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A stone falls into a pond, and ripples spread across the water. If the pond is choppy, with waves coming from all directions, the pattern gets more complicated. That's roughly how spacetime behaves around a black hole when something perturbs it: it vibrates, emitting gravitational waves. Until now, astrophysicists have mostly studied the individual 'bursts'—quasinormal modes, which resemble the fading ring of a bell. But any disturbance also has a continuous tail that persists for a long time, like the hum that follows a strike.

The authors of the new study applied the complex scaling method to describe this hum for black holes in de Sitter space: that is, in a universe where dark energy is driving accelerated expansion. The idea is akin to a sound trick: to hear a faint noise, you turn up the gain. Here, the coordinates are mathematically 'rotated,' turning outgoing waves into decaying ones, so that all perturbations, both discrete and continuous, can be analyzed in a unified spectrum.

The researchers tested the method on scalar, electromagnetic, and gravitational perturbations around a Schwarzschild–de Sitter black hole. They found that a nonzero cosmological constant noticeably shifts both the quasinormal frequencies and the density of the continuous spectrum. Moreover, the approach proved applicable to black holes in higher-dimensional spacetimes — at least for tensor and vector types of oscillations. This opens the door to studying more exotic objects, including those predicted by certain string theories.

This approach could be useful for analyzing data from gravitational wave detectors. Instead of searching only for known quasinormal mode templates, we can also account for the continuous response — which means we can weigh black holes more accurately and test whether the space around us is truly permeated by dark energy.