# Can small-scale baryon inhomogeneities resolve the Hubble tension? An
investigation with ACT DR4

Leander Thiele<EMAIL_ADDRESS>Department of Physics, Princeton
University, Princeton, NJ, USA 08544  Yilun Guan  Department of Physics and
Astronomy, University of Pittsburgh, Pittsburgh, PA, USA 15260  J. Colin Hill
Department of Physics, Columbia University, 538 West 120th Street, New York,
NY, USA 10027  Center for Computational Astrophysics, Flatiron Institute, 162
5th Avenue, New York, NY, USA 10010  Arthur Kosowsky  Department of Physics
and Astronomy, University of Pittsburgh, Pittsburgh, PA, USA 15260  David N.
Spergel  Department of Astrophysical Sciences, Princeton University, Peyton
Hall, Princeton, NJ, USA 08544  Center for Computational Astrophysics,
Flatiron Institute, 162 5th Avenue, New York, NY, USA 10010

###### Abstract

Small-scale inhomogeneities in the baryon density around recombination have
been proposed as a solution to the tension between local and global
determinations of the Hubble constant. These baryon clumping models make
distinct predictions for the cosmic microwave background anisotropy power
spectra on small angular scales. We use recent data from the Atacama Cosmology
Telescope to test these predictions. No evidence for baryon clumping is found,
assuming a range of parameterizations for time-independent baryon density
probability distribution functions. The inferred Hubble constant remains in
significant tension with the SH0ES measurement.

##  I Introduction

One cornerstone of the $\Lambda$CDM standard model of cosmology is the Hubble
constant $H_{0}$. We may define it in two ways. In a model-agnostic way,
$H_{0}$ measures the rate at which space is expanding at the present day. This
definition enables us to determine $H_{0}$ locally by measuring the apparent
speed of recession of galaxies in the Hubble flow. The other definition
depends on the assumed cosmological model: it interprets $c/H_{0}$ as a
measurement of the angular diameter distance to a given redshift, modulo the
values of the background density parameters in the Friedmann equation.
Measurements of $H_{0}$ according to this definition can be performed from the
cosmic microwave background (CMB) power spectra or the late-time matter
distribution. For both measurements, the baryonic acoustic oscillation (BAO)
feature sets a characteristic length scale.

In the correct cosmological model the two definitions of the Hubble constant
should be consistent, i.e., barring unknown systematics low- and high-redshift
measurements should give consistent results. This, however, appears not to be
the case, as is illustrated in Tab. 1. Roughly, a $>4\sigma$ discrepancy
between the low- and high-redshift inferences is observed.

| dataset | $H_{0}[\text{km}/\text{s}/\text{Mpc}]$ | type
---|---|---|---
local | SH0ES Riess  _et al._ (2021) | $73.2\pm 1.3$ | SNIa+Cepheids
CSP Freedman  _et al._ (2020) | $69.6\pm 0.8\pm 1.7$ | DEB + TRGB
MCP Reid  _et al._ (2009) | $73.9\pm 3.0$ | VLBI geometric
H0LiCOW Wong  _et al._ (2020) | $73.3^{+1.7}_{-1.8}$ | lensing time delay
TDCOSMO Birrer  _et al._ (2020) | $67.4_{-3.2}^{+4.1}$ | lensing time delay
global |  Planck Planck Collaboration (2020a) | $67.4\pm 0.5$ | CMB
ACT ACT Collaboration  _et al._ (2020a) | $67.9\pm 1.5$ | CMB
ACT+WMAP ACT Collaboration  _et al._ (2020a); Bennett  _et al._ (2013) | $67.6\pm 1.1$ | CMB
BOSS Philcox  _et al._ (2020) | $68.6\pm 1.1$ | BAO
Table 1: Selection of recent $H_{0}$ measurements. See Sec. II for comment on
the H0LiCOW measurement.

Currently, the observed discrepancy is the most severe anomaly in the
$\Lambda$CDM standard model, and hence the most promising avenue for the
discovery of new physics in cosmology. Thus, it has spurred a wide range of
attempts at an explanation (so many, in fact, that we now have dedicated
reviews to keep track of them, e.g., Knox and Millea (2020)). Pre-
recombination modifications include the Early Dark Energy models Karwal and
Kamionkowski (2016); Poulin  _et al._ (2019); Agrawal  _et al._ (2019); Lin
_et al._ (2019); Sakstein and Trodden (2020), which however are strongly
constrained by large-scale structure data Hill  _et al._ (2020); Ivanov  _et
al._ (2020); D’Amico  _et al._ (2020). Additional light states may also be
able to accommodate higher $H_{0}$, as long as their dynamics is non-trivial
D’Eramo  _et al._ (2018); Kreisch  _et al._ (2020) (but see Brinckmann  _et
al._ (2020); Choudhury  _et al._ (2020)). Post-recombination modifications
such as dynamical dark energy and modified gravity Huang and Wang (2016); Renk
_et al._ (2017); Zhao  _et al._ (2017); Nunes (2018); Di Valentino  _et al._
(2018); Wang  _et al._ (2018); Khosravi  _et al._ (2019); Raveri (2020);
Banihashemi  _et al._ (2020); Cai  _et al._ (2020); Di Valentino  _et al._
(2020) or a fifth force Desmond  _et al._ (2019) approach the tension from the
opposite point of view, being not always physically well-motivated however,
and often encountering problems with low-redshift expansion history data
Efstathiou (2021).

