Measurements of the Hubble Constant: Tensions in Perspective^*

The nearby spiral galaxy NGC 4258, at a distance of 7.6 Mpc, is an excellent target for providing a high-accuracy calibration of the TRGB. It is host to a sample of H₂O megamasers rotating within a highly inclined (87°) accretion disk about a supermassive black hole from which a geometric distance to the galaxy can be measured (see Humphreys et al. 2013; Reid et al. 2019). The most recent geometric distance to NGC 4258 is μ₀ = 29.397 ± 0.024 (stat) ± 0.022 (sys) mag (Reid et al. 2019), a 1.5% measurement.

The most extensive study of the TRGB in NGC 4258 was published by Jang et al. (2021). This measurement is based on a set of 15 archival HST/ACS fields covering 54 arcmin² located near the minor axis in the dust- and gas-free outer halo of the galaxy. The analysis was further confined primarily to regions at a deprojected semimajor axis distance of >14' (∼30 kpc) from the center of the galaxy. The RGB stars at this large distance are well separated from each other and demonstrably free from crowding/blending effects. Moreover, these halo RGB stars are relatively blue and metal-poor and do not exhibit a wide range in color/metallicity. The wide areal coverage results in a well-populated giant branch with about 3000 red giant stars 1 mag below the tip itself. As described in detail in Jang et al., extensive tests for systematics were undertaken, for example, using artificial stars, comparing DOLPHOT and DAOPHOT photometry, and comparing results using different point-spread functions, sky-fitting parameters, and radial spatial cuts. Moreover, the HST/ACS data used for this study are on the F814W flight magnitude system used in the F19 study and thus do not require a photometric transformation, as for the case of the LMC zero-point. Jang et al. obtained a TRGB zero-point of ${M}_{814}^{\mathrm{TRGB}}$ = −4.050 ± 0.028 ± 0.048 using the maser distance determined by Reid et al. (2019). A detailed description of the error budget and the adopted statistical and systematic uncertainties are given in Section 6 and Table 4 of Jang et al. This independent TRGB calibration agrees to better than 1% with the value of M₈₁₄ = −4.054 mag found earlier by F20, as well as that of −4.045 mag measured by Hoyt (2021), as described in Section 2.3.

Alternatively, if we instead determine the distance to NGC 4258 based on the LMC TRGB calibration of Hoyt (2021), given the measured apparent TRGB magnitude of ${m}_{814,o}^{N4258}=25.347\pm 0.014\pm 0.005$ (Jang et al. 2021), we find a distance modulus of μ_o = 29.392 ± 0.018 ± 0.032 mag. The agreement with the maser distance of 29.397 ± 0.033 mag (Reid et al. 2019) is at a level of better than 1%, differing by <0.2σ. In contrast, we note that a Cepheid calibration of the distance to NGC 4258 does not yield as good agreement with that of the maser distance. As recently described in Efstathiou (2020), a calibration of the Cepheid distance to NGC 4258 based on the LMC differs from the maser distance by 2.0σ–3.5σ, depending on the adopted correction for metallicity. The Milky Way and NGC 4258 metallicities are very similar, however, and should be independent of a metallicity effect. If instead, the Milky Way is adopted as the anchor galaxy to determine the Cepheid distance to NGC 4258, a distance modulus of 29.242 ± 0.052 is obtained, which differs from the maser distance by 7% at a 2σ level of significance. We defer a discussion of the implications of these differences to Section 5.

Finally, we note that the location of the fields studied by Jang et al. (2021) in the outer halo of NGC 4258 is optimal for avoiding dust and gas, as well as being separated from the high surface brightness galactic disk, thereby minimizing the level of systematic effects that plague efforts to measure the TRGB in the star-forming region of the disk of this galaxy, issues not considered, for example, in Macri et al. (2006) and Reid et al. (2019).

A second and completely independent method for calibrating the TRGB uses photometry of well-measured giant branches in globular clusters within our own Milky Way. Collectively, the Milky Way globular clusters span a wide range in metallicity, which overlaps well with those measured for giant stars in the halos of nearby resolved galaxies.

This approach to calibrating the TRGB was first carried out by Da Costa & Armandroff (1990) using CCD imaging data for six globular clusters. That calibration, for which distances were obtained using theoretical horizontal branch models from Lee et al. (1990; to calibrate the luminosities of RR Lyrae stars), formed the basis of the Lee et al. (1993) early application of the TRGB method to the extragalactic distance scale. A decade later, Ferraro et al. (1999) assembled a homogenous sample of 60 globular clusters, adopting the level of the theoretical zero-age horizontal branch as the basis from which to measure absolute distances. As these authors noted, the advantage of the horizontal branch is the simplicity of the measurement as compared to RR Lyrae stars, for which variability and evolutionary effects need to be accounted for and uncertainties due to metallicity still remain. Bellazzini et al. (2001, 2004) based their calibration on observations of the two populous globular clusters, ω Centauri and 47 Tucanae, calibrated using a DEB distance to ω Cen (Thompson et al. 2001), and an average of literature distances for 47 Tuc. Subsequently, Rizzi et al. (2007) based their distances on the well-developed horizontal branches of five Local Group galaxies (IC 1613, NGC 185, Fornax, Sculptor, and M33) spanning a range in metallicities of −1.74 dex < [Fe/H] < −1.02 dex.

In a recent study, Cerny et al. (2020) analyzed a sample of 46 low-reddening (E(B − V) < 0.25 mag) Milky Way globular clusters with uniformly reduced photometry available from Stetson et al. (2019) and through the Canadian Astronomy Data Center (CADC). ³ This 46-cluster catalog was then cross-matched to the Gaia Data Release 2 (DR2) database, and membership for these clusters was determined using the DR2 proper-motion data and a Gaussian mixture model clustering algorithm. Preliminary E(B − V) reddening estimates and initial distance estimates were taken from Harris (1996, 2010).

