Linear Stability Analysis of Slowly Rotating Kerr-de Sitter Black Holes under Gravitational Perturbations: A Spectral Approach

Mathematical Applications of Science Fiction We present a rigorous derivation of the linear mode stability of slowly rotating Kerr–de Sitter black holes under massless spin-two gravitational perturbations. Using the Teukolsky formalism adapted to the presence of a positive cosmological constant, we analyze the separability of the master equation into radial and angular components. We construct a globally defined energy functional on the intersection of the exterior domain with the causal diamond bounded by the event and cosmological horizons. Through a Whiting-type integral transformation and a detailed analysis of the spectral properties of the radial operator, we prove that for sufficiently small rotation and cosmological parameters, the spectrum of quasinormal modes contains no unstable frequencies in the upper complex half-plane. Eight key lemmas are established concerning the coercive structure of the angular operator and the asymptotic behavior of the effective potential, culminating in a proof of uniform boundedness for the gravitational perturbation scalars.