A 2.4% Determination of the Local Value of the Hubble Constant
We use the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST) to reduce the uncertainty in the local value of the Hubble constant from 3.3% to . The bulk of this improvement comes from new, near-infrared observations of Cepheid variables in 11 host galaxies of recent type Ia supernovae (SNe Ia), more than doubling the sample of reliable SNe Ia having a Cepheid-calibrated distance to a total of 19; these in turn leverage the magnitude-redshift relation based on SNe Ia at . All 19 hosts as well as the megamaser system NGC4258 have been observed with WFC3 in the optical and near-infrared, thus nullifying cross-instrument zeropoint errors in the relative distance estimates from Cepheids. Other noteworthy improvements include a 33% reduction in the systematic uncertainty in the maser distance to NGC4258, a larger sample of Cepheids in the Large Magellanic Cloud (LMC), a more robust distance to the LMC based on late-type detached eclipsing binaries (DEBs), HST observations of Cepheids in M31, and new HST-based trigonometric parallaxes for Milky Way (MW) Cepheids.
We consider four geometric distance calibrations of Cepheids: (i) megamasers in NGC4258, (ii) 8 DEBs in the LMC, (iii) 15 MW Cepheids with parallaxes measured with HST/FGS, HST/WFC3 spatial scanning and/or Hipparcos, and (iv) 2 DEBs in M31. The Hubble constant from each is , , , and km s Mpc, respectively. Our best estimate of H km s Mpc combines the anchors NGC4258, MW, and LMC, yielding a determination (all quoted uncertainties include fully-propagated statistical and systematic components). This value is higher than km s Mpcpredicted by CDM with 3 neutrino flavors having a mass of 0.06 eV and the new Planck data, but the discrepancy reduces to relative to the prediction of km s Mpcbased on the comparably precise combination of WMAP+ACT+SPT+BAO observations, suggesting that systematic uncertainties in cosmic microwave background radiation measurements may play a role in the tension.
If we take the conflict between Planck high-redshift measurements and our local determination of H at face value, one plausible explanation could involve an additional source of dark radiation in the early Universe in the range of N. We anticipate further significant improvements in H from upcoming parallax measurements of long-period MW Cepheids.
The Hubble constant (H) measured locally and the sound horizon observed from the cosmic microwave background radiation (CMB) provide two absolute scales at opposite ends of the visible expansion history of the Universe. Comparing the two gives a stringent test of the standard cosmological model. A significant disagreement would provide evidence for fundamental physics beyond the standard model, such as time-dependent or early dark energy, gravitational physics beyond General Relativity, additional relativistic particles, or nonzero curvature. Indeed, none of these features has been excluded by anything more compelling than a theoretical preference for simplicity over complexity. In the case of dark energy, there is no simple explanation at present, leaving direct measurements as the only guide among numerous complex or highly tuned explanations.
Recent progress in measuring the CMB from WMAP (Hinshaw et al. 2013; Bennett et al. 2013) and Planck (Planck Collaboration et al. 2016) have reduced the uncertainty in the distance to the surface of last scattering () to below 0.5% in the context of CDM, motivating complementary efforts to improve the local determination of H to percent-level precision (Suyu et al. 2012; Hu 2005). Hints of mild tension at the level with the measurements of H stated by Riess et al. (2011), Sorce et al. (2012), Freedman et al. (2012), and Suyu et al. (2013) have been widely considered and in some cases revisited in great detail (Efstathiou 2014; Dvorkin et al. 2014; Bennett et al. 2014; Spergel et al. 2015; Becker et al. 2015), with no definitive conclusion except for highlighting the value of improvements in the local observational determination of H.
1.1 Past Endeavors
Considerable progress in the local determination of H has been made in the last 25 years, assisted by observations of water masers, strong-lensing systems, SNe, the Cepheid period-luminosity (–) relation (also known as the Leavitt law; Leavitt & Pickering 1912), and other sources used independently or in concert to construct distance ladders (see Freedman & Madore 2010; Livio & Riess 2013, for recent reviews).
A leading approach utilizes Hubble Space Telescope (HST) observations of Cepheids in the hosts of recent, nearby SNe Ia to link geometric distance measurements to other SNe Ia in the expanding Universe. The SN Ia HST Calibration Program (Sandage et al. 2006) and the HST Key Project (Freedman et al. 2001) both made use of HST observations with WFPC2 to resolve Cepheids in SN Ia hosts. However, the useful range of that camera for measuring Cepheids, Mpc, placed severe limits on the number and choice of SNe Ia which could be used to calibrate their luminosity (e.g., SNe 1937C, 1960F, 1974G). A dominant systematic uncertainty resulted from the unreliability of those nearby SNe Ia which were photographically observed, highly reddened, spectroscopically abnormal, or discovered after peak brightness. Only two objects (SNe 1990N and 1981B) used by Freedman et al. (2001, 2012) and four by Sandage et al. (2006) (the above plus SN 1994ae and SN1998aq) were free from these shortcomings, leaving a very small set of reliable calibrators relative to the many hundreds of similarly reliable SNe Ia observed in the Hubble flow. The resulting ladders were further limited by the need to calibrate WFPC2 at low flux levels to the ground-based systems used to measure Cepheids in a single anchor, the Large Magellanic Cloud (LMC). The use of LMC Cepheids introduces additional systematic uncertainties because of their shorter mean period ( dex) and lower metallicity ( dex, Romaniello et al. 2008) than those found with HST in the large spiral galaxies that host nearby SNe Ia. Despite careful work, the estimates of H by the two teams (each with 10% uncertainty) differed by 20%, owing in part to the aforementioned systematic errors.
More recently, the SH0ES (Supernovae, H, for the Equation of State of dark energy) team used a number of advancements to refine this approach to determining H. Upgrades to the instrumentation of HST doubled its useful range for resolving Cepheids (leading to an eight-fold improvement in volume and in the expected number of useful SN Ia hosts), first with the Advanced Camera for Surveys (ACS; Riess et al. 2005, 2009b) and later with the Wide Field Camera 3 (WFC3; Riess et al. 2011, hereafter R11) owing to the greater area, higher sensitivity, and smaller pixels of these cameras. WFC3 has other superior features for Cepheid reconnaissance, including a white-light filter (F350LP) that more than doubles the speed for discovering Cepheids and measuring their periods relative to the traditional F555W filter, and a 5 square arcmin near-infrared (NIR) detector that can be used to reduce the impact of differential extinction and metallicity differences across the Cepheid sample. A precise geometric distance to NGC4258 measured to 3% using water masers (Humphreys et al. 2013, hereafter H13) has provided a new anchor galaxy whose Cepheids can be observed with the same instrument and filters as those in SN Ia hosts to effectively cancel the effect of photometric zeropoint uncertainties in this step along the distance ladder. Tied to the Hubble diagram of 240 SNe Ia (now SNe Ia; Scolnic et al. 2015; Scolnic & Kessler 2016), the new ladder was used to initially determine H with a total uncertainty of 4.7% (Riess et al. 2009a, hereafter R09). R11 subsequently improved this measurement to 3.3% by increasing to 8 the number of Cepheid distances to reliable SN Ia hosts, and formally including HST/FGS trigonometric parallaxes of 10 Milky Way (MW) Cepheids with distance kpc and individual precision of 8% (Benedict et al. 2007). The evolution of the error budget in these measurements is shown in Figure 1.
Here we present a broad set of improvements to the SH0ES team distance ladder including new near-infrared HST observations of Cepheids in 11 SN Ia hosts (bringing the total to 19), a refined computation of the distance to NGC4258 from maser data, additional Cepheid parallax measurements, larger Cepheid samples in the anchor galaxies, and additional SNe Ia to constrain the Hubble flow. We present the new Cepheid data in §2 and in S. L. Hoffmann et al. (2016, in prep.; hereafter H16). Other improvements are described throughout §3, and a consideration of analysis variants and systematic uncertainties is given in §4. We end with a discussion in §5.
2 HST Observations of Cepheids in the SH0ES Program
Discovering and measuring Cepheid variables in SN Ia host galaxies requires a significant investment of observing time on HST. It is thus important to select SN Ia hosts likely to produce a set of calibrators that is a good facsimile of the much larger sample defining the modern SN Ia magnitude-redshift relation at (e.g., Scolnic et al. 2015; Scolnic & Kessler 2016). Poor-quality light curves, large reddening, atypical SN explosions, or hosts unlikely to yield a significant number of Cepheids would all limit contributions to this effort. Therefore, the SH0ES program has been selecting SNe Ia with the following qualities to ensure a reliable calibration of their fiducial luminosity: (1) modern photometric data (i.e., photoelectric or CCD), (2) observed before maximum brightness and well thereafter, (3) low reddening (implying mag), (4) spectroscopically typical, and (5) a strong likelihood of being able to detect Cepheids in its host galaxy with HST. This last quality translates into any late-type host (with features consistent with the morphological classification of Sa to Sd) having an expectation of Mpc, inclination , and apparent size . To avoid a possible selection bias in SN Ia luminosities, the probable distance of the host is estimated via the Tully-Fisher relation or flow-corrected redshifts as reported in NED
The occurrence of SNe Ia with these characteristics is unfortunately quite rare, leading to a nearly complete sample of 19 objects observed between 1993 and 2015 (see Table 1). Excluding supernovae from the 1980s, a period when modern detectors were rare and when suitable SNe Ia may have appeared and gone unnoticed, the average rate of production is year. Regrettably, it will be difficult to increase this sample substantially (by a factor of ) over the remaining lifetime of HST. We estimate that a modest augmentation of the sample (at best) would occur by removing one or more of the above selection criteria, but the consequent increase in systematic uncertainty would more than offset the statistical gain.
