# A Cepheid systematics-free test of $H_{0}$ to $\lesssim 2.5\%$ accuracy
using SH0ES photometry

Doron Kushnir1 and Amir Sharon1
1Dept. of Particle Phys. & Astrophys., Weizmann Institute of Science, Rehovot
76100, Israel
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The recent SH0ES determination of the Hubble constant, $H_{0}=73.04\pm 1.04$
km/s/Mpc, deviates significantly by $\mathord{\approx}5\sigma$ from the Planck
value, stimulating discussions on cosmological model extensions. To minimize
statistical uncertainty and mitigate sensitivity to systematic errors in any
single anchor distance determination, SH0ES combines Cepheids from various
observations, including those from Type Ia supernova (SNe Ia) host galaxies,
NGC 4258, and closer galaxies (MW, LMC, SMC, and M31), although this mixed
sample may introduce unknown or subtle systematic errors due to comparing
distant and closer Cepheids. To address this, we propose a subset excluding
Cepheids from the closer galaxies, retaining only the NGC 4258 water
megamasers as a single anchor, circumventing potential systematic errors
associated with observational methods and reduction techniques. Focusing
solely on these Cepheids yields competitive statistical errors, approximately
$2.5\%$, sufficient to identify a $\mathord{\approx}3\sigma$ tension with the
Planck $H_{0}$ value. Our approach offers an opportunity to utilize optical
photometry with systematic uncertainty smaller than the statistical
uncertainty, potentially achieving higher precision than NIR photometry, given
the lower optical background. However, currently the optical photometry
sample’s fidelity does not match that of NIR photometry. The significant
Hubble tension obtained is unrelated to Cepheids and we discuss other options.

###### keywords:

cosmological parameters – distance scale – stars: variables: Cepheids–
supernovae: general

††pubyear: 2020††pagerange: A Cepheid systematics-free test of $H_{0}$ to
$\lesssim 2.5\%$ accuracy using SH0ES photometry–A

##  1 Introduction

The most recent determination of the Hubble constant by the SH0ES
collaboration (Riess et al., 2022, hereafter R22), $H_{0}=73.04\pm 1.04$ (in
units of $\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$ hereafter), exhibits a
significant deviation of $\mathord{\sim}5\sigma$ from the Planck value (Planck
Collaboration et al., 2020), $H_{0}=67.4\pm 0.5$, commonly referred to as the
Hubble tension. This discrepancy between the Cepheid- and Type Ia supernovae
(SNe Ia)-based SH0ES measurement and the cosmic microwave background
temperature and polarization anisotropies Planck measurement has spurred
numerous proposals for extensions of the standard $\Lambda$CDM cosmology model
(see Di Valentino et al., 2021, for a review).

The absolute distance scale utilized by SH0ES is grounded on the period-
luminosity relation of Cepheids ($P-L$ relation; Leavitt & Pickering, 1912),
measured in the Hubble Space Telescope (HST) F160W filter (similar to the NIR
$H$ band). These Cepheids are situated in 37 SNe Ia host galaxies (hereafter
referred to as host Cepheids) along with other anchor galaxies with absolute
distance measurements. The Hubble tension manifests as a discrepancy of
$\mathord{\sim}0.1-0.2\,\textrm{mag}$ in the magnitudes of SH0ES Cepheids
(Riess, 2019b; Efstathiou, 2020), wherein the host Cepheids appear brighter
than the $\Lambda$CDM prediction.

The approach adopted by the SH0ES collaboration to determine $H_{0}$ involves
combining the host Cepheids with a diverse large sample of Cepheids obtained
from various observations. This strategy offers several advantages, including
minimizing the statistical uncertainty in $H_{0}$ and utilizing multiple
independent geometrical distance anchors, which reduces sensitivity to
potential systematic errors in any single anchor distance determination. The
additional Cepheids reside at the anchor galaxy NGC 4258 (at a distance of
$\mathord{\approx}7.5\,\textrm{Mpc}$, hereafter referred to as N4258 Cepheids;
absolute distance from VLBI observations of water megamasers orbiting its
central supermassive black hole; Humphreys et al., 2013; Reid, Pesce, & Riess,
2019), as well at much closer galaxies such as the SMC and LMC (at
$\mathord{\approx}50\,\textrm{kpc}$; absolute distance from double eclipsing
binaries observed using long-baseline NIR interferometry; Pietrzyński et al.,
2019; Graczyk et al., 2020) and the Milky Way (MW, at
$\mathord{\sim}1\,\textrm{kpc}$; absolute distance from Gaia EDR3 parallaxes;
Riess et al., 2021). Cepheids from M31 (at
$\mathord{\approx}750\,\textrm{kpc}$) are also incorporated to better
constrain the $P-L$ relation. However, the comparison of distant host Cepheids
(typically at distances of $\mathord{\approx}15-50\,\textrm{Mpc}$ and M101 at
a distance of $\mathord{\approx}6.5\,\textrm{Mpc}$) to Cepheids much closer to
us involves a measurement range of $\geq$ 10 magnitudes which could in
principle result in systematic errors between such measurements.

The measurement of host and N4258 Cepheids involves HST imaging of galaxies at
multiple epochs using visual and NIR filters. This imaging data is utilized to
identify and select Cepheids, determine their periods, extract their
photometry at each epoch and filter, and amalgamate all epochs to derive a
single mean magnitude value for each filter. These mean magnitudes are then
employed to construct the Wesenheit index (Madore, 1982)111We follow the
convention that a single Cepheid magnitude $x$ is the magnitude of intensity
mean, $\langle x\rangle$, and colors $(x-y)$ stand for $\langle
x\rangle-\langle y\rangle$.:

$\displaystyle
W_{H}=\textrm{F160W}-R_{H}^{VI}\left(\textrm{F555W}-\textrm{F814W}\right),$
(1)

where F555W and F814W are the observed magnitudes in the corresponding HST
filters (similar to the optical $V$ and $I$ band, respectively) and
$R_{H}^{VI}\approx 0.4$ is chosen to achieve extinction-independent
measurements (for some specific extinction law, see a detailed discussion in
Section 5.1). On the other hand, the measurement of close Cepheids involves a
distinct process (Riess et al., 2019a, 2021; Li et al., 2021):

  1. 1.

Close Cepheids are identified and selected based on archival observations.

  2. 2.

These Cepheids, which are significantly brighter (by a factor of
$10^{3}-10^{9}$) than host Cepheids, necessitate a different observing
technique with HST, often involving a rapid spatial scan or DASH mode, along
with a correction for count-rate nonlinearity.

  3. 3.

Due to the narrow field of view of HST, a limited number of close Cepheids can
be observed in one orbit, typically resulting in observations at a single
epoch, which are then phase-corrected to mean light using archival (usually
ground-based) observations.

  4. 4.

The blending of host Cepheids with their environment (either chance
superposition of Cepheids on crowded backgrounds or light from stars
physically associated with Cepheids) due to the HST’s angular resolution222
$None$ corresponds to $\mathord{\approx}7-25\,\textrm{pc}$ for the hosts (and
$\mathord{\approx}3\,\textrm{pc}$ for M101). requires a correction to
photometry known as crowding correction, which is not applied to close
Cepheids.

The SH0ES collaboration extensively studied the impact of these differences on
the comparison between close and host Cepheids and compensated for them
through various corrections (R22).

Additionally, the large distance to host Cepheids biases their selection
towards longer periods and lower extinction. Since long-period Cepheids are
rarer, the close Cepheid sample is biased towards shorter period Cepheids, see
Figure 1. This discrepancy in period distributions is mitigated by assuming a
specific shape of the $P-L$ relation, which may include breaks. Furthermore,
the extinction of close Cepheids (in the MW and M31) tends to be higher than
that of host Cepheids (see discussion in Section 5.2). To address this,
observations are conducted in the NIR and the Wesenheit index is utilized.
Finally, the metallicity of LMC and SMC Cepheids is lower than that of host
Cepheids, which reside in galaxies specifically selected to have similar,
roughly solar, metallicity. This difference is accounted for by assuming a
metallicity-dependent $P-L$ relation.

Figure 1: The distribution of SH0ES Cepheid periods across different galaxies.
Each $0.1$ $\log P$ bin showcases the count of Cepheids (multiplied by 4 in
some cases for better visibility), with the total number displayed in the
legend. Cepheids from various galaxy types are distinguished by color: Black
represents faraway host galaxies, red indicates nearby galaxies (M101 and NGC
4258), blue denotes M31, green signifies the LMC, and brown represents MW
Cepheids sourced from Table 1 in Riess et al. (2021). Notably, the majority of
extragalactic (beyond M31) Cepheids exhibit $\log P>1$. Furthermore, Cepheid
periods in nearby galaxies (M101 and NGC 4258) tend to be shorter on average
compared to those in faraway galaxies, where the majority of Cepheids have
$\log P>1.2$.

