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\section{Introduction}
\label{sec:intro}
One of the main goals of the \PS~(PS1) Medium Deep survey is to detect and monitor thousands of SN\,Ia in order to measure the equation of state parameter of dark energy, $w=P/\rho c^2$ (where $P$ is pressure and $\rho$ is density). The first results of this effort are reported in the companion paper by Rest et al. (2013, hereafter R14). For PS1 and other new surveys to advance our understanding of dark energy, the flood of new SNe must be accompanied by similar improvement in the reduction of systematic uncertainties.
Since the initial discovery of cosmic acceleration (\citealp{Riess98}, \citealp{Saul99}), there have been many supernova surveys utilizing multiple passbands and dense time-sampling at both low-z (e.g., CSP,CfA1-4, LOSS, SNFactory\footnote{Carnegie Supernova Project (CSP), Center for Astrophysics (CfA), Lick Observatory Supernova Search (LOSS), Nearby Supernova Factory (NSF)}) and at intermediate and higher-z (e.g., SDSS, ESSENCE, SNLS\footnote{Sloan Digital Sky Survey (SDSS),Equation of State: SupErNovae trace Cosmic Expansion (ESSENCE), SuperNova Legacy Survey (SNLS)}). While the sample sizes have increased, the systematic uncertainties of these samples now are of nearly equal value to the statistical uncertainties (\citealp{Conley_etal_2011}; hereafter C11). Nearly all of the systematic uncertainties in the analysis of these samples fall into a small handful of categories: calibration, selection effects, correlated flows, extinction corrections and light curve modeling. There has been significant recent progress in understanding each of them. For example, recent studies suggest that properties of host galaxies of SNe appear to be correlated with distance residuals relative to a best fit cosmology (e.g., \citealp{2010ApJ...715..743K}, \citealp{2010MNRAS.406..782S}, \citealp{2010ApJ...722..566L}). Other studies have shown that supernova colors and brightnesses, long thought to be inconsistent with a Milky Way (MW)-like reddening law, can be explained by a MW-like dust model (\citealp{Folatellietal:2010}, \citealp{Fo11a}, \citealp{Mandel/etal:2011}, \citealp{Chotard_2011}, Scolnic et al. 2013).
%Ideally, a single survey can be used to determine SNe distances at both low redshifts ($z<0.1$) and medium/high redshifts ($z\sim0.5$) so that one can bypass tension between various SN samples. The PS1 sample consists of SNe between $0.03
The PS1 Medium Deep Survey has discovered over 1700 SN candidates in its first 1.5 years. Of these, \SNIAPSall ~SNe were spectroscopically identified as Type Ia. Well-sampled multi-band light curves with near-peak observations were measured for~\SNIAPSused~of the spectroscopically confirmed sample. We include a low-z sample of \SNIAlowzused~SNe to improve our cosmological constraints. The companion paper by R14 analyzes the photometry of the PS1 light curves, presents the light curve fit parameters and derives constraints on $w$ from a combined data set of PS1 SNe and low-z SNe (hereafter PS1+lz). In this paper, we augment the work of R14 with a more comprehensive analysis of the systematic uncertainties of $w$. Values of the matter density $\Omega_m$ and equation-of-state $w$ are recovered with constraints from SNe alone and when we include constraints from measurements of the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillation (BAO) and the Hubble Constant.
In section \S\ref{sec:overall}, we present an overview of the major systematic uncertainties in our sample, and detail the two approaches towards quantifying these uncertainties. In section \S\ref{sec:calib_sdss} we analyze the photometric calibration of PS1 and attempt to reconcile the reported calibration discrepancies (\citealp{Tonry12}, hereafter T12) between PS1 and SDSS. We also discuss the data sets in PS1+lz and tension between the various samples. Accurate simulations of the PS1 survey and expected selection effects for each of the surveys in the combined PS1+lz are given in \S\ref{sec:Malmquist}. In \S\ref{sec:lcfits} we probe the validity of the two major assumptions of the SALT2 light curve fitter for determining distances to SNe. In \S\ref{sec:pecvel} we analyze coherent flows of the combined sample for the PS1+lz sample. Changes to Milky Way extinction maps are presented in \S\ref{sec:extinction}. Our review and discussion of the dominant uncertainties is given in \S\ref{sec:system} and our conclusions are in \S\ref{sec:conclus}.
