A 3% SOLUTION: DETERMINATION OF THE HUBBLE CONSTANT WITH THE HUBBLE SPACE TELESCOPE AND WIDE FIELD CAMERA 3^*

Measurements of the expansion history, H(z), from Type Ia supernovae (SNe Ia) provide crucial, empirical constraints to help guide the emerging cosmological model. While high-redshift SNe Ia reveal that the universe is now accelerating (Riess et al. 1998; Perlmutter et al. 1999), nearby ones provide the most precise measurements of the present expansion rate, H₀.

Recently, high-redshift measurements from the cosmic microwave background radiation (CMB), baryon acoustic oscillations (BAOs), and SNe Ia have been used to derive a cosmological model-dependent prediction of the value of H₀ (e.g., Komatsu et al. 2011). They are not, however, a substitute for its measurement in the local universe. Such forecasts of H₀ from the high-redshift universe also make specific assumptions about unsettled questions: the nature of dark energy, the global geometry of space, and the basic properties of neutrinos (number and mass). Instead, we can gain insights into these unknowns from a precise, local measurement of H₀. The most precise measurements of H₀ have come from distance ladders which calibrate the luminosities of nearby SNe Ia through Hubble Space Telescope (HST) observations of Cepheids in their host galaxies (see Freedman & Madore 2010 for a review).

In the early Cycles of HST, the SN Ia HST Calibration Program (Sandage et al. 2006, hereafter SST) and the HST Key Project (Freedman et al. 2001, hereafter KP) each calibrated H₀ via Cepheids and SNe Ia using the Wide Field Planetary Camera 2 (WFPC2) and the Large Magellanic Cloud (LMC) as the first rung on their distance ladder. Unfortunately, the LMC was not an ideal anchor for the cosmic ladder because its distance was constrained to only 5% to 10% (Gibson 2000); its Cepheids (observed from the ground) are of shorter mean period (Δ〈P〉 ≈ 25 days), and lower metallicity (Δ[O/H] = 0.4) than those of the spiral galaxies hosting nearby SNe Ia. These differences and uncertainties between ground-based and space-based photometric zero points introduced a 7% systematic error in the determinations of H₀ obtained by those teams (see Section 4). Additional uncertainty arose from the unreliability of the measurements from several of the SNe Ia selected by SST, which were photographically observed, highly reddened, atypical, or discovered after peak brightness. Only three SNe Ia (SNe 1990N, 1981B, and 1998aq) from the SST sample lacked these shortcomings, defining only a small set of nearby SNe suitable to calibrate H₀. Despite careful work, the teams' estimates of H₀, each with an uncertainty of ∼10%, differed from each other's by 20%, due to the aforementioned systematic errors. Additional progress required rebuilding the distance ladder to address these systematic errors.

The installation of the Advanced Camera for Surveys (ACS) extended the range of HST for observing Cepheids, reduced their crowding with finer pixel sampling, and increased their rate of discovery by doubling the field of view. In Cycle 11, members of our team began using ACS to measure Cepheids at optical wavelengths in the hosts of more modern SNe Ia (SN 1994ae by Riess et al. 2005; SN 1995al and SN 2002fk by Riess et al. 2009b) and in a more ideal anchor galaxy (NGC 4258 by Macri et al. 2006).

In HST Cycle 15, we began the "Supernovae and H₀ for the Equation of State" (SH0ES) project to measure H₀ to better than 5% precision by addressing the largest remaining sources of systematic error. The SH0ES program constructed a refurbished distance ladder from high-quality light curves of SNe Ia, a geometric distance to NGC 4258 determined through radio (very long baseline interferometry, VLBI) observations of megamasers, and Cepheid variables observed with HST in NGC 4258 and in the hosts of recent SNe Ia. The reduction in systematic errors came from additional observations of NGC 4258 and from our use of purely differential measurements of the fluxes of Cepheids with similar metallicities and periods throughout all galaxies in our sample. The latter rendered our distance scale insensitive to possible changes in Cepheid luminosities as a function of metallicity or to putative changes in the slope of the period–luminosity relations from galaxy to galaxy. We measured H₀ to 4.7% precision (Riess et al. 2009a, hereafter R09), a factor of two better than previous measurements with HST and WFPC2. An alternative analysis using the Benedict et al. (2007) parallax measurements of Milky Way (MW) Cepheids in lieu of the megamaser distance to NGC 4258 showed good agreement, with comparable 5.5% precision.

This result formed a triumvirate of constraints in the Wilkinson Microwave Anisotropy Probe (WMAP) 7 year analysis (i.e., BAO + H₀ + WMAP-7yr) which were selected as the combination most insensitive to systematic errors with which to constrain the cosmological parameters (Komatsu et al. 2011). Together with the WMAP constraint on Ω_Mh², this measurement of H₀ provides a constraint on the nature of dark energy, w = −1.12 ± 0.12 (R09; Komatsu et al. 2011), which is comparable to but independent of the use of high-redshift SNe Ia. It also improves constraints on the properties of the elusive neutrinos, such as the sum of their masses and the number of species (Komatsu et al. 2011).

In HST Cycle 17, we used the newly installed Wide Field Camera 3 (WFC3) to increase the sample sizes of both the Cepheids and the SN Ia calibrators along the ladder used by SH0ES to determine H₀. The near-infrared (IR) channel of WFC3 provides an order-of-magnitude improvement in efficiency for follow-up observations of Cepheids over the Near-Infrared Camera and Multi-Object Spectrograph (NICMOS), while the finer pixel scale of the visible channel (relative to ACS) is valuable for reducing the effects of crowding when searching for Cepheids. We present these new observations in Section 2, the redetermination of H₀ in Section 3, and an analysis of the error budget including systematics in Section 4. In Section 5, we address the use of this new measurement along with external data sets to constrain properties of dark energy and neutrinos.

