# The Dark Energy Survey Supernova Program: Cosmological Biases from Host
Galaxy Mismatch of Type Ia Supernovae

H. Qu Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA  M. Sako Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, PA 19104, USA  M. Vincenzi
Department of Physics, Duke University Durham, NC 27708, USA  C. Sánchez
Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona, Spain  Institut de Física d’Altes Energies (IFAE), The Barcelona
Institute of Science and Technology, Campus UAB, 08193 Bellaterra, Barcelona,
Spain  D. Brout Department of Astronomy, Boston University, 725 Commonwealth
Ave., Boston, MA 02215, USA  R. Kessler Department of Astronomy and
Astrophysics, University of Chicago, Chicago, IL 60637, USA  Kavli Institute
for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA  R.
Chen  Department of Physics, Duke University Durham, NC 27708, USA  L. Galbany
Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain  L. Kelsey  Institute of Cosmology and
Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK  School of
Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
J. Lee Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA  C. Lidman Centre for Gravitational Astrophysics,
College of Science, The Australian National University, ACT 2601, Australia
The Research School of Astronomy and Astrophysics, Australian National
University, ACT 2601, Australia  B. Popovic Department of Physics, Duke
University Durham, NC 27708, USA  B. Rose Department of Physics, Duke
University Durham, NC 27708, USA  M. Smith Univ Lyon, Univ Claude Bernard Lyon
1, CNRS, IP2I Lyon / IN2P3, IMR 5822, F-69622 Villeurbanne, France  M.
Sullivan School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK  P. Wiseman  School of Physics and Astronomy,
University of Southampton, Southampton, SO17 1BJ, UK  T. M. C. Abbott  Cerro
Tololo Inter-American Observatory, NSF’s National Optical-Infrared Astronomy
Research Laboratory, Casilla 603, La Serena, Chile  M. Aguena  Laboratório
Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de
Janeiro, RJ - 20921-400, Brazil  O. Alves  Department of Physics, University
of Michigan, Ann Arbor, MI 48109, USA  D. Bacon  Institute of Cosmology and
Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK  E. Bertin
CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique
de Paris, F-75014, Paris, France  D. Brooks Department of Physics & Astronomy,
University College London, Gower Street, London, WC1E 6BT, UK  D. L. Burke
Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  A. Carnero Rosell Instituto de
Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain  Laboratório
Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de
Janeiro, RJ - 20921-400, Brazil  Universidad de La Laguna, Dpto. Astrofísica,
E-38206 La Laguna, Tenerife, Spain  J. Carretero Institut de Física d’Altes
Energies (IFAE), The Barcelona Institute of Science and Technology, Campus
UAB, 08193 Bellaterra (Barcelona) Spain  L. N. da Costa  Laboratório
Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de
Janeiro, RJ - 20921-400, Brazil  M. E. S. Pereira  Hamburger Sternwarte,
Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany  H. T. Diehl
Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA
P. Doel  Department of Physics & Astronomy, University College London, Gower
Street, London, WC1E 6BT, UK  S. Everett  Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109,
USA  I. Ferrero  Institute of Theoretical Astrophysics, University of Oslo.
P.O. Box 1029 Blindern, NO-0315 Oslo, Norway  J. Frieman Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA  Kavli Institute
for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA  J.
García-Bellido Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de
Madrid, 28049 Madrid, Spain  G. Giannini Institut de Física d’Altes Energies
(IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193
Bellaterra (Barcelona) Spain  D. Gruen University Observatory, Faculty of
Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany
R. A. Gruendl  Center for Astrophysical Surveys, National Center for
Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W.
Green Street, Urbana, IL 61801, USA  G. Gutierrez Fermi National Accelerator
Laboratory, P. O. Box 500, Batavia, IL 60510, USA  S. R. Hinton  School of
Mathematics and Physics, University of Queensland, Brisbane, QLD 4072,
Australia  D. L. Hollowood  Santa Cruz Institute for Particle Physics, Santa
Cruz, CA 95064, USA  K. Honscheid Center for Cosmology and Astro-Particle
Physics, The Ohio State University, Columbus, OH 43210, USA  Department of
Physics, The Ohio State University, Columbus, OH 43210, USA  D. J. James
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  K. Kuehn Australian Astronomical Optics, Macquarie
University, North Ryde, NSW 2113, Australia  Lowell Observatory, 1400 Mars
Hill Rd, Flagstaff, AZ 86001, USA  O. Lahav Department of Physics & Astronomy,
University College London, Gower Street, London, WC1E 6BT, UK  J. L. Marshall
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, and Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA  J. Mena-Fernández Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain  F.
Menanteau Center for Astrophysical Surveys, National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA  Department of
Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street,
Urbana, IL 61801, USA  R. Miquel Institució Catalana de Recerca i Estudis
Avançats, E-08010 Barcelona, Spain  Institut de Física d’Altes Energies
(IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193
Bellaterra (Barcelona) Spain  R. L. C. Ogando Observatório Nacional, Rua Gal.
José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil  A. Palmese
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania
15312, USA  A. Pieres Laboratório Interinstitucional de e-Astronomia - LIneA,
Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil  A. A. Plazas Malagón Department of Astrophysical Sciences,
Princeton University, Peyton Hall, Princeton, NJ 08544, USA  M. Raveri
Department of Physics, University of Genova and INFN, Via Dodecaneso 33,
16146, Genova, Italy  E. Sanchez Centro de Investigaciones Energéticas,
Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain  I. Sevilla-Noarbe
Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain  M. Soares-Santos Department of Physics, University of
Michigan, Ann Arbor, MI 48109, USA  E. Suchyta Computer Science and
Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831  G.
Tarle Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
N. Weaverdyck  Department of Physics, University of Michigan, Ann Arbor, MI
48109, USA  Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley,
CA 94720, USA Helen Qu<EMAIL_ADDRESS>
###### Abstract

Redshift measurements, primarily obtained from host galaxies, are essential
for inferring cosmological parameters from type Ia supernovae (SNe Ia).
Matching SNe to host galaxies using images is non-trivial, resulting in a
subset of SNe with mismatched hosts and thus incorrect redshifts. We evaluate
the host galaxy mismatch rate and resulting biases on cosmological parameters
from simulations modeled after the Dark Energy Survey 5-Year (DES-SN5YR)
photometric sample. For both DES-SN5YR data and simulations, we employ the
directional light radius method for host galaxy matching. In our SN Ia
simulations, we find that 1.7% of SNe are matched to the wrong host galaxy,
with redshift difference between the true and matched host of up to 0.6. Using
our analysis pipeline, we determine the shift in the dark energy equation of
state parameter ($\Delta w$) due to including SNe with incorrect host galaxy
matches. For SN Ia-only simulations, we find $\Delta w=0.0013\pm 0.0026$ with
constraints from the cosmic microwave background (CMB). Including core-
collapse SNe and peculiar SNe Ia in the simulation, we find that $\Delta w$
ranges from 0.0009 to 0.0032 depending on the photometric classifier used.
This bias is an order of magnitude smaller than the expected total uncertainty
on $w$ from the DES-SN5YR sample of $\sim 0.03$. We conclude that the bias on
$w$ from host galaxy mismatch is much smaller than the uncertainties expected
from the DES-SN5YR sample, but we encourage further studies to reduce this
bias through better host-matching algorithms or selection cuts.

##  1 Introduction

Type Ia supernovae (SNe Ia) enabled the discovery of accelerating cosmic
expansion (Perlmutter et al., 1999; Riess et al., 1998) and have since been an
important probe of the dark energy thought to cause it. To constrain
cosmology, each SN Ia must have an accurate estimate of cosmological redshift
as well as physical distance. Distance estimates are obtained from the
standardized luminosities of SNe Ia, earning them the title of standardizable
candles, but redshift is difficult to determine without spectroscopy. The
depth of modern photometric surveys has resulted in orders of magnitude more
photometrically observed SNe than can be followed up spectroscopically. Thus,
the vast majority of SNe Ia used for estimation of cosmological parameters are
assigned the redshift of their matched host galaxies. Moreover, galaxy
redshifts are more precise than redshifts measured from SN spectroscopy due to
the broadened features and phase-dependent nature of SN spectra.

