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\section{Overall Systematics Review}  \label{sec:overall}  \nobreak  \subsection{Data}  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: Cal\'an/Tololo [\survCalan~SNe] \citep{1996AJ....112.2408H}, CfA1 [\survCFa~SNe] \citep{1999AJ....117..707R}, CfA2 [\survCFb~SNe] \citep{2006AJ....131..527J}, CfA3 [\survCFc~SNe] \citep{2009ApJ...700..331H}, CSP [\survCSP] \citep{2010AJ....139..519C} and CfA4 [\survCFd~SNe] \citep{2009ApJ...700.1097H}. We also include supernovae not discovered in these surveys but collected as part of the JRK07 \citep{2007ApJ...659..122J} paper [\survOther~SNe]. The PS1 sample contains \SNIAPSused~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 \citep{Ho08} and SNLS \citep{2010guylightcurves} 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\footnote{http://ps1sc.org/transients/}.   External constraints from CMB, BAO and $H_0$ measurements are described in detail in R14. For all these measurements, we use the Markov chains derived by \cite{Planck13XVI}. The Planck data set that is quoted includes data from the Planck temperature power spectrum data, Planck temperature data, Planck lensing, and WMAP polarization at low multipoles. The BAO measurement quoted is from the aggregate of BAO measurements of different surveys, as compiled by \cite{Planck13XVI} and listed in R14. The $H_0$ measurement is from \cite{Riess1103.2976}.  \subsection{Potential Sources of Systematic Errors}  Here, we briefly enumerate the dominant systematic uncertainties in the PS1+lz sample.   \textit{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 \cite{1996AJ....111.1743B}\footnote{http://www.stsci.edu/hst/observatory/cdbs/calspec.html}. In this paper, we increase the size of the sample of Calspec standards that underpin the HST flux scale from 7 to 10 (adding five, but eliminating two) 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 zeropoints. We also explore noted discrepancies between the PS1 and SDSS photometric systems.   \textit{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 simulator\footnote{ SNANA\_v10\_23~@~ \wwwSNANA} \citep{SNANA09} 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.  \textit{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 \citep{phillips93} 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 \citep{2010guylightcurves} as the primary light curve fitter, recently Scolnic et al. (hereafter, S13) 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 afterwards 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.  \textit{Host Galaxy Relations}. Multiple studies have shown relations between various host galaxy properties and Hubble residuals (e.g., \citealp{2010ApJ...715..743K}, \citealp{2010MNRAS.406..782S}, \citealp{2010ApJ...722..566L}). However, there is no consensus about which host galaxy property is directly linked to luminosity \citep{Ch13}, or whether these correlations may be artifacts of light curve corrections \citep{kimgp}.   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.  \textit{MW Extinction Corrections.} For each SN, we correct for the MW extinction at its specific sky location. Our primary fit uses values from \cite{1998ApJ...500..525S}, with corrections from \cite{Schlafly11} 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 non-linearity of extinction corrections \citep{Schlafly11}.   \textit{Coherent Velocity Flows.} To account for density fluctuations, we correct the redshift of each SN for coherent flows \citep{2004MNRAS.352...61H}.   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.  \textit{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 $\sigma{\mu_{\textrm{lens}}}=0.055z$ \citep{2010MNRAS.405..535J}, 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 \citep{Planck13XVI}, to address systematics in external data sets we compare the results when we include constraints from WMAP \citep{Hinshaw+12}.  \subsection{Error Propagation}  To determine the entire systematic uncertainty of $w$ from the PS1+lz sample, we follow two different approaches towards 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 \cite{Riess1103.2976}, 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}} . \label{eqn:cdef}  \end{equation}  where $\mathbf{D}_{\mathrm{stat}}$ is a diagonal matrix with each element consisting of the square of the intrinsic dispersion of the sample $\sigma_{\textrm{int}}^2$ and the square of the noise error $\sigma_n ^2$ for each SN. $\mathbf{C}_{\mathrm{sys}}$ is the systematic covariance matrix. C11 further separates $\mathbf{C}_{\mathrm{sys}}$ 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 \citep{tripp} and using SALT2 to fit the light curve,   \begin{equation}  \mu=m_B+\alpha \times x_1 -\beta \times c -M,  \end{equation}  where $m_B$ is the peak brightness of the SN, $x_1$ is the stretch of the light curve, $c$ is the color of the light curve, and $\alpha$, $\beta$ and M are nuisance parameters. Explanation of the derivations of $\alpha$ and $\beta$ is given in R14. The systematic covariance, for a vector of distances $\vec{\mu}$, between the i'th and j'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)  \left( \sigma_{S_k} \right)^2,  \label{eqn:covar}  \end{equation}  where the sum is over the $K$ systematics $S_k$, $\sigma_{ S_k}$ is the  magnitude of each systematic error, and  $ \partial \mu$ 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 ~\rps~filter, we refit all of the SNe light curves after adding $0.01$ mag to the zeropoint of all observed \rps~values. Following C11, we do not fix $\alpha$ and $\beta$ when we propagate the systematic covariance matrix. $\alpha$ and $\beta$ are derived with SALT2mu \citep{Mar11} 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 ($\sigma_{\textrm{int}}=0.115$), $\alpha$ and $\beta$ are found to be $0.147\pm0.010$ and $3.13\pm0.12$ respectively.   Given a vector of distance residuals for the SN sample $\Delta  \vec{\mathbf{\mu}} = \vec{\mu} -  \vec{\mu}(H_0,\Omega_M,\Omega_{\Lambda},w,\vec{z}) $ then $\chi^2$ may be expressed as  \begin{equation}  \chisq = \Delta \vec{\mu}^T \cdot \mathbf{C}^{-1} \cdot \Delta \vec{\mu} .  \label{eqn:chieqn}  \end{equation}  We minimize Eqn. \ref{eqn:chieqn} to determine cosmological parameters that include $H_0, \Omega_M, \Omega_{\Lambda}$ and $w$. The cosmological parameters are defined in R14 - Eq. 3. All cosmological parameters quoted in this pair of papers are of the marginalized values and not the minimum $\chi^2$ 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 $\Omega_M$ 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 $w$ vs. $\Omega_M$ 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. \citealp{Riess1103.2976}). Unlike the method by C11, there is no systematic error component to the error matrix in Eqn.~\ref{eqn:covar}. 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.