Phantom Origin for f(R) Inflation

We argue that for $f(R)=R+\alpha R^n$ model of the early universe, the de Sitter point appears as partial attractor for phantom paths. The phantom trajectories then bend their way to the oscillatory period passing near the de Sitter saddle point. Surprisingly, this vicinity is minimized for $1<n\le 3$ where a trajectory could cross the phantom border tangential to the de Sitter point. The only integers in this interval are $2$ and $3$ where $n=2$ reproduces also the most acceptable $f(R)$ model of inflation with strong motivations from quantum theory and also avoiding the vacuum meta-stability. Therefore, one may ask about a possible phantom pre-inflationary scenario. Combining with the increasing chance of falling in a phantom era in our time, proposing a singularity-free cyclic universe seems appealing, too. We use the dynamical system approach throughout this work which allows us to provide numerical support for the analytical discussions.