This work investigates the impact of mismatched host galaxies and the
resulting incorrect SN redshifts on the measurement of cosmological
parameters. Using images alone, host galaxy matching is a nontrivial problem
when there are multiple galaxies near the SN location. Two-dimensional images
have little distance information, and galaxies in a crowded field can be
difficult to disentangle without distance measurements to each. In addition,
the large scale structure of the universe dictates that many galaxies occur in
pairs, groups, or clusters, making it difficult to determine which galaxy is
the host. Finally, some “hostless” SNe explode in extremely faint or distant
galaxies that fall below the threshold of detection, even in the deep coadded
images created for this work described in Section 3.2. These SNe could be
incorrectly matched to brighter, nearby galaxies that are close in projected
distance and thus assigned an incorrect redshift. Figure 1 illustrates a
challenging example of host galaxy matching, where the larger and more likely
host galaxy is further in terms of SN-galaxy separation than the smaller
galaxy on the right.

Though automated methods for host galaxy matching have been in use since the
SuperNova Legacy Survey (SNLS) analysis (Sullivan et al., 2006, S06), accurate
characterization of systematic uncertainties such as the effect of mismatched
host galaxies has become more pressing with the advent of wider and deeper SN
surveys. Future surveys such as the Legacy Survey of Space and Time (LSST) at
the Vera Rubin Observatory will observe hundreds of thousands more SNe in the
coming decade (LSST Science Collaboration et al., 2009), further shrinking
statistical uncertainties and necessitating accurate measurements of
systematic uncertainties. This work represents the first thorough exploration
of systematics related to the host galaxy mismatch problem and its effect on
cosmological parameter estimates as part of an ongoing cosmology analysis.

A commonly used method for matching SNe with a host galaxy is the directional
light radius (DLR) method, initially developed to characterize host properties
for the SNLS survey by S06 and tested extensively on simulations by Gupta et
al. (2016). More details about the DLR method can be found in Section 2.1.
Recently, Popovic et al. (2020, P20) performed the first retrospective
estimate of cosmological biases resulting from host galaxy mismatches by
applying the DLR method on simulations based on SDSS data. P20 found a
mismatch rate of 0.6% with a resulting bias on $w$ of $\Delta w=0.0007$.

In this work, we focus on understanding the cosmological biases arising from
host galaxy mismatch using the DLR method on the full Dark Energy Survey
5-Year photometric SNe Ia sample (DES-SN5YR; Vincenzi et al., 2021; Möller et
al., 2022). Section 2 details the DLR host matching algorithm and an overview
of our analysis strategy. Section 3 describes the DES-SN5YR data sample along
with the host galaxy catalog used in this analysis. Section 4 reviews the
simulations used to model the host mismatch rate and characterize its effects
on the cosmological parameters, and Section 5 describes the methods and
results used to ensure consistency between the DES-SN5YR data and our
simulations. Section 6 reviews the framework used for cosmological parameter
estimation in this work, and Section 7 describes the cosmological biases
resulting from host galaxy mismatch.

Figure 1: An illustration adapted from Gupta et al. (2016) showing an example
of a challenging host galaxy matching problem. The supernova (labeled “SN”) is
closer in angular separation (red arrows) to the smaller galaxy on the right,
but it is closer to the edge of the larger galaxy on the left. The directional
light radius (DLR) of each galaxy is shown in the blue arrows. According to
the DLR method, the most likely host galaxy is the one with minimal
$d_{\mathrm{DLR}}$ value, or ratio of the angular separation to the DLR, which
would correctly identify the larger galaxy on the left as the more likely
host.

##  2 Host Galaxy Matching

###  2.1 The DLR Method

The directional light radius (DLR) method for host galaxy matching, developed
in S06, involves computing a dimensionless distance ($d_{\mathrm{DLR}}$) for
each potential host galaxy measured between the SN position and the centroid
of the galaxy normalized by the galaxy size in the direction of the SN.
Explicitly, $d_{\mathrm{DLR}}$ is defined as

$d_{\mathrm{DLR}}=\frac{\mathrm{\Delta\theta}}{\mathrm{DLR}}$ (1)

where $\Delta\theta$ is the angular separation (arcsec) between the galaxy
centroid and the SN, and DLR (arcsec) is the radius of the galaxy in the
direction of the SN. We require all galaxies in our catalog (see Section 3.2)
to be well modeled by ellipses parameterized by a semi-major axis $a$, a semi-
minor axis $b$, and a position angle $\phi$, which is defined relative to the
positive RA axis. Thus, the DLR value for each SN-galaxy pair is computed as
follows:

$\textrm{DLR}=\frac{ab}{\sqrt{(a\,\textrm{sin}\,\phi)^{2}+(b\,\textrm{cos}\,\phi)^{2}}}.$
(2)

For our DLR calculations, the values for the $a,b,\phi$ parameters are the
A_IMAGE, B_IMAGE (converted into arcsec), and THETA_IMAGE parameters output by
Source Extractor (Bertin, 2011, sextractor). The galaxy with the lowest
$d_{\mathrm{DLR}}$ value is chosen as the host galaxy.

###  2.2 Analysis Overview

We quantify the cosmological biases resulting from host galaxy mismatch by
generating two sets of simulated SNe, one with and one without host galaxy
mismatches, and comparing the fitted cosmological parameters. Details about
the DES data can be found in Section 3 and the simulations are described in
Section 4.

To ensure that our results from simulations are applicable to real DES data,
we show consistency between simulations and data across relevant parameter
distributions, following Popovic et al. (2020):

  * •

angular SN-galaxy separation ($\Delta\theta$),

  * •

directional light radius (DLR),

  * •

the DLR-normalized SN-galaxy separation,
$d_{\mathrm{DLR}}=\frac{\Delta\theta}{\mathrm{DLR}}$

  * •

$r$-band host galaxy magnitude ($m_{r,\mathrm{gal}}$),

  * •

the ratio between smallest and second smallest $d_{\mathrm{DLR}}$ values,
$r_{\mathrm{DLR}}=\frac{d_{\mathrm{DLR,HOSTGAL1}}}{d_{\mathrm{DLR,HOSTGAL2}}}$,
where $d_{\mathrm{DLR,HOSTGAL}\\{i\\}}$ is the $i^{\text{th}}$ smallest
$d_{\mathrm{DLR}}$ value, i.e. the $d_{\mathrm{DLR}}$ value of the
$i^{\text{th}}$ most likely host galaxy.

Further description of these parameters and the results of our consistency
checks can be found in Section 5. Host matching is performed on one set of
simulated SNe to model mismatches as well as the DES-SN5YR data with the DLR
method using the catalog of galaxies generated from deep DES imaging stacks,
as described in Section 3.2. Finally, we estimate biases by comparing the
fitted cosmological parameters from simulations with matched hosts and an
identical set of simulations with true hosts, i.e. those assigned by the
simulation.

##  3 Data and host galaxy catalog

###  3.1 Dark Energy Survey 5-Year Photometric Sample

DES is an optical imaging survey designed to deliver precision cosmological
results and constraints on dark energy by combining the probing power of weak
gravitational lensing, baryon acoustic oscillations, galaxy clusters, and SNe
Ia (Abbott et al., 2019). DES imaged 5000 $\text{deg}^{2}$ of the southern sky
for 6 years using the Dark Energy Camera (Flaugher et al., 2015) mounted on
the 4m Blanco Telescope at the Cerro Tololo Inter-American Observatory. The
time-domain survey component of the DES survey strategy covers a smaller area
on the sky (10 supernova fields covering 27 $\text{deg}^{2}$), but exposures
were repeated approximately weekly over the course of the survey. Eight of the
ten fields were surveyed to a depth of $\sim 23.5$ mag per visit (‘shallow
fields’), and the remaining two to a deeper limit of $\sim 24.5$ mag per visit
(‘deep fields’), thus extending the SN detection limit to $z\sim 1.2$.
Transient identification from images was performed using difference imaging
(DiffImg; Kessler et al., 2015, hereafter K15). Spectroscopic follow-up of SN
Ia candidates as well as their host galaxies was performed as described in
Smith et al. (2020); Lidman et al. (2020).