The SH0ES program was developed to improve upon the calibration of the luminosity of SNe Ia in order to better measure the Hubble constant. To ensure a reliable calibration sample, we selected SNe Ia having the following qualities: (1) modern photometric data (i.e., photoelectric or CCD), (2) observed before maximum brightness, (3) low reddening (implying A_V < 0.5 mag), (4) spectroscopically normal, and (5) optical HST-based observations of Cepheids in its host galaxy. In addition to providing robust distance measures, these qualities are crucial for producing a calibration sample which is a good facsimile of the SN Ia sample they are used to calibrate—i.e., those defining the modern SN Ia magnitude–redshift relation at 0.01 < z < 0.1 (e.g., Hicken et al. 2009b).

In HST Cycles 16 and 17, we used WFC3, ACS, and WFPC2 to discover Cepheids in two new SN Ia hosts: NGC 5584 (host of SN 2007af; L. M. Macri et al. 2011b, in preparation) and NGC 4038/9 ("the Antennae," host of SN 2007sr; L. M. Macri et al. 2011c, in preparation) whose light curves were presented in Hicken et al. (2009b).⁸ We also employed the optical channel of WFC3 to reobserve all previous SN Ia hosts in the calibration sample and NGC 4258. This provided for the first time a calibration of all Cepheid optical and infrared photometry using the same zero points. In the case of some hosts, the additional epoch (obtained well after the prior ones) allowed us to discover previously unidentified, longer period (P>60 days) variables. We also used these observations to search for additional Cepheids in the hosts which previously had the smallest numbers of Cepheids: NGC 3021, NGC 3982, NGC 4536, and NGC 4639 (L. M. Macri et al. 2011a, in preparation). The new observations, together with those from Riess et al. (2009b), 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 730 Cepheids in eight hosts with reliable SN Ia data as well as NGC 4258. The Cepheids in each host were typically imaged on 14 epochs in F555W and 2–5 epochs with F814W (except for NGC 4258, which has 12 epochs of F814W data). An illustration of the entire data set used to observe the Cepheids is shown in Figure 1. Having previously determined the positions, periods, and optical magnitudes of these Cepheids, it is highly advantageous to observe their near-IR magnitudes with a single photometric system in order to (1) reduce the differential extinction by a factor of five over visual bands, (2) reduce the possible dependence of Cepheid luminosities on chemical composition (Marconi et al. 2005), and (3) negate zero-point errors. This was previously done with NICMOS on HST in Cycle 15 by R09 for a subset of these Cepheids.

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The near-IR channel on WFC3 provides a tremendous gain over NICMOS for the study of extragalactic Cepheids. Photometry of comparable signal-to-noise ratio can be obtained in a quarter of the exposure time. More significant for Cepheid follow-up observations is the factor of 40 increase in the area of WFC3–IR over NICMOS/Camera 2 (NIC2), the channel which offered the best compromise between area and uniform pixel sampling. The one advantage of NIC2 over WFC3–IR is better sampling of the point-spread function (PSF); the 013 pixels of WFC3–IR undersample the HST PSF by a factor of 1.6 at 1.6 μm. However, the finer sampling of NIC2 is largely offset by the numerous photometric anomalies unique to that camera, whose subsequent correction leads to correlated noise among neighboring pixels which reduces the independence of the NICMOS pixel sampling. In contrast, the detector of WFC3–IR is much better behaved and pixel sampling noise can be mitigated with dithering.

2.1. WFC3 Data Reduction

Each host galaxy was observed for 2–7 ks with individual exposures 400–700 s in length, using integer and half-pixel dithering between exposures to improve sampling of the PSF (see Table 1). The WFC3 images of the two new hosts, NGC 5584 and NGC 4038, are shown in Figures 2 and 3. Figure 4 shows an example of a host previously observed with NIC2 by R09 and with WFC3–IR in this study.

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Table 1. Hosts Observed with WFC3–IR F160W by GO-11570

Host	SN Ia	Exp. Time (s)	Optical Data Source
NGC 4536	SN 1981B	2564	Saha et al. (1996)
NGC 4639	SN 1990N	5377	Saha et al. (1997)
NGC 3982	SN 1998aq	4016	Saha et al. (2001)
NGC 3370	SN 1994ae	4374	Riess et al. (2005)
NGC 3021	SN 1995al	4424	Riess et al. (2009b)
NGC 1309	SN 2002fk	6988	Riess et al. (2009b)
NGC 4038/9	SN 2007sr	6794a	L. M. Macri et al. (2011c, in preparation)
NGC 5584	SN 2007af	4926	L. M. Macri et al. (2011b, in preparation)
NGC 4258	⋅⋅⋅	2011b	Macri et al. (2006)

Notes. ^aData in GO-11577. ^bDepth per pointing; galaxy covered in 16 pointing mosaic.

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We developed an automated pipeline to calibrate the raw WFC3 F160W frames. The first step was to pass the data through the STScI-supported calwf3 pipeline in the STSDAS suite of routines in PyRAF to remove the bias and dark current, reject cosmic rays through up-the-ramp sampling, and flat-field the data. A small correction to the standard flat-field frame was used to correct the WFC3 "blobs," which are 10% depressions in flux covering ∼1% of the area due to spotting on the surface of the WFC3 Channel Select Mechanism (CSM). Next, we used multidrizzle to combine the exposures from each visit into a master image, resampling onto a finer pixel scale while correcting for the known geometric distortions in the camera. We utilized a final pixel scale of 008 pixel⁻¹ and an input-to-output fraction of 0.6.

We identified the positions of Cepheids in the WFC3–IR images by deriving geometric transformations from the F814W images to those in F160W, successively matching fainter sources to improve the registration. This procedure empirically determined the difference in plate scale between ACS–WFC, WFPC2, WFC3–UVIS, and WFC3–IR. We typically identified more than 100 sources in common, resulting in an uncertainty in the mean position of each Cepheid below 0.03 pixels (<2.4 mas).

