SYSTEMATIC UNCERTAINTIES ASSOCIATED WITH THE COSMOLOGICAL ANALYSIS OF THE FIRST PAN-STARRS1 TYPE Ia SUPERNOVA SAMPLE

The sample analyzed in this paper includes SN Ia discovered by PS1 and observed in low-z follow-up programs. We apply the same selection criteria for the quality and coverage of the light curve observations to these samples as was done in R14. As detailed in R14, the low-z SN sample is selected from six different samples: Calán/Tololo [16SNe] (Hamuy et al. 1996), CfA1 [5SNe] (Riess et al. 1999), CfA2 [19SNe] (Jha et al. 2006), CfA3 [85SNe] (Hicken et al. 2009a), CSP [45] (Contreras et al. 2010), and CfA4 [43SNe] (Hicken et al. 2009b). We also include supernovae not discovered in these surveys but collected as part of the JRK07 (Jha et al. 2007) paper [8SNe]. The PS1 sample contains 113 SNe after selection cuts. While the focus of this paper will be on the PS1+low-z sample, we will compare results with the SDSS (Holtzman et al. 2008) and SNLS (Guy et al. 2010) samples. For these samples, we apply the same selection criteria from R14. We make all data used in this analysis publicly available, including light curve fit parameters.²⁰

External constraints from CMB, BAO and H₀ measurements are described in detail in R14. For all these measurements, we use the Markov chains derived by Planck Collaboration et al. (2013). The Planck data set that is quoted includes data from the Planck temperature power spectrum data, Planck temperature data, Planck lensing, and Wilkinson Microwave Anisotropy Probe (WMAP) polarization at low multipoles. The BAO measurement quoted is from the aggregate of BAO measurements of different surveys, as compiled by Planck Collaboration et al. (2013) and listed in R14. The H₀ measurement is from Riess et al. (2011).

Here, we briefly enumerate the dominant systematic uncertainties in the PS1+lz sample.

Calibration. Flux calibration errors are typically the largest source of systematic uncertainty in any supernova sample (C11). The original PS1 photometric system (T12) is based on accurate filter measurements obtained in situ. T12 adjusts the throughput of these measurements on a <3% scale for better agreement between synthetic and photometric observations of Hubble Space Telescope (HST) Calspec standards (Bohlin 1996).²¹ In this paper, we increase the size of the sample of Calspec standards that underpin the HST flux scale from 7 to 10 (adding 5, but eliminating 2) to reduce the uncertainty in the calibration.

We call the improved photometric system used throughout this paper the PS1_14 calibration system. For the low-z samples, we follow the C11 treatment of photometric systems. For our total calibration uncertainty, we combine uncertainties from the HST Calspec and Landolt standards, as well as the uncertainties in measurements of the bandpasses and zero points. We also explore noted discrepancies between the PS1 and SDSS photometric systems.

Selection Effects. Selection effects can bias a magnitude-limited survey, due either to detection limits or selection of the objects for spectroscopic follow-up. The SNANA simulator²² (Kessler et al. 2009b) allows us to use actual observing conditions, cadence, and spectroscopic efficiency to mimic our survey. The spectroscopic efficiency of a survey is particularly difficult to formalize if the survey does not have a single consistent follow-up program that is based on well-defined criteria for selecting targets.

We correct for PS1 selection effects by incorporating the observing history into a simulation and identifying the effective selection criteria that best match the data. For the low-z sample, we follow the same approach. For our systematic uncertainty, we explore how well our simulations match the data.

Light Curve Fitting. To optimize the use of SN Ia as standard candles to determine distances, most light curve fitters correct the observed peak magnitude of the SN using the width and color of the light curve. While each fitter accounts in some way for a light curve shape–luminosity relation (Phillips 1993) to correct for the width of the light curve, there is a disagreement between fitters about the best manner to correct for the color of the light curve. Using SNANA's fitter with the SALT2 model (Guy et al. 2010) as the primary light curve fitter, recently Scolnic et al. (2014 hereafter, S14) showed that there is a degeneracy between models of SN color when the intrinsic scatter of SN Ia is mostly composed of luminosity variation or color variation.

The primary method for determining distances uses SALT2 to find light curve parameters and afterward corrects the distances with the average bias from simulations based on these two models of SN color. For our systematic uncertainty, we explore the difference in distances between the two models from when we take the average.

Host Galaxy Relations. Multiple studies have shown relations between various host galaxy properties and Hubble residuals (e.g., Kelly et al. 2010; Sullivan et al. 2010; Lampeitl et al. 2010). However, there is no consensus about which host galaxy property is directly linked to luminosity (Childress et al. 2013), or whether these correlations may be artifacts of light curve corrections (Kim et al. 2013).

The primary fit does not include any corrections to SN Ia distances for host galaxy properties. For our systematic uncertainty, we explore whether correcting the distances of the SNe in the PS1+lz sample by including information about host galaxies properties is statistically significant.

MW Extinction Corrections. For each SN, we correct for the MW extinction at its specific sky location. Our primary fit uses values from Schlegel et al. (1998), with corrections from Schlafly & Finkbeiner (2011) and the restriction that E(B − V) < 0.5 in the direction of the SN. We include systematic uncertainties in the extinction correction from uncertainties in the subtraction of the zodiacal light, temperature corrections, and the nonlinearity of extinction corrections (Schlafly & Finkbeiner 2011).

Coherent Velocity Flows. To account for density fluctuations, we correct the redshift of each SN for coherent flows (Hudson et al. 2004). The primary fit corrects all redshifts for coherent flows starting at z_min = 0.01. For our systematic uncertainty, we measure the change in recovered cosmological parameters when we vary the minimum redshift of the sample.

Other. Uncertainties not analyzed in this paper, but considered in other studies include contamination by other types of SN, SN evolution, and gravitational lensing. Contamination by other types of SN was already discussed in R14, and we apply the same treatment here. While gravitational lensing should increase the amount of dispersion of the SN Ia distances at high-z σμ_lens = 0.055z (Jönsson et al. 2010), selection effects dominate any trend seen at high-z in the PS1 sample. For SN evolution, this uncertainty is already included in the light curve modeling uncertainty.

Finally, while we include cosmological constraints from the Planck survey (Planck Collaboration et al. 2013) to address systematics in external data sets, we compare the results when we include constraints from WMAP (Hinshaw et al. 2012).

To determine the entire systematic uncertainty of w from the PS1+lz sample, we follow two different approaches toward error accounting. The first approach follows C11, determining a covariance matrix that includes uncertainties from multiple sources. The second approach is similar to that of Riess et al. (2011), which finds the variations of cosmological constraints due to variants in the analysis. For example, in the second method one may find the different values of w using the PS1 calibration system as stated, or when modified to match the SDSS photometric system. In the first method, the errors from the PS1 calibration system are propagated. Since there are a number of discrete choices of how to do steps of the analysis in this paper, we incorporate both methods of error accounting. Each of these methods is different from the conventional method that adds all the systematic errors in quadrature at the end of the analysis.