Reliable SNe Ia from early-type hosts could augment the sample, with distance estimates based on RR Lyrae stars or the tip of the red-giant branch (TRGB) for their calibration. Unfortunately, the reduced distance range of these distance indicators for HST compared to Cepheids (2.5 mag or Mpc for TRGB, 5 mag or Mpc for RR Lyrae stars) and the factor of smaller sample of SNe Ia in early-type hosts limits the sample increase to just a few additional objects (SN 1994D, SN 1980N, 1981D, and SN 2006dd with the latter 3 all in the same host; Beaton et al. 2016), a modest fraction of the current sample of 19 SNe Ia calibrated by Cepheids.
Figure 2 shows the sources of the HST data obtained on every host we use, gathered from different cameras, filters, time periods, HST programs and observers. All of these publicly available data can be readily obtained from the Mikulski Archive for Space Telescopes (MAST; see Table 1). The utility of the imaging data can be divided into two basic functions: Cepheid discovery and flux measurement. For the former, a campaign using a filter with central wavelength in the visual band and epochs with nonredundant spacings spanning –90 days will suffice to identify Cepheid variables by their unique light curves and accurately measure their periods (Madore & Freedman 1991; Saha et al. 1996; Stetson 1996). Revisits on a year timescale, although not required, will yield increased phasing accuracy for the longest-period Cepheids. Image subtraction can be very effective for finding larger samples of variables (Bonanos & Stanek 2003), but the additional 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).
Flux measurements are required in order to use Cepheids as standard candles for distance measurement and are commonly done with HST filters at known phases in optical (F555W, F814W) and NIR (F160W) bands to correct for the effects of interstellar dust and the nonzero width in temperature of the Cepheid instability strip. We rely primarily on NIR “Wesenheit” magnitudes (Madore 1982), defined as
where , , in the HST system, and . We note that the value of due to the correlation between Cepheid intrinsic color and luminosity is very similar to that due to extinction (Macri et al. 2015), so the value of derived for the latter effectively also reduces the intrinsic scatter caused by the breadth of the instability strip. However, to avoid a distance bias, we include only Cepheids with periods above the completeness limit of detection (given in H16) in our primary fit. (In future work we will use simulations to account for the bias of Cepheids below this limit to provide an extension of the Cepheid sample.)
In HST observations, Cepheid distances based on NIR measurements have somewhat higher statistical uncertainties than those solely based on optical photometry owing to the smaller field of view, lower spatial resolution, and greater blending from red giants. However, as characterized in §4.2, this is more than offset by increased robustness to systematic uncertainties (such as metallicity effects and possible breaks in the slope of the – relation) as well as the reduced impact of extinction and a lower sensitivity to uncertainties in the reddening law. The latter is quantified by the value of in Equation 1, ranging from 0.3 to 0.5 at depending on the reddening law, a factor of lower than the value at . At the high end, the Cardelli et al. (1989) formulation with yields . The Fitzpatrick (1999) formulation with and 2.5 yields and , respectively. At the low end, a formulation appropriate for the inner Milky Way (Nataf et al. 2016) yields . We analyze the sensitivity of H to variations in in §4.
|Galaxy||SN Ia||Exp. time (s)||Prop IDs||UT Date|
Note. – (a) Data obtained with WFC3/IR and F160W. (b) Data obtained with WFC3/UVIS and F555W, F814W, or F350LP used to find and measure the flux of Cepheids. (c) Date of first WFC3/IR observation. (d) Includes time-series data from an earlier program and a different camera — see Fig. 2.
2.1 Cepheid Photometry
The procedure for identifying Cepheids from time-series optical data (see Table 1 and Figure 2) has been described extensively (Saha et al. 1996; Stetson 1996; Riess et al. 2005; Macri et al. 2006); details of the procedures followed for this sample are presented by H16, utilize the DAO suite of software tools for crowded-field PSF photometry, and are similar to those used previously by the SH0ES team. The complete sample of Cepheids discovered or reanalyzed by H16 in these galaxies (NGC4258 and the 19 SN Ia hosts) at optical wavelengths contains 2113 variables above the periods for completeness across the instability strip (with limits estimated using the HST exposure-time calculators and empirical tests as described in that publication). There are 1566 such Cepheids in the 19 SN Ia hosts within the smaller WFC3-IR fields alone. The positions of the Cepheids within each target galaxy are shown in Figure 3. For hosts in which we used F350LP to identify Cepheid light curves, additional photometry was obtained over a few epochs in F555W and F814W. These data were phase-corrected to mean-light values using empirical relations based on light curves in both F555W and F350LP from Cepheids in NGC5584. Figure 4 shows composite Cepheid light curves in for each galaxy. Despite limited sampling of the individual light curves, the composites clearly display the characteristic “saw-toothed” light curves of Population I fundamental-mode Cepheids, with a rise twice as fast as the decline and similar mean amplitudes across all hosts.
For every host, optical data in F555W and F814W from WFC3 were uniformly calibrated using the latest reference files from STScI and aperture corrections derived from isolated stars in deep images to provide uniform flux measurements for all Cepheids. In a few cases, F555W and F814W data from ACS and WFC3 were used in concert with their well-defined cross-calibration to obtain photometry with a higher signal-to-noise ratio (S/N). The cross-calibration between these two cameras has been stable to mag over their respective lifetimes.
As in R11, we calculated the positions of Cepheids in the WFC3 F160W images using a geometric transformation derived from the optical images using bright and isolated stars, with resulting mean position uncertainties for the variables pix. We used the same scene-modeling approach to F160W NIR photometry developed in R09 and R11. The procedure is to build a model of the Cepheid and sources in its vicinity using the superposition of point-spread functions (PSFs). The position of the Cepheid is fixed at its predicted location to avoid measurement bias. We model and subtract a single PSF at that location and then produce a list of all unresolved sources within 50 pixels. A scene model is constructed with three parameters per source (, , and flux), one for the Cepheid (flux) and a local sky level in the absence of blending; the best-fit parameters are determined simultaneously using a Levenberg-Marquardt-based algorithm. Example NIR scene models for each of the 19 SN Ia hosts are shown in Figure 5.
Care must be taken when measuring photometry of visible stellar sources in crowded regions as source blending can alter the statistics of the Cepheid background (Stetson 1987). Typically the mean flux of pixels in an annulus around the Cepheid is subtracted from the measured flux at the position of the Cepheid to produce unbiased photometry of the Cepheid. This mean background or sky would include unresolved sources and diffuse background. However, we can improve the precision of Cepheid photometry by correctly attributing some flux to the other sources in the scene, especially those visibly overlapping with the Cepheid. The consequence of differentiating the mean sky into individual source contributions plus a lower constant sky level is that the new sky level will underestimate the true mix of unresolved sources and diffuse background superimposed with the Cepheid flux (in sparse regions without blending, the original and new sky levels would approach the same value). This effect may usefully be called the sky bias or the photometric difference due to blending and is statistically easily rectified. To retrieve the unbiased Cepheid photometry from the result of the scene model we could either recalculate the Cepheid photometry using the original mean sky or correct the overestimate of Cepheid flux based on the measured photometry of artificial stars added to the scenes. The advantage of the artificial star approach is that the same analysis also produces an empirical error estimate and can provide an estimate of outlier frequency.
Following this approach, we measure the mean difference between input and recovered photometry of artificial Cepheids added to the local scenes in the F160W images and fit with the same algorithms. As in R09 and R11, we added and fitted 100 artificial stars, placed one at a time, at random positions within 5 arcseconds of (but not coincident with) each Cepheid to measure and account for this difference. To avoid a bias in this procedure, we initially estimate the input flux for the artificial stars from the Cepheid period and an assumed – relation (iteratively determined), measure the difference caused by blending, refine the – relation, and iterate until convergence. Additionally, we use the offset in the predicted and measured location of the Cepheid, a visible consequence of blending, to select similarly affected artificial stars to customize the difference measurements for each Cepheid. The median difference for the Cepheids in all SN hosts hosts observed with HST is 0.18 mag, mostly due to red-giant blends, and it approaches zero for Cepheids in lower-density regions such as the outskirts of hosts. The Cepheid photometry presented in this paper already accounts for the sky bias. We also estimate the uncertainty in the Cepheid flux from the dispersion of the measured artificial-star photometry around the 2.7-clipped mean. The NIR Cepheid – relations for all hosts and anchors are shown in Figure 6.
