The late-time response of vacuum black holes in general relativity is notoriously governed by power-law tails arising from the wave scattering off the curved spacetime geometry far from the black hole. While it is known that such tails are universal to a certain extent, a precise characterization of their key ingredients is missing. Here we provide an analytical proof that the tail fall-off is universal for any effective potential asymptotically decaying as $1/r^2$, while the power-law decay is different if the potential decays as $1/r^{\alpha }$ with $1<\alpha <2$. This result extends and revises some previous work and is in agreement with numerical analyses. Our proof is based on an analytical evaluation of the branch cut contribution to the Green’s function, and includes charged black holes, different kinds of perturbations, the Teukolsky equation for the Kerr metric, exotic compact objects, extensions of general relativity, and environmental effects. In the latter case, our results indicate that tails are largely insensitive to a wide range of physically motivated matter distributions around black holes, including the Navarro-Frenk-White profile commonly used to model dark matter.

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• Dark matter
• General relativity
• Gravitational waves