The advantage of the approach shown in C11 is that it properly accounts for covariances between SNe and also for interactions between systematic uncertainties. The full covariance error matrix is given as

$$\begin{equation} \mathbf {C} = \mathbf {D}_{\mathrm{stat}} + \mathbf {C}_{\mathrm{sys}}, \end{equation} \tag{ 1 }$$

where D_stat is a diagonal matrix with each element consisting of the square of the intrinsic dispersion of the sample $\sigma _{{\rm int}}^2$ and the square of the noise error $\sigma _n ^2$ for each SN. C_sys is the systematic covariance matrix. C11 further separates C_sys into two components, only one of which may be further reduced with more SNe. For simplicity, we do not separate these components. Given the Tripp estimator (Tripp 1998) and using SALT2 to fit the light curve,

$$\begin{equation} \mu =m_B+\alpha \times x_1 -\beta \times c -M, \end{equation} \tag{ 2 }$$

where m_B is the peak brightness of the SN, x₁ is the stretch of the light curve, c is the color of the light curve, and α, β and M are nuisance parameters. Explanation of the derivations of α and β is given in R14. The systematic covariance, for a vector of distances $\boldsymbol {\mu }$, between the i'th and i'th SN is calculated as

$$\begin{equation} \mathbf {C}_{ij,\mathrm{sys}} = \sum _{k=1}^K \left(\frac{ \partial \mu _{i} }{ \partial S_k } \right) \left(\frac{ \partial \mu _{j} }{ \partial S_k } \right) (\sigma _{S_k})^2, \end{equation} \tag{ 3 }$$

where the sum is over the K systematics S_k, $\sigma _{ S_k}$ is the magnitude of each systematic error, and ∂μ is defined as the difference in distance modulus values after changing one of the systematic parameters. For example, in order to determine the covariance matrix due to a systematic error of 0.01 mag in the transmission function of r_p1 filter, we refit all of the SNe light curves after adding 0.01 mag to the zero point of all observed r_p1 values. Following C11, we do not fix α and β when we propagate the systematic covariance matrix. α and β are derived with SALT2mu (Marriner et al. 2011) in the statistical case, though when including the covariance matrix, we write a compatible routine that allows off-diagonal elements. As given in R14 (Table 4), when attributing the remaining intrinsic scatter to luminosity variation (σ_int = 0.115), α and β are found to be 0.147 ± 0.010 and 3.13 ± 0.12, respectively.

Given a vector of distance residuals for the SN sample $\Delta \boldsymbol {\mathbf {\mu }} = \boldsymbol {\mu } - \boldsymbol {\mu }(H_0,\Omega _{\scriptsize M},\Omega _{\Lambda },w,\boldsymbol {z})$ then χ² may be expressed as

$$\begin{equation} \chi ^2= \Delta \boldsymbol {\mu }^T \cdot \mathbf {C}^{-1} \cdot \Delta \boldsymbol {\mu }. \end{equation} \tag{ 4 }$$

We minimize Equation (4) to determine cosmological parameters that include $H_0, \Omega _{\scriptsize M}, \Omega _{\Lambda }$ and w. The cosmological parameters are defined in R14—Equation (3). All cosmological parameters quoted in this pair of papers are of the marginalized values and not the minimum χ² values. We assess the impact of each systematic uncertainty by examining the shift it produces in the inferred cosmological parameters. We also compute the "relative area" which we define as the area of the contour that encloses 68.3% of the probability distribution between w and $\Omega _{\scriptsize M}$ compared to when only including statistical uncertainties. For this analysis, we assume that the universe is flat. It is worth clarifying that the relative area may decrease as the contours shift in w versus $\Omega _{\scriptsize M}$ space, so relative area alone does not quantify the entire effect of a systematic error.

In the second approach, we redetermine distances based on variations (often binary) in the analysis methods (e.g., Riess et al. 2011). Unlike the method by C11, there is no systematic error component to the error matrix in Equation (3). Instead, cosmological parameters are found for each difference in analysis approaches. The overall systematic uncertainty of w from this method is the standard deviation of values for w from the variants to the primary fit.

Uncertainties in the calibration of the various samples comprise the largest systematic uncertainty in our analysis. In Figure 1, we show a schematic describing the calibration of the various subsamples. The overall systematic uncertainty in the calibration of our combined PS1+lz sample may be expressed as the combination of three uncertainties. The first component encompasses systematic uncertainties in the nightly photometry and how well the filter bandpasses are measured. For the PS1 sample, R14 presents analysis of the systematic uncertainty due to spatial and temporal uncertainties in the nightly photometry. T12 presents the uncertainty in how well the bandpasses are measured (uncertainty of filter edges <7 Å). T12 also analyzes uncertainty in the atmospheric attenuation compensation for the filter zero points.

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Spatial and temporal variation of the filter bandpasses propagate into our total calibration uncertainty in three ways: how the catalog photometry is determined, how the photometry of the Calspec standards is determined, and how the photometry of the supernovae is determined. We expect that the uncertainty in the nightly zero points to be small. We find by comparing Pan-STARRs and SDSS photometry that any variation of the PS1 photometry across the focal plane for colors 0.4 < g − i < 1.5 is less than 3mmag and is difficult to detect because of noise. The effect of variation of the filter bandpasses on photometry of the Calspec standards and the supernovae are significantly larger because of the very blue colors of a large fraction of the Calspec standards and the narrow spectral features of these supernovae. These effects are both considered.

The second major component of the total calibration uncertainty is in determining the flux zero points of each filter based on observations of astronomical standards. Since the accuracy of the internal PS1 measurements of the flux zero points is not better than 1%, the zero points are adjusted so that the observed photometry of HST Calspec standards (e.g., AB–HST Calspec; Bohlin 1996) matches the synthetic photometry of these standards. Analogously, for the low-z sample, this uncertainty encompasses the accuracy of the color transformation of Landolt standards. For PS1, the adjustment of the photometry to agree with the synthetic photometry dominates the uncertainty in the filter measurements by T12. Both uncertainties are included in our analysis.

The third main component of the total calibration uncertainty is the accuracy in the measurements of the standard stars (e.g., HST Calspec or Landolt). This is composed of errors in the color of the standard stars and the absolute flux of the standards. For the PS1 sample and select measurements in the low-z samples, this uncertainty is due to possible errors in measurements of the HST Calspec standards. For most of the low-z sample, this uncertainty is due to color and absolute flux errors from the realization of the Vega²³ magnitude system as implemented in the standard catalogs of Landolt. A common flux scale for the PS1 supernovae and low-z SNe can be achieved by the binding between the Landolt catalog and Calspec standards (Landolt & Uomoto 2007). In a future analysis, we plan to cross-calibrate the Landolt catalog to the PS1 catalog to further improve the flux scales of the different samples. If we limited our analysis to SNe from a single survey, the overall absolute flux calibration would be degenerate with the absolute peak magnitude of SNe. However, combining distance moduli of SNe from multiple surveys requires that there is a common absolute flux scale.

