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 fine-structure constant, and we find that these changes are dominated by the change in the redshift of recombination, rather than by the change in the Thomson scattering cross-section. We estimate the accuracy with which the next generation of cosmic microwave background experiments might constrain any variation in the fine-structure constant at z of roughly 1000. We find that such experiments could potentially give $|\Delta \alpha/\alpha| < 10^{-2} - 10^{-3}$.