence of a positive cosmological constant. Assuming vanishing radial pressure, we derive the constraints imposed by the null, weak, strong, and dominant energy conditions on the halo matter. The halo is confined to a finite radial interval separated from both horizons, and the total mass is bounded above by the Nariai value. We construct a Hernquist-type profile and study black hole images, showing that a positive cosmological constant can substantially change the images. We also compute scalar quasinormal modes of the Hernquist-type spacetime. When the dark matter mass is sufficiently small, the relative difference of a selected mode is approximately linear in the compactness of the matter halo, i.e., the ratio of mass parameter $M_{\text{DM}}$ and scale parameter $a$, with a coefficient that depends on the cosmological constant.