What It Takes to Solve the Hubble Tension through Modifications of Cosmological Recombination.

We construct data-driven solutions to the Hubble tension which are perturbative modifications to the fiducial ΛCDM cosmology, using the Fisher bias formalism. Taking as proof of principle the case of a time-varying electron mass and fine structure constant, and focusing first on Planck CMB data, we demonstrate that a modified recombination can solve the Hubble tension and lower S_{8} to match weak lensing measurements. Once baryonic acoustic oscillation and uncalibrated supernovae data are included, however, it is not possible to fully solve the tension with perturbative modifications to recombination.