The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample

The PS1 data presented here are from the PS1 Medium Deep (MD) Survey, which observes SNe in gri${z}_{{\rm{p}}1}$ with an average cadence of 7 days per filter.²² This cadence provides well-sampled, multiband light curves. The description of the PS1 survey is given in Kaiser et al. (2010). The PS1 Image Processing Pipeline system (Magnier et al. 2013) performs flat-fielding on each individual image and determines an initial astrometric solution. The full description of these algorithms is given in Chambers et al. (2016), Magnier et al. (2016a, 2016b, 2016c), Flewelling et al. (2016), and Waters et al. (2016). Once done, images are processed in Photpipe (Rest et al. 2005) with updated methodology given in R14.

The discovery pipeline is explained in R14. The main difference between the pipeline in the first and second half of the survey is that, as the survey went along, the average nightly seeing improved by 012 (due to camera/operation improvements) and better templates (>0.5 mag deeper) were used for the transient search. The improved templates also had better artifact removal, which significantly reduced the number of false positives in the transient candidate lists.

The spectroscopic selection over the full survey is similar to that outlined in R14. Spectroscopic observations of PS1 targets were obtained with a variety of instruments: the Blue Channel Spectrograph (Schmidt et al. 1989) and Hectospec (Fabricant et al. 2005) on the 6.5 m MMT, the Gemini Multi-Object Spectrographs (GMOS; Hook et al. 2004) on both Gemini North and South, the Low Dispersion Survey Spectrograph-3 (LDSS3²³ ) and the Magellan Echellette (MagE; Marshall et al. 2008) on the 6.5 m Magellan Clay telescope, the Inamori-Magellan Areal Camera and Spectrograph (IMACS; Dressler et al. 2011) on the 6.5 m Magellan Baade telescope, the ISIS spectrograph on the WHT,²⁴ and DEIMOS (Faber et al. 2003) on the 10 m Keck telescope.

Since there were a multitude of spectroscopic programs without a well-defined algorithm to determine which candidates to observe, an empirical algorithm is retroactively determined that best describes our selection of spectroscopic targets. This is discussed further in Section 3, but we note here that the spectroscopic selection for the full survey is very similar to that of the first 1.5 yr of the PS1 survey described in R14. The one exception was a program by PI Kirshner (GO-13046) to observe HST candidates for infrared follow-up specifically at z ∼ 0.3. A table of the spectroscopically confirmed SNe Ia that includes the dates of the observations and the telescopes used is given in Appendix A.

The distribution of redshifts of the confirmed SNe Ia is shown in Figure 2. The median redshift is 0.3, which is Δz ∼ 0.05 smaller than the median redshift of likely photometric SNe Ia discovered during the survey (Jones et al. 2017). As shown in Figure 2, the observed candidates are well dispersed over the focal plane with no systematic grouping at one focal position. It is also shown in Figure 2 that the majority of candidates are discovered before peak. A discovery (defined as three detections with signal-to-noise ratio (S/N) > 4) after peak does not exclude the possibility that there were pre-explosion images acquired, only that the object was not detected at that time.

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The photometry pipeline used in this analysis is a modified version of that described in R14. The overall process used is summarized as such:

• 1.  
  Template Construction. For each PS1 chip, templates are constructed from stacking multiple, nightly, variance-weighted images from all but a single survey year around the SN explosion date. The seasonal templates are made of ∼60+ images and reach 5σ depths of 25.00, 25.1, 25.15, and 24.80 mag in gri${z}_{{\rm{p}}1}$. Excluding a particular year removes the possibility that >1 mmag of SN flux is included in the template. We develop a scene-modeling pipeline (e.g., Holtzman et al. 2008) as an independent cross-check on the template construction. This is presented in Appendix B.
• 2.  
  Astrometric Alignment. All nightly images and templates are astrometrically aligned with an initial catalog provided by the PS1 survey. For all bright stars and galaxies observed on each CCD with an SN observation, an astrometry catalog is recreated with the average locations of each of the stars and galaxies over the full survey. Then an astrometric solution for each nightly image and template is determined to match the improved catalog.
• 3.  
  Stellar Zero-points. Point-spread function (PSF) photometry is performed on the image at the positions of the stars from the final catalog; there is no re-centroiding of the star's position per image, such that "forced" photometry is done. The PSF module is based on a Python implementation (Jones et al. 2015) of the DAOPhot package (Stetson 1987). A comparison of the photometry of these stars to updated PS1 stellar catalogs is used to find the zero-point of each image. Forced photometry on the stars is necessary so that a consistent procedure is done for both the stars and the SNe. The PSF is determined for each epoch from neighboring stars of the SN. Due to the fast-varying PSF on CCDs near the center of the focal plane (<04), the region cutout to find neighboring stars is roughly 1/4 the area of the chip. For CCDs away from the center of the focal plane (>04), the full area of the 125 chip is used.
• 4.  
  Template Matching. Templates are convolved with a PSF to match the nightly images (Becker 2015). The convolved templates are then subtracted from the nightly images.
• 5.  
  Forced SN Photometry. A flux-weighted centroid is found for each SN position. Forced photometry is performed at the position of the SN. The nightly zero-point is applied to the photometry to determine the brightness of the SN for that epoch. Small adjustments are made to the SN photometry based on the expectation value from the astrometric uncertainty of the SN centroid. Forced photometry is also applied to random positions in the difference image to empirically determine the amount of correlated noise in the image. The SN photometry uncertainties are then increased to account for this correlated noise.
• 6.  
  Flux Adjustment. The errors and the baseline flux of the SN measurements are adjusted so that the mean pre-explosion baseline flux level is 0 and the reduced χ² is near unity. The prescription for this step is described in R14.

The most significant changes relative to R14 are the additions of iterative astrometric alignment, forced photometry of stars with an updated PSF fitting routine, an updated Ubercal catalog, and a reduction in the area from which neighboring stars are drawn for building PSF models. These steps improve the accuracy of the astrometric solution, alleviate systematic uncertainties in the photometry due to uncertainties in the astrometry, and account for the fast-varying PSF near the center of the focal plane, respectively.

Improvements to understand the systematic uncertainties in this process are discussed below. The systematic uncertainties in the photometry analysis are given in Table 1.

2.2.1. Astrometry

The recovered position of an SN detection can be different from the true SN centroid for the following reasons: accuracy of the WCS for a given image, the limited number of observations of the SN, Poisson noise from sky, host galaxy, and SN and difference image artifacts. Unlike in R14, forced photometry is performed on both stars and SNe, so the errors on SN positions and stellar position are similar and do not propagate to additional biases. However, uncertainties that affect only the SN position are treated separately.

R14 shows that the astrometric uncertainty of objects depends on both the FWHM and S/N of the object. Because of the S/N dependence, the astrometric uncertainty of the higher-redshift SNe is larger than the astrometric uncertainty of the lower-redshift SNe. This astrometric uncertainty will propagate to a photometric bias because the expected average offset from the true centroid value causes biased photometric measurements. To understand this trend, the astrometric uncertainty of the individual detections is quantified. This is done in R14 by first finding the linear relation between astrometric uncertainty (e.g., ${\sigma }_{{\rm{\Delta }}x}^{2},{\sigma }_{{\rm{\Delta }}y}^{2}$, here denoted as ${\sigma }_{a}^{2}$) and ${(\mathrm{FWHM}/\mathrm{S/N})}^{2}$:

$$\begin{eqnarray}&&{\sigma }_{a}^{2}={\sigma }_{a1}^{2}+{\sigma }_{a2}^{2}{\left(\displaystyle \frac{\mathrm{FWHM}}{\mathrm{S/R}}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

where the astrometric uncertainty σ_a in pixels of a given detection has a floor mostly due to pixelization (${\sigma }_{a1}$), and in addition a random error σ_a2. R14 conservatively uses σ_a1 =0.20 pixels and σ_a2 = 1.5 to calculate the astrometric uncertainty of a single detection. In Figure 3, it is clear that the quantified relation from R14 is too high by a factor of 2 for our sample owing to our improved astrometry such that we find the uncertainty of astrometry as we find a σ_a1 = 0.1 and σ_a2 = 0.75. Much of this improvement is from the iterative astrometric alignment discussed above.

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The relation in Figure 3 is used to determine the astrometric uncertainty of each SN observation to properly determine the centroid accuracy of the SN. With a more appropriate estimate of the astrometric uncertainty, the centroids and centroid errors are recalculated for each SN detection. To remove the expected bias in the photometry from the centroid error, a conversion from R14 is applied between the astrometric uncertainty ${\sigma }_{\mathrm{SN},\mathrm{cent},a}$ and the bias in photometry ${\rm{\Delta }}{m}_{\mathrm{Corr}}$ from Equation (1) such that

$$\begin{eqnarray}&&{\rm{\Delta }}{m}_{\mathrm{Corr}}=\int {{ht}}^{2}\mathrm{PDF}({\sigma }_{\mathrm{SN},\mathrm{cent},a},t){dt},\end{eqnarray} \tag{ 2 }$$

where PDF(σ_a, t) is the probability density function with sigma ${\sigma }_{a}$ and pixel variable t, assuming that h is constant and independent of S/N. The value of h (0.043) is taken from R14 and is found to be a reliable first-order approximation. The corrections to the photometry for each SN are shown in the bottom panel of Figure 3. The maximum correction is 6 mmag.