###  3.2 Host Galaxy Catalog

To ensure sufficient depth and density of potential host galaxies as well as
consistency between the galaxy catalogs used for real data matching and
simulation, we produce a deep galaxy catalog by coadding DES images,
identifying sources, and estimating photometric redshifts and galaxy
parameters for each source.

The coadd procedure is similar to those described in Wiseman et al. (2020)
with several updates. First, we apply stricter selection requirements (cuts)
on the quality of the single-epoch images. Images with effective exposure time
ratio
111$\tau_{\mathrm{eff}}=\Big{(}\frac{FWHM_{fid}}{FWHM}\Big{)}^{2}\Big{(}\frac{B_{fid}}{B}\Big{)}F_{trans}$,
where $FWHM$ and $FWHM_{fid}$ are the measured and fiducial PSF full width
half max, respectively; $B$ and $B_{fid}$ are the measured and fiducial sky
background; $F_{trans}$ is the atmospheric transmission relative to a nearly
clear night. See Morganson et al. (2018) for details. $\tau_{\mathrm{eff}}$
$\leq 0.3$ and those with point spread function full width at half maximum
(PSF FWHM) $\geq 1.3\arcsec,1.2\arcsec,1.1\arcsec,1.0\arcsec$ in griz,
respectively, are excluded to ensure higher quality images across the focal
plane and to mitigate source confusion. Second, each image is scaled to a
common zeropoint using the same framework adopted in DiffImg (see K15). Third,
rather than excluding images from the season in which the supernova candidate
was discovered, we instead use all images from the 5-year survey and perform a
median coaddition with swarp (Bertin, 2011). Although median coaddition is not
statistically optimal, it loses only $\sim 0.1$ mag in depth compared to other
weighted average methods and is a more robust way to exclude light from
transients and image artifacts when adding hundreds of images. Finally, we
determine the PSF model of the coadded images with psfex (Bertin, 2011) using
our tertiary standard stars. The PSFs are used by sextractor (Bertin, 2011) to
fit for the true (unblurred) Sérsic profiles and derive their parameters,
which are necessary for the placement of simulated SNe in their host galaxies.

Sources from all 10 DES SN fields are identified using sextractor. Magnitudes
are corrected for Milky Way dust using $E(B-V)$ values from Schlegel et al.
(1998) and extinction coefficients for DECam filters from Schlafly &
Finkbeiner (2011) and assuming $R_{V}=3.1$. These coefficients are
$A/E(B-V)=3.237,2.176,1.595,$ and $1.217$ in DECam $griz$, respectively.

The galaxies were matched to a spectroscopic redshift catalog compiled from
multiple surveys including OzDES (Lidman et al., 2020), SDSS (Sako et al.,
2008), 6dF (Jones et al., 2004), ATLAS (Cappellari et al., 2011), GAMA (Driver
et al., 2009), VVDS (Le Fèvre, O. et al., 2005), VIPERS (Guzzo et al., 2014),
ACES (Cooper et al., 2012), DEEP2 (Newman et al., 2013), 2dFGRS (Colless et
al., 2001), UDS/VANDELS (McLure et al., 2018), and PRIMUS (Coil et al., 2011).
Among the 7.8 million galaxies, a total of 124,824 have secure spectroscopic
redshifts. Although we require a spectroscopic redshift for the DES data, the
simulation needs a complete galaxy catalog to accurately model mis-matches. We
therefore use photometric redshifts when spectroscopic redshifts are not
available.

For the vast majority of galaxies without a spectroscopic redshift, we
estimate photometric redshifts of galaxies that are brighter than $i=25.5$ mag
using the method described in Section 3.2.1. Sources fainter than this limit
do not have reliable photo-$z$ estimates and are not included in the galaxy
catalog. Figure 2 shows the magnitude distributions of the remaining sources.

Figure 2: $i$-band MAG_MODEL magnitude distribution of all galaxies in our
deep DES galaxy catalog with a spectroscopic or photometric redshift estimate.
The catalog cuts off at $i\sim 26$ because photometric redshifts are
unavailable for galaxies fainter than $i=25.5$ mag.

####  3.2.1 Host Galaxy Photometric Redshifts

Host galaxy photometric redshift estimation is performed independently for the
deep and shallow SN fields, consisting of two and eight fields, respectively,
as described in Section 3.2. For each galaxy in each field, we have a set of
$griz$ photometric flux measurements and their corresponding uncertainties.
Using this data, we train a _Self-Organizing Map_ (SOM) to characterize and
discretize the photometric space of host galaxies, using the SOM algorithm
described in the Appendix of Sánchez et al. (2020). This SOM algorithm uses
unsupervised learning to project the 4-dimensional photometric data ($griz$)
onto a lower-dimensional grid, in our case a 2-dimensional grid, while
attempting to preserve the topology of the 4-dimensional space. This means
that similar objects in the 4-D space will be grouped together in the SOM,
enabling a visual understanding of features.

The particular SOM algorithm used in this work differs from the SOM algorithm
used in previous DES analyses (such as Myles et al., 2021; Giannini et al.,
2022) in order to improve the classification of galaxies with low- and
modest-S/N photometry, which is relevant given the faint nature of many SN
host galaxies. First, we alter the distance metric used by the SOM algorithm
to incorporate flux uncertainties. Next, we include flux information, not just
colors, and we do not impose periodic boundary conditions on the map. For the
application in this work, we build a SOM of size $22\times 22$, with a total
of 484 cells. These alterations are described in detail in Sánchez et al.
(2020).

After the SOM is constructed, we use the subset of SN host galaxies with
spectroscopic redshifts to populate the SOM and compute the redshift
distribution of each SOM cell. The spectroscopic subset for deep SN fields has
a total of 45,937 galaxies, while the one for shallow SN fields has a total of
78,887 galaxies. Once we populate the SOMs with redshift information, we
assign to each galaxy the redshift distribution of the cell it is assigned to.
Even if these spectrosopic subsets provide a good coverage of the
corresponding photometric spaces, some SOM cells do not contain redshift
information and therefore we do not estimate redshifts for galaxies in them.
This is the case for 3.9% of galaxies in the deep SN fields, and 6.7% of
galaxies in the shallow SN fields.

To validate this procedure, we split the spectroscopic samples into separate
training and validation samples, with a random 90% of galaxies going into
training and the remaining 10% into validation. Next, we compare the estimated
and spectroscopic redshifts for galaxies in the validation sample to assess
the quality of the reconstruction. For this purpose, and to enable comparisons
with previous DES estimates, we use photo-$z$ metrics from Sánchez et al.
(2014). We construct the $\Delta z\equiv z_{phot}-z_{spec}$ distribution, and
we compute two different metrics:

  1. 1.

We estimate the photo-$z$ precision by calculating the 68-percentile width
$\sigma_{68}$ of the $\Delta z$ distribution around its median value. This
estimator, $\sigma_{68}$, measures the width of the core of the $\Delta z$
distribution. In particular, it is defined as half of the width of the
distribution, measured with respect to the median, where 68% of the data are
enclosed.

  2. 2.

We estimate the fraction of photo-$z$ outliers by calculating the fraction of
objects with 3$\sigma$ deviations in $\Delta z$,
$\mathrm{out}_{3\sigma}\equiv|\Delta z|>3\sigma_{z}$, where $\sigma_{z}$ is
the standard deviation of the $\Delta z$ distribution.

