The value of the Hubble-Lemaître constant queried by Type Ia Supernovae: A journey from the Calán-Tololo Project to the Carnegie Supernova Program

After one century of research, the advances of recent years both in the field of theory and experimentation have allowed us to witness remarkable progress in our understanding of the Universe on large scales. A concordant $\Lambda$CDM cosmological model is able to reproduce the evolution of the Universe from the epoch of recombination, characterized by the remnants effects of density fluctuations of quantum origin, to its complex current large scale structure. Such a model is geometrically flat, composed of cold dark matter, and has a dominant component of dark energy that is responsible for the current acceleration of the Universe. Remarkably, one requires only six cosmological parameters to define the basic cosmology as has been observationally demonstrated by the WMAP and Planck missions.

Within the $\Lambda$CDM model, the Hubble-Lemaître constant ($H_{0}$) is arguably the most important cosmological parameter. By definition it corresponds to the expansion rate of the Universe at the present time. It sets the size, age, and critical density of the Universe, and pervades virtually all models in extra-galactic research. Ever since the discovery of the cosmic expansion in 1927-29 (Lemaître, 1927; Hubble, 1929), there has been a continuous effort from the astronomical community to measure its value, with the range of experimentally measured values of $H_{0}$ decreasing over time, from $\sim$500 $\,\rm km\,s^{-1}\,Mpc^{-1}$ to a narrow interval of only 67-74 $\,\rm km\,s^{-1}\,Mpc^{-1}$.

The traditional method has consisted in measuring luminosity distances to galaxies in the smooth Hubble flow from bright astronomical sources with properly calibrated luminosities. This approach has required the calibration of a series of increasingly brighter astrophysical sources, which, altogether is known as the “cosmic distance ladder” (CDL). Many different techniques have been attempted in order to build the ladder and determine the value of $H_{0}$. In this work we will focus in a particularly successful architecture, which is based on enormous improvements over the past three decades in: (1) improving our ability to measure precise (5-7%) distances to individual Type Ia supernovae (SNe Ia), (2) establishing Cepheid or Tip of the Red Giant Branch (TRGB) distances, with the Hubble Space Telescope (HST), to a growing sample of galaxies having hosted SNe Ia, and (3) improving the determination of the distance to the Large Magellanic Cloud (LMC) or other very nearby galaxies. The concatenation of these techniques creates a three-rung ladder where all links are essential for the purpose of determining the value of $H_{0}$ and none is less important than the other. Several authors claim today that this concatenation of methods can lead to a $\sim$1% precision in the measurement of $H_{0}$ (Riess et al., 2016; Burns et al., 2018; Riess et al., 2019; Freedman et al., 2019, R16, B18, R19 & F19). However, the reported values range between 74.22$\pm$1.82 (R19) and 69.8$\pm$0.8 $\,\rm km\,s^{-1}\,Mpc^{-1}$(F19), which shows that the 1% precision is a future goal for the CDL.

Other experimental approaches sensitive to $H_{0}$ but independent of the cosmic ladder in the local Universe have been advocated in recent years such as the measurement of temperature anisotropies in the cosmic microwave background (CMB). The exceptional data provided by the WMAP and Planck satellites have allowed precise determinations of the Hubble-Lemaître constant using the CMB data alone, namely $H_{0}$=70.0$\pm$2.0 and $H_{0}$=67.4$\pm$0.5 $\,\rm km\,s^{-1}\,Mpc^{-1}$, respectively (Hinshaw et al., 2013; Planck Collaboration et al., 2018). It must be kept in mind that these values are indirect constraints based on a flat $\Lambda$CDM cosmological model.

The measurement of the angular diameter of the baryon acoustic oscillation (BAO) feature is also sensitive to the expansion history. However, the BAO depends on the sound horizon measured by the CMB, so the two $H_{0}$ results are not independent. The parameters yielded by the BAO experiment using the Sloan Digital Sky Survey III data anchored to the Planck CMB data leads to $H_{0}$=67.6$\pm$0.5 $\,\rm km\,s^{-1}\,Mpc^{-1}$(Alam et al., 2017), thus providing further evidence for the six parameter cosmological model fit by the Planck collaboration. As shown by Addison et al. (2018), when Ly$\alpha$ BAO data are combined with CMB data, the solutions for $H_{0}$ obtained from WMAP and Planck agree even better and with smaller uncertainties, namely, $H_{0}$=68.3$\pm$0.7 and $H_{0}$=68.1$\pm$0.6 $\,\rm km\,s^{-1}\,Mpc^{-1}$, respectively.