The absolute calibration of the PS1 photometric system has been improved in a series of PS1 analyses. The basis for the PS1 absolute calibration is first presented in Tonry et al. (2012), and a full review of subsequent improvements is given in Scolnic et al. (2015). The relative calibration across the sky of the PS1 survey is determined by the Ubercal process (Schlafly et al. 2012; Finkbeiner et al. 2016). For the MD fields, we made a custom Ubercal star catalog of all of the data from the MD fields in the same way as those produced in Schlafly et al. (2012) but with a higher-resolution nightly flat field, a lower threshold for masking of problematic areas of the focal plane, and a per-image zero-point. These catalogs are released with results from this paper at doi:10.17909/T95Q4X and are consistent with Schlafly et al. (2012) in relative calibration on degree scales to the 1 mmag level. Scolnic et al. (2015), using the same catalogs as in this current analysis, show in comparisons with SDSS and SNLS that the likely systematic uncertainty of the zero-points of each MD field for each filter is ∼3 mmag.

The nightly photometry is transferred onto the PS1 system using a zero-point measured by comparing the photometry with stellar magnitudes from the Ubercal catalog. The result of this process is shown in Figure 4, which presents differences between our PS1 catalogs and the final nightly photometry. As the Ubercal catalogs are used to determine both the calibration of the HST Calspec standards and the MD fields, Figure 4 demonstrates the consistency of the nightly photometry with that used to create the PS1 calibration across 3π of the sky. We find that the nightly photometry and Ubercal catalogs are consistent across 4 mag to levels of ∼2 mmag, though with a trend in the discrepancy of ∼1 mmag per mag as shown with a linear fit overlaid on the left panel of Figure 4. It is unclear what is causing this trend, and it is possible that this small trend is partly due to selection effects in the cuts made to make the catalogs. This trend is therefore included as part of our systematic error budget. Image zero-points for each observation are determined using stars brighter than 21.5, 21.0, 21.0, and 21.0 in $\mathrm{gri}{z}_{{\rm{p}}1}$, respectively. Future analyses may try to use an S/N cut instead of a magnitude cut to reduce the Malmquist bias. Possible nonlinearity has been tested in Scolnic et al. (2015) in comparisons of PS1 with SDSS, SNLS, and multiple low-z surveys: using our PS1 stellar catalogs, linearity behaves to better than 3 mmag in gri${z}_{{\rm{p}}1}$ between 15 and 21 mag. Further discussions of PS1 detector nonlinearity are in Waters et al. (2016).

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Systematic uncertainties in our photometry, due to spatial variation of the throughput across the focal plane, as well as temporal variation of the filters over the entire survey, are examined here as well. There is no evidence (<1 mmag) of differences in the system photometry over the full course of the survey. There is also excellent agreement (<1 mmag) between the stellar photometry from our pipeline and the Ubercal catalogs across the focal plane. A much larger effect (>0.15 mag) was seen in R14 owing to the fast-varying PSF (change of 1 pixel in FWHM over 04) near the center of the focal plane that was not accounted for. Therefore, in R14, SNe near the center (r < 04) of the focal plane were not used in the analysis. This problem has been fixed by reducing the area for choosing neighboring stars from which to build a PSF near the center of the focal plane. Furthermore, there is little dependence (<2 mmag) on the airmass of the nightly observations.

In S14, the filters used to measure the SN light curves are those given at the median radial position across the field of view. From measuring the expected photometry of synthetic SN spectra integrated through the known PS1 passbands at various focal positions, differences in the photometry of the SN dependent on focal plane position increase scatter by 0.01 mag. However, S14 showed that there is only a 2 mmag bias with redshift due to the different passbands. Further corrections based on the airmass of each observation, as done in Li et al. (2016), may be implemented in the future; however, it is shown in Figure 4 that the impact is on the 1 mmag scale. All uncertainties are summarized in Table 1.

It is difficult to fully blind an analysis of this sort, since any update in photometry, calibration, etc., of a sample has a direct, and sometimes obvious, impact on recovered cosmological parameters. As discussed later in this section, we use the BEAMS with Bias Corrections (BBC) method (Kessler & Scolnic 2017) to recover binned Hubble residuals with redshift, with respect to a reference cosmology. Therefore, to blind the analysis, the reference cosmology is randomly chosen. So that a full analysis can be completed without introducing any further SN systematics based on a highly unlikely cosmology, the reference cosmology is randomly chosen from a Gaussian distribution of values of the matter density Ω_m and equation of state of dark energy w (discussed in Section 5) centered around the recovered values in the B14 analysis of w = −1.02 and Ω_m = 0.307 with a standard deviation of σ_w = 0.06 and ${\sigma }_{{{\rm{\Omega }}}_{m}}=0.02$, where the standard deviation is determined from the uncertainties on the cosmological parameters in the B14 analysis. We use the B14 analysis to choose the blinding parameters rather than R14 because the full sample that this paper will analyze is more consistent with that in B14 than that in R14 and has lower uncertainties.

While multiple light-curve fitters can be used to determine accurate distances (e.g., Jha et al. 2007; Guy et al. 2010; Burns et al. 2011; Mandel et al. 2011), we use SALT2 (Guy et al. 2010) for this analysis, as it has been trained on the JLA sample (B14) and it is easy to assess various systematic uncertainties with this fitter. We use the most up-to-date published version of SALT2 presented in B14 and implemented in SNANA²⁵ (Kessler et al. 2009b). Differences between the SALT2 spectral model in Guy et al. (2010) and that in B14 are mainly due to calibration errors in the light curves used for the model training. The models differ the most at rest-frame wavelengths <4000 Å and are described in detail in B14. Three values are determined in the light-curve fit that are needed to derive a distance: the color c, the light-curve shape parameter x₁ and the log of the overall flux normalization m_B. The solid lines in Figure 5 show the respective light-curve fits with SALT2 for three representative PS1 SNe Ia.

The SALT2 light-curve fit parameters are transformed into distances using a modified version of the Tripp formula (Tripp 1998),

$$\begin{eqnarray}&&\mu ={m}_{B}-M+\alpha {x}_{1}-\beta c+{{\rm{\Delta }}}_{M}+{{\rm{\Delta }}}_{B},\end{eqnarray} \tag{ 3 }$$

where μ is the distance modulus, Δ_M is a distance correction based on the host galaxy mass of the SN, and Δ_B is a distance correction based on predicted biases from simulations. Furthermore, α is the coefficient of the relation between luminosity and stretch, β is the coefficient of the relation between luminosity and color, and M is the absolute B-band magnitude of a fiducial SN Ia with x₁ = 0 and c = 0. Motivated by Schlafly & Finkbeiner (2011) and following S14, we modify SALT2 by replacing the "CCM" (Cardelli et al. 1989) Milky Way (MW) reddening law with that from Fitzpatrick (1999).

The total distance error of each SN is

$$\begin{eqnarray}&&{\sigma }^{2}={\sigma }_{{\rm{N}}}^{2}+{\sigma }_{\mathrm{Mass}}^{2}+{\sigma }_{\mu -z}^{2}+{\sigma }_{\mathrm{lens}}^{2}+{\sigma }_{\mathrm{int}}^{2}+{\sigma }_{\mathrm{Bias}}^{2},\end{eqnarray} \tag{ 4 }$$

where ${\sigma }_{{\rm{N}}}^{2}$ is the photometric error of the SN distance, ${\sigma }_{\mathrm{Mass}}^{2}$ is the distance uncertainty from the mass step correction, ${\sigma }_{\mathrm{Bias}}^{2}$ is the uncertainty from the distance bias correction, ${\sigma }_{\mu -z}^{2}$ is the uncertainty from the peculiar velocity uncertainty and redshift measurement uncertainty in quadrature, ${\sigma }_{\mathrm{lens}}^{2}$ is the uncertainty from stochastic gravitational lensing, and ${\sigma }_{\mathrm{int}}^{2}$ is the intrinsic scatter. For this analysis, ${\sigma }_{\mathrm{lens}}=0.055z$ as given in Jönsson et al. (2010).

For this analysis, we require every SN Ia to have adequate light-curve coverage to accurately constrain light-curve fit parameters, as well as properties that limit systematic biases in the recovered distance. We follow the light-curve requirements in B14 such that the only SNe allowed in the sample have −3 < x₁ < 3, −0.3 < c < 0.3, ${\sigma }_{({pkmjd})}\lt 2$, and σ_x1 < 1 (where ${\sigma }_{(t0)}$ is the uncertainty on the rest-frame peak date and ${\sigma }_{x1}$ is the uncertainty on x₁). Most of the cuts, as shown in Table 2, are motivated by B14. The cuts are somewhat different than those used in R14 that require observations before and after the peak brightness date. These updated requirements are more stringent than those used in R14, though three of the SNe Ia that do not pass the R14 cuts do pass these new cuts. These three SNe Ia are all at low z, where it was unclear if there were observations taken before peak owing to uncertainty in the peak date, though the i-band peak was measured accurately. A related issue due to uncertainty in the peak date was pointed out in Dai & Wang (2016), which finds ∼10 SNe with double-peak probability distribution functions of the light-curve parameters of the SALT2 fits. We find that many (8/10) of these SNe would be removed from our set if we place an additional cut enforcing observations after post-maximum brightness. Therefore, we include a cut such that there is an observation at least 5 days after peak brightness. B14 also places a requirement for $E{(B-V)}_{\mathrm{MW}}\lt 0.15$. This does not apply to the PS1 SN sample but will apply to other samples, as all the MD fields have low extinction, and as discussed in S14, this constraint is loosened to $E{(B-V)}_{\mathrm{MW}}\lt 0.20$ owing to improved nonlinear modeling of high extinction (Schlafly & Finkbeiner 2011). Furthermore, in B14, a cut on the fit likelihood is placed on the SDSS SNe Ia but not on the SNLS sample in B14. We follow the strategy for the SDSS sample and place a cut on the χ²/NDOF < 3.0. Finally, there is one last additional cut from the BBC method that removes three of the SNe because their x₁ and c parameters do not fall in the expected distribution—this is discussed in Section 3.5. Applying these cuts, only $279$ SNe Ia from the initial sample of $365$ spectroscopically confirmed Pan-STARRS1 objects remain in our sample for a cosmological analysis.