One attempted resolution of the $H_{0}$-tension involves modifications of
$\Lambda$CDM around the time of recombination that decrease the sound horizon
$r_{s}$. The peaks and troughs in the CMB power spectra are located at wave
numbers scaling as $(H_{0}r_{s})^{-1}$. Thus, a decrease in the sound horizon
allows for higher values of $H_{0}$ as inferred from the CMB. It should be
noted that this picture is somewhat simplistic, since only changing $r_{s}$
introduces other tensions Jedamzik  _et al._ (2020). The model considered in
this work, however, has other secondary effects on the CMB power spectra that
may be able to circumvent this constraint.

The mechanism to achieve the change in $r_{s}$ that we will focus on in this
paper is the introduction of $O(1)$ inhomogeneities in the baryon density
around the time of recombination. These inhomogeneities are required to be on
comoving scales well below a Mpc so that they would not be directly visible in
the currently available CMB data. Even in simple models of recombination
physics it is clear that such small-scale baryon inhomogeneities will shift
the surface of last scattering to higher redshift, thus decreasing the sound
horizon. In fact, Peebles’s model Peebles (1976) contains a term
$\dot{n}_{e}\propto-n_{e}^{2}$, which upon averaging gives faster
recombination in inhomogeneous plasma on account of $\langle
n_{e}^{2}\rangle>\langle n_{e}\rangle^{2}$ Jedamzik and Abel (2011).

One attractive proposal to achieve such a scenario is the introduction of
primordial magnetic fields (PMFs) Banerjee and Jedamzik (2004); Jedamzik and
Abel (2011); Jedamzik and Saveliev (2019); Jedamzik and Pogosian (2020). This
mechanism would have the advantage that it naturally sustains the required
amount of small-scale inhomogeneity through a dynamical process, which in many
other conceivable models would be entirely erased through Silk damping.
Furthermore, theoretically well-motivated PMF configurations would lead to
inhomogeneities of roughly the required amount on kpc scales. The necessary
field strengths would be such that present-day large-scale magnetic fields
could be explained with limited invocation of galactic dynamo effects. One
open question that could be answered through MHD simulations is the effect of
an inverse energy cascade from the turbulent small-scale magnetic fields and
baryon inhomogeneities towards larger scales. If such an energy flow was
appreciable, it would add additional constraints. We emphasize that this work
deals with baryon inhomogeneities in an agnostic way and makes no reference to
PMFs; however, it seems difficult to generate the required inhomogeneities
without invoking PMFs. Furthermore, most other changes to recombination
physics that would lead to appreciable shifts in $H_{0}$ are difficult to
reconcile with atomic physics constraints Liu  _et al._ (2020).

While previous work – Ref. Jedamzik and Pogosian (2020), hereafter JP20 –
restricted CMB data to the results from the Planck satellite Planck
Collaboration (2020a), this paper will include more recent data from the
Atacama Cosmology Telescope (ACT), specifically their data release 4 (DR4) ACT
Collaboration  _et al._ (2020b, a). Although the angular scales covered by
Planck enable tight constraints on the location of the acoustic peaks (and
thus on $H_{0}r_{s}$), they are less constraining with regard to a secondary
prediction of the inhomogeneous models, namely the change in amplitude in the
Silk damping tail Jedamzik and Abel (2011). While faster recombination
decreases the time photons have to diffuse (thus suppressing Silk damping),
helium recombination is also sped up which decreases the ionization fraction
and therefore increases the photon mean free path (thus enhancing Silk
damping). Generically, these two competing effects will not completely cancel;
hence the inhomogeneous models have a characteristic signature on small
angular scales that is not degenerate with the shift in peak positions. For
this reason, the inclusion of small-scale ACT data provides valuable
additional constraints on the proposed resolution of the $H_{0}$ tension. We
may go even further and posit that the new ACT data covering scales that were
not considered in JP20 provide an independent test of the secondary small-
scale effects predicted by this model. If the ACT data do not show evidence of
these effects, then the probability of this model resolving the Hubble tension
is accordingly lessened, compared to a hypothetical additional data set that
covers the same scales as Planck.

As the baryon inhomogeneities are on scales $10^{3}$ times below current
resolution, their effect is completely described by the baryon density
probability distribution function (PDF). While future MHD simulations assuming
PMFs could provide constraints on the shape of this PDF, in this work we treat
it as physical input. This has the advantage that we can treat the baryon
inhomogeneities agnostically with regard to their origin, but we will have to
make certain assumptions on the shape of the PDF in order to retain a
predictive model. Future work will hopefully be able to give physically better
motivated parameterizations of the PDF; however, we believe that the range of
PDFs considered in this work provides strong motivation to believe that our
results are in fact generic.

The rest of this paper is structured as follows. In Sec. II, we review the
three-zone model for the baryon density PDF from JP20 that we adopt in this
work. Futhermore, we explain our choice of likelihoods and summarize the
methods used. Sec. III contains the main results of this work, namely
posteriors on $H_{0}$ and various parameters of the three-zone model for a
variety of likelihoods. In Sec. IV we add local $H_{0}$ measurements to the
CMB likelihoods in order to fully excite the baryon clumping mode and derive
some intuition from the resulting best-fit models. We conclude in Sec. V.

