SPECTRA AND HUBBLE SPACE TELESCOPE LIGHT CURVES OF SIX TYPE Ia SUPERNOVAE AT 0.511 < z < 1.12 AND THE UNION2 COMPILATION^*

Type Ia supernovae (SNe Ia) are an excellent tool for probing the expansion history of the universe. About a decade ago, combined observations of nearby and distant SNe Ia led to the discovery of the accelerating universe (Perlmutter et al. 1998, 1999; Garnavich et al. 1998; Schmidt et al. 1998; Riess et al. 1998).

Following these pioneering efforts, the combined work of several different teams during the past decade has provided an impressive increase in both the total number of SNe Ia and the quality of the individual measurements. At the high redshift end (z ≳ 1), the Hubble Space Telescope (HST) has played a key role. It has successfully been used for high-precision optical and infrared follow-up of SNe discovered from the ground (Knop et al. 2003; Tonry et al. 2003; Barris et al. 2004; Nobili et al. 2009), and, by using the Advanced Camera for Surveys (ACS), to carry out both search and follow-up from space (Riess et al. 2004, 2007; Kuznetsova et al. 2008; Dawson et al. 2009). At the same time, several large-scale ground-based projects have been populating the Hubble diagram at lower redshifts. The Katzman Automatic Imaging Telescope (Filippenko et al. 2001), the Nearby Supernova Factory (Copin et al. 2006), the Center for Astrophysics SN group (Hicken et al. 2009b), the Carnegie Supernova Project (Hamuy et al. 2006; Folatelli et al. 2010), and the Palomar Transient Factory (Law et al. 2009) are conducting searches and/or follow-up for SNe at low redshifts (z < 0.1). The SN Legacy Survey (SNLS; Astier et al. 2006) and ESSENCE (Miknaitis et al. 2007; Wood-Vasey et al. 2007) are building SN samples over the redshift interval 0.3 < z < 1.0, and the Sloan Digital Sky Survey (SDSS) SN Survey (Holtzman et al. 2008; Kessler et al. 2009) is building an SN sample over the redshift interval 0.1 < z < 0.3, a redshift interval that has been relatively neglected in the past. These projects have discovered ∼700 well-measured SNe. The number of well-measured SNe beyond z ∼ 1 is approximately 20 and is comparatively small.

Kowalski et al. (2008, hereafter K08) provided a framework to analyze these and future data sets in a homogeneous manner and created a compilation, called the "Union" SNe Ia compilation, of what was then the world's SN data sets. Recently, Hicken et al. (2009a, hereafter H09) added a significant number of nearby SNe to a subset of the "Union" set to create a new compilation, and similarly the SDSS SN survey (Kessler et al. 2009, hereafter KS09) carried out an analysis of a compilation including their large intermediate-z data set (Holtzman et al. 2008). When combined with baryon acoustic oscillations (Eisenstein et al. 2005), the H09 compilation leads to an estimate of the equation-of-state (EOS) parameter that is consistent with a cosmological constant, while KS09 get significantly different results depending on which light-curve fitter they use.

An important role for SNe Ia beyond z ∼ 1, in addition to constraining the time evolution of w, is their power to constrain astrophysical effects that would systematically bias cosmological fits. Most evolutionary effects are expected to monotonically change with redshift and are not expected to mimic dark energy over the entire redshift interval over which SNe Ia can be observed. Evolutionary effects might also have additional detectable consequences, such as a shift in the average color of SNe Ia or a change in the intrinsic dispersion about the best-fit cosmology.

Interestingly, the most distant SNe Ia in the Union compilation (defined here as SNe Ia with z ≳ 1.1) are almost all redder than the average color of SNe Ia over the redshift interval 0.3 < z < 1.1. The result is unexpected as bluer SNe Ia at lower redshifts are also brighter (Tripp 1998; Guy et al. 2005) and should therefore be easier to detect at higher redshifts. Possible explanations for the redder than average colors of very distant SNe Ia range from the technical, such as an incomplete understanding of the calibration of the instruments used for obtaining the high redshift data, to the more astrophysically interesting, such as a real lack of bluer SNe at high redshifts.

The underlying assumption in using SNe Ia in cosmology is that the luminosity of both near and distant events can be standardized with the same luminosity versus color and luminosity versus light-curve shape relationships. While drifts in SN Ia populations are expected from a combination of the preferential discovery of brighter SNe Ia and changes in the mix of galaxy types with redshift (Howell et al. 2007)—effects that will affect different surveys by differing amounts—a lack of evolution in these relationships with redshift has not been convincingly demonstrated given the precision of current data sets. This assumption needs to be continuously examined as larger and more precise SN Ia data sets become available.

In this paper, we report on work to increase the number of well-measured distant SNe Ia by presenting SNe Ia that were discovered in ground-based searches during 2001 and then followed with Wide Field Planetary Camera 2 (WFPC2) on HST. Two of the new SNe Ia are at z ∼ 1.1 and have high-quality ground-based infrared observations that were obtained with ISAAC on the Very Large Telescope (VLT) and NIRI on Gemini. This paper is the first paper in a series of papers that will provide a comparable sample of z > 1 SNe Ia to the SNe now available in the literature. The SNe Ia in this series of papers were discovered in 2001 (this paper), 2002 (N. Suzuki et al. 2010, in preparation) and from 2005 to 2006 during the Supernova Cosmology Project (SCP) cluster survey (Dawson et al. 2009).

This paper is organized as follows. In Section 2, we describe the SN search and the spectroscopic confirmation, while Sections 3, 4, and 5 contain a description of the follow-up imaging and the SN photometry. The light-curve fitting is described in Section 6. In Section 7, we update the K08 analysis both by adding new data and by improving the analysis chain. The paper ends with a discussion and a summary.

The SNe were discovered during two separate high-redshift SN search campaigns that were conducted during the Northern Spring of 2001. The first campaign (hereafter Spring 2001) consisted of searches with the CFH12k (Cuillandre et al. 2000) camera on the 3.4 m Canada–France–Hawaii Telescope (CFHT) and the MOSAIC II (Muller et al. 1998) camera on the 4.0 m Cerro-Tololo Inter-American Observatory (CTIO) Blanco telescope. The second campaign (hereafter Subaru 2001) was done with SuprimeCam (Miyazaki et al. 2002) on the 8.2 m Subaru telescope. All searches were "classical" searches (Perlmutter et al. 1995, 1997), i.e., the survey region was observed twice with a delay of approximately 1 month between the two observations, and the two epochs were then analyzed to find transients. Details of the search campaigns can be found in Lidman et al. (2005) and Morokuma et al. (2010).

The Spring 2001 data were processed to find transient objects and the most promising candidates were given an internal SCP name and a priority. The priority was based on a number of factors: the significance of the detection, the relative increase in the brightness, the distance from the center of the apparent host, the brightness of the candidate, and the quality of the subtraction. Note however that these factors were not applied independently of each other, but the priorities were rather based on a combination of factors. For example, candidates on core were only avoided if they had a small relative brightness increase over the span of 1 month. The AGN structure function shows that AGNs rarely have strong changes over 1 month.

The candidates discovered in the Spring 2001 campaign were distributed to teams working at Keck and Paranal observatories for spectroscopic confirmation. The distribution was based on the likely redshifts. Candidates that were likely to be SNe Ia at z ≳ 0.7 were sent to Keck, while candidates that were thought to be nearer were sent to the VLT. In later years, when FORS2 was upgraded with a CCD with increased red sensitivity, the most distant candidates would also be sent to the VLT. All Subaru 2001 candidates were sent to FOCAS (Faint Object Camera And Spectrograph) on Subaru for spectroscopic confirmation (Morokuma et al. 2010).

