Gravitationally-decoupled hairy black holes: probing geometric, optical, and quasinormal mode signatures
Gravitational-decoupling provides a systematic framework for constructing black hole geometries from known (seed) solutions (such as the Schwarzschild), whose exterior properties closely resemble those of the seed solution while still retaining potentially distinguishable features. Using this approach, we study a class of static, spherically symmetric hairy black holes and examine how such deformations affect their horizon structure, photon spheres, shadow scales, and scalar quasinormal modes (QNMs). We first determine the physically controlled regions of the parameter space by imposing the relevant horizon and energy condition requirements, while retaining configurations outside these domains only for exploratory comparison. To isolate geometric effects, we compare solutions at a common horizon radius, while accounting explicitly for the distinction between the mass parameter entering the seed geometry and the ADM mass measured asymptotically. The resulting ADM normalized shadow scales are used for an illustrative comparison with the characteristic angular scales of M87* and Sgr A*. We also compute massless scalar QNMs using the sixth-order Wentzel-Kramers-Brillouin approximation and the improved Asymptotic Iteration Method, finding close agreement over the modes considered. Our results show that fixing the horizon radius does not uniquely determine either the optical or perturbative properties of the spacetime, and that mass normalization can significantly alter the interpretation of geometric trends. Horizon structure, null geodesic observables, and scalar QNMs therefore provide complementary probes of gravitationally-decoupled black hole geometries.