For the photo-$z$ estimation procedure described above, we find
$\sigma_{68}=0.124$ and $\mathrm{out}_{3\sigma}=0.017$ for the deep SN fields,
and $\sigma_{68}=0.124$ and $\mathrm{out}_{3\sigma}=0.015$ for the shallow SN
fields. For the wide-field DES survey, the photo-$z$ requirements set prior to
the start of the survey specified $\sigma_{68}<0.12$ and
$\mathrm{out}_{3\sigma}<0.015$ for 90% of the sample of galaxies. The values
we find are slightly above these requirements, but it is important to point
out that galaxy samples from the SN fields reach significantly fainter
magnitudes than wide-field DES galaxies, and hence it is more difficult to
satisfy the wide-field requirements. However, the SOM model is constructed
such that it is primarily sensitive to color, and the color-redshift relation
should be agnostic to galaxy brightness. In addition, the numbers we find are
similar to those reported by several photo-$z$ codes in Sánchez et al. (2014),
demonstrating the comparable performance of our method when applied to fainter
galaxies.

####  3.2.2 Profile Fitting

Both the measured profile, which was used to calculate DLR values and
described in Section 2.1, and the intrinsic galaxy light profile are used in
this work.

The intrinsic light profile is used to determine the location of each
simulated SN within its assigned host galaxy (see Section 4.1.2). To calculate
the intrinsic light profile, each galaxy in the catalog is fit with a Sérsic
profile (Sérsic, 1963) that describes the variation of galaxy intensity $I$
with radius $R$:

$I(R)=I_{e}\;\mathrm{exp}\Bigl{\\{}-b_{n}\Bigl{[}\Bigl{(}\frac{R}{R_{e}}\Bigr{)}^{1/n}-1\Bigr{]}\Bigr{\\}}$
(3)

$I_{e}$ is the galaxy intensity at the half-light radius $R_{e}$, $n$ is the
Sérsic index describing the cuspiness of the profile, and $b_{n}$ is a known
function of $n$. The fitting was performed by sextractor, which fits for
$R_{e}$ as well as additional parameters theta and aspect, which describe the
angle relative to the positive RA axis and the ellipticity of the galaxy,
respectively. These additional parameters enable us to model each galaxy as an
ellipse and determine its semimajor and semiminor axes, which we denote $a,b$.
These quantites are calculated with
$R_{e}=\sqrt{ab},\;\texttt{aspect}=\frac{a}{b}$. The fitted Sérsic half-light
radii were scaled by a factor of 0.8 to obtain better data-simulation
agreement (see Section 5.2 for details on the data-simulation matching
procedure and Section 7.3.3 for comparisons with and without this scaling
factor). This is notably the same scaling factor found to best match the
DES3YR data when comparing distributions of host galaxy surface brightness at
the SN position (see Figure 6 of Kessler et al., 2019).

####  3.2.3 Catalog Cuts and Parameter Fitting

Table 1: Cuts applied to the host galaxy library and resulting galaxy counts. Cut Requirement | Galaxies Remaining
---|---
Full catalog | 8,401,139
Removed duplicate galaxies | 7,860,305
Has photo-$z$ or spec-$z$ | 5,118,585
Has CIGALE galaxy parameter fit | 5,108,812
$5<\mathrm{log}(M_{\star}/M_{\odot})<14$ | 4,938,372
Has reasonable Sérsic fit | 4,221,001
Final | 4,221,001

Several selection criteria are applied to select sextractor sources that
balance the trade-off between preserving realistic catalog depth/density, and
maintaining the quality of the galaxy photometry. The list of cuts as well as
the number of galaxies remaining after each is shown in Table 1.

First, duplicate observations of galaxies are removed, prioritizing deep field
observations when possible. The duplication is due to the slight overlap of
certain DES SN fields, causing galaxies in the overlapping regions to appear
in multiple coadded images. Next, galaxies without a spectroscopic redshift
determination or photometric redshift estimate are removed, because simulated
SNe cannot be associated with such galaxies. As stated in Section 3.2,
galaxies fainter than $i=25.5$ mag do not have reliable photometric redshift
estimates and are removed. In addition to the $i=25.5$ mag cut, 3.9% of
galaxies in the deep SN fields and 6.7% of galaxies in the shallow SN fields
were not successfully assigned a redshift estimate by the SOM and are removed
as well.

Certain galaxy properties are known to be correlated with SN Ia rate (S06;
Mannucci et al., 2005; Graur et al., 2017; Wiseman et al., 2021), and we use
these known correlations to assign suitable host galaxies in the simulation.
We estimate the total stellar mass $(M_{\star})$ and the star formation rate
(SFR) for galaxies in our catalog using CIGALE (Boquien et al., 2019). CIGALE
uses grid search to find the best-fit (lowest-$\chi^{2}$) combination of user-
specified model parameters given galaxy photometry and redshift estimates. For
the galaxy parameter fits, we assume a delayed star formation history, where
SFR is defined by

$\mathrm{SFR}(t)\propto\frac{t}{\tau^{2}}\mathrm{exp}(-t/\tau),\;0\leq t\leq
t_{0}$ (4)

with $t_{0}$ the age of the onset of star formation and $\tau$ the time at
which the SFR peaks. The bc03 (Bruzual & Charlot, 2003) library of single
stellar populations with a Salpeter initial mass function is used to compute
the intrinsic stellar spectrum. Attenuation from dust and other sources is
parameterized by the Calzetti et al. (2000) starburst attenuation curve
extended with the Leitherer et al. (2002) curve. Nebular emission is modeled
by templates from Inoue (2011). We remove a small subset of galaxies with
poorly constrained CIGALE parameter fits by restricting our library to
galaxies with $5<\mathrm{log}(M_{\star}/M_{\odot})<14$.

We found by manual inspection that in some cases, such as very diffuse
galaxies with low Sérsic index, the Sérsic profile fit fails catastrophically
and produces greatly exaggerated estimates of $R_{e}$. This will result in SNe
placed very far from the galaxy center in simulations, potentially creating a
clear mismatch with SN-galaxy separations measured from the DES data. To
remove galaxies with these pathological fits, we calculate ellipse areas from
sextractor parameters
($A_{\texttt{sextractor}}=\texttt{A\\_IMAGE}*\texttt{B\\_IMAGE}$) as well as
from Sérsic ellipse parameters ($A_{\text{Sérsic}}=ab$). We select galaxies
with a “reasonable” Sérsic fit, which we define to be
$\frac{A_{\mathtt{sextractor}}}{A_{\text{Sérsic}}}\geq 0.25$.

###  3.3 DES Data Host Matching

Table 2: Summary of host galaxy matching for the DES Y5 sample. | Number of SNe | Percent of Total
---|---|---
Total sample size | 2,186 | 100%
Has $\geq 1$ host match | 2,047 | 94%
Has $\geq 2$ host matches | 126 | 6%

Host galaxy matching for the DES Y5 sample is performed using the DLR method
with the host galaxy catalog described in Section 3.2. $d_{\mathrm{DLR}}$
values are computed for all galaxies in the catalog within 15″of the SN
position. Galaxies with $d_{\mathrm{DLR}}$ $>4$ are discarded, following the
SDSS convention (Sako et al., 2018), and those with lowest and second lowest
$d_{\mathrm{DLR}}$ values are identified as the most likely and next most
likely host match (HOSTGAL1 and HOSTGAL2). If fewer than two galaxies have
$d_{\mathrm{DLR}}$ $\leq 4$, those SNe are considered to be missing a host
match or (if one galaxy has $d_{\mathrm{DLR}}$ $\leq 4$) missing a second host
match. Summary statistics for the DES data with host matches is shown in Table
2.

