Assessing the robustness of sound horizon-free determinations of the Hubble constant

The Hubble tension can be addressed by modifying the sound horizon (rₛ) before recombination, triggering interest in early universe estimates of the Hubble constant, H₀, independent of rₛ. Constraints on H₀ from an rₛ-free analysis of the full shape BOSS galaxy power spectra within ΛCDM were recently reported and used to comment on the viability of physics beyond ΛCDM. Here we demonstrate that rₛ-free analyses with current data depend on both the model and the priors placed on the cosmological parameters, such that ΛCDM analyses cannot be used as evidence for or against new physics. We find that beyond-ΛCDM models which introduce additional energy density with significant pressure support, such as early dark energy (EDE) or additional neutrino energy density (ΔNeff), lead to rₛ-free values of H₀ which are larger by 3--4 km/s/Mpc. On the other hand, models which only affect the time of recombination, such as a varying electron mass (Δmₑ), produce H₀ constraints similar to ΛCDM. Using BOSS data, constraints from light element abundances, cosmic microwave background (CMB) lensing, a CMB-based prior on the primordial scalar amplitude (Aₛ), spectral index (nₛ), and Ωₘ from the Pantheon+Type Ia supernovae dataset, we find that in ΛCDM, H₀=64.9±2.2 km/s/Mpc; in EDE, H₀=68.7_-3.9⁺³; in ΔNeff, H₀=68.1_-3.8+2.7; and in Δmₑ, H₀=64.7_-2.3+1.9. Using a prior on the angular size of the sound horizon at baryon drag from BAO and CMB measurements, these values become in ΛCDM, H₀=67.9±1.7; in EDE, H₀=72.2_-3.8+2.9; in ΔNeff, H₀=71.5_-3.3+2.5; and in Δmₑ, H₀=68.0±1.7. With current data, none of the models are in significant tension with SH₀ES, and consistency tests based on comparing H₀ posteriors with and without rₛ marginalization are inconclusive with respect to the viability of beyond ΛCDM models.