forgerard.tex

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\defcitealias{1999PASP..111...63F}{F99}
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\begin{document}

\title{Evidence for a Color Parameter Unassociated With Dust Within the Type~Ia Supernovae of the Nearby Supernova Factory}
\author{A.~G.~Kim}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}
    
\author{     G.~Smadja}
\affiliation{    Universit\'e de Lyon, F-69622, Lyon, France ; Universit\'e de Lyon 1, Villeurbanne ; 
    CNRS/IN2P3, Institut de Physique Nucl\'eaire de Lyon}

\author{     G.~Aldering}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}

\author{     P.~Antilogus}
\affiliation{    Laboratoire de Physique Nucl\'eaire et des Hautes \'Energies,
    Universit\'e Pierre et Marie Curie Paris 6, Universit\'e Paris Diderot Paris 7, CNRS-IN2P3, 
    4 place Jussieu, 75252 Paris Cedex 05, France}
    
\author{     S.~Bailey}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}

\author{     C.~Baltay}
\affiliation{    Department of Physics, Yale University, 
    New Haven, CT, 06250-8121}

\author{     K.~Barbary}
\affiliation{
    Department of Physics, University of California Berkeley,
    366 LeConte Hall MC 7300, Berkeley, CA, 94720-7300}

\author{    D.~Baugh}
\affiliation{   Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China }

\author{     K.~Boone}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}
\affiliation{
    Department of Physics, University of California Berkeley,
    366 LeConte Hall MC 7300, Berkeley, CA, 94720-7300}

\author{     S.~Bongard}
\affiliation{    Laboratoire de Physique Nucl\'eaire et des Hautes \'Energies,
    Universit\'e Pierre et Marie Curie Paris 6, Universit\'e Paris Diderot Paris 7, CNRS-IN2P3, 
    4 place Jussieu, 75252 Paris Cedex 05, France}

\author{     C.~Buton}
\affiliation{    Universit\'e de Lyon, F-69622, Lyon, France ; Universit\'e de Lyon 1, Villeurbanne ; 
    CNRS/IN2P3, Institut de Physique Nucl\'eaire de Lyon}
    
\author{     J.~Chen}
\affiliation{   Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China }

\author{     N.~Chotard}
\affiliation{    Universit\'e de Lyon, F-69622, Lyon, France ; Universit\'e de Lyon 1, Villeurbanne ; 
    CNRS/IN2P3, Institut de Physique Nucl\'eaire de Lyon}
    
\author{     Y.~Copin}
\affiliation{    Universit\'e de Lyon, F-69622, Lyon, France ; Universit\'e de Lyon 1, Villeurbanne ; 
    CNRS/IN2P3, Institut de Physique Nucl\'eaire de Lyon}
    
\author{     P.~Fagrelius}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}
\affiliation{
    Department of Physics, University of California Berkeley,
    366 LeConte Hall MC 7300, Berkeley, CA, 94720-7300}

\author{     H.~K.~Fakhouri}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}
  \affiliation{
    Department of Physics, University of California Berkeley,
    366 LeConte Hall MC 7300, Berkeley, CA, 94720-7300}

\author{     U.~Feindt}
\affiliation{The Oskar Klein Centre, Department of Physics, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden}

\author{     D.~Fouchez}
\affiliation{    Centre de Physique des Particules de Marseille, 
    Aix-Marseille Universit\'e , CNRS/IN2P3, 
    163 avenue de Luminy - Case 902 - 13288 Marseille Cedex 09, France}
    
\author{     E.~Gangler}  
\affiliation{    Clermont Universit\'e, Universit\'e Blaise Pascal, CNRS/IN2P3, Laboratoire de Physique Corpusculaire,
    BP 10448, F-63000 Clermont-Ferrand, France}
    
\author{     B.~Hayden}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}

\author{     W.~Hillebrandt}
\affiliation{    Max-Planck-Institut f\"ur Astrophysik, Karl-Schwarzschild-Str. 1,
D-85748 Garching, Germany}

\author{     M.~Kowalski}
\affiliation{    Institut fur Physik,  Humboldt-Universitat zu Berlin,
    Newtonstr. 15, 12489 Berlin}
\affiliation{ DESY, D-15735 Zeuthen, Germany}

\author{     P.-F.~Leget}
\affiliation{    Clermont Universit\'e, Universit\'e Blaise Pascal, CNRS/IN2P3, Laboratoire de Physique Corpusculaire,
    BP 10448, F-63000 Clermont-Ferrand, France}
    
\author{     S.~Lombardo}
\affiliation{    Institut fur Physik,  Humboldt-Universitat zu Berlin,
    Newtonstr. 15, 12489 Berlin}
    
\author{     J.~Nordin}
\affiliation{    Institut fur Physik,  Humboldt-Universitat zu Berlin,
    Newtonstr. 15, 12489 Berlin}
    
\author{     R.~Pain}
\affiliation{    Laboratoire de Physique Nucl\'eaire et des Hautes \'Energies,
    Universit\'e Pierre et Marie Curie Paris 6, Universit\'e Paris Diderot Paris 7, CNRS-IN2P3, 
    4 place Jussieu, 75252 Paris Cedex 05, France}
     
\author{     E.~Pecontal}
\affiliation{   Centre de Recherche Astronomique de Lyon, Universit\'e Lyon 1,
    9 Avenue Charles Andr\'e, 69561 Saint Genis Laval Cedex, France}
    
\author{    R.~Pereira}
 \affiliation{    Universit\'e de Lyon, F-69622, Lyon, France ; Universit\'e de Lyon 1, Villeurbanne ; 
    CNRS/IN2P3, Institut de Physique Nucl\'eaire de Lyon}
 
 \author{    S.~Perlmutter}
 \affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720} 
\affiliation{
    Department of Physics, University of California Berkeley,
    366 LeConte Hall MC 7300, Berkeley, CA, 94720-7300}
    
 \author{    D.~Rabinowitz}
 \affiliation{    Department of Physics, Yale University, 
    New Haven, CT, 06250-8121}
    
 \author{    M.~Rigault} 
\affiliation{    Institut fur Physik,  Humboldt-Universitat zu Berlin,
    Newtonstr. 15, 12489 Berlin}
     
 \author{    D.~Rubin}
 \affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}
    \affiliation{    Department of Physics, Florida State University,
    315 Keen Building, Tallahassee, FL 32306-4350}
 
 \author{    K.~Runge}
 \affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}
 
 \author{    C.~Saunders}
 \affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}

\author{    C.~Sofiatti}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720} 
\affiliation{
    Department of Physics, University of California Berkeley,
    366 LeConte Hall MC 7300, Berkeley, CA, 94720-7300}

\author{    N.~Suzuki}
\affiliation{    Physics Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road, Berkeley, CA, 94720}

\author{     S.~Taubenberger}
\affiliation{    Max-Planck-Institut f\"ur Astrophysik, Karl-Schwarzschild-Str. 1,
D-85748 Garching, Germany}

\author{     C.~Tao}
\affiliation{   Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China }
\affiliation{    Centre de Physique des Particules de Marseille, 
    Aix-Marseille Universit\'e , CNRS/IN2P3, 
    163 avenue de Luminy - Case 902 - 13288 Marseille Cedex 09, France}
   
\author{     R.~C.~Thomas}
\affiliation{    Computational Cosmology Center, Computational Research Division, Lawrence Berkeley National Laboratory, 
    1 Cyclotron Road MS 50B-4206, Berkeley, CA, 94720}
    
\collaboration{(The Nearby Supernova Factory)}


\begin{abstract}
\color{red}
An empirical model for SN~Ia peak magnitudes with two color parameters and dependence on the equivalent widths of CaII, SiII, and SiII velocity
is applied to the supernova sample of the Nearby Supernova Factory.  The peak magnitudes in synthetic
broadband photometry and their colors are found to be 
dependent on the spectral equivalent widths and the color parameters, to better than 
%-----
$(1-5\times10^{-5})$
%------
confidence.
The two color parameters exhibit the behavior expected from the two-parameter
\citet{1999PASP..111...63F} dust model, yielding a per-supernova scatter in
total-to-selective extinction, an effective
%-----
$R^F_V=2.24 \pm 0.16$
%------
for the full sample, and a difference in the average $R^F_V$ between extremely blue and red supernovae .  The derived dust law is able to reproduce the optical colors of the highly reddened SN~2014J better than the \citet{1999PASP..111...63F} dust model.
Extending the model to three color parameters gives almost identical results for
the two parameters
%-----
$R^F_V=2.23 \pm 0.16$,
%------
plus a detection of a third parameter at  $>98$\% confidence; this new parameter is responsible for supernova color diversity
beyond those tracked by the spectral features and dust.
\color{black}
\end{abstract}

\keywords{supernovae: general; supernovae: SN 2014J}

\section{Introduction}
Type~Ia supernovae (SNe~Ia) form a homogenous set of exploding stars and as such were early recognized and utilized as a powerful distance indicator 
and probe of cosmology \citep[e.g.][]{1992ARA&A..30..359B, 1993ApJ...415....1S}.  After further careful consideration of supernova data, it was recognized
that SN~Ia light-curve shapes \citep{1993ApJ...413L.105P} and colors \citep{1996ApJ...473...88R, 1998A&A...331..815T} exhibit subtle signs of heterogeneity,
which being correlated with absolute magnitude
need to be considered when inferring distances.  Empirical models parameterizing SNe~Ia by their light-curve shape \citep{1996ApJ...473...88R,
1997ApJ...483..565P,
1999ApJ...517..565P}
and color  \citep{1996ApJ...473...88R}  were developed that enabled absolute magnitude corrections
and accurate distance measurements of cosmological supernovae,
which 
were subsequently used in the discovery of the accelerating expansion of the Universe \citep{1998AJ....116.1009R,1999ApJ...517..565P}.

The two most commonly used supernova-cosmology light-curve fitters today are SALT2 \citep{2007A&A...466...11G} and MLCS2k2
\citep{2007ApJ...659..122J}.\footnote{Light-curve fitters with more flexible degrees of freedom
\citep[e.g.][]{2008ApJ...681..482C, 2011AJ....141...19B, 2011ApJ...731..120M} are available and have for
the most part been used to study SN~Ia heterogeneity.}
They remain two-parameter models, with one parameter characterizing light-curve shape and the other
 color.
 \textcolor{red}{In SALT2 the light curve shapes are described by a multiplicative scaling (``stretch'')  of the time evolution,
 whereas MLCS2k2 varies shapes through additive corrections to magnitudes.}
 The physical cause of the color diversity is interpreted differently by the two sets of authors: 
\citet{2007A&A...466...11G} pragmatically extract color variation empirically from SNe that span a wide range of colors, with no attribution
to either intrinsic or extrinsic origins;
\citet{2007ApJ...659..122J}
attribute changes in color
\textcolor{red}{partially to intrinsic variations linked to light-curve shape, and partially}
to the attenuation of light from host-galaxy dust.


There is evidence that supports the expectation that a single parameter beyond light-curve shape  cannot describe the full range
of colors seen in the SN~Ia population.  One approach to look for color diversity is to find correlations between color and spectral features.
\citet{2009ApJ...699L.139W, 2011ApJ...729...55F} find two subpopulations distinguished
by Si velocity with differing $B_{max}-V_{max}$; this color correlation, \textcolor{red}{ in addition to one with $B-R$, is confirmed by
\citet{2014ApJ...797...75M}.
}
\citet{2015MNRAS.451.1973S}
find that high-velocity SiII$\lambda$6355 is found in objects that have red ultraviolet/optical colors near maximum brightness.
\textcolor{red}{
\citet{2011MNRAS.413.3075M} show evidence that supernova asymmetry and viewing angle,
traced by wavelength shifts in nebular emission lines, is an important determinant in controlling supernova color; such correlations are also seen by \citet{2011A&A...534L..15C}.
}

\textcolor{red}{
Another approach to probe color diversity is through multiple colors (at least 3 bands)
of individual supernovae.  Color ratios are sensitive to processes of the responsible physics.   For example,
relative dust absorption varies as a function of wavelength depending on grain size and shape,
independent (to first order) of the amount of dust along the line of sight.
\citet{2013ApJ...779...23M} find diversity in SNe~Ia UV emission, which they use to distinguish subclasses based on UV-optical colors.}
\textcolor{red}{
Such measurements of $R_V$ are being advanced with the development of flexible empirical light curve models that accommodate flexibility in multi-band colors
\citep[e.g.][]{2011ApJ...731..120M}.}
\citet{2014ApJ...789...32B, 2015MNRAS.453.3300A} find wide
ranges of total-to-selective extinction with average values significantly lower than $R_V = 3.1$,
the canonical value for diffuse Milky Way dust.
They also confirm the \citet{2011ApJ...731..120M, 2011ApJ...729...55F} finding that low $R_V$ is associated with high-extinction supernovae.
In contrast, \citet{2011A&A...529L...4C} argue that after accounting for the diversity of spectral features,
the standard $R_V=3.1$ is recovered on average.


\textcolor{red}
{
Hierarchical modeling has recently opened
the study of intrinsic supernova color based on SN~Ia Hubble diagrams. Latent parameters that are not directly tied to observables,
such as multiple parameters that simultaneously influence color, can be included in the model.
\citet{2016arXiv160904470M} take the approach of modeling the distribution of their parameters, to find that
scatter in the Hubble diagram is better explained by a combination of intrinsic color dispersion and
$R_B=3.7$ dust, rather than by dust with no color dispersion.
They draw these conclusions by using only the SALT2 $x_1$ parameter as the summary statistic that describes color.
}

The Nearby Supernova Factory \citep[SNfactory;][]{2002SPIE.4836...61A} has systematically observed the
spectrophotometric time series of hundreds of Hubble-flow $0.03<z<0.08$ SNe~Ia.   The $3200$--$10000$~\AA\ spectral coverage
provides measurements of an array of supernova spectral features while also providing synthetic broadband photometry
spanning near-UV to near-IR SN-frame wavelengths.  \textcolor{red}{SNfactory specifically targeted objects
early in their temporal evolution, so} that well over a hundred of these supernovae have  coverage over
peak brightness.  This dataset provides a homogenous sample with which to study SN~Ia colors and spectral features together.

In this article we use the idea that spectral indicators carry information on intrinsic supernova colors at peak magnitude.
We allow for \textcolor{red}{three independent color parameters, one attributed to intrinsic supernova
properties and two  attributed} to
extrinsic processes, which we attribute to dust.  The data used in this analysis are described in \S\ref{data:sec}.  \textcolor{red}{
In \S\ref{model:sec} we present a
first analysis with two extrinsic parameters and omitting the intrinsic parameter.  The properties of the two extrinsic parameters
are consistent with the expectations of parameterized dust designed for the Milky Way, and do an excellent job of fitting the data of the out-of-sample
supernova SN2014J that has an extreme total-to-selective extinction.
In \S\ref{model2:sec} we add to analysis the intrinsic parameter, and find strong evidence for its influence on observed magnitudes.
We summarize our findings, and relate our results with light-curve shape, SiII velocity, and the ``mass step'' in SN~Ia Hubble
residuals in \S\ref{discussion:sec}.}


\section{Data}
\label{data:sec}

Our analysis uses the spectrophotometric data set obtained by
the SNfactory with the SuperNova Integral Field
Spectrograph \citep[SNIFS,][]{2004SPIE.5249..146L}.  SNIFS is a fully integrated
instrument optimized for automated observation of point sources on a
structured background over the full ground-based optical window at
moderate spectral resolution ($R \sim 500$).  It consists of a
high-throughput wide-band lenslet integral field spectrograph, a
multi-filter photometric channel to image the field in the vicinity of
the IFS for atmospheric transmission monitoring simultaneous with
spectroscopy, and an acquisition/guiding channel.  The IFS possesses a
fully-filled $6\farcs 4 \times 6\farcs 4$ spectroscopic field of view
subdivided into a grid of $15 \times 15$ spatial elements, a
dual-channel spectrograph covering 3200--5200~\AA\ and 5100--10000~\AA\
simultaneously, and an internal calibration unit (continuum and arc
lamps).  SNIFS is mounted on the south bent Cassegrain port of the
University of Hawaii 2.2~m telescope on Mauna Kea, and is operated
remotely.  Observations are reduced using the SNfactory's dedicated data
reduction pipeline, similar to that presented in \S4 of \citet{2001MNRAS.326...23B}.
A discussion of the software pipeline is presented in
\citet{2006ApJ...650..510A} and is updated in \citet{2010ApJ...713.1073S}. 
\textcolor{red}{The flux calibration is presented in \citet{2013A&A...549A...8B}. }
A detailed
description of host-galaxy subtraction is given in \citet{2011MNRAS.418..258B}.

A recent description of the data is presented in \citet{2015ApJ...815...58F}.
We provide a brief summary of the points important for this analysis.
The spectral time-series  are corrected for Milky Way dust
extinction \citep{1989ApJ...345..245C,1998ApJ...500..525S}.  
Each spectral time series is
blue-shifted to rest-frame
based on the systemic redshift of the host \citep[c.f.][]{2013ApJ...770..107C}, and the fluxes are converted to luminosity assuming
distances expected for the supernova redshifts given a flat
$\Lambda$CDM cosmology with $\Omega_M = 0.28$ (with an arbitrarily selected
$H_0$ since the current analysis does not depend on the absolute magnitude scale).

