The Hubble Constant from Infrared Surface Brightness Fluctuation Distances^*

Our sample comprises 63 galaxies with WFC3/IR imaging in the F110W filter that we have used to measure SBF distances based on the calibration by Jensen et al. (2015). The majority of the observations come from a program (GO-14219) to obtain SBF distances to all galaxies in the MASSIVE survey priority sample (Ma et al. 2014) out to 6000 km s⁻¹. MASSIVE is a volume-limited survey to study the structure, stellar populations, internal dynamics, and central black holes of the most massive early-type galaxies within 100 Mpc (e.g., Greene et al. 2015; Veale et al. 2018; Ene et al. 2020; Liepold et al. 2020). Uncertainties in supermassive black hole masses for nearby galaxies are often dominated by distance errors (Kormendy & Ho 2013). Recognizing this, McConnell & Ma (2013) used updated distances for 44 galaxies in their black hole mass compilation, 41 of which were SBF values. Goullaud et al. (2018) present an analysis of the surface brightness profiles for the galaxies targeted in GO-14219.

The 6000 km s⁻¹ limit (∼80 Mpc) for the initial WFC3/IR SBF sample was chosen as the point where the typical peculiar velocities of 300 km s⁻¹ drop below 5% of the Hubble velocity, meaning that the relative distance error from redshifts is comparable to the expected SBF distance error. In addition, we calculated that in F110W we could reach far enough along the globular cluster luminosity function (GCLF) at this distance to reduce the contamination in the SBF power spectrum to ∼10% in only one orbit. The ability to remove contamination from globular clusters is usually our limiting factor, rather than the signal-to-noise ratio of the SBF signal itself, and this level of detection completeness means that the error in this correction will be ≲0.03 mag.

Another large fraction of the galaxies in our data set come from a program (GO-14654) to measure SBF distances to host galaxies of well-studied SNe Ia with the goal of investigating possible luminosity differences among subgroups with different ultraviolet colors (e.g., Milne et al. 2013, 2015; Brown et al. 2017, 2019; Foley et al. 2020). We selected early-type hosts, or in some cases, disk galaxies that appeared to have useful regions for the SBF measurement, based on the imaging data that were available when we were designing the program. These galaxies are naturally less luminous than those in the MASSIVE survey. For three targets in this program that were expected to be near or beyond 80 Mpc, we conservatively obtained two orbits of WFC3/F110W integration to ensure adequate depth along the GCLF.

In a follow-up to GO-14219, program GO-15265 obtained second-band WFC3/UVIS imaging of a subset of the previously targeted MASSIVE galaxies to study the metallicity distributions of their globular clusters. As part of this program, we were also awarded time for single-orbit WFC3/IR F110W imaging of an additional six MASSIVE galaxies beyond our initial 6000 km s⁻¹ limit. We have found that we can reliably characterize the GCLFs, and thus correct for their residual contributions to the variance, in these very luminous ellipticals despite the somewhat larger distances. Thus, we have been able to measure the SBF distances for these six galaxies and include them in our sample as well. To these, we add a two-orbit WFC3/F110W observation of the cD galaxy in Coma from another of our programs and single-orbit observations of two galaxies for which we previously published WFC3/F110W SBF distances.

The following list summarizes the HST programs that contributed to the current data set. All of the data were reduced homogeneously by our team.

• 1.  
  GO-11711: NGC 4874, the cD galaxy in the Coma cluster (PI: J. Blakeslee).
• 2.  
  GO-12450: NGC 3504 (PI: C. Kochanek), published previously by Nguyen et al. (2020).
• 3.  
  GO-14219: 35 early-type galaxies selected from MASSIVE Survey (PI: J. Blakeslee).
• 4.  
  GO-14654: 19 mainly early-type host galaxies of SNe Ia (PI: P. Milne).
• 5.  
  GO-15265: 6 additional MASSIVE Survey galaxies (PI: J. Blakeslee).
• 6.  
  GO-14771, GO-14804, GO-15329: NGC 4993, host of the binary neutron star merger that produced GW 170817 (PIs: N. Tanvir, A. Levan, E. Berger), published previously by Cantiello et al. (2018b).

In addition, we also reprocessed the observations from GO-11712 (PI: Blakeslee) that were used by Jensen et al. (2015) to calibrate the WFC3/IR SBF method. This reprocessing was done in order to verify consistency between the calibration and target samples. Although Jensen et al. (2015) presented SBF calibrations for both F110W and F160W, we use only F110W data in the present study and apply the same analysis for all program galaxies. This enables an extremely homogeneous and self-consistent set of distance measurements.

