11institutetext:  Department of Physics, Duke University, Durham, NC 27708,
USA 22institutetext: Department of Astronomy and Astrophysics, University of
Chicago, Chicago, IL 60637, USA 33institutetext: Kavli Institute for
Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
44institutetext: Lawrence Livermore National Laboratory, 7000 East Ave.,
Livermore, CA 94550-9234, USA 55institutetext: Stockholm University,
Universitetsvägen 10, 114 18 Stockholm, Sweden 66institutetext: SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA, 94025, USA
77institutetext: Kavli Institute for Particle Astrophysics and Cosmology,
Stanford University, Stanford CA 94305, USA 88institutetext: Laboratoire
Univers et Particules de Montpellier, Place Eugène Bataillon - CC 72, F-34095
Montpellier Cedex 05, France 99institutetext: Centre de Physique des
Particules de Marseille, 163, avenue de Luminy Case 902 13288 Marseille cedex
09, France 1010institutetext: LPC, IN2P3/CNRS, Université Clermont Auvergne,
F-63000 Clermont-Ferrand, France 1111institutetext: HEP Division, Argonne
National Laboratory, 9700 S Cass Ave, Lemont, IL 60439, USA 1212institutetext:
University of Toronto, 27 King’s College Cir, Toronto, ON M5S, Canada
1313institutetext: Department of Physics and Astronomy, Rutgers University,
Piscataway, NJ 08854, USA 1414institutetext: Department of Physics and
Astronomy, University of Pittsburgh, 4200 Fifth Ave, Pittsburgh, PA 15260, USA
1515institutetext: Pittsburgh Particle Physics, Astrophysics, and Cosmology
Center (PITT PACC). 1616institutetext: Department of Astronomy, University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA 1717institutetext: Center
for AstroPhysical Surveys (CAPS), National Center for Supercomputing
Applications (NCSA), University of Illinois at Urbana-Champaign, Urbana IL
61801, USA 1818institutetext: School of Physics and Astronomy, University of
Southampton, University Rd, Southampton SO17 1BJ, United Kingdom.

# SNIa-Cosmology Analysis Results from Simulated LSST Images: from Difference
Imaging to Constraints on Dark Energy

B. Sánchez 11 R. Kessler 2233 D. Scolnic 11 B. Armstrong 44 R. Biswas 55 J.
Bogart 66 J. Chiang 66 7 7 J. Cohen-Tanugi 88 10 10 D. Fouchez 99 Ph. Gris
1010 K. Heitmann 1111 R. Hložek 1212 S. Jha 1313 H. Kelly 66 7 7 S. Liu 1414
15 15 G. Narayan 1616 17 17 B. Racine 99 E. Rykoff M. Sullivan 661818 C.
Walter 11 M. Wood-Vasey  The LSST Dark Energy Science Collaboration (DESC)
1414 15 15

(Received - ; accepted - )

The Vera Rubin Observatory Legacy Survey of Space and Time (LSST) is expected
to process ${\sim}10^{6}$ transient detections per night. To use these
transients for precision measurements of cosmological parameters and rates
studies, it is critical to understand the detection efficiency, magnitude
limits, artifact contamination levels, and biases in the selection and
photometry. Here we rigorously test the LSST Difference Image Analysis (DIA)
pipeline using simulated images from the Rubin Observatory LSST Dark Energy
Science Collaboration (DESC) Data Challenge (DC2) simulation for the Wide-
Fast-Deep (WFD) survey area. DC2 is the first large-scale (300 deg2) image
simulation of a transient survey that includes realistic cadence, variable
observing conditions, and CCD image artifacts. We analyze ${\sim}$15 deg2 of
DC2 over a 5-year time-span in which artificial point-sources from Type Ia
Supernovae (SNIa) light curves have been overlaid onto the images. We measure
the detection efficiency as a function of Signal-to-Noise Ratio (SNR) and find
a $50\%$ efficiency at $\rm{SNR}=5.8$ averaged over all bands. The
corresponding magnitude limits for each filter are: $u=23.66$, $g=24.69$,
$r=24.06$, $i=23.45$, $z=22.54$, $y=21.62$ $\mathrm{\,mag}$. The artifact
contamination levels is $\sim 90\%$ of all detections, corresponding to $\sim
1000$ artifacts/deg2n $g$ band, and falling to 300/deg2n $y$ band. The
recovered photometry has biases $<1\%$ for magnitudes $19.5<m<23$. We show
that our DIA performance on simulated images is similar to that of the Dark
Energy Survey difference-imaging pipeline applied to real images. We also
characterize DC2 image properties to produce catalog-level simulations needed
for distance bias corrections. We find good agreement between DC2 data and
simulations for distributions of SNR, redshift, and fitted light-curve
properties. Applying a realistic SNIa-cosmology analysis for redshifts $z<1$,
we recover the input cosmology parameters to within statistical uncertainties.
Finally, we discuss further applications of this dataset and analysis, and we
suggest pipeline improvements before LSST operations begins.

###### Key Words.:

image processing – cosmology – Supernovae

##  1 Introduction

The Vera C. Rubin Observatory Legacy Survey of Space and Time
(LSST111http://www.lsst.org, Ivezić et al., 2019) is expected to soon begin
operations and acquire images over 10 years. This survey will use the Simonyi
Survey Telescope at Rubin Observatory, which is a 8.4m class2226.7m of
effective collecting area telescope with a 3.2 Gigapixel camera, yielding a
9.6 deg2 field of view. The Rubin Observatory LSST Camera design includes
ugrizy filters, and the expected $5\sigma$ $r$ band depth is $>24$ (AB system)
in a single 30 second visit, where each visit is comprised of two 15 second
exposures. The instrument and the survey strategy have been optimized towards
obtaining repeated observation of ${\sim}20,000$ deg2 of the sky over 10
years.

LSST will explore a broad range of research fields in astrophysics (LSST
Science Collaboration et al., 2009); the main science objectives are the study
of solar system dynamics, mapping the Milky Way structure, and probing dark
matter and dark energy. Many of these science goals rely on the discovery of
transient sources, and the expected number of transient detections from all
astrophysical variability sources is ${\sim}10^{6}$ per
night333https://www.lsst.org/scientists/keynumbers (Ivezić et al., 2019;
Ridgway et al., 2014; Graham et al., 2020), an unprecedented rate when
comparing to precursor surveys. Past transient surveys have either focused on
low redshift ($z<0.1$) using shallow/wide area strategies, or higher redshift
using deep/limited area strategies (see Scolnic et al., 2018 for a review).
The unique capabilities of LSST enable survey strategies using wide areas with
deep images. To discover transients and measure their light curves, the LSST
Project has developed Difference Image Analysis (DIA) software components. The
Rubin Observatory LSST Dark Energy Science Collaboration
(DESC444http://lsstdesc.org) used these components to develop an orchestration
software layer called dia_pipe555https://github.com/LSSTDESC/dia_pipe. In this
paper, we make the first evaluation of this pipeline by analyzing simulated
images.

Type Ia supernovae (SNIa) are transient events that are used as cosmological
probes to measure the expansion history of the universe and in particular the
dark energy equation of state $w$ (and its cosmic evolution parameterized by
$w_{a}$; The LSST Dark Energy Science Collaboration (2019), DESC, SRM). LSST
is expected to increase the SNIa sample size by up to a factor of 100 compared
to previous samples (Betoule et al., 2014; Sako et al., 2018; Scolnic et al.,
2018; Jones et al., 2019). Furthermore, the survey will yield the discovery of
SNIa using a single instrument with redshifts up to $z\sim 1.2$. The
requirements on systematic uncertainties from the SNIa-cosmology analysis are
detailed in The LSST Dark Energy Science Collaboration et al. (2018); these
requirements include photometric precision at the few mmag-level and
accurately determined selection biases.

