Probing spatial variation of the fine-structure constant using the CMB

The fine-structure constant, $\alpha$, controls the strength of the electromagnetic interaction. There are extensions of the standard model in which $\alpha$ is dynamical on cosmological length and time-scales. The physics of the cosmic microwave background (CMB) depends on the value of $\alpha$. The effects of spatial variation in $\alpha$ on the CMB are similar to those produced by weak lensing: smoothing of the power spectrum, and generation of non-Gaussian features. These would induce a bias to estimates of the weak-lensing potential power spectrum of the CMB. Using this effect, Planck measurements of the temperature and polarization power spectrum, as well as estimates of CMB lensing, are used to place limits (95% C. L.) on the amplitude of a scale-invariant angular power spectrum of $\alpha$ fluctuations relative to the mean value ($C_L^{\alpha}=A^\alpha_{\rm SI}/[L(L+1)]$) of $A^\alpha_{\rm SI}\leq 1.6 \times 10^{-5}$. The limits depend on the assumed shape of the $\alpha$-fluctuation power spectrum. For example, for a white noise angular power spectrum ($C_L^{\alpha}=A^\alpha_{\rm WN}$), the limit is $A^\alpha_{\rm WN}\leq 2.3 \times 10^{-8}$. It is found that the response of the CMB to $\alpha$ fluctuations depends on a separate-universe approximation, such that theoretical predictions are only reliable for $\alpha$ multipoles with $L\lesssim 100$ . An optimal trispectrum estimator can be constructed and it is found that it is only marginally more sensitive than lensing techniques for Planck but significantly more sensitive when considering the next generation of experiments. For a future CMB experiment with cosmic-variance limited polarization sensitivity (e.g., CMB-S4), the optimal estimator could detect $\alpha$ fluctuations with $A^\alpha_{\rm SI}>1.9 \times 10^{-6}$ and $A^\alpha_{\rm WN} > 1.4 \times 10^{-9}$.