Ideally, matching should be done with the full catalog, including galaxies
without a known redshift. This would allow for us to remove SNe hosted by
galaxies without known redshifts, rather than match them incorrectly to a
galaxy with known redshift. However, we find that $\sim 98\%$ of DES SNe have
the same host match when matched with the full catalog, and those that do not
mostly become “hostless” when the cuts are employed. Thus, we conclude that
matching with the cut catalog is a valid method, as these SNe are ineligible
for the analysis regardless of the cuts.

##  4 Simulations and event selection

###  4.1 Simulations

All simulations for this work are produced with the SuperNova ANAlysis (SNANA)
software (Kessler et al., 2009). The SNANA simulation starts with a SN
spectral template and generates survey-specific photometry under realistic
observing conditions by utilizing a cadence library containing zero points,
sky noise, and PSF information for each telescope pointing on each observing
night.

We note that overlaying simulated SNe on actual galaxy images is the ideal
method of performing a host matching analysis; however, it would be very
computationally expensive to run the volume of simulations needed to develop
and constrain the host mismatch systematic. For improved computational
efficiency, we use the catalog-level simulation from SNANA.

####  4.1.1 SN Models

SN Ia simulations for this work are created using the SALT2 model (Guy et al.,
2007) with training parameters determined from the Joint Lightcurve Analysis
(JLA, Betoule et al., 2014). The SALT2 model defines several restframe
parameters for SN lightcurves: the time of SN peak brightness $t_{0}$, a
stretch-like parameter $x_{1}$, a color parameter $c$ and the lightcurve
normalization parameter $x_{0}$. Nuisance parameters $\alpha,\beta$ are
determined according to Scolnic & Kessler (2016), while the color ($c$) and
stretch ($x_{1}$) populations follow Popovic et al. (2021). The SNe Ia are
simulated with a redshift-dependent volumetric rate, using measured rates from
Dilday et al. (2010) and Perrett et al. (2012) and recomputed by Frohmaier et
al. (2018). To model empirically measured Hubble scatter after the SN Ia
standardization procedure, we use the spectral-variation intrinsic scatter
model in Kessler et al. (2013) that is based on the model uncertainties
determined in Guy et al. (2010). The simulations include two separate
populations: the DES-like photometric sample and a spectroscopically confirmed
low-$z$ anchor. For simplicity, the low-$z$ anchor is an ad-hoc DES simulation
applied to $0<z<0.1$ with an inflated rate to match the true number of low-$z$
events in the DES-SN5YR sample.

Two types of core-collapse SNe as well as two types of peculiar SNe Ia are
simulated alongside the SNe Ia to evaluate the effects of host galaxy mismatch
on redshift-dependent photometric classification: SNII, SNIbc, SNIax, and
SNIa-91bg. These simulations use the SED templates introduced in Vincenzi et
al. (2019) with luminosity functions and rates following Vincenzi et al.
(2021). We use relative rates as measured by Shivvers et al. (2017) anchored
by an overall rate following the cosmic star formation history presented in
Madau & Dickinson (2014) normalized by the local SN rate from Frohmaier et al.
(2021).

All simulations are generated assuming a flat $\Lambda$CDM cosmology with
$H_{0}=70\;\mathrm{km\;s^{-1}\;Mpc^{-1}}$ and $\Omega_{\mathrm{M}}=0.311$.

Figure 3: Redshift distribution of matched host galaxies in one realization of
the SNIa-only simulations. Since we find the mismatch rate resulting from the
DLR method to be quite low (see Table 3), the true redshift distribution is
not visibly different and was not included in this plot.

####  4.1.2 Host Association and Matching

To associate a simulated SN with a host galaxy, the simulation first generates
properties for each SN, including color, stretch, redshift, and sky location
(RA, DEC). Subsequently, all galaxies whose redshifts match the true SN
redshift within a small tolerance
($dz_{\mathrm{tol}}=\text{max}|z_{\mathrm{SN}}-z_{\mathrm{GAL}}|=0.002+0.04z$)
are selected. The assigned host galaxy is chosen from this subset using the
host mass-dependent weighting shown in Wiseman et al. (2021). The simulated
host association and matching use the same deep coadded DES galaxy catalog
used for the data analysis.

Note that the process of assigning a host for a simulated SN does not take
into account the locations of the host or the SN, only the redshifts of the SN
and galaxy as well as the galaxy stellar mass. The host and its neighboring
galaxies are moved near the SN such that it satisfies the simulated SN-host
separation as well as models host confusion in the analysis. This strategy is
efficient for modeling SN-host correlation and incorrect host matches, but it
does not model large scale structure.

The SN-host separation is determined by placing the SN at a radial distance
$R$ from the center of the assigned host galaxy according to the probability
distribution $p(R)\sim I(R)$, where $I(R)$ is the galaxy intensity at radius
$R$ described by the galaxy’s fitted Sérsic profile (see Equation 3). This
approach follows past work using SNANA such as Kessler et al. (2019).

To replicate the host matching procedure used for real data, we apply host
galaxy matching with the DLR method to the simulated SNe. The simulation
computes $d_{\mathrm{DLR}}$ values for up to 10 galaxies within a 10″ radius
of the assigned host and saves the galaxies with the lowest and second lowest
$d_{\mathrm{DLR}}$ values as the most and next most likely host match
(HOSTGAL1 and HOSTGAL2), as long as $d_{\mathrm{DLR}}\leq 4$. The redshift
distribution of the matched host galaxies for one realization of a SNIa-only
simulation is shown in Figure 3. The host matching procedure is run for the
full set of simulations, including the low-$z$ population, though low-$z$
mismatches are rare.

###  4.2 Event Selection

####  4.2.1 Lightcurve Fitting

All DES-SN observed and simulated SN lightcurves are fit with the same SALT2
model with JLA parameters that was used to simulate the SNe Ia. The fit is
performed with a $\chi^{2}$-minimization program included in SNANA and
determines several parameters under the assumption that the event is a SN Ia:
the time of SN peak brightness $t_{0}$, a stretch-like parameter $x_{1}$, a
color parameter $c$ and the lightcurve normalization parameter $x_{0}$, as
well as their uncertainties and covariances (i.e., $\sigma_{t_{0}}$, etc.).
These parameters are used to calculate the distance modulus $\mu$, allowing
the SNe to be placed on the Hubble diagram.

We apply selection cuts on these fitted parameters and select SN lightcurves
well described by the SALT2 model. Specifically, we restrict our sample to SNe
satisfying the following criteria:

  * •

$|x_{1}|<3$,

  * •

$|c|<0.3$,

  * •

$\sigma_{x_{1}}<1$, and

  * •

$\sigma_{t_{0}}<2$ days.

A summary of these cuts on our simulations can be found in Table 3. The right
panel of Figure 4 shows the effect of these cuts on the redshift error
distribution between the matched and true hosts. These data show that the cuts
not only reduce the average mismatch rate over 25 realizations from 2.5% to
1.7%, but also significantly reduce the spread in redshift error.
Quantitatively, the cuts reduced the middle 90% of the
$z_{\text{match}}-z_{\text{true}}$ distribution for mismatched pairs from 1.1
(left panel of Figure 4) to 0.6 after cuts (right panel).