The potential systematic errors arising from the aforementioned differences
may introduce uncertainties as these effects lack a comprehensive theoretical
understanding. Consequently, several studies have scrutinized the potential
impact of such systematic errors on the determination of $H_{0}$ (Follin &
Knox, 2018; Anderson & Riess, 2018; Efstathiou, 2020; Riess et al., 2020;
Javanmardi et al., 2021; Mörtsell et al., 2022a, b; Perivolaropoulos & Skara,
2021; Sharon et al., 2024). However, none of these works have presented solid
evidence of the existence of systematic biases. Nonetheless, it remains
challenging to conclusively rule out a systematic bias (though see the
comprehensive assessments in R22), thereby directing many investigations into
potential systematic errors in the SH0ES measurement towards the Cepheids. In
this paper, we propose a methodology to utilize the SH0ES Cepheids for $H_{0}$
measurement in a manner that is robust against Cepheid-related systematic
errors. Our approach focuses solely on the hosts and N4258 Cepheids,
capitalizing on the fact that the relative distances between the N4258
Cepheids and the hosts Cepheids are less susceptible to many systematic errors
that may arise when comparing the hosts Cepheids to other anchor Cepheids. In
essence, the NGC 4258 anchor stands out because it is the sole anchor galaxy
for which the Cepheids are measured (both in terms of observation technique
and data reduction) precisely as the hosts Cepheids.

For instance, consider the correction for blending photometry applied to the
hosts Cepheids. Previous works found no evidence that this correction
introduces systematic biases, both employing HST (see R22 and the amplitude
tests in Sharon et al., 2024, which are relevant for $\log P<1.7$333$\log
P\equiv\log_{10}(P\,[\textrm{d}])$) and JWST observations (Riess et al., 2024,
who observed NGC 4258 and five hosts and found no significant difference in
the mean distance measurements determined from HST and JWST). Nevertheless, it
is challenging to rule out the possibility that a significant portion of long-
period Cepheids (that are more massive and younger) are physically associated
with stars. For example, Anderson & Riess (2018) demonstrated by observing
Cepheids in M31 that long period Cepheids have a higher chance of being in
open clusters (see their Figure 13). The available data, however, are limited
to Cepheid ages older than $\mathord{\sim}50\,\rm{Myr}$ (see also Breuval et
al., 2023, with similar age limitations). The ages of the long-period Cepheids
are $\lesssim 20\,\rm{Myr}$ (see, e.g, Table A1 of Anderson et al., 2016),
probably shorter than the dispersing time of open clusters. It is, therefore,
reasonable to assume that a significant fraction of long-period Cepheids
reside in open clusters. In such scenarios, the photometry of these long-
period Cepheids would be biased, as the open clusters are not resolved for the
hosts, and the crowding corrections applied by the SH0ES collaborations cannot
rectify this effect. When comparing these Cepheids to the close Cepheids,
where the cluster is resolved, their relative distances would be biased. While
we do not claim the existence of such an effect (and existing estimates
suggest it is small; Anderson & Riess, 2018), it inherently cancels out when
comparing the hosts Cepheids to the N4258 Cepheids, as the open clusters
affect both in the same manner. Other potential systematic errors, such as
those in identification, selection, period determination, photometry, and mean
magnitude determination, are likewise eliminated or minimized. Therefore, we
propose utilizing a subsample of SH0ES Cepheids that excludes the MW, LMC,
SMC, and M31 Cepheids, effectively making the determination of $H_{0}$
impervious to Cepheid-related systematic errors.

By excluding the MW, LMC, SMC, and M31 Cepheids, the majority of Cepheids in
the sample exhibit $\log P>1$ (see Figure 1). Consequently, all Cepheids with
$\log P<1$ can be disregarded, effectively mitigating the sensitivity to
breaks in the $P-L$ relation. Moreover, Cepheids in the remaining sample
exhibit similar metallicities, and the extinction towards them is minimal.
Although systematic differences persist between M101 and NGC 4258 (hereafter
nearby galaxies) and other (hereafter faraway) galaxies, such as the average
shorter periods of Cepheids in nearby galaxies compared to those in faraway
galaxies (see Figure 1), we demonstrate in Section 3 that these systematic
differences have negligible effects on the determination of $H_{0}$. With this
refined sample, the remaining statistical errors in determining the Cepheid
relative distance scale, specifically those associated with the zero point of
the Cepheid brightness (attributed to intrinsic scatter, measurement error,
and finite number of Cepheids) and the SNe Ia brightness (due to intrinsic
scatter and measurement error), are sufficiently small to be competitive
(Section 3). The statistical error is approximately $\mathord{\approx}2.3\%$
(or about 1.65 km/s/Mpc), which is adequate to identify a
$\mathord{\approx}3\sigma$ tension with the Planck $H_{0}$ value.

The systematic errors associated with Cepheids in our approach are
significantly smaller than the statistical uncertainty. Specifically, we
illustrate that uncertainties related to the reddening law and its free
parameters, metallicity sensitivity, and breaks in the $P-L$ relation (or even
the existence of a global $P-L$ relation) are either eliminated or
substantially reduced. Since these sensitivities are the rationale for opting
for NIR over optical photometry, there exists an opportunity to utilize
optical photometry with significantly reduced systematic errors stemming from
these effects. In this case, the relevant Wesenheit index becomes:

$\displaystyle
W_{I}=\textrm{F814W}-R_{I}^{VI}\left(\textrm{F555W}-\textrm{F814W}\right),$
(2)

where $R_{I}^{VI}\approx 1.3$ is chosen for $W_{I}$ to be independent of
extinction (e.g., Riess et al., 2019a). In Section 4, we demonstrate that the
systematic uncertainty with optical photometry is markedly smaller than the
statistical uncertainty, potentially allowing for higher precision than that
achieved with NIR photometry, as the optical background is nearly an order of
magnitude lower than that in the NIR, owing to higher resolution, smaller
pixels, and lower flux from red giants (R22). The reduced sensitivity to
extinction correction is attributed to the minimal extinction of Cepheids in
our sample (Section 5). Upon scrutinizing the optical optical datasets for the
19 hosts of Riess et al. (2016, hereafter R16), we find that, in practice, the
photometry of the optical bands is not much improved compared to the NIR
photometry and the fidelity of the optical photometry sample is not at the
level of the NIR photometry sample (see Section 4).

The key advantage of our approach is that the significant Hubble tension
obtained is unrelated to Cepheids. Given the absence of any indications of SNe
Ia-related systematic errors following detailed scrutiny (R22, Carr et al.,
2022; Brout et al., 2022; Peterson et al., 2022, but see also Steinhardt,
Sneppen, & Sen 2020; Wojtak & Hjorth 2022, 2024), our analysis suggests that
either the standard cosmology is flawed or there exists a systematic error in
the current distance determination to NGC 4258. We delve into this matter and
summarize our findings in Section 6.

##  2 Fitting method and data

In this section, we describe our fitting technique (subsection 2.1) and the
dataset (subsection 2.2) used for our main fits. These fits specifically
exclude Cepheids from the MW, LMC, SMC and M31, along with those with periods
$\log P<1$.

