improve SN distance estimates now or in the future, the matching of SN and host samples employed here and the large sample size for each would mitigate any significant effect on the determination of $H_0$.

### 7.4. State of the Hubble Tension

Our baseline determination of $H_0$ is $73.04 \pm 1.04$ km s$^{-1}$Mpc$^{-1}$ (with systematics), which exceeds the Planck+$\Lambda$CDM result by $5\sigma$. It may be of interest and it is straightforward to calculate a combined value of $H_0$ that is free of measurement interdependencies and has lower uncertainty using additional redshift–magnitude relation data. We start with the combined Cepheid and TRGB result, $72.53 \pm 0.99$ km s$^{-1}$Mpc$^{-1}$ (or Fits 31 and 32), and to this we add two recent measures that provide enough information that allow us to have a consistent calibration and redshift frame while avoiding double use of data. Pesce et al. (2020) measured six masers in the Hubble flow; excluding the nearest from that set, NGC 4258 (because its maser distance is used here), and quoting their result in the 2M++ frame (same as our baseline) gives $H_0 = 72.1 \pm 2.7$ km s$^{-1}$Mpc$^{-1}$. Blakeslee et al. (2021) measured the IR surface brightness fluctuation distances with HST in 63 galaxies calibrated by the TRGB in the 2M++ frame. Accounting for the small difference in TRGB zero-point ($M_I = -4.014 \pm 0.025$ mag for the mean of CCHP and EDD) and used there ($M_I = -4.03$ mag) yields $H_0 = 74.0 \pm 3.0$ km s$^{-1}$Mpc$^{-1}$. The combination of these independent and independently consistent measures gives $H_0 = 72.61 \pm 0.89$ km s$^{-1}$Mpc$^{-1}$ (or $H_0 = 72.42 \pm 0.89$ km s$^{-1}$Mpc$^{-1}$ with the CCHP TRGB in Fit 32 and $H_0 = 72.80 \pm 0.89$ km s$^{-1}$Mpc$^{-1}$ from the EDD TRGB with Fit 31), a local determination with 1.2% precision, which is also $5.2\sigma$ greater than the Planck+$\Lambda$CDM result. Other combinations may be determined but require care to avoid measurement inconsistencies in calibration, redshift frame, or double use of any data.

There has been a wide variety of ideas proposed to resolve the Hubble tension, including (but not limited to) an episode of scalar-field dark energy before recombination, the presence of additional species of neutrinos (perhaps with interactions), decaying dark matter, the presence of primordial magnetic fields, a changing electron mass, decaying or interacting dark matter, a breakdown of the Friedmann–Lemaître–Robertson–Walker metric or general relativity, and so on; we direct the reader to recent reviews, such as Di Valentino et al. (2021) and Schöneberg et al. (2021). These proposals range from moderately successful to unsuccessful with no clear resolution. Some of the more successful ideas mitigate the tension through a similar mechanism, such as increasing $H(z)$ in the early universe so that recombination occurs earlier, thereby shrinking the sound horizon which is the fundamental scale of the CMB (and also of baryon acoustic oscillations). In many scenarios this will produce additional features in the CMB, which are either incompatible with the data or make them appear more plausible (Hill et al. 2021; Poulin et al. 2021). The presence of unaccounted systematics in early- or late-universe measurements have also been suggested, but in Section 6 we comprehensively reviewed those pertaining to the route presented here, with none showing indications of validity.

Both the late- and early-universe data present formidable obstacles to hypotheses involving “new physics” or new systematics owing to the rigor and redundancy of the measurements; any proposals require specificity to see which

may be viable. However, opportunities for progress on this problem exist on many fronts. We anticipate gains from improved characterization of $H(z)$, the use of new facilities to refine the local measurements (e.g., LIGO and JWST) and the early-universe measurements (e.g., CMB Stage 4 and the Simons Observatory), neutrino experiments, as well as from new theoretical insights.

## 8. Conclusions

1. Our baseline determination of $H_0$ is $73.04 \pm 1.04$ km s$^{-1}$Mpc$^{-1}$ (with systematics) from a Cepheid-only calibration of 42 SNe Ia with good SN data quality, or $72.53 \pm 0.99$ km s$^{-1}$Mpc$^{-1}$ combining Cepheid and TRGB for a total of 46 SN Ia calibrators with good SN data quality.

2. The measurement exceeds the Planck+$\Lambda$CDM result by $5\sigma$ (one in 3.5 million), making it implausible to reconcile the two by chance.

3. An exhaustive study of variations in the analysis and systematic uncertainties including 67 variants of analyses reveals no indication of significant inconsistencies within the measurement or promising sources of unrecognized error. The dispersion of the 59 NIR variants is $0.3$ km s$^{-1}$Mpc$^{-1}$ and is conservatively adopted as an additional systematic uncertainty. The mean of the variants is $73.25$ km s$^{-1}$Mpc$^{-1}$, which is higher than the baseline by $0.2$ km s$^{-1}$Mpc$^{-1}$.

4. We find the dispersion between 42 SN Ia and Cepheid relative distance measures is $\sigma = 0.130$ mag, similar (albeit lower) than the $\sigma = 0.135$ mag dispersion of SNe Ia in the Hubble-flow sample and yielding no evidence of excess noise in Cepheid distance measurements.

