A Minimal and Stable Vacuum Bounce in Exponential $f(R)$ Gravity

TL;DR

This paper demonstrates a minimal, stable vacuum bounce in exponential $f(R)$ gravity by extending the Starobinsky model, ensuring perturbative stability and well-behaved scalar and tensor perturbations.

Contribution

It introduces a minimal extension to exponential $f(R)$ gravity that achieves a nonsingular vacuum bounce with stability beyond background solutions.

Findings

A no-go result for positive-curvature vacuum bounce with the original model.

A constant term extension restores the bounce exactly.

Scalar and tensor perturbations remain finite and well behaved.

Abstract

We investigate the realization of a nonsingular cosmological bounce in metric $f(R)$ gravity using a controlled exponential deformation of the Starobinsky $R^{2}$ model. Adopting a smooth Gaussian-type bouncing scale factor, we first demonstrate a no-go result showing that a positive-curvature vacuum bounce cannot be supported by the model $f(R)=R+\alpha R^{2}(1-e^{-R/R_b})$ alone. We then show that a minimal extension obtained by introducing a constant term restores the bounce exactly, with the constant fixed algebraically by the bounce condition. A systematic parameter-space scan is performed to identify regions free of ghost and tachyonic instabilities. Working in the Einstein frame, we study the evolution of scalar and tensor perturbations across the bounce and show that both remain finite and well behaved. Our results establish a minimal, perturbatively stable realization of a vacuum bounce in $f(R)$ gravity that goes beyond background-level constructions.