The SBF analysis has been described many times (e.g., Tonry et al. 1990; Jensen et al. 1998; Blakeslee et al. 1999a; Cantiello et al. 2005; Mei et al. 2005). For these HST WFC3/IR data, we follow the procedure documented for the calibration sample by Jensen et al. (2015). Details specific to the present sample of 63 galaxies, including some refinements to the masking process and consistency tests with the calibration sample, are presented in a companion data paper (Jensen et al. 2021, hereafter J21). In brief, we model and subtract the mean galaxy surface brightness, mask contaminating sources (mainly globular clusters and background galaxies) down to some completeness threshold, and measure the Fourier power spectrum of the remaining image. The normalization of the power spectrum on the scale of the point-spread function (PSF) includes contributions from the stellar fluctuations and contaminating sources. We fit and extrapolate the magnitude distribution of the sources and use this to calculate the background variance, which we subtract from the normalization of the PSF component of the power spectrum to get the variance from the stellar fluctuations. These fluctuations are measured in a series of concentric annuli, normalized by the local surface brightness, and converted into the SBF magnitude, labeled ${\overline{m}}_{110}$ for the F110W bandpass. Applying a calibration for ${\overline{M}}_{110}$ (see the following section) then gives the distance modulus.

We adopted a photometric zero point of 26.822 AB mag for WFC3/F110W from the STScI website ⁶ at the time our data were obtained and processed. In December 2020, this value was revised by −0.004 mag based on Bajaj et al. (2020). Although this would change the numerical value of the ${\overline{m}}_{110}$ measurements tabulated in J21 by a small amount, it would not change our distances, which are referenced to the WFC3/IR SBF calibration of Jensen et al. (2015, see also Cantiello et al. 2018b), who used the former photometric zero point. We correct the photometry for Galactic extinction according to Schlafly & Finkbeiner (2011), using the values provided for individual galaxies by the NASA Extragalactic Database (NED). Following that work, we adopted a 10% uncertainty on the extinction correction and included it in quadrature in our photometric error estimates.

The absolute SBF magnitude $\overline{M}$ in a given bandpass depends on stellar population properties such as age, metallicity, initial mass function, alpha-element enhancement, etc. To calibrate this dependence, one generally uses broadband color as a distance-independent proxy for a galaxy's stellar population (e.g., Tonry et al. 1997; Blakeslee et al. 2001; Jensen et al. 2003; Cantiello et al. 2007). Jensen et al. (2015) derived high-quality calibrations of the F110W SBF magnitude ${\overline{M}}_{110}$ using data for 16 Virgo and Fornax cluster early-type galaxies. For the sake of flexibility, the calibrations were provided using both ACS (g₄₇₅ − z₈₅₀) and WFC3 (J₁₁₀ − H₁₆₀) colors.

Because the broad-baseline optical color is more sensitive to metallicity, the slope of the ${\overline{M}}_{110}$−(g₄₇₅ − z₈₅₀) calibration is shallower and therefore less susceptible to photometric errors; thus, it is the preferred calibration. However, for galaxies with large amounts of foreground extinction (e.g., Cantiello et al. 2018b), or lacking in optical data, the ${\overline{M}}_{110}$−(J₁₁₀ − H₁₆₀) calibration can be used. For the current sample, we adopt the calibration based on optical color for nearly all galaxies, but instead of ACS colors (unavailable for most of the sample), we use colors derived from Pan-STARRS images (Magnier et al. 2020; Waters et al. 2020), with sky estimation and object masking as in Jensen et al. (2015). The photometric transformation of Pan-STARRS (g − z) to ACS (g₄₇₅ − z₈₅₀) is described in detail by J21. We include the scatter in this transformation in our estimate of the color error, which contributes to the uncertainty in the calibrated ${\overline{M}}_{110}$ for each galaxy.

One galaxy in our sample (ESO 125-G006) lacks Pan-STARRS data; for this, we use 2MASS (J − H) color transformed to (J₁₁₀ − H₁₆₀). As with (g − z), we include the scatter in this transformation in the estimated ${\overline{M}}_{110}$ error. This galaxy also suffers from the highest amount of Galactic extinction in our sample, 20% higher than the next highest, and more than three times the sample average. Although the color measurement error is amplified by the steeper slope of the (J − H) calibration, the extinction is much lower in the IR, resulting in an error for ${\overline{M}}_{110}$ only slightly larger than for the rest of the galaxies. The value of H₀ derived from ESO 125-G006 is very close to the best-fit value for the full sample; there is no evidence for any systematic difference resulting from the use of (J − H) for the calibration. Likewise, Cantiello et al. (2018b) found closely consistent distances for the gravitational wave event host NGC 4993 using the two different calibrations; J21 provide further tests illustrating the consistency.

