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ZTF Early Observations of Type Ia Supernovae I: Properties of the 2018 Sample

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In this paper, the authors presented high-quality light curves of 127 SNe Ia discovered by the Zwicky Transient Facility (ZTF) in 2018, which can be used to study the shape and color evolution of the rising light curves in unprecedented detail.
Abstract
Early-time observations of Type Ia supernovae (SNe Ia) are essential to constrain their progenitor properties. In this paper, we present high-quality light curves of 127 SNe Ia discovered by the Zwicky Transient Facility (ZTF) in 2018. We describe our method to perform forced point spread function (PSF) photometry, which can be applied to other types of extragalactic transients. With a planned cadence of six observations per night ($3g+3r$), all of the 127 SNe Ia are detected in both $g$ and $r$ band more than 10\,d (in the rest frame) prior to the epoch of $g$-band maximum light. The redshifts of these objects range from $z=0.0181$ to 0.165; the median redshift is 0.074. Among the 127 SNe, 50 are detected at least 14\,d prior to maximum light (in the rest frame), with a subset of 9 objects being detected more than 17\,d before $g$-band peak. This is the largest sample of young SNe Ia collected to date; it can be used to study the shape and color evolution of the rising light curves in unprecedented detail. We discuss six peculiar events in this sample, including one 02cx-like event ZTF18abclfee (SN\,2018crl), one Ia-CSM SN ZTF18aaykjei (SN\,2018cxk), and four objects with possible super-Chandrasekhar mass progenitors: ZTF18abhpgje (SN\,2018eul), ZTF18abdpvnd (SN\,2018dvf), ZTF18aawpcel (SN\,2018cir) and ZTF18abddmrf (SN\,2018dsx).

