\section{Introduction}\label{sec: intro}
How do galaxy surveys measure the Hubble constant? \new{Recent} analyses have determined $H_0$ by comparing the angular scale of Baryon Acoustic Oscillations (BAO) with the theoretical size of the sound horizon scale at decoupling, $r_d$. A second `standard ruler' exists however; the equality scale, \textit{i.e.}, the horizon wavenumber
at matter radiation equality, \resub{whose angular scale can be measured from the power spectrum shape and physical scale predicted by theory}.
In this work, we explore the extent to which galaxy surveys can use this scale to place constraints on $H_0$ that are independent of \newww{the sound horizon}.

Until recently, precise $H_0$ constraints have stemmed from two sources:
the Cepheid-calibrated local distance ladder \citep[e.g.,][]{2019ApJ...876...85R,2020AJ....160...71S} and anisotropies in the cosmic microwave background (CMB) \citep[e.g.,][]{2020A&A...641A...6P}. Today, a host of additional constraints are available, arising from data-sets such as galaxy and Lyman-alpha (Ly$\alpha$\xspace) BAO \citep[e.g.,][]{2017MNRAS.470.2617A,2019JCAP...10..044C,2018MNRAS.480.3879A,2019ApJ...874....4A,2019JCAP...10..029S,2018ApJ...853..119A,2020arXiv200708991E}, strong gravitational lensing \citep[e.g.,][]{2019MNRAS.498.1420W,2020arXiv200702941B}, and gravitational wave observations
\citep{2017Natur.551...85A,2019NatRP...2...10R}. 
Broadly, these fall into two camps: `indirect' measurements, which require a full cosmological model for interpretation; and `direct' probes, independent of early-universe physics.\footnote{These are sometimes classified as `early' and `late' measurements respectively, but the terminology can be \resub{confusing}, since an `early' measurement might not involve high-redshift data-sets.} Probes in the former category, including CMB and calibrated BAO, usually derive information from the sound-horizon at recombination,\footnote{For the purposes of this work, there is little difference between `recombination' and `decoupling'; we thus use the terms interchangeably.} calculated assuming $\Lambda$CDM\xspace. Previously, a tension between direct and indirect measurements seemed apparent; however, \resub{this distinction has become less clear} with the latest results from the TRGB-calibrated distance ladder \citep{2019ApJ...882...34F}, strong-lensing \citep{2020arXiv200702941B} and \resub{recalibrated mega-maser results \citep{2020arXiv201001119B}}. Nevertheless, there remains significant disagreement between indirect probes and the SH0ES distance ladder measurements \citep{2019ApJ...876...85R}, reaching a significance of $\sim 5\sigma$ \citep{2019NatAs...3..891V}.








Two primary possibilities exist to resolve this: (a) unresolved systematics \citep[e.g.,][]{2020arXiv200710716E}; or (b) incompleteness of the cosmological model. For the latter, a wide variety of new-physics models have been proposed; many of these
resolve the tension by providing mechanisms to reduce the sound horizon at recombination.
As yet, there is no generally accepted solution.



Ref.\,\citep{2020arXiv200704007B} proposed a new method to shed light on the discrepancy, using CMB lensing to measure $H_0$ without the sound horizon scale; constraints were instead derived from the 
angular equality scale, $L_\mathrm{eq}$. Being simply the projected wavenumber of modes entering the horizon at $z_\mathrm{eq}\sim 3400$, this produces a definitive feature in the convergence spectrum, \newww{and can be used as a standard ruler}. 
Importantly, the equality scale is sensitive to different redshifts than those of CMB and BAO analyses ($z\sim 1100$).  
This yields an important test: inconsistency of equality- and recombination-based $H_0$ constraints would give evidence for physics beyond $\Lambda$CDM\xspace operating \newww{at $z\gtrsim 10^3$}.
Combining \textit{Planck} lensing with cosmological priors on $\Omega_m$ and $A_s$, Ref.\,\citep{2020arXiv200704007B} obtained $H_0 = 73.8\pm 5.1\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$; unfortunately, the projected improvements from future surveys were modest \new{owing to the intrinsically large cosmic variance.