Strong gravitational lenses afford another route for the cosmic ladder, yet model-dependent. This method consists in measuring time delays between different images of a background quasar lensed by a foreground galaxy and modeling the lens mass distribution. Recently, Wong et al. (2019) presented a measurement of the Hubble-Lemaître constant of 73.3$\pm$1.8 $\,\rm km\,s^{-1}\,Mpc^{-1}$ from six lens systems.

The Megamaser Cosmology Project has recently obtained another measurement of the Hubble-Lemaître constant independent from the cosmic ladder. Their analysis yielded distances from six magamaser-hosting galaxies, which led to a constraint to the Hubble-Lemaître constant of $H_{0}$=73.9$\pm$3.0 $\,\rm km\,s^{-1}\,Mpc^{-1}$ (Reid et al., 2019).

The value of the Hubble-Lemaître constant is a long standing controversy. Thirty years ago the debate was between values of 50 and 100 $\,\rm km\,s^{-1}\,Mpc^{-1}$. Since then we have seen a notable progress but, as the precision of our measurements has increased, we find ourselves once again with two camps advocating significant, although small, differences between 67-74 $\,\rm km\,s^{-1}\,Mpc^{-1}$. This $\sim$ 10% difference is 5$\sigma$ beyond their internal uncertainties, a difference too large for the precision astronomy era, that could be explained for our current inability to identify and handling the systematics in the distance ladder, or for the lack of a complete understanding of the early Universe physics or to later variations in the behaviour of dark energy. The latter makes the $H_{0}$ problem even more interesting to solve.

The purpose of this paper is to make a thorough revision of the setting of the CDL which, as shown above, is currently delivering internally discrepant values between 74.22$\pm$1.82 (R19) and 69.8$\pm$0.8 $\,\rm km\,s^{-1}\,Mpc^{-1}$ (F19). Our goal is to focus the attention into the heart of the distance ladder method, that is, we will not discuss the Rung 1 (the determination of the LMC distance), which we assume well determined, but we will reanalyze in detail the second and third rungs of the distance ladder. For the Rung 2 we will investigate the impact on the value of $H_{0}$ using recent Cepheid and TRGB relative distances anchored to the LMC. For Rung 3 we will employ different samples of SNe Ia, starting with the first set of digital light curves obtained in the early 90s, combined with different methodologies to standardize their luminosities, which will allow us to assess the consistency and the systematics uncertainties of the SNe Ia technique.

This paper is organized as follows. In section 2 we review the latest advances in the establishment of the cosmic distance ladder. In section 3 we analyze the systematics of the CDL. First, we focus on Rung 2 assessing the implications of adopting the Cepheid and TRGB distances for the calibrations of several SN Ia samples. Then we assess the robustness of Rung 3 employing 30 combinations of SN Ia samples observed in optical and near-infrared (NIR) bandpasses, and six different methodologies for the standardization of the SN peak luminosities. Finally, in section 4 we summarize the main conclusions of this paper.

The traditional method to measure $H_{0}$ consists in establishing a cosmic distance ladder whose highest third rung provides a direct measurement of the cosmic expansion rate from galaxies in the smooth Hubble flow¹¹1Defined as the limit where the distance modulus errors are roughly matched to the peculiar velocity. For a peculiar velocity of 200 $\,\rm km\,s^{-1}$ and a 0.1 mag error in distance modulus, the limit of the smooth Hubble flow is $z$=0.014.. This last step has been approached using various types of objects, but the most precise methods remain those involving SNe Ia (Freedman et al., 2001). Thus, the modern determination of $H_{0}$ involves the following three steps (or rungs), namely, (1) the measurement of the distance to a nearby galaxy such as the LMC, NGC 4258, M31 or parallaxes in the Milky Way; (2) distance determinations to other nearby SN Ia host galaxies (distance modulus $\mu$ $<$ 33), relative to the first anchor, via the traditional Cepheid method or the most recent TRGB technique; and (3) the measurement of distances to SNe Ia in the Hubble flow (range of redshifts $z$=0.01-0.1, or $\mu$=33-38), applying the inverse square law to their apparent magnitudes and their intrinsic luminosities calibrated via Cepheids or TRGB stars.

