  The cosmology of the $F(R)$ gravity rebuilding by the Cartan formalism is
investigated. This is called Cartan $F(R)$ gravity. The well-known $F(R)$
gravity has been introduced to extend the standard cosmology, e.g. to explain
the cosmological accelerated expansion as the inflation. Cartan $F(R)$ gravity
is based on the Riemann-Cartan geometry. The curvature $R$ can separate to two
parts, one is derived from the Levi-Civita connection and the other from the
torsion. Assuming the matter-independent spin connection, 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. This feature simplifies building and analysis of new
model of inflation. In this paper, we study two models, the power-law model and
logarithmic model, and evaluate fluctuations in the cosmological microwave
background (CMB) radiation. We found the robustness of CMB fluctuation by the
analytical computation and confirm this feature by the numerical calculation.