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ZTF Early Observations of Type Ia Supernovae. I. Properties of the 2018 Sample
Yuhan Yao
1
, Adam A. Miller
2,3
, S. R. Kulkarni
1
, Mattia Bulla
4
, Frank J. Masci
5
, Daniel A. Goldstein
1
,
Ariel Goobar
4
, Peter Nugent
6,7
, Alison Dugas
1
, Nadia Blagorodnova
8
, James D. Neill
1
, Mickael Rigault
9
,
Jesper Sollerman
10
, J. Nordin
11
, Eric C. Bellm
12
, S. Bradley Cenko
13,14
, Kishalay De
1
, Suhail Dhawan
4
, Ulrich Feindt
4
,
C. Fremling
1
, Pradip Gatkine
15
, Matthew J. Graham
16
, Melissa L. Graham
17
, Anna Y. Q. Ho
1
, T. Hung
18
,
Mansi M. Kasliwal
1
, Thomas Kupfer
19
, Russ R. Laher
5
, Daniel A. Perley
20
, Ben Rusholme
5
, David L. Shupe
5
,
Maayane T. Soumagnac
21
, K. Taggart
20
, Richard Walters
1,22
, and Lin Yan
1
1
Cahill Center for Astrophysics, California Institute of Technology, MC 249-17, 1200 E California Boulevard, Pasadena, CA 91125, USA; yyao@astro.caltech.edu
2
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, 2145 Sheridan
Road, Evanston, IL 60208, USA
3
The Adler Planetarium, Chicago, IL 60605, USA
4
The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden
5
IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA
6
Computational Cosmology Center, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
7
Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA
8
Department of Astrophysics/IMAPP, Radboud University, Nijmegen, The Netherlands
9
Université Clermont Auvergne, CNRS/IN2P3, Laboratoire de Physique de Clermont, F-63000 Clermont-Ferrand, France
10
The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden
11
Institute of Physics, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany
12
DIRAC Institute, Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA
13
Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA
14
Joint Space-Science Institute, University of Maryland, College Park, MD 20742, USA
15
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
16
Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
17
University of Washington, Department of Astronomy, Box 351580, Seattle, WA 98195-1580, USA
18
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA
19
Kavli Institute for Theoretical Physics, University of California Santa-Barbara, Santa Barbara, CA 93106, USA
20
Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK
21
Benoziyo Center for Astrophysics, Weizmann Institute of Science, Rehovot, Israel
22
Caltech Optical Observatories, California Institute of Technology, MC 249-17, 1200 E California Boulevard, Pasadena, CA 91125, USA
Received 2019 July 22; revised 2019 September 27; accepted 2019 October 5; published 2019 December 3
Abstract
Early-time observations of Type Ia supernovae (SNe Ia) are essential to constrain the properties of their progenitors. In
this paper, we present high-quality light curves of 127 SNe Ia discovered by the Zwicky Transient Facility (ZTF) in
2018. We describe our method to perform forced point-spread function photometry, which can be applied to other
types of extragalactic transients. With a planned cadence of six observations per night (three g+three r), all of the 127
SNe Ia are detected in both g and r bands more than 10 days (in the rest frame) prior to the epoch of g-band maximum
light. The redshifts of these objects range from z=0.0181 to 0.165; the median redshift is 0.074. Among the 127 SNe,
50 are detected at least 14 days prior to maximum light (in the rest frame), with a subset of nine objects being detected
more than 17 days before g-band peak. This is the largest sample of young SNe Ia collected to date; it can be used to
study the shape and color evolution of the rising light curves in unprecedented detail. We discuss six peculiar events in
this sample: one 02cx-like event ZTF18abclfee (SN 2018crl), one Ia-CSM SN ZTF18aaykjei (SN 2018cxk),andfour
objects with possible super-Chandrasekhar mass progenitors: ZTF18abhpgje (SN 2018eul), ZTF18abdpvnd
(SN 2018dvf),ZTF18aawpcel(SN 2018cir), and ZTF18abddmrf (SN 2018dsx).
Unied Astronomy Thesaurus concepts: Type Ia supernovae (1728); Sky surveys (1464); Catalogs (205);
Supernovae (1668); Surveys (1671
); Photometry (1234)
Supporting material: machine-readable tables, FITS le
1. Introduction
Despite being used as standardizable candles to study
cosmology, the origin of Type Ia supernovae (SNe Ia) is not
settled (see review by Maoz et al. 2014). Two major formation
channels have been proposed: single degenerate (SD), where a
carbon/oxygen white dwarf (WD) accretes matter from a non-
degenerate star and triggers an explosion near the Chandrase-
khar mass (M
ch
, Whelan & Iben 1973), and double degenerate
(DD), in which the primary WD accretes material from (or
merges with) another WD (Woosley & Weaver 1994; Tutukov
& Yungelson 1996; Shen 2015).
Observations obtained in the hours to days after explosion
(i.e., earl y-time ) provide a path toward diagnosing the
various explosion mechanisms (Maoz et al. 2014).Early
photometry can constrain the radii of the possible companion
and the progenitor sta r (Kasen 2010;Nugentetal.2011;
Bloom et al. 2012;Goobaretal.2014, 2015).Theshapeand
duration of the risi ng light curves probe the radial distributi on
of rad ioacti ve
56
Ni in the exploding core, as well as the
existence of circumstellar material (Dessart et al. 2014;Piro
& Nakar 2014; Firth et al. 2015;Piro&Morozova2016;
Miller et al. 2018).
The Astrophysical Journal, 886:152 (22pp), 2019 December 1 https://doi.org/10.3847/1538-4357/ab4cf5
© 2019. The American Astronomical Society. All rights reserved.
1

