Chameleon perfect scalar field as a geometric correction in $f(R)$ gravity

In this work, we derive the analytical form for a $f(R)$ model that describes a perfect scalar field $\phi$ by assuming the existence of a chameleon mechanism. Based on four statements, at the background and perturbative level, it is possible to relate the extra terms from this theory as a geometrical perfect fluid term, whose has been expressed as possible candidates to explain the nature of the dark sector, and possibly, in the case of a perfect scalar chameleon during inflation, satisfy the big bang nucleosynthesis (BBN) constraints until late times.