The First Rung. Rung 1 (R1) has been established with four methods: (1) the modelling of masers in the galaxy NGC 4258 which yields a distance modulus of 29.40$\pm$0.02 (Reid et al., 2019); (2) Detached Eclipsing Binary stars (DEBs) which yield a distance modulus for the LMC of 18.48$\pm$0.02 (precision of 1% in distance; Pietrzyński et al., 2019); (3) trigonometric parallaxes of Milky Way Cepheids (van Leeuwen et al., 2007; Benedict et al., 2007) and; (4) DEBs in M31 (Ribas et al., 2005; Vilardell et al., 2010).

The calibration of Rung 2 has been anchored to one or more of these four calibrations. For instance, R16 adopted all four of these calibrations for the determination of the Cepheid luminosities in 19 galaxies which have hosted SNe Ia. Instead, F19 adopted solely the LMC distance calibration measured from DEBs for the measurement of the TRGB in 18 SNe Ia host galaxies. This complicates a direct comparison of both methods and their associated systematic uncertainties.

R19 updated the Cepheid calibration to be anchored solely to the LMC DEB distance but did not provide the revised individual galaxy distances. They found that the net effect of changing the zero point (R1) of the CDL is an increase in $H_{0}$ from 73.24 to 74.22 $\,\rm km\,s^{-1}\,Mpc^{-1}$, 1.34% relative to their 2016 value. This increase in the value of $H_{0}$ can be translated into a global correction of -0.029 magnitudes to their 2016 distance moduli catalog. This allows us to establish a common ground for the first rung, leaving the R1 calibration out of this discussion, to focus on the assessment of the R2 calibration using Cepheid and TRGB techniques, and the R3 calibration using SNe Ia.

The Second Rung. The calibration of Rung 2 (R2) has been historically approached using the classical Cepheid Leavitt law (the P-L relation), ever since the pioneering work of Sandage et al. and Freedman et al. in the 1990 decade using the Hubble Space Telescope (HST). This approach has been improved significantly as the sensitivity of HST has allowed the discovery and characterization of Cepheid stars in a greater and more distant sample of SN Ia host galaxies and through the addition of new SNe Ia that have exploded in nearby galaxies over recent years. There are 19 SNe Ia possessing Cepheid-calibrated distances in the range $\mu$=29.1-32.9 (R16, R19). In the last years an alternative technique has matured allowing the determination of precise distances to nearby galaxies by identifying the locus of the TRGB stars in a colour magnitude diagram (Lee et al., 1993; Beaton et al., 2016). The TRGB technique affords a competitive and independent method for the calibration of R2. This work has been vigorously championed by the Carnegie-Chicago Hubble Project (CCHP) in a series of eight papers that introduce the method, measure distances to 13 SNe Ia host galaxies, bring to a common scale five additional galaxies measured by Jang & Lee (2015, 2017), thus raising to 18 the total number of SNe Ia host galaxies with TRGB distances (F19).

The Cepheid and TRGB sets of distance calibrators overlap in ten galaxies, thus allowing a measurement of the systematic differences in the calibration of the SNe Ia luminosities and of R2, and the implications in the determination of the value of $H_{0}$, as the reader will see in section 3.1.

The Third Rung. The third and highest rung of the cosmic distance ladder (R3) has been reached with different methods such as galaxies themselves using the Tully-Fisher (Giovanelli et al., 1997), Surface Brightness Fluctuations (Tonry et al., 1997), Planetary Nebulae (Feldmeier et al., 2007), and Faber-Jackson (Faber & Jackson, 1976) techniques. However, it has been demonstrated that supernovae play the most competitive role in this endeavour. Several approaches have been developed for Type II SNe such as the Standardized Candle Method (SCM; Olivares E.F et al., 2010), the Photometric Colour Method (PCM; de Jaeger et al., 2015; de Jaeger et al., 2020), and the Photospheric Magnitude Method (PMM; Rodríguez et al., 2014). Undoubtedly, the most precise approach to measure the local expansion rate of the Universe has been achieved using SNe Ia, thanks to their enormous brightness and the standardization of their peak luminosities to reach an unrivaled level of 5-7% precision in distance.