Since the number of modes available to a 3D galaxy survey is typically much greater than for CMB lensing, one might expect stronger constraints on $H_0$ from this avenue: indeed, this was the primary source of $H_0$ information from galaxy surveys two decades ago (\citep[e.g.,][]{1997PhRvL..79.3806T,2001MNRAS.327.1297P} \resub{and Ref.\,\citep{2013MNRAS.429.1902P} for a more recent attempt}). In this work, we perform such a measurement with modern surveys


\resub{This paper has the following structure. In Sec.\,\ref{sec: physics} we discuss the physics behind our approach, before considering the data-sets and analysis pipeline in Sec.\,\ref{sec: data}. Results are presented in Sec.\,\ref{sec: results}, both for the BOSS data and idealized mock catalogs, before we present a discussion in Sec.\,\ref{sec: discussion}.}

\section{Equality and The Sound Horizon}\label{sec: physics}
A glance at the matter power spectrum reveals two features: the broadband peak at wavenumber $k_\mathrm{eq}\sim 10^{-2}h\,\mathrm{Mpc}^{-1}$ and the oscillatory behavior with period $\Delta k \sim 0.05h\,\mathrm{Mpc}^{-1}$. The behavior around $k_\mathrm{eq}$ is well known, arising from the transition between modes that enter the horizon in radiation-dominated and matter-dominated epochs.
In galaxy surveys, resolving the peak is difficult (though possible with experiments such as SPHEREx \citep{2014arXiv1412.4872D}), due to relativistic effects and integral constraints \citep[e.g.,][]{2019JCAP...08..036D},
alongside cosmic variance \resub{and imaging systematics}. More generally, $k_\mathrm{eq}$ information is encoded in the shape of the power spectrum, and can be inferred from smaller \resub{(though still linear)} scales,
as seen by the approximate scaling solution to the linearly-biased perturbation equations:
\begin{eqnarray}\label{eq: meszaros}
    P_g(k) \approx \begin{cases}b_1^2A_s\left(\frac{k}{k_\mathrm{eq}}\right)^{n_s} & k < k_\mathrm{eq}\\ b_1^2A_s\left(1+\log\left(\frac{k}{k_\mathrm{eq}}\right)\right)^2 \left(\frac{k}{k_\mathrm{eq}}\right)^{n_s-4} & k > k_\mathrm{eq}.\end{cases}
\end{eqnarray}
This is facilitated in part by the addition of amplitude and bias information from redshift-space distortions or priors.\footnote{\resub{If the transfer function was a pure power-law we would expect full $A_s-k_\mathrm{eq}$ degeneracy; the logarithmic shape for $k>k_\mathrm{eq}$ reduces this, though we note that measuring $k_\mathrm{eq}$ in this manner is inherently model-dependent.}}


In $\Lambda$CDM\xspace, the equality scale is simply related to cosmological parameters;
\begin{eqnarray}
    k_\mathrm{eq} = \left(2\Omega_{cb}H_0^2z_\mathrm{eq}\right)^{1/2},\,\, z_\mathrm{eq} = 2.5\times 10^{4}\Omega_{cb} h^2\Theta_{2.7}^{-4}
\end{eqnarray}
\citep{1998ApJ...496..605E,2019JCAP...11..034C}, where $\Theta_{2.7}\equiv T_\mathrm{CMB}/(2.7\,\mathrm{K})$ is the temperature of the CMB monopole, $\Omega_{cb}\equiv \Omega_{cdm} + \Omega_b$ (assuming neutrinos to be relativistic at $z_\mathrm{eq}$) \resub{is the CDM+baryon density fraction} and $h\equiv H_0\,/\,(100\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1})$. Measuring $k_\mathrm{eq}$ in $h\,\mathrm{Mpc}^{-1}$ units probes the combination $\Omega_{cb} h\equiv (\omega_{cdm}+\omega_b)/h$, or, marginalizing over $\omega_{b}$, $\omega_{cdm}/h$. \neww{Given $k_\mathrm{eq}$ and a probe of $\Omega_{cb}$ \resub{(or, more commonly, $\Omega_m$)} we can thus solve for the Hubble constant.} 