Table 2.  Impact of Various Cuts Used for Cosmology Analysis. Both the Number Removed from Each cut and the Number Remaining After Each Cut are Shown. The "Quality fit" Includes Both the Light Curves that are Rejected by the SNANA Fitter Owing to Poorly Converged fits and those with a fit ${\chi }^{2}/\mathrm{NDOF}\lt 3.0$

	Discarded	Remaining
Initial	⋯	$365$
Quality fit	$33$	$332$
$\sigma ({x}_{1})\lt 1$	$29$	$303$
${\sigma }_{({pkmjd})}\lt 2$	$0$	$303$
$-0.3\lt c\lt 0.3$	$10$	$293$
−3 < x1 < 3	$5$	$288$
$E{(B-V)}_{\mathrm{MW}}\lt 0.20$	0	$288$
${T}_{\mathrm{Max}}\gt 5$	$6$	$282$
BBC cut	3	$279$

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The SALT2 parameters for the entire set of cosmologically useful SNe Ia from the PS1 sample are presented in Appendix A. A table with the full output of each SNANA fit with SALT2 is included at doi:10.17909/T95Q4X.

To correct for biased distance estimates, the PS1 SN survey must be accurately simulated. Following S14, we simulate the PS1 survey with SNANA using cadences, observing conditions, spectroscopic efficiency, etc., from the data. Following Jones et al. (2017), we include a complete library of observations of the PS1 survey, noise contributions from the host galaxies of the SNe Ia, and a newly modeled SN discovery efficiency. The noise contributions from the host galaxies were modeled for the PS1 photometric sample, which is a good approximation for the confirmed sample, as the average host galaxy magnitudes for the confirmed and photometric samples are within 0.1 mag.

To model the spectroscopic selection of the PS1 SN survey, an efficiency function must be empirically determined. Similar to R14, we find that it is well modeled with a dependence on the peak r-band magnitude of the SN. The function is shown in the top panel of Figure 6. The method to determine this function is analogous to the approach taken in Scolnic & Kessler (2016, hereafter SK16) for determining the underlying color populations. We simulate PS1 without a spectroscopic efficiency function and divide the distribution of r magnitudes at the peak of the light curves from the data by that from the simulation. The ratio is the spectroscopic efficiency function, and the final curve shown in Figure 6 is smoothed from the recovered function. A coherent shift of the selection function by 0.25 mag in one direction is found to reduce the match between the predicted and actual redshift distribution by 1σ, and this error is overlaid in Figure 6.

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The underlying population of the stretch and color of the PS1 light curves is redetermined for the full data sample according to the process described in SK16. These are given in Table 3 for two different models of the Gaussian intrinsic scatter of SNe Ia: the "C11" model that is composed of 75% chromatic variation and 25% achromatic variation (Chotard et al. 2011), and the "G10" model that is composed of 30% chromatic variation and 70% achromatic variation (Guy et al. 2010). Besides scatter models that have 100% of one type of variation, the C11 and G10 models are the only two published models available for this type of analysis, and either of them may accurately represent the PS1 SN population. To use these models in simulations, Kessler et al. (2013) convert broadband models into spectral variation models. The color population parameters in Table 3 show agreement within 1σ between this analysis and that derived for the PS1 R14 sample (SK16). The stretch population parameters appear to be slightly discrepant, though this difference is exaggerated because we do not report covariances between $\bar{{x}_{1}}$, ${\sigma }_{-}$, and ${\sigma }_{+}$: the mean of the distribution of recovered x₁ values for the full PS1 sample is Δx₁ ∼ 0.03 from the mean of the x₁ distribution from R14.

Table 3.  Underlying Populations of SN Ia x₁ and c Parameters for the Full PS1 Sample and those Found in SK16. The First Column Shows the Analysis, and the Second Column Shows the Scatter Model Used in the Simulation. The First Part of the Table Shows the Recovered Values of the Underlying Color (c) Population, and the Second Part of the Table Shows the Recovered Values of the Underlying Stretch (x₁) Distribution. These Parameters Define the Asymmetric Gaussian for the Color and Light-Curve Shape Distributions: ${e}^{[-{(x-\bar{x})}^{2}/2{\sigma }_{-}^{2}]}$ for $x\lt \bar{x}$ and ${e}^{[-{(x-\bar{x})}^{2}/2{\sigma }_{+}^{2}]}$ for $x\gt \bar{x}$

Analysis	Scat.	$\bar{c}$	${\sigma }_{-}$	${\sigma }_{+}$
This work	G10	−0.068 ± 0.023	0.034 ± 0.016	0.123 ± 0.022
This work	C11	−0.100 ± 0.004	0.003 ± 0.003	0.134 ± 0.016
SK16	G10	−0.077 ± 0.023	0.029 ± 0.016	0.121 ± 0.019
SK16	C11	−0.103 ± 0.003	0.003 ± 0.003	0.129 ± 0.014
		$\bar{{x}_{1}}$	${\sigma }_{-}$	${\sigma }_{+}$
This work	G10	0.365 ± 0.208	0.963 ± 0.162	0.514 ± 0.140
This work	C11	0.384 ± 0.200	0.987 ± 0.155	0.505 ± 0.135
SK16	G10	0.604 ± 0.183	1.029 ± 0.138	0.363 ± 0.121
SK16	C11	0.589 ± 0.179	1.026 ± 0.137	0.381 ± 0.117

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The population parameters given here are derived from simulations that assume a ${\rm{\Lambda }}\mathrm{CDM}$ model. SK16 found that changes in the input cosmology within typical statistical uncertainties have a <0.2σ effect on the recovered populations. R. C. Wolf et al. (2018, in preparation) improves on the analysis of SK16 by attempting to fit for cosmological parameters and these population parameters simultaneously.

Figure 7 shows how well simulations model the data by comparing the distribution of redshift, constraint on time of maximum light, color error, and peak S/N distribution compared to the data. Comparisons of the color and stretch distributions, as well as their trends with redshift, are also shown. There is substantial improvement from S14 in how well the simulations and data match owing to more statistics in our sample and better modeling methods.

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SK16 and Kessler & Scolnic (2017, hereafter KS17) show that the Tripp estimator does not account for distance biases due to intrinsic scatter and selection effects. KS17 introduces the BBC method to properly correct these expected biases and simultaneously fit for the α and β parameters from Equation (3). The method relies heavily on Marriner et al. (2011) but includes extensive simulations to correct the SALT2 fit parameters m_B, c, and x₁. In Equation (1), this correction is expressed as Δ_B, which is actually a function of α, β, Δ_mB, Δ_c, and Δ_x1 that follows the same Tripp format such that ${{\rm{\Delta }}}_{B}={{\rm{\Delta }}}_{{mB}}-\beta \times {{\rm{\Delta }}}_{c}+\alpha \times {{\rm{\Delta }}}_{x1}$. Furthermore, the measurement uncertainty σ_N in Equation (2) is similarly corrected according to predictions from simulations because KS17 shows that the fit with SALT2 regularly overestimates the uncertainties of the fit parameters. Finally, the BBC method requires that the properties of an SN in the data sample are well represented in a simulation of 500,000 SNe, so any SNe with z, c, and x₁ properties that are not found within 99.999% of the simulated sample will not pass the BBC cut. The impact of the BBC cut is given in Table 2. The three SNe that are cut have x₁ and/or c values removed from the distribution as shown in Figure 7. They have (x₁, c) values of (−2.915, 0.083), (−1.702, 0.271), and (−0.893, 0.298). A simpler cut would be to shrink the current x₁ range from (−3, 3) and the c range of (−0.3, 0.3) to narrower ranges, and this will be studied in the future.

Given accurate simulations of the survey, the BBC method retrieves the nuisance parameters α and β from Equation (3) and derives the distances of each SN. As discussed in KS17, the recovered nuisance parameters depend on assumptions about the intrinsic scatter model. The method returns α = $0.167\pm 0.012$, β = $3.02\pm 0.12$, and σ_int = $0.08$ when assuming the G10 scatter model and $\alpha =0.167\pm 0.012$, $\beta =3.51\pm 0.16$, and ${\sigma }_{\mathrm{int}}=0.10$ when assuming the C11 scatter model. The difference in σ_int values is related to the assumed variation in the scatter model; the dispersion of Hubble residuals from both of these fits is about equal at σ_tot = 0.14 mag.