In total, four instruments (FORS1 on the ESO-VLT, ESI and LRIS on Keck, and FOCAS on Subaru) were used to determine redshifts and to spectroscopically confirm the SN type. The dates of the spectroscopic runs are listed in Table 1 and the observations of individual candidates are listed in Table 2.

Table 2. Summary of the Spectroscopic Observations

IAU Name	Search	α(2000)	δ(2000)	E(B − V)	MJD	Instr./Tel.	Exp.
					(days)		(s)
SN 2001cw	Subaru 01	15h23m063	+29°39'32''	0.024	52056.6	FOCAS/Subarua	4200
SN 2001gn	Spring 01	14h01m599	+05°05'00''	0.028	52023.1	ESI/Keck IIb	9700
SN 2001go	Spring 01	14h02m009	+05°00'59''	0.027	52021.3	FORS1/Antua	2400
SN 2001gq	Spring 01	14h01m514	+04°53'12''	0.027	52020.3	LRIS/Kecka	3600
SN 2001gy	Spring 01	13h57m045	+04°31'00''	0.030	52021.3	FORS1/Antua	2400
SN 2001hb	Spring 01	13h57m119	+04°20'27''	0.032	52024.3	ESI/Keck IIb	3600

Notes. For completeness, the observations that are reported in Lidman et al. (2005) and Morokuma et al. (2010) are also included. The Galactic extinction, E(B − V), from Schlegel et al. (1998), is presented together with the coordinates for each SN. ^aLong slit. ^bEchellette.

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Only those candidates that were confirmed as SNe Ia were then scheduled for follow-up observations from the ground and with HST. In total, six SNe were sent for HST follow-up: one from the Subaru 2001 campaign and five from the Spring 2001 campaign. The SNe are listed in Tables 2 and 3. Finding charts are provided in Figure 1.

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Here, we describe the analysis of data that were taken with ESI and LRIS. The analyses of the spectra taken with FORS1 and FOCAS (SN 2001cw, SN 2001go, and SN 2001gy) are described in Lidman et al. (2005) and Morokuma et al. (2010), respectively. The spectra of these SN are shown in these papers and will not be repeated here.

2.1. ESI

2.2. LRIS

2.3. Spectral Fitting and Supernova Typing

A total of nine different instruments, listed in Table 4, were used for the photometric follow-up of the SNe described in this work.

All observations are listed in Table B1. Here, the modified Julian Date (MJD) is the weighted average of all images taken during a given night except for the NIR data where data taken over several nights were combined. We do not report the MJD for combined reference images that were taken over several months.

3.1. Ground-based Optical Observations and Reductions

We obtained ground-based optical follow-up data of the SNe through different combinations of passbands, shown in Figure 2, similar to Bessel R and I (Bessell 1990), and SDSS i (Fukugita et al. 1996). Here we used FORS1 (Appenzeller et al. 1998) at VLT and SuSI2 (D'Odorico et al. 1998) at NTT in addition to the search instruments. The SuSI2, MOSAIC II, CFH12k, and SuprimeCam data were obtained in the visitor mode, while the FORS1 observations were carried out in the service mode.

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All optical ground data were reduced (Raux 2003) in a standard manner including bias subtraction, flat fielding, and fringe map subtraction using the IRAF²⁹ software.

3.2. Ground-based IR Observations and Reduction

3.3. HST Observations and Reduction

The photometry technique applied to the optical ground-based data is the same as the one applied in Amanullah et al. (2008), except for the SuprimeCam i-band data. This is also very similar to the method (Fabbro 2001) used in Astier et al. (2006) and is briefly summarized here.

• 1.  
  Each exposure of a given SN in a given passband was aligned to the best seeing photometric reference image.
• 2.  
  In order to properly compare images of different image quality we fitted convolution kernels, K_i, modeled by a linear decomposition of Gaussian and polynomial basis functions (Alard & Lupton 1998; Alard 2000), between the photometric reference and each of the remaining images, i, for the given passband. The kernels were fitted by using image patches centered on fiducial objects across the field.
• 3.  
  The background sky level for each image, i, is not expected to have any spatial variation for a small patch, I_i(x, y), centered on the SN with a radius of the worst seeing FWHM. We assume that the patch can be modeled by a point-spread function (PSF) at the location of the SN, a model of the host galaxy and a constant offset for the sky,

  $$\begin{eqnarray*} I_i(x,y) &=& f_i\cdot \left[K_i\otimes \mathrm{PSF}\right](x - x_0, y - y_0)\nonumber\\ &&+ \left[K_i\otimes G\right](x,y)\, + S_i\,. \end{eqnarray*}$$

  For a time series of such patches we simultaneously fit the SN position, (x₀, y₀), and brightnesses, f_i, the host model, G(x, y), and background sky levels, S_i. We use a non-analytic host model with one parameter per pixel. This means that the model will be degenerated with the SN and the sky background. We break these degeneracies by fixing the SN flux to zero for all reference images and fix the sky level to zero in one of the images. Figure 3 shows an example of image patches, galaxy model residuals, and resulting residuals when the full model has been subtracted, for increasing epochs.

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The SN light curves were obtained in this manner for one filter and instrument at a time where SN-free reference images were available. In the cases where SN-free references did not exist for a given telescope, reference images obtained with other telescopes were used instead.

For the IR image of 2001hb, we performed aperture photometry directly on the images, assuming the SN to be hostless for the purpose of J-band photometry, since no host could be detected at the limit of the ACS references (see below). For the IR image of 2001gn, we carried out steps (1)–(2) above, and then subtracted the reference image (Figure 4) and did aperture photometry on the resulting image. In both cases, the SN positions from the HST images were used for centroiding the aperture and the diameter was chosen to maximize the signal-to-noise ratio. The fluxes are corrected to larger apertures by analyzing bright stars in the same image.

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The i-band data for 2001cw were analyzed together with a larger sample of SNe discovered at Subaru. The details of this analysis are given in N. Yasuda et al. (2010, in preparation).

4.1. Calibration of the Ground-based Data

Nightly observations of standard stars were not available for all optical instruments and filters. Instead the recipe from Amanullah et al. (2008), using SDSS (Adelman-McCarthy et al. 2007) measurements of the field stars was applied. However, the SDSS filter system (Fukugita et al. 1996) differs significantly from the filters used in this search except for the SuprimeCam i band. To overcome this difference, we fitted relations between the SDSS filter system and the Landolt system (Landolt 1992) in a similar manner to Lupton (2005), using stars with SDSS and Stetson photometry (Stetson 2000). Stetson has been publishing photometry of a growing list of faint stars that is tied to the Landolt system within 0.01 mag (Stetson 2000). The most up-to-date version can be obtained from the Canadian Astronomy Data Center.³² In contrast to Lupton (2005), we applied magnitude, uncertainty, and color cuts (15 < r_SDSS < 18.5, σ_m < 0.03 mag in r_SDSS and i_SDSS and −0.5 < r_SDSS − i_SDSS < 1.0) for the stars that went into the fit. We also applied a 5σ outlier cut after our initial fit (and lost about ∼6% of the sample). After refitting, the following relations were derived for the Landolt R and I filters

$$\begin{eqnarray*} R-r_\mathrm{SDSS }& = -0.13 - 0.32\cdot (r_\mathrm{SDSS }-i_\mathrm{SDSS })\\ I-i_\mathrm{SDSS }& = -0.38 - 0.26\cdot (r_\mathrm{SDSS }-i_\mathrm{SDSS }), \end{eqnarray*}$$

which are also shown in Figure 5. The results for R − r_SDSS are close to the ones derived by Lupton (2005) as well as to Tonry et al. (2003) who performed a similar operation. Neither Tonry et al. (2003) nor Lupton (2005) present fits for I − i_SDSS versus r_SDSS − i_SDSS.

