THE HUBBLE SPACE TELESCOPE CLUSTER SUPERNOVA SURVEY. V. IMPROVING THE DARK-ENERGY CONSTRAINTS ABOVE z > 1 AND BUILDING AN EARLY-TYPE-HOSTED SUPERNOVA SAMPLE^*

As described in Dawson et al. (2009), the survey produced a total of 39 likely SNe during the active phase of the search. In Barbary et al. (2012a), types are determined for 29 of these candidates. (The remaining 10 do not have enough light-curve information to determine their types, since they lie outside of our fiducial search time window or our signal-to-noise cuts.) Twenty SNe are classified as SNe Ia, with confidence levels (CL) of secure, probable, or plausible. A secure SN Ia is one that either has a spectrum that directly confirms it to be an SN Ia or one that satisfies two conditions: (1) it occurred in a host whose spectroscopic, photometric, and morphological properties are consistent with those of an early-type galaxy with no detectable signs of recent star formation, and (2) it has a light-curve shape consistent with that of an SN Ia and inconsistent with all other known SN types. A probable SN Ia is one that does not have a secure spectrum but satisfies one of the two non-spectroscopic conditions that are required for a secure classification. A plausible SN Ia is one that has an indicative light curve but we do not have enough data to rule out other types. Details of the classification scheme can be found in Barbary et al. (2012a) and details of the galaxy typing can be found in Meyers et al. (2012).

Sixteen SNe are classified as either secure or probable. We use these SNe in the cosmological analysis. We include the photometry and light curves of an additional four plausible SNe Ia to illustrate the quality of the data and the potential for a similar sample with complete classification (and because additional host galaxy data may later bring one of these into the larger sample). Secure, probable, and plausible SN Ia are listed in Table 1, together with their position, redshift, and typing. Postage stamp images of the SNe and host galaxies are shown in Figure 1.

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We labeled each of our 25 clusters with a letter from "A" to "Z" (excluding "O" to avoid confusion with zero) and assigned SN names as "SCP"+[discovery year]+ [discovered cluster]+[SN ID]. The cluster IDs, coordinates, and redshifts are found in tables in Dawson et al. (2009), Barbary et al. (2012a), and Meyers et al. (2012). The cluster membership is discussed in detail in Meyers et al. (2012) and summarized in Table 1 along with host type information.

Several SNe Ia deserve special mention.

SN SCP06C1, an SN discovered in 2006 in cluster C (XMMU J2205.8−0159) and numbered as "1" among other transient candidates, could not be clearly associated with a galaxy in the cluster but was spectroscopically confirmed as an SN Ia at the cluster redshift. It might be an example of an SN Ia that comes from a progenitor in the intracluster stellar population (Sand et al. 2011). SN SCP06C1 is discussed in greater detail in Barbary et al. (2012a). As there was a bright background galaxy near the position of the SN, had we not obtained a spectrum of the SN, we would have misidentified the redshift. We note that experiments that do not obtain spectroscopy of their SNe and instead assume the nearest visible galaxy is the host will have to factor cases like SN SCP06C1 into their analyses.

SN SCP05D6, SN SCP06G4, and SN SCP06N33 occur behind the cluster and are therefore gravitationally lensed by the cluster. In most surveys, the lensing for SNe Ia hosted by field galaxies averages to nearly zero. In this survey, we target regions with larger than average magnification, so we must make the correction for SNe behind the clusters. To estimate the amount of magnification, we use the virial mass, M₂₀₀, from our weak lensing measurements (Jee et al. 2011). We assume a spherical Navarro–Frenk–White profile (Navarro et al. 1997) with the concentration parameters determined by the M₂₀₀–c relation in Duffy et al. (2008). The lensing magnifications for SN SCP05D6, SN SCP06G4, and SN SCP06N33 are estimated to be 1.021^+0.012_−0.008, 1.015^+0.005_−0.004, and 1.066^+0.017_−0.014, respectively. The magnification of SN SCP06N33 is larger than the others because the cluster is massive and the host galaxy is located at approximately half the distance from cluster center as the others. We apply these corrections when using these SNe Ia in the cosmological fits and propagate the uncertainties accordingly.

SN SCP06C0 is a more interesting case. The host is also behind the lensing cluster, which means that SN SCP06C0 will be lensed as well. Following the methodology used to correct SN SCP06G4, SCP SN05D6, and SCP SN06N33, we find the magnification of SN SCP06C0 to be 1.030^+0.007_−0.005. Upon closer inspection of the host in the stacked ACS data and in more recent WFC3 data, a second, much fainter object, projected only 06 from the center of the host and about 02 from SN SCP06C0, was detected. The object could be a satellite of the host, or it could be an unrelated galaxy along the line of sight. A spectrum of SN SCP06C0 was taken when it was about six days after maximum light; however, the signal-to-noise ratio was insufficient to allow a clear detection of light from the SN given the observing conditions and the relative brightnesses of the hosts and the SN. The classification and redshift therefore rely on correctly assigning the SN to the brighter galaxy. In Section 3.2.3, we model the surface brightness distribution of both galaxies. At the location of the SN, the surface brightness of the large galaxy is four times greater than the surface brightness of the small galaxy. Since [O ii] was not detected in spectra that were taken after the SN had faded from view, neither galaxy is actively forming stars. The relative SN rate is therefore directly related to the relative surface mass density of the two galaxies. Using the surface brightness as a proxy for the surface mass density, we therefore assign SN SCP06C0 to the larger of the two galaxies with ∼80% confidence and include this SN Ia in the cosmological fits.

In general, the search consisted of four z₈₅₀ exposures per epoch and a single i₇₇₅-band exposure. All exposures are geometrically corrected (and multiple exposures are stacked) using MultiDrizzle (Fruchter & Hook 2002; Koekemoer et al. 2002). The photometry is performed on the stacked z₈₅₀-band image and the single i₇₇₅-band drizzled image. In addition to the data that were taken during the search, five clusters, CL1604+4304 (Postman et al. 2005), RDCS0910+54 (Mei et al. 2006), RDCS0848+44 (Postman et al. 2005), RDCS1252−29 (Blakeslee et al. 2003), and XMMU 2235.3−2557 (Jee et al. 2009), were observed with ACS prior to our program (PID9290 and PID9919). We have included all of these data and processed them and the search data in a uniform way. In total, 1006 ACS exposures were processed.

After processing the ACS data with the standard STScI pipeline using the most up-to-date calibration files, we removed the spatially variable background by masking all objects and artifacts, and subtracting a heavily smoothed, median version of what remained. Given the time baseline of our observations, guide stars changed between epochs. We must therefore use objects in the images—typically 20–30 objects per image—to tie the relative astrometry between epochs. The size of the residuals was typically 0.2 pixels, which is larger than one usually expects from point sources with good signal-to-noise ratios. We attribute this to temporal and spatial uncertainties in the distortion correction that are applied to images that are both temporally separated by many months and rotated with respect to one another (Anderson 2007). The absolute astrometry is tied to the Guide Star Catalogue, version 2.3.2,⁴⁵ and has an uncertainty of 03.