Complicating this is the second scale:
the sound horizon at $z_d$, the redshift of photon-baryon decoupling. This is given by
\begin{eqnarray}\label{eq: r-d-approx}
    r_d \equiv r_s(z_d) &=& \int_{z_d}^{\infty} \frac{c_s(z)}{H(z)}dz\\\nonumber
    &\approx& \frac{55.154h\exp\left[-72.3(\omega_\nu+0.0006)^2\right]}{\omega_{cb}^{0.25351}\omega_b^{0.12807}}\,h^{-1}\mathrm{Mpc}
\end{eqnarray}
\citep{2015PhRvD..92l3516A}, where $H(z)$ and $c_s(z)$ are the Hubble parameter and sound-speed. $r_d$ sources two main features: the BAO wiggles with $\Delta k \approx 0.05h\,\mathrm{Mpc}^{-1}$, and a small-scale suppression of power \newww{on the baryonic Jeans scale} \citep{2006PhR...429..307L}. Both have amplitudes scaling as $\omega_b/\omega_{cb}$ and could be used to infer the \new{physical scale} of the sound horizon.
\neww{In combination with the measured angular scale, this constrains $H_0$.}
From \eqref{eq: r-d-approx}, such measurements carry the degeneracy $\omega_{cdm}\propto h^{4}$ (after $\omega_b$ marginalization), and a measurement using the Jeans suppression is degenerate with $n_s$. 

An $H_0$ measurement from the full-shape (FS) of the galaxy power spectrum will include information from both $r_d$ and $k_\mathrm{eq}^{-1}$ standard rulers, whilst BAO analyses are sensitive only to $r_d$. To extract only information deriving from equality,
one may wish to `marginalize over the sound horizon'; this is non-trivial 
since $r_d$ is not a direct input to any Boltzmann code, emerging only following simplifications such as tight-coupling.\footnote{An \textit{ad-hoc} rescaling of $r_d$ would be dangerous, as it would not conserve the stress-energy $T_{\mu\nu}$; instead, one should self-consistently add any new physics model to the perturbation equations.} Here, we limit the $H_0$ information arising from the sound horizon simply by removing the usual informative prior on $\omega_b$, and thus the external $r_d$ calibration \resub{even if $\omega_{cb}$ is known precisely}.
For future data this may be insufficient: the BAO features and Jeans suppression can, in principle, calibrate each other, sourcing an effective sound-horizon prior





\section{Data-sets and Analysis}\label{sec: data}
\subsection{Redshift-Space Power Spectrum}\label{subsec: pk-data}
Our main observational data-set is the twelfth data release (DR12) \citep{2017MNRAS.470.2617A} of the Baryon Oscillation Spectroscopic Survey (BOSS), part of SDSS-III \citep{2011AJ....142...72E,2013AJ....145...10D}. Split across two redshift bins \resub{(at $z = 0.38, 0.61$)} in each of the Northern and Southern galactic caps, the survey contains $\sim 1.2\times 10^6$ galaxy positions with a total volume of $5.8h^{-3}\mathrm{Gpc}^3$. Here, we use the
(unreconstructed) power spectrum monopole and quadrupole,\footnote{\href{https://fbeutler.github.io/hub/boss_papers.html}{fbeutler.github.io/hub/boss\_papers.html}} each in 48 $k$-bins for $k\in[0.01,0.25]h\,\mathrm{Mpc}^{-1}$, with
covariances generated from a suite of 2048 MultiDark-Patchy mocks \citep{2016MNRAS.456.4156K,2016MNRAS.460.1173R}, using 
the cosmology $\{\Omega_m = 0.307115,\Omega_b = 0.048,\sigma_8 = 0.8288, h= 0.6777, \sum m_\nu = 0\,\mathrm{eV}\}$.\footnote{\resub{We caution that the spectra may have non-trivial imaging systematics at low-$k$; investigation of these is beyond the scope of this work.}} 

To extract maximal shape information, we model $P_\ell(k)$ with
the Effective Field Theory of Large Scale Structure, following
Ref.\,\citep{2020JCAP...05..042I} (see also Ref.\,\citep{2020JCAP...05..005D}). This includes
one-loop perturbation theory, infra-red resummation of long-wavelength modes, and counterterms parametrizing the impact of small-scale physics. The model is convolved with the survey window function, and incorporates Alcock-Paczynski (AP) effects \citep{1979Natur.281..358A}.\footnote{We use a fiducial value of $\Omega_m = 0.31$ to apply the AP rescaling to the BOSS data.}
The procedure has been used in a number of works \citep{2020PhRvD.101h3504I,2019JCAP...11..034C,2020JCAP...05..032P,2020arXiv200611235I,om4}, including a rigorous test on huge volume simulations \citep{2020arXiv200308277N}, \resub{showing any theory error to be strongly subdominant to the BOSS statistical error}. Here, we utilize the \texttt{CLASS-PT} implementation \citep{2020PhRvD.102f3533C}, with MCMC performed using \texttt{montepython} v3.3 alongside heavily optimized public likelihoods,\footnote{\href{https://github.com/michalychforever/lss_montepython}{github.com/michalychforever/lss\_montepython}.} \resub{with convergence assumed once the Gelman-Rubin diagnostic is below $1.05$}.