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By forcing χ²/dof = 1, we determined that there is a scatter of 0.03 mag coming from intrinsic spectral distributions. This is a systematic uncertainty for individual stars, but will average out when a big sample is used, assuming that the color distribution of the sample is similar to the stars used to derive the relations.

The transformations were applied to the SDSS stellar photometry of our SN field, and these were then used as tertiary standard stars in order to tie the SN photometry to the Landolt system. The flux of the stars was determined on the photometric reference for each light curve build using the same method as for the SN and with the same PSF model that was used for fitting the SN fluxes. A zero-point relation of the form

$$\begin{equation} m + 2.5\log _{10}f = \mathrm{ZP} + c_X\cdot (R-I) \end{equation} \tag{ 1 }$$

could then be fitted between the measured stellar fluxes, f, and their Landolt magnitudes, m. Here ZP is the zero point and c_X is the color term for the filter. We applied the same color cuts (−0.5 < r_SDSS − i_SDSS < 1.0) to the stars that went into the fit.

Unfortunately we did not have enough stars over a wide enough color range to accurately fit the color term. Instead, we used values from the literature or from the observatories, which are summarized in Table 5. The zero points could then be derived from Equation (1). The values obtained this way are the sums of three components; the instrumental zero point, the aperture correction to the PSF normalization radius, and the atmospheric extinction for the given airmass, and are shown along with the SN fluxes in Table B1.

Table 5. Measured and Synthetic Color Terms for the Three Instruments that were Used as Photometric References

Instr./Tel.	Measured		Synthetic
	cR	cI	cR	cI
CFH12k/CFHT	0.031a	0.107a	0.066	−0.023
MOSAIC II/CTIO	...	0.030b	...	0.007
FORS1/VLT	0.034c	-0.050c	0.040	−0.076

Notes. ^ahttp://www.cfht.hawaii.edu/Instruments/Elixir/filters.html ^bMiknaitis et al. (2007). ^chttp://www.eso.org/observing/dfo/quality/FORS1/qc/photcoeff/photcoeffs_fors1.html

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We also calculated color terms synthetically by using Landolt standard stars that have extensive spectrophotometry from Stritzinger et al. (2005). The synthetic Vega magnitudes of the stars were calculated by multiplying the spectra and the Vega spectrum (Bohlin 2007) with the filter and instrument throughputs provided by the different observatories. The color terms were then fitted by assuming a linear relation for the deviation between the synthetic and the Landolt magnitudes as a function of the Landolt color. The resulting fitted synthetic terms are also presented in Table 5.

The difference is ≲0.03 mag for all but the CFH12k I band, where there is a significant discrepancy. A mismatch between the effective filter transmission curves we use and the true photometric system is most likely the origin of this. We have investigated if the deviation could be explained by the differences in quantum efficiencies between the different chips of the detector which are presented on the CFH12k Web page. However, these differences propagated to the color term result in relatively low scatter and deviate significantly from the measured value.

For the purpose of fitting the zero points, we can use the measured color terms, but for light-curve fitting erroneous filter transmissions could introduce systematic effects. In order to study the potential impact on the light-curve fits we modified the filters to match the measured color terms. The modifications were implemented by either shifting or clipping the filters until the synthetic color terms matched all measured values for a given filter (e.g., R or I versus R − I, V − I, and B − I for the ELIXIR measurements for the I band). The light curves were then refitted using the modified filter transmissions and the resulting values were compared. Since the light curves are tightly constrained by the high-precision HST data, the modified filter only leads to negligible differences (less than 30% of the statistical uncertainty) in the fitted parameters.

The J-band data, on the other hand, were calibrated using G-type standard stars from the Persson LCO standard star catalog (Persson et al. 1998). Since the ISAAC and NIRI J-band filters are slightly redder and narrower than the Persson J-band filter, we subtracted 0.012 mag from the ISAAC and NIRI zero points to place the ISAAC and NIRI IR photometry onto the natural system. Since the data were taken over many nights, the photometry was carefully cross-checked. Differences in the absolute photometry usually amount to less than 0.02 mag, which we conservatively adopt as our zero-point uncertainty.

A modified version of the photometric technique from Knop et al. (2003) was used for the HST data presented here. This is similar to the method used for the ground-based optical data above, but instead of aligning and resampling all images to a common frame, the host+SN model is resampled to each individual image. A procedure like this is preferred when the PSF FWHM is of the same order as the pixel scale, and it also preserves the image noise properties.

Linear geometric transformations were first fitted from each image, k, in a given filter to the deepest image of the field using field objects. Due to the similar orientation of the WFPC2 images, linear transformations were sufficient for these, and the geometric distortion of WFPC2 could be ignored. This was however not the case for the transformations between the ACS and WFPC2 images and the distortion was then hardcoded into the fitting procedure. Due to the sparse number of objects in the tiny PC field the accuracy of all transformations was only good to ≲1 pixel (∼0.5 FWHM). Unfortunately, using objects from the remaining three chips for the alignment did not lead to improved accuracy, which is probably due to small movements of the chips between exposures (Anderson & King 2003). This alignment precision was not enough for PSF photometry, and we therefore allowed the SN position, (x_k0, y_k0) to float for the individual images which increased the alignment precision by a factor of 10. Allowing this extra degree of freedom could bias the results toward higher fluxes since the fit will favor positive noise fluctuations. However, as in Knop et al. (2003), this was shown to be of minor importance by studying the covariance between the fitted flux and SN position.

The full model used to describe each image patch can be expressed as

$$\begin{equation} I_k(x_k,y_k) = f_k\cdot \mathrm{PSF}_k(x_k-x_{k0},y_k-y_{k0}) + G(x_k,y_k,a_j) + S_k \,. \end{equation} \tag{ 2 }$$

Here I_k is the value in pixel (x_k, y_k) on image k, f_k is the SN flux, PSF_k is the point spread function, G is the host galaxy model that is parameterized by a_j, and S_k is the local sky background. The fits were carried out using a χ² minimization approach using MINUIT (James & Roos 1975).

Four SNe in the sample had ACS reference images, and for these cases we used field objects to fit non-analytic convolution kernels, K, between the PC chip and the drizzled ACS image. These accounted for the difference in quantum efficiency and PSF shape between the two instruments. When kernels were used, the uncertainties of individual pixels were propagated and the correlation between pixels produced by the convolution were taken into account. Also, in this case Equation (2) above was modified so with both the PC images and the PC PSF being convolved with the fitted kernel.

The PSF, PSF_k, of the WFPC2 PC chip was simulated for each filter and pixel position using the Tiny Tim software (Krist 2001) and normalized to the WFPC2 calibration radius 05. We also did an extensive test where we iteratively updated the simulated PSF based on the knowledge of the SN epoch, and therefore the approximate spectral energy distribution (SED), but this did not have any significant effect on the fitted fluxes. The Tiny Tim PSF was generated to be subsampled by a factor of 10. For each iteration in the fitting procedure, any shift of the PSF position was first applied in the subsampled space. The PSF was then rebinned to normal sampling and convolved with a charge diffusion kernel (Krist 2001) before it was added to the patch model.

