In this contribution, we consider the well-known equivalence between [Formula: see text] gravity and Brans–Dicke-type scalar–tensor theories to study the evolution of scalar cosmological perturbations for a class of shear-free cosmological dust models with irrotational fluid flows. We use the [Formula: see text] covariant formalism to present the covariant linearized evolution and constraint equations. We then derive the integrability conditions describing a consistent evolution of the linearized field equations of quasi-Newtonian universes in the modified (scalar–tensor) theory 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 explore the behavior of the matter density contrast by considering [Formula: see text] toy models. 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. We then apply the so-called quasi-static approximation to obtain exact solutions on small scales, but the results show that this approximation is not applicable here. Moreover, any small deviation from general relativity and any small change in the initial conditions of the perturbations causes huge orders-of-magnitude deviations from limiting general relativistic results, potentially putting constraints on the modified theory in the quasi-Newtonian cosmologies treatment. Our current work differs from other works in the literature, in that it is the first such work to show quasi-Newtonian cosmologies are unstable to linearized perturbations in modified gravity.

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