A Cosmological Tale of Two Varying Constants

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We formulate a simple extension of general relativity which incorporates space-time 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.