##  II Setup

Following JP20, we parameterize the baryon density PDF as a three-zone model,
where each zone is described by its density $\Delta_{i}=n_{b}^{i}/\langle
n_{b}\rangle$ and its volume fraction $f_{i}$ ($i=1\ldots 3$). We can think of
the comoving coordinates being split into small regions, each of which
realizes one of the three zones.

We introduce the ‘clumping factor’ $b$ as

$1+b=\sum_{i}f_{i}\Delta_{i}^{2}$ (1)

and remind the reader of the constraints $\Sigma_{i}f_{i}=1$ and
$\Sigma_{i}f_{i}\Delta_{i}=1$. The baryon clumping models are defined by four
parameters, because there are three zones, each with density and volume
fraction, and two constraints between these quantities. For physical
convenience, we choose the 4-parameter model space to be spanned by $f_{2}$,
$\Delta_{1}$, $\Delta_{2}$, and the clumping factor $b$ defined in Eq. (1).

In this paper, for ease of calculation and comparison with JP20, we consider
several subsets of these models, with either one or two of the four parameters
being free and the others fixed to specific values, as listed in Tab. 2. These
likely bracket all of the reasonable physical cases, but a more extensive
analysis could consider 4-parameter models instead.

While models M1 and M2 were chosen to facilitate direct comparison with JP20,
the more extended models ($f_{2}=\ldots$) are constructed so as to cover a
relatively wide range of possible baryon density PDFs while also being
convenient computationally111Not all parameterizations are equally good, some
will only allow narrow windows in the free parameters in which the constraint
system has solutions at all, which complicates the sampling and
interpretation.. They also constrain the variations of the baryon density to
$O(1)$ excursions, consistent with the prediction by the PMF model.

model | $b$ | $f_{2}$ | $\Delta_{1}$ | $\Delta_{2}$
---|---|---|---|---
$\Lambda$CDM | 0 | 1 | 0 | 1
M1 | free | 1/3 | 0.1 | 1
M2 | free | 1/3 | 0.3 | 1
$f_{2}=1/3$ | free | 1/3 | free | 1
$f_{2}=1/2$ | free | 1/2 | free | 1
$f_{2}=2/3$ | free | 2/3 | free | 1
Table 2:  Versions of the three-zone model used in this work. M1 and M2 are
identical to the ones from JP20, while the other three have one more degree of
freedom.

We use two combinations of Boltzmann codes, recombination models, and Monte
Carlo samplers to independently arrive at our results, namely CAMB Lewis  _et
al._ (2000) \+ RECFAST Seager  _et al._ (1999, 2000); Wong  _et al._ (2008) \+
Cobaya Torrado and Lewis (2020, 2019) and CLASS Lesgourgues (2011); Blas  _et
al._ (2011) \+ HyRec Ali-Haïmoud and Hirata (2011) \+ MontePython Brinckmann
and Lesgourgues (2018); Audren  _et al._ (2013)222the latter with the flag
MODEL=FULL instead of the default RECFAST that ships with CLASS.. As previous
work on our topic had relied on RECFAST which was calibrated for models close
to $\Lambda$CDM, and certainly not for $O(1)$ deviations from the average
baryon density, an independent check with the more sophisticated HyRec code is
reassuring. All figures in this work were produced with the second toolchain.
We run CLASS with the provided cl_permille.pre file instead of the default
precision settings, and second Ref. McCarthy  _et al._ (2021) in recommending
this as good practice, given the high precision of current and upcoming CMB
data. Our convergence criterion is for the Gelman-Rubin $R_{\alpha}$ to
achieve $\max_{\alpha}(R_{\alpha}-1)<0.05$ ($\alpha$ labels the parameters),
and we generally achieve $R_{H_{0}}-1\lesssim 0.02$.

In both codes, we separately compute the recombination history $x_{e}^{i}(z)$
in each of the three zones and then perform the average to arrive at the
large-scale ionization fraction

$\langle x_{e}(z)\rangle=\sum_{i}f_{i}\Delta_{i}x_{e}^{i}(z)\,.$ (2)

We use the following likelihoods:

  * •

Planck 2018: high-$\ell$ TTTEEE (lite), low-$\ell$ TT, low-$\ell$ EE Planck
Collaboration (2020b);

  * •

ACT DR4: high-$\ell$ TT, EE, TE ACT Collaboration  _et al._ (2020a, b);

  * •

SMH: SH0ES Reid  _et al._ (2019), MCP Reid  _et al._ (2009), H0LiCOW Wong  _et
al._ (2020);

  * •

BAO: SDSS DR7 Ross  _et al._ (2015) & BOSS DR9 SDSS-III Collaboration (2012),
6dFGS Beutler  _et al._ (2011).

We found that working with the extended Planck likelihood (as opposed to the
lite version) did not give appreciable changes in the posteriors in our
parameter space.

The ACT likelihood is chosen such that covariance with Planck is negligible,
with $\ell<1800$ discarded in TT but no cuts in TE and EE ACT Collaboration
_et al._ (2020a).

The H0LiCOW measurement used in the SMH likelihood is not the most recent one;
an updated lens modeling found substantially lower $H_{0}$ Birrer  _et al._
(2020). Similarly, we do not use the most recent SH0ES measurement Riess  _et
al._ (2021) but rather the one used in JP20 from Ref. Reid  _et al._ (2019).
The reason for this is that we use the SMH combination of likelihoods only for
direct comparison to JP20 (equal to their “H3”) as well as in contexts in
which we simply want to excite the baryon clumping mode to a considerable
extent (i.e., not in posteriors).

