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Einstein static Universe in non-minimal kinetic coupled gravity

We study the stability of Einstein static Universe, with FLRW metric, by considering linear homogeneous perturbations in the kinetic coupled gravity. By taking linear homogeneous perturbations, we find that the stability of Einstein static Universe, in the kinetic coupled gravity with quadratic scalar field potential, for closed ($K=1$) isotropic and homogeneous FLRW Universe depends on the coupling parameters $κ$ and $\varepsilon$. Specifically, for $κ=L_P^2$ and $\varepsilon=1$ we find that the stability condition imposes the inequality $a_0>\sqrt{3}L_P$ on the initial size $a_0$ of the closed Einstein static Universe before the inflation. Such inequality asserts that the initial size of the Einstein static Universe must be greater than the Planck length $L_P$, in consistency with the quantum gravity and quantum cosmology requirements. In this way, we have determined the non-minimal coupling parameter $κ$ in the context of Einstein static Universe. Such a very small parameter is favored in the inflationary models constructed in the kinetic coupled gravity. We have also studied the stability against the vector and tensor perturbations and discussed on the acceptable values of the equation of state parameter.