Measuring α in the early Universe:: cosmic microwave background polarization, re-ionization and the Fisher matrix analysis

We present a detailed analysis of present and future cosmic microwave background (CMB) constraints of the value of the fine-structure constant, alpha. We carry out a more detailed analysis of the WMAP first-year data, deriving state-of-the-art constraints on alpha and discussing various other issues, such as the possible hints for the running of the spectral index. We find, at 95 per cent confidence level, that 0.95<alpha(dec)/alpha(0)<1.02. Setting dn(S)/d ln k=0 yields 0.94<alpha(dec)/alpha(0)<1.01 as previously reported. We find that a lower value of alpha(dec)/alpha(0) makes a value of dn(S)/d ln k=0 more compatible with the data. We also perform a thorough Fisher matrix analysis (including both temperature and polarization, as well as alpha and the optical depth tau), in order to estimate how future CMB experiments will be able to constrain alpha and other cosmological parameters. We find that Planck data alone can constrain tau with an accuracy of the order 4 per cent and that this constraint can be as small as 1.7 per cent for an ideal cosmic variance limited (CVL) experiment. Constraints on alpha are of the order of 0.3 per cent for Planck and can in principle be as small as 0.1 per cent using CMB data alone: tighter constraints will require further (non-CMB) priors.