Figure 4: Redshift differences ($\Delta z\equiv z_{\mathrm{match}}-z_{\mathrm{true}}$) for simulated SNe Ia with mismatched hosts. (left) $\Delta z$ as a function of $z_{\mathrm{true}}$ for the full simulated sample prior to applying any selection cuts. (right) $\Delta z$ as a function of $z_{\mathrm{true}}$ after all selection cuts (described in Section 4.2.1) and removal of SNe without a valid bias correction (described in Section 6.2). Our selection process not only reduces the mismatch rate (see Table 3) but also removes SNe with extremely biased redshift estimates from mismatched hosts, reducing the spread in $\Delta z$. Quantitatively, the spread in $z_{\mathrm{match}}-z_{\mathrm{true}}$ characterized by the middle 90% of the distribution is reduced from 1.1 before cuts to 0.6 after cuts. Table 3: Summary of the mismatch rate averaged over 25 realizations after each selection cut on the SNIa+CC simulated dataset, split by Ia vs. non-Ia SNe. The non-Ia SNe include SNIax, SNIa-91bg, SNII, and SNIbc. All results using SNe Ia only are using the SN Ia subset of this dataset. Cut | SN Ia | non-Ia SNe
---|---|---
| Mismatches | Total | Mismatch Rate | Mismatches | Total | Mismatch Rate
No cuts | 233 | 9,356 | 2.5% | 151 | 5,957 | 2.5%
$|x_{1}|<3$, $|c|<0.3$ | 95 | 5,562 | 1.7% | 22 | 800 | 2.8%
$\sigma_{x_{1}}<1$ | 83 | 4,886 | 1.7% | 15 | 545 | 2.8%
$\sigma_{t_{0}}<2$ | 83 | 4,870 | 1.7% | 15 | 543 | 2.8%

####  4.2.2 Photometric Classification

In the absence of SN spectra, we rely on photometric classifiers to remove
non-Ia contaminants from the cosmological sample. We choose two recent neural-
network-based classifiers with high demonstrated accuracies for this work:
SuperNNova (SNN, Möller & de Boissière, 2020) and SCONE (Qu et al., 2021).
High quality SN redshifts have been shown to improve SNN accuracy classifying
SN Ia vs. non-Ia, so we test SNN in both redshift-dependent and redshift-
independent configurations to evaluate the impact of misidentified redshifts.
SNN models were trained in both the redshift-independent and dependent
configurations following Vincenzi et al. (2022). SCONE is redshift-independent
and performs classification based on SN lightcurves alone, so these results
should not be affected by host misidentification. Each classifier outputs
$P_{\mathrm{Ia}}$ values, the predicted probability of each SN to be a type
Ia. The SNIa-only simulations are not run through photometric classification;
all objects are simply labeled as SNe Ia.

##  5 Comparing Data with Simulations

###  5.1 Host Matching Rates and Hostless SNe

First, we compare the fraction of simulated and observed DES SNe that pass our
selection cuts with one or more matched hosts. Since we have reliable SN Ia
photometric classifiers, we compare simulated true SNe Ia with DES SNe Ia as
predicted by SCONE (Qu et al., 2021). Since most of the redshifts for the DES
SNe come from host galaxies, we chose to use SCONE for this comparison as it
does not require redshift information.

After applying the cuts specified in Table 3 to both data and simulations, the
DLR algorithm is able to find at least one host galaxy match in 98.3% of SNe
in our SN Ia-only simulations compared to 98.2% of SCONE-classified SNe Ia in
the DES data. The remaining $\sim 2$% of SNe in our data and simulations is
the rate of “hostless” SNe. SNe can appear hostless because their host
galaxies are too faint to be detected and our imposed $d_{\mathrm{DLR}}$ $\leq
4$ cut on potential host matches avoids matching to unrealistically faraway
galaxies.

SNe in galaxy-dense regions have multiple potential host matches with
$d_{\mathrm{DLR}}$ $\leq 4$. We find that ($9.35\pm 0.09$)% of simulated SN Ia
have more than one host match, compared to ($8.71\pm 1.06$)% of SCONE-
classified DES SNe Ia. The agreement in the fraction of hostless and multiple-
host events provides confidence in our simulated data sample.

Figure 5: Histograms comparing our simulated SNe Ia against DES Y5
photometrically confirmed SNe Ia for angular SN-galaxy separation
($\Delta\theta$, arcsec), directional light radius (DLR), the DLR-normalized
SN-galaxy separation ($d_{\mathrm{DLR}}=\frac{\Delta\theta}{\mathrm{DLR}}$),
and $r$-band host galaxy magnitude ($m_{r,\mathrm{gal}}$) (see Section 5 for
explanations of each). For ease of comparison, the simulation histograms are
normalized to match the integral of the data histogram and the $x$ axis limits
of each histogram are determined by the middle 90% of the data distribution to
remove outliers. In both rows, points with error bars represent parameter
distributions measured from the DES data and filled or unfilled histograms
represent the analogous quantities for simulations. (top row) Histograms of
parameter values for closest host galaxy match (HOSTGAL1) for SNe with only
one host galaxy match. (bottom row) Histograms of parameter values for closest
(HOSTGAL1) and second closest (HOSTGAL2) host galaxy match for SNe with 2 or
more host galaxy matches.

###  5.2 Comparing Parameter Distributions

To further verify that the host mismatch rate estimates and resulting
cosmological biases derived from our simulations are representative of the DES
sample, we compare simulations and data over five relevant parameter
distributions: SN-galaxy separation ($\Delta\theta$), DLR, $d_{\mathrm{DLR}}$,
$r$-band host galaxy magnitude ($m_{r,\mathrm{gal}}$), and the HOSTGAL1 to
HOSTGAL2 $d_{\mathrm{DLR}}$ ratio ($r_{\mathrm{DLR}}$), following Popovic et
al. (2020). A comparison of these distributions for the DES-SNY5YR sample
(shown in points) and the simulations (shown in filled/unfilled bars) used for
this analysis is shown in Figure 5.

The top row of Figure 5 shows the parameter distributions of data and
simulations for SNe with a single host galaxy match (i.e. only one galaxy with
$d_{\mathrm{DLR}}$ $\leq 4$). The second row shows the same parameter
distributions for SNe with at least two host galaxy matches, where green show
distributions for the closest host galaxy match (smallest $d_{\mathrm{DLR}}$
value, labeled HOSTGAL1) and orange denotes the second closest host galaxy
match (labeled HOSTGAL2).

The angular separation between the center of the galaxy and the SN position,
$\Delta\theta$, is shown in panels a) and e) of Figure 5. Good agreement in
the data vs. simulation $\Delta\theta$ distribution validates that (1) the
algorithm used by the simulation to place SNe within their host galaxies is
representative of real observations, and (2) the Sérsic ellipse parameters
$a,b$ used to place SNe within their host galaxies are well estimated. Note
that, as described in Section 3.2.2, the fitted Sérsic parameters were scaled
by a factor of 0.8. The distributions from data and simulations match very
well overall, but the simulations slightly underestimate HOSTGAL1 matches at
the very low end of the $\Delta\theta$ distribution.

DLR, shown in panels b) and f), corresponds to the size of the matched host
and is measured as in Section 2.1. Agreement in this parameter verifies that
matched hosts are similar in size between data and simulations. The data and
simulations agree well for both HOSTGAL1 and HOSTGAL2.

The distribution of $d_{\mathrm{DLR}}$ values is shown in panels c) and g).
$d_{\mathrm{DLR}}$ agreement is an important quantity, since it is used to
determine which galaxies are host matches, and shows that simulated SNe are
placed at reasonable distances from the host center. These distributions
largely show the same trend as the $\Delta\theta$ distributions, where the
HOSTGAL2 distributions match very well but the HOSTGAL1 distributions from
simulations appear to be skewed slightly higher than those of the DES data.

$m_{r,\text{gal}}$, the host galaxy $r$-band magnitude, is shown in panels d)
and h). Agreement in this parameter is an additional validation of similarity
in the overall populations of host galaxies between data and simulations. The
slight discrepancy between these distributions can likely be attributed to the
fact that the spectroscopic efficiency is defined using MAG_AUTO, but we
applied the efficiency to MAG_MODEL magnitudes, which tend to be slightly
fainter.

Finally, we define

$r_{\mathrm{DLR}}=\frac{d_{\mathrm{DLR,HOSTGAL1}}}{d_{\mathrm{DLR,HOSTGAL2}}}.$
(5)

Figure 6 shows the distributions of this parameter for DES data and our
simulations. Since the definition requires $d_{\mathrm{DLR,HOSTGAL2}}$, only
SNe with at least 2 host galaxy matches are included. Consistency in
$r_{\mathrm{DLR}}$ and the HOSTGAL2 distributions indicate that the galaxy
catalog is sufficiently dense, since the spacing between galaxies will affect
the HOSTGAL2 parameter distributions much more than HOSTGAL1.