Finally, we note that the dependence of ${\overline{M}}_{110}$ on galaxy color has some intrinsic scatter due to stellar population effects. For example, galaxies with the same colors may have slightly different ${\overline{M}}_{110}$ values because age and metallicity are not completely degenerate in their effects on ${\overline{M}}_{110}$ and broadband color. In the z band, this intrinsic scatter is 0.06 mag for red galaxies (Blakeslee et al. 2009). Although the observed scatter in the ${\overline{M}}_{110}$ calibration from Jensen et al. (2015) is consistent with measurement error, suggesting a negligible contribution from intrinsic scatter, we conservatively adopt the same 0.06 mag intrinsic scatter for F110W. With this added in quadrature, our median distance modulus error is 0.083 mag, or ∼4% in distance.

The I-band SBF distance zero point was tied to Cepheids by Tonry et al. (2000) using six spirals that had both SBF and Cepheid distances. This calibration was revised by +0.06 mag by Blakeslee et al. (2002) using the final Key Project Cepheid distances from Freedman et al. (2001), which were based on an LMC distance modulus of 18.50 mag. Additional discussion of this distance zero point, including checks for consistency with HST/ACS SBF distances, is given in Appendix A of Blakeslee et al. (2010). For WFC3/IR, the F110W and F160W SBF zero points were determined by Jensen et al. (2015) using 16 Virgo and Fornax cluster galaxies with previously measured SBF distances by Blakeslee et al. (2009). Cantiello et al. (2018b) revised these zero points by 0.05 ± 0.02 mag following improved PSF characterization resulting from extensive tests with a library of template stars that were used to reanalyze the power spectra of the calibration sample. We use the same set of PSF templates (see J21 for details) and therefore adopt this zero-point shift and its associated scatter.

The above Cepheid calibration assumed a distance modulus of 18.50 mag for the LMC. Based on a sample of 20 DEBs, Pietrzyński et al. (2019) present an improved LMC distance of 49.59 ± 0.55 kpc (combining random and systematic error), or 0.023 ± 0.024 mag less than the value used previously. For consistency with other recent studies (e.g., Riess et al. 2019; Freedman et al. 2019, 2020), we also apply this shift in zero point. The fully revised calibration is presented by J21.

The systematic uncertainty in the ${\overline{M}}_{110}$ zero point was estimated by Cantiello et al. (2018b) to be 0.10 mag, including contributions of 0.03 mag from the tie between WFC3/IR and optical SBF distances, 0.08 mag from the tie between SBF and Cepheids, and 0.06 mag for the Cepheid zero point, dominated by the uncertainty in the LMC distance. With the improved precision of the revised LMC distance, the Cepheid zero-point error drops to 0.028 mag (Riess et al. 2019), reducing the SBF zero-point uncertainty to 0.09 mag, or 4.2% in distance. This is the current limit of our precision in measuring H₀ using the Cepheid-based SBF calibration alone.

Stellar population models provide confidence in this zero point. Comparisons between SBF predictions for evolved stellar populations and the empirical zero point show agreement in optical bandpasses to better than the 0.09 mag uncertainty (e.g., Blakeslee et al. 2010; Cantiello et al. 2018a; Greco et al. 2021), although there is more discrepancy among models for bluer populations (e.g., Trujillo et al. 2019). The models are less constraining in their predictions for near-IR SBF magnitudes, but they encompass the range of the observations (Jensen et al. 2003, 2015). Empirically, the tie between WFC3/IR and ACS SBF distances is very tight.

Constraints on the zero point can be improved using the TRGB method for red galaxies with well-measured SBF distances. Cohen et al. (2018) show excellent agreement between SBF and TRGB distances for a sample of 12 dwarf galaxies with blue colors observed by HST. However, obtaining an overlapping sample of distances for massive red ellipticals requires reaching the Virgo cluster; there are few TRGB distances for early-type galaxies at this distance. Two exceptions are M60 (NGC 4649; Lee & Jang 2017) and M87 (Bird et al. 2010). There is also a TRGB distance to the massive merger remnant NGC 1316 (Arp 154) in Fornax, which has multiple independent SBF distances (e.g., Blakeslee et al. 2009; Cantiello et al. 2013; Jensen et al. 2015). In the Appendix, we compare the SBF and TRGB distances for these three galaxies using an absolute magnitude for the TRGB itself based on the maser distance to NGC 4258 so that it is fully independent of the LMC-based Cepheid calibration. Excluding the problematic calibrator NGC 1316, we find that the mean offset between the SBF and TRGB distances is −0.01 ± 0.08 mag. In other words, SBF distances would be very slightly longer if calibrated from the TRGB. Including systematic effects, the uncertainty on the TRGB-based SBF zero point becomes 0.10 mag, or 4.6%. Full details are provided in the Appendix.

Clearly, there is no significant difference between the Cepheid-based and TRGB-based SBF zero points. However, the complete independence of the two approaches reduces the final systematic uncertainty on the jointly constrained zero point from >4% for each method independently to 3.1% when combined. Ultimately, we hope to anchor the SBF method using a much larger sample of giant ellipticals with TRGB distances tied to Gaia parallaxes. We return to this prospect in Section 6.

