A REDETERMINATION OF THE HUBBLE CONSTANT WITH THE HUBBLE SPACE TELESCOPE FROM A DIFFERENTIAL DISTANCE LADDER*

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Published 2009 June 12 Copyright is not claimed for this article. All rights reserved.
, , Citation Adam G. Riess et al 2009 ApJ 699 539 DOI 10.1088/0004-637X/699/1/539

0004-637X/699/1/539

ABSTRACT

This is the second of two papers reporting results from a program to determine the Hubble constant to ∼5% precision from a refurbished distance ladder based on extensive use of differential measurements. Here we report observations of 240 Cepheid variables obtained with the Near-Infrared Camera and Multi-Object Spectrometer (NICMOS) Camera 2 through the F160W filter on the Hubble Space Telescope (HST). The Cepheids are distributed across six recent hosts of Type Ia supernovae (SNe Ia) and the "maser galaxy" NGC 4258, allowing us to directly calibrate the peak luminosities of the SNe Ia from the precise, geometric distance measurements provided by the masers. New features of our measurement include the use of the same instrument for all Cepheid measurements across the distance ladder and homogeneity of the Cepheid periods and metallicities, thus necessitating only a differential measurement of Cepheid fluxes and reducing the largest systematic uncertainties in the determination of the fiducial SN Ia luminosity. In addition, the NICMOS measurements reduce the effects of differential extinction in the host galaxies by a factor of ∼5 over past optical data. Combined with a greatly expanded set of 240 SNe Ia at z < 0.1 which define their magnitude–redshift relation, we find H0 = 74.2 ± 3.6 km s−1 Mpc−1, a 4.8% uncertainty including both statistical and systematic errors. To independently test the maser calibration, we use 10 individual parallax measurements of Galactic Cepheids obtained with the HST fine guidance sensor and find similar results. We show that the factor of 2.2 improvement in the precision of H0 is a significant aid to the determination of the equation-of-state parameter of dark energy, w = P/(ρc2). Combined with the Wilkinson Microwave Anisotropy Probe five-year measurement of ΩMh2, we find w = −1.12 ± 0.12 independent of any information from high-redshift SNe Ia or baryon acoustic oscillations (BAO). This result is also consistent with analyses based on the combination of high-redshift SNe Ia and BAO. The constraints on w(z) now including high-redshift SNe Ia and BAO are consistent with a cosmological constant and are improved by a factor of 3 due to the refinement in H0 alone. We show that future improvements in the measurement of H0 are likely and should further contribute to multi-technique studies of dark energy.

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1. INTRODUCTION

The Hubble Space Telescope (HST) established a cornerstone in the foundations of cosmology by observing Cepheid variables beyond the Local Group, leading to a measurement of the Hubble constant (H0) with 10%–15% precision (Freedman et al. 2001; Sandage et al. 2006). This measurement resolved decades of extreme uncertainty about the scale and age of the universe. The discovery of cosmic acceleration and the dark energy that drives it (Riess et al. 1998; Perlmutter et al. 1999; see Frieman et al. 2008 and Filippenko 2005 for reviews) has intensified the need for ever-higher-precision measurements of H0 to constrain and test the new cosmological models. Observations are essential to determine, empirically, aspects of the new model including its geometry, age, mass density, and the dark energy equation-of-state parameter, w = P/(ρc2), where P is its pressure, and ρ is its energy density. Perhaps the most fundamental question is whether dark energy is a static, cosmological constant or a dynamical, inflation-like scalar field—or whether it can be accommodated at all within the framework of general relativity.

While measurements of the high-redshift universe from the cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and Type Ia supernovae (SNe Ia) in concert with a fully parameterized cosmological model can be used to predict the Hubble constant (e.g., Spergel et al. 2007; Komatsu et al. 2009), they are not a substitute for its measurement in the local universe. Using all of these measures and the assumptions that w = −1 and that space is flat, a predicted precision of 2% in the Hubble constant may be inferred—see Table 1 for ΛCDM—(Komatsu et al. 2009). However, significant tension (at the 3σ level) exists in the value of H0 predicted from CMB+BAO and CMB+high-z SNe Ia when the other cosmological parameters such as curvature and w are constrained only by data (see Table 1; OWCDM). This suggests that something interesting about the model or the measurements would be learned from an independent determination of H0 of comparable precision.

Table 1. H0 Inferred from Five-Year WMAP Combined With the Most Constraining Data

Data Set ΛCDM (i.e., ΩK ≡ 0, w ≡ −1) OWCDM (i.e., ΩK = free, w = free)
WMAP5 71.9+2.6−2.7 47+14−12
WMAP5 + BAO 70.9 ± 1.5 81.7+6.5−6.4
WMAP5 + high-z SNe 69.6 ± 1.7 57.5 ± 4.8
WMAP5 + BAO + high-z SNe 70.1 ± 1.3 68.7+1.9−2.0

Note. The constraints on H0 are based on the WMAP team's analysis of the five-year WMAP data combined with other data sets (Komatsu et al. 2009), as listed; see http://lambda.gsfc.nasa.gov/product/map/dr3/parameters.cfm. High-z SNe refers to measurements of the magnitude−z relation of SNe without reference to their distance scale.

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Increasing the precision of the measurement of the Hubble constant requires reducing systematic uncertainties which dominate the error budget along the conventional distance ladder (Freedman et al. 2001; Leonard et al. 2003). As the Hubble diagram of SNe Ia establishes the relative expansion rate to an unprecedented uncertainty of <1% (e.g., Hicken et al. 2009), the calibration of the luminosity of SNe Ia affords the greatest potential for precision in measuring H0. As we show in Section 4, the largest sources of systematic error along this route come from the use of uncertain transformations to meld heterogeneous samples of Cepheids observed with different photometric systems in the anchor galaxy and SN Ia hosts.

1.1. The SHOES Program

The goal of the Supernovae and H0 for the Equation of State (SHOES) Program (HST Cycle 15, GO-10802) is to measure H0 to <5% precision by mitigating the dominant systematic errors.8 To obviate the limited accuracy of photographic SN data, we have been calibrating recent SNe Ia recorded with modern detectors and acquiring uniform samples of Cepheids observed in the SN Ia hosts and in the anchor galaxy. Progress in the former was presented by Riess et al. (2005, 2009), more than doubling the sample of high-quality calibrators by providing reliable calibration for four modern SNe Ia. Here we address the latter, reporting the results of infrared (IR) observations of Cepheids which are homogeneous in their periods, metallicities, and measurements in both the anchor (NGC 4258) and the SN hosts.

NGC 4258 offers attractive benefits over the use of the Large Magellanic Cloud (LMC) or the Milky Way Galaxy as an anchor of the distance ladder: (1) all of its Cepheids can be treated as being at a single distance determined geometrically from the Keplerian motion of its masers as 7.2 ± 0.5 Mpc (Herrnstein et al. 1999); (2) more than a decade of tracking its masers has resulted in little change to its distance while steadily increasing its precision from 7% (Herrnstein et al. 1999) to 5.5% (Humphreys et al. 2005) to 3% (Humphreys et al. 2008; E. M. L. Humphreys 2009, in preparation; Greenhill et al. 2009); (3) the geometric distance measurement can be internally crosschecked via proper motion and centripetal acceleration, and the method can be externally tested by measurements of other maser systems (Braatz et al. 2008; Greenhill et al. 2009); (4) its Cepheids have a metallicity similar to those found in the hosts of SNe Ia (Riess et al. 2009); (5) HST observations of NGC 4258 from Cycles 12, 13, and 15 provide the largest sample of extragalactic long-period (P > 10 d) Cepheids (Macri et al. 2006, 2009); and (6) its Cepheids can be observed with the HST in exactly the same manner as those in SN Ia hosts. In Section 4, we independently test the use of the distance to NGC 4258 by adopting the individual parallax measurements of Galactic Cepheids from Benedict et al. (2007).

The IR observations of Cepheids presented in Section 3 provide additional advantages over those in the optical: (1) reducing the differential extinction by a factor of 5 over visual data, and (2) reducing the dependence of Cepheid magnitudes on chemical composition (Marconi et al. 2005). The resulting refurbished distance ladder builds on past work while removing four of the largest systematic sources of uncertainty in H0. In Sections 3 and 4, we show that the total uncertainty in the measurement of H0 has been reduced from 11% (Freedman et al. 2001) to 4.8%.

2. NICMOS CEPHEID OBSERVATIONS OF THE SHOES PROGRAM

In Riess et al. (2009), we used HST/ACS+WFPC2 observations to discover Cepheids in two new SN Ia hosts and to expand previous samples in four other SN Ia hosts with newly discovered, longer period (P > 60 d) variables. In Macri et al. (2009), we used HST/ACS+WFPC2 observations to augment the Cepheid sample in NGC 4258. These new observations, together with those from Saha et al. (1996, 1997, 2001), Gibson et al. (2000); Stetson & Gibson (2001), and Macri et al. (2006), provide the position, period, and phase of 450 Cepheids in six hosts with reliable SN Ia data and NGC 4258, each with typically 14 epochs of HST imaging with F555W and one to five epochs with F814W (except for NGC 4258, which has 12 epochs of F814W data).

The Near-Infrared Camera and Multi-Object Spectrometer (NICMOS) on HST provides the means to obtain near-IR measurements of optically identified Cepheids. Macri et al. (2001) used short exposures (∼1 ks) with NICMOS to measure 70 extragalactic Cepheids in 14 galaxies at an average distance of 5 Mpc (including two Cepheids in NGC 4536) to verify the Galactic extinction law.

In HST Cycle 15, the SHOES program obtained deep (10–35 ks), near-IR observations of the Cepheids in these SN Ia hosts. The SN Ia in each host was chosen for meeting the following criteria: (1) has modern data (i.e., photoelectric or CCD), (2) was observed before maximum brightness, (3) has low reddening, (4) is spectroscopically typical, and (5) has optical HST-based observations of Cepheids in its host. The resulting sample consists of six SN Ia hosts given in Table 2. The six members and their SNe are shown in Figure 1. In Cycle 15, we also obtained 2 ks NICMOS imaging of Cepheids in NGC 4258 to augment that obtained in Cycle 13 by GO 10399 (P.I. Greenhill).

Figure 1.

Figure 1. Optical images of SNe Ia near peak (see Figures 27 for orientations and scales). These images show the objects used to calibrate the SN Ia fiducial luminosity. The images were obtained with CCDs. The exception is SN 1981B which was observed photoelectrically and with the Texas Griboval electrographic camera (image shown here) which has better sensitivity and linearity that photographic plates.

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Table 2. Cepheid Hosts Observed by SHOES

Host SN Ia Initial Optical HST Cycle Reobservation, Cycle 15 Observation Near-IR, Cycle 15
NGC 4536 SN 1981B WFPC2 4 WFPC2 NIC2
NGC 4639 SN 1990N WFPC2 5 ACS NIC2
NGC 3982 SN 1998aq WFPC2 8 ACS NIC2
NGC 3370 SN 1994ae ACS 11 ACS NIC2
NGC 3021 SN 1995al ACS 14 ACS NIC2
NGC 1309 SN 2002fk ACS 14 ACS NIC2
NGC 4258  ⋅⋅⋅  ACS 12 ACS/WFPC2 NIC2a

Note. aSome NIC2 data obtained in Cycle 13.

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2.1. NICMOS Data Reduction

Groupings of optically characterized Cepheids were observed using the NICMOS Camera 2 and the F160W filter. This camera offers the best compromise of area and sampling of the point-spread function (PSF) of the three NICMOS cameras. For each SN host galaxy, we selected four to five 0.1 arcmin2 pointings of 3–14 orbit depth (10–35 ks) to contain multiple previously identified long-period Cepheids. The pointing centers and total integration times are given in Table 3. The imaging configurations are shown in Figures 27. The observations were obtained in single-orbit visits spread over ∼2 months.

Figure 2.

Figure 2. HST ACS F555W image of NGC 3370. The positions of Cepheids with periods in the range P > 60 d, 30 < P < 60 d, and 10 < P < 30 d are indicated by red, blue, and green circles, respectively. A yellow circle indicates the position of the host's SN Ia. The orientation is indicated by the compass rose whose vectors have lengths of 15''. The fields of view for the NIC2 follow-up fields in Table 2 are indicated.

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Figure 3.

Figure 3. As Figure 2 for NGC 1309.

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Figure 4.

Figure 4. As Figure 2 for NGC 3021.

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Figure 5.

Figure 5. As Figure 2 for NGC 4639.

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Figure 6.

Figure 6. As Figure 2 for NGC 4536, image from HST WFPC2.

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Figure 7.

Figure 7. As Figure 2 for NGC 3982.

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Table 3. NIC2 F160W Cepheid Observations

Target α(J2000) δ(J2000) Exp Time (s)
NGC1309-BLUE 3h22m3fs499 −15°24'4farcs242 10,239.5
NGC1309-YELLOW 3 22 7.040 −15 24 26.800 23,031.0
NGC1309-CYAN2 3 22 8.790 −15 24 46.148 34,174.3
NGC1309-GREEN 3 22 9.862 −15 23 48.548 10,239.5
NGC3021-BLUE 9 50 55.493 33 32 48.598 10,239.5
NGC3021-CYAN 9 50 54.079 33 33 28.190 10,239.5
NGC3021-GREEN 9 50 59.984 33 33 5.202 20,478.9
NGC3021-RED 9 50 57.330 33 33 36.883 20,158.9
NGC3370-BLUE 10 47 4.334 17 17 3.726 20,478.9
NGC3370-CYAN 10 47 6.432 17 15 26.429 10,239.5
NGC3370-GREEN 10 47 1.673 17 16 56.221 26,238.7
NGC3370-RED 10 47 8.354 17 15 48.264 10,239.5
NGC3982-BLUE 11 56 24.458 55 7 13.644 17,919.0
NGC3982-CYAN-COPY 11 56 32.368 55 7 30.329 7,679.6
NGC3982-GREEN 11 56 21.375 55 7 23.696 7,679.6
NGC3982-RED-COPY 11 56 23.150 55 6 38.776 17,919.0
NGC3982-YELLOW 11 56 30.253 55 7 54.595 10,239.5
NGC4536-BLUE 12 34 18.360 2 11 35.570 17,919.0
NGC4536-CYAN 12 34 21.395 2 13 10.230 12,799.3
NGC4536-GREEN 12 34 17.313 2 13 6.429 17,919.0
NGC4536-RED 12 34 21.371 2 12 6.289 17,919.0
NGC4639-BLUE 12 42 53.114 13 14 54.342 10,239.5
NGC4639-BLUE-LATEa 12 42 52.946 13 14 55.207 10,239.5
NGC4639-CYAN 12 42 49.300 13 16 9.272 19,966.9
NGC4639-GREEN 12 42 52.163 13 16 30.699 20,478.9
NGC4639-RED 12 42 56.438 13 15 20.369 10,239.5
NGC4258-NIC-POS4 12 18 50.76 47 19 24.2 2559.8
NGC4258-NIC-POS5 12 18 57.92 47 20 35.5 2559.8
NGC4258-NIC-POS3 12 18 47.52 47 20 05.1 2559.8
NGC4258-NIC-POS2 12 18 50.09 47 20 43.1 2559.8
NGC4258-NIC-POS1 12 18 50.77 47 21 09.3 2559.8
NGC4258-NIC-POS13 12 18 54.90 47 21 47.9 2559.8
NGC4258-NIC-POS11 12 19 20.28 47 14 54.0 2559.8
NGC4258-NIC-POS10 12 19 22.72 47 14 44.6 2559.8
NGC4258-NIC-POS9 12 19 08.93 47 12 25.2 2559.8
NGC4258-NIC-POS8 12 19 12.02 47 12 21.1 2559.8
NGC4258-NIC-POS12 12 19 25.32 47 13 44.2 2559.8
NGC4258-NIC-POS6 12 19 21.03 47 10 21.5 2559.8
NGC4258-NIC-POS7 12 19 25.39 47 09 41.2 2559.8
NGC4258-INNER-NIC-09 12 18 53.21 47 18 43.5 2559.8
NGC4258-INNER-NIC-10 12 18 54.50 47 19 00.9 2559.8
NGC4258-INNER-NIC-08 12 18 51.38 47 18 42.1 2559.8
NGC4258-INNER-NIC-12 12 18 54.72 47 19 16.5 2559.8
NGC4258-INNER-NIC-05 12 18 48.98 47 19 13.2 2559.8
NGC4258-INNER-NIC-13 12 18 55.88 47 20 17.8 2559.8
NGC4258-INNER-NIC-06 12 18 48.99 47 19 47.4 2559.8
NGC4258-INNER-NIC-04 12 18 48.21 47 20 10.1 2559.8
NGC4258-INNER-NIC-11 12 18 54.55 47 20 41.7 2559.8
NGC4258-INNER-NIC-01 12 18 45.28 47 20 02.2 2559.8
NGC4258-INNER-NIC-07 12 18 50.33 47 21 06.3 2559.8
NGC4258-INNER-NIC-02 12 18 47.64 47 20 58.2 2559.8
NGC4258-INNER-NIC-03 12 18 48.07 47 21 19.7 2559.8
NGC4258-OUTER-NIC-04 12 19 20.62 47 13 12.4 2559.8
NGC4258-OUTER-NIC-02 12 19 15.93 47 12 00.0 2559.8
NGC4258-OUTER-NIC-07 12 19 21.22 47 11 41.4 2559.8
NGC4258-OUTER-NIC-05 12 19 20.18 47 11 25.6 2559.8
NGC4258-OUTER-NIC-10 12 19 26.03 47 12 02.2 2559.8
NGC4258-OUTER-NIC-08 12 19 24.35 47 11 34.2 2559.8
NGC4258-OUTER-NIC-13 12 19 20.73 47 10 54.6 2559.8
NGC4258-OUTER-NIC-06 12 19 20.95 47 10 20.1 2559.8
NGC4258-OUTER-NIC-09 12 19 24.82 47 10 14.0 2559.8

