The Phase Space of $k$-Essence $f(R)$ Gravity Theory

Among the remaining viable theories that can successfully describe the late-time era is the $k$-Essence theory and in this work we study in detail the phase space of $k$-Essence $f(R)$ gravity in vacuum. This theory can describe in a viable way the inflationary era too, so we shall study the phase space in detail, since this investigation may reveal general properties regarding the inflationary attractors. By appropriately choosing the dimensionless variables corresponding to the cosmological system, we shall construct an autonomous dynamical system, and we find the fixed points of the system. We focus on quasi-de Sitter attractors, but also to radiation and matter domination attractors, and study their stability. As we demonstrate, the phase space is mathematically rich since it contains stable manifold and unstable manifold. With regard to the inflationary attractors, these exist and become asymptotically unstable, a feature which we interpret as a strong hint that the theory has an inherent mechanism for graceful exit from inflation. We describe in full detail the underlying mathematical structures that control the instability of the inflationary attractors, and also we also address the same problem for radiation and matter domination attractors. The whole study is performed for both canonical and phantom scalar fields, and as we demonstrate, the canonical scalar $k$-Essence theory is structurally more appealing in comparison to the phantom theory, a result also demonstrated in the related literature on $k$-Essence $f(R)$ gravity.