As a first approach, we use the group-averaged velocities in the CMB frame as described above. In doing this, it is necessary to adopt some value for the random dispersion of the groups and isolated galaxies within the general velocity field. Estimates of this small-scale peculiar velocity "noise" σ_p range from 100 to 300 km s⁻¹, depending on the sample considered (Masters et al. 2006; Hoffman et al. 2015; Graziani et al. 2019). Here we assume σ_p = 240 km s⁻¹, which we add in quadrature with the generally small error in the group mean velocity (taken as zero for isolated galaxies). This results in a mean total velocity error of 250 km s⁻¹, typical of many peculiar velocity surveys involving early-type galaxies in groups (e.g., Zaroubi et al. 2001; Hudson et al. 2004). It corresponds to a 5.4% velocity error at the median distance of our sample.

Table 1 presents the results of many different trial fits of H₀, beginning with the full sample of 63 galaxies with group-averaged velocities. For this case, we find H₀ = 73.53 ± 0.66 km s⁻¹ Mpc⁻¹, but with a χ² per degree of freedom ${\chi }_{\nu }^{2}=1.19$, which means that either σ_p or our distance errors are underestimated for at least a portion of the sample. Because morphological irregularities can cause problems for the SBF method, we experimented by keeping only galaxies with morphological type T ≤ −3 in the HyperLEDA database (Makarov et al. 2014). This selects 45 galaxies classified as ellipticals or early-type S0s; the fitted H₀ is virtually identical, but with a significantly lower ${\chi }_{\nu }^{2}=1.02$. This is an acceptable fit for 44 degrees of freedom, indicating that the errors are a reasonable description of the scatter for this sample. However, with our high-resolution HST data, we can do better than cataloged types derived from ground-based surveys. In reducing the data, one of us (J.B.J.) made consistent note of the presence of dust, spiral structure, bars, and shells. This was done prior to deriving the distances so that knowledge of discrepant results could not bias the classification. Eleven galaxies show evidence of spiral structure or dust. However, one of these is NGC 4993, which has deep high-resolution ACS g₄₇₅ imaging that clearly reveals the localized dust features, allowing them to be excised to give a clean area for the SBF measurement. A distance for this galaxy was already published by Cantiello et al. (2018b), and we keep it in our resulting sample of 53 "clean" galaxies while rejecting the other ten. Line 3 of Table 1 shows that this sample gives H₀ = 73.44 ± 0.71 with ${\chi }_{\nu }^{2}=0.97$, again indicating a good fit (further cuts of galaxies with shells, bars, etc., do not yield additional improvement). We also show results from a fit using only clean galaxies that were selected as part of MASSIVE, a well-defined mass-selected sample. Again, the value of H₀ is consistent, but ${\chi }_{\nu }^{2}$ increases, likely because the adopted σ_p works well on average but slightly underestimates the small-scale dispersion of galaxy groups embedded within rich environments. A value of σ_p = 275 km s⁻¹ would give ${\chi }_{\nu }^{2}=1.0$ with a negligible change to H₀.

Table 1. Hubble Constants for Various Selections

Selected Sample a	vb	Ngxy	${\chi }_{\nu }^{2}$	H0
All galaxies	grp	63	1.19	73.53 ± 0.66
Ellipticals	grp	45	1.02	73.52 ± 0.74
All clean	grp	53	0.97	73.44 ± 0.71
MASSIVE, clean	grp	37	1.16	73.86 ± 0.82
d > 60, clean	grp	34	0.88	73.33 ± 0.82
d < 70, clean	grp	33	0.88	74.08 ± 0.96
d < 80, clean	grp	46	0.96	72.78 ± 0.77
SN Ia hosts, clean	grp	20	0.68	73.31 ± 1.26
All galaxies	ind	63	1.53	73.31 ± 0.67
All clean	ind	53	0.95	73.27 ± 0.73
MASSIVE, clean	ind	37	0.86	73.79 ± 0.85
d < 80, clean	ind	46	1.02	72.96 ± 0.76
All galaxies	cf3	63	1.14	73.32 ± 0.71
All clean	cf3	53	1.05	73.30 ± 0.76
MASSIVE, clean	cf3	37	1.16	73.62 ± 0.88
d < 80, clean	cf3	46	1.07	72.67 ± 0.83
All clean, −1% c	cf3	53	1.03	72.54 ± 0.76
All clean, +1% c	cf3	53	1.06	74.07 ± 0.77
All galaxies	2M++	63	0.99	73.90 ± 0.65
All clean	2M++	53	0.89	73.78 ± 0.69
Massive, clean	2M++	37	1.02	74.09 ± 0.80
d < 80, clean	2M++	46	0.94	73.42 ± 0.75

Notes.