For three of the SNe, 2001cw, 2001gq, and 2001hb we used an analytical model for the host galaxy. Both 2001cw and 2001gq are offset from the core of their respective host galaxies and we therefore chose not to obtain SN-free reference images for these. Instead the hosts could be modeled by a second-order polynomial and constrained by the galaxy light in the vicinity of the SNe. For 2001hb, we did obtain a deep ACS reference image but no host could be detected. For this SN we used a simple plane to model the host which was further constrained by fixing the SN flux to zero for the ACS reference. One caveat with analytical host modeling in general is that the patch size must be chosen with care. The model will only work if there is no dramatic change in the background across the patch, which is an assumption that is likely to fail if the patch is too large and includes the host galaxy core. On the other hand, the patch cannot be too small either in order to successfully break the degeneracy between the SN and the background.

To make sure that the choice of the host model does not bias the fitted SN fluxes we required, in addition to clean residuals once the SN + host was subtracted from the data, that the fitted SN fluxes were insensitive to variations in the patch size of a few pixels. Further, for 2001gq, we tested the host model by putting fake SNe at different positions around the core of the host. The distances to the core and the fluxes of the fakes were always chosen to match the corresponding values for the real SN, and the retrieved photometry was always within the expected statistical uncertainty.

The recipe described above could not be used for the remaining three SNe in the sample, since they were located too close to the cores of their host galaxies. Additionally, the hosts have small angular sizes and the light gradients in the vicinity of the SNe were steep enough to lead to biased photometry due to the coarse geometric alignment. To overcome this we changed the photometric procedure slightly. Instead of fitting the SN position on each image, we chose to fit it only on the geometric reference image, and then introduce a free shift for the whole model. That is, in this case we used the galaxy model + SN for the patch alignment. However, this procedure did force us to apply some constraints on the galaxy modeling. Using a non-analytic pixel model, the approach used for the ground-based optical data was not feasible. It slowed down the fits considerably and rarely converged. Instead we chose to use the ACS images of the host galaxies directly. The ACS references were much deeper then the WFPC2 data and choosing this procedure did not increase the uncertainty of the fitted fluxes.

A general problem with doing photometry on HST images is that CCD photometry of faint objects over a low background suffers from an imperfect charge transfer, which will lead to an underestimate of the flux. We used the charge transfer efficiency (CTE) recipe for point sources from Dolphin (2009). The correction for our data is usually around 5%–8% but it can be as large as ∼17%. The uncertainties of the corrections were propagated to the flux uncertainties. The corrected SN fluxes are given in Table B1 together with the instrumental zero-points, which were also obtained from Andrew Dolphin's Web page.

SN Ia that has bluer colors or broader light curves tends to be intrinsically brighter (Phillips 1993; Tripp 1998). Several methods of combining this information into an accurate measure of the relative distance have been used (Riess et al. 1996; Goldhaber et al. 2001; Wang et al. 2003; Guy et al. 2005, 2007; Jha et al. 2007; Conley et al. 2008).

K08 consistently fitted all light curves using the SALT (Guy et al. 2005) fitter, which is built on the SN Ia SED from Nugent et al. (2002). In this paper, we use SALT2 (Guy et al. 2007), which is based on more data.

Conley et al. (2008) compared the performances of different light-curve fitters while also introducing their own empirical fitter, SiFTO, and concluded that SALT2 along with SiFTO perform better than both SALT (which is conceptually different from its successor SALT2) and MLCS2k2 (Jha et al. 2007) when judged by the scatter around the best-fit luminosity–distance relationship. Furthermore, SALT2 and SiFTO produce consistent cosmological results when both are trained on the same data. Recently KS09 made a thorough comparison between SALT2 and their modified version of MLCS2k2 for a compilation of public data sets, including the one from the SDSS SN survey. The two light-curve fitters result in an estimate of w (for a flat wCDM cosmology) that differs by 0.2. The difference exceeds their statistical and systematic (from other sources) error budgets. They determine that this deviation originates almost exclusively from the difference between the two fitters in the rest-frame U-band region, and the color prior used in MLCS2k2. They also noted that MLCS2k2 is less accurate at predicting the rest-frame U band using data from filters at longer wavelengths.

This difference in U-band performance is not surprising: observations carried out in the observer-frame U band are in general associated with a high level of uncertainty due to atmospheric variations. While the training of MLCS2k2 is exclusively based on observations of nearby SNe, the SiFTO and SALT2 trainings address this difficulty by also including high redshift data where the rest-frame U band is observed at redder wavelengths. This approach also allows these fitters to extend blueward of the rest-frame U band.

In addition, for this paper, we have conducted our own test validating the performance of SALT2 by carrying out the Monte Carlo simulation described in Section 7.3.6, where we compare the fitted SALT2 parameters to the corresponding real values for mock samples with poor cadence and low signal to noise drawn from individual well-measured nearby SNe.

Given these tests that have been carried out on SALT2, and its high redshift source for rest-frame U band, we have chosen to use SALT2 in this paper.

6.1. SALT2

The SALT2 SED model has been derived through a pseudo-principal component analysis based on both photometric and spectroscopic data. Most of these data come from nearby SN Ia data, but SNLS SNe are also included. To summarize, the SALT2 SED, F(SN, p, λ), is a function of both wavelength, λ, and time since B-band maximum, p. It consists of three components; a model of the time-dependent average SN Ia SED, M₀(p, λ), a model of the variation from the average, M₁(p, λ), and a wavelength-dependent function that warps the model, CL(λ). The three components have been determined from the training process (Guy et al. 2007) and are combined as

$$\begin{eqnarray*} F(\mathrm{SN },p,\lambda) &=& x_0\times \left[M_0(p,\lambda) + x_1\times M_1(p,\lambda)\right]\nonumber\\ &&\times \exp \left[c\times \mbox{CL}(\lambda)\right], \end{eqnarray*}$$

where x₀, x₁, and c are free parameters that are fit for each individual SN.

Here, x₀, describes the overall SED normalization, x₁, the deviation from the average decline rate (x₁ = 0) of an SN Ia, and c, the deviation from the mean SN Ia B − V color at the time of B-band maximum. These parameters are determined for each observed SN by fitting the model to the available data. The fit is carried out in the observer frame by redshifting the model, correcting for Milky Way extinction (using the CCM-law from Cardelli et al. 1989 with R_V = 3.1), and multiplying by the effective filter transmission functions provided by the different observatories. All synthetic photometry is carried out in the Vega system using the spectrum from Bohlin (2007). Following Astier et al. (2006) we adopt the magnitudes (U, B, V, R_C, I_C) = (0.020, 0.030, 0.030, 0.030, 0.024) mag (Fukugita et al. 1996) for Vega. For the near-infrared we adopt the values J = 0 and H = 0.

In the fit to our data, we take into account the correlations introduced between different light-curve points from using the same host galaxy model. We also chose to run SALT2 in the mode where the diagonal of the covariance matrix is updated iteratively in order to take model and K-correction uncertainties into account. See Guy et al. (2007) for details on this. The Milky Way reddening for our SNe from the Schlegel et al. (1998) dust maps is given in Table 2. The results of the fits are shown in Table 6 and plotted in Figure 6 together with the data.