A composite M_I versus (V − I)_o CMD is shown in Figure 1 for the 46 low-reddening clusters from Cerny et al. (2020). This composite shows a well-defined giant branch, sampling a wide range of metallicities from −2.4 dex< [Fe/H] < −1.0 dex. As described in more detail in Cerny et al., high signal-to-noise and low-extinction clusters were used to define a fiducial lower envelope to the blue and red horizontal branches, and a maximum-likelihood grid search technique was used to align the remaining clusters onto a common calibration. The zero-point of the calibration was set by the geometric DEB distance to ω Cen, measured by Thompson et al. (2001). The resultant blue and red horizontal branches are shown in Figure 1. ⁴ Applying a Sobel edge-detection filter to the composite luminosity function for the TRGB, Cerny et al. determined an absolute I-band TRGB magnitude of −4.056 mag, which, following F19, transforms to flight magnitudes as M_814W = −4.063 ± 0.022 ± 0.101 mag.

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2.2.1. Gaia Early Data Release 3 Calibration of Galactic Globular Clusters

Using the OGLE "Shallow" survey data of Ulaczyk et al. (2012), F20 measured the TRGB for the LMC. ⁷ In order to avoid crowding/blending effects within the high surface brightness bar, the sample of stars analyzed was confined to stars outside of a circle of 1° radius, centered on the bar of the LMC. The LMC reddening and extinction were measured using VIJHK photometry, differentially with respect to two low-reddening galaxies, IC 1613 and the SMC. Based on the DEB (Pietrzyński 2019) distance modulus to the LMC of 18.477 mag, the extinction-corrected absolute magnitude of the TRGB for the I band was found to be ${M}_{I}^{\mathrm{TRGB}}$ = −4.047 ± 0.022 (stat) ± 0.039 (sys) mag. The Pietrzyński measurement is based on the surface brightness–color calibration for late-type giant stars, from which the angular diameters of giant stars can be measured to an accuracy of 0.8%.

Recently, Hoyt (2021) undertook a detailed remeasurement of the LMC TRGB based on OGLE-III photometry, isolating regions where the edge-detection measurements are sharp and single-peaked. He illustrated that these same regions are also low in dust content and located away from regions of star formation. He incorporated the new reddening and extinction maps of Skowron et al. (2021) determined from the colors of red clump stars based on OGLE-IV photometry. Adopting the 1% distance to the LMC based on DEBs (Pietrzyński 2019), he found ${M}_{I}^{\mathrm{TRGB}}$ = −4.038 ± 0.012 (stat) ± 0.032 (sys) mag, consistent to within 1% with the earlier results. A detailed description of the error budget and the adopted statistical and systematic uncertainties is given in his Table 3. The systematic uncertainty includes a ±0.01 mag term on the OGLE photometric zero-point. An additional ±0.01 mag systematic uncertainty is included in the ground-to-HST calibration, resulting in ${M}_{814}^{\mathrm{TRGB}}$ = −4.045 ± 0.012 ± 0.034 mag.

As an aside, we note that Yuan et al. (2019) argued that the F19 calibration of H₀ based on the distance to the LMC was in error. However, Freedman et al. (2020) and Hoyt (2021) described in some detail a number of incorrect assumptions that were made by Yuan et al. The excellent agreement found here between the completely independent LMC, NGC 4258, SMC, and Galactic globular cluster calibrations argues even more strongly against the claims made in Yuan et al. Moreover, even if the LMC were to be excluded from the TRGB calibration altogether, the resulting change in the overall value of H₀ is insignificant (<1%).

The interaction of the LMC and SMC has resulted in a tidally extended structure to the SMC, which has historically complicated the measurement of the SMC distance. The TRGB was measured by F20 using published OGLE data ⁸ for the inner region of the SMC, thereby avoiding confusion with the more extended tidal tails. They measured an I-band magnitude for the TRGB of ${m}_{I}^{\mathrm{TRGB}}$ = 14.93 mag, adopting a foreground extinction value of A_I = 0.056 mag. ⁹

Mapping out the inclined system with very high precision, Graczyk et al. (2020) recently measured a new DEB distance to the central region of the SMC to an accuracy of better than 2% based on the surface brightness–color calibration of Pietrzyński (2019). Augmenting the sample of measured DEBs from their previously published sample (from 5 to 15, a threefold increase), Graczyk et al. (2020) determined a distance modulus of μ₀ = 18.977 ± 0.016 (stat) ± 0.028 (sys) mag. The SMC thus provides another opportunity for an updated and independent calibration of the TRGB. An advantage of the SMC is its low star formation rate and dust content.

Hoyt (2021) undertook a reanalysis of the SMC OGLE-III data incorporating the updated Skowron et al. (2021) reddening maps. He measured an apparent tip magnitude of ${m}_{I}^{\mathrm{TRGB}}$ = 14.93 mag. A detailed description of the adopted statistical and systematic uncertainties is given in his Table 3. Based on the new Graczyk et al. (2020) true DEB distance modulus, he found ${M}_{I}^{\mathrm{TRGB}}$ = −4.050 ± 0.030 (stat) ± 0.040 (sys) mag, in excellent agreement with the NGC 4258, Milky Way globular cluster, and LMC calibrations discussed above. An additional ±0.01 mag systematic uncertainty is included in the ground-to-HST calibration, resulting in ${M}_{814}^{\mathrm{TRGB}}$ = −4.057 ± 0.030 ± 0.040 mag.

In the cases described in this section, we do not use these systems to calibrate H₀ but rather note their excellent consistency with the other calibrations presented here, lending further confidence to the overall calibration of the TRGB.

Two recent studies of the Sculptor (Tran et al. 2021) and Fornax (Oakes et al. 2021) dwarf spheroidal companions to the Milky Way provide additional calibrations of the TRGB, constituting consistency checks on the geometric calibrations (for the LMC, the Milky Way, NGC 4258, and the SMC) described above. Wide-field Magellan IMACS VI data were obtained for each galaxy, from which the positions of the apparent TRGB and the horizontal branch were measured. Tran et al. measured an extinction-corrected value of the apparent TRGB I-band magnitude for Sculptor of ${m}_{{I}_{o}}^{\mathrm{TRGB}}$ = 15.487 ± 0.057 ± 0.014 mag. For Fornax, Oakes et al. found ${m}_{{I}_{o}}^{\mathrm{TRGB}}$ = 16.75 ± 0.03 ± 0.01 mag. Adopting the absolute calibration of the horizontal branch from Cerny et al. (2020), as described in Section 2.2 above and shown plotted in Figure 1, yields true distance moduli of 19.56 ± 0.03 ± 0.10 and 20.79 ± 0.02 ± 0.10 mag for Sculptor and Fornax, respectively. These measurements yield absolute I-band calibrations of the TRGB (based on the horizontal branch) of −4.07 ± 0.06 ± 0.10 and −4.04 ± 0.04 ± 0.10 mag, again in excellent agreement with the independent calibrations based on NGC 4258, the LMC, and the SMC.