The success of SNe Ia as precise distance indicators goes back to the pioneering work of Kowal (1968) using photographic photometry. The potential of supernovae was also emphasized by Sandage (1970) in his famous paper “Cosmology: A search for two numbers", which was later confirmed using high-precision digital photometry. The first large, multiband CCD sample of distant SNe was produced by the Calán-Tololo survey carried out from the Cerro Tololo Inter-American Observatory between 1989-1993 (Hamuy et al., 1993b) which ended with the publication of 29 $BVRI$ optical SNe Ia light curves (Hamuy et al., 1996b). In 1993 astronomers of the Center for Astrophysics (CfA) started a photometric monitoring campaign of SNe Ia using CCD detectors at the Fred Lawrence Whipple Observatory, which yielded a first release of 22 SNe Ia optical ($BVRI$) light curves (Riess et al., 1999), a second release of 44 SNe Ia (Jha et al., 2006) and more recently a third release (CfA 3) of 185 SNe Ia observed between 2001-2008 (Hicken et al., 2009). The extensive Lick Observatory Supernova Search (LOSS) program carried out since 1998 has produced more than 200 $BVRI$ SNe Ia light curves (Li et al., 2000; Filippenko et al., 2001; Ganeshalingam et al., 2010; Stahl et al., 2019). The Carnegie Supernova Program (CSP) carried out between 2004-2009 from Las Campanas Observatory (LCO) meant a significant advance in the quality of the SN Ia optical light curves, thanks to the use of a uniform photometric system, in situ measurements of the full transmission curves for the telescope/filter/CCD system, and by expanding the survey to NIR $YJHK$ bands (Hamuy et al., 2006). The CSP released two initial datasets (Contreras et al., 2010; Stritzinger et al., 2011), and a third final data release published by Krisciunas et al. (2017) which contains the overall CSP I dataset of 134 SNe Ia. Since 2017, the Foundation Supernova Survey has been obtaining $griz$ light curves with the Pan-STARRS telescope on the peak of Haleakala on the island of Maui. A first data release of 225 SNe Ia was recently published by Foley et al. (2018) ²²2We omit from this summary the high-z surveys designed for the measurement of dark energy..

Along with obtaining larger samples of SNe Ia with increasing precision, observing cadence and wavelength coverage, the success of the SNe Ia method has critically relied on the developments of novel techniques for the standardization of their peak luminosities, such as the correction light curve decline rate (Phillips, 1993; Riess et al., 1995), host-galaxy extinction (Riess et al., 1996; Phillips et al., 1999), and, most recently, host-galaxy mass (Kelly et al., 2010; Sullivan et al., 2010; Burns et al., 2018). The great variety of distant SN Ia samples and the different techniques implemented in the standardization of their peak magnitudes afford an opportunity to study possible systematic differences in the SNe Ia method, as will be seen in the following section.

The purpose of this section is to derive an additional evaluation of the systematic uncertainties in the determination of $H_{0}$ from the CDL approach to that addressed by Freedman et al. (2019) and Riess et al. (2019). As mentioned in the previous section, our strategy consists in leaving the R1 calibration out of this discussion, to focus on the assessment of R2 applying both the Cepheid and the TRGB calibrations to several modern samples of nearby SNe Ia, and study the R3 calibration using various datasets and methodologies for standardizing the luminosities of distant SNe Ia. Having multiple datasets/methodologies affords a novel opportunity to empirically assess the internal consistency, possible systematic differences in the SNe Ia technique, and derive a more precise value of $H_{0}$ by combining independent datasets.

Ever since the work of Rust (1974), Pskovskii (1977) and Phillips (1993), it was unambiguously demonstrated that SNe Ia were not perfect standard candles in the optical bands, and that their peak magnitudes were correlated with the width of their light curves. The gathering of the first dataset of digital photometry for SNe in the Hubble flow by the Calán-Tololo survey confirmed such correlation and proved that it was possible to successfully standardize their peak magnitudes to unrivaled levels of 0.14 mag, or 7% in distance (Hamuy et al., 1995, 1996a). As the distant samples became more numerous, it was possible to identify additional parameters such as SN colour (Lira, 1996; Riess et al., 1996; Tripp, 1998; Phillips et al., 1999), or host galaxy properties to further standardize the SN luminosities (Kelly et al., 2010; Sullivan et al., 2010). Several novel methodologies were developed for the analysis of greater and higher quality datasets, and improve the usefulness of SNe Ia as distance indicators, as will be summarized below.