We use apertures of 3 pixel radius to measure fluxes and report the i₇₇₅ and z₈₅₀ photometry for each epoch of each SN in the online version of this paper. An example is shown in Table 2. The background noise is found empirically by randomly placing apertures within regions that are free of objects and then measuring the dispersion in the integrated counts within these apertures. We have compared the signal-to-noise ratios obtained for aperture and point-spread function (PSF) photometry and found that the difference between the two is small if we use such small apertures. We use aperture photometry here, as it allows for a more robust correction of CTE and red-halo scattering (described below).

The photometry is corrected for variable CTE and for flux that is outside the aperture (referred to here as the aperture correction (AC)). CTE depends on the position of the source, the level of the background sky, and the flux of the object that is being corrected. It also degrades with time. We follow the formulation of Riess & Mack (2004) and apply the correction factor that corresponds to the 3 pixel radius case and the dates of our observations. On average, we applied a 4.3% correction to the SN Ia flux. When calculating CTE corrections, we include the local background that had been previously subtracted.

We also apply a color-dependent AC on the ACS z₈₅₀-band measurements. In the ACS z₈₅₀-band data, long-wavelength photons scatter off the back side of the CCD, causing a degradation in the PSF. The effect is commonly known as red-halo scattering (Sirianni et al. 1998). The TinyTim⁴⁶ PSF does not account for this effect, and hence it does not reproduce the observed z₈₅₀-band PSF. We studied how the PSF changes with wavelength using standard stars taken with a series of narrowband filter observations from the HST calibration programs PID9020 and PID10720. We measured the red-halo scattering correction factor as a function of aperture radius and wavelength. For a given aperture, we then can treat the correction as a modification of the F850LP throughput and zero point. We discuss the details of this procedure in the Appendix.

We have used the updated STScI ACS Vega zero points for light-curve fitting. The latest zero points are from the STScI web site⁴⁷ which were posted on 2009 May 19. The ACS zero points changed on 2006 July 4 due to a change in the detector temperature. The STScI definition assigns Vega (α Lyr) a magnitude of 0.0 in every filter. However, in the Landolt system, to which the SN Ia photometry in the literature refers, Vega is not zero. To be consistent with the literature about SNe Ia, we introduce this non-zero Vega magnitude correction to our zero points. For both the i₇₇₅ and z₈₅₀ bands, the correction is 0.024 mag (Fukugita et al. 1996). We therefore adjust the STScI zero points by this amount (Astier et al. 2006) and use "non-zero Vega magnitude" system zero points of i₇₇₅ = 25.291(STScI) + 0.024 = 25.315 and z₈₅₀ = 24.347(STScI) + 0.024 = 24.371 for data taken before 2006 July (corresponding to a detector temperature of −77°C), and i₇₇₅ = 25.277(STScI) + 0.024 = 25.301 and z₈₅₀ = 24.323(STScI) + 0.024 = 24.347 for data taken after 2006 July (corresponding to a detector temperature of −81°C).

All NICMOS science frames were processed with the latest CALNICA pipeline (version 4.4.1; Dahlen et al. 2008). This pipeline includes accurate weighting of each readout and optimal removal cosmic rays, as recommended in Fadeyev et al. (2006). CALNICA does not account for the effect of cosmic-ray hits on neighboring pixels (Fadeyev et al. 2006), but these were found to have no appreciable impact for the data reported here. Subsequently, the science frames were corrected for three well-known anomalies: the offset between amplifiers, which affects all NICMOS exposures and is removed using the STSDAS PyRAF task PEDSKY; persistence after the passage of the telescope through the SAA; and persistence after exposing the detectors to the limb of the Earth. Nine exposures are affected by the SAA, which leaves persistent signals from SAA cosmic rays. We applied the STSDAS PyRAF task SAACLEAN (Barker et al. 2007) to remove SAA persistence effects from the images. When a NICMOS observation is immediately preceded by an ACS data dump, the process could delay the NICMOS placement of the filter blank, subjecting the detectors to the bright limb of the Earth, which imprints a persistent pattern on subsequent exposures (Riess & Bergeron 2008). Four exposures were affected in this way and were corrected using the STSDAS software NIC_REM_PERSIST. At this point, the mode of the flux distribution in each image is measured and recorded. These values are used as the sky levels for the count-rate nonlinearity correction.

Even after correcting NICMOS data for these well-studied anomalies, significant large-scale background non-uniformities remain. We developed methods to extract and remove the background structures; these are detailed in Hsiao et al. (2011). Briefly, the models for the background structures are studied and characterized using approximately 600 NICMOS exposures observed through the F110W filter and processed with the procedures described above. Principal component analysis applied to these images revealed that the intensity of the residual corner amplifier glow depends on the exposure sequence. The amount of residual glow decays exponentially and resets every orbit. With exposure times on the order of 1000 s, the exposures can be separated into two glow groups, each with approximately constant intensity. This makes it possible to extract the residual glow algebraically. The structured background is modeled as a combination of a constant component and a component that scales with the sky level and exposure time. The models are derived from the algebraic manipulation of stacked images for each glow group. The resulting constant component of the model is dominated by residual amplifier glow at the corners and residual persistent structure at the center. The model component which scales with sky level and exposure time displays a curious fringe pattern whose origin is unknown. The model components are fit to individual exposures via scale parameters to create the customized background models to be subtracted from the individual exposures. In a final step, the bias offsets apparent in the middle column and middle row are removed. Additional details can be found in Hsiao et al. (2011).

3.2.1. NICMOS Count-rate Nonlinearity

3.2.2. Galaxy Models

3.2.3. Photometry Testing

The photometry of z > 1.2 SNe has been almost exclusively measured from HST images (Knop et al. 2003; Riess et al. 2004, 2007; Amanullah et al. 2010). At these redshifts, the rest-frame B band is redshifted beyond 9000 Å and falls in the near-IR (NIR). NIR observations are typically much easier from space, not only because of the higher spatial resolution of HST compared to ground-based seeing-limited systems, but also because of the lower NIR sky noise in space. However, we show here that AO on large ground-based telescopes can overcome these limitations and allow high-z SNe to be studied from the ground.

We observed the z = 1.315 SN SCP05D6 with the Keck Laser Guide Adaptive Optics (LGS AO) system. These observations were made in the H band (1.6 μm), which corresponds to the rest-frame R band. The diffraction-limited resolution of the Keck AO images was ∼005, or a factor of three better than the spatial resolution of HST at these wavelengths. The high spatial resolution meant a much better separation of the SN from the galaxy core compared to HST. It also allowed greater contrast between the SN and the sky background. We obtain a photometric precision of 0.14 mag at H ∼ 24 mag in a 1 hr exposure with Keck AO, showing the potential of AO in SN Ia cosmology.

Melbourne et al. (2007) report the details of the Keck AO photometry; here we briefly summarize the observations. The Keck LGS AO observing runs for the Center for Adaptive Optics Treasury Survey (Melbourne et al. 2005) coincided with the HST Cluster SN Survey program, and we successfully observed SN SCP05D6 at three epochs, before, near, and after the light-curve maximum.

A 14 mag star 25'' away was chosen to provide AO tip-tilt correction. The ∼11 mag sodium LGS was pointed at the galaxy to provide higher-order AO corrections. The observations were sampled with a 001 pixel scale, allowing the diffraction-limited Keck PSF to be fully resolved. Individual exposures of 60 s were taken with five dithered pattern positions. The sequence was repeated until sufficient depth was reached. Total exposure times varied from 30 minutes to 1 hr per epoch.