To rigorously test analysis pipelines before the pre-survey commissioning
period, DESC has generated a large and comprehensive set of image simulations
known as Data Challenge 2 (DC2) (Sánchez et al., 2020; LSST Dark Energy
Science Collaboration et al., 2020). Each DC2 image is based on models of the
LSST instrument and expected observing conditions at the summit (sky noise and
Point Spread Function - PSF), along with realistic catalogs of galaxies and
supernovae light-curves. The full DC2 area covers 300 deg2 of the Wide-Fast-
Deep (WFD) survey and includes injected point sources of Type Ia supernovae
with an average cadence of $3$ days ($15$ days in each filter). Using a DC2
subset of 15 deg2, we have processed the raw CCD pixels with DIA to
characterize SNIa transient finding, photometric precision, and selection
effects. In addition, we treat DC2 like real data and perform a cosmology
analysis that includes light-curve fitting, bias-correcting distances, and
fitting for cosmological parameters $w$ and $\Omega_{M}$. This pixel-to-
cosmology test is a critical part of evaluating dia_pipe readiness for survey
operations.

The layout of this work is the following. In Section 2 we explain the DC2
dataset used and in Section 3 we give details on the analysis and the
techniques implemented. Section 4 shows our results on DC2 data processing,
and Section 5 lists performance metric scores obtained for transient detection
as well as cosmology fitting. In Section 6 we discuss our results and compare
them to previously reported analysis. The final discussion and conclusions are
presented in Section 7.

##  2 The DC2 Dataset

The Data Challenge 2 (DC2) is a broad DESC effort to create and process
simulated LSST images based on modelling galaxies and transients in the
universe (LSST Dark Energy Science Collaboration et al., 2020). The simulation
is composed of observations spanning a sky area of 300 deg2 during 5 years of
survey operations. The simulated data includes the expected instrumental
signatures from the LSST Camera as well as the atmospheric effects in all six
optical bands ugrizy. DC2 contains stars, galaxies, and astrophysical effects
such as clustering, cosmic web/structure formation, and gravitational lensing
effects such as cosmic shear. DC2 also includes variable stars, transient
variability from SNe Ia666No other SN types are included, Active Galactic
Nuclei (AGN) galaxies, and strong lensed SNe Ia.

DC2 used the state-of-the-art N-body simulation Outer Rim (Heitmann et al.,
2019). The cosmological parameters used to create Outer Rim are consistent
with WMAP-7 (Komatsu et al., 2011). From this gravity-only simulation, the
cosmoDC2mock catalog (Korytov et al., 2019) is created; it covers 440 deg2 of
sky area up to a redshift of $z=3$. CosmoDC2 contains more than 500 properties
for each galaxy, including stellar and halo mass, shape, Spectral Energy
Density (SED), central black hole parameters, AGN activity, as well as
environment related quantities such as the full gravitational shear and
convergence maps of the sky, which gives the observed shape of each galaxy.

In this work we use Run 2.2i Wide Fast Deep (WFD) images with an average
transient cadence of 3 days between observations. DC2 images were produced
with the image simulation software imSim777https://github.com/LSSTDESC/imSim
that imprints observing conditions and instrumental signatures using a model
of the LSSTCam. The observing conditions, which include sky noise, Point
Spread Function (PSF), zeropoint, and dithering are based on the
minion_1016888http://ls.st/Collection-4604 observing strategy produced with
the operations simulator software
OpSim999https://github.com/lsst/sims_operations. Each object SED is attenuated
from a calculation of Galactic dust extinction and atmospheric effects that
includes differential chromatic refraction. Effects from the CCD readout
electronics are also simulated, including charge repulsion effects and
saturation. Each simulated visit illuminates 189 $4k\times 4k$ CCD detectors
(3 billion pixels) covering almost 10 deg2, with a plate scale of 0.2
$\mathrm{\,arcsec}$/px.

A summary of the DC2 SNIa properties is shown in Table 1. For SNIa, the rest
frame SED is computed with the SALT2 model (Guy et al., 2010; Betoule et al.,
2014). Since the original SED model covers only the $g$ and $r$ bands in the
rest-frame, we use a wavelength-extended model (Pierel et al., 2018) that
covers all of the LSST bands. For imSim to run properly, we include an
additional modification that prevents negative UV spectral fluxes. The
properties of each SNIa in DC2 are determined by the following SALT2
parameters: redshift ($z$), time at peak brightness ($t_{0}$), stretch
($x_{1}$), color ($c$), and amplitude ($x_{0}$). Each redshift is randomly
selected from a volumetric rate, $r_{v}(z)=2.5\times
10^{-5}(1+z)^{1.5}\rm{Mpc}^{3}\rm{yr}^{-1}$ (Dilday et al., 2008). The $t_{0}$
value is randomly selected within the 5 year DC2 time span. Each SNIa includes
an intrinsic scatter drawn from a Gaussian distribution with $\sigma=0.15$
$\mathrm{\,mag}$; a coherent $\mathrm{\,mag}$ fluctuation is applied at all
SNIa phases and wavelengths. The SALT2 parameters $x_{1}$ and $c$ were each
drawn from an asymmetric Gaussian distribution with parameters shown in Table
1. The amplitude $x_{0}$ is computed from the SALT2 parameters and the
luminosity distance. The luminosity parameters $\alpha$ and $\beta$ were set
to $0.137$ and $3.21$ respectively.

Table 1: SNIa properties used in DC2 simulations.

DC2 SNIa property
---
Light curve model | SALT2 Extended + 0.15 mag offset*
| (Pierel et al., 2018)
Rate model | $r_{v}(z)=2.5\times 10^{-5}(1+z)^{1.5}\rm{Mpc}^{3}\rm{yr}^{-1}$
| (Dilday et al., 2008)
Intrinsic scatter |  $\sigma_{\rm int}=0.15$$\mathrm{\,mag}$
Host correlation | None
Stretch population |  $\overline{x}_{1}=0.873$,
|  $\sigma_{+}=1.43$, $\sigma_{-}=0.359$,
| Range$=[-3.0,2.0]$
Color population |  $\overline{c}=-0.048$,
|  $\sigma_{+}=0.043$, $\sigma_{-}=0.097$,
| Range$=[\pm 0.3]$
Luminosity parameters |  $\alpha=0.137$, $\beta=3.21$

  * *

This offset was a mistake in the DC2 generation, and is included in the
simulations for bias corrections.

Correlations between the SN Ia and host-galaxy properties were not included in
the simulations. Supernovae were assigned to a galaxy using an occupation
probability proxy of stellar mass. For host galaxies of SNe Ia, the cosmoDC2
stellar mass distribution peaks at $\sim 10^{11}\rm{M}_{\odot}$, which is
$\sim 5$ dex higher than the typical mass of field galaxies (See Fig. 5 in
(LSST Dark Energy Science Collaboration et al., 2020)). $10\%$ of SNe Ia are
assigned to be ‘hostless’ in order to provide a control sample of isolated
transients.

DC2 images are processed by the LSST pipelines, which organize image data in
sets of tracts and patches for operational purposes. Each 2.56 deg2 tract is a
square containing $7\times 7$ patches that share a common World Coordinate
System (WCS) projection. This sky map organization is used for image-
coaddition grid resampling, database access and image data manipulation, DIA
processing, and template creation. Each DC2 patch contains $4100\times 4100$
pixels (roughly the size of a CCD) with a scale of $0.{\arcsec}2$. To avoid
missing area due to edge effects, patches overlap with their neighbors by 100
pixels and tracts overlap by 1 $\mathrm{\,arcmin}$.

In this work, we select a 15 deg2 area from the DC2 WFD (hereafter called
“DC2-SNIa” area) which includes 1967 LSSTcam visit observations and $105,942$
CCD images. We select SNe Ia with redshifts $z\leq 1.0$, which includes the
full range of cosmologically useful SNe Ia that will be discovered in the WFD
(The LSST Dark Energy Science Collaboration et al., 2018). We discard objects
$<65\arcsec$ from the edges of the DC2-SNIa area in order to avoid subtraction
artifacts from template overlapping issues. DC2-SNIa contains 5884 Type Ia
SNe, and Fig. 1 shows distributions of redshift and SALT2 parameters. Fig. 2
shows the DC2-SNIa sky area used in this work, illustrating  the tracts and
patches used, and the locations of analyzed SNe Ia.

Figure 1: Distribution of redshift and SALT2 model parameters for DC2 Type Ia
Supernovae. The light grey regions (bottom panel) are for events whose peak
brightness occurs outside a season of observations.  Figure 2:  DC2-SNIa sky
area. Small squares indicate patches, and large squares are tracts. Detected
and undetected SNe are shown as red stars and grey crosses, respectively. In
light-blue we show used patches.