Note. aSame region as NGC4639-BLUE with different orientation.

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We developed an automated pipeline to calibrate the raw NICMOS frames. The first step subtracted one of two "superdarks" produced from archival data obtained after the installation of the NICMOS cooling system, corresponding to the closest temperature state (of two primary temperature regimes) at which the data were obtained. Next, the data were processed through the STScI-supported CALNICA pipeline with the following additions. The STSDAS routine BIASEQ (Bushouse et al. 2000) was used after the corrections for bias, dark counts, and linearity to account for stochastic changes in quadrant bias level. After flat-fielding, cosmic-ray rejection, and count-rate conversion, the images were corrected for the count-rate nonlinearity as calibrated by de Jong et al. (2006). The remaining quadrant-dependent linear DC bias was fit and removed using the PEDSUB task. Any data obtained soon after a passage by HST through the South Atlantic Anomaly (SAA) were corrected for the persistence of cosmic rays using a post-SAA dark frame and the routine SAACLEAN (Bergeron & Dickinson 2003).

Approximately 10% of our images were contaminated by charge persistence after the detector was exposed to the bright limb of the Earth in the preceding orbit. The structure of the persistence image is time independent and is a map of the density of charge traps saturated by the Earth light. A persistence image was produced from the data which was then scaled and subtracted from the affected data as described by Riess & Bergeron (2008).

Residual amplifier glow and its persistence were removed by subtracting a model of the sky image from the combination of all exposures in a visit. The model was smoothed with a ring filter (larger in diameter than the PSF) to ensure that stellar sources in the data were not present in this sky model.

Next we combined the exposures from each visit to produce a full image combination for each pointing listed in Table 3. We first registered the exposures within a visit using the dither positions indicated in the image headers. To register images between visits, we used between 30 and 100 bright sources to empirically measure the shifts and rotations between visits (we also verified that scale variations between orbits were negligible). The typical rms deviation of sources between our visit-to-visit registration solutions was 0.2–0.3 pixels, yielding an error in the mean of less than 0.05 pixels. The final image combination was resampled on a pixel scale of 0farcs038 using the drizzle algorithm (Fruchter & Hook 2002).

Because the PSF of NIC2 with F160W is well sampled, our image combination should cause little broadening of the PSF. To test this, we measured the difference in the photometry of nonvariable supergiants in single epochs and in the full image combinations. The median difference was ∼0.003 mag (in the sense of the combination being brighter, opposite the expected direction if the effect were real) and consistent with zero to within the statistical uncertainty. Thus, we concluded there was no loss in accuracy of the photometry obtained by combining images from individual visits.

To identify the precise positions of Cepheids in the NICMOS image combinations, we derived the geometric transformation from the HST F814W images to the F160W images, iteratively matching bright-to-faint sources to find sources in common. This registration empirically determined the difference in plate scale among ACS, WFPC2, and NIC2. Typically, we identified more than 100 sources in common, resulting in an uncertainty in the mean Cepheid position of <0.03 pixels (1 mas).

2.2. NICMOS Cepheid Photometry

We developed software to measure Cepheid photometry in crowded NICMOS images based on the procedures established for HST optical photometry (Stetson 1994; Saha et al. 1996). Since we know a priori the precise position of the Cepheids in our NICMOS data, we can fix the positions in the NICMOS images to mitigate the measurement bias which can arise naturally for flux measurements of sources made from the same data used for their discovery (Hogg & Turner 1998).

We derived a model of the PSF in our NICMOS images using observations of the bright solar analogue, P330E, averaged over several visits and processed in the same way our host images. P330E provides a fundamental standard for the NICMOS Vega magnitude zero point (F160W = 11.45 mag, Vega system), and our natural system magnitudes are measured relative to this zero point. However, the difference between the photometry of Cepheids in NGC 4258 and the SN hosts, employed to measure H0 in Section 3, are independent of the adopted zero point.

For each known Cepheid, we produced a list of neighboring stars in the NICMOS images within its "critical" radius (4 × FWHM, where FWHM is the full width at half-maximum intensity) or within that of one of its neighbors. Together, these stars and the Cepheid define a "crowded group" whose members must be modeled together. Initially, we subtracted a PSF model at the location of the Cepheid (as determined from the optical data) and then used the algorithm DAOFIND to identify neighboring stars within the critical radius but at least 0.75 × FWHM beyond each Cepheid. Stellar sources within and beyond the group were modeled and subtracted, and the background level for the group was determined from the mode of the pixels in an annulus around the Cepheid with an inner radius of 15 pixels and an outer radius of 20 pixels.

We then used a Levenberg–Maquardt-based algorithm to find the most likely values and uncertainties of the group parameters by minimizing the χ2 statistic between the image and model pixels within the critical radii of the modeled sources. For all non-Cepheid sources, their positions were allowed to vary within 0.5 pixels of their original detected position, and the amplitudes were allowed to vary. For the Cepheids, only the amplitude of the PSF was varied. The Cepheid position determined from the optical images was fixed, as was the group sky level. Our typical group had 5–15 unresolved, modeled sources besides the Cepheids producing 3 times this number of free parameters plus one additional parameter for the Cepheid brightness. The individual pixel noise was relatively uniform, resulting from a combination of sky, dark current and read noise.

After identifying the optimal solution, we subtracted the model from the data and inspected the residuals to determine the best set of global photometry parameters for all images. In Figure 8, we show as an example the image, model, and residuals of the groups for one of the richest NICMOS pointings, NGC3370-GREEN, with 14 Cepheids over a wide range of periods.

Figure 8.

Figure 8. Example of scene modeling for the arcsec surrounding each Cepheid in one NIC2 field, NGC3370-GREEN. For each Cepheid, the stamp on the left shows the region around the Cepheid, the middle stamp shows the model of the stellar sources, and the right stamp is the residual of the image minus the model. The position of the Cepheid as determined from the optical data is indicated by the circle.

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2.3. Sky Determination and Bias Correction

The surroundings of the Cepheids are mottled, with unresolved sources and surface brightness fluctuations whose fluxes are generally fainter but occasionally brighter than our target Cepheids (see Figure 10, NGC 4258 and NGC 3370 GREEN field). These scenes pose a challenge to estimating the correct background level for the Cepheids. Simply measuring the mean flux in an annulus centered on the Cepheid would provide an unbiased but very noisy estimate of the background.

Instead, we follow the conventional approach of determining the sky level from the sky annulus after first subtracting models of the stellar sources within it. Because we would expect a similar number of background sources coincident with (yet inseparable from) the Cepheid, we would naturally underestimate the sky level for the Cepheid. Though this bias is ameliorated somewhat by the use of the mode statistic from the residual image as discussed by Stetson (1987), a bias still remains.

In previous work, it has been shown that this photometric bias in optically selected Cepheid samples is reduced by the act of selecting Cepheids with strong amplitudes and statistically significant variations in flux (Ferrarese et al. 2000). The addition of significant, blended flux would reduce the amplitude of the Cepheid, increase the model uncertainty, and reduce the significance of true variations. However, this mechanism does not apply to the NICMOS images as they were not used to select Cepheids. Indeed, Cepheids are bluer than a common source of blending, red giants, so the blending bias in the NICMOS data can be significant. Macri et al. (2001) found this photometric bias to vary from negligible to 0.1 mag for the Cepheids discovered with WFPC2 and re-observed with NICMOS, and measured the impact to artificial stars injected in the vicinity of each Cepheid to correct for this effect. We adopt the same approach here.

In addition, we can improve our estimate of the individual Cepheids' blending bias. On average, the displacement of a Cepheid's centroid in the NICMOS data relative to its optically determined position correlates with the degree of blending in the NICMOS data. For randomly located sources of blending, brighter blended sources cause larger Cepheid displacements and bias. For artificial stars rediscovered within ∼0.1 pixels of their injected position, we find <0.1 mag of blending bias. The photometric bias grows linearly with the displacement of the centroid, rising to ∼0.3 mag for a full pixel (0farcs038) displacement. Beyond a pixel, the recovered star is often not the same as the one injected (as occurs when the injected star is too faint to be found), and any relation between the displacement and bias dissipates. Rarely, a Cepheid will be exactly coincident with a bright source causing it to be an outlier in the period–luminosity (PL) relation. Such complete blends are later eliminated from both our sample (and the artificial star simulations) with a 2.5σ rejection from the mean, a threshold based on Chauvenet's criterion (i.e., less than half a Cepheid would be expected to exceed the outlier limit for a Gaussian distribution of residuals).

To determine the individual photometric correction for each Cepheid, we added 1000 artificial stars at random positions within a radius of 0farcs05–0farcs75 from each Cepheid. The magnitudes of the artificial stars were given by the Cepheid magnitude predicted by its period using an initial fit (i.e., uncorrected) to the period–magnitude relations. After correcting the Cepheid magnitudes for the measured bias, these relations were refit and the process was repeated until convergence. The dispersion of the artificial stars was used to estimate the uncertainty in the magnitude of the Cepheid by adding this term in quadrature to the Cepheid measurement uncertainty. An example of this artificial-star analysis is shown in Figure 9 for a Cepheid in our second-most-distant galaxy, NGC 3021.

Figure 9.

Figure 9. Example of the artificial stars tests in the region around a Cepheid in the field NGC3021-GREEN with P = 82.0 days. A thousand artificial stars of the brightness of the Cepheid (as determined from its period) were randomly added to the image. The magnitudes of the artificial stars are measured at their known positions (in the same way as the Cepheids). The difference between the input and measured star magnitudes (i.e., the bias) is shown as a function of the displacement between the injected position and the centroid of the star found nearest this position. The photometric bias (brighter) increases with the displacement, a direct consequence of blending. For displacements beyond a pixel, the recovered star is no longer the same as the one injected and the relation between bias and displacement dissipates. Averages and dispersions in bins of the displacement are indicated by the filled dots. For an individual Cepheid, the displacement between the NICMOS and optical position is used to predict and correct for the bias as shown in the vertical dotted line. The uncertainty is derived from the dispersion of the artificial stars.

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For NGC 4258, our anchor galaxy, the median bias correction was 0.14 mag (±0.014), and for the SN hosts the median was 0.16 (±0.021) mag. Thus, we conclude that the photometric corrections for the Cepheids in NGC 4258 and the SN hosts are extremely similar. This is not surprising as the apparent stellar density of the fields is also quite similar as seen in Figure 10. Although NGC 4258 is closer than the SN hosts, reducing its relative crowding, the NGC 4258 inner fields (Macri et al. 2006) are closer to the nucleus where the true stellar density is greater.

Figure 10.

Figure 10. Example NIC2 fields for the anchor galaxy, NGC 4258 (NIC-POS3, right), and a SN field, NGC3370-GREEN (left). The Cepheid positions are indicated. The artificial star tests show that the mean photometric bias in the Cepheid magnitudes due to blending is very similar for these fields (0.14 mag for the anchor and 0.16 mag for the SN host), not surprising from the apparent similarity of their stellar density. Thus, the difference in Cepheid magnitudes in these hosts, the quantity used to construct the distance ladder, is quite insensitive to blending.

Standard image High-resolution image

Because the luminosity calibration of SNe Ia depends only on the difference in the magnitudes of the Cepheids in the anchor galaxy and the SN hosts, the net effect of blending even without correction is quite small: ∼0.02 mag, or about 1% in the distance scale. However, our corrections account for this small difference as further addressed in Section 4.2.

Artificial-star simulations cannot account for blending which is local to the Cepheids (i.e., binarity or cluster companions). However, we expect little net effect from such blending (after the removal of outliers) as such blending is likely to occur with similar frequency in the anchor and SN hosts and thus would largely cancel in their difference.

Because the amplitudes of IR light curves are <0.3 mag, even magnitudes measured at random phases provide comparable precision to the mean flux for determining the PL relation (Madore & Freedman 1991). As our exposures were obtained over ∼2 months, the magnitude measured from the mean image will have a dispersion of <0.08 mag around the mean flux. To account for this error, we correct the measured magnitude to the mean-phase magnitude using the Cepheid phase, period, and amplitude from the optical data, the dates of the NICMOS observations, and the Fourier components of Soszyski et al. (2005) which quantify the relations between Cepheid light curves in the optical and near-IR. These phase corrections were found to be insignificant in the subsequent analysis.

Table 4 contains the aforementioned parameters for each Cepheid. The Cepheid's NIC2 field, position, identification number (from Riess et al. 2009 and Macri et al. 2006), period, mean VI color, F160W mag, and its uncertainty are given in the first eight columns. Column 9 contains the displacement of the Cepheid in the NICMOS data from its optical position in pixels of 0farcs038 size. Column 10 gives the photometric bias determined from the artificial star tests for the Cepheid's environment and displacement and are already added to determine Column 7. Column 11 contains the correction from the sampled phase to the mean and has already been subtracted to determine Column 7. Column 12 contains the metallicity parameter, 12+log[O/H], inferred at the position of each Cepheid. Columns 13 and 14 contain the rejection flag employed and the source of the Cepheid detection as noted in Table 4, respectively.