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Table 6. Light-curve Fit Results Using SALT2

SN	MJDmax	mcorrB	mmaxB	c	x1	s
2001cw	52066.60 ± 0.86	24.88 ± 0.42	24.71 ± 0.13	−0.10 ± 0.36	0.140 ± 0.844	0.993 ± 0.077
2001gn	52030.33 ± 0.16	25.16 ± 0.23	25.37 ± 0.05	0.07 ± 0.05	0.648 ± 0.649	1.040 ± 0.061
2001go	52010.99 ± 0.35	23.13 ± 0.11	23.08 ± 0.07	−0.11 ± 0.07	−1.149 ± 0.259	0.881 ± 0.022
2001gq	52028.81 ± 0.42	23.62 ± 0.19	23.79 ± 0.03	0.06 ± 0.04	0.459 ± 0.287	1.022 ± 0.027
2001gy	52031.66 ± 0.25	22.97 ± 0.07	23.05 ± 0.03	−0.03 ± 0.03	−0.312 ± 0.265	0.952 ± 0.024
2001hb	52038.02 ± 0.97	24.84 ± 0.13	24.79 ± 0.03	0.00 ± 0.04	1.292 ± 0.410	1.101 ± 0.039

Notes. The SALT2 parameters are MJD_max, m_B, x₀, x₁, and c. Here, the light-curve shape, x₁, and color, c, corrected peak B-band magnitude are obtained from Equation (3) as m^corr_B = m^max_B + α · x₁ − β · c, where the values α = 0.12 and β = 2.51 have been used. For comparison we also present the SALT stretch, s, which has been derived from x₁ using the relation provided in Guy et al. (2007).

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The three parameters m^max_B = −2.5log₁₀[∫_BF(SN, 0, λ). .λ dλ], x₁ and c, can for each SN be combined to form the distance modulus (Guy et al. 2007),

$$\begin{equation} \mu _B = m_B^\mathrm{corr} - M_B = m_B^\mathrm{max} + \alpha \cdot x_1 - \beta \cdot c - M_B, \end{equation} \tag{ 3 }$$

where M_B is the absolute B-band magnitude. The resulting color and light-curve shape corrected peak B-band magnitudes, m^corr_B, are presented in the third column of Table 6. The parameters α, β, and M_B are nuisance parameters which are fitted simultaneously with the cosmological parameters.

K08 presented an analysis framework for combining different SN Ia data sets in a consistent manner. Since then two other groups (H09 and KS09) have made similar compilations, using different fitters. In this work, we carry out an improved analysis, using and refining the approach of K08. We extend the sample with the six SNe presented here, the SNe from Amanullah et al. (2008), the low-z and intermediate-z data from Hicken et al. (2009b) and Holtzman et al. (2008), respectively.³³

First, all light curves are fitted using a single light-curve fitter (the SALT2 method) in order to eliminate differences that arise from using different fitters. For all SNe going into the analysis we require

• 1.  
  data from at least two bands with rest-frame central wavelengths between 2900 Å and 7000 Å, the default wavelength range of SALT2;
• 2.  
  that there is at least one point between −15 days and 6 rest-frame days relative to the B-band maximum;
• 3.  
  that there are in total at least five valid data points available;
• 4.  
  that the fitted x₁ values, including the fitted uncertainties, lie between −5 < x₁ < 5. This is a more conservative cut than that used in K08 and results in several poorly measured SNe being excluded. Part of the discrepancy observed by KS09 when using different light-curve models could be traced to poorly measured SNe; and
• 5.  
  that the CMB-centric redshift is greater than z > 0.015.

We also exclude one SN from the Union compilation that is 1991bg-like, which neither the SALT nor the SALT2 models are trained to handle. Note that another 1991bg-like SN from the Union compilation was removed by the outlier rejection. All SNe Ia considered in this compilation are listed in Table B2. For each SN, the redshift and fitted light-curve parameters are presented as well as the failed cuts, if any.

It should be pointed out that the choice of the light-curve model also has an impact on the sample size. Using SALT2 will allow more SNe to pass the cuts above, since the SALT2 model covers a broader wavelength range than SALT. This is particularly important for high-z data that heavily rely on rest-frame UV data. For example, two net SNe would have been cut from the Riess et al. (2007) sample with the SALT model.

7.1. Revised HST Zero Points and Filter Curves

7.2. Fitting Cosmology

Following Conley et al. (2006) and K08, we adopt a blind analysis approach for cosmology fitting where the true fitted values are not revealed until the complete analysis framework has been settled. The blind technique is implemented by adjusting the magnitudes of the SNe until they match a fiducial cosmology (Ω_M = 0.25, w = −1). This procedure leaves the residuals only slightly changed, so that the performance of the analysis framework can be studied. The best-fitted cosmology with statistical errors is obtained through an iterative χ²-minimization of

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

where

$$\begin{equation} \sigma _{\mathrm{lc}}^2 = V_{m_B} + \alpha ^2 V_{x_1} + \beta ^2 V_{c} + 2\alpha V_{m_B,x_1} - 2\beta V_{m_B,c} - 2\alpha \beta V_{x_1,c}\, \end{equation} \tag{ 5 }$$

is the propagated error from the covariance matrix, V, of the light-curve fits, with α and β being the x₁ and color correction coefficients of Equation (3). Uncertainties due to host galaxy peculiar velocities of 300 km s⁻¹ and uncertainties from Galactic extinction corrections and gravitational lensing as described in Section 7.3 are included in σ_ext. A floating dispersion term, σ_sys, which contains potential sample-dependent systematic errors that have not been accounted for and the observed intrinsic SN Ia dispersion, is also added. The value of σ_sys is obtained by setting the reduced χ² to unity for each sample. Computing a separate σ_sys for each sample prevents samples with poorer-quality data from increasing the errors of the whole sample. This approach does however still assume that all SNe within a sample are measured with roughly the same accuracy. If this is not the case there is a risk in degrading the constraints from the sample by down weighting the best-measured SNe. It should also be pointed out that the fitted values of σ_sys will be less certain for small samples and can therefore deviate significantly from the average established by the larger samples (in particular, the six high-z SNe presented in this work are consistent with σ_sys = 0), as are three other samples.

A number of systematic errors are also being considered for the full cosmology analysis. These are taken into account by constructing a covariance matrix for the entire sample which will be described below in Section 7.3. The terms in the denominator of Equation (4) are then added along the diagonal of this covariance matrix.

Following K08, we carry out an iterative χ² minimization with 3σ outlier rejection. Each sample is fit for its own absolute magnitude by minimizing the sum of the absolute residuals from its Hubble line (rather than the sum of the squared residuals). The line is then used for outlier rejection. This approach was investigated in detail in K08, and it was shown with simulations that the technique is robust and that the results are unaltered from the Gaussian case in the absence of contamination and that in the presence of a contaminating contribution, its impact is reduced. Table 7 summarizes the effect of the outlier cut on each sample. We also note that the residuals have a similar distribution to a Gaussian in that ∼5% of the sample is outside of 2σ.

