Extended Metric-Affine $f(R)$ Gravity with Dynamical Connection in Vacuum

TL;DR

This paper extends vacuum metric-affine $f(R)$ gravity by adding quadratic invariants in torsion and non-metricity, resulting in a theory with an additional scalar degree of freedom linked to spacetime geometry.

Contribution

It introduces a new extended theory of gravity that propagates an extra scalar mode due to quadratic invariants, connecting geometric features to scalar-tensor formulations.

Findings

The extended theory propagates an additional scalar degree of freedom.

The theory can be reformulated as a scalar-tensor theory with specific potential and kinetic terms.

The added invariants significantly alter the dynamical structure of the original gravity model.

Abstract

We extend the usual vacuum Metric-Affine $f(R)$ Gravity by supplementing it with all parity even quadratic invariants in torsion and non-metricity. As we show explicitly this supplementation drastically changes the status of the Theory which now propagates an additional scalar degree of freedom on top of the graviton. This scalar degree of freedom has a geometric origin as it relates to spacetime torsion and non-metricity. The resulting Theory can be written equivalently as a metric and torsionless Scalar-Tensor Theory whose potential and kinetic term coupling depend on the choice of the function $f(R)$ and the dimensionless parameters of the quadratic invariants respectively.