###  2.1 Fitting method

We adopt a fitting methodology similar to that of Mörtsell et al. (2022a),
which bears resemblance to the approach outlined in R16. We express the
Wesenheit magnitude, elaborated upon in Section 5.1, in the $k$ band (either
$I$ for F814W or $H$ for F160W) of the $j$th Cepheid within the $i$th galaxy
as follows:

$\displaystyle m_{k,i,j}^{W}$
$\displaystyle=\mu_{i}+M_{k}^{W}+b^{W}_{k}\left[P\right]_{i,j}+Z^{W}_{k}\Delta\log_{10}\left(\rm{O}/\rm{H}\right)_{i,j}.$
(3)

Here, $\left[P\right]_{i,j}=\log P_{i,j}-1$, $M_{k}^{W}$ denotes the absolute
Cepheid magnitude normalized to a period of $\log P=1$ and a fiducial
metallicity of $\log_{10}\left(\rm{O}/\rm{H}\right)=8.9$,
$\Delta\log_{10}\left(\rm{O}/\rm{H}\right)_{i,j}=\log_{10}\left(\rm{O}/\rm{H}\right)_{i,j}-\log_{10}\left(\rm{O}/\rm{H}\right)$
represents the difference between the estimated Cepheid metallicity and the
fiducial value, and $\mu_{i}=5\log_{10}D_{i}+25$ stands for the distance
modulus to the $i$th galaxy, with $D_{i}$ being the luminosity distance in
Mpc. The nuisance parameters $b^{W}_{k}$ and $Z^{W}_{k}$ define the
relationship between Cepheid period, metallicity, and luminosity. An
alternative analysis employing period bins is conducted, where for each
Cepheid in the $l$ period bin, we express:

$\displaystyle m_{k,i,j}^{W}$
$\displaystyle=\mu_{i}+M_{k}^{W,l}+b^{W,l}_{k}\left[P\right]_{i,j}+Z^{W,l}_{k}\Delta\log_{10}\left(\rm{O}/\rm{H}\right)_{i,j}.$
(4)

Here, $M_{k}^{W,l}$, $b^{W,l}_{k}$ and $Z^{W,l}_{k}$ define the relation in
the $l$ period bin. An analysis incorporating a break in the $P-L$ relation
entails two period bins with $M_{k}^{W,1}=M_{k}^{W,2}$ and
$Z_{k}^{W,1}=Z_{k}^{W,2}$.

The magnitudes of SNe Ia in the calibration sample are expressed as:

$\displaystyle m_{B,i,j}=\mu_{i}+M_{B},$ (5)

Where $m_{B,i,j}$ represents the maximum-light apparent $B$-band brightness of
the $j$th SNe Ia in the $i$th host444or, possibly, to the $j$th measurement of
the same SN Ia. at the time of $B$-band peak, corrected to the fiducial color
and luminosity. The distance to NGC 4258, $\mu_{\rm{N}2458}=29.397\pm
0.032\,\rm{mag}$ (Reid, Pesce, & Riess, 2019), is incorporated as an
additional data point.

We simultaneously fit for $b^{W}_{k}$, $Z^{W}_{k}$, $M_{k}^{W}$ (or for
$b^{W,l}_{k}$, $Z^{W,l}_{k}$, $M_{k}^{W,l}$ in the case of the binned
analysis), $M_{B}$, and the galaxy distances $\mu_{i}$. As the system of
equations is linear, the fit can be conducted analytically (see Appendix A of
Mörtsell et al., 2022a). We employ a global outlier rejection threshold of
$2.7\sigma$. Our findings in Sections 3-4 demonstrate that our results are
relatively insensitive to the method of outlier rejection. Given $M_{B}$, we
determine $H_{0}$ through:

$\displaystyle H_{0}=10^{M_{B}/5+a_{B}+5},$ (6)

where $a_{B}=0.71273\pm 0.00176\,\rm{mag}$ represents the intercept of the SNe
Ia magnitude-redshift relation (R16).

###  2.2 Cepheid Data

Since the latest extragalactic Cepheid optical dataset from R22 has not yet
been made publicly accessible, we perform our analysis using two separate
datasets, as made available through the data release of R22
555https://github.com/PantheonPlusSH0ES/DataRelease. The tables exclusively
include Cepheids that pass the $3.3\sigma$ outlier rejection criterion of
R22.:

  * •

R22-$W_{H}$ – This dataset incorporates $m_{H,i,j}^{W}$,
$\left[P\right]_{i,j}$, $\log_{10}\left(\rm{O}/\rm{H}\right)_{i,j}$, and
$\sigma_{H,i,j}^{W}$ (representing the total statistical uncertainty in
$m_{H,i,j}^{W}$; we use the provided covariance matrix as well) for the full
($37$ hosts; $42$ SNe Ia) sample.

  * •

R22-$W_{I}$ – Here, we utilize $m_{I,i,j}^{W}$, $\left[P\right]_{i,j}$,
$\log_{10}\left(\rm{O}/\rm{H}\right)_{i,j}$, and $\sigma_{I,i,j}^{W}$
(covariance matric is not provided in this case) for the $19$ R16 hosts (and
$19$ SNe Ia; hosts distances $\mathord{\lesssim}30\,\textrm{Mpc}$). We scale
$\sigma_{I,i,j}^{W}$ by a factor of $0.85$, as recommended in the data release
notes (see the discussion in Section 4).

The R22 datasets incorporate several improvements compared to earlier data
releases, as outlined in Section 3.4 of R22. R22 data release includes
$m_{B,i,j}$ along with its corresponding covariance matrix.

##  3 $W_{H}$ fitting results

For our primary fit using the R22-$W_{H}$ dataset, we adopt $R_{H}^{VI}=0.386$
(corresponding to $R_{V}=3.3$ of Fitzpatrick, 1999, for consistency with R22;
see Section 5.1), resulting in $H_{0}=72.68\pm 1.67$ (refer to case 1 in Table
1 for detailed information). Figure 2 illustrates the individual Cepheid $P-L$
relations, where the solid red line indicates the best fit, and the dashed red
lines represent one standard deviation of the residuals around the best fit
within each galaxy. Cepheids failing the global $2.7\sigma$ outlier rejection
criterion are highlighted in red. Our outcome closely aligns with the result
of fit 10 from R22, which yielded $H_{0}=72.51\pm 1.54$, utilizing NGC 4258 as
the sole anchor while incorporating all ($\mathord{\sim}3500$) Cepheids in the
analysis.

The obtained error of approximately $\mathord{\approx}2.3\%$ in $H_{0}$ is
linked to the error in $M_{B}$ through $\mathord{\approx}\delta
M_{B}\ln(10)/5$, where

$\displaystyle\delta
M_{B}^{2}\approx\delta\mu_{\rm{N}4258,\rm{anc}}^{2}+\delta\Delta\mu_{\rm{N}4258}^{2}+\delta(\Delta\mu_{\rm{host}}+m_{B})^{2}.$
(7)

Here, $\delta\mu_{\rm{N}4258,\rm{anc}}\approx 0.032\,\rm{mag}$ denotes the
absolute distance error to NGC 4258 from megamaser observations (Reid, Pesce,
& Riess, 2019). $\delta\Delta\mu_{\rm{N}4258}$ represents the NGC 4258
contribution to the error of the relative distance between NGC 4258 and the
host galaxies based on the Cepheids $P-L$ relation.
$\delta(\Delta\mu_{\rm{host}}+m_{B})$ signifies the weighted error in the mean
of each host’s contribution to the relative distance and $m_{B}$. For each
galaxy, we can estimate
$\delta\Delta\mu_{i}\approx(\sum_{m,n}C^{-1}_{mn})^{-1/2}$, where $C$ is the
covariance matrix for $\sigma_{H,i,j}^{W}$, and $\delta
m_{B,i}\approx(\sum_{m,n}D^{-1}_{mn})^{-1/2}$, where $D$ is the covariance
matrix for $\delta m_{B,i,j}$. This estimate leads to
$\delta\Delta\mu_{\rm{N}4258}\approx 0.028\,\rm{mag}$, which can be
approximated by $\delta\Delta\mu_{\rm{N}4258}\approx 0.42/\sqrt{223}\approx
0.028\,\rm{mag}$, where $\mathord{\approx}0.42\,\rm{mag}$ is the mean
statistical uncertainty of the 223 N4258 Cepgeids’ Wesenheit magnitude.
Similarly, we obtain
$\delta(\Delta\mu_{\rm{host}}+m_{B})=[\sum(\delta\Delta\mu_{i}^{2}+\delta
m_{B,i}^{2})^{-1}]^{-1/2}\approx 0.021\,\rm{mag}$, where the SNe Ia
contribution to the error budget, $\delta_{m_{B}}=[\sum\delta
m_{B,i}^{-2}]^{-1/2}\approx 0.016\,\rm{mag}$, can be approximated by the
$\mathord{\approx}0.11\,\rm{mag}$ SNe Ia intrinsic scatter,
$\delta_{m_{B}}\approx 0.11/\sqrt{37}\approx 0.018\,\rm{mag}$. Our estimate
yields $\delta M_{B}\approx 0.047\,\rm{mag}$, resulting in an
$\mathord{\approx}2.2\%$ error in $H_{0}$, consistent with the fit results.