Supernova-frame synthetic supernova-frame photometry is generated for a top-hat filter system
comprised of five 
\color{red}
bands with the following wavelength ranges: $U$ $[3300.00 - 3978.02]$\AA;
$B$ $[3978.02-4795.35]$\AA;
$V$ $[4795.35-5780.60]$\AA;
$R$ $[5780.60-6968.29]$\AA;
$I$ $[6968.29-8400.00]$\AA.
For each supernova, the data magnitudes within 5-days of peak brightness are used to regress the peak magnitude
using single-band SALT2 fits.
\color{black}
The equivalent widths of SiII$\lambda 4141$ and the CaII H\&K features are computed as
in \citet{2008A&A...477..717B} and the 
\textcolor{red}{wavelength of the SiII$\lambda 6355$ feature}
as in \citet{chotard:thesis}.
Equivalent widths and the
\textcolor{red}{SiII$\lambda 6355$ wavelength are taken from spectra  within $\pm 2.5$ days from $B$-band maximum;
the average is used  in cases where there are multiple spectral measurements within that time window.}

Our analysis sample is comprised by the
168
supernovae that have the data coverage to 
give photometric and spectroscopic statistics described above.
The data are given in Table~\ref{data:tab} \textcolor{red}{[TABLE CAN GO IF THERE IS A NICO PAPER]}.
The peak magnitude uncertainties do have covariance, which is accounted
for in the analysis; only the standard deviation is included in the table.