Simulations of the double detonation of a helium shell on the
surface of a WD predict an unusually red excess well before peak
luminosity (Noebauer et al. 2017; Maeda et al. 2018;Polinetal.
2019), which was observed in SN 2016jhr (Jiang et al. 2017) and
SN 2018byg (De et al. 2019). Should the SD channel hold, a
collision between the SN ejecta and the stellar companion will
give rise to strong ultraviolet (UV) emission at early times
(Hayden et al. 2010a;Kasen2010). The detection of a declining
UV pulse in the peculiar SN Ia iPTF 14atg reasonably favors this
scenario (Cao et al. 2015), although Kromer et al. (2016) argue
that its spectral evolution is more consistent with a merger
product. The power of well-sampled early-time photometry has
also been demonstrated in single-object studies of normal SNe Ia
(Marion et al. 2016; Hosseinzadeh et al. 2017; Dimitri adi s et al.
2019; Shappee et al. 2019,Lietal.2019), where the clearly
resolved early bumps in their light curves pose challenges to
simple explosion models.
Up to now there has not been a large (
>100
objects) uniform
data set of SN Ia light curves with both multi-band photometry and
dense early-time sampling. With the Zwicky Transient Facility
(ZTF, Bellm et al. 2019b;Grahametal.2019), we are undertaking
a high-cadence survey with six epochs per night (th ree g+three r;
Bellm et al. 2019a). This experiment is conducted over a large area
of the sky (2500 deg
2
) and thus enables large-number statistics.
In this study, we focus on a special subset of SNe Ia that were
discovered more than 10 days prior to maximum light. Our large
(127 objects), hom ogeneous sample of youn g SNe Ia was
constructed within the rstyearofoperationsbyZTF.
As the rst in a series of three papers, we present the light
curves and sample properties of 127 SNe Ia. A detailed analysis of
the early evolution of these SNe will be addressed will be
addressed in A.A. Miller et al.(2019, in preparation) and M. Bulla
et a l.(2019, in preparation). Throughout this paper, we assume a
at ΛCDM cosmology with
=
--
H
73.24 km s Mpc
0
11
(Riess
et al. 2016) and Ω
m
=0. 275 (Amanullah et al. 2010).
2. The ZTF-2018 High-cadence Sample of SNe Ia
2.1. Observations
The ZTF camera is mounted on the 48 inch Samuel Oschin
Telescope (P48) at Palomar Observatory (Dekany et al. 2016).
At a limiting magnitude of r20.5 mag, three custom lters
(g
ZTF
, r
ZTF
, and i
ZTF
; hereafter g, r, and i) are designed to
maximize throughput by avoiding major skylines at Palomar
(Bellm et al. 2019b). ZTF divides its observing time between
public surveys (40%), partnership surveys (40%), and Caltech
surveys (20%). Bellm et al. (2019a) provide details of the ZTF
surveys. In brief, 85% of the public time was allocated to a
Northern Sky Survey with a cadence of three days in g and r,
and the remaining 15% to a Galactic Plane Survey with two
visits to the Galactic plane (one g+one r) every night. The
bulk of the partnership time in 2018 (MayDecember) was
dedicated to two experiments, including an extragalactic high-
cadence experiment covering 2500 deg
2
with six visits
(three g+three r) every night, and a lower-cadence, wide-
eld i-band survey. The Caltech time was conducted in a one-
day cadence in both g and r with a total footprint of
3000 deg
2
. Each nights schedule is arranged by the survey
scheduler (Bellm et al. 2019a) to optimize volumetric survey
speed (Bellm 2016). In this work, we only focus on the g- and
r-band observations within the partnership high-cadence elds.
The current ZTF alert distribution system (Patterson et al.
2019) generates a source packet once a transient
23
is detected.
By denition, a detection means that the observed ux is ve
times larger than the ux uncertainty (see Masci et al. 2019,
Section 6). For each transient the alert packet includes a rolling
30 days history of detections and non-detections.
Following the association of all alerts generated at the same
position, the GROWTH Marshal (Kasliwal et al. 2019)
compiles a complete historical record of variability, which is
further used to aggregate and visualize follow-up observations.
2.2. Initial Sample Selection
The sample selection process is summarized in Table 1.Intotal,
therewere336SNeIaclassied in the partnership elds in 2018,
247 of which were observed as part of the high-cadence
partnership survey.
24
For the sample of 247 SNe with high-
cadence observations, we performed a preliminary t to their
light curves using the SALT2 software package (Guy et al.
2007) implemented in the sncosmo Python package (Barbary
et al. 2016)
25
to estimate the time of maximum light. The
sample was further reduced to include only those sources with
more than one detection in either the g or r band obtained at
least ve days before the SALT2-estimated time of B-band
maximum, t
B,max
. This resulted in a selection of 191 SNe.
Note that although the 191 SNe were all discovered in the
partnership high-cadence elds, some of them also have
observations during the public or Caltech time. We retained
those observations in the following analysis.
The ZTF Science Data System (ZSDS) constructed reference
images for each eld and lter by taking the stack-average of
154 0 historical images
26
(Mascietal.2019). Observations for
each target could be covered by multiple elds. To ensure that the
reference image s do not contain con tamination from SN ux, we
need to treat eac h eld (with specic CCD-quadrant therein) for a
given lter separately. Hereafter we use fcqf ID, dened by
()( ) ( )
()()()
+ ´
+
fcqf ID field ID 10000 CCD ID 100
quadrant ID 10 filter ID 1
as an identier of the reference images.
We grouped observations by fcqf ID, and excluded those where
the time of the latest exposure used to create the reference product
was within 25 days of
t
B,max
(in the SN rest frame).Sincethe
typical rise time of SNe Ia is 17 days (Firth et al. 2015),25daysis
a conservative choice. For each target, we further required that the
remaining number of observations in both g and r must be no less
than 35. 154 SNe met this criterion.
3. Data Analysis
We perform forced point-spread function (PSF) photometry
to extract precise ux measurements of the SN in all ZTF images,
including those that were obtained prior to explosion. Forced PSF
light curves are obtained by measuring the PSF ux at the position
of the SN in all epochs. Figure 1 demonstrates the difference
between the light curves generated by the ZTF alert packets and
forced-PSF photometry. The improvement is clear as highlighted
23
For the purpose of this paper, moving objects are ignored.
24
A measurement of the detection efciency and completeness of this sample
is beyond the scope of this study and will be addressed in a future paper
(J. Nordin et al. 2019, in preparation).
25
https://sncosmo.readthedocs.io/en/v2.0.x/models.html
26
The ZTF camera has 16 CCDs, with each CCD divided into four quadrants.
2
The Astrophysical Journal, 886:152 (22pp), 2019 December 1 Yao et al.