R14 presents an error budget for the PS1 photometric system. Here we explain many of these uncertainties, along with uncertainties of the low-z calibration. We also detail the derivation of new zero point offsets for the PS1 calibration that are used in R14. The total uncertainty in the recovery of cosmological parameters due to calibration for the PS1+lz sample is given in Table 1. For each uncertainty described, this uncertainty is independently added to each observed magnitude of the SN, and the light curves are refit. Afterward, cosmological constraints are redetermined.

Table 1. Calibration Systematics

Systematic	$\Delta \Omega _{\scriptsize M}$	Δw	Rel. Area	$\Delta \Omega _{\scriptsize M}$	Δw	Rel. Area
		SN Only			SN+BAO+CMB+H0
Stat. Only	0.000	0.000	1.000	0.000	0.000	1.000
	$(\Omega _{\scriptsize M}=0.223^{+0.209}_{-0.221})$	$(w=-1.010^{+0.360}_{-0.206})$		$(\Omega _{\scriptsize M}=0.284^{+0.010}_{-0.010})$	($w=-1.131^{+0.049}_{-0.049}$)
PS1 ZP+bandpass	0.005	−0.038	1.285	−0.003	−0.025	1.221
PS1 g	0.008	−0.038	1.074	−0.001	−0.013	1.081
PS1 r	0.004	−0.006	1.028	0.001	0.001	1.000
PS1 i	0.000	0.002	1.119	0.000	−0.005	1.079
PS1 z	−0.004	−0.001	1.068	−0.001	−0.009	1.080
Low-z ZP+bandpass	−0.001	−0.012	1.070	−0.001	−0.010	1.085
Landolt color	−0.000	0.004	1.013	0.001	0.001	1.004
Calspec uncertainty	−0.001	−0.025	1.089	−0.002	−0.020	1.126
Abs. ZP	−0.000	0.004	1.004	0.001	0.001	1.001
SALT2 calibration	0.029	−0.063	1.202	0.000	−0.002	1.033
All cal. systematics	0.024	−0.093	1.566	−0.004	−0.035	1.309
	$(\Omega _{\scriptsize M}=0.248^{+0.210}_{-0.165})$	$(w=-1.105^{+0.435}_{-0.305})$		$(\Omega _{\scriptsize M}=0.280^{+0.013}_{-0.012})$	($w=-1.166^{+0.067}_{-0.069}$)

Notes. Individual systematic uncertainties for each of the PS1 passbands as well as the systematic uncertainties for each low-z sample. Relative area is the size of the contour that encloses 68.3% of the probability distribution between w and $\Omega _{\scriptsize M}$ compared with that of statistical-only uncertainties.

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The Pan-STARRS AB magnitude system, as described in T12, is based on small (<0.03 mag) adjustments to highly accurate measurements of the PS1 system throughput and filter transmissions measured in situ. Perturbations to each filter transmission are optimized so that "synthetic" photometry, using measurements of filter transmission throughputs and stellar spectra, agrees with observations of HST Calspec standard stars. To do this, T12 analyzed the PS1 observations of seven Calspec standards, all observed on the same photometric night. The error in how well the Calspec spectral energy distributions (SEDs) are defined on the AB system as well as the offsets between the observed and synthetic Calspec magnitudes represent the two largest errors in the PS1 calibration system. T12 finds that the entire systematic uncertainty from absolute calibration in each filter is ∼0.017 mag. We recalculate that value here.

Once the PS1 calibration is defined to be on the AB system, there is an uncertainty from the relative calibration between the fields with Calspec standards to the rest of the sky. To do the relative calibration, Schlafly et al. (2012) use repeat PS1 observations of stars. Star catalogs created by this process are used in our supernova pipeline. Internal consistency tests show Schlafly et al. (2012) achieve field-to-field relative precision of <0.01 mag in g_p1, r_p1, and i_p1 and ∼0.01 mag in z_p1. These errors are included in our overall zero point uncertainties, after dividing by $\sqrt{10}$—the number of fields. While a following discussion will focus on agreement between the absolute calibration of PS1 and SDSS, it is worth mentioning that Schlafly et al. (2012) find greater internal inconsistencies at the ∼0.01 mag level in the SDSS photometry than in the PS1 photometry.

To improve the PS1 absolute calibration, we analyze a larger sample of Calspec standards that have been observed throughout the PS1 survey. In total, there are 12 Calpsec standards that have been observed by PS1 in griz_p1 that are not saturated in the observations. These standards were observed so that they avoided the direct center of the focal plane, where there are some unresolved discrepancies as described by R14. We measure the PS1 magnitudes of the observed Calspec standards in the same way as T12. We then apply a zero point offset obtained by computing the difference of the magnitudes of stars in the fields with the stars in the full-sky star catalogs set by Schlafly et al. (2012). To avoid a Malmquist bias in the determination of the zero point, we empirically determine the magnitude limit at which stars can be used for this comparison. We also remove any observations of Calspec standards where there is greater than a 0.03 mag difference between the aperture and point-spread function (PSF) photometry, as we found this adequately removes any saturated observations. As the Calspec standards observed are so bright, we follow T12 and include a 0.005 mag uncertainty to account for how some of the standards may be near the saturation limit.

To make full use of the observations of Calspec standards, we must consider how the filter functions change across the focal plane. In Figure 2, we show the change in synthetic magnitudes of stars in the Pickles' library (Pickles 1998), Calspec standards and supernovae for a given color and position on the focal plane. We find the variation in synthetic magnitudes for supernovae and Calspec standards may be significantly larger than the variation of stars in the narrow color range used to define the stellar zero points. Therefore, given the measurements of filter functions across the focal plane (measured at Δr = 0.15 deg), we transform the observed magnitudes of the Calspec standards to a uniform system defined at the center of the focal plane. As shown in Figure 2, this correction for the blue Calspec standards may be as large as 5 mmag. This approach is similar to that of Betoule et al. (2012).

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For each observation of a Calspec standard, we follow Schlafly et al. (2012) and assign an observational error of 0.015 + σ_mag-psf, which Schlafly et al. find adequately describes the scatter seen in their star catalogs. To determine the net adjustment needed for each passband, we find the weighted difference of the observed and synthetic magnitudes of the Calspec standards. For this process, we add an additional uncertainty of 0.008 mag to each difference in order to represent the uncertainty in the ubercal process. In Appendix A, we present the entire set of Calspec standards observed and their synthetic magnitudes in the PS1 system,²⁴ as well as the observed magnitudes. From Figure 3, we find that corrections should be added to the zero points of observations in each filter (given by T12 and Schlafly et al. 2012) such that Δg_PS1 = −0.008, Δr_PS1 = −0.0095, Δi_PS1 = −0.004, Δz_PS1 = −0.007. These adjustments represent the weighted difference of the observed and synthetic magnitudes of the Calspec standards. R14 includes these offsets in their light curve fits; we call the new calibration "PS1₁₃." We determine the uncertainties in the mean for all four passbands to be [0.0085,0.0050,0.0060,0.0025] mag. These uncertainties are included in the overall calibration uncertainty table of R14 (T).

