Updated fundamental constant constraints from Planck 2018 data and possible relations to the Hubble tension.

We present updated constraints on the variation of the fine structure constant, $\alpha_{\rm EM}$, and effective electron rest mass, $m_{\rm e}$, during the cosmological recombination era. These two fundamental constants directly affect the ionization history at redshift $z\simeq 1100$ and thus modify the temperature and polarisation anisotropies of the cosmic microwave background (CMB) measured precisely with {\it Planck }. The constraints on $\\alpha_{\rm EM}$ tighten slightly due to improved {\it Planck} 2018 polarisation data but otherwise remain similar to previous CMB analysis. However, a comparison with the 2015 constraints reveals a mildly discordant behaviour for $m_{\rm e}$, which from CMB data alone is found below its local value. Adding baryon acoustic oscillation data brings $m_{\rm e}$ back to the fiducial value, $m_{\rm e}=(1.0078\pm0.0067) m_{\rm e,0}$, and also drives the Hubble parameter to $H_0=69.1\pm 1.2$ [in units of ${\rm km \, s^{-1} \, Mpc^{-1} }$]. Further adding supernova data yields $m_{\rm e}=(1.0190\pm0.0055) m_{\rm e,0}$ with $H_0=71.24\pm0.96$. We perform several comparative analyses using the latest cosmological recombination calculations to further understand the various effects. Our results indicate that a single-parameter extension allowing for a slightly increased value of $m_{\rm e}$ ($\simeq 3.5\sigma$ above $m_{\rm e,0}$) could play a role in the Hubble tension.