Cartan $F(R)$ Gravity and Equivalent Scalar-Tensor Theory

TL;DR

This paper explores Cartan formalism in $F(R)$ gravity, rewriting it as a scalar-tensor theory without conformal transformations, and applies it to cosmological models like Starobinsky's to analyze fluctuations.

Contribution

It demonstrates rewriting Cartan $F(R)$ gravity into a scalar-tensor form directly from the action, avoiding conformal transformations, and applies this to cosmological inflation models.

Findings

Successfully reformulated Cartan $F(R)$ gravity as a scalar-tensor theory.

Applied the framework to the Starobinsky model for cosmological fluctuations.

Provided a new approach for analyzing $F(R)$ gravity in cosmology.

Abstract

We investigate the Cartan formalism in $F(R)$ gravity. $F(R)$ gravity has been introduced as a theory to explain cosmological accelerated expansion by replacing the Ricci scalar $R$ in the Einstein-Hilbert action with a function of $R$. As is well-known, $F(R)$ gravity is rewritten as a scalar-tensor theory by using the conformal transformation. Cartan $F(R)$ gravity is described based on the Riemann-Cartan geometry formulated by the vierbein. In the Cartan formalism, the Ricci scalar $R$ is divided into two parts, one derived from the Levi-Civita connection and the other from the torsion. Assuming the spin connection independent matter action, we have successfully rewritten the action of Cartan $F(R)$ gravity into the Einstein-Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. The resulting scalar-tensor theory is useful in applying the usual slow-roll scenario. As a simple case, we employ the Starobinsky model and evaluate fluctuations in the cosmological microwave background radiation.