We carried out the photometry of Cepheids using the algorithms developed by R09; they employ PSF fitting to model the crowded regions around Cepheids, fixing their positions to those derived from optical data and using artificial-star tests to determine photometric errors and crowding biases. As an example, we show in Figure 5 the HST optical image, near-IR image, model, and residuals for eight typical Cepheids spanning a wide range of periods in the SN host NGC 5584. We used the same approach to determine zeropoints as in R09 from the Persson et al. (1998) standard star P330E.

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Due to the low amplitudes of their near-IR light curves (<0.3 mag), Cepheid magnitudes determined at random phases provide nearly the same precision as mean fluxes for determining the intercept of their P–L relations (Madore & Freedman 1991).⁹

Since we had previously observed with NIC2 many of the Cepheids now observed with WFC3–IR, we can directly compare their F160W photometry on these two systems. Figure 6 shows the magnitude differences for the Cepheids utilized in the P–L relations in both R09 and Section 2.2. The mean difference is 0.036 ± 0.027 mag (in the sense that photometry with WFC3 is brighter), with no apparent dependence on Cepheid brightness. While the difference in photometry between instruments may include differences in system zero points, the subsequent determination of H₀ via Cepheids observed with a single instrument in the SN Ia hosts and in NGC 4258 will be independent of instrument zero points. Thus, for the determination of H₀ it is more relevant to calculate the change in magnitudes between Cepheids in NGC 4258 and the SN hosts between WFC3 and NIC2; the measurement of this change is 0.019 ± 0.054 mag.

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Table 2 contains relevant parameters for each Cepheid observed with WFC3 F160W. The first eight columns give the Cepheid's host, position, identification number (from Macri et al. 2006; Riess et al. 2009b; L. M. Macri et al. 2011a, 2011b, 2011c, in preparation), period, mean V−I color, WFC3 F160W magnitude, and the magnitude uncertainty. Column 9 contains the displacement of the flux centroid in the near-IR data relative to the optical Cepheid position, expressed in units of pixels (1 pixel = 008), a quantity used to refine the determination of the crowding bias. Column 10 gives the photometric crowding bias determined using the artificial-star tests for each Cepheid's environment (see Section 2.3 of R09) and the displacement tabulated in the previous column; this correction has already been applied to the magnitudes listed in Column 7. Column 11 contains the rms of the residual image, weighted by the inverse distance from the Cepheid position, useful for determining the quality of the crowded-scene fit. Column 12 contains the metallicity parameter, 12 + log [O/H] (Zaritsky et al. 1994), derived from the deprojected galactocentric radii of each Cepheid and the abundance gradient of its host.¹⁰ Column 13 contains a rejection flag used for the P–L relations.

Table 2. WFC3–IR Cepheids

Field	α	δ	ID	P	V−I	F160W	σ	Offset	Bias	IMrms	[O/H]	Flag
	(J2000)	(J2000)		(days)	(mag)	(mag)	(mag)	(pixel)	(mag)
n4536	188.590	2.16830	27185	13.00	0.97	24.91	0.31	1.64	0.13	2.16	8.54
n4536	188.604	2.18312	42353	13.07	0.73	26.29	0.74	3.32	0.37	4.30	8.97	Rej
n4536	188.584	2.18070	50718	13.73	0.88	24.51	0.42	0.88	0.28	11.4	8.64
n4536	188.583	2.19700	72331	13.91	0.89	24.84	0.44	0.07	0.22	1.40	8.81
n4536	188.590	2.19545	65694	14.38	0.98	25.26	0.38	2.91	0.39	30.8	8.90
n4536	188.587	2.18864	58805	14.44	1.13	23.41	0.35	3.94	0.26	47.9	8.78	Rej
n4536	188.586	2.18406	53703	14.53	0.72	25.38	0.47	0.63	0.27	14.9	8.72
n4536	188.592	2.20025	70938	14.62	0.64	25.81	0.58	2.39	0.30	17.4	8.94	Rej
n4536	188.594	2.17693	40098	14.64	0.95	25.12	0.52	4.40	0.63	12.9	8.72
n4536	188.597	2.18489	48539	15.03	0.90	23.53	0.31	0.46	0.29	7.28	8.89	Rej

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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2.2. Near-infrared Cepheid Relations

The nine individual P–L relations measured with WFC3 F160W and fit with a common slope are shown in Figure 7. Intercepts relative to NGC 4258 are given in Table 3. While 636 Cepheids previously identified at optical wavelengths were measurable¹¹ in the WFC3–IR F160W images, ∼20% appeared as outliers in the IR P–L 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, (2) a poor model reconstruction of a crowded group when the Cepheid is a small component of the group's flux, (3) objects misidentified as classical Cepheids in the optical (e.g., blended Type II Cepheids), and (4) Cepheids with the wrong period (aliasing or incomplete sampling of a single cycle). As expected, the outlier fraction is greater in WFC3 images than in NIC2 ones because the former contain a larger fraction of Cepheids from crowded regions (such as the nucleus) which yield more outliers and were intentionally avoided in the small, selective NIC2 pointings (see Figure 4).

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Table 3. Distance Parameters

Host	SN Ia	Filters	m0v,i + 5av	σa	μ0,i − μ0,4258	μ0 Best
n4536	SN 1981B	UBVR	15.147	0.145	1.567 (0.0404)	30.91 (0.07)
n4639	SN 1990N	UBVRI	16.040	0.111	2.383 (0.0630)	31.67 (0.08)
n3370	SN 1994ae	UBVRI	16.545	0.101	2.835 (0.0284)	32.13 (0.07)
n3982	SN 1998aq	UBVRI	15.953	0.091	2.475 (0.0460)	31.70 (0.08)
n3021	SN 1995al	UBVRI	16.699	0.113	3.138 (0.0870)	32.27 (0.08)
n1309	SN 2002fk	BVRI	16.768	0.103	3.276 (0.0491)	32.59 (0.09)
n5584	SN 2007af	BVRI	16.274	0.122	2.461 (0.0401)	31.72 (0.07)
n4038	SN 2007sr	BVRI	15.901	0.137	2.396 (0.0567)	31.66 (0.08)
Weighted mean	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	0.0417	⋅⋅⋅  (0.0133)	⋅⋅⋅

Note.^a For MLCS2k2, 0.08 mag added in quadrature to fitting error.

