Perturbations of quasi-Newtonian universes in scalar-tensor gravity

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.