We can calculate the dependence of the bias in recovered distance on redshift by simulating the survey with both scatter models and measuring the difference between the true and recovered distances. In the bottom panel of Figure 6, we show the distance bias when we simulate the two different scatter models with their associated β values (G10: $3.02\pm 0.12;$ C11: $3.51\pm 0.16$), but we assume that the true scatter model was the G10 model (effectively determining distances with β = 3.0). The biases calculated using the two different scatter models are within 5 mmag for almost the entire redshift range until z ∼ 0.6. The uptick at high z is due to the interplay between color and brightness selection and is discussed further in Section 5.

Figure 8 shows a comparison of the mean distances in redshift bins between the R14 sample and our sample. This is shown for before and after distance bias corrections are applied. For the BBC method, the average of the corrections from the two scatter models is used. Our sample has roughly twice as many SNe as the R14 sample, so the comparisons show both statistical differences and systematic differences between the two samples. Relative to a reference cosmology, one indication of a trend is observed in the R14 sample but not in our sample: an increasing positive distance bias with redshift. This trend appears significant in the R14 sample owing to the highest-z bin, which has a positive residual of ∼0.3 mag. That residual is driven by only two SNe, both with high (>0.25 mag) residuals. In our sample, one of these SNe is cut owing to the selection cuts, and one of them has significantly changed photometry by 0.3 mag owing to low S/N and poor astrometry. Smaller differences are driven by changes in the calibration of both the PS1 system and the SALT2 model.

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Multiple SN Ia analyses (discussed below) show that there is a correlation between luminosity of the SNe and properties of the host galaxies of the SNe Ia. This effect is important for measuring cosmological parameters, as the demographics of SNe with certain host galaxy properties may change with redshift, and also because correcting for the effect may reduce the scatter of the distances. Correlations between luminosity and the host galaxy mass (e.g., Kelly et al. 2010; Lampeitl et al. 2010; Sullivan et al. 2010), age, metallicity, and star formation rate (e.g., Hayden et al. 2013; Roman et al. 2017) have all been shown. So far, the relation with mass appears to be the strongest of the correlations, possibly because it is easier to measure the mass of galaxies than the other properties, so in this analysis we use the mass dependence. In S14, the difference in the mean Hubble residual for SNe in galaxies with high versus low masses (at a split of $\mathrm{log}({M}_{\odot })=10$) was found to be 0.037 ± 0.032 mag, which is consistent with a null result, as well as with the B14 statistical result of 0.06 ± 0.012 mag. In this analysis, our statistics are a factor of 2 larger and allow us to better measure the step.

The masses of PS1 host galaxies are derived similarly to S14 and follow the approach in Pan et al. (2014). Here we use the seasonal templates, discussed in Section 2, to measure host galaxy photometry, and we combine PS1 observations with u-band data from SDSS (Alam et al. 2015) where available. SExtractor's FLUX_AUTO (Bertin & Arnouts 1996) was used to determine the flux values in ${g}_{{\rm{p}}1}$, ${r}_{{\rm{p}}1}$, ${i}_{{\rm{p}}1}$, ${z}_{{\rm{p}}1}$, ${y}_{{\rm{p}}1}$. The measured magnitudes are analyzed with the photometric redshift code Z-PEG (Le Borgne & Rocca-Volmerange 2002), which is based on the PEGASE.2 spectral synthesis code (Fioc & Rocca-Volmerange 1999) and follows da Cunha et al. (2012) to calculate the stellar masses of host galaxies. This is a very similar analysis code to what is used to determine the masses of the host galaxies in the JLA sample (B14). Further details of the assumptions used to run the code are discussed in Pan et al. (2014).

Masses are determined by the Z-PEG code for all but four of the $279$ PS1 host galaxies. For the four SNe without matched host galaxies, no host galaxy was detected near the SN. The masses of the hosts of these SNe are placed in the lowest mass bin (as done in B14). The mass step is typically placed at 10¹⁰ M_⊙, and we find that there are $116$ host galaxies with masses higher than the split value and $163$ with masses lower than the split value. In Figure 9, we show relations between the stretch, color, and Hubble residuals of the SNe Ia with the mass of the host galaxies. We find no trend with color such that the mean color of SNe in low-mass hosts is −0.001 ± 0.004 mag smaller than the color for SNe with high-mass hosts. We also recover the typical trend such that SNe with lower stretch values are more often found in high-mass hosts, with a median difference in stretch values between high- and low-mass hosts of Δx₁ = 0.210 ± 0.041.

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The split in luminosity with mass is determined by three parameters: a relative offset in luminosity, a mass step for the split, and an exponential transition term in a Fermi function that describes the relative probability of masses being on one side or the other of the split:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{M}=\gamma \times {[1+{e}^{(-(m-{m}_{\mathrm{step}})/\tau )}]}^{-1}.\end{eqnarray} \tag{ 5 }$$

The Fermi function that is chosen here is used to allow for both uncertainty in the mass step and uncertainty in the host masses themselves. For the PS1 sample, $\gamma =0.039\pm 0.016$ mag, m_step = 10.02 ± 0.06, and τ = 0.134 ± 0.05. The step $\gamma =0.039\pm 0.016$ is similar to that found in S14, although with a smaller uncertainty. Interestingly, if we did not apply the BBC method, as done in S14, we find for this sample ${\rm{\Delta }}\mu =0.064\pm 0.018$ mag. This is roughly 1σ larger than with the BBC method and is more consistent with the mass step recovered in B14 of 0.06 ± 0.012 mag, which also did not implement the BBC method.

To test how the BBC method accounts for a relation between mass and luminosity, we created new simulations with host galaxy mass properties assigned to every SN. We assigned a host mass to each SN so that the simulated sample replicates the trends of mass with c and x₁ seen in Figure 9. Applying Equation (5) to the simulations using the BBC method, we see a bias of only 0.0035 mag in the recovered value of γ given an input value of γ = 0.08 mag. Furthermore, we find that including a mass step has only a 0.001 mag effect on the distance bias corrections shown in Figure 6 and therefore has a limited impact on our analysis.

The "Supercal" calibration of all the samples in this analysis is presented in S15. S15 takes advantage of the sub-1% relative calibration of PS1 (Schlafly et al. 2012) across 3π sr of sky to compare photometry of tertiary standards from each survey. S15 measures percent-level discrepancies between the defined calibration of each survey by determining the measured brightness differences of stars observed by a single survey and PS1 and comparing this with predicted brightness differences of main-sequence stars using a spectral library. The largest calibration discrepancies found were in the B band of the Low-z photometric systems: CfA3 and CfA4 showed calibration offsets relative to PS1 of 2%–4%.

In S15, calibration offsets from each system relative to PS1 were given. However, since there is uncertainty in the AB zero-points of PS1 and both SDSS and SNLS attempt to tie their calibration to HST Calspec standards, we adjust the PS1 zero-points to reduce discrepancies in the cross-calibration with SDSS and SNLS; we average the absolute calibration offsets that are given in S15 from SNLS, SDSS, and PS1 (PS1 has by definition offsets of 0.0). Doing so, we subtract the following calibration offsets from PS1 catalog magnitudes: Δg =−0.004, Δ_r = −0.007, Δ_i = −0.004, and Δ_z = 0.008. Calibration offsets of every other survey are corrected accordingly. These calibration offsets are shown in Table 5, in addition to the uncertainties in the S15 zero-points. The uncertainties on the mean effective wavelength of the transmission functions of each system are unchanged, except for SNLS r band, which in B14 has a stated uncertainty of 3.7 nm. The recovered calibration discrepancy found in S15 for the SNLS r band is <1 nm, so we conservatively prescribe a 1 nm uncertainty to this band.

This work does not achieve the maximum possible reduction of systematic biases from the Supercal approach, because the SALT2 model was trained and calibrated using an SN sample that was not recalibrated using the Supercal method. Therefore, the SALT2 model itself propagates calibration uncertainties with values that are assigned by B14. To account for the possibility of calibration biases in the SALT2 model, we fitted our SN sample with multiple iterations of the SALT2 model that were made in B14 by propagating systematic uncertainties in the calibration of each sample used for training. Furthermore, we estimate an additional systematic uncertainty of the whole Supercal process to be 1/3 of the Supercal correction, as the correction is dominated by discrepancies of B − V to 3%, and we are confident to roughly 1%.

The calibration uncertainty from the HST Calspec standards is described in Bohlin et al. (2014). A relative flux uncertainty as a function of wavelength is determined by a comparison of pure hydrogen models of different white dwarfs to observed spectra and is set such that the relative flux uncertainty is 0 at 5556 Å. Roughly, the uncertainty is 5 mmag for every 7000 Å. There is an additional absolute uncertainty from Bohlin et al. (2014) of 5 mmag coherent across all wavelengths; however, this uncertainty has no impact since all subsamples are tied to the same system. In follow-up analyses, we will include a new network of WD standards from Narayan et al. (2016).

Following the method described in Section 3.5, to model the dependence of distances on assumptions about SN color and selection effects, the BBC method is applied with two different intrinsic scatter models to determine distances. The population parameters for each non-PS1 sample are given in SK16. For this baseline analysis, we do not allow for any evolution in α, β, or γ.

The simulations for SDSS and SNLS are described in B14 and S15, and the Low-z simulations and selection effects are described in Appendix C. The HST simulations are made in the same way as the PS1 and Low-z simulations, so that they directly represent the data in the SCP Cluster survey, GOODS, and CANDELS/CLASH surveys, but the spectroscopic selection efficiency was set equal to unity for these surveys.