\startlongtable
\begin{deluxetable}{crrrrrrrr}
\tabletypesize{\tiny}
\tablecaption{Supernova Spectral-Feature and Peak-Magnitude Data
\label{data:tab}}
\tablehead{
\colhead{Name} & \colhead{$EW_{Ca}$ (\AA)} & \colhead{$EW_{Si}$ (\AA)} & \colhead{$\lambda_{Si}$ (\AA)} & \colhead{$U$} & \colhead{$B$} & \colhead{$V$} & \colhead{$R$} & \colhead{$I$}
}
\startdata
SN2007bd & $109.7 \pm 5.9$ & $ 17.5 \pm 0.7$& $ 6098 \pm   4$ & $-29.31 \pm   0.01$ & $-29.12 \pm   0.01$& $-28.60 \pm   0.01$& $-28.35 \pm   0.01$& $-27.60 \pm   0.01$ \\
PTF10zdk & $149.7 \pm 1.2$ & $ 14.3 \pm 0.6$& $ 6150 \pm   3$ & $-28.61 \pm   0.02$ & $-28.69 \pm   0.02$& $-28.32 \pm   0.02$& $-28.08 \pm   0.02$& $-27.40 \pm   0.02$ \\
SNF20080815-017 & $ 63.8 \pm 21.5$ & $ 27.6 \pm 3.8$& $ 6132 \pm   6$ & $-29.04 \pm   0.07$ & $-28.79 \pm   0.07$& $-28.32 \pm   0.07$& $-28.12 \pm   0.07$& $-27.41 \pm   0.07$ \\
PTF09dnl & $129.9 \pm 0.9$ & $  9.5 \pm 0.7$& $ 6093 \pm   3$ & $-29.23 \pm   0.01$ & $-29.07 \pm   0.01$& $-28.72 \pm   0.01$& $-28.44 \pm   0.01$& $-27.69 \pm   0.01$ \\
SN2010ex & $114.4 \pm 0.9$ & $  8.4 \pm 0.4$& $ 6129 \pm   6$ & $-29.26 \pm   0.01$ & $-28.99 \pm   0.01$& $-28.50 \pm   0.01$& $-28.20 \pm   0.01$& $-27.44 \pm   0.01$ \\
PTF09dnp & $ 64.9 \pm 4.5$ & $ 16.5 \pm 0.7$& $ 6098 \pm   4$ & $-29.55 \pm   0.02$ & $-29.19 \pm   0.02$& $-28.68 \pm   0.02$& $-28.48 \pm   0.02$& $-27.93 \pm   0.02$ \\
PTF11bnx & $151.4 \pm 3.0$ & $ 13.9 \pm 1.1$& $ 6142 \pm   5$ & $-28.63 \pm   0.02$ & $-28.57 \pm   0.01$& $-28.20 \pm   0.01$& $-27.99 \pm   0.01$& $-27.34 \pm   0.01$ \\
PTF12jqh & $151.9 \pm 1.5$ & $  7.9 \pm 0.7$& $ 6116 \pm  10$ & $-29.37 \pm   0.01$ & $-29.14 \pm   0.01$& $-28.71 \pm   0.01$& $-28.40 \pm   0.01$& $-27.64 \pm   0.01$ \\
SNF20080802-006 & $108.2 \pm 6.0$ & $ 20.6 \pm 1.9$& $ 6122 \pm   5$ & $-29.02 \pm   0.06$ & $-28.80 \pm   0.06$& $-28.40 \pm   0.06$& $-28.20 \pm   0.06$& $-27.50 \pm   0.06$ \\
PTF10xyt & $123.7 \pm 6.6$ & $ 16.4 \pm 4.3$& $ 6101 \pm   4$ & $-28.26 \pm   0.02$ & $-28.20 \pm   0.02$& $-27.93 \pm   0.02$& $-27.74 \pm   0.02$& $-27.22 \pm   0.04$ \\
PTF11qmo & $101.7 \pm 1.1$ & $  7.7 \pm 0.7$& $ 6150 \pm   8$ & $-29.77 \pm   0.02$ & $-29.43 \pm   0.02$& $-28.97 \pm   0.02$& $-28.64 \pm   0.02$& $-27.93 \pm   0.02$ \\
SNF20070331-025 & $119.8 \pm 7.4$ & $ 14.2 \pm 2.7$& $ 6120 \pm  10$ & $-28.94 \pm   0.02$ & $-28.75 \pm   0.02$& $-28.32 \pm   0.02$& $-28.07 \pm   0.02$& $-27.31 \pm   0.02$ \\
SNF20070818-001 & $157.5 \pm 7.5$ & $ 16.7 \pm 1.8$& $ 6115 \pm   5$ & $-28.97 \pm   0.02$ & $-28.96 \pm   0.01$& $-28.61 \pm   0.01$& $-28.37 \pm   0.01$& $-27.62 \pm   0.01$ \\
SNBOSS38 & $ 57.1 \pm 0.4$ & $ 17.9 \pm 0.3$& $ 6127 \pm   3$ & $-29.20 \pm   0.01$ & $-28.84 \pm   0.01$& $-28.47 \pm   0.01$& $-28.23 \pm   0.01$& $-27.73 \pm   0.04$ \\
SN2006ob & $ 90.0 \pm 16.5$ & $ 26.5 \pm 1.5$& $ 6112 \pm   5$ & $-29.11 \pm   0.02$ & $-28.82 \pm   0.01$& $-28.42 \pm   0.01$& $-28.19 \pm   0.01$& $-27.54 \pm   0.01$ \\
PTF12eer & $165.6 \pm 10.7$ & $ 12.7 \pm 2.8$& $ 6150 \pm  10$ & $-28.76 \pm   0.01$ & $-28.76 \pm   0.01$& $-28.40 \pm   0.01$& $-28.17 \pm   0.01$& $-27.45 \pm   0.02$ \\
PTF10ops & $ 38.7 \pm 9.9$ & $  7.2 \pm 8.7$& $ 6141 \pm   5$ & $-27.93 \pm   0.38$ & $-27.76 \pm   0.38$& $-27.73 \pm   0.38$& $-27.59 \pm   0.38$& $-27.21 \pm   0.38$ \\
SNF20080514-002 & $ 83.2 \pm 0.7$ & $ 19.4 \pm 0.6$& $ 6131 \pm   3$ & $-29.30 \pm   0.01$ & $-28.95 \pm   0.01$& $-28.44 \pm   0.01$& $-28.17 \pm   0.01$& $-27.49 \pm   0.01$ \\
PTF12evo & $129.2 \pm 2.8$ & $  9.1 \pm 1.3$& $ 6156 \pm   4$ & $-29.14 \pm   0.02$ & $-28.98 \pm   0.01$& $-28.56 \pm   0.01$& $-28.28 \pm   0.01$& $-27.61 \pm   0.01$ \\
SNF20080614-010 & $125.4 \pm 5.1$ & $ 26.9 \pm 1.6$& $ 6128 \pm   3$ & $-29.04 \pm   0.04$ & $-28.81 \pm   0.04$& $-28.38 \pm   0.04$& $-28.16 \pm   0.04$& $-27.57 \pm   0.04$ \\
PTF10icb & $104.8 \pm 0.9$ & $ 12.7 \pm 0.3$& $ 6138 \pm   3$ & $-28.58 \pm   0.02$ & $-28.36 \pm   0.02$& $-27.98 \pm   0.02$& $-27.77 \pm   0.02$& $-27.17 \pm   0.02$ \\
SNNGC4424 & $109.0 \pm 0.3$ & $  8.6 \pm 0.1$& $ 6138 \pm   2$ & $-28.35 \pm   0.01$ & $-28.15 \pm   0.01$& $-27.79 \pm   0.01$& $-27.58 \pm   0.01$& $-26.97 \pm   0.01$ \\
SNF20080516-022 & $100.1 \pm 2.1$ & $ 13.7 \pm 1.1$& $ 6158 \pm   3$ & $-29.46 \pm   0.01$ & $-29.19 \pm   0.01$& $-28.71 \pm   0.01$& $-28.39 \pm   0.01$& $-27.77 \pm   0.01$ \\
PTF12hwb & $ 21.1 \pm 78.0$ & $ -1.8 \pm 8.9$& $ 6090 \pm  14$ & $-28.32 \pm   0.02$ & $-28.24 \pm   0.02$& $-28.03 \pm   0.02$& $-27.79 \pm   0.02$& $-27.05 \pm   0.04$ \\
PTF10qyz & $106.4 \pm 2.1$ & $ 23.0 \pm 1.0$& $ 6120 \pm   5$ & $-29.05 \pm   0.17$ & $-28.92 \pm   0.17$& $-28.41 \pm   0.17$& $-28.14 \pm   0.17$& $-27.30 \pm   0.17$ \\
SNF20060907-000 & $106.1 \pm 10.4$ & $ 17.0 \pm 0.9$& $ 6149 \pm   4$ & $-29.54 \pm   0.02$ & $-29.28 \pm   0.01$& $-28.76 \pm   0.01$& $-28.42 \pm   0.01$& $-27.74 \pm   0.04$ \\
LSQ12fxd & $122.9 \pm 1.7$ & $ 11.4 \pm 0.8$& $ 6119 \pm   4$ & $-29.62 \pm   0.07$ & $-29.39 \pm   0.07$& $-28.95 \pm   0.07$& $-28.64 \pm   0.07$& $-27.91 \pm   0.07$ \\
SNF20080821-000 & $105.1 \pm 2.2$ & $  8.6 \pm 1.3$& $ 6121 \pm   4$ & $-29.34 \pm   0.01$ & $-29.10 \pm   0.01$& $-28.73 \pm   0.01$& $-28.46 \pm   0.01$& $-27.82 \pm   0.01$ \\
SNF20070802-000 & $158.3 \pm 3.3$ & $ 16.3 \pm 1.7$& $ 6102 \pm   5$ & $-28.90 \pm   0.01$ & $-28.81 \pm   0.01$& $-28.45 \pm   0.01$& $-28.22 \pm   0.01$& $-27.52 \pm   0.01$ \\
PTF10wnm & $105.8 \pm 2.3$ & $  6.5 \pm 1.0$& $ 6124 \pm   3$ & $-29.38 \pm   0.01$ & $-29.07 \pm   0.01$& $-28.68 \pm   0.01$& $-28.37 \pm   0.01$& $-27.69 \pm   0.01$ \\
PTF10mwb & $116.5 \pm 1.2$ & $ 19.8 \pm 0.9$& $ 6138 \pm   2$ & $-29.02 \pm   0.07$ & $-28.84 \pm   0.07$& $-28.40 \pm   0.07$& $-28.14 \pm   0.07$& $-27.52 \pm   0.07$ \\
SN2010dt & $116.2 \pm 14.9$ & $ 15.5 \pm 0.7$& $ 6138 \pm   6$ & $-29.30 \pm   0.01$ & $-29.15 \pm   0.01$& $-28.64 \pm   0.01$& $-28.35 \pm   0.01$& $-27.63 \pm   0.01$ \\
SNF20080623-001 & $149.1 \pm 1.4$ & $ 14.9 \pm 0.7$& $ 6131 \pm   3$ & $-29.11 \pm   0.01$ & $-28.97 \pm   0.01$& $-28.50 \pm   0.01$& $-28.22 \pm   0.01$& $-27.46 \pm   0.01$ \\
LSQ12fhe & $ 42.8 \pm 1.2$ & $  4.0 \pm 3.1$& $ 6108 \pm   4$ & $-29.76 \pm   0.02$ & $-29.40 \pm   0.02$& $-29.04 \pm   0.02$& $-28.74 \pm   0.02$& $-28.11 \pm   0.02$ \\
PTF11bju & $ 30.2 \pm 4.4$ & $  4.0 \pm 3.0$& $ 6139 \pm   5$ & $-29.47 \pm   0.02$ & $-29.10 \pm   0.01$& $-28.75 \pm   0.01$& $-28.45 \pm   0.01$& $-27.87 \pm   0.01$ \\
PTF09fox & $117.6 \pm 2.7$ & $  9.1 \pm 1.0$& $ 6116 \pm   3$ & $-29.44 \pm   0.03$ & $-29.21 \pm   0.03$& $-28.72 \pm   0.03$& $-28.42 \pm   0.03$& $-27.68 \pm   0.03$ \\
PTF13ayw & $104.6 \pm 2.4$ & $ 26.6 \pm 3.2$& $ 6114 \pm   9$ & $-29.16 \pm   0.02$ & $-28.82 \pm   0.02$& $-28.43 \pm   0.02$& $-28.20 \pm   0.02$& $-27.55 \pm   0.02$ \\
SNF20070810-004 & $126.7 \pm 1.8$ & $ 21.1 \pm 1.1$& $ 6118 \pm   7$ & $-29.22 \pm   0.01$ & $-29.10 \pm   0.01$& $-28.63 \pm   0.01$& $-28.34 \pm   0.01$& $-27.62 \pm   0.01$ \\
PTF11mty & $111.4 \pm 2.3$ & $ 10.6 \pm 1.5$& $ 6138 \pm   5$ & $-29.54 \pm   0.01$ & $-29.23 \pm   0.01$& $-28.80 \pm   0.01$& $-28.46 \pm   0.01$& $-27.82 \pm   0.01$ \\
SNF20080512-010 & $ 95.3 \pm 3.5$ & $ 23.3 \pm 1.5$& $ 6129 \pm   5$ & $-29.22 \pm   0.08$ & $-28.96 \pm   0.08$& $-28.50 \pm   0.08$& $-28.26 \pm   0.08$& $-27.56 \pm   0.08$ \\
PTF11mkx & $ 31.5 \pm 3.7$ & $  4.5 \pm 1.3$& $ 6168 \pm   6$ & $-29.50 \pm   0.45$ & $-29.25 \pm   0.45$& $-28.89 \pm   0.45$& $-28.61 \pm   0.45$& $-27.97 \pm   0.45$ \\
PTF10tce & $135.7 \pm 1.1$ & $ 11.2 \pm 1.5$& $ 6090 \pm   4$ & $-29.13 \pm   0.02$ & $-28.99 \pm   0.01$& $-28.59 \pm   0.01$& $-28.31 \pm   0.01$& $-27.55 \pm   0.01$ \\
SNF20061020-000 & $ 95.4 \pm 18.8$ & $ 24.1 \pm 1.0$& $ 6120 \pm   5$ & $-29.01 \pm   0.03$ & $-28.78 \pm   0.03$& $-28.35 \pm   0.03$& $-28.17 \pm   0.03$& $-27.54 \pm   0.03$ \\
SN2005ir & $115.6 \pm 2.8$ & $ 13.5 \pm 6.9$& $ 6069 \pm   5$ & $-29.33 \pm   0.02$ & $-29.12 \pm   0.02$& $-28.84 \pm   0.02$& $-28.49 \pm   0.02$& $-27.77 \pm   0.02$ \\
SNF20080717-000 & $ 93.3 \pm 2.6$ & $  8.3 \pm 2.2$& $ 6104 \pm   3$ & $-28.58 \pm   0.01$ & $-28.47 \pm   0.01$& $-28.29 \pm   0.01$& $-28.05 \pm   0.01$& $-27.50 \pm   0.01$ \\
PTF12ena & $101.1 \pm 1.6$ & $  7.4 \pm 1.0$& $ 6129 \pm   4$ & $-28.01 \pm   0.01$ & $-28.00 \pm   0.01$& $-27.85 \pm   0.01$& $-27.77 \pm   0.01$& $-27.31 \pm   0.01$ \\
PTF13anh & $166.8 \pm 1.8$ & $ 21.8 \pm 1.2$& $ 6175 \pm   4$ & $-28.67 \pm   0.20$ & $-28.74 \pm   0.20$& $-28.30 \pm   0.20$& $-28.05 \pm   0.20$& $-27.28 \pm   0.20$ \\
CSS110918\_01 & $110.6 \pm 1.0$ & $  8.0 \pm 1.3$& $ 6101 \pm   2$ & $-29.88 \pm   0.76$ & $-29.58 \pm   0.76$& $-29.09 \pm   0.76$& $-28.71 \pm   0.76$& $-27.91 \pm   0.76$ \\
SNF20070506-006 & $ 94.1 \pm 1.3$ & $  6.7 \pm 0.6$& $ 6153 \pm   3$ & $-29.72 \pm   0.01$ & $-29.39 \pm   0.01$& $-28.97 \pm   0.01$& $-28.64 \pm   0.01$& $-27.96 \pm   0.01$ \\
SNF20070403-001 & $105.9 \pm 5.4$ & $ 18.3 \pm 1.8$& $ 6124 \pm   4$ & $-29.23 \pm   0.02$ & $-29.04 \pm   0.01$& $-28.63 \pm   0.01$& $-28.35 \pm   0.01$& $-27.62 \pm   0.01$ \\
PTF10hmv & $109.6 \pm 1.3$ & $  8.9 \pm 0.7$& $ 6143 \pm   3$ & $-28.54 \pm   0.01$ & $-28.40 \pm   0.01$& $-28.11 \pm   0.01$& $-27.89 \pm   0.01$& $-27.31 \pm   0.01$ \\
SNF20071015-000 & $105.0 \pm 3.2$ & $  6.9 \pm 1.1$& $ 6124 \pm   7$ & $-27.89 \pm   0.02$ & $-27.82 \pm   0.02$& $-27.69 \pm   0.02$& $-27.63 \pm   0.02$& $-27.16 \pm   0.04$ \\
SNhunt89 & $ 88.0 \pm 2.7$ & $ 32.2 \pm 1.9$& $ 6111 \pm   7$ & $-28.37 \pm   0.03$ & $-28.26 \pm   0.03$& $-27.92 \pm   0.03$& $-27.77 \pm   0.03$& $-27.13 \pm   0.03$ \\
SNF20070902-021 & $108.9 \pm 3.5$ & $ 17.1 \pm 1.0$& $ 6131 \pm   6$ & $-29.25 \pm   0.02$ & $-29.02 \pm   0.02$& $-28.56 \pm   0.02$& $-28.32 \pm   0.01$& $-27.65 \pm   0.02$ \\
PTF09dlc & $143.5 \pm 2.2$ & $ 10.2 \pm 0.9$& $ 6143 \pm   3$ & $-29.38 \pm   0.01$ & $-29.17 \pm   0.01$& $-28.69 \pm   0.01$& $-28.40 \pm   0.01$& $-27.62 \pm   0.01$ \\
PTF13ajv & $150.5 \pm 8.9$ & $ 46.3 \pm 8.6$& $ 6110 \pm  21$ & $-28.70 \pm   0.02$ & $-28.61 \pm   0.02$& $-28.16 \pm   0.02$& $-27.91 \pm   0.02$& $-27.07 \pm   0.04$ \\
SNF20080919-000 & $114.7 \pm 2.8$ & $  9.4 \pm 0.9$& $ 6145 \pm   5$ & $-28.53 \pm   0.02$ & $-28.41 \pm   0.01$& $-28.11 \pm   0.01$& $-27.99 \pm   0.01$& $-27.38 \pm   0.01$ \\
SNF20080919-001 & $ 85.0 \pm 1.1$ & $  6.0 \pm 0.4$& $ 6150 \pm   5$ & $-29.73 \pm   0.01$ & $-29.43 \pm   0.01$& $-29.04 \pm   0.01$& $-28.72 \pm   0.01$& $-28.07 \pm   0.01$ \\
SN2010kg & $ 95.1 \pm 28.5$ & $ 21.7 \pm 0.7$& $ 6077 \pm   5$ & $-28.85 \pm   0.01$ & $-28.74 \pm   0.01$& $-28.41 \pm   0.01$& $-28.20 \pm   0.01$& $-27.47 \pm   0.01$ \\
SNF20080714-008 & $134.8 \pm 15.7$ & $ 19.7 \pm 3.7$& $ 6100 \pm   6$ & $-28.56 \pm   0.02$ & $-28.63 \pm   0.01$& $-28.32 \pm   0.01$& $-28.13 \pm   0.01$& $-27.42 \pm   0.01$ \\
SNF20070714-007 & $129.6 \pm 5.6$ & $ 31.1 \pm 23.8$& $ 6146 \pm   5$ & $-27.88 \pm   0.02$ & $-28.12 \pm   0.01$& $-28.02 \pm   0.01$& $-27.86 \pm   0.01$& $-27.24 \pm   0.03$ \\
SNF20080522-011 & $122.1 \pm 1.7$ & $  8.3 \pm 0.5$& $ 6125 \pm   3$ & $-29.63 \pm   0.01$ & $-29.38 \pm   0.01$& $-28.92 \pm   0.01$& $-28.60 \pm   0.01$& $-27.88 \pm   0.01$ \\
SNF20061111-002 & $110.8 \pm 10.7$ & $ 20.4 \pm 1.0$& $ 6145 \pm   6$ & $-29.16 \pm   0.01$ & $-28.99 \pm   0.01$& $-28.59 \pm   0.01$& $-28.29 \pm   0.01$& $-27.61 \pm   0.01$ \\
SNNGC6343 & $ 87.0 \pm 1.4$ & $ 20.7 \pm 0.7$& $ 6136 \pm   3$ & $-28.78 \pm   0.01$ & $-28.66 \pm   0.01$& $-28.30 \pm   0.01$& $-28.08 \pm   0.01$& $-27.41 \pm   0.01$ \\
SNF20080825-010 & $102.4 \pm 13.4$ & $ 19.2 \pm 0.6$& $ 6116 \pm   4$ & $-29.46 \pm   0.01$ & $-29.17 \pm   0.01$& $-28.71 \pm   0.01$& $-28.47 \pm   0.01$& $-27.83 \pm   0.01$ \\
PTF10ufj & $141.1 \pm 3.4$ & $ 11.7 \pm 1.2$& $ 6131 \pm   6$ & $-29.28 \pm   0.15$ & $-29.16 \pm   0.15$& $-28.72 \pm   0.15$& $-28.41 \pm   0.15$& $-27.65 \pm   0.15$ \\
PTF10wof & $129.6 \pm 2.7$ & $ 17.3 \pm 1.0$& $ 6102 \pm   2$ & $-28.91 \pm   0.01$ & $-28.84 \pm   0.01$& $-28.46 \pm   0.01$& $-28.18 \pm   0.01$& $-27.43 \pm   0.01$ \\
SNF20080918-000 & $146.8 \pm 3.5$ & $  7.5 \pm 2.5$& $ 6110 \pm   5$ & $-28.79 \pm   0.02$ & $-28.65 \pm   0.02$& $-28.35 \pm   0.02$& $-28.12 \pm   0.02$& $-27.46 \pm   0.02$ \\
SNF20080516-000 & $117.4 \pm 2.2$ & $  9.0 \pm 1.2$& $ 6127 \pm  10$ & $-29.50 \pm   0.01$ & $-29.23 \pm   0.01$& $-28.80 \pm   0.01$& $-28.47 \pm   0.01$& $-27.74 \pm   0.01$ \\
SN2005cf & $159.1 \pm 0.7$ & $ 15.7 \pm 0.8$& $ 6144 \pm   2$ & $-29.37 \pm   0.02$ & $-29.16 \pm   0.02$& $-28.68 \pm   0.02$& $-28.41 \pm   0.02$& $-27.69 \pm   0.02$ \\
CSS130502\_01 & $ 91.5 \pm 10.9$ & $ 15.6 \pm 0.5$& $ 6128 \pm   3$ & $-29.43 \pm   0.02$ & $-29.09 \pm   0.02$& $-28.60 \pm   0.01$& $-28.30 \pm   0.01$& $-27.62 \pm   0.04$ \\
SNF20080620-000 & $107.8 \pm 14.1$ & $ 20.0 \pm 0.7$& $ 6132 \pm   4$ & $-28.82 \pm   0.02$ & $-28.78 \pm   0.01$& $-28.32 \pm   0.01$& $-28.09 \pm   0.01$& $-27.39 \pm   0.01$ \\
SNPGC51271 & $ 92.1 \pm 16.5$ & $ 21.1 \pm 0.7$& $ 6121 \pm   2$ & $-29.28 \pm   0.02$ & $-28.95 \pm   0.02$& $-28.46 \pm   0.02$& $-28.20 \pm   0.02$& $-27.62 \pm   0.04$ \\
PTF11pdk & $128.6 \pm 2.8$ & $ 15.6 \pm 1.7$& $ 6153 \pm   5$ & $-29.35 \pm   0.02$ & $-29.11 \pm   0.02$& $-28.61 \pm   0.02$& $-28.32 \pm   0.02$& $-27.67 \pm   0.02$ \\
SNF20060511-014 & $102.6 \pm 2.8$ & $ 15.6 \pm 1.1$& $ 6141 \pm   8$ & $-29.16 \pm   0.07$ & $-29.04 \pm   0.06$& $-28.56 \pm   0.06$& $-28.30 \pm   0.06$& $-27.63 \pm   0.06$ \\
SNF20080612-003 & $120.0 \pm 1.1$ & $  7.3 \pm 0.6$& $ 6123 \pm   3$ & $-29.64 \pm   0.02$ & $-29.41 \pm   0.02$& $-28.99 \pm   0.02$& $-28.70 \pm   0.02$& $-28.00 \pm   0.02$ \\
SNF20080626-002 & $130.0 \pm 1.0$ & $  6.1 \pm 4.2$& $ 6111 \pm   3$ & $-29.42 \pm   0.01$ & $-29.24 \pm   0.01$& $-28.84 \pm   0.01$& $-28.52 \pm   0.01$& $-27.76 \pm   0.01$ \\
SNF20060621-015 & $111.9 \pm 1.3$ & $  9.8 \pm 0.7$& $ 6144 \pm   3$ & $-29.63 \pm   0.01$ & $-29.36 \pm   0.01$& $-28.88 \pm   0.01$& $-28.54 \pm   0.01$& $-27.81 \pm   0.01$ \\
SNF20080920-000 & $135.2 \pm 1.4$ & $  5.6 \pm 1.6$& $ 6085 \pm   3$ & $-29.44 \pm   0.02$ & $-29.19 \pm   0.02$& $-28.79 \pm   0.02$& $-28.49 \pm   0.02$& $-27.74 \pm   0.02$ \\
SN2007cq & $ 65.8 \pm 4.1$ & $ 10.2 \pm 0.9$& $ 6137 \pm   3$ & $-29.53 \pm   0.02$ & $-29.30 \pm   0.02$& $-28.89 \pm   0.02$& $-28.56 \pm   0.02$& $-27.90 \pm   0.02$ \\
SNF20080918-004 & $ 87.8 \pm 7.2$ & $ 21.5 \pm 0.9$& $ 6141 \pm   4$ & $-29.00 \pm   0.22$ & $-28.82 \pm   0.22$& $-28.37 \pm   0.22$& $-28.13 \pm   0.22$& $-27.43 \pm   0.22$ \\
CSS120424\_01 & $138.1 \pm 2.1$ & $ 11.7 \pm 0.7$& $ 6138 \pm   3$ & $-29.40 \pm   0.02$ & $-29.23 \pm   0.02$& $-28.77 \pm   0.01$& $-28.45 \pm   0.02$& $-27.68 \pm   0.02$ \\
SNF20080610-000 & $119.9 \pm 10.4$ & $ 16.4 \pm 1.7$& $ 6131 \pm   6$ & $-29.05 \pm   0.07$ & $-28.92 \pm   0.07$& $-28.50 \pm   0.07$& $-28.22 \pm   0.07$& $-27.55 \pm   0.07$ \\
