Recombination-independent determination of the sound horizon and the Hubble constant from BAO

The Hubble tension and attempts to resolve it by modifying the physics of (or at) recombination motivate finding ways to determine $H_0$ and the sound horizon at the epoch of baryon decoupling $r_{\rm d}$ in ways that neither rely on a recombination model nor on late-time Hubble data. In this work, we investigate what one can learn from the current and future BAO data when treating $r_{\rm d}$ and $H_0$ as independent free parameters. It is well known that BAO gives exquisite constraints on the product $r_{\rm d}H_0$. We show here that imposing a moderate prior on $\Omega_{\rm m} h^2$ breaks the degeneracy between $r_{\rm d}$ and $H_0$. Using the latest BAO data, including the recently released eBOSS DR16, along with a $\Omega_{\rm m} h^2$ prior based on the Planck best fit $\Lambda$CDM model, we find $r_{\rm d} =143.7 \pm 2.7$ Mpc and $H_0 = 69.6 \pm 1.8$ km/s/Mpc. BAO data therefore prefers somewhat lower $r_{\rm d}$ and higher $H_0$ than those inferred from Planck data in a $\Lambda$CDM model. We find similar values when combing BAO with the Pantheon supernovae, DES galaxy weak lensing, Planck or SPTPol CMB lensing and the cosmic chronometers data. We perform a forecast for DESI and find that, when aided with a moderate prior on $\Omega_{\rm m} h^2$, DESI will measure $r_{\rm d}$ and $H_0$ without assuming a recombination model with an accuracy surpassing the current best estimates from Planck.