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 background expansion can be obtained even in a pure matter dominated universe. This happens due to the dynamic character of the effective gravitational coupling which is sourced by a time evolving scalar 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.