##  3 Analysis

Here we describe the DIA LSST pipeline framework, and SNIa-cosmological
analysis.

###  3.1 DIA pipeline framework

The Rubin Observatory Data-Management team has developed a state-of-the-art
set of software tools for CCD data reduction that contains several routines
for image processing, such as image coaddition, flux measurements, etc. This
image processing framework, named “LSST Science
Pipelines”101010https://pipelines.lsst.io, is open source and can be used on
any optical and infra-red survey data set. DESC uses the LSST pipelines system
to remove instrumental signatures from DC2 images, (e.g., electronic readout
bias, dark current, illumination gradients), to calibrate images, and to
obtain a World Coordinate System (WCS) solution.

For transient detection, DESC has developed a specific pipeline package
dia_pipe that uses LSST Science Pipelines’ image processing tools, including
Difference Image Analysis (DIA) routines. The central concept of DIA is to
compare two images of the same sky area taken at different times, and detect
sources that change in brightness. Each image has different properties (such
as PSF, sky noise and zeropoint, etc.) and the subtraction accounts for these
effects. DIA uses a co-added reference image (template) and one recently
observed search image on which we want to find variability. The reference
image is constructed by stacking a subset of archival images taken in
exceptional observing conditions with low sky noise, small PSF size, and high
atmosphere transparency (more details in Sec. 4.1). Since these templates are
built from many individual observations, pixels with artifacts (e.g. moving
object trails, cosmic rays, CCD blooming and bleeding) are rejected during the
co-addition process. In this work, DIA is based on the Alard & Lupton (1998)
(herafter A&L) technique, which uses a kernel to transform the template image
such that its pixel locations, orientation, and PSF match the search image.

The dia_pipe stages are illustrated in the upper panel of Fig. 3 and briefly
described below:

Figure 3: Diagram of dia_pipe and LSST Science Pipelines (upper box). Each
processing step (yellow boxes) takes an image (green cylinders) or catalog
data (gray tables), and produces new images or catalogs. Additionally we show
connections to the cosmology analysis (bottom box).

  * •

Instrumental signature removal and calibration: Simulated images, including
calibration frames (dark current, bias and flat field exposures) are ingested
to perform Instrumental Signature Removal (or ISR) and image calibration,
resulting in a repository of calibrated image exposures called “Calexp’s.”

  * •

Template Coaddition: From the Calexp repository, 40 images are selected from
the first season (Y1) with good seeing and low sky noise. Among these images,
those that overlap each DC2 patch by a significant fraction are co-added to
build templates. Pixel weight was estimated using the inverse variance.

  * •

DIA: The A&L image subtraction algorithm is run on images from seasons 2-5,
which produces difference images. Next, source detection is run on each
difference image to obtain a catalog of DIA single-visit detections, called
diaSource. To avoid artifacts near CCD edges, detections within 16 pixels of
an image edge are discarded; this cut is about ${\times}3$ larger than the
typical PSF-FWHM size. At least one corner of each search image is required to
overlap a template image within $65\arcsec$ of the edge. The DIA kernel basis
is composed of 3 Gaussians, with an adaptive spatially varying size to
accommodate varying PSF sizes. The basis components also vary spatially.

  * •

Association: a candidate-association algorithm creates diaObjects from one or
more diaSources that match spatially within $0\arcsec.5$. This cut radius is
much larger than the average astrometric precision of DC2 calibration (see
Fig. 13 of (LSST Dark Energy Science Collaboration et al., 2020)). As each
diaSource is added to a diaObject, the average RA and Dec coordinate of the
diaObject is updated.

  * •

Forced photometry: For each diaObject, forced PSF photometry is performed at
the location of the object for all overlapping images, regardless of whether
there was a diaSource detection. The collection of forced photometry fluxes
and uncertainties for each diaObject comprises the light-curve used in the
cosmology analysis (Sec. 3.3).

###  3.2 DIA validation

Before using the light curves for the cosmology analysis we perform several
validation checks on the performance of the DIA steps explained in Sec. 3.1.
These validation checks include:

  * •

Template quality (depth and PSF size)

  * •

Efficiency vs. SNR

  * •

Detection depth per band

  * •

Artifact contamination level

  * •

Photometric precision for fluxes

  * •

Photometric uncertainty correlation with Surface Brightness

  * •

Photometric flux-outlier fractions

###  3.3 Cosmology analysis

Here we describe a cosmology analysis that combines the DC2 light-curves,
obtained with the DIA pipeline framework, with a simulated low redshift
(SimLow-z) sample generated with the SNANA simulation. The SimLow-z sample
covers a redshift range of $z<0.08$ with an assumed spectroscopic-selection
efficiency of $100\%$. The “DC2+SimLow-z” analysis includes light curve
fitting to standardize the SNIa brightness, a Monte Carlo simulation to
correct for selection effects, a global fit to produce a bias-corrected Hubble
diagram, and a $w$CDM fit to estimate $w$ and $\Omega_{M}$ (see bottom panel
in Fig. 3). We closely follow the procedures used in the analyses for Pantheon
(Scolnic et al., 2018), PS1 (Rest et al., 2014; Jones et al., 2018), and DES
(Brout et al., 2019).

To the extent possible, we treat DC2 light curve data as real data by not
using underlying truth information. However, there are four caveats where
truth information is used. (1) We do not use DC2 data to train the SALT2
model, nor to measure the true color and stretch populations; instead, we use
the known SALT2 model for light curve fitting, and we use the known SALT2 and
population model for the bias-correction simulations. (2) We select light
curves from diaObjects that match true DC2 SNe Ia, and thus our DC2 sample
corresponds to a spectroscopically confirmed sample without contamination from
other SNe types. (3) We use the true DC2 redshifts, and thus assume accurate
redshifts from either the SN or correctly-matched host-galaxy. (4) To
characterize DIA detection efficiency vs. Signal-to-Noise Ratio (SNR) we use
the same DC2 light curve data as in the analysis; for future LSST analysis of
real data, there will be a separate data stream of fake sources to measure
this DIA property.

The analysis stages described below use programs from the Supernova Analysis
(SNANA: Kessler et al. 2009) software
package111111https://github.com/RickKessler/SNANA:

  * •

Light curve fitting on data: we use the SALT2-Extended (Pierel et al., 2018)
light curve model, the same model used to generate DC2 SNIa, and fit for
$t_{0}$, $x_{0}$, $x_{1}$ and $c$ parameters and their covariances. We impose
the following selection requirements (cuts) based on previous cosmology
analyses:

    * –

at least one detection (Sec. 5.1) in any passband

    * –

maximum $\rm{SNR}>4$ in at least 3 separate passbands

    * –

fitted $|x_{1}|<3$, and $0<\sigma_{x_{1}}<2$

    * –

fitted $|c|<0.3$

    * –

fitted peak MJD uncertainty $<3$ days

    * –

fit probability ($P_{\rm{fit}}$), computed from $\chi^{2}$ and the number of
degrees of freedom (NDOF), satisfies $P_{\rm{fit}}>0.05$.

    * –

At least one observation before $t_{0}$, and another 10 days after $t_{0}$ in
the rest frame.

    * –

require valid bias correction in BBC (see Hubble Diagram determination below)

In addition to the SN selection cuts, we select observations that satisfy:

    * –

no interpolated pixels near the center

    * –

rest frame time is between $t_{0}-15$ and $t_{0}+45$ days

  * •

Simulation for bias corrections: to prepare for distance bias corrections, we
generate a catalog level simulation as described in Kessler et al. (2019). We
use DC2 DIA data to determine a cadence library (Section 6.1 of Kessler et
al., 2019), detection efficiency vs SNR for each band, and flux uncertainty
corrections. The same cadence library and detection efficiency were used to
simulate both the DC2 and the SimLow-z samples.

  * •

Hubble Diagram: we use “BEAMS with Bias Corrections” (Kessler & Scolnic, 2017,
BBC) to determine a bias corrected Hubble diagram in 9 redshift bins, and to
determine nuisance parameters: stretch-luminosity correlation $\alpha$, color-
luminosity correlation $\beta$, and intrinsic scatter $\sigma_{\rm{int}}$.

