Constraining variations in the fine-structure constant with the cosmic microwave background

Any time variation in the fine-structure constant alters the ionization history of the universe and therefore changes the pattern of cosmic microwave background fluctuations. We calculate the changes in the spectrum of these fluctuations as a function of the change in $\alpha$, and we find that these changes are dominated by the change in the redshift of recombination due to the shift in the binding energy of hydrogen. We estimate the accuracy with which the next generation of cosmic microwave background experiments might constrain any variation in $\alpha$ at $z\sim 1000$. We find that such experiments could potentially be sensitive to $|\Delta \alpha /\alpha |\sim {10}^{-2}-{10}^{-3}$.