A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s⁻¹ Mpc⁻¹ Uncertainty from the Hubble Space Telescope and the SH0ES Team

The Hubble constant is the most accessible parameter in the cosmological model. It can be estimated with a wide range of approaches and accuracies from limited knowledge of many types of astronomical sources, nearly all of which have been utilized in this endeavor over the past century. There have been >1000 estimates published since 1980, with one-third of those in the last five years and 20% in the last two years—a recent quadrupling of the effort indicating the accelerating interest in H₀ (Steer 2020). However, past discrepancies internal to the body of local measurements reveal that systematic errors can dominate determinations of H₀, and that there is no reason to believe all efforts will regress to the mean or that a more accurate result can be derived from their median (Chen & Ratra 2011). Rather, to keep systematic uncertainties in check, it is necessary to pursue the most powerful, simplest, and most reliable tools, with strict attention paid to understanding, mitigating, and accounting for sources of measurement error.

Since the launch of the Hubble Space Telescope (HST) with a design goal of achieving a 10% determination of H₀, the leading approach to measuring it in the local universe (as indicated by the observing time competitively awarded by the community) has relied on imaging of Cepheid variable stars in the host galaxies of recent, nearby Type Ia supernovae (SNe Ia). Cepheids have been favored as primary distance indicators because they are very luminous (M_V ≈ −6 mag at P ≈ 30 days), are easy to identify thanks to their periodicity (Leavitt & Pickering 1912), obey a tight period–luminosity relation (P–L, the "Leavitt Law") that yields extremely precise distances (3% per source; Riess et al. 2019a, hereafter R19), and have well-understood physics (Eddington 1917; Bono et al. 1999). Other primary distance indicators that have been measured in the hosts of SNe Ia to determine H₀ include the tip of the red giant branch (TRGB; Freedman et al. 2019, hereafter F19) and Mira variable stars (Huang et al. 2020).

Cepheids also offer the most opportunities for obtaining strictly differential flux measurements—i.e., the use of the same facility to measure calibrator and source, a key requirement for eliminating zero-point errors. This is feasible through the use of HST to directly observe Cepheids in a large set of SN hosts and in geometric calibrators of Cepheid luminosities: the megamaser host NGC 4258, the Milky Way (hereafter MW) with plentiful parallaxes, and the Large Magellanic Cloud (hereafter LMC) via detached eclipsing binaries (DEBs). SNe Ia are favored to measure the Hubble expansion owing to their high precision (5% in distance per source), ubiquity, and deep reach, which reduces the impact of local flows.

The SH0ES program (Supernovae and H₀ for the Equation of State of dark energy) began in 2005 with a proposal in HST Cycle 15 to break the degeneracy among cosmological parameters used to model CMB data and the equation-of-state parameter w = P/(ρ c²), where P is the pressure and ρ is the mass density of dark energy. Its stated ambitious goal, based on the recommendation by Hu (2005), was to eventually reach a percent-level measurement of H₀, a goal approached if not fully reached in this work. This project was a "second-generation" effort to measure H₀ with HST from a distance ladder of Cepheids and SNe Ia using the then recently installed ACS (and later WFC3) instruments, following successful efforts during the 1990s by the "first-generation" HST Key Project on the Extragalactic Distance Scale (Freedman et al. 2001) and the SNe Ia Luminosity Calibration Program (Sandage et al. 2006), both of which primarily used WFPC2. The former searched for Cepheids in the hosts of numerous secondary distance indicators (excluding SNe Ia) while the latter focused on SN Ia hosts. The types of targets suitable for Cepheid searches and the observing sequences for use with HST were developed and first implemented by these ground-breaking programs.

The precision of the first-generation programs was ultimately limited by the lack of a precise geometric calibration of Cepheid variables, by the limited practical range of WFPC2 to measure Cepheids in SN Ia hosts (distance D ≲ 20–25 Mpc), by the impact of reddening in the optical, and by the limited characterization of the Cepheid metallicity dependence at those wavelengths. The ability of SNe Ia to measure individual distances with ∼5%–10% precision and to sharply delineate the Hubble flow began with the use of light-curve versus luminosity relations (Phillips 1993), SN Ia colors (Riess et al. 1996; Tripp 1998; Phillips et al. 1999), and modern, digital samples (Hamuy et al. 1996; Riess et al. 1999).

Unfortunately, SNe Ia at D ≤ 20 Mpc are rare, occurring about once per decade, with most of the few objects in this range observed up to a century ago using photographic technology. Such observations lacked the photometric precision, well-characterized bandpasses, and accurate determinations of host-galaxy backgrounds, SN light-curve shapes, and SN colors to take advantage of the new standardization methods. The tendency for intrinsically brighter SNe Ia with broader light curves to occur in (late-type) Cepheid hosts would also bias H₀ lower without light-curve standardization. A number of systematic differences in the first-generation calibration of SNe Ia by Sandage et al. (2006) were quantified by Riess et al. (2005, Table 16). These differences, totaling about 20%, arose from several effects which were amplified by small sample statistics: problematic SN Ia data such as photographic photometry, highly extinguished objects, and poorly sampled light curves; from photometric anomalies in WFPC2, such as the "long versus short" effect (Holtzman et al. 1995) and charge transfer efficiency (CTE; e.g., Whitmore et al. 1999); and from limited knowledge of the slope of the Cepheid P–L relation. The present geometric calibration of the distance to the LMC by Pietrzyński et al. (2019) using DEBs is also 7% smaller than the value assumed by Sandage et al. (2006) to calibrate Cepheids.

The SH0ES program has been designed to improve upon past determinations of H₀ by (1) extending the range of Cepheid observations with ACS and WFC3 to reach the hosts of a large sample of "ideal" SNe Ia, free from the preceding problems; (2) using near-infrared (NIR) observations of all Cepheids in SN Ia hosts with NICMOS and WFC3 to reduce the systematic uncertainty associated with the reddening laws for Cepheids and their hosts and the Cepheid metallicity dependence; and (3) calibrating Cepheids with new, geometric distances tied directly with HST to the Cepheids in SN Ia hosts to nullify zero-point uncertainties. "Ideal" or suitable SNe Ia for calibrating H₀ (given limited HST time) were defined by Riess et al. (2005) to be (1) observed before maximum light, (2) through low interstellar extinction (A_V < 0.5 mag), (3) with the same instruments and filters as the SNe Ia in the Hubble flow (at that time obtained by the Calán/Tololo and CfA surveys), and (4) to have typical light-curve shapes. ¹¹ These characteristics are necessary to provide low dispersion in the Hubble flow, but they applied to only three Cepheid-calibrated SNe Ia from the first-generation projects (SNe 1981B, 1990N, and 1998aq).

Unlike the first-generation programs, which were granted long-term status and a large initial allocation of observing time with HST, the SH0ES program was proposed to the STScI time-allocation committee year by year. The cumulative result, after 15 yr (cycles), has been to collect Cepheid observations in 37 hosts of 42 SNe Ia and calibrate them geometrically to Cepheids in the MW, the LMC, and NGC 4258, with a total of 18 individual HST proposals utilizing >1000 orbits. The source of the distance-ladder data is shown in Figure 1, and the source and sequence of all observations in SN Ia hosts are shown in Figure 2 and listed in Table 1. Observing Cepheids in hosts at D ≈ 40–50 Mpc, double the range of first-generation observations and a factor of 8 increase in targets, was feasible owing to two features of WFC3: significantly better sampling of the point-spread function (PSF); with pixel sizes a factor of 2.5 smaller than the wide channel of WFPC2, which greatly reduced the impact of crowded backgrounds; and a white-light filter (F350LP) that, combined with the better sensitivity of WFC3, reduced the observing time required to identify Cepheids and measure their periods by a factor of ∼4. Calibrating those Cepheids differentially to bright Cepheids in the MW and LMC became feasible through the development of spatial scanning of HST and a rapid slew and guiding mode under gyroscopic control.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The first SH0ES results (Riess et al. 2009, hereafter R09) were based on Cepheids observed in the hosts of six ideal SN Ia calibrators using ACS (optical) and NICMOS (NIR), and one geometric anchor (NGC 4258) with a maser-based distance of 3% precision (Humphreys et al. 2013) and a large sample of Cepheids (Macri et al. 2006). The result was a 5% measurement of H₀, 74.2 ± 3.6 km s⁻¹ Mpc⁻¹, which, combined with the 5 yr WMAP results (Komatsu et al. 2009), yielded w = −1.12 ± 0.12 and was consistent with a value of H₀ = 71.9 ± 2.6 determined from WMAP and ΛCDM alone. ¹² The second iteration (Riess et al. 2011, hereafter R11) increased the calibrator sample to eight SNe Ia, observed and measured all Cepheids with WFC3 (both optical and NIR), and expanded the geometric calibration of Cepheids beyond NGC 4258 by including two additional independent anchors: the LMC, through various DEB-based distances (e.g., Pietrzyński et al. 2009), and the MW via parallaxes measured with the Fine Guidance Sensor on HST (Benedict et al. 2007). This resulted in H₀ = 73.8 ± 2.4, which, coupled with the 7 yr WMAP results (Komatsu et al. 2011), yielded w =−1.08 ±0.10, or an estimate of the effective number of relativistic species of N_eff = 4.2 ± 0.7. This result was closely matched by a recalibration of the final HST Key Project results using the same MW parallaxes, different Cepheid measurements, and an updated Hubble diagram of SNe Ia, which yielded H₀ = 74.4 ± 2.2 (Freedman et al. 2012, hereafter F12). Indications of any tension between the early and late universe at that time were <2σ in significance.

The first release of CMB data from the ESA Planck mission (Planck Collaboration et al. 2014) yielded H₀ = 67.2 ± 1.2 in the context of ΛCDM, a then 2σ reduction relative to WMAP and a difference of 3σ from the results of R11 and F12. Reanalyses of the R11 data (Fiorentino et al. 2013; Efstathiou 2014; Zhang et al. 2017) produced essentially the same results as R11, with H₀ ranging from 72.5 to 76.0. The third iteration of SH0ES (Riess et al. 2016, hereafter R16) more than doubled the calibrator sample to 19 SNe Ia, used refined distance estimates to NGC 4258 and the LMC (Pietrzyński et al. 2013), and new Cepheid parallaxes measured by the SH0ES team using spatial scanning with WFC3 (Riess et al. 2014; Casertano et al. 2016) to reach a 2.4% determination of H₀ = 73.2 ± 1.7, 3.4σ greater than the refined value from Planck+ΛCDM of 66.9 ± 0.6 (Planck Collaboration et al. 2016), a difference of ∼9% or 0.2 mag in units of 5 log H₀. An extensive number of reanalyses of the R16 data with many variations were undertaken (Cardona et al. 2017; Feeney et al. 2019; Follin & Knox 2018; Bovy 2018; Burns et al. 2018; Dhawan et al. 2018; Avelino et al. 2019), resulting in H₀ values ranging from 73 to 74 and uncertainties from 2% to 2.5%. These analyses explored varying reddening laws, use of NIR SN data, use of alternative SN light-curve fitting, and hierarchical Bayesian statistics for data fitting, resulting in little change in H₀ or its uncertainty. Javanmardi et al. (2021) selected the Cepheid host from R16 with the largest sample of Cepheids (NGC 5584, also near the median sample host distance) and remeasured these starting from the archived HST pixels and using different methods to photometry, finding agreement with the R16 measurement to 1% precision and ruling out a significant methodological error in these measurements.

Since then, additional sources of MW parallax calibration of Cepheid luminosities have come from further use of HST WFC3 spatial scanning (Riess et al. 2018b), from Gaia DR2 Cepheid parallaxes with HST photometry (Riess et al. 2018a), from Gaia DR2 Cepheid binary companions and cluster hosts (Breuval et al. 2020), and from Gaia EDR3 parallaxes coupled with additional HST photometry (Riess et al. 2021, hereafter R21). Likewise, HST observations of 70 long-period Cepheids in the LMC (Riess et al. 2019b) and improved distance estimates to the LMC (Pietrzyński et al. 2019) and NGC 4258 (Reid et al. 2019) resulted in H₀ = 73.2 ± 1.3, raising the difference with Planck+ΛCDM to 4.2σ. Other precise measures of H₀ in the local universe from the distance–redshift relation generally range from 70 to 75, and those grounded in the pre-recombination version of ΛCDM range from 67 to 68, and have been extensively reviewed (Verde et al. 2019; Di Valentino et al. 2021; Shah et al. 2021). Because the tension is seen between different routes, which are comparable only via an accurate cosmological model, numerous possible theoretical explanations for the emergent "Hubble tension" have been proposed but no consensus has yet emerged (Di Valentino et al. 2021). Indeed, theoretical priors weigh heavily on these proposals or whether it may be considered "extraordinary" for the ΛCDM model to fail or pass this cosmic test. The test itself, however, is empirical and few would conclude it has yet been satisfactorily passed.

In this publication, we more than double the sample of SN Ia calibrators from 19 in R16 to reach 42 objects in 37 hosts. This increase is a milestone for a sample whose size has limited the precision to which H₀ can be locally measured. It provides the largest increase in size we can anticipate in the remaining lifetime of HST as it now includes all suitable SNe Ia (of which we are aware) observed between 1980 and 2021 at z < 0.011 and which slowly accrue at ∼1 yr⁻¹. We have also reprocessed and reanalyzed the NIR observations of Cepheids in the previous 19 hosts reported by R16 for consistency with the new sample. We make use of an automated pipeline (W. Yuan et al. 2022, in preparation, hereafter Y22b) to find the Cepheids in 18 new hosts (and reanalyze past hosts), which follows the steps developed manually for the first 19 hosts (Hoffmann et al. 2016, hereafter H16). We benefit from a factor of 3 increase in the sample of Cepheids within NGC 4258 discovered by observing four new fields with HST, fully reanalyzed using the same pipeline by (Yuan et al. 2022, hereafter Y22a). Extensive details concerning the analysis of SN Ia data are given by Scolnic et al. (2021) and Brout et al. (2021).

We present the formalism for measuring H₀ from the distance ladder in Section 2; new Cepheid data in Section 3 and in Y22a+Y22b; ancillary data used to measure H₀ in Section 4; our baseline, local determination of H₀ and a simultaneous measurement of H₀ and the expansion history with high-redshift SNe Ia in Section 5; extensions to the baseline and variants of the local measurement of H₀ in Section 6; discussion in Section 7; and conclusions in Section 8. Appendices provide further details on characterizing the spectral and photometric properties of the 42 calibrator SNe Ia (A), independent tests of the accuracy of Cepheid photometry (B), the Cepheid metallicity scale (C), and alternative applications of "Wesenheit" magnitudes (D).

The Hubble constant is the present relation between redshift and distance, cz = H₀ D, measured at cosmological distances where expansion is the dominant source of redshift. Here it is measured via a three-step (or three-rung) distance ladder employing a single, simultaneous fit between (1) geometric distance measurements to standardized Cepheid variables, (2) standardized Cepheids and colocated SNe Ia in nearby galaxies, and (3) SNe Ia in the Hubble flow. The fit is accomplished simultaneously by optimizing a χ² statistic to determine the most likely values of the parameters in the relevant relations. The data include measurements, their uncertainties, and their covariances as described in the next section. The parameters are the distances of all hosts and five additional parameters: the fiducial luminosity of SNe Ia and Cepheids, two parameters standardizing Cepheid luminosities (their dependence on period and metallicity), and H₀. This parameterization of the distance ladder can be expressed as a simple system of linear equations in a compact set of matrices as given below, useful for transmission, and for which the maximum-likelihood solution is easily found. The distance-ladder data and scheme are displayed in approximate form in Figure 1.