We vary the parameter set
\begin{eqnarray}
    &&\{h, \omega_{cdm}, \omega_b, A_s/A_{s,\mathrm{Planck}}, n_s, \sum m_\nu\}\\\nonumber
    &&\times\quad\{b_1,b_2,b_{G_2},b_4,c_{s,0},c_{s,2},P_\mathrm{shot}\}
\end{eqnarray}
where $A_{s,\mathrm{Planck}} = 2.0989\times 10^{-9}$, and $\sum m_\nu$ is the summed mass of three degenerate neutrinos. \newww{To aid convergence, we add a Gaussian prior on $\omega_b$ of width $50\%$ \resub{centered at the \textit{Planck} best-fit} and flat priors of $[0,0.18]\,\mathrm{eV}$ and $[0.87,1.07]$ to $\sum m_\nu$ and $n_s$ respectively.}\footnote{\resub{Initial testing showed that these do not significantly affect the $H_0$ constraints; indeed we can increase the $\omega_b$ prior width by a factor of $2$ without changing $\sigma_{H_0}$, though this leads to slower convergence. This is further discussed in Sec.\,\ref{subsec: mock-constraints}.}} The second line gives nuisance parameters of the EFT model, which are allowed to vary independently in each of the four data \neww{patches} \resub{necessitated by their differing redshifts and calibrations},
subject to the weak Gaussian priors of Ref.\,\citep{2020JCAP...05..042I}. Those entering the likelihood linearly ($b_4$, $c_{s,0}$, $c_{s,2}$ and $P_\mathrm{shot}$) are marginalized analytically \citep{2002MNRAS.335.1193B,2010MNRAS.408..865T,om4}, reducing the total number of
sampled parameters to $6 + 3\times 4 = 18$.



Later, we will require mock data \resub{to explore the information content of our model}.
This is generated from the theory model using the baseline cosmology $\{\omega_{cdm} = 0.118, \omega_b = 0.022, A_s/A_{s,\mathrm{Planck}} = 1.025, h = 0.6777, n_s = 0.9649, \sum m_\nu = 0.06\,\mathrm{eV}\}$, similar to the MultiDark-Patchy parameters, but with massive neutrinos. Three samples
are created: (1) fiducial; (2) with negligible BAO wiggles; and (3) with fewer baryons (to reduce both BAO and baryon damping effects). For each, we fit nuisance parameters to the observed $P_\ell(k)$ and do not include \neww{noise}, \resub{such that all data-sets may be simply compared.}
Set (2) is generated by \neww{increasing the BAO damping scale by}
$1000\times$,
whilst (3) reduces $\Omega_b$ by $10\times$ \neww{relative to the fiducial value},
keeping
$\omega_{cdm}$ and $A_s$ fixed. \resub{When performing parameter inference, all mock data-sets are analyzed using the MultiDark-Patchy covariance matrix described above.}


\subsection{Cosmological Priors}\label{subsec: cosmo-priors}
Equality based measurements of $H_0$ are assisted by information on $A_s$ (to constrain $k_\mathrm{eq}$) and $\Omega_{cb}$ (to break the $H_0-\Omega_{cb}$ degeneracy).\footnote{Knowledge of $b_1$ is also useful; this is provided by redshift-space distortions.} For the former, we employ a weak Gaussian prior of $A_s = (2.11\pm 0.36)\times 10^{-9}$, centered on the \textit{Planck} best-fit \citep{2020A&A...641A...6P}. The $r_d$-dependence of this is minimal, since the CMB measurement is limited by the optical depth and hence derives from very large scales; however, to be maximally conservative, we choose the prior width to be $10\times$ that of the \textit{Planck} constraint.