Table 4. NICMOS Cepheids

Field α (J2000) δ (J2000) Id P (days) VI (mag) F160W (mag) σ (mag) Offset (pix) Bias (mag) Phase (mag) [O/H] Flaga Src*
N4536-B 188.57747 2.193860 9491 19.79 1.10 24.20 0.24 0.52 0.06 0.06 8.69   psl
N4536-B 188.57760 2.192130 9935 42.81 1.05 23.59 0.19 0.35 0.04 −0.09 8.68   psl
N4536-B 188.57383 2.194050 12075 49.70 1.17 23.26 0.22 0.48 0.15 0.02 8.64   psl
N4536-C 188.58908 2.221220 3633 23.37 0.82 23.91 0.20 0.57 0.06 0.14 8.75   psl
N4536-G 188.57352 2.219870 6827 24.54 0.74 23.86 0.31 0.70 0.28 0.19 8.71   psl
N4536-G 188.57237 2.216330 5630 55.24 1.03 22.55 0.19 0.10 0.05 −0.09 8.72   psl
N4536-R 188.58741 2.202540 3571 20.98 0.78 24.02 0.39 0.95 0.29 0.07 8.89   psl
N4536-R 188.58842 2.204110 2692 31.59 1.09 23.71 0.20 0.57 0.20 −0.04 8.90   psl
N4536-R 188.59047 2.200050 6914 34.63 1.22 22.72 0.22 0.37 0.13 −0.03 8.92 rej psl
N4536-R 188.59103 2.202740 4786 37.45 0.93 23.67 0.22 0.46 0.13 −0.03 8.93   psl
N4536-B 188.57908 2.191510 9023 19.90 0.51 24.40 0.30 0.72 0.22 0.04 8.69   lm
N4536-C 188.58946 2.220100 3607 80.17 0.69 22.79 0.25 0.34 0.03 0.01 8.76   lm
N4536-G 188.57149 2.218370 6582 38.84 1.43 23.57 0.20 0.24 0.05 0.00 8.70   lm
N4536-R 188.58811 2.199800 6233 25.30 0.64 23.31 0.23 0.65 0.25 0.09 8.89   lm
N4536-B 188.57932 2.199800 13 18.54 0.72 24.56 0.38 0.60 0.16 0.00 8.71   ps
N4536-R 188.58854 2.192987 5 30.25 1.14 23.64 0.22 0.52 0.15 0.00 8.89   ps
N4639-B 190.72103 13.24735 11893 26.59 1.09 25.83 0.47 1.15 0.29 0.06 8.98 rej psl
N4639-B 190.72412 13.24921 8651 27.54 0.91 24.58 0.54 0.74 0.44 0.08 9.05   psl
N4639-B 190.71891 13.25026 16601 42.20 1.18 23.54 0.56 0.54 0.18 0.01 9.09   psl
N4639-B 190.72047 13.24661 12394 54.82 1.06 24.10 0.47 2.69 0.53 0.05 8.95   psl
N4639-B-L 190.72412 13.24921 8651 27.54 0.91 24.68 0.47 0.36 0.59 0.11 9.05   psl
N4639-B-L 190.71891 13.25026 16601 42.20 1.18 23.84 0.34 0.41 0.29 −0.02 9.09   psl
N4639-B-L 190.72047 13.24661 12394 54.82 1.06 23.80 0.33 8.76 0.06 −0.03 8.95   psl
N4639-C 190.70392 13.26827 40321 37.27 1.24 23.89 0.25 0.20 0.11 0.02 8.83   psl
N4639-C 190.70509 13.26793 39829 39.41 0.91 23.79 0.22 0.24 0.07 −0.07 8.86   psl
N4639-C 190.70451 13.26849 40158 52.16 1.09 24.26 0.27 0.32 0.16 −0.09 8.84   psl
N4639-C 190.70539 13.26932 61786 56.31 1.18 24.04 0.24 0.87 0.11 0.13 8.84   psl
N4639-G 190.71845 13.27400 30160 51.11 1.11 24.21 0.19 0.16 0.04 −0.00 8.69   psl
N4639-R 190.73302 13.25647 4481 34.24 0.97 24.08 0.19 0.08 0.16 0.00 8.88   psl
N4639-B 190.72165 13.25058 12430 39.53 0.93 25.23 0.67 2.33 0.70 0.01 9.11 rej lm
N4639-B 190.72020 13.24847 13602 47.27 1.44 23.92 0.31 0.56 0.16 0.04 9.02   lm
N4639-B-L 190.72020 13.24847 13602 47.27 1.44 24.06 0.30 0.50 0.03 −0.02 9.02   lm
N4639-R 190.73314 13.25609 4383 42.43 1.09 23.98 0.22 0.38 0.09 −0.03 8.88   lm
N3370-B 161.76928 17.28204 24497 16.78 0.87 25.89 0.68 2.63 0.39 −0.05 8.86 low P lm
N3370-B 161.76766 17.28206 21444 19.64 0.92 26.21 0.42 3.98 0.03 −0.06 8.91 low P lm
N3370-B 161.76554 17.28584 15081 32.56 1.22 25.56 0.39 0.98 0.34 0.00 8.80   lm
N3370-B 161.76923 17.28640 21445 37.10 1.07 24.67 0.20 0.47 0.14 −0.14 8.68   lm
N3370-B 161.76844 17.28412 21506 38.54 0.75 24.63 0.30 0.60 0.25 0.13 8.80   lm
N3370-B 161.76774 17.28324 20732 41.55 1.15 24.48 0.27 4.33 0.17 −0.06 8.86   lm
N3370-B 161.76869 17.28313 22612 69.35 1.04 23.83 0.22 0.37 0.14 0.01 8.83   lm
N3370-C 161.77931 17.25660 50670 20.52 0.84 25.50 0.41 2.96 0.13 0.00 8.64 low P lm
N3370-C 161.77441 17.25595 47059 24.49 0.93 24.95 0.34 1.10 0.31 −0.01 8.68   lm
N3370-C 161.77627 17.25957 47494 24.43 1.22 24.57 0.40 0.07 0.28 −0.01 8.77 rej lm
N3370-C 161.78015 17.25611 51334 28.79 0.98 24.88 0.32 1.31 0.25 −0.06 8.61   lm
N3370-C 161.77485 17.25741 46992 29.60 0.95 24.94 0.27 2.27 −0.02 −0.03 8.72   lm
N3370-C 161.77587 17.25844 47492 39.41 1.15 24.84 0.22 0.48 0.08 −0.05 8.75   lm
N3370-C 161.77827 17.26024 48903 51.68 1.09 24.76 0.28 0.99 0.09 −0.00 8.74   lm
N3370-C 161.77799 17.26004 48741 96.49 0.96 23.90 0.25 0.44 0.11 −0.01 8.75   lm
N3370-R 161.78414 17.26088 52428 33.48 1.04 24.95 0.32 0.74 0.15 −0.06 8.54   lm
N3370-R 161.78547 17.26580 52279 33.69 1.03 24.93 0.26 0.63 0.13 −0.03 8.48   lm
N3370-G 161.75400 17.28417 2638 17.46 0.76 26.08 0.63 6.05 0.47 0.00 8.71 low P lm
N3370-G 161.75882 17.28007 8807 23.72 1.04 26.46 0.41 0.87 0.61 −0.08 8.92 rej lm
N3370-G 161.75875 17.28389 61720 25.43 0.90 26.21 0.47 1.70 0.61 −0.02 8.86 rej lm
N3370-G 161.75791 17.28025 62219 29.43 1.10 25.25 0.49 1.29 0.61 −0.07 8.89   lm
N3370-G 161.75647 17.28052 5744 27.74 1.06 25.35 0.28 5.16 0.18 −0.09 8.83   lm
N3370-G 161.75620 17.28353 4345 34.07 1.21 25.85 0.31 1.05 0.37 −0.12 8.79 rej lm
N3370-G 161.75647 17.28320 4710 32.62 1.07 25.30 0.39 0.63 0.39 0.07 8.80   lm
N3370-G 161.75710 17.28387 59919 36.99 0.88 25.22 0.25 3.02 0.21 −0.04 8.81   lm
N3370-G 161.76072 17.28069 10677 35.24 1.11 23.49 0.36 10.5 0.29 −0.00 8.97 rej lm
N3370-G 161.75713 17.28309 5439 45.82 0.98 23.66 0.28 0.95 0.31 0.05 8.83 rej lm
N3370-G 161.75761 17.28213 6440 43.94 1.17 24.37 0.28 0.32 0.31 0.08 8.86   lm
N3370-G 161.75980 17.28234 9014 45.10 1.12 24.60 0.27 1.47 0.28 −0.06 8.91   lm
N3370-G 161.75525 17.28084 4367 52.72 1.22 24.94 0.28 0.09 0.05 0.04 8.78   lm
N3370-G 161.75695 17.28274 5361 50.60 0.90 23.99 0.27 0.28 0.05 −0.06 8.83   lm
N3370-G 161.75874 17.28466 6706 64.79 0.96 25.00 0.40 1.65 0.31 0.09 8.84 rej lm
N3370-G 161.75677 17.28193 5501 62.71 1.26 23.58 0.25 0.24 0.04 0.02 8.83   lm
N3982-B 179.09710 55.12080 9531 25.83 0.87 25.45 0.35 0.76 0.00 −0.11 8.70   psl
N3982-B 179.09800 55.11996 32398 45.43 0.87 24.33 0.19 0.36 0.04 −0.10 8.72   psl
N3982-B 179.10593 55.12011 9075 37.04 0.80 24.32 0.33 8.02 0.23 −0.00 8.97   psl
N3982-B 179.10563 55.11935 9114 53.86 1.07 23.52 0.34 0.73 0.13 −0.09 8.93   psl
N3982-C 179.13277 55.12384 12134 19.08 0.86 24.92 0.67 0.80 0.64 −0.03 8.90   psl
N3982-C 179.13148 55.12278 627 38.86 1.15 24.28 0.39 0.47 0.21 0.00 8.93   psl
N3982-C 179.13803 55.12526 281 40.88 1.31 24.36 0.25 0.52 0.23 0.03 8.72   psl
N3982-C 179.13666 55.12775 43380 44.22 0.99 23.97 0.33 0.21 0.21 0.10 8.77   psl
N3982-G 179.09010 55.12513 32813 30.04 1.16 25.22 0.41 4.81 −0.09 −0.01 8.48   psl
N3982-G 179.08546 55.12250 33304 32.02 0.97 24.56 0.21 0.38 0.11 −0.05 8.32   psl
N3982-R 179.09379 55.10950 9610 38.11 1.23 24.40 0.20 0.24 0.06 −0.13 8.28   psl
N3982-R 179.09561 55.11319 32634 41.00 0.89 24.85 0.34 0.69 0.03 −0.11 8.46   psl
N3982-C 179.13316 55.12718 11942 25.57 1.49 25.28 0.37 1.59 0.56 −0.02 8.89   lm
N3982-R 179.09459 55.11199 9584 61.79 1.16 24.59 0.24 0.48 0.15 −0.07 8.39 rej lm
N3982-Y 179.12839 55.13052 12782 24.11 1.11 25.47 0.46 1.53 0.85 −0.14 8.99   lm
N3982-Y 179.12240 55.13018 2002 37.57 1.35 22.81 0.37 1.07 0.42 −0.08 9.14 rej lm
N3982-Y 179.12400 55.12933 1434 75.40 1.86 22.33 0.25 0.69 0.21 −0.06 9.14 rej lm
N3982-Y 179.13114 55.13153 32 29.53 0.89 23.90 0.50 10.4 0.13 0.00 8.88 rej ps
N3021-B 147.72778 33.54702 30672 13.92 0.48 25.88 0.37 0.87 0.24 −0.04 8.29 low P lm
N3021-B 147.73211 33.54878 26946 26.84 0.85 25.04 0.32 0.95 0.22 0.04 8.76   lm
N3021-B 147.72812 33.54750 30428 32.60 0.78 25.03 0.30 1.08 0.17 −0.01 8.37   lm
N3021-B 147.73249 33.54885 26545 39.57 0.87 24.79 0.34 4.08 0.08 0.05 8.79   lm
N3021-C 147.72678 33.55614 32088 25.77 0.98 26.35 0.59 1.79 0.59 −0.03 8.89 rej lm
N3021-C 147.72586 33.55581 32375 24.01 0.82 24.46 0.45 1.37 0.11 −0.01 8.82 rej lm
N3021-C 147.72645 33.56000 32380 25.18 0.72 25.27 0.31 0.91 0.08 −0.09 8.69   lm
N3021-C 147.72789 33.55893 31803 37.27 0.89 25.30 0.28 1.04 −0.16 −0.05 8.83   lm
N3021-G 147.74838 33.55002 8621 15.37 0.75 25.94 0.64 3.49 0.70 −0.06 8.94 low P lm
N3021-G 147.74935 33.55170 8102 18.71 0.62 26.10 0.47 2.77 0.48 0.08 8.90 low P lm
N3021-G 147.74871 33.55237 8636 24.36 0.72 24.91 0.60 4.36 0.10 −0.02 8.93   lm
N3021-G 147.74791 33.55032 9028 31.89 0.72 24.70 0.32 1.53 0.60 0.13 8.98   lm
N3021-G 147.74757 33.55109 9402 39.77 1.15 23.98 0.33 5.43 0.13 0.12 9.02 rej lm
N3021-G 147.74740 33.55142 9611 40.49 0.63 25.30 0.49 3.20 0.05 0.05 9.04   lm
N3021-G 147.74683 33.55170 10203 95.91 0.85 24.05 0.25 5.58 0.13 0.00 9.08   lm
N3021-G 147.75116 33.55414 7098 82.66 0.72 24.18 0.25 0.55 0.12 0.00 8.67   lm
N3021-G 147.74734 33.55075 9558 88.18 1.43 23.96 0.25 1.28 0.33 −0.05 9.03   lm
N3021-R 147.73688 33.55930 23149 32.52 0.92 25.59 0.40 2.34 0.37 0.06 8.92   lm
N3021-R 147.73982 33.56093 19817 68.61 1.06 23.47 0.24 0.66 0.18 −0.02 8.61   lm
N1309-B 50.513220 −15.40390 52566 47.41 0.53 24.79 0.43 0.64 0.12 −0.01 8.70   lm
N1309-B 50.513500 −15.39881 52170 47.99 0.97 24.63 0.24 0.39 0.07 −0.01 8.73   lm
N1309-B 50.512020 −15.39909 53187 59.75 0.57 24.91 0.35 0.81 0.14 −0.00 8.68   lm
N1309-B 50.516480 −15.40236 49485 74.19 0.39 23.72 0.37 0.24 −0.05 0.01 8.83   lm
N1309-C 50.535980 −15.41296 6737 25.45 0.71 26.03 0.33 3.53 0.03 0.03 8.77 low P lm
N1309-C 50.535850 −15.41538 7224 30.90 0.82 25.23 0.31 0.71 0.06 0.09 8.70 low P lm
N1309-C 50.535240 −15.41099 7989 39.41 0.90 24.70 0.24 1.33 0.17 −0.06 8.85   lm
N1309-C 50.535740 −15.41413 59151 32.61 0.59 24.85 0.22 4.19 0.01 −0.02 8.74 low P lm
N1309-C 50.537010 −15.41209 4882 48.91 0.76 25.25 0.19 0.11 0.12 −0.05 8.78   lm
N1309-C 50.536060 −15.41233 6542 59.12 0.92 24.57 0.22 0.45 0.12 −0.03 8.79   lm
N1309-C 50.535980 −15.41154 6581 58.98 0.79 24.82 0.22 0.36 0.11 0.12 8.82   lm
N1309-C 50.535540 −15.41410 7702 73.76 0.86 24.36 0.25 0.06 0.18 −0.04 8.74   lm
N1309-G 50.540170 −15.39411 2032 42.53 0.72 24.91 0.26 0.74 0.11 −0.03 8.85   lm
N1309-G 50.541640 −15.39645 1166 41.11 1.10 24.84 0.44 4.43 −0.23 0.05 8.83   lm
N1309-Y 50.528160 −15.40843 23076 30.66 0.82 25.99 0.49 2.33 0.52 0.00 9.00 low P lm
N1309-Y 50.525250 −15.40856 30349 33.51 0.61 26.04 0.68 3.78 −0.22 0.00 8.96 rej,low P lm
N1309-Y 50.531480 −15.40689 15346 46.85 0.81 25.53 0.51 1.63 0.13 0.01 9.04   lm