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As in R09, we eliminated outliers >0.75 mag or >2.5σ (following Chauvenet's criterion) from an initial fit of the P–L relations, refitted the relations and repeated these tests for outliers until convergence. The objects rejected in this way are indicated in the last column of Table 2. This resulted in a reduction of the sample to 484 objects; the next section considers the effect of this rejection on the determination of H₀ and an alternative method for contending with outliers.

The determination of the Hubble constant follows from the relations given in Section 3 of R09. To summarize, we perform a single, simultaneous fit to all Cepheid and SN Ia data to minimize the χ² statistic and measure the parameters of the distance ladder. We express the jth Cepheid magnitude in the ith host as

$$\begin{eqnarray} m_{W,i,j}&=&(\mu _{0,i}-\mu _{0,4258})+zp_{W,4258}+b_W \ {\rm log} \ P_{i,j}\nonumber\\ && +Z_W \ \Delta {\rm log[O/H]}_{i,j}, \end{eqnarray} \tag{ 1 }$$

where the "Wesenheit reddening-free" mean magnitude (Madore 1982) is given as

$$\begin{equation} m_{W,i,j} =m_{H,i,j} - R(m_{V,i,j} - m_{I,i,j}), \end{equation} \tag{ 2 }$$

and R ≡ A_H/(A_V − A_I). The Cepheid parameters with subscripts i, j are given in Table 2. For a Cardelli et al. (1989) reddening law, a Galactic-like value of R_V = 3.1, and the H band corresponding to the WFC3 F160W band, we have R = 0.410. In the next section, we consider the sensitivity of H₀ to the value of R_V.

We determine the values of the nuisance parameters b_W and Z_W—which define the relation between Cepheid period, metallicity, and luminosity—by minimizing the χ² for the global fit to all Cepheid data. The reddening-free distances, μ_0,i, for the hosts relative to NGC 4258 are given by the fit parameters μ_0,i − μ_0,4258, while zp₄₂₅₈ is the intercept of the P–L relation simultaneously fit to the Cepheids of NGC 4258.

The SN Ia magnitudes in the SH0ES hosts are simultaneously expressed as

$$\begin{equation} m_{v,i}^0=(\mu _{0,i}-\mu _{0,4258})+m^0_{v,4258}, \end{equation} \tag{ 3 }$$

where the value m⁰_v,i is the maximum-light apparent V-band brightness of an SN Ia in the ith host at the time of B-band peak corrected to the fiducial color and luminosity. This quantity is determined for each SN Ia from its multi-band light curves and a light curve fitting algorithm, either from the MLCS2k2 (Jha et al. 2007) or from the SALT-II (Guy et al. 2005) prescription (see Section 4.2 for further discussion).

A minor change from R09 is the inclusion of a recently identified, modest relationship between host-galaxy mass and the calibrated SN Ia magnitude. Several studies (Hicken et al. 2009a; Kelly et al. 2010; Lampeitl et al. 2010; Sullivan et al. 2010) have shown the existence of a correlation between the corrected SN magnitude and the mass of its host, with a value of 0.03 mag dex⁻¹ in M_stellar, in the sense that more massive (and metal-rich) hosts produce more luminous SNe. This correlation has been independently detected using both low- and high-redshift samples of SNe Ia, as well as with multiple fitting algorithms. The effect on H₀ is quite small, a decrease of 0.75%, due to the modest difference in mean masses for the nearby hosts (Neill et al. 2009; mean log M_stellar = 10.0) and for those that define the magnitude–redshift relation (Sullivan et al. 2010; mean log M_stellar = 10.5). We include these corrections based on host-galaxy mass in our present determination of m⁰_v,i, given in Table 3 and Figure 8, normalizing to a fiducial host mass of log M_stellar = 10.5 as appropriate for the objects used to measure the Hubble flow.

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The simultaneous fit to all Cepheid and SN Ia data via Equations (1) and (3) results in the determination of m⁰_v,4258, which is the expected reddening-free, fiducial, peak magnitude of an SN Ia appearing in NGC 4258. Finally, the Hubble constant is determined from

$$\begin{equation} {\rm log} \ H_0={\big(m_{v,4258}^0-\mu _{0,4258}\big)+5a_v+25 \over 5}, \end{equation} \tag{ 4 }$$

where μ_4258,0 is the independent, geometric distance estimate to NGC 4258 obtained through VLBI observations of water megamasers orbiting its central supermassive black hole (Herrnstein et al. 1999; Humphreys et al. 2005, 2008; Argon et al. 2007; Greenhill 2009). The term a_v is the intercept of the SN Ia magnitude–redshift relation, approximately log cz − 0.2m⁰_v but given for an arbitrary expansion history as

$$\begin{eqnarray} a_v &=& {\rm log} \ cz \Bigg\lbrace 1 + {1\over 2}\left[1-q_0\right]{z} -{1\over 6}\left[1-q_0-3q_0^2+j_0 \right] z^2\nonumber\\ && +\, O(z^3) \Bigg\rbrace - 0.2m_v^0, \end{eqnarray} \tag{ 5 }$$

measured from the set of SN Ia (z, m⁰_v) independent of any absolute (i.e., luminosity or distance) scale. As in R09, we determine a_v from a Hubble diagram for 240 SNe Ia from Hicken et al. (2009b) using MLCS2k2 (Jha et al. 2007) or the SALT-II (Guy et al. 2005) prescription to determine m⁰_v. Limiting the sample to 0.023 < z < 0.1 (to avoid the possibility of a local, coherent flow; z is the redshift in the rest frame of the CMB) leaves 140 SNe Ia. (In the next section, we consider a lower cut of z > 0.01.) Ganeshalingam et al. (2010) recently published light curves of a large sample of SNe Ia from the Lick Observatory Supernova Search, but there is a large overlap with those given by Hicken et al. (2009b). There are only 13 SNe Ia at z > 0.023 not already included in our sample and we have added these to raise the total to 253 SNe Ia. Together with the present acceleration q₀ = −0.55 and prior deceleration j₀ = 1 (Riess et al. 2007), we find a_v = 0.697 ± 0.00201.

