Solving the trans-Planckian censorship problem with a power-law tail in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> inflation: A dynamical system

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,ϕ) 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,ϕ) 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,ϕ) system. To this end, we propose a well-motivated phenomenological F(R) gravity model for which the R2 term is switched off below a critical curvature near the end of the R2 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 R2 slow-roll era.