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The Hubble tension can be addressed by modifying the sound horizon (rₒ) before recombination, triggering interest in early universe estimates of the Hubble constant, H₀, independent of rₒ. Constraints on H₀ from an rₒ-free analysis of the full shape BOSS galaxy power spectra within were recently reported and used to comment on the viability of physics beyond. Here we demonstrate that rₒ-free analyses with current data depend on both the model and the priors placed on the cosmological parameters, such that analyses cannot be used as evidence for or against new physics. We find that beyond- models which introduce additional energy density with significant pressure support, such as early dark energy (EDE) or additional neutrino energy density (N₄₅₅), lead to rₒ-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 (m₄), produce H₀ constraints similar to. Using BOSS data, constraints from light element abundances, cosmic microwave background (CMB) lensing, a CMB-based prior on the primordial scalar amplitude (Aₒ), spectral index (nₒ), and ₌ from the Pantheon+Type Ia supernovae dataset, we find that in, H₀=64. 92. 2 km/s/Mpc; in EDE, H₀=68. 7-₃. ₉^+3; in N₄₅₅, H₀=68. 1-₃. ₈^+2. 7; and in m₄, 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, H₀=67. 91. 7; in EDE, H₀=72. 2-₃. ₈^+2. 9; in N₄₅₅, H₀=71. 5-₃. ₃^+2. 5; and in m₄, H₀=68. 01. 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 rₒ marginalization are inconclusive with respect to the viability of beyond models.
Smith et al. (Tue,) studied this question.