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 $\mathrm{\Lambda }\mathrm{CDM}$ 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.

Figure 1

Solutions for $(\mathrm{\Delta }{m}_e/m_e)(z)$ given target values of the CMB-only best-fit Hubble constant $H_0$, using Planck data [1] alone. All solutions are constructed to keep the Planck best-fit chi-squared unaffected. The solution with a best-fit consistent with SH0ES [2] $H_0^{\mathrm{BF}}=H_0^{\mathrm{SH}0\mathrm{ES}}\equiv 73.04\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ (black curve), is denoted as $\mathrm{\Lambda }\mathrm{CDM}+m_e(z)$ model in the text.

Figure 2

Posteriors of $H_0$ (top), $S_8$ (middle), and ${\mathrm{\Omega }}_m$ (bottom) inferred from Planck full likelihood, together with SH0ES, KiDS-1000, and PantheonPlus results, shown as gray bands.