A black hole is not silent. When perturbed, it rings like a bell, and that ring is its quasinormal modes — damped oscillations. Physicists delved into the world of charged black holes in anti-de Sitter space (a universe with negative curvature, like a saddle). The holes are wrapped in dark matter — invisible stuff that emits no light but pulls with gravity.

Dark matter acts like a thermostat. As its parameter (a measure of density) increases, the critical temperature and pressure of the black hole rise, while the critical horizon radius (the boundary beyond which there is no return) shrinks. In other words, dark matter makes the black hole boil at higher temperatures. Along isobars (lines of constant pressure) below the critical point, small and large black holes sing differently. Their quasinormal modes diverge like two trains at a switch, sharply changing slope near a first-order phase transition (an abrupt change, like ice melting).

At the critical point, where the difference between small and large holes vanishes, things are different. The frequencies of quasinormal modes change smoothly, without sharp jumps. The second-order phase transition leaves no trace in the ring. Along the coexistence curve, small and large holes live side by side, and their voices converge. This happens as the 'swallowtail' of Gibbs free energy (a measure of available energy) shrinks and disappears at the critical point. Dark matter is not just a backdrop: it shifts the entire thermodynamic stage and changes the black hole's acoustics.