Figure 6: Distributions of the ratio
$r_{\mathrm{DLR}}=\frac{d_{\mathrm{DLR,HOSTGAL1}}}{d_{\mathrm{DLR,HOSTGAL2}}}$

for DES Y5 photometrically confirmed SNe Ia and our simulated SNe Ia.

##  6 Cosmological Parameter Estimation

###  6.1 Cosmology Analysis Overview

The cosmology analysis pipeline is orchestrated end-to-end by the Pippin
framework (Hinton & Brout, 2020), beginning with simulations or data as input
and concluding with cosmological parameter estimates. After SNe are simulated,
the resulting lightcurves are fit with an empirical model that outputs
characteristics for each SN, such as color and stretch (see Section 4.2.1).
SNe that are fit successfully are then assigned a SN Ia probability by a
photometric classifier (see Section 4.2.2) and bias corrections on distance
moduli are computed for each SN based on its fitted parameters as well as SN
Ia probability. Finally, the bias corrected distance moduli are used to
determine constraints on cosmological parameters.

###  6.2 Bias Corrections

Biases due to core collapse contamination and survey selection are modeled and
corrected for with the BEAMS with Bias Corrections framework (BBC, Kessler &
Scolnic, 2017), an extension of BEAMS (Kunz et al., 2007), which allows
photometrically classified SNe Ia to be used for cosmology. The primary output
of the BBC framework applied on a SN sample is a redshift-binned Hubble
diagram corrected for biases from selection effects and non-SNIa
contamination.

First, systematic biases due to selection effects are modeled by a large
simulation of SNe Ia ($\sim 800,000$ SNe Ia). We found empirically that our
current framework for modeling and correcting for biases, typically used for
e.g. the Malmquist bias, is not suited for including wrong hosts in our bias
correction simulations. Thus, our bias correction simulation includes only
assigned hosts for our primary results. In Section 7.3.2, we explore
incorporating host matching (and thus mismatched hosts) into the bias
correction simulations.

Using this large SN sample, distance moduli are calculated using the Tripp
formula (Tripp, 1998),

$\mu_{\mathrm{obs}}=m_{B}+\alpha x_{1}-\beta
c+M_{B}+\Delta\mu_{\mathrm{bias}}$ (6)

for each SN. $m_{B}=-2.5\mathrm{log_{10}}(x_{0})$ and $M_{B}$ is the absolute
magnitude of a SN Ia with $x_{1}=c=0$ and $\alpha,\beta$ are nuisance
parameters determined according to Scolnic & Kessler (2016). Biases from a
reference cosmology, $\Delta\mu_{\mathrm{bias}}$, are calculated in a
3-dimensional grid of $\\{z,x_{1},c\\}$ bins using the method described above.
We use this grid of estimated biases to correct all of our distance moduli
prior to cosmology fitting. A small percentage of SNe with parameter values
that do not fit into the grid are discarded. Finally, the BEAMS method is used
to estimate binned distance moduli from the bias-corrected distance moduli
from the previous step in the presence of core-collapse contamination. This is
done by minimizing the BEAMS likelihood, which models the SNe Ia population
and a population of contaminants separately. These terms are weighted by
$P_{\mathrm{Ia}}$, the probability of each SN to be a type Ia as output by a
photometric classifier. We omit the mass step correction in Equation 6;
details on the mass step and the impact of host mismatches can be found in
Section 7.1.1.

###  6.3 Cosmological Parameters

We fit for $w$ and $\Omega_{m}$ using wfit, a fast cosmology grid-search
program in SNANA, assuming a diagonal covariance matrix
$\mathcal{C}_{\text{stat}}$ and an approximate CMB prior computed with the
$R$-shift parameter (see e.g. Equation 69 in Komatsu et al. (2009)) from the
same cosmological parameters used to generate the SNe Ia. The $R$ uncertainty
is $\sigma_{R}=0.006$, tuned to have the same constraining power as Planck
Collaboration et al. (2020). As we are only interested in the impact of host
mismatches on cosmology, the approximation of a diagonal covariance matrix is
sufficient for our purposes. wfit calculates the $\chi^{2}$ of the SN
likelihood to compute our final cosmological fit,

$\chi^{2}=\Delta\mu_{\text{model}}^{T}\cdot\mathcal{C}_{\text{stat}}^{-1}\cdot\Delta\mu_{\text{model}}$
(7)

where

$\Delta\mu_{\text{model}}=\mu-\mu_{\text{model}}(\Omega_{m},w).$ (8)

##  7 Results and Discussion

We use the cosmological parameter estimation framework described in Section 6
to evaluate the effects of host galaxy mismatch on the resulting best fit
cosmology. We focus primarily on shifts in the best fit value for $w$, the
dark energy equation of state parameter. We quantify this shift by creating
two sets of identical simulations that differ only in whether or not the DLR
method is run to determine the matched hosts using the procedure described in
Section 4.1.2. We define $S_{\mathrm{match}}$ as simulations with matched
hosts and $S_{\mathrm{truehost}}$ as simulations with perfect host matching.
In $S_{\mathrm{truehost}}$, we do not calculate matched hosts using the DLR
method; we instead force a match to the true hosts assigned by the simulation.
This ensures that there will be no mismatches and serves as a baseline for
comparison.

We define the $w$ shift as the difference between the inferred $w$ values from
$S_{\mathrm{match}}$ ($w_{\text{match}}$) and $S_{\mathrm{truehost}}$
($w_{\text{truehost}}$). We average over 25 realizations with the same
simulation parameters. Explicitly, we define the $w$ shift as

$\Delta w=\langle
w_{\textrm{match}}-w_{\textrm{truehost}}\rangle_{\mathrm{(25\;realizations)}}$
(9)

where $\langle\rangle$ denotes the inverse-variance weighted average. We
calculate the associated uncertainty on $\Delta w$ as follows:

$\sigma_{\Delta
w}=\sqrt{\frac{\sum_{i}(w_{\textrm{match},i}-w_{\textrm{truehost},i})^{2}}{25}}.$
(10)

###  7.1 Cosmological Biases with SNe Ia Only

For simulations with SNe Ia only, we find $\Delta w=0.0013\pm 0.0026$, which
is consistent with zero. In Figure 7, we show the biases on the binned Hubble
diagram comparing distance moduli $\mu$ from simulations with DLR matched
hosts ($\mu_{\text{match}}$) and true hosts ($\mu_{\text{truehost}}$). We
define

$\Delta\mu=\mu_{\textrm{match}}-\mu_{\textrm{truehost}}.$ (11)

We see that the bias is consistent with 0 until $z\sim 1$, where the sample
becomes very sparse (see Figure 3 for the redshift distribution of the
sample).

Figure 8 shows the Hubble diagram and Hubble residuals for a single
realization of simulations with mismatches. In this particular realization, 79
SNe were matched to the wrong host out of 5,811 total simulated SNe, which
translates to a 1.4% mismatch rate. The mismatched SNe, shown in red circles,
show similar Hubble residuals compared to the correctly matched SNe, which is
consistent with the small recovered bias on $w$. The observed similarity in
Hubble residuals is likely due to the fact that catastrophic outliers in
redshift are removed by the selection criteria described in Table 3 and shown
in Figure 4.

The binned Hubble residuals for all 25 simulations with mismatch are shown in
Figure 9. This plot shows
$\Delta\mu=\mu_{\mathrm{match}}-\mu_{\mathrm{model}}$, as opposed to Figure 7,
which shows $\mu_{\mathrm{match}}-\mu_{\mathrm{truehost}}$. This allows us to
compare residuals from the subpopulations of SNe with wrong and correct host
match with respect to a fiducial cosmology. Aggregated over all 25
simulations, the bias from SNe with the wrong host match (shown in orange) is
clearly distinct from the nearly unbiased subset of SNe with the correct host
match (teal). Although wrong hosts clearly lead to biases on the Hubble
diagram, SNe with the wrong host match make up $<2\%$ of the sample, resulting
in the small $\Delta w$ value we observe.