SNF20070701-005 & $101.8 \pm 2.6$ & $ 12.4 \pm 1.0$& $ 6158 \pm   5$ & $-29.46 \pm   0.02$ & $-29.27 \pm   0.02$& $-28.87 \pm   0.02$& $-28.60 \pm   0.02$& $-27.96 \pm   0.02$ \\
SN2007kk & $128.5 \pm 1.4$ & $ 10.6 \pm 1.0$& $ 6098 \pm   4$ & $-29.48 \pm   0.02$ & $-29.31 \pm   0.02$& $-28.87 \pm   0.01$& $-28.54 \pm   0.01$& $-27.77 \pm   0.02$ \\
SNF20060908-004 & $114.4 \pm 1.2$ & $ 12.6 \pm 0.6$& $ 6136 \pm   3$ & $-29.59 \pm   0.23$ & $-29.34 \pm   0.23$& $-28.91 \pm   0.23$& $-28.58 \pm   0.23$& $-27.87 \pm   0.23$ \\
SNF20080909-030 & $ 93.7 \pm 1.0$ & $  7.8 \pm 0.4$& $ 6171 \pm   3$ & $-29.38 \pm   0.02$ & $-29.12 \pm   0.01$& $-28.74 \pm   0.01$& $-28.44 \pm   0.01$& $-27.78 \pm   0.01$ \\
PTF11bgv & $ 79.4 \pm 3.2$ & $ 12.6 \pm 0.7$& $ 6146 \pm   3$ & $-28.90 \pm   0.02$ & $-28.62 \pm   0.01$& $-28.27 \pm   0.01$& $-28.08 \pm   0.01$& $-27.54 \pm   0.01$ \\
SNNGC2691 & $ 39.0 \pm 22.2$ & $  4.5 \pm 0.2$& $ 6139 \pm   8$ & $-29.46 \pm   0.02$ & $-29.06 \pm   0.02$& $-28.75 \pm   0.02$& $-28.49 \pm   0.02$& $-27.93 \pm   0.02$ \\
PTF13asv & $ 75.6 \pm 1.1$ & $  2.2 \pm 0.4$& $ 6148 \pm   4$ & $-29.92 \pm   0.32$ & $-29.49 \pm   0.32$& $-29.02 \pm   0.32$& $-28.63 \pm   0.32$& $-27.90 \pm   0.32$ \\
SNF20070806-026 & $ 98.8 \pm 12.1$ & $ 25.9 \pm 0.7$& $ 6114 \pm   7$ & $-29.14 \pm   0.02$ & $-28.91 \pm   0.02$& $-28.44 \pm   0.02$& $-28.21 \pm   0.02$& $-27.49 \pm   0.02$ \\
SNF20070427-001 & $ 81.3 \pm 2.3$ & $  6.3 \pm 0.9$& $ 6142 \pm   5$ & $-29.89 \pm   0.02$ & $-29.46 \pm   0.02$& $-28.97 \pm   0.02$& $-28.62 \pm   0.02$& $-27.97 \pm   0.02$ \\
SNF20061108-004 & $129.5 \pm 5.6$ & $  6.3 \pm 2.5$& $ 6110 \pm   6$ & $-29.53 \pm   0.02$ & $-29.31 \pm   0.02$& $-28.95 \pm   0.02$& $-28.60 \pm   0.02$& $-27.96 \pm   0.02$ \\
SNF20060912-000 & $106.5 \pm 1.8$ & $ 21.4 \pm 1.7$& $ 6163 \pm   7$ & $-28.98 \pm   0.02$ & $-28.92 \pm   0.02$& $-28.66 \pm   0.02$& $-28.42 \pm   0.02$& $-27.77 \pm   0.02$ \\
CSS110918\_02 & $109.1 \pm 9.4$ & $ 15.0 \pm 0.6$& $ 6137 \pm   3$ & $-29.36 \pm   0.02$ & $-29.14 \pm   0.01$& $-28.69 \pm   0.01$& $-28.41 \pm   0.01$& $-27.70 \pm   0.01$ \\
SNF20080918-002 & $ 97.7 \pm 2.8$ & $ 12.6 \pm 1.4$& $ 6141 \pm   6$ & $-29.50 \pm   0.02$ & $-29.11 \pm   0.02$& $-28.61 \pm   0.02$& $-28.34 \pm   0.02$& $-27.71 \pm   0.02$ \\
SNIC3573 & $102.7 \pm 1.8$ & $ 11.9 \pm 1.0$& $ 6142 \pm   5$ & $-29.28 \pm   0.02$ & $-29.14 \pm   0.02$& $-28.74 \pm   0.02$& $-28.46 \pm   0.01$& $-27.76 \pm   0.03$ \\
SNF20080725-004 & $133.6 \pm 2.1$ & $  6.9 \pm 0.9$& $ 6131 \pm   6$ & $-29.09 \pm   0.01$ & $-28.93 \pm   0.01$& $-28.59 \pm   0.01$& $-28.31 \pm   0.01$& $-27.55 \pm   0.03$ \\
SNF20050728-006 & $127.8 \pm 2.5$ & $ 15.8 \pm 1.3$& $ 6124 \pm   6$ & $-28.80 \pm   0.02$ & $-28.68 \pm   0.02$& $-28.37 \pm   0.02$& $-28.18 \pm   0.02$& $-27.55 \pm   0.02$ \\
SN2012fr & $134.2 \pm 0.5$ & $  7.4 \pm 0.2$& $ 6102 \pm   1$ & $-29.91 \pm   0.01$ & $-29.70 \pm   0.01$& $-29.31 \pm   0.01$& $-28.94 \pm   0.01$& $-28.10 \pm   0.01$ \\
SNF20060512-002 & $100.2 \pm 2.8$ & $ 13.4 \pm 1.1$& $ 6107 \pm   8$ & $-29.33 \pm   0.02$ & $-29.11 \pm   0.02$& $-28.77 \pm   0.02$& $-28.52 \pm   0.02$& $-27.80 \pm   0.02$ \\
SNF20060512-001 & $ 88.4 \pm 1.2$ & $  5.4 \pm 0.4$& $ 6169 \pm   3$ & $-29.33 \pm   0.01$ & $-29.05 \pm   0.01$& $-28.68 \pm   0.01$& $-28.40 \pm   0.01$& $-27.79 \pm   0.01$ \\
SNF20071003-016 & $125.2 \pm 4.6$ & $ 17.1 \pm 2.0$& $ 6124 \pm  11$ & $-28.58 \pm   0.02$ & $-28.54 \pm   0.02$& $-28.19 \pm   0.02$& $-27.99 \pm   0.02$& $-27.31 \pm   0.02$ \\
SNF20050821-007 & $141.7 \pm 2.6$ & $  7.7 \pm 1.0$& $ 6140 \pm   9$ & $-29.38 \pm   0.02$ & $-29.20 \pm   0.02$& $-28.77 \pm   0.02$& $-28.46 \pm   0.02$& $-27.67 \pm   0.02$ \\
SNF20070803-005 & $ 22.7 \pm 21.4$ & $  0.9 \pm 0.6$& $ 6157 \pm  27$ & $-29.87 \pm   0.01$ & $-29.43 \pm   0.01$& $-29.04 \pm   0.01$& $-28.74 \pm   0.01$& $-28.11 \pm   0.01$ \\
PTF09foz & $127.2 \pm 1.9$ & $ 21.7 \pm 1.2$& $ 6136 \pm   4$ & $-29.14 \pm   0.01$ & $-29.00 \pm   0.01$& $-28.59 \pm   0.01$& $-28.35 \pm   0.01$& $-27.65 \pm   0.01$ \\
PTF12grk & $162.3 \pm 9.8$ & $ 19.6 \pm 1.4$& $ 6085 \pm   8$ & $-28.86 \pm   0.02$ & $-28.87 \pm   0.01$& $-28.42 \pm   0.01$& $-28.19 \pm   0.01$& $-27.50 \pm   0.03$ \\
SNF20080720-001 & $138.5 \pm 4.0$ & $ 14.0 \pm 2.0$& $ 6112 \pm   2$ & $-27.59 \pm   0.02$ & $-27.78 \pm   0.01$& $-27.73 \pm   0.01$& $-27.71 \pm   0.01$& $-27.19 \pm   0.02$ \\
SNF20080810-001 & $ 88.4 \pm 21.6$ & $ 22.3 \pm 1.1$& $ 6145 \pm   5$ & $-29.11 \pm   0.01$ & $-28.89 \pm   0.01$& $-28.45 \pm   0.01$& $-28.23 \pm   0.01$& $-27.60 \pm   0.01$ \\
SNF20050729-002 & $109.4 \pm 2.2$ & $ 11.5 \pm 1.7$& $ 6142 \pm   6$ & $-29.35 \pm   0.13$ & $-29.17 \pm   0.13$& $-28.68 \pm   0.13$& $-28.38 \pm   0.13$& $-27.56 \pm   0.13$ \\
SN2008ec & $103.7 \pm 17.0$ & $ 23.1 \pm 0.4$& $ 6125 \pm   3$ & $-28.67 \pm   0.01$ & $-28.52 \pm   0.01$& $-28.18 \pm   0.01$& $-28.03 \pm   0.01$& $-27.47 \pm   0.01$ \\
SNF20070902-018 & $ 93.8 \pm 12.2$ & $ 23.8 \pm 3.0$& $ 6120 \pm   8$ & $-28.87 \pm   0.02$ & $-28.70 \pm   0.01$& $-28.26 \pm   0.01$& $-28.08 \pm   0.01$& $-27.41 \pm   0.02$ \\
SNF20070424-003 & $122.5 \pm 3.8$ & $ 12.7 \pm 1.6$& $ 6132 \pm   6$ & $-29.10 \pm   0.01$ & $-28.96 \pm   0.01$& $-28.51 \pm   0.01$& $-28.25 \pm   0.01$& $-27.57 \pm   0.01$ \\
SN2006cj & $101.7 \pm 1.3$ & $  4.8 \pm 0.8$& $ 6127 \pm   3$ & $-29.43 \pm   0.01$ & $-29.14 \pm   0.01$& $-28.74 \pm   0.01$& $-28.43 \pm   0.01$& $-27.76 \pm   0.01$ \\
SN2007nq & $ 89.8 \pm 9.9$ & $ 23.4 \pm 1.1$& $ 6109 \pm   5$ & $-29.11 \pm   0.02$ & $-28.91 \pm   0.02$& $-28.50 \pm   0.02$& $-28.27 \pm   0.02$& $-27.57 \pm   0.02$ \\
SNF20070817-003 & $ 93.9 \pm 2.4$ & $ 18.5 \pm 1.3$& $ 6116 \pm   6$ & $-29.19 \pm   0.02$ & $-29.03 \pm   0.01$& $-28.59 \pm   0.01$& $-28.30 \pm   0.01$& $-27.55 \pm   0.02$ \\
SNF20070403-000 & $ 61.8 \pm 6.5$ & $ 27.1 \pm 1.8$& $ 6154 \pm   8$ & $-28.37 \pm   0.02$ & $-28.27 \pm   0.02$& $-27.97 \pm   0.02$& $-27.80 \pm   0.02$& $-27.24 \pm   0.02$ \\
SNF20061022-005 & $ 64.6 \pm 3.8$ & $  3.7 \pm 1.4$& $ 6148 \pm   4$ & $-29.49 \pm   0.02$ & $-29.06 \pm   0.02$& $-28.71 \pm   0.02$& $-28.42 \pm   0.02$& $-27.93 \pm   0.02$ \\
SNNGC4076 & $127.3 \pm 2.4$ & $ 15.5 \pm 1.2$& $ 6152 \pm   4$ & $-28.77 \pm   0.01$ & $-28.66 \pm   0.01$& $-28.37 \pm   0.01$& $-28.15 \pm   0.01$& $-27.52 \pm   0.01$ \\
SNF20070727-016 & $ 77.5 \pm 2.5$ & $  5.1 \pm 0.8$& $ 6140 \pm   4$ & $-29.96 \pm   0.06$ & $-29.56 \pm   0.06$& $-29.06 \pm   0.06$& $-28.75 \pm   0.06$& $-28.01 \pm   0.06$ \\
PTF12fuu & $105.5 \pm 3.0$ & $  6.2 \pm 1.2$& $ 6124 \pm   5$ & $-29.54 \pm   0.01$ & $-29.23 \pm   0.01$& $-28.74 \pm   0.01$& $-28.40 \pm   0.01$& $-27.64 \pm   0.01$ \\
SNF20070820-000 & $107.2 \pm 3.5$ & $ 18.6 \pm 1.3$& $ 6132 \pm  14$ & $-28.80 \pm   0.02$ & $-28.69 \pm   0.02$& $-28.34 \pm   0.02$& $-28.13 \pm   0.02$& $-27.52 \pm   0.02$ \\
SNF20070725-001 & $108.4 \pm 2.0$ & $ 11.1 \pm 1.5$& $ 6140 \pm   7$ & $-29.61 \pm   0.02$ & $-29.32 \pm   0.02$& $-28.84 \pm   0.02$& $-28.50 \pm   0.02$& $-27.76 \pm   0.02$ \\
SNF20071108-021 & $ 99.1 \pm 2.7$ & $  5.8 \pm 0.8$& $ 6164 \pm   5$ & $-29.67 \pm   0.01$ & $-29.34 \pm   0.01$& $-28.94 \pm   0.01$& $-28.60 \pm   0.01$& $-27.96 \pm   0.01$ \\
SNF20080914-001 & $126.5 \pm 1.2$ & $ 15.4 \pm 1.1$& $ 6159 \pm   3$ & $-28.67 \pm   0.02$ & $-28.60 \pm   0.02$& $-28.31 \pm   0.02$& $-28.13 \pm   0.02$& $-27.58 \pm   0.02$ \\
SNF20060609-002 & $ 87.7 \pm 3.6$ & $  7.3 \pm 1.3$& $ 6132 \pm   4$ & $-28.60 \pm   0.02$ & $-28.42 \pm   0.02$& $-28.19 \pm   0.02$& $-28.05 \pm   0.02$& $-27.53 \pm   0.02$ \\
SNF20050624-000 & $121.0 \pm 5.3$ & $  9.3 \pm 3.1$& $ 6129 \pm   7$ & $-29.75 \pm   0.01$ & $-29.42 \pm   0.01$& $-28.99 \pm   0.01$& $-28.68 \pm   0.01$& $-27.97 \pm   0.01$ \\
SNF20080531-000 & $133.0 \pm 1.5$ & $ 17.6 \pm 0.8$& $ 6114 \pm   5$ & $-29.12 \pm   0.01$ & $-28.98 \pm   0.01$& $-28.54 \pm   0.01$& $-28.28 \pm   0.01$& $-27.51 \pm   0.01$ \\
SN2006do & $106.4 \pm 2.1$ & $ 26.7 \pm 1.3$& $ 6101 \pm   2$ & $-29.00 \pm   0.01$ & $-28.83 \pm   0.01$& $-28.42 \pm   0.01$& $-28.20 \pm   0.01$& $-27.53 \pm   0.04$ \\
PTF12ikt & $110.3 \pm 1.6$ & $ 14.2 \pm 0.7$& $ 6141 \pm   4$ & $-29.34 \pm   0.01$ & $-29.04 \pm   0.01$& $-28.57 \pm   0.01$& $-28.32 \pm   0.01$& $-27.66 \pm   0.01$ \\
SN2006dm & $ 99.5 \pm 1.6$ & $ 30.0 \pm 0.7$& $ 6118 \pm   3$ & $-28.81 \pm   0.01$ & $-28.65 \pm   0.01$& $-28.23 \pm   0.01$& $-28.02 \pm   0.01$& $-27.33 \pm   0.01$ \\
PTF13azs & $138.0 \pm 5.1$ & $ 16.2 \pm 1.6$& $ 6125 \pm  10$ & $-27.84 \pm   0.02$ & $-27.92 \pm   0.02$& $-27.69 \pm   0.02$& $-27.60 \pm   0.02$& $-26.99 \pm   0.02$ \\
SN2005hj & $ 80.8 \pm 2.4$ & $  4.3 \pm 0.8$& $ 6138 \pm   4$ & $-29.54 \pm   0.02$ & $-29.16 \pm   0.01$& $-28.87 \pm   0.01$& $-28.54 \pm   0.01$& $-28.01 \pm   0.01$ \\
PTF12iiq & $150.4 \pm 2.2$ & $ 22.5 \pm 0.8$& $ 6041 \pm   6$ & $-28.60 \pm   0.01$ & $-28.77 \pm   0.01$& $-28.41 \pm   0.01$& $-28.10 \pm   0.01$& $-27.29 \pm   0.01$ \\
PTF10ndc & $124.2 \pm 2.4$ & $  6.8 \pm 1.1$& $ 6119 \pm   3$ & $-29.52 \pm   0.01$ & $-29.25 \pm   0.01$& $-28.80 \pm   0.01$& $-28.49 \pm   0.01$& $-27.76 \pm   0.01$ \\
SNF20080919-002 & $103.6 \pm 7.2$ & $ 27.2 \pm 1.9$& $ 6133 \pm   8$ & $-28.74 \pm   0.02$ & $-28.46 \pm   0.01$& $-28.09 \pm   0.01$& $-27.87 \pm   0.01$& $-27.26 \pm   0.04$ \\
SNPGC027923 & $ 85.5 \pm 0.6$ & $  5.9 \pm 0.3$& $ 6130 \pm   4$ & $-29.87 \pm   0.02$ & $-29.45 \pm   0.02$& $-28.94 \pm   0.02$& $-28.57 \pm   0.02$& $-27.85 \pm   0.02$ \\
SNF20070330-024 & $118.1 \pm 2.1$ & $  4.6 \pm 2.2$& $ 6101 \pm   3$ & $-29.77 \pm   0.02$ & $-29.52 \pm   0.02$& $-29.08 \pm   0.02$& $-28.74 \pm   0.01$& $-27.94 \pm   0.02$ \\
SNF20061030-010 & $131.4 \pm 2.2$ & $ 17.4 \pm 1.1$& $ 6116 \pm   4$ & $-28.60 \pm   0.02$ & $-28.55 \pm   0.02$& $-28.25 \pm   0.02$& $-28.03 \pm   0.02$& $-27.34 \pm   0.02$ \\
SNhunt46 & $ 94.1 \pm 2.0$ & $ 11.2 \pm 0.6$& $ 6132 \pm   4$ & $-29.50 \pm   0.02$ & $-29.11 \pm   0.02$& $-28.67 \pm   0.02$& $-28.37 \pm   0.02$& $-27.71 \pm   0.02$ \\
SN2005hc & $126.9 \pm 2.5$ & $ 10.0 \pm 0.7$& $ 6123 \pm   3$ & $-29.38 \pm   0.01$ & $-29.13 \pm   0.01$& $-28.69 \pm   0.01$& $-28.37 \pm   0.01$& $-27.61 \pm   0.01$ \\
LSQ12dbr & $106.9 \pm 0.6$ & $  7.1 \pm 0.7$& $ 6138 \pm   4$ & $-29.29 \pm   0.73$ & $-29.00 \pm   0.73$& $-28.51 \pm   0.73$& $-28.15 \pm   0.73$& $-27.38 \pm   0.73$ \\
LSQ12hjm & $ 82.6 \pm 17.5$ & $ 12.2 \pm 1.4$& $ 6144 \pm   5$ & $-29.51 \pm   0.02$ & $-29.14 \pm   0.01$& $-28.60 \pm   0.01$& $-28.30 \pm   0.01$& $-27.71 \pm   0.02$ \\
SNF20060521-001 & $ 78.9 \pm 20.2$ & $ 21.1 \pm 1.4$& $ 6123 \pm  10$ & $-29.37 \pm   0.05$ & $-29.04 \pm   0.05$& $-28.54 \pm   0.05$& $-28.30 \pm   0.05$& $-27.57 \pm   0.05$ \\
SNF20070630-006 & $125.5 \pm 3.2$ & $ 10.1 \pm 1.6$& $ 6126 \pm   4$ & $-29.34 \pm   0.01$ & $-29.12 \pm   0.01$& $-28.65 \pm   0.01$& $-28.38 \pm   0.01$& $-27.66 \pm   0.01$ \\
PTF11drz & $132.6 \pm 1.4$ & $ 15.2 \pm 1.0$& $ 6116 \pm   5$ & $-29.12 \pm   0.01$ & $-28.95 \pm   0.01$& $-28.53 \pm   0.01$& $-28.27 \pm   0.01$& $-27.55 \pm   0.01$ \\
SNF20080323-009 & $ 95.9 \pm 2.3$ & $ 10.6 \pm 1.1$& $ 6143 \pm   6$ & $-29.59 \pm   0.02$ & $-29.22 \pm   0.02$& $-28.68 \pm   0.02$& $-28.42 \pm   0.02$& $-27.77 \pm   0.02$ \\
SNF20071021-000 & $167.5 \pm 2.2$ & $ 20.4 \pm 0.6$& $ 6112 \pm   4$ & $-28.75 \pm   0.02$ & $-28.78 \pm   0.02$& $-28.40 \pm   0.02$& $-28.18 \pm   0.02$& $-27.41 \pm   0.02$ \\
SNNGC0927 & $155.2 \pm 1.4$ & $ 11.0 \pm 0.7$& $ 6109 \pm   4$ & $-28.87 \pm   0.02$ & $-28.81 \pm   0.01$& $-28.46 \pm   0.01$& $-28.22 \pm   0.01$& $-27.48 \pm   0.01$ \\
SNF20060526-003 & $112.1 \pm 2.5$ & $  9.8 \pm 1.0$& $ 6121 \pm   3$ & $-29.34 \pm   0.01$ & $-29.09 \pm   0.01$& $-28.68 \pm   0.01$& $-28.39 \pm   0.01$& $-27.70 \pm   0.01$ \\
SNF20080806-002 & $135.8 \pm 1.8$ & $  7.5 \pm 0.9$& $ 6135 \pm   4$ & $-29.22 \pm   0.02$ & $-29.02 \pm   0.02$& $-28.61 \pm   0.02$& $-28.35 \pm   0.01$& $-27.71 \pm   0.02$ \\
SNF20080803-000 & $117.6 \pm 2.6$ & $  8.9 \pm 2.0$& $ 6125 \pm   4$ & $-28.84 \pm   0.01$ & $-28.70 \pm   0.01$& $-28.35 \pm   0.01$& $-28.16 \pm   0.01$& $-27.50 \pm   0.01$ \\
SNF20080822-005 & $ 78.5 \pm 1.8$ & $  6.3 \pm 0.9$& $ 6138 \pm   4$ & $-29.71 \pm   0.01$ & $-29.34 \pm   0.01$& $-28.93 \pm   0.01$& $-28.61 \pm   0.01$& $-27.92 \pm   0.01$ \\
SNF20060618-014 & $137.2 \pm 2.5$ & $  9.3 \pm 1.1$& $ 6112 \pm   7$ & $-29.27 \pm   0.03$ & $-29.09 \pm   0.03$& $-28.73 \pm   0.03$& $-28.38 \pm   0.03$& $-27.68 \pm   0.03$ \\
PTF12ghy & $ 99.3 \pm 3.6$ & $ 16.8 \pm 0.7$& $ 6134 \pm   3$ & $-28.29 \pm   0.02$ & $-28.27 \pm   0.01$& $-28.05 \pm   0.01$& $-27.95 \pm   0.01$& $-27.40 \pm   0.01$ \\
SNF20070531-011 & $122.4 \pm 2.7$ & $ 21.2 \pm 0.8$& $ 6114 \pm   4$ & $-29.07 \pm   0.01$ & $-28.94 \pm   0.01$& $-28.50 \pm   0.01$& $-28.26 \pm   0.01$& $-27.53 \pm   0.03$ \\
SNF20070831-015 & $112.2 \pm 2.7$ & $  7.8 \pm 1.0$& $ 6145 \pm   6$ & $-29.42 \pm   0.01$ & $-29.17 \pm   0.01$& $-28.78 \pm   0.01$& $-28.46 \pm   0.01$& $-27.78 \pm   0.01$ \\
SNF20070417-002 & $104.5 \pm 5.5$ & $ 24.4 \pm 2.2$& $ 6123 \pm   9$ & $-29.20 \pm   0.05$ & $-29.01 \pm   0.05$& $-28.48 \pm   0.05$& $-28.23 \pm   0.05$& $-27.54 \pm   0.05$ \\
PTF11cao & $143.3 \pm 1.6$ & $ 18.9 \pm 1.3$& $ 6104 \pm   5$ & $-28.78 \pm   0.02$ & $-28.79 \pm   0.02$& $-28.44 \pm   0.02$& $-28.18 \pm   0.02$& $-27.45 \pm   0.02$ \\
SNF20080522-000 & $ 61.8 \pm 3.5$ & $  3.3 \pm 0.9$& $ 6131 \pm   7$ & $-29.86 \pm   0.01$ & $-29.41 \pm   0.01$& $-29.03 \pm   0.01$& $-28.70 \pm   0.01$& $-28.06 \pm   0.01$ \\
PTF10qjq & $ 73.9 \pm 2.4$ & $ 12.8 \pm 0.8$& $ 6133 \pm   3$ & $-29.29 \pm   0.02$ & $-28.94 \pm   0.02$& $-28.53 \pm   0.01$& $-28.35 \pm   0.01$& $-27.76 \pm   0.01$ \\
PTF12dxm & $ 95.4 \pm 41.8$ & $ 35.7 \pm 2.8$& $ 6136 \pm   4$ & $-28.71 \pm   0.01$ & $-28.58 \pm   0.01$& $-28.19 \pm   0.01$& $-27.99 \pm   0.01$& $-27.34 \pm   0.01$ \\
SNF20061021-003 & $122.8 \pm 2.3$ & $  9.7 \pm 1.7$& $ 6131 \pm   4$ & $-29.04 \pm   0.02$ & $-28.86 \pm   0.02$& $-28.56 \pm   0.02$& $-28.30 \pm   0.02$& $-27.64 \pm   0.02$ \\
SNF20080510-005 & $111.6 \pm 2.6$ & $  6.4 \pm 1.1$& $ 6115 \pm   4$ & $-29.41 \pm   0.01$ & $-29.15 \pm   0.01$& $-28.70 \pm   0.01$& $-28.38 \pm   0.01$& $-27.73 \pm   0.04$ \\
SNF20080507-000 & $ 98.1 \pm 1.6$ & $ 10.6 \pm 2.1$& $ 6143 \pm   5$ & $-29.23 \pm   0.01$ & $-29.05 \pm   0.01$& $-28.71 \pm   0.01$& $-28.45 \pm   0.01$& $-27.79 \pm   0.01$ \\
SNF20080913-031 & $118.2 \pm 1.5$ & $ 11.3 \pm 1.8$& $ 6158 \pm   5$ & $-29.13 \pm   0.08$ & $-29.01 \pm   0.07$& $-28.62 \pm   0.07$& $-28.32 \pm   0.07$& $-27.68 \pm   0.07$ \\
SNF20080510-001 & $118.8 \pm 2.1$ & $ 15.3 \pm 1.3$& $ 6115 \pm   4$ & $-29.35 \pm   0.01$ & $-29.15 \pm   0.01$& $-28.69 \pm   0.01$& $-28.38 \pm   0.01$& $-27.68 \pm   0.01$ \\
SNF20070712-003 & $108.8 \pm 2.7$ & $ 13.5 \pm 0.9$& $ 6155 \pm   6$ & $-29.44 \pm   0.02$ & $-29.19 \pm   0.01$& $-28.74 \pm   0.01$& $-28.42 \pm   0.01$& $-27.78 \pm   0.01$ \\
\enddata
\end{deluxetable}