N1309-Y 50.528240 −15.40865 22918 42.03 0.81 24.97 0.38 0.76 0.54 0.02 8.99   lm
N1309-Y 50.528300 −15.40526 68817 49.93 0.53 24.63 0.62 10.6 0.40 −0.01 9.11   lm
N1309-Y 50.526610 −15.40578 71911 51.99 0.75 24.07 0.44 4.48 −0.18 −0.02 9.07   lm
N1309-Y 50.526040 −15.40768 28132 52.24 1.00 24.89 0.35 1.29 0.24 0.01 9.00   lm
N1309-Y 50.528080 −15.40923 69494 60.17 0.93 24.97 0.46 6.65 −0.05 −0.00 8.97   lm
N1309-Y 50.529580 −15.40892 19918 64.94 0.80 24.48 0.33 0.26 0.08 0.02 8.98   lm
N1309-Y 50.531070 −15.40794 64757 65.03 1.09 24.27 0.29 1.12 0.24 0.02 9.01   lm
IN-NIC-01 184.68938 47.33554 118961 12.82 1.26 22.58 0.28 0.76 0.16 −0.06 8.90 rej lm
IN-NIC-02 184.70089 47.34927 110213 11.60 1.18 23.20 0.32 0.71 0.20 −0.01 8.91   lm
IN-NIC-02 184.69963 47.35114 113982 11.64 0.84 24.81 0.48 3.12 −0.14 0.04 8.90 rej lm
IN-NIC-05 184.70311 47.31951 83857 28.13 0.76 22.05 0.25 4.61 0.32 −0.04 8.93   lm
IN-NIC-05 184.70506 47.32093 80885 65.23 1.13 20.91 0.20 0.32 0.14 −0.07 8.94   lm
IN-NIC-06 184.70488 47.33184 91209 10.80 0.88 23.59 0.37 1.14 0.35 −0.09 8.94   lm
IN-NIC-06 184.70320 47.32795 91129 11.16 1.00 23.25 0.39 1.40 0.40 −0.03 8.93   lm
IN-NIC-07 184.71071 47.34988 95403 11.58 1.03 24.41 0.56 0.99 0.38 −0.00 8.91 rej lm
IN-NIC-07 184.70793 47.34997 100093 12.02 0.71 24.78 0.59 7.06 0.82 0.12 8.91 rej lm
IN-NIC-08 184.71066 47.31184 57246 10.90 0.89 22.91 0.63 0.33 0.66 −0.05 8.94 rej lm
IN-NIC-08 184.71169 47.30965 51416 11.09 0.77 22.81 0.44 5.43 0.40 0.06 8.94 rej lm
IN-NIC-08 184.71509 47.31114 43119 23.81 0.85 22.56 0.32 5.17 0.65 0.15 8.96   lm
IN-NIC-08 184.71112 47.31241 56661 23.83 0.68 23.04 0.36 0.22 0.05 −0.09 8.94   lm
IN-NIC-08 184.71746 47.31136 36357 29.48 0.86 22.33 0.43 1.78 0.11 −0.10 8.96   lm
IN-NIC-08 184.71239 47.30964 49279 36.12 0.94 21.44 0.19 1.74 0.03 0.27 8.94   lm
IN-NIC-08 184.71440 47.31272 47358 50.89 0.85 21.63 0.20 0.30 0.16 −0.04 8.96   lm
IN-NIC-08 184.71707 47.31160 37841 66.89 1.13 21.61 0.25 0.89 0.21 0.02 8.96   lm
IN-NIC-08 184.71324 47.31224 50193 93.23 0.83 21.17 0.25 0.34 0.06 −0.00 8.95   lm
IN-NIC-08 184.71528 47.31138 42837 95.92 0.93 20.93 0.25 0.34 0.14 0.03 8.96   lm
IN-NIC-09 184.71932 47.31409 34408 22.42 0.85 22.18 0.43 0.63 0.49 −0.08 8.97   lm
IN-NIC-09 184.71746 47.31136 36357 29.48 0.86 22.16 0.29 0.58 0.42 −0.04 8.96   lm
IN-NIC-09 184.71962 47.31407 33434 34.48 0.78 20.99 0.38 0.98 0.20 −0.01 8.98 rej lm
IN-NIC-09 184.72344 47.31211 19435 39.53 0.82 20.70 0.33 0.39 0.18 −0.00 8.99 rej lm
IN-NIC-09 184.71707 47.31160 37841 66.89 1.13 21.48 0.25 6.30 0.79 −0.10 8.96   lm
IN-NIC-10 184.72821 47.31332 6616 16.99 0.93 22.83 0.37 5.96 0.64 0.02 9.00   lm
IN-NIC-10 184.72937 47.31691 8052 20.76 0.90 22.17 0.49 6.21 0.23 0.02 9.00   lm
IN-NIC-10 184.72616 47.31461 14643 22.04 1.92 22.05 0.63 2.99 −0.30 −0.04 8.99   lm
IN-NIC-10 184.72389 47.31625 23741 22.68 1.43 22.77 0.32 1.78 0.03 −0.16 8.99   lm
IN-NIC-10 184.72948 47.31746 8361 23.79 1.22 21.97 0.33 0.89 0.35 0.11 8.99   lm
IN-NIC-10 184.72728 47.31776 15470 50.70 1.48 22.19 0.41 1.83 0.24 −0.04 8.99   lm
IN-NIC-10 184.72834 47.31586 9633 69.46 1.19 20.75 0.26 0.48 0.12 0.13 9.00   lm
IN-NIC-11 184.72657 47.34519 54398 15.73 1.06 22.48 0.50 1.82 0.47 0.10 8.91 rej lm
IN-NIC-12 184.72567 47.32182 25760 9.979 0.92 24.06 0.44 5.12 0.69 −0.07 8.98   lm
IN-NIC-12 184.72505 47.32063 25811 10.30 0.90 23.26 0.68 2.82 0.37 −0.07 8.98   lm
IN-NIC-12 184.72545 47.32166 26176 18.28 1.67 22.49 0.59 0.46 0.15 0.21 8.98   lm
IN-NIC-12 184.72759 47.31971 17151 22.35 1.65 22.46 0.39 0.08 0.47 −0.10 8.99   lm
IN-NIC-12 184.72948 47.31746 8361 23.79 1.22 22.41 0.40 0.98 0.35 −0.01 8.99   lm
IN-NIC-12 184.73086 47.32120 9241 27.25 0.72 22.96 0.47 3.40 0.57 −0.01 8.98   lm
IN-NIC-12 184.73086 47.32065 8480 37.63 0.76 21.29 0.26 0.36 0.04 −0.08 8.98 rej lm
IN-NIC-12 184.72728 47.31776 15470 50.70 1.48 22.49 0.28 2.30 0.42 −0.06 8.99 rej lm
IN-NIC-01 184.68739 47.33468 121078 18.42 0.97 22.61 0.21 0.55 0.00 0.18 8.89   lm
IN-NIC-01 184.69005 47.33265 116159 21.87 0.89 23.09 0.21 0.17 0.11 −0.13 8.90   lm
IN-NIC-02 184.70103 47.35073 111064 14.59 0.98 22.99 0.34 2.11 0.30 0.16 8.90   lm
IN-NIC-05 184.70263 47.31955 84934 15.64 0.97 23.35 0.45 1.28 0.42 −0.12 8.93   lm
IN-NIC-05 184.70646 47.32085 77610 42.82 1.00 22.14 0.25 0.71 0.17 −0.15 8.94   lm
IN-NIC-06 184.70453 47.32912 89618 12.47 1.01 23.06 0.31 0.77 0.39 −0.01 8.94   lm
IN-NIC-06 184.70347 47.33143 93585 18.19 0.97 22.90 0.36 1.15 0.48 −0.00 8.93   lm
IN-NIC-07 184.70981 47.35380 99783 14.31 0.80 23.68 0.35 3.80 0.13 0.14 8.90   lm
IN-NIC-07 184.71165 47.35150 95003 20.57 1.21 23.00 0.26 0.78 0.14 0.23 8.90   lm
IN-NIC-07 184.70864 47.35115 99756 29.05 1.04 22.70 0.23 0.59 0.10 −0.00 8.90   lm
IN-NIC-08 184.71348 47.31023 46762 12.65 0.71 23.65 0.56 6.64 −0.38 −0.02 8.95   lm
IN-NIC-08 184.71404 47.31339 49332 16.00 0.71 23.63 0.68 5.62 0.45 0.10 8.96   lm
IN-NIC-08 184.71249 47.31037 49942 16.34 1.03 23.68 0.39 0.63 0.62 −0.03 8.95   lm
IN-NIC-08 184.71407 47.31168 46945 25.12 0.90 22.83 0.31 0.61 0.35 −0.00 8.95   lm
IN-NIC-08 184.71470 47.30831 189390 34.41 0.95 21.96 0.22 0.27 0.02 −0.02 8.95   lm
IN-NIC-09 184.71988 47.31322 31615 23.98 1.32 22.07 0.45 2.34 0.40 0.22 8.98   lm
IN-NIC-09 184.72036 47.31321 30136 26.07 1.22 22.36 0.40 1.02 0.35 −0.13 8.98   lm
IN-NIC-09 184.72467 47.31105 14316 29.63 1.13 22.26 0.50 2.13 0.08 0.11 8.99   lm
IN-NIC-09 184.72402 47.31193 17423 34.57 1.28 22.79 0.33 1.54 0.19 0.24 8.99   lm
IN-NIC-09 184.72228 47.31203 22927 33.99 1.04 22.70 0.49 2.72 −0.09 0.19 8.98   lm
IN-NIC-09 184.71972 47.31095 29058 40.54 0.95 22.56 0.32 4.97 −0.28 −0.19 8.97   lm
IN-NIC-09 184.71854 47.31213 34159 41.57 1.01 21.91 0.24 0.38 0.13 −0.17 8.97   lm
IN-NIC-10 184.72694 47.31762 16299 12.28 1.28 22.86 0.53 6.31 0.54 0.17 8.99   lm
IN-NIC-10 184.72795 47.31605 11066 15.91 1.21 23.80 0.65 13.7 0.60 0.16 9.00   lm
IN-NIC-10 184.72657 47.31756 17357 22.45 1.21 22.39 0.58 4.39 −0.11 −0.06 8.99   lm
IN-NIC-10 184.72375 47.31687 24960 25.49 1.06 22.19 0.50 5.02 0.14 −0.07 8.99   lm
IN-NIC-12 184.72694 47.31762 16299 12.28 1.28 23.33 0.60 2.37 0.83 0.03 8.99   lm
IN-NIC-12 184.73048 47.31999 8723 35.57 0.96 21.46 0.29 0.33 0.10 0.03 8.99   lm
IN-NIC-13 184.73382 47.33895 24365 22.38 0.93 23.12 0.39 0.59 0.44 −0.10 8.91   lm
IN-NIC-13 184.73069 47.33826 32759 42.31 0.95 22.14 0.26 0.99 0.19 −0.10 8.92   lm
IN-NIC-03 184.69895 47.35497 117710 14.30 0.84 23.56 0.26 0.51 0.21 −0.06 8.89   lm
IN-NIC-03 184.69870 47.35607 118782 25.56 0.89 22.34 0.20 0.41 0.09 −0.03 8.89   lm
IN-NIC-04 184.69961 47.33411 102255 12.25 1.04 24.23 0.42 1.68 0.28 −0.10 8.92 rej lm
IN-NIC-04 184.69929 47.33694 105183 23.00 1.11 22.85 0.29 1.18 0.26 −0.14 8.92   lm
IN-NIC-04 184.69827 47.33338 104131 22.89 0.87 22.21 0.23 3.02 0.14 0.24 8.92   lm
IN-NIC-04 184.70092 47.33803 103070 24.86 1.12 22.88 0.25 0.47 0.24 −0.05 8.92   lm
NIC-POS10 184.84596 47.24667 310420 45.40 0.86 21.34 0.18 0.17 0.03 0.19 8.74   lm
NIC-POS11 184.83328 47.24912 312665 39.09 0.80 23.20 0.19 0.25 0.05 0.11 8.78 rej lm
NIC-POS12 184.85726 47.22978 307758 32.40 1.02 22.23 0.18 0.20 0.00 0.22 8.71   lm
NIC-POS13 184.72838 47.36342 999999 92.00 1.00 20.26 0.25 0.23 0.02 0.20 8.84   lm
NIC-POS1 184.70981 47.35380 99783 14.31 0.80 23.41 0.36 5.59 0.16 −0.01 8.90   lm
NIC-POS1 184.71165 47.35150 95003 20.57 1.21 23.24 0.26 0.92 0.13 −0.00 8.90   lm
NIC-POS1 184.70864 47.35115 99756 29.05 1.04 22.38 0.21 0.48 0.06 0.06 8.90   lm
NIC-POS1 184.71263 47.35496 95995 36.79 1.05 22.25 0.21 0.21 −0.00 −0.15 8.89   lm
NIC-POS2 184.71006 47.34746 94632 16.03 1.14 23.17 0.38 2.89 0.02 −0.12 8.91   lm
NIC-POS2 184.70643 47.34573 99411 17.02 0.98 22.97 0.28 0.59 0.16 −0.12 8.92   lm
NIC-POS2 184.70905 47.34317 92950 16.70 1.01 23.39 0.35 0.71 0.09 0.00 8.92   lm
NIC-POS2 184.70773 47.34559 97135 31.78 1.20 22.43 0.28 0.57 0.15 −0.11 8.92   lm
NIC-POS3 184.69961 47.33411 102255 12.25 1.04 23.66 0.30 0.46 0.52 −0.13 8.92   lm
NIC-POS3 184.69929 47.33694 105183 23.00 1.11 22.85 0.28 1.06 0.15 −0.12 8.92   lm
NIC-POS3 184.69827 47.33338 104131 22.89 0.87 22.03 0.25 0.98 0.09 0.21 8.92   lm
NIC-POS3 184.69648 47.33310 106960 28.26 0.97 22.51 0.19 0.40 0.08 0.18 8.92   lm
NIC-POS4 184.70931 47.32443 74725 11.99 0.81 23.18 0.68 0.14 0.15 −0.08 8.95   lm
NIC-POS4 184.71440 47.32211 59576 12.65 1.10 23.43 0.36 7.48 0.70 0.15 8.96   lm
NIC-POS4 184.70898 47.32425 75254 16.52 0.98 23.02 0.26 0.34 0.42 −0.04 8.95   lm
NIC-POS4 184.71341 47.32261 62769 33.02 0.77 21.93 0.25 0.66 0.09 −0.05 8.96   lm
NIC-POS5 184.74252 47.34238 2686 44.05 1.06 21.57 0.19 0.28 0.05 0.10 8.88   lm
NIC-POS6 184.83891 47.17151 107069 83.40 1.58 20.56 0.25 0.06 0.01 −0.10 8.67   lm
NIC-POS7 184.85593 47.16107 104251 33.29 1.04 22.56 0.18 0.03 0.01 −0.13 8.64   lm
NIC-POS8 184.79990 47.20616 220789 21.29 1.04 23.31 0.19 0.35 0.04 0.16 8.77   lm
NIC-POS8 184.79992 47.20734 220887 31.29 1.26 22.70 0.18 0.15 0.00 −0.14 8.77   lm
NIC-POS8 184.80002 47.20401 220576 101.9 1.92 20.22 0.25 0.11 0.01 −0.02 8.76   lm
OUT-NIC-02 184.81866 47.19866 34729 14.92 1.12 23.44 0.23 0.22 0.07 −0.01 8.75   lm
OUT-NIC-04 184.83594 47.22012 28606 53.88 1.03 21.76 0.18 0.25 0.03 0.09 8.76   lm
OUT-NIC-05 184.83415 47.19030 21109 8.503 0.86 23.98 0.27 8.81 0.34 0.04 8.72   lm
OUT-NIC-06 184.83914 47.17078 12705 9.942 0.84 24.23 0.25 0.31 0.07 −0.08 8.67   lm
OUT-NIC-06 184.83644 47.17427 14709 10.97 0.84 23.68 0.22 0.42 −0.00 0.10 8.68   lm
OUT-NIC-06 184.83530 47.17378 15276 16.43 0.87 23.62 0.21 0.62 0.03 −0.07 8.68   lm
OUT-NIC-07 184.83867 47.19483 19312 13.55 0.82 23.32 0.21 0.52 0.07 −0.08 8.72   lm
OUT-NIC-08 184.85030 47.19245 11990 8.024 0.65 23.78 0.22 0.34 0.09 0.10 8.71   lm
OUT-NIC-09 184.85468 47.16904 5713 31.74 0.96 22.41 0.18 0.14 0.02 −0.15 8.66   lm
OUT-NIC-10 184.85857 47.20057 9786 8.920 0.72 23.85 0.24 0.28 0.14 −0.05 8.70   lm
OUT-NIC-13 184.83969 47.18170 14656 8.779 0.80 24.23 0.35 1.52 0.06 0.03 8.70   lm

