Constraints on Spatial Variations in the Fine-Structure constant from Planck

We use the Cosmic Microwave Background (CMB) anisotropy data from Planck to constrain the spatial fluctuations of the fine-structure constant \alpha. Through Thompson scattering of CMB photons, spatial anisotropies of \alpha lead to higher-order correlations in the CMB anisotropies. We use a quadratic estimator based on the four-point correlation function of the CMB temperature anisotropy to extract the angular power spectrum of the spatial variation of the fine-structure constant projected along the line of sight at the last scattering surface. At tens of degree angular scales and above, we constrain the rms fluctuations of the fine structure constant to be \delta \alpha/\alpha_0= (1.34 +/- 5.82) x 10^-2 at the 95% confidence level with respect to the standard value \alpha_0. We find no evidence for a spatially varying \alpha at a redshift of 10^3.