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

We investigate the Cartan formalism in F(R) gravity. F(R) gravity has been introduced as a theory to explain cosmologically accelerated expansions 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-associated local Lorenz symmetry. 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. red Thus, symmetries in Cartan F(R) gravity are clearly conserved. The resulting scalar-tensor theory is useful in applications of the usual slow-roll scenario. As a simple case, we employ the Starobinsky model and evaluate fluctuations in cosmological microwave background radiation.