^a "Ellipticals" refers to morphological type T ≤ −3; "clean" indicates galaxies with no discernible dust or spiral structure; "MASSIVE" means limited to MASSIVE Survey galaxies. ^b Velocities used for the fit: grp for group-averaged; ind for individual galaxy; cf3 for the flow model of Graziani et al. (2019); 2M++ for the flow model of Carrick et al. (2015). ^c Velocities from the CF3 linear flow model rescaled by ±1%

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Figure 1 (left panel) presents the Hubble diagram combining our SBF distances with the group-averaged velocities, along with the best-fit H₀ value for the clean sample. The figure also shows the values of H₀ derived from individual galaxies. The scatter decreases at larger distance as the velocity errors become a smaller fraction of the recession velocity. Overall, the fit is good, but by eye, there is a suggestion of small-scale coherent trends, with several galaxies near ∼75 Mpc lying below the line and three near ∼90 Mpc scattering above. Features in the Hubble diagram can be associated with small-scale coherent flows. There is an ongoing HST/WFC3 SBF program exploring this issue in more detail.

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Alternatively, if we were inadequately accounting for the effects of globular clusters in the power spectra of the most distant galaxies, this could cause those galaxies to scatter to high H₀. We have explored a number of cuts in distance for our clean sample; the table shows three examples: excluding galaxies within 60 Mpc, excluding those beyond 70 Mpc, or beyond 80 Mpc. In all these cases, ${\chi }_{\nu }^{2}\lt 1$. The nearer galaxies have little leverage on H₀, and removing them has little effect. Changing the upper distance limit can change H₀ by up to ±1%, but this is still within the statistical error.

Finally, we also show the results from a fit using only clean galaxies that are identified as SN Ia hosts. These galaxies range in distance from 19 to 91 Mpc, with the few most distant ones having two orbits of integration. The best-fit H₀ is similar to that derived for the full clean sample, but the statistical uncertainty is much larger because the fit includes only 20 galaxies.

We performed a second set of fits using individual galaxy velocities from the 2MRS referenced to the CMB frame. However, we know that two-thirds of our sample consist of massive ellipticals that preferentially reside in rich groups and clusters that can have velocity dispersions up to 900 km s⁻¹. Thus, the assumption of a fixed random velocity error is insufficient. The Tully (2015) group catalog provides line-of-sight velocity dispersions for all groups, and these can be used as estimates of the random velocity errors for the member galaxies (neglecting the random motion of the groups themselves). We adopt this approach: using the individual galaxy velocities with the errors taken to be the dispersions of the parent groups. For galaxies not in groups, we adopt an error of 150 km s⁻¹, consistent with the quoted velocity field error in isolated regions from Graziani et al. (2019).

The second section of Table 1 shows example fits using this approach. Interestingly, the subsample of clean MASSIVE galaxies now gives the lowest ${\chi }_{\nu }^{2}$. This is likely because the most massive members of groups and clusters tend to deviate from the mean by less than one standard deviation, although there are exceptions in our sample such as NGC 1272 in Perseus, which has a velocity 1600 km s⁻¹ below the cluster mean. Overall, the H₀ values from these fits track those from the group velocity fits to within ∼0.2 km s⁻¹ Mpc⁻¹.

The lines labeled "cf3" in the second column of Table 1 show results obtained using the flow-corrected recessional velocities from the CF3 linear velocity model of Graziani et al. (2019), implemented as described in Section 3. For the velocity error in this case, we use the model's best-fit nonlinear dispersion value of 280 km s⁻¹, which represents the unmodeled small-scale motion in the velocity field. The results of these tests are very similar to those found using the individual CMB-frame velocities. The ${\chi }_{\nu }^{2}$ values are a bit higher in most cases, but there is no significant change in H₀.

We perform two other fits for this model, varying the scale factor of the velocity field within the reported ±1% uncertainty. Unsurprisingly, this changes the H₀ we derive by ±1% as well. Because the result given by the baseline scale factor agrees well with that found using the uncorrected CMB-frame velocities, this change in H₀ would represent a net inward or outward flow of our sample with respect to the comoving volume (a small Hubble bubble or sinkhole). Until we have more complete high-precision distance mapping of the local volume extending beyond 100 Mpc, we take this ±1% as the systematic uncertainty in the velocity scale.

Finally, the fits in Table 1 identified as "2M++" use the flow-corrected recessional velocities from the density-based model of Carrick et al. (2015), also described in Section 3. As for the group velocity fits above, we again use σ_p = 240 km s⁻¹. We find that the corrected velocities from this model result in significant improvements in ${\chi }_{\nu }^{2}$ with respect to the other sets of velocities. For instance, the unreduced χ² for the full sample drops by Δχ² > 12 compared to the fit using the group-averaged velocities. The resulting H₀ values are ∼0.4 km s⁻¹ Mpc⁻¹ higher, again consistent within the errors. The Hubble diagram in the right panel of Figure 1 illustrates the slightly higher value of H₀ and reduced scatter when using the 2M++ model velocities.