To briefly summarize the relevant relations, the distance modulus of a source is ${\mu }_{0}={m}_{0}-{M}_{0}=5\,\mathrm{log}\,D+25$, with D the luminosity distance in Mpc, m the apparent magnitude (flux), M the absolute magnitude (luminosity), and the subscript 0 denoting a magnitude free of (or corrected for) intervening absorption by interstellar dust. The form of the dependence of Cepheid or SN Ia luminosity on observed characteristics (i.e., "standardization") has been well determined by prior work and is only briefly reviewed here. The dereddened Cepheid apparent magnitudes (also called "Wesenheit" magnitudes; Madore 1982) at mean phase will be described in Section 3.3 and are identified here as ${m}_{H}^{W}$ (in a specific host like NGC 4258, those Cepheids would have magnitudes given as ${m}_{H,N4258}^{W}$). For the jth such Cepheid magnitude in the ith host given the period P_{i, j} in days and metallicity [O/H]_i,j relative to the Sun, we have

$$\begin{eqnarray}&&{m}_{H,i,j}^{W}={\mu }_{0,i}+{M}_{H,1}^{W}+{b}_{W}\,(\mathrm{log}\,{P}_{i,j}-1)+{Z}_{W}{[{\rm{O}}/{\rm{H}}]}_{i,j},\end{eqnarray} \tag{ 1 }$$

where ${M}_{H,1}^{W}$ is the fiducial absolute magnitude (in the Wesenheit magnitude system of 3.4) of a Cepheid with $\mathrm{log}\,P=1$ (P = 10 days) and solar metallicity, and the parameters b_W and Z_W (sometimes called γ in the literature) define the empirical relation between Cepheid period, metallicity, and luminosity. The [O/H]_i,j is inferred at its galactocentric radial position as described in Section 3.5.

Any number of Cepheid hosts may have an independent geometric distance that contributes to the calibration of the Cepheid luminosity; e.g., for variables observed in the maser host NGC 4258 we adopt a distance modulus μ_0,N4258, the best estimate of the distance, with formal uncertainty σ(μ_0,N4258) from Reid et al. (2019). In this case the individual host distance parameter μ_0,i above is replaced with the external constraint, converting the apparent magnitudes to absolute,

$$\begin{eqnarray}&&{M}_{H,j}^{W}={m}_{H,N4258,j}^{W}-{\mu }_{0,{\rm{N}}4258}+{\rm{\Delta }}{\mu }_{{\rm{N}}4258},\end{eqnarray} \tag{ 2 }$$

and introducing a new parameter Δμ_N4258 as the difference from the measured and true distance with the additional, simultaneous constraint equation 0 = Δμ_N4258 ± σ(μ_0,N4258). This definition allows the simultaneous use of multiple geometric "anchors" to calibrate the distance ladder; as we later show, it also allows the use of additional distance indicators for the same anchor, such as the tip of the red giant branch (TRGB) or Mira variable stars, while keeping track of their mutual dependence on the same geometric constraint.

A set of hosts of both SNe Ia and Cepheids connects the two distance indicators. Thus, for an SN Ia in the ith Cepheid host,

$$\begin{eqnarray}&&{m}_{B,i}^{0}={\mu }_{0,i}+{M}_{B}^{0},\end{eqnarray} \tag{ 3 }$$

where ${m}_{B,i}^{0}$ is its maximum-light apparent magnitude that has been standardized (i.e., corrected for variations around the fiducial color, luminosity, and any host dependence; see Scolnic et al. 2021), ${M}_{B}^{0}$ is the fiducial SN Ia luminosity, and μ_0,i is the same parameter as in Equation (1). For SNe Ia, unlike Cepheids, the convention for keeping track of covariance in the standardization, as described by Scolnic et al. (2021), is to employ a set of standardized ${m}_{B,i}^{0}$, their uncertainties, and the covariance between any pair. This is an equally mathematically valid approach as keeping track of covariance through the standardizing relation for Cepheids in Equation (1).

The ladder is completed with a set of SNe Ia that measure the expansion rate quantified as the intercept, a_B , of the distance (or magnitude)–redshift relation. This is simply ${a}_{B}=\mathrm{log}\,{cz}-0.2{m}_{B}^{0}$ in the low-redshift limit (z ≈ 0) but given for an arbitrary expansion history and for z > 0 as

$$\begin{eqnarray}&&{a}_{B}=\mathrm{log}\,{cz}\left\{1+\displaystyle \frac{1}{2}\left[1-{q}_{0}\right]z-\displaystyle \frac{1}{6}\left[1-{q}_{0}-3{q}_{0}^{2}+{j}_{0}\right]{z}^{2}+O({z}^{3})\right\}-0.2{m}_{B}^{0},\end{eqnarray} \tag{ 4 }$$

measured from a set of SNe Ia ($z,{m}_{B}^{0}$), where z is the redshift due to expansion, q₀ is the deceleration parameter, and j₀ is the jerk (see Visser 2004 for definitions). The determination of H₀ follows from

$$\begin{eqnarray}&&\mathrm{log}\,{H}_{0}=0.2{M}_{B}^{0}+{a}_{B}+5.\end{eqnarray} \tag{ 5 }$$

If the set of standardized SN Ia magnitudes in the hosts of Cepheids, which serve to calibrate ${M}_{B}^{0}$ (hereafter "calibrators" or CC SNe Ia), and those in the Hubble flow used to measure a_B (hereafter HF SNe Ia) have no common sources of uncertainty (i.e., no covariance), then Equations (3) and (4) and the ladder parameters they provide (${M}_{B}^{0}$ and a_B ) can be determined independently. This was the approach taken by R16. However, an increasingly thorough quantification of systematic uncertainties in SN Ia measurements, and the standardization undertaken as common practice for the determination of w (Scolnic et al. 2018), has demonstrated nontrivial covariance of SN Ia data, quantified following the approach of Conley et al. (2011) and Dhawan et al. (2020). We therefore undertake the optimization of Equations (3) and (5) simultaneously.

It is useful to expand the same set of Equations (1)–(5) in the form of matrices that organize the data into a vector of magnitude measurements y, the covariance matrix of standard errors of the magnitude measurements C, the equation matrix L, and the vector of free parameters q (hence the model, Lq) as follows:

$$\begin{eqnarray*}&&y=\begin{array}{l}\left(\begin{array}{c}{m}_{H,1}^{W}\\ ..\\ \frac{\,}{{m}_{H,\mathrm{nh}}^{W}}\\ {m}_{H,{\rm{M}}31}^{W}\\ \frac{\,{m}_{H,\mathrm{LMC}}^{W}-{\mu }_{0,\mathrm{LMC}\,}}{{m}_{B,1}^{0}}\\ ..\\ \frac{\,}{{m}_{B,\mathrm{ncc}}^{0}}\\ {M}_{H,1,\mathrm{Gaia}}^{W}\\ 0\\ 0\\ \frac{\,0\,}{{m}_{B,1}^{0}-5\mathrm{log}{{cz}}_{1}\{\}-25}\\ ..\\ {m}_{B,\mathrm{nhf}}-5\mathrm{log}{{cz}}_{\mathrm{nhf}}\{\}-25\end{array}\right)\,\begin{array}{l}\left.,\Space{0ex}{3.25ex}{0ex}\right\}\ \mathrm{Cepheids}\ \mathrm{in}\ \mathrm{SN}\ \mathrm{Ia}\ \mathrm{hosts}\\ \left.,\Space{0ex}{4.55ex}{0ex}\right\}\ \mathrm{Cepheids}\ \mathrm{in}\ \mathrm{anchors}\ \mathrm{or}\ \mathrm{non}-\mathrm{SN}\,\mathrm{Ia}\ \mathrm{hosts}\\ \left.,\Space{0ex}{3.25ex}{0ex}\right\}\ \mathrm{SNe}\,\mathrm{Ia}\ \mathrm{in}\ \mathrm{Cepheid}\ \mathrm{hosts}\\ \left.,\Space{0ex}{6.25ex}{0ex}\right\}\ \mathrm{External}\ \mathrm{constraints}\\ \left.,\Space{0ex}{3.25ex}{0ex}\right\}\ \mathrm{SNe}\,\mathrm{Ia}\ \mathrm{in}\ \mathrm{the}\ \mathrm{Hubble}\ \mathrm{flow}\end{array}\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}C=\left(\begin{array}{ccccccccccccccccc}{\rm\small{}}{\sigma }_{\mathrm{tot},1}^{2} & .. & {Z}_{\mathrm{cov}} & {Z}_{\mathrm{cov}} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & ..\\ {Z}_{\mathrm{cov}} & .. & {\rm\small{}}{\sigma }_{\mathrm{tot},\mathrm{nh}}^{2} & {Z}_{\mathrm{cov}} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {Z}_{\mathrm{cov}} & .. & {Z}_{\mathrm{cov}} & {\rm\small{}}{\sigma }_{\mathrm{tot},{\rm{N}}4258}^{2} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{\mathrm{tot},{\rm{M}}31}^{2} & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{\mathrm{tot},\mathrm{LMC}}^{2} & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{{m}_{{\rm\small{B}}},1}^{2} & .. & {\mathrm{SN}}_{\mathrm{cov}} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\mathrm{SN}}_{\mathrm{cov}} & .. & {\mathrm{SN}}_{\mathrm{cov}}\\ .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & ..\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\mathrm{SN}}_{\mathrm{cov}} & .. & {\rm\small{}}{\sigma }_{{m}_{{\rm\small{B}},\mathrm{ncc}}}^{2} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\mathrm{SN}}_{\mathrm{cov}} & .. & {\mathrm{SN}}_{\mathrm{cov}}\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{\mathrm{HST}}^{2} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{\mathrm{Gaia}}^{2} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{\mathrm{grnd}}^{2} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{\mu ,{\rm{N}}4258}^{2} & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{\mu ,\mathrm{LMC}}^{2} & {\rm\small{}}0 & .. & {\rm\small{}}0\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\mathrm{SN}}_{\mathrm{cov}} & .. & {\mathrm{SN}}_{\mathrm{cov}} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}{\sigma }_{{m}_{{\rm\small{B}}},z,1}^{2} & .. & {\mathrm{SN}}_{\mathrm{cov}}\\ .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & ..\\ {\rm\small{}}0 & .. & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\mathrm{SN}}_{\mathrm{cov}} & .. & {\mathrm{SN}}_{\mathrm{cov}} & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\rm\small{}}0 & {\mathrm{SN}}_{\mathrm{cov}} & .. & {\rm\small{}}{\sigma }_{{m}_{{\rm\small{B}},z,\mathrm{nhf}}}^{2},\end{array}\right)\end{eqnarray*}$$

where ${\sigma }_{\mathrm{tot},1}^{2}$ representing the n × n covariance matrix for the Cepheids in the first host is expanded as

$$\begin{eqnarray*}{\sigma }_{\mathrm{tot},j}^{2}=\left(\begin{array}{ccc}{\sigma }_{\mathrm{tot},1,1}^{2} & .. & {C}_{\mathrm{1,1},n,\mathrm{bkgd}}\\ .. & .. & ..\\ {C}_{1,n,1,\mathrm{bkgd}} & .. & {\sigma }_{\mathrm{tot},1,n}^{2}\end{array}\right);\end{eqnarray*}$$

$$\begin{eqnarray*}L=\begin{array}{l}\left(\begin{array}{cccccccccccc}1 & .. & 0 & 0 & 1 & 0 & 0 & \mathrm{log}{P}_{N,1}-1 & 0 & {[{\rm{O}}/{\rm{H}}]}_{N,1} & 0 & 0\\ .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & 0\\ 0 & .. & 1 & 0 & 1 & 0 & 0 & \mathrm{log}{P}_{N,\mathrm{nh}}-1 & 0 & {[{\rm{O}}/{\rm{H}}]}_{N,\mathrm{nh}} & 0 & 0\\ 0 & .. & 0 & 1 & 1 & 0 & 0 & \mathrm{log}{P}_{{\rm{N}}4258}-1 & 0 & {[{\rm{O}}/{\rm{H}}]}_{{\rm{N}}4258} & 0 & 0\\ 0 & .. & 0 & 0 & 1 & 0 & 1 & \mathrm{log}{P}_{{\rm{M}}31}-1 & 0 & {[{\rm{O}}/{\rm{H}}]}_{{\rm{M}}31} & 0 & 0\\ 0 & .. & 0 & 0 & 1 & 1 & 0 & \mathrm{log}{P}_{\mathrm{LMC}}-1 & 0 & {[{\rm{O}}/{\rm{H}}]}_{\mathrm{LMC}} & 1 & 0\\ 1 & .. & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & 0\\ 0 & .. & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & .. & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & .. & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & .. & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & .. & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & .. & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & .. & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1\\ .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & ..\\ 0 & .. & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1\end{array}\right)\begin{array}{l}\left.,\Space{0ex}{4.25ex}{0ex}\right\}\ \mathrm{Cepheids}\ \mathrm{in}\ \mathrm{SN}\,\mathrm{Ia}\ \mathrm{hosts}\\ \left.,\Space{0ex}{4.55ex}{0ex}\right\}\ \mathrm{Cepheids}\ \mathrm{in}\ \mathrm{anchors}\ \mathrm{or}\ \mathrm{non}-\mathrm{SN}\,\mathrm{Ia}\ \mathrm{hosts}\\ \left.,\Space{0ex}{4.25ex}{0ex}\right\}\ \mathrm{SNe}\,\mathrm{Ia}\ \mathrm{in}\ \mathrm{Cepheid}\ \mathrm{hosts}\\ \left.,\Space{0ex}{6.25ex}{0ex}\right\}\ \mathrm{External}\ \mathrm{constraints}\\ \left.,\Space{0ex}{3.25ex}{0ex}\right\}\ \mathrm{SNe}\,\mathrm{Ia}\ \mathrm{in}\ \mathrm{the}\ \mathrm{Hubble}\ \mathrm{flow}\\ \end{array}\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}&&q=\left(\begin{array}{c}{\mu }_{\mathrm{0,1}}\\ ..\\ {\mu }_{0,\mathrm{nh}}\\ {\rm{\Delta }}{\mu }_{{\rm{N}}4258}\\ {M}_{H,1}^{W}\\ {\rm{\Delta }}{\mu }_{\mathrm{LMC}}\\ {\mu }_{M31}\\ {b}_{W}\\ {M}_{B}^{0}\\ {Z}_{W}\\ {\rm{\Delta }}\mathrm{zp}\\ 5\,\mathrm{log}\,{{\rm{H}}}_{0}\end{array}\right).\end{eqnarray*}$$

The number of Cepheid hosts is nh, the number of SNe Ia in these hosts is ncc, and the number of SNe Ia in the Hubble flow is nhf. Period uncertainties are comparatively negligible (Yuan et al. 2021). σ_HST and σ_Gaia denote the uncertainties in ${M}_{H,1,\mathrm{Gaia}}^{W}$ and ${M}_{H,1,\mathrm{HST}}^{W}$ as derived from parallaxes, respectively, while σ_grnd denotes the uncertainty in ground-based photometry. The term SN_cov is the covariance between SNe, Z_cov is the metallicity covariance given later in Equation (9), and C_i,j,k,bkgd is the background covariance given later in Equation (8). The χ² statistic is given as

$$\begin{eqnarray}&&{\chi }^{2}={\left(y-{Lq}\right)}^{T}{C}^{-1}(y-{Lq})\end{eqnarray} \tag{ 6 }$$

and maximum-likelihood parameters are given as ${q}_{\mathrm{best}}\,={({L}^{T}{C}^{-1}L)}^{-1}{L}^{T}{C}^{-1}y$, while the standard errors and the covariance matrix of the parameters come from the matrix ${cq}={({L}^{T}{C}^{-1}L)}^{-1}$. The value of H₀ is derived from the final entry of q, $5\,\mathrm{log}\,$ H₀, and its error from the square root of the corner entry of cq. This provides a compact form for storing and transmitting the full data set used to determine H₀, to edit or augment it, and to enable others to determine its value. The above formalism is the same as used by R16 with the additions of SN and Cepheid covariance. We will also derive the parameters independently of the analytical solution in Section 5.1 by sampling the χ² statistic using a Markov Chain Monte Carlo (MCMC) approach to verify the analytical result with a different methodology.

The SH0ES program has been selecting the SNe Ia that are most suitable for calibrating their fiducial luminosity (with selection criteria given in Section 1.1) through observations of Cepheids in their hosts. The results here include a complete sample of all such SNe Ia of which we are aware within z < 0.011 (40 objects), with the addition of two beyond this limit that are useful for testing the range of Cepheid distance measurements, for a total of 42 SN Ia calibrators.