  * •

Cosmology fitting: we fit for $w$ and $\Omega_{M}$ using a fast minimization
program that combines the DC2+SimLow-z SN Ia Hubble diagram with a cosmic
microwave background (CMB) prior using the $R(z_{*})$ shift parameter (e.g.,
see Eq. 69 in Komatsu et al., 2009). To avoid bias from a measured prior,
$R(z_{*})$ is computed from the DC2 cosmology parameters. To have CMB
constraining power similar to that of Planck (Ade et al., 2016), we tuned the
uncertainty $\sigma_{R}=0.007$ as follows. We use the publicly available DES
3-year SN Ia Hubble diagram,121212https://des.ncsa.illinois.edu/releases/sn
which resulted in $\sigma_{w}({\rm stat})=0.042$ when combined with Planck
constraints. We run our fast cosmology fitting program on this DES 3-year
sample, and tune $\sigma_{R}$ to achieve the same $\sigma_{w}(\rm{stat})$.

##  4 Data processing results

We processed DC2 image data using six tracts, which comprises a sky area of 15
deg2 square degrees during 5 observing year-seasons. We used the first year
for template creation (Fig. 4) and the remaining 4 years (hereafter called DIA
seasons) for image subtractions, creating a total of ${\sim}106\rm{k}$ DIA
images in all LSST filters $ugrizy$.

###  4.1 Template creation

We show the distributions of the number of images used per template in Fig. 5
in a boxplot format. The distributions peak near 40, and they all have a tail
extending down to ${\sim}15$ images due to insufficient overlap with the
corresponding patch area.

Figure 4: Distribution of observation dates (MJD) from DC2 images spanning 5
seasons. The first season (shaded) is used for templates. Figure 5:
Distribution boxplots of the number of CCD sensors used per template image
(one template per patch, of roughly 14 arc minutes a side). Boxes and whiskers
represent $1\sigma$ and $3\sigma$ width of distribution, with center line
indicating the mean value. Outliers are marked with open circles.

For the template and DIA season images, photometric properties of PSF and
$5\sigma$ limiting magnitude depth ($m_{5\sigma}$) are listed in Table 2.

Set | Property | Statistic | $u$ | $g$ | $r$ | $i$ | $z$ | $y$
---|---|---|---|---|---|---|---|---
Template | $m_{5\sigma}$ | mean | $24.94$ | $26.24$ | $25.69$ | $25.03$ | $24.25$ | $23.23$
|  | RMS | $0.11$ | $0.14$ | $0.16$ | $0.09$ | $0.14$ | $0.05$
| PSF | mean | $1.03$ | $0.76$ | $0.72$ | $0.71$ | $0.78$ | $1.06$
| [FWHM, arcsec.] | RMS | $0.03$ | $0.06$ | $0.03$ | $0.02$ | $0.03$ | $0.02$
DIA seasons | $m_{5\sigma}$ | mean | $23.89$ | $24.82$ | $24.22$ | $23.59$ | $22.66$ | $21.80$
|  | RMS | $0.18$ | $0.22$ | $0.27$ | $0.31$ | $0.25$ | $0.15$
| PSF | mean | $0.96$ | $0.95$ | $0.90$ | $0.87$ | $0.98$ | $1.21$
| [FWHM, arcsec.] | RMS | $0.14$ | $0.17$ | $0.16$ | $0.15$ | $0.15$ | $0.13$
Table 2: Mean and RMS for properties of images used for template co-addition
and for DIA season images.

We compare $m_{5\sigma}$ for templates and DIA seasons in Fig. 6. The template
depth is ${\sim}1$ $\mathrm{\,mag}$ deeper than DIA season images.

Figure 6: The $m_{5\sigma}$ distribution for templates, and for visits
corresponding to DIA seasons.

Fig. 7 shows that the PSF distribution for templates is generally smaller and
narrower compared to DIA season visits. The exception is $u$-band, where the
template PSF is larger than for DIA season visits because weather fluctuations
caused the  first season PSF distribution to be larger than the other seasons.

Figure 7: PSF size distribution for template coadds (dashed histogram), year 1
(histogram) and DIA season visits (years 2-5, smooth shaded curve), for each
LSST filter. Right y-axis correspond to template distribution. For $grizy$,
the average template seeing is smaller than for the DIA seasons; for $u$ band
the average template seeing is larger.

###  4.2 Image differences

Our final DIA sample contains a total number of $1967$ visits, or $105,942$
individual images. In Fig. 8, we show an example of the DIA process from DC2.

Figure 8:  Example DIA calexp image (top), template image (middle), and
difference image (bottom) from dia_pipe. Red circles show diaSource
detections, white circles show found transients and yellow circles show missed
transients. On top of the circles we include the SNR value of each true
transient point source.

##  5 Performance metric results

Here we characterize the performance of DIA detections, matching, photometric
precision, and level of non-astrophysical detections (subtraction artifacts).
For the cosmology analysis, we define metrics based on data-sim distribution
overlays of properties of the light-curves, $z$-dependent Hubble diagram bias,
and fitted cosmology parameters.

###  5.1 DIA on single detections: diaSources

We crossmatch to the truth catalog in two independent steps. First we use the
diaSource catalog as a reference, and find the closest true SN location on the
image. Next we use the true SN catalog as the reference, and find the closest
diaSource. Finally, we compare these two sets of matches and define a True
Positive Match (TP) if there is mutual agreement in both matches, and their
separation is below $0\arcsec.5$. If a true SN Ia doesn’t match a diaSource we
flag it as a False Negative (FN).

From this procedure, we find a total of 17,719 matches, spanning a wide range
of SNR and true magnitudes in all six filters. For the bias-correction
simulation, we measure the detection efficiency
($\epsilon=\rm{TP}/(\rm{TP}+\rm{FN})$) as a function of SNR, and characterize
this efficiency distribution with $\rm{SNR}_{1/2}$ defined as
$\epsilon(\rm{SNR}_{1/2})=0.5$. To better connect the measured $\epsilon$ to
simulations, we don’t use measured SNR, but instead we compute SNR from the
true flux and the true noise, where the latter is computed from the zero
point, PSF and sky noise (see Eq 11 in Kessler et al. (2019)).

We estimate the value for $\rm{SNR}_{1/2}$ by fitting a sigmoid function,
$\epsilon=(1+e^{-{\rm SNR}})^{-1}$. Fig. 9 shows $\epsilon({\rm SNR})$ for all
filters combined; $\epsilon$ increases with SNR as expected, and the filter-
averaged $\rm{SNR}_{1/2}=5.82$.

Figure 9: Detection efficiency ($\epsilon$) vs. calculated SNR. The color scale shows the number of objects per bin, and the dashed vertical line shows $\rm{SNR}_{1/2}$.  Property | $u$ | $g$ | $r$ | $i$ | $z$ | $y$
---|---|---|---|---|---|---
$\rm{SNR}_{1/2}$ | 5.76 | 5.57 | 5.87 | 5.84 | 5.59 | 5.60
$m_{1/2}$ | 23.66 | 24.69 | 24.06 | 23.45 | 22.54 | 21.62
Table 3: Measured $\rm{SNR}_{1/2}$ and $m_{1/2}$ vs. filter.

We also estimate $\epsilon$ as a function of magnitude for each bandpass as
shown in Fig. 10, which shows a clear correlation with $5\sigma$ limiting
magnitudes (see Fig.6). For each LSST filter, $\rm{SNR}_{1/2}$ and $m_{1/2}$
are listed in Table 3. The $\rm{SNR}_{1/2}$ values are between 5.5 and 6 in
each band. The $m_{1/2}$ values range from 21.6 in $y$ band to $>24$ in the
$g$ and $r$ bands.

Figure 10: Detection efficiency vs. true magnitude for each bandpass. Open
circles indicate $m_{1/2}$.