Finally, we have also examined the F814W and F555W HST/ACS data obtained by Olsen et al. (1998) for a number of globular clusters in the LMC, specifically, NGC 1754, NGC 1835, NGC 2005, and NGC 2019. Table 2 lists the reddenings and extinctions measured for each cluster by Olsen et al. and the true distance moduli based on the horizontal branch calibration of Cerny et al. (2020). We show a composite CMD for these objects in Figure 2. Adopting the Cerny et al. calibration results in a measured TRGB magnitude of −4.085 ± 0.05 ± 0.10 mag.

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As noted previously, these systems are not of comparable accuracy (or independence) to yield an independent calibration of H₀, but their consistency, to within the uncertainties, already provides a further test of the robustness of the TRGB calibration. In future, when parallaxes accurate to 1% become available for a large sample of Milky Way globular clusters, these horizontal branch measurements will become a powerful independent route to a calibration of the TRGB.

Table 3 lists the TRGB absolute magnitude at F814W for the geometric calibrations described above. Where the calibration was carried out for ground-based data (as for the LMC, SMC, and Milky Way clusters), these have been transformed to the HST/ACS F814W flight magnitude system. The NGC 4258 calibration was carried out entirely with HST and is already on the F814W flight magnitude system. As discussed in F19 and F20, the transformation from the I band to F814W results in a zero-point that is brighter by −0.0068 mag. As can be seen from this table, the good agreement of the TRGB zero-point based on the calibrations for many anchors means that the adoption or rejection of a particular galaxy does not significantly impact the overall result.

Table 3. TRGB Zero-point Calibration

Object	${M}_{{\rm{F}}814{\rm{W}}}^{\mathrm{TRGB}}$ (mag)	σstat	σsys	References
NGC 4258 a	−4.050	0.028	0.048	Jang et al. (2021)
Milky Way globular clusters b	−4.063 c	0.022	0.101	Cerny et al. (2020)
LMC d	−4.045 c	0.012	0.034	Hoyt et al. (2021)
SMC e	−4.057 c	0.030	0.040	Hoyt et al. (2021)
Sculptor f	−4.08 c	0.06	0.10	Tran et al. (2021)
Fornax g	−4.05 c	0.04	0.10	Oakes et al. (2021)
LMC globular clusters h	−4.085	0.05	0.10	This paper
Adopted value (MW, NGC 4258, LMC, SMC)	−4.049	0.015	0.035	This paper (Section 2.6)

Notes.

^a H₂O megamaser distance calibration. ^b Optical data, Gaia proper-motion selection; ω Cen DEB calibration; M_I = −4.056 mag. ^c Transformation to ${M}_{{\rm{F}}814{\rm{W}}}^{\mathrm{TRGB}}$ = M_I − 0.0068 mag following Freedman et al. (2019). ^d LMC DEB calibration; M_I = −4.038 mag. An additional ± systematic uncertainty is included in the ground-to-HST calibration. ^e SMC DEB calibration; M_I = −4.050 mag. An additional ± systematic uncertainty is included in the ground-to-HST calibration. ^f ω Cen DEB calibration; M_I = −4.07 mag. ^g ω Cen DEB calibration; M_I = −4.04 mag. ^h ω Cen DEB calibration.

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Figure 3 shows the relative probability density functions (PDFs) for the absolute TRGB F814W magnitudes discussed in Section 2 above. Here we separate the contributions of the statistical and systematic errors in each case, so that the contribution of both types of uncertainties can be clearly seen. In Figure 3(a)), the widths of the Gaussians represent the individual statistical errors in each determination only, whereas the systematic uncertainties are illustrated separately by the error bars at the top of the plot (using the same color coding) for each object. Conversely, in Figure 3(b), the widths of the PDFs represent the individual systematic errors in each determination only, whereas the statistical uncertainties are illustrated separately by the error bars at the top of the plot. The integrals of the PDFs for the LMC, the Milky Way, NGC 4258, and the SMC each have unit area. The statistical and systematic errors for each individual determination, σ_i , are given by the 16th and 84th percentiles of the Gaussians in Figures 3(a) and (b), respectively. The frequentist sums of the probability distributions are shown in both cases by the black lines. For the total sample, σ_mean = $\sum {\sigma }_{{\boldsymbol{i}}}/\sqrt{(N-1)}$, where N = 4.

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As we have seen, the Milky Way TRGB magnitude is based on a sample of 46 clusters calibrated to the DEB distance to ω Cen. The calibrations of Sculptor, Fornax, and the LMC clusters are not independent, however, since they all rely on the Milky Way calibration of the horizontal branch. Moreover, Sculptor and Fornax are single objects, and the TRGB for the LMC clusters is sparsely populated. For illustrative purposes, in Figures 3(a) and (b), the areas for these Gaussians have thus been down-weighted by a factor 1/f, as shown in Equation (1),

$$\begin{eqnarray}&&\displaystyle \frac{1}{f\sqrt{2\pi {\sigma }^{2}}}{e}^{-0.5\left(\displaystyle \frac{x-\langle x\rangle }{{\sigma }^{2}}\right)},\end{eqnarray} \tag{ 1 }$$

where f = $\sqrt{(N)}$, and N = 46, the size of the Milky Way globular cluster sample. Thus, Sculptor, Fornax, and the LMC clusters do not contribute to the adopted overall calibration, but they do provide a consistency check on the horizontal branch–to–TRGB distance scale.

The frequentist sums of the probability distributions are shown by the black lines in Figure 4(a). The mode of the summed distribution for the four primary TRGB calibrators is −4.049 mag. As shown in Figure 4(b), an identical result is obtained for a Bayesian analysis (albeit with smaller uncertainty), in which a uniform prior is adopted, and the product of the distributions is determined. In addition, a simple weighted average for the LMC, the Milky Way, NGC 4258, and the SMC also gives a result to within 0.001 mag of −4.049 mag. We adopt this robust value, ${M}_{{\rm{F}}814{\rm{W}}}^{\mathrm{TRGB}}$ = −4.049 ± 0.015 (stat) ±0.035 (sys) mag, for the absolute magnitude of the TRGB. The (exact) agreement of the various means of combining the four calibrations lends confidence to the overall result; i.e., it is independent of the choice of statistical approach adopted to combine the results. Finally, we note that this value agrees to better than 1% with that given by F20, who found ${M}_{{\rm{F}}814{\rm{W}}}^{\mathrm{TRGB}}$ = −4.054 ± 0.022 ± 0.039 mag.