Likewise, in the optical images, we used as many as 200 measurements of randomly placed stars in the vicinity of each Cepheid in F555W and F814W images to measure and account for the photometric difference due to the process of estimating the sky in the presence of blending. Only 10 stars at a time were added to each simulated image to avoid increasing the stellar density. These tests show that similarly to the NIR measurements, uncertainty in the Cepheid background is the leading source of scatter in the observed – relations of the SN hosts. The mean dispersions at F555W and F814W, with values for each host listed in Table 2 in columns 6 and 7, are 0.19 and 0.17 mag, respectively. All SN hosts and NGC4258 display some difference in their optical magnitudes due to blending, with mean values of and mag (bright) in F555W and F814W, respectively. The most crowded case ( and mag) is NGC4424, a galaxy whose Cepheids are located in a circumnuclear starburst region with prominent dust lanes. We tabulate the mean photometric differences due to blending for each host in Table 2, columns 2 and 3. However, the effect of blending largely cancels when determining the color F555WF814W used to measure Cepheid distances via equation (1) since the blending is highly correlated across these bands. Indeed, the estimated change in color across all hosts given in Table 2, column 4 has a mean of only mag (blue) and a host-to-host scatter of 0.01 mag, implying no statistically significant difference from the initial measurement and thus we have not applied these to the optical magnitudes in Table 4. Even the additional scatter in the – relation owing to blending in the optical color measurement is a relatively minor contribution of 0.07 mag. The small correction due to blending in the optical bands does need to be accounted for when using a conventional optical Wesenheit magnitude, F814WF555WF814W, because (unlike the color) the cancellation in is not complete. We find a small mean difference for in our SN hosts of mag (bright) with a host-to-host dispersion in this quantity of 0.03 mag. If uncorrected, this would lead to a 1% underestimate of distances and an overestimate of H for studies that rely exclusively on . The more symmetric effect of blending on than magnitudes results from the mixture of blue blends (which make faint) and red blends (which make bright). These results are consistent with those found from simulations by Ferrarese et al. (2000), who drew similar conclusions. We will make use of these results for in §4.2. Although the net effect of blending for is typically small, the uncertainty it produces is the dominant source of dispersion with a mean of 0.36 mag for the SN hosts, similar in impact and scatter to what was found for .
Note. – median magnitude or color offset derived from tests; dispersion around ; stands for F555W; stands for F814W; ct, with for and the Fitzpatrick (1999) extinction law; defined in text.
Note. – FoV: located within the WFC3/IR field of view. Meas.: good quality measurement within allowed color range and with period above completeness limit. Fit: after global outlier rejection, see §4.1. : median period of the final NIR sample used in this analysis; time interval between first and last NIR epochs; median value of (uncertainties) for Cepheids in each host (see text for definition); apparent dispersion of NIR – relation after outlier rejection.
Note. – stands for F555WF814W and stands for F160W.
Although we quantify and propagate the individual measurement uncertainty for each Cepheid, we conservatively discard the lowest-quality measurements. As in R11, scene models of Cepheids were considered to be useful if our software reported a fitted magnitude for the source with an uncertainty mag, a set of model residual pixels with root-mean square (rms) lower than from the other Cepheid scenes, and a measured difference from the artificial star analyses of mag. In addition, we used a broad (1.2 mag) allowed range of F814WF160W colors centered around the median for each host, similar to the color selection common to optical studies (see H16), to remove any Cepheids strongly blended with redder or bluer stars of comparable brightness. As simulations in §4.1 show, most of these result from red giants but also occasionally from blue supergiants.
1028 of the 1566 Cepheids present in the F160W images of the SN Ia hosts with periods above their respective completeness limits yielded a good quality photometric measurement within the allowed color range. Excessive blending in the vicinity of a Cepheid in lower-resolution and lower-contrast NIR images was the leading cause for the failure to derive a useful measurement for the others. The number of Cepheids available at each step in the measurement process is given in Table 3.
2.2 Statistical Uncertainties in Cepheid Distances
We now quantify the statistical uncertainties that apply to Cepheid-based distance estimates. As described in the previous section, the largest source of measurement uncertainty for (defined in equation 1) arises from fluctuations in the NIR sky background due to variations in blending, and it is measured from artificial star tests; we refer to this as . For SN Ia hosts at 20–40 Mpc and for NGC4258, the mean for Cepheids in the NIR images is mag, but it may be higher or lower depending on the local stellar density. The next term which may contribute uncertainty in equation 1 is . While blending does not change the mean measured optical colors (discussed in §2.1), it does add a small amount of dispersion. The artificial star tests in the optical data yield a mean value for of 0.07 mag across all hosts, with values for each host given in Table 2, column 8). There is also an intrinsic dispersion, , resulting from the nonzero temperature width of the Cepheid instability strip. It can be determined empirically using nearby Cepheid samples which have negligible background errors. We find mag for (0.12 mag for ) using the LMC Cepheids from Macri et al. (2015) over a comparable period range (see Figure 6). This agrees well with expectations from the Geneva stellar models (R. I. Anderson et al. 2016, in prep.). We use this value as the intrinsic dispersion of mean magnitudes. The last contribution comes from our use of random- or limited-phase (rather than mean-phase) F160W magnitudes. Monte Carlo sampling of complete -band light curves from Persson et al. (2004) shows that the use of a single random phase adds an error of 0.12 mag
Thus, we assign a total statistical uncertainty arising from the quadrature sum of four terms: NIR photometric error, color error, intrinsic width and random-phase:
We give the values of for each Cepheid in Table 4. These have a median of 0.30 mag (mean of 0.32 mag) across all fields; mean values for each field range from 0.23 mag (NGC3447) to 0.47 mag (NGC4424). The mean for NGC4258 is 0.39 mag. We also include in Table 4 an estimate of the metallicity at the position of each Cepheid based on metallicity gradients measured from optical spectra of H II regions obtained with the Keck-I 10 m telescope and presented by H16.
3 Measuring the Hubble Constant
The determination of H follows the formalism described in §3 of R09. To summarize, we perform a single, simultaneous fit to all Cepheid and SN Ia data to minimize one statistic and measure the parameters of the distance ladder. We use the conventional definition of the distance modulus, , with a luminosity distance in Mpc and measured as the difference in magnitudes of an apparent and absolute flux, . We express the th Cepheid magnitude in the th host as
where the individual Cepheid parameters are given in Table 4 and was defined in Equation 1. We determine the values of the nuisance parameters and — which define the relation between Cepheid period, metallicity, and luminosity — by minimizing the for the global fit to the sample data. The reddening-free distances for the hosts relative to NGC4258 are given by the fit parameters , while zp is the intercept of the – relation simultaneously fit to the Cepheids of NGC4258.
Uncertainties in the nuisance parameters are due to measurement errors and the limited period and metallicity range spanned by the variables. In R11 we used a prior inferred from external Cepheid datasets to help constrain these parameters. In the present analysis, instead, we explicitly use external data as described below to augment the constraints.
Recent HST observations of Cepheids in M31 provide a powerful ancillary set of Cepheids at a fixed distance to help characterize NIR – relations. Analyses of the HST PHAT Treasury data (Dalcanton et al. 2012) by Riess et al. (2012), Kodric et al. (2015), and Wagner-Kaiser et al. (2015) used samples of Cepheids discovered from the ground with NIR and optical magnitudes from HST to derive low-dispersion – relations. We used the union set of these samples and their WFC3 photometry in F160W measured with the same algorithms as the previous hosts to produce a set of 375 Cepheids with d as shown in Figure 6. We add Equation 2 (actually, a set of such equations) for these data to those from the other hosts, requiring the addition of one nuisance parameter, the distance to M31, but providing a large range of ( dex) for the determination of the – relation slopes. These M31 Cepheids alone constrain the slope to an uncertainty of 0.03 mag dex, a factor of 3 better than the prior used by R11. They also hint at the possible evidence of a break in the – relation at the confidence level (Kodric et al. 2015) if the location of a putative break is assumed a priori to be at 10 days as indicated by optical – relations (Ngeow & Kanbur 2005). To allow for a possible break, we include two different slope parameters in Equation 2 in the primary analysis, one for Cepheids with d and another for d. We will consider alternative approaches for dealing with nonlinear – relations in §4.1.
The SN Ia magnitudes in the calibrator sample are simultaneously expressed as
where the value is the maximum-light apparent -band brightness of a SN Ia in the ith host at the time of -band peak, corrected to the fiducial color and luminosity. This quantity is determined for each SN Ia from its multiband light curves and a light-curve fitting algorithm. For the primary fits we use SALT-II (Guy et al. 2005, 2010). For consistency with the most recent cosmological fits we use version 2.4 of SALT II as used by Betoule et al. (2014) and more recently from Scolnic & Kessler (2016)
The simultaneous fit to all Cepheid and SN Ia data via Equations 2 and 3 results in the determination of , which is the expected reddening-free, fiducial, peak magnitude of a SN Ia appearing in NGC4258. The individual Cepheid – relations are shown in Figure 6. Lastly, H is determined from
where is the independent, geometric distance modulus estimate to NGC4258 obtained through VLBI observations of water megamasers orbiting its central supermassive black hole (Herrnstein et al. 1999; Humphreys et al. 2005; Argon et al. 2007; Humphreys et al. 2008, 2013).
Observations of megamasers in Keplerian motion around a supermassive blackhole in NGC 4258 provide one of the best sources of calibration of the absolute distance scale with a total uncertainty given by H13 of 3%. However, the leading systematic error in H13 resulted from limited numerical sampling of the multi-parameter model space of the system, given in H13 as 1.5%. The ongoing improvement in computation speed allows us to reduce this error.
Here we make use of an improved distance estimate to NGC4258 utilizing the same VLBI data and model from H13 but now with a 100-fold increase in the number of Monte Carlo Markov Chain (MCMC) trial values from in that publication to for each of three independent “strands” of trials or initial guesses initialized near and at % of the H13 distance. By increasing the number of samples, the new simulation averages over many more of the oscillations of trial parameters in an MCMC around their true values. The result is a reduction in the leading systematic error of 1.5% from H13 caused by “different initial conditions” for strands with only MCMC samples to 0.3% for the differences in strands with MCMC samples. The smoother probability density function (PDF) for the distance to NGC4258 can be seen in Figure 7. The complete uncertainty (statistical and systematic) for the maser distance to NGC4258 is reduced from 3.0% to 2.6%, and the better fit also produces a slight 0.8% decrease in the distance, yielding
equivalent to mag.