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T12 finds consistency of ∼0.01 mag among their seven Calspec stars used to define the AB system. For two of the standards, 1740346 and P177D, T12 noticed disagreement at the 0.02 mag level in i_p1. T12 explained that this disagreement may be due to the discontinuity at 800 nm where the STIS spectra gives way to NICMOS in the Calspec SEDs. With our larger sample of Calspec standards, we can see that the disagreement T12 noticed is most likely not due to 1740346 and P177D, but rather the three "KF" stars (KF08T3, KF01T5, KF06T2), which are red stars with r − i ∼ 0.3. In the analysis done by T12, the spectra of the KF stars did not include STIS data, which covers the optical spectrum. We find that with an updated STIS spectrum of KF06T2, the synthetic photometry is corrected by ∼0.02 mag, in better agreement with observations. While we present the entire set of HST Calspec standards observed, we exclude the two KF stars without STIS measurements from our absolute calibration.

Including the uncertainties described above, R14 finds the combined uncertainty for each filter in quadrature is ∼0.012 mag. Uncertainty in g_p1 appears to have the largest effect on the cosmological constraints compared to the other passbands. Interestingly, a calibration error in r_p1 appears to have a different effect than the other passbands because the change in distance due to peak brightness in this filter cancels out the change in distance due to color (for >50% of redshift range). The effect on recovered cosmological parameters from griz_p1 together (labelled "PS1 ZP +bandpasses" in Table 1) increase the relative area of the constraints by 40% (SN only).

We also consider the effects of the third component of the total systematic uncertainty in calibration (bottom level of Figure 1): that of the calibration of the HST Calspec standards to the AB system. T12 states this uncertainty is 0.013 mag for all filters. We find a more appropriate solution is to take the uncertainty as the inconsistency between the synthetic photometry of the STIS measurement of BD17 and observed photometry from Advanced Camera for Surveys (ACS) given in Bohlin & Gilliland (2004). This error is explained by the STIS flux for BD17 that continuously drops from a multiplicative factor times the flux of 1.005 at 4000 Å to 0.985 at 9500 Å.²⁵ This is similar to the 0.5% slope uncertainty stated by Bohlin & Hartig (2002). Additionally, there is a measurement error in the repeatability of the individual measurements with STIS spectra, on the order of 0.005 mag (Betoule et al. 2012). The impact of these uncertainties of the Calspec standards is given in Table 1 and increase the relative area by ∼5%. Finally, the absolute flux of the Calspec system itself must be taken into account. This uncertainty will be considered as part of the low-z discussion later in this section (given as "Abs. ZP" in Table 1).

Surveys like SDSS, CSP, and SNLS have recently undertaken large, collaborative efforts (Mosher et al. 2012; Betoule et al. 2012) to improve the agreement between their respective calibration systems. Here we focus on the consistency of the absolute calibration between PS1/SDSS as the absolute calibration differences between SDSS/SNLS (Betoule et al. 2012) and SDSS/CSP (Mosher et al. 2012) have been shown to be less than 1%. As SDSS photometry has been defined to be on the AB system, this analysis is an alternate diagnostic to quantify the accuracy of the PS1 photometric system itself.

T12 compares the PS1 magnitudes of stars in the MD09 field with those tabulated by SDSS as part of Stripe 82. They note ∼0.02 mag offsets at 3–4σ after transforming the SDSS DR8 catalogs into the PS1 system with linear terms in color.²⁶ T12 conclude that discrepancies are most likely due to errors within the SDSS calibration system.

For comparisons between SDSS and PS1, we repeat the analysis in T12, now using the most up to date S82 catalogs from Ivezić et al. (2007) with AB offsets from Betoule et al. (2012). Discrepancies between these two systems are shown in Figure 4. The offsets between the calibration zero points of PS1 and SDSS are given in Table 2 and are up to ∼0.02 mag in r_p1. Betoule et al. (2012) redefines the SDSS AB system using SDSS PT observations of seven Calspec standards. While T12 explains that the absolute calibration of SDSS DR8 may be biased from using SDSS SEDs, Betoule et al. (2012) uses HST Calspec spectra to define the flux system so there should not be an issue.

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To further probe the inconsistency between the PS1 and SDSS calibration systems, in Appendix A, we compare synthetic and observed magnitudes for the SDSS standards in the same way we did for PS1. In both Figure 3 and Appendix A, we show how, for PS1 and SDSS, the zero points should be shifted into agreement with the color-transformed system of the other. We note that for SDSS, the dispersion of the differences between synthetic and observed photometry is smaller than that for PS1, and likely does not explain the difference in absolute zero points. In the comparison of PS1 and SDSS catalogs shown in Figure 4, the differences have a very small dependence on the color g − r (<5 mmag for g − r < 1.2, highest in the z band). This result is encouraging that while the absolute zero points of the filters are in disagreement, the filter transmission curves appear to be well measured for both systems. Also, the zero point discrepancies do not appear to be correlated across filters.

There are three Calspec standards observed by both PS1 and SDSS: P177D, GD71, and GD153. The discrepancies between the PS1 and SDSS observations of these standards are very similar to the overall discrepancies in the calibration of these two systems and do not provide enough leverage to understand the source of the differences. Therefore, more work must be done to understand the disagreement between the PS1 and SDSS calibration. One possible cause may be due to nonlinearities with these particular observations of very bright standards, which T12 estimated for the PS1 observations to be up to ∼0.005 mag. We conclude that when combining data from the PS1 and SDSS surveys that the AB offsets between the two must be taken into account. We find that the change in w when the PS1 calibration system is chosen to be in agreement with SDSS is Δw = +0.018 with constraints only from SN measurements, and Δw = −0.006 when including CMB, BAO, and H₀ constraints (due to how constraints combine). This difference for the SN-only constraints is the largest variant in our analysis.

We rely on analysis of past studies, in particular C11 and Kessler et al. (2009a), for our understanding of the calibration systematics of low-z surveys. We discuss our additions to the growing low-z sample: the CfA4 survey, a recalibration of the CfA3 survey, and a larger set of CSP SNe. Given the U-band systematic error discussed in Kessler et al. (2009a), we follow the C11 decision to not use rest-frame observations in the U band.

Each of the newly added nearby samples has photometry on its natural system. For each sample, the absolute calibration is defined by the magnitudes of the fundamental flux standard BD17. For CSP, the magnitudes of BD17 are given in the natural system. For CfA3 and CfA4, we use the linear transformations from the Landolt (Landolt 1992) and Smith (Smith et al. 2002) colors to the natural system to determine the magnitude of BD 17°4708. These transformations are given in Hicken et al. (2009a). The magnitudes of BD17 given in Landolt (1992) are transformed to calibrate the BV bands, and the magnitudes of BD17 given in Smith et al. (2002) are transformed to calibrate the r'i' bands.