For the preferred H₀ from our analysis, we adopt the result in Table 1 for the full "clean" sample using the group velocities. Although the 2M++ model velocities gave lower values of ${\chi }_{\nu }^{2}$, we suspect that the model-independent group velocities may be more robust against systematic errors in scaling. We note that the same approach was used by Pesce et al. (2020) in deriving H₀ from their maser distances, for which the 2M++ model similarly gave the best fit. Readers who prefer the H₀ result based on the 2M++ model should increase the values given in this section by 0.5%. Thus, we adopt H₀ = 73.44 ± 0.71 km s⁻¹ Mpc⁻¹, statistical error only, for our Cepheid-calibrated value of H₀. The revised Cepheid-based SBF calibration has a systematic uncertainty of 0.09 mag, or 4.2% in distance (Section 2.4), translating to ±3.11 km s⁻¹ Mpc⁻¹ for H₀.

In the Appendix, we find that the TRGB-based calibration differs by 0.007 ± 0.099 mag from the Cepheid calibration, in the sense that the SBF distances increase by ∼0.3%. The TRGB calibration therefore gives H₀ = 73.20 km s⁻¹ Mpc⁻¹, with a systematic uncertainty of 4.7%, or ±3.41 km s⁻¹ Mpc⁻¹. Averaging these two independent calibrations gives 73.33 km s⁻¹ Mpc⁻¹ with a systematic error in the distance scale of 3.1%. We combine this in quadrature with the estimated 1% systematic error in the velocity scaling (Section 4.3) to give a total systematic error of 3.3%, or 2.41 km s⁻¹ Mpc⁻¹. Our final result, to single decimal-point precision, is then H₀ = 73.3 ± 0.7 ± 2.4 km s⁻¹ Mpc⁻¹. This calculation is summarized in Table 2.

Before tying SBF to the TRGB method, we must enforce that the galaxies all use the same TRGB absolute magnitude calibration. Numerous TRGB calibrations have been published recently; we list them for reference in Table 3. Some of the calibrations are quoted in the HST F814W bandpass, ${M}_{814}^{\mathrm{TRGB}}$, while others are in the standard Cousins I band, ${M}_{I}^{\mathrm{TRGB}}$. To convert between the two, we use the following transformation from Riess et al. (2016) with the TRGB color assumed by Freedman et al. (2019):

$$\begin{eqnarray}&&{M}_{814}^{\mathrm{TRGB}}={M}_{I}^{\mathrm{TRGB}}+0.02-0.018(V-I);\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}&&{M}_{814}^{\mathrm{TRGB}}\approx {M}_{I}^{\mathrm{TRGB}}-0.009.\end{eqnarray} \tag{ A2 }$$

Thus, ${M}_{I}^{\mathrm{TRGB}}\approx {M}_{814}^{\mathrm{TRGB}}+0.01$. For ease of comparison, the fourth column of Table 3 shows the value of ${M}_{I}^{\mathrm{TRGB}}$, rounded to the nearest hundredth, for each of the recent TRGB calibrations. Several of these take their zero point from the LMC, using the DEB distance of Pietrzyński et al. (2019), as we did in adjusting the Cepheid zero point for SBF. In order to keep the two SBF calibrations independent, we prefer to use a TRGB calibration based on another geometric method, the maser distance to NGC 4258. For this purpose, we take a simple average of the recent results of Reid et al. (2019) and Jang et al. (2020) using this approach. Because these studies use the same basic distance and photometric data, averaging them does not reduce the error, and we adopt

$$\begin{eqnarray}&&{M}_{I}^{\mathrm{TRGB}}=-4.02\pm 0.05\mathrm{mag}.\end{eqnarray} \tag{ A3 }$$

This is very close to the value derived by Jang & Lee (2017) and the reassessment by Capozzi & Raffelt (2020) combining the distance from Reid et al. (2019) with the photometric measurements of Jang & Lee (2017).

Table 3. Ten Recent TRGB Absolute Calibrations

		${M}_{\mathrm{Band}}^{\mathrm{TRGB}}$	${M}_{I}^{\mathrm{TRGB}}$	σtot	σstat	σsys
Reference	Band	(mag)	(mag)	(mag)	(mag)	(mag)	Anchoring Method
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
Freedman et al. (2019)	F814W	−4.049	−4.04	0.045	0.022	0.039	DEB distance to LMC a
Yuan et al. (2019)	F814W	−3.970	−3.96	0.046	0.038	0.026	DEB distance to LMC b
Freedman et al. (2020)	I	−4.047	−4.05	0.045	0.022	0.039	DEB distance to LMC a
Soltis et al. (2021)	I	−3.961	−3.96	0.040	0.011	0.038	DEB distance to LMC c
Reid et al. (2019)	F814W	−4.012	−4.00	0.044	0.030	0.032	Maser distance to NGC 4258
Jang et al. (2020)	F814W	−4.050	−4.04	0.056	0.028	0.048	Maser distance to NGC 4258
Capozzi & Raffelt (2020)	I	−4.027	−4.03	0.055	0.045	0.032	Maser distance to NGC 4258
Capozzi & Raffelt (2020)	I	−3.960	−3.96	0.067	0.064	0.021	GAIA EDR2 kinematic d to ω Cen
Soltis et al. (2021)	I	−3.970	−3.97	0.062	0.041	0.047	GAIA EDR3 parallax d to ω Cen
Cerny et al. (2020)	I	−4.056	−4.06	0.10	0.022	0.101	HB for 46 GCs + DEB in ω Cen d