The term in Equation 4 is the intercept of the SN Ia magnitude-redshift relation, approximately in the low-redshift limit but given for an arbitrary expansion history and for as
measured from the set of SN Ia () independent of any absolute (i.e., luminosity or distance) scale. We determine from a Hubble diagram of up to SNe Ia with a light-curve fitter used to find the individual as shown in Figure 8. Limiting the sample to (to avoid the possibility of a coherent flow in the more local volume; is the redshift in the rest frame of the CMB corrected for coherent flows, see §4.3) leaves 217 SNe Ia (in the next section we consider a lower cut of ). Together with the present acceleration and prior deceleration which can be measured via high-redshift SNe Ia (Riess et al. 2007; Betoule et al. 2014) independently of the CMB or BAO, we find for the primary fit , with the uncertainty in contributing 0.1% uncertainty (see §4.3). Combining the peak SN magnitudes to the intercept of their Hubble diagram as provides a measure of distance independent of the choice of light-curve fitter, fiducial source, and measurement filter. These values are provided in Table 5.
We use matrix algebra to simultaneously express the over 1500 model equations in Equations 2 and 3, along with a diagonal correlation matrix containing the uncertainties. We invert the matrices to derive the maximum-likelihood parameters, as in R09 and R11.
Individual Cepheids may appear as outliers in the – relations owing to (1) a complete blend with a star of comparable brightness and color, (2) a poor model reconstruction of a crowded group when the Cepheid is a small component of the total flux or a resolved cluster is present, (3) objects misidentified as classical Cepheids in the optical (e.g., blended Type II Cepheids), or (4) Cepheids with the wrong period (caused by aliasing or incomplete sampling of a single cycle). For our best fit we identify and remove outliers from the global model fit which exceed (see §4.1 for details), comprising 2% of all Cepheids (or 5% from all SN hosts). We consider alternative approaches for dealing with these outliers and include their impact into our systematic uncertainty in §4.1.
Our best fit using only the maser distance to NGC4258 in Equation 4 to calibrate the Cepheids yields a Hubble constant of km s Mpc (statistical uncertainty only; hereafter “stat”), a determination compared to 4.0% in R11. The statistical uncertainty is the quadrature sum of the uncertainties in the three independent terms in Equation 4. We address systematic errors associated with this and other measurements in §4.
3.1 Additional Anchors
We now make use of additional sources for the calibration of Cepheid luminosities, focusing on those which (i) are fundamentally geometric, (ii) have Cepheid photometry available in the , , and bands, and (iii) offer precision comparable to that of NGC4258, i.e., less than 5%. For convenience, the resulting values of H are summarized in Table 6.
Milky Way Parallaxes
Trigonometric parallaxes to Milky Way Cepheids offer one of the most direct sources of geometric calibration of the luminosity of these variables. As in R11, we use the compilation from van Leeuwen et al. (2007), who combined 10 Cepheid parallax measurements with HST/FGS from Benedict et al. (2007) with those measured at lower precision with Hipparcos, plus another three measured only with significance by Hipparcos. We exclude Polaris because it is an overtone pulsator whose “fundamentalized” period is an outlier among fundamental-mode Cepheids. In their analysis, Freedman et al. (2012) further reduced the parallax uncertainties provided by Benedict et al. (2007), attributing the lower-than-expected dispersion of the – relation of the 10 Cepheids from Benedict et al. (2007) as evidence for lower-than-reported measurements errors. However, we think it more likely that this lower scatter is caused by chance (with the odds against ) than overestimated parallax uncertainty, as the latter is dominated by the propagation of astrometry errors which were stable and well-characterized through extensive calibration of the HST FGS. As the sample of parallax measurements expands, we expect that this issue will be resolved, and for now we retain the uncertainties as determined by Benedict et al. (2007).
We add to this sample two more Cepheids with parallaxes measured by Riess et al. (2014) and Casertano et al. (2015) using the WFC3 spatial scanning technique. These measurements have similar fractional distance precision as those obtained with FGS despite their factor of 10 greater distance and provide two of only four measured parallaxes for Cepheids with d. The resulting parallax sample provides an independent anchor of our distance ladder with an error in their mean of 1.6%, though this effectively increases to 2.2% after the addition of a conservatively estimated =0.03 mag zeropoint uncertainty between the ground and HST photometric systems (but see discussion in §5).
We use the parallaxes and the , , and -band photometry of the MW Cepheids by replacing Equation 2 for the Cepheids in SN hosts and in M31 with
where is the absolute magnitude for a Cepheid with d, and simultaneously fitting the MW Cepheids with the relation
where and is the distance modulus derived from parallaxes, including standard corrections for bias (often referred to as Lutz-Kelker bias) arising from the finite S/N of parallax measurements with an assumed uncertainty of 0.01 mag (Hanson 1979). The , , and -band photometry, measured from the ground, are transformed to match the WFC3 F160W, F555W, and F814W as discussed in the next subsection. Equation 3 for the SNe Ia is replaced with
The determination of for SNe Ia together with the previous term then determines H,
The statistical uncertainty in is now derived from the quadrature sum of the two independent terms in equation 9, and .
For Cepheid photometry not derived directly from HST WFC3, we assume a fully-correlated uncertainty of 0.03 mag included as an additional, simultaneous constraint equation, , to the global constraints with mag. The free parameter, , which expresses the zeropoint difference between HST WFC3 and ground-based data, is now added to Equation 7 for all of the MW Cepheids. This is a convenience for tracking the correlation in the zeropoints between ground-based data and providing an estimate of its size. In future work we intend to eliminate and its uncertainty by replacing the ground-based photometry with measurements from HST WFC3 enabled by spatial scanning (Riess et al. 2014).
Using these 15 MW parallaxes as the only anchor, we find H = km s Mpc(stat). In order to use the parallaxes together with the maser distance to NGC4258, we recast the equations for the Cepheids in NGC4258 in the form of Equation 7 with in place of and the addition of the residual term to these as a convenience for keeping track of the correlation among these Cepheids and the prior external constraint on the geometric distance of NGC 4258. We then add the simultaneous constraint equation with mag. Compared to the use of the maser-based distance in §3, has moved from Equation 4 to the a priori constraint on . This combination gives H = km s Mpc (stat), a measurement that is consistent with the value from NGC4258 to considering only the distance uncertainty in the geometric anchors.
LMC Detached Eclipsing Binaries
In R11 we also used photometry of Cepheids in the LMC and estimates of the distance to this galaxy based on detached eclipsing binaries (DEBs) to augment the set of calibrators of Cepheid luminosities. DEBs provide the means to measure geometric distances (Paczynski & Sasselov 1997) through the ability to determine the physical sizes of the member stars via their photometric light curves and radial velocities. The distance to the LMC has been measured with both early-type and late-type stars in DEBs. Guinan et al. (1998), Fitzpatrick et al. (2002), and Ribas et al. (2002) studied three B-type systems (HV 2274, HV 982, EROS 1044) which lie close to the bar of the LMC and therefore provide a good match to the Cepheid sample of Macri et al. (2015). In R11 we used an average distance modulus for these of mag
The approach using DEBs composed of late-type stars is more reliable and fully empirical because their surface brightness can be estimated from empirical relations between this quantity and color, using interferometric measurements of stellar angular sizes to derive surface brightnesses (Di Benedetto 2005). Pietrzyński et al. (2013) estimated the distance to the center of the LMC to 2% precision using 8 DEBs composed of late-type giants in a quiet evolutionary phase on the helium burning loop, located near the center of the galaxy and along its line of nodes. The individual measurements are internally consistent and yield (sys) mag, with the uncertainty dominated by the accuracy of the surface brightness vs. color relation.
Recently, Macri et al. (2015) presented NIR photometry for LMC Cepheids discovered by the OGLE-III project (Soszynski et al. 2008), greatly expanding the sample size relative to that of Persson et al. (2004) from 92 to 785, although the number of Cepheids with d increased more modestly from 39 to 110. Similarly to the M31 Cepheids, the LMC Cepheids provide greater precision for characterizing the – relations than those in the SN Ia hosts, and independently hint at a change in slope at d (Bhardwaj et al. 2016).
We transform the ground-based , and -band Vega-system photometry of Macri et al. (2015) into the Vega-based HST/WFC3 photometric system in F555W, F814W and F160W, respectively, using the following equations:
where the color terms were derived from synthetic stellar photometry for the two systems using SYNPHOT (Laidler et al. 2008). To determine any zeropoint offsets (aside from the potentially different definitions of Vega) for the optical bands we compared photometry of 97 stars in the LMC observed in and by OGLE-III and in WFC3/F555W and F814W as part of HST-GO program #13010 (P.I.: Bresolin). The latter was calibrated following the exact same procedures as H16, which uses the UVIS 2.0 WFC3 Vegamag zeropoints. The uncertainties of the zeropoints in the optical transformations were found to be only 4 mmag. The change in color, is quite small, at 0.014 mag or a change (decrease) in H of 0.3 % for a value determined solely from an anchor with ground-based Cepheid photometry (LMC or MW). For -band transformed to F160W, the net offset besides the aformentioned color term is zero after cancellation of an 0.02 mag offset measured between HST and 2MASS NIR photometry (Riess 2011) and the same in the reverse direction from the very small count-rate non-linearity of WFC3 at the brightness level of extragalactic Cepheids (Riess 2010). The mean metallicity of the LMC Cepheids is taken from their spectra by Romaniello et al. (2008) to be dex.