We note two peculiarities with the CfA3 and CfA4 samples. To analyze these two samples, we take passbands defined by C. Cramer et al. (in preparation) in which highly precise measurements are obtained of the telescope-plus-detector throughput by direct monochromatic illumination. This method, like that done in T12, is based on Stubbs & Tonry (2006). However, the CfA4 survey must be broken into two separate time periods because it was found that a warming of the CCDs of KeplerCam to remove contamination in 2011 May "produced a dramatic difference" in the response function of the camera (Hicken et al. 2009b). This difference is quantified by measurements of the V, V − i', U − B and u' − B color coefficients between the Landolt/Smith measurements and the natural system. Therefore, we use a set of transmission functions for before 2009 August and after 2011 May, when the system was measured to be consistent, and a separate set of transmission functions between these two dates (Hicken et al. 2009a). The second peculiarity is that when analyzing the CfA4 light curves, we found the uncertainties for each observation to be on average roughly $\sqrt{3}$ larger than that of CfA3, a surprising result considering the similarity of the surveys. We discovered this was due to a change in uncertainty accounting in the software pipeline based on the number of image subtractions done for each observation (M. Hicken 2013, private communication). We have returned the uncertainty propagation method to that used for CfA3, which we believe to be correct.

For our uncertainties in the low-z bandpasses and zero points (top level of Figure 1), we follow the analysis of C11. We use the shifted Bessell bandpasses found empirically by C11 with uncertainties of 12 Å (edge locations) for the JRK07 sample and adopt zero point uncertainties of 0.015 mag. For CSP, the uncertainties in the bandpasses are taken from Contreras et al. (2010) and the uncertainties in the zero points for each filter are 0.008 mag, as given in C11. While more work must be done to better determine this zero point uncertainty, this result is consistent with the small discrepancies of ∼0.01 mag seen between the CSP and SDSS samples (Mosher et al. 2012). We take the zero point uncertainties for the CfA4 sample given in Hicken et al. (2009a) of 0.014, 0.010, 0.012, 0.014, 0.046 mag in BVr'i'u', which are larger than those found by C11 for the CfA3 sample of 0.011, 0.007, 0.007, 0.007 mag for BVRr'. Since the uncertainty of the CfA4 bandpasses measured by C. Cramer et al. (in preparation) has not yet been given, we fix this uncertainty to be that found by T12 for the PS1 passbands (3 Å), as Cramer et al. and T12 perform very similar measurements to determine the instrument response.

While the absolute flux of the HST Calspec standards is defined by the AB system, the absolute flux of the Landolt standards is not well defined. Although the Landolt measurements are self-consistent, it is not known exactly how the absolute flux was defined. Therefore, there may be discrepancies between the absolute flux of these different sets of standards (see bottom level of Figure 1). We follow the analysis of Landolt & Uomoto (2007) of the calibration agreement between the Landolt catalog and HST observations of Calspec standards for an uncertainty of 0.006 mag between the absolute fluxes of the two samples. For the difference between the Smith and AB systems, we take the uncertainty in determining the AB offsets for the SDSS sample of ∼0.004 mag (Betoule et al. 2012). We also account for uncertainties in the colors of Landolt measurement of BD17 itself of ∼0.002 mag (Regnault et al. 2009). This last uncertainty could be reduced by defining the low-z samples using more standards besides BD17 (a subdwarf star), which will be done in a future work.

While we have discussed the entirety of calibration errors that affect the measurements of the supernova in our sample, we must also propagate how calibration errors affect the SALT2 light curve model that we use to fit distances. To do so, we refit our entire SN sample with 100 variants of the SALT2 model based on the calibration errors of the training sample used to determine the model (Guy10). These variants were provided by the SALT2 team. For the total systematic from the SALT2 calibration error, we sum the covariance matrices over all of the iterations and then divide by the total number of iterations. This impact of this uncertainty is quite large with respect to our other calibration uncertainties as it increases our w versus $\Omega _{\scriptsize M}$ constraints for the SN-only case by >15%.

The impact on the recovery of cosmological parameters from all of the uncertainties discussed above are presented in Table 1. Uncertainties in the low-z transmission measurements are significant (SN-only relative area ∼1.07), though do not have as great an increase on the relative area as the uncertainty in the PS1 transmission throughputs. We present the distance residuals from the best-fit cosmology for each low-z survey in Figure 5. We refer here to R14 (Section 7.2), which details the quality culls on the light curves and reduces the number of light curves significantly. The intrinsic dispersion (σ_int), rms and effects on retrieved cosmology from removing a particular subsample are all shown in Table 3. We note that the σ_int of the PS1 sample (σ_int = 0.07) is lower than in other samples, though is closest to the CSP sample. We also note that the CfA4 sample appears to have a larger scatter (σ_int = 0.22) than the other samples. The various values of σ_int may be due to over or under-estimation of calibration uncertainties. Part of this trend may also be due to a low-z Malmquist bias, which will be discussed in a later section. The maximum tension between the low-z subsamples is about <2σ from the mean. REF27: There are 23 SNe observed by both CSP and CfA3, and the mean difference in distances for these SNe is 0.026 ± 0.03 mag (CSP-CfA3) with an rms of 0.16 mag. We only allow a single distance for a given supernova, and choose based on which has better cadence near peak. Following C11, we add different σ_int values to the photometric uncertainty of the SN distances for our high and low-z samples, though not for the individual low-z subsamples.

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To make use of SNe discovered near the magnitude limits of the PS1 supernova survey, we account for the selection bias toward brighter SNe. These SNe may be brighter, even after the light curve shape and color corrections, due to the intrinsic dispersion of SN Ia and/or noise fluctuations. To determine the correction for this selection bias, we simulate the PS1 Medium Deep Survey and the spectroscopic follow-up. We apply the SNANA Monte Carlo (MC) code to generate realistic SN Ia light curves with the same cadence, observing conditions, non-Gaussian PSFs (R14), and zero points as our actual data. All simulations are based on a standard ΛCDM cosmology with w = −1, Ω_M = 0.3, Ω_Λ = 0.7, H₀ = 70 and an absolute brightness of a fiducial SN Ia of M_B = −19.36 mag Kessler et al. (2009b). Varying these initial conditions has negligible impact on our derived Malmquist bias. To simulate the intrinsic scatter of SN Ia, we use the SALT2 model (Guy et al. 2010, hereafter called Guy10), which will be discussed in detail in the next section.