Figure 2 and Table 1 show the sources of the HST data obtained for every SN Ia and host we measured, gathered from the indicated HST cameras, filters, and time periods. All of these publicly available data are readily obtained from the Mikulski Archive for Space Telescopes (MAST). The imaging data are used for both Cepheid discovery and their flux measurement. For the former, a campaign using a filter with a central wavelength in the visual band and ∼11–12 epochs with nonredundant spacings spanning ∼60–100 days is optimal to identify Cepheid variables by their unique light curves and large amplitudes (∼1 mag peak to trough) and accurately measure their periods (Madore & Freedman 1991; Saha et al. 1996; Stetson 1996). Image subtraction may find additional Cepheids (Bonanos et al. 2003), but these objects will be subject to greater photometric biases owing to blends which suppress their amplitudes and chances of discovery in time-series data (Ferrarese et al. 2000). The data were collected from ∼150 HST orbits with WFC3-NIR and ∼700 orbits obtained for the optical identification (350 from WFC3, 170 from ACS, and 180 from WFPC2), including ∼200 orbits from NICMOS superseded by WFC3-NIR, for a total of ∼1050 HST orbits. Additional observations of Cepheids in the MW and LMC anchors utilized ∼200 orbits or snapshots.

Earlier efforts contribute ∼200 orbits of imaging of these hosts: 35 orbits from the HST Key Project, 105 from the SNe Ia Luminosity Program, 36 from Mager et al. (2013), and 50 from Macri et al. (2006), with the remaining ∼800 from SH0ES.

The procedure for identifying Cepheids from time-series optical data in visual or white-light bands has been described extensively (Saha et al. 1996; Stetson 1996; Riess et al. 2005; Macri et al. 2006; Hoffmann et al. 2016); details of the procedures followed for this sample are presented by W. Yuan et al. (2022, in preparation). These procedures utilize the DAO suite of software tools (Stetson 1987, 1994) for crowded-field PSF photometry and are similar to those used previously by the SH0ES team (Hoffmann et al. 2016) and to a large extent by the first generation of HST-based H₀ measurements. Past work has demonstrated that the use of different photometry algorithms yields a largely overlapping list of Cepheids with similar periods and photometry (Ferrarese et al. 1998).

The end result is a set of high-confidence Cepheids, which have passed the selection and quality cuts described by Y22b, with periods and mean colors (F555W – F814W) measured in the HST WFC3 photometric system. For each Cepheid, we estimate a precise position in the WFC3/NIR F160W images using a geometric transformation derived from the optical images using bright and isolated stars, with resulting mean position uncertainties for variables <0.03 pix. The positions of the Cepheids are indicated in Figure 3. Figure 4 shows color images of each Cepheid host.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We present composite light curves of all Cepheids with 10 < P < 80 days in Figure 5 based on ∼60,000 individual F555W or F350LP photometric measurements, which identify these as bona fide classical Cepheids with the characteristic sawtooth light-curve shape of fundamental-mode pulsators. There are more subtle Cepheid light-curve features that are not apparent in individual examples at these distances. However, we can leverage the statistical power of the sample to look for these features as a strong validation test of the universality of Cepheids, as presented below.

Zoom In Zoom Out Reset image size

The shapes of Cepheid light curves change in subtle but characteristic ways (often referred to as the "Hertzsprung progression"; Hertzsprung 1926) as a function of period, visible only with high signal-to-noise-ratio (S/N) observations. Hertzsprung (1926) noted the characteristic "quick rise and slow decrease" for most periods but also a small range of periods (9–13 days) where the light curves are quite symmetric, and the presence of a "bump" or small local maximum whose phase occurs earlier with increasing period, appearing on the descending phase at P ≈ 6 days, merging with the main peak at P ≈ 10 days, and appearing on the rising phase at P ≈ 10–15 days before disappearing (variables where the bump is apparent are sometimes referred to as "bump" Cepheids). While the general fast rise and slow decline are characteristics of a star pulsating in the fundamental mode, the "bump" is understood through modeling to arise from a 2:1 resonance between the fundamental and second overtone (Cox 1993; Bono et al. 2002) and is a unique feature of classical Cepheids. Cepheids with P > 40 days occur at the high-mass end of the luminosity function and are thus rare, with the MW hosting only a few examples that show a broadening of the light curve and a decrease in amplitude.

By binning all of the extragalactic light curves for the 37 SN Ia hosts and NGC 4258 by period as shown in Figure 6, a striking display of this more subtle light-curve structure emerges. The resonance-induced bump is indicated on the descending phase at P = 5–9 days with the MW Cepheid DL Cas (P = 7.6 days) shown for comparison, the narrow-peaked symmetric curve at P = 9–14 days as in SS CMa (P = 12.4 days), and the resonance bump on the rising phase in the P = 14–19 days as in XZ Car (P = 16.6 days), before transitioning to sawtoothed curves in the next 3 bins (P = 19–35 days). Beyond this range, we see the gradual transition to flatter and more sinusoidal curves matching the MW sequence of SV Vul (P = 45 days), GY Sge (P = 52 days), and S Vul (P = 69 days).

Zoom In Zoom Out Reset image size

While our pipeline selection requires candidate Cepheids to have amplitudes in the characteristic range of 0.2–1 mag, we impose no requirement regarding the presence of these more subtle light-curve features, which are not detectable at the data quality typical of single extragalactic Cepheids (Hoffmann et al. 2016). The presence of these features in aggregate light curves at the appropriate periods offers an additional level of scrutiny and validation of these sources as classical Cepheids, as well as the means for a more detailed comparison to those in the MW. The impressive similarity of these light-curve sequences demonstrates the universality of the physics producing these pulsating standard candles and provides an important test of their consistency along the distance ladder. The appearance of the Hertzsprung progression among the extragalactic sample demonstrates these objects are bona fide Cepheid variables.

Following the same procedures described by R16, we NIR photometry of the Cepheids using their positions derived and fixed from the higher-resolution optical data. The NIR images of all Cepheids were reprocessed for this analysis using revised calibrations by the STScI pipeline (including new flat fields and geometric-distortion tables), and we were able to predict the expected NIR positions of the Cepheids found in the optical data with enhanced precision compared to our previous reduction (Hoffmann et al. 2016). Additional sources found in the NIR images are fit simultaneously with the Cepheids using an empirical model of the PSF derived from the solar-analog star P330E. The background calibration ¹³ and photometric uncertainties are determined from the retrieval of 100 locally placed artificial stars per Cepheid as described by R11 and R16.

In Figure 7 we show the distributions of the artificial-star measurements divided by their standard deviation, which demonstrates that they are well approximated by Gaussian distributions in magnitude space around their mean out to 3σ, a consequence of the log-normal distribution of underlying surface brightness fluctuations (primarily red giants, with larger and brighter collections increasingly rare) in the NIR. The mean difference between the mean and median of the artificial-star distributions, which vanishes for a true Gaussian, is 0.03σ. Averaged across all SN hosts, the difference between the mean and median of the magnitude distribution drops to 0.01σ or 4 mmag, showing no apparent correlation between its third moment (skewness) and the host distance. The distribution for the geometric calibrator, NGC 4258, appears similar to that of the others, with a difference between the mean and median of 0.01 mag. These measurements justify the use of Gaussian statistics in magnitude space in the calculations to follow.

Zoom In Zoom Out Reset image size

The accuracy of the background estimates owes to Cepheids being randomly superimposed on scenes, a consequence of our perspective whose local levels can be measured statistically. A caveat to this approach would be the presence of associated flux (colocated with the Cepheid), which becomes important only if it is then resolved for nearby Cepheids but not for distant ones. The level of such "associated flux" has been measured statistically from hundreds of Cepheids in M31 by Anderson & Riess (2018) to be ∼7 mmag at distances beyond a few Mpc, and is due to associated open clusters that would not be resolved at those distances. This term is explicitly included here in the background estimates in order to compensate for this effect. An additional and direct consequence of a potential miscalibration of the background, independent of the Cepheid mean flux, would be a change in the apparent light-curve amplitude. Riess et al. (2020) determined that the NIR amplitudes of Cepheids in three SN hosts are fully consistent with those in the Milky Way, yielding an independent upper limit of 0.03 mag for the possible misestimation of the background if representative. The sensitivity of H₀ to the unresolved background can be further mitigated by the use of a distant anchor with similar background as the SN Ia hosts such as NGC 4258; this will be addressed in Section 4.

In Figure 8 we provide a strong test of the background estimation by comparing the Cepheid photometry in dense (inner) and sparse (outer) regions of NGC 4258. Because the Cepheids in both fields are at the same distance from us and the metallicity gradient in NGC 4258 is small (Bresolin et al. 2016), an apparent difference of the dereddened magnitudes would be a consequence of misestimating the background in the dense field. The difference in intercept (i.e., distance) is 0.01 mag and well within the indicated errors of the means, demonstrating that Cepheid PSF photometry is accurate in the presence of the same level of crowded backgrounds seen in the SN hosts.

Zoom In Zoom Out Reset image size

In Appendix B we provide an independent validation of the PSF photometry using aperture photometry, a method that is accurate when the background is measured from the mean of pixels in concentric annuli and uncomplicated to apply, albeit less precise than the standard approach of using PSFs to model photometry. This test validates the mean PSF photometry in SN hosts to σ ≈ 0.02 mag. In that appendix we further perform an additional "null test" of the background estimates by regressing them against the distance-ladder fit residuals, finding a dependence of 0.010 ± 0.014 mag per magnitude of source background (in the sense of overestimating the background but with no significance).

In Section 7.1 we review multiple tests of Cepheid PSF photometry in addition to six strong tests of background estimates in the presence of crowded backgrounds, all of which indicate that the Cepheid measurements are accurate.

The SH0ES program uses observations in HST filters at known Cepheid phases in optical (F555W, F814W) and NIR (F160W) bands to correct for the effects of interstellar dust and the finite width in temperature of the Cepheid instability strip. We employ NIR "Wesenheit" magnitudes (Madore 1982) to deredden Cepheids throughout, defined as

$$\begin{eqnarray}&&{m}_{H}^{W}={m}_{H}-R\,({m}_{V}-{m}_{I}),\end{eqnarray} \tag{ 7 }$$

where m_H = F160W, m_V = F555W, m_I = F814W and V − I =m_V − m_I in the HST system, and R ≡ A_H /(A_V − A_I ). Wesenheit magnitudes are not conventional magnitudes, which compare the brightness of one star to another; rather, they are used to compare the brightness of one standardized candle to another through the removal of their unequal extinction, as reviewed in Appendix D. While the value of R obtained from well-characterized extinction laws for these bands is ∼0.4, we note that the correlation between Cepheid intrinsic color and luminosity at a fixed period has the same sense as extinction (cooler is fainter), and is similar in size with an intrinsic value of ∼0.6 (as discussed in Section 6.2). Therefore, the value of R derived for extinction effectively also reduces the intrinsic scatter caused by the breadth of the instability strip. We analyze the sensitivity of H₀ to values of R derived from different extinction laws in Section 6.3. In Appendix D, we discuss pitfalls associated with varying R in Equation (7) between galaxies if the intrinsic color is not first subtracted from the observed color ¹⁴ (Follin & Knox 2018; Mortsell et al. 2021).

To avoid a magnitude bias, we include only Cepheids with periods above the completeness limit of detection in our primary fit for each host (Y22b). The measurements of Cepheids in SN Ia hosts are provided in Table 2, while Table 3 summarizes the properties of the resulting NIR P–L relations. We identify a number of improvements realized here since our previous Cepheid measurements in SN hosts ∼6 yr ago (R16, H16) and for NGC 4258, ∼16 yr ago (Macri et al. 2006).

• 1.  
  Sky annuli: The size of the annulus used to estimate the level of the sky value of the region around the Cepheid (after subtracting all detected sources) in F160W was reduced in size to inner and outer radii of 024 and 08 (from 096 and 144 in R16) based on extensive simulations. This determination of the sky level is more precise, although it requires taking into account the contribution to the sky from the wings of the PSF; the resulting offset of 0.008 mag is robustly determined from bright stars and corrected in the final photometry. The random sky error is propagated to the photometry error through the sampling of sky values in the artificial-star analyses).
• 2.  
  Determination of the background and covariance: As in R16, artificial stars are added in the F160W images in the vicinity of each Cepheid at the apparent magnitude expected from its period and trial fits of the P–L relation. The difference between the input and recovered magnitudes is used to refine the initial estimate of the sky background and revise the Cepheid photometry. This process is necessarily iterative as the P–L relation used to predict the magnitude of a given Cepheid from its period is determined after correcting the photometry based on the retrieval of the artificial stars. Thanks to greater processing power, we increased the number of iterations since R16 and found that in some cases the iterations in our previous work were not adequate (as they had not fully converged). The new process fully converges with improved determination of the trial intercepts and slopes, and we use the final iteration to estimate a systematic uncertainty of 20% of the background (in units of the Cepheid magnitude correction) as the covariance "error floor" of any pair of Cepheids, j, k in the same ith host, given by

  $$\begin{eqnarray}&&{C}_{i,j,k,\mathrm{bkgd}}={0.2}^{2}({\mathrm{bkgd}}_{i,j})({\mathrm{bkgd}}_{i,k}).\end{eqnarray} \tag{ 8 }$$

  This provides a systematic uncertainty in the range of 0.03–0.06 mag for all Cepheids within each host, with the term "bkgd" representing the change in Cepheid magnitude due to the addition of the mean level of the crowded background from unresolved sources derived from the artificial stars. The artificial-star magnitudes are determined from the trial P–L relations of each host independently from every other host so there is no source of background covariance between different hosts.
• 3.  
  Reference Files: The photometry benefits from the latest STScI data pipelines, including new flat fields (with better "blob" mapping), bias frames, long-history dark frames, pixel-based CTE corrections for optical data, and geometric-distortion corrections, yielding improved alignment between the optical and NIR frames.
• 4.  
  Count Rate Nonlinearity (CRNL): We adopt a new calibration of the WFC3/NIR CRNL (Riess et al. 2019b). By convention, this is applied to Cepheids in anchors between their flux and the background level of Cepheids in SN hosts.
• 5.  
  M101 long-period Cepheids: A reexamination of the Cepheids in M101, together with simulations, revealed that the baseline of the original monitoring campaign (carried out in 2006 and reported by Mager et al. 2013) was too short to provide reliable periods for apparent Cepheids with P > 35 days (equivalent to 1.2× the time span of the observations). Two additional epochs obtained 7 yr later and separated by a week are insufficient to resolve the issue because at these periods the two epochs provide effectively only one phase measurement, and the prevailing period uncertainty of >0.5 days makes the phasing of the two sets unreliable. For that reason we exclude M101 Cepheids with P > 35 days (about 10% of the sample used by R16).
• 6.  
  We correct Cepheid periods to rest-frame values owing to (1+z) time dilation (Anderson 2019), a small (given our typical z ≈ 0.005) but one-sided effect.

We note that the Cepheid color measurements, V − I, employed in Equation (7) to determine the baseline value of H₀ here and in R16 (as well as most variants) are relatively insensitive to the previously noted improvements to the calibration of the Cepheid optical measurements realized in the last 6 or 16 years and included in Y22a and Y22b. These include the use of artificial stars to estimate the crowded backgrounds, revised flat fields, archival dark frames, updated geometric-distortion maps to rectify frames, and pixel-based CTE corrections, most of which were not implemented by H16 or by Macri et al. (2006), from which the N4258 data in H16 were derived. These improvements cancel to first order in the difference, V − I, as do changes in CCD sensitivity, which can decline on orbit or change abruptly when the electronics are refurbished as occurred for ACS in 2009. However, the use of optical Wesenheit magnitudes to determine H₀ without bias (as we explore in Section 6.13) requires fully calibrated optical magnitudes rather than only accurate optical colors (as in H16 and provided in R16). The fully calibrated optical magnitudes available in Y22a and Y22b are suitable for this purpose.

We do not attempt to quantify the impact of each of these improvements individually; however, in the aggregate, matching Cepheids within 1'' of those in R16, the net change to Cepheid photometry in F160W is that 63% (37%) of Cepheids are fainter (brighter). We provide additional details of this comparison in Appendix B.