5. We find that Cepheid and TRGB distance measures are consistent when starting and ending from the same hosts (i.e., between rung one and two). We highlight a net difference of $1.3$ km s$^{-1}$Mpc$^{-1}$ (or $0.04 \pm 0.02$ mag) between measurements by two groups of the location of the tip in NGC 4258 and the resulting calibration of the TRGB (we use the mean of both), and a net $1$ km s$^{-1}$Mpc$^{-1}$ (0.03 mag) higher value of $H_0$ from the change (tripling) of the SN calibrator sample, which is consistent with the $\pm 0.05$ mag shot noise of the subsamples.

6. We find that each of the three independent geometric anchors is consistent with the distance predicted by its Cepheids and the other two anchors. Improvements in the calibration of the Cepheid metallicity dependence have tightened this conclusion.

7. The SNe between the second and third rungs of the ladder are hosted by galaxies of the same late type (i.e., spiral) with the same or similar mean SFR and mass. Their color and shape distributions are also highly consistent. The calibrator set contains a complete sample of all suitable SNe Ia (i.e., with good data quality) at $z < 0.011$ in the last four decades. We see no indication of differences between the mean properties of the samples' hosts, nor a reason with matched selection that such would exist and impact $H_0$.

8. Extragalactic Cepheids appear to have a uniform relation between period and luminosity consistent with a single slope and fine structure in their light curves that
resembles those in the MW at the same period (i.e., the Hertzsprung progression).

9. The constraint provided by the distance ladder presented here is well approximated by the derived value of $H_0$, except in the case of models that introduce rapid, unexpected, late-time changes in $H(z)$ (relative to either $\Lambda$CDM or low-order fits to $H(z)$), or perhaps some forms of new physics. For such models, we advise replacing $H_0$ with the absolute SN Ia host distances derived from the first two rungs, the set of SN Ia standardized magnitudes in these hosts, SN Ia magnitudes in the Hubble flow, and their covariance, and provide an example that yields a joint constraint of $H_0 = 73.30 \pm 1.04 \text{ km s}^{-1} \text{ Mpc}^{-1}$ and $q_0 = -0.51 \pm 0.024$.

10. The source of this long-standing, significant discrepancy between the local and cosmological routes to determining the Hubble constant remains unknown.

We thank Graeme Addison, Fabio Bresolin, George Efstathiou, and Doron Kushnir for helpful conversations related to this work. We are grateful to Peter Challis for sharing an unpublished spectrum of SN 2008fv. The ESA Hubble office provided most of the color composites shown in Figure 4. An anonymous referee provided a thoughtful report that led to improvements in this paper.

This research was supported by NASA/HST grants GO-12879, GO-12880, GO-13334, GO-13335, GO-13344, GO-15145, GO-15146, and GO-15640 from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the STScI. The specific observations analyzed can be accessed at doi:10.17909/fkay-8z97.

L.M.M. acknowledges additional support from the Mitchell Institute for Fundamental Physics & Astronomy at Texas A&M University. D.O.J. acknowledges support from NASA Hubble Fellowship grant HF2-51462.001 awarded by the Space Telescope Science Institute. A.V.F.'s group at UC Berkeley is also grateful for financial assistance from NSF grant AST-1211916, the TABASGO Foundation, the Christopher R. Redlich Fund, the Miller Institute for Basic Research in Science (in which A.V.F. is a Miller Senior Fellow), and numerous individual donors.

Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA; the observatory was made possible by the generous financial support of the W. M. Keck Foundation. We thank Brad Tucker for designing the multislit masks used for the Keck LRIS spectroscopy of H\ II regions in the host galaxies.

## Appendix A
### Properties of Calibrator SNe Ia

The members of the SN Ia calibrator sample were first identified when selecting hosts for observing Cepheids with HST. We have attempted to follow the criteria for selection given by Riess et al. (2005) to provide reliable SN Ia distances: objects observed before maximum light, through low interstellar extinction ($A_V < 0.5$ mag), with modern (i.e., nonphotographic) photometry, and with typical light-curve shapes. Here

we analyze the spectra of the calibrators to confirm their initial selection.

#### A.1. Spectral Properties Using DeepSIP

We consider two samples of calibrators. The first contains 40 SNe Ia and is a complete, volume-limited ($z < 0.011$) sample of all suitable SNe Ia seen over the years 1980–2021 in spiral hosts (a requirement for finding Cepheids, with results for $H_0$ provided for these alone in Fit 38). A second sample contains just two SNe Ia (SN 1999dq and SN 2007A) from a program targeting hosts that are located at greater redshifts and are more luminous, in an effort to reach the Hubble flow directly with Cepheids. Using the following quantitative spectroscopic analysis, we find that all 42 SNe Ia are in the normal range.