While the photometric selection bias may be inferred from survey conditions, the spectroscopic selection bias must be estimated with an external model for selection efficiency. We first find the minimum signal-to-noise ratio (S/N) values of all PS1 SNe in the photometric and spectroscopic samples. For photometric detection of a SN, we require that there are two measurements in any filter with S/N > 5. For spectroscopic follow-up, we find from our sample that there are always at least two measurements in any filter with S/N > 7 and one measurement in any filter with S/N > 9. The spectroscopic efficiency function, E_spec, is applied after the S/N cuts are in place and is split into two parts because there were separate PS1 follow-up programs for high and low-z SNe:

$$\begin{eqnarray} &&E_{{\rm spec}} (z\le 0.1)=e^{-z/z1}; \nonumber\\ &&E_{{\rm spec}} (z>0.1)=1.0/[1 + (r_{\rm p1}-19.0)]^{r1}. \end{eqnarray} \tag{ 5 }$$

Here, z is the redshift and r_p1 is the observed peak magnitude. We vary z1 and r1 to optimize agreement between the number of observed SNe with redshift for the data and MC. We find that z₁ = 0.05 and r₁ = 2.05.

A comparison of properties of the SNe recovered (e.g., redshift distribution, S/N distribution) from the PS1 simulation to the actual data is shown in Figure 6. Overall we find very good agreement between our data and MC. Discrepancies in these comparisons, especially that of cadence, may be explained by observational effects like masking and saturation which we do not model in the simulation. To determine the systematic uncertainty of the selection bias, we vary the spectroscopic selection function at high-z and find where the data/MC comparison of the redshift distributions is worse than our best fit by 2σ. We conservatively choose 2σ here because our spectroscopic follow-up program was done with multiple telescopes and assuming one follow-up program may lead to a bias. For this test, we exclude z < 0.25 as the PS1 selection bias at low-z should be negligible, and the low-z sample dominates at z < 0.1. Results of this simulation (r₁ = 2.5 in Equation (5)) are overplotted in Figure 6. We fix the uncertainty at a given redshift in the selection bias to be the difference between the selection bias for these two simulations. The Malmquist bias is shown in the following section. The uncertainty in the selection bias is included in the overall uncertainty budget at the end of the analysis in Table 5.

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For the low-z sample, it is much more difficult to simulate the individual subsamples because of the smaller statistics, discoveries that are external to the survey, and in some cases, multiple telescopes used for observations. Selection effects in these surveys result from the discovery threshold and the selection for spectroscopic follow-up, not from limitations in S/N, because the S/N reach ∼10 even for the faintest observations in the sample. To quantify the Malmquist bias in the low-z sample, we find the spectroscopic/selection efficiency of the aggregate of the low-z surveys. Similar to how we found the spectroscopic efficiency of the PS1 survey, we find the selection efficiency function that best describes the combined low-z sample:

$$\begin{equation} {\rm Spec.\, Sel.\, Eff.} = e^{-\frac{(m_B-14)}{1.5}}, \end{equation} \tag{ 6 }$$

where m_B is the peak B-band magnitude and is greater than 14. This efficiency function assumes a flat volumetric rate at low-z. A comparison of the simulation to the data, including trends of the SALT2 light curve parameters color (c) and light curve shape (x₁) with redshift, is shown in Figure 7. For the entire low-z sample, we find a mean difference in colors for z > 0.04 and z < 0.04 of Δc = −0.031 ± 0.011 and a mean difference in light curve shape of Δx₁ = +0.53 ± 0.16. We analyze these same trends for each low-z subsample in Appendix B. The color and stretch trends with redshift appear to change more significantly in CfA3 and CfA4 than in JRK07 and CSP.

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We further probe whether the survey is magnitude-limited in Appendix B by comparing the redshift distribution of the low-z sample to the distribution of galaxy redshifts in the New General Catalogue (NGC). We find that the redshift distribution of the SNe is actually fairly well represented by the redshifts in the NGC, though higher redshift galaxies are slightly overrepresented. This suggests that any preference for brighter targets at a given redshift should be negligible. Rather, the only selection bias of the low-z sample should be the selection bias of the NGC galaxies themselves. In Figure 7, we compare simulations of a NGC-based survey, where the redshift distribution matches the NGC distribution, to the data. The redshift distribution of the simulations of the magnitude limited and galaxy-targeted surveys are similar, though the NGC-based simulation cannot replicate the trends of color and stretch with redshift. Therefore, like we did for the PS1 selection bias, we determine the systematic uncertainty on the selection bias empirically rather than using the NGC-follow-up to determine the systematic uncertainty. A survey that reproduces the trends with color and stretch less well than our optimal method by 2σ is also shown in Figure 7.

This uncertainty for the low-z sample is roughly 0.003 mag for the entire low-z sample. The uncertainty due to generalizing all of the low-z surveys is included in the overall selection effect uncertainty in Table 5. The change in w from not accounting for any low-z Malmquist bias is large: Δw ≈ −0.035 (SN-only). By opting to determine the error on the selection bias empirically rather than allowing for a systematic based on a model in which there is no selection bias, we significantly reduce the systematic uncertainty of the Malmquist bias.

SN Ia light curve fit parameters determined using the SNANA's light curve fitter with a SALT2 model (Guy et al. 2010) are presented in R14. Recently, S14 found that a model of the intrinsic brightness variation of SN Ia that is dominated by achromatic variation may lead to a similar color–luminosity relation as one in which there is a large amount of chromatic variation. Here, we find the different values of w when the dispersion of SN Ia distances is attributed to a model that contains mostly (⩾70%) luminosity variation or a model that contains mostly color variation.

R14 shows the values of the color–luminosity relation, β, depend on different assumptions about the source of intrinsic scatter. They find that β = 3.10 ± 0.12 when scatter is attributed to luminosity variation and β = 3.86 ± 0.15 when scatter is attributed to color variation. The latter value is only <2σ from a MW like reddening law, though R14 notes this high value is strongly pulled by the low-z sample only. To understand the consequences of these two assumptions about intrinsic scatter, we first match simulations, as explained in the previous section, of two variation models to the data. We create simulations using SNANA with two models, one in which the majority of scatter is due to luminosity variation (Guy10, color-luminosity variation–30%/70%) and the other in which the majority of scatter is due to color variation (Chotard 2011, hereafter Ch11, color-luminosity variation—75%/25%). For the model with a majority of luminosity variation, the color–luminosity relation, β is set to be 3.1, the value found after attributing all of the Hubble residual scatter to luminosity variation (R14). For the model with mostly color variation, SN Ia color is composed of a dust component that correlates with luminosity via a MW like reddening law (β = 4.1) and a variation component (Ch11). SNANA provides these two models. For Ch11, SNANA converts a covariance matrix among bands into a model of SED variations. The Ch11 model used is denoted "C11₀" in SNANA.

S14 showed that trends in Hubble residuals versus color depend not only on the intrinsic scatter but also the underlying color distribution. Parameters for the underlying color and stretch distributions that best match simulations to the data are given in Table 4. These values are found using a grid-based search of the x₁ and c asymmetric Gaussian parameters. S14 showed that simulations with a Guy10 variation model, combined with a slightly asymmetric underlying color distribution (Kessler et al. 2013), cannot reproduce the significant asymmetry around c ∼ −0.1 seen in the trends of Hubble residuals versus color (similarly shown for PS1+lz sample, given in Figure 8, top). Some correlation between color and Hubble residuals should be expected from the Tripp distance estimator, though the trend seen here can be best explained by a narrow and asymmetric underlying color distribution (Table 4). To improve the consistency of the Guy10 model with observations, we find that the underlying color distributions for this variation model should be significantly asymmetric. The distribution presented here is much more asymmetric than that given in Kessler et al. (2013) for the SDSS or SNLS surveys.

