Black holes in space do not sit in empty vacuum. They live inside clouds of dark matter and ordinary matter. This surrounding matter changes the black hole’s gravitational field. The authors present a new way to describe that environment that is easier to connect to observations of gravitational waves.

Instead of starting with a dark-matter density at every point, the paper builds the spacetime around a black hole from the enclosed mass as a function of radius. “Enclosed mass” means the total mass inside a spherical shell at a given distance from the black hole. This choice more directly reflects what astronomy measures, such as stellar motions, rotation curves, and lensing, because those probes respond to the total gravitational pull rather than to the local density alone.

The enclosed-mass formulation also makes it simple to combine several components — different kinds of dark matter and ordinary matter — by adding their mass contributions. The authors impose basic physical requirements on their models, like smoothness, causality, and the right behavior far away from the black hole. They show that common halo shapes used in astrophysics (for example Navarro–Frenk–White, Einasto, Hernquist or Burkert profiles) appear as limiting cases within this framework.

To test how the environment affects observables, the paper studies three kinds of gravitational-wave signatures. First, quasinormal modes (QNM) — the “ringing” frequencies of a disturbed black hole — are mainly sensitive to mass close to the black hole. Second, tidal Love numbers (TLN) — which measure how a compact object deforms in an external tidal field — respond more to mass at larger radii. Third, extreme-mass-ratio inspiral (EMRI) signals — produced when a small object orbits a much larger black hole — probe the environment through how the mass distribution alters the orbit and the propagation of waves. The authors find that these probes are complementary: each is sensitive to different parts of the surrounding mass.