We assume flat priors on $\omega_{b}$, $\omega_{\text{CDM}}$, $n_{s}$,
$A_{s}$, $H_{0}$, $\tau_{\text{reio}}$, $A_{\text{\sf Planck}}$ and the hard
prior $0.9<y_{\text{p}}<1.1$ for the only ACT nuisance parameter (polarization
efficiency). When they are used, the three-zone model parameters have the wide
priors $0<b<10$ and $0<\Delta_{1}<0.7$ (the upper bound on $\Delta_{1}$ is
important as it removes the spurious second posterior mode that would
otherwise be allowed by inverting the first and third zone). We assume a fixed
neutrino mass sum of $0.06\,\text{eV}$ (this helps us focus on the degeneracy
between $H_{0}$ and baryon clumping, rather than having to worry about the
$H_{0}-m_{\nu}$ degeneracy in addition).

##  III Results: posteriors

Figure 1:  Posteriors in the same parameterizations and with the same
likelihod as the one used in JP20. The solid lines are our results, while the
dashed lines are JP20. Here and in all subsequent posterior plots, the grey
shaded region is the $1\sigma$ confidence interval from the latest SH0ES
measurement Riess  _et al._ (2021).

As a first step, we reproduce the results from JP20, staying within their
M1/M2 parameterization of the three-zone model and working solely with the
Planck+SMH likelihood (which we adopt only in order to reproduce previous
results and assess the accuracy of RECFAST in the baryon clumping models). The
resulting posteriors are shown in Fig. 1. We observe that our posteriors on
$H_{0}$ are almost identical with those found in JP20 (which are plotted as
dashed lines). There are some minor discrepancies in the posterior on the
clumping parameter $b$, which we believe are most likely explained by
differences between HyRec and RECFAST.

In the remainder of this section, we will not include SMH (i.e. SH0ES, MCP,
and the old H0LiCOW value) in the likelihoods. As we will see, the CMB(+BAO)
likelihoods are not compatible with SMH even after allowing for baryon
clumping, which in our view renders a formally combined likelihood problematic
for interpretation.

Figure 2:  Posteriors in the same parameterizations as in JP20, comparing
Planck alone with ACT added. Here and in the following posteriors, the line
color indicates the model while the line style differentiates between
different likelihoods.

In Fig. 2, we compare posteriors in the JP20 parameterizations fitted to
Planck and Planck+ACT. For these posteriors, as well as for all the others
plotted in this section, central values and 1-$\sigma$ uncertainties are
listed in Tab. 5 in Appendix B. First, we observe that within $\Lambda$CDM
both likelihoods yield very similar posteriors on $H_{0}$ (this is to be
expected, as ACT DR4 itself has only about 1/3 the statistical power of
Planck, and moreover their $\Lambda$CDM constraints independently agree well
ACT Collaboration  _et al._ (2020a)). However, in the models allowing baryon
clumping, M1 and M2, the situation is different. While the $H_{0}$ posterior’s
lower end is virtually identical for both likelihoods, the inclusion of ACT
significantly reduces the extent of the high-$H_{0}$ tail (due to this
asymmetry, the posterior plot is more illuminating than the bare 1-$\sigma$
uncertainties). The effect is more pronounced for model M1 which yields a less
severe tension with SH0ES than M2 does. Similarly, adding ACT reduces the
$95\,\%$ upper limits on the clumping parameter $b$ (c.f. Tab. 5). Although M1
and M2 consistently lead to slightly higher central values of $H_{0}$, the
shift is not nearly enough to get in reasonable agreement with the SH0ES
measurement. In summary, at least in the JP20 parameterizations M1/M2, adding
ACT to Planck increases the tension with SH0ES within the baryon clumping
models.

Figure 3:  Posteriors in the parameterizations with one more degree of
freedom, $\Delta_{1}$.

We proceed by adding an additional degree of freedom to the baryon density
PDF, treating $\Delta_{1}$ as a free parameter. The models considered here may
be considered as spanning the entire range of which M1 and M2 sample only two
discrete points, while also sampling $f_{2}$ (which was identical for M1 and
M2) at three different points. Posteriors for the Planck and Planck+ACT
likelihoods are shown in Fig. 3. We observe extremely similar posteriors in
$H_{0}$ and $b$ for the three flavors of this extended parameterization.
Again, no detection of the clumping effect (i.e., non-zero $b$) is made.
Similarly to the posteriors shown in Fig. 2, the addition of ACT has the most
pronounced effect in the high-$H_{0}$ tails, where it visibly reduces the
leeway the baryon clumping models have in getting into somewhat decent
agreement with SH0ES.

The extra degree of freedom introduced in these models, $\Delta_{1}$, is
virtually unconstrained (for this reason, we do not include it in Tab. 5).
This may be a consequence of a poor choice of parameterization, however, it
seems more likely that we are already close to covering the entire range of
CMB spectra compatible with the data and allowed by the three-zone model. This
view is supported by the fact that all three values of $f_{2}$ lead to
extremely similar posteriors. If this is indeed the case, the $H_{0}$
posteriors shown in Fig. 3 should be similar to those that could be obtained
with any reasonable baryon density PDF. As we discuss in the conclusions, time
dependence may prove to be an additional mode that could alter this picture.