Using the late-type DEB distance to the LMC as the sole anchor and the Cepheid sample of Macri et al. (2015) for a set of constraints in the form of Equation 7 yields H = km s Mpc (stat). As in the prior section, these fits include free parameters and , with additional constraint . The Appendix shows how the system of equations is arranged for this fit. The last few equations (see Appendix) express the independent constraints on the external distances (i.e., for NGC4258 and the LMC) with uncertainties contained in the error matrix. Using the anchor combination of NGC 4258 and the LMC, the optimal set for TRGB calibration, gives H= km s Mpc (stat).
Using all three anchors, the same set used by R11 and by Efstathiou (2014), results in H = km s Mpc(stat), a determination. The fitted parameters which would indicate consistency within the anchor sample are mag, within the range of its 0.0568 mag prior, and mag, within range of its 0.0452 mag prior. The metallicity term for the NIR-based Wesenheit has the same sign but only about half the size as in the optical (Sakai et al. 2004) and is not well-detected with mag dex including systematic uncertainties.
DEBs in M31
As discussed in §3, we make use of a sample of 375 Cepheids in M31 in order to help characterize the Cepheid – relations. In principle, we can also use M31 as an anchor in the determination of H by taking advantage of the two DEB-based distance estimates to the galaxy (Ribas et al. 2005; Vilardell et al. 2010) which have a mean of mag.
Yet, there are several obstacles with the use of M31 as an anchor. The PHAT HST program (Dalcanton et al. 2012), which obtained the HST data, did not use the F555W filter, nor did it include time-series data, so we cannot use the same individual, mean-light F555WF814W colors to deredden the Cepheids in F160W as for other SH0ES galaxies (or the individual mean colors to deredden -band data with a 0.03 mag uncertainty as for LMC and MW Cepheids as individual ground-based colors are too noisy). The best available color for measuring the individual reddenings of the M31 Cepheids is F110WF160W so we must recalibrate these colors to match the reddening in the data. Following Riess et al. (2012), we add a constant to these colors so that their mean measured F160W extinction is the same as derived from the mean Cepheid colors in M31 based on data from the ground-based DIRECT program (Kaluzny et al. 1998)
On the other hand, as previously discussed, DEB distances for early-type stars (the only ones currently measured in M31) include significant inputs from non-LTE stellar model atmospheres with systematic uncertainties that are hard to assess. It is somewhat reassuring to note that in the LMC, where both types of DEBs have been measured, the difference in the distance moduli obtained from either type is only mag, a test with the same precision as the early-type DEB distance to M31. Future measurements of late-type DEBs or water masers in M31 (Darling 2011) would place M31 as an anchor on equal footing with the others.
To be conservative, we use as our primary determination of H the result from the combination of NGC4258 masers, MW parallaxes, and LMC late-type DEBs (the same set of anchors used by R11): H = km s Mpc (stat). Note, however, the consistency of our primary result with the result using M31 alone. If M31 were included together with the other anchors, the resulting value of H would be km s Mpc (stat).
While the global model accounts for the covariance between all distances and model parameters, we can explore the internal agreement of the Cepheid and SN distance estimates by deriving approximate Cepheid-only distances for the 19 hosts. For each host, we remove only its SN distance from the global fit and derive its Cepheid distance, based on the remaining data. The result is a set of Cepheid distances to each host which are independent of their SN distances (although these distances are slightly correlated with each other and thus do not provide a substitute for the full analysis which accounts for such covariance). The results are listed in Table 5, column 5 as approximate Cepheid distances (i.e., ignoring the covariance) and Figure 9 shows the SN distances versus those from Cepheid optical and NIR magnitudes. Figure 10 shows an approximation to the full distance-ladder fit to provide a sense of the sampling using the previously described approximations. These approximations should be good to 0.01–0.02 mag. The resulting relation between the SN and Cepheid-based distances will be considered in the next section. The Cepheid-based distances for 7 of the 8 hosts used in R11 have a mean difference of 0.01 mag and a dispersion of 0.12 mag. The eight host, N4038, shifted from to relative to the SN-inferred distances (, closer in R16). The shift primarily arises because we conservatively excluded a unique set of 10 variables from R11 with ultra-long periods ( days) due to very sparse phase coverage and the poorly constrained properties of the P-L relation for these intrinsically rare objects (Bird et al. 2009; Fiorentino et al. 2012).
Note. – (a) For SALT-II, 0.1 mag added in quadrature to fitting error. (b) Approximate, SN-independent Cepheid-based distances as described at the end of §3.
4 Analysis Systematics
The statistical uncertainties quoted thus far include the full propagation of all known contributions as well as the degeneracies resulting from simultaneous modeling and characterization of the whole dataset of Cepheids ( 1000 in SN hosts), 19 SNe Ia, 15 MW parallaxes, the DEB-based distance to the LMC, and the maser distance to NGC4258. Our model formally contains parameters used to propagate what were considered sources of systematic uncertainties in other analyses (Freedman et al. 2001, 2012; Sandage et al. 2006) such as zeropoint errors, metallicity dependences, and the slopes and breaks in the – relation, therefore our statistical uncertainties incorporate many effects that others consider among systematics (see Appendix).
Following the approach of R09 and R11, we therefore explore reasonable alternatives to the global determination of H which are not easily parameterized for inclusion in the framework of §3, and we use these to determine an additional systematic error component. While truly unknown systematic errors can never be ruled out, we address this possibility in §4 by comparing our measurement to independent measurements of H which do not utilize SN-based distance measurements.
4.1 Cepheid Systematics
The Cepheid outlier fraction in §3 is 2% for all hosts (or 5% across all SN hosts), smaller than the 15%–20% in R11. This reduction in the outlier fraction results largely from the use of a color selection in F814WF160W around 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 distance- and period-independent, insensitive to reddening, and anchored to the physical properties of Cepheids (i.e., stars with spectral types F–K). The well-characterized LMC Cepheids from Macri et al. (2015) have a mean of 0.96 mag with a dispersion of just 0.10 mag, much smaller than the allowed 1.2 mag breadth which alone would exclude only stars hotter than early-F or cooler than late-K (i.e., colors which cannot result from Cepheids). Because measurement errors owing to blending are correlated across bands, the uncertainty in this color is smaller than either band and a factor of 6 smaller than the allowed range, so colors outside the range primarily result from color blends rather than noise. Doubling the breadth of the color cut decreased H by 0.9 km s Mpcand removing a color cut altogether lowered H by an additional 0.2 km s Mpc, both shifts much smaller than the statistical uncertainty.
We further tested the use of our color cut by simulating the appearance of a distribution of Cepheids in a galaxy at Mpc using star catalogues of the LMC. Cepheids with low optical blending (hence identifiable by amplitude and allowed range in F555WF814W; see H16 and Ferrarese et al. 2000) but with significant NIR blending are most often blended with red giants. This shifts their colors redward in F814WF160W to a degree, on average, that is proportional to their local surface brightness. While we account for this mean, blended sky level in our photometry, the “direct hits” by red or blue sources are removed by the color cut. However, blending may still occur with stars of a similar color, such as the (less common) yellow supergiants, or the sample may include a small number of objects erroneously identified as Cepheids. For these reasons we still identify and remove a small fraction of the sample (%) as outliers from the – relations.
A number of reasonable approaches would likely suffice for identifying these outliers as demonstrated for the R11 sample (Becker et al. 2015; Efstathiou 2014; Kodric et al. 2015). R11 used a threshold to identify outliers from the individual -band – relations for their primary H analysis, while evaluating the impact of no outlier rejection to determine the sensitivity of H to this step. Efstathiou (2014) used a similar threshold but applied to outliers of the final, global fit. Kodric et al. (2015) used a global rejection as well but recalculated the global fit after removing the single most deviant point until none remained above the threshold. Becker et al. (2015) applied a Bayesian characterization of outliers, attributing them to a second, contaminating distribution with uniform properties. However, the artificial-star tests and LMC analysis indicate that the outliers are well described by the tails of the blending distribution. For our primary fit we use a global rejection of , the threshold where the of our global fit matches that of a normal distribution with the same rejection applied. Following Kodric et al. (2015) we recalculated the global fit after removing the single most deviant point until none remain above the threshold. We also performed as variants a single-pass, global rejection and a rejection from individual – relations, both applied at the aforementioned threshold and a larger threshold, as well as no outlier rejection. The results of all these variants are presented in Table 8. These variants of outlier rejection changed H by less than 0.6 km s Mpc. Because the outlier fraction of 2% is quite small here and the Cepheid slope is better constrained relative to that of R11, we conclude that the outlier analysis does not warrant further consideration. The Cepheids in Table 4 are those that passed the best-fit, global outlier rejection.
We consider a number of variants related to the Cepheid reddening law. Besides the primary fits, which use a Fitzpatrick (1999) law with , we also use and alternative formulations of the reddening law from Cardelli et al. (1989) and Nataf et al. (2016). We also explore variants related to a possible break in the Cepheid – relation near 10 days. Our primary fit allows for a break or discontinuity (while not requiring one) by providing two independent slope parameters: one for Cepheids at d and one for d. The allowance for a break only increases the uncertainty in H by 0.01 km s Mpcwhich is negligible. We also evaluate changes in H arising from a single-slope formulation for all periods, as well as from removing all Cepheids with d, or removing those with d as shown in Table 8. Interestingly, we see no evidence of a change in slope at d in the – relation to a precision of 0.02 mag dex in the global fit to all Cepheids. Hints of an increasing (LMC) or decreasing (M31) slope with period are not confirmed in this broader analysis with many more hosts. We further included variants that ignored the possibility of a Cepheid metallicity dependence and another based on a recalibration of nebular oxygen abundances (Bresolin 2011). We also included a variant foregoing the use of optical colors to correct for NIR reddening as it tends to be low. The results of all these variants are presented in Table 8.