The recovered nuisance parameters α and β from the BBC method are given in Table 6 for both scatter models. For the G10 and C11 models, values from each survey of α and β are within 1σ of the combined Pantheon sample. The recovered β values are slightly less consistent using the C11 model, with a range of $\beta =3.59\pm 0.17$ from SNLS and $\beta =4.04\pm 0.18$ from SDSS, but are all still near 1σ of the mean. These higher values of β, when using the C11 model, are consistent with recent analyses (Mosher et al. 2014; Scolnic et al. 2014b; Mandel et al. 2017) that find larger β dependent on various assumptions about the intrinsic scatter of SNe Ia.

Table 6.  Nuisance Parameters α and β of each Sample, as well as the Derived Mass Step, the Intrinsic Scatter ${\sigma }_{\mathrm{int}}$, and the Total Hubble Residual rms ${\sigma }_{\mathrm{tot}}$. Values are given for each Sample, as well as the Combined Pantheon Sample. Furthermore, Values are given for the two Different Scatter Model Assumptions, G10 and C11, Using the BBC Method, as well as for the Conventional Method Assuming that all Residual Scatter after the SALT2 fits is due to Luminosity Variation

Survey	α	β	γ	${\sigma }_{\mathrm{int}}$[${\sigma }_{\mathrm{tot}}]$
G10 with BBC
Pantheon	$0.154\pm 0.006$	$3.02\pm 0.06$	$0.053\pm 0.009$	$0.09$[$0.14]$
Low-z	$0.154\pm 0.011$	$2.99\pm 0.15$	$0.076\pm 0.030$	$0.10$[$0.15]$
SDSS	$0.159\pm 0.010$	$3.08\pm 0.13$	$0.057\pm 0.015$	$0.09$[$0.14]$
PS1	$0.167\pm 0.012$	$3.02\pm 0.12$	$0.039\pm 0.016$	$0.08$[$0.14]$
SNLS	$0.139\pm 0.013$	$3.01\pm 0.14$	$0.045\pm 0.020$	$0.09$[$0.14]$
C11 with BBC
Pantheon	$0.156\pm 0.005$	$3.69\pm 0.09$	$0.054\pm 0.009$	$0.11$[$0.14]$
Low-z	$0.156\pm 0.011$	$3.53\pm 0.20$	$0.067\pm 0.030$	$0.12$[$0.15]$
SDSS	$0.156\pm 0.009$	$4.04\pm 0.18$	$0.059\pm 0.015$	$0.11$[$0.14]$
PS1	$0.167\pm 0.012$	$3.51\pm 0.16$	$0.041\pm 0.016$	$0.10$[$0.14]$
SNLS	$0.139\pm 0.013$	$3.59\pm 0.17$	$0.037\pm 0.020$	$0.10$[$0.14]$
G10 with Conventional Fitting
Pantheon	$0.148\pm 0.005$	$3.02\pm 0.06$	$0.072\pm 0.010$	$0.10$[$0.17]$
Low-z	$0.147\pm 0.011$	$3.00\pm 0.13$	$0.077\pm 0.032$	$0.11$[$0.16]$
SDSS	$0.149\pm 0.009$	$3.11\pm 0.12$	$0.078\pm 0.016$	$0.09$[$0.15]$
PS1	$0.161\pm 0.011$	$2.93\pm 0.11$	$0.064\pm 0.018$	$0.09$[$0.16]$
SNLS	$0.128\pm 0.013$	$3.08\pm 0.14$	$0.054\pm 0.023$	$0.09$[$0.18]$

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The predicted distance bias for each survey, using simulations of >100,000 SNe for each, is shown in Figure 12. For display purposes, these biases are shown after simulating both assumptions about the scatter model but then assuming that the "G10" scatter model is correct in the analysis and assuming a β value of 3.1. It is instructive to compare the biases in m_B and c for the different scatter models (the distance bias from x₁ is typically <10% of the total distance bias). A key difference due to the scatter models is whether the selection effects at relatively high z favor SNe with bluer color values or brighter peak brightness values (e.g., the SNLS m_b panel and c panel in Figure 12). For the G10 scatter model, the relative color bias with redshift is small compared to the relative m_B bias with redshift. For the C11 scatter model, the opposite is true. The distance biases, when applying the two scatter models, agree to within 1% for PS1, SNLS, and HST, but as shown in Figure 12, they diverge more significantly for SDSS at high z. This difference for SDSS is likely due to its preferential selection of SNe based on both the magnitude and color of the SNe, rather than just the magnitude, as is the case for the other surveys.

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The most significant difference in the distance biases between the two scatter models is for the Low-z sample, which, as seen in Figure 12, has a ∼0.03 mag difference in the predicted distance bias for the two scatter models. This large offset, relative to the other surveys, is due to the different distribution of color for this sample, as shown in Figure 10: the mean color in the Low-z sample is Δc ∼ 0.025 mag redder than the other three samples. The difference in bias between C11 and G10 for the Low-z sample comes from the difference in β of Δβ = 0.7 from the two models multiplied by the difference in the mean color of the sample relative to the higher-z samples of Δc ∼ 0.035.

Table 6 also shows the nuisance parameters and Hubble residual dispersion (σ_tot, different from the intrinsic scatter σ_int) for each subsample using both the BBC and the conventional method from B14 and S14. It is clear from the dispersion values given in Table 6 that the BBC method reduces the dispersion of each subsample significantly. A comparison with the conventional model is discussed more in Section 7. The impact of the BBC method is higher when the measurement noise is greater. For example, the rms of SNLS distance residuals decreases from 0.18 to 0.14 mag, and this reduction can be seen higher on the higher-z SNe. Furthermore, the rms reduction from bias corrections is least significant for the Low-z sample because the widths of the underlying c and x₁ distributions are larger than the noise and intrinsic scatter of these parameters.

To correct the distances for this analysis, we take the average of the G10 and C11 bias corrections. The systematic uncertainty is half the difference between the two models. For each survey, there is an additional systematic uncertainty in the distance bias corrections due to the uncertainty of the selection function of each survey. As shown for the PS1 simulations in Figure 6, this uncertainty is determined by varying the selection function so that the χ² agreement of the simulated redshift distribution and observed redshift distribution is reduced by 1σ. Understanding the uncertainty of the Low-z selection is most difficult because it is unclear to what extent the discoveries were magnitude limited or volume limited (see S14, Appendix). The differences in distance biases with redshift for the volume-limited and magnitude-limited assumptions are shown for the different cases in Figure 13. For the volume-limited case, we prescribe a mean c and x₁ dependence on z in our simulations to mimic the trends seen in the data. The differences in bias corrections at the high end of the redshift range can be as much as 0.03 mag. We use the magnitude-limited case in our baseline analysis and the volume-limited case as our systematic, and this is discussed further in Appendix C.

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For the global analysis, we include mass estimates for all of the host galaxies of the SNe in each of the samples. For the PS1 sample, these estimates are discussed in Section 3.7, and the estimates for the SDSS and SNLS sample are provided in B14. We select 70 galaxies in the SDSS sample to compare our mass procedure with that done in B14, and we find a difference of ${\rm{\Delta }}{\mathrm{log}}_{10}({M}_{\mathrm{stellar}}/{M}_{\odot })=0.08\pm 0.04$ with no dependence on galaxy mass. For the Low-z sample, we redetermine masses for all the host galaxies using the same procedure as done for the PS1 sample and with photometry in ugrizBVRIJHK that is available from the 2MASS data release (Skrutskie et al. 2006) and the SDSS Photometric Catalog, Data Release 12 (Alam et al. 2015). For hosts that are too faint for survey depth, we assign them a mass in the lowest mass bin.

The correlation between host mass and Hubble residual is shown in Figure 14. Fitting Equation (5) to the Pantheon sample, we find that ${m}_{\mathrm{step}}=10.13\pm 0.02$ and τ = 0.001 ±0.071. We assume these values for this analysis and the difference with the fiducial m_step = 10.0 as a systematic uncertainty. The inferred value of γ and its uncertainty given the change in location of mass step is <1%. We find a mass step of $\gamma =0.053\pm 0.009$ mag and $\gamma =0.054\pm 0.009$ mag for the Pantheon sample using the G10 and C11 scatter models, respectively. This is smaller than the offset of $\gamma =0.072\,\pm 0.010$ mag using no bias corrections.

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As shown in Table 6, the HR mass steps found for each subsample are all consistent to ∼1σ. For the Pantheon sample, there are $411$ host galaxies with ${\mathrm{log}}_{10}({M}_{\mathrm{stellar}}/{M}_{\odot })\lt 10$and $611$ host galaxies with ${\mathrm{log}}_{10}({M}_{\mathrm{stellar}}/{M}_{\odot })\gt 10$. The relative splits of low/high-mass galaxies for each subsample are as follows: PS1 ($116$/$163$), SNLS ($140$/ $96$), SDSS ($126$/$209$), and Low-z ($29$/$143$). While the Low-z sample has the highest offset of $0.076\pm 0.030$ mag, the significance is ∼2σ because there is a large imbalance in the number of high- and low-mass galaxies. As shown in Figure 14, the relative numbers of high- and low-mass galaxies are within a factor of 2 for all redshift bins z > 0.1. A change in host galaxy demographics with redshift is expected owing to galaxy evolution, though the galaxy-targeted nature of the Low-z sample exaggerates the effect seen in Figure 14. This is discussed further in Appendix C.

