Cosmological Consequences of Varying Couplings in Gravity Action

We develop a Lagrangian formulation for gravity with matter where the gravitational couplings are universally treated as being field-dependent. The solutions for FLRW geometries and the associated time evolution of the Newton and cosmological couplings are found. The distance-redshift relations are shown to prefer a slowly growing Newton's coupling along with a negative equation of state ($w\geq -\frac{1}{3}$) for the matter fluid at the present epoch of accelerated expansion, while ruling out Dirac's large number hypothesis. We obtain an improved bound on $\dot{G}(t)$ as: $1.67\times 10^{-11} yr^{-1}<\frac{\dot{G}}{G}<3.34\times 10^{-11} yr^{-1}$ in the context of supernova cosmology, as well as a new constraint on $\dot{\Lambda}(t)$ as: $-0.67\times 10^{-11} yr^{-1}<\frac{\dot{\Lambda}}{\Lambda}<-1.34\times 10^{-11} yr^{-1}$. Based on this formulation, we also present a dynamical solution to the `cosmic coincidence' problem.