A Cosmological Tale of Two Varying Constants

We formulate a simple extension of general relativity which incorporates space-time variations in the Newtonian gravitation 'constant', $G$, and the fine structure 'constant', $\alpha $, which generalises Brans-Dicke theory and our theory of varying $\alpha .$ We determine the behaviour of Friedmann universes in this theory. In the radiation and dust-dominated eras $\alpha G$ approaches a constant value and the rate of variation of $\alpha $ 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 $\alpha .$ Time variations in $\alpha $ are extinguished by the domination of the expansion by spatial curvature or quintessence fields, as in the case with no $G$ variation.