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The possible variation of the fine structure constant may be due to thenon-minimal coupling of the electromagnetic field to a light scalar field whichcan be the candidate of dark energy. Its dynamical nature renders the finestructure constant varies with time as well as space. In this paper we pointout the spatial fluctuation of the fine structure will modify the power spectraof the temperature and the polarization of the cosmic microwave background. Weshow explicitly that the fluctuations of the coupled scalar field generate newtemperature anisotropies at the linear order and induce a $B$ mode to thepolarization at higher order in general.

Spacetime variation of $\alpha$ and the CMB power spectra after the recombination