The full statistical error in H₀ is the quadrature sum of the uncertainty in the three independent terms in Equation (4): μ_4258,0, m⁰_v,4258, and 5a_V, where μ_4258,0 is the geometric distance estimate to NGC 4258 by Herrnstein et al. (1999), claimed by Greenhill (2009) to currently have a 3% uncertainty.

Hui & Greene (2006) point out that the peculiar velocities of SN Ia hosts and their correlations can produce an additional systematic error in the determination of the SN Ia m–z relation used for cosmography. However, by making use of a map of the matter density field, it is possible to correct individual SN Ia redshifts for these peculiar flows (Riess et al. 1997). Neill et al. (2007) made use of the IRAS PSCz density field (Branchini et al. 1999) to determine the effect of the density field on the low-redshift SN Ia m–z relation and its impact on the equation-of-state parameter of dark energy, w = P/(ρc²) (where P is the pressure and ρc² is the energy density). Using their results for a light-to-matter bias parameter β = 0.5 and the dipole from Pike & Hudson (2005) results in an increase of the mean velocity of the low-redshift sample and in the Hubble constant by 0.4% over the case of uncorrelated velocities at rest with respect to the CMB. We use a new estimate of this mean peculiar velocity for the Hicken et al. (2009b) SN sample which is a slightly larger value of 0.5%. We account for this and assume an uncertainty of 0.1% resulting from a ±0.2 error in the value of β.

The result is H₀ = 74.8 ± 3.0 km s⁻¹ Mpc⁻¹, a 4.0% measurement (top line, Table 4). It is instructive to deconstruct the individual sources of uncertainty to improve our insight into the measurement. In principle, the covariance between the data and parameters does not allow for an exact and independent allocation of propagated error for each term toward the determination of H₀. However, in our case, the diagonal elements of the covariance matrices provide a very good approximation to the individual components of error. These are given in Table 5 and shown in Figure 9 for past and present determinations of H₀.

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Table 5. H₀ Error Budget for Cepheid and SN Ia Distance Ladders^a

Term	Description	Previous	R09	Here	Here
		LMC	N4258	N4258	All Threeb
σanchor	Anchor distance	5%	3%	3%	1.3%
σanchor−PL	Mean of P–L in anchor	2.5%	1.5%	1.4%	0.7%c
$\sigma _{{\rm host}-PL}/\sqrt{n}$	Mean of P–L values in SN hosts	1.5%	1.5%	0.6 %	0.6%
$\sigma _{\rm SN}/\sqrt{n}$	Mean of SN Ia calibrators	2.5%	2.5%	1.9%	1.9%
σm−z	SN Ia m–z relation	1%	0.5%	0.5%	0.5%
Rσλ,1,2	Cepheid reddening, zero points, anchor-to-hosts	4.5%	0.3%	0.0%	1.4%
σZ	Cepheid metallicity, anchor-to-hosts	3%	1.1%	0.6 %	1.0%
σPL	P–L slope, Δlog P, anchor-to-hosts	4%	0.5%	0.4%	0.6%
σWFPC2	WFPC2 CTE, long-short	3%	0%	0%	0%
Subtotal, $\sigma _{H_0}$		10%	4.7 %	4.0%	2.9%
Analysis systematics		NA	1.3%	1.0%	1.0%
Total, $\sigma _{H_0}$		10%	4.8 %	4.1%	3.1%

Notes. ^aDerived from diagonal elements of the covariance matrix propagated via the error matrices associated with Equations (1), (3), (7), and (8). ^bUsing the combination of all three calibrations of the Cepheid distance scale, LMC, MW parallaxes, and NGC 4258. ^cFor MW parallax, this term is already included with the term above.

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A number of improvements since R09 are evident by comparing Columns 2 and 3 in Table 5 and as shown in Figure 9. The uncertainty in H₀ from all of the terms independent of the megamaser distance to NGC 4258 is 2.3%, 50% smaller than these same terms in R09, a result of the increased sample of Cepheids and SN calibrators. This term includes uncertainties due to the form of the P–L relation, Cepheid metallicity dependences, photometry bias, and zero points—all of which were important systematic uncertainties in past determinations of the Hubble constant (see Column 1, which contains the values from Freedman et al. 2001). In this analysis, as in R09, these uncertainties have been reduced by the collection of samples of Cepheids whose measures (i.e., metallicity, periods, and photometric systems) are a good match between NGC 4258 and the SN hosts. Here the contribution from an unknown dependence of Cepheid luminosity on metallicity has been furthered reduced by 40% owing to a better match between the metallicity of the Cepheid samples in NGC 4258 and the expanded SN host sample. In R09, the mean metallicity of the NGC 4258 Cepheid sample on the ZKH abundance scale was 12 + log [O/H] = 8.91, nearly the same as the present mean of 8.90. However, the mean metallicity of the Cepheid sample in the SN hosts has risen from 8.81 to 8.85. Some of this change can be attributed to the inclusion of Cepheids closer to the nuclei of the hosts and some to the inclusion of two new hosts, NGC 5584 and NGC 4038/9, with higher-than-average metallicities. The reduction in the mean abundance difference between NGC 4258 and the SN Ia hosts from 0.077 to 0.045 dex results in a decrease of the error propagated into H₀ from 1.1% to 0.6%. A similar reduction is seen with the use of MW Cepheids whose mean metallicity of 8.9 is closer to the mean of the new Cepheid sample in the SN hosts. We consider an alternative calibration of abundances from Bresolin (2011) in Section 4.1.

