Inhomogeneous perturbations and stability analysis of the Einstein static universe in $f(R,T)$ gravity

The purpose of this paper is to analyze the existence of static stable Einstein universe using inhomogeneous linear perturbations in the context of $f(R,T)$ gravity ($R$ and $T$ denote the scalar curvature and trace of the stress-energy tensor, respectively). The static and perturbed field equations are constructed for perfect fluid parameterized by linear equation of state parameter. We obtain solutions manifesting the Einstein static state by considering peculiar $f(R,T)$ forms for vanishing and non-vanishing conservation of the stress-energy tensor. It is observed that stable static Einstein regions exist for both closed as well as open FLRW universe models for an appropriate choice of parameters. We conclude that this theory is efficient for presenting such cosmological solutions leading to emergent universe scenario.