Table 1: Summary of fitting results for different cases. The Wesenheit index ($W_{I}$ or $W_{H}$) is presented in the second column. For $W_{H}$ we use the R22-$W_{H}$ dataset ($37$ hosts; $42$ SNe Ia; full covariance matrix for Cepheid Wesenheit index) and for $W_{I}$ we use the R22-$W_{I}$ dataset ($19$ hosts; $19$ SNe Ia; covariance matrix for Cepheid Wesenheit index was not provided). Anchor galaxies are specified in the third column (N for NGC 4258 and M for MW). The number of Cepheids included in the fit (after outlier rejection) is listed in the fifth column. The reduced chi-squared value and the corresponding degrees of freedom are provided in the eighth and ninth columns, respectively. Case | Wesenheit | Anc | $H_{0}$ | $N$ | $Z^{W}_{k}$ | $b^{W}_{k}$ | $\chi^{2}_{\nu}$ | dof | comments
---|---|---|---|---|---|---|---|---|---
| index |  | $[\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}]$ |  | $[\rm{mag}\,\rm{dex}^{-1}]$ | $[\rm{mag}\,\rm{dex}^{-1}]$ |  |  |
|  |  |  $W_{H}$ fits, Section 3 |  |  |  |
1 | $W_{H}$ | N | $72.68\pm 1.67$ | $2306$ | $-0.07\pm 0.12$ | $-3.22\pm 0.05$ | $0.96$ | 2342 | primary
2 | $W_{H}$ | N | $72.50\pm 1.69$ | $2302$ |  |  | $0.97$ | 2311 |  $\log P$ bins
|  |  |  $W_{I}$ fits, Section 4 |  |  |  |
3 | $W_{I}$ | N | $76.16\pm 1.69$ | $1285$ | $-0.43\pm 0.12$ | $-3.20\pm 0.04$ | $0.90$ | 1299 | primary
4 | $W_{I}$ | N | $75.25\pm 1.71$ | $1275$ |  |  | $0.88$ | 1262 |  $\log P$ bins
|  |  | Section 5.2 |  |  |  |
5 | $W_{H}$ | N | $72.39\pm 1.66$ | $2312$ | $0.05\pm 0.12$ | $-3.15\pm 0.05$ | $0.92$ | 2348 | $R_{H}^{VI}=0$
6 | $W_{I}$ | N | $74.42\pm 1.66$ | $1290$ | $-0.11\pm 0.11$ | $-2.84\pm 0.04$ | $0.91$ | 1304 | $R_{I}^{VI}=0$
|  |  | Section 5.3 |  |  |  |
7 | $W_{H}$ | M | $74.43\pm 2.21$ | $2632$ | $-0.17\pm 0.09$ | $-3.25\pm 0.03$ | $0.96$ | 2666 |
8 | $W_{I}$ | M | $75.67\pm 2.24$ | $1561$ | $-0.28\pm 0.09$ | $-3.16\pm 0.03$ | $0.88$ | 1573 |
9 | $W_{H}$ | M | $74.28\pm 2.21$ | $2628$ | $-0.01\pm 0.09$ | $-3.14\pm 0.03$ | $0.92$ | 2662 | $R_{H}^{VI}=0$
10 | $W_{I}$ | M | $83.10\pm 2.60$ | $1547$ | $-0.05\pm 0.09$ | $-2.76\pm 0.03$ | $0.89$ | 1559 | $R_{I}^{VI}=0$
Figure 2: The individual Cepheid $P-L$ relations derived from the primary
$W_{H}$ fit. In each panel, the solid red line indicates the best fit, while
the dashed red lines represent one standard deviation of the residuals within
each galaxy around the optimal fit. Black symbols denote Cepheids that passed
the outlier rejection, whereas red symbols represent those that did not.

We proceed to demonstrate that the systematic error associated with
determining $H_{0}$ using our primary R22-$W_{H}$ fit is considerably smaller
than the statistical error. Following the methodology of R22, we evaluate the
systematic uncertainty by investigating the impact of various fit variants on
the derived value of $H_{0}$. Most of the examined variants exhibit a
negligible effect on $H_{0}$ ($|\delta H_{0}|<0.6$). These include variants
such as incorporating Cepheids with $\log P<1$ (with or without a different
$P-L$ slope; the effect is minimal due to the limited number of Cepheids in
this range, approximately 250), exclusion of the outlier rejection, adjustment
of the outlier rejection to eliminate the single largest outlier at a time,
raising the outlier rejection threshold to $3.5\sigma$ (with either global or
single-outlier-at-a-time rejection), adjusting the period cutoff to $\log
P>1.2$ (which aligns the period distributions of nearby and faraway galaxies
more closely), disregarding the metallicity term (setting $Z_{H}^{W}=0$),
adopting $R_{H}^{VI}=0.3$ (smaller than $R_{V}=2.5$ of Fitzpatrick, 1999),
employing $R_{H}^{VI}=0.5$ (larger than $R_{V}=3.3$ of Cardelli, Clayton, &
Mathis, 1989), and including M31 Cepheids (the effect is minimal due to the
limited number of Cepheids in M31 with $\log P>1$, approximately 50). The sole
variant exhibiting a non-negligible effect on the value of $H_{0}$ is the
adjustment of the period cutoff to $\log P<1.6$ ($\delta H_{0}\approx-0.88$).
Given the minimal impact of all fit variants, we can safely disregard the
systematic uncertainty in this case.

The robustness of our primary R22-$W_{H}$ fit stems from the uniformity of the
Cepheid dataset utilized. The consistent observational methodology employed to
determine photometry across this sample mitigates many potential systematic
errors associated with photometry determination. Generally, any systematic
biases that are power-laws of the period are absorbed within the $P-L$
relation, rendering them inconsequential. However, other functional forms or
other systematic differences between nearby and faraway galaxies may still
exert some influence. For instance, the average periods of Cepheids in nearby
galaxies tend to be shorter compared to those in faraway galaxies (refer to
Figure 1). To scrutinize such effects rigorously, we propose a robust,
overarching test. This test directly compares Cepheids with identical periods
across nearby and faraway galaxies, without presupposing a universal $P-L$
relation. Before applying this test, it is insightful to revisit our primary
fit within a narrow period bin. The outcomes of such fits within $0.1$-wide
$\log P$ bins are depicted in Figure 3. Evidently, the fit results are
distributed around the primary fit outcome, exhibiting no discernible period
dependence, and all are consistent with the primary fit (though the precision
within each bin is constrained by the number of Cepheids and SNe Ia available,
indicated for each bin). Subsequently, we conduct a joint fit across all bins
simultaneously, as illustrated in Figure 4, encompassing ten $\log P$ period
bins spanning the range $[1,2]$. This fit assumes only a local power-law $P-L$
relation within each $0.1$-wide $\log P$ bin and remains unaffected by
potential period distribution disparities between nearby and faraway galaxies.
The resultant determination of $H_{0}=72.50\pm 1.69$ fully aligns with our
primary fit outcome (case 2 in Table 1), with no compelling evidence for a
substantially better fit.

Figure 3: Results of the fitting procedure using $W_{H}$ in $0.1-\log P$ bins.
The fits’ outcomes (depicted in black; errors denote one standard deviation)
are dispersed around the primary fit result (shown in blue, where the solid
line represents the best fit and dashed lines indicate one standard
deviation), without exhibiting a clear dependence on period, and all align
with the primary fit result (though the precision within each period bin is
constrained by the quantity of Cepheids and SNe Ia, indicated for each bin).
The red lines denote the Planck results.  Figure 4: The individual $W_{H}$
Cepheid $P-L$ relations for the simultaneous fit in $0.1-\log P$ bins. Each
panel shows the best fit (in each period bin) represented by the broken solid
red line. Black (red) symbols denote Cepheids that passed (did not pass)
outlier rejection for the respective galaxy under examination. This fit
remains unbiased by the varying distributions of periods in nearby and faraway
galaxies. The derived value of $H_{0}$ is $72.09\pm 1.75$, fully consistent
with the primary fit result.

In conclusion, our primary $W_{H}$ fit enables the determination of $H_{0}$
with an accuracy of approximately $2.3\%$, with a significantly reduced impact
from systematic uncertainties.

##  4 $W_{I}$ fitting results

The minimal systematic uncertainties associated with extinction, metallicity,
and the $P-L$ relation shape, as demonstrated for our primary $W_{H}$ fit
(Section 3), suggest that comparable accuracy can also be achieved with
$W_{I}$. In this section, we validate this assertion. Leveraging optical
photometry offers the potential for enhanced precision compared to NIR
photometry, given the significantly lower optical background attributed to
higher resolution, smaller pixels, and diminished flux from red giants (R22).
While $\sigma_{I}^{W}$ is indeed smaller than $\sigma_{H}^{W}$ (refer to
Figure 5), the magnitude of this effect is not as substantial as anticipated
(with the mean of the error distribution being approximately
$0.45\,\textrm{mag}$ for $W_{H}$ and $0.35\,\textrm{mag}$ for $W_{I}$).
Further discussion on the fidelity of the R22-$W_{I}$ dataset is provided
towards the end of this section.