3.1. Buttressing the First Rung

In our present determination of H₀, the 3% uncertainty in the distance to NGC 4258 claimed by Greenhill (2009) is now greater than all other sources combined (in quadrature). The next largest term, the uncertainty in mean magnitude of the eight nearby SNe Ia, is 1.9%. To significantly improve upon our determination of H₀, we would need an independent calibration of the first rung of the distance ladder as good as or better than the megamaser-based measurement to NGC 4258 in terms of precision and reliability. Independent calibration of the first rung is also valuable as an alternative to NGC 4258, should future analyses reveal previously unidentified systematic errors affecting its distance measurement.

A powerful alternative has recently become available through high signal-to-noise ratio measurements of the trigonometric parallaxes of MW Cepheids using the fine guidance sensor (FGS) on HST. Benedict et al. (2007) reported parallax measurements for 10 Cepheids, with mean individual precision of 8% and an error in the mean of the sample of 2.5%. These were used in R09 as a test of the distance scale provided by NGC 4258, but the improvement in precision beyond the first rung in the previous section suggests greater value in their use to enhance the calibration of the first rung.

van Leeuwen et al. (2007) reanalyzed Hipparcos observations and determined independent parallax measurements for the same 10 Cepheids (albeit with half the precision of HST/FGS) and for three additional Cepheids (excluding Polaris which is an overtone pulsator and whose estimated fundamental period is an outlier among the Cepheids pulsing in the fundamental mode). The resulting sample can be considered an independent anchor with a mean, nominal uncertainty of just 1.7%. We use the combined parallaxes tabulated by van Leeuwen et al. (2007) and their H-band photometry as an alternative to the Cepheid sample of NGC 4258 by replacing Equation (1) for the Cepheids in the hosts of SNe Ia with

$$\begin{equation} m_{W,i,j}=\mu _{0,i}+M_{W,1}+b_W {\rm log} P_{i,j}+Z_W \Delta {\rm log[O/H]}_{i,j}, \end{equation} \tag{ 6 }$$

where M_W,1 is the absolute Wesenheit magnitude for a Cepheid with P = 1 day and simultaneously fitting the MW Cepheids with the relation

$$\begin{equation} M_{W,i,j}=M_{W,1}+b_W {\rm log} P_{i,j}+Z_W \Delta {\rm log[O/H]}_{i,j}. \end{equation} \tag{ 7 }$$

Equation (3) for the SNe Ia is replaced with

$$\begin{equation} m_{v,i}^0=\mu _{0,i}-M_V^0. \end{equation} \tag{ 8 }$$

The determination of M⁰_V for SNe Ia together with the previous term a_v then determines the Hubble constant,

$$\begin{equation} {\rm log}\, H_0={M_V^0+5a_v+25 \over 5}. \end{equation} \tag{ 9 }$$

Since the near-IR magnitudes of these MW Cepheids have not been directly measured with WFC3, the use of these variables requires an additional allowance for possible differences in their photometry. These may arise from differences in instrumental zero points, crowding, filter transmission functions, and detector well depth at which the sources are measured together with an uncertainty in detector linearity. Analysis of the absolute photometry from WFC3–IR (Kalirai et al. 2009) and the ground system (e.g., Two Micron All Sky Survey; Skrutskie et al. 2006) claim absolute precision of 2%–3%. We therefore assume a systematic uncertainty in the relative magnitudes between HST WFC3 F160W Cepheid photometry and the ground-based measurements of MW Cepheids on the H-band system of Persson et al. (1998) of 4%. This reduces the effective precision of the parallax distance scale from 1.7% to 2.6%. The ground-based photometry of these MW Cepheids is tabulated by Groenewegen (1999) and R09. This systematic error is included in the global fit as an additional calibration equation with uncertainty given in the error correlation matrix.

When using the MW Cepheids, we now include an external constraint on the slope of the near-IR P–L relation. No such constraint was necessary or even of significant value in the previous section because the Cepheid periods in NGC 4258 (mean log P = 1.51) are so similar to those in the SN Ia host (mean log P = 1.63). In contrast, the mean period of the MW sample (mean log P = 1.0) is substantially lower, giving an unconstrained slope of the P–L relation a greater and unrealistically large lever arm. Following analyses of optical and near-IR Cepheid data in the MW (Fouqué et al. 2007) and the LMC (Persson et al. 2004; Udalski et al. 1999), we adopt a conservative constraint on the slope of the Wesenheit relation of −3.3 ± 0.1 mag dex⁻¹ in log P.

Using the MW Cepheids instead of NGC 4258 as the first rung of the distance ladder gives 75.7 ± 2.6 km s⁻¹ Mpc⁻¹, in good agreement with (and even greater precision than) the NGC 4258-based value. However, an overall improvement in precision is realized by the simultaneous use of both the MW parallaxes and the megamaser-based distance to NGC 4258, yielding 74.5 ±  2.3 km s⁻¹ Mpc⁻¹, a remarkably small uncertainty of 3.0%.

Another opportunity to improve upon the first rung on the distance ladder comes from the sample of H-band observations of Cepheids in the LMC by Persson et al. (2004). Recent studies of detached eclipsing binaries (DEBs) by different groups provide claims of a reliable and precise distance to the LMC. Guinan et al. (1998), Fitzpatrick et al. (2002), and Ribas et al. (2002) studied three B-type systems (HV2274, HV982, and EROS1044) which lie close to the bar of the LMC and therefore provide a good match to the Cepheid sample of Persson et al. (2004). The error-weighted mean of these is 49.2 ± 1.6 kpc.¹² Pietrzyński et al. (2009) analyzed OGLE-051019.64-685812.3, an eclipsing binary system comprised of two giant G-type stars also located near the barycenter of the LMC, and found a distance of 50.2 ±  1.3 kpc. The average result, 49.8 kpc, provides a good estimate of the distance to the LMC.¹³ Here we retain the larger of the two previous uncertainties to estimate the distance modulus as 18.486 ± 0.065 mag or an effective error of ±0.076 mag when including the aforementioned 0.04 mag uncertainty between the ground-based and HST-based near-IR photometric systems. Using this distance to the LMC and the Cepheid sample of Persson et al. (2004) yields 71.3 ± 3.8 km s⁻¹ Mpc⁻¹, as seen in Table 4.