Table 7. Statistics of Each Sample with No Outlier Rejection or 3σ Outlier Rejection (Used in This Paper)

Set	No Outlier Cut			σcut = 3
	SNe	σsys(68%)	rms (68%)	SNe	σsys(68%)	rms (68%)
Hamuy et al. (1996)	18	0.15+0.05−0.03	0.17+0.03−0.03	18	0.15+0.05−0.03	0.17+0.03−0.03
Krisciunas et al. (2005)	6	0.01+0.14−0.01	0.11+0.02−0.03	6	0.04+0.13−0.04	0.11+0.03−0.03
Riess et al. (1999)	11	0.16+0.07−0.03	0.17+0.03−0.04	11	0.15+0.07−0.03	0.17+0.03−0.04
Jha et al. (2006)	15	0.20+0.07−0.04	0.22+0.04−0.04	15	0.21+0.07−0.04	0.22+0.04−0.04
Kowalski et al. (2008)	8	0.04+0.08−0.04	0.14+0.03−0.04	8	0.07+0.09−0.06	0.15+0.03−0.04
Hicken et al. (2009b)	104	0.18+0.02−0.02	0.21+0.01−0.01	102	0.15+0.02−0.01	0.19+0.01−0.01
Holtzman et al. (2008)	133	0.19+0.02−0.01	0.23+0.01−0.01	129	0.10+0.01−0.01	0.15+0.01−0.01
Riess et al. (1998) + HZT	11	0.31+0.19−0.09	0.53+0.10−0.12	11	0.31+0.19−0.09	0.52+0.10−0.12
Perlmutter et al. (1999)	33	0.41+0.12−0.09	0.64+0.07−0.08	33	0.41+0.12−0.09	0.64+0.07−0.08
Barris et al. (2004)	19	0.19+0.13−0.10	0.39+0.06−0.07	19	0.18+0.13−0.10	0.38+0.06−0.07
Amanullah et al. (2008)	5	0.18+0.21−0.06	0.20+0.05−0.07	5	0.19+0.21−0.06	0.21+0.05−0.07
Knop et al. (2003)	11	0.04+0.10−0.04	0.15+0.03−0.03	11	0.05+0.10−0.05	0.15+0.03−0.03
Astier et al. (2006)	73	0.18+0.03−0.03	0.25+0.02−0.02	72	0.13+0.03−0.02	0.21+0.02−0.02
Miknaitis et al. (2007)	77	0.25+0.04−0.03	0.34+0.03−0.03	74	0.19+0.04−0.03	0.29+0.02−0.02
Tonry et al. (2003)	6	0.17+0.21−0.10	0.24+0.06−0.07	6	0.15+0.21−0.12	0.23+0.05−0.07
Riess et al. (2007)	33	0.27+0.07−0.04	0.49+0.06−0.06	31	0.16+0.06−0.05	0.45+0.05−0.06
This paper	6	0.00+0.000.00	0.12+0.03−0.04	6	0.00+0.000.00	0.12+0.03−0.04
Total	569			557

Notes. Here, σ_sys has the same meaning as in Equation (4). Both σ_sys and the rms are also plotted for each sample in Figure 8. Although each sample is independently fit for its σ_sys and rms, a global α and β are always used. This explains the minor shifts in parameters for samples where no SNe are cut. A 2σ cut removes 34 more SNe, so going from 3σ to 2σ is consistent with Gaussian residuals.

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Figure 7 shows the individual residuals and pulls from the best-fit cosmology together with the fitted SALT2 colors for the different samples. The photometric quality is illustrated by the last column in the figure showing the color uncertainty. It is notable how the photometric quality on the high redshift end has improved from the K08 analysis. This is due to the extended rest-frame range of the SALT2 model compared to SALT.

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Figure 8 shows the diagnostics used for studying the consistency between the different samples. The left panel shows the fitted σ_sys values for each sample together with the rms around the best-fitted cosmology. The intrinsic dispersion associated with all SNe can be determined as the median of σ_sys as long as the majority of the samples are not dominated by observer-dependent uncertainties that have not been accounted for. The median σ_sys for this analysis is 0.15 mag, indicated by the leftmost dashed vertical line in the figure. The two mid-panels show the tensions for the individual samples, by comparing the average residuals from the best-fit cosmology. The two panels show the tensions without and with systematic errors (described in Section 7.3) being considered. Most samples fall within 1σ and no sample exceeds 2σ. The right panel shows the tension of the slopes of the residuals as a function of redshift. This test may not be very meaningful for sparsely sampled data sets, but could reveal possible Malmquist bias for large data sets.

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The SNe introduced in this work show no significant tension in any of the panels. The Hubble residuals for these are also presented in Figure 9. Here, the individual SNe are consistent with the best-fit cosmology.

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All tables and figures, including the complete covariance matrix for the sample, are available in electronic format on the Union Web page.³⁴ We also provide a CosmoMC module for including this supernova compilation with other data sets.

7.3. Systematic Errors

The best-fit cosmological parameters for the compilation are presented in Table 11 with constraints from CMB and BAO. The confidence regions in the (Ω_M, Ω_Λ) and (Ω_M, w) planes for the last fit in the table are shown in Figures 10 and 11, respectively.

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In the cosmology analysis presented here, the statistical errors on Ω_M have decreased by a significant 24% over the K08 Union analysis, while the estimated systematic errors have only improved by 13%. When combining the SN results with BAO and CMB constraints, statistical errors on w have improved by 16% over K08, though the quoted systematic errors have increased by 7%. Figure 12 shows a comparison between the constraints from K08 and this compilation in the (Ω_M–w) plane. Even with some improvement on the understanding of systematic errors, it is clear that the data set is dominated by systematic error (at least at low- to mid-z).

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For the CMB data we implement the constraints from the seven year data release of the Wilkinson Microwave Anisotropy Probe (WMAP) as outlined in Komatsu et al. (2010). We take their results on z_⋆ (the redshift of last scattering), l_A(z_⋆), and R(z_⋆), updating the central values for the cosmological model being considered. Here, l_A(z_⋆) is given by

$$\begin{equation*} l_A(z_{\star }) \equiv (1+z_{\star })\frac{\pi D_A(z_{\star })}{r_s(z_{\star })}, \end{equation*}$$

where D_A is the angular distance to z_⋆, while

$$\begin{equation*} R(z_{\star }) \equiv \frac{\sqrt{\Omega _MH_0^2}}{c} (1 + z_{\star })D_A(z_{\star })\,. \end{equation*}$$

Percival et al. (2010) measure the position of the BAO peak from the SDSS DR7 and 2dFGRS data, constraining d_z ≡ r_s(z_d)/D_V(0.275) to 0.1390 ± 0.0037, where r_z(z_d) is the comoving sound horizon and D_V(z) ≡ [(1 + z)²D²_Acz/H(z)]^1/3.

For the SNe + BAO fit in Table 11, we add an H₀ measurement of 74.2 ± 3.6 km s⁻¹ Mpc⁻¹ from Riess et al. (2009), creating a constraint without the CMB that is therefore largely independent of the high-redshift behavior of dark energy (as long as the dark energy density contribution is negligible in the early universe). Note that the H₀ constraint relies on most of the nearby SNe used in this compilation. However, the effect on w through H₀ from these SNe is several times smaller than the effect through the Hubble diagram. Alternatively, adding a CMB constraint on Ω_mh² of 0.1338 ± 0.0058 from the WMAP7 Web page³⁷ allows us to create a constraint that is independent of H₀. This final result for SNe+BAO+CMB does not improve significantly if the current H₀ constraint is added.

8.1. Time Variation of the Dark Energy Equation of State

The constraints shown in Figure 11 were obtained assuming that the dark energy EOS is redshift independent. SNe Ia are useful for constraining a redshift dependent w(z) since, unlike e.g., CMB, their measured distances at a given redshift are independent of the behavior of dark energy at higher redshifts. A common method to parameterize w(z) is

$$\begin{equation*} w(z) = w_0 + w_a\frac{z}{1+z}, \end{equation*}$$

where a cosmological constant is described by (w₀, w_a) = (−1, 0). It can be shown (Linder 2003) that this parameterization provides an excellent approximation to a wide variety of dark energy models. The constraints from the current SN data together with the CMB and BAO data are presented in Figure 13. In terms of the figure of merit,³⁸ introduced by the dark energy task force (Albrecht et al. 2006), these constraints correspond to 1.2 and 1.8 with and without systematics, respectively. The flattening of the contours in this diagram at w₀ + w_a = 0 comes from the implicit constraint of matter domination in the early universe imposed by the CMB and BAO data. Only modest constraints can currently be placed on w_a.