For the $\Omega_m$ prior, we principally use the marginalized result from Pantheon supernovae: $\Omega_m = 0.298\pm 0.022$ \citep{2018ApJ...859..101S}. 
This cannot constrain $H_0$ directly, since the supernova absolute magnitudes are unknown. An alternative source of $\Omega_m$ information is given by
\textit{uncalibrated} BAO measurements. A standard BAO analysis proceeds by comparing the radial and angular oscillatory scales to the $\Lambda$CDM\xspace sound horizon,
providing information on $\Omega_m$ and $H_0r_d$ through \newww{the evolution of} the angular diameter distance and Hubble parameter
\citep[e.g.,][]{2019JCAP...10..044C}.
To remove the dependence on $r_d$,
we rescale the sound horizon by a free parameter $\alpha_{r_d}$ \resub{when including BAO $H(z)r_d$ and $D_A(z)/r_d$ measurements}. In this formalism, no knowledge of recombination physics is required, just the existence of a time-independent correlation function peak. Here, we use a range of galaxy BAO measurements
from BOSS DR7 \resub{(6dF\resub{GS} and Main Galaxy Samples)} \citep{2011MNRAS.416.3017B,2015MNRAS.449..835R}, and eBOSS DR14 Lyman-alpha measurements, (including cross-correlations with quasars) \citep{2019A&A...629A..85D,2019A&A...629A..86B}. We exclude the BOSS DR12 BAO measurements, since they are covariant with the FS data-set, which would cause additional complications. Alone, the uncalibrated BAO \resub{are found to} give the constraint $\Omega_m = 0.308^{+0.025}_{-0.030}$. \resub{For analyses using mock data, we center the priors on the true parameter values, keeping the same fractional width.}







\subsection{Additional Data-sets}\label{subsec: add-data}








$H_0$ constraints from CMB lensing were demonstrated in Ref.\,\citep{2020arXiv200704007B}.
Due to the presence of projection integrals, the measurements are relatively free from $r_d$-calibration, 
\new{even with a restrictive prior on $\omega_b$.}
The lensing power spectrum measures the combination $L_\mathrm{eq} \sim \Omega_m^{0.6}h$, with a different scaling than that of $k_\mathrm{eq}$, thus we may expect some degeneracy breaking \new{when this is combined with FS measurements.}
Here, we use the public \textit{Planck} 2018 CMB-marginalized lensing likelihood \citep{2020A&A...641A...5P}, assuming zero covariance between this and the BOSS data.\footnote{Technically, some correlation will be present since the probes partially overlap. 
Since the lensing kernel is much broader in redshift space than the BOSS selection function, and CMB lensing is only sensitive to
modes that are perpendicular to the line-of-sight, we expect this to be small.} For analyses including the lensing data-set, we impose a twice tighter prior on $A_s$ of $(2.11\pm 0.18)\times 10^{-9}$ (as in Ref.\,\citep{2020arXiv200704007B}), to break the significant $A_s-L_\mathrm{eq}$ degeneracy.



\section{Results}\label{sec: results}

\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{obs_data3b.pdf}
  \caption{\small Parameter constraints from analyses of BOSS DR12 power spectra and \textit{Planck} lensing (as in Ref.\,\citep{2020arXiv200704007B}).
  Data-sets are combined with either Pantheon supernovae or uncalibrated BAO to provide $\Omega_m$ information, but no $\omega_b$ prior is assumed, such that the $H_0$ constraints do not derive information from the sound horizon (as evidenced by subsequent figures). Dashed lines show linear relationships and vertical bands give the SH0ES $H_0$ constraint \citep{2019ApJ...876...85R}. We omit the posteriors for $n_s$, $\sum m_\nu$ and the 12 nuisance parameters for clarity. \neww{Following the caption ordering}, $68\%$ $H_0$ confidence intervals are $65.1^{+3.0}_{-5.4}$, $65.6^{+3.4}_{-5.5}$, $70.6^{+3.7}_{-5.0}$ and $73.4\pm 6.1$ respectively, in $\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ units.
 
  }
  \label{fig: obs-data}
\end{figure}%

\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{mock_data3.pdf}
  \caption{\small As Fig.\,\ref{fig: obs-data}, but for three mock data-sets: fiducial; with BAO wiggles removed; and with
  $\omega_b$ reduced by a factor of 10. All analyses include Pantheon priors on $\Omega_m$. 
  Thin lines show the true parameter values, and the $1\sigma$ constraints on $H_0$ are $65.6^{+3.7}_{-5.3}$, $65.9^{+3.7}_{-4.6}$ and $66.2^{+4.1}_{-5.6}$ in $\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ units. The insensitivity of $H_0$ constraints to the removal of BAO wiggles and baryon damping implies that information is not being sourced from the sound horizon scale
 
