Abstract
We investigate the extent to which modifying the ionization history at cosmological recombination can relieve the Hubble tension, taking into account all relevant datasets and considering the implications for the galaxy clustering parameter $S_8$ and the matter density fraction $\Omega_m$. We use the linear response approximation to systematically search for candidate ionization histories parameterized with a cubic-spline that provide good fits to the Planck CMB and DESI BAO data while relieving the $H_0$ tension, followed by MCMC fits of the most promising candidate models to the data. We also fit to the data a physically motivated phenomenological model of ionization history that has four parameters. Our main result is that models of modified recombination can reduce the Hubble tension to below 2$\sigma$ while improving the fit to the current CMB and BAO data and reducing the $S_8$ tension. The promising candidate ionization histories have simple shapes, with no need for an oscillatory dependence on redshift. Our study also demonstrates the importance of the high-resolution CMB temperature and polarization anisotropies for constraining modified recombination, with the candidate models in this study showing varying levels of agreement with the current ACT DR4 and SPT-3G data.