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It can be illuminating to study w(z) in redshift bins, where w is assumed constant in each bin. This method has the advantage that w(z) can be studied without assuming a specific form for the relation (Huterer & Starkman 2003). We carry out the analysis following Kowalski et al. (2008) and fit a constant w in each bin, while the remaining cosmological parameters are fit globally for the entire redshift range. Figure 14 shows three such models for the combined constraints from SNe, BAO, CMB, and H₀ measurements where we assume a flat universe. In these scenarios, the H₀ measurement does not contribute much, but due to its small improvement on the CMB constraints it gives a small (∼10%) improvement on the errorbar of the highest redshift bin.

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The left panel shows constraints on w for three bins. The first bin (0 < z < 0.5) shows a well-constrained w. The middle bin (0.5 < z < 1) shows a poorly constrained w, though one that is distinct from −∞ (which would drive ρ to 0 above z = 0.5, resulting in a matter-only universe) at high confidence, indicating the detection of some kind of dark energy in this redshift range. For z > 1, there is little constraint on w and only a weak constraint on the existence of dark energy.

The middle panel shows the effect of dividing the highest redshift bin. The constraints on w for z > 1 get much weaker, showing that most of the (weak) constraint on the highest bin in the left panel comes from a combination of the CMB with the well-constrained low-redshift supernova data. Current SNe at z > 1 offer no real constraint on w(z > 1). Providing a significant constraint at these redshifts requires significantly better supernova measurements. As in the left panel, w in the highest redshift bin is constrained to be less than zero by the requirement from BAO and CMB constraints that the early universe has a matter-dominated epoch.

The right panel shows the effect of dividing the lowest redshift bin. While no significant change in w with redshift is detected, there is still considerable room for evolution in w, even at low redshift.

Figure 15 shows dark energy density constraints, assuming the same redshift binning as in Figure 14. Note that this is not equivalent to the left and center panels of Figure 14; only in the limit of an infinite number of bins do binned ρ and binned w give the same model. Dark energy can be detected at high significance in the middle bin (redshift 0.5–1), but there is only weak evidence for dark energy above redshift 1 (left panel). When the bin above redshift 1 is split at a redshift greater than the supernova sample (right panel), it can be seen that the current small sample of SNe cannot constrain the existence of dark energy above redshift 1.

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8.2. SNe with Ground-based Near-IR Data

8.3. Comparison of KeplerCam and SDSS Photometry

We have presented the light curves of SNe Ia in the redshift range 0.511 < z < 1.12. The SNe were discovered as part of a search conducted by the SCP using the Subaru, CFHT, and CTIO Blanco telescopes in 2001. The fitted light-curve shapes and colors at maximum light are all consistent with the corresponding distributions from previous SN surveys.

Following K08, we add these six SNe and other SN Ia data sets to the Union compilation. We have also improved the Union analysis in a number of respects, creating the new Union2 compilation. The most important improvements are (1) systematic errors are directly computed using the effect they have on the distance modulus and (2) all SN light curves are fitted with the SALT2 light-curve fitter.

We determine the best-fit cosmology for the Union2 compilation, and the concordance ΛCDM model remains an excellent fit. The new analysis results in a significant improvement over K08 in constraining w over the redshift interval 0 < z < 1. Above z ≳ 1, evidence for dark energy is weak. This will remain the case until there is much more high redshift data, with better signal-to-noise and wavelength coverage.

Based, in part, on observations obtained at the ESO La Silla Paranal Observatory (ESO programs 67.A-0361 and 169.A-0382). Based, in part, on observations obtained at the CTIO, National Optical Astronomy Observatory, which are operated by the Association of Universities for Research in Astronomy, under contract with the National Science Foundation. Based, in part, on observations obtained at the Gemini Observatory (Gemini programs GN-2001A-SV-19 and GN-2002A-Q-31), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (USA), the Particle Physics and Astronomy Research Council (UK), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil), and CONICET (Argentina). Based, in part on observations obtained at the Subaru Telescope, which is operated by the National Observatory of Japan. Some of imaging data taken with Suprime-Cam were obtained during the commissioning phase of the instrument, and we thank all members of Suprime-Cam instrument team. Based, in part, on data that were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. Based on observations obtained at the CFHT which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The authors recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.

Based, in part, on observations made with the NASA/ESA Hubble Space Telescope, obtained at the STScI, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs HST-GO-08585 and HST-GO-09075.

Support for programs HST-GO-08585.14-A and HST-GO-09075.01-A was provided by NASA through a grant from the STScI, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.

This work is supported in part by a Japan Society for the Promotion of Science (JSPS) core-to-core program "International Research Network for Dark Energy" and by JSPS research grants (20040002).

This work was supported by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE- AC02-05CH11231.

T.M. is financially supported by the JSPS through the JSPS Research Fellowship. C.L. acknowledges the support provided by the Oskar Klein Centre at Stockholm University. The authors thank the anonymous referee for helpful comments and suggestions.

Facilities: HST (WFPC2,ACS) - Hubble Space Telescope satellite, Keck:I (LRIS) - KECK I Telescope, Keck:II (ESI) - KECK II Telescope, CFHT (CFH12k) - Canada-France-Hawaii Telescope, Gemini:Gillett (NIRI) - Gillett Gemini North Telescope, Subaru (SuprimeCam) - Subaru Telescope, Blanco (MOSAIC II) - Cerro Tololo Inter-American Observatory's 4 meter Blanco Telescope, NTT (SuSI2) - New Techology Telescope, VLT:Antu (FORS1,ISAAC) - Very Large Telescope (Antu)

The spectra of six SNe are presented in this paper. Details of the spectra for 2001go and 2001gy can be found in Lidman et al. (2005), while the spectrum for 2001cw is described in Morokuma et al. (2010).

The spectrum of each candidate is plotted twice (see Figures A1–A3). In the upper panel, the spectrum of the candidate is plotted in the observer frame and is uncorrected for host galaxy light. Telluric absorption features are marked with the symbol ⊕. In the lower panel, contamination from the host (if any) is removed, and the resulting spectrum is rescaled and rebinned. This spectrum is plotted in black as a histogram and it is plotted in both the rest frame (lower axis) and the observer frame (upper axis). For comparison, low redshift SNe are plotted as a blue continuous line.

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Tabulated below are light-curve data for the six SNe presented in this paper. The data are listed in chronological order for one SN at a time and correspond to data points plotted in Figure 6.

All photometry is given as the measured flux, f, in the instrumental system, i.e., no color correction (S-correction) to a standard system has been applied. The instrumental Vega magnitude may be calculated using the standard formula: m = −2.5log f + ZP, where ZP is the associated zero point. For the HST data these are the updated values from Dolphin (2000) given on Andrew Dolphin's Web page.³⁹ The zero points have then been converted to e/s by using a gain of 7.12. The zero points for the ground-based data on the other hand are always given for the photometric reference image, and therefore include exposure time, gain, and for the PSF fitted data, the aperture correction. Note that there are correlated errors between the data points that come from the same light-curve build, since the same host galaxy model and zero point have been used. This is also reflected by small offsets between different data sets in Figure 6. The covariances are taken into account for the light-curve fitting and the full covariance matrices used will be available from the SCP Web site.