Notes. *Source of Optical Cepheid parameters. lm=L.M. from Riess et al. 2009 and Macri et al. 2009, psl = P.B.S. from Stetson & Gibson 2001. aCepheid Rejection Flag. <P indicates that the period is shorter than the optical completeness from Riess et al. 2009, rej is σ clipped.

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2.4. Near-Infrared Cepheid Relations

In the ith galaxy, for a set of Cepheids with periods P, with mean magnitudes mX, the pulsation equation leads to a PL relation of the form

Equation (1)

where zpX,i is the intercept of the PL relation, and bX is its slope for the passband X. We will make use of multi-linear regressions to simultaneously fit the Cepheid data (and in the following section the SN data) and to propagate the covariance of the data and model to the fitted parameters.

It is convenient to express the PL relation for the jth Cepheid in the ith host as

Equation (2)

Although the H-band PL relation is expected to be relatively insensitive to metallicity as compared to the visible, where metal-line blanketing influences opacity (Marconi et al. 2005), we will not assume the LMC slope applies to our more metal-rich Cepheid sample.9 Instead, we will determine the slope for the narrow range of solar-like metallicity of our sample.

We rewrite Equation (2) in the matrix form to allow a single, unknown value of bH,

Equation (3)

Referring to this matrix equation symbolically as y = Lq, we define: y is the column of measured magnitudes, L is the two-dimensional "design matrix" with entries that arrange the operations, and q is the set of free parameters. With these definitions and C as the matrix of measurement errors, we write the χ2 statistic as

Equation (4)

The minimization of χ2 with respect to q gives the following expression for the maximum likelihood estimator of q:

Equation (5)

The standard errors for the parameters in ${\bf \hat{q}}$ are given by the covariance matrix, (LTC−1L)−1 (Rybicki & Press 1992).

The seven individual PL relations fitted with a common slope are shown in Figure 11. While 240 Cepheids previously identified in the optical (Riess et al. 2009) could be measured in the NICMOS data, it is apparent from Figure 11 that ∼10% appear as outliers in the relations. This is not surprising as we expect outliers to occur from (1) a complete blend with a bright, red source such as a red giant or (2) objects misidentified as Cepheids in the optical or with the wrong period. To reject these outliers, we performed an iterative rejection of objects > 0.75 mag from the PL relations, resulting in a reduction of the sample to 209. In the following section, we consider the effect of this rejection on the determination of H0.

Figure 11.

Figure 11. Near-infrared Cepheid period–luminosity relations. For the 6 SN Ia hosts and the distance-scale anchor, NGC 4258, the Cepheid magnitudes are from the same instrument and filter combination, NIC2 F160W. This uniformity allows for a significant reduction in systematic error when utlizing the difference in these relations along the distance ladder. The measured metallicity for all the Cepheids is solar-like (12+log [O/H] ∼ 8.9). A single slope has been fit to the relations and is shown as the solid line. 10% of the objects were outliers from the relations (open diamonds) and are flagged as such for the subsequent analysis. Filled points with asterisks indicate Cepheids whose periods are shorter than the incompleteness limit identified from their optical detection. The lower right panel shows the near-IR PL relation derived from 10 Milky Way Cepheids with precise, individual parallax measurements from Benedict et al. (2007).

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For the sample, we find bH = −3.09 ± 0.11, in good agreement with the value of −3.23 ± 0.04 from the LMC (Persson et al. 2004).

To determine the difference in distances between the anchor galaxy and the SN hosts, we now account for interstellar extinction of the Cepheids. Although such extinction is a factor of ∼5 smaller in the H band than in the optical and might be ignored (an option we consider in Section 4), the difference in what remains directly impacts the determination of H0 at the few percent level.

The use of two or more passbands allows for the measurement of reddening and the associated correction for extinction. For each Cepheid, we use the measurement of its mean VI color from WFPC2 or ACS. Following Madore (1982), we define a "Wesenheit reddening-free" mean magnitude

Equation (6)

where RAH/(AVAI). For a Cardelli et al. (1989) reddening law and a Galactic-like value of RV = 3.1, R = 0.479. In the following section, we consider the sensitivity of H0 to the value of RV.

To account for possible differences in the Cepheid photometry measured with ACS and WFPC2, we compared the photometry of 8711 nonvariable stellar sources in the field of NGC 3982 observed with both cameras through F555W and F814W. Using the master catalog of WFPC2 photometry used by Gibson et al. (2000) and Stetson & Gibson (2001), we find the mean difference between our WFPC2 and ACS VI colors of these sources to be 0.054 ± 0.005 mag (WFPC2 is bluer), with no dependence on source color or magnitude. The origin of this difference largely resides in the specific zero points adopted by Stetson & Gibson (2001) and those from Riess et al. (2009) and Macri et al. (2006, 2009). Because our goal is limited to placing the WFPC2 Cepheid colors on the same photometric scale as the ACS data to measure distances relative to NGC 4258, we corrected the WFPC2 Cepheid data of Gibson et al. (2000) for NGC 4639, NGC 4536, and NGC 3982 to the ACS color scale. The mean VI colors of the Cepheids are given in Table 4. We propagate a systematic 0.02 mag error in the difference between VI colors measured with WFPC2 and ACS in the following section, though the net effect on mW amounts to only 0.02R ≈ 0.01 mag.10

Substituting the values of mW for mH in Equation (3), we find bW = −3.23 ± 0.11.

Differences in zpW between galaxies are equivalent to differences in distances, which follows from Equation (1) and μ0 = mWMW. Therefore, we can now substitute (zpW,izpW,4258) = (μ0,i − μ0,4258) to derive reddening-free distances, μ0,i, for the SN hosts relative to NGC 4258 from the Cepheids, μ0,i − μ0,4258. The results are given in Table 5, Column 6.

Table 5. Distance Parameters

      SN Only Cepheids Only
Host SN Ia Filters m0v,i + 5av σa μ0,i − μ0,4258
NGC 4536 SN 1981B UBVR 15.156 0.145 1.145 (0.0845)
NGC 4639 SN 1990N UBVRI 16.059 0.111 2.185 (0.0963)
NGC 3982 SN 1998aq UBVRI 15.976 0.091 2.473 (0.101)
NGC 3370 SN 1994ae UBVRI 16.578 0.102 2.831 (0.0771)
NGC 3021 SN 1995al UBVRI 16.726 0.113 2.914 (0.101)
NGC 1309 SN 2002fk BVRI 16.806 0.103 3.261 (0.0861)
Weighted mean  ⋅⋅⋅   ⋅⋅⋅   ⋅⋅⋅  0.0448  ⋅⋅⋅  (0.0367)

Note. aFor MLCS2k2, 0.08 mag added in quadrature to fitting error.

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To account for the possible dependence of Cepheid magnitude on metallicity even over the narrow range of metallicity in our Cepheid sample, we express mW as

Equation (7)

where the individual values of Δlog[O/H]i,j were derived from the metallicity values and gradients for the Cepheid hosts given by Riess et al. (2009). These values are listed in Table 4.

For the parameter ZW, we find −0.27 ± 0.18 in the sense that metal-rich Cepheids have a brighter value of mW, though this relation is not significant. Indeed, the benefit of using Cepheids across the distance ladder with similar metallicities is that, as shown in the following section, their relative distance measures are insensitive to the uncertainty in their metallicity relation.

We now move to the joint use of the Cepheid and SN Ia data for deriving the Hubble constant.

3. MEASURING THE HUBBLE CONSTANT

3.1. Type Ia Supernova Magnitudes

Distance estimates from SN Ia light curves are derived from the luminosity distance

Equation (8)

where L and ${\cal F}$ are the intrinsic luminosity and the absorption-free flux within a given passband, respectively. Equivalently, logarithmic measures of the flux luminosity in a passband (i.e., apparent magnitudes, mV) and luminosity (absolute magnitude, MV) are used to derive extinction-corrected distance moduli

Equation (9)

(dL in units of Mpc), where m0V derives from mV corrected for selective absorption through the use of colors and a reddening law.

We may relate the observables of SN Ia distance and redshift, z, to the scale factor of the universe, a, by expanding a(t) using the definitions

Equation (10)

(cf. Visser 2004). For z ≈ 0,

Equation (11)

where H0 is the present expansion rate (z = 0) of the universe.

Allowing for changes in the expansion rate at z > 0:

Equation (12)

or

Equation (13)

Using empirical relations between SN Ia light curve shape and luminosity allows for a modest correction of individual SN Ia magnitudes to relate them to a fiducial luminosity, M0V, at a fiducial epoch (by convention, B-band peak). For the multi-color light-curve shape (MLCS; Riess et al. 1996,) method of fitting SN Ia light curves, M0V is the V-band peak absolute magnitude for a SN Ia matching the template light-curve shape (i.e., the light-curve parameter Δ = 0). The value m0V is the maximum light apparent V-band brightness of the fiducial SN Ia at the time of B-band peak if it had AV = 0 and Δ = 0. This quantity is determined from a full light-curve fit, so that it is a weighted average, not a measurement at a single epoch.

We can rewrite Equation (13) to move the intercept of the magnitude–redshift relation to the left,

Equation (14)

and define the intercept of the log cz–0.2 m0V relation, av,

Equation (15)

The intercept, av, is an apparent quantity which is measured from the set of (z, m0V) independent of any absolute (i.e., luminosity or distance) scale. We use the kinematic expansion of av to include terms of order z2 and z3 rather than the Friedmann relation (i.e., ΩM, ΩΛ or w = P/(ρc2)) to retain its conventional definition (and measurement) as an apparent (not inferred) quantity. In practice, the difference between the kinematic and Friedmann relations is negligible in the range z < 0.1 where we determine av.11

Figure 12 shows a Hubble diagram for 240 SNe Ia from Hicken et al. (2009) whose intercept determines the value of av. The magnitude–z relation was determined with the fiducial parameters in MLCS2k2 (Jha et al. 2007). Limiting the sample to 0.023 < z < 0.1 (to avoid the possibility of a local, coherent flow) leaves 140 SNe Ia, where z is the redshift in the rest frame of the CMB. The present acceleration q0 = −0.55 and the prior deceleration j0 = 1 (Riess et al. 2007) yield av = 0.698 ±  0.00225. The sensitivities of av to the cosmological model, the minimum redshift and the MLS2k2 parameters are discussed in the following section.

Figure 12.

Figure 12. Magnitude–redshift relation of nearby (z < 0.1) SNe Ia. The term m0V is the peak apparent magnitude in V corrected for extinction and to the fiducial luminosity using a light-curve fitter. The intercept of the relation Equation (15) is used for the determination of H0. SNe Ia with redshifts in the range z > 0.01 or z > 0.0233 are used in the analysis.

Standard image High-resolution image

For the ith member of a set of nearby SNe Ia whose luminosities are calibrated by independent estimates of the distances to their hosts, the Hubble constant is given from Equation (14) and (15) as

Equation (16)

The terms μ0,i, determined from Cepheid data, were discussed in the previous section (e.g., Equation (7)).

Because the selection of the fiducial SN Ia along the luminosity versus light-curve shape relation is arbitrary, the value of av is also arbitrary. However, the inferred value of H0 is independent of this choice because the luminosity of the fiducial cancels in the sum m0v,i + 5av in Equation (16). For each SN Ia, the sum m0v,i + 5av in Equation (16) is a fundamental measure of its distance (in magnitudes) in the sense that it is independent, in principle, of the various approaches used to relate SN Ia light curves and their luminosity. It is also independent of bandpass. This sum makes it clear that the measurement of H0 depends only on the apparent differences between SN Ia distances in the calibration set and the Hubble-flow set.12 Systematic errors may arise from a combination of inaccuracies in the light-curve fitter and differences in the mean properties of the calibration and Hubble-flow samples. We will explore the size of these errors in Section 4.1 by varying the assumptions of the light-curve fitter and by using a different one, SALT II (Guy et al. 2005).

In Table 5, we give the quantities m0v,i + 5av for each of the SHOES SNe Ia.

In Figure 13, we compare the relative distances determined strictly from Cepheids, μ0,i − μ0,4258, and from SNe Ia, m0v,i + 5av. These quantities are relative in the sense that they both involve purely differential measurements of like quantities and benefit from the cancelation of systematic errors associated with the determination of absolute quantities. The dispersion between these relative distances is 0.08 mag, somewhat smaller than the mean SN distance error of 0.11 mag.

Figure 13.

Figure 13. Relative distances from Cepheids and SNe Ia. The x-axis (bottom) shows the peak apparent visual magnitude of each SN Ia (red points) corrected for reddening and to the fiducial brightness (using the luminosity–light-curve-shape relations), m0V. The upper x-axis includes the intercept of the m0V–log cz relation for SNe Ia, av to provide SN Ia distance measures, m0V + 5av, which are independent of the light curve shape relations. The y-axis (right), shows the relative distances between the hosts determined from the Cepheid VIH Wesenheit relations. The left y-axis shows the same with the addition of the independent geometric distance to NGC 4258 (blue point) based on its circumnuclear masers. The contribution of the nearby SNe Ia and Cepheid data to H0 can be expressed as a determination of m0V,4258, the theoretical mean of 6 fiducial SNe Ia in NGC 4258.

