# Evolution of scalar and vector cosmological perturbations through a bounce in metric $f(R)$ gravity in flat FLRW spacetime

## TL;DR

This paper analyzes scalar and vector cosmological perturbations in a non-singular bouncing universe within metric $f(R)$ gravity, revealing that vector perturbations can remain bounded during the bounce, unlike in general relativity.

## Contribution

It provides a comprehensive treatment of perturbations in $f(R)$ gravity using the Einstein frame, correcting previous scalar perturbation calculations and exploring vector perturbation behavior.

## Key findings

- Vector perturbations remain bounded during the bounce.
- The Einstein frame is effective for perturbation calculations in most cases.
- Scalar perturbations are correctly computed in the Einstein frame, completing previous work.

## Abstract

In the present work we present the full treatment of scalar and vector cosmological perturbations in a non-singular bouncing universe in the context of metric $f(R)$ cosmology. Scalar metric perturbations in $f(R)$ cosmology were previously calculated in the Jordan frame, in the present paper we successfully use the Einstein frame to calculate the scalar metric perturbations where the cosmological bounce takes place in the Jordan frame. The Einstein frame picture presented corrects and completes a previous calculation of scalar perturbations and adds new information. Behavior of fluid velocity potential and the pure vector fluid velocity terms are elaborately calculated for the first time in $f(R)$ cosmology for a bouncing universe in presence of exponential gravity. It is shown that the vector perturbations can remain bounded and almost constant during the non-singular bounce in $f(R)$ gravity unlike general relativistic models where we expect the vector perturbations to be growing during the contracting phase and decaying during the expanding phase. The paper shows that the Einstein frame can be used for calculation of scalar and vector metric perturbations in a bouncing universe for most of the cases except the case of asymmetric non-singular bounces.