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The Cosmic Microwave Background anisotropies provide a unique opportunity toconstrain simultaneous variations of the fine-structure constant $\alpha$ andNewton's gravitational constant $G$. Those correlated variations are possiblein a wide class of theoretical models. In this brief paper we show that thecurrent data, assuming that particle masses are constant, gives no clearindication for such variations, but already prefers that any relativevariations in $\alpha$ should be of the same sign of those of $G$ forvariations of $\sim 1 %$. We also show that a cosmic complementarity is presentwith Big Bang Nucleosynthesis and that a combination of current CMB and BBNdata strongly constraints simultaneous variations in $\alpha$ and $G$. Wefinally discuss the future bounds achievable by the Planck satellite mission.

Varying couplings in the early universe: correlated variations of $\alpha$ and $G$