Perturbing the perturbed: Stability of quasinormal modes in presence of a positive cosmological constant

In this work, we wish to address the question -- whether the quasi-normal modes, the characteristic frequencies associated with perturbed black hole spacetimes, central to the stability of these black holes, are themselves stable. Though the differential operator governing the perturbation of black hole spacetimes is self-adjoint, the boundary conditions are dissipative in nature, so that the spectral theorem becomes inapplicable, and there is no guarantee regarding the stability of the quasi-normal modes. We have provided a general method of transforming to the hyperboloidal coordinate system, for both asymptotically flat and asymptotically de Sitter spacetimes, which neatly captures the dissipative boundary conditions, and the differential operator becomes non-self-adjoint. Employing the pseudospectrum analysis and numerically implementing the same through Chebyshev's spectral method, we present how the quasi-normal modes will drift away from their unperturbed values under external perturbation of the scattering potential. Intriguingly, for strong enough perturbation, even the fundamental quasi-normal mode drifts away from its unperturbed position for asymptotically de Sitter black holes, in stark contrast to the case of asymptotically flat black holes. This feature holds irrespective of the nature of the perturbing field and is also independent of black hole hairs. Besides presenting several other interesting results, specifically for asymptotically de Sitter black holes, we also discuss the implications of the instability of the fundamental quasi-normal mode on the strong cosmic censorship conjecture.