%Absolute photometric calibration is of extreme importance for the supernova cosmology.
%This analysis, however, is confined to the relative colors for a low-redshift set of supernovae
%and is thus insensitive to uncertainties in these zeropoints.


\section{Supernova Model I: Two Extrinsic Parameters}
\label{model:sec}

We hypothesize that at peak brightness
SN~Ia broadband magnitudes and colors are correlated with
spectral features: equivalent widths and line positions are considered as statistics localized in wavelength that are insensitive to variations in
broadband colors.
%Spectra may not efficiently capture all variations in broadband magnitude, so an additional source
%of intrinsic variation is accommodated.
These intrinsic spectral  parameters may not deterministically predict magnitudes, but rather do so with some intrinsic dispersion,
\textcolor{red}{which in this section is not yet associated with a parameter. The intrinsic magnitudes are then
modified by an extrinsic physical process (i.e.\ dust) to produce apparent magnitudes.}
\subsection{Model}
We assume 
that  peak intrinsic $UBVRI$ magnitudes are linearly dependent
on the
 equivalent widths of the CaII H\&K and SiII$\lambda$4141 spectral features
$EW_{Ca}$, $EW_{Si}$,
\textcolor{red}{ and $\lambda_{Si}$ the wavelength of the minimum of 
the SiII$\lambda6355$ feature}:
these spectral features are associated with SN~Ia  spectroscopic diversity  
\citep{2006PASP..118..560B, 2008A&A...492..535A, 2009A&A...500L..17B, 2009PASP..121..238B, 2009ApJ...699L.139W, 2011ApJ...729...55F}.
\textcolor{red}{
The explicit omission of light-curve shape in our model is compensated by its proxy
$EW_{Si}$ at peak brightness}
\citep{2008A&A...492..535A, 2011A&A...529L...4C}. 
% An intrinsic-color parameter $D$ describes  intrinsic magnitude/color variation unaccounted for by the spectral features.
Residual dispersion is described by a Normal distribution with a parameterized covariance matrix
$C_c$.  A grey magnitude offset, $\Delta$, is included for each supernova
to capture intrinsic dispersion and peculiar-velocity errors introduced when converting fluxes to luminosity.
The observed magnitudes are linearly dependent on the
extrinsic-color parameters $k_0$ and $k_1$.  
The observables
$U_o, B_o, V_o, R_o, I_o$, $EW_{Ca,o}$, $EW_{Si,o}$, $\lambda_{Si,o}$
shown in Table~\ref{data:tab} have Gaussian measurement uncertainty with covariance $C$.
\textcolor{red}{Unlike the parameters associated with
the spectral measurements, $EW_{Ca}$, $EW_{Si}$ and $\lambda_{Si}$,  the latent
parameters $k_0$ and $k_1$ are not directly associated
with observables but are rather drawn from the set of peak magnitudes.}
The model is written as
\begin{equation}
\begin{pmatrix}
U\\B\\V\\R\\I
\end{pmatrix}
\sim \mathcal{N}
\left(
\Delta +
\begin{pmatrix}
c_0+\alpha_0 EW_{Ca} + \beta_0 EW_{Si} + \eta_0 \lambda_{Si} \\
c_1+\alpha_1 EW_{Ca} + \beta_1 EW_{Si} + \eta_1 \lambda_{Si}  \\
c_2+\alpha_2 EW_{Ca} + \beta_2 EW_{Si} + \eta_2 \lambda_{Si} \\
c_3+\alpha_3 EW_{Ca} + \beta_3 EW_{Si} + \eta_3 \lambda_{Si} \\
c_4+\alpha_4 EW_{Ca} + \beta_4 EW_{Si}+ \eta_4 \lambda_{Si}
\end{pmatrix}
,C_{c}
\right)
\label{ewsiv:eqn}
\end{equation}
\begin{equation}
\begin{pmatrix}
U_o\\B_o\\ V_o\\R_o\\I_o\\EW_{Si, o}\\ EW_{Ca, o} \\ \lambda_{Si, o}
\end{pmatrix}
\sim \mathcal{N}
\left(
\begin{pmatrix}
U +\gamma^0_{1} k_0 +\gamma^1_{1} k_1 \\B +\gamma^0_{1} k_0 +\gamma^1_{1} k_1 \\
V+\gamma^0_{2} k_0+\gamma^1_{2} k_1\\R+\gamma^0_{3} k_0 + \gamma^1_{3} k_1\\I+\gamma^0_{4} k_0+\gamma^1_{4} k_1\\
EW_{Si}\\ EW_{Ca} \\ \lambda_{Si}
\end{pmatrix}
,C
\right).
\label{dust:eqn}
\end{equation}
The global parameters $c$, $\alpha$, $\beta$,  and $\eta$,  are the intercepts and slopes of the linear relationships that
relate intrinsic magnitudes with the spectral-feature parameters.
(In this article we freely interchange the parameter indices with  $01234$ and $UBVRI$.)
The global parameters $\gamma^0$, $\gamma^1$  are the slopes that connect the extrinsic-color
parameters to observed magnitudes.

To constrain the degrees of freedom and degeneracies inherent in the model we impose that
\color{red}
each of the vectors $k_0+1/N$ and $k_1+1/N$ are simplexes (all elements are $\ge 0$ and sum to one),
where $N$ is the number of objects in our sample: these
conditions break the 
degeneracies  $\gamma ^0\rightarrow a\gamma^0$, $k_0 \rightarrow a^{-1}k_0$ and $\gamma^1 \rightarrow a\gamma^1$, $k_1 \rightarrow a^{-1}k_1$
up to a $\pm$ sign.
Furthermore we impose
\color{black} 
\begin{equation}
\langle \Delta \rangle=0, \langle k_0 \rangle=0, \langle k_1 \rangle=0, \gamma^0_0 > 0, \gamma^1_0 < 0.
\end{equation}
The first two conditions specify the definition of zero color relative to which the color excess is measured:
\textcolor{red}{in a linear model this value is arbitrary.} 
The latter two exclude degenerate posterior space
associated
with the $\pm$ sign, i.e.\ simultaneous sign flips of
$\gamma^0$--$k_0$ and $\gamma^1$--$k_1$
\color{red}
(when running our fits, $k_0$ and $k_1$ are set to
identical initial conditions, so the difference in the signs of $\gamma^0_0$ and $\gamma^1_0$ distinguishes the two.)
As  $\gamma^0$--$k_0$ and $\gamma^1$--$k_1$ are degenerate with
each other,
the different initial and boundary conditions for $\gamma^0_0$ and $\gamma^1_0$ break the degeneracy of our mixture model;
otherwise identical initial conditions for $\gamma^0$, $\gamma^1$ produce indistinguishable posteriors
with relatively broad credible intervals and lower maximum likelihood.

\color{black}

Having the intrinsic dispersion, $C_c$, as fit parameters seemingly introduces degeneracy in the model, as magnitude and color variation
ascribed to $\Delta$, $\gamma^0 k_0$, and $\gamma^1 k_1$ could also be attributed to intrinsic dispersion.  There are several features of the model
that drive the assignation of variations away from $C_c$:  Maximizing the posterior disfavors the increase of $\det{(C_c)}$;
The distributions of $\Delta$, $k_0$, and $k_1$ turn out to
be non-Gaussian, and so are not well described by the Normal covariance we impose in $C_c$; the grey magnitude offsets, $\Delta$, would appear as a constant
in all elements of the covariance matrix, which is disfavored for the Bayesian prior selected for $C_c$.

In a Bayesian analysis such as this the priors must be described.  A flat prior is used for all parameters except
for the covariance matrix $C_c$, which is constructed from a correlation matrix with  $\nu=4$  LKJ prior\footnote{
Visualization of the LKJ correlation distribution can be found in \url{http://www.psychstatistics.com/2014/12/27/d-lkj-priors/}.}
\citep{Lewandowski20091989} and standard
deviations $\sigma_i = \sqrt{C_{c,ii}}$ with a  Cauchy distribution prior with location
 $0.1$ and scale $0.1$ mag restricted to positive values.
 \textcolor{red}{This covariance matrix prior is recommended by STAN, the Monte Carlo we use to evaluate the model.}
 We find that imposing a stricter assumption of a
 diagonal covariance matrix, with no reference to a correlation matrix with LKJ prior, produces little change in the posteriors of
 the other parameters.

\subsection{Results}
\label{results:sec}
The posterior of the model parameters is evaluated using Hamiltonian Monte Carlo as implemented in
STAN \citep{stan}.  \textcolor{red}{We run four chains, each with 10000 iterations of which
half are used for warmup.}
STAN provides output statistics to assess
the convergence of the output Markov chains.
\textcolor{red}{The 
potential scale reduction statistic, $\hat{R}$ \citep{Gelman92}, measures the convergence of the target distribution
in iterative simulations 
by using multiple independent sequences to estimate how much that distribution would sharpen if the simulations were run longer.
$N_{eff}$ is an estimate of the number of independent draws. The output for our data and model gives $\hat{R} \sim 1.0$ for all parameters, meaning there is no evidence for non-convergence.  The
output also gives  $N_{eff} \gg 100$ for all parameters, indicating that are all densely sampled.}
Empirically, the confidence regions are localized and unimodal as is seen in  Figures~\ref{global1:fig}, \ref{global2:fig}, \ref{global3:fig}, \ref{global4:fig},
\ref{global5:fig}.  We find no evidence that
the Monte Carlo chains have not converged to the stationary posterior distribution.
We have rurun the analysis with a variety of initial conditions, including one with all the $\gamma$'s equal to zero, except for a small positive 
$\gamma^0_0$ and small negative $\gamma^1_0$ to satisfy our limit conditions.  Parameter credible intervals
remain consistent within the 68\% credible intervals.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output11/coeff0.pdf} 
            \caption{Posterior contours for $c$, $\alpha$, $\beta$, $\eta$, $\gamma^0$, $\gamma^1$, and $\sigma$ in the $U$ band.
            The contours shown here and in future plots represent 1-$\sigma$ in the parameter distribution (i.e.\ they should be
            projected onto the corresponding 1-d parameter axis), not to 68\%, 95\%, etc.\
            enclosed probability.  \label{global1:fig}}
\end{figure}

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output11/coeff1.pdf} 
            \caption{Posterior contours for $c$, $\alpha$, $\beta$, $\eta$, $\gamma^0$, $\gamma^1$, and $\sigma$ in the $B$ band.
 \label{global2:fig}}
\end{figure}

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output11/coeff2.pdf} 
            \caption{Posterior contours for $c$, $\alpha$, $\beta$, $\eta$, $\gamma^0$, $\gamma^1$, and $\sigma$ in the $V$ band.
 \label{global3:fig}}
\end{figure}

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
      \includegraphics[width=5.5in]{output11/coeff3.pdf} 
            \caption{Posterior contours for $c$, $\alpha$, $\beta$, $\eta$, $\gamma^0$, $\gamma^1$, and $\sigma$ in the $R$ band.
 \label{global4:fig}}
\end{figure}

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
         \includegraphics[width=5.5in]{output11/coeff4.pdf} 
            \caption{Posterior contours for $c$, $\alpha$, $\beta$, $\eta$, $\gamma^0$, $\gamma^1$, and $\sigma$ in the $I$ band.
 \label{global5:fig}}
\end{figure}

\color{red}

The model-predicted versus observed supernova colors are shown in Figure~\ref{residual:fig}.
The points lie along a line with slope equal to one, and show no evidence of a catastrophic fit nor of extreme
outlying data.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output11/residual.pdf} 
            \caption{\textcolor{red}{Predicted versus observed colors.  Overplotted is a line with slope equal to one.}
            \label{residual:fig}}
\end{figure}

\color{black}

For each of the five filters, the 68\%  equal-tailed credible intervals for the global parameters $\alpha$, $\beta$, $\eta$, $\gamma^0$, $\gamma^1$, and $\sigma$
are given in Table~\ref{global:tab}.
\color{red}
As the $\gamma$ parameters are free up to a multiplicative constant and have non-zero values of $\gamma^i_V$,
their results are shown in terms of $\gamma^i_X/\gamma^i_V-1$.
\color{black}
Contours of the posterior surface for parameter pairs grouped by filter are shown in Figures~\ref{global1:fig} -- \ref{global5:fig} .