We were also fortunate to have a 17.9 mag natural PSF star only 4'' away from SN SCP05D6, so the PSF near the location of the SN was well determined. From the star, we measured an FWHM of 0055 while we had a mean H-band seeing of 04. Using the observed PSF, the host galaxy was modeled by GALFIT (Peng et al. 2002) and subtracted from the image, providing a clean measurement of the SN diffraction-limited core. Relative photometry with respect to the nearby PSF star was performed at each epoch and calibrated by the UKIRT standard star FS6 (the photometry is reported in Table 2). The photometric uncertainty was estimated by simulations of model PSFs embedded into the AO image at the same galactocentric radius as the actual SN.

We fit the SN light curve with the photometric data from HST/ACS F775W, F850LP, HST/NICMOS F110W, and this Keck/AO observation. We found that the Keck AO data were consistent with the HST observations (Figure 2). Although the uncertainty in the AO measurement was larger than that of HST, including it reduces the color uncertainty by more than 10%, and reduces the sensitivity of the fit to the NICMOS zero-point uncertainty by more than a one-third.

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There is evidence that SN Ia luminosity correlates with the mass of the host galaxy, even after the corrections for color and light-curve width have been applied (Kelly et al. 2010; Sullivan et al. 2010; Lampeitl et al. 2010). Since low-redshift SNe Ia are predominantly from surveys that target cataloged galaxies, the host galaxies of SNe Ia in these surveys are, on average, more massive than the host galaxies of distant SNe Ia from untargeted surveys. SNe Ia from low-redshift samples therefore have brighter absolute magnitudes. Left uncorrected, the correlation biases cosmological results (Sullivan et al. 2010).

Sullivan et al. (2010) find that the correlation can be corrected by fitting a step in absolute magnitude at m^threshold_⋆ = 10¹⁰ m_☉. There are two complications with making this correction: most of the SNe in the Union2 compilation do not have host-mass data available in the literature and SN Ia hosts with masses close to the cutoff may scatter across, decreasing the fitted size of the step. To address these problems, we adopt a probabilistic approach to determining the proper host-mass correction to apply to each SN, correcting each SN by the probability that it belongs in the low-host-mass category. (The low-host-mass category was chosen because most of the low-redshift SNe are from high-mass galaxies, so correcting the low-host-mass SNe minimizes the correlation between M_B and the correction coefficient.)

Suppose we have a mass measurement m^obs_⋆ and we would like to estimate the probability that the true mass m^true_⋆ is less than the mass threshold. We begin by noting that

$$\begin{eqnarray} &&P\big(m^{\mathrm{obs}}_{\star }, m^{\mathrm{true}}_{\star }\big) = P\big(m^{\mathrm{obs}}_{\star }| m^{\mathrm{true}}_{\star }\big) P\big(m^{\mathrm{true}}_{\star }\big) \;. \end{eqnarray} \tag{ 1 }$$

We can then integrate this probability over all true host masses less than the threshold:

$$\begin{eqnarray} &&P\big(m^{\mathrm{true}}_{\star }< m^{\mathrm{threshold}}_{\star }| m^{\mathrm{obs}}_{\star }\big)\nonumber\\ &&= \int _{m^{\mathrm{true}}_{\star }= 0}^{m^{\mathrm{threshold}}_{\star }} P\big(m^{\mathrm{obs}}_{\star }| m^{\mathrm{true}}_{\star }\big) P\big(m^{\mathrm{true}}_{\star }\big) \end{eqnarray} \tag{ 2 }$$

up to a normalization constant found by requiring the integral to be unity when integrating over all possible true masses. P(m^true_⋆) is estimated from the observed distribution for each type of survey. The Supernovae Legacy Survey (SNLS; Sullivan et al. 2010) and SDSS (Lampeitl et al. 2010) host masses were assumed to be representative of untargeted surveys, while the mass distribution in Kelly et al. (2010) was assumed to be typical of nearby targeted surveys. As these distributions are approximately log-normal, we use this model for P(m^true_⋆) using the mean and rms from the log of the host masses from these surveys (with the average measurement errors subtracted in quadrature), giving $\log _{10} P(m^{\mathrm{true}}_{\star }) = \mathcal {N}(\mu = 9.88, \sigma ^2 = 0.92^2)$ for untargeted surveys and $\log _{10} P(m^{\mathrm{true}}_{\star }) = \mathcal {N}(10.75, 0.66^2)$ for targeted surveys. When host-mass measurements are available, P(m^obs_⋆|m^true_⋆) is also modeled as a log-normal; when no measurement is available, a flat distribution is used.

For an SN from an untargeted survey with no host-mass measurement (including SNe presented in this paper which are not in a cluster), P(m^true_⋆ < m^threshold_⋆) is the integral of P(m^true_⋆) up to the threshold mass: 0.55. Similarly, nearby SNe from targeted surveys without host galaxy mass measurements are given a P(m^true_⋆ < m^threshold_⋆) of 0.13. (Very similar numbers of 0.50 and 0.09 are derived from the observed distribution, without using the log-normal approximation.) We must make the correction for SNe in clusters, as these are from a targeted survey. We take advantage of the simpler SEDs of early-type galaxies to precisely measure these masses.⁵⁰

The best-fit mass-correction coefficient, δ, is much smaller in magnitude (−0.03) than that found in other studies (≈ − 0.08). This may be due to the small value for δ from the first-year SNLS data, as shown in Table 6. We include the difference in these two δs as a systematic, as discussed in Section 4.5. For this analysis, we assumed that the host-mass correction does not evolve with redshift.

In Amanullah et al. (2010), we use SALT2 (Guy et al. 2007) to fit SN light curves. The SALT2 model fits three parameters to each SN: an overall normalization, x₀, to the time-dependent SED of an SN Ia, the deviation, x₁, from the average light-curve shape, and the deviation, c, from the mean SN Ia B − V color. The three parameters, x₁, c, and integrated B-band flux of the model SALT2 SED at maximum light, m^max_B, are then combined with the host mass to form the distance modulus:

$$\begin{equation} \mu _B = m_B^\mathrm{max} + \alpha \cdot x_1 - \beta \cdot c + \delta \cdot P\big(m^{\mathrm{true}}_{\star }< m^{\mathrm{threshold}}_{\star }\big) - M_B\,,\quad \end{equation} \tag{ 3 }$$

where M_B is the absolute B-band magnitude of an SN Ia with x₁ = 0, c = 0, and P(m^true_⋆ < m^threshold_⋆) = 0. The parameters α, β, δ, and M_B are nuisance parameters that are fitted simultaneously with the cosmological parameters. The SN Ia photometry data and SALT2 light-curve fits are shown in Figure 2. The fitted SALT2 parameters are listed in Table 3 as well as the host galaxy host stellar mass and lensing magnification factor.