We discuss a possible prior volume effect in the more extended $f_{2}=\ldots$
models in Appendix A.

Figure 4:  Posteriors in the extended parameterization, illustrating the
effect of adding a BAO likelihood.

As a final result of this section, we show in Fig. 4 how adding $z\lesssim 1$
BAO data affects the posteriors in the extended models. As for $H_{0}$ the BAO
likelihood has comparable statistical power to the CMB data, the observation
that the low-$H_{0}$ tail is reduced by the addition of BAO is no surprise. On
the other hand, the high-$H_{0}$ tail is almost invariant under adding BAO,
which indicates that the CMB data alone are almost entirely responsible for
constraining the baryon clumping modes of the model. In fact, the posterior on
$b$ is slightly widened once BAO is included. Thus, we conclude that the BAO
likelihood is not particularly useful in constraining the baryon clumping
models, at least as long as the CMB-only likelihood does not allow appreciable
increases in $H_{0}$.

##  IV Results: best-fit models

likelihood |  Planck+ACT |  Planck+ACT+SMH
---|---|---
model |  $\Lambda$CDM |  $\Lambda$CDM | M1 | $f_{2}=1/2$
$H_{0}$ | 67.200 | 68.703 | 70.342 | 72.178
$100\omega_{b}$ | 2.2316 | 2.2561 | 2.2511 | 2.2520
$\omega_{\text{cdm}}$ | 0.1204 | 0.1171 | 0.1203 | 0.1246
$n_{s}$ | 0.9660 | 0.9742 | 0.9652 | 0.9523
$10^{9}A_{s}$ | 2.1086 | 2.1282 | 2.0940 | 2.0617
$\tau_{\text{reio}}$ | 0.0522 | 0.0687 | 0.0533 | 0.0484
$b$ | – | – | 0.4071 | 1.1822
$f_{1}$ | – | – | 0.2865 | 0.3524
$\Delta_{1}$ | – | – | 0.1000 | 0.0048
$f_{2}$ | – | – | 0.3333 | 0.5000
$\Delta_{2}$ | – | – | 1.0000 | 1.0000
$f_{3}$ | – | – | 0.3801 | 0.1476
$\Delta_{3}$ | – | – | 1.6784 | 3.3757
$A_{\text{\sf Planck}}$ | 1.0043 | 1.0042 | 1.0030 | 1.0018
$y_{\text{p}}$ | 1.0012 | 1.0016 | 1.0026 | 1.0016
Table 3:  Best-fit models used in Sec. IV. The first group of parameters are
$\Lambda$CDM, the second group describes the three-zone model (not all of them
are independent), and the third group are the nuisance parameters for our
likelihoods.

In order to gain some further intuition, in this section we attempt to fully
excite the baryon clumping mode and explore its consequences for the CMB
spectra. In order to do this, we now add the previously discussed SMH
likelihood to Planck+ACT. For this combined likelihood, we compute three best-
fit models, listed in Tab. 3 (the model in the first column, Planck+ACT
$\Lambda$CDM, was not fitted to SMH and serves as the reference point). The
models including baryon clumping, M1 and $f_{1}=1/2$, were chosen because they
are the ones that had the highest central values for $H_{0}$ in the posteriors
discussed in the previous section. We emphasize that it can be difficult to
infer general trends from individual best-fit models, so the results in this
section should be taken as starting points to develop intuition. The main
results of this work have been presented in the previous section.

Figure 5:  The ionization fraction as a function of redshift for the best-fit
models listed in Tab. 3.

For each of the best-fit models, we compute the average ionization fraction
$x_{e}(z)$, shown in Fig. 5. In the figure, we also list $z_{0.5}$ and $\Delta
z_{0.9\rightarrow 0.1}$ as proxies for the location and width of the last
scattering surface. First, we observe that without inclusion of baryon
inhomogeneities the recombination history is essentially fixed. As expected,
once the additional degrees of freedom in the baryon density PDF are
introduced, recombination is pushed to higher redshift. If we write
$A\,\delta\log\Delta z_{0.9\rightarrow 0.1}=\delta\log z_{0.5}$ (where
$\delta$ is with respect to the Planck+ACT $\Lambda$CDM reference model), we
find $A=-1.60$ and $A=-3.53$ for M1 and $f_{2}=1/2$ respectively. This is, to
order of magnitude, consistent with $A=-2$, the expectation from the argument
that the duration of recombination should scale as $v_{\text{rms}}^{-1}\propto
T^{-1/2}\propto z^{-1/2}$. However, it is interesting to note that both M1 and
$f_{2}=1/2$ yield almost identical values for $\Delta z_{0.9\rightarrow 0.1}$.

