Determining the Hubble Constant without the Sound Horizon: Perspectives with Future Galaxy Surveys

$H_0$ constraints from galaxy surveys are sourced by the geometric properties of two standardisable rulers: the sound horizon scale, $r_s$, and the matter-radiation equality scale, $k_{\rm eq}$. While most analyses over the last decade have focused on the first scale, recent work has emphasised that the second can provide an independent source of information about the expansion rate of the universe. Recent approaches to obtain a sound-horizon-independent measurement of $H_0$ from the equality scale have avoided $r_s$-based information by removing the sound-horizon-calibrating prior on the baryon density. We present a new method to marginalise over $r_s$; this allows baryon information to be retained enabling tighter parameter constraints. For a Euclid-like spectroscopic survey, we forecast sound-horizon-independent $H_0$ constraints of $\sigma_{H_0} = 0.7\rm{\ km\ s^{-1}\ Mpc^{-1}}$ for our method using the equality scale, compared with $\sigma_{H_0} = 0.5\rm{\ km\ s^{-1}\ Mpc^{-1}}$ from the sound horizon. Upcoming equality scale $H_0$ measurements thus can be highly competitive, although we caution that the impact of observational systematics on such measurements still needs to be investigated in detail. Applying our new approach to the BOSS power spectrum gives $H_0 = 69.5^{+3.0}_{-3.5}\rm{\ km\ s^{-1}\ Mpc^{-1}}$ from equality alone, somewhat tighter than previous constraints. Consistency of $r_s$- and $k_{\rm eq}$-based $H_0$ measurements can provide a valuable internal consistency test of the cosmological model; as an example, we consider the change in $H_0$ created by early dark energy. Assuming the \textit{Planck}+SH0ES best-fit EDE model we find a $2.6\sigma$ shift ($\Delta H_0 = 2.6\rm{\ km\ s^{-1}\ Mpc^{-1}}$) between the two measurements for Euclid; if we instead assume the ACT best-fit model, this increases to $9.0\sigma$ ($\Delta H_0 = 7.8\rm{\ km\ s^{-1}\ Mpc^{-1}}$).