We obtain spectra directly from the Open Supernova Catalog<fnref n="25" /> (OSC; Guillochon et al. 2017), which aggregates data from numerous sources (including notable low-redshift SN Ia spectroscopy releases; e.g., Silverman et al. 2012; Blondin et al. 2012; Folatelli et al. 2013; Stahl et al. 2020b). From the OSC-retrieved spectra, we select—per SN Ia—the nearest-to-maximum-light spectrum having (i) full coverage<fnref n="26" /> of the characteristic Si II $\lambda$ 6355 absorption feature, and (ii) an S/N of at least 10 per pixel.<fnref n="27" /> If these criteria cannot be met by the available spectra, we reduce the S/N threshold to 5 pixel$^{-1}$; if still no spectra satisfy the criteria, we remove the Si II $\lambda$ 6355 coverage requirement. If no spectra are available even after this relaxation, our automated algorithm flags the SN in question for manual intervention. Following this approach, 37 spectra are obtained with our full criteria satisfied, one is obtained with our most relaxed criteria, and two fail.

We resolve these two failures as follows.

1. SN 2008fv was spectroscopically classified as a normal SN Ia in CBET 1522 (Challis 2008). Though never published, we have obtained this spectrum from the author.

2. SN 2021hpr has three spectra available on the Transient Name Server.<fnref n="28" /> We take the spectrum that best satisfies our full criteria as stated above.

We also override successful OSC acquisitions in several cases where more suitable spectra (i.e., those that better match our criteria) are available elsewhere. The spectra in our final set have a median phase of −0.1 days, with the earliest at −11.7 days and the latest at 14.5 days. A full accounting of relevant metadata is available upon reasonable request.

A key concern in studies that utilize SN Ia distances (such as this one) is ensuring that the objects used are indeed “normal” SNe Ia in the sense that they can be standardized using width-luminosity relations (Phillips 1993). Here we take the conservative approach of excluding subluminous SN 1991bg-like (Filippenko et al. 1992b; Leibundgut et al. 1993) and overluminous SN 1991T-like objects (Filippenko et al. 1992a; Phillips et al. 1992), thereby defining “normal” SNe Ia as those that fall within the central, most well-studied and precise region of the Phillips relation. Moreover, we confirm the normalcy of our selected objects spectroscopically by employing the

<footnote n="25">²⁵ sne.space</footnote>

<footnote n="26">²⁶ As in Stahl et al. (2020b), we define “full coverage” as having a wavelength minimum below 5750 Å and a maximum above 6600 Å.</footnote>

<footnote n="27">²⁷ Ties when two spectra have exactly the same phase are broken by taking the one with broader wavelength coverage.</footnote>

<footnote n="28">²⁸ https://www.wis-tns.org/</footnote>
<figure><img src="image_29.png" /><figcaption>**Figure A1.** Spectra (scaled $f_\lambda$) of all 42 SNe Ia in the calibrator sample, color-coded by their deepSIP score calibrated to the photometric parameter, $\Delta m_{15}$. As discussed in the text, we find all calibrator SNe to be in the normal range. Peculiar objects SN 1991T and SN 1991bg bracket the extremes and are shown for comparison.</figcaption></figure>

deepSIP package (Stahl et al. 2020a), which provides highly effective, trained convolution neural networks that, among other things, can (i) classify if a spectrum belongs to an SN Ia with a rest-frame phase between −10 days and 18 days and light-curve shape (parameterized by SNOOPY's $\Delta m_{15}$ parameter; see Burns et al. 2011 for more details) between 0.85 and 1.55 mag, a conservatively narrow window corresponding to “normal” objects, and (ii) predict (with uncertainties) quantitative $\Delta m_{15}$ values. Because we know the rest-frame phase of each spectrum in our sample from the light-curve-derived times of maximum brightness, distinctions made by the aforementioned classifier provide a direct probe if the SNe Ia in our sample are spectroscopically normal. The calibrator SNe Ia have DeepSIP $\Delta m_{15}$ values between 0.84 and 1.37 mag.

We find that only two spectra do not satisfy this test, one of which (SN 2008fv) is expected owing to it being the sole case in our sample where the spectrum is at a phase earlier than −10 days. The other case (SN 1997bq) is most certainly spectroscopically normal (see Blondin et al. 2012). The fact that this single false negative is obtained is not unexpected because in developing deepSIP, Stahl et al. (2020a) tuned the decision threshold of the model with an eye to their subsequent scientific use case in which false positives (i.e., classifying an SN Ia as normal when it actually is not) represent a far worse

error than false negatives (i.e., failing to classify an SN Ia as normal when it actually is). We visualize the entire spectral sample along with representative near-maximum-light spectra of SN 1991bg and SN 1991T for reference, color-coded by deepSIP-predicted $\Delta m_{15}$ value, in Figure A1.

## A.2. Photometric Properties

In Figure A2 we show the distributions of the SALT II color (c) and shape (x1) parameters, as well as the host masses, for the calibrator sample and for three Hubble-flow samples: (1) the baseline 0.0233 < z < 0.15 spiral sample and the tighter quality cuts |c| < 0.15, |x1| < 2; (2) the same redshift range for all host types and the Pantheon+ standard quality cuts |c| < 0.3, |x1| < 3; and (3) a sample of all types to z < 0.8 and standard quality cuts. It can be seen that the samples are well matched in the mean, with the baseline sample better matched in breadth to the calibrator sample owing to the tighter quality cuts. As a result of this investigation, we conclude that each SN in the calibrator sample is unambiguously normal and thus can be reliably standardized to match well the baseline Hubble-flow sample.
<figure><img src="image_30.png" /><figcaption>**Figure A2.** SN Ia and host properties for the calibrator sample (green) and three Hubble-flow samples. The color (left panel) and light-curve shape (middle) distributions are in good agreement as discussed in the text, particularly for the baseline sample (Lo-z Spiral, orange) whose quality cuts limits are $|c| < 0.15$ and $|x1| < 2$. The right panel shows the host mass including a truncation of the Lo-z sample (dashed line) with values for this and other host properties in Table 9.</figcaption></figure>