\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Parameter & $X=U$ &$B$&$V$&$R$&$I$\\ \hline
$\alpha_X$
&
$0.0043^{+0.0009}_{-0.0009}$
&
$0.0016^{+0.0007}_{-0.0007}$
&
$0.0015^{+0.0006}_{-0.0006}$
&
$0.0015^{+0.0005}_{-0.0005}$
&
$0.0026^{+0.0005}_{-0.0004}$
\\
${\alpha_X/\alpha_V-1}$
&
$   1.9^{+   1.0}_{  -0.5}$
&
$   0.1^{+   0.1}_{  -0.2}$
&
\ldots
&
$  -0.0^{+   0.1}_{  -0.1}$
&
$   0.7^{+   0.7}_{  -0.3}$
\\
$\beta_X$
&
$ 0.032^{+ 0.003}_{-0.003}$
&
$ 0.025^{+ 0.002}_{-0.003}$
&
$ 0.025^{+ 0.002}_{-0.002}$
&
$ 0.020^{+ 0.002}_{-0.002}$
&
$ 0.019^{+ 0.002}_{-0.002}$
\\
${\beta_X/\beta_V-1}$
&
$  0.26^{+  0.05}_{ -0.05}$
&
$ -0.01^{+  0.03}_{ -0.03}$
&
\ldots
&
$ -0.19^{+  0.01}_{ -0.01}$
&
$ -0.24^{+  0.03}_{ -0.03}$
\\
$\eta_X$
&
$-0.0002^{+0.0012}_{-0.0011}$
&
$-0.0000^{+0.0010}_{-0.0009}$
&
$0.0005^{+0.0008}_{-0.0008}$
&
$0.0005^{+0.0007}_{-0.0007}$
&
$-0.0003^{+0.0006}_{-0.0006}$
\\
${\eta_X/\eta_V-1}$
&
$ -0.40^{+  2.33}_{ -1.90}$
&
$ -0.34^{+  1.60}_{ -1.20}$
&
\ldots
&
$ -0.11^{+  0.29}_{ -0.27}$
&
$ -0.84^{+  1.66}_{ -1.27}$
\\
${\gamma^0_X/\gamma^0_V-1}$
&
$  0.66^{+  0.06}_{ -0.05}$
&
$  0.35^{+  0.03}_{ -0.03}$
&
\ldots
&
$ -0.23^{+  0.01}_{ -0.01}$
&
$ -0.45^{+  0.03}_{ -0.03}$
\\
${\gamma^1_X/\gamma^1_V-1}$
&
$ -0.31^{+  0.17}_{ -0.18}$
&
$ -0.17^{+  0.09}_{ -0.10}$
&
\ldots
&
$ -0.07^{+  0.05}_{ -0.04}$
&
$ -0.17^{+  0.10}_{ -0.09}$
\\
$\sigma_X$
&
$ 0.060^{+ 0.012}_{-0.012}$
&
$ 0.033^{+ 0.007}_{-0.007}$
&
$ 0.021^{+ 0.004}_{-0.006}$
&
$ 0.012^{+ 0.007}_{-0.008}$
&
$ 0.044^{+ 0.005}_{-0.004}$
\\
\hline
\end{tabular}
\caption{68\% credible intervals for the Global Fit Parameters of the 2-Parameter Extrinsic Model in \S\ref{model:sec}.\label{global:tab}}
\end{table}

\textcolor{red}{
We find significant non-zero values for $\alpha$ and $\beta$, indicating that $EW_{Ca}$ and $EW_{Si}$ are indicators of broadband
peak magnitudes.}
This validates our hypothesis that spectral indicators
are tracers of supernova absolute magnitude.  On the other hand, the values of $\eta$ (the coefficients attached to $\lambda_{Si}$) are consistent with zero
to  within one standard deviation.
The effect of spectral parameters on color is shown in the rows of $\alpha_X/\alpha_V-1$,  $\beta_X/\beta_V-1$, and  $\eta_X/\eta_V-1$
in Table~\ref{global:tab}:
values of zero signify no color changes associated with magnitude changes.
Both $EW_{Ca}$ and $EW_{Ca}$ are associated with color changes, though not in $B-V$ specifically.
We do not detect a significant association between
$\lambda_{Si}$ and color.


The ideogram for the grey offsets, $\Delta$, for all supernovae is shown in Figure~\ref{hist:fig}.  The distribution is non-Gaussian, 
has a standard deviation of
%-----
$0.10$
%-----
mag, and a broad tail in the positive (fainter) direction.
\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output11/Delta_hist.pdf} 
   \caption{Ideogram for the grey offset $\Delta$.  This and other ideograms plotted in this article are normalized to have unit area.
   \label{hist:fig}}
\end{figure}


Non-trivial residual magnitude dispersions are captured in $C_c$.  The diagonal elements are captured by the $\sigma$ parameters;
% Figure~\ref{sigma:fig} shows the confidence regions for $\sigma$, the square root of the diagonal elements of $C_c$. 
the residual intrinsic dispersion ranges from
$\sim 0.01$ to 0.06 mag, significantly smaller
than the dispersion in $\Delta$.  Given that the 
off-diagonal elements of $C_c$ are parameterized by the Cholesky factors of a correlation matrix rather than the matrix elements themselves,
it is not straightforward to present correlation matrix confidences:
the average over Monte Carlo links of the
Cholesky factors will not  generally yield a correlation matrix.  
To characterize a typical posterior draw of $C_c$ we use the matrix that is the mean of all covariance realizations in the
chain, element by element.
For $UBVRI$ the matrix is
\begin{equation}
C_c(U,B,V,R,I)=
\begin{pmatrix}
\begin{array}{rrrrr}
0.0038 & 0.0010 & -0.0002 & -0.0000 & 0.0003 \\
0.0010 & 0.0011 & 0.0002 & -0.0000 & -0.0006 \\
-0.0002 & 0.0002 & 0.0004 & 0.0000 & -0.0002 \\
-0.0000 & -0.0000 & 0.0000 & 0.0002 & 0.0002 \\
0.0003 & -0.0006 & -0.0002 & 0.0002 & 0.0020
\end{array}
 \end{pmatrix} \text{mag}^2.
 \label{mag_cov:eqn}
 \end{equation}
The  covariance of the colors $U-V$, $B-V$, $V-R$, and $V-I$ is
expressed as the standard deviations and
 correlation matrix
 \begin{equation}
 \sigma(U-V, B-V, V-R, V-I)=
 \begin{pmatrix}
0.068 & 0.034& 0.023 & 0.053
  \end{pmatrix} \text{mag}
 \label{color_sd:eqn}
   \end{equation}
 \begin{equation}
 Cor(U-V, B-V, V-R, V-I)=
\begin{pmatrix}
\begin{array}{rrrr}
1.000 & 0.615 & -0.334 & -0.296 \\
0.615 & 1.000 & -0.193 & 0.099 \\
-0.334 & -0.193 & 1.000 & 0.634 \\
-0.296 & 0.099 & 0.634 & 1.000 
\end{array}
\end{pmatrix}.
  \label{color_cor:eqn}
 \end{equation}
 \textcolor{red}{
These color variances are smaller than those derived by \citet{2003A&A...404..901N, 2007ApJ...659..122J}, for cases where direct comparison can be made.
} 
% \begin{figure}[htbp] %  figure placement: here, top, bottom, or page
%   \centering
%   \includegraphics[width=5.5in]{output11/sigma_corner.pdf} 
%   \caption{Posterior contours for the parameters $\sigma$, the square root of the diagonal elements of $C_c$.
%   \label{sigma:fig}}
%\end{figure}

Each supernova is described by its parameters $EW_{Ca}$, $EW_{Si}$, $\lambda_{Si}$, $E_{\gamma^0}(B-V)=(\gamma^0_B-\gamma^0_V)k_0$, and
$E_{\gamma^1}(B-V)=(\gamma^1_B-\gamma^1_V)k_1$, as well as its grey offset
$\Delta$: their distributions for all Monte Carlo links for all supernovae are shown in Figure~\ref{perobject:fig}.
There is a core concentration in the  parameter-space, with around ten objects that occupy its outskirts.
Many outliers appear in the red tail of $E_{\gamma^0}(B-V)$, as would be expected for the (infrequent) selection of supernovae
heavily extinguished by host-galaxy dust.
Outliers  are also clearly distinguishable in  $EW_{Ca}$--$\lambda_{Si}$ space.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output11/perobject_corner.pdf} 
   \caption{Distributions for the supernova parameters $EW_{Ca}$, $EW_{Si}$, $\lambda_{Si}$, $E_{\gamma^0}(B-V)$, and $E_{\gamma^1}(B-V)$, as well as the grey offset
$\Delta$.  All Monte Carlo links are plotted, so that each supernova contributes a cloud of points.
   \label{perobject:fig}}
\end{figure}

The Pearson correlation coefficients for $\Delta$, $EW_{Ca}$, $EW_{Si}$, $\lambda_{Si}$, $E_{\gamma^0}(B-V)$, and $E_{\gamma^0}(B-V)$ are given in the matrix
\begin{multline}
Cor(\Delta, EW_{Ca}, EW_{Si}, \lambda_{Si}, E_{\gamma^0}(B-V), E_{\gamma^1}(B-V)) =\\
\begin{pmatrix}
\begin{array}{rrrrrr}
1.00 & 0.00 & -0.06 & -0.05 & 0.11 & 0.01 \\
0.00 & 1.00 & 0.11 & -0.25 & -0.07 & -0.00 \\
-0.06 & 0.11 & 1.00 & -0.14 & -0.16 & -0.03 \\
-0.05 & -0.25 & -0.14 & 1.00 & 0.04 & 0.03 \\
0.11 & -0.07 & -0.16 & 0.04 & 1.00 & 0.06 \\
0.01 & -0.00 & -0.03 & 0.03 & 0.06 & 1.00
\end{array}
\end{pmatrix}.
\end{multline}
\color{red}
We find weak correlations between our color-excess parameters and the input features, the strongest being $-0.16$ between
$E_\gamma(B-V)$ and $EW_{Si}$.
\color{black}
Recall that $EW_{Si}$ is correlated with light-curve shape, which is correlated with host-galaxy (including dust) properties 
\citep{2000AJ....120.1479H, 2003MNRAS.340.1057S}.
An extensive discussion of the correlations between the spectral features of the SNfactory data set can be found in \citet{chotard:thesis}
and \citet{leget:thesis}.

\color{red}
The range of SiII$\lambda$4130 equivalent widths is $\pm 20$~\AA\ whereas the width of the $B$-band is 851~\AA, so that its affect on magnitude
is
$2.5 \log{(20/850)} \sim 0.03$ mag.  
The implied span in $B$ magnitude based on $\beta_1$ is 0.54~mag.  Therefore $\beta_1$ cannot wholly be attributed to the flux deficit
from the line itself.
Similarly, the CaII H\&K equivalent widths have range $\pm 50$~\AA, while the width of the $U$ band is
701~\AA, so that its affect on magnitude
is
$2.5 \log{(50/701)} \sim 0.08$ mags.   The implied span in $U$ magnitude is  0.21~mag, so $\alpha_0$ cannot be completely due to the line itself.
\color{black}

\subsection{The Two Extrinsic Parameters $k_0$ and $k_1$}
\color{red}
The two sets of  $\gamma^i$ parameters  are significantly non-zero. 
\color{black}
None of the 20000 links of 
our Monte Carlo chains for $\gamma$ exceed 0 (see Figures~\ref{global1:fig}--\ref{global5:fig}), we thus claim detection of the
influence of $k_0$ and $k_1$  on supernova magnitudes
with probability $(1-5\times 10^{-5})$.

\color{red}
Combining the effects of both parameters, we can define a total $X$-band extinction $A_X=\gamma^0_X k_0 + \gamma^1_X k_1$
and a corresponding $E(B-V) = A_B-A_V$.  Figure~\ref{avebv:fig} shows the relationship between $A_V$ and $E(B-V)$, from
which a linear fit gives a slope of $R_V=2.51^{+0.23}_{-0.21}$.  The fit also gives the zeropoint, which we use to 
estimate $R_V=(A_V-zeropoint)/E(B-V)$ per supernova, shown in Figure~\ref{avebv:fig}.
Extremely blue supernovae ($E(B-V) + const <-0.05$ on the plot) have an average $R_V=1.88 \pm   0.13$,
whereas the extreme red objects ($E(B-V) + const >0.1$) have $R_V=  2.96 \pm   0.11$, indicating that different physical mechanisms
operate in these regimes.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=2.8in]{output11/avebv.pdf}
   \includegraphics[width=2.8in]{output11/rv.pdf}
   \caption{\textcolor{red}{Left: $A_V$ versus $E(B-V)$ and Right: $R_V=A_V/(E(B-V)-zeropoint$\
   versus $E(B-V)$ for the supernova sample.}
   \label{avebv:fig}}
\end{figure}

Until now we have considered $k_0$ and $k_1$ without association with any specific dust model, but they could be interpreted as
parameters of a single  dust model.  Indeed the \citet{1989ApJ...345..245C} dust law can be expressed as a two-parameter linear model
\begin{equation}
A_X = a(X)  A_V + b(X) E(B-V).
\end{equation}
The \citet{1999PASP..111...63F} (hereafter referred to as \citetalias{1999PASP..111...63F}) model is approximately described by a linear model with
the above form, with
$a(U,B,V,R,I)=\{0.96,   1.00,   1.00,   0.97,   0.78\}$ and $b(U,B,V,R,I)=\{  1.64,   1.06,   0.16,  -0.40,  -0.52\}$ for the case of
$R^F_V=2.5$ and the updated
\citet{2007ApJ...663.1187H} SN~Ia template at $B$-band peak.  This linear approximation
is good in the ranges of $R^F_V$ and wavelength under consideration, in that these vectors vary by $<5$\% over the range $1.9<R^F_V<3.1$.


Our model can be expressed as
\begin{equation}
A_X = \frac{\gamma^0_X}{\gamma^0_1-\gamma^0_2}  E_{\gamma^0}(B-V) +  \frac{\gamma^1_X}{\gamma^1_1-\gamma^1_2}  E_{\gamma^1}(B-V).
\end{equation}
Our best-fit values for the $\gamma$ coefficients (from Table~\ref{global:tab}) can expressed in terms of the $a(X)$ and $b(X)$ vectors:
\begin{equation}
\begin{pmatrix}
 \frac{\gamma^0_X}{\gamma^0_1-\gamma^0_2} \\
\frac{\gamma^1_X}{\gamma^1_1-\gamma^1_2} 
\end{pmatrix}=
\begin{pmatrix}
\begin{array}{rr}
1.22 & 2.75 \\
0.81 & -5.50
\end{array}
\end{pmatrix} 
\begin{pmatrix}
a(X) \\
b(X)
\end{pmatrix}+
\begin{pmatrix}
\vec{\epsilon}_a \\
\vec{\epsilon}_b
\end{pmatrix},
\end{equation}
where the two residual $\vec{\epsilon}$ vectors are perpendicular to both $a(X)$ and $b(X)$ and
contribute  3\% and 4\% respectively to the total  length of the $\gamma$ vectors.
This result indicates that
the allowed variation of $UBVRI$ allowed by the \citetalias{1999PASP..111...63F} model and our best-fit model are almost identical.

The model parameters $E_{\gamma^0}(B-V)$ and  $E_{\gamma^1}(B-V)$ for the supernovae in our sample
are projected onto the  \citetalias{1999PASP..111...63F} values of $E^F(B-V)$ and $A^F$, up to an additive constant:  these are shown in Figure~\ref{avebv:fig}.
The points are confined to a tight locus around a line, with slope that corresponds to $R^F_V=2.24 \pm 0.16$.  
The linear fit also produces the zeropoint, allowing us to determine individual $R^F_V$ for all supernovae,
which are shown in Figure~\ref{avebv_synth:fig}.   The relative values of the projected parameters are similar to
the native $R_V$, $A_V$, and $E(B-V)$  shown in Figure~\ref{avebv:fig}.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=2.8in]{output11/avebv_synth.pdf}
   \includegraphics[width=2.8in]{output11/avrv_synth.pdf}
    \caption{\textcolor{red}{Left: Projections of our model parameters $E_{\gamma^0}(B-V)$ and  $E_{\gamma^1}(B-V)$ for the supernovae in our sample
onto the  \citetalias{1999PASP..111...63F} dust-model plane.  Note that the error bars do not capture the strong
covariance between the parameters. Right: $R^F_V$ for each supernova.}
   \label{avebv_synth:fig}}
\end{figure}


The near coplanarity of the plane defined by our parameters and the plane of  \citetalias{1999PASP..111...63F}, together with the fact that our the individual supernova
parameters translate to  \citetalias{1999PASP..111...63F} parameters consistent with a distribution expected for a narrow range of $R^F_V$, lead us to conclude that 
$\{k_0, k_1\}$, describe, for the most part the $\{ E(B-V), R_V\}$ parameters used
in standard models of Milky Way dust.

