  $H_0$ 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 $9.0\sigma$ ($\Delta H_0 = 7.8\rm{\ km\ s^{-1}\
Mpc^{-1}}$).

Cosmology and Nongalactic Astrophysics