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Observations of the CMB today allow us to answer detailed questions about theproperties of our Universe, targeting both standard and non-standard physics.In this paper, we study the effects of varying fundamental constants (i.e., thefine-structure constant, $\alpha_{\rm EM}$, and electron rest mass, $m_{\rme}$) around last scattering using the recombination codes CosmoRec andRecfast++. We approach the problem in a pedagogical manner, illustrating theimportance of various effects on the free electron fraction, Thomson visibilityfunction and CMB power spectra, highlighting various degeneracies. Wedemonstrate that the simpler Recfast++ treatment (based on a three-level atomapproach) can be used to accurately represent the full computation of CosmoRec.We also include explicit time-dependent variations using a phenomenologicalpower-law description. We reproduce previous Planck 2013 results in ouranalysis. Assuming constant variations relative to the standard values, we findthe improved constraints $\alpha_{\rm EM}/\alpha_{\rm EM,0}=0.9993\pm 0.0025$(CMB only) and $m_{\rm e}/m_{\rm e,0}= 1.0039 \pm 0.0074$ (including BAO) usingPlanck 2015 data. For a redshift-dependent variation, $\alpha_{\rmEM}(z)=\alpha_{\rm EM}(z_0)\,[(1+z)/1100]^p$ with $\alpha_{\rmEM}(z_0)\equiv\alpha_{\rm EM,0}$ at $z_0=1100$, we obtain $p=0.0008\pm 0.0025$.Allowing simultaneous variations of $\alpha_{\rm EM}(z_0)$ and $p$ yields$\alpha_{\rm EM}(z_0)/\alpha_{\rm EM,0} = 0.9998\pm 0.0036$ and $p = 0.0006\pm0.0036$. We also discuss combined limits on $\alpha_{\rm EM}$ and $m_{\rm e}$.Our analysis shows that existing data is not only sensitive to the value of thefundamental constants around recombination but also its first time derivative.This suggests that a wider class of varying fundamental constant models can beprobed using the CMB.

New constraints on time-dependent variations of fundamental constants using Planck data