As in R16 and H16, we measured radial gradients of the strong-line abundance ratios (R₂₃) of oxygen to hydrogen in H ii regions in the Cepheid hosts. The optical spectra were obtained with the Low-Resolution Imaging Spectrometer (LRIS; Oke et al. 1995)) on the Keck I 10 m telescope on Maunakea, Hawaii. See R16 and H16 for details regarding the observations and data reduction.

We define the metallicity of each Cepheid to be the value of this linear function at its galactocentric radius. We have revised the calibration of these strong-line abundance measurements relative to those from Zaritsky et al. (1994, hereafter Z94) used by R16, taking advantage of more recent calibrations between R₂₃ and the metallicity 12 + log [O/H]. We adopt the average of nine recent literature calibrations, with the transformations between the Z94 system and the newer ones given by Teimoorinia et al. (2021). Further details about the metallicity measurements and their uncertainties are given in Appendix C.

We also use direct abundance measurements derived from high-resolution spectra of Cepheids in the MW, LMC, and Small Magellanic Cloud (SMC). In Appendix C, we evaluate the consistency of the direct and radial (strong-line) abundance measures by comparing H ii regions and Cepheid spectral abundances in the MW. While we use the average of nine strong-line abundance calibrations for our baseline results, we select the Pettini & Pagel (2004, hereafter PP04) calibration for comparison to the mean, as it is known to provide a good match to measured extragalactic stellar abundances (Bresolin et al. 2016, and F. Bresolin, 2021, private communication), and to estimate the uncertainty of the strong-line metallicity estimates. Specifically, we propagate a systematic uncertainty in the strong-line abundance scale as well as its covariance by including in our fit covariance matrix (given in Section 3) the product of the difference between the mean calibration and the PP04 calibration for the ith and jth Cepheids,

$$\begin{eqnarray}&&{C}_{i,j,\mathrm{syst}}={Z}_{W}^{2}({[{\rm{O}}/{\rm{H}}]}_{i,\mathrm{avg}}-{[{\rm{O}}/{\rm{H}}]}_{i,\mathrm{PP}04})({[{\rm{O}}/{\rm{H}}]}_{j,\mathrm{avg}}-{[{\rm{O}}/{\rm{H}}]}_{j,\mathrm{PP}04}).\end{eqnarray} \tag{ 9 }$$

This approach requires an iteration to use the same value of Z_W in Equation (9) that is determined from the optimization of the global χ². The mean difference of ∼0.05 dex between the average and PP04 scale represents a systematic uncertainty in the abundance scale, propagated here in the covariance matrix, which is consistent with the empirical assessment in Appendix C

Trigonometric parallaxes to MW Cepheids offer a direct source of geometric calibration of their luminosities. We employ two samples with precise parallaxes and fluxes measured on the same HST system as the extragalactic Cepheids and with direct spectroscopic metallicity measurements. The first is a sample of eight MW Cepheids from Riess et al. (2018a) with parallaxes measured with HST WFC3 spatial scanning. The other contains 75 MW Cepheids with parallaxes from Gaia EDR3 as given by Riess et al. (2021). We do not use the sample from Benedict et al. (2007) that was part of R16 because their fluxes are too bright (0 < m_H < 3 mag) to be measured directly with HST nor have they been measured with good accuracy from the ground; the parallax sample from Gaia EDR3 provides superior information on MW Cepheids.

Because the distance uncertainties are measured (and Gaussian) in parallax space jointly and are not insignificant (>5%), the direct transformation of parallax to distance modulus (i.e., distance to magnitudes) would yield a bias (often called the "Lutz–Kelker" bias, following Lutz & Kelker 1973). One can compensate for it by estimating approximate statistical corrections between the parallax and distance moduli for individual Cepheids, assuming the form of their spatial distribution in the MW using Bayesian inference (Bailer-Jones et al. 2021). However, it is simpler and more reliable to analyze the MW Cepheid data directly in parallax space, in order to retain the Gaussian parallax errors, and to derive their joint constraint on the Cepheid absolute-magnitude zero-point (${M}_{H,1}^{W}$) following Riess et al. (2021), or similarly through the use of "astrometric-based luminosity" (Arenou & Luri 1999).

Given that the constraints from the parallaxes across a Cepheid sample are related by the P–L relation, the combined constraint is evaluated in parallax space and the resulting error in the mean is greatly reduced (for our two samples to 1% and 3%) so that the resulting mean constraint on the zero-point in magnitudes can then be approximated as Gaussian to better than 0.1%. We note that the 1% calibration from Gaia EDR3 includes marginalization over the Gaia parallax offset term as described in Riess et al. (2021) and that the measured σ = 6 μas uncertainty in the offset is the source of 0.9% uncertainty for the MW Cepheid sample due to its mean 650 μas parallax. These constraints are given in Table 4 for the P–L relation determined here. The constraints are not identical to those provided in Riess et al. (2021) because they employ the same P–L parameters, b and Z_W , that optimizes the global χ² for all Cepheids, not just MW Cepheids. Riess et al. (2021) showed that the global P–L parameters, including slope and metallicity dependence, are good fits for the MW Cepheids on their own.

Table 4. Ancillary Cepheid Data

Sample	Reference	N	〈P〉	〈[O/H]〉	Photometry	Selection	Notes
			[days]	[dex]
MW Gaia EDR3	Riess et al. (2021)	66	12.5	0.13	HSTmH , mV , mI	see ref.	${M}_{H,1,\mathrm{Gaia}}^{W}=-5.903,$ σGaia = 0.024 a ,
+HST							ZW = −0.20 ± 0.12
MW WFC3 SS	Riess et al. (2018a)	8	22.6	0.05	HST mH , mV , mI	see ref.	${M}_{H,1,\mathrm{HST}}^{W}=-5.810,$ σHST = 0.054 a
LMC HST	Riess et al. (2019b)	70	16.0	−0.29 b	HST mH , mV , mI	see ref.
LMC ground	Macri et al. (2015)	272	12.6	−0.29 b	ground mH , mV , mI	P > 5 days	mH transformed to 2MASS5
SMC ground	Kato et al. (2007)	145	9.9	−0.70 b	ground mH , mV , mI	P > 5 days,	mH transformed to 2MASS5
						r < 0.6°
M31 SH0ES	Li et al. (2021)	55	19.1	−0.11	HST mH , mV , mI	P > 4 d
M31 PHAT	Kodric et al. (2018)	463	10.5	0.12	HST mH , mJ	P > 4 days	mH , mJ transformed to mH , mV , mI

Notes.

^a Measured following Riess et al. (2021) with global fit P–L parameters. ^b From Romaniello et al. (2021) and Romaniello et al. (2008) and 〈[Fe/H]〉~〈[O/H]〉−0.06.

Download table as:  ASCIITypeset image

The R16 analysis used 143 Cepheids observed with HST in the maser host NGC 4258 with a geometric distance from Reid et al. (2019). With new HST imaging campaigns in four fields shown in Figure 3, we have more than tripled that sample to 443 Cepheids; details of their identification are given by Yuan et al. (2022). These Cepheids are included in Table 2.

We include in the joint constraint the sample of 70 Cepheids in the LMC measured on the same HST three-band photometric system given by R19. These have also been corrected with the best-fit planar geometry model of the LMC to be at a single distance as described by R19. The distance to the center of the LMC is given by Pietrzyński et al. (2019) from a sample of 20 DEBs as μ₀ = 18.477 ± 0.0263 mag. Romaniello et al. (2021) has obtained spectra for 68 of these variables, demonstrating that they are consistent within the errors of having a single, common abundance of [Fe/H] = −0.40 dex. Sixty percent of the variables have sufficiently measured lines to further provide a mean spectroscopic abundance of [O/H] = −0.29 ± 0.02 dex, which we adopt for the full LMC sample. LMC Cepheids are corrected for the WFC3 CRNL across the 5 dex between the Cepheid flux and the SN-host background (Riess et al. 2019a). The LMC Cepheids from HST also set the intrinsic scatter in ${m}_{H}^{W}$ owing to the finite width of the instability strip to be 0.07 mag, which we include for all Cepheids.

The LMC also provides ancillary data, as defined above, that can be used independently of their inferred zero-point to refine the characterization of the P–L relation. For this purpose we use the ground-based sample of 785 LMC Cepheids from Macri et al. (2015) with m_H , m_V , and m_I photometry limited to 270 with P > 5 days. This photometry has been transformed to the HST system as described by Equations (10)–(12) in R16; however, we assign a common, systematic uncertainty of σ_grnd = 0.10 mag to the transformed magnitudes, hence the simultaneous constraint takes the form 0 = Δzp ± σ_grnd, where Δzp is a parameter describing the difference between the ground and HST zero-points to account for possible systematics associated with ground-based observations. Because the assigned systematic uncertainty is a factor of ∼10 larger than the mean of the smaller HST LMC sample (which has no such photometric system difference), the consequence is that the ground-based sample has negligible (100× less) weight in the tie to the LMC distance; therefore, the ground sample only helps constrain the slope of the P–L relation, still an invaluable contribution.

An important development in support of the distance ladder is the recent measurement of a geometric distance to the SMC from 10 DEBs (Graczyk et al. 2020) with a precision of better than 2%. While this precision and method would make the SMC suitable as an additional anchor, two issues present limitations. The first is the considerable line-of-sight depth of the SMC, which can cause a 10% dispersion in star distances across the full SMC structure and an offset between the DEBs and the Cepheids in the SMC. Here we follow the approach of Breuval et al. (2021) to make use of a sample of SMC Cepheids by (1) correcting for the depth by adopting the same geometry of the SMC as used by Graczyk et al. (2020) to characterize the DEBs and (2) limiting the Cepheids to the inner core of the SMC, a radius of 06. This combination yields a Cepheid sample that, based on the SMC geometric model, is offset in depth by 2 mmag from the mean DEB distance (with an uncertainty of a small fraction of this) and has a modeled dispersion in depth of σ = 0.024 mag, while still providing a sample of 145 Cepheids (with P > 5 days) that can yield valuable constraints on the Cepheid metallicity term. Earlier studies (Gieren et al. 2018) were unable to employ the DEB-based distance from Graczyk et al. (2020) and did not focus on the core of the SMC (Wielgorski et al. 2017).

The other limitation is the lack of HST photometry for SMC Cepheids. Elsewhere we have been able to negate zero-point uncertainties in the distance scale by using Cepheids observed with the same instruments. However, the SMC Cepheids can still provide a powerful constraint on the Cepheid metallicity term, which is well-constrained using only differential, cross-calibrated ground-based photometry and the differential DEB distance measurement between the SMC and LMC, the latter of which is known better than the simple difference in the individual DEB cloud distances.

As discussed by Graczyk et al. (2020), most of the uncertainty in the DEB distance estimates to the LMC and SMC is systematic and propagates from the uncertainty in the surface brightness versus color calibration of red giants, the zero-point of the V-band and K-band photometry, and the uncertainty in the extinction law. Because the aforementioned SMC and LMC DEB measurements utilized the same relations, observational setup, and reduction methodologies, their differential distance from DEBs as given by Graczyk et al. (2020) is far better constrained to 0.500 ± 0.017 mag, independent of the absolute calibrations and their uncertainties in the DEB method. The use of consistently calibrated LMC and SMC ground photometry and this differential distance, even without reference to the absolute DEB distances, helps constrain the metallicity term as we will show in Section 6.2. Here we use the 2MASS Point Source Catalog (Cutri et al. 2003) to produce a consistent calibration ¹⁵ of the H-band LMC Cepheid data from Macri et al. (2015) and the H-band SMC data from Kato et al. (2007), and m_V and m_I data from OGLE III (Soszynski et al. 2008), with data provided in Table 2. The precision of these transformations allows us to fully leverage the constraint on the distance difference from Graczyk et al. (2020).

We take the mean SMC Cepheid metallicity to be Δ[Fe/H] = −0.41 dex relative to the LMC or [O/H] = −0.70 dex (Romaniello et al. 2021, 2008) ([Fe/H] = −0.76). We add to this a systematic uncertainty (common covariance) in the mean LMC and SMC metallicity of 0.05 dex, as in Equation (9), each, separately, and relative to other hosts.

In Li et al. (2021) we presented measurements of 55 Cepheids in M31 using the same three-filter HST system adopted elsewhere. Owing to their low dispersion and broad range in period, these Cepheids provide additional constraints on the slope of the P–L relation, independent of any prior knowledge of the distance to M31; this is how we employ these data for our baseline analysis. Given the M31 metallicity gradient of −0.023 dex kpc⁻¹ (Zurita & Bresolin 2012), these Cepheids span only a narrow range of abundances with a mean [O/H] = −0.1 dex and a standard deviation of 0.05 dex.

A much larger sample of M31 Cepheids is available from the HST PHAT Treasury program (Dalcanton et al. 2012), but the filters used to observe it do not correspond to the three used here, limiting its utility. The PHAT program observed these Cepheids with WFC3 F160W and a "wide-J" filter (F110W; Riess et al. 2012) defined a transformation to the ${m}_{H}^{W}$ system. In R16 we included measurements for 375 PHAT Cepheids before the availability of those from Li et al. (2021). An expanded compilation from Kodric et al. (2018) includes 522 Cepheids from the PHAT program with 3 < P < 78 days. We use the latter sample as an alternative to Li et al. (2021) in some variants of our baseline analysis in Section 6.5 because of its powerful leverage to examine evidence of a possible break in the P–L relation near P ≈ 10 days.

Figure 9 shows the 40 individual Cepheid host ${m}_{H}^{W}$ P–L relations (not including the MW as explained above) for a period range of 5 < P < 120 days fitted with a common slope. Before combining the results of many hosts in a global analysis, we examine in Figure 10 the independently fitted slopes of the ${m}_{H}^{W}$ P–L relations across all Cepheid hosts.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The slopes are all consistent (at the ∼2σ level) with a mean in the range of −3.27 to −3.30 mag dex⁻¹. The most tightly constrained slope comes from the LMC with −3.284 ±0.017 mag dex⁻¹, mostly from the ground sample. Overall, we see no evidence to reject the null hypothesis of a single slope. There have been claims in the past of a break at P ≈ 10 days, which we will consider as a variant of the primary analysis in the next section. However, the mean slopes from these data below and above P = 10 days are −3.33 ± 0.02 and −3.21 ± 0.06 mag dex⁻¹ (respectively), a difference of 1.8σ; individual hosts with the strongest constraint (LMC and the M31 PHAT sample) show slope changes at P = 10 days in opposite directions. Furthermore, formal uncertainties in these slopes are somewhat underestimated because of the uneven sampling of periods between hosts. A Monte Carlo analysis (bootstrap resampling with replacement) from hosts with the largest samples of Cepheids shows variations owing to uneven period sampling increases the formal slope uncertainty typically by 10% and up to 35% for the LMC. We find that the metallicity dependence has a negligible effect on the mean slope. Therefore, in the following we will consider a single slope for 5 < P < 120 days in our baseline analysis, but we analyze the impact of a break or limited period range on the determination of H₀ as variants of the baseline analysis.

We make use of the geometric distances to NGC 4258 from masers (Reid et al. 2019) of μ _n4258 = 29.398 ± 0.032 mag, the LMC from DEBs (Pietrzyński et al. 2019) of μ _lmc =18.477 ± 0.0263 mag, and the SMC from DEBs (Graczyk et al. 2020) in the form of Δμ _lmc−smc = 0.500 ±0.017 mag, as well as the MW Gaia EDR3 and HST spatial-scan parallaxes discussed in Section 4.1. These are all given in Table 4.