Notes. Columns list: (1) calibration paper; (2) reference band used in the study (Vega-based calibrations); (3) derived TRGB absolute magnitude in reference band; (4) absolute TRGB magnitude in standard Cousins I, assuming where needed I = m_814W + 0.009, and rounded to the nearest hundredth; (5) total error quoted from the study, or quadrature sum of quoted random and systematic errors; (6) quoted statistical error or derived from information provided; (7) quoted systematic error or derived from information provided; (8) distance method used for anchoring the zero point.

^a Extinction determined from observed TRGB color differences. ^b Extinction from the Haschke et al. (2011) OGLE reddening map. ^c Extinction from the Skowron et al. (2021) OGLE reddening map. ^d "HB" refers to the horizontal branch, used by Cerny et al. (2020) to shift the 46 globular clusters (GCs) into agreement before setting the distance zero point based on a DEB in ω Cen (Thompson et al. 2001).

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Ultimately, the best calibration of ${M}_{I}^{\mathrm{TRGB}}$ will come from Gaia, and the recent result of Soltis et al. (2021) in Table 3, using Gaia Early Data Release 3 (EDR3) parallaxes, is promising. Thus far, however, it is based on only the single large globular cluster ω Cen, likely the remnant nucleus of a stripped dE (e.g., Hilker & Richtler 2000; Majewski et al. 2000). Capozzi & Raffelt (2020) use the same ω Cen photometric data but adopted the kinematic distance from Baumgardt et al. (2019) based on Gaia EDR2 data. Cerny et al. (2020) use Gaia EDR2 proper motion data to assign membership for a sample of 46 globular clusters and determine their relative distances from the horizontal branch; the zero point is set from the DEB distance to ω Cen. We look forward to the analysis of the TRGB in an expanded sample of globular clusters using Gaia parallaxes. This will also enable a determination of the intrinsic scatter in the TRGB absolute magnitude, an error term that is generally ignored.

Blakeslee et al. (2009) tabulate high-quality HST/ACS SBF distances for 134 early-type galaxies mainly in the Virgo and Fornax clusters, and Jensen et al. (2015) present similar quality WFC3/IR SBF measurements for 16 of these galaxies. These samples enabled the calibration of the color dependence of the SBF magnitude in the giant ellipticals that we measure at large distances. We wish to tie these calibration samples to the TRGB method in order to improve the SBF zero-point calibration.

Unfortunately, there are few TRGB distances to giant ellipticals. Two exceptions are M60/NGC 4649 (Lee & Jang 2017) and M87/NGC 4486 (Bird et al. 2010). While not ideal, the giant, dusty merger remnant NGC 1316 (Arp 154, Fornax A) is another galaxy that has a high-quality HST-based SBF distance and a published TRGB distance (Hatt et al. 2018). Table 4 displays the SBF and TRGB distances for the three galaxies. The SBF distances in this table are the homogeneous set from Blakeslee et al. (2009), adjusted to the revised LMC distance based on DEBs (see Section 2.4). The TRGB distances have been adjusted for consistency with our adopted calibration in Equation (A3). For NGC 1316, we use the revised TRGB distance given by Freedman et al. (2019) before adjusting to our adopted calibration.

Table 4. Homogenized SBF–TRGB Distance Comparisons

	(m − M)SBF a	σSBF	(m − M)TRGB b	σTRGB	Δ(m − M) c	σΔ c
Galaxy	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)	Reference for TRGB
NGC 4486/M87	31.088	0.079	31.09	0.10	−0.002	0.128	Bird et al. (2010)
NGC 4649/M60	31.059	0.076	31.07	0.07	−0.011	0.103	Lee & Jang (2017)
NGC 1316	31.583	0.073	31.44	0.04	+0.143	0.083	Hatt et al. (2018); Freedman et al. (2019)
	weighted average for Virgo galaxies: 〈Δ(m − M)〉 = −0.007 ± 0.080
	weighted average for all three galaxies: 〈Δ(m − M)〉 = +0.065 ± 0.058

Notes.

