  In this contribution, we consider the equivalence between $f(R)$ gravity and
scalar-tensor theories to study the evolution of scalar cosmological
perturbations in the $1 + 3$ covariant formalism for the classes of shear-free
cosmological dust models with irrotational fluid flows. The $f(R)$ gravity is
considered to be a subclass of Brans-Dicke models, we gave an overview on the
equivalence between $f(R)$ gravity and scalar-tensor theories. We use the $1 +
3$ covariant formalism to present the covariant linearised evolution and
constraint equations. We then derive the integrability conditions describing a
consistent evolution of the linearised field equations of quasi-Newtonian
universes in the scalar-tensor theories of gravity. Finally, we derive the
evolution equations for the density and velocity perturbations of the
quasi-Newtonian universe. We apply the harmonic decomposition and we explore
the behaviour of the matter density contrast by considering $R^{n}$ models. We
introduce the so-called quasi-static approximation to study the approximated
solutions on small scales. The growth of the matter density contrast for both
short- and long- wavelength modes has been examined by applying certain
assumptions of the initial conditions.
