Linear Stability of Metric Perturbations in the Context of Extended Gravitational Models

This article presents a full theoretical framework for analyzing the linear stability of metric perturbations in extended gravitational models with anisotropic, matter-free backgrounds. Using a generalized variational principle and a modified scalar sector, the study derives the full linearized system, casts it into matrix form, and applies spectral analysis to identify stability domains. The results reveal parametric resonance bands and demonstrate the stabilizing role of geometric anisotropy, offering new insights into early-universe dynamics and scalar-tensor gravity.