# Appendix B
## Independent Tests of Cepheid Photometry

### B.1. Aperture Photometry

The photometry of Cepheids presented here is derived from a set of standard procedures referred to in the astronomical literature as “scene modeling” or “crowded-field photometry.” These methods use an empirical description of the PSF to model unresolved sources (i.e., stellar profiles) by comparing image pixels to a model constructed from the superposition of PSFs, each with its own X and Y coordinate and amplitude as well as the constant level of the background. The initial set of sources and their positions may be derived from the image or a catalog.

This approach offers greater precision than fixed-aperture photometry (i.e., summing flux in an aperture), as it can separate the blended flux of distinct sources and additionally improves the S/N by the optimal weighting of source pixels. Bias resulting from the inability to resolve nearly coincident background sources from the Cepheid can be determined statistically by adding artificial stars of known flux to the scene. This bias is really just the mean level of the fluctuating background. Uncertainties are measured from the distributions of recovered artificial stars. Frequently used software packages that enable this approach include DAOphot, DoPHOT, DolPhot, HSTphot, and Romaphot. In general, the use of these different packages has been shown to yield similar results subject to the settings for which these packages are employed.

However, it is valuable to have a robust cross-check of the accuracy of the photometry measured with these techniques using a simpler approach that is easily replicated by others without reference to any specific piece of software. Such a method is aperture photometry, which, though less precise, provides a strong test of the accuracy the Cepheid photometry reported above.

Since aperture photometry cannot readily separate superimposed sources in dense regions, a few considerations described here are necessary to produce accurate aperture

photometry. (1) Small apertures are required, here set to 1.4 drizzled pixels (0.′11) in radius, along with an aperture correction derived from our PSF model to estimate the flux outside the aperture. (2) We limit the comparison to one-third of the Cepheid sample with the lowest surface brightness, a sample large enough to measure $H_0$ but which still has less background and thus greater precision. (3) We use a mean of the pixels in an annulus around each Cepheid to determine the sky, rather than the mode or median, which is more commonly used in sparse regions. This last step is very important for deriving accurate aperture photometry in the presence of a fluctuating background. In dense regions, the mean of the sky is an unbiased estimator of the level beneath a source, as it includes both the spatially constant background as well as the mean level of superimposed sources. In contrast, the mode or median of the crowded sky or any statistic calculated after fitting and subtracting visible stars necessarily underestimates the sky under the target source where we cannot resolve or subtract blended sources. We assume the position of the Cepheid was previously identified from sparse, high-contrast optical images, as is the case for the SHOES program.

We measured aperture photometry of the Cepheids using the same images (before removing any background level) used for the PSF-based photometry and the same, fixed Cepheid positions. The sky value was set to the simple mean of the sky pixels in an annulus between radii of 15 and 25 pixels from each Cepheid.<fnref n="29" /> Figure B1 displays a comparison of the aperture and PSF photometry for the Cepheids as a function of

<footnote n="29">²⁹ If anomalously bright Cepheids are removed as outliers from the final sample it is necessary to apply the same threshold to the sky pixels before calculating their mean to avoid a measurement bias. In practice, it is common to exclude Cepheids brighter than the $P-L$ by $\le 3\sigma$ or $> 1$ mag to provide a more robust $P-L$ relation. Thus, we remove sky pixels from the determination of the mean, which would cause an outlier of this size, i.e., those $\ge 2.5 \times$ the level of the central Cepheid pixel. For the images of Cepheids studied here this fraction is typically a few percent of the brightest Cepheids and sky pixels. To avoid a similar bias in PSF photometry, it is necessary to apply this same threshold when measuring the simple mean of artificial stars to determine the mean background correction.</footnote>
<figure><img src="image_31.png" /><figcaption>**Figure B1.** Comparison of PSF photometry (with backgrounds from artificial stars) for Cepheids in SN hosts (black) and NGC 4258 (blue) used here and basic small-aperture photometry with backgrounds from the mean pixel in a local annulus as discussed in the text.</figcaption></figure>

PSF magnitude, showing agreement within the sample mean errors.

We can make a few additional observations about the comparison that clarifies the relationship between the statistics used to measure the sky in aperture photometry and the crowded background corrections used to account for bias in PSF photometry. Because PSF photometry sets the sky as a “floor” level added to modeled sources, the PSF sky level will be similar to the mode (i.e., the most common value or peak of the distribution of sky annulus pixels) of the sky pixels used in (uncrowded) aperture photometry, the uniform level of pixels without apparent sources. Therefore, the difference in aperture photometry calculated from a sky level using the mode or mean of the sky pixels will be similar to the crowded background correction in PSF photometry. In sparse fields, the mode and mean are equivalent, and the crowded background bias is negligible.