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Table 4. Asymmetric Gaussian Parameters to Describe the Parent Distribution of x₁ and c

Parameter	Sample	Intr. Variation	$\bar{x}$	σ−	σ+
c	PS1	Ch11	−0.1	0.0	0.095
c	PS1	Guy10	−0.08	0.04	0.13
c	Low-z	Ch11	−0.09	0.0	0.12
c	Low-z	Guy10	−0.05	0.04	0.13
x1	PS1	Ch11	0.5	1.0	0.5
x1	PS1	Guy10	−0.3	1.2	0.8
x1	Low-z	Ch11	0.5	1.0	0.5
x1	Low-z	Guy10	−0.3	1.2	0.8

Notes. The parameters defining the asymmetric Gaussian for the color and light curve shape distributions: $e^{ [-(x - \bar{x})^2/2\sigma _{-}^2] }$ for $x < \bar{x}$ and $e^{ [-(x - \bar{x})^2/2\sigma _{+}^2] }$ for $x > \bar{x}$. The optimized parameters for the variation models with a majority due to color variation (Ch11-Chotard et al., color-luminosity variation—75%/25%) and luminosity variation (Guy10-Guy et al., color-luminosity variation—30%/70%) are given.

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We present our distances biases with redshift in Figure 9. Distances included here are found once intrinsic dispersion of SN Ia is attributed to luminosity variation. We find differences in the distance corrections to be up to 0.03 mag and note that the offsets are fairly constant with redshift for the PS1 sample. We find overall a mean offset of −0.01 mag for both models with the low-z sample, and this offset is subtracted out from both the low-z and high-z samples as they are combined. This offset is due to the asymmetric underlying color distribution and covariances between m_b and c. We further understand the predictions by our two models by analyzing trends between Hubble residual and color at separate redshift bins in Figure 10. While there are discrepancies in the predictions for the two models of SN variation, the statistics of the PS1+lz sample do not favor either model; the data cannot break the degeneracy. In Figure 6, we also overplot a simulation with our color variation model, and find no noticeable differences from our simulation with luminosity variation.

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Recent analyses (e.g., Campbell et al. 2013; Kessler et al. 2013) attempt to remove any fitter bias (including the Malmquist bias) by using simulations to find the mean distance residual for a given redshift, like the ones we have done here. We may use our two different simulations to find the systematic uncertainty in our distance corrections from our incomplete understanding of the true variation model. For our primary analysis, we correct for the average distance residual at all redshifts from the two simulations. These average distance residuals are shown in Figure 9. The difference in w after applying the corrections from one model or the other is Δw ≈ 0.055 (Lum.-Col.). By taking the average, we reduce the systematic uncertainty due to the color models by a factor of two.

A separate way to determine the systematic uncertainty from SN color is to compare the values of w when using SALT2 in the conventional manner (attribute intrinsic scatter to luminosity variation) as well as using BaSALT (S14), a Bayesian approach that separates the color of each SN into components of color variation and dust. To retrieve the component of color that correlates with luminosity (c_dust), BaSALT applies a Bayesian prior to the observed color (c_obs) such that

$$\begin{equation} c_{{\rm dust}}=\frac{1}{P} \int _{c>\bar{c}} c e^{-(c-c_{{\rm obs}})/2\sigma _{c_n}^2}e^{-(c-\bar{c})^2/\tau _S(z)^2} \partial c. \end{equation} \tag{ 7 }$$

Here, c_n is the noise from the color measurement and P is a normalization constant. The second part of Equation (7) describes the Bayesian prior (Riess et al. 1996) for the dust distribution where $\bar{c}$ is the blue cutoff of the distribution. τ_S(z) describes the shape of the one-sided Gaussian due to extinction for a given redshift z for each survey S; the dependence of τ on survey and redshift allows selection effects to be modeled. As discussed in the previous section, for the low-z sample we do not expect selection effects to significantly vary with redshift and therefore we do not change τ with redshift. For the PS1 sample, however, τ varies with redshift.²⁷ The results are shown in Figure 8 (bottom). The BaSALT method does not allow the color fits to be bluer than c = −0.1 and therefore the particular nonlinearity between Hubble residual and color is much weaker with this method. However, given the BaSALT method, we introduce with the color prior additional correlations between color and distance that are dependent on the uncertainty of the prior. Therefore, we only use the BaSALT method as a way to confirm our results using the simulations discussed above. We find a change in w of Δw = w_B − w_S = −0.06 when the BaSALT method is used instead of the SALT2 method. This is nearly equal to the difference found for w from using distance corrections from simulations of the different scatter models. With the BaSALT method, we still must use simulations to correct for any further biases due to covariances between color and the other light curve fit parameters. If information about the underlying color distribution was included during the light curve fit itself, this would not be needed.

The difference in recovered cosmological parameters due to variation models largely depends on the mean color of the sample. If the mean color of the sample is constant with redshift, we should not expect significant (|Δw| > 0.01) discrepancies due to finding the wrong β. In the PS1+lz sample, the PS1 subsample is bluer by ∼0.03 mag than the low-z sample, and thus the effect of an incorrect β is significant.

We now explore whether the relation between the light curve shape and luminosity is adequately described by a linear model. Similar to the analysis of color, we observe the trend between Hubble residuals and light curve shape, known in SALT2 as "stretch" (x₁). Doing so, we find that a second-order polynomial ($\alpha _1 \times x_1 + \alpha _2 \times x_1^2$, where [a₁, a₂] = [0.160 ± 0.010, 0.017 ± 0.007]) appears to better fit the data than the conventional linear model (α × x₁). The reduction in χ² when including the second order is from 312 to 301.8 for 312 SNe (after uncertainties have been inflated such that $\chi ^2_\nu =1$ and not including the α₂ term in the uncertainty). The significance found here is larger than that found in Sullivan et al. (2010) or Sullivan et al. (2011) when they examine a second order stretch correction. In the Sullivan et al. papers, they find two α values for high- and low-stretch values, both of which were larger than the value of α found for the whole sample. We find a discrepancy in Hubble residuals for high (x₁ > 0.5) and low stretch (x₁ < 0.5) values to be Δμ = 0.042 ± 0.020 (after Malmquist correction).