Figure 5: The histograms depict the distributions of the total error in
magnitudes and colors provided by the R22 data release. In each histogram set,
black represents $\sigma_{H}^{W}$, red represents $\sigma_{I}^{W}$, and blue
represents the error of the color term (F555W-F814W). Solid histograms
correspond to faraway galaxies, while dashed histograms correspond to nearby
galaxies. The legend provides the means and standard deviations of the
distributions (in magnitudes). We applied a correction of $15\%$, as
recommended by the SH0ES collaboration due to the correlation between F555W
and F814W background errors, to both $\sigma_{I}^{W}$ and (F555W-F814W).
Errors in faraway galaxies are slightly larger than those in nearby galaxies.
$\sigma_{I}^{W}$ is smaller than $\sigma_{H}^{W}$, although the extent of this
effect is not as substantial as expected.

For our primary R22-$W_{I}$ fit, we adopt $R_{I}^{VI}=1.18$ (corresponding to
$R_{V}=3.3$ of Fitzpatrick, 1999, which closely aligns with $R_{I}^{VI}=1.19$
of R22, see Section 5.1), yielding $H_{0}=76.16\pm 1.69$ (case 3 in Table 1).
It is noteworthy that we obtain $\chi_{\nu}^{2}\approx 0.901$ for $1299$
degrees of freedom, with a probability of $\mathord{\approx}5\times 10^{-3}$
to achieve a lower value of $\chi_{\nu}^{2}$. Using the initially published
$\sigma_{I,i,j}^{W}$ without the $0.85$ multiplier (see Section 2.2) resulted
in $\chi_{\nu}^{2}\approx 0.750$ for $1321$ degrees of freedom, with an
insignificant chance of obtaining such a low value of $\chi_{\nu}^{2}$666After
we communicated this discrepancy to the SH0ES collaboration, they introduced
the $15\%$ correction.. This relatively low value of $\chi_{\nu}^{2}$ might be
attributed to an overestimate of $\sigma_{I,i,j}^{W}$ due to the correlation
between F555W and F814W background errors777SH0ES collaboration, private
communication., which is presently estimated at $\mathord{\approx}15\%$ (and
accounted for in our analysis), although not explicitly discussed in any SH0ES
publication. If we increase the assumed error overestimation to $20\%$, we
obtain $\chi_{\nu}^{2}\approx 0.97$ that aligns perfectly with $1291$ degrees
of freedom, without altering any of our conclusions below. Consequently, we
proceed with the recommended $15\%$ error overestimation for the optical data,
mindful that further quality checks may be necessary before achieving the same
level of reliability as the NIR data.

The obtained $\mathord{\approx}2.2\%$ error in $H_{0}$ can be interpreted
similarly to how it was estimated for the primary $W_{H}$ fit. We find
$\delta\Delta\mu_{\rm{N}4258}\approx 0.019\,\rm{mag}$, which can be
approximated as $\delta\Delta\mu_{\rm{N}4258}\approx 0.33/\sqrt{220}\approx
0.022\,\rm{mag}$, where the mean statistical uncertainty of the 220 N4258
Cepheids’ Wesenheit magnitude is $\mathord{\approx}0.33\,\rm{mag}$. Similarly,
we derive $\delta(\Delta\mu_{\rm{host}}+m_{B})\approx 0.027\,\rm{mag}$, with
SNe Ia dominates the error budget, where $\delta_{m_{B}}\approx
0.024\,\rm{mag}$, which can be approximated as $\delta_{m_{B}}\approx
0.11/\sqrt{19}\approx 0.025\,\rm{mag}$. Although the photometry of the
Cepheids offers higher accuracy, reducing the error of the relative distance
between NGC 4258 and the host galaxies based on the Cepheids $P-L$ relation,
the smaller number of SNe Ia increases the error, resulting in a total error
similar to the $W_{H}$ case. Our estimate yields $\delta M_{B}\approx
0.046\,\rm{mag}$, leading to an error of $\mathord{\approx}2.1\%$ in $H_{0}$,
consistent with the fit results. It is expected that the $H_{0}$ error with
the complete 42 SNe Ia sample of R22 would be $\mathord{\approx}1.9\%$,
although inclusion of the full covariance matrix for the optical Wesenheit
magnitude (not accounted for in our analysis) might slightly alter this
estimate.

We proceed to assess the systematic error in determining $H_{0}$ with our
primary R22-$W_{I}$ fit. Most of the examined variants exhibit negligible
effects on $H_{0}$ ($|\delta H_{0}|<0.55$). These include incorporating
Cepheids with $\log P<1$, with or without a different $P-L$ slope (the impact
is minor due to the limited number of Cepheids, $\mathord{\approx}200$, with
$\log P<1$), exclusion of the outlier rejection, adjustment of the outlier
rejection to eliminate the single largest outlier at a time, raising the
outlier rejection threshold to $3.5\sigma$ (with either global or single-
outlier-at-a-time rejection), implementing a period cutoff $\log P<1.6$,
disregarding the metallicity term (i.e., setting $Z_{I}^{W}=0$), adopting
$R_{I}^{VI}=0.95$ (smaller than $R_{V}=2.5$ of Fitzpatrick, 1999), adopting
$R_{I}^{VI}=1.4$ (larger than $R_{V}=3.3$ of Cardelli, Clayton, & Mathis,
1989), and including M31 Cepheids (with minimal impact due to the limited
number of $\mathord{\approx}50$ Cepheids in M31 with $\log P>1$). The sole
variant exhibiting a non-negligible effect on the value of $H_{0}$ is the
adjustment of the period cutoff to $\log P>1.2$ ($\delta H_{0}\approx-1.26$).
Given the marginal effects of all fit variants, we conclude that the
systematic uncertainty can be disregarded in this case as well. One
particularly surprising outcome is the observed low sensitivity to
$R_{I}^{VI}$, considering the conventional rationale for employing $W_{H}$ due
to the presumed high sensitivity of $W_{I}$ to extinction correction. We
elucidate the reason for this diminished sensitivity with our sample in the
subsequent section.

We next replicate our binned analysis for the $W_{I}$ scenario. The outcomes
of the fits in $0.1-\log P$ bins are displayed in Figure 6. As depicted in the
figure, although the fit outcomes in each bin align with the primary fit
result (albeit with accuracy limited by the number of Cepheids and SNe Ia in
each period bin, indicated in the figure), there appears to be a discernible
trend with period in this case. Upon fitting $H_{0}$ in each bin as a linear
function of $\log P$, we ascertain a slope of $11.4\pm
3.5\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}\,\rm{dex}^{-1}$ (a significance of
$\mathord{\approx}3.3\sigma$). This should be considered an upper limit to the
trend, as we have not accounted for the correlation among some bins stemming
from shared hosts, and all bins are influenced by the NGC 4258 distance error.
Fitting only for $\log P>1.2$, where each bins contains a significant fraction
of the hosts (to minimize the variance of the SNe Ia sample compared with the
Cepheid sample), we obtain a slope of $10.1\pm
4.7\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}\,\rm{dex}^{-1}$ (a significance of
$\mathord{\approx}2.2\sigma$). This observation might signify additional
concerns with the optical photometry (in addition to the overestimated
$\sigma_{I,i,j}^{W}$). Subsequently, upon fitting all bins collectively, we
arrive at $H_{0}=75.25\pm 1.71$, aligning with the primary fit result (case 4
in Table 1). We discern no substantial evidence for a superior fit, with a
$\chi^{2}$ smaller by approximately 59 from a single, $[1,2]$, $\log P$-bin
fit and 37 fewer degrees of freedom.

Figure 6: Results of fitting using $W_{I}$ in $0.1-\log P$ bins. The primary
fit result (blue, solid line represents the best fit, and dashed lines
indicate one standard deviation) aligns with the fit in each bin (though the
accuracy in each period bin is restricted by the number of Cepheids and SNe
Ia, indicated for each bin). An apparent trend in the fit outcomes (black;
errors denote one standard deviation) with period is observable. By fitting
$H_{0}$ in each bin as a linear function of $\log P$, we derive a slope of
$11.4\pm 3.5\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}\,\rm{dex}^{-1}$ (a
significance of $\mathord{\approx}3.3\sigma$). This should be considered an
upper limit to the trend, as we have not accounted for the correlation among
some bins stemming from shared hosts, and all bins are influenced by the NGC
4258 distance error. Fitting only for $\log P>1.2$, where each bins contains a
significant fraction of the hosts (to minimize the variance of the SNe Ia
sample compared with the Cepheid sample), we obtain a slope of $10.1\pm
4.7\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}\,\rm{dex}^{-1}$ (a significance of
$\mathord{\approx}2.2\sigma$). Red lines denote the Planck results.