Combining all three first rungs (MW, LMC, and NGC 4258) provides the most precise measurement of H₀: 73.8 ± 2.1 km s⁻¹ Mpc⁻¹, a slightly smaller uncertainty of 2.9%. As expected, the use of all three anchors for the distance ladder instead of just one has the largest impact on the overall uncertainty, reducing the total contribution of the first rung to the error from 3.3% to 1.5%. However, a substantial penalty is paid for the mixing of ground-based and space-based photometric systems and the resultant uncertainties in Wesenheit or dereddened magnitudes, adding a 1.4% error to H₀ where for NGC 4258 alone none pertained. Modest increases in error also result from the larger difference in mean Cepheid metallicity (LMC) and period (LMC and MW). See Figure 9 for the full error profile.

Past determinations of the absolute distance scale have had a checkered history, with revisions common. Thus, it may be prudent to rely on no more than any two of the three possible anchors of the distance scale in the determination of H₀. The omission of NGC 4258, MW parallaxes, or the LMC yields a precision in H₀ of 3.3%, 3.2%, and 3.0%, respectively. We thus adopt as our best determination 73.8 ± 2.3 km s⁻¹ Mpc⁻¹, the measurement from all three sources of the distance scale, but with the larger error associated with only two independent origins of the distance scale.

Should future work revise the distance to any one of the absolute distance-scale determinations, we provide the following recalibration: H₀ decreases by 0.25, 0.30, and 0.14 km s⁻¹ Mpc⁻¹ for each increase of 1% in the distance to either NGC 4258, the MW parallax scale, or the distance to the LMC.

In the last column of Table 3, we also give the best estimate of the distance to each host from the global fit to all first rungs, Cepheid, and SN data. These are useful to compare to alternative methods of measuring distances to these hosts or to place a sample of relative measures of SNe Ia distances onto an absolute scale. For example, there has been recent disagreement on the distance modulus of the Antennae (NGC 4038/9); Saviane et al. (2008) claim a value of μ₀ = 30.62 ± 0.17 mag based on the apparent tip of the red giant branch (TRGB), while Schweizer et al. (2008) obtain μ₀ = 31.74 ± 0.27 mag from SN 2007sr and μ₀ = 31.51 ± 0.16 mag from a different determination of the TRGB, in agreement with previous estimates by Whitmore et al. (1999) and Tonry et al. (2000) based on flow-field models. Our result of μ₀ = 31.66 ± 0.08 mag (with the uncertainty based on the global fit) strongly favors the "long" distance to the Antennae.

Although we have been careful to propagate our statistical errors, as well as past sources of systematic error such as metallicity dependence, system zero point, and instrumental uncertainties, we now consider a broader range of systematic uncertainties relating to alternative approaches to the analysis of the data.

In the preceding section, we presented our preferred approach to analyzing the Cepheid and SN Ia data, incorporating uncertainties within the framework used to model the data. Here we follow the same approach used by R09 to quantify the systematic uncertainty in the determination of H₀, by measuring the impact of a number of variants in the modeling of the Cepheid and SN Ia data.

In Table 4, we show 15 variants of the previously described analysis for every combination of choices of distance anchors (NGC 4258, MW, or LMC), any two of the preceding or all three; these amount to a total of 105 combinations. Our primary analysis for any anchor choice is given in the first row (shown in bold) for which that choice initially appears. Column 1 gives the value of χ²_ν, Column 2 the number of Cepheids in the fit, Column 3 the value and total uncertainty in H₀, and Column 4 whether the near-IR data for Cepheids with periods shorter than the completeness limit from the optical selection were included. Column 5 gives the SN Ia magnitude–redshift intercept parameter, Column 6 gives the determination of M⁰_V which is specific to the light curve fitter employed, Column 7 the calibration system for the metal abundances, Column 8 the value and uncertainty in the metallicity dependence, and Column 9 the value and uncertainty of the slope of the Cepheid P–L or P–W relation. Column 10 gives the minimum SN Ia redshift used to define the m–z relation, Column 11 encodes aspects of the SN fitting routine and assumptions therein addressed below, and Column 12 is the choice of anchors to set the distance scale. Column 13 gives the type of P–L relation employed, either Wesenheit (H, V, I) or the H band only. Column 14 is the reddening law value used for the Cepheids. Column 15 lists the filters allowed for fitting the SN Ia light curves and Column 16 gives the value of R_V used to fit the SN light curves.

4.1. Cepheid Systematics

4.2. SN Systematics

An independent and precise measurement of H₀ is an important complement to the determination of cosmological model parameters. Alternatively, it serves as a powerful test of model-constrained measurements at higher redshifts. It is beyond the scope of this paper to provide a complete analysis of the impact of the measurement of H₀ on the cosmological model from all extant data. We encourage others to do so. However, one such example using the present measurement of H₀ can be illustrative.

Making use of the simplest present hypothesis for the cosmological model (namely Λ-cold-dark-matter without curvature, exotic neutrino physics, or specific early-universe physics), and using the single most powerful cosmological data set (the 7 year WMAP results from Komatsu et al. 2011), results in a predicted value of H₀ = 71.0 ± 2.5 km s⁻¹ Mpc⁻¹. This value agrees well with our determination of 73.8 ± 2.4 km s⁻¹ Mpc⁻¹ at better than the combined 1σ confidence level.