The asymmetric distribution of pixel levels in the sky annulus also explains the useful feature that in log-normal or magnitudes, the Cepheid uncertainties that are dominated by the asymmetric distribution of sky pixels are relatively Gaussian to a few standard deviations, as shown in Section 3.3.

## B.2. Impacts of Improvements in Photometry

As discussed in Section 3.4, we itemize half a dozen improvements in Cepheid photometry from the last time these Cepheids were measured between ~6 and ~16 yr ago. Matching Cepheids by position, the error-weighted mean of the matched Cepheids are fainter in SN hosts by 0.06 mag and in NGC 4258 by 0.04 mag (a net difference of 0.02 mag when NGC 4258 calibrates the Cepheids and $H_0$). For the LMC, Cepheids became fainter by 0.03 mag between the ground sample used by R16 and their replacement by direct HST observations in R19. The mean change in $F555W-F814W$ was below 0.01 mag for Cepheids in SN hosts and in the outer field of NGC 4258 but 0.06 mag in the inner field relative to the

previous calibration undertaken by Macri et al. (2006; and provided in H16), which lacked accounting for the crowded background level, a larger effect in $F814W$ than in $F555W$, and pixel-based CTE rectification. We cannot report the change in photometry for MW Cepheids because we use different MW parallax samples here than those in R16.

For context, R16 propagated a zero-point uncertainty between ground and HST system photometry of $\sigma = 0.03$ mag, which is the same size as the changed observed for the LMC sample upon direct observation with HST. The net change between the Cepheids in the SN hosts and NGC 4258 or the LMC, the quantity that determines $H_0$, is 0.02, comparable to the systematic uncertainty in R16 of 0.026 mag for the NGC 4258 anchor and smaller than the overall uncertainty in $H_0$ of 0.052 mag. A more noteworthy change appears between the MW-only anchor results where the replacement with the Gaia EDR3 results plus HST photometry with the sample from Benedict et al. (2007) and their ground-based photometry reduced $H_0$ by 0.086 mag, with some of this change related to the aforementioned update of the Cepheid photometry in SN hosts, though the quadrature sum error for only the parallax samples is ~0.06 mag, making this change not surprising. For additional context for the LMC, we note that the DEB distance also decreased by ~0.015 mag and the metallicity term by 0.033 mag (same sense) since R16. These factors explain the net increase in $H_0$ for the LMC.

## B.3. Background Dependence of Fit Residuals

An accurate assessment of the background flux is critical to the use of standard candles. The Cepheid backgrounds are determined statistically by adding and measuring artificial stars in random positions local to the Cepheid scenes as described above. Here we provide an additional null test of the background estimates.

Because the individual backgrounds are determined locally as part of measuring the photometry, they are not part of the model. A null test of the level of background is to analyze the
<figure><img src="image_32.png" /></figure>

**Figure B2.** Mean differences between Cepheid photometry and distance-ladder fit versus the local background measured by artificial stars (in units of the Cepheid magnitudes), binned in equal numbers of Cepheids. Red is for Cepheids in SN Ia hosts and green for Cepheids in the anchor NGC 4258, with vertical lines indicating the mean background of each set. An underestimate of Cepheid backgrounds in SN Ia hosts at a level of 0.2 mag for the mean Cepheid background or an overestimate of Cepheid backgrounds in NGC 4258 by 0.2 mag for the mean Cepheid (indicated), or half that level for both simultaneously, would be required to solve the tension, but these possibilities are inconsistent with the data, which are consistent with no overestimates or underestimates for either set. Furthermore, it is hard to imagine how an overestimate or underestimate would not similarly affect (and therefore cancel) Cepheids in both NGC 4258 and SN Ia hosts.

correlation of background with the model residual of each Cepheid. If backgrounds are significantly systematically overestimated or underestimated, we would expect a correlation in this space as the residual would be a function of the (misestimated) background. We show the results of this test in Figure B2 for the baseline fit, binning the residuals in small ranges of background values. These bins show no significant trend with background. A linear fit gives a relation with slope $0.015 \pm 0.014$ mag per mag of background in the sense of overestimating the background by $0.006 \pm 0.005$ mag at the mean background level in the SN hosts and in NGC 4258 (as displayed in that figure). The figure also shows the expected trend if the background were underestimated enough to produce a 0.2 mag change in $H_0$ for the mean host (i.e., to solve the tension), which is very far from the data (and would also apply to the Cepheids in NGC 4258 and thus not address the discrepancy for that anchor in Fit 10).

For those who would like to check the accuracy of independently-measured extragalactic Cepheid PSF photometry, we can provide (upon reasonable request) images of hosts with artificial Cepheids of known brightness and position added to the frames. Reproducing the known photometry would be an important test of any photometry algorithm that can be completed blindly and in advance of an independent determination of $H_0$.