To the Union2 SN Ia compilation (Amanullah et al. 2010), we add 16 SNe Ia from this paper that were classified as either secure or probable, including 6 SNe Ia hosted by high-z cluster elliptical galaxies. The four SNe Ia that were classified as possible are not used. We also add 18 SNe Ia from the low-redshift sample of Contreras et al. (2010), 9 of which were not in Union2 (the others had published data from CfA). As in Union2, for all SNe we require

• 1.  
  that the CMB-centric redshift is greater than 0.015;
• 2.  
  that there is at least one point between −15 and 6 rest-frame days from B-band maximum light;
• 3.  
  that there are at least five valid data points;
• 4.  
  that the entire 68% confidence interval for x₁ lies between −5 and +5;
• 5.  
  data from at least two bands with rest-frame central wavelength coverage between 2900 Å and 7000 Å; and
• 6.  
  at least one band redder than rest-frame U band (4000 Å). This cut is new to this analysis, but only affects SN 2002fx, a GOODS SN that is very poorly measured.

In addition to these quality cuts, we removed any SN spectroscopically classified as SN 1991bg-like. These SNe Ia are a distinct subclass that is not modeled well by SALT2. In cases when spectroscopic sub-typing is not possible or not available, we screen for these SNe photometrically by searching for any SNe with red (c > 0.2) and narrow-width (x₁ < −3) light curves. In the current data set, none are cut by this screening. When fit with SALT2, and color-corrected and shape-corrected (as though they were normal SNe Ia), spectroscopically identified members of this class have an average absolute magnitude only 0.2 mag fainter than normal SNe Ia; any contamination from the handful of SNe near this cut will have only a small impact (and one well accounted for by our contamination systematic; see Amanullah et al. 2010).

From the 16 SNe Ia that were classified as either secure or probable (see Table 1), SN SCP06U4 and SN SCP06K18 fail to pass these cuts. SN SCP06K18 lacks good enough light-curve coverage and SN SCP06U4 fails the x₁ cut.⁵¹ This leaves 14 SNe Ia that are used to constrain the cosmology.

Following Amanullah et al. (2010), the best-fit cosmology is determined by minimizing

$$\begin{equation} \chi _{\mathrm{stat}}^2 = \sum _\mathrm{SNe}\frac{\left[\mu _B(\alpha, \beta, \delta, M_{\rm B}) - \mu (z;\Omega _{m},\Omega _w,w)\right]^2}{\sigma _{\mathrm{lc}}^2 + \sigma _{\rm ext}^2 + \sigma _{\mathrm{sample}}^2}. \end{equation} \tag{ 4 }$$

A detailed discussion of the terms in this equation can be found in Amanullah et al. (2010). We only comment on the final term in the denominator, σ²_sample, which is computed by setting the reduced χ² of each sample to unity. This term was referred to as "σ²_systematic" in Kowalski et al. (2008) and Amanullah et al. (2010). We note that σ²_sample includes intrinsic dispersion as well as sample-dependent effects. This term effectively further de-weights samples with poorer-quality data that has sources of error which have not been accounted for. As noted in Amanullah et al. (2010), this may occasionally deweight an otherwise well-measured SN.

Following Conley et al. (2006), Kowalski et al. (2008), and Amanullah et al. (2010), we hide our cosmology results until the full analysis approach is settled. As in previous Union analysis, we carry out an iterative χ² minimization with outlier rejection. Each sample is fit for a flat ΛCDM cosmology independently of the other samples (but with α, β, and δ set to their global values). An M_B is chosen for each sample by minimizing the absolute variance-weighted sum of deviations, minimizing the effects of outliers. We then reject any SN more than 3σ from this fit. All of the SNe Ia in our new sample pass the outlier rejection. As each sample is fit independently with its own Hubble line, systematic errors and the choice of cosmological model are not relevant in this selection.

4.4.1. Diagnostics

A diagnostic plot, which is used to study possible inconsistencies between SN Ia samples, is shown in Figure 3. The median of σ_sample can be used as a measure of the intrinsic dispersion associated with all SNe Ia. The intrinsic dispersion is a reflection of how well our empirical models correct for the observed dispersion in SN luminosities. The median σ_sample for this paper is 0.15 mag and is indicated with the leftmost dashed vertical line in the left panel.

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The variance-weighted rms about the best-fit cosmology gives an indication of the quality of the photometry. A sample with more accurate photometry will have a smaller rms. For SNe Ia from our survey, the rms is 0.19 ± 0.04, which is only slightly larger than that measured for the first-year SN Ia sample from SNLS, and equal to the median of all samples (shown as the rightmost dashed line in Figure 3, left panel).

The two middle panels show the tension between data sets, the first with statistical errors only and the second with statistical and systematic errors (see Section 4.5). Most samples land within 1σ of the mean defined by all samples and about one-third lie outside 1σ, as expected for a normal distribution. No sample exceeds 2σ. The right panel shows the slope of the residuals, which, for larger data sets, can be used to reveal Malmquist-like biases or calibration errors.

The SNe from our sample are 1.5σ brighter than the average sample. While the source of the difference may certainly be a simple statistical fluctuation, part of the difference might be attributable to errors in the filter responses of the ACS filters. (The difference is largely driven by the SNe Ia that have only ACS i₇₇₅ and z₈₅₀ data to constrain their light curves.) Based on photometric observations of spectrophotometric standards, Bohlin (2007) reports possible blueward shifts of 94 Å for the z₈₅₀ filter and 57 Å for the i₇₇₅ filter (with smaller shifts in bluer filters). The red triangle in the sample residual panel shows the effect of applying these shifts. The shifts also affect the GOODS SNe. The green triangle shows the effect of applying the filter shifts to those data. Bohlin (2007) notes that more data to confirm the filter shifts are needed, so we do not apply them in our primary analysis. Instead, we include the uncertainty in the filter curves as a systematic error, as described in Section 4.5.

Part of the difference could also be due to the correction that we apply for the recently discovered correlation between host galaxy mass and the luminosity of SNe Ia after the light-curve width and color corrections have been applied. Many of the hosts in our sample are massive early-type galaxies. In this analysis, the correction we use is smaller than the correction that has been noted by others. We add this difference as a systematic error, as described in Section 4.5.

Figure 4 shows the Hubble Diagram with SNe from the updated Union2 sample and the best-fit ΛCDM model. We add 14 SNe Ia from this current paper. (As discussed above, SN SCP06U4 is likely to be included in future analyses so it is included on the plot with a different symbol.) Ten (11 with SN SCP06U4) are above a redshift of one, significantly increasing the number of well-measured SNe above this redshift.

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In this paper, we follow the systematics analysis we presented in Amanullah et al. (2010). Systematic errors that directly affect SN distance measurements (calibration and galactic extinction, for example) are treated as nuisance parameters to be fit simultaneously with the cosmology. Minimizing over these nuisance parameters gives additional terms to add to the distance modulus covariance matrix:

$$\begin{equation} U_{ij} = \sum _{\epsilon } \frac{d\mu _i(\alpha, \beta)}{d \epsilon }\frac{d\mu _j(\alpha, \beta)}{d \epsilon } \sigma _\epsilon ^2\;, \end{equation} \tag{ 5 }$$

where the sum is over each of these distance systematic errors in the analysis. (Although the distance modulus depends on δ as well as α and β, the derivatives with respect to the zero points do not.) In this analysis, α and β have little interaction with cosmological parameters. When computing cosmological constraints, we therefore freeze the covariance matrix in order to avoid multiple matrix inversions.⁵² Only when the α and β may vary significantly from the global best fit (Table 6) do we update α and β.

