Gravitational-Wave Echoes from a Hayward Black Hole under Minimal Geometric Deformation
Abstract
Abstract We study a regular black-hole geometry obtained by applying a radial minimal geometric deformation (MGD) of dimensionless strength $\alpha$ to the Hayward seed, characterized by its ADM mass $M$ and its regularity scale $g$,and then trace the construction through the odd-parity perturbation problem. The background sector is kept exact: the deformation function is written in closed form, the Einstein equations separate into seed and decoupler blocks (the decoupler being the additional effective source that deforms the seed), asymptotic flatness is preserved, and the deformed metric remains curvature regular at the center while inheriting the seed horizon structure in an explicit admissible range of the deformation parameter $\alpha$. In the axial sector we work under the frozen-source odd-parity closure appropriate to a static anisotropic background without an independent propagating odd matter mode. Within that framework the master equation acquires an exact effective potential whose seed contribution retains the nonvacuum Hayward correction proportional to the radial derivative $m_H'(r)$ of the Hayward mass function $m_H(r)$ and whose decoupling contribution is isolated analytically. We show that this potential passes from a single-barrier profile to a genuine max--min--max structure in an open region of the $(g,\alpha)$ plane, so the exact geometry itself supplies a trapping cavity without any externally imposed mirror or phenomenological reflecting surface. A…
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