## Appendix C
### Cepheid Metallicity

Cepheid abundances beyond the Magellanic Clouds are generally derived from radial metallicity gradients measured from the ratios of strong emission lines in H II regions, such as

the $R_{23}$ strong-line diagnostic based on oxygen and hydrogen (Zaritsky et al. 1994, hereafter Z94). The conversion of $R_{23}$ to 12 + log [O/H] in the Z94 formula was based on photoionization models that themselves were calibrated at a time when the standard solar metallicity was believed to be 12 + log [O/H] = 8.93 (Anders & Grevesse 1989). Since Asplund et al. (2005, 2009), the solar abundance was revised downward by 0.24 dex to $8.69 \pm 0.05$. Over the same period, empirical transformations between strong-line measures like $R_{23}$ were also revised downward by ~0.2–0.3 dex, making them more consistent with stellar measures referenced to the Sun (Pettini & Pagel 2004; Kewley & Ellison 2008; Tremonti et al. 2004; Dopita et al. 2016; Curti et al. 2017). Bresolin et al. (2016) have shown good agreement between extragalactic oxygen abundance measurements from young stars and from H II regions using these more modern systems (particularly for Pettini & Pagel 2004, hereafter, PP04 O3N2; F. Bresolin, private communication). Collisionally excited lines generally underestimate stellar abundances by 0.2 dex for reasons discussed by Carigi et al. (2019). Using 30,000–50,000 galaxy spectra from SDSS DR7, Teimoorinia et al. (2021) derived third-order polynomial conversions between the Z94 scale and nine different strong-line metallicity calibrations developed since 2004.

For our measurement of $H_0$, we employ two strong-line abundance systems. The first is the simple mean of all nine recent (since 2004) diagnostics of Teimoorinia et al. (2021). This revises the Z94 scale down by a mean of 0.28 dex. The change is not quite constant, as the mean produces flatter galaxy gradients than Z94, revising the Z94 values down an additional 0.05 dex at >9.0 and less at <8.8 by 0.04 dex.
<figure><img src="image_33.png" /><figcaption>**Figure C1.** Comparison of oxygen abundance gradients in the MW inferred by spectra of Cepheids (blue) and from the PP04 calibration of H II regions (black) with data sources indicated in the plot. The mean levels are found to be consistent to $\sigma < 0.1$ dex, and we propagate systematic differences in scales in the covariance matrix (as discussed in the text).</figcaption></figure>

The second is the PP04 O3N2 diagnostic, which uses the dereddened abundances of [O III] $\lambda5007$ relative to H$\beta$ $\lambda4861$ and [N II] $\lambda6583$ relative to H$\alpha$ $\lambda6563$ to measure the log of the ratio of [O III] to [N II], O3N2, with the calibration $12 + \log[\text{O/H}] = 8.73 - 0.32 \times \text{O3N2}$. The difference between the Z94 diagnostic and the PP04 O3N2 diagnostic is very nearly constant, reducing the oxygen abundance scale by $0.23 \text{ dex} (0.235 \text{ dex at Z94} > 9.0$ and $0.222 \text{ at } < 8.8)$. The PP04 conversions of Teimoorinia et al. (2021) are valid for oxygen abundance on the Z94 scale $> 8.35$, encompassing all Cepheid hosts, with a typical stated precision of 0.03 dex. This change not surprisingly mirrors the drop in solar abundance of 0.24 dex from the revisions of Asplund et al. (2005, 2009) and the related recalibration of strong-line diagnostics. For this reason, it has made little difference for Cepheid measurements whether to use the prior Z94 calibration referenced to the Anders & Grevesse (1989) solar abundance or the revised (mean of) the recent systems referenced to the Asplund et al. (2005, 2009) solar abundance.

When calibrating Cepheids in SN Ia hosts to those in NGC 4258, both sets of metallicities are derived using the same strong-line abundance scale so that differences in abundance calibration do not affect the difference in abundance. However, LMC and MW Cepheid abundances are generally measured through direct spectroscopy of the stars, so when these calibrators are employed it is important to test the consistency of direct and indirect methods for measuring Cepheid abundances. This is most readily accomplished in the MW where both measures are available. Luck & Lambert (2011) provide oxygen abundances for 219 MW Cepheids on the Asplund et al. (2005) solar calibration of 8.69 from which they measured a gradient of $12 + \log[\text{O/H}] = 9.303 (\pm 0.028) - 0.056(\pm 0.003) \text{ dex kpc}^{-1}$ as shown in Figure C1.

To compare to these, we derived the strong-line abundances from the three largest samples of MW H II regions from Esteban et al. (2017), Esteban & García-Rojas (2018), and

Arellano-Córdova et al. (2021). We used the tabulated values of [O III] $\lambda5007$, H$\beta$, H$\alpha$, and [N II] $\lambda6583$ in those references to calculate the ratio O3N2 and from that the PP04 calibration to determine $12 + \log[\text{O/H}]$ as shown in Figure C1. The abundance gradient from the Cepheids as seen in that figure is a good match to the H II regions. A formal fit gives the PP04 stellar scale $= 0.035 \pm 0.06$ dex. From the Teimoorinia et al. (2021) transformations we find the mean difference between the PP04 and the nine recent systems, PP04-ave $= 0.047$ dex, demonstrating that the average of the nine systems is right on the stellar scale to a precision of $\sim 0.06$ dex. In Section 3.5, Equation (9), we employ a metallicity covariance in the fit covariance matrix measured as the difference between the mean of the metallicity diagnostics and the PP04 O3N2 diagnostic, which propagates both a 0.05 dex scale uncertainty, and also the differences that depend on the range of metallicity and rise to 0.1 dex at $12 + \log[\text{O/H}] = 8.8$ (i.e., $[\text{O/H}] = +0.11$) and up to 0.15 dex at 9.0. We also present the determination of $H_0$ on both reference systems as well for the exclusion of any dependence of Cepheid luminosity on metallicity to bracket the full range.