Figure 6:  Fractional changes in CMB power spectra for the best-fit models
listed in Tab. 3, with respect to the Planck+ACT $\Lambda$CDM baseline. In the
second panel (TE), because the data have zero crossings, we computed the
normalizing factor $\overline{|C_{\ell}^{\text{ref}}|}$ as a convolution of
the absolute value of the reference TE power spectrum with a Gaussian filter
of width $\sigma=100$ (in $\ell$). This gives a useful normalization without
artifacts from the roots. Note that for clarity the experimental data curves
for ACT and Planck have been rescaled by the given factors. While the Planck
data points are relatively close to what we use in our likelihood (except for
binning and $\ell$-cuts to improve clarity, in particular for TE and EE), the
ACT data points are for illustration purposes only and were generated as
explained in the footnote.33footnotemark: 3 22footnotetext: The ACT DR4 data
only contain band powers, which are related to the underlying CMB power
spectra by $C_{b}=W_{b\ell}C_{\ell}$, where $W_{b\ell}$ is an $m\times n$
matrix with $m<n$. Thus, it is not possible to uniquely infer the $C_{\ell}$
from the band powers. In order to generate power spectra that can be directly
compared to Planck and the theory models, we first find some solution
$C_{\ell}^{0}$ of the above equation (which is, of course, completely
unphysical). Then, writing $K_{\ell\alpha}$ for the $n\times(n-m)$ matrix
whose columns are a basis for the kernel of $W_{b\ell}$, we can generate
additional solutions
$C_{\ell}(a_{\alpha})=C_{\ell}^{0}+K_{\ell\alpha}a_{\alpha}$, where
$a_{\alpha}$ is an arbitrary vector. We now find the $a_{\alpha}$ that
minimizes $||\Lambda_{\ell^{\prime}\ell}C_{\ell}(a_{\alpha})||$, where
$\Lambda_{\ell^{\prime}\ell}$ implements a discrete second derivative. We
perform this procedure for both the ‘deep’ and ‘wide’ parts of the ACT data
set, associating approximate error bars with the resulting $C_{\ell}$
according to the normalized diagonal elements of the covariance matrix of the
$C_{b}$. Finally, we take the weighted average of the ‘deep’ and ‘wide’
$C_{\ell}$.

Next, we consider the CMB power spectra arising from the listed best-fit
models, shown in Fig. 3, where we compare them to Planck and ACT data. As
already alluded to in Sec. I, the models with baryon inhomogeneities show
significant deviations from $\Lambda$CDM in the high $\ell$ part of the
spectra, particularly for TT and EE. These deviations are partly caused by
shifts in other parameters, as is already evident from the orange curve
depicting $\Lambda$CDM fit to Planck+ACT+SMH. However, in simple experiments
where we put baryon inhomogeneities of magnitude comparable to the models
shown here into otherwise unchanged $\Lambda$CDM cosmologies, similar trends
occur (the same happens if $H_{0}$ is increased so as to neutralize the shift
in peak positions). In particular, we always observe an _increase_ in TT power
at high-$\ell$ once the baryon inhomogeneities are introduced. Thus, we
conclude that the effect of Silk damping is in fact reduced in the
inhomogeneous models. This is contrary to previous statements in the
literature, which presumably rested on numerical experiments made with much
higher amounts of inhomogeneity.

It is interesting to disentangle how the suppression in the $\ell<1000$ part
of the EE spectrum is generated. The ratio of EE to TT for $\ell\gg 1$ can be
approximated as Zaldarriaga and Harari (1995)

$\frac{C_{\ell}^{EE}}{C_{\ell}^{TT}}=\left(\frac{\eta_{*}\Delta\eta_{*}}{r^{2}(1+R)}\right)^{2}B_{\ell}\,,$
(3)

where $\eta_{*}$ and $\Delta\eta_{*}$ are the location and thickness of the
last scattering surface in conformal time respectively, $r$ is its distance
from us, $R=3\rho_{b}/4\rho_{\gamma}$, and $B_{\ell}$ is a geometric factor
independent of cosmology. Comparing, for example, the best-fit models
Planck+ACT $\Lambda$CDM and Planck+ACT+SMH $f_{2}=1/2$, we find that the
differences in the various quantities entering Eq. (3) are all $\sim 2\ldots
4\,\%$. On aggregate, however, the cancellation gives a $\sim 1\,\%$ effect.

The ACT data do not seem to show a preference for the general trends in the
spectra introduced by the baryon inhomogeneities.

likelihood |  Planck+ACT+SMH
---|---
model |  $\Lambda$CDM | M1 | $f_{2}=1/2$
Planck high-$\ell$ TTTEEE | 5.16 | 3.35 | 1.82
Planck low-$\ell$ TT | -1.24 | 0.31 | 2.86
Planck low-$\ell$ EE | 1.53 | 0.02 | -0.18
ACT TTTEEE | 1.12 | 2.13 | -0.20
ACT TT | -1.82 | -3.57 | -3.72
ACT TE | 3.07 | 5.52 | 3.29
ACT EE | -0.33 | 1.08 | 2.17
ACT high-$\ell$ TT | -1.66 | -3.09 | -2.77
ACT high-$\ell$ TT+TEEE | 1.29 | 3.46 | 1.79
SH0ES | -9.06 | -16.39 | -21.43
MCP | -1.99 | -3.58 | -4.66
H0LiCOW | -5.25 | -9.29 | -11.74
CMB likelihood | 6.74 | 7.14 | 6.29
SMH likelihood | -16.30 | -29.26 | -37.83
Table 4:  Values of $\Delta\chi^{2}$ for three best-fit models from Tab. 3,
compared to the Planck+ACT $\Lambda$CDM reference. The final two lines show
the combined Planck+ACT and SMH likelihoods used in this work.