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We turn now to a determination of H₀ based on the TRGB calibration discussed in Section 2.6. This is an update of the calibration of H₀ presented in F19 and F20, applied to the CSP sample of 99 SNe Ia observed at high cadence and multiple wavelengths (Krisciunas et al. 2017). That measurement of H₀ was based on HST/ACS observations of the halos of 15 galaxies that were hosts to 18 SNe Ia. The measured absolute I-band magnitude of the TRGB from Freedman et al. (2020) was M_F814W = −4.054 ± 0.022 (stat) ± 0.039 (sys) mag, tied to the geometric DEB distance modulus to the LMC from Pietrzyński (2019) of 18.477 mag. We now use four independent calibrations (NGC 4258 and the Milky Way, LMC, and SMC), superseding the single calibration based on the LMC alone.

To briefly summarize, in F19, the CSP analysis was undertaken with the SNooPy package (Burns et al. 2018), which characterizes the SN Ia light-curve shape using a color-stretch parameter, s_BV . Magnitudes were computed using two different approaches to the reddening, where

$$\begin{eqnarray}\begin{array}{rcl}{B}^{{\prime} } & = & B-{P}^{1}({s}_{{BV}}-1)-{P}^{2}{\left({s}_{{BV}}-1\right)}^{2}-{\rm{CT}}\\ & & -\ {\alpha }_{M}({\mathrm{log}}_{10}{M}_{* }/{M}_{\odot }-{M}_{0}),\end{array}\end{eqnarray} \tag{ 2 }$$

where P¹ is the linear coefficient and P² is the quadratic coefficient in (s_BV − 1), B and V are the apparent k-corrected peak magnitudes, α_M is the slope of the correlation between peak luminosity and host stellar mass M_*, and CT denotes the color term for the two approaches. In the first case, CT = β(B − V), where a color coefficient β results in a reddening-free magnitude, an approach originally proposed by Tripp (1998). In the second approach, CT = R_B E(B − V), where R_B , the ratio of total-to-selective absorption, and the reddening, E(B − V), are solved for explicitly using both optical and near-infrared colors for each SN Ia. Using the Markov Chain Monte Carlo (MCMC) fitter described in Burns et al. (2018) and F19, which uses the "No U-Turn Sampler" from the data-modeling language STAN (Carpenter et al. 2017), and solving for the parameters in Equation (2), the value of H₀ and its error was obtained using both approaches.

As described in Hoyt et al. (2021), two new galaxies with directly measured TRGB distances have been added to the Carnegie Chicago Hubble Project (CCHP) sample since F19 and F20. Object NGC 5643 is host to SN 2013aa and SN 2017cbv (Burns et al. 2020), and NGC 1404, a member of the Fornax cluster, is host to SN 2007on and SN 2011iv (Gall et al. 2018). Hoyt et al. (2021) found that SN 2007on appears to be significantly underluminous, and it is therefore excluded from the current analysis. In F19, the distance to NGC 1404 was taken to be the average value given by the two other Fornax galaxies in the CCHP sample, NGC 1316 and NGC 1365. In this paper, we adopt the new direct distance to NGC 1404 (Hoyt et al. 2021) for SN 2011iv and add the two additional SNe Ia in NGC 5643, augmenting the sample of 18 SNe Ia described in F19 to an updated sample of 19. All distances are calibrated adopting ${M}_{{\rm{F}}814{\rm{W}}}^{\mathrm{TRGB}}$ = −4.049 ± 0.015 (stat) ± 0.035 (sys) mag and used as new input to the MCMC analysis described in F19 (C. Burns 2021, private communication).

In Table 4, we give the values of H₀ and the uncertainties obtained adopting the new calibration of ${M}_{{\rm{F}}814{\rm{W}}}^{\mathrm{TRGB}}$. Listed are H₀ values based on both the CSP B band and H-band SNe Ia magnitudes for different color and dust-reddening constraints. The uncertainties (for the SN Ia analysis alone) are determined from a diagonal covariance matrix with respect to the TRGB distances. Following F19, we present results applying both the Tripp and explicit E(B − V) reddening corrections. In addition, we present the results adopting host-galaxy masses from Burns et al. (2018) originally used in F19, as well as those measured in the more recent study of Uddin et al. (2020). Figure 5 shows these results in flowchart form.

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The values presented in Table 4 and Figure 5 represent different choices for the SN Ia sample (color and stretch), dealing with dust (Tripp versus E(B − V)), the bandpass for the SN Ia magnitudes (B versus H), and the host-galaxy mass peak SN Ia luminosity correlation (Burns versus Uddin). The various choices result in a full range in H₀ values from 68.47 to 71.50 km s⁻¹ Mpc⁻¹. In selecting a best value of H₀ from those listed in Table 4, we select (following F19) the sample that minimizes the difference between the calibrator sample and the distant sample in terms of the nuisance variables: color, stretch, and host mass. In the histograms in Figure 6, we illustrate the characteristics of the SNe Ia in the distant galaxy sample compared with those for the TRGB calibrators. The TRGB sample is shown in green; the overall CSP sample is divided such that the blue, slow decliners (with s_BV > 0.5 and (B − V) < 0.5) are shown in blue (and labeled "slowblue") and those with fast decline rates and redder colors are shown in orange. In terms of stretch and color, the tails seen in orange (extending to s_BV < 0.5 and (B − V) > 0.5) are absent in the TRGB calibrating sample. Unambiguously, in terms of stretch and color, the sample with the best overlap of calibrator and distant SNe Ia is that of "slowblue." This sample also overlaps well in terms of host-galaxy mass, an advantage of the TRGB method, which can be applied to both early- and late-type galaxies. (Cepheid variables are young objects found only in star-forming, e.g., spiral, galaxies and cannot calibrate the SNe Ia found in elliptical or S0 galaxies.)

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We note the following.