by the dotted box, where the forced photometry recovers
detections that are otherwise missed by the real-time pipeline.
It is also the case that the forced photometry provides deeper
pre-explosion upper limits. Thus, forced-PSF photometry can
(i) provide sub-threshold ux measurements, (ii) reveal structure
in the early-time light curves, and (iii) allow more stringent
constraints to be placed on the epoch of explosion.
3.1. Astrometry
The position of ZTF transients reported on the GROWTH
Marshal is based on the initial detection of the source, which
is often at low signal-to-noise ratio (S/N).Astherst step of
forced-PSF photo metry, we need to determine the positio n of the
transient more accurately. To this end, for each SN, we obtained
the coordinates in all epochs where the SN is detected using
Kowalski,
27
a ZTF database system. The typical scatter in both
R.A. and decl. is 0
088 (0.09 pixel size). This uncertainty is
small, and thus we do not incorporate it into the PSF modeling.
We took the median R.A. and decl. as the true position of each SN.
3.2. Data Description
The pixel scale of the ZTF camera is 1
012 pixel
1
. The
typical seeing-limited FWHM of the PSF is 2 pixel
1
.
Figure 2 shows an example of the point-source cutouts of a
difference image and its normalized PSF-model template. The
PSF is normalized such that the sum of all pixel response
values is equal to 1. These images are available at the NASA/
IPAC Infrared Science Archive (IRSA).
28
The difference image PSF is a product of the ZOGY image
subtraction algorithm (Zackay et al. 2016). ZOGY generates
this by combining the input PSF templates from the science and
reference images prior to subtraction. The science and reference
image PSF templates were generated using an automated
version of the classic DAOPhot/AllStar software (Stet-
son 1987), with further optimizations for ZTF (Masci et al.
2019; Sections 3.5 and 4). A linearly spatially varying PSF
model consisting of a Gaussian core modulated by corrections
is t to a set of pre-ltered (uncontaminated and unsaturated)
stars in the science and reference images separately. Only PSF
estimates at the center of the science and reference CCD-
quadrants are used for input to ZOGY. Prior to use in ZOGY,
the PSF templates are further regularized to suppress pixel
outliers in their outer regions. ZOGY then combines the PSFs
using a Fourier inversion method to generate a single PSF
template for the difference image. This single PSF therefore
represents an effective PSF for the entire quadrant image. Its
spatial variation on quadrant scales (inherent in the science and
reference images) is <1%. This is not signicant enough to
impact the accuracy of our PSF-t photometry in our
magnitude range of interest (17 mag), where measurements
are dominated by sky background noise.
Hereafter we denote the pixel values of model image and
difference image by x
i
and
¢
y
i
, respectively. x is unitless and y has
the unit of detector data number (DN), which is analogous to
analog digital units (ADU). We estimate the background noise
σ
bkg
(in the unit of DN) from all pixels inside an annulus centered
at the location of the target, with an inner radius r
in
=10 pixels
and an outer radius r
out
=15 pixels (indicated by the dashed
Table 1
Steps in Sample Selection
Step Criteria Total # SNe
1 Spectroscopically classied SNe Ia observed by the ZTF partnership survey 336
2 Observed by the high-cadence elds 247
3 At least one detection on the Marshal light curve earlier than 5 days prior to
t
B,max
191
4 Remove observations where the reference images are obtained after t
B,max
25 days 154
Extract forced PSF photometry light curves
5 Before
t
B,max
, the target must be detected in both g and r over at least ve nights 140
6 The rst 3σ detection in both g and r must be earlier than t
B,max
10(1+z) 129
7 Must be detected at least once in both g and r in [t
B,max
, t
B,max
+20(1+z)] 127
Figure 1. Upper panel: the r-band light curve of ZTF18abxxssh generated by
the alert distribution system. Bottom panel: forced-PSF photometry light curve
of the same object. The dotted box highlights additional early-time r-band
detections recovered by forced photometry.
27
https://github.com/dmitryduev/kowalski
28
https://irsa.ipac.caltech.edu
3
The Astrophysical Journal, 886:152 (22pp), 2019 December 1 Yao et al.

