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 alongwith a negative equation of state (w ≥ -1/3) for the matter fluid at the present epoch of accelerated expansion, while ruling out the evolution of G predicted by Dirac's large number hypothesis. We obtain an improved bound on Ġ(t) as: 1.67 × 10-11 yr-1 < Ġ/G < 3.34 × 10-11 yr-1 in the context of supernova cosmology, as well as a new constraint on Λ̇(t) as: -0.67 × 10-11 yr-1 < Λ̇/Λ < -1.34 × 10-11 yr-1. Based on this formulation, we also present a dynamical solution to the `cosmic coincidence' problem.