Trans-Planckian Effect in $f(R)$ Cosmology

TL;DR

This paper investigates how a finite initial quantum state, specifically the $ ext{alpha}$-vacuum with a momentum cutoff, affects the scalar power spectrum in $f(R)$ gravity, including the Starobinsky model, revealing oscillatory corrections consistent with observations.

Contribution

It extends the analysis of trans-Planckian effects to $f(R)$ gravity, showing that initial quantum states induce oscillatory modifications in the power spectrum similar to general relativity.

Findings

Oscillatory corrections to scalar power spectrum in $f(R)$ gravity.

Results align with $ ext{Lambda}$CDM model.

Trans-Planckian effects are significant in $f(R)$ models.

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.