We adopt standardized SN Ia magnitudes ${m}_{B}^{0}$ from the Pantheon+ analysis (Scolnic et al. 2021; Brout et al. 2021), where the value ${m}_{B,i}^{0}$ is a measure of the maximum-light apparent B-band brightness of an SN Ia in the ith host at the time of B-band peak, corrected to the fiducial color and luminosity determined for each SN Ia from its multiband light curves and a light-curve-fitting algorithm. We use the uncertainties and covariance of ${m}_{B}^{0}$ as given by the Pantheon+ analysis. The SN Ia covariance matrix has substantial off-diagonal terms and is displayed in Figure 11. An improvement in the current analysis of calibrator SNe Ia over R16 is our use of multiple SN light-curve data sets for most calibrators, 77 sets in all for 42 SNe Ia, a mean of ∼2 independent sets per SN, reducing measurement errors (not intrinsic scatter, which is covariant among multiple samples of the same SN) by a mean factor of 1.4. These data sets are given in Scolnic et al. (2021).

Zoom In Zoom Out Reset image size

To check the preceding calculation using a different methodology, we performed Markov Chain Monte Carlo (MCMC) sampling. Our likelihood function is

$$\begin{eqnarray}&&p(q| L,C,y)\propto p(q)\ p(L,C,y| q),\end{eqnarray} \tag{ 11 }$$

where the p(L, C, y∣q) is a Gaussian likelihood based on the covariance matrix C; if C is diagonal, ${C}_{{ij}}={\sigma }_{i}^{2}{\delta }_{{ij}}$, it has the form

$$\begin{eqnarray}&&p(L,C,y| q)=\prod _{i}\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{y}}\exp \left[\displaystyle \frac{-{\left({y}_{i}-{y}_{\mathrm{fit},i}\right)}^{2}}{2{\sigma }_{y}^{2}}\right].\end{eqnarray} \tag{ 12 }$$

Because the covariance matrix C is independent of q in our setup, Equation (11) can be simplified to

$$\begin{eqnarray}&&p(q| L,C,y)\propto p(q)\exp \left(-\displaystyle \frac{1}{2}\sum _{i}{\chi }_{i}^{2}\right).\end{eqnarray} \tag{ 13 }$$

For the calculation of χ², we followed the method described in Equation (6). For numerical accuracy, we substituted C⁻¹ with an inverse of Cholesky-decomposed lower triangular matrix ${\tilde{C}}_{L}^{-1}$ (which satisfies ${\tilde{C}}_{L}{\tilde{C}}_{L}^{-1}=I$ and ${\tilde{C}}_{L}{\tilde{C}}_{L}^{T}=C$):

$$\begin{eqnarray}&&{\chi }^{2}=\left(y-L\cdot q\right){\tilde{C}}_{L}^{-1}\left(y-L\cdot q\right).\end{eqnarray} \tag{ 14 }$$

As a "neutral" prior, we chose the uniform distribution

$$\begin{eqnarray}&&{q}_{i}\sim \mathrm{Uniform}({\mu }_{i}-10{\sigma }_{i},{\mu }_{i}+10{\sigma }_{i}),\end{eqnarray} \tag{ 15 }$$

which is centered at the results from the analytical result. Its width is set to be sufficiently (20×) larger than the estimated standard deviation from linear regression σ. We find that this range is more than enough to allow the determination of median and standard deviation of the resulting population from μ and σ.

We used the Python package emcee (Foreman-Mackey et al. 2013) to perform sampling under conditions mentioned above. The initial states are uniformly distributed in the prior range (Equation (14)) for a total of 100 walkers. The convergence is monitored using emcee's recommended method to estimate the autocorrelation time τ, which employs a method originally proposed by Goodman & Weare (2010). The burn-in time is set to be 5τ to allow chains to fully converge. ¹⁶

The convergence of the sampled distribution is checked visually and by the estimated autocorrelation time value. We required the total number of chains after burn-in N to satisfy N > 100τ before any visual inspection for convergence is performed. The samples used in the final analysis are contained well within the inner region of the initial states for all parameters, indicating that our choice of prior size did not affect the final result.

The results for selected parameters are shown in Figure 14. All parameters exhibit Gaussian-like posterior probability density functions (PDFs), which indicates that the results from the linear-regression method are a good representation of the results from this project. The locations of Gaussian-equivalent percentiles (i.e., the locations at which the same relative volume is met) in our sample suggest that the samples in H₀ are more broadly distributed toward larger values (right-hand side; RHS) than smaller ones (left-hand side; LHS), a consequence of Gaussian errors in magnitudes of 5 log H₀, with the mean distances between each line ${\bar{\sigma }}_{R}=1.06$ on the RHS and ${\bar{\sigma }}_{L}=0.98$ on the LHS (toward lower H₀), similar to the analytical results.

Zoom In Zoom Out Reset image size

More than 120 million samples were used for the baseline fit to delineate the position of the 5σ confidence interval for H₀, as shown in Figure 15. The calculated values of σ, whose mean is ${\bar{\sigma }}_{\mathrm{all}}=1.02$, contains the estimated uncertainty from the linear-regression method σ_reg = 1.01, hence showing a full consistency between two methods. Using these three values (${\bar{\sigma }}_{L}$, ${\bar{\sigma }}_{\mathrm{all}}$, and σ_reg), and regarding the 50th percentile value as the most probable value for the first two cases, our results are H_0,MCMC−L = 73.04 ± 0.98, H_0,MCMC−all = 73.42 ± 1.02, and H_0,reg = 73.04 ± 1.01 km s⁻¹ Mpc⁻¹.

Zoom In Zoom Out Reset image size

The significance of the discrepancy between our results and those of Planck Collaboration et al. (2020), H_0,Planck =67.4 ±0.5, are 5.1σ, 4.9σ, and 5.0σ, respectively.

This section illustrates an approach to simultaneously measure H₀ and the expansion history for an arbitrary form of H(z), using the full covariance of the data sets.

The "near-field" determination of H₀ in the preceding section is quite insensitive to specific knowledge of the recent expansion history because the mean redshift at which the measurement is made, 〈z〉 = 0.055, is low. Yet to make the measurement more precise at even these small redshifts, in the prior section we accounted for the derivative of the expansion history in Equation (4) through the empirical derivative of H(z) measured from higher-redshift SNe Ia, q₀. We set q₀ = −0.55 as this is historically a good fit to high-redshift SNe Ia. ¹⁷ This empirical approach is fully independent of the CMB, so an independent comparison of H₀ to the CMB with ΛCDM is appropriate.

However, to consider less-conventional expansion histories such as a rapid change in H(z) together with the measured value of H₀, as may be undertaken in the effort to find a resolution of the Hubble tension, it might be necessary and it is certainly more reliable to explicitly account for the dependence of H₀ on the form of the expansion history, H(z), at low redshifts (Camarena & Marra 2021; Efstathiou 2021). Furthermore, there is covariance in the measurements of SNe Ia, whether in Cepheid hosts or at moderate redshifts (Scolnic et al. 2018; Dhawan et al. 2020), and it is necessary to account for this when SNe Ia are used simultaneously to measure H₀ and the recent expansion history.

It is important to recognize that standardized SN Ia data can provide only relative distance measurements between all SNe measured within the same standardization scheme, with SN parameters, uncertainties, and covariance with values relevant within the context of the algorithm used to standardize the SNe. Therefore, to avoid inconsistencies between SN standardization schemes or loss of knowledge of measurement covariance, one would ideally make full use of all relevant SN data simultaneously to determine absolute quantities. A straightforward and reliable path is to use the set of absolute distances to SN hosts (their uncertainties and covariance) derived from only the first two rungs without the use of any SN data, together with a consistently standardized set of SNe (in these hosts and in the Hubble flow), to determine H₀ and H(z) simultaneously, the so-called "forward" direction. Or, one could alternatively follow an "inverse" approach, starting with CMB data and using the best-fit ΛCDM model to calibrate the distance–redshift relation of SNe including the predicted distances of nearby hosts of SNe Ia and Cepheids (or TRGB). ¹⁸ Here we follow the forward approach.

For the set of SN hosts with calibrated distances (using only the first two rungs of the distance ladder), the dereddened absolute distance modulus μ₀ is given by

$$\begin{eqnarray}&&{\mu }_{0,\mathrm{host}}-{m}_{B}={M}_{B},\end{eqnarray} \tag{ 16 }$$

with terms given in Equation (3). We then have a second set of SNe with cosmological redshifts z > 0,

$$\begin{eqnarray}&&{m}_{B}=5\,\mathrm{log}\,{cz}\left\{1+\displaystyle \frac{1}{2}\left[1-{q}_{0}\right]z-\displaystyle \frac{1}{6}\left[1-{q}_{0}-3{q}_{0}^{2}+{j}_{0}\right]{z}^{2}+O({z}^{3})\right\}-5\,\mathrm{log}\,{{\rm{H}}}_{0}+{M}_{B}+25.\end{eqnarray} \tag{ 17 }$$

The two leading terms on the RHS of Equation (17) can be replaced with any empirical or cosmological model for H(z), such as the example of ΛCDM with the dark energy equation-of-state parameter w and mass density Ω_M ,

$$\begin{eqnarray}&&{m}_{B}=5\,\mathrm{log}\left[c{{\rm{H}}}_{0}^{-1}(1+z){\int }_{0}^{z}\displaystyle \frac{{dz}^{\prime} }{E(z^{\prime} )}\right]+25+{M}_{B},\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}&&E(z)\equiv {\left\{{{\rm{\Omega }}}_{m}{\left(1+z^{\prime} \right)}^{3}+(1-{{\rm{\Omega }}}_{m})\times \exp \left[+3{\int }_{0}^{\mathrm{ln}(1+z)}d\mathrm{ln}(1+z^{\prime} )(1+w(z^{\prime} ))\right]\right\}}^{1/2}.\end{eqnarray} \tag{ 19 }$$

The LHS of Equations (16) and (17) (or 16 and 18) are measured quantities, either μ_0,host or m_B , and the free parameters M_B , H₀, and q₀ (or w and Ω_M for Equation (18) instead of Equation (17)) are determined by simultaneously optimizing these.

Following this approach, the set of 37 values of μ_0,host for 42 SNe Ia given in Table 6 (or 40 values for 46 SNe Ia including the TRGB data as in Section 6.7) and their covariance matrix, derived from the first two rungs (i.e., without the use of any SN data), are taken as input. The covariance matrix of μ_0,host is shown in Figure 16. In addition, the values of m_B for the SNe in these hosts and in the Hubble flow (including their redshifts) and the covariance matrix between all SN measurements are taken as additional data and the two constraining relations are solved simultaneously.

Zoom In Zoom Out Reset image size

As an example, in Figure 17 we follow this approach, marginalizing over the SN standardization parameter, M_B , and simultaneously measuring H₀ and q₀. As expected, H₀ will be quite uncorrelated with q₀ (or other cosmological parameters) unless a cosmological model produces a much more rapid change in H(z) at z ≪ 1 than the polynomial in Equation (17). For this case we add SNe from Pantheon+ at 0.15 < z < 0.8 to the prior analysis, finding q₀ = −0.51 ±0.024 and H₀ =73.30 ± 1.04 km s⁻¹ Mpc⁻¹. The result is very similar to our baseline result, and the added freedom in the expansion history has had little impact on the uncertainty. Brout et al. (2021) give results for a Friedman–Robertson–Walker (FRW) expansion history governed by w and Ω_M . We provide the SN-independent host distances and their covariance matrix at pantheonplussh0es.github.io to allow for other forms of H(z).

Zoom In Zoom Out Reset image size

The optical selection of our Cepheid sample is discussed by H16 and Y22b. As in R16, we include only Cepheids with colors F814W–F160W within 0.8 mag of the median color in each host to remove blends with unresolved sources of comparable luminosity and different color (e.g., red giants, blue supergiants, unresolved star clusters). This is a useful criterion as it is independent of distance and period, insensitive to reddening, and anchored to the physical properties of Cepheids (i.e., stars with spectral types F–K). We still may expect a small number of outliers owing to fully blended yellow supergiants, or the sample may include a small number of objects erroneously identified as Cepheids.

Our baseline analysis removes Cepheids that deviate from the global fit at >3.3σ (Chauvenet's criterion), iteratively discarding the single largest outlier (i.e., MAD algorithm) until none remain above the threshold. The fraction of such objects is 1.2% (these outliers are available upon reasonable request). Fits 2–9 explore other rejection approaches: global (i.e., all objects above the threshold are removed as opposed to the most deviant followed by recalculation of the fit), from the individual P–L relations, and with tighter or looser thresholds (3σ or 5σ), as well as no rejection. The median H₀ of these alternatives to the baseline clipping is larger by 0.2 km s⁻¹ Mpc⁻¹. Because the global outlier removal (Fit 2) is faster to calculate and gives results within 0.15 km s⁻¹ Mpc⁻¹ of the MAD baseline, we use this approach for most other fits unless we explicitly state the use of MAD. For the number of degrees of freedom here, fits with ${\chi }_{{dof}}^{2}\gt 1.07$ are considered not good (probability to exceed >3σ), which applies to Fits 6, 7, and 9 that remove few or no Cepheid outliers.

Fits 10–15 provide the results of including the geometric distance measurement(s) of only one or at most two anchors, rather than all three. The goal here is to explore the possibility of an unexpected error in one of the external geometric distance constraints. In Figure 19 we compare the geometric distance estimate to each anchor with the value modeled using only its Cepheids and the geometric distance of the other two anchors. As shown, the external measured distances are consistent with their Cepheids at −0.2σ for NGC 4258, +0.4σ for the LMC, and −0.5σ for the MW. Their internal consistency (despite having different mean abundances) is a consequence of the Cepheid metallicity dependence.

Zoom In Zoom Out Reset image size

The consistency of the anchors is most readily seen by comparing their intercepts and metallicity as shown in Figures 20 and 21. Here we determine the intercepts from the Cepheid data table for each host, NGC 4258, LMC, SMC, and the MW using their geometric distances for an absolute measure; see also Breuval et al. (2021) for a similar analysis. It is clear that the consistency of the anchors is a direct consequence of the Cepheid metallicity dependence, which has been greatly refined since R16. In our primary analysis we find a metallicity dependence on [O/H] of −0.217 ± 0.046 mag/dex. In R16 our primary result was γ = −0.13 ± 0.07 (with variants values ranging from −0.24 to −0.08) and in R19 it was –0.17 ± 0.06. The total χ² for the three anchors in 21 is 0.18, which for a line fit leaves one remaining degree of freedom, and the likelihood to find agreement this good or better is 33% and thus nominal but not surprising.

Zoom In Zoom Out Reset image size

A number of recent developments have tightened this constraint considerably while broadening its range. The DEB distance for the SMC from Graczyk et al. (2020) as discussed in Section 4.4 provides a differential measurement between the Cepheids in the LMC and SMC, which constrains the metallicity dependence. As shown in Figure 21, this constraint alone gives γ = −0.22 ± 0.05, similar to the values and uncertainties found by Breuval et al. (2021) for [Fe/H], and falling along the line that joins the other two anchors. It is one of the strongest constraints available because it comes from the difference in the DEB distances to each Cloud, a measure that has a small uncertainty owing to calibration cancellation and that does not depend on the comparison between the LMC and the other two anchors (Graczyk et al. 2020). In addition, the constraint internal to the MW Gaia EDR3 parallaxes alone (−0.22 ± 0.09) indicates a similar value. ¹⁹ The global fit also makes use of the internal metallicity gradients in the SN hosts to constrain Z_W . Individually, these are not constraining, with a median uncertainty greater than 1 and a minimum of 0.3. However, combined, these are supportive of the results from the local galaxies albeit less constraining, yielding −0.13 ±0.11 mag dex⁻¹ by combining their independent fits with uncertainties in both axes.

Zoom In Zoom Out Reset image size

Abundance measurements for 68 of the 70 LMC Cepheids used here (Romaniello et al. 2021) show that they are consistent with a single value, and the lack of any measurable breadth in metallicities negates the ability to measure an abundance dependence internal to the LMC as claimed in prior analyses (Freedman & Madore 2011) and further discussed by Romaniello et al. (2021). Together, these developments provide a consistent result of a ∼−0.2 mag dex⁻¹ metallicity dependence in the NIR that also provides accord among the anchors and is little changed by excluding any anchor as shown in Figure 20.