Comparing the individual SN distances to the previously discussed approximate, independent Cepheid distances, we find none of the hosts to be an outlier. There is also no evidence () for a trend between SN and Cepheid NIR distances over a 3.8 mag range in distance modulus (equivalent to a factor of 5.8 in distance). This suggests that Cepheids are not associated with significant unresolved luminosity overdensities across the range of 7–38 Mpc spanned by our sample of SN hosts and one of our anchors (NGC4258). This agrees well with Senchyna et al. (2015), who used HST to determine that only 3% of Cepheids in M31 are in parsec-scale clusters. Further, only a small fraction of these would alter Cepheid photometry at the resolution available from the ground or the similar resolution of HST at the distance of the SN hosts.
Lastly, we test for a dependence of the measured Cepheid distance with the level of blending by comparing the six hosts with blending higher than the inner region of NGC4258 to the remaining 13. The difference in the mean model residual distances of these two subsamples is mag, providing no evidence of such a dependence.
4.2 Optical Wesenheit Period-Luminosity Relation
The SH0ES program was designed to identify Cepheids from optical images and to observe them in the NIR with F160W to reduce systematic uncertainties related to the reddening law, its free parameters, sensitivity to metallicity, and breaks in the – relation. However, some insights into these systematics may be garnered by replacing the NIR-based Wesenheit magnitude, , with the optical version used in past studies (Freedman et al. 2001), , where and the value of here is times larger than in the NIR. The advantage of this change is the increase in the sample by a little over 600 Cepheids in HST hosts owing to the greater field of view (FoV) of WFC3/UVIS. Of these additional Cepheids, 250 come from M101, 140 from NGC4258, and the rest from the other SN hosts. In Table 8 we give results based on Cepheid measurements of instead of for the primary fit variant with all 4 anchors, the primary fit anchor set of NGC4258, MW and LMC and for NGC4258 as the sole anchor.
The fits for all Cepheids with data generally show a significantly steeper slope for d than for d, with our preferred variant giving a highly significant slope change of mag dex. We also see strong evidence of a metallicity term with a value of mag dex for our preferred fit, also highly significant and consistent with the value from Sakai et al. (2004) of mag dex. The constraint on the metallicity term is nearly unchanged when using NGC4258 as the sole anchor, mag dex, demonstrating that the metallicity constraint comes from the metallicity gradients and SN host-to-host distance variations and not from improving the consistency in the distance scale of different anchors.
The dispersion between the individual SN and Cepheid distances (see Figure 9 and the next subsection) is mag for , somewhat smaller than mag from . Some reduction may be expected because a larger number of Cepheids are available in the optical relative to the NIR. However, the SNe have a mean distance uncertainty of 0.12 mag and the sets of magnitudes in each host have a typical mean uncertainty of 0.06 mag, indicating that the dispersion between SN and Cepheid distances is already dominated by the SN error and leaving little room for improvement with additional Cepheids. The one exception is NGC4424, where the paucity of variables with valid NIR measurements results in a Cepheid-dominated calibration error which is reduced by a third by adding Cepheids only available in the optical. Based on the good agreement between the relative SN and Cepheid distances and uncertainties, we conclude that the intrinsic SN dispersion of 0.1 mag from SALT-II is reasonable.
Using the three primary anchors and the optical Wesenheit – relation, we find H = km s Mpc (stat), extremely similar to the NIR-based result and with a statistical error just 0.05 km s Mpcsmaller. We determined the systematic error for the optical Wesenheit from the dispersion of its variants after eliminating those expected to perform especially poorly in the optical: no allowance for reddening, no metallicity term, and no lower-period cutoff. Even without these variants, the systematic error in the optical of is still considerably worse than its NIR counterpart and is also larger than the statistical error. The reason is that changes to the treatment of reddening, metallicity, – relation breaks, and outlier rejection cause larger changes in H for the optical Wesenheit magnitudes than for the NIR counterparts. This is a fairly uniform result, not driven by any one or two variants. For example, changing from the preferred Fitzpatrick (1999) reddening law to the alternative formulations by Cardelli et al. (1989) or Nataf et al. (2016) changes H by and 0.15 km s Mpc for , respectively. These same variants change H by -2.15 and 3.82 km s Mpc for the data. This increased sensitivity to the reddening law is a natural consequence of the larger value of . Changing the two-slope – formulation to a single slope or restricting the period range to or d changes H by , , and km s Mpc, respectively, for the optical formulation. These changes are generally smaller for the NIR Wesenheit at 0.03, -1.59, and -0.18 km s Mpc, respectively. Finally, changing the outlier clipping from one-at-a-time to a single pass changes H by 0.01 and km s Mpc for the NIR and optical approaches, respectively.
Using the three primary anchors with the optical Wesenheit and now including systematic errors, we find H = km s Mpc, equivalent to an uncertainty of . This result is somewhat less precise than the 3.3% total error of R11, which used the NIR Wesenheit but only 8 SN-Cepheid hosts instead of the present 19. Until or unless additional studies can improve our understanding of Cepheid reddening, metallicity sensitivity, and the scale of – breaks at optical wavelengths, our analysis shows that improvements in the determination of H via Cepheids must primarily rely on the inclusion of NIR observations.
Similar conclusions are reached when using only NGC4258 as an anchor: H = km s Mpc without systematic errors, so the statistical error is slightly better than the equivalent NIR result at . However, the systematic error of is considerably worse, leading to a combined value of H = km s Mpc. While the use of strictly optical Wesenheit magnitudes can be informative, our best results for H with lowest systematics consistently come from using the NIR data in concert with optical observations.
4.3 Supernova Systematics
The SALT-II SN light-curve fits, the composition of the Hubble-flow sample, and sources of SN photometry used to determine in Equation 5 are described in Scolnic et al. (2015); Scolnic & Kessler (2016). These take advantage of the “Supercal” procedure (Scolnic et al. 2015) which uses reference stars in the fields of the SNe and the homogeneous set of star photometry over 3 steradians from Pan-STARRS to remove small photometric inconsistencies between SN photometry obtained across multiple observatories and systems. As is common in recent analyses of SN Ia distances (e.g., Betoule et al. 2014), to determine we use “quality cuts” to include only SN Ia light curves for which the SALT color parameter () is within , the light-curve parameter () is within (error ), the of the light-curve fit is “good” (fitprob ), the peak time of the light curve is constrained to better than 2 days, and the uncertainty in the corrected peak magnitude is mag. All of the 19 calibrators pass these quality cuts as well. The SN redshifts are corrected for coherent (peculiar) flows based on density maps (Scolnic et al. 2014, 2015) which reduces correlated deviations from expansion caused by visible large-scale structure and empirical residuals determined from simulations (Scolnic & Kessler 2016). A residual velocity (peculiar) error of 250 km s is assumed. As a final step, we exclude SNe Ia which deviate from the form of Equation 5 by more than ; this excludes 3% of the sample for the primary fit with , leaving 217 SNe Ia (or 281 SNe Ia for variants with ). These have a dispersion of 0.128 mag around Equation 5 with a mean error of 0.129 mag and a per degree of freedom of 0.91, and yield for SALT-II ( for MLCS2k2). As an alternative to the SALT-II light-curve fitter, we used the MLCS2k2 fitter (Jha et al. 2007) with a value of for the SN host galaxy, the same as the primary fits of R11. The resulting value of H is higher by 1.9 km s Mpcor of the total error as given in Table 8.
As in R11, we make use of several studies (Hicken et al. 2009a; Kelly et al. 2010; Lampeitl et al. 2010; Sullivan et al. 2010) which have shown the existence of a small step brighter for the corrected SN magnitude for hosts more massive than . We use the same value of 0.06 mag used by Betoule et al. (2014) for the size of the mass step to account for this effect. The net effect on H is a small decrease of 0.7% because of the modest difference in masses of the nearby hosts (mean ) and of those that define the magnitude-redshift relation (Sullivan et al. 2010, mean ). We include these corrections based on host-galaxy mass in our present determination of given in Table 5 and for , correcting those with hosts above and below by 0.03 fainter and brighter, respectively.
An alternative host dependence on SN Ia distance has been proposed by Rigault et al. (2013, 2015) based on the local star-formation rate (LSFR) measured at the site of the SN. The results from Rigault et al. (2015) suggested a 3 correlation between SN distance residual and LSFR inferred from ultraviolet photometry measured with GALEX for a set of 82 SNe Ia from Hicken et al. (2009b), with somewhat higher significance for distances from MLCS2k2 and somewhat lower for SALT-II. Jones et al. (2015) repeated the LSFR analysis using a larger sample of SNe Ia which better matched the samples and light-curve quality selection used in the cosmological analyses of R11 and Betoule et al. (2014) as well as the more recent version of SALT II. Using 179 GALEX-imaged SN Ia hosts from the JLA SN sample (Betoule et al. 2014) and the Pan-STARRS sample (Scolnic et al. 2015), or 157 used by R11, the significance of a LSFR effect diminished to due to two differences from Rigault et al. (2015): (1) the increase in the sample statistics, and (2) use of the JLA or R11 quality criteria. Because we employ both the larger local SN sample as well as the quality cuts used by Jones et al. (2015), we include only the mass-based correction whose significance has remained in cosmological SN samples.