One possible systematic uncertainty from our baseline analysis is if the mass–luminosity relation itself changes with redshift, as predicted by Rigault et al. (2013) and Childress et al. (2014). These predictions are due to the inference that Δ_M from Equation (5) is due to a dependence of SN luminosity on the age of the SN progenitor. Since the correlation between host mass and progenitor age evolves with cosmic time, Δ_M should also change with redshift. The modeled change in Hubble residuals due to this transition for Childress et al. (2014) is shown in Figure 14—the magnitude of the Hubble step decreases with redshift. We find that the best-fit line, solved simultaneously with the other nuisance parameters in the BBC fit, is $(0.075\pm 0.015)+(-0.079\pm 0.041)\times z$ for the G10 scatter model, and it is roughly the same for the C11 scatter model. Since our measured slope appears to be roughly consistent (2σ) with the prediction of Childress et al. (2014) and Rigault et al. (2013), we include it as a systematic uncertainty.

As done for PS1, we simulate host galaxy masses for each sample to determine whether there are any biases in the recovered host mass–luminosity relation. We do not see any discrepancies between input and output values of γ from simulations beyond 3 mmag, much smaller than the uncertainty on γ values reported in Table 6.

Any change in the standardization parameters can produce systematic uncertainties in the measurements of cosmological parameters (Conley et al. 2011). We define β(z) = β₀ + β₁ × z, and similarly for α(z). Before discussing the recovered α and β values with redshift, we note that in our simulations of the Pantheon sample, using 30 simulations of ∼1000 SNe, when we input β₁ = 0, we recover a biased value of β₁ (β₁ = −0.35 ± 0.06 for the G10 model and β₁ =−0.7 ± 0.10 for the C11 model). This bias is not present when we just fit for $\beta (z)={\beta }_{0}$, and the problem is predicted in Kessler et al. (2013) owing to problems with using the correct intrinsic scatter matrix in the fit. We find similar issues with or without the BBC method. Therefore, we subtract out the evolution bias predicted from the simulations in our fits for evolution.

In Figure 15, the values of α, β, and σ_int are all shown for discrete redshift bins (only for the G10 model for simplicity). Table 7 reports the parameters of the best-fit lines to the evolution shown in Figure 15. We do not see convincing evidence of α or β evolution except in the highest redshift bin in the β evolution plot, where the SNe have the largest uncertainties. We therefore choose for the baseline analysis to have β₁ = 0 and α₁ = 0, though we allow for there to be a β₁ systematic equal to the size of the uncertainty in the β₁ measurement, treating it like a statistical uncertainty. In past analyses (e.g., Conley et al. 2011), different values of the σ_int are used for different samples; however, as shown in Figure 15, we find consistency across samples and fix one value.

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Table 7.  Recovered Evolution Values From the Data Assuming the G10 Scatter Model. α, β, and γ Evolutions are each Fit Separately, as well as Together. The Zeroth- and First-Order Component of each Term is given, Such that, e.g., $\alpha (z)={\alpha }_{0}+{\alpha }_{1}\times z$

	α0	α1	β0	β1	γ0	γ1
Baseline	$(0.154\pm 0.005)$	0	$(3.030\pm 0.063)$	0	$(0.053\pm 0.009)$	0
α evol.	$(0.156\pm 0.007)$	$(-0.007\pm 0.024)$	$(3.030\pm 0.064)$	0	$(0.053\pm 0.009)$	0
β evol.	$(0.154\pm 0.006)$	0	$(3.139\pm 0.099)$	$(-0.348\pm 0.289)$	$(0.052\pm 0.009)$	0
γ evol.	$(0.155\pm 0.005)$	0	$(3.028\pm 0.063)$	0	$(0.075\pm 0.015)$	$(-0.079\pm 0.041)$
α, β, γ evol.	$(0.158\pm 0.008)$	$(-0.015\pm 0.024)$	$(3.138\pm 0.098)$	$(-0.348\pm 0.285)$	$(0.076\pm 0.015)$	$(-0.082\pm 0.041)$

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One related issue to the parameter evolution is possible population drift of the underlying c and x₁ populations with redshift. As shown in Rubin & Hayden (2016), not accounting for this drift yields very large differences in the inferences of cosmological parameters. This drift is accounted for using the BBC method, since it accounts for selection effects and allows for different underlying light-curve parameter distributions for each subsample as presented in SK16.

For each SN, we use an estimate of the extinction from dust along the line of sight determined from Schlafly & Finkbeiner (2011). Following S14, for the systematic uncertainty we adopt a global 5% scaling of $E(B-V)$ as the systematic uncertainty. A systematic bias in the SN distances due to uncertainty in the MW extinction is partially mitigated from fitting of the light curve, as the light-curve color parameter may absorb some of the effects of the uncorrected MW extinction. However, the impact on recovered cosmology may still be significant because the average MW $E(B-V)$ (mag) value per survey is different: 0.033 for Low-z, 0.037 for SDSS, 0.029 for PS1, 0.018 for SNLS, 0.010 for HST. Additionally, extinction is treated differently in rest frame versus redshift frame. Finally, we note that the SALT2 model was trained with the Schlegel et al. (1998) extinction model, which has been improved. It is unclear how large of a systematic another retraining would yield, though we estimate that this is subdominant to the calibration and intrinsic scatter systematics.

The motions of SN host galaxies from coherent flows, like dipole or bulk flows, are corrected to reduce biases in cosmological parameters. Past analyses, like S14 and B14, use the velocity field in Hudson et al. (2004), which is derived using the galaxy density field from the IRAS PSCz redshift survey (Branchini et al. 1999). The same method is applied here using a map of the matter density field calibrated by the 2M++ catalog²⁶ out to z ∼ 0.05, with a light-to-matter bias parameter²⁷ of β = 0.43 and a dipole as described in Carrick et al. (2015).

The impact on the dispersion of distance residuals due to the coherent flow corrections is relatively small. The dispersion of distance residuals for SNe with z < 0.1 is $0.144$ mag after the corrections but $0.149$ mag before the corrections. Carrick et al. (2015) state that the uncertainty in galaxy velocities after the corrections is 150 km s⁻¹, though it is unclear whether that magnitude fully describes the uncertainty in redshift estimates of the SN host galaxies. To test this issue, one may compare the intrinsic scatter of SN distances for low-z SNe to those for high-z SNe. However, this test is complicated because there are many key differences between these subsamples, as shown in Figure 10 and discussed throughout this analysis. Instead, we can compare the intrinsic scatter of distances of SNe in the range of 0.01 < z < 0.03 to distances of SNe in the range of 0.03 < z < 0.06 so that the subsamples are much more consistent. Doing so, we find that the difference in intrinsic scatter between these two subsamples indicates an individual redshift uncertainty of 250 km s⁻¹. This is higher than the estimate from Carrick et al. (2015), but we use it for our analysis. Furthermore, if we did not apply the coherent flow corrections, we find that the individual redshift uncertainty must be 260 km s⁻¹ so that the sample has the same intrinsic scatter as the sample with coherent flow corrections.

The systematic uncertainty of the coherent flow corrections should account for covariance between the velocities of the SN host galaxies (Hui & Greene 2006). This covariance matrix is modeled explicitly in Huterer et al. (2017) using the Low-z sample from Scolnic et al. (2015), which is nearly identical to the Low-z sample used here. Analyzing the SN and galaxy data from 6dFGS separately, Huterer et al. (2017) show that both samples are consistent with the peculiar velocity signal of a fiducial ${\rm{\Lambda }}\mathrm{CDM}$ model. Instead of implementing the full covariance matrix from Huterer et al. (2017), which does not account for corrections of bulk flows, following Zhang et al. (2017), we account for a systematic uncertainty in the coherent flow corrections by shifting the light-to-matter bias parameter β by 10% and redetermining the velocity corrections.

There is a large list of systematic uncertainties associated with the various analysis steps in this section. In total, there are 85 separate systematic uncertainties, though 74 are related to calibration. All of the main systematic uncertainties on the binned distances are shown in Figure 16; here we show the change in binned distances if we vary a given systematic by 1σ. For the survey-calibration uncertainties, we only show two of them, a systematic from SNLS and PS1, and there are roughly eight of these uncertainties per survey as described in Table 5. The full systematic covariance matrix is shown in Figure 17.

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To determine cosmological parameters, each measured distance modulus (μ) from Equation (3) is compared to a model distance that depends on redshift and cosmological parameters, ${\mu }_{\mathrm{model}}=+5\mathrm{log}({d}_{L}/10\,\mathrm{pc})$, such that

$$\begin{eqnarray}&&{d}_{L}(z)=\displaystyle \frac{c}{{H}_{0}}\ \mathop{\mathrm{lim}}\limits_{{{\rm{\Omega }}}_{k}^{{\prime} }\to {{\rm{\Omega }}}_{k}}\displaystyle \frac{1+z}{\sqrt{{{\rm{\Omega }}}_{k}^{{\prime} }}}\,\sinh \left[\sqrt{{{\rm{\Omega }}}_{k}^{{\prime} }}{\int }_{0}^{z}\displaystyle \frac{{dz}^{\prime} }{E(z^{\prime} )}\right]\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&E(z)=\sqrt{{{\rm{\Omega }}}_{m}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}{(1+z)}^{3(1+w)}+{{\rm{\Omega }}}_{k}{(1+z)}^{2}}.\end{eqnarray} \tag{ 10 }$$

A grid of cosmological models is fitted over to minimize the χ² in Equation (8). We explore four cosmological models: a flat ${\rm{\Lambda }}\mathrm{CDM}$ model ($w=-1$, Ω_K = 0), a nonflat $o\mathrm{CDM}$ model (w = −1, Ω_K varies), a flat wCDM model (w₀ varies, w_a = 0), and a flat ${w}_{0}{w}_{a}$CDM model (w₀ and w_a both vary, Ω_K = 0). All of the calculations are done with CosmoMC (Lewis & Bridle 2002). While the BBC method produces binned distances over redshift, for cosmological fitting we use the unbinned, full SN data set. We use the unbinned data set mainly to be in line with general community reproducibility. We still use the binned distances to generate the systematic covariance matrix, which is used as a 2D 40-bin interpolation grid to create a covariance matrix for the full SN data set. Diagonal uncertainties from the individual distances can be added together with the full systematic matrix following Equation (6). Differences in w between the binned and unbinned data sets are at a <1/16σ level for the statistical measurements, and <1/8σ when including the systematic covariance matrix.