A more practical approach to understand the source of the second-order trend of Hubble residuals with stretch is to reproduce the observed effect in simulations. We find (Figure 11) that the trend seen in the data may be replicated with two different α parameters for high (x₁ > 0) and low (x₁ < 0) stretch values. We use this split function as we do not yet have the tools to simulate a second order polynomial of the stretch-luminosity relation. A comparison of results of various simulations with the data is presented in Figure 11. We show that the quadratic trend between Hubble residuals and stretch that is seen in the data is not seen in simulations with only one α (α = 0.14). However, for the 2α model, where for x₁ < 0, α = 0.08 and for x₁ > 0, α = 0.17, the quadratic trend is observed. More work must be done to determine if this effect is real, and if a continuous, but nonlinear, Phillips relation is empirically favorable to the discontinuous relation presented here. We find the change in the determined value of w when accounting for a second-order light curve shape correction is small: Δw = +0.01. Since there is only mild evidence (∼2σ) that a second-order light curve shape correction is beneficial, it is not included in our overall uncertainty budget.

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We examine here whether Hubble residuals of the SNe in the PS1+low-z sample correlate with properties of the host galaxies of SNe. These correlations have been shown for other SN Ia samples in multiple recent studies (e.g., Kelly et al. 2010; Sullivan et al. 2010; Lampeitl et al. 2010). While age, metallicity, and star formation rate of host galaxies all also have been observed to correlate with Hubble residuals (e.g., Gupta et al. 2011; Hayden et al. 2013), here we focus on the masses of the host galaxies.

To determine the masses of host galaxies of PS1 SN Ia, we combine PS1 observations with u-band data from SDSS. Since the PS1 Medium Deep Survey has now observed each of the SN Ia host galaxies for over two years, we stacked deep (S/N of 10 at r_p1 ∼ 24) SN-free templates and used SExtractor's FLUX_AUTO (Bertin & Arnouts 1996) to determine the flux values in g_p1,r_p1,i_p1,z_p1,y_p1. The measured magnitudes are analyzed with the SED fitting software Multi-wavelength Analysis of Galaxy Physical Properties (da Cunha et al. 2011) to calculate the stellar masses of host galaxies. To verify both our galaxy photometry and the mass fitting routine, we ran our mass fitter on PS1 photometry of host galaxies in the Gupta et al. (2011) sample, and found differences with Gupta et al. (2011) to be <log (M_☉) ∼ 0.05.

The mass distributions of the host galaxies from the combined PS1+low-z sample analyzed for this paper are shown in Figure 12. We currently have masses for 110 of the 113 PS1 host galaxies and 61 of the 197 low-z host galaxies. The host galaxy masses of the low-z sample are from Sullivan et al. (2010) and the incompleteness limits a full analysis. Because most low-z supernovae were found in cataloged galaxies, and cataloged galaxies are generally brighter and more massive on average, the host galaxies in the low-z surveys are log (M_☉) ∼ 0.5 more massive than those in the PS1 survey. We find from Figure 13 a trend such that supernovae with more massive host galaxies have lower stretch values, though quantifying this trend is obscured by a number of supernovae with very low stretch errors (dx₁ < 0.1). We find a relation between Hubble residual and color with a slope of 0.0054 ± 0.0049. While the significance of this relation is weak, it is in agreement with the trend observed by Childress et al. (2013).

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In Figure 14, we show the trend between Hubble residuals and the host galaxy mass. We find a step-size in Hubble residuals, for a mass split of log(M_☉) = 10, of Δμ = 0.037 ± 0.032 mags. While this is consistent with the step size seen in previous studies (∼0.08 mag; e.g., Sullivan et al. 2010; Gupta et al. 2011), it is also consistent with zero. For the PS1 SNe alone, we find only a difference of 0.019 ± 0.025, weaker than the typical value seen. Part of the impact of adding in the low-z sample may be due to the fact that low-z hosts have higher masses, on average, than high-z hosts. Therefore, any difference in luminosity is degenerate with discrepancies in calibration for the low-z and high-z samples. It is also possible that the low value of intrinsic scatter seen in the PS1 sample (σ_int = 0.07) found for the PS1 sample is related to the low significance of the host galaxy relation. One concern is that the mass fits are likely subject to selection biases as the number of observations in separate filters decreases at higher redshift, (e.g., Galex and UKIDSS detect more nearby hosts). However, Smith et al. (2012) finds discrepancies in the determination of mass for SNe with high and low redshift to be negligible for the SDSS+SNLS sample.

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As we do not understand why the difference in luminosity between low- and high-mass galaxies is weaker in the PS1+lz sample than in other samples, we include as a systematic uncertainty the possibility that the host galaxy relation is as seen in other surveys (e.g., Childress et al. 2013) versus the size seen in our sample. We implement for the PS1 sample the same mass-distribution split done in Sullivan et al. (2010) at (log(M_☉) = 10) and apply a difference in luminosities between SNe with high and low massive host galaxies at ∼0.075 mag (Sullivan et al. 2010). When we do not have sufficient host galaxy photometry to find masses, we do not correct the SN distance. If, however, the host is not visible, we assume the mass is low. We find that the correction based on host galaxy masses changes our value of w by Δw = 0.026 (SN+BAO+CMB+H0: host correction—no-host correction). The overall uncertainty for the host galaxy correction is shown in Table 5. Its effect on the recovery of cosmological parameters is not as large as some of the other uncertainties, which may be partly due to the incompleteness of host galaxy measurements for the low-z sample.

Table 5. Overall Systematics

Systematic	$\Delta \Omega _{\scriptsize M}$	Δw	Rel. Area	$\Delta \Omega _{\scriptsize M}$	Δw	Rel. Area
		SN Only			SN+BAO+CMB+H0
Stat. Only	0.000	0.000	1.000	0.010	0.000	1.000
	$(\Omega _{\scriptsize M}=0.223^{+0.209}_{-0.221})$	$(w=-1.010^{+0.360}_{-0.206})$		$(\Omega _{\scriptsize M}=0.284^{+0.010}_{-0.010})$	($w=-1.131^{+0.049}_{-0.049}$)
Calibration	0.024	−0.093	1.566	0.006	−0.035	1.309
Color model	0.004	−0.023	1.117	0.009	−0.012	1.106
Host gal.	0.006	0.000	1.035	0.011	0.006	0.977
Malmquist	0.004	−0.011	1.024	0.010	−0.002	1.015
zmin	0.000	−0.004	1.020	0.010	−0.003	1.023
MW extinction	0.001	−0.007	1.020	0.010	−0.004	1.019
Syst.+Stat.	0.032	−0.108	1.697	0.006	−0.035	1.333
	$(\Omega _{\scriptsize M}=0.256^{+0.201}_{-0.174})$	$(w=-1.120^{+0.450}_{-0.357})$		$(\Omega _{\scriptsize M}=0.280^{+0.013}_{-0.012})$	($w=-1.166^{+0.072}_{-0.069}$)

Notes. The dominant systematic uncertainties in the PS1+lz sample with respect to Ω_m and w. Ω_m and w are given for a flat universe and ΔΩ_m and Δw values are the relative values such that, e.g., w_{stat + sys} − w_stat. Individual systematic uncertainties for each of the PS1 passbands as well as the systematic uncertainties for each low-z sample. Relative area is the size of the contour that encloses 68.3% of the probability distribution between w and $\Omega _{\scriptsize M}$ compared with that of statistical only uncertainties.

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