Spacetime variation of $\alpha$ and the CMB power spectra after the
  recombination
  The possible variation of the fine structure constant may be due to the
non-minimal coupling of the electromagnetic field to a light scalar field which
can be the candidate of dark energy. Its dynamical nature renders the fine
structure constant varies with time as well as space. In this paper we point
out the spatial fluctuation of the fine structure will modify the power spectra
of the temperature and the polarization of the cosmic microwave background. We
show explicitly that the fluctuations of the coupled scalar field generate new
temperature anisotropies at the linear order and induce a $B$ mode to the
polarization at higher order in general.
