Abstract
Analyses of the full shape of BOSS DR12 power spectrum using the one-loop prediction from the Effective Field Theory of Large-Scale Structure (EFTBOSS) have led to new constraints on extensions to the $\Lambda$CDM model, such as Early Dark Energy (EDE) which has been suggested as a resolution to the "Hubble tension". In this paper, we re-assess the constraining power of the EFTBOSS on EDE in light of a correction to the normalization of BOSS window functions. Overall we find that constraints from EFTBOSS on EDE are weakened, and represent a small change compared to constraints from Planck and the conventional BAO/$f\sigma_8$ measurements. The combination of Planck data with EFTBOSS provides a bound on the maximal fractional contribution of EDE $f_{\rm EDE}<0.083$ at 95% C.L. (compared to $<0.054$ with the incorrect normalization, and $<0.088$ without full-shape data) and the Hubble tension is reduced to $2.1\sigma$. However, the more extreme model favored by an analysis with just data from the Atacama Cosmology Telescope is disfavored by the EFTBOSS data. We also show that the updated Pantheon+ Type Ia supernova analysis can slightly increase the constraints on EDE. Yet, the inclusion of the SN1a magnitude calibration by SH0ES strongly increases the preference for EDE to above $5\sigma$, yielding $f_{\rm EDE}\sim 0.12^{+0.03}_{-0.02}$ around the redshift $z_c=4365^{+3000}_{-1100}$. Our results demonstrate that EFTBOSS data (alone or combined with Planck data) do not exclude the EDE resolution of the Hubble tension.