Alternatively, we can use the WMAP data together with the measured value of H₀ to constrain added complexity to the model. In Figure 10, we show the use of this data combination for constraining a redshift-independent dark energy equation-of-state parameter (w), the number of relativistic species (e.g., neutrino number), and the sum of neutrino masses. The result for dark energy is w = −1.08 ± 0.10, about 20% more precise than the same result derived from the determination of H₀ in R09. If we had perfect knowledge of the CMB, our overall 30% increase in the precision of H₀ would yield the same-sized improvement in the determination of w. However, the fractional uncertainty in Ω_Mh² from the WMAP 7 year analysis is comparable to our measurement of H₀; thus, greater precision in w may still be wrung from future higher-precision measurements of the CMB by WMAP or Planck.

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The enhanced precision in measuring H₀ also provides a strong rebuff to recent attempts to explain accelerated expansion without dark energy but rather by our presence in the center of a massive void of gigaparsec scale. Already such models are hard to fathom as they require an exotic location for the observer, at the center of the void to within a part in a million (Blomqvist & Mörtsell 2010) to avoid an excess dipole in the CMB. It is also not yet apparent if such a model is consistent with other observables of the CMB or the late-time-integrated Sachs–Wolfe effect. However, using measurements of H(z > 1) to constrain void models of the Lemaitre, Tolmon, and Bondi variety already predicts slower-than-observed local expansion with values of H₀ = 60 (Nadathur & Sarkar 2010) or 62 (Wiltshire 2007) km s⁻¹ Mpc⁻¹, more than 5σ below our measurement.

Comparable improvements to cosmological constraints on relativistic species are also realized from R09, as shown in Figure 10. Most interesting may be the effective number of relativistic species, N_eff = 4.2 ± 0.75, which is nominally higher than the value of 3.046 expected from the three known neutrino species plus tau-neutrino heating from e⁺e⁻ collisions (Mangano & Serpico 2005). While this nominal excess of relativistic species has been noted previously (e.g., Reid et al. 2010; Komatsu et al. 2011; Dunkley et al. 2010), and even interpreted as a possible indication of the presence of a sterile neutrino (Hamann et al. 2010), we caution that the cosmological model provides other avenues for reducing the significance of this result including additional degrees of freedom for curvature, dark energy, primordial helium abundance, and neutrino masses. The 30% improvement in the present constraint on H₀ combined with improved high-resolution CMB data (e.g., Dunkley et al. 2010) and ultimately with Planck satellite CMB data should reduce the present uncertainty in N_eff by a factor of ∼ 3 which may provide a more definitive conclusion on the presence of excess radiation in the early universe.

Examination of the complete error budget for H₀ in the last two columns of Table 5 indicates additional approaches for improved precision in future measurements of H₀. Expanding the sample of well-measured parallaxes to MW Cepheids (especially those at log P > 1) with the GAIA satellite could drive the precision of the first rung of the distance ladder well under 1%. However, as we have found with the "baker's dozen" of present MW parallaxes, much of this precision would be lost without better cross-calibration between the space and ground photometric systems used to measure Cepheids, near and far.

The largest remaining term comes from the quite limited sample of ideal SN Ia calibrators, just eight objects. The occurrence of an ideal SN Ia in the small volume within which HST can measure Cepheids (R ≈ 30 Mpc) is rare, on average only once every 2–3 years. Given the recent proliferation of SN surveys and instances of multiple, independent discoveries, we are confident that all such SNe Ia within this volume are being found. Collecting more will require extending the range of Cepheid measurements—without introducing new systematics—and patience. The forthcoming JWST offers a promising route to extend Cepheid observations out to 50 Mpc and to redder wavelengths, where uncertainties due to possible variations in the extinction law and the dependence of Cepheid luminosities on metallicity are further reduced. This extension would increase the SN sample suitable for calibration by a factor of ∼5, reaching ∼40 ideal SNe Ia observed over the past 20 years. Based on a 5% distance precision per ideal SN, such a sample would enable a determination of H₀ to better than 1%. However, discovering these Cepheids may require imaging at optical wavelengths where the amplitude of the variations is significant, a requirement which will challenge the short-wavelength capabilities of the JWST.

We have improved upon the precision of the measurement of H₀ from Riess et al. (2009a) by (1) more than doubling the sample of Cepheids observed in the near-IR in SN Ia host galaxies, (2) expanding the SN Ia sample from six to eight with the addition of SN 2007af and SN 2007sr, (3) increasing the sample of Cepheids observed in NGC 4258 by 20%, (4) reducing the difference in metallicity for the observed sample of Cepheids between the calibrator and the SN hosts, and (5) calibrating all optical Cepheid colors with WFC3 to remove cross-instrument zero-point errors. Further improvements to the precision and reliability of the measurement of H₀ come from the use of additional sources of calibration for the first rung; foremost of these are the trigonometric parallaxes of 13 Cepheids in the MW.

Our primary analysis gives H₀ = 73.8 ± 2.4 km s⁻¹ Mpc⁻¹ including systematic errors determined from varying assumptions and priors used in the analysis. The combination of this result alone with the WMAP 7 year constraints yields w = −1.08 ± 0.10 and improves constraints on a possible but still uncertain excess in relativistic species above the number of known neutrino flavors. The measured H₀ is also highly inconsistent with the simplest inhomogeneous matter models invoked to explain the apparent acceleration of the universe without dark energy. Given that statistical errors still dominate over systematic errors, future work is likely to further improve the precision of the determination of H₀.

We are grateful to William Januszewski for his help in executing this program on HST. We are indebted to Mike Hudson for assisting with the peculiar-velocity calculations from the PSCz survey, to David Larson for contributions to the WMAP MCMC analysis, to Daniel Scolnic for donating some useful routines, and to Mark Huber for an analysis of pre-discovery observations of SN 2007sr. Financial support for this work was provided by NASA through programs GO-11570 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. L.M. acknowledges support from a Texas A&M University faculty startup fund. The metallicity measurements for NGC 5584 and NGC 4038/9 presented herein were obtained with the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA; the observatory was made possible by the generous financial support of the W. M. Keck Foundation. We thank Chris Kochanek and Kris Stanek for their support of GO-11570.