Anisotropic Perturbations and Cosmic Stability via Modified Gravity

This work explores the stability of the Einstein Universe in the framework of [Formula: see text] gravity, where matter and geometry are directly coupled. Using the closed Bianchi IX geometry, we derive and linearize the field equations under minimally coupled model, i.e., [Formula: see text]. This leads to coupled fourth-order differential equations that describe the behavior of small anisotropic perturbations. To capture different coupling effects, we examine four models, a linear model [Formula: see text], representing the simplest extension of GR, a quadratic model [Formula: see text], which incorporates higher-order matter contributions, an inverse trace model [Formula: see text], with the inverse depending on the trace [Formula: see text] and a logarithmic model [Formula: see text], with an interplay between curvature and matter. Both conserved and non-conserved forms of the energy momentum tensor are analyzed. The results, illustrated through graphs, show that stability depends on the matter-geometry coupling, the equation of state, and the chosen model. Conserved cases are more restrictive, while non-conserved cases allow broader regions of stability. Overall, the study indicates that matter geometry interaction can act as a stabilizing mechanism beyond GR and supports [Formula: see text] gravity as a meaningful alternative to dark energy in explaining cosmic acceleration.