Varying couplings in the early universe: correlated variations of
  $\alpha$ and $G$
  The Cosmic Microwave Background anisotropies provide a unique opportunity to
constrain simultaneous variations of the fine-structure constant $\alpha$ and
Newton's gravitational constant $G$. Those correlated variations are possible
in a wide class of theoretical models. In this brief paper we show that the
current data, assuming that particle masses are constant, gives no clear
indication for such variations, but already prefers that any relative
variations in $\alpha$ should be of the same sign of those of $G$ for
variations of $\sim 1 %$. We also show that a cosmic complementarity is present
with Big Bang Nucleosynthesis and that a combination of current CMB and BBN
data strongly constraints simultaneous variations in $\alpha$ and $G$. We
finally discuss the future bounds achievable by the Planck satellite mission.