The instruments used are as given below. MOSAIC II is the multichip imager on the CTIO 4 m telescope. CFH12k is the multichip imager on the 3.6 m CFHT telescope on Mauna Kea in Hawaii. SuprimeCamis the wide-field imager on the 8.2 m Subaru telescope. SuSI2is the imager on the NTT 3.6 m telescope at ESO. FORS1 is the imager on the 8 m UT2 (Kueyen) at VLT, ESO. ISAAC and NIRI are the near-infrared imagers on the 8 m UT1 (Antu) at VLT, ESO, and Gemini North telescopes, respectively. Finally, WFPC2 indicates data obtained with the PC CCD and ACS is the Advanced Camera for Surveys, both on HST.

Suppose that observations $\boldsymbol{y}$ are modeled by $\boldsymbol{F}(\boldsymbol{\theta })$, where some of the parameters enter into the model linearly ($\boldsymbol{\theta }^L$) while some enter nonlinearly ($\boldsymbol{\theta }^N$). The χ² can be written as

$$\begin{equation} \chi ^2(\boldsymbol{\theta }) = (\boldsymbol{y}- (\boldsymbol{F}_N(\boldsymbol{\theta }^N) + H\boldsymbol{\theta }^L))^T V^{-1} (\boldsymbol{y}- (\boldsymbol{F}_N(\boldsymbol{\theta }^N) + H\boldsymbol{\theta }^L)), \end{equation} \tag{ C1 }$$

where V is the covariance matrix of the observations and H is the Jacobian matrix of the model with respect to the $\boldsymbol{\theta }^L$. Taking the derivative of the χ² and setting it to zero gives the analytic formula for the best-fit $\boldsymbol{\theta }^L$

$$\begin{equation} \hat{\boldsymbol{\theta }^L} = (H^T V^{-1} H)^{-1} H^T V^{-1} (\boldsymbol{y}- \boldsymbol{F}_N(\boldsymbol{\theta }^N)) \equiv D(\boldsymbol{y}- \boldsymbol{F}_N(\boldsymbol{\theta }^N)). \end{equation} \tag{ C2 }$$

The likelihoods for the $\boldsymbol{\theta }^N$ can be found from this restricted parameter space where $\boldsymbol{\theta }^L = \hat{\boldsymbol{\theta }^L}$. Thus, the restricted χ² is given by

$$\begin{eqnarray} {\chi}^2 &=& ((I - HD)(\bm{y}- \bm{F}_N(\bm{\theta }^N)))^T V^{-1}\nonumber\\ && \times (I - HD)(\bm{y}- \bm{F}_N(\bm{\theta }^N)) \end{eqnarray} \tag{ C3 }$$

$$\begin{eqnarray} \qquad &\equiv & {(\bm{y}- \bm{F}_N(\bm{\theta }^N))^T U^{-1}(\bm{y}- \bm{F}_N(\bm{\theta }^N))} \end{eqnarray} \tag{ C4 }$$

with U⁻¹ given by

$$\begin{eqnarray} \hspace{-1pc}U^{-1} & = & (I - HD)^T V^{-1} (I - HD) \end{eqnarray} \tag{ C5 }$$

$$\begin{eqnarray} \quad\quad \quad \qquad\;\; & = & {V^{-1} - V^{-1} H (H^T V^{-1} H)^{-1} H^T V^{-1}}.\nonumber \\ \end{eqnarray} \tag{ C6 }$$

In this way, all of the ${\btheta }^L$ do not have to be explicitly included in the χ², as long as the weight matrix, U⁻¹, is updated appropriately. Note that U⁻¹ may depend on $\boldsymbol{\theta }^N$.

C.1. Minimization Over x^true₁ and c^true

When errors in the independent variable are present, the true values must be solved for as part of the fit (see discussion in K08). For one supernova, $\boldsymbol{y}= (m_B, x_1, c)$, while the model is given by

$$\begin{equation} \big(M_B + \mu (z, \mathrm{cosmology}) - (\alpha x_1^{\mathrm{true}}- \beta c^{\mathrm{true}}), x_1^{\mathrm{true}}, c^{\mathrm{true}}\big). \end{equation} \tag{ C7 }$$

Using $\boldsymbol{\theta }^L = (x_1^{\mathrm{true}}, c^{\mathrm{true}})$ (M_B, α and β are global parameters and cannot be handled one supernova at a time), we have

$$\begin{equation} H = \left(\begin{array}{@{}ll@{}} -\alpha & \beta \\ 1 & 0\\ 0 & 1 \end{array} \right). \end{equation} \tag{ C8 }$$

The new χ² for this specific example (equation (C4)) is given by Equations (4) and (5).

C.2. Minimization Over Systematic Errors

Suppose that we have two kinds of measurements: supernova distance measurements (N SNe) and zero-point measurements (M zero points). The true value of each zero point is given by ZP_true = ZP_observed + ΔZP. Because the uncertainties on the supernova distances are unrelated to the uncertainties on the zero points,

$$\begin{eqnarray*} V = \left(\begin{array}{@{}ll@{}} V_{\mu }&\\ & V_{\Delta \mathrm{ZP}}\end{array} \right), \end{eqnarray*}$$

where V_μ is an N × N block and V_ΔZP is an M × M block. Similarly, $\boldsymbol{y}$ can be split into distances and zero points $\boldsymbol{y}= (\boldsymbol{y}_{\mu }, \boldsymbol{y}_{\Delta \mathrm{ZP}} = \boldsymbol{0})$. The model for each supernova distance is given by

$$\begin{equation} M_B + \mu (z, \mathrm{cosmology}) - \sum _{\lambda } \frac{\partial (m_B + \alpha x_1 - \beta c)}{\partial \mathrm{ZP}_{\lambda }}\Delta \mathrm{ZP}_{\lambda }. \end{equation} \tag{ C9 }$$

It is derived in the same way as in the previous section, with the substitutions $x_1 \rightarrow x_1 + \sum _{\lambda } \frac{\partial x_1}{\partial \mathrm{ZP}_{\lambda }} \Delta \mathrm{ZP}_{\lambda }$ and $c \rightarrow c + \sum _{\lambda } \frac{\partial c}{\partial \mathrm{ZP}_{\lambda }} \Delta \mathrm{ZP}_{\lambda }$. The model for the true offset of each zero point is simply ΔZP. Thus,

$$\begin{equation*} H = \left(\begin{array}{@{}l@{}} H_{\mu }\\ I \end{array} \right)\,. \end{equation*}$$

The upper block of U⁻¹ is given by

$$\begin{equation} U^{-1}_{\mathrm{upper\;block}} = V_{\mu }^{-1} - V_{\mu }^{-1}H_{\mu }\big(V_{\Delta \mathrm{ZP}}^{-1} + H_{\mu }^{T}V_{\mu }^{-1} H_{\mu }\big)^{-1} H_{\mu }^{T}V_{\mu }^{-1}\; ; \end{equation} \tag{ C10 }$$

none of the other blocks enter into the χ². Inverting this block (see Hager 1989) gives

$$\begin{equation} U = V_{\mu }+ H_{\mu }V_{\Delta \mathrm{ZP}}H_{\mu }^{T}. \end{equation} \tag{ C11 }$$

Equation (C11) gives us the terms we must add to the supernova distance modulus errors to correctly take the zero points into account.