• 1.  
  Using B-band photometry, restricting the sample to that with s_BV > 0.5 and (B − V) < 0.5 ("slowblue"), and basing the analysis on the more recent Uddin et al. (2020) host-galaxy masses results in a value of H₀ = 69.57 ± 1.24 km s⁻¹ Mpc⁻¹ using the Tripp method and H₀ = 70.04 ± 1.05 km s⁻¹ Mpc⁻¹ when explicitly correcting for dust (E(B − V)). Using H-band photometry, respectively, results in similar values of H₀ = 70.00 ± 1.25 and 69.47 ± 1.42 km s⁻¹ Mpc⁻¹. The corresponding values based on the Burns et al. (2018) masses are slightly lower. The difference arises primarily because the slope of the mass correlation in the optical is steeper for the Burns masses, whereas for the Uddin masses, the relation is nearly flat for all filters.
• 2.  
  The H-band data have the advantage of a smaller dependence on the reddening, as the correction (R_λ ) is smaller, but they have the disadvantage of a larger variance because the sample of SNe Ia having H-band photometry is smaller. (The "full sample" has 147 objects, "slowblue" restricts the sample to 129 objects, and restricting the sample to those with H-band photometry results in 102 objects.)
• 3.  
  The largest value of H₀ (71.5) is obtained when the redder, faster decliners are included in the analysis (the "full sample"). However, as noted above, these solutions are strongly disfavored, since there are no redder, faster decliners in the more distant sample. In a broader context, no solution here reaches a value as high as 74 km s⁻¹ Mpc⁻¹.

Although the differences in these H₀ values are small (a total range of only 3 km s⁻¹ Mpc⁻¹), they illustrate the effect of different choices in the host-galaxy mass correlation and method/filters adopted to correct for dust.

Our adopted best-fit value is based on (1) the sample of SNe Ia for which the nuisance variables (color, stretch, and host mass) are comparable for the calibrating TRGB galaxies and the distant SNe Ia, (2) an average of the Tripp/E(B − V) determinations, and (3) the recent host-galaxy masses measured by Uddin et al. (2020). We choose the B-band measurements because the sample of SNe Ia is largest, and the scatter for the H-band measurements is 40% larger (or a factor of 2 in the variance, in the case of the E(B − V) correction). We adopt a best-fit value of 69.8 ± 1.2 (sys) km s⁻¹ Mpc⁻¹. This latter uncertainty takes into account the systematic uncertainties in the SN Ia analysis alone, without yet combining it with the TRGB zero-point systematic error. As discussed in Burns et al. (2018) and Freedman et al. (2019), all of the correction factors to the SN Ia light curves (P¹, P², s_BV − 1, α_M , β, E(B − V), R_B ) as described in Section 2 are computed; these then provide corrected magnitudes and a full covariance matrix used to determine H₀ and the uncertainty given in Table 4. The total uncertainty adopted for H₀, including the uncertainty in the TRGB calibration, is discussed below.

The H₀ values and uncertainties based individually on the new TRGB calibrations for NGC 4258 (Section 2.1), Galactic globular clusters (Section 2.2), the SMC (Section 2.4), and the LMC (Section 2.3) are listed in Table 5. For comparison, also listed are the H₀ values, their uncertainties, and their published references from the SHoES team, based on the Cepheid calibrations for the LMC, NGC 4258, and the Milky Way.

Both statistical and systematic uncertainties are given for the TRGB H₀ determinations in Table 5. The error bars include both the uncertainties for the TRGB calibration discussed in Section 2.6 above and those arising from the calibration of the SNe Ia, as discussed above and in F19. For the SNe Ia, the statistical uncertainty amounts to ±0.5% with a systematic uncertainty of ±1.7%. The final percentage errors are summarized in Table 6, with a final adopted value of H₀ = 69.8 ± 0.6 (stat) ± 1.6 (sys) km s⁻¹ Mpc⁻¹.

Figure 7 shows the PDFs for the values of H₀ based on the seven calibrations of the TRGB discussed in Section 2. The width of each Gaussian is based on the statistical uncertainties alone for each individual determination. The error bars at the top of the plot (using the same color coding) represent the corresponding systematic uncertainties in each case. The 1σ uncertainties are determined from the 16th and 84th percentiles for the frequentist sum of the distributions, adding the statistical and systematic errors in quadrature: ${\sigma }_{{\boldsymbol{i}}}=\sqrt{\sigma {{}_{\mathrm{stat},i}}^{2}+\sigma {{}_{\mathrm{sys},i}}^{2}}$ and σ_mean = $\sum {\sigma }_{{\boldsymbol{i}}}/\sqrt{(N-1)}$, where N = 4. The four objects with independent geometric distances (the LMC, the Milky Way, NGC 4258, and the SMC) are represented by Gaussians with unit area. The secondary calibrations of Sculptor, Fornax, and the LMC clusters are based on the Milky Way calibration of the horizontal branch and therefore not completely independent. Once again, their areas have been scaled following Equation (1) and are shown for illustrative purposes only. Thus, Sculptor, Fornax, and the LMC clusters do not contribute to the adopted overall calibration, but they do provide a consistency check on the horizontal branch–to–TRGB distance scale.

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From Figure 7, it can also be seen that the range in the values of H₀ for the various calibrators is small relative to the published systematic error bars. The small χ² value may be indicating that the systematic errors have been overestimated or, alternatively, that statistical fluctuations have resulted in a fortuitously tight grouping of H₀ values. In either case, a conservative estimate of the overall uncertainty still seems warranted; that is, we do not consider this (better than 1% statistical) agreement to be indicating that H₀ has now been measured to a level of 1%.

In Figure 8, we show the normalized relative PDFs for the values of H₀ based on the different calibrators (the LMC, NGC 4258, the Milky Way, and the SMC), comparing both the TRGB and Cepheid calibrations in a self-consistent manner. For comparison with the SHoES results (where the statistical and systematic uncertainties are not treated independently), only the total uncertainties are considered. The TRGB calibrations are shown at the top (in red) and the Cepheid calibrations in the middle (in blue). In this case, we follow a Bayesian approach, assuming that each anchor is equally valid and adopting a uniform prior. The bottom panel shows the product of the PDFs. In the case of the Milky Way, the H₀ values are based on the calibration from Cerny et al. (2020) for the TRGB and R21 for the Cepheids. (The earlier Cepheid results for the Milky Way based on HST/WFC3 scanning parallaxes (R16) resulted in a much higher value of H₀ = 76.18 ± 2.17 km s⁻¹ Mpc⁻¹.) The resulting values of H₀ for the TRGB and Cepheids, respectively, are shown as solid lines. The difference between the TRGB calibration with H₀ = 69.8 ± 0.6 (stat) ± 1.6 (sys) km s⁻¹ Mpc⁻¹ (this paper) and the Cepheid calibration with H₀ = 73.2 ± 1.3 km s⁻¹ Mpc⁻¹ (R21) represents a 1.6σ tension between the TRGB and Cepheid calibrations.