Systematic errors that affect sample composition or the color and shape correction coefficients cannot be parameterized SN by SN in this way. These are incorporated by assigning each data set its own constant covariance. This is an adequate treatment, as these systematic errors are subdominant.

There are two systematic errors that were not included in Amanullah et al. (2010) but are included in this analysis for the first time: a systematic error on the host-mass-correction coefficient, δ (which might affect δ at the ∼0.05 level), and uncertainties in the effective wavelengths of the ACS i₇₇₅ and z₈₅₀ filters.

In addition to updating the NICMOS F110W zero point and uncertainty, as described in Section 3.2.1, we revise the uncertainty assigned to the zero point for NICMOS F160W to account for the uncertainty in the count-rate nonlinearity at this wavelength (de Jong et al. 2006). Table 4 gives the assumed zero-point error for each filter.

We note that the nearby SNe from targeted searches are sensitive to δ (relative to the untargeted searches) at the level of (0.55–0.13)Δδ ≈ 0.02 mag, while the covariance-weighted mean of the cluster SNe varies with δ as 0.24Δδ ≈ 0.01 mag. We cannot propagate this systematic on an SN-by-SN basis, as this would be equivalent to fitting for δ, which we already do. Therefore, we include this error by adding a covariance of 0.02² to the nearby targeted SN surveys, a covariance of 0.01² to our new data set, and 0.02 × 0.01 between these data sets.

Including uncertainties in filter effective wavelength is not as straightforward as including zero-point uncertainties. Effective wavelength is only the first-order method of describing a filter. For a simple filter shift, as implemented here, dμ(α, β)/dλ will undergo significant variations as SN spectral features shift in and out of the filter. These are likely to be worse than the actual effect of simply reweighting filter throughput. Although in general these variations will be averaged out with different phases, redshifts, and additional filters, we have modeled a worst case in accounting for this systematic (and even then it only affects the SNe most dependent on z₈₅₀).

Table 5 shows the impact of each type of systematic error on our cosmological constraints, in combination with BAO, CMB, and H₀ data (see Section 5). For the purpose of constructing Table 5, for each systematic error in the table we add the contribution from just that systematic to the statistical-only covariance matrix. The confidence interval for constant w where the χ² is within 1 of the minimum χ² (the edges of this confidence interval are hereafter referred to with the notation Δχ² = 1) is found iteratively; the plus and minus errors for constant w are averaged. The statistical-only constant w error bar is subtracted in quadrature, leaving the effect of each systematic on constant w. We also quote the effect of each systematic error of the Δχ² = 5.99 confidence contour in the (w₀, w_a) plane; as this is two dimensional, we subtract the area (not in quadrature) of the statistical-only contour.

Since the derived cosmology errors vary with the best-fit cosmology, after a given systematic error has been added, the SN magnitudes are shifted so that the best-fit cosmology including that systematic matches the best fit with statistical errors only. This magnitude adjustment (which is the same adjustment we use for blinding ourselves to the best-fit cosmology) consists of repeatedly computing the difference in distance modulus between the best-fit cosmology and fiducial value and adding it to the SNe.

As with the Union2 compilation, calibration systematics represent the largest contribution to the error on constant w. Here, we see that they are also the dominant systematics for (w₀, w_a). As noted by Amanullah et al. (2010), significantly smaller systematic errors are derived by adding each covariance in the covariance matrix rather than adding the cosmological impacts together. This is due to the different redshift dependence of each systematic error, as well as some self-calibration that occurs as described in Amanullah et al. (2010).

Some potential systematic errors can be investigated by dividing the whole data set into subsets. Table 6 shows many of these divisions. All of the numbers are computed including SN systematics; the cosmological constraints are computed including BAO and CMB data. In short, we do not see any evidence of unknown systematic errors, requiring the cosmological impact to be smaller than the current errors.

Table 6. Constraints on Standardization and Cosmological Parameters for Subsets

Subset	Number	MB(h = 0.7)	α	β	δ	Ωm	w
Whole Sample
z ⩾ 0.015	580	−19.321+0.030−0.030	0.121+0.007−0.007	2.47+0.06−0.06	−0.032+0.031−0.031	0.271+0.015−0.014	−1.013+0.068−0.074
Correction Coefficients, Split by Redshift
0.015 ⩽ z ⩽ 0.10	175	−19.328+0.037−0.038	0.118+0.011−0.011	2.57+0.08−0.08	−0.027+0.054−0.054	0.270 (fixed)	−1.000 (fixed)
0.100 ⩽ z ⩽ 0.25	75	−19.371+0.054−0.054	0.146+0.019−0.019	2.56+0.18−0.17	−0.087+0.060−0.060	0.270 (fixed)	−1.000 (fixed)
0.250 ⩽ z ⩽ 0.50	152	−19.317+0.046−0.046	0.116+0.014−0.013	2.46+0.12−0.12	−0.042+0.066−0.066	0.270 (fixed)	−1.000 (fixed)
0.500 ⩽ z ⩽ 1.00	137	−19.307+0.048−0.049	0.124+0.019−0.019	1.46+0.19−0.19	0.023+0.060−0.060	0.270 (fixed)	−1.000 (fixed)
z ⩾ 1.000	25	−19.289+0.217−0.254	−0.019+0.072−0.076	3.48+1.13−0.89	−0.151+0.384−0.446	0.270 (fixed)	−1.000 (fixed)
Effect of δ on w
z ⩾ 0.015	580	−19.340+0.026−0.026	0.123+0.007−0.007	2.47+0.06−0.06	−0.080 (fixed)	0.272+0.015−0.014	−1.004+0.067−0.072
z ⩾ 0.015	580	−19.303+0.031−0.031	0.120+0.007−0.007	2.47+0.06−0.06	$\hphantom{-} 0.000$ (fixed)	0.271+0.015−0.014	−1.013+0.069−0.075
Cosmological Results, Split by Light-curve Color and Shape
c ⩾ 0.05	256	−19.387+0.037−0.038	0.118+0.011−0.011	2.77+0.09−0.09	−0.057+0.052−0.052	0.269+0.015−0.014	−1.028+0.077−0.084
c ⩽ 0.05	321	−19.323+0.030−0.030	0.125+0.011−0.010	1.29+0.32−0.33	−0.057+0.038−0.038	0.275+0.015−0.014	−0.982+0.069−0.075
x1 ⩾ −0.25	311	−19.366+0.041−0.041	0.020+0.026−0.025	2.58+0.10−0.10	−0.004+0.047−0.047	0.269+0.015−0.014	−1.037+0.077−0.085
x1 ⩽ −0.25	269	−19.386+0.044−0.045	0.152+0.021−0.020	2.43+0.08−0.08	−0.087+0.050−0.050	0.267+0.015−0.014	−1.045+0.077−0.084
Correction Coefficients and MB for the Large Data Sets
Hicken et al. (2009b)	94	−19.314+0.055−0.055	0.115+0.015−0.015	2.74+0.11−0.11	−0.053+0.098−0.099	0.270 (fixed)	−1.000 (fixed)
Holtzman et al. (2008)	129	−19.336+0.051−0.051	0.149+0.014−0.013	2.40+0.15−0.14	−0.061+0.050−0.050	0.270 (fixed)	−1.000 (fixed)
Miknaitis et al. (2007)	74	−19.325+0.078−0.080	0.113+0.037−0.035	2.49+0.17−0.16	$\hphantom{-}0.000$ (fixed)	0.270 (fixed)	−1.000 (fixed)
Astier et al. (2006)	71	−19.292+0.047−0.048	0.145+0.019−0.018	1.70+0.18−0.18	−0.023+0.040−0.040	0.270 (fixed)	−1.000 (fixed)
z > 0.9, Split by Galaxy Host
Early type z > 0.9	13	−19.388+0.139−0.186	0.112+0.139−0.151	3.16+1.84−1.26	$\hphantom{-} 0.000$ (fixed)	0.270 (fixed)	−1.000 (fixed)
Late type z > 0.9	15	−19.141+0.067−0.067	0.094+0.049−0.041	0.49+0.85−0.69	$\hphantom{-} 0.000$ (fixed)	0.270 (fixed)	−1.000 (fixed)

