Solving the Trans-Planckian Censorship Problem with a Power-law Tail in $R^2$ Inflation: A Dynamical System Approach

In this work we elaborate on solving the trans-Planckian censorship problem of standard slow-roll inflation by using a power-law inflationary tail generated by a scalar field with an exponential potential. We use a quantitative approach by studying in detail the phase space of a combined $F(R,\phi)$ cosmological system, focusing on the de Sitter and power-law subspaces of the total phase space. As we show, the de Sitter subspace of the $F(R,\phi)$ system shares the same fixed points as the vacuum $F(R)$ gravity system and the trajectories in the phase space tend to these fixed points. However, the power-law subspace is not stable and cannot be realized by the combined $F(R,\phi)$ system. To this end, we propose a well-motivated phenomenological $F(R)$ gravity model for which the $R^2$ term is switched off below a critical curvature near the end of the $R^2$ slow-roll inflationary era, and below that critical curvature, only the Einstein-Hilbert gravity term and the scalar field remain in the effective inflationary Lagrangian. The remaining system can successfully realize a power-law tail of the $R^2$ slow-roll era.