##  5 Limited Impact of Extinction Corrections

In the preceding section, we illustrated the minimal impact of extinction
correction on the $W_{I}$ fitting results. Here, we delve into the rationale
behind this observation. We commence with a discussion on extinction
corrections (subsection 5.1), followed by an evaluation of the extinction
distribution of extragalactic Cepheids (subsection 5.2). Subsequently, in
subsection 5.3, we showcase a substantial sensitivity to extinction when
incorporating MW Cepheids.

###  5.1 Correction for extinction

The rationale behind employing the Wesenheit index lies in its minimal
susceptibility to extinction (Madore, 1982). Defined for three bands, $X$,
$Y$, and $Z$, the index is expressed as

$\displaystyle W=X-R_{X}^{YZ}\left(Y-Z\right),$ (8)

measured in magnitudes. In the presence of extinction, this can be expressed
as

$\displaystyle W$ $\displaystyle=$
$\displaystyle(X-X_{0})+X_{0}-R_{X}^{YZ}E(Y-Z)-R_{X}^{YZ}\left(Y-Z\right)_{0}$
(9) $\displaystyle=$ $\displaystyle X_{0}-R_{X}^{YZ}\left(Y-Z\right)_{0}\equiv
W_{0},$

where the subscript $0$ represents the extinction-free magnitude,
$E(Y-Z)=(Y-Z)-(Y-Z)_{0}$ denotes the selective extinction, and the final
equality holds if
$R_{X}^{YZ}=(X-X_{0})/E(Y-Z)=A_{X}/E(Y-Z)\equiv\tilde{R}^{X}_{YZ}$, with
$A_{X}$ representing the total extinction in band $X$. An often-overlooked
aspect is that the Wesenheit index doesn’t correct $X$ for extinction;
instead, with the appropriate choice of $R_{X}^{YZ}$, it yields a quantity
insensitive to extinction (also noted in R22). For any arbitrary value of
$R_{X}^{YZ}$, the deviation of $W$ from $W_{0}$ is given by

$\displaystyle W-W_{0}$ $\displaystyle=$ $\displaystyle
A_{X}-R_{X}^{YZ}E(Y-Z)$ (10) $\displaystyle=$
$\displaystyle\left(\tilde{R}^{X}_{YZ}-R_{X}^{YZ}\right)E(Y-Z),$

such that even if $\tilde{R}^{X}_{YZ}$ is not precisely known or varies across
different lines-of-sight towards the Cepheids, the deviation in $W-W_{0}$
remains minimal as long as the selective extinction $E(Y-Z)$ remains small.
With zero extinction, $E(Y-Z)=0$, any value of $R_{X}^{YZ}$ can be chosen.
However, this value must be consistent across all Cepheids; otherwise,
disparate quantities are being compared. Mörtsell et al. (2022a);
Perivolaropoulos & Skara (2021) selected different values of $R_{X}^{YZ}$ for
different Cepheids, rendering their conclusion invalid for that analysis (also
highlighted in R22).

An alternative approach involves directly adjusting $X$ for extinction (Follin
& Knox, 2018):

$\displaystyle F$ $\displaystyle=$ $\displaystyle X-R_{X}^{YZ}E(Y-Z),$ (11)
$\displaystyle=$
$\displaystyle(X-X_{0})+X_{0}-R_{X}^{YZ}\left[(Y-Z)-(Y-Z)_{0}\right]$
$\displaystyle=$ $\displaystyle X_{0},$

where the last equality holds if $R_{X}^{YZ}=\tilde{R}^{X}_{YZ}$. Similar to
the $W$ case, for any given value of $R_{X}^{YZ}$, the deviation of $F$ from
$X_{0}\equiv F_{0}$ is given by

$\displaystyle F-F_{0}$
$\displaystyle=\left(\tilde{R}^{X}_{YZ}-R_{X}^{YZ}\right)E(Y-Z).$ (12)

The distinction lies in the allowance for choosing different $R_{X}^{YZ}$
values for different Cepheids, as the color term vanishes under zero
extinction. However, this method assumes knowledge of $(Y-Z)_{0}$, which is
not always available, such as for $(V-I)_{0}$ in the range $\log P>1.72$
(Sharon et al., 2024), where a significant fraction of the extragalactic
Cepheids is situated (see Figure 1). It is worth noting that if $(Y-Z)_{0}$
follows a linear relationship with $\log P$, the analysis with $F$ is
identical to that with $W$ (for a single global $R_{X}^{YZ}$ value), as the
difference between $E(Y-Z)$ and $(Y-Z)$ gets absorbed into $M_{k}^{W}$ and
$b_{k}^{W}$ (Follin & Knox, 2018).

We infer that minimizing sensitivity to extinction corrections involves
selecting Cepheids with minimal extinction. As demonstrated in Section 5.2,
this holds true for both nearby and faraway galaxies, elucidating the reduced
susceptibility of our dataset to extinction correction.

###  5.2 The extinction distribution

We initiate the analysis by assessing the reddening distribution of
extragalactic Cepheids (utilizing the R22-$W_{H}$ sample unless specified
otherwise). The top panel of Figure 7 displays the (F555W-F814W) color of
extragalactic Cepheids against their period. We confine this examination to
$\log P<1.72$, where an accurate determination of the intrinsic MW Cepheids
colors is available (Sharon et al., 2024). In Appendix A, we compute the
intrinsic color for $1<\log P<1.72$ (employing methods from Tammann, Sandage,
& Reindl, 2003):

$\displaystyle(\textrm{F555W-F814W})_{0}$ $\displaystyle\approx$
$\displaystyle 0.156\log P+0.75,$ (13)

represented by a solid black line. Additionally, we distinguish between nearby
(M101 and NGC 4258; black circles) and faraway (the other hosts; red circles)
galaxies, considering that Cepheids in nearby galaxies are measured with
slightly higher precision (see Figure 5). The errors in (F555W-F814W) color
(not depicted) primarily stem from dispersion due to blending, amounting to
approximately $0.15,\rm{mag}$ for the nearby sample and $0.2,\rm{mag}$ for the
faraway sample.

Figure 7: Top panel: the $P-C$ relation for the (F555W-F814W) color of the
R22-$W_{H}$ sample. Nearby (faraway) Cepheids from M101 and NGC 4258 (the
other hosts) are denoted by black (red) circles, while M31 Cepheids are
represented by blue circles. The color errors (not depicted) are primarily
influenced by dispersion due to blending, with values of approximately
$\mathord{\approx}0.15\,\rm{mag}$ for the nearby sample and
$\mathord{\approx}0.2\,\rm{mag}$ for the faraway sample. The intrinsic MW
$P-C$ relation, Equation (13), is illustrated by the black line. Bottom panel:
The distributions of the host $E(B-V)$, derived using Equation (5.2) and
corrected for (the small, $\mathord{\lesssim}0.1$ mag) Galactic extinction
(Schlafly & Finkbeiner, 2011). Nearby (faraway) galaxies are represented by
solid black (red) lines, while M31 Cepheids are shown in blue. The solid
(dashed) green lines denote MW Cepheids from the sample of Riess et al.
(2021), with MW $E(B-V)$ values sourced from the catalog of Sharon et al.
(2024), without (with) a $\log P>1$ restriction. The $E(B-V)$ distributions of
the nearby sample exhibit a pronounced change at $E(B-V)\approx 0$, indicative
of a blue edge. The small number of Cepheids with negative $E(B-V)$ values
(physically implausible) are in line with the scatter of
$\sigma_{\rm{int}}\approx 0.1\,\rm{mag}$ around the intrinsic $P-C$ relation
(see Appendix A), as well as the color error attributed to blending. The blue
edge of the faraway sample’s $E(B-V)$ distribution appears somewhat less
distinct, likely due to larger color errors in faraway galaxies that can
obscure the blue edge. Cepheids in nearby and faraway galaxies experience
relatively low extinction, while M31 and MW Cepheids are subject to more
substantial extinction.