Notes. M_B is the B-band corrected absolute magnitude; α, β, and δ are the light-curve shape, color, and host-mass-correction coefficients, respectively. The outlier rejection is redone each time, so the totals may not add up to the whole sample. The constraints are computed including BAO, CMB, and H₀ constraints and supernova systematic errors.

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The first subsets are subsets in redshift. These can be used to study possible evolution of correction coefficients for shape, color, and host mass. The redshift range 0.5–1 seems to show β and δ smaller in magnitude, but the revised SNLS sample (Guy et al. 2010) which uses a newer version of the calibration and light-curve fitting (as well as many more SNe) shows no signs of this. As we have already budgeted these systematic uncertainties, these updates will be within our error bars.

The next rows show the effect of changing δ from 0 to −0.08 (the size of the correction in Sullivan et al. 2010). Because a large error on δ is already included in the systematic error covariance matrix, this has less than a 0.01 effect on w, about 10 times smaller than it would have if we did not include this systematic.

Next, we consider systematics caused by potentially different populations of SNe. We perform a cut on the best-fit true x₁ or c of each SN (see Amanullah et al. 2010 for details). The cosmology in each case is compatible with the cosmology derived from the whole sample.

We now look at each of the four largest data sets for evidence of tension. The only tension found is in the first-year SNLS sample (Astier et al. 2006). Here, β and δ are both at odds with the whole sample, but as noted above, we do not believe this is a cause for concern.

The final two rows show the high-redshift sample split by host type; this is discussed in Section 6.2.

In the ΛCDM model, the equation-of-state parameter is exactly −1 and does not vary with time. In a flat universe, SNe Ia alone constrain the dark-energy density, Ω_Λ, to be Ω_Λ = 0.705^+0.040_−0.043. In Figure 5, we show the confidence intervals on Ω_m and Ω_Λ from SNe, CMB, and BAO. Both the individual constraints and the combined constraint are shown (the BAO constraints are computed with an Ω_mh² prior from the CMB). The SN constraint is almost orthogonal to that of the CMB. Adding the constraints from CMB, BAO, and H₀ reduces the uncertainty. Under the assumption of a flat universe, the four probes yield

$$\begin{equation*} \Omega _{\Lambda }={0.729}^{+0.014}_{-0.014}{\rm \hspace{8.53581pt} (\Lambda CDM : SN+CMB+BAO+{\it H_{0}})}. \end{equation*}$$

In this ΛCDM model, the expansion of the universe switched from deceleration to acceleration at z = 0.752 ± 0.041, which corresponds to a look-back time of 6.62 ± 0.22 Gyr, about the half of the age of the universe. Equality between the energy density of dark energy and matter occurred later, at z = 0.391 ± 0.033 or a look-back time of 4.21 ± 0.27 Gyr.

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If we remove the flatness prior (labeled as oΛCDM in Table 7), the best-fit Ω_m and Ω_Λ change by a fraction of their errors with Ω_k = 0.002^+0.005_−0.005.

In wCDM models, w is constant but is allowed to be different from −1. While few dark-energy theories give w ≠ −1 and yet constant (Copeland et al. 2006), constraints on the constant w model are still useful. The wCDM model contains fewer parameters than the dynamical dark-energy models considered in the following section, yet a value different from w = −1 would still provide insights for alternative theories for dark energy.

In a flat universe ($\mbox{$\Omega _k$}=0$), SNe Ia alone give w = −1.001^+0.348_−0.398. Adding the constraints from the other three probes tightens the constraint on w considerably, as the constraints from SNe Ia in the Ω_m– w parameter plane are almost orthogonal to those provided by BAO and the CMB (Figure 6).

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In principle, a constraint on H₀ helps to break the degeneracy between Ω_m and h for CMB, which measures Ω_mh² (Spergel et al. 2003). However, in this case adding SN data helps more, as narrowing the degeneracy between Ω_m and w allows the CMB itself to constrain H₀. By combining all four probes, we find w = −1.013^+0.068_−0.073. As seen in Table 7, neither BAO nor H₀ currently make much of a difference in the error bars for this model.

Inflation models generally predict that the curvature of the universe, Ω_k, is ∼10⁻⁵ (Guth 1981; Liddle & Lyth 2000). In curved universes, SNe Ia play the critical role in constraining w, while CMB+BAO constrain Ω_k and Ω_m. By combining all four probes, we find Ω_k = 0.002^+0.007_−0.007 and w = −1.003^+0.091_−0.095. Even with the additional freedom for non-zero curvature, a flat universe is supported from observations. Among many cosmological parameters, the curvature of the universe is the most well-determined parameter.

We note that CMB alone does not place a tight constraint on curvature,⁵³$\mbox{$\Omega _k$}=-0.102^{+0.085}_{-0.097}$ (Komatsu et al. 2011). In order to break the degeneracy between Ω_m,  Ω_k, and H₀ that exists in the CMB constraints, we need to add constraints either from BAO or H₀; this reduces the curvature uncertainty by a factor of 10. However, the combination of these three probes does not place a tight constraint on the equation-of-state parameter w. SNe improve the constraint on w from CMB+BAO+H₀ by more than a factor of three.

We next examine models in which dark energy changes with time. For a wide range of dark-energy models, it can be shown (Linder 2003) that, to good approximation, the dark-energy equation of state can be parameterized by

$$\begin{equation} w(a)=w_{0}+w_{a}(1-a), \end{equation} \tag{ 7 }$$

where a = 1/(1 + z) is a scale factor. The ΛCDM model is recovered when w₀ = −1 and w_a = 0. The constraints on w₀ and w_a are shown in Figure 7 and Table 7.

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The Dark Energy Task Force (Albrecht et al. 2006) proposed a figure of merit (FoM) for cosmological measurements equal to the inverse of the area of the 95% confidence contour in the w₀ − w_a plane. When we make this measurement, using the Δχ² = 5.99 contour, we find an FoM of 1.84 (statistical-only) and 1.04 (including systematics). Frequently, the FoM is also defined in terms of the 1σ errors (Δχ² = 1); this FoM is 39.3 (statistical-only) and 22.6 (including systematics). Surprisingly, even with w_a floating, we still find an interesting constraint on Ω_k of ∼0.02.