 
  }
  \label{fig: mock-data}
\end{figure}

\subsection{Constraints from current data-sets}\label{subsec: current-constraints}
Fig.\,\ref{fig: obs-data} shows the cosmological constraints obtained.
Combining BOSS power spectra with Pantheon $\Omega_m$ priors, 
we obtain $H_0 = 65.1^{+3.0}_{-5.4}\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$, below the best-fit SH0ES value at a 95\% confidence level (including non-Gaussianity of the posterior), even though our analysis \resub{is not based on the sound horizon}
\footnote{\newww{This corresponds to $k_\mathrm{eq} = \left(1.40^{+0.10}_{-0.14}\right)\times 10^{-2}\,h\,\mathrm{Mpc}^{-1}$.}} 
As expected, the $\omega_b$ posterior is broad, since, unlike in previous analyses, we have not imposed a restrictive prior. There is a strong $h-\omega_{cdm}-\omega_b$ degeneracy, close to the expected linear relationship, rather than the $\omega_{cdm}\propto h^{4}$ scaling of $r_d$-calibration. The $\omega_b-\omega_{cdm}$ degeneracy indicates that a small amount of $\omega_{cdm}$ information comes from the BAO wiggles, though, as argued below, we do not expect this to inform our $H_0$ constraints. \newww{We note little dependence on the $A_s$ prior, with $< 10\%$ degradation in $\sigma_{H_0}$ if this is removed; this is expected since $A_s$ can be measured from the power spectrum through the loop corrections.} 


Using uncalibrated BAO instead of the Pantheon sample gives a similar posterior; $H_0 = 65.6^{+3.4}_{-5.5}\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$. This is unsurprising; the sound horizon rescaling parameter, $\alpha_{r_d}$, removes the $H_0$ information, and the marginalized $\Omega_m$ \neww{constraint} from BAO alone is similar to Pantheon.
Interestingly, the $\alpha_{r_d}$ posterior is $0.993\pm 0.016$; the combination of equality-based power spectra and (independent) uncalibrated BAO prefer a sound horizon consistent with $\Lambda$CDM\xspace.


Combination with \textit{Planck} lensing shifts the $H_0$ posterior to larger values, with a marginalized limit of $70.6^{+3.7}_{-5.0}\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$. Due to the addition of galaxy information, this is somewhat tighter than the lensing-only constraints of Ref.\,\citep{2020arXiv200704007B}, though there is no improvement relative to the $P_\ell(k)$ posteriors, due to the broad error bars \neww{on the CMB lensing measurements} and similar degeneracy directions. Note that the lensing-only constraint \new{($H_0 = 73.4\pm 6.1\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1} $)} is shifted somewhat from that of Ref.\,\citep{2020arXiv200704007B}, due to slightly different prior choices.





\subsection{Dependence on $r_d$}\label{subsec: mock-constraints}


We now demonstrate
that our $H_0$ constraints do not receive significant information from the sound horizon, using three tests: repeating the analysis on mock data-sets without baryon oscillations and damping; employing scale cuts; and performing Fisher forecasts, where we can explicitly marginalize over $r_d$.

First, we turn to the synthetic data-sets discussed above.
As shown in Fig.\,\ref{fig: mock-data}, our $H_0$ constraints are negligibly impacted by removing BAO wiggles or reducing baryonic damping. Since the mock data are generated to match the BOSS spectra, this is a strong indication that our $H_0$ constraints are independent of sound horizon physics.\footnote{\resub{That the constraints are not affected by the removal of \textit{all} baryon information also indicates that the weak prior placed on $\omega_b$ in the real analysis is not affecting the $H_0$ posterior in the fiducial analysis.}} Note that the best-fit values of $\omega_{cdm}$ and $h$ are shifted \new{by $\sim 0.5\sigma$} from the truth;
this indicates a (modest) prior-volume effect \resub{due to non-Gaussianity of the high-dimensional posterior}, confirmed by its removal when reanalyzing the data with a covariance appropriate for a $10\times$ larger survey. Whilst this could be ameliorated by stricter nuisance parameter priors, given that the offsets are small, we do not include these.
The mocks also highlight the importance of $\Omega_m$ priors; since the FS likelihood sources $\omega_{cdm}$ information from BAO wiggles, the no-wiggle constraints on $H_0$ would degrade if an external prior was not present.