###  5.2 Artifact contamination level

To quantify the number of detected artifacts from DIA, we first match
diaSource detections to all transients in the DC2 truth catalog, using a
tolerance of $1^{\prime\prime}$. We define artifacts as unmatched diaSources.
Using two bins in PSF seeing size for each filter, Table 4 shows the number of
diaSource detections, number of matches, and number of artifacts. We show the
number of matches for all true astrophysical variables
($N_{\texttt{diaSrc}}{}^{\rm{var}}{}$) and true SNe Ia
($N_{\texttt{diaSrc}}{}^{\rm{SN}}{}$), along with the percentage of the total
number of diaSources detections. We characterize artifacts by computing the
mean and RMS of the density per square degree ($D_{\rm art}$). $D_{\rm art}$
is larger for images with smaller PSF, and may in part be due to increased
depth for smaller PSF. This effect is most pronounced in $u$ band: $D_{\rm
art}{\sim}2000$ for PSF$>1\arcsec$, and increases to ${\sim}4000$ for
PSF$<1\arcsec$. In $g$ band, $D_{\rm art}{\sim}1000$ with a 20% difference
between the PSF bins. $D_{\rm art}$ falls with increasing wavelength, and is
correlated with search depth; in $y$ band, $D_{\rm art}{\sim}300$ and the PSF
difference is $<10\%$. From Table 4, ${\sim}90\%$ of the diaSource detections
are artifacts in the $grizy$ bands; in $u$ band the artifact fraction is 99%
due to template seeing size, which is relatively broader than search image PSF
size. In previous surveys, machine learning methods have significantly reduced
artifacts (Goldstein et al., 2015; Kessler et al., 2015; Mahabal et al.,
2019), and similar methods are under development within LSST.

Table 4: Number of diaSources detections for SN and artifacts, split into two
PSF bins

.  Filter $N_{\rm{CCD}}$* $\rm{PSF}$ $\overline{\rm{PSF}}$
$N_{\texttt{diaSrc}}{}$+ $N_{\texttt{diaSrc}}{}^{\rm{SN}}{}$**
$N_{\texttt{diaSrc}}{}^{\rm{var}}{}$$\dagger$
$N_{\texttt{diaSrc}}{}^{\rm{art}}{}$$\ddagger$ $\overline{D_{\rm art}}$
$\sigma_{D_{\rm art}}$ [arcsec] [$\rm{deg}^{-2}$] $u$ 3901 ¿1.0 1.1 1384171 61
$(<0.01\%)$ 6298 $(0.5\%)$ 1377812 ($99.5\%$) 2190 3092 3902 ¡1.0 0.9 2999820
71 $(<0.01\%)$ 7815 $(0.3\%)$ 2991934 ($99.7\%$) 4189 6620 $g$ 5870 ¿1.0 1.1
411704 692 $(0.17\%)$ 41423 $(10.1\%)$ 369589 ($89.8\%$) 956 372 5870 ¡1.0 0.8
502003 922 $(0.18\%)$ 53411 $(10.6\%)$ 447670 ($89.2\%$) 1207 453 $r$ 13905
¿0.9 1.0 743402 3146 $(0.42\%)$ 76986 $(10.4\%)$ 663276 ($89.2\%$) 725 244
13852 ¡0.9 0.8 823973 3679 $(0.45\%)$ 90830 $(11.0\%)$ 729464 ($88.5\%$) 841
288 $i$ 14346 ¿0.8 1.0 652462 2695 $(0.41\%)$ 70934 $(10.9\%)$ 578835
($88.7\%$) 614 227 13495 ¡0.8 0.7 724098 3472 $(0.48\%)$ 76858 $(10.6\%)$
643768 ($88.9\%$) 736 250 $z$ 7375 ¿1.0 1.0 279164 361 $(0.13\%)$ 30040
$(10.8\%)$ 248764 ($89.1\%$) 497 189 6579 ¡1.0 0.9 292950 491 $(0.17\%)$ 31543
$(10.8\%)$ 260916 ($89.1\%$) 591 227 $y$ 8533 ¿1.2 1.3 177009 66 $(0.04\%)$
20275 $(11.5\%)$ 156668 ($88.5\%$) 293 115 8105 ¡1.2 1.1 177733 68 $(0.04\%)$
22226 $(12.5\%)$ 155439 ($87.5\%$) 312 122

  * *

Number of CCD images

  * **

Number of diaSource detections

  * +

Number of diaSources matching with SNe (and percentage relative to diaSources)

  * $\dagger$

Number of diaSources matching with other variable sources (and percentage
relative to diaSources)

  * $\ddagger$

Number of artifacts (and percentage relative to diaSources)

###  5.3 DIA on multiple detections: diaObjects

We cross-match the diaObject catalog with the true DC2 SNe Ia using a two step
procedure (as explained in Sec. 5.1) with a tolerance radius of $0\arcsec.5$,
finding a total of $2186$ matched SNe. Fig. 2 shows the DC2-SNIa area, and
true SNe that were matched and not matched to a diaObject. We define SN
detection efficiency ($\epsilon_{\rm SN}$) as the probability of associating a
true SNIa with a diaObject. Fig. 11 shows $\epsilon_{\rm SN}$ vs. redshift for
a subsample of true SNe Ia which have been observed more than 5 times, have at
least one observation before $t_{0}$, and have at least one observation after
$t_{0}+10$ days in the rest-frame. Fitting this distribution to a sigmoid
model (as in Sec. 5.1), $\epsilon_{\rm SN}=0.5$ at $z=0.72$.

Figure 11: $\epsilon_{\rm SN}$ vs redshift. The dashed vertical line shows
where $\epsilon_{\rm SN}=0.5$. Gray color scale reflects the number of events
per bin.

###  5.4 DIA photometry: flux measurements

Forced PSF photometry is measured at the diaObject location on all DIA images.
Using the set of diaObjects matched with DC2 SNe, we measure flux and
magnitude residuals. Fig. 12 shows the fractional photometric bias as a
function of true SN magnitude ($m_{\rm true}$), and the RMS in each bin is
illustrated by the $\pm 1\sigma$ envelope. The shaded region shows low-
statistics bins with 20 observations, but only 7 events. While there is a hint
of bias for bright events, note that correlated residuals among observations
from the same event would result in under-estimated uncertainties.

Figure 12: Mean fractional photometric bias, $F/F_{\rm true}-1$, as a function
of true SN magnitude. Error bars show uncertainty on the mean, and solid gray
lines show the standard deviation ($\pm 1\sigma$) in each bin, as well as the
$\pm 1\%$ margin. The shaded area shows bins with less than 20 epochs per bin
(a total of 75 observations); these epochs are all from 7 SNe.

We accurately measure fluxes for $19<m_{\rm true}<23.25$, where the mean
fractional photometric bias values are $<1\%$. For magnitudes $m_{\rm
true}>23.25$ the photometry is biased towards faint values, suggesting a
slight bias in the sky subtraction.

###  5.5 DIA photometry: flux uncertainties

To evaluate the flux uncertainties, we measure the pull distribution in each
band (Fig.13), $(F-F_{\rm true})/\sigma_{F}$, where $F$ is the forced
photometry flux, $\sigma_{F}$ is the uncertainty, and $F_{\rm true}$ is the
true flux. Defining

${\rm RMS}_{\rm pull}\equiv{\rm RMS}[(F_{\rm
true}-F)/\sigma_{F}]\leavevmode\nobreak\ ,$ (1)

we expect ${\rm RMS}_{\rm pull}=1$ if the uncertainties are accurate. We find
that the distributions are nearly Gaussian, but ${\rm RMS}_{\rm pull}>1$. For
$u$-band ${{\rm RMS}_{\rm pull}\sim}1.5$, indicating a significant
underestimate of the flux uncertainties. For the other bands, ${\rm RMS}_{\rm
pull}{\sim}1.1$.

Figure 13: Distribution of Forced photometry pull values as described in the
text. Red curve shows best fit Gaussian model, and black curve is a zero-mean
unit-dispersion Normal distribution for comparison.