We next consider a model in which the dark-energy equation-of-state parameter is constant inside fixed redshift bins. This model has more parameters (and thus more freedom) than w₀ − w_a. The results are shown in Figure 8 and Table 7. We adopt the redshift bins used in Amanullah et al. (2010) so that a direct comparison can be made.

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In the left panel with broad bins, we show a reasonably good measurement of the equation-of-state parameter from redshift 0 to 0.5. From redshift 0.5 to 1, there is no real constraint. For example, any scalar field model (|w| < 1) is reasonably compatible with the data. Above redshift 1, the constraints are weaker. w ≳ 0 is ruled out, as this violates early matter domination.

We separate the SN and early universe constraints by defining a bin at redshift 1.6, as shown in the middle panel. This shifts the confidence interval for w(1.0 < z < 1.6) toward higher w. Eliminating this division, and instead adding more bins up to redshift 0.5 (right panel), gives three constraints of moderate quality with a possible crossing of w = −1. No matter the binning, we will need more data extending above redshift 1 to investigate the dark-energy equation-of-state parameter where the uncertainty is still very large.

To examine constraints on the existence of dark energy at different epochs, we study ρ(z), which is the density of the dark energy and is allowed to have different values in fixed redshift bins. Within each bin, ρ is constant. (Note that the discontinuities in ρ(z) at the bin boundaries introduce discontinuities in H(z).) We choose the same binning as above, but note that binned ρ and binned w models give different expansion histories. Our results are shown in Figure 9 and Table 8.

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Table 8. Constraints on Redshift Binned Equation-of-state w and Density ρ (Normalized by the Current Critical Density)

			z < 0.5	0.5 < z < 1.0	1.0 < z < 1.6	z > 1.6 a
w(z)	Stat Only:		−1.013+0.067−0.069	−0.78+0.58−0.68	−3.7+2.2−4.4	<0.18
	w/Sys:		−1.006+0.110−0.113	−0.69+0.80−0.98	−3.9+3.2−8.2	<0.24
ρDE(z)/ρc0	Stat Only:		0.732+0.013−0.014	0.85+0.18−0.17	$\hphantom{-} 0.23^{+1.29}_{-0.79}$	0.9+1.9−1.5
	w/Sys:		0.731+0.014−0.015	0.88+0.24−0.21	0.33+1.90−1.00	0.7+2.4−1.8

Notes. Constraints on binned ρ_DE(z)/ρ_c0 and w(z). This redshift binning corresponds to the middle panel of Figure 8 and the right panel of Figure 9. The constraints are computed including SNe, BAO, H₀, and CMB data. ^aWe note that the weak constraints in these bins come mostly from the CMB (which tells us that the early universe was matter dominated) and are only indirectly constrained by supernovae.

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Although there is no real constraint on the equation-of-state parameter at redshift 0.5–1, dark energy is seen at high significance in both panels. There is weak evidence for the existence of dark energy above redshift 1, as can be seen in the left panel. However, if we again separate the SN data and early universe constraints (right panel) we see neither probe has any constraint on the existence of dark energy above redshift 1.

Beyond z = 1, we add 10 new well-measured SNe Ia to the Hubble diagram. The variance-weighted rms scatter of the new sample is 0.20 ± 0.05 mag. As a comparison, the 15 z > 1 SNe Ia from the GOODS survey that pass our Union2 selection cuts have a variance-weighted rms scatter of 0.25 ± 0.05  mag. The new sample almost doubles the weight of HST-discovered SNe Ia beyond z = 1. The increase provides improvements on the most difficult-to-measure parameters, those that describe the time-varying properties of dark energy: ρ(z) and w(z) at the higher redshifts. In particular, the SNe from this search improve the constraint on ρ(z) at redshifts 1.0 < z < 1.6 by 28% (statistical errors only) and 18% (including SN systematics) after adding the constraints from the CMB, BAO, and H₀ (using the binning illustrated in the right panel of Figure 9). (It is more difficult to compare binned w results, as the constraints are much less Gaussian and more sensitive to the location of the best fit.)

The new sample is also obtained with greater observing efficiency with HST. Considering the number of z > 1 SNe Ia that satisfy the Union2 selection cuts, the yield of SNe Ia increases from a rate of one SN Ia per 43 HST orbits in the GOODS survey to one SN Ia per 22 HST orbits in this survey.

SNe Ia are well standardized with a small dispersion in magnitudes across the whole class. Any clues to heterogeneous characteristics therefore offer exciting possibilities to further improve standardization, enhancing the use of SNe as a cosmological probe. There is now evidence from studies of large samples of SNe Ia at both low and intermediate redshifts (0 < z ≲ 0.8) that SN Ia properties are related to the properties of the host. The clearest of these is the relation between light-curve width and the specific star formation rate. SNe Ia in passive galaxies tend to have narrower light curves than SNe Ia that are in galaxies that are actively forming stars.

More than two-thirds of our new SNe Ia beyond z = 0.9 are hosted by early-type galaxies (Meyers et al. 2012). In field surveys, such as the GOODS survey, this ratio is inverted. By combining SNe Ia from our HST Cluster SN Survey and GOODS, together with our z > 0.9 SNe Ia in Amanullah et al. (2010), which have HST images of the host, we can create a sample of SNe Ia that has roughly equal numbers when split according to host type. When split this way, we find that z > 1 SNe Ia in early galaxies rise and fall more quickly than SNe Ia in later host types, thus extending the redshift interval over which the effect is now detected. Finding that low- and high-redshift SNe Ia follow similar trends gives us confidence that we can use very distant events to constrain cosmological parameters. This finding is reported in more detail in Meyers et al. (2012).

There is also evidence from SNe Ia at low and intermediate redshifts for other correlations with host type. Sullivan et al. (2010) find that both β and the rms scatter about the Hubble diagram are smallest for SNe Ia in passive galaxies. These trends suggest that dust plays a greater role in reddening and dimming SNe Ia in late-type galaxies. We examined our z > 0.9 sample for evidence of similar correlations using our host classification from Tables 3 and 4 of Meyers et al. (2012).

After correcting SN Ia luminosities for light-curve shape, SN Ia color and host galaxy mass (with the global values of these correction coefficients), we measure a sample dispersion of 0.14^+0.11_−0.08 mag for SNe Ia in early-type galaxies and 0.14^+0.06_−0.05 mag for SNe Ia in late-type galaxies. In terms of the rms, we find 0.23 ± 0.05 mag and 0.26 ± 0.05 mag for early- and late-type samples, respectively. The uncertainties are currently too large to distinguish between the two samples. Similarly for β, the errors are larger than the difference between the two samples, as seen in Table 6. Clearly, higher quality data of a larger number of z > 0.9 SNe Ia in both early- and late-type galaxies are required before the trends that are seen at low redshift can be detected in high-redshift samples.

We also examined the error-weighted difference in the brightness of SNe Ia in the two samples after correcting for light-curve shape and color, but without correcting for the host-mass–luminosity relation (setting δ = 0) and find that SNe Ia in early-type galaxies are 0.18  ±  0.09 mag brighter. Since early-type galaxies are typically more massive than late-type galaxies, this 2σ difference, if confirmed with larger statistics, may be related to host galaxy mass.