%
%
%The extrinsic parameters $\gamma$ attributed to dust are fit for without reference to a specific dust model:
%we now do so in a fit to the 
%model of \citet{1999PASP..111...63F},
%taking the fixed 3693,  4369,  5287,  6319, 7611 \AA\  for the effective wavelengths of the supernova flux through the bands.
%Figure~\ref{fitz:fig} shows the   68\% credible intervals of $\gamma$ and our result
%$R_V=2.97^{+0.18}_{-0.17}$..  
%We do the analysis in $\gamma$, as opposed to $R_V$ say, because the posteriors of $\gamma$ are well-approximated by a normal distribution that is used for the fit.
%The posterior distributions for $R_{\gamma X}=\frac{\gamma_X}{\gamma_B-\gamma_V}$ are shown in Figure~\ref{rx:fig}.
%\color{black}
%
%\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
%   \centering
%   \includegraphics[width=3.2in]{output11/fitz.pdf}
%   \caption{\textcolor{red}{68\% credible intervals of $\gamma_X$ and the best-fit to the  \citet{1999PASP..111...63F}  model $R^F_V=2.97$.}
%   \label{fitz:fig}}
%\end{figure}
%
%
%\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
%   \centering
%   \includegraphics[width=2.8in]{output11/rx_corner.pdf}
%      \includegraphics[width=2.8in]{output11/rxdelta_corner.pdf} 
%   \caption{Posterior contours for Left:  $R_X=\frac{\gamma_X}{\gamma_B-\gamma_V}$;  Right:  $R_{\delta X}=\frac{\delta_X}{\delta_B-\delta_V}$.
%   \label{rx:fig}}
%\end{figure}


%
%The best-fit distributions for $R_X=\frac{\gamma_X}{\gamma_1-\gamma_2}$ are shown in Figure~\ref{rx:fig}.  
%Our measurements of
%$R_X$ and $\frac{\gamma_X}{\gamma_2}$, placed at the effective wavelength of the supernova flux that passes
%through the synthetic bands, are shown in
%Figure~\ref{ccm:fig}.
%
%

%
%\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
%   \centering
%   \includegraphics[width=2.8in]{output11/ccm.pdf}
%      \includegraphics[width=2.8in]{output11/ccm2.pdf} 
%   \caption{68\% credible intervals of Left: $R_X=\frac{\gamma_X}{\gamma_1-\gamma_2}$; Right: $\frac{\gamma_X}{\gamma_2}$. 
%   \label{ccm:fig}}
%\end{figure}


\subsection{SN 2014J in the Context of Our Model}
\label{sn2014j:sec}

SN~2014J   is one of several SNe~Ia that exhibits colors that imply a low $R_V<2.0$ \citep{2014ApJ...788L..21A, 2014MNRAS.443.2887F, 
2014arXiv1411.3332J,
2014ApJ...795L...4K, 2015ApJ...805...74B}.
The colors of this supernova can be studied in comparison to SN~2011fe, an object with similar
spectral evolution that 
suffers very low galactic and host reddening
\citep[this technique has been used in][]{2006MNRAS.369.1880E,2007AJ....133...58K,2008MNRAS.384..107E,2010AJ....139..120F, 2014ApJ...788L..21A}.
The overall higher photospheric velocities of
SN~2014J are unimportant  based on the findings of  \S\ref{results:sec}.


Data of SN~2014J are  analyzed using our model and using the parameter values found with the SNfactory sample.
The data for the color excesses  in $UBRi$  relative to $V$ at peak brightness  are taken from \citet{2014ApJ...788L..21A},
following their prescription of averaging measurements within 5-days of peak $B$ magnitude.
Their values 
$E_o(U-V) =   2.23 \pm   0.03$,
$E_o(B-V) =   1.28 \pm   0.04$,
$E_o(R-V) =  -0.47 \pm   0.03$,
$E_o(i-V) =  -0.92 \pm   0.03$
are plotted in Figure~\ref{sn2014j:fig}.
These data are fit to the model
\begin{equation}
E_o(X-V) =  \left(\frac{\gamma^0_X}{\gamma^0_V}-1\right)x_0 +  \left(\frac{\gamma^1_X}{\gamma^1_V}-1\right)x_1,
\end{equation}
where we use the median values of the $\gamma$- and $\delta$-terms from Table~\ref{global:tab} and the fit
parameters $x_i $.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output11/sn2014j.pdf} 
   \caption{Measured color excess for SN~2014J and the best-fit predictions from this article and from  \citet{2014ApJ...788L..21A}.
   \label{sn2014j:fig}}
\end{figure}

The best-fit parameters for SN~2014J are 
%----
$x_0= 2.65$, $ x_1=-1.64$ with covariance
%----
\begin{equation}
\begin{pmatrix}
\begin{array}{rr}
0.048 & 0.047 \\
0.047 & 0.248
\end{array}
\end{pmatrix}.
\end{equation}
The predicted values of observed $E_o(B-V)$ from the fit are shown in Figure~\ref{sn2014j:fig}, where they are found to
overlap with the data within the error bars.   For comparison, the best-fit model determined by  \citet{2014ApJ...788L..21A} model using
UV through NIR data,
a  \citet{1999PASP..111...63F} dust with $R_V^F=1.4$ and $E(B-V)=1.37$, 
is also plotted.  Our model predictions provide a significantly better match to the data.

In the fit to our model, the observed color excess is attributed to 
%----
$E_{\gamma^0}(B-V)=  0.92 \pm   0.08$ and  $E_{\gamma^1}(B-V)=  0.28 \pm   0.08$
%-----
contributions.
Among the supernovae in the SNfactory  set used to determine $\gamma^0$ and $\gamma^1$, the
\color{red}
objects with the extreme median values have 
%---
$E_{\gamma^0}(B-V)$ are $-0.07 \pm 0.01$ and  $  0.33 \pm 0.04$,
and for $E_{\gamma^1}(B-V)$ $-0.01 \pm 0.01$  and
$  0.06 \pm 0.03$ 
%---
(see Figure~\ref{ebv:fig}).
The deduced parameters for SN~2014J, which are relative to SN~2011fe, live well outside the 
span of supernovae used to train the coefficients of the model, which is not unexpected if
attributed to dust. 
%The projection of our median parameter values onto the Fitzpatrick plane gives
%$A^F=1.27$, $E^F(B-V)=1.07$, and thus $R^F_V=1.18.$



\section{Supernova Model II: Two Extrinsic Parameters, One Intrinsic Parameter}
\label{model2:sec}
In \S\ref{model:sec} we show that the SNfactory data analyzed with an agnostic model with
two extrinsic parameters produce
magnitudes consistent
with two-parameter dust models that have been established for stars in the Milky Way, LMC, and SMC.
An extension to our model is now added to explore the possibility of
a possible additional
color parameter that could have its origins in supernova astrophysics.
The extended model is written as
\begin{equation}
\begin{pmatrix}
U\\B\\V\\R\\I
\end{pmatrix}
\sim \mathcal{N}
\left(
\Delta +
\begin{pmatrix}
c_0+\alpha_0 EW_{Ca} + \beta_0 EW_{Si} + \eta_0 \lambda_{Si} + \delta_0 D\\
c_1+\alpha_1 EW_{Ca} + \beta_1 EW_{Si} + \eta_1 \lambda_{Si} + \delta_1 D \\
c_2+\alpha_2 EW_{Ca} + \beta_2 EW_{Si} + \eta_2 \lambda_{Si} + \delta_2 D\\
c_3+\alpha_3 EW_{Ca} + \beta_3 EW_{Si} + \eta_3 \lambda_{Si} + \delta_3 D\\
c_4+\alpha_4 EW_{Ca} + \beta_4 EW_{Si}+ \eta_4 \lambda_{Si} + \delta_4 D
\end{pmatrix}
,C_{c}
\right)
\label{ewsiv:eqn}
\end{equation}
\begin{equation}
\begin{pmatrix}
U_o\\B_o\\ V_o\\R_o\\I_o\\EW_{Si, o}\\ EW_{Ca, o} \\ \lambda_{Si, o}
\end{pmatrix}
\sim \mathcal{N}
\left(
\begin{pmatrix}
U +\gamma^0_{1} k_0 +\gamma^1_{1} k_1 \\B +\gamma^0_{1} k_0 +\gamma^1_{1} k_1 \\
V+\gamma^0_{2} k_0+\gamma^1_{2} k_1\\R+\gamma^0_{3} k_0 + \gamma^1_{3} k_1\\I+\gamma^0_{4} k_0+\gamma^1_{4} k_1\\
EW_{Si}\\ EW_{Ca} \\ \lambda_{Si}
\end{pmatrix}
,C
\right).
\label{dust:eqn}
\end{equation}
In all aspects it is identical to the previous model except for the addition of the term $\delta D$,
which affects the mean
absolute magnitudes:
each supernova has an intrinsic $D$ parameter that is linearly related to
absolute magnitudes through global coefficients $\delta$.  As with $k_0$ and $k_1$,  $D$ is a latent supernova parameter whose existence is inferred from
its effects on broadband magnitudes.
Although
this term is introduced as part of the  absolute rather than observed magnitudes,
practically identical results are obtained when it is placed alongside the extrinsic $\gamma k$ terms.  Despite this
degeneracy, for convenience this term is referred to as ``intrinsic''.


To constrain the new degrees of freedom and degeneracies inherent in the model we impose that
$D+1/N$ be a simplex to break the 
degeneracy  $\delta \rightarrow a\delta$, $D \rightarrow a^{-1}D$.
Furthermore we impose
\begin{equation}
\langle D\rangle=0,  \delta \cdot \gamma^0=0, \delta \cdot \gamma^1=0.
\end{equation}
The first constraint specifies the definition of zero color.   The latter two confine possible magnitude
variations to those that cannot be attributed to 
the $\gamma k$ terms (i.e.\ dust); this condition breaks the degeneracies between $\delta D$  
and $\gamma k$. 
Unlike for the $\gamma$'s, there is no condition on the sign of $\delta_0$,
so that the posterior has degenerate solutions with sign flips of $\delta$ and $D$.
In the running of the Monte Carlo, the initial conditions for
$\gamma^0$ and $\gamma^1$ were set to be in the neighborhood of the best fit of \S\ref{model:sec}
to maintain their correspondence between the two models. 

The resulting credible intervals of our parameters are given in Table~\ref{global2:tab}.  Comparison with the
results of the 2-parameter extrinsic model given in Table~\ref{global:tab} shows that  parameters that are common to both
are consistent well within 1$\sigma$ uncertainties.  The most significant changes of note are in the standard
deviations of the residuals in the $V$ and $I$ bands, which are now reduced.
The newly fitted $\gamma$ parameters have an almost identical association with the  \citetalias{1999PASP..111...63F} dust law, in that now
\begin{equation}
\begin{pmatrix}
 \frac{\gamma^0_X}{\gamma^0_1-\gamma^0_2} \\
\frac{\gamma^1_X}{\gamma^1_1-\gamma^1_2} 
\end{pmatrix}=
\begin{pmatrix}
\begin{array}{rr}
1.22 & 2.75 \\
0.80 & -5.51
\end{array}
\end{pmatrix} 
\begin{pmatrix}
a(X) \\
b(X)
\end{pmatrix}+
\begin{pmatrix}
\vec{\epsilon}_a \\
\vec{\epsilon}_b
\end{pmatrix},
\end{equation}
where the residual vectors $\vec{\epsilon}$ are perpendicular to the  \citetalias{1999PASP..111...63F} plane and
contribute similar 3\% and 4\% to the total  length of the vector.
The effective total-to-selective extinction for the supernova sample has a slight shift to $R^F_V=2.23 \pm 0.16$.

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Parameters& $X=U$ &$B$&$V$&$R$&$I$\\ \hline
$\alpha_X$
&
$0.0044^{+0.0009}_{-0.0009}$
&
$0.0016^{+0.0007}_{-0.0007}$
&
$0.0016^{+0.0006}_{-0.0006}$
&
$0.0015^{+0.0005}_{-0.0005}$
&
$0.0026^{+0.0005}_{-0.0005}$
\\
${\alpha_X/\alpha_V-1}$
&
$   1.8^{+   0.9}_{  -0.4}$
&
$   0.0^{+   0.1}_{  -0.2}$
&
\ldots
&
$  -0.0^{+   0.1}_{  -0.1}$
&
$   0.6^{+   0.6}_{  -0.3}$
\\
$\beta_X$
&
$ 0.032^{+ 0.003}_{-0.003}$
&
$ 0.025^{+ 0.002}_{-0.003}$
&
$ 0.025^{+ 0.002}_{-0.002}$
&
$ 0.020^{+ 0.002}_{-0.002}$
&
$ 0.019^{+ 0.002}_{-0.002}$
\\
${\beta_X/\beta_V-1}$
&
$  0.26^{+  0.05}_{ -0.05}$
&
$ -0.01^{+  0.03}_{ -0.03}$
&
\ldots
&
$ -0.19^{+  0.01}_{ -0.02}$
&
$ -0.24^{+  0.03}_{ -0.03}$
\\
$\eta_X$
&
$-0.0002^{+0.0012}_{-0.0011}$
&
$0.0000^{+0.0010}_{-0.0009}$
&
$0.0005^{+0.0008}_{-0.0008}$
&
$0.0005^{+0.0007}_{-0.0007}$
&
$-0.0003^{+0.0006}_{-0.0006}$
\\
${\eta_X/\eta_V-1}$
&
$ -0.40^{+  2.19}_{ -1.84}$
&
$ -0.36^{+  1.57}_{ -1.19}$
&
\ldots
&
$ -0.12^{+  0.27}_{ -0.25}$
&
$ -0.89^{+  1.72}_{ -1.35}$
\\
${\gamma^0_X/\gamma^0_V-1}$
&
$  0.66^{+  0.06}_{ -0.05}$
&
$  0.35^{+  0.03}_{ -0.03}$
&
\ldots
&
$ -0.23^{+  0.01}_{ -0.01}$
&
$ -0.45^{+  0.03}_{ -0.03}$
\\
${\gamma^1_X/\gamma^1_V-1}$
&
$ -0.30^{+  0.18}_{ -0.19}$
&
$ -0.17^{+  0.09}_{ -0.10}$
&
\ldots
&
$ -0.06^{+  0.05}_{ -0.04}$
&
$ -0.16^{+  0.10}_{ -0.08}$
\\
${{\delta_X/\delta_I-1}}$
&
$ -0.61^{+  0.47}_{ -0.18}$
&
$ -1.21^{+  0.25}_{ -0.77}$
&
$ -1.76^{+  0.16}_{ -0.20}$
&
$ -1.17^{+  0.29}_{ -0.20}$
&
\ldots
\\
$\sigma_X$
&
$ 0.059^{+ 0.013}_{-0.013}$
&
$ 0.031^{+ 0.007}_{-0.008}$
&
$ 0.017^{+ 0.005}_{-0.007}$
&
$ 0.012^{+ 0.006}_{-0.007}$
&
$ 0.039^{+ 0.005}_{-0.005}$
\\
\hline
\end{tabular}
\caption{68\% credible intervals for the Global Fit Parameters of the Extrinsic--Intrinsic Model in \S\ref{model2:sec}.\label{global2:tab}}
\end{table}


The distributions of both sources of color excess $E(B-V)$ and
 $E_\delta(B-V) = (\delta_B-\delta_V)D$ are shown in the ideograms in Figure~\ref{ebv:fig}.
The standard deviations of $E_\gamma(B-V)$ and $E_\delta(B-V)$ are
%-----
0.081
and 0.010
%-----
mag respectively.
The range of $B-V$ colors is $\sim 8$ times larger for the external contribution; variation in $B-V$ color due to the intrinsic term is dwarfed by the extrinsic dust
component. 
Both distributions are non-Gaussian, that of $E(B-V)$ having a sharp rise in the blue and an extended tail in the red, consistent
with the expectation of dust considering the spatial distribution of supernovae within galaxies and the distribution of galaxy orientations with respect to the observer \citep{1998ApJ...502..177H}.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=2.8in]{output15/ebv.pdf}
   \includegraphics[width=2.8in]{output15/ebv_delta.pdf}
      \caption{Left: Ideograms of the external $E(B-V)$ and
   internal $E_\delta(B-V) = (\delta_B-\delta_V)D$  contributions to color excess  for the supernovae in our sample.
   Right: Ideogram of $E_\delta(B-V)$ on an enlarged scale.
   \label{ebv:fig}}
\end{figure}
The significance of the $\delta$ parameters is shown pictorially.
Figure~\ref{deltacorner:fig} shows the $\delta$ posteriors, which are bimodal due to the $\pm$ sign degeneracy.  The probability
of the null result $\delta_X=0$ for all $X$ is $<98\%$, as determined through
the population of its bin in a 5-dimensional histogram.
Figure~\ref{deltaratio:fig} shows  values of $\delta_X/\delta_I-1$ that  have significant non-zero values.   An interesting feature
is that the shift in magnitude is not monotonic in wavelength, but rather decreases from $U$ to $V$, and then increases from $V$
to $I$.
These are not the behaviors expected from normal dust,
circumstellar dust suggested as a source of SN extinction \citep{2005ApJ...635L..33W,2008ApJ...686L.103G,
2015ApJ...807L..26G} and so may be indicative of underlying supernova physics beyond that captured by $EW_{Ca}$, $EW_{Si}$ and
$\lambda_{Si}$.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=4.5in]{output15/delta_corner.pdf}
   \caption{Significance of the non-zero values of $\delta$ seen through the posterior contours for $\delta$.  The bimodality is due to the degenerate solution due to sign change.
   \label{deltacorner:fig}}
\end{figure}


