A tale of two (or more) $h$'s

We use the large-scale structure galaxy data (LSS) from the BOSS and eBOSS surveys, in combination with abundances information from Big Bang Nucleosynthesis (BBN) to measure two values of the Hubble expansion rate, $H_0=100h\,[{\rm km}\, {\rm s}^{-1}\,{\rm Mpc}^{-1}]$, each of them based on very different physical processes. One is a (traditional) late-time-background measurement based on determining the BAO scale and using BBN abundances on baryons for calibrating its absolute size (BAO+BBN). This method anchors $H_0$ to the (standard) physics of the sound horizon scale at pre-recombination times. The other is a newer early-time based measurement associated with the broadband shape of the power spectrum. This second method anchors $H_0$ to the physics of the matter-radiation equality scale, which also needs BBN information for determining the suppression of baryons in the power spectrum shape (shape+BBN). Within the $\Lambda$CDM model, we find very good consistency among these two $H_0$'s: BAO+BBN (+growth) delivers $H_0=67.42_{-0.94}^{+0.88}$ $(67.37_{-0.95}^{+0.86})$ km s$^{-1}$Mpc$^{-1}$ , whereas the shape+BBN (+growth) delivers $H_0 = 70.1_{-2.1}^{+2.1}$ $(70.1_{-2.1}^{+1.9})$ km s$^{-1}$ Mpc$^{-1}$, where "growth" stands for information from the late-time-perturbations captured by the growth of structure parameter. These are the tightest sound-horizon free $H_0$ constraints from LSS data to date. As a consequence to be viable, any $\Lambda$CDM extension proposed to address the so-called "Hubble tension" needs to modify consistently not only the sound horizon scale physics, but also the matter-radiation equality scale, in such a way that both late- and early-based $H_0$'s return results mutually consistent and consistent with the high $H_0$ value recovered by the standard cosmic distance ladder (distance-redshift relation) determinations.