^a SBF distance moduli from Blakeslee et al. (2009), reduced by 0.023 mag as described in Section 2.4; σ_SBF is the statistical error as published. ^b TRGB distance moduli from references in the last column, corrected by −0.03, +0.02, and −0.02 mag (M87, M60, and NGC 1316, respectively) for consistency with our adopted zero point of ${M}_{I}^{\mathrm{TRGB}}=-4.02\,\mathrm{mag}$ (${M}_{814}^{\mathrm{TRGB}}=-4.03\,\mathrm{mag}$), which is an average of two recent TRGB calibrations based on the NGC 4258 maser distance (Reid et al. 2019; Jang et al. 2020). The statistical errors σ_TRGB are as published; unlike the SBF errors, they include no allowance for intrinsic scatter in the absolute magnitude of the standardized candle. ^c Difference in distance moduli: (m − M)_SBF−(m − M)_TRGB, and error in this difference σ_Δ.

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We note that Freedman et al. (2019), with the goal of increasing the number of TRGB–SN Ia calibrators, also tabulate a "TRGB distance," with a purported precision of 0.05 mag, for NGC 1404, a giant elliptical just 50 kpc from the cD NGC 1399 in the core of the Fornax cluster. We have excellent SBF distances for these two galaxies. However, the quoted TRGB distance is actually an average of the TRGB distances to the spiral galaxy NGC 1365 and the merger remnant NGC 1316. NGC 1365 lies 420 kpc in projection from the Fornax core, has an angular size larger than any of the Fornax ellipticals and has been suggested to lie significantly in the foreground (Saha et al. 1997). NGC 1316 is projected 1.25 Mpc from the core; if the separation along the line of sight is the same, the offset would be 0.14 mag. There is no evidence that these two late-type galaxies at substantial separations from the core have distance moduli within 0.05 mag of NGC 1404. Thus, we cannot use it as a TRGB calibrator.

Table 4 reports the weighted average offsets between the SBF and TRGB distances when using just M60 and M87 in Virgo, and when using NGC 1316 as well. These averages differ by 0.07 mag. While this is in statistical agreement, given that we have so few calibrators, we want to ensure that all individual measurements are reliable. NGC 1316 is an irregular galaxy with prominent dust features, Hα filaments, tidal loops, and shells indicative of recent merging (Schweizer 1980). Based on the ages of the centrally concentrated population of bright, metal-rich star clusters, the galaxy is believed to have undergone a major merger, with an associated starburst, 2–3 Gyr ago (Goudfrooij et al. 2001; Sesto et al. 2018). The IR SBF magnitude for NGC 1316 is 0.25 mag brighter than expected based on the color calibration defined by more evolved early-type galaxies. This is consistent with predictions from stellar population models if there is a 3 Gyr old population present (Jensen et al. 2015). In contrast, optical SBF distances are less affected by younger populations because their effect is to move the galaxy both brighter and bluer along the SBF–color relation, rather than simply scattering the SBF magnitude brighter. For this reason, we have more confidence in the optical SBF distance (shown in Table 4) for this galaxy than the IR one.

At present, the star formation in NGC 1316 is mainly occurring at larger radius. Based on the mid-IR emission, Temi et al. (2005) suggest that "young, luminous stars are forming in the infalling dusty gas, raising the dust temperature sufficiently to emit at 15 μm." Thus, unlike in spiral galaxies, where the star formation occurs in an orderly fashion in the disk while the halo hosts only ancient metal-poor stars, NGC 1316 appears to have a significant population of young stars at large radius. Notably, the field used for the TRGB measurement was about 10' from the galaxy's center, in an area of intense radio emission (Ekers et al. 1983). We suspect that a dispersed population of asymptotic giant branch stars associated with the prominent 3 Gyr, and perhaps younger, component may have biased the TRGB measurement. More dramatic biases have occurred in TRGB estimates for more recent mergers at roughly the same distance (Schweizer et al. 2008). The young component at large radius in NGC 1316 could account for the observed difference in the TRGB and optical SBF distances for this galaxy.

In short, we lack confidence in NGC 1316 as a reliable calibrator. We therefore adopt the mean offset of −0.007 ± 0.080 mag between the TRGB and SBF distances, determined from the two giant ellipticals in Virgo. Including the 0.03 mag error in the tie between the optical and IR SBF distances and the 0.05 mag error in the TRGB calibration, the final offset and uncertainty in the SBF zero point from this TRGB comparison is −0.007 ± 0.099 mag, which we round to −0.01 ± 0.10 mag in Section 2.5 of the present work.

Ideally, we would make a direct tie between geometrically calibrated TRGB distances to giant ellipticals and the IR SBF measurements to the same galaxies, avoiding the 0.03 mag error incurred from the tie through the more extensive optical SBF data set. However, at present, M60 is the only IR SBF calibrator from Jensen et al. (2015) with a TRGB distance. The result would be similar, but with a larger error. Section 6 presents a plan for improving this situation.