For DES, Kessler et al. (2015) reported a “Surface Brightness (SB) anomaly” in
which the true scatter was larger than the reported uncertainties, and this
effect was strongly correlated with SB at the SN location. Here we repeat this
analysis for DC2 and measure the local surface brightness magnitude ($m_{\rm
SB}$) in template coadds at each SN location, using aperture photometry with a
radius of $0^{\prime\prime}.9$. We find that the SB anomaly is present in DC2
simulated images. Fig. 14 shows ${\rm RMS}_{\rm pull}$ vs $m_{\rm SB}$ for
each filter. ${\rm RMS}_{\rm pull}$ is near 1 for faint SB, and increases with
increasing SB for $grizy$ bands. For $u$ band, ${\rm RMS}_{\rm pull}\sim 1.5$
for all $m_{\rm SB}$. ${\rm RMS}_{\rm pull}$ reaches a maximum of $\sim 3$ in
the $g$ and $r$ bands with $m_{\rm SB}\sim 21$ and $m_{\rm SB}\sim 20$
$\mathrm{\,mag}$ respectively. ${\rm RMS}_{\rm pull}$ values are consistent
for SNe of all peak brightness. This effect is not understood and we therefore
apply an empirical scale correction (Fig. 14) to the flux uncertainties.
Comparisons with DES are presented in Sec.6.

Figure 14:  RMS of forced photometry pull as a function of $m_{\rm SB}$ at the
location of each SN. Dashed line shows the unit dispersion reference. The
distribution of $m_{\rm SB}$ is also shown for each band.

We show in Table 5 the robust mean and ${\rm RMS}_{\rm pull}$ for each filter
flux pull distribution, as well as the percent of $5\sigma$ and $10\sigma$
outliers. The $u$ band outlier fraction (${\sim}1\%$) is roughly an order of
magnitude higher than in the other bands.

Table 5: Flux pull distribution parameters

Filter | N |  $\overline{x}$a | ${\rm RMS}_{\rm pull}$ |  $f_{5\sigma}$ (%)b |  $f_{10\sigma}$ (%)c
---|---|---|---|---|---
$u$ | 3936 | -0.35 | 1.48 | 4.9 | 2.0
$g$ | 8480 | -0.03 | 1.18 | 0.24 | 0.09
$r$ | 22366 | 0.15 | 1.12 | 0.22 | 0.05
$i$ | 22530 | 0.06 | 1.10 | 0.16 | 0.04
$z$ | 10221 | 0.10 | 1.06 | 0.06 | 0.01
$y$ | 11892 | 0.05 | 1.07 | 0.07 | 0.02

  * a

Robust mean of the pull

  * b

Percent of $5\sigma$ outliers.

  * c

Percent of $10\sigma$ outliers.

We found $u$ band bias and ${\rm RMS}_{\rm pull}$ to be anti-correlated with
PSF size; the smallest PSF bin has the largest bias and ${\rm RMS}_{\rm
pull}$.

###  5.6 Cosmology Analysis Results

Following the steps for cosmology analysis described in Sec 3.3 and shown
visually in Fig. 3, we apply the selection requirements and fit light curves
with the SALT2 light curve model; a total of 655 events pass cuts (504 for
DC2, and 151 for SimLow-z). We also create two catalog-level simulations that
have the same DC2/SimLow-z proportion as the data, and undergo the same cuts
and light curve fitting as the data: (1) a DATA-like simulation with 2061
events (1560 for DC2 and 501 for SimLow-z) is used to compare data-sim
distributions and to crosscheck the analysis, and (2) a large ($6.8\times
10^{5}{}$ events) bias-correction simulation is used in BBC. The difference
between the two simulations is that the latter is generated on a $2\times 2$
grid of $\alpha$ and $\beta$ to enable interpolating the bias-correction
during the BBC fit.

Since the SimLow-z sample is generated by the same catalog simulation used for
bias-correction, there is no need to validate this bias-correction. However it
is important to validate the bias-correction for DC2 by comparing several
distributions between the DC2 data and the DATA-like simulation: the observed
magnitude for the brightest flux in each filter (Fig. 15), and SALT2 fit
parameters (Fig. 16). All distributions show excellent agreement except for
brightest $u$-band mag in Fig. 15.

Figure 15:  Distribution of brightest observed $\mathrm{\,mag}$ in each filter
for DC2 data (black circles) and DATA-like simulation (red histogram). Each
simulated distribution is scaled to match the DC2 sample size. The $\chi^{2}$
per degrees of freedom ($\nu$) quantifies the data-sim agreement.  Figure 16:
Distribution of SALT2 fit parameters for DC2 data (black circles) and DATA-
like simulation (red histogram). Each simulated distribution is scaled to
match the DC2 sample size. The $\chi^{2}$ per degrees of freedom ($\nu$)
quantifies the data-sim agreement.

In Fig. 17 we compare the DC2 detection efficiency vs. redshift, as well as
the analysis efficiency vs redshift using the requirements listed in Sec. 3.3.
Fitting a sigmoid function to the detection efficiency curves,
$z_{1/2}=0.730\pm 0.0051$ for DC2 data, and $z_{1/2}=0.730\pm 0.003$ for the
DATA-like simulation.  For the analysis efficiency, $z_{1/2}=0.58\pm 0.02$ for
DC2 data, and $z_{1/2}=0.61\pm 0.01$ for the DATA-like simulation.

Figure 17:  Detection efficiency vs. redshift in DC2 data (black solid curve),
and in DATA-like simulation (red solid). Analysis efficiency vs. redshift for
events satisfying cuts and SALT2 fits in DC2 data (black dashed curve) and in
DATA-like simulation (red dashed curve).

The distance bias correction for SimLow-z averages to zero, with small
${\sim}0.01$ mag fluctuations. For DC2, the distance bias correction vs.
redshift is shown in Fig. 18 for all events (black circles), where the average
bias increases rapidly for $z>0.6$. The subset of blue ($c<0.05$) events,
which are brighter than average, has a smaller bias in the intermediate
redshift range. The fainter subset of red ($c\leq 0.05$) events has a much
larger bias at lower redshifts. Accurate simulations and bias corrections are
essential for the cosmology analysis.

Figure 18:  Average DC2 bias correction vs. redshift for BBC fitted distances.
Black dots are for all events, blue curve is for fitted SALT2 color $c<0.05$,
red curve is for $c>0.05$.

For the full DC2+SimLow-z sample, Fig. 19 shows the bias corrected Hubble
diagram, and the Hubble residuals with respect to the reference DC2 cosmology,
for both DC2+SimLow-z data and DATA-like simulation. Using 9 bins of redshift
we estimate mean Hubble residuals and error on the mean, shown in the lower
panels of Fig. 19; the binned residuals are $<0.04$ $\mathrm{\,mag}$, and
consistent with 0, and no clear trend is seen with redshift.

After applying the bias correction analysis to the DATA-like simulation, we
measure nuisance and cosmological parameters. Defining the bias on $x$ as
$\Delta x\equiv x-x_{\rm true}$, the biases with respect to input values (Tab.
1) are shown in the “DATA-like Sim” row of Table 6, and these biases are
consistent with zero: $\Delta w=-0.002\pm 0.026$ and
$\Delta\Omega_{M}=0.001\pm 0.009$. After this validation, we apply the same
treatment to DC2+SimLow-z data and obtain nuisance and cosmological parameters
(DC2 row in Table 6). We find $\Delta w=-0.032\pm 0.046$ and
$\Delta\Omega_{M}=-0.007\pm 0.013$. The nuisance and cosmological parameter
biases are consistent with zero.

Table 6: Bias corrected parameter estimation for DC2 sample and simulations. |  | From Cosmology Fit | From BBC Fit
---|---|---|---
Data Set | $N_{\rm{Events}}$ | $\Delta w=w-w_{\rm{true}}$ | $\Delta\Omega_{M}=\Omega_{M}-\Omega_{M}^{\rm{true}}$ | $\chi^{2}/\nu$ | $\Delta\alpha=\alpha-\alpha_{\rm{true}}$ | $\Delta\beta=\beta-\beta_{\rm{true}}$ | $\sigma_{\rm int}/\sigma_{\rm int}^{\rm{true}}$
DC2+SimLow-z | 655 | $-0.032\pm 0.046$ | $-0.007\pm 0.013$ | 11/8 | $-0.004\pm 0.010$ | $-0.15\pm 0.14$ | $1.02$
DATA-like Sim | 2061 | $-0.002\pm 0.026$ | $0.001\pm 0.009$ | 12/8 | $0.004\pm 0.005$ | $-0.12\pm 0.06$ | $0.94$
Figure 19: Bias corrected Hubble Diagram for DC2 data (top) and redshift-
binned Hubble residuals for DC2 data and DC2-like simulation (bottom).