green circles in Figure 2). Thus, σ
bkg
=0.5×[(the 84th
percentile of
¢
y
bkg
) (the 16th percentile of
¢
y
bkg
)]. Ideally the
median of all pixels in this background annulus should be around
zero, i.e., median(
)¢»y 0
bkg
, assuming that image subtraction is
perfect. However, it was found that the background level is
sometimes far from zero in regions close to the center of galaxies.
Therefore, we subtracted the local background from y to get a
more robust estimate of the excess ux relative to background:
(
)
- ¢yy ymedian
i
i bkg
.They thus derived also has units
of DN.
3.3. The PSF Fitting Method
In ZSDS, image subtraction is performed using the ZOGY
algorithm (Zackay et al. 2016). The PSF template image (left
panel of Figure 2) is generated such that the difference image
(right panel of Figure 2) can be modeled by the PSF image
multiplied by a number m, plus some random noise ò, i.e.,
y=mx+ò. Here, m is the PSF-t ux in the unit of DN, and ò
is a noise term:
(
)
s~
0,
2
. The statistical pixel uncertainty
for y
i
is
()ss=+
y
gain
2
i
i2
bkg
2
where the gain is the electronic detector-gain (in the unit of
electron per DN).
Although our task is simply to t a straight line to a set of (x
i
,
y
i
) pairs, there is no consensus on how to derive the best
measurement of m (see Hogg et al. 2010 or Sharma 2017
(Section 2) for a recipe on this problem). The commonly
adopted maximum likelihood estimate has the advantage of
being fast, but is only optimal for the background-dominated-
noise limit (Zackay et al. 2016). In principle we expect
measurements of intra-night observations to be consistent with
each other, but we found that our initially adopted maximum
likelihood method did not provide such a result. Instead, a
Bayesian method was attempted whereby we implemented a
Markov chain Monte Carlo (MCMC) t, which was found to
give the smallest variance of intra-night observations in the
same band. Therefore, we adopted the MCMC approach, and
utilized emcee, which is an afne-invariant MCMC ensemble
sampler that uses multiple walkers to sample the posterior
probability distribution (Goodman & Weare 2010; Foreman-
Mackey et al. 2013).
Assuming that the uncertainties in Equation (2) are under-
estimated by a constant systematic factor σ
0
, the probability of
y
i
given (x
i
, σ
i
, m, σ
0
) is
(∣ )
()
()
()
()
ss
ps s
ss
=
+
-
-
+
pym x
ymx
,,,
1
2
exp
2
.3
i
ii
i
i
i
i
0
2
0
2
2
2
0
2
From (3) it follows that the log-likelihood is
()
()
()
å
ps s
ss
=
+
-
-
+
ymx
ln ln
1
2
2
.4
i
N
i
i
i
i
2
0
2
2
0
2
2
We only include the central 7×7 cutout ( indicated by the
dotted red square in Figure 2) in the t, so N=49 is the
number of pixels that were taken into consideration.
The posterior probability distribution function of the model
parameters (m, σ
0
) for each observation can be obtained from
the following equation according to Bayes theorem:
(∣{} )
({ } ) ( ) ( )
ss
ss s=
=
=
pm y x
Z
py m x pm
,,,
1
,,, , 5
i
i
N
ii
i
i
N
ii
0
1
1
00
where Z is a normalization factor and p(m, σ
0
) is the prior.
We adopted wide and at priors: (i) m was uniformly
distributed in the range [10
6
,10
6
]; (ii) σ
0
was logarithmically
uniformly distributed in the range [e
10
, e
10
]. The two-
dimensional parameter space was investigated using 250
walkers. All models were run to convergence as determined
by the evolution of the autocorrelation of the individual
MCMC chains (see https://emcee.readthedocs.io/en/latest/
tutorials/autocorr/). A demonstration of this step is given in
Figure 3.
We obtained the posterior probability distributions for m and
σ
0
as the output from the MCMC tting, and marginalized over
σ
0
to estimate the slope, m. Throughout this paper, we take the
median value of the distribution as the measured ux, f
mcmc
,
whereas the uncertainty on this value,
f
mcmc
, was estimated as
half of the difference between the 84th and 16th percentiles of
Figure 2. An example of the cutouts of the PSF-model image (left, 25 × 25 pixels), the difference image (middle, 31 × 31 pixels), and the residual (right, 25 × 25
pixels) centered on the position of the target. The central 7×7 pixel cutouts are marked by the dotted red squares. The background region is marked by the dashed
green annulus, with inner radius=10 pixels and outer radius=15 pixels. Note that 10 pixels 5 FWHM.
4
The Astrophysical Journal, 886:152 (22pp), 2019 December 1 Yao et al.

