Axial gravitational perturbations and echo-like signals of a hairy black hole from gravitational decoupling

TL;DR

This paper analyzes axial gravitational perturbations of a hairy black hole, revealing that certain parameter regions produce echo-like signals due to a double-peak potential structure, offering insights into black hole hair detection via gravitational waves.

Contribution

It demonstrates that echo-like signals can arise dynamically from the black hole's geometry without artificial boundary conditions, expanding understanding of black hole perturbations and gravitational wave signatures.

Findings

Double-peak potential supports echo-like waveforms.

Echoes occur in parameter regions not satisfying all energy conditions.

The study links black hole hair to observable gravitational wave features.

Abstract

We study axial gravitational perturbations of a hairy black hole constructed in the framework of gravitational decoupling and investigate the geometric origin of echo-like late-time signals in this spacetime. We derive the odd-parity master equation and the corresponding effective potential, and we compute the quasinormal-mode spectrum by using frequency-domain and time-domain methods. We show that, in a suitable region of parameter space, the axial potential develops a double-peak structure that supports a trapping cavity and gives rise to echo-like late-time waveforms. Rather than imposing near-horizon reflectivity by hand, the delayed pulses therefore arise dynamically from the geometry of the effective potential. We also clarify that the parameter region exhibiting echoes need not coincide with the region in which the weak energy condition is satisfied everywhere outside the event horizon, and this distinction must be taken into account when interpreting the physical status of the solution. Our results provide a useful framework for probing black-hole hair through gravitational-wave ringdown and for exploring possible observational departures from the standard no-hair paradigm.