##  6 Discussion

Our DIA performance study is similar to that in Kessler et al. (2015) for the
Dark Energy Survey (DES), where they used fake light curves injected onto real
images during DES operations. In Table 7 we compare several difference-image
properties for DC2 and DES. To avoid confusion related to different search
depths, we only compare DES bands with similar DC2 depth in Table 3: DES
$g$-band in their deep fields ($m_{1/2}{\sim}24.5$ mag) and DES $i$-band in
their shallow fields ($m_{1/2}{\sim}23.5$ mag).

For SNR at 50% detection efficiency, the $\rm{SNR}_{1/2}$ values are almost
identical in $g$-band, while the DES $i$-band value is ${\sim}10$% smaller. To
compare the excess scatter on bright galaxies, we evaluate the SB magnitude
where the flux-uncertainty scale factor is 2: $m_{\rm SB}({\rm RMS}_{\rm
pull}{=}2)$. These $m_{\rm SB}$ values are nearly identical in $g$-band (22.1
and 22.0 mag for DC2 and DES, respectively), but differ by 0.7 mag in the
$i$-band (19.8 and 21.5 mag). The $m_{\rm SB}({\rm RMS}_{\rm pull}{=}2)$
values for all DC2 bands are shown in Table 8. The number of detection
artifacts per deg2 is also similar; DC2 is a factor of 2 higher for $g$-band,
and 7% smaller in $i$-band. Finally, the $5\sigma$ flux-outlier fractions for
DC2 are about a factor of 2 smaller compared to DES; the corresponding
$10\sigma$ outlier fractions are much more similar. Assessing the overall
performance, DC2 appears to be slightly better because of the improved $m_{\rm
SB}({\rm RMS}_{\rm pull}{=}2)$ in the $i$-band and the reduced $5\sigma$ flux-
outliers. While this DC2-vs-DES comparison is encouraging for dia_pipe, the
processing of simulated DC2 images may be giving somewhat optimistic results
compared to real data.

The SB anomaly is present in both DES and DC2, but still not understood.
Follow-up to DC2 simulations would enable modifying various atmospheric or
detector effects to trace the origin of the anomaly. Improving DIA is
important for studies of transients near cores of bright galaxies.

A similar DIA efficiency study was done by the Palomar Transient Factory (PTF,
Frohmaier et al., 2017) using $\sim 7$ million artificial point sources
overlaid on PTF images. They characterized their real-time detection
efficiency as a function source magnitude, host galaxy surface brightness, and
various observing conditions. Their efficiencies are mostly parameterized by
surface brightness, and thus cannot be directly compared with our DC2 results
using SNR. Nevertheless, they report a $m_{1/2}\simeq 20.3$ in $R$ band for
PTF 48 inch class instrument. As shown in DES, PTF, and this DC2 analysis, the
detection efficiency has not been analytically modelled and was therefore
determined empirically with fake sources.

DIA performance depends critically on using template images with exceptional
quality, in particular a narrower PSF with respect to search images. We
created templates using Y1 data, and found that poor $u$-band seeing in Y1
(Fig. 7) degraded DIA for $u$ band, where we find a drop in detection
efficiency, as well as biases in photometric flux and uncertainty. The $u$
band filter transmission is much lower than for the other bands, and it is
unlikely to discover SNIa because they are faint in the UV; nonetheless, $u$
band is useful for photometric classifiers to distinguish between SNe Ia and
core collapse SNe.

The level of artifact detections from DIA is consistent with DES (Table 7).
Machine Learning (ML) methodologies are expected to reduce this contamination
significantly, according to results obtained by several collaborations (Brink
et al., 2013; Goldstein et al., 2015; Mahabal et al., 2019; Duev et al.,
2019). We find that the density of artifact detections decreases with
increasing PSF size in the search image. This effect might be caused by the
DIA kernel transformation, which performs better when the search and template
PSF difference becomes larger (Liu et al. in prep.).

Table 7: Difference image properties for DC2 and DES.

|  DC2-$g$ |  DES-$g$* |  DC2-$i$ |  DES-$i$**
---|---|---|---|---
$m_{1/2}$ | 24.7 | 24.5 | 23.5 | 23.5
$\rm{SNR}_{1/2}$ | 5.57 | 5.61 | 5.84 | 5.36
$m_{\rm SB}({\rm RMS}_{\rm pull}{=}2)$ | 22.1 | 22.0 | 19.8 | 21.5
$\overline{D_{\rm art}}$ | 1080 | 520 | 680 | 730
$f_{5\sigma}(\%)$ | 0.24 | 0.49 | 0.16 | 0.25
$f_{10\sigma}(\%)$ | 0.09 | 0.09 | 0.04 | 0.06

  * *

From DES SN Deep fields, to match DC2-$g$ depth.

  * **

From DES SN Shallow fields, to match DC2-$i$ depth.

Table 8: $m_{\rm SB}({\rm RMS}_{\rm pull}{=}2)$a for each DC2 band.

$u_{\rm{SB}}$ | $g_{\rm{SB}}$ | $r_{\rm{SB}}$ | $i_{\rm{SB}}$ | $z_{\rm{SB}}$ | $y_{\rm{SB}}$
---|---|---|---|---|---
–b | 22.1 | 20.5 | 19.8 | 19c | –b

  * a

Surface brightness $[\mathrm{\,mag}/\mathrm{\,arcsec}^{2}]$ where flux-
uncertainty scale is $\sim 2$

  * b

Scale is always $<2$

  * c

Estimated from extrapolation

The DC2 baseline cadence is sub-optimal with respect to the recent
developments in LSST cadence studies (Scolnic et al., 2018; Lochner et al.,
2018, 2021). Repeating this DC2 image simulation and analysis on alternative
cadences is impractical from both a computational and human-effort
perspective. To rigorously evaluate alternative cadences, however, this DC2
analysis demonstrates that the SNANA simulation can rapidly generate light
curve samples that accurately model a full DIA analysis on images. The SNANA
simulation uses meta-data from images and DIA that includes cadence, zero
point, PSF, sky noise, detection efficiency vs. SNR, and flux-uncertainty vs.
SB. A recommended simulation upgrade is to model catastrophic flux-outliers
shown in Table 5.

##  7 Conclusions

In this work, we show results of an integrated Difference Image Analysis
pipeline, built using DESC’s dia_pipe and LSST pipelines, to analyze simulated
images that include SN Ia light curves. Using a light curve catalog compiled
from dia_pipe results, we applied a commonly used SN Ia standardization method
to measure cosmic distances and cosmological parameters. This is the first
time that a survey team has carried out such a pixel-to-cosmology test before
commissioning operations begin. This analysis is an important stepping stone,
enabling monitoring of pipeline performance evolution from survey simulations
to real-time analysis during operations.

We have analyzed 15 deg2 of DC2 WFD images using LSST DESC’s dia_pipe pipeline
framework for difference-imaging and transient discovery. The detection
efficiency is ${\sim}100\%$ for point sources with $\rm SNR\geq 8$, and is
$50\%$ efficient for events with $\rm SNR{\sim}5.8$.  Comparing DC2 and DES in
bands with the same search depth, the difference-image properties are quite
similar (Table 7). To the extent that the simulated DC2 images are realistic,
this comparison shows that dia_pipe is already performing at the level of a
stage III precursor survey that was focused on precision measurements of
cosmological parameters.

We apply a cosmology analysis using a SALT2+BBC framework, resulting in 655 SN
Ia light curves (504 for DC2, 151 for SimLow-z). To correct for distance
biases in BBC, we used SNANA to generate a DATA-like simulation of SN Ia light
curves using measured DC2 image properties (PSF, zero point, sky noise) and
measured DIA properties (efficiency vs. SNR, flux-uncertainty scale vs. SB).
Both the DC2 and DATA-like simulated samples were used to measure $w$ and
$\Omega_{M}$ from a bias-corrected Hubble diagram; in both cases we recovered
the true cosmological and nuisance parameter values within statistical
uncertainties.

We emphasize that the pipeline system is still in active development and may
improve by the time LSST starts operations. The pre-commissioning analysis of
DC2 is a central contribution for operational readiness.