It is important to note that excluding knowledge of the geometric distance to an anchor (e.g., the DEB distance to the LMC) as we do in Fits 10–15 does not exclude the Cepheids in that anchor, which remain extremely valuable (at any distance) for constraining the global properties of Cepheids. Rather, when excluding knowledge of an anchor distance, we allow that distance to become a free parameter which may subsequently be compared to the external geometric estimate. In this case, a parameter such as the slope of the P–L remains constrained by the excluded anchor Cepheids because these Cepheids are only consistent with a single distance to the host with an accurate value of their slope.

Here we explore variants in how the Cepheid color, V − I, is used in their distance determination. Our baseline analysis derives the Wesenheit parameter R (sometimes refereed to as R_H to avoid confusion with the optical reddening law parameter, R_V ) from the Fitzpatrick (1999) law with reddening parameter R_V = 3.3, assuming MW-like reddening for the sample of late-type hosts and thus R = 0.386. Fits 16, 17, and 18 (respectively) change the reddening parameter to R_V = 2.5 or adopt different laws—Cardelli et al. (1989, with R_V = 3.1) or Nataf et al. (2016, appropriate for the inner halo). The relationship between R, R_V , and the reddening laws is shown in the upper-left panel of Figure 22. Fit 19 allows the value of R for all hosts to be a common but free parameter. The value derived from MW Cepheids for R is 0.36 ± 0.04 (Riess et al. 2021) and for the full sample of Cepheids here is 0.34 ± 0.02. These fits yield a very similar value for H₀; including R as a free parameter has little impact on the uncertainty.

Zoom In Zoom Out Reset image size

If one wishes to allow for differences in the reddening law or the value of R across different hosts, it is necessary to first subtract the intrinsic color of Cepheids using their empirical period–color (P–C) relation in order to separate the component of the color that results from dust reddening; see Appendix D for details. Sandage et al. (2004) dereddened MW Cepheids to determine $\langle {(V-I)}_{0}\rangle =0.256\mathrm{log}P+0.497$. We used the LMC reddening maps from Skowron et al. (2021) to deredden the Cepheids in the catalog of Macri et al. (2015) to derive a P–C relation of $\langle {(V-I)}_{0}\rangle =0.238\mathrm{log}P+0.513$, very similar to the MW one. From the same data we derived the intrinsic P–C relation from the LMC Cepheids as above to be Δm_H =0.635(±0.021) Δ(V − I).

We use a mean relation of $\langle {(V-I)}_{0}\rangle =0.25\mathrm{log}P+0.50$ to first subtract the intrinsic color from all Cepheids using their periods, and then in Fit 20 we substitute the Wesenheit magnitudes in Equation (7) for m_H − R E(V − I) with R =0.386, which reduces H₀ by 0.1 km s⁻¹ Mpc⁻¹. In Fit 21 we allow this reddening ratio to be a free parameter with little change from the result given in Fit 19.

We can also consider different values for the reddening ratio, R, for different hosts. One might derive these as the value that optimizes the relation between colors and magnitudes within each host. However, after subtracting the intrinsic P–C relation, the small residual color span, coupled with relatively large color measurement uncertainties, does not provide any meaningful constraint on the individual values of R for a given host beyond the few nearest galaxies as demonstrated in Appendix D. In addition, Appendix D shows that determining an unbiased estimate of R requires accounting for uncertainties in both axes (in this case color and brightness with the statistical issue discussed by Tremaine et al. 2002), and failing to do so leads to large underestimates of R and its uncertainty, as seen (for example) in Perivolaropoulos & Skara (2021) and Mortsell et al. (2021). We find typical uncertainties in individual values of R to be ∼1 and thus uninformative. Likewise, Follin & Knox (2018) also concluded that such color data are "insufficient to make a completely data-driven inference on [individual] R," and they used a "wide prior" of 0.39 ± 0.1, finding distant hosts consistent with that prior and little impact on H₀. It is also crucial to recognize that the empirical reddening law is only valid for use at R_V > 2.5, as stated by Fitzpatrick et al. (2019), because sight lines with R_V < 2 are not seen among the hundreds of massive MW stars used to determine the law as shown in Figure 22 (indeed, the Rayleigh-scattering limit for absorption corresponds to R_V ≥ 1.5). Thus, there is no empirical support for R < 0.3 from these laws or from the stars (significantly, of the same B type that later become Cepheids) that inform them.

An alternative approach with better grounding is to derive individual values of R for each host based on its specific properties and the empirically determined correlation of these properties with dust attenuation. We use the empirical dust attenuation (EDA) framework from Hahn et al. (2022), which derives individual extinction laws for hosts as a function of their mass, type, and star formation rate (SFR) as determined to best match colors to SDSS galaxy observations. These individual estimates of R are given in Table 6 and shown in Figure 22, and have a mean of 0.42 and dispersion of 0.10, matching well the prior used by Follin & Knox (2018). The smallest value for any host is 0.30 for NGC 7541, demonstrating that values lower than this become unrealistic as discussed further in Appendix D. Fit 22 uses these modeled values of R from Hahn et al. (2022) and results in an increase in H₀ of 0.8 km s⁻¹ Mpc⁻¹.

It is clear why varying the reddening ratio in hosts is not effective at changing H₀. The mean value of E(V − I) in the SN hosts is 0.35 mag, as shown in Figure 22. For the Cepheids in the anchors MW (Gaia EDR3), LMC, and NGC 4258, it is 0.58, 0.23, and 0.24 mag (respectively), for a mean anchor value (weighted by the precision of the anchor distance) of 0.43 mag. Thus, the net difference in E(V − I) between anchors and SN hosts is 0.08 mag, and hence the full impact on H₀ of correcting for Cepheid reddening is 0.03 mag or ∼1.5%, with perturbations to this procedure changing the result by a smaller amount. As shown in Figure 22, the strongest empirical evidence has the characteristic value of R less than 0.1 from the baseline; hence, e.g., ΔH₀ = 0.08ΔR (mag) <0.01 mag. In Fit 23 we discard the use of color altogether, representing a reasonable assumption that the Cepheid extinction at 1.6 μm is modest and to first approximation cancels along the distance ladder, and find that H₀ goes up by 2%.

In Section 4.6 and Figure 10 we measure the slopes of the P–L relations of each host as well as the mean slope above and below P = 10 days, finding no clear evidence of a break. It is important to note that we have not included Cepheids with P < 5 days (either in the anchors or SN hosts) in our analyses. Such Cepheids could provide additional support for a break; however, Cepheids with overtone pulsations become much more common below this period, obey a different P–L relation, and may be confused with fundamental Cepheids.

However, in R16 the baseline was to allow for a break at P = 10 days, in accordance with prior claims of a break in the optical in the LMC (Sandage et al. 2004; although no evidence for one was found by R16). To allow for additional comparison to R16, in Fit 24 we allow for a break at P = 10 days with independent slopes above and below this pivot. The two slopes have a slope difference of 0.10 ± 0.05 as given in Table 5, which lowers H₀ by 0.45 km s⁻¹ Mpc⁻¹, with the difference largely driven by the ground-based samples from the LMC and SMC, whose Cepheids at lower periods have a shallower slope by 0.12 ± 0.08 and 0.34 ± 0.20 (respectively). To further explore the evidence for a break, we expand the sample of Cepheids in M31 by including those from Kodric et al. (2018, sample III) in Fit 28. Because in M31 the opposite occurs (shorter-period Cepheids have a steeper slope by 0.20 ± 0.12), the inclusion of both samples reduces the difference in slope to 0.07 ± 0.04 and H₀ is lower than the baseline by only 0.3 km s⁻¹ Mpc⁻¹. A free-form analysis of a slope change (allowing each host to have two individual slopes) reduces the sample evidence of a break to <1σ, with only three hosts (the LMC, SMC, and M31) providing any significant weight at P < 10 days (MW Cepheids with Gaia parallaxes provide no indication of a change in slope). The lack of significant evidence of a break (or change in slope) is why the baseline analysis uses the single-slope parameterization. Another option here is to exclude Cepheids with P < 10 days, which we do in Fit 25, resulting in an 0.1 unit increase in H₀.

Fits 27 and 28 exchange the Li et al. (2021) sample of M31 Cepheids for the PHAT sample from Kodric et al. (2018) with a 10 fold increase in the number of variables, but with filters that are transformed to the set of three used elsewhere rather than directly observed with them. The main value of this change is to gain further traction of a possible break at P = 10 days as discussed in the prior section.

In Fit 29 we ignore the metallicity term and find that it has little impact, with H₀ rising by 0.3 km s⁻¹ Mpc⁻¹. The metallicity term has little impact on H₀, as shown in the marginal confidence region of Figure 20. The reason can be seen in Figure 21, where the distribution of Cepheid metallicities in SN hosts matches well the range of the three anchors; thus, the metallicity term does not move the SN hosts with respect to the mean of the anchors. It does, however, provide for consistency of the Cepheid and anchor distances.

Appendix C gives further details about the Cepheid metallicity scale we adopt, the average of nine recent and well-characterized relations (Teimoorinia et al. 2021). In Fit 30, we replace the use of the mean metallicity scale with one of these systems (Pettini & Pagel 2004) ²⁰ based on [O iii] and [N ii] lines, which we show in Appendix C provides consistent metallicities for MW H ii regions with those at the same radius derived from spectra of MW Cepheids.

The TRGB offers an additional, independent rung between anchors and SN hosts (Freedman et al. 2019; Anand et al. 2021, and references therein). The inclusion of TRGB distances can in principle improve the calibration of SN Ia hosts and add to the number of such objects on the second rung. However, before including TRGB distances, it is important to determine if they are consistent with those employed here for Cepheids. ²¹ We use the TRGB samples measured by the CCHP (Freedman 2021, hereafter F21) and EDD (Anand et al. 2021) groups.

To determine the consistency of the two distance indicators, we calibrate both using the same geometric anchor, NGC 4258, whose Cepheids and TRGB were both observed with HST, making this a purely differential comparison to a set of the same SN hosts. In Figure 23 we compare the distances from each method for all seven SN Ia hosts with Cepheid and TRGB measurements, which are available from both CCHP and EDD. The mean difference between Cepheids and TRGB is −0.014 ± 0.030 and 0.002 ± 0.030 mag for EDD and CCHP, respectively (we have not included the four more-distant SN Ia hosts measured by Jang & Lee 2017 and included in F19 because Anand et al. 2021 could not identify any reliable TRGB for these, but we plot them in Figure 23), and conclude these are consistent (see Section 7.2 for further discussion). For consistency with the preceding Cepheid-only analyses, we do not include SNe Ia with TRGB measures that do not pass the SN quality cuts employed above (∣c∣ < 0.15 and ∣x1∣ < 2), which affects four objects (SNe 1989B and 1998bu with c = 0.3 corresponding to A_V ≈ 1 mag, and SN 1981D with c = 0.2, also with high reddening; SN 2007on with x1 = −2.2). Ten TRGB distances for 11 SNe Ia are thus added: the 7 hosts with Cepheid distances (and 8 SNe) in Figure 23 (M101 and NGC 1365, 1448, 4038, 4424, 4536, and 5643) and 3 additional hosts without Cepheids with 4 additional SN calibrators (SNe 1980N and 2006dd in NGC 1316, SN 2011iv in NGC 1404, and SN 1994D in NGC 4526).

Zoom In Zoom Out Reset image size

As we did for the Cepheids, we use TRGB distances where the anchor and SN hosts made use of the same telescope and instrument to negate telescope zero-points. In this case we first use Table 2 from Anand et al. (2021), which includes the calibration based on new observations of NGC 4258 from our program (GO 16198) that also employ the same filters (F606W and F814W), ACS electronics, and similar level of CTE to the SN hosts. We do not include TRGB zero-points in the Magellanic Clouds or the MW because these have not been measured directly on the HST system (owing to the impracticable area). For the joint analysis we include an additional parameter for the TRGB luminosity, M_I , which is optimized in the fit. We note that the TRGB constraint is included as available for an SN host simultaneously with the Cepheid constraint through the addition of the relation for the ith SN host,

$$\begin{eqnarray}&&{m}_{I,\mathrm{TRGB},i}={\mu }_{0,i}+{M}_{I,\mathrm{TRGB}},\end{eqnarray} \tag{ 20 }$$

and the calibrating relation

$$\begin{eqnarray}&&{m}_{I,{\rm\small{N}}4258}-{\mu }_{0,{\rm\small{N}}4258}={\rm{\Delta }}{\mu }_{{\rm\small{N}}4258}+{M}_{I,\mathrm{TRGB}},\end{eqnarray} \tag{ 21 }$$

which adds a single free parameter, M_I,TRGB.

Including TRGB (EDD), Fit 31 lowers H₀ by 0.3 units to 72.76 ± 0.95 km s⁻¹ Mpc⁻¹ and reduces the overall error by 5%, yielding a value of M_I,TRGB = −4.003 ± 0.025 mag, similar to that found by Anand et al. (2021). The TRGB SN host and Pantheon SN data yield 71.5 km s⁻¹ Mpc⁻¹ before the sample is tripled by combining with the SN-Cepheid hosts with the increase in H₀ of ∼1 km s⁻¹ Mpc⁻¹, consistent with the shot noise of each subsample as discussed in detail in Section 7.2. We note that because the EDD TRGB parameterization includes a color dependence, the value of M_I,TRGB quoted here corresponds to their fiducial, blue TRGB with V − I = 1.23 mag. For Fit 32 we replace the EDD TRGB measurements with the CCHP set as given by F19 and F21 for the same SN hosts and by Jang et al. (2021) for NGC 4258 TRGB ²² and without a color dependence to match the CCHP implementation. This reduces H₀ by 0.45 units from the EDD-based result (0.75 units below the baseline). The difference between the EDD- and CCHP-based result is a direct consequence of the 0.04 mag brighter measurement of the tip in NGC 4258 by Jang et al. (2021) compared to EDD, the significance of which is 1.9σ as further discussed in Section 7.2. The Jang et al. (2021) measurement in NGC 4258 in concert with the other constraints yields M_I,TRGB = −4.025 ± 0.023 mag for the above parameter. (We note that a direct comparison of the M_I,TRGB parameter to that observed for NGC 4258 requires adding to this Δμ_N4258 in Equation (2), which for the baseline is −0.013 mag.) The mean result from the two TRGB implementations is 72.53 ±0.95 km s⁻¹ Mpc⁻¹, which we adopt as a reference value for TRGB inclusion.

Here we explore fits utilizing different selection criteria for inclusion in the Hubble-flow sample. The goal of the selection of SNe and their hosts for the Hubble-flow sample is to match as well as possible the same criteria used to collect the calibrator sample to guard against the presence and imbalance between samples of additional host or SN characteristics currently known or unknown that may correlate with Hubble residuals. The enhanced size of the calibrator sample, now with 42 SNe Ia, reduces the likelihood of a chance imbalance of such properties.

The host selection for the baseline Hubble-flow samples requires visual identification from the best available optical imaging to be a spiral type in the full range of Sa–Sd, making them likely hosts of massive star formation in the last 0.1 Gyr—the primary criterion for our targeting them for Cepheid observations. This excludes highly inclined hosts (>75°) as these are complex targets and were not considered for finding Cepheids nearby. The host requirements are in addition to the SN Ia quality cuts, which are relatively tight in order to match the calibrators (∣c∣ ≤ 0.15, ∣x1∣ < 2), with the first observation earlier than 5 days after maximum light, distance measurement error of ≤0.2 mag, with outliers >3.5σ from the Hubble flow removed. The redshift range, as in R16, is 0.0233 < z < 0.15. We refer to this sample as "LZSPI" (low-z, spiral). In Section 7.2 we will further analyze the host properties of these samples to confirm their balance in mass, SFR, and specific star formation rate. In Appendix A we compare the colors and light-curve shapes of the SN samples, confirming the similarity of the calibrator and selected Hubble-flow sample.

In Fit 33 we expand the Hubble-flow sample by removing any limitation on the host and including all SNe that pass the Pantheon+ quality cuts (which allow ∣c∣ ≤ 0.3 and ∣x1∣ < 3), nearly doubling the sample to 482 SNe. It is noteworthy that there is no change in H₀, suggesting that after Pantheon+ standardization, SNe in different host types provide consistent distances. In Fit 34, we expand the sample to high redshift (z < 0.8) and more than 1300 SNe, which raises H₀ by 0.5 km s⁻¹ Mpc⁻¹. The use of such high-redshift SNe requires knowledge of q₀ as discussed in Sections 5.2 and 6.12. Fits 35 and 36 use these two expanded samples but exclude z < 0.06 to circumvent concerns about the presence of a hypothesized local void structure. These raise H₀ by ∼0.3 km s⁻¹ Mpc⁻¹.