Datasets and Methodologies. The selection of the methodologies employed for this study is driven by the a priori decision to not alter the original analysis of the distant SNe performed by their authors and consistently apply such formalisms to the nearby SNe. Given this constraint we are able to employ six prescriptions for the standardization of the SN luminosities. For four of them we calculate ourselves the standardized peak magnitudes for the nearby SNe, and for two methodologies such magnitudes are available in the literature:

• •

  the Hamuy et al. (1996a, H96) technique which used $BVI$ photometry for a subsample of 26 Calán-Tololo distant SNe

• •

  the Phillips et al. (1999, P99) approach which employed $BVI$ light curves for a subsample of 40 Calán-Tololo+CfA distant SNe

• •

  the Folatelli et al. (2010, F10) implementation based on $JH$ photometry from 31 CSP distant SNe

• •

  the Kattner et al. (2012, K12) model which is based on $JH$ data for the 24 best-observed CSP distant SNe

• •

  the Freedman et al. (2019, F19) method based on the $BiJH$ photometry from 99 CSP distant SNe

• •

  the Riess et al. (2019, R19) method based on the $u^{\prime}$$g^{\prime}$$r^{\prime}$$i^{\prime}$$UBVRI$ Supercal dataset, a combination of 217 CSP, LOSS, and CfA SNe.

Table 1 summarizes the six prescriptions employed for this work. In the case of the first four methods (H96, P99, F10, K12), we remeasure the light curve parameters for each of the nearby SNe, such as peak magnitude, colour, and decline rate directly from the data, that is, without attempting to apply a light curve fitter. These parameters are obtained from a simple Legendre polynomial fit performed around maximum light and the scatter around the fit yields the peak magnitude error (with an adopted minimum of 0.02 mag). The magnitude decline, $\Delta m_{15}(B)$, is computed by interpolating the $B$ magnitude directly from the data at an epoch of 15 days past maximum, and subtracting the $B$ peak magnitude. A minimum uncertainty of 0.04 mag is adopted for $\Delta m_{15}(B)$. In this manner we maintain a uniform method of calibration. In these first four papers, a calibration recipe is provided to correct the peak magnitude to a standard candle (absolute magnitude) value. We use this calibration as given in these papers but apply it to our directly measured light curve parameters. In all cases we apply Galactic reddening corrections from Schlafly & Finkbeiner (2011, SF11), although H96 used Burstein & Heiles (1982, BH82), while P99 employed Schlegel et al. (1998, SFD98). To ensure internal consistency, we compute the differences among BH82, SFD98 and SF11 for each of the samples of distant SNe and apply the corresponding corrections. All of the technical details employed for computing the standardized absolute magnitudes can be found in the Appendix, for each of these four methods.

Table 5 presents the resulting light curve parameters for the nearby SNe with TRGB distances, for which we are able to apply the H96 method. For each SN we present their $BVI$ absolute magnitudes standardized to an equivalent decline rate of $\Delta m_{15}(B)$=1.1. The uncertainties quoted for the individual absolute magnitudes are, by choice, the quadrature sum of the uncertainties in the measured parameters and the standardization coefficients, without attempting to estimate systematic errors. Table 6 presents the same information for the nearby SNe with Cepheid distances, for which we are able to apply the H96 method. Likewise, the pair of Tables 7 and 8 present the results for the $BVI$ filters using the P99 method. In Tables 9 and 10 we present the results for the $JH$ filters applying the F10 technique. Similarly, Tables 11 and 12 summarize the same but using the K12 method. At the bottom of these tables we provide, for each filter, the weighted mean absolute magnitude for the whole sample of nearby SNe that we are able to employ in each case, the weighted standard deviation, the standard error of the mean, the error of the weighted mean, and the number of SNe employed³³3Not surprisingly, the dispersion and mean error decreased over time from H96 to K12 as a result improvements in the photometry.. In the following analysis we also use mean absolute magnitudes for different subsamples of the nearby SNe listed in such tables.

For the remaining two cases, namely, F19 and R19, the standardized peak magnitudes of the nearby SNe were derived with a light curve fitter by their own authors. They qualify for this study because both the nearby and distant SN corrected peak magnitudes were analyzed in a consistent manner and the data are publicly available. In the Appendix we summarize each of these two methodologies and the relevant parameters drawn from each of them.

Since we apply each of these six prescriptions to one or more bandpasses, we are able to study a total of 12 methodology/bandpass combinations, namely, H96($B$), H96($V$), H96($I$), P99($B$), P99($V$), P99($I$), F10($J$), F10($H$), K12($J$), K12($H$), F19($B$), and R19($B$). In the case of H96 we derive two sets of solutions, as explained in the Appendix, raising to 15 the number of cases studied. For each of these possible combinations we compute absolute magnitudes and $H_{0}$ values. We perform this analysis independently for the TRGB and Cepheid calibrations, that is, we obtained a total of 30 sets of absolute magnitudes and $H_{0}$ values. For each of these methodology/bandpass combinations we attempt to include in the analysis as many of the 18 and 19 nearby SNe with with TRGB and Cepheid distances, respectively. But in some cases, we have to exclude objects that lack the indispensable data required for standardizing their magnitudes in an identical manner as their corresponding distant counterparts. Given that each methodology draws a different sample of nearby SNe, the resulting absolute magnitudes and $H_{0}$ values are subject to different systematic uncertainties. Hence, care has to be exercised when comparing these techniques, as explained below.

