Evolution of matter density perturbations in f(R) theories of gravity with non-minimal coupling between matter and geometry

In the context of f(R) theories of gravity with non-minimal coupling between matter and geometry, we study the evolution of scalar cosmological perturbations in the metric formalism. We not only derive the perturbation equations in the longitudinal gauge, but also obtain the equation of matter density perturbations by using both the perturbed covariant conservation equations which are subject to taking the Lagrangian density of matter as opposite to the energy density of a perfect fluid, and the subhorizon approximation. Furthermore, in order to investigate the behavior of matter density perturbations, we apply the obtained perturbation equation to a class of models. In these models, it is shown that 1) the stronger the non-minimal coupling between matter and geometry is, the more the growth of matter density perturbations slows down; 2) the small scale perturbations grows faster than the large ones; and 3) the more the deviation of parameter n or λ from zero gets, the more evident the scale dependence of the growth of matter density perturbations becomes.