\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{cut_data3b.pdf}
  \caption{\small As Fig.\,\ref{fig: obs-data}, but restricting the $k$ range of the BOSS power spectrum analysis from the fiducial value of $k_\mathrm{max} = 0.25h\,\mathrm{Mpc}^{-1}$. All analyses include the Pantheon $\Omega_m$ prior. For the $k_\mathrm{max} = 0.1h\,\mathrm{Mpc}^{-1}$ data-sets, $H_0$ constraints are $65.6^{+3.8}_{-5.8}$ (BOSS) and $71.2^{+4.3}_{-5.7}$ (BOSS + \textit{Planck}) in $\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ units. The lack of dependence of $H_0$ on $k_\mathrm{max}$ again indicates that information is not arising from the BAO scale
  }
  \label{fig: cut-data}
\end{figure}%

Scale cuts provide further evidence to support our conclusions. Fig.\,\ref{fig: cut-data} shows the effect of reducing $k_\mathrm{max}$ from $0.25h\,\mathrm{Mpc}^{-1}$ to $0.10h\,\mathrm{Mpc}^{-1}$, which, if information were coming from BAO wiggles, would be expected to \resub{significantly} inflate the $H_0$ posterior \resub{\citep[e.g.,][]{2020JCAP...05..042I}}. Notably, the reduction in constraining power is slight \new{($\sim 10\%$)}, though the nuisance parameters of the one-loop model suffer significant posterior inflation. Again, this indicates that the primary information is sourced by $k_\mathrm{eq}^{-1}$ rather than $r_d$ \resub{(and thus relatively large scales, though we note the lowest $k$-modes have limited impact due to their large statistical error)}.


\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{fisher_data.pdf}
  \caption{\small Posteriors from a simplified Fisher forecast mimicking the BOSS + Pantheon results, including explicit marginalization over the sound horizon, $r_d$, using the Eisenstein-Hu transfer function \citep{1998ApJ...496..605E}. 
  We obtain $\sigma_{H_0} = 5.9$ ($5.5$) $\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1} $ with (without) $r_d$-marginalization; the small size of this degradation supplies further evidence that our constraints are insensitive to the sound horizon scale.}
 
 
 
 
  \label{fig:fisher_results}
\end{figure}%

Finally, we consider a simplified Fisher analysis in which $r_d$ can be marginalized over exactly. This is made possible by using the Eisenstein-Hu transfer function \citep{1998ApJ...496..605E} in \texttt{CLASS-PT}, rather than the usual output from \texttt{CLASS}. 
In addition to the five cosmological parameters $\{h, \omega_{cdm}, \omega_b, A_s/A_{s,\mathrm{Planck}}, n_s\}$ (the Eisenstein-Hu approximation does not allow for massive neutrinos) we vary a sound horizon rescaling parameter $\beta_{r_d}$, \resub{which rescales $r_d$ within the transfer function}.
For simplicity, a single redshift \new{bin}
(centered at $z = 0.51$) with the total BOSS volume is used, and window function effects are ignored.
As seen in Fig.\,\ref{fig:fisher_results}, the $H_0$ posteriors are broader than those found in Fig.\,\ref{fig: obs-data}; this is due to the assumptions of an Eisenstein-Hu model and exclusion of redshift evolution, yet the model retains qualitative utility. Marginalization over $r_d$ has little effect, reducing $\sigma_{H_0}$ by $< 10\%$, reinforcing our conclusions that the $H_0$ constraints are insensitive to $r_d$. Without an $\Omega_m$ prior the marginalization gives significant degradation, with $\sigma_{H_0} = 11\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$; in this case, $\Omega_m$ information enters from the BAO wiggle \textit{amplitudes} which are washed-out by the marginalization.

\resub{The Fisher formalism may also be used to test the dependence of the $H_0$ constraints on the range of $k$-modes included in the analysis. Given that data systematics are concentrated in the first few $k$-bins, this is a useful probe of our sensitivity to such effects. Rerunning the above forecast (without marginalization over $\beta_{r_d}$) increasing $k_\mathrm{min}$ from $0.01h\,\mathrm{Mpc}^{-1}$ to $0.03h\,\mathrm{Mpc}^{-1}$ ($0.05h\,\mathrm{Mpc}^{-1}$) gives an $H_0$ posterior inflated by $14\%$ ($35\%$). We thus stress that the constraints found in this work have greater dependence on large-scale modes (and thus any present systematics) than for most BAO-only analyses. This is as expected since most of the $k_\mathrm{eq}$ information is wavenumbers in the linear regime, yet large enough to avoid excessive cosmic variance. We also note that the constraints are not affected by removal of the weak $\omega_b$ prior for $k_\mathrm{min} = 0.01h\,\mathrm{Mpc}^{-1}$ and $0.03h\,\mathrm{Mpc}^{-1}$, but suffer $\sim 10\%$ inflation if $k_\mathrm{min} = 0.05h\,\mathrm{Mpc}^{-1}$. This indicates that the results are prior-limited only if most of the large-scale power is removed.}





