Covariant gauge-invariant perturbations in multifluid f(R) gravity

We study the evolution of scalar cosmological perturbations in the (1+3)- covariant gauge-invariant formalism for generic $f(R)$ theories of gravity. Extending previous works, we give a complete set of equations describing the evolution of matter and curvature fluctuations for a multi-fluid cosmological medium. We then specialize to a radiation-dust fluid described by barotropic equations of state and solve the perturbation equations around a background solution of $R^n$ gravity. In particular we study exact solutions for scales much smaller and much larger than the Hubble radius and show that $n>2\mathrm{/}3$ in order to have a growth rate compatible with the M'esz'aros effect.