A new framework for cosmological expansion in a reformulated Newtonian-Like gravity with variable G

A Newtonian-like theory inspired by the Brans-Dicke gravitational Lagrangian has been recently proposed in Ref. arXiv:2009.04434(v4). We propose here a new variant of this theory such that the usual Newtonian second law is preserved. The cosmological solutions are analysed and accelerated cosmic expansion can be obtained in pure matter dominated universes. This happens due to the dynamical gravitational coupling of the introduced time varying field. We also analyse the matter density perturbations and find they exhibit an enhanced growth in comparison with the usual Newtonian like behavior in Einstein-de Sitter model.