Varying fundamental constants meet Hubble

Abstract: Fundamental physical constants need not be constant, neither spatially nor temporally. -- This seeming simple statement has profound implications for a wide range of physical processes and interactions, and can be probed through a number of observations. In this chapter, we highlight how CMB measurements can constrain variations of the fine-structure constant and the electron rest mass during the cosmological recombination era. The sensitivity of the CMB anisotropies to these constants arises because they directly affect the cosmic ionization history and Thomson scattering rate, with a number of subtle atomic physics effects coming together. Recent studies have revealed that variations of the electron rest mass can indeed alleviate the Hubble tension, as we explain here. Future opportunities through measurements of the cosmological recombination radiation are briefly mentioned, highlighting how these could provide an exciting avenue towards uncovering the physical origin of the Hubble tension experimentally.