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3.2. Global Fit for H0

For convenience, we define a parameter (m0v,4258) which is the expected reddening-free, fiducial, peak magnitude of a SN Ia appearing in NGC 4258. We then express m0v for the ith SN Ia as

Equation (17)

Combining the two equations for apparent magnitudes; for SNe Ia, Equation (17), and for Cepheids, Equation (7), we write one matrix equation

Equation (18)

for n SN host galaxies (n = 0 will correspond to NGC 4258), with host i having ri Cepheids. Thus, we have t = n + ∑ni = 0ri equations to solve simultaneously. The only term of significance for the determination of H0 is m0v,4258 and its uncertainty derived from the covariance matrix of fitted parameters, which propagates the uncertainties in Cepheid nuisance parameters such as the slope and metallicity relations for the Cepheid.13 The meaning of m0v,4258 can be readily seen from Figure 13 as it connects the Cepheid and SN Ia relative distance measures, i.e., m0v,4258 = m0v,i − (μ0,i − μ0,4258).

From Equation (16), we derive our best estimate of H0 using

Equation (19)

Derived this way, the full statistical error in H0 is the quadrature sum of the uncertainty in the three independent terms (μ0,4258, m0v,4258, and 5aV), where μ0,4258 is the previously discussed geometric distance estimate to NGC 4258 (Herrnstein et al. 1999; E. M. L. Humphreys 2009, in preparation). More than a decade of tracking the Keplerian motion of its water masers supports an uncertainty of 3% (σ = 0.06 mag; Humphreys et al. 2008; E. M. L. Humphreys 2009, in preparation; Greenhill et al. 2009).

Our result is H0 = 74.2 ± 3.4 km s−1 Mpc−1, a 4.6% measurement. The uncertainty from the terms independent of the maser distance to NGC 4258, e.g., errors due to the form of the PL relation, metallicity dependences, photometry bias, and zeropoint errors as well as the SN Ia mz relation results in a ±3.4% uncertainty in H0. In past determinations of the Hubble constant, these sources of uncertainty have been the leading systematic uncertainties. In this analysis, these uncertainties have been reduced by matching the distribution of Cepheid measurements (i.e., metallicity, periods, and photometric systems) between NGC 4258 and the SN hosts. However, given the small uncertainty in H0, it is important to consider a broader exploration of systematic uncertainties of the type now under examination for large-scale high-redshift SN Ia surveys (e.g., Astier et al. 2006; Wood-Vasey et al. 2007; Sullivan et al. 2007).

4. SYSTEMATICS

In Table 6, we show 22 variants of the previously described analysis which we use to estimate the systematic error on our measurement of H0. Our primary analysis in Row 1 of Table 6 is based on our estimation of the best approach. Column 1 gives the value of χ2ν, Column 2 the number of Cepheids in the fit, Column 3 the value and total uncertainty in H0, and Column 4 the uncertainty without including the uncertainty in the maser distance for NGC 4258. Column 5 gives the determination of M0V, a parameter specific to the light-curve fitter employed, Column 6 the value and uncertainty in the metallicity dependence, and Column 7 the value and uncertainty of the slope of the Cepheid PL or PW relation. The next seven parameters are used to indicate variants in the analysis whose impact we now consider.

Table 6. Fits for H0

χ2dof No. H0 σfit av M0V δM/δ[O/H] b zmin Fit Scale PLW CRV SNe SN RV
0.85 209 74.16(3.41) 2.56 0.698 −19.13 −0.23(0.17) −3.14(0.10) 0.023 37 4258 HV,I 3.1 UBVRI 2.5
0.84 209 73.97(3.44) 2.60 0.702 −19.16 −0.22(0.17) −3.15(0.10) 0.023 20 4258 HV,I 3.1 UBVRI 3.1
0.83 209 75.12(3.43) 2.56 0.702 −19.13 −0.22(0.17) −3.12(0.10) 0.010 37 4258 HV,I 2.5 UBVRI 2.5
0.82 209 75.10(3.46) 2.61 0.707 −19.15 −0.21(0.17) −3.12(0.10) 0.010 20 4258 HV,I 2.5 UBVRI 3.1
0.85 209 73.27(3.30) 2.43 0.698 −19.16  ⋅⋅⋅  −3.17(0.10) 0.023 37 4258 HV,I 3.1 UBVRI 2.5
1.41 240 77.24(4.05) 3.30 0.698 −19.05 −0.72(0.20) −3.10(0.13) 0.023 37 4258 HV,I 3.1 UBVRI 2.5
0.86 198 74.62(3.49) 2.64 0.698 −19.12 −0.31(0.18) −3.13(0.11) 0.023 37 4258 HV,I 3.1 UBVRI 2.5
0.85 209 73.19(3.45) 2.63 0.701 −19.18 −0.24(0.17) −3.14(0.10) 0.023 61 4258 HV,I 3.1 BVRI 2.5
0.81 209 74.66(3.37) 2.49 0.692 −19.09 −0.22(0.17) −3.09(0.10) 0.023 28 4258 HV,I 2.0 UBVRI 2.0
0.84 209 73.51(3.30) 2.35  ⋅⋅⋅   ⋅⋅⋅  −0.23(0.17) −3.14(0.10) 0.023 42 4258 HV,I 3.1 UBVRI  ⋅⋅⋅ 
0.82 209 73.79(3.29) 2.33  ⋅⋅⋅   ⋅⋅⋅  −0.23(0.17) −3.12(0.10) 0.023 42 4258 HV,I 2.5 UBVRI  ⋅⋅⋅ 
0.84 209 74.84(3.37) 2.40  ⋅⋅⋅   ⋅⋅⋅  −0.23(0.17) −3.15(0.10) 0.010 42 4258 HV,I 3.1 UBVRI  ⋅⋅⋅ 
0.79 209 75.64(3.41) 2.52 0.698 −19.09 −0.19(0.17) −3.01(0.10) 0.023 37 4258 H 3.1 UBVRI 2.5
0.79 209 74.88(3.31) 2.39 0.698 −19.11  ⋅⋅⋅  −3.03(0.10) 0.023 37 4258 H 3.1 UBVRI 2.5
0.83 209 73.20(3.47) 2.65 0.690 −19.13 −0.19(0.17) −3.15(0.10) 0.023 26 4258 HV,I 3.1 UBVRI 3.1
0.84 209 74.05(3.50) 2.60 0.699 −19.15 −0.22(0.17) −3.15(0.10) 0.023 27 4258 HV,I 3.1 UBVRI 3.1
0.85 209 74.10(3.38) 2.53 0.692 −19.11 −0.24(0.17) −3.14(0.10) 0.023 28 4258 HV,I 3.1 UBVRI 2.0
0.85 209 74.11(3.36) 2.50 0.687 −19.08 −0.25(0.17) −3.14(0.10) 0.023 29 4258 HV,I 3.1 UBVRI 1.5
0.84 219 74.91(4.13) 4.11 0.698 −19.11 −0.21(0.17) −3.20(0.09) 0.023 37 MW HV,I 3.1 UBVRI 2.5
0.85 219 73.68(3.93) 3.91 0.698 −19.15  ⋅⋅⋅  −3.22(0.09) 0.023 37 MW HV,I 3.1 UBVRI 2.5
0.84 219 74.68(4.15) 4.13 0.702 −19.14 −0.20(0.17) −3.21(0.09) 0.023 20 MW HV,I 3.1 UBVRI 3.1
0.78 219 74.56(3.97) 3.95 0.698 −19.12 −0.18(0.17) −3.06(0.09) 0.023 37 MW H 2.5 UBVRI 2.5
1.01 262 73.27(4.57) 2.71 0.698 −19.16  ⋅⋅⋅  −3.17(0.04) 0.023 37 LMC HV,I 2.5 UBVRI 2.5

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4.1. SN Systematics

Following Wood-Vasey et al. (2007), the leading sources of systematic uncertainty in the cosmological use of SNe Ia relevant to our analysis are addressed here.

Lower Limit in SN redshift used to measure Hubble flow. The minimum redshift beyond which SNe Ia measure the Hubble flow has been an ongoing source of debate. Zehavi et al. (1998) and later Jha et al. (2006) claimed to see a local "Hubble bubble" with an increased outflow of ∼5% within a local void ending at z = 0.023. Conley et al. (2007) demonstrated that the evidence for the bubble rested on a set of SNe Ia at 0.01 < z < 0.023 with more than average reddening and that the reality of the bubble depended on the form of their extinction, whether RV is Galactic in nature (RV = 3.1) or empirically determined by minimizing the scatter in the Hubble flow (RV = 2–2.5). We consider both approaches to estimating the extinction in the range of 0.01 < z < 0.023 when we consider the value of RV used for the SNe.

We think the safest choice is to begin the measurement of the Hubble flow at z > 0.023 to avoid the uncertainty of the Bubble or other coherent large-scale flows. A number of authors (Hui & Greene 2006; Cooray & Caldwell 2006) have shown that coherent flows like a Hubble bubble are likely to induce bias at lower redshifts, and we maintain our view that it is better to restrict our analysis to z > 0.023 and avoid this possible bias. The penalty is a reduction in the statistical precision of the measurement of the Hubble flow, but this term remains subdominant in the determination of H0. However, we also include a number of analyses with zmin = 0.01, as indicated by Column 8 of Table 6. These have the effect of raising the Hubble constant by 1.0–1.2 km s−1 Mpc−1 depending on the aforementioned treatment of extinction. An alternate selection of the Hubble-flow set would be to consider all SNe Ia at z > 0.01 but limit the selection to those with AV < 0.5, making the Hubble-flow sample a good match to the calibrators and avoiding the degeneracy between the Hubble Bubble and the extinction law at z < 0.023. This approach results in a value of H0 from SNe Ia at z > 0.01, which is only 0.7 km s−1 Mpc−1 greater than the nominal fit at z > 0.023.

SN-host RV. In our primary analysis, we account for the difference in SN Ia extinction between the calibration and Hubble-flow samples using the UBVRI colors of the SNe and the MLCS2k2 prescription. For the extinction due to host-galaxy dust, our primary analysis uses a recent "consensus" value of RV = 2.5 (fit parameter 37 in Column 9 of Table 6) for the lines of sight of SNe Ia (Kessler et al. 2009), but we also consider values for RV of 1.5, 2.0, and 3.1 with fit parameters of 29, 28, and 20, respectively. The change in H0 is 0.2 km s−1 Mpc−1 across the range of 1.5 < RV < 3.1 for the SNe. The effect is so small because the SN colors for the calibration sample and those in our nominal Hubble-flow sample are well matched, so altering RV for the SNe provides little change.

Distribution of host-galaxy extinction. The observed distribution of SN Ia host-galaxy extinction is used as a prior in the determination of the extinction of individual SNe Ia (Riess et al. 1996) and is particularly important in the absence of precise color measurements (e.g., at high redshifts). However, the prior has little effect on the present analysis because the SN colors at low redshifts are well measured. To determine the sensitivity to this prior, we varied its functional form across two extremes, using either a simulation of the lines of sight through galaxies (Hatano et al. 1998; "glos") which anticipates less extinction on average than the default or no extinction prior at all, (fit parameters 27 and 26), respectively. The difference in H0 is only 0.8 km s−1 Mpc−1. We also changed the algorithm used to fit and compare the SN Ia light and color curves from MLCS2k2 to the SALT II (Guy et al. 2005) approach. These fits (fit parameter 42) reduce H0 by 0.5 km s−1 Mpc−1 with other variants held fixed. Overall we find the determination of H0 is insensitive to assumptions about the relation between SN Ia colors and extinction.

SN Ia U-band. We also perform an analysis of the SN data discarding the U band, fit 61, as it should be most sensitive to the form of the extinction law, changes in SN Ia metallicity, and errors in calibration. This decreases H0 by 1.0 km s−1 Mpc−1. Because both the nearby and Hubble-flow samples make use of the same U-band calibration (Jha et al. 2006), our results are insensitive to the parameters of the U band (Kessler et al. 2009).

Other sources of systematic error listed in Wood-Vasey et al (2007) arise from a large change in redshift between two samples of SNe Ia (i.e., cross-filter K-corrections and the possibility of SN Ia evolution) and are not significant in our analysis as all SN data are at z < 0.1. In general, changes to the treatment of the SN Ia light curves affect both the calibration and Hubble-flow sample similarly, largely canceling in the sum m0v + 5av and their impact on H0.

4.2. Cepheid Systematics

For systematic errors relevant to the analysis of Cepheid data, Table 14 in Freedman et al. (2001) lists the dominant terms. The largest terms relevant to our analysis are considered here.

Cepheid metallicity. Metallicity was addressed in Section 3. The critical conclusion is that the range in metallicity for our Cepheid data is small (Δ[O/H] ∼ 0.1), a factor of 4 times smaller than if LMC Cepheids are used to calibrate SNe Ia. In addition, the metallicity sensitivity should be further reduced by a significant factor by observing Cepheids in the near-IR (Marconi et al. 2005). However, we formally include and marginalize over a first-order metallicity dependence for the Cepheids using our previous measurements of the host metallicities. It may be of interest to remove the metallicity term in the analyses to determine its impact on H0. We include a few such entries in Table 6, indicated by "..." in the entry for this term. The result is that the nominal uncertainty in H0 decreases by 5% and its value is reduced by ∼1.2 km s−1 Mpc−1.

Cepheid reddening. Reddening of the Cepheids is largely mitigated over optical-based analyses by the use of H-band photometry, which reduces the net by a factor of 5 over the V band. The use of Wesenheit magnitudes should account for what extinction remains. However, our knowledge of the reddening law is imperfect, perhaps resulting in systematic errors. Previous work has shown that a Galactic value of RV = 3.1 is appropriate for extragalactic Cepheids (Macri et al. 2001), and this is used in our primary analysis. We also fit the Cepheids with RV = 2.0 and 2.5 as indicated in Table 6. The result is an increase in H0 by 0.5–1.0 km s−1 Mpc−1 when the value of RV for Cepheids is decreased from 3.1 to 2.0. As an alternative, we fit the Cepheids with only their H-band magnitudes, which increases H0 by 1.6 km s−1 Mpc−1, indicating that the differential extinction of the Cepheids in the H-band between NGC 4258 and the SN hosts is ∼0.04 mag; we think it prudent to account for this difference using the colors of the Cepheids.

Short-end limit of Cepheid periods. Because the Cepheids were selected at bluer wavelengths, the bias of selecting brighter Cepheids at shorter periods due to a magnitude limit does not necessarily apply to the H-band magnitudes. The dispersion in magnitude at a given period arising from the width of the instability strip will be significantly reduced in the near-IR as we view Cepheids on their Rayleigh–Jeans tail. In addition, the use of Wesenheit magnitudes mitigates the contribution to the selection bias due to the color variation on the instability strip. However, 11 of the Cepheids used in our primary analysis have periods which are shorter than the low-period limits determined in Riess et al. (2009) for the onset of optical selection bias and these are indicated in Figure 11. Rejecting these (resulting in the entry in Table 6 with 199 Cepheids) results in an increase in H0 of 0.4 km s−1 Mpc−1 and a 2% increase in its uncertainty.

Other. Other significant terms in Freedman et al. (2001) include bulk flows, crowding, and zero points. We addressed bulk flows in Section 4.1. Errors due to crowding were discussed in Section 2.3. The key points regarding crowding are: (1) we correct each Cepheid statistically for crowding bias; (2) H0 is only sensitive to a difference in crowding between NGC 4258 and the SN hosts; and (3) artificial-star tests indicate that this difference is only 0.02 mag in the photometry of the Cepheids, even before a statistical correction is applied. To test for any remaining dependence on H0 on the degree of crowding, we analyzed subsets of Cepheids with the least apparent crowding. We found that truncating the Cepheid sample to the objects in the lower 40% or 60% of the crowding bias (<0.12 or <0.20 mag) results in a reduction in the Hubble constant by 2.3% and 0.8%, respectively. The overall uncertainty in H0 naturally increases as the Cepheid sample is reduced, rising by 25% when retaining only 40% of the original sample. Thus, we find the net effect on H0 due to crowding is contained within the statistical uncertainties.14 This is an advantage of the use of NGC 4258 over the LMC and the Galaxy, as this and other difficulties in achieving accurate photometry of Cepheids (such as the determination of photometric zeropoints) largely cancel in the determination of H0.

