# Tensor $f(R)$ theory of gravity

## TL;DR

This paper introduces a novel $f(R)$ gravity theory applying the function to the Ricci tensor, leading to modified Einstein equations with potential cosmological implications like singularity avoidance and exponential expansion.

## Contribution

It proposes a new $f(R)$ gravity framework based on the Ricci tensor, extending standard $f(R)$ theories and deriving corresponding field equations for various connection types.

## Key findings

- Modified Einstein equations derived for metric and nonmetric connections.
- Cosmological models exhibit non-singular initial states and exponential expansion.
- Potential observational tests for the new gravity theory are provided.

## Abstract

I propose an alternative $f(R)$ theory of gravity constructed by applying the function $f$ directly to the Ricci tensor instead of the Ricci scalar. The main goal of this study is to derive the resulting modified Einstein equations for the metric case with Levi-Civita connection, as well as for the general nonmetric connection with torsion. The modification is then applied to the Robertson-Walker metric so that the cosmological evolution corresponding to the standard model can be studied. An appealing feature is that even in the vacuum case, scenarios without initial singularity and exponential expansion can be recovered. Finally, formulae for possible observational tests are given.