## Appendix D

### Use and Misuse of Wesenheit Magnitudes

The Wesenheit magnitude used to measure distances in Equation (7) is not a conventional magnitude (i.e., not a ratio of flux to a standard star), but rather a “dereddened magnitude” (see Appendix C of Madore & Freedman 1991 for a review), the ratio of two standardized candles with the same fiducial luminosity would be expected to have in the absence of reddening. The Wesenheit formalism takes advantage of the candles having the same mean luminosity and color (before being reddened by dust) to preserve their difference in distance (in magnitudes) without requiring knowledge of either the (absolute) intrinsic luminosity or color.
This method, and potential pitfalls with alternative uses of these magnitudes, can be better understood by decomposing the Cepheid color into its constituent parts. We can define the apparent Cepheid color,

$$V - I = (V - I)_0 + \Delta(V - I) + E(V - I), \quad (22)$$

where $(V - I)_0$ is the color (at a given period) in the middle of the Cepheid instability strip, $\Delta(V - I)$ is the displacement in color from the midline of the instability strip (with an accompanying displacement in mean luminosity owing to Stefan's law projected into the observed bandpasses, redder is fainter) and is quantified empirically in the Cepheid P-L-C relation. For a statistical sample of Cepheids beyond period completeness (which provides Cepheids uniformly distributed over the instability strip, mixed by the comparatively small evolution time to cross the strip compared to the larger differences in formation times), the mean $\langle\Delta(V - I)\rangle = 0$ mag. Hence, the relative difference in distance (in magnitudes) between samples of Cepheids is statistically preserved by subtracting from the apparent magnitudes of each any fixed number, $R$, which multiplies the (same) intrinsic color of the midlines of their instability strips (i.e., $R(V - I)_0$). If the value of $R$ is chosen to be the reddening ratio derived from an accurate reddening law of dust, then the subtracted term will also remove the absorption by dust (i.e., $R E(V - I)$) from each Cepheid for dissimilar values of $E(V - I)$. This is the function of a Wesenheit magnitude. A “happy coincidence” is that Stefan's law is quite parallel to the reddening law (redder is fainter or dustier), so this operation will also reduce the small dispersion due to the width of the instability strip (Madore & Freedman 1991). As given in Section 6.3, we derived the intrinsic P-L-C (Period-Luminosity-Color) relation from the LMC Cepheids as above to be $\Delta H = 0.635 (\pm 0.021) \Delta(V - I)$, the same sense and similar in scale to reddening by dust for which $R \approx 0.4$.

It is not valid to use a different reddening ratio (law) in different hosts by simply varying the value of $R$ for each host using the same Wesenheit system. Using a different value of $R$ for different Cepheids (or for Cepheids in different hosts) in Equation (7) without first removing the intrinsic color from the apparent color produces a change in the measured differences in the distance of Cepheid samples of size $R_1(V - I)_0 - R_2(V - I)_0$, where $R_1 \neq R_2$, which is unrelated to dust but is rather a large, ad hoc change in the standard candle luminosity itself as a fraction of its color. Because the typical $(V - I)_0$ color is a factor of a few times larger than the typical $E(V - I)$, this change in Cepheid luminosity will dominate the dust correction and produce a bias that is a rather arbitrary fraction of a Cepheid's color. This is a shortcoming in the analysis presented by Mortsell et al. (2021).

It is also important to recognize that Cepheids are well-understood yellow supergiant stars that will only pulsate over a very narrow range in surface temperature of $\log T = 3.7$ ($T = 5000$ K), which sets their intrinsic color of $(V - I)_0 \approx 0.8$ mag (at $P = 10$ days) at the instability strip of the temperature-magnitude diagram of stars. This is the surface temperature where its internal structure supports pulsations by the kappa mechanism (change in internal opacity in a deep partial ionization zone from ionization and recombination of He+) so

that the value of $(V - I)_0$ is well constrained by the physics that causes a Cepheid to be formed.

Thus, varying $R$ in the Wesenheit definition does not appear to follow any specific hypothesis. To consider the hypothesis of different reddening laws in different hosts, it is necessary to first subtract the intrinsic color as described in Section 6.3 and by Follin & Knox (2018).

### D.1. Errors in Color and Fitting

Because the value of $R$ is tightly constrained by the sample of MW Cepheids to $0.363 \pm 0.038$ (and better together with the Cepheids in other nearby hosts such as M31 and M101), the error in color $(V - I)$ can be propagated to an error in $H$ through $H \propto R(V - I)$ ignoring the small uncertainty in $R$ and added in quadrature in the covariance matrix of errors. However, this is no longer true if $R$ is poorly constrained by the data and the errors in color are large, as would be the case if we tried to determine $R$ independently for each individual SN host after subtracting the intrinsic color. Because the measurement errors in color are significant, to determine individual host values of $R$ without bias, it is necessary to consider errors in both axes ($H$-band magnitude and $V - I$ color) to optimize $R$ as discussed for a similar problem by Tremaine et al. (2002) or follow the constrained approach of Follin & Knox (2018).