Here we explore variations of the calibrator sample. A complete, volume-limited sample is desirable, as it has the simplest and most unbiased selection function. Such a sample is limited by the volume z ≤ 0.011 of suitable calibrators between 1980 and 2021 and can be selected by excluding two SNe Ia from the baseline sample at higher redshifts (SNe 1999dq and 2007A), with results given in Fit 38. Fit 39 contains the same with the added TRGB distance measures. Fit 40 limits the hosts to only high-mass galaxies, $\mathrm{log}(M/{M}_{\odot })\gt 10$.

In Section 7 we discuss tests of the Cepheid background. Here we undertake another test of the background with Fits 41 and 42, which divide the calibrator sample into halves with lower-than and higher-than-median background. The difference in H₀ is 0.0 km s⁻¹ Mpc⁻¹, which is less than the independent shot noise of each half (1.3% or 0.9 units for each), and thus there is no indication of a difference in H₀ for high and low backgrounds. Fits 43 and 44 respectively use only the same 19 SNe from R16 or only the 23 SNe Ia added here, yielding a difference of 0.4 units. Fit 45 includes only the nearer half, defined from SNe as having m_B < 13 mag (the median of the sample, corresponding to D < 28 Mpc), as nearer hosts offer greater spatial resolution, lowering H₀ by 0.1 units.

The SN standardized magnitudes have been drawn from the Pantheon+ sample, which is based on more than a dozen past SN surveys. The Pantheon+ sample recalibrates each survey photometrically to a common reference using the standard-star measurements in the fields of each SN to negate the impact of survey calibration errors. In addition, most SNe observed in the Cepheid host sample are matched by SNe in the Hubble-flow sample observed by the same SN survey with no one survey having a dominant share. As demonstrated by Brownsberger et al. (2021), "gray" photometric survey errors strongly and beneficially cancel by populating both samples with SNe from the same surveys. Brownsberger et al. (2021) find that survey miscalibration and incomplete cancellation would affect the H₀ measured from the survey mix used here at σ = 0.15 km s⁻¹ Mpc⁻¹ even for extremely large survey zero-point errors of ∼0.1 mag, and more likely 0.06 units for realistic survey calibration errors of 0.025 mag.

To further explore the sensitivity of H₀ to survey errors, Fits 46–54 present the results excluding each of the major surveys contributing to the sample. While none of these exclusions change the baseline H₀ by more than ∼0.3 units, we note that excluding the CSP sample raises H₀ by 0.3 units, with most of this (0.2) seen in the change in the Hubble intercept (a_B ). Using a Hubble-flow sample exclusively from the CSP (while using a mix of surveys for the calibrator sample) yields a lower value of H₀ by 0.5 units. While this is consistent with the sample shot noise, it will reduce H₀ for the CCHP results, which use only the CSP sample for the Hubble flow. Brownsberger et al. (2021) find that this sample asymmetry is expected to produce errors of 0.8 units for realistic calibration errors of 0.025 mag; it is discussed further in Section 7.2.

Fit 55 changes the way the intrinsic scatter of SN colors is modeled from an empirical approach that includes both a component intrinsic to SNe Ia and another due to host dust as given by Brout & Scolnic (2021) and is further described in the Pantheon+ sample (Brout et al. 2021; Scolnic et al. 2021). It is a simpler description, where the intrinsic scatter of SNe Ia is monochromatic (i.e., dispersion only in the luminosity, not the color) from Guy et al. (2010) and used in the JLA analysis (Betoule et al. 2014) and in the first Pantheon compilation (Scolnic et al. 2018). This fit raises H₀ by 0.3 units. The calibrator sample has no preference for either method, yielding similar dispersion between SN and Cepheid distances.

Here we provide variations related to values of the redshifts as implemented in Equation (4), which benefit from the combined and improved values from Pantheon+ (Carr et al. 2021). Peterson et al. (2021) provide a comprehensive overview and comparative analyses of various predictions of empirical cosmic flows (or peculiar velocities). They found important improvements to the Hubble-flow residuals by (1) replacing SN-host redshifts with their host-galaxy group redshift (when available), and (2) using local density maps to account for motions induced by local gravity. The latter are provided by constrained realizations of the peculiar-velocity field by Carrick et al. (2015), Said et al. (2020), and Lilow & Nusser (2021) based on 2M++ (Lavaux & Hudson 2011) and 2MRS (Huchra et al. 2012; Macri et al. 2019), respectively. The baseline included both the group-redshift replacement and the 2M++ corrections from Peterson et al. (2021). Fit 56 exchanges the 2M++ values for those from 2MRS, which provide a comparable improvement in residuals and lower H₀ by 0.05 units. A noteworthy change is seen in Fit 57, which forgoes the flow corrections and leaves the redshifts in the CMB frame, reducing H₀ by 0.5 units. However, as shown by Peterson et al. (2021) for 585 SNe Ia with z < 0.08, the tightening of the Hubble diagram (from σ = 0.17 mag to <0.15 mag, or a decrease in χ² of 100) gives evidence in favor of these corrections which is too strong to ignore. Further, the increase in H₀ that comes with the decrease in residuals runs counter to the hypothesis that the SN sample lives inside a large-scale void that artificially raises the local value of H₀ (Kenworthy et al. 2019).

While peculiar flows cause a small perturbation in H₀ measured from SNe Ia at 0.0233 < z < 0.15, they would produce a greater uncertainty if we forgo the use of SNe and measure H₀ directly from only the first two rungs—that is, from the Cepheid-host redshifts (which are not used in the three-rung distance ladder). The baseline sample host redshifts have a mean of cz = 2000 km s⁻¹, with many <1000 km s⁻¹. Elsewhere (W. D. Kenworthy et al. 2022, in preparation), we present an analysis of H₀ from this two-rung ladder, which importantly accounts for the spatial covariance of the local peculiar flows, largely limiting the available precision from this route to 3%–4% and demonstrating the value of SNe Ia for the third rung for measuring H₀.

Fits 58 and 59 raise q₀ from −0.55 to −0.52 (equivalent to raising Ω_M in flat ΛCDM from 0.30 to 0.32) for either the local sample of spiral hosts or the sample with all hosts and for z < 0.8, with little impact on H₀ relative to these samples at q₀ = −0.55. In Section 5 we considered a free-form fit for H(z) using q₀ as a free parameter simultaneous to the determination of H₀.

Fits 60–67 use an optical-only Wesenheit magnitude, substituting for Equation (7), m^W _I = m_I − R(V − I), and thus discarding the NIR observations. The Fitzpatrick (1999) reddening law with R_V = 3.3 yields R = 1.19 in the HST passband system (m_V = F555W, m_I = F814W). The optical Wesenheit has the advantage of lower "sky" backgrounds (and their fluctuations) but the disadvantage of higher reddening (and sensitivity to the form of the reddening law). The baseline fit with the optical Wesenheit yields 72.70 ± 1.07 km s⁻¹ Mpc⁻¹, similar to the baseline fit. However, the optical Wesenheit is somewhat noisier when compared to the SN distances with a relative dispersion of 0.16 mag (versus ∼0.13 mag with the NIR data). We also see larger variations in the anchors and the color variants as seen in Figure 18, with the scatter among optical-based variants that is three times greater than the NIR-based results and comparable to the statistical uncertainties. This illustrates the rationale by the SH0ES program for pursuing NIR observations for Cepheids.

Both of these differences are expected consequences of variations in the reddening law in the optical. For example, for hosts whose Cepheids have a mean E(V − I) = 0.4 mag, a difference between a Fitzpatrick (1999) reddening law with R_V = 2.5 and R_V = 3.3 causes a difference in distance of only 0.01 mag for the NIR Wesenheit but 0.09 mag for the optical one, which can explain the aforementioned noise. In R16 we concluded that future improvements must rely on NIR data until additional studies of variations in reddening laws in the optical were available. The situation has not improved in that regard. While optical-only Wesenheit data have yielded similar values of H₀ (Freedman et al. 2012; Riess et al. 2016), their larger systematic uncertainties make them unsuitable to pursue the percent-level determination of H₀ we approach, and their further analysis is not pursued here. Nevertheless, none of the optical Wesenheit fits result in a noteworthy change to H₀.

Our baseline determination of H₀ lies 0.2 km s⁻¹ Mpc⁻¹ (20% of the uncertainty) below the median of all analysis variants, indicating that it is a good proxy for the set. In R16 we measured the dispersion of 23 variants and identified that as a systematic error. In this analysis we have moved previous sources of systematic uncertainty into the covariance matrix to include them formally, and thus most of the variants presented here were intended to gauge sensitivities in the analysis (e.g., excluding a data source) rather than true uncertainties. Nevertheless, we measure the dispersion of the NIR variants (see Figure 18) around a 3σ-clipped mean to be 0.3 km s⁻¹ Mpc⁻¹ and conservatively add this in quadrature as characterizing additional systematic uncertainties to yield a full uncertainty in H₀ of 1.04 km s⁻¹ Mpc⁻¹, or 1.4%.

None of the variants appear to offer a particularly promising route to solving the Hubble tension, with none shifting the value of H₀ much below the full error interval. The lowest value of H₀ comes from Fit 65, 71.93 ± 1.19 km s⁻¹ Mpc⁻¹, from discarding the NIR Cepheid data and two anchors (leaving only the Milky Way); the highest is from Fit 22, 74.85 ± 2.33 km s⁻¹ Mpc⁻¹ and comes from discarding the use of Cepheid colors to account for extinction. Both of these fits represent suboptimal accounting of dust.

Table 7 and Figure 24 present the full error budget derived (in approximate form) from the global fit and in comparison to prior analyses from the SH0ES team. The fractional reduction presented here is the largest seen since that between Riess et al. (2011) and Riess et al. (2016).

Zoom In Zoom Out Reset image size

Table 7.  H₀ Error Budgets (%), Terms Approximated from the Global Fit

Term	Description	Riess+ (2016)			Riess+ (2019)			This work
		LMC	MW	4258	LMC	MW	4258	LMC	MW	4258
σμ,anchor	Anchor distance	2.1	2.1	2.6	1.2	1.5	2.6	1.2	1.0 a	1.5 b
σPL,anchor	Mean of P–L in anchor	0.1	...	1.5	0.4	...	1.5	0.4	...	1.0
Rσλ,1,2	zero-points, anchor-to-hosts	1.4	1.4	0.0	0.1	0.7	0.0	0.1	0.1 a	0.0
σZ	Cepheid metallicity, anchor-hosts	0.8	0.2	0.2	0.9	0.2	0.2	0.5	0.15	0.15
	subtotal per anchor	2.6	2.5	3.0	1.5	1.7	3.0	1.4	1.0	1.8
		$\underbrace{\,}$			$\underbrace{\,}$			$\underbrace{\,}$
All Anchor subtotal			1.6			1.0			0.7
${\sigma }_{\mathrm{PL}}/\sqrt{n}$	Mean of P–L in SN Ia hosts		0.4			0.4			0.4
${\sigma }_{\mathrm{SN}}/\sqrt{n}$	Mean of SN Ia calibrators (# SN)	1.3 (19)			1.3 (19)			0.9 (42−46)
σm−z	SN Ia m–z relation		0.4			0.4			0.4
σPL	P–L slope, Δ log P, anchor-hosts		0.6			0.3			0.3
statistical error, ${\sigma }_{{{\rm{H}}}_{0}}$			2.2			1.8			1.3
Analysis systematics c			0.8			0.6			0.3
Total uncertainty on ${\sigma }_{{{\rm{H}}}_{0}}$ [%]			2.4			1.9			1.35

Notes.

^a Riess et al. (2021). ^b Reid et al. (2019). ^c Uncertainties labeled in past analyses as "systematics" related to the metallicity scale, Cepheid background/crowding corrections, and SN systematics are formally included here in the covariance matrix in Figure 11 and thus propagate there as part of the complete uncertainty. Following past work, we measure the remaining systematic errors as the standard deviation of analysis variants presented in each work as in the dispersion in Figure 18 and as discussed in Section 6.14. All terms here are approximations derived from the global fit.

Download table as:  ASCIITypeset image

The accuracy of Cepheid photometry is important to the measurement of H₀. The dependence of photometry on calibration has been negated by the use of the same photometric system throughout the above measurements. Here we review a number of tests of the relative accuracy for extragalactic Cepheids.

• 1.  
  Replication of PSF photometry by others: Measurements of NIR PSF Cepheid photometry using the same raw pixels but different software and methods have been directly compared to those used here by Javanmardi et al. (2021) in NGC 5584 and by Yuan et al. (2021) in NGC 4051, both finding good agreement with reported differences in distance moduli of 0.024 ± 0.046 and 0.00 ± 0.04 mag (respectively). ²³ While this is short of a full replication of all hosts, it is sufficient to exclude a large methodological error in Cepheid photometry as a primary source of the ∼0.2 mag H₀ tension.
• 2.  
  Replication of PSF photometry with apertures: In Appendix B we use aperture photometry, an independent method that is simple, highly reproducible, and accurate, albeit less precise than the standard approach of using PSFs to model photometry in crowded fields. For this validation, we employ aperture photometry for which the background is measured from the mean pixel value (not the mode of the background pixels) in an annulus centered on the Cepheids. This approach does not depend on artificial-star measurements to determine the variable background because the pixels in the annulus include the mean source contribution to the background. We compare the photometry and find good agreement in their means with a difference (PSF minus aperture) of 0.008 ±0.010 mag for the Cepheids in SN Ia hosts and 0.002 ±0.030 mag for the Cepheids in NGC 4258.

There are a number of strong tests of the accuracy of the background estimates presented here.

• 1.  
  The Cepheids in NGC 4258 have a similar mean level of crowded backgrounds as in the SN Ia hosts (see Figure B2), almost fully negating a systematic underestimate or overestimate of the background on the determination of H₀ when NGC 4258 is the sole anchor. The crowded background is similar because although NGC 4258 is three to four times closer than the mean SN host, its Cepheids have been mined from fields that are closer to the dense center by a similar factor. This results in H₀ = 72.51 ±1.54 km s⁻¹ Mpc⁻¹, similar to the baseline result.
• 2.  
  In Figure 8, we compare the P–L relations of Cepheids in the dense, inner (high background) and sparse, outer region (low background) of NGC 4258, finding a negligible difference of 0.01 mag.
• 3.  
  In Appendix B we regress the background with the distance-ladder fit residuals and find a dependence of 0.010 ± 0.014 mag per magnitude of source background (in the sense of overestimating the background and consistent with no misestimate). The background misestimate trends required to explain the tension are strongly excluded as shown in Figure B2.
• 4.  
  The optical background is nearly an order of magnitude smaller than that in the NIR owing to the higher resolution, smaller pixels, and lower flux from red giants. The baseline results are consistent with those from the optical Wesenheit, which do not include the NIR data, H₀ = 72.70 ± 1.30 km s⁻¹ Mpc⁻¹.
• 5.  
  There is no significant trend with distance and distance difference between Cepheids and SNe Ia (zero at <1.5σ); see Figures 12 and 13.
• 6.  
  Splitting the calibrator sample by background (Fits 41 and 42) yields no difference in H₀. Splitting in distance (determined by the SN Ia, Fit 45) yields a difference of 0.1 units. Both are consistent with no trend based on the shot noise of half the sample (1.0 units).

Finally, external to this paper, an additional and unavoidable consequence of the miscalibration of the background, independent of Cepheid mean flux, would be a change in apparent light-curve amplitude. Riess et al. (2020) compared the NIR amplitudes of Cepheids in three SN hosts and in the MW, found them to be consistent, and provided a quantitative limit of any misestimate of background to be 0.03 mag.