The cosmological fits to the SN-only sample are shown in Table 8 with and without systematic uncertainties. Using our full SN sample with systematic uncertainties, with no external priors, we find ${{\rm{\Omega }}}_{m}=0.298\pm 0.022$. Without systematic uncertainties, the uncertainty on Ω_m is roughly 2× smaller. When not assuming a flat universe, we combine various probes together to constrain the $o\mathrm{CDM}$ model. When using SNe alone, we find that ${{\rm{\Omega }}}_{m}=0.319\pm 0.070$ and ${{\rm{\Omega }}}_{L}=0.733\,\pm 0.113$. We find that the evidence for nonzero Ω_Λ from the SN-only sample is >6σ when including all systematic uncertainties. As shown in Figure 18, this is a factor of ∼20 improvement over the Riess et al. (1998) constraints in this plane. Furthermore, the significance for nonzero Ω_Λ is much higher than the <3σ effect quoted by Nielsen et al. (2016), which re-analyzed the B14 sample, though their analysis technique is disputed by Rubin & Hayden (2016). A study using the Pantheon sample and null tests done in this analysis to examine nonstandard cosmological results like those from Nielsen et al. (2016) and Dam et al. (2017) is currently in preparation (D. L. Shafer et al. 2018, in preparation).

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Table 8.  Cosmological Constraints for the SN-only Sample with and Without Systematic Uncertainties. Values are given for Three Separate Cosmological Models: ${\rm{\Lambda }}\mathrm{CDM}$, $o\mathrm{CDM}$, and wCDM

Analysis	Model	w	Ωm	ΩΛ
SN-stat	${\rm{\Lambda }}\mathrm{CDM}$	⋯	0.284 ± 0.012	0.716 ± 0.012
SN-stat	$o\mathrm{CDM}$	⋯	0.348 ± 0.040	0.827 ± 0.068
SN-stat	wCDM	−1.251 ± 0.144	0.350 ± 0.035	⋯
SN	${\rm{\Lambda }}\mathrm{CDM}$	⋯	0.298 ± 0.022	0.702 ± 0.022
SN	$o\mathrm{CDM}$	⋯	0.319 ± 0.070	0.733 ± 0.113
SN	wCDM	−1.090 ± 0.220	0.316 ± 0.072	⋯

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To evaluate the impact of the systematic uncertainties, we combine constraints from the Pantheon SN sample with those from the compressed likelihood of the CMB from Planck Collaboration et al. (2016b) and measure Ω_m and w in the wCDM model. Constraints from BAO and H₀ measurements are included later in this section. The impact of systematic uncertainties is shown in terms of the relative size of the uncertainty of w in Table 9. We find that the systematic uncertainty (${\sigma }_{w}=0.025$) is smaller than the statistical uncertainty (${\sigma }_{w}=0.031$). Unlike previous analyses (e.g., B14 and S14) that found that calibration uncertainties made up >80% of the systematic error budget, we find a more even split between the various systematics. The calibration uncertainties are due to uncertainties of the individual photometric systems of each sample, as well as the calibration uncertainties propagated through the SALT2 model. We find that the SALT2 calibration uncertainty is larger in magnitude than the combined impact from all the various systems, which are reduced by S15 and are independent of each other. Still, all of the systematic uncertainties related to calibration have a net effect of roughly 66% of the total systematic error.

Table 9.  The Dominant Systematic Uncertainties in the Pantheon SN Sample with Respect to w While Solving for a wCDM Model. The w Shift is Defined Relative to the Statistical Value, and ${\sigma }_{w}^{\mathrm{syst}}$ is Defined to be $\sqrt{{\sigma }_{w}^{2}-{\sigma }_{w-\mathrm{stat}}^{2}}$ When a Specific Systematic Uncertainty is Applied

	w Shift	${\sigma }_{w}^{\mathrm{syst}}$	Fraction of ${\sigma }_{w}^{(\mathrm{stat})}$
Stat. uncertainty	+0.000	0.031	1.000
Total sys. uncertainty	+0.031	0.025	0.814
Calibration
SALT2 cal	−0.002	0.014	0.457
Survey cal	+0.006	0.009	0.285
HST cal	−0.006	0.006	0.177
Supercal	+0.002	0.003	0.098
SN modeling
Selection	+0.010	0.007	0.233
Intrinsic scatter	+0.019	0.005	0.170
β evol.	−0.001	0.007	0.238
γ evol.	−0.002	0.000	0.000
mstep shift	−0.002	0.002	0.064
External
MW extinction	+0.010	0.008	0.262
Pec. vel.	+0.000	0.003	0.103

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The systematic uncertainties increase the uncertainties of the best-fit parameters and also shift the best-fit parameters by reweighting the pulls of each SN in the fit. These two impacts are shown in Table 9 as both the best-fit value of w is shifted and the uncertainty on w is increased. The shifts are mainly due to systematic uncertainties that most strongly affect the low-z sample: calibration, MW extinction, intrinsic scatter, and selection.

Of the noncalibration uncertainties listed in Table 9, the uncertainties due to selection, MW extinction, intrinsic scatter, and β evolution are all similarly large at ∼0.25 of the statistical error. While the impact of the systematic uncertainty due to intrinsic scatter is not the largest, including it shifts the best-fit value of w by Δw = 0.019. The systematic uncertainty effectively deweights the low-z sample and would have been even larger if we had not reduced the impact by a factor of 2 by averaging the distance biases from the G10 and C11 models. We find little impact on w from the location of the mass step or the possibility that the magnitude of the host mass–luminosity relation is changing with redshift. We see negligible impact from the peculiar velocity corrections.

A useful guide to understand the possible scale of systematic uncertainties is to redetermine the cosmological parameters without the main sample corrections. This is shown in Table 10 for an analysis only considering statistical uncertainties for cases where no distance bias correction is applied, no mass correction is applied, no Supercal correction is applied, and no peculiar velocity correction is applied. While the size of these shifts is larger than any of the systematics, they give a sense of where possible systematic uncertainties could reside. We find that the distance bias correction changes the value of w by ∼7%, larger than any of the other corrections. The mass and Supercal corrections are both large (Δw ∼ 0.024), due to their impact on the Low-z sample. As shown in Figure 14, the change in demographics of the host galaxies with redshift makes the recovered cosmological values from the sample sensitive to the mass correction. The Supercal correction is Δw = 0.024 because there is a recalibration of B − V zero-points in the low-z sample that shifts the average distances by ∼0.02 mag. The peculiar velocity correction is small, on the order of 1% in w.

Furthermore, it is instructive to compare the relative pulls on the distances and recovered w values from each subsample. As shown in Figure 19, we find that the mean distance residual relative to the best-fit cosmology is 0.02 mag or lower. Upon removing any single subsample from the analysis, the Low-z sample has the largest impact on w and causes a change of Δw ∼ 0.07 if only the statistical uncertainties are included. When including systematic uncertainties, the Low-z sample causes a change of Δw ∼ 0.04. The pulls of the other samples are all within $| {\rm{\Delta }}w| =0.02$. Finally, we also compare the impact on the uncertainty on w when each subsample is removed for the both the statistical and statistical+systematic analyses. The Low-z sample has the strongest impact on the statistical uncertainty, but not on the total uncertainty, because of its large systematics. The SNLS has the strongest impact on the total uncertainty, likely because it is at high redshift and has small systematic uncertainties. We find that the small high-z sample from HST has a minimal impact on our measurement of w. This is likely due to the sample size and this parameterization of dark energy, which assumes that w(z) is constant; this is discussed in detail in Riess et al. (2018), which uses the Pantheon sample and varies the parameterization of dark energy.

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The Low-z sample has an outsized impact on a number of the variants shown in Table 10. Interestingly, we find that when applying the C11 scatter model with the BBC method, we recover a difference in w of 0.065 depending on whether we include SNe with z < 0.1, but when we apply the G10 scatter model with the BBC method, we recover a difference in w of 0.010 depending on whether we include SNe with z < 0.1. This effect can be traced back to the 3% offset in distance biases for the Low-z sample shown in Figure 12. This issue must be resolved in future analyses to continue using the Low-z sample.

To better determine cosmological parameters, we include constraints from measurements of the CMB from Planck Collaboration et al. (2016a), measurements of the local value of H₀ from Riess et al. (2016), and measurements of baryon acoustic oscillations from the SDSS Main Galaxy Sample (Ross et al. 2015), the Baryon Oscillation Spectroscopic Survey, and the CMASS survey (Anderson et al. 2014). These BAO measurements set the BAO scale at z = 0.106, 0.35, and 0.57. For all CMB constraints, we include data from the Planck temperature power spectrum and low-ℓ polarization (Planck TT + lowP).