\subsection{Forecasting for Future Surveys}\label{subsec: future-forecast}
\newww{To estimate the potential of future surveys to constrain $H_0$ without the sound horizon, we perform a simplistic Fisher analysis, similar to that presented above. In particular, we consider a Euclid-like survey in eight redshift bins, taking the volumes and fiducial bias parameters from the forecast of Ref.\,\citep{Chudaykin2019}. For consistency, we slightly expand our $k$-range up to $k_\mathrm{max}=0.3h\,\mathrm{Mpc}^{-1}$ and do not impose nuisance parameter priors. Adopting the $A_s$ and $\Omega_m$ priors of this work, and marginalizing over $r_d$, we obtain $\sigma_{H_0}\sim 1.7\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$; this tightens to $\sim 1.6$ $\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ with the more optimistic $\sigma_{\Omega_m} = 0.012$ prior of Ref.\,\citep{2020arXiv200704007B}.
For future surveys, it is unclear whether removing the $\omega_b$ prior will be sufficient to ensure $r_d$-independence; this will be discussed in future work alongside a more complete forecast.




\section{Discussion}\label{sec: discussion}
In the past decade, galaxy surveys have focused on measuring BAO. In this work, we make use of the fact 
that an additional standard ruler is present; the horizon size at matter-radiation equality, $k^{-1}_\mathrm{eq}$. Combining galaxy power spectra from BOSS with cosmological priors on $\Omega_m$ gives equality-based constraints of $H_0 = 65.1^{+3.0}_{-5.4}\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ (power spectrum only) and $70.6^{+3.7}_{-5.0}\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$ (adding \textit{Planck} lensing). For BOSS such a measurement can be obtained simply by analyzing the data without use of an informative $\omega_b$ prior; we demonstrate this using mock catalogs, scale cuts and Fisher forecasts. For the next generation of surveys, simple forecasts indicate that sound horizon independent constraints of \newww{$\sigma_{H_0} \simeq 1.6\,\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$} should be possible; more sophisticated techniques may be required to remove $r_d$ information, however.






To close, we consider implications for the `Hubble tension'. Most proposed mechanisms for its resolution rely on modifying the sound horizon at recombination, and thus altering the BAO scale. Given that equality-based measurements are sensitive to higher redshifts than BAO measurements, $H_0$ constraints anchored at $z_d$ and $z_\mathrm{eq}$ may differ if new physics is at work,
making this a valuable test of new physics prior to recombination. Here, we find good agreement between galaxy-only $H_0$ measurements derived from the sound horizon \resub{\citep[e.g.,][]{2020JCAP...05..042I,2020JCAP...05..032P}} and equality scales, both of which favor lower values than those of SH0ES. If this consistency holds to much higher precision, it will \newww{place strong bounds on many}
beyond-$\Lambda$CDM\xspace 
resolutions of the Hubble tension.\\




\begin{acknowledgments}
\footnotesize
We thank Mikhail Ivanov, Julien Lesgourgues and Marko Simonovi\'c for insightful discussions; we are additionally grateful to Jo Dunkley, Dragan Huterer, Eiichiro Komatsu and Matias Zaldarriaga for comments on a draft of this manuscript, \resub{and to \resub{Adam Riess} and the anonymous referees for feedback on the original submission}. OHEP would like to thank the Max Planck Institute for Astrophysics for hospitality when this work was being finalized. OHEP acknowledges funding from the WFIRST program through NNG26PJ30C and NNN12AA01C.
GSF acknowledges support through the Isaac Newton Studentship and the Cambridge Trust Vice Chancellor's Award. BDS acknowledges support from an Isaac Newton Trust Early Career Grant, from a European Research Council (ERC) Starting Grant under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 851274), and from an STFC Ernest Rutherford Fellowship.
\end{acknowledgments}




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