In Figure 23 we presented a comparison of distances measured with Cepheids and TRGB to seven SN Ia hosts—the set that allows for a purely differential and direct comparison by employing the same geometric calibration source (NGC 4258), with data in both host and calibrator obtained with the same telescope (HST) and setup to negate zero-point and geometric calibration errors. This comparison further employed the two most widely used methods for measuring the TRGB, edge-detection (F19) and luminosity-function fit (EDD). These Cepheids and TRGB measures are consistent with each other, with a mean difference of −0.002 ± 0.03 mag (CCHP; Freedman 2021) and −0.014 ±0.03 mag (EDD; Anand et al. 2021). With no mean difference between methods as a starting point, we explore the broader question of the sources of the difference in the value of H₀ from distance ladders measured through the use of either method.

In Table 8 we expand on the results from Anand et al. (2021), who compared the value of H₀ from EDD, 71.5 km s⁻¹ Mpc⁻¹, and CCHP, 69.8 km s⁻¹ Mpc⁻¹, derived using the same images to measure the TRGB and the same geometric calibration source, NGC 4258. For consistency with the comparison presented by Anand et al. (2021) of EDD and CCHP, which was provided in units of magnitudes (for differences in 5 log ΔH₀), we retain this unit below and in Table 8 in which a difference of 0.03 mag corresponds to a difference of ∼1.0 km s⁻¹ Mpc⁻¹.

The largest contribution (0.04 mag of the full 0.05 mag difference) arises from the difference in the calibration of the TRGB measured in NGC 4258 applicable to (i.e., in relation to) the mean SN host, where CCHP derived M_I,TRGB = −4.05 mag for a blue TRGB and EDD derived M_I,TRGB = −4.01 mag, color-corrected to the fiducial, blue TRGB with F606W − F814W =1.2 mag. Although additional sources in the LMC and MW have been used to support the TRGB calibration by F21, we use NGC 4258 as the reference here because it is the only source where the TRGB calibration is available directly on the HST system measured in a manner consistent with the TRGB in SN hosts. ²⁴ The difference in the EDD and CCHP measurement of the tip in NGC 4258 is persistent, having been found for two different fields of NGC 4258, being significant at the ∼2σ level and is readily evident in Figure 3 of Anand et al. (2021), where the apparent location of the TRGB edge of m_F814W =25.372 ± 0.014 mag given by Jang et al. (2021) appears much brighter than the edge highlighted at 25.43 ± 0.025 mag. A comparison of each group's color–magnitude diagrams (i.e., photometry) might identify the cause of the difference, but these data are only available from Anand et al. (2021). Nevertheless, we offer the option for either TRGB result in Fits 31 and 32.

The second half of Table 8 provides the sources of difference between the EDD TRGB analysis (based on 16 SNe Ia) for which H₀ = 71.5 km s⁻¹ Mpc⁻¹ and our baseline analysis with TRGB (based on 46 SNe Ia), starting with the same geometric calibrator (NGC 4258). The primary change is in the makeup of the Cepheid and TRGB study calibrator samples with 34 SNe Ia added here and 4 subtracted, a net change of 30 objects producing a net increase in H₀ of 0.03 mag or 1 km s⁻¹ Mpc⁻¹. The first three rows break out changes related to calibrators available from both TRGB analyses, which together lower H₀ by 0.03 mag. The first change is due to the addition of SN 2021pit, which recently appeared in NGC 1448 and is added here. SN 2007on has a light-curve shape (x1 = −2.2), which falls outside the quality range of ∣x1∣ < 2 imposed in our baseline analysis and is excluded. There are three SNe Ia (SNe 1981D, 1989B, and 1998bu) used by CCHP and EDD that are redder (c > 0.15 and A_V > 0.5 mag) than our baseline quality range cut and have not been included in this or any past SH0ES analyses. The next row adds 33 SNe Ia in 30 Cepheid hosts, tripling the sample from the preceding line's 13 SNe Ia in 10 hosts (with EDD and CCHP TRGB measurements and not excluded by SN quality cuts) to 46 SNe Ia in 40 hosts (for Fit 30, which includes TRGB-only hosts). This step raises H₀ by 0.06 mag or 2 km s⁻¹ Mpc⁻¹. This difference is fully seen internal to the set of Cepheid-only SN Ia calibrations (i.e., independent of absolute anchors or TRGB distances) in the baseline fit as shown in Figure 25, which compares the two sets of SNe as measured only by Cepheids. The 9 SNe in 7 EDD TRGB hosts are brighter than the 33 SNe Ia in 30 hosts without EDD and CCHP TRGB by 0.08 ± 0.05 mag as measured by Cepheids, which is consistent at 1.6σ with the sampling noise of the two sets (0.04 mag and 0.02 mag random error, respectively). As seen in Fits 38–45, this difference does not correlate with potentially relevant changes to the calibrator sample including splitting in distance, background level, or newness of the measurements, so we conclude that this difference of 1.6σ is a not-unexpected result of the increase in sample statistics.

Zoom In Zoom Out Reset image size

Combining all of the additions and subtractions to the calibrator sample, the net change between the calibrator samples with both TRGB measures versus the baseline+TRGB here is 0.03 mag and the combined result of all membership changes is less than 1σ from the shot noise of the two samples. We note that if we use only the (average) TRGB results of the 13 good-quality SNe, discarding 33 good-quality SNe with only Cepheid measurements, we get 71.0 ± 2.5 km s⁻¹ Mpc⁻¹ as expected (near the mean of CCHP and EDD), but this uses less than a third of the SN sample and has a higher shot noise, while still consistent with the baseline.

As the sample size from both methods increases, we would expect them to regress to an increasingly similar mean. Including the two other anchors (MW and LMC) raises H₀ by 0.02 mag (and results in M_I,TRGB = −4.00 ± 0.024 mag), though each of the three anchors is consistent with the others at the 0.5σ level, as shown in Figure 19.

Finally, there are two developments based on recent analyses of SNe in the Hubble flow, which yield an increase in the value of H₀ relative to the CCHP TRGB analysis in F19 and F21, independent of the calibrator sample. The CCHP measurement relies on a calibrator SN sample from several SN surveys, but selects a Hubble-flow SN sample from only the CSP SN survey. Regardless of whether the CSP survey is better calibrated than others in an absolute sense, Brownsberger et al. (2021) show that the use of similar surveys for both SN samples reduces errors in H₀ arising from survey miscalibration to 0.1 km s⁻¹ Mpc⁻¹ owing to error cancellation, but to only 0.8 km s⁻¹ Mpc⁻¹ for the CCHP reliance on one survey compared to many. Scolnic et al. (2021) find that the CSP sample measures fainter compared to the mean of all other surveys by 0.025 mag, which matches what we find when we remove CSP as shown by Fit 47 in Table 5, where H₀ then increases by 0.4 km s⁻¹ Mpc⁻¹. With only 5 of 19 calibrators in F19 observed with the CSP survey, the incomplete cancellation of this survey difference would be 0.6 km s⁻¹ Mpc⁻¹. In addition, the CCHP analysis of SNe Ia does not account for cosmic flows expected from local density maps, the best of which (2M++ or 2MRS) as shown by Peterson et al. (2021) reduces the Hubble diagram χ² for ∼500 SNe Ia by >50 units, decreases the scatter, and raises H₀ by ∼0.5 km s⁻¹ Mpc⁻¹. These two terms at the bottom of Table 8 should be included in a direct comparison between the results here and those from CCHP. The combination of both balancing SN survey errors in both samples and accounting for peculiar velocities would raise the CCHP value by 1.1 units and is hard to ignore. As expected, if we use only CSP for the Hubble flow and also do not account for cosmic flows, the baseline H₀ goes down by 1.5 km s⁻¹ Mpc⁻¹.

To summarize, we find that TRGB and Cepheids give consistent distances when using the same anchors and consistent procedures, so that comparisons are meaningful. There are two main sources of the difference in H₀ between the CCHP implementation in F21 and the baseline here. (1) As given by Anand et al. (2021), a (net) difference of 0.04 mag between the EDD and CCHP implementation of TRGB in F21 arises from the difference in the apparent location of the tip in NGC 4258 and is the source of the difference between H₀ = 69.8 and 71.5 km s⁻¹ Mpc⁻¹ (i.e., TRGB-only results) in these two studies. (2) The other, corresponding to a net increase of 1 km s⁻¹ Mpc⁻¹ (0.03 mag) between the EDD TRGB and the baseline, is due to differences in the SN Ia calibrator samples including a reduction of 1 km s⁻¹ Mpc⁻¹ owing to our exclusion of 4 SNe Ia that fail quality cuts and an increase of 2 km s⁻¹ Mpc⁻¹ owing to the fainter mean seen for 30 SNe in 33 Cepheid-only hosts compared to 9 SNe in 7 TRGB+Cepheid hosts, a difference seen internal to Cepheid measurements and consistent with a statistical fluctuation due to the combined changes at the 1σ level (see Figure 25). We expect that additional TRGB data will result in these distributions agreeing, as we see no reason for a difference besides shot noise. Relative to F21, the sample difference change is 1.3 km s⁻¹ Mpc⁻¹ (0.04 mag) with other differences from the baseline owing to the absence of an accounting for peculiar flows as described by Peterson et al. (2021) and not including SNe from multiple surveys in the Hubble flow to cancel zero-point differences among the calibrator surveys as described by Brownsberger et al. (2021). Combined, these raise H₀ by ∼1.1 km s⁻¹ Mpc⁻¹, and we see no reason not to include these in an SN Ia–derived measurement of H₀ given their strong empirical support. The combined Fit 30 yields 72.76 ± 0.95 km s⁻¹ Mpc⁻¹ for EDD and 72.29 ±0.94 km s⁻¹ Mpc⁻¹ for CCHP, and we cite the mean of the two (72.53 ± 0.95 km s⁻¹ Mpc⁻¹) as representative of the combination of Cepheids and TRGB.

In Appendix A we show that the calibrator SNe Ia are spectroscopically all prototypical and photometrically have a distribution of light-curve shapes and colors that are well matched to the selected Hubble-flow sample. The use of tighter-than-typical quality cuts (∣c∣ < 0.15, ∣x1∣ < 2) ensures that the sample comparison is insensitive to the standardization method, as it makes little difference in the sample means.

In R11 the Hubble-flow sample consisted of SNe Ia without limitations placed on the properties of their hosts (with about two-thirds coming from spirals). Rigault et al. (2015) suggested that the calibrator sample, all with spiral hosts and thus greater mean SFR, could introduce a bias in H₀ if Hubble-flow residuals (even after accounting for an empirical host-mass dependence as done by R11 and R16) presented a residual correlation with host SFR (either the global rate of the host or local to the SN). R16 addressed this sample difference by including as an analysis variant a Hubble-flow sample composed of only spiral or globally star-forming hosts, a precaution (regardless of whether such a correlation exists) conservatively adopted here as the baseline. Because the size of the Hubble-flow sample is so much larger than the calibrator sample, it is sensible to cull the former to match the selection of the latter to control even yet-undiscovered systematics with little cost to the precision of H₀. It is also important to recognize that the size of a correlation of Hubble residuals with a host property depends on the method of SN standardization and the SN sample host selection and that it is quite possible for a specific combination of method and sample to show a significant correlation that does not exist for different methods and samples.

Jones et al. (2018) measured the correlation of the local and global SFR and specific SFR (sSFR) with the Hubble residuals from Pantheon SNe (Scolnic et al. 2018) used by R16 and found little or no significant correlation with implied corrections (if significant) at the 0.3 km s⁻¹ Mpc⁻¹ level. Here we use two SN standardization methodologies from Pantheon+, one of which has no correlation with host mass (Brout & Scolnic 2021) in the baseline, and Fit 55, which has a 0.045 mag step at $\mathrm{log}(M/{M}_{\odot })=10$, increasing H₀ by 0.3 km s⁻¹ Mpc⁻¹.

In Table 9 we compare the global properties of galaxies in the calibrator and Hubble-flow samples including mean mass, SFR, and SSFR. The baseline Hubble-flow and calibrator samples have the same mean SFR (0.3), consistent with their matched selection (the mean SFR of the early-type hosts in Pantheon+ is −0.6). The mean mass is also similar, with the calibrators higher by 0.5 dex—a difference smaller than the dispersion of either sample. In Appendix A we compare the distributions of host masses for various samples. To produce a late-type Hubble-flow sample with mass exceeding the calibrator sample, Fit 37 limits the LZSPI to $\mathrm{log}(M/{M}_{\odot })\gt 10$ and lowers H₀ by 0.3 km s⁻¹ Mpc⁻¹. We conclude that the calibrator and baseline Hubble-flow samples are well matched in mass and SFR.

Table 9. Host Properties of SN Ia Samples

Sample	N	Log(M/M⊙)	Log SFR	Log sSFR
Baseline Calibrator	37	10.3 SD = 0.6	0.3 SD = 0.5	−9.8 SD = 0.7
LZSPI HF	276	9.8 SD = 0.8	0.3 SD = 0.7	−9.3 SD = 0.4
LZBRD HF	482	10.1 SD = 0.8	0.1 SD = 0.9	−9.6 SD = 0.7
HZBRD HF	1354	N/A	N/A	N/A
LZSPI HF $\mathrm{log}(M/{M}_{\odot })\gt 10$	132	10.4 SD = 0.5	0.4 SD = 0.7	−9.7 SD = 0.6

Download table as:  ASCIITypeset image

We would not expect any other host properties, especially any local to the SN that was not a selection criterion, to significantly differ between samples. For example, Anderson et al. (2015) measured the relative strength of Hα at the sites of 98 SNe Ia in exclusively late-type, star-forming hosts, including by chance 20 of the 38 selected for Cepheid measurements, and the fraction with detected local Hα is similar for calibrators and Hubble-flow hosts (30% versus 45%). Indeed, it would be very hard to understand how a difference in local SN-host properties could occur between the two sets of hosts with matched selection. While additional host or SN properties beyond the ones we have used to measure H₀ may be used to 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 H₀.

Our baseline determination of H₀ is 73.04 ±1.04 km s⁻¹ Mpc⁻¹ (with systematics), which exceeds the Planck+ΛCDM result by 5σ. It may be of interest and it is straightforward to calculate a combined value of H₀ 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 ±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 H₀ = 72.1 ± 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 ± 0.025 mag for the mean of CCHP and EDD) and used there (M_I = −4.03 mag) yields H₀ = 74.0 ±3.0 km s⁻¹ Mpc⁻¹. The combination of these independent and independently consistent measures gives H₀ = 72.61 ±0.89 km s⁻¹ Mpc⁻¹ (or H₀ = 72.42 ± 0.89 km s⁻¹ Mpc⁻¹ with the CCHP TRGB in Fit 32 and H₀ = 72.80 ±0.89 km s⁻¹ Mpc⁻¹ from the EDD TRGB with Fit 31), a local determination with 1.2% precision, which is also 5.2σ greater than the Planck+ΛCDM 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.

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 H₀ 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 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 ²⁶ of the characteristic Si ii λ 6355 absorption feature, and (ii) an S/N of at least 10 per pixel. 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 λ 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. 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 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 Δm₁₅ 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 Δm₁₅ 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 Δm₁₅ 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 Δm₁₅ value, in Figure A1.

Zoom In Zoom Out Reset image size

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.

Zoom In Zoom Out Reset image size

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 (011) 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 H₀ 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 SH0ES 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. ²⁹ Figure B1 displays a comparison of the aperture and PSF photometry for the Cepheids as a function of PSF magnitude, showing agreement within the sample mean errors.

Zoom In Zoom Out Reset image size

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.

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 H₀). 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 σ = 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 H₀, 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 H₀ 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 H₀ 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 H₀ for the LMC.

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 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 ± 0.014 mag per mag of background in the sense of overestimating the background by 0.006 ± 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 H₀ 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).

Zoom In Zoom Out Reset image size

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 H₀.

Because the value of R is tightly constrained by the sample of MW Cepheids to 0.363 ± 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 ∝ 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.

Zoom In Zoom Out Reset image size

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 properties used to determine R above but now including the host inclination. These values range from A_H = 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_H 〉= 0.14 mag, indicating that the level of extinction inferred from the Cepheid reddening is the expected amount. ³⁰