Finally, we present the changes in $\chi^{2}$ relative to the Planck+ACT
$\Lambda$CDM baseline for the other three best-fit models, listed in Tab. 4.
The model M1 increases $\chi^{2}$ by about equal amounts in Planck and ACT,
while the model $f_{2}=1/2$ with one extra degree of freedom in the baryon
density PDF is comparably more disfavoured by Planck than by ACT. ACT TT
actually appears to favour the baryon clumping models, which holds true for
both the entire $\ell$-range covered by ACT and $\ell>1800$, the part used in
our likelihoods. On the other hand, ACT TE and, to a lesser extent, EE,
disfavour baryon clumping. This may be explained by the inability of those
models to keep the last scattering surface’s thickness fixed (c.f. Fig. 5).
Another possibility is simply that the steeper slopes in the polarization
spectra translate into higher sensitivity to $H_{0}$. It should be mentioned
that there have been hints of inconsistencies between ACT TE and other CMB
data ACT Collaboration  _et al._ (2020a). As no mechanism has been proposed
that could explain these trends, and the tensions are still relatively mild,
we do not see this work as the appropriate place to investigate them further.

##  V Conclusions

The introduction of inhomogeneities in the baryon density around the time of
recombination is an attractive proposal to resolve the Hubble tension. While
we do not constrain ourselves to a particular mechanism that is supposed to
generate the required small-scale power, PMFs appear to be the most natural
candidate.

A key prediction of the inhomogeneous models is, besides the geometric shift
in acoustic peak positions, a change in the high-$\ell$ part of the CMB power
spectra due to different amount of Silk damping. We have argued that the
addition of small-scale ACT data to the previously considered Planck
likelihood constitutes a powerful way to test this prediction. The higher
statistical power of the thus enlarged likelihood also enables us to extend
the parameter space; specifically, we are able to add an additional degree of
freedom to the baryon density PDF.

Fitting the baryon clumping models to CMB-only likelihoods, we have shown that
the inclusion of ACT narrows the posterior for the clumping parameter $b$
around zero compared to Planck alone. Likewise, although ACT does not
appreciably alter the low-$H_{0}$ tails in the posteriors, it significantly
shrinks the high-$H_{0}$ tails, indicating that the added small-scale
information disfavours the introduction of baryon clumping. The CMB data still
leave room for mild baryon clumping and some upward shift in $H_{0}$, thus
somewhat relieving the Hubble tension, but values as high as the SH0ES
1-$\sigma$ lower bound are strongly disfavoured.

Although the absence of theoretical predictions for the baryon density PDF
restricts us to relatively simple parameterizations, we have argued that there
is evidence that the posteriors on $H_{0}$ presented in this work are in fact
generic for any time-independent baryon density PDF. An intriguing possibility
that could evade the constraints presented would be the introduction of time
dependence. Assuming the existence of PMFs, the decrease in magnetic field
strength during recombination would lead to a homogenization of the baryon
distribution (the relaxation time for kpc-scale baryon clumps is considerably
smaller than recombination’s duration). It is possible that, in the large-
scale average, this would lead to a faster onset of recombination while
keeping the thickness of the last scattering surface approximately fixed,
perhaps leading to a better fit with the polarization data.

Besides our primary results, we have also shown that while the previously used
RECFAST recombination code is probably sufficiently accurate for the
modifications considered in this work, there are some systematic differences
to the more recent HyRec which we recommend should be used going forward.
Furthermore, we have shown that, contrary to previous claims, baryon
inhomogeneities of the amount necessary to appreciably shift $H_{0}$ do not
generically lead to enhanced Silk damping.

We have found no evidence that the addition of small-scale ACT data to the
likelihood favours the introduction of baryon inhomogeneities around
recombination over $\Lambda$CDM. Looking forward, the increased small-scale
CMB information from future ACT releases, SPT-3G SPT-3G Collaboration  _et
al._ (2014), Simons Observatory Simons Observatory Collaboration (2019), and
CMB-S4 CMB-S4 Collaboration (2019) will provide more constraining power. As
long as no currently unknown systematics in ACT are discovered, however,
smaller error bars will only serve to exclude the class of time-independent
three-zone models considered in this paper with higher significance. Parallel
to this improvement in data quality, theoretical advancements in constructing
viable models of the baryon density PDF, for example through MHD simulations,
will be essential in increasing the predictivity of the theory.

Modifications to CLASS used in this work are publicly available here.

###### Acknowledgements.

We thank Karsten Jedamzik and Levon Pogosian for useful discussions during the
final stages of this work. JCH and DNS thank the Simons Foundation for
support. LT thanks Zack Li for help with the ACT DR4 likelihood.

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##  Appendix A Assessment of prior volume effects

The more extended parameterizations considered in this paper, $f_{2}=\ldots$,
exhibit a pathology in that $f_{1}=0$ is equivalent to $\Lambda$CDM regardless
of the value of $\Delta_{1}$. Thus, there may be a prior volume effect in
these models that underestimates the posterior for non-vanishing baryon
clumping (and thus increased $H_{0}$). We assess this effect for the example
of the $f_{2}=1/2$ model; for other choices of $f_{2}$ the conclusions are
similar. Our methods are similar to those used in Ref. Ivanov  _et al._
(2020).

First, we consider changing the model such that a constraint $f_{1}>0.1$ is
enforced, thereby avoiding $\Lambda$CDM entirely. The resulting posteriors are
shown in Fig. 8. It is expected that the posteriors for $b$ and $H_{0}$ move
upward to some extent since the model, by construction, contains at least some
baryon clumping now. The shift in the central value of $H_{0}$ is about
$\sigma/3$ and we do not observe a substantial enlargement of the high-$H_{0}$
tail.

