Trans-Planckian effect in f(R) cosmology

Abstract

Apart from the assumption that the inflation started at an infinite time in the past, the more realistic initial state of the quantum fluctuations is described by a mixed quantum state imposed at a finite value of the initial time.
One of the most important non-trivial vacua is the $\alpha$-vacuum, which is specified by a momentum cutoff $\Lambda$ \cite{Danielsson:2002kx}. As a consequence, 
the initial condition is imposed at different initial times for the different $k$-modes. This modifies the amplitude of the quantum fluctuations, and thus the corresponding power spectra. 
In this paper, we consider the imprint of the $\alpha$-vacuum state on the power spectrum of scalar perturbations in a generic $f(R)$ gravity by assuming an ultraviolet cutoff $\Lambda$. As a specific model, we consider the Starobinsky model and find the trans-Planckian power spectrum. We find that the leading order corrections to the scalar power spectra in $f(R)$ gravity have an oscillatory behavior as in general relativity \cite{Lim}, and furthermore, the results are in sufficient agreement with the $\Lambda$CDM model.