Subsequently, we estimate the host galaxy $E(B-V)$ toward each Cepheid using
(see Appendix A):

$\displaystyle E(B-V)$ $\displaystyle\approx$ $\displaystyle
1.42[(\textrm{F555W-F814W})-(\textrm{F555W-F814W})_{0}],$

and then we correct for (the small, $\mathord{\lesssim}0.1$ mag) Galactic
extinction (Schlafly & Finkbeiner, 2011). The resulting $E(B-V)$ distribution
is depicted in the bottom panel of Figure 7. As illustrated in the figure, the
$E(B-V)$ distribution of the nearby sample (solid black line) exhibits a sharp
change at $E(B-V)\approx 0$, indicative of a blue edge. This blue edge
corroborates both Equation (13) and the photometry of nearby Cepheids in
(F555W-F814W), which were determined through entirely independent
methods888For instance, deriving Equation (13) necessitates knowledge of the
extinction toward MW Cepheids, sourced from Fernie et al. (1995); Turner
(2016); Groenewegen (2020).. The small number of Cepheids with negative
$E(B-V)$ (physically implausible values) aligns with the scatter of
$\sigma_{\rm{int}}\approx 0.1\,\rm{mag}$ around the intrinsic $P-C$ relation
(refer to Appendix A, corresponding to the width of the instability strip;
Tammann, Sandage, & Reindl, 2003), as well as with color errors stemming from
blending. The blue edge of the faraway sample’s $E(B-V)$ distribution (solid
red line) appears somewhat less distinct, which is understandable given the
larger color errors associated with faraway galaxies that can blur the blue
edge.

The distributions’ shapes are influenced by various selection biases, notably
favoring Cepheids with minimal extinction ($E(B-V)\lesssim 0.5$), resulting in
a negligible impact of extinction corrections. To further illustrate this
minimal impact, we perform our primary fit without the color term, setting
$R_{I}^{VI}=R_{H}^{VI}=0$. For $W_{H}$, we derive $H_{0}=72.39\pm 1.66$ (case
5 in Table 1), compared to $H_{0}=72.68\pm 1.67$ with $R_{H}^{VI}=0.386$.
Similarly, for $W_{I}$, we obtain $H_{0}=74.42\pm 1.66$ (case 6 in Table 1),
contrasting with $H_{0}=76.16\pm 1.69$ with $R_{I}^{VI}=1.18$. The overall
effect on $H_{0}$ amounts to $\delta H_{0}\approx 0.3$ for $W_{H}$ and $\delta
H_{0}\approx 1.7$ for $W_{I}$, validating our assertion of small systematic
uncertainty in Sections 3 and 4.

###  5.3 Incorporating MW Cepheids

Here, we illustrate the significant sensitivity of the $W_{I}$ fitting results
to extinction that arises when including MW Cepheids. Initially, we repeat our
primary $W_{H}$ fit with the inclusion of MW Cepheids, employing the MW anchor
instead of the NGC 4258 anchor. Following the methodology of Mörtsell et al.
(2022a), we transform the residual Gaia parallax calibration offset, $zp$,
into a linear parameter and fit for it. The data for MW Cepheids are sourced
from Table 1 in Riess et al. (2021). Since a significant fraction of the MW
Cepheids have $\log P<1$, we do not restrict the Cepheid sample to $\log P>1$
in this case (for all galaxies). We obtain $H_{0}=74.43\pm 2.21$999We
incorporate an intrinsic dispersion of $\sigma_{\rm{int}}=0.069\,\rm{mag}$ for
$W_{H}$ of the MW Cepheids (Riess et al., 2019a). (case 7 in Table 1),
consistent with the result using the NGC 4258 anchor, $H_{0}=72.68\pm 1.67$.
Similarly small impact is observed for our primary $W_{I}$ fit101010Here, we
adopt $\sigma_{\rm{int}}=0.085\,\rm{mag}$ for the intrinsic dispersion (upper
limit from Table 3 of Riess et al., 2019a). ($H_{0}=75.67\pm 2.24$ for the MW
anchor, case 8 in Table 1, compared to $H_{0}=76.16\pm 1.69$ for the NGC 4258
anchor). Subsequently, we demonstrate the considerable systematic uncertainty
of the $W_{I}$ fitting results related to the extinction law, attributed to
the substantial extinction associated with MW Cepheids.

The bottom panel of Figure 7 illustrates the selective extinction affecting
the MW Cepheids utilized in our analysis. These data are sourced from the
catalog of Sharon et al. (2024), with detailed references therein outlining
the methods employed to estimate the extinction. As depicted in the figure, MW
Cepheids, particularly those with $\log P>1$, endure substantial extinction
due to their location within the Galactic disc, where massive stars are
prevalent, relative to the position of the Sun. When using the MW anchor
instead of the NGC 4258 anchor and setting $R_{I}^{VI}=R_{H}^{VI}=0$, we
obtain $H_{0}=74.28\pm 2.21$ for $W_{H}$ (case 9 in Table 1) and
$H_{0}=83.1\pm 2.6$ for $W_{I}$ (case 10 in Table 1). While a negligible
effect is obtained for the $W_{H}$ case, the effect is notably pronounced in
the $W_{I}$ case ($\delta H_{0}\approx 7.5$), indicating that even minor
uncertainties in correcting for this effect would result in significant
systematic uncertainties. The heightened effect observed for $W_{I}$ primarily
motivates observations with the F160W filter. However, as demonstrated in
Section 5.2, this effect is considerably mitigated for our sample.

A comparable scenario arises for M31. The selective extinction affecting M31
Cepheids is depicted in the bottom panel of Figure 7. This distribution is
derived using the methodologies outlined in Section 5.2. As illustrated in the
figure, M31 Cepheids experience substantial extinction due to the nearly edge-
on orientation of M31’s disk, but close enough to allows for the measurement
of obscured Cepheids.

##  6 Discussion and Summary

The SH0ES collaboration employs a comprehensive approach to ascertain $H_{0}$,
combining host Cepheids with a diverse array of Cepheids from various
observations. This strategy minimizes statistical uncertainty in $H_{0}$ and
harnesses multiple independent geometrical distance anchors, mitigating
sensitivity to potential systematic errors in any single anchor distance
determination. Cepheids from NGC 4258, as well as from closer galaxies like
the MW, LMC, SMC, and M31, are incorporated to allow the use of multiple
anchors and to enhance the precision of the $P-L$ relation. However, this
Cepheid sample with a wide-distance range is susceptible to systematic errors
due to the comparison of distant host Cepheids with those much closer. Host
and NGC 4258 Cepheids are measured via HST imaging, while closer Cepheids are
identified through archival observations and observed using different
techniques. Corrections are applied to account for differences in observing
methodologies and biases in period distributions and extinction levels between
close and host Cepheids. Additionally, the metallicity discrepancy between
LMC/SMC and host Cepheids is addressed by considering a metallicity-dependent
$P-L$ relation.

We proposed utilizing a subset of SH0ES Cepheids to derive $H_{0}$
independently of potential systematic errors associated with Cepheids. This
subset excludes Cepheids from the MW, LMC, SMC, and M31, retaining only the
water megamasers in NGC 4258 as a single anchor. By omitting these Cepheids,
which are subject to different observational methods and reduction techniques
compared to those in host galaxies and NGC 4258, potential systematic errors
in Cepheid measurement are circumvented, encompassing identification,
selection, period determination, photometry, template usage, and crowding
correction. We demonstrate the elimination or substantial reduction of
uncertainties pertaining to reddening law, metallicity sensitivity, and breaks
in the $P-L$ relation, facilitating precise determination of Cepheid relative
distances. Remaining statistical errors in Cepheid relative distance
determination, primarily associated with Cepheid and SNe Ia brightness zero
points, are sufficiently small for competitiveness, $\mathord{\approx}2.3\%$
(or about 1.65 km/s/Mpc) using the R22 data release. The Cepheid-related
systematic errors are significantly smaller than the statistical uncertainty,
rendering the accuracy adequate to identify $\mathord{\approx}3\sigma$ tension
with the Planck $H_{0}$ value.

Once the decision is made to forego the local anchor points, there is no
inherent requirement to shift to the NIR. As demonstrated in Section 4, we
established that the systematic uncertainty associated with optical photometry
is considerably smaller than the statistical uncertainty. This implies the
potential for achieving greater accuracy compared to NIR photometry, primarily
because the optical background is approximately an order of magnitude lower
than that in the NIR, owing to factors such as higher resolution, smaller
pixels, and reduced flux from red giants (R22). The observed low sensitivity
to extinction correction stems from the minimal extinction experienced by
Cepheids within our sample (Section 5). However, in practice, the quality of
photometry in optical bands does not exhibit substantial improvement over NIR
photometry, and the reliability of the optical photometry sample does not
match that of the NIR photometry sample, as demonstrated in Section 4.

