Dark matter as scalaron in $f(R)$ gravity models

We explore the scalar field obtained under the conformal transformation of the spacetime metric $g_{\mu\nu}$ from the Jordan frame to the Einstein frame in $f(R)$ gravity. This scalar field is the result of the modification in the gravitational part of the Einstein's general relativistic theory of gravity. For $f(R)=\frac{R^{1+\delta}}{R_{c}^{\delta}}$, we find the effective potential of the scalar field and calculate the mass of the scalar field particle "scalaron". It is shown that the mass of the scalaron depends upon the energy density of standard matter in the background. The interaction between standard matter and scalaron is weak in the high curvature regime. This linkage between the mass of the scalaron and the background leads to the physical effects of dark matter and is expected to reflect the anisotropic propagation of scalaron in moving baryonic matter fields as in merging clusters (Bullet cluster, the Abell 520 system, MACS etc.). Such scenario also satisfies the local gravity constraints of $f(R)$ gravity. We further calculate the equation of state of the scalar field in the action-angle variable formalism and show its distinct features as the dark matter and dark energy with respect to energy density of the scalar field at different values of the model parameter $\delta$.