  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.