We assess the robustness of the two highest rungs of the Cosmic Distance Ladder (CDL) for Type Ia supernovae and the corresponding determination of the Hubble-Lemaître constant. In this analysis we hold fixed the first rung of the CDL (R1) as the distance modulus to the LMC, 18.48$\pm$0.02, determined to a 1% precision level using DEB stars (Pietrzyński et al., 2019). For the second rung (R2) we analyze the two currently most competitive methods, the TRGB and Cepheid luminosity calibration of Type Ia supernovae in nearby galaxies. Finally, for the third rung of the CDL (R3) we analyze various modern digital samples of SNe Ia in the smooth Hubble flow, such as the Calán-Tololo, CfA, CSP, Supercal datasets, and six prescriptions to standardize their optical and NIR peak luminosities. We apply each of these six prescriptions to one or more bandpasses, leading to a total of 15 determinations of $H_{0}$ from all possible combinations of bandpasses and methodologies when using the TRGB calibration, and 15 additional determinations for the Cepheid calibration. This metadata analysis allowed us to draw the following conclusions:

$\bullet$ No matter which SN sample, bandpass or methodology is employed for standardizing the SN luminosities, in all cases the F19 TRGB calibration yields smaller $H_{0}$ values than the R19 Cepheid calibration, a direct consequence of the systematic difference in the distance moduli calibrated from the TRGB and Cepheid methods. From the TRGB calibration we obtain a mean value of $H_{0}$=69.5$\pm$1.5 $\,\rm km\,s^{-1}\,Mpc^{-1}$ ($\sigma$), whereas from the Cepheid method we find $H_{0}$=73.5$\pm$1.5 $\,\rm km\,s^{-1}\,Mpc^{-1}$. Adding in quadrature the systematic uncertainty in the TRGB method of 0.033 mag (Freedman et al., 2019) and in the Cepheid technique of 0.030 mag (Riess et al., 2019) (excluding the systematic uncertainty in the adopted LMC distance modulus), the systematic offset between the TRGB and Cepheid calibrations can be clearly seen, with a significance of 1.6 $\sigma$ (see Fig. 3).

$\bullet$ Selecting the three most independent possible methodologies/bandpasses (the $V$ band by Phillips et al. (1999), the $J$ band by Kattner et al. (2012), and the $B$ band by Riess et al. (2019)), we obtain $H_{0}$=69.9$\pm$0.8 and $H_{0}$=73.5$\pm$0.7 $\,\rm km\,s^{-1}\,Mpc^{-1}$from the TRGB and Cepheid calibrations, respectively. Adding in quadrature the systematic uncertainty in the TRGB method of 0.033 mag (Freedman et al., 2019) and in the Cepheid technique of 0.030 mag (Riess et al., 2019) (excluding the systematic uncertainty in the adopted LMC distance modulus), this subset reveals a significant 2.0 $\sigma$ systematic difference in the calibration of R2.

$\bullet$ If R1 and R2 are held fixed, the different formalisms developed for standardizing the SN peak magnitudes yield consistent results, with a standard deviation of 1.5 $\,\rm km\,s^{-1}\,Mpc^{-1}$  that is, SNe Ia are able to anchor R3 to a level of 2% precision. This internal agreement yielded by SNe Ia, either using the TRGB or Cepheid calibrations, is remarkable as it comprises light curves of increasingly quality, starting with the Calán-Tololo $BVI$ sample, the first digital survey carried out in the early 90s, various releases of the CfA project, and the most modern CSP dataset obtained over recent years with a uniform photometric system over a wide range of optical and NIR bandpasses. This study demonstrates that SNe Ia have provided a remarkably robust calibration of R3 for over 25 years.

We thank Chris Burns for his kind and detailed responses to our inquiries about his analysis of the CSP data. We are grateful to Adam Riess and Chuck Bennett for sending us valuable comments on a previous version of this paper, which were incorporated into our study. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. MH acknowledges support from the Hagler Institute of Advanced Study at Texas A&M University. NBS has been supported by the NSF grant AST-1613455 and the Mitchell/Heep/Munnerlyn Chair in Observational Astronomy. Additional support has come from the George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy.

All data used in this paper are available on request to the corresponding author.