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The tension between the TRGB and Cepheid calibrations is perhaps not a serious problem, given that systematic uncertainties can be difficult to identify, and 2σ indicates generally good agreement, given those challenges. However, unlike the tension between the early universe (CMB results) and the local value of H₀, the true distances to galaxies are fixed with unique values. Rather than signifying potential new physics in the early universe, this "local" tension is unambiguously signaling that the uncertainties in one or both distance scales (out to and including the SNe Ia) have been underestimated.

Gaia EDR3, as described in Section 2.2.1, also presents the opportunity to derive a new zero-point calibration for Milky Way Cepheids (e.g., R21; Breuval et al. 2021; Owens et al. 2021). (The R21 results were shown in the third panel of Figure 8.) We discuss the Owens et al. Gaia EDR3-based calibration of a multiwavelength sample of field Cepheids below and compare these calibrations with the sample of field Cepheids analyzed by R21.

Owens et al. (2021) analyzed Gaia EDR3 data for 49 Milky Way field Cepheids in an attempt to provide a multiwavelength calibration of the Leavitt law. In early anticipation of the Gaia mission, Freedman et al. (2011) and Monson et al. (2012) undertook a program to augment the sample of published optical photometry for Milky Way Cepheids with Spitzer mid-infrared (3.6 and 4.5 μm) photometry, providing a multiwavelength (BVRIJHK[3.6][4.5]) database for 37 Cepheids, located both in the field and in open clusters.

Adopting the photogeometric distances obtained from the EDR3 parallax measurements by Bailer-Jones et al. (2021), Owens et al. (2021) derived optical–to–mid-infrared Leavitt law relations for the Milky Way sample. The Bailer-Jones et al. measurements include correction for the zero-point offset in Gaia EDR3 parallaxes (Lindegren et al. 2021a). A challenge at present is that this sample of Milky Way Cepheids is very bright in apparent magnitude (4 mag < G < 11 mag). As already discussed in Section 2.2.1, the corrected Gaia EDR3 parallaxes have large uncertainties and have been shown to be underestimates. Moreover, they are significantly underestimated at brighter magnitudes (e.g., El-Badry et al. 2021), up to 30% for isolated sources with small quoted astrometric uncertainties (and up to 80% for those with companions). It was found by R21 that a −14 μas correction to their Cepheid parallaxes was indicated, obtained by minimizing the scatter in their Wesenheit Leavitt law.

In a comparison with HST parallaxes and published infrared Baade–Wesselink distances, as well as the DEB distances to the LMC and SMC, Owens et al. (2021) concluded that the current uncertainty in their sample of EDR3 parallaxes is conservatively at a level of ∼±5%, much larger than the 1% or better accuracy anticipated from future (DR4 and DR5) Gaia releases. Owens et al. also explored adding a constant offset to the Leavitt law but found that there is no single offset that minimizes the scatter (as would be expected for distance errors) for their multiwavelength sample. They instead used the DEB distances measured for the LMC and SMC by Pietrzyński (2019) and Graczyk et al. (2020) to provide an external estimate of the offset in the Milky Way sample, finding a value of +17.5 μas, similar in magnitude but opposite in sign to that found by R21. (The sense of the offset found by Owens et al. is in the same sense as that found by Maíz Apellániz et al. (2021).) However, as Owens et al. emphasized, the adoption of the DEB distances does not then provide an independent Gaia EDR3 zero-point calibration, and uncertainty in the required correction to the Gaia EDR3 parallaxes remains.

Although the uncertainties are not yet at a level of 1%, there is still internal consistency at a few percent level in the Cepheid zero-points obtained using different Cepheid samples, parallax measurements, and external constraints and analyzed by different authors.

• 1.  
  Calibration of the TRGB zero-point. As shown in Section 2 of this paper, direct geometric calibrations of the TRGB method for the LMC (F19, F20; Hoyt 2021), NGC 4258 (Jang et al. 2021), the globular clusters in the Milky Way (Cerny et al. 2020; Freedman et al. 2020), and the SMC (Hoyt 2021) all agree to within ±1%.
• 2.  
  The TRGB as a standard candle. The strikingly sharp and flat definition of the TRGB at F814W (comparable to the ground-based I band) for Milky Way globular clusters (see Figure 9) provides growing direct evidence that old, blue, metal-poor giant branch stars at the TRGB are actual standard candles, distinctive from other commonly employed standardizable candles (for example, SNe Ia and Cepheids). The fact that this sharp cutoff is not simply an empirical feature but rather the result of a well-understood physical mechanism (the core helium flash) lends confidence to the use of these stars as reliable distance indicators.
• 3.  
  Photometric errors due to crowding/blending effects. The TRGB method is best applied in the outer halos of galaxies (e.g., see the discussion in Jang et al. 2021, and references therein), where the surface brightness of the galaxy is low and the overlapping of stellar point-spread functions is minimal. Crowding/blending effects are not currently a significant source of uncertainty for the TRGB method if carefully applied to stars in the outer halos of galaxies.
• 4.  
  Effects of dust: foreground and internal. Foreground Milky Way reddening corrections are obtained from the all-sky extinction maps of Schlafly & Finkbeiner (2011). Beyond the Milky Way, for the application of the TRGB method targeted in the halos of galaxies, the effects of dust are small (e.g., Ménard et al. 2010). For the four current anchors (the LMC, the Milky Way, NGC 4258, and the SMC), the local line-of-sight circumstances are different for each case, and extinction and reddening corrections have been investigated in detail on a case-by-case basis as described in Jang et al. (2018), Cerny et al. (2020), Freedman et al. (2020), and Hoyt (2021). For an individual anchor, the distance uncertainty attributed to this correction contributes to its systematic uncertainty; however, for the determination of H₀ based on several anchors, it contributes only to the overall statistical error, and not to the final systematic uncertainty.
• 5.  
  Metallicity effects. For RGB stars, there is a metallicity (and concomitant color) dependence of the luminosity that is both predicted by theory and independently confirmed by observation (e.g., Freedman et al. 2020). A significant advantage of the TRGB method is that it has long been known that the color of a star on the RGB is a direct indicator of the metallicity of the star (e.g., Da Costa & Armandroff 1990; Carretta & Bragaglia 1998).Given a known (flat) TRGB slope in the I band, the corresponding slope of the giant branch luminosity with color, at any other given wavelength, is not arbitrary; it is a priori mathematically defined for the other wavelengths (Madore & Freedman 2020). Empirically, the slope and zero-points of the VIJHK RGB terminations determined for the LMC and SMC agree with those measured for Milky Way globular clusters to within their 1σ uncertainties (F20; Cerny et al. 2020).For the purposes of the I-band (F814W) calibration presented in this paper, the effects of metallicity are negligible, given that only the bluest (metal-poor) stars enter the calibration, and that the flat (color-independent) nature of the TRGB in this restricted color regime is well established (see, for example, Figure 9 above and Figure 6 of Jang et al. 2021).
• 6.  
  Future prospects for the TRGB distance scale.