Second, we compute a frequentist mean likelihood profile in $b$, which is the
exponential of the average log-likelihood extracted from the Markov chain in
20 $b$-bins between 0 and 1, shown as the solid blue line in Fig. 8. We
observe that this profile is somewhat broader than the posterior shown in Fig.
3. Furthermore, at the $95\,\%$ upper limit on $b$ from Tab. 5, the averaged
likelihood is reduced by about $\Delta\chi^{2}\sim 2.4$. This value would
correspond to only an $\sim 88\,\%$ upper bound.

We also compute the more common likelihood profile in which we minimize the
likelihood at a range of values of $b$, shown as the solid green line in Fig.
8. In this case, we observe that the maximum is at non-zero $b$, but then the
likelihood profile drops off more steeply than the marginalized Bayesian
posterior. For a discussion of the relative merits of the two ways to compute
a likelihood profile, see Ref. Ivanov  _et al._ (2020).

We conclude that we observe evidence for a mild prior volume effect. However,
the posteriors in Fig. 8 indicate that our main conclusions regarding $H_{0}$
are only marginally affected.

Figure 7:  Comparison between the posterior in the unconstrained $f_{2}=1/2$
model, same as in Fig. 3, and the one with the constraint $f_{1}>0.1$. The
latter should substantially reduce the prior volume effect arising from the
degeneracy with $\Lambda$CDM at $f_{1}=0$. In the dashed posterior, the non-
vanishing support at $b=0$ is an artifact of MontePython’s posterior smoothing
routine.

Figure 8:  Likelihood profiles as a function of $b$ for the $f_{2}=1/2$ Markov
chain. The ‘mean’ and ‘best-fit’ lines are computed as described in the text.
The orange line is the same marginalized likelihood that was shown in Fig. 3.
The blue dashed line is a Gaussian fit with only the standard deviation
varied, while the green dashed line is a 3-parameter Gaussian fit. The green
solid line was rescaled so that the Gaussian fit attains its maximum at one.
These fits should only be treated as visual guides.

##  Appendix B Central values and uncertainties

likelihood | model | $H_{0}\,[{\rm km/s/Mpc}]$ | $100\omega_{b}$ | $\omega_{\text{cdm}}$ | $n_{s}$ | $10^{9}A_{s}$ | $\tau_{\text{reio}}$ | $b$
---|---|---|---|---|---|---|---|---
Planck |  $\Lambda$CDM | $67.24\pm 0.66$ | 2.23(2) | 0.120(1) | 0.964(5) | 2.10(4) | 0.054(9) | –
M1 | $68.45\pm 1.08$ | 2.24(2) | 0.122(2) | 0.960(5) | 2.10(4) | 0.053(8) | <0.57
M2 | $67.86\pm 0.83$ | 2.24(2) | 0.121(2) | 0.964(5) | 2.10(4) | 0.055(9) | <0.32
$f_{2}=1/3$ | $68.04\pm 0.97$ | 2.24(2) | 0.121(2) | 0.963(6) | 2.10(4) | 0.054(9) | <0.42
$f_{2}=1/2$ | $68.12\pm 1.04$ | 2.24(2) | 0.121(2) | 0.963(6) | 2.10(4) | 0.054(9) | <0.53
$f_{2}=2/3$ | $68.03\pm 0.96$ | 2.24(2) | 0.121(2) | 0.964(6) | 2.10(4) | 0.054(9) | <0.52
Planck+ACT |  $\Lambda$CDM | $67.26\pm 0.60$ | 2.23(1) | 0.120(1) | 0.967(4) | 2.12(4) | 0.055(8) | –
M1 | $68.18\pm 0.87$ | 2.23(1) | 0.121(2) | 0.962(5) | 2.11(4) | 0.053(8) | <0.42
M2 | $67.74\pm 0.71$ | 2.24(1) | 0.121(1) | 0.967(4) | 2.12(4) | 0.054(8) | <0.26
$f_{2}=1/3$ | $67.99\pm 0.86$ | 2.23(1) | 0.121(2) | 0.964(6) | 2.11(4) | 0.054(8) | <0.39
$f_{2}=1/2$ | $68.06\pm 0.88$ | 2.23(1) | 0.121(2) | 0.964(6) | 2.11(4) | 0.054(8) | <0.47
$f_{2}=2/3$ | $68.01\pm 0.89$ | 2.24(2) | 0.121(2) | 0.965(5) | 2.11(4) | 0.054(8) | <0.49
Planck+ACT+BAO | $f_{2}=1/3$ | $68.18\pm 0.77$ | 2.24(2) | 0.121(2) | 0.964(6) | 2.11(4) | 0.054(8) | <0.38
$f_{2}=1/2$ | $68.24\pm 0.83$ | 2.24(1) | 0.121(2) | 0.964(6) | 2.11(4) | 0.054(8) | <0.46
$f_{2}=2/3$ | $68.22\pm 0.79$ | 2.24(1) | 0.121(2) | 0.966(5) | 2.11(4) | 0.054(9) | <0.50
Table 5:  Central values and 1-$\sigma$ errors for the posteriors in this
section. For conciseness, for $\Lambda$CDM parameters except $H_{0}$, the
numbers in parentheses are the 1-$\sigma$ uncertainties on the last digit.
Since we do not observe a detection of the clumping effect, for the clumping
parameter $b$ $95\,\%$ upper limits are quoted.