We also consider the effect on H0 of the rejection of outliers on the Cepheid PL relations discussed in Section 2.4. Including the rejected objects naturally has a severe impact on the value of χ2ν (where ν is the number of degrees of freedom), increasing it from 0.84 to 1.38, with each rejected object contributing an average of χ2 = 5. This variant is indicated in Table 6 by the increase in the sample of Cepheids from 209 to 240. The change in H0 is an increase of 3 km s−1 Mpc−1, the largest change, but still within the 1σ of the statistical error. However, as discussed in Section 2.4, such outliers are expected, and we think it is sensible to reject them as they may pull the global solution well beyond their merit. They are included in Table 4 for those who want to consider them further. We also considered a less stringent outlier cut of ±1.0 mag resulting in the retention of 229 out of 240 Cepheids, increasing H0 by 1.0 km s−1 Mpc−1 and demonstrating that most of the change in H0 results from a handful of the most extreme of outliers.

Historically, the determination of H0 through the Cepheid and SN Ia distance ladder has been significantly altered by choices made in the analysis, with different authors making different (if all reasonable) choices leading to different results. Thanks to the greater homogeneity of the data we are using and the smaller number of steps needed to proceed from a direct geometric distance determination to the final measurement of H0, we could expect a priori that different choices would not have a major impact on our results. We have already shown that no single variant described above causes a significant change in H0. However, to propagate the systematic uncertainty from variants in the analysis and to consider combinations of analysis variants, we developed a number of plausible scenarios in which different choices are made and the full analysis is completed to determine H0. The results are presented in Table 6.

Although systematic errors are notoriously difficult to quantify, our approach is to use the variation in H0 in the previous analyses to determine the systematic error. The variation in the inferred value of H0 is relatively small, with a median and dispersion of 74.2 ± 1.0 km s−1 Mpc−1. The median is the same as our primary determination (thus the changes scatter fairly equally between increases and decreases), and all inferred values lie within a range of about ±3 km s−1 Mpc−1. We take the formal dispersion of 1.0 km s−1 Mpc−1 as an estimate of the systematic uncertainties in our determination, which we then add in quadrature to the statistical uncertainty of the value derived with our preferred approach, yielding a final estimate of H0 = 74.2 ± 3.6 km s−1 Mpc−1.

4.3. Anchor Systematics

The use of NGC 4258 in lieu of the LMC or the Galaxy as an anchor to the distance ladder provides a significant enhancement to the precision and accuracy in the measurement of H0. Indeed, a 3% uncertainty in the distance to NGC 4258 does not even dominate the current total uncertainty. The natural advantages of NGC 4258, including the sample size, period range, and typical metallicity of its Cepheids, and the ability to measure them in the same way as those in SN Ia hosts, provide for extensive use of differential measurements of the Cepheids in the distance ladder and the means to measure H0 to <5%. In the following section, we discuss the use of additional maser hosts which can serve to test and improve the maser distance estimates. However, at present there is only one thoroughly measured system, and use of the LMC or the Galaxy as an anchors can still provide a test of the distance scale set by NGC 4258.

A set of 10 parallax measurements to Galactic Cepheids was recently obtained by Benedict et al. (2007) using the Fine Guidance Sensor on HST. Parallax measurements remain the "gold standard" of distance measurements, and unlike previous HIPPARCOS measurements, the individual precision of this set of measurements is high, averaging σ = 8% for each. We have not made use of additional distance measures to Galactic Cepheids based on the Baade–Wesselink method or stellar associations as they are much more uncertain than well-measured parallaxes, and the former appear to be under refinement due to uncertainties in their projection factors, as discussed by Fouqué et al. (2007) and van Leeuwen et al. (2007).

Considered as a set, the Cepheids in Benedict et al. (2007) have an uncertainty in their mean distance measure of only 2.5%, comparable to the precision of the measurement of NGC 4258. These Galactic Cepheids also have metallicities which are very similar to that of Cepheids in the SN hosts as discussed by Sandage et al. (2006). Using the values of μ0 (including corrections for interstellar extinction and Lutz–Kelker–Hanson bias) and V-, I-band magnitudes given by Benedict et al. (2007), as well as H-band magnitudes compiled by Groenewegen (1999),15 we determined the absolute Wesenheit magnitudes of this set of 10 variables. Their PL relation is shown in Figure 11. Their inclusion in the global fit is achieved by altering Equation (7) for the NICMOS Cepheids to be

Equation (20)

and Equation (17) for the SNe Ia to be

Equation (21)

Moreover, for the Galactic Cepheids,

Equation (22)

where MW is the absolute Wesenheit magnitude for a Cepheid with P = 1 d. The key parameters in the determination of the H0 change from m0v,4258 and μ0,4258 in Equation (19) to M0V,

Equation (23)

As before, the statistical error in M0V includes all Cepheid-related uncertainties such as the nuisance parameters like the slope and metallicity relations, and the uncertainty in H0 comes from the two independent terms (M0V, 5aV). The Cepheids in NGC 4258 still contribute to the global analysis as they help determine the slope of the PL relation, though their distance estimate is immaterial to the determination of H0. We now include a σ = 0.04 mag uncertainty in the photometry (i.e., zero points and relative crowding) between the space-based Cepheid data and the ground-based Cepheid data. These analyses are indicated in Table 6 with the scale given as "MW." Compared to the primary analysis based on the independent distance measurement to NGC 4258, use of the Benedict et al. (2007) parallaxes reduces H0 by 0.9 km s−1 Mpc−1 with an increase in the uncertainty of 15%.

However, there are some "risks" in taking this route over that based on the distance to NGC 4258. The magnitudes of these Galactic Cepheids, unlike the distant Cepheids, suffer little crowding, and so we must fully rely on the statistical crowding corrections of mean 0.16 mag in Section 2.3 rather than the more modest difference in the correction between NGC 4258 and the SN hosts of 0.02 mag. (We assumed a systematic uncertainty of 0.03 mag for use of the full corrections.) Errors along the magnitude scale from Galactic Cepheids of 〈H〉 = 2 mag to those in SN hosts of 25 mag pose another risk in this route. We estimate 0.03 mag systematic uncertainty for the magnitude scale which is included in the values in Table 6. In addition, the mean period of the Benedict et al. (2007) Cepheids, 〈P〉 = 10 d is significantly lower than the 〈P〉 ≈ 35 d in the SN hosts. The use of the Cepheids in NGC 4258, even without the use of its distance, provides an empirical bridge across this period range. Still, the assumption of the linearity of the PL relation, even in the H band and even for a Wesenheit relation, is another weakness along this route. We derive an 0.04 mag systematic uncertainty from this mismatch in mean periods. Including systematics, the total uncertainty in H0 is 5.8%, only moderately worse than the NGC 4258 route but the result carries more caveats. Future measurements from GAIA of precise parallaxes for ∼103 Cepheids over a wide range of periods will provide increased precision while removing the reliance on the form of the PL relation to yield a great improvement to the pursuit of H0 if accompanied with a more precise calibration of the near-IR magnitude scale.

The use of the LMC as our anchor for the distance scale carries similar risks as those discussed for Milky Way Cepheids with two significant additions: the metallicity of the LMC differs substantially from the SN hosts and the distance to the LMC is uncertain at the >5% level. Nevertheless, the LMC has a long history of use as an anchor, and for comparison to previous work it is valuable to again cast the LMC in that role. We use the set of H-band Cepheid measurements from Persson et al. (2004) and the optical measurements of Sebo et al. (2002) to extract the 53 Cepheids with measurements of their mean magnitudes in VIH. Due to the significant difference in metallicity between the LMC and the SN hosts of Δ[O/H] ≈ 0.4 dex and our lack of constraint on or detection of a metallicity parameter, we made no metallicity correction. This approach is supported by theory, in which the zeropoints of near-IR PL relations are found to vary with chemical composition by a factor of ∼3 less than those of optical zeropoints (Marconi et al. 2005). Assuming that for the LMC, μ0 = 18.42 mag based on a set of 4 detached eclipsing binaries (Fitzpatrick et al. 2003) and with a generous ±0.10 mag uncertainty to allow for the wide range of estimates for the LMC distance, we find H0 = 73.3 ± 4.6 km s−1 Mpc−1 as shown in Table 6, in good accord with the previous two anchors.

In summary, we find that a full propagation of statistical error and the inclusion of the systematic error gives H0 = 74.2 ± 3.6 km s−1 Mpc−1, based on the cleanest route through NGC 4258, but also consistent with independent though riskier distance-scale anchors from Milky Way Cepheid parallaxes and the LMC.

4.4. Error Budget

As discussed in Section 4, our total error is the sum of the uncertainty in the three measured terms on the right-hand side of Equation (19) and the systematic error derived from considering alternatives to the primary analysis. To illuminate how error propagates along our (and other) distance ladders, we itemize the contributions in Table 7.

Table 7. Error Budget for H0 for Cepheid and SN Ia Distance Ladders

Term Description Previous Here
σanchor Anchor distance 5% 3%
σanchor−PL Mean of PL in anchor 2.5% 1.5%
$\sigma _{{\rm host}-PL}/\sqrt{n}$ Mean of PL values in SN hosts 1.5% 1.5%
$\sigma _{\rm SN}/\sqrt{n}$ Mean of SN Ia calibrators 2.5% 2.5%
σmag−z SN Ia mz relation 1% 0.5%
Rσλ,1,2 Cepheid reddening, zeropoints, anchor-to-hosts 4.5% 0.3%
σZ Cepheid metallicity, anchor-to-hosts 3% 0.8%
σPL PL slope, Δ log P, anchor-to-hosts 4% 0.5%
σWFPC2 WFPC2 CTE, long–short 3% 0%
Total, $\sigma _{H_0}$   10% 4.8%

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The first term is the distance precision of the anchor, followed by the mean of its set of Cepheids (i.e., the zero point of its PL relation). The next two terms are the mean of the set of Cepheids in each SN host and the precision of a single SN, each divided by the number (n) of hosts. For this calculation we use n = 6. Next is the uncertainty in the SN Ia apparent magnitude versus z relation; SNe Ia in the Hubble flow now provide a Hubble diagram with 240 published SNe Ia out to z ≈ 0.1, yielding an uncertainty of 0.5% (Hicken et al. 2009). The next term arises from the uncertainty in the difference between the photometric calibration used to observe Cepheids in the anchor and in the SN hosts in two or more passbands. These photometric calibration errors are then amplified by the need to deredden Cepheids with a reddening law, R, of size 2.1 and 0.48 for VI and VIH Cepheid measurements, respectively. The next two terms arise from the difference in the mean metallicities and the mean periods of the Cepheids in the anchor and hosts, and the uncertainty in their respective correlation with Cepheid luminosity. The last term contains the uncertainty from the photometric anomalies of WFPC2, charge transfer efficiency (CTE), and the "long versus short effect." (Holtzman et al. 1995).

The reduction in total uncertainty in $\sigma _{H_0}$ from 10% to 5% is a consequence of a number of improvements along the ladder. Most come from greater homogeneity in zero points, metallicity, and periods of the samples of Cepheids collected in the anchor and the SN hosts. Changing from the optical to the near-IR reduces the reddening term, R, by a factor of 4.4. NGC 4258 also provides greater distance precision than the LMC, and a larger sample of long-period Cepheids. The recent increase in the sample of SNe Ia at 0.01 < z < 0.1 (Hicken et al. 2009) provides a modest improvement.

5. DARK ENERGY

An independent measurement of H0 is a powerful complement to the measurement of the cosmological term ΩMH20 derived from the power spectrum of the CMB. In the context of a flat universe, the fractional uncertainty in the value of an (assumed constant) equation-of-state parameter (w) of dark energy is approximately twice the fractional uncertainty in H0 $({\sigma _w}/w \approx 2 \sigma _{H_0}/{H_0})$, as long as the fractional uncertainty in H0 is greater than or equal to that in ΩMH20 (Hu 2005). A marked improvement in the precision of ΩMH20 has been realized in the recent five-year WMAP analysis from the localization of the third acoustic peak (Komatsu et al. 2009). The result is a model-insensitive measurement of ΩMH20 to better than 5% precision.

Using the output of the WMAP five-year Monte Carlo Markov Chain (MCMC) from Komatsu et al. (2009)16 in a flat, wCDM cosmology (i.e., dark energy with constant w) yields the degenerate confidence regions in the H0w plane shown in Figure 14. Combined with our measurement of H0, we find w = −1.12 ± 0.12, a value consistent with a cosmological constant (Λ). This result is similar in value and precision to those found from the combination of baryon acoustic oscillations (BAO) and high-redshift SNe Ia (Wood-Vasey et al. 2007; Astier et al. 2006). The important difference from the prior measurements is that this one is independent of the systematic uncertainties associated with the use of high-redshift SNe Ia. Since such measurements are now dominated by their systematic errors (Wood-Vasey et al. 2007; Kessler et al. 2009; Hicken et al. 2009; Kowalski et al. 2008), independent measurements are a route to progress. For comparison, the combination of the WMAP and BAO data alone gives w = −1.15 ±  0.22 and that from WMAP and the Freedman et al. (2001) measurement of H0 yields w = −1.01 ± 0.23.

Figure 14.

Figure 14. Confidence regions in the plane of H0 and the equation of state of dark energy, w. The localization of the third acoustic peak in the WMAP five-year data (Komatsu et al. 2009) produces a confidence region which is narrow but highly degenerate in this space. The improved measurement of H0, 74.2 ± 3.6 km s−1 Mpc−1, from the SHOES program is complementary to the WMAP constraint resulting in a determination of w = −1.12 ± 0.12 for a constant equation of state. This result is comparable in precision to determinations of w from baryon acoustic oscillations and high-redshift SNe Ia, but is independent of both. The inner regions are 68% confidence and the outer regions are 95% confidence. The modest tilt of the SHOES measurement of 0.2% in H0 for a change in w = 0.1 results from the mild dependence of av on w at the mean z = 0.04. The measurement of H0 employs the emprically determined higher order terms q0 = −0.55 and j0 = 1.

Standard image High-resolution image

The H0 + WMAP measurement of w is quite insensitive to the effect of w on the determination of av because the mean redshift of the Hubble-flow sample is only z = 0.04. Specifically, the change in H0 for a change in w of 0.1 (evaluated at z = 0.04) is only 0.2%, far less than the total 4.8% uncertainty in H0 and justifying our use of a kinematic expansion to determine av. The very mild degeneracy between av and w is shown (as a tilt) in Figure 14.

However, fitting a cosmological model with the assumption of a constant equation of state (EOS) is itself limiting to the investigation of dark energy. It obscures our ability to detect evolution of w, an important test of the presence of a cosmological constant. An alternative approach is to use a variant of principal-component analysis (Huterer & Starkman 2003; Huterer & Cooray 2005) to extract discrete, decorrelated estimates of w(z), binned in redshift. This method was used by Riess et al. (2007) and Sullivan et al. (2007) to constrain multiple independent measures of w(z). With the improved constraint on H0, we can use this approach to determine the effect on the constraints on the components of w(z). In the following, we employ the implementation of the component analysis from Sarkar et al. (2008) using N + 3 free parameters in the MCMC corresponding to H0, Ωm, ΩK, and the N independent estimates of w.