• 1.  
  Calibration of the Cepheid zero-point. Direct geometric calibrations of the Cepheid Leavitt law for the LMC are based on (a) the DEB distance to the LMC (Pietrzyński 2019), (b) HST and Gaia parallaxes for field Cepheids in the Milky Way (Benedict et al. 2007; Riess et al. 2018, 2021), and (c) the maser distance to NGC 4258 (Reid et al. 2019). The resulting values of H₀ for these three calibration methods currently span a range of 72–74 km s⁻¹ Mpc⁻¹. ¹¹
• 2.  
  Cepheids as standardizable candles. The well-defined relationship between period, luminosity, and color can, in principle, produce a standardizable candle of high precision. Then, including a metallicity term, the Leavitt law can be expressed as

  $$\begin{eqnarray}&&\,{M}_{{\lambda }_{1}}=\alpha \ \mathrm{log}P+\beta {\left({m}_{{\lambda }_{1}}-{m}_{{\lambda }_{2}}\right)}_{o}+\gamma [{\rm{O}}/{\rm{H}}]+\delta ,\,\end{eqnarray} \tag{ 3 }$$

  where the Cepheid magnitude at a given wavelength λ₁ is a function of the logarithm of the period (P), a color term with coefficient β, and a term with coefficient γ that allows for a metallicity effect (where [O/H] represents the logarithmic oxygen-to-hydrogen ratio for H ii regions in the vicinity of the Cepheids relative to the solar value), and δ is the zero-point. It has long been recognized that the decreasing scatter in the correlation between period and luminosity with increasing wavelength (e.g., Madore & Freedman 1991), as well as the decreasing effect of reddening and metallicity with increasing wavelength, motivates the application of the Leavitt law at near-infrared (or longer) wavelengths (McGonegal et al. 1982; Freedman et al. 1991, 2008; Madore & Freedman 1991; Macri et al. 2001). ¹²
• 3.  
  Photometric errors due to crowding/blending effects. Cepheid variables are yellow supergiants, generally found in relatively high surface brightness areas in the star-forming disks of late-type galaxies. For nearby galaxies, the crowding and blending of Cepheids is not a serious practical issue for the brightest long-period Cepheids, but the problem worsens as the distance increases and the angular resolution decreases. Using artificial star tests, R16 concluded that these crowding/blending effects do not induce systematic effects. In addition, Riess et al. (2020) tried to infer the quantitative effects of crowding by comparing the amplitudes of Cepheids in four galaxies out to a distance of 20 Mpc. They concluded that the erroneous measurements of Cepheid backgrounds alone cannot explain the Hubble tension. Future work is still needed to assess the implications for the even more distant galaxies in the SHoES program, which extend out to 40–50 Mpc. The effects of crowding/blending also become more severe with increasing wavelength, where, for a given aperture telescope, the resolution is poorer in the infrared than in the optical. Disk red giants and the even brighter asymptotic giant branch (AGB) stars (both of which are redder than Cepheids) are the main, and unavoidable, contaminants. Thus, although both dust and metallicity effects are decreasing functions of wavelength, there is a trade-off to be made with the decreasing (wavelength-dependent) resolution and the increasing challenges of overlapping images dominated by red stars, particularly as the distance increases. In the case of HST and WFC3, the longest wavelength available, the F160W filter (comparable to the ground-based H band), has an advantage in reducing the effects of dust and metallicity, but it is at a disadvantage in dealing with the effects of increased crowding and blending.
• 4.  
  Effects of dust. As a consequence of their relative youth, Cepheid variables are unavoidably located close to the regions of dust and gas out of which they formed. In practice, however, Cepheid reddening can be dealt with in a straightforward manner. With accurate colors, Madore (1976, 1982) showed that a reddening-free magnitude can be constructed; for example,