Unfortunately, beyond the nearest few galaxies, the data are quite inadequate for constraining $R$ owing to the combination of a small range of color and comparatively large color uncertainties. Figure D1 shows the available Cepheid data for a nearby host with considerable measurable extinction (M31), one of the two nearest SN Ia hosts (NGC 5643) for which the data are just sufficient, and for a typical SN Ia host (UGC 9391) where the data are insufficient. Accounting only for uncertainty in the dependent variable produces a biased result and underestimates the uncertainty when the data quality is low, as is readily seen in Monte Carlo simulations compared to the input value of $R$.

In these cases, the breadth of colors is largely attributed to the measurement errors rather than simply extinction. Accounting for uncertainty in both axes, we find that the value of $R$ is quite unconstrained beyond the nearest few hosts. A bootstrap resampling of the data shows that the constraints on $R$ become uninformative beyond the nearest two SN Ia hosts with a mean uncertainty for the rest of ~1.5. The lower-right panel of Figure D1 shows the likelihoods for different host $R$ values. We find that the nearest few large spirals have $R$ consistent with the MW, but there is no meaningful constraint or well-defined peak for the rest. We can combine all such constraints as shown in the figure with a result that is unsurprisingly not well defined. Mortsell et al. (2021) and Perivolaropoulos & Skara (2021) did not include the color errors, which our analysis shows leads to an underestimate of the values of $R$ and their uncertainties for nearly all SN hosts. From a Bayesian perspective, the addition of a free parameter, $R$, for each host (~40 free parameters) is not justified by the improvement in the fit over the global $R$, as also suggested in the analysis by Mortsell et al. (2021) and Follin & Knox (2018). An informed approach to modeling individual host $R$ values without so many poorly constrained free parameters can be found in Section 6.3, Fit 22, and Hahn et al. (2022).

The EDA framework also provides estimates of the mean extinction seen midway through the SN hosts (i.e., the average location of a Cepheid) in our NIR bandpass from the same host
<figure><img src="image_34.png" /></figure>

**Figure D1.** Measuring the reddening ratio *R* in each host. For each host we subtract the intrinsic period–color relation as given in the text to provide *E(V − I)* and correct the *H*-band magnitudes for the *P*–*L* relation (period and metallicity). The ideal case, M31, has a large span of extinction (because it is highly inclined) and small measurement errors (because it is nearby) to produce a good constraint on the slope, *R*, whether we consider errors in only the dependent variable (red line and dotted for uncertainty) or both axes (green line). *R* = 0 is shown as a black line. NGC 5643 is one of the two nearest SN Ia hosts and has the most Cepheids; it yields some constraint on *R* along with M101. In contrast, the rest of the SN hosts have data similar to UGC 9391, which produces an uninformative constraint on *R* and shows a large difference in its value and uncertainty by ignoring uncertainties in both axes. The lower right shows the bootstrap resampling for all hosts, with a different color for each, showing that a meaningful constraint is only seen for the few nearest cases, with the others relatively flat. The measurable cases are consistent with the MW *R* shown as a black dashed line, as are the sum of all SN hosts. Ignoring errors in *E(V − I)* underestimates *R* and its uncertainty, as discussed in the text.

properties used to determine *R* above but now including the
host inclination. These values range from *A*<sub>H</sub> = 0.07 mag
(NGC 4424, which at type Sa is the earliest spiral in our
sample) to 0.30 mag with a mean of 0.18 mag and a dispersion
of 0.05 mag. This mean is similar to the empirical result of the
product of the SN-host mean *E*(V − I) and *R*, which
yields ⟨*A*<sub>H</sub>⟩ = 0.14 mag, indicating that the level of extinction
inferred from the Cepheid reddening is the expected amount.<fnref n="30" />

ORCID iDs

Adam G. Riess https://orcid.org/0000-0002-6124-1196

Wenlong Yuan https://orcid.org/0000-0001-9420-6525

Lucas M. Macri https://orcid.org/0000-0002-1775-4859

Dan Scolnic https://orcid.org/0000-0002-4934-5849

Dillon Brout https://orcid.org/0000-0001-5201-8374

David O. Jones https://orcid.org/0000-0002-6230-0151

Yukei Murakami https://orcid.org/0000-0002-8342-3804

Gagandeep S. Anand https://orcid.org/0000-0002-5259-2314

Louise Breuval https://orcid.org/0000-0003-3889-7709

Thomas G. Brink https://orcid.org/0000-0001-5955-2502

Alexei V. Filippenko https://orcid.org/0000-0003-3460-0103

Samantha Hoffmann https://orcid.org/0000-0002-4312-7015

Saurabh W. Jha https://orcid.org/0000-0001-8738-6011

W. D'arcy Kenworthy https://orcid.org/0000-0002-5153-5983

John Mackenty https://orcid.org/0000-0002-4312-7015

Benjamin E. Stahl https://orcid.org/0000-0002-3169-3167

WeiKang Zheng https://orcid.org/0000-0002-2636-6508

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