Before combining constraints from different probes, we can compare constraints on Ω_m when we assume that the universe is flat, w₀ = −1, and w_a = 0. Using our full SN sample with systematic uncertainties, with no external priors except flatness, we find ${{\rm{\Omega }}}_{m}=0.298\pm 0.022$. This is similar to the value determined from Planck Collaboration et al. (2016a) of 0.315 ± 0.013 and the value from BAO of 0.310 ± 0.005 (Alam et al. 2017). Using only SNe, there is no constraint on H₀ since H₀ and ${ \mathcal M }$ from Equation (3) are degenerate. Constraints on H₀ from data that include SN measurements only come indirectly from the SN component in that the SN measurements constrain parameters like Ω_m and w that have covariance with H₀. Since the low-z SNe in this sample and the one used in Riess et al. (2016) are very similar, there may be some common systematics that affect both probes, though this is likely to be small, as Riess et al. (2016) compare SNe in the Hubble flow to SNe with z < 0.01, whereas our analysis compares SNe in the Hubble flow to SNe with z > 0.1. If we relax the assumption that the universe is flat, we can see the results of combined probes in Table 11.

Relaxing the assumption of a cosmological constant, we measure w, the dark energy equation-of-state parameter. For these wCDM models, we assume a flat universe (${{\rm{\Omega }}}_{k}=0$). In Table 12, we compare how the different cosmological probes impact the constraints on Ω_m and w. As shown in Figure 20, combining Planck and SN measurements, we find ${{\rm{\Omega }}}_{m}=0.307\pm 0.012$ and $w=-1.026\pm 0.041$. This is to date the tightest constraint on dark energy, and we find that it is consistent with the cosmological constant model. These values are more precise than, though consistent with, the values from combining Planck and BAO measurements, which are ${{\rm{\Omega }}}_{m}=0.312\pm 0.013$ and $w=-0.991\pm 0.074$. Combining SN, BAO, Planck, and H0 measurements yields ${{\rm{\Omega }}}_{m}=0.299\,\pm 0.007$ and $w=-1.047\pm 0.038$, similar to the results of just SN+Planck. If we replace constraints from Planck with those from WMAP9 (Hinshaw et al. 2013), we see a shift of Δw ∼ +0.04 seen in past studies (e.g., B14 or R14), which does not change any of our conclusions.

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In Table 13, we compare how the different cosmological probes impact the constraints on w₀ and w_a. We show in Figure 21 the constraints of various combinations of the different probes given the ${w}_{0}{w}_{a}$CDM model. We find that combining SN, BAO, Planck, and H0 measurements, ${w}_{0}=-1.007\pm 0.089$ and ${w}_{a}=-0.222\pm 0.407$. These values are consistent with the cosmological constant model of dark energy such that w₀ is consistent with −1 and w_a is consistent with 0, or no evolution of the equation of state of dark energy.

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Comparisons of the results from R14 and B14 with the results from this analysis are shown in Table 14. R14 used a sample of 112 PS1 SNe and 180 Low-z SNe to measure cosmological parameters and found for the wCDM model a ∼2σ deviation from w = −1 when combining SN and Planck measurements. With a larger sample of PS1 SNe and an improved analysis, we find no hints of tension with a cosmological constant from the parameters derived for the PS1+Low-z sample.

As can be seen in Table 14, the statistical-only constraints from the improved PS1+Low-z sample are consistent with those from R14, and the constraints on Ω_m and w are tighter. However, accounting for systematic uncertainties causes the best-fit parameters of this analysis to diverge from R14. One of the main reasons for this is that compared to the analysis of S14, the systematics of the PS1 sample are smaller but the systematics of the Low-z sample are larger, thereby effectively down-weighting the Low-z sample with respect to the PS1 sample.

There are no large differences between the constraints from our full Pantheon sample and those from the B14 analysis. The reason for this is shown in Figure 19—even though our Low-z sample is much larger, our systematic uncertainties on the Low-z bias correction are also much larger. Furthermore, the addition of the PS1 sample does not have much pull, as it is consistent with SNLS and SDSS. This subsample also occupies a redshift range in between those of the SNLS and SDSS subsamples. Still, we note the 30% decrease in total uncertainties from B14 and our analysis.

Each aspect of the analysis from R14 and S14 has been improved for the present analysis, though we find here that the Low-z sample must be better modeled in order to realize the significant gains from the larger statistics and smaller systematics in the high-z SN samples. Since there are ∼180 Low-z SNe each with a distance modulus precision of 0.15 mag, the standard error on the sample is 0.011 mag. Therefore, systematics that affect the Low-z sample relative to the high-z sample at the 1% will significantly diminish the impact of the Low-z sample. There are a series of systematics on this level that affect the Low-z sample more than other samples: intrinsic scatter, selection, MW extinction, and calibration. The impact is higher for the Low-z sample because the Low-z sample has redder SNe on average (by 0.03) than each of the higher-z samples, the MW extinction at the location of the SNe is higher (by 0.05) on average than in the higher-z samples, the selection effects are more difficult to model because there is uncertainty in whether the selection was volume or magnitude limited, and the calibration uncertainties are 2× as large as the high-z samples.

While there are some Low-z data samples not included here (e.g., Ganeshalingam et al. 2013), other low-z samples face similar issues and will likely not improve the cosmological constraint without improving the systematic uncertainties. The most helpful Low-z sample would be one that was based off a rolling survey, so that selection effects are well understood and the color distribution is similar to that of the high-z samples, and one in which the calibration of the sample is on the level of the high-z samples. This can be expected from the Foundation SN sample (Foley et al. 2018), which uses the PS1 telescope to follow up SNe discovered by rolling surveys. Other possible low-z samples based on rolling surveys, like ATLAS (Tonry 2011), may further help this issue.

There's a fundamental difference in the approach of applying bias corrections between this analysis and that of B14 and S14. Both B14 and S14 use a redshift-dependent distance bias correction as shown in Figure 12, though both these analyses use inaccurate underlying c and x₁ populations for their simulations (see SK16 for a review). KS17 showed that with very large statistics and the same underlying populations, only very slight mmag-level differences are expected between a redshift-dependent distance bias correction and the more complex BBC method if α and β are known a priori. However, using incorrect α and β introduces small biases in the method of B14 and S14 because α and β are not solved simultaneously when measuring the distance biases.

Both B14 and S14 consider the G10 and C11 scatter models, though while S14 and our analysis average the two, B14 chooses the G10 scatter model for its baseline analysis. We do not choose one model or the other, as there is insufficient empirical evidence to favor either model. Somewhat implicit in the choice of scatter model is the assumption of a single σ_int value. Both B14 and S14 determine separate σ_int values for the high-z and low-z samples, but we find that a single value characterizes the full sample. This is likely due to our higher value of the peculiar velocity uncertainty used in our analysis (S14 and B14 used 150 km s⁻¹; we use 250 km s⁻¹) and the fact that the BBC method corrects for the overestimation of the fit parameter errors, which has led to an incorrect assessment of σ_int in past analyses (Kessler & Scolnic 2017).

All of these analyses are still limited in measurements of the evolution of standardization parameters. In this analysis, we find a 1σ signal for evolution of the β parameter. Interestingly, the analysis of Jones et al. (2018) finds β evolution of −1.28 ± 0.49 with a sample 3× larger (though with its own systematic uncertainties from contamination). PS1 is not ideal for determining this evolution because its maximum redshift is ∼0.6. Additionally, we find ∼2σ evolution of the γ parameter. If the BBC method is not applied, we recover a measured slope of $(-0.067\pm 0.049)$, which is a ∼1.5σ effect. B14 saw no evidence of evolution.

New releases from SNLS and DES should help settle this question of parameter evolution. High-z SNe observed by HST should provide excellent leverage to determine evolution of the nuisance parameters. However, there are currently not enough data at high z to provide tight constraints. There is a similar issue in trying to use HST SNe for constraining ${w}_{0}-{w}_{a}$ (Riess et al. 2018). However, WFIRST (Spergel et al. 2015; Hounsell et al. 2017) should provide subpercent-level distance constraints of SNe at z ∼ 1.5 and significantly improve constraints on both evolution systematics and dark energy models.

There could be further evolution in the mean of the color variation of the intrinsic scatter model; for both C11 and G10 scatter models, the color scatter is centered around c = 0, and this is assumed to not change with redshift. Various analyses (Foley & Kasen 2011; Mandel et al. 2014) have hypothesized that evolution of the mean of the color scatter may be possible; however, it is unclear with what significance and how well current data already constrain it. It is not included as a systematic here, but studies like the one by Mandel et al. (2017) may be able to isolate the effect so that we can put it into our simulations.

A related evolution uncertainty is due to a possible bimodal population of SNe when considering their UV flux (Milne et al. 2015). If the relative fractions of the different UV subclasses change with redshift, it would propagate to systematic biases in the recovery of SN color with redshift, which would itself propagate to errors in the recovered cosmology. Cinabro et al. (2017) simulated simplistic models of different UV subclasses of SNe inferred from Milne et al. (2015) and compared the output light curves of real SDSS and SNLS samples and did not see consistency. We did not include it as a systematic uncertainty, but more UV data would serve the double purpose of clarifying this issue and helping with SALT2 training.