  $$\begin{eqnarray}&&W=V-{R}_{V}\times (B-V),\end{eqnarray} \tag{ 4 }$$

  where R_V = A_V /E(B − V) is the ratio of total-to-selective absorption, and W has been widely applied to the Cepheid distance scale (e.g., Freedman et al. 2001; Riess et al. 2016). An advantage of W is that it simultaneously corrects for all line-of-sight absorption, including both host-galaxy (internal) and Galactic (foreground) reddening. ¹³
• 5.  
  Metallicity effects. The effects of metallicity on the Cepheid Leavitt law are still being actively debated in the literature (e.g., for a recent summary, see Ripepi et al. 2020). One of the immediate challenges in constraining any metallicity effect for Cepheids is the difficulty of determining abundances for the individual Cepheids themselves. Spectroscopic abundances have been measured for Cepheids in the Milky Way and LMC (e.g., Romaniello et al. 2008); however, more distant Cepheids are generally too faint to measure abundances from spectroscopy. Three decades of empirical tests for a Cepheid abundance effect (the measurement of γ in Equation (3); e.g., Freedman & Madore 1990; Kennicutt et al. 1998; Romaniello et al. 2008; Fausnaugh et al. 2015; Riess et al. 2016; Ripepi et al. 2020; Breuval et al. 2021) have not yet led to a consensus view on the magnitude of the effect or even its sign, or indeed, whether there is an effect at any given wavelength. Most of these studies have had to rely on the use of [O/H] abundances for nearby H ii regions as a proxy for the Cepheid metallicities, which cannot generally be measured directly. Theoretical models suggest that the effect of metallicity will be smaller at longer wavelengths, but there also remain significant differences in the predicted effects on both the slope and intercept of the period–luminosity relations with wavelength (Bono et al. 2008a; Ripepi et al. 2020), even at the long wavelength of the K band (2.2 μm). Ripepi et al. found that the slope of the metallicity term ranges from −0.04 to −0.36 mag dex⁻¹ for fundamental pulsators and +0.23 to −0.30 mag dex⁻¹ for overtone Cepheids. Recently, incorporating Gaia EDR3 data for the Milky Way and comparing to the LMC and SMC, Breuval et al. (2021) found that the metallicity effect is negligible in the optical (V band) and moreover, contrary to previous studies, concluded that the effect increases through IJHK, with the largest effect being in the near-infrared. As we enter an era where 1%–2% accuracies are required to resolve whether there is an H₀ tension, it is critical that the long-standing uncertainties due to metallicity be better understood and calibrated. Authors R16 computed a Wesenheit function of the form

  $$\begin{eqnarray}&&\begin{array}{l}{M}_{H}^{W}={m}_{H}-{R}_{H,VI}\times (V-I),\\ {\rm{w}}{\rm{h}}{\rm{e}}{\rm{r}}{\rm{e}}\,\,R\equiv {A}_{H}/({A}_{V}-{A}_{I}),\end{array}\end{eqnarray} \tag{ 5 }$$

  and solved for a metallicity correction on a star-by-star basis. Their conclusion is that metallicity contributes to their total H₀ uncertainty of 2.4% only at the 0.5% level. Given the long-standing disagreement in the literature (from both theory and observations), further work is clearly warranted to confirm this assertion. This issue is best addressed with multiwavelength, high signal-to-noise data for nearby galaxies where covariant crowding effects are less severe.
• 6.  
  Summary and future prospects for the Cepheid distance scale. Cepheids have many strengths that make them good distance indicators. However, they still face a number of challenges, particularly when it comes to applying them under conditions at the limits of current telescopes and detectors, with the goal of achieving distances that are accurate and precise to a level of 1%–2%. The main challenge for Cepheid standardization is that several wavelengths, each of equally high precision, are required: first, to correct for reddening; second, to correct for a possible metallicity effect (the wavelength dependence and sign of which remain under debate); and third, to ensure that the effects of crowding/blending are not systematically influencing the results. All three of the above systematic effects (reddening, metallicity, and crowding) increase toward the centers of galaxies. Since Cepheids are being crowded/blended, particularly by red giant and red (even brighter) AGB stars, all three effects will also act in the sense of causing Cepheids to appear redder in regions of coincidentally higher metallicity. Put another way, the corrections for reddening, metallicity, and crowding/blending are covariant; for example, if the currently applied metallicity or crowding corrections are incorrect, then the reddening corrections will also be in error because they all involve the same limited sets of colors, making it difficult to break the degeneracy. These issues will continue to pose a serious challenge for 1% accuracy, especially when the scatter in the observed Wesenheit Leavitt law can be 20%–25% in distance or 0.4–0.5 mag in distance modulus (R16), even for (anchor) galaxies as close as 7.6 Mpc (e.g., NGC 4258). There are many areas where future tests could further constrain the uncertainties in the Cepheid distance scale. (a) High signal-to-noise, multicolor, time-averaged (BVIJHK) photometry and spectroscopy for nearby galaxies with a range of metallicities can help resolve the question of the magnitude, sense, and wavelength dependence of metallicity corrections. The inclusion of additional distant galaxies will not lead to better constraints on the systematic effects, such as metallicity; obtaining larger samples of galaxies will simply reduce the statistical uncertainties alone. (b) As further SNe Ia are discovered in the nearby universe, the numbers of SN Ia host galaxies with observable Cepheids will also slowly be increased. (c) The JWST/NIRCam in the J band has four times the angular resolution of HST/WFC3 in the H band, where the longest-wavelength SHoES Cepheid measurements have been made, and thus can allow the effects of crowding/blending in the HST photometry to be assessed directly.

At present, the systematic accuracies of the TRGB and the Cepheid distance scale zero-points are constrained by the small number of available geometric calibrators providing high-accuracy distances. Below, we outline the degree to which the two distance scales are codependent (or not) on the same (or different) zero-point calibrators.

As illustrated in Table 7, there are four galaxies with geometric measurements that have been used to calibrate the local distance scale: the LMC, NGC 4258, the Milky Way, and the SMC. There are several important points to take away from this table.

• 1.  
  Both the TRGB and the Cepheids adopt the same distances for the LMC and NGC 4258, therefore sharing any systematic errors that may have been incurred in those measurements. The current total uncertainties quoted for these measurements are at a level of 1% and 1.5%, respectively (Pietrzyński 2019; Reid et al. 2019).
• 2.  
  The only galaxy sufficiently nearby for which an accurate maser distance can be measured is NGC 4258, which is also close enough for the calibration of the TRGB and Cepheids. A "sample of one" precludes rigorous testing for potential systematic errors in this galaxy's geometric distance.
• 3.  
  In the case of the TRGB method, the LMC and SMC calibrations share the systematic uncertainties of the Skowron et al. (2021) reddening maps. The dominant uncertainty is that of the zero-point, estimated by Hoyt et al. (2021) to be ±0.014 mag (0.6%) and ±0.018 mag (0.8%), respectively. In addition, their DEB distances are both based on the surface brightness–color relation from Pietrzyński (2019), estimated to be 0.8%.
• 4.  
  Finally, it should be noted that for both the TRGB and the Cepheids, the same reddening law is adopted and assumed to be universal; moreover, the same ratio of total-to-selective absorption, R_V , is adopted in both applications. However, the TRGB method is less susceptible to the assumption of the universality of the reddening law because the dust content in the halos of galaxies is generally negligible compared to that in the disks.

6.3.1. The SN Ia Host Galaxies

6.3.2. Distant SNe Ia in the Hubble Flow