5.1. Current Data

We first examine how the errors on w(z) improve from using the improved constraint on H0. For this, we use the Davis et al. (2007) compilation of 192 SNe, 2 BAO estimates from Percival et al. (2007), and the WMAP five-year constraint (Komatsu et al. 2009) on the distance to the last-scattering surface (RCMB) in the H0-independent form. We also use the WMAP five-year constraint on ΩmH20 (Komatsu et al. 2009) and allow curvature to be free. We use the publicly available wzBinned17 code and analyze the data using an MCMC likelihood approach to estimate w(z) in each redshift bin. We take a total of three bins between z = 0 and z = 1.8 (see Table 8 for the redshift ranges) and assume that dark energy at z > 2 was subdominant by fixing w to a constant value of −1 from that redshift to the last-scattering surface (z = 1089).

Table 8. Decorrelated Estimates of w from Available Data Sets (68% Uncertainty)

Data Set Prior on w1 w2 w3
Used H0 z = [0-0.2] z = [0.2-0.5] z = [0.5-1.8]
192 SNe + 2 BAO 72 ± 8.0 −0.976+0.142−0.162 −0.944+0.230−0.235 −0.471+0.327−1.515
  74 ± 3.5 −0.940+0.102−0.139 −0.948+0.175−0.160 −0.692+0.301−0.759

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We analyze the data using the value of H0 from both HST Cepheids (Freedman et al. 2001) and the present work. Our results are summarized in Table 8. Using the new constraints on the Hubble constant, we get a significant improvement on the 1σ errors of the EOS parameters. The improvement in the inverse product of the uncertainties in w(z), widely referred to as a dark energy "figure of merit," improves by a factor of 3 due to the increased precision in H0, a result of the degeneracy between w and H0. The data remain consistent with Λ within 1σ with [w1, w2, w3] = [74 ± 3.5 − 0.940+0.102−0.139 − 0.948+0.175−0.160 − 0.692+0.301−0.759] for the ranges z = [0 − 0.2], [0.2 − 0.5], [0.5 − 1.8], respectively. The data continue to indicate the presence of a dark-energy component (i.e., w < 0) when it was a subdominant part of the universe, in agreement with Riess et al. (2007); see also Kowalski et al. (2008).

5.2. Future Surveys

We now consider the constraints on w(z) from future surveys in three different scenarios under frequent consideration:

Case (1). An aggressive set of 17 BAO distance measurements. This includes two BAO estimates (as before) at z = 0.2 and z = 0.35, with 6% and 4.7% uncertainties, respectively (Percival et al. 2007); five BAO constraints at z = [0.6, 0.8, 1.0, 1.2, 3.0] from SDSS III and HETDEX with respective precisions of [1.9, 1.5, 1.0, 0.9, 0.6]% (Seo & Eisenstein 2003, scenario V5N5); and 10 BAO estimates from a space mission with precisions of [0.36, 0.33, 0.34, 0.33, 0.31, 0.33, 0.32, 0.35, 0.37, 0.37]% from z = 1.05 to 1.95 in steps of 0.05.

Case (2). An aggressive SN Ia data set of 2300 SNe with 300 SNe uniformly distributed out to z = 0.1, as expected from ground-based low-redshift samples, and an additional 2000 SNe uniformly distributed in the range 0.1 < z < 1.7, as expected from future space mission (Kim et al. 2004). We bin the Hubble diagram into 32 redshift bins (corresponding to a width of the relevant redshift bin of Δz = 0.05). The error in the distance modulus for each SN bin is given by σm = ((σint/N1/2bin)2 + δm2)1/2, where σint = 0.1 mag is the intrinsic error for each SN, Nbin is the number of SNe in the redshift bin, and δm is the irreducible systematic error. We take the systematic error to have the form δm = 0.02(0.1/Δz)1/2(1.7/zmax)(1 + z)/2.7, where zmax is the redshift of the most distant SNe. This is equivalent to the form in Linder & Huterer (2003). In generating the SN catalog, we do not include the effect of gravitational lensing, as it is expected to be small (Sarkar et al. 2008) and should not affect our results much.

Case (3). A combination of the above: 2300 SNe and 17 BAO estimates.

For each of the above-mentioned scenarios, we also use the WMAP five-year constraint on RCMB (Komatsu et al. 2009). Since we are considering future surveys, we marginalize over an Ωm prior obtained from the Planck prior on ΩmH20 and different priors on H0 (see Table 9 for details). As before, we allow the curvature to be free. In this case, we take a total of six bins between z = 0 and z = 2 (see Table 9 for the redshift ranges) and fix w(2 < z < 1089) = −1. The sixth bin (extending from z = 1.2 to z = 2.0) is suppressed as it is not well constrained.

Table 9. 68% Error in the Decorrelated Binned Estimates of w from Upcoming Surveys

Mocks Used H0 74± Δw1 z = [0-0.07] Δw2 z = [0.07-0.15] Δw3 z = [0.15-0.30] Δw4 z = [0.3-0.6] Δw5 z = [0.6-1.2] FoM (×104)
17 BAO 8.0 0.549 0.462 0.323 0.202 0.158 0.038
  6.0 0.389 0.374 0.255 0.196 0.166 0.083
  4.0 0.342 0.340 0.238 0.174 0.150 0.138
  3.5 0.331 0.329 0.224 0.163 0.143 0.176
  2.0 0.203 0.203 0.144 0.118 0.131 1.090
  1.0 0.130 0.134 0.096 0.081 0.093 7.938
2300 SNe 8.0 0.128 0.137 0.162 0.308 7.999 0.014
  6.0 0.127 0.132 0.156 0.294 8.799 0.015
  4.0 0.105 0.098 0.105 0.193 1.622 0.296
  3.5 0.098 0.085 0.088 0.145 1.334 0.705
  2.0 0.091 0.070 0.064 0.083 0.291 10.16
  1.0 0.078 0.052 0.043 0.048 0.124 96.33
2300 SNe 8.0 0.064 0.054 0.045 0.048 0.104 129
+ 6.0 0.063 0.051 0.042 0.045 0.099 166
17 BAO 4.0 0.063 0.049 0.041 0.043 0.097 189
  3.5 0.062 0.049 0.041 0.043 0.092 203
  2.0 0.061 0.047 0.038 0.039 0.086 274
  1.0 0.057 0.043 0.034 0.034 0.065 543

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Table 9 and Figure 15 summarize our results. A significant improvement of the 68% error in the decorrelated binned estimates of w is apparent as we make use of better constraints on the Hubble constant.

Figure 15.

Figure 15. Projected constraints on five principal components of w(z) as a function of the future precision of H0. Three future scenarios are considered: an aggressive BAO experiment (black), an aggressive high-z SN Ia experiment (red), or both (blue) along with a Planck-based prior on Ωmh2. Panels 1–5 show the expected constraints in different redshift ranges. Panel 6 shows a figure of merit, the inverse product of the uncertainties of the 5 components. As seen, a ∼1% measurement of H0 can compensate for either BAO of high-z SNe Ia being limited by systematic errors or can aid their joint use.

Standard image High-resolution image

Further improvement in the measurement of H0 should allow for the measurement of a fourth independent parameter of the EOS to an accuracy better than 10%, even without making use of any BAO estimate. A combination of next generation surveys will most likely be able to measure five independent parameters of the EOS to better than 10% accuracy.

An alternative use of a precise measurement of H0 is as an "end-to-end" test of the best constraints on the cosmological model from all other data. As shown in Table 1, the combination of measurements from WMAP, BAO, and high-redshift SNe Ia, together with the assumption of a constant value for w, predict H0 to greater precision than measured here. This prediction is in good agreement with our measurement, but belies tension between the predictions of H0 from BAO and high-redshift SNe Ia. Either of these combined with WMAP results in a 3σ difference in their prediction of H0. Although our present measurement lies between these two combinations, it is significantly closer to BAO and inconsistent with WMAP and high-redshift SNe Ia at the 2.8σ confidence level. Improvements in all data sets should reveal whether this tension results from systematic error or is indicative of the need for a more complex description of dark energy.

6. DISCUSSION

Ever more precise measurements of the Hubble constant can contribute to the determination of the even more elusive nature of dark energy. The Planck CMB mission is expected to measure ΩMH20 to 1%. A complementary goal would be to reach the same for H0. We show in Figure 15 that a measurement of H0 approaching 1% would be competitive with "next generation" measurements of BAO and high-redshift SNe Ia (Albrecht et al. 2006) for constraining the evolution of w, and could buttress either tool should they encounter insurmountable systematic errors before reaching their goals. Attempts to explain accelerated expansion without dark energy by an unexpected failure of the cosmological principle also benefit from improved measurements of H0. For example, an approach by (Wiltshire 2007) in this vein predicts H0 = 62 ± 2 which is already inconsistent with the present measurement at the 3σ confidence level.

How realistic is a measurement of H0 to 1%? In most respects, the measurement of H0 to 1% is no more ambitious than the plans to push high-redshift SN Ia measurements to their next level of precision. Indeed, the dominant sources of systematic uncertainty in measuring distant SNe Ia do not pertain to H0 as they result from large redshifts: cross-filter, cross-detector flux calibration, K-corrections, and evolution of SNe Ia and dust over large changes in redshift (Wood-Vasey et al. 2007).

Following Table 7, we consider the two biggest challenges to a 1% measurement of H0: the precision of the distance measurement of the anchor and the size of the calibrator sample of SNe Ia. The other terms are near or below 1% and can be reduced with the collection of more data.

Further improvements in the distance measurement to NGC 4258 require understanding and modeling remaining complexity in its inner disk, including eccentricity and the possible presence of a spiral structure (Humphreys et al. 2008; E. M. L. Humphreys 2009). We expect progress with future work. Though 1% or better would be challenging, with the present route it may be possible to measure H0 to 2% or 3%.

More maser hosts of comparable quality could further reduce the uncertainty in the anchor through averaging. The Maser Cosmology Project (MCP; Braatz et al. 2008) is a large project at NRAO with the goal of measuring 10 more hosts in the next five years (Greenhill et al. 2009). Of the 112 extragalactic maser galaxies now known, 30% show the required high-velocity features on their limbs and 10% are disks and are good candidates for distance measurements. Two of these, UGC 3789 and NGC 6323, have already yielded initial distance estimates with 15% uncertainty and which, combined with their redshifts, are consistent with the value of H0 inferred here (J. A. Braatz et al. 2008, private communication; F. Lo 2008, private communication). Reaching 1% will require the 10 new MCP maser hosts to each be measured to the 3% uncertainty of NGC 4258 (Greenhill et al. 2009), or some other combination of number of systems and individual precision. Considering that the majority of maser hosts have been found in just the last five years, there is reason for optimism in the future. If such a sample of maser hosts is collected, it would then be necessary to correct their recession velocities for peculiar and coherent flows (Hui & Greene 2006) to a mean of 1% or observe their Cepheids to tie their distance scale to the present 0.5% calibration of the Hubble flow from SNe Ia.

Another promising route is offered by GAIA which should collect a few hundred high-precision parallax measurements for long-period Cepheids in the Galaxy. The resulting PL relations would be more than sufficient to support a 1% measurement of H0. However, the comparison of bright Galactic Cepheids and faint ones in SN hosts raises the challenge of measuring fluxes over a range of 20 mag to better than 1% precision. Though formidable, this appears still easier than the challenge facing future high-redshift SN Ia studies because the Cepheid measurements may all be obtained at the same wavelengths. Accounting for the difference in crowding between Galactic and extragalactic Cepheids is also a concern.

The size of the sample of reliable SNe Ia close enough to resolve Cepheids in their hosts, those within ∼30 Mpc, presently limits the determination of their mean fiducial luminosity to 2.5%. At least 30 SNe Ia are needed in this sample. At a rate of ∼1 new object appearing every three years we cannot wait on nature. A factor of 2 increase in distance (factor of 8 in volume), and hence 1.5 mag in the range of resolving Cepheids, is needed. Ultra-long-period Cepheids with 80 < P < 180 d (Bird et al. 2008) and MV = −7 mag are ∼2 mag brighter than the typical, P = 30 d Cepheids observed in SN hosts. Though these Cepheids appear to obey different PL relations than their shorter period brethren and are rare, their use when intercompared between galaxies is promising (Bird et al. 2008; Riess et al. 2009). The James Webb Space Telescope (JWST) is expected to routinely resolve Cepheids at ∼50 Mpc and could be enlisted to help measure H0 to 1%.

7. SUMMARY AND CONCLUSIONS

  • 1.  
    We have observed 240 long-period Cepheids in six SN Ia hosts and NGC 4258 using NICMOS in F160W.
  • 2.  
    Unprecedented homogeneity in the periods and metallicities of these Cepheids greatly reduces systematic uncertainties along the distance ladder.
  • 3.  
    Use of the same telescope, instrument, and filters for all Cepheids markedly reduces the systematic uncertainty related to flux calibration.
  • 4.  
    Our primary analysis gives H0 = 74.2 ± 3.6 km s−1 Mpc−1 including statistical and systematic uncertainties.
  • 5.  
    A wide range of alternative analyses yield consistent results and are used to quantify the systematic uncertainty which is subdominant to the statistical uncertainty.

We are grateful to William Januszewski, William Workman, Neil Reid, Howard Bond, Louis Bergeron, Rodger Doxsey, Craig Wheeler, Malcolm Hicken, Robert Kirshner, Peter Challis, Elizabeth Humphreys, Lincoln Greenhill, and Ken Sembach for their help in realizing this measurement.

Financial support for this work was provided by NASA through programs GO-9352, GO-9728, GO-10189, GO-10339, GO-10497, and GO-10802 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. A.V.F.'s supernova group at U.C. Berkeley is also supported by NSF grant AST–0607485 and by the TABASGO Foundation.

Footnotes

  • Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.

  • The HST observations were also designed to find SNe Ia at z > 1 with coordinated ACS parallel observations. Two high-z SNe were found before the failure of ACS on 2007 February 1.

  • The slope in the H band, bH, has been measured by Persson et al. (2004) to be −3.234 ± 0.042 based on 88 Cepheids in the LMC. Limiting the sample to 75 variables with P > 10 d yields the same result.

  • 10 

    For analysis using the optical relation mW = mV − 2.45(mVmI), differences in color measurements between different photometric systems are ∼5 times larger with additional uncertainties due to the difficulty in cross-calibrating ground-based and space-based systems. The resulting systematic uncertainty is typically 0.10 mag, one of the leading systematic errors in the determination of H0.

  • 11 

    It is worth noting that the terms of order z2 were not included in the use of av and SNe Ia by Freedman et al. (2001) from Suntzeff et al. (1999) and Phillips et al. (1999), tantamount to setting q0 = 1 or ΩM = 2 and reducing H0 by ∼3%.

  • 12 

    This SN difference measurement is similar to the way SNe are used at high redshift to measure dark energy, but without the complexity of significant SN evolution, reddening-law evolution, K-corrections, time dilation changes in demographics, or gravitational lensing.

  • 13 

    However, unlike one such real event, the precision of our estimate of m0v,4258 is equivalent to measuring n such SN Ia events (requiring a millennium to accomplish!), though modestly diminished by the noise in the Cepheids measurements.

  • 14 

    Implicit in this analysis is that local blending of Cepheids with binary companions or cluster companions would also cancel between NGC 4258 and the SN hosts.

  • 15 

    For η Gem and W Sgr, we determined H = 2.18 ± 0.05 and 2.87 ± 0.05, respectively, based on J and K data from Berdnikov et al. (1996).

  • 16 
  • 17 
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10.1088/0004-637X/699/1/539