\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
      \includegraphics[width=5.5in]{output15/deltaratio.pdf}
   \caption{Significance of the non-zero values of $\delta$ seen through $\delta_X/\delta_I-1$, which is insensitive to the $\pm$ sign degeneracy.
   \label{deltaratio:fig}}
\end{figure}

The Pearson correlation coefficients for $\Delta$, $EW_{Ca}$, $EW_{Si}$, $\lambda_{Si}$, $E_{\gamma^0}(B-V)$, $E_{\gamma^1}(B-V)$,  $A_{\delta I}$ are given in the matrix.
\begin{multline}
Cor(\Delta, EW_{Ca}, EW_{Si}, \lambda_{Si}, E_{\gamma^0}(B-V), E_{\gamma^1}(B-V),  A_{\delta I}) =\\
\begin{pmatrix}
\begin{array}{rrrrrrr}
1.00 & 0.01 & -0.06 & -0.05 & 0.11 & 0.01 & -0.03 \\
0.01 & 1.00 & 0.11 & -0.26 & -0.07 & -0.00 & 0.05 \\
-0.06 & 0.11 & 1.00 & -0.14 & -0.17 & -0.03 & -0.02 \\
-0.05 & -0.26 & -0.14 & 1.00 & 0.04 & 0.03 & 0.01 \\
0.11 & -0.07 & -0.17 & 0.04 & 1.00 & 0.06 & 0.03 \\
0.01 & -0.00 & -0.03 & 0.03 & 0.06 & 1.00 & -0.00 \\
-0.03 & 0.05 & -0.02 & 0.01 & 0.03 & -0.00 & 1.00
\end{array}
\end{pmatrix}.
\end{multline}
\color{red}
The intrinsic parameter $A_{\delta I}$ has correlations $<0.05$ with respect to all the other parameters, including those attributed
to extrinsic dust.


\section{Summary and Discussion}
\label{discussion:sec}
\subsection{Summary}
To summarize, we model SNe~Ia broadband optical peak magnitudes allowing for correlations with spectral features at peak and
extrinsic color parameters.  Analyzing SNfactory data with the model, we find significant evidence that the above parameters do
affect supernova magnitudes and colors.  The external color parameters are consistent with \citet{1999PASP..111...63F} dust
with $R^F_V=2.24 \pm 0.16$.  This model  does an excellent job in
describing SN~2014J, a supernova with extreme colors that was not used in the training.  
This work represents the first determination of the dust extinction curve outside the Local Group
derived entirely independently of assumptions about the shape of the extinction curve and/or assumptions about the
distribution of $A_V$.  An extended version
of the model retains the features of the first model, and gives the first identification of  a new parameter that affects supernova
colors in a manner that is distinct from  the expectations from dust or the spectral features $EW_{Ca}$, $EW_{Si}$, and $\lambda_{Si}$.

\subsection{Light Curve Shape}
Our models do not explicitly include light-curve shape, which is known to track SN~Ia diversity.
We now examine this in more depth using the intrinsic--extrinsic model of \S\ref{model2:sec}.
The correlation between $EW_{Si}$ and the SALT2 $x_1$ light-curve shape parameter is verified
for our sample.
On the other hand, there is no such correlation seen between $A_{\delta I} = \delta_I D$ and $x_1$, as shown in Figure~\ref{x1:fig}. 
Our magnitude residuals, $\sigma_X$, are comparable to the residuals after SALT2 training
\citep{2010A&A...523A...7G}, showing that our model exhibits little loss in its ability to standardize
magnitudes although $x_1$ is neglected.
There is no apparent correlation
between $x_1$ and the residuals between observed and model-predicted colors, as seen in
Figure~\ref{x1res:fig}. 
We  run an extension of our model to include light-curve shape by adding a new linear term $\zeta x_1$ to the 
intrinsic magnitude; the credible intervals
from this analysis are given in Table~\ref{globalx1:tab}; there are no significant changes with respect to the reference model. 
Light-curve shape is not a strong predictor of supernova magnitude relative to the spectral features: only $\zeta_R/\zeta_V$
has a non-zero value at $>2\sigma$.
The evidence for a non-zero $\delta$ remains, meaning that it is not an artifact of
having neglected light-curve shape.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=2.8in]{output15/x1si.pdf}
   \includegraphics[width=2.8in]{output15/x1D.pdf}
    \caption{\textcolor{red}{Left: SALT2 $x_1$ versus $EW_{Si}$.  Right: $x_1$ versus $A_{\delta I}  = \delta_I D$.}
   \label{x1:fig}}
\end{figure}


\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=5.5in]{output15/residualx1.pdf}
    \caption{\textcolor{red}{Differences between observed colors and the colors predicted from the analysis, as a function
            of the SALT2 light-curve shape parameter $x_1$.  For reference a dotted line is plotted at zero difference.}
   \label{x1res:fig}}
\end{figure}


\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Parameters & $X=U$ &$B$&$V$&$R$&$I$\\ \hline
$\alpha_X$
&
$0.0044^{+0.0010}_{-0.0010}$
&
$0.0016^{+0.0008}_{-0.0008}$
&
$0.0017^{+0.0006}_{-0.0007}$
&
$0.0015^{+0.0005}_{-0.0005}$
&
$0.0028^{+0.0005}_{-0.0005}$
\\
${\alpha_X/\alpha_V-1}$
&
$   1.6^{+   0.8}_{  -0.4}$
&
$  -0.0^{+   0.1}_{  -0.2}$
&
\ldots
&
$  -0.1^{+   0.1}_{  -0.1}$
&
$   0.7^{+   0.7}_{  -0.3}$
\\
$\beta_X$
&
$ 0.033^{+ 0.006}_{-0.006}$
&
$ 0.025^{+ 0.005}_{-0.005}$
&
$ 0.022^{+ 0.004}_{-0.004}$
&
$ 0.019^{+ 0.004}_{-0.003}$
&
$ 0.016^{+ 0.003}_{-0.003}$
\\
${\beta_X/\beta_V-1}$
&
$  0.52^{+  0.13}_{ -0.12}$
&
$  0.14^{+  0.07}_{ -0.07}$
&
\ldots
&
$ -0.13^{+  0.03}_{ -0.03}$
&
$ -0.27^{+  0.06}_{ -0.06}$
\\
$\eta_X$
&
$0.0004^{+0.0011}_{-0.0011}$
&
$0.0005^{+0.0010}_{-0.0010}$
&
$0.0009^{+0.0008}_{-0.0008}$
&
$0.0009^{+0.0007}_{-0.0007}$
&
$0.0001^{+0.0006}_{-0.0006}$
\\
${\eta_X/\eta_V-1}$
&
$ -0.34^{+  0.58}_{ -1.20}$
&
$ -0.36^{+  0.35}_{ -0.86}$
&
\ldots
&
$ -0.10^{+  0.20}_{ -0.14}$
&
$ -0.74^{+  0.35}_{ -0.75}$
\\
$\zeta_X$
&
$  0.00^{+  0.04}_{ -0.04}$
&
$ -0.01^{+  0.03}_{ -0.03}$
&
$ -0.03^{+  0.03}_{ -0.03}$
&
$ -0.01^{+  0.02}_{ -0.02}$
&
$ -0.03^{+  0.02}_{ -0.02}$
\\
${\zeta_X/\zeta_V-1}$
&
$ -0.76^{+  0.78}_{ -1.78}$
&
$ -0.50^{+  0.47}_{ -1.06}$
&
\ldots
&
$ -0.46^{+  0.21}_{ -0.50}$
&
$ -0.16^{+  0.45}_{ -0.34}$
\\
${\gamma^0_X/\gamma^0_V-1}$
&
$  0.65^{+  0.06}_{ -0.05}$
&
$  0.35^{+  0.03}_{ -0.03}$
&
\ldots
&
$ -0.24^{+  0.01}_{ -0.01}$
&
$ -0.46^{+  0.03}_{ -0.03}$
\\
${\gamma^1_X/\gamma^1_V-1}$
&
$ -0.39^{+  0.16}_{ -0.17}$
&
$ -0.24^{+  0.09}_{ -0.10}$
&
\ldots
&
$ -0.06^{+  0.04}_{ -0.03}$
&
$ -0.15^{+  0.09}_{ -0.08}$
\\
${{\delta_X/\delta_I-1}}$
&
$  0.04^{+  0.48}_{ -0.31}$
&
$ -2.16^{+  0.36}_{ -0.64}$
&
$ -1.94^{+  0.23}_{ -0.42}$
&
$ -0.63^{+  0.56}_{ -0.29}$
&
\ldots
\\
$\sigma_X$
&
$ 0.050^{+ 0.014}_{-0.012}$
&
$ 0.027^{+ 0.008}_{-0.008}$
&
$ 0.015^{+ 0.004}_{-0.005}$
&
$ 0.008^{+ 0.006}_{-0.005}$
&
$ 0.041^{+ 0.005}_{-0.004}$
\\
\hline
\end{tabular}
\caption{68\% credible intervals for the Global Fit Parameters including the light-curve shape parameter $x_1$ \label{globalx1:tab}}
\end{table}

\color{black}

%The trend in Figure~\ref{rveff:fig} showing an increased range of brighter-than-expected magnitudes  with increasing $E_{eff}(B-V)$  can
%be compared
%to Figure~\ref{betoule:fig}, which shows the $B$-band  Hubble residuals as a function of the color $C$ parameter (directly
%comparable to $E(B-V)$) from the Hubble diagram analysis of \citet{2014A&A...568A..22B}.
%A linear fit for the relationship between color and Hubble residual has a significant slope ($-0.57 \pm 0.07$), consistent with the slope
%found for the SNfactory data.
%
%\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
%   \centering
%   \includegraphics[width=3.2in]{output11/betoule.pdf} 
%   \caption{Best-fit SN~Ia  $B$-band Hubble residuals as a function of the best-fit color parameter $C$ from the cosmology analysis of \citet{2014A&A...568A..22B}.
%   \label{betoule:fig}}
%\end{figure}

\subsection{SiII Line Velocity}

\citet{2009ApJ...699L.139W, 2011ApJ...729...55F} find a connection between $v_{Si}$ and color, and  
\citet{2015MNRAS.447.1247S} show that $v_{Si}$ is an important spectral classifier within the SNfactory data themselves.
\textcolor{red}{The $\lambda_{Si}$ treated in this article varies linearly with $v_{Si}$.}
The values of $\eta$ in this article are consistent with zero.  $EW_{Ca}$ and $EW_{Si}$ have a significant effect on color,
and in turn $\lambda_{Si}$ is correlated with $EW_{Ca}$.
Removing from our model the dependence on equivalent widths (eliminating the  $\alpha$ and $\beta$ parameters), we recover
non-zero $\eta$ values at  $\gtrsim 2\sigma$.  We conclude that $v_{Si}$ is correlated with color, 
\textcolor{red}{but this correlation
is absorbed into the equivalent-width corrections, which give higher likelihood as they give lower residual dispersions $C_c$}.

\subsection{Host-Galaxy Mass}
\color{red}
A correlation between Hubble residual and host-galaxy mass
was first noted by \citet{2010ApJ...715..743K,2010MNRAS.406..782S}, a signal confirmed to exist in the SNfactory
sample \citep{2013ApJ...770..108C}.
This host-mass bias could be the result of a parameter that was not accounted for in the inference of SN~Ia absolute magnitude.
\citet{2016arXiv160904470M} find that when introducing intrinsic color scatter as a latent parameter, the null mass-step effect falls on the 95\%-level of their posterior.
To explore whether the Hubble residual may be due to unaccounted color,
we plot in Figure~\ref{childress:fig} our intrinsic parameter  $A_{\delta I}$  versus host mass 
\citep{2016rigault}, and their ideograms for two samples divided by a host mass of  $\log{(M/M_\sun)}=10$
used by  \citet{2013ApJ...770..108C}.
Although by eye it appears that the high-mass hosts tend the have lower $A_{\delta I}$, the difference is
not significant; we calculate  average $\langle A_{\delta I} \rangle=   0.0042 \pm    0.0014$,
$\langle A_{\delta I} \rangle=   0.0031 \pm    0.0017 $ for low- and high-mass hosts respectively. 
Using the SNfactory sample,
\citet{2013A&A...560A..66R} find that step in Hubble residuals is better related to local star formation rate, rather than host mass.
A deeper exploration of the possible connection between our intrinsic parameter and local star formation and other host properties is deferred to
\citet{2016rigault}.
\color{black}
\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=2.8in]{output15/rigault3.pdf}
s     \includegraphics[width=2.8in]{output15/rigault2.pdf}
      \caption{\textcolor{red}{Left: Intrinsic parameter $A_{\delta I}=\delta_I D$  versus host mass.  Hosts too faint for a mass measurement
      are assigned a mass of 5.  Overplotted are the mean and 1$\sigma$ values for supernovae with hosts
      less than and greater than  $\log{(M/M_\sun)}=10$.
Right: Ideograms of  $A_{\delta I}$ for supernovae in $\log{(M/M_\sun)}<10$ and $\log{(M/M_\sun)}>10$ hosts. 
}
   \label{childress:fig}}
\end{figure}


\subsection{Physical Motivation}

We find an intrinsic supernova parameter that does not have a monotonic influence on supernova magnitudes as a function
of wavelength but rather has an inflection in the $V$ band.  In addition, red $B-V$ colors imply brighter magnitudes
for this component.  Absorption and re-emission of light
over broad bands are the most obvious candidate for producing this behavior.
\citet{2006ApJ...649..939K} points out that at a temperature of 7000~K, the iron/cobalt gas in the ejecta transition
from doubly and singly ionized states, making the gas phosphorescent and efficient in redistributing energy from the UV/blue to redder
wavelengths.  Qualitatively this would result in the structure for $\gamma_X/\gamma_V-1$ that we measure.
We conjecture that varying $^{56}$Ni mass affects the rise-time to peak $B$ brightness and hence the optical depth
of the reionization front with respect to the core of the ejecta at peak, yielding the wavelength-dependent color behavior
detected in this analysis.

\subsection{Improving SNe~Ia as Standard Candles}
The model and results presented here
carry information on the calibration of absolute magnitude.  The grey parameter $\Delta$ represents Hubble residuals after
application of the model, and its  $0.10$ mag dispersion has contributions from intrinsic dispersion, peculiar velocities, and
measurement uncertainties.  It is also subject to overtraining.
The  analysis of this article was intended to model the color behavior of supernovae in our dataset.
\color{red}
A new analysis insensitive to overtraining is required to robustly determine how
well this model can be used to measure supernova distances.
\color{black}

\acknowledgments
We thank the STAN team for providing the statistical tool without which this analysis would not have been possible,
and Michael Betancourt specifically for his helpful guidance.  We thank Danny Goldstein for useful discussions.
We thank Dan Birchall for observing assistance, the technical and
scientific staffs of the Palomar Observatory, the High Performance
Wireless Radio Network (HPWREN), and the University of Hawaii 2.2~m
telescope.  We recognize the significant cultural role of Mauna Kea
within the indigenous Hawaiian community, and we appreciate the
opportunity to conduct observations from this revered site.  This
work was supported in part by the Director, Office of Science,
Office of High Energy Physics, of the U.S. Department of Energy
under Contract No. DE-AC02- 05CH11231.  Support in France was
provided by CNRS/IN2P3, CNRS/INSU, and PNC; LPNHE acknowledges
support from LABEX ILP, supported by French state funds managed by
the ANR within the Investissements d'Avenir programme under reference
ANR-11-IDEX-0004-02.  NC is grateful to the LABEX Lyon Institute
of Origins (ANR-10-LABX-0066) of the Universit\'e de Lyon for its
financial support within the program ``Investissements d'Avenir''
(ANR-11-IDEX-0007) of the French government operated by the National
Research Agency (ANR).  Support in Germany was provided by the DFG
through TRR33 ``The Dark Universe;'' and in China from Tsinghua
University 985 grant and NSFC grant No~11173017.  Some results were
obtained using resources and support from the National Energy
Research Scientific Computing Center, supported by the Director,
Office of Science, Office of Advanced Scientific Computing Research,
of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
HPWREN is funded by National Science Foundation Grant Number
ANI-0087344, and the University of California, San Diego.


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