The primary drawback of our approach lies in the pronounced reliance of
$H_{0}$ on the distance to NGC 4258, $D_{4258}$, as expressed by the equation:

$\displaystyle\frac{dH_{0}}{H_{0}}\approx-f\frac{dD_{425}}{D_{4258}},$ (15)

where $f\approx 1$, in contrast to $f\approx 1/3$ with the inclusion of three
anchors (NGC 4258, LMC, and MW). This heightened dependency renders our method
more susceptible to systematic uncertainties in $D_{4258}$. Nonetheless, given
the resilience of our method against Cepheid-related systematic errors and the
absence of discernible signs of SNe Ia-related systematic errors in detailed
examination (R22, Carr et al., 2022; Brout et al., 2022; Peterson et al.,
2022, but see also Steinhardt, Sneppen, & Sen 2020; Wojtak & Hjorth 2022,
2024), our analysis suggests two potential scenarios: either standard
cosmology requires revision or there exists a systematic error in the current
determination of the distance to NGC 4258, $D_{4258}\approx 7.57\pm
0.24\,\textrm{Mpc}$. To fully alleviate the Hubble tension, a distance of
$D_{4258}\approx 8.15\,\textrm{Mpc}$ is necessary, and considering the
prevailing uncertainties, a distance (with a central value) exceeding
$D_{4258}\gtrsim 7.8\,\textrm{Mpc}$ is needed to reduce the tension below
$2\sigma$. Moreover, the scenario of systematic error in the distance to NGC
4258 to resolve the Hubble tension would also necessitate an unrelated
systematic error in the comparison of distant host Cepheids with those much
closer to explain the $H_{0}$ measurement of R22. However, solid evidence
supporting this assertion has not been presented so far.

Our investigation underscores the importance of conducting a meticulous
analysis of optical observations of extragalactic Cepheids to enable the
application of our method with an optical Cepheid sample. Looking ahead,
observations with JWST hold promise for further reducing optical crowding
corrections (Riess et al., 2024). Moreover, the heightened sensitivity of JWST
may facilitate the measurement of Cepheids in additional megamaser galaxies
with precise geometrical distances, such as UGC 3789 ($D=49.6\pm
5.1\,\rm{Mpc}$; Reid et al., 2013), thereby enabling the extension of our
method with supplementary anchors.

## Acknowledgements

We thank members of the SH0ES collaboration: Adam Riess, Dan Scolnic, Lucas
Macri, Stefano Casertano, and Wenlong Yuan for extensive correspondences and
discussions that significantly improved the manuscript. We thank Andy Gould,
Eli Waxman, Boaz Katz, Barak Zackay, Subo Dong, and Eran Ofek for useful
discussions. DK is supported by a research grant from The Abramson Family
Center for Young Scientists, and by the Minerva Stiftung.

## Data availability

All data used in this study are publicly available through other publications.

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##  Appendix A The period-color relation

This appendix utilizes the catalog provided by Sharon et al. (2024) to
investigate the intrinsic period-color ($P-C$) relationship of Cepheids
falling within the range of $1<\log P<1.72$ across various colors, as
pertinent to the analysis discussed in Section 5. We adopt the methodology
outlined by Tammann, Sandage, & Reindl (2003). Initially, we compute
$(B-V)_{0}$ by subtracting $E(B-V)$111111$E(B-V)$ values in the catalog are
sourced from Fernie et al. (1995); Turner (2016); Groenewegen (2020). from
$(B-V)$, as illustrated in the upper left panel of Figure 8 (following the
exclusion of outliers VZ-Pup and ER-Aur). We fit $(B-V)_{0}$ against $\log P$
linearly and observe a relatively high reduced $\chi^{2}$, suggesting that the
scatter around the fit is intrinsic, so we calibrate an intrinsic scatter of
approximately $0.07$ mag to obtain a unity for the reduced $\chi^{2}$. Upon
incorporating the calibrated intrinsic scatter, the resulting fit is
$(B-V)_{0}=(0.387\pm 0.048)\log P+(0.331\pm 0.059)\,\rm{mag}$ (depicted by
black lines, with a scatter of $0.10$ mag). In the bottom left panel, we
depict the deviation of each Cepheid from the best-fit line, $\Delta(B-V)$,
against the estimated $E(B-V)$. No discernible trend is observed (the slope is
$-0.010\pm 0.027$, indicated by black lines), indicating the absence of a
systematic scaling error in the determined $E(B-V)$ values. Notably, our
derived intrinsic $(B-V)_{0}$ color closely resembles the findings of Tammann,
Sandage, & Reindl (2003, illustrated by the red line), albeit with updated
$E(B-V)$ values sourced from Turner (2016) for $56$ Cepheids and a limitation
of Cepheid periods to $1<\log P<1.72$.

Figure 8: Determining the intrinsic $P-C$ relation of Cepheids with $1<\log
P<1.72$. The upper left panel displays the direct determination
$(B-V)_{0}=(B-V)-E(B-V)$ (black symbols). Fitting $(B-V)_{0}$ as a linear
function to $\log P$ yields a large reduced $\chi^{2}$, indicating intrinsic
scatter around the fit (with a calibrated intrinsic scatter of
$\mathord{\approx}0.07\,\rm{mag}$). Incorporating this calibrated intrinsic
scatter, the resulting fit is $(B-V)_{0}=(0.387\pm 0.048)\log P+(0.331\pm
0.059)\,\rm{mag}$ (black lines, with a scatter of $0.10\,\rm{mag}$). In the
bottom left panel, we present the distance of each Cepheid from the best-fit
line, $\Delta(B-V)$, plotted against the estimated $E(B-V)$. No trend is
observed (the slope is $-0.010\pm 0.027$, black lines), indicating no
systematic scale error in the determined $E(B-V)$ values. Our derived
intrinsic $(B-V)_{0}$ color closely matches the result of Tammann, Sandage, &
Reindl (2003, red line), albeit utilizing updated $E(B-V)$ values from Turner
(2016) for $56$ Cepheids and restricting Cepheid periods to $1<\log P<1.72$.
Middle panels: We calibrate $E(V-I)=1.33E(B-V)$, and repeating the analysis of
$(B-V)$ for $(V-I)$ reveals no dependence of $\Delta(V-I)$ on $E(V-I)$. The
best fit we obtain is $(V-I)_{0}=(0.225\pm 0.044)\log P+(0.517\pm
0.054)\,\rm{mag}$ (black lines, with a scatter of $0.11\,\rm{mag}$), and the
ratio between $E(V-I)$ and $E(B-V)$ closely resembles the findings of Tammann,
Sandage, & Reindl (2003, red line). Right panels: Repeating the procedure for
$(\textrm{F555W}-\textrm{F814W})$, we calibrate
$E(\textrm{F555W}-\textrm{F814W})=1.42E(B-V)$. The resulting fit is
$(\textrm{F555W}-\textrm{F814W})_{0}=(0.156\pm 0.083)\log P+(0.750\pm
0.102)\,\rm{mag}$ (black lines, with a scatter of $0.08\,\rm{mag}$).

We continue to adhere to the methodology outlined by Tammann, Sandage, &
Reindl (2003) for the determination of $(V-I)_{0}$. By calibrating
$E(V-I)=1.33E(B-V)$, and subsequently analyzing $(V-I)$ in a manner akin to
that of $(B-V)$, we find no discernible dependence of $\Delta(V-I)$ on
$E(V-I)$ (depicted in the middle panels of Figure 8, following the exclusion
of outlier V0396-Cyg). Our best-fit result is $(V-I)_{0}=(0.225\pm 0.044)\log
P+(0.517\pm 0.054)\,\rm{mag}$ (represented by black lines, with a scatter of
$0.11$ mag). Additionally, the ratio between $E(V-I)$ and $E(B-V)$, as well as
the obtained best fit, closely resemble the findings of Tammann, Sandage, &
Reindl (2003, highlighted by the red line).

Finally, we calibrate $E(\textrm{F555W}-\textrm{F814W})=1.42E(B-V)$ (as
depicted in the right panels of Figure 8), resulting in
$(\textrm{F555W}-\textrm{F814W})_{0}=(0.156\pm 0.083)\log P+(0.750\pm
0.102)\,\rm{mag}$ (illustrated by black lines, with a scatter of $0.08$ mag).