In this contribution, classes of shear-free cosmological dust models with irrotational fluid flows will be investigated in the context of scalar-tensor theories of gravity. In particular, the integrability conditions describing a consistent evolution of the linearised field equations of quasi-Newtonian universes are presented. We also derive the covariant density and velocity propagation equations of such models and analyse the corresponding solutions to these perturbation equations.DOI: 10.31526/LHEP.1.2018.05Although general relativity theory (GR) is a generalization of Newtonian Gravity in the presence of strong gravitational fields, it has no properly defined Newtonian limit on cosmological scales. Newtonian cosmologies are an extension of the Newtonian theory of gravity and are usually referred to as quasi-Newtonian, rather than strictly Newtonian formulations [1,2,3]. The importance of investigating the Newtonian limit for general relativity on cosmological contexts is that, there is a viewpoint that cosmological studies can be done using Newtonian physics, with the relativistic theory only needed for examination of some observational relations [1]. General relativistic quasi-Newtonian cosmologies have been studied in the context of large-scale structure formation and non-linear gravitational collapse in the late-time universe. This despite the general covariant inconsistency of these cosmological models except in some special cases such as the spatially homogeneous and isotropic, spherically symmetric, expanding (FLRW) spacetimes. Higher-order or modified gravitational theories of gravity such as f (R) theories of gravity have been shown to exhibit more shared properties with Newtonian gravitation than does general relativity [4,5]. In [1], a covariant approach to cold matter universes in quasi-Newtonian cosmologies has been developed and it has been applied and extended in [2] in order to derive and solve the equations governing density and velocity perturbations. This approach revealed the existence of integrability conditions in GR. In this work, we derive the evolution of the velocity and density perturbations in the comoving (Lagrangian) and quasi-Newtonian frames. We investigate the existence of integrability conditions of a class of irrotational and shear-free perfect fluid cosmological models in the context of scalar-tensor gravity. Such work has been done in the context of f (R) gravity [6], where some models of f (R) gravity have been shown to exhibit Newtonian behaviour in the shear-free regime.The so-called f (R) theories of gravity are among the simplest modification of Einstein's GR. These theories come about by a straightforward generalisation of the Lagrangian in the Einstein-Hilbert action [7,8] as where L m is the matter Lagrangian and g is the determinant of the metric tensor g µν . Another modified theory of gravity is the scalar-tensor theory of gravitation. This is a broad class of gravitational models that tries to explain the gravitational interaction through both a scalar ...