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