Due to the much improved sensitivity of the WFC3 IR detector, it will be feasible to measure z > 1 SNe with much better precision. The color measurement errors (∼0.03 in B − V) can be made comparable to the color measurement errors in the SDSS SN survey (Smith et al. 2002; Holtzman et al. 2008). Assuming that the intrinsic dispersion of SN Ia luminosities does not change with redshift, the variance-weighted rms of the WFC3 sample should be similar to that measured for the SDSS, i.e., ∼0.14 mag. A well-observed SN Ia with WFC3 should have a statistical weight of two to three SNe Ia from the Cluster and GOODS surveys.

With a sufficient number of well-measured z > 1 SNe Ia with WFC3, it should be possible to search for the correlations between the properties of SNe Ia and their hosts that are seen at lower redshifts. As discussed above, current samples at z > 1 are too small to detect most of these differences. With the improved WFC3 photometry, only 40 SNe Ia, split evenly between early- and late-type hosts, would be needed to constrain a difference in β to an uncertainty of 0.4, which is about half the difference found for lower redshift SNe Ia (Sullivan et al. 2010). These samples would be just enough to see evidence of the lower rms for passive hosts seen by Sullivan et al. (2010).

Current WFC3 SN Ia surveys target empty fields, which means that there will be few SNe Ia in passive host galaxies. A WFC3 SN Ia survey that spends part of its time targeting z ≳ 1 clusters would ensure a better balance between host types while increasing the overall yield.

In order to investigate the FoM constraints possible with WFC3, we simulate a sample of 40 SNe at redshift 1.2 and add this sample into the current compilation. As there is a hard wall at w₀ + w_a = 0 when including BAO and CMB data, we simply fix Ω_m, rather than including BAO and CMB data (the alternative would be to adjust the SN magnitudes to a cosmology model far away from the wall). When adding these SNe, the statistical FoM improves by 39%. By the same metric, the current cluster sample improves the FoM by 10%.

As has been stressed by several authors, systematic errors are now larger than statistical errors. To fully utilize the potential of current and future SN Ia surveys to constrain cosmology, it will be necessary to reduce these errors significantly.

The largest current source of systematic uncertainty is calibration. Calibration uncertainties can be split into uncertainties related to the primary standard and uncertainties in instrumental zero points and bandpasses. In principle, all of these uncertainties can be reduced by establishing a network of well-calibrated standard stars and monitoring telescope system throughputs (Regnault et al. 2009). The SDSS demonstrated that a 1% relative photometric calibration is possible with the current standard star network and system throughput monitoring (Doi et al. 2010).

The ongoing Nearby Supernova Factory (SNf) project (Aldering et al. 2002) is aiming to provide the network of standard stars. SNf will also address the systematic uncertainty due to host-mass correction since the range of host masses would become comparable to that of high redshift for the first time. Additionally, the comprehensive SN Ia spectral time series from the SNf will allow one to tackle systematic uncertainties related to modeling of the light curves.

In the future, recently approved experiments such as Absolute Color Calibration Experiment for Standard Stars (Kaiser et al. 2010) and the proposed National Institute for Standards and Technology STARS project (McGraw et al. 2010; Zimmer et al. 2010) are aiming to achieve sub-percent absolute flux calibration for the network of stars in the wavelength range of visible to NIR. With this network of stars and with new techniques for monitoring throughput of the telescopes (Stubbs et al. 2007), we will be able to cross-calibrate systems and reduce the systematic errors below the statistical errors.

The magnitude of an object in the Vega magnitude system is defined (Fukugita et al. 1995; Sirianni et al. 2005) as

$$\begin{equation} m=-2.5 \log _{10} \left(\frac{\int \lambda R f_{\lambda } d\lambda }{\int \lambda R f_{\lambda {\rm Vega}} d\lambda } \right), \end{equation} \tag{ A2 }$$

where λ is wavelength, R is the system response, f_λ is the SED of the object (e.g., SN Ia), and f_λVega is the SED of Vega. We define the Vega magnitude zero point (Zpt) as

$$\begin{equation} {\rm Zpt_{{\rm Vega}}}= 2.5 \log _{10} \left(\int \lambda R f_{\lambda {\rm Vega}} d\lambda \right). \end{equation} \tag{ A3 }$$

Equation (A2) can then be rewritten as

$$\begin{equation} m=-2.5 \log _{10} \left({\int \lambda R f_{\lambda } d\lambda } \right) + {\rm Zpt_{{\rm Vega}}}. \end{equation} \tag{ A4 }$$

The observed magnitude, m_obs, within a given aperture is

$$\begin{equation} m_{\rm obs}=-2.5 \log _{10} \left({\int \lambda \, R \, \mbox{EE}(\lambda) \, f_{\lambda } d\lambda } \right) + {\rm Zpt_{{\rm Vega}}}, \end{equation} \tag{ A5 }$$

where EE(λ) is the wavelength-dependent EE, which can be derived from Equation (A1).

We define the SED-dependent AC (Δm_corr) as

$$\begin{equation} \Delta m_{\rm corr}= -2.5 \log _{10} \left(\frac{\int \lambda \, R \, \mbox{EE}(\lambda) \, f_{\lambda } d\lambda }{\int \lambda R f_{\lambda } d\lambda } \right). \end{equation} \tag{ A6 }$$

In practice, we do not know f_λ in advance, so we cannot compute the AC directly. Instead, we derive it iteratively using the SED derived from fitting the SN Ia light curve with SALT2 as input to the next iteration. With this method, we use the STScI filter response function and zero points.

We can rewrite Equation (A5) as

$$\begin{eqnarray} m_{\rm obs}&=&{-}2.5 \log _{10} \left(\frac{\int \lambda \, R \, \mbox{EE}(\lambda) f_{\lambda } d\lambda }{\int \lambda \, R \, \mbox{EE}(\lambda) \, f_{\lambda {\rm Vega}} d\lambda } \right) + {\rm Zpt_{{\rm Vega}}}\nonumber\\ && {-}2.5 \log _{10} \left({\int \lambda \, R \, \mbox{EE}(\lambda) \, f_{\lambda {\rm Vega}} d\lambda } \right). \end{eqnarray} \tag{ A7 }$$

Effectively, the last term serves as a zero-point offset for a given aperture radius.

We rewrite Equation (A7) as

$$\begin{eqnarray} m_{\rm obs}&=&-2.5 \log _{10} \left(\frac{\int \lambda \, R \, \mbox{EE}(\lambda) \, f_{\lambda } d\lambda }{\int \lambda \, R \, \mbox{EE}(\lambda) \, f_{\lambda {\rm Vega}} d\lambda } \right)\nonumber\\ && + {\rm Zpt_{{\rm Vega}}} - \Delta {\rm Zpt_{{\rm Vega}}}; \end{eqnarray} \tag{ A8 }$$

$\Delta {\rm Zpt_{{\rm Vega}}}=0.438$ for a 3 pixel radius aperture and the z₈₅₀ filter. We interpret Equation (A8) as a magnitude measurement that uses a modified filter response, R EE(λ), and an adjusted zero point, ${\rm Zpt_{{\rm Vega}}} - \Delta {\rm Zpt_{{\rm Vega}}}$. We then run SALT2 using the counts in a fixed aperture, the modified filter response, and the adjusted zero point.