A cosmological tale of two varying constants

We formulate a simple extension of general relativity which incorporates spacetime variations in the Newtonian gravitation ‘constant’, G, and the fine structure ‘constant’, α, which generalises Brans–Dicke theory and our theory of varying α. We determine the behaviour of Friedmann universes in this theory. In the radiation and dust-dominated eras αG approaches a constant value and the rate of variation of α is equal to the magnitude of the rate of variation in G. The expansion dynamics of the universe are dominated by the variation of G but the variation of G has significant effects upon the time variation of α. Time variations in α are extinguished by the domination of the expansion by spatial curvature or quintessence fields, as in the case with no G variation.