###### Acknowledgements.

Author contributions are listed below.
B. Sánchez: Lead design, performed DIA, writing.
R. Kessler: Co-lead project, SNANA simulations and analysis, writing.
D. Scolnic: Lead design and writing.
B. Armstrong: Main dia_pipe software developer; draft reviewer.
R. Biswas: DC2 simulation; analysis and data preparation.
J. Bogart: DC2 database storage, access and curation.
J. Chiang: Computing infrastructure, DC2 data access and analysis.
J. Cohen-Tanugi: DC2 image processing and data retrieval.
D. Fouchez: Internal reviewer; suggested paper text and figures edits.
P. Gris: SNWG convener; SNIa discussions.
K. Heitmann: DC2 simulations, paper comments.
R. Hložek: Early discussions, DC2 analysis software.
S. Jha: Internal reviewer; suggested edits to paper text and figures.
H. Kelly: Computing infrastructure; data processing.
S. Liu: DIA analysis tests and debugging.
G. Narayan: DIA analysis discussion, SNIa discussions
B. Racine: DIA analysis discussion; data visualization.
E. Rykoff: DC2 metadata extraction.
M. Sullivan: TD Working Group convener.
C. Walter: DIA discussion and comments.
M. Wood-Vasey: Consulted on several DIA aspects; DESC builder.

B.S and D.S. are supported by DOE grant DE-SC0010007. D.S. is also supported
by DOE grant DE-SC0021962 and the David and Lucile Packard Foundation. The
DESC acknowledges ongoing support from the Institut National de Physique
Nucléaire et de Physique des Particules in France; the Science & Technology
Facilities Council in the United Kingdom; and the Department of Energy, the
National Science Foundation, and the LSST Corporation in the United States.
DESC uses resources of the IN2P3 Computing Center (CC-IN2P3–Lyon/Villeurbanne
- France) funded by the Centre National de la Recherche Scientifique; the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded
by UK BIS National E-infrastructure capital grants; and the UK particle
physics grid, supported by the GridPP Collaboration. This work was performed
in part under DOE Contract DE-AC02-76SF00515.
This work was completed in part with resources provided by the University of
Chicago’s Research Computing Center. This work has gone through DESC internal
review process, and main authors would like to explicitly thank Saurabh Jha,
Dominique Fouchez, and Bob Armstrong for their comments.
This research has made use of the following Python software packages: Astropy
(Astropy Collaboration et al., 2013, 2018), Matplotlib (Hunter, 2007), Pandas
(McKinney et al., 2010), NumPy (van der Walt et al., 2011), Seaborn (Waskom et
al., 2014), SciPy (Virtanen et al., 2020).

## References

  * Ade et al. (2016) Ade, P. A., Aghanim, N., Arnaud, M., et al. 2016, Astronomy & Astrophysics, 594, A13
  * Alard & Lupton (1998) Alard, C. & Lupton, R. H. 1998, ApJ, 503, 325
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Betoule et al. (2014) Betoule, M., Kessler, R., Guy, J., et al. 2014, A&A, 568, A22
  * Brink et al. (2013) Brink, H., Richards, J. W., Poznanski, D., et al. 2013, MNRAS, 435, 1047
  * Brout et al. (2019) Brout, D., Scolnic, D., Kessler, R., et al. 2019, The Astrophysical Journal, 874, 150
  * Dilday et al. (2008) Dilday, B., Kessler, R., Frieman, J. A., et al. 2008, The Astrophysical Journal, 682, 262
  * Duev et al. (2019) Duev, D. A., Mahabal, A., Masci, F. J., et al. 2019, Monthly Notices of the Royal Astronomical Society, 489, 3582
  * Frohmaier et al. (2017) Frohmaier, C., Sullivan, M., Nugent, P. E., Goldstein, D. A., & DeRose, J. 2017, ApJS, 230, 4
  * Goldstein et al. (2015) Goldstein, D. A., D’Andrea, C. B., Fischer, J. A., et al. 2015, The Astronomical Journal, 150, 82
  * Graham et al. (2020) Graham, M. L., Bellm, E., Guy, L., & Slater, C. T. 2020, LSST Alerts : Key Numbers
  * Guy et al. (2010) Guy, J., Sullivan, M., Conley, A., et al. 2010, Astronomy & Astrophysics, 523, A7
  * Heitmann et al. (2019) Heitmann, K., Finkel, H., Pope, A., et al. 2019, The outer rim simulation: A path to many-core supercomputers
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, The Astrophysical Journal, 873, 111
  * Jones et al. (2019) Jones, D. O., Scolnic, D. M., Foley, R. J., et al. 2019, ApJ, 881, 19
  * Jones et al. (2018) Jones, D. O., Scolnic, D. M., Riess, A. G., et al. 2018, ApJ, 857, 51
  * Kessler et al. (2019) Kessler, R., Brout, D., D’Andrea, C. B., et al. 2019, Mon. Not. R. Astron. Soc., 485, 1171
  * Kessler et al. (2015) Kessler, R., Marriner, J., Childress, M., et al. 2015, The Astronomical Journal, 150, 172
  * Kessler & Scolnic (2017) Kessler, R. & Scolnic, D. 2017, Astrophys. J., 836, 56
  * Komatsu et al. (2009) Komatsu, E., Dunkley, J., Nolta, M. R., et al. 2009, ApJS, 180, 330
  * Komatsu et al. (2011) Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18
  * Korytov et al. (2019) Korytov, D., Hearin, A., Kovacs, E., et al. 2019, ApJS, 245, 26
  * Lochner et al. (2021) Lochner, M., Scolnic, D., Almoubayyed, H., et al. 2021, arXiv e-prints, arXiv:2104.05676
  * Lochner et al. (2018) Lochner, M., Scolnic, D. M., Awan, H., et al. 2018, arXiv e-prints, arXiv:1812.00515
  * LSST Dark Energy Science Collaboration et al. (2020) LSST Dark Energy Science Collaboration, Abolfathi, B., Alonso, D., et al. 2020, 13 KEVIN REIL, 12, 13
  * LSST Science Collaboration et al. (2009) LSST Science Collaboration, Abell, P. A., Allison, J., et al. 2009, LSST Science Book, Version 2.0
  * Mahabal et al. (2019) Mahabal, A., Rebbapragada, U., Walters, R., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 038002
  * McKinney et al. (2010) McKinney, W. et al. 2010, in Proceedings of the 9th Python in Science Conference, Vol. 445, Austin, TX, 51–56
  * Pierel et al. (2018) Pierel, J. D. R., Rodney, S., Avelino, A., et al. 2018, PASP, 130, 114504
  * Rest et al. (2014) Rest, A., Scolnic, D., Foley, R. J., et al. 2014, ApJ, 795, 44
  * Ridgway et al. (2014) Ridgway, S. T., Matheson, T., Mighell, K. J., Olsen, K. A., & Howell, S. B. 2014, Astrophysical Journal, 796, 53
  * Sako et al. (2018) Sako, M., Bassett, B., Becker, A. C., et al. 2018, Publications of the Astronomical Society of the Pacific, 130, 064002
  * Sánchez et al. (2020) Sánchez, F. J., Walter, C. W., Awan, H., et al. 2020, arXiv e-prints, arXiv:2001.00941
  * Scolnic et al. (2018) Scolnic, D., Kessler, R., Brout, D., et al. 2018, ApJ, 852, L3
  * Scolnic et al. (2018) Scolnic, D. M., Jones, D. O., Rest, A., et al. 2018, The Astrophysical Journal, 859, 101
  * Scolnic et al. (2018) Scolnic, D. M., Lochner, M., Gris, P., et al. 2018, arXiv e-prints, arXiv:1812.00516
  * The LSST Dark Energy Science Collaboration et al. (2018) The LSST Dark Energy Science Collaboration, Mandelbaum, R., Eifler, T., et al. 2018, arXiv e-prints, arXiv:1809.01669
  * The LSST Dark Energy Science Collaboration (2019) (DESC) The LSST Dark Energy Science Collaboration (DESC). 2019, LSST DESC Science Roadmap
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * Waskom et al. (2014) Waskom, M., Botvinnik, O., Hobson, P., et al. 2014, seaborn: v0.5.0 (November 2014)