Nevertheless, if we were to assume the existence of a LSFR (despite the preceding lack of significance), we can select a Hubble-flow sample to match the LSF of the calibrator sample and thus nullify the possible impact on H. In the calibrator sample, 17 of 19 hosts (or 89%) are above the LSFR threshold adopted by Rigault et al. (2013, 2015), which is a larger fraction than the 50–60% in the Hubble-flow sample (Jones et al. 2015). To determine an upper limit on a LSFR mismatch and thus H across sets, we selected all Hubble-flow SNe significantly above the LSFR threshold (i.e., a purely LSFR sample), requiring these SNe Ia to have good GALEX detections. By changing the Hubble flow selection, only the term is affected. For this all-LSFR sample, is higher than for the primary fit by 0.00446 at and lower by 0.0010 at . Thus the Hubble constant from the primary fit increased by 0.8 or 0.2 km s Mpc, respectively (see Table 8). Thus even if a relation existed, we find that a LSFR in SN hosts would have no significant impact on the determination of H here.
However, to address the possibility of host-galaxy dependence that arises from sample selection, we also recalculated the intercept of the Hubble-flow SNe () using only those found in spiral hosts. Because the 19 hosts selected for Cepheid observations were chosen on the basis of their appearance as spirals (as well as their proximity and modest inclination), this selection would be expected to match the two samples if global star formation or its history had an impact on the measured SN distance. Because the Hubble-flow sample is so much larger than the nearby sample, such a cut has a modest effect on the uncertainty in H. Doing so raised H for the SALT-II fitter and lowered H for MLCS2k2 each by km s Mpc. We note that the spiral-host sample has a mean LSFR of dex, similar to the mean of the 19 calibrators at dex and higher than the full Hubble-flow set of dex.
We also changed the lower redshift cutoff of the Hubble diagram from to , originally adopted to mitigate the impact of a possible local, coherent flow. This raised H by 0.2 km s Mpc for the primary fit. Changing the deceleration parameter used to fit the SNe Ia at from (as expected for , ) to (, ; or , , ) decreases H by 0.2%. More generally, an uncertainty in of 0.02 (Betoule et al. 2014) produces an uncertainty in of 0.03 resulting in an uncertainty in of 0.1%. As expected, the sensitivity of to knowledge of is very low as the mean SN redshift is only 0.07. As a further test, we reduced the upper range of redshifts used to measure the intercept from 0.15 to 0.07. This reduces the sample by nearly half, increases the uncertainty in the intercept by 40% and increases the intercept and by 0.7%. We do not use this more limited redshift range because it introduces the potential for larger peculiar flows and the sensitivity to knowledge of is already very low.
Two of the SNe in the calibrator sample (SN 1981B and SN 1990N) were measured before the Hubble-flow sample was acquired. Relative to the global fit, SN 1990N is faint by mag and SN 1981B is bright by mag, so this older digital photometry does not appear to bias the value of H in a significant way.
|[km s Mpc]|
|MW: 15 Cepheid Parallaxes|
|LMC: 8 Late-type DEBs|
|M31: 2 Early-type DEBs|
|NGC4258 + MW|
|NGC4258 + LMC|
|Three anchors (preferred)|
|NGC4258 + MW + LMC|
|NGC4258 + MW + LMC + M31|
|Optical only (no NIR), three anchors|
|NGC4258 + MW + LMC|
|LMC||N4258||All 3||N4258||All 3|
|Anchor distance, mean||5%||3%||1.3%||2.6%||1.3%|
|Mean of – in anchor||2.5%||1.5%||0.8%||1.2%||0.7%|
|Mean of – values in SN Ia hosts||1.5%||1.5%||0.6%||0.4%||0.4%|
|Mean of SN Ia calibrators||2.5%||2.5%||1.9%||1.2%||1.2%|
|SN Ia – relation||1%||0.5%||0.5%||0.4%||0.4%|
|Cepheid reddening & colors, anchor-to-hosts||4.5%||0.3%||1.4%||0%||0.3%|
|Cepheid metallicity, anchor-to-hosts||3%||1.1%||1.0%||0.0%||0.5%|
|– slope, log , anchor-to-hosts||4%||0.5%||0.6%||0.2%||0.5%|
|WFPC2 CTE, long-short||3%||N/A||N/A||N/A||N/A|
Note. – (*) Derived from diagonal elements of the covariance matrix propagated via the error matrices associated with Equations 1, 3, 7, and 8. (a) For MW parallax, this term is already included with the term above. (b) For R09, R11, and this work, calculated with covariance included. (c) One anchor not included in R11 estimate of to provide a crosscheck.
A budget for the sources of uncertainty in the determination of H is given in Table 7. These are necessarily marginalized approximations, as they do not show the (small) covariance between terms included in the full global fit.
Our systematic error is estimated based on the variations in H resulting from the reasonable, alternative fits. These alternatives are, by their nature, difficult to formally include in the global fit. All of the discussed NIR variants, 207 in total including combinations of anchors, are listed in Table 8. As shown in Figure 11, the histogram for the primary-fit anchors (NGC4258, MW, and LMC) is well fit by a Gaussian distribution with km s Mpc, a systematic uncertainty that is a little less than half of the statistical error. None of the variants is a noteworthy outlier from this distribution. The complete error in using multiple anchors can be traced to the quadrature sum of three terms, the two independent terms in equation 9 and the systematic error. The error in for any variant, derived from the global fit, is given in third column of Table 8 and the error in the intercept, , was given in §3.
Including the systematic error, we arrive at a complete result of H = km s Mpc, corresponding to a total uncertainty, combining statistical and systematic contributions, of . The two largest remaining uncertainties are the mean geometric distance calibration (1.3%) and the mean of the 19 SN Ia calibrators (1.2%).
Our primary fit of H = km s Mpc is higher than the value of km s Mpc predicted by Planck Collaboration et al. (2016) based on CDM with 3 neutrino flavors having a mass of 0.06 eV and the Planck CMB data (TT,TE,EE+SIMlow; for TT+SIMlow). Assuming the difference is not a fluke (99.9% confidence), possible explanations include systematic errors in the local H or CMB measurements, or an unexpected feature in the cosmological model that connects them. Previous indications of tension from the less-precise measurements of H and the CMB (Riess et al. 2011; Planck Collaboration et al. 2014) elicited a number of new studies, many of which were addressed above and helped improve the present analysis.
The analysis of the R11 dataset by Efstathiou (2014) yielded a value of H km s Mpc, similar to the primary result of km s Mpcfound by R11 using the same three anchors (MW, LMC, and NGC4258, including the same H13 distance for NGC4258 for both) and resulting in a 1.9 tension with Planck and CDM. Efstathiou (2014) also found H km s Mpcwith NGC4258 as the only anchor, and the Planck team adopted this value instead of the three anchor result with its reduced precision and tension. The main difference in the analysis with R11 was the use of a global instead of –-specific outlier rejection. Our use here of F814WF160W colors to identify blends as discussed in §4.1 has significantly reduced the need for outlier rejection, and we have adopted a global outlier rejection for the 2% that remain. The internal model constraints on the slope and metallicity parameters have also improved substantially over the R11 dataset with no need for the priors set by R11 or Efstathiou (2014). We find the difference in H between the use of three anchors and just NGC4258 to be 1.0 km s Mpc, less than the 1.9 km s Mpcfound by Efstathiou (2014) with the R11 dataset, a consequence of the tightened constraints on the Cepheid relations, and we conclude that use of the three anchors provides our best determination of H.
In the previous section we addressed systematic errors related to Cepheids and SNe used in our determination of H. A third component comes from our use of geometric distances to calibrate Cepheids. We used four sets: masers in NGC4258, parallaxes to MW Cepheids, DEBs in the LMC, and DEBs in M31. The four values of H using each as the sole anchor (see Table 6) are in good relative agreement, with none more than from the primary fit considering only their mean geometric distance error of 2.8%. Thus we see no basis for excluding any of these four as outliers. Among the four, NGC4258 has the advantage of a Cepheid sample with mean period closer to those in the SN hosts and with all their photometry on the same HST system.
However, in our analyses we parameterize the difference in zeropoints for non-HST data, and the a posteriori result of 0.013 mag for the primary fit is well below the estimated a priori constraint of mag, indicating no unexpected inconsistency with zeropoints. Our use of Cepheid samples in M31 and the LMC, which sample the short- and long-period range as well as the allowance in the fits for a – break, strictly limits the impact of a difference in sample mean periods on H. The residuals among the anchor distances for our primary fit are , within the range of its 0.0568 mag prior, and , within range of its 0.0452 mag prior. To be conservative, we removed M31 from the anchor set of our primary fit for the reasons discussed in §3.1.3 — but we discourage any additional winnowing or editing of the anchor set as it is unwarranted by the data and is likely to give a false sense of reducing the tension merely by inflating the present uncertainties.