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Frequently Asked Questions (11)
Q1. What have the authors contributed in "Ztf early observations of type ia supernovae. i. properties of the 2018 sample" ?

In this paper, the authors present high-quality light curves of 127 SNe Ia discovered by the Zwicky Transient Facility ( ZTF ) in 2018. The authors describe their method to perform forced point-spread function photometry, which can be applied to other types of extragalactic transients. This is the largest sample of young SNe Ia collected to date ; it can be used to study the shape and color evolution of the rising light curves in unprecedented detail. The authors discuss six peculiar events in this sample: one 02cx-like event ZTF18abclfee ( SN 2018crl ), one Ia-CSM SN ZTF18aaykjei ( SN 2018cxk ), and four objects with possible super-Chandrasekhar mass progenitors: ZTF18abhpgje ( SN 2018eul ), ZTF18abdpvnd ( SN 2018dvf ), ZTF18aawpcel ( SN 2018cir ), and ZTF18abddmrf ( SN 2018dsx ). 

For the five overluminous events (one Ia-CSM and four SC(∗)), the time range used in the fit is from −10 to +16 days (in the rest frame) relative to maximum light. 

The fact that at z0.1, the majority of SNe (22/27) have positive values of the light-curve shape parameter (x1) suggests that their sample is biased toward overluminous,slowly declining SNe at higher redshift. 

Ideally the median of all pixels in this background annulus should be around zero, i.e., median( )¢ »y 0bkg , assuming that image subtraction is perfect. 

For others who would like to model these light curves in the future, it is also advised to perform such a baseline validation and uncertainty scaling, or to remove observations associated with cn 42 or ∣ ∣ C 15 from the sample. 

At +81 days, the Hα emission line profiles have a narrow component on top of a broad (FWHM ≈ 1020 km s−1) base, much greater than the Hα FWHM of the host-only spectrum (97 km s−1). 

Despite being used as standardizable candles to study cosmology, the origin of Type Ia supernovae (SNe Ia) is not settled (see review by Maoz et al. 2014). 

The authors note that the correlation between x1 and Δm15(g) is sufficiently strong that x1 can be used as a proxy for the rate of decline of the light curve and thus the peak luminosity (see Figure 15). 

The inconsistency may result from the lack of early 91T-like templates in the SNID database, which suggests that the “normal” typing from early-time spectra (phase <−10 days) of seven events32 in Table 2 may be questionable. 

the authors tentatively classified them as normal SNe Ia (denoted by “normal∗”), and identified those with Mg,max−19.6 as potentially 91T-like events (denoted by “91T-like∗”). 

Assuming that the uncertainties in Equation (2) are underestimated by a constant systematic factor σ0, the probability of yi given (xi, σi, m, σ0) is( ∣ )( )( ) ( ) ( ) ⎛ ⎝⎜ ⎞ ⎠⎟s sp s s s s = + --+p y m xy mx, , ,12 exp2 .