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.
as written by the authors arXiv:2608.01200v1
We investigate how thermodynamic phase transitions are reflected in the quasinormal modes (QNMs) of charged anti-de Sitter (AdS) black holes surrounded by perfect fluid dark matter (PFDM). In the extended phase space, increasing the positive PFDM parameter raises the critical temperature and pressure while reducing the critical horizon radius. We compute the fundamental QNMs of a massless scalar perturbation using a Chebyshev pseudospectral method and analyze their evolution along isobaric and isothermal processes below the critical point. The small and large black hole branches trace clearly separated QNM trajectories and display sharply different slopes near the first-order transition, providing a dynamical signature of the branch change. Along isotherms, this evolution results from the competing effects of the horizon radius and pressure, or equivalently the AdS radius, rather than from the horizon radius alone. At the critical point, however, the QNM frequencies vary smoothly with the horizon radius and show no sharp signature of the second-order transition. Along the coexistence curve, the separation between the small and large black hole QNMs decreases as the Gibbs free energy swallowtail shrinks and vanishes at criticality. These results show that PFDM shifts both the thermodynamic phase structure and the associated QNM response, while the fundamental scalar QNM spectrum remains sensitive to the first-order small/large black hole transition.
The quasinormal modes of black holes can be compared to a voice: each hole has its own timbre and damping time, and from them you can infer its mass and charge.