We may consider whether the local determination of H is different than the global (i.e., cosmological) value. In a homogeneous and isotropic Universe the two have the same expectation value. However, we live in an unusual place (a dark matter halo), and the inhomogeneity of matter on our measurement scale could lead to important variations in H. We currently account for flows induced by visible structures using host redshift corrections derived from a map of the matter density field calibrated by the 2M++ catalogue (with a light-to-matter bias parameter of and a dipole from Carrick et al. 2015). This produces a small net increase in H of a few tenths of a percent over the case of uncorrelated velocities at rest with respect to the CMB as discussed by R11. We also account for the cosmological change in expansion rate using and as discussed in §4.2. Because the Hubble diagram of SNe Ia is continuously sampled from to , a percent-level change in the local expansion rate at would be empirically evident in the distance residuals. In Figure 12 we show the relative change in H starting at and decreasing the influence of the local volume by gradually increasing the redshift cutoffs for determining H. As shown, the value of H never changes by more than 1.3 times the statistical uncertainty in the fit of the intercept over a range of 0.2 and a factor of 5 increase in volume.
Odderskov et al. (2016) simulated the effect of inhomogeneities on the local value of H using mock sources in N-body simulations using the GADGET code with a box size of 700 Mpc and dark matter particles with cosmological parameters in agreement with Planck Collaboration et al. (2014) from to the present. In the simulation, halos are resolved using the halo-finder ROCKSTAR and realistic SN sampling is obtained from the redshift distribution of the samples with . Cosmic variance is taken into account by varying the location of the observer. The uncertainty in the local measurement of H is found to be 0.27% for the case of a typical SN sample, observer in a Local Group halo and the maximum redshift of for our primary fit (I. Odderskov, priv. comm.). This analysis is in good agreement with our empirical result of 0.4% uncertainty in , which shows that such convergence to the sub-percent level has occurred within the SN sample at . We conclude that the uncertainty in H owing to inhomogeneities is adequately taken into account by the procedure of empirically correcting the redshifts for expected flows, testing for convergence of H on large scales, and comparing the propagated uncertainty to simulations. A difference in H at even the % level caused by inhomogeneities would be triple the empirical or theoretical uncertainty and thus appears exceedingly unlikely.
Could the difference result from a systematic error in the Planck measurement? To explore this possibility, we consider an independent set of CMB data from the combination of WMAP9, ACT, and SPT observations. Based on the analysis by Calabrese et al. (2013) using CDM but including the same neutrino mass of 0.06 eV used in the Planck analysis yields H km s Mpc, a difference from our local measurement of and thus quite consistent. While some of the improved agreement comes from the lower precision of this CMB dataset, most comes from a change in the central value of H itself; the WMAP9+ACT+SPT value, even with the uncertainty of the Planck data, would still be consistent at the 1.3 level. The difference in CMB datasets appears to play some role in the perceived tension with the local value of H. Addison et al. (2015) has reported a parallel 2.5 tension (size and significance) internally within the Planck data based on H parameters determined from multipoles with and , with the two halves of the data producing H and km s Mpc, respectively. Considering the two Planck halves with the R11 measurement of H, BAO, WMAP9, and SPT, Addison et al. (2015) finds 5 of the 6 consistent with H km s Mpc, with only the Planck data pulling toward significantly lower values. Because the SPT and Planck data cover similar ranges in , their disagreement should be independent of the cosmological model and thus could indicate the presence of a systematic error and a role in the present tension with local H measurements.
However, some degree of the previous tension remains, even without Planck, after including other datasets explicitly to constrain the cosmological model. Bennett et al. (2014) used WMAP9+ACT+SPT with BAO from BOSS DR11 and 6dFGS (their Table 2, column G) and find H km s Mpc, which has a tension with our determination of H (and a tension with Planck in the other direction). A lower value of is given by Aubourg et al. (2015) for WMAP9, BAO, and high-redshift SNe — but this neglects SPT, which pulls toward higher H (Addison et al. 2015; Story et al. 2013). More direct comparisons and analyses of CMB data may be expected to resolve the tensions between them and the local value of H.
It is useful to compare our result with recent measurements of the local Hubble constant which are independent of SNe Ia
We may consider the simplest extensions of CDM which could explain a difference between a local and cosmological Hubble constant of –6 km s Mpc. We are not the first to look for such a resolution, though the roster of datasets examined has varied substantially and evolves as measurements improve (Wyman et al. 2014; Leistedt et al. 2014; Aubourg et al. 2015; Cuesta et al. 2015; Dvorkin et al. 2014). The simplest parameterizations of dark energy with or with and can alleviate but not fully remove tension with H (see Figure 13) due to support for signal from high-redshift SNe Ia and BAO (Cuesta et al. 2015; Aubourg et al. 2015, see Figure 14). A very recent () and dramatic decrease in or an episode of strong dark energy at may evade detection and still produce a high value of H. Whether such a model creates additional tensions will depend on its prescription and still, if empirically motivated, is likely to suffer from extreme fine-tuning.
A synthesis of the studies cited above indicates a more fruitful avenue is found in the “dark radiation” sector. An increase in the number of relativistic species (dark radiation; e.g., neutrinos) in the early Universe increases the radiation density and expansion rate during the radiation-dominated era, shifting the epoch of matter-radiation equality to earlier times. The resulting reduction in size of the sound horizon (which is used as a standard ruler for the CMB and BAO) by a few percent for one additional species (N) increases H by about 7 km s Mpc for a flat Universe, more than enough to bridge the divide between the local and high-redshift scales. A fractional increase (i.e., less than unity) is also quite plausible for neutrinos of differing temperatures or massless bosons decoupling before muon annihilation in the early Universe (e.g., Goldstone bosons; Weinberg 2013), producing N or depending on the decoupling temperature. An example of such a fit comes from Aubourg et al. (2015) using a comprehensive set of BAO measurements and Planck data, finding N and H km s Mpc. A similar result from WMAP9+SPT+ACT+SN+BAO gives N and H km s Mpc (Hinshaw et al. 2013). Thus, a value of N in the range 0.4–1.0 would relieve some or all of the tension. Although fits to the Planck dataset (Planck Collaboration et al. 2015) do not indicate the presence of such additional radiation, they do not exclude this full range either.
Allowing the N degree of freedom triples the uncertainty in the cosmological value of H from Planck Collaboration et al. (2015), BAO and high-redshift SNe and modestly raises its value to H km s Mpc, reducing the tension to 2.1 and demonstrating that a local measurement of H appears to offer a powerful aid to determining N. A cosmologically constrained value of can be used to diagnose the nature of the new particle and its decoupling temperature (Brust et al. 2013).
Including the present measurement of H with the Planck (Planck Collaboration et al. 2015) data (including lensing), the full BAO set of measurements (including the Lyman-alpha QSO’s) and the Betoule et al. (2014) SN sample pulls N higher to a value of (and H km s Mpc), a result favoring (though not requiring) additional dark radiation. This fit provides the lowest value of the best-fit log likelihood among standard extensions to CDM we considered (lower than CDM by 2) and the result is shown in Figure 15. If Planck CMB, BAO, SN, and H data are taken at face value, this extension of CDM remains an intriguing avenue toward their resolution and highlights the need for additional improvements in local determinations of H. More broadly, the present discrepancy in the measured Hubble constant may provide a clue to one of the many enigmas contained in the 95% of the Universe within the dark sector.
Fortunately, the prospects for near-term improvements in the local determination of the Hubble constant are quite promising. We have begun obtaining a new sample of parallax measurements of long-period MW Cepheids using the spatial scanning technique with WFC3 on HST (Riess et al. 2014; Casertano et al. 2015). These improvements alone would reduce the total uncertainty in H to 1.8% based on the terms in Table 7. In a parallel effort, we are obtaining spatial-scan photometry of a larger sample of MW Cepheids slated for even higher-precision Gaia parallax determinations in a few years. With additional progress from this route and others, the goal of 1% (Suyu et al. 2012) is not far-fetched and has the potential, in concert with Stage-IV CMB experiments (see Figure 16), to provide new leverage on the dark Universe.
We thank Alessandro Manzotti, Ariel Goobar, Mike Hudson, WeiKang Zheng, Bill Januszewski, Robert Kirshner, Licia Verde, Liz Humphreys, Dan Shafer, the PHAT collaboration, and Peter Stetson for valuable discussions and other contributions, as well as Doug Welch for providing the optimal temporal spacings used in the optical Cepheid searches. We are grateful to Melissa L. Graham for assistance with the Keck observations.
This research was supported by NASA/HST grants GO-12879, 12880, 13334, 13335 & 13344 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. L.M.M.’s group at Texas A&M University acknowledges additional support from the Mitchell Institute for Fundamental Physics & Astronomy. D.S. acknowledges support from the Kavli Institute for Cosmological Physics at the University of Chicago through grant NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder Fred Kavli. A.V.F.’s group at UC Berkeley is also grateful for financial assistance from NSF grant AST-1211916, the TABASGO Foundation, Gary and Cynthia Bengier, and the Christopher R. Redlich Fund. P.C. is supported by NSF grant AST-1516854 to the Harvard College Observatory. J.M.S. is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under under award AST-1302771. R.J.F. gratefully acknowledges support from the Alfred P. Sloan Foundation.
Some of the results presented herein are based on data obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA; the observatory was made possible by the generous financial support of the W. M. Keck Foundation. We thank the Keck staff for their expert help.
Appendix A Setup of System of Equations
Equations 1 through 8 describe the relationships between the measurements and parameters with additional constraint equations given in §3. To improve clarity we explicitly show the system of equations we solve to derive the value of which together with the independent determination of provides the measurement of H via Equation 9. Here we refer to the vector of measurements as , the free parameters as , the equation (or design) matrix as , and the error matrix as with and maximum likelihood parameters given as