Figure 7: Biases in binned Hubble residuals for the SN Ia-only sample between samples with and without mismatched host galaxies, $\Delta\mu=\mu_{\text{match}}-\mu_{\text{truehost}}$, as a function of redshift. Uncertainties are shown as the shaded region and calculated from the binned standard deviations of $\mu_{\text{match}}$ and $\mu_{\text{truehost}}$. Lines showing $\Delta w=\pm 0.03$ are also plotted for reference. Figure 8: Hubble diagram for a single realization of our SNIa-only simulation with matched hosts. This realization has 79 SNe matched to an incorrect host out of 5,811 total SNe, a 1.4% mismatch rate. The low-$z$ sample is omitted from this plot, since it is not part of the main DES sample. Figure 9: Binned Hubble residuals ($\mu_{\text{match}}-\mu_{\text{model}}$) for all 25 realizations of our SNIa-only simulation with mismatch. The percentages in the legend show the fraction of the sample represented by each population in the plot. Classifier | SNe | $\Delta w$ | Ia vs. non-Ia Classification Accuracy
---|---|---|---
|  |  | with host mismatch | no host mismatch | change
Perfect | Ia only | $0.0013\pm 0.0026$ | 100% | 100% | 0%
Perfect | Ia+CC | $0.0011\pm 0.0027$ | 100% | 100% | 0%
$\text{SNN}_{+Z}$ | Ia+CC | $0.0032\pm 0.0040$ | 97.94% | 98.06% | -0.1%
$\text{SNN}_{NoZ}$ | Ia+CC | $0.0009\pm 0.0028$ | 97.05% | 97.05% | 0%
SCONE | Ia+CC | $0.0016\pm 0.0032$ | 96.13% | 96.13% | 0%
Table 4: $\Delta w$ and classification accuracies for each classifier.
Accuracy change is only expected for the $\text{SNN}_{+Z}$ classifier, as it
is the only classifier tested that requires redshift information. Accuracy
change is defined as $\textrm{Accuracy (with mismatch)}-\textrm{Accuracy (no
mismatch)}$.

####  7.1.1 Host Mismatch and the Mass Step

The ‘mass step’ is the observed correlation between SNe Ia intrinsic
luminosity and host galaxy stellar mass, $M_{\star}$. Specifically, SNe Ia in
more massive galaxies are more luminous after lightcurve corrections than
their counterparts occurring in galaxies with lower stellar mass, with the
average corrected luminosity distribution following a two-part step function
with a break at log($M_{\star}/M_{\odot})\sim 10$ (Kelly et al., 2010;
Sullivan et al., 2011; Betoule et al., 2014; Smith et al., 2020; Kelsey et
al., 2021, 2023). Though the underlying astrophysical cause is unknown, recent
cosmological analyses have incorporated a correction in which SN luminosities
in hosts with log($M_{\star}/M_{\odot}$) $\geq 10$ and those in hosts with
log($M_{\star}/M_{\odot}$) $<10$ are fit for separately. When this two-part
fit is employed, incorrect host matches will produce additional bias through
incorrect host mass estimates, leading to different best fit values for SN
luminosities. We observe that 34.3% of our simulated SNe Ia with the wrong
host match “switch sides”, i.e. the correct host is on one side of the mass
step but the matched host is on the other. Given that the rate of mismatches
averaged over 50 realizations is 1.7% after selection cuts for our Ia-only
sample, the overall fraction of SNe Ia that switch sides of the mass step is
$\sim 0.6\%$. We assume that such a small percentage of the sample switching
sides makes little impact and we do not pursue this aspect of the analysis
further.

###  7.2 Cosmological Biases with Photometric Classification

Some photometric classifiers, such as SuperNNova (SNN), rely on SN redshift
information to improve classification accuracy. To evaluate the impact of
incorrect redshifts from host galaxy mismatches on the predictions from
photometric classifiers, we jointly simulate SNe Ia, two types of peculiar SNe
Ia, and two types of core collapse SNe: SNII, SNIbc, SNIax, and SNIa-91bg.
Details on these simulations can be found in Section 4.1.1 and Table 3.

We tested 4 different photometric classifiers on our SNIa+CC simulations: the
baseline perfect classification, SNN with redshift information
($\text{SNN}_{+Z}$), SNN without redshift information ($\text{SNN}_{NoZ}$),
and SCONE. SNN is typically used with SN redshift information and has been
shown to produce highly accurate Ia vs. non-Ia classification results in this
paradigm, so testing both redshift-dependent and redshift-independent
configurations will show which effect is more detrimental to performance:
incorrect redshift information or lack of redshift information altogether.
SCONE uses SN lightcurves alone without the need for redshift information, so
its predictions are not affected by host matching. For this analysis, we
choose to define an SN Ia classification as $P_{\text{Ia}}\geq 0.5$.

Following the same approach as the Ia-only analysis, $w$ shifts were
calculated for each photometric classifier by comparing two sets of identical
simulations with and without mismatches. The results are shown in Table 4.

Figure 10: Visualization of the impact of incorrect redshifts on
$\text{SNN}_{+Z}$ predictions. $\Delta
P_{\mathrm{Ia}}=P_{\mathrm{Ia,wronghost}}-P_{\mathrm{Ia,correcthost}}$ is the
difference between $P_{\mathrm{Ia}}$ values output by the $\text{SNN}_{+Z}$
classifier given the wrong host redshift ($P_{\mathrm{Ia,wronghost}}$) and the
correct host redshift ($P_{\mathrm{Ia,correcthost}}$) for SNe with mismatched
hosts. $\Delta P_{\mathrm{Ia}}$ values are plotted against the difference
between the distance modulus $\mu$ calculated at the wrong
($\mu(z_{\text{match}})$) and correct host redshifts ($\mu(z_{\text{true}})$).
As $\mu(z_{\text{match}})-\mu(z_{\text{true}})$ deviate from 0, we would
expect larger deviations in $P_{\mathrm{Ia}}$ values, i.e. $|\Delta
P_{\mathrm{Ia}}|>0$. Non-Ia SNe (orange points) more likely to be
misclassified as Ia with the wrong redshift will appear in the upper half of
the plot ($\Delta P_{\text{Ia}}>0$), whereas SNe Ia (blue points) more likely
to be misclassified as non-Ia with the wrong redshift will appear in the lower
half. 7% of mismatched SNe are incorrectly classified as a result of wrong
host redshifts, leading to a overall 0.1% reduction in classification accuracy
compared to a simulation with correct host redshifts. Figure 11: Biases in
binned Hubble residuals for all 4 photometric classifiers between samples with
and without host galaxy mismatch. Uncertainties are shown as the shaded region
and calculated from the binned standard deviations of $\mu_{\mathrm{match}}$
and $\mu_{\mathrm{truehost}}$. Lines showing $\Delta w=\pm 0.03$ are also
plotted for reference. Figure 12: Binned Hubble residuals,
$\Delta\mu=\mu_{\mathrm{match}}-\mu_{\mathrm{model}}$, for the 4 photometric
classifiers for our SNIa+CC simulation with mismatch. (top) SN populations
classified as SNe Ia by each photometric classifier, including true SNe Ia
with mismatched hosts as well as core collapse contamination. (bottom) SN
populations classified as CC SNe, including SNe with mismatched hosts.

$\text{SNN}_{+Z}$ is the only classifier that should be impacted by host
galaxy mismatches, and this relationship is observed in the larger $\Delta w$
value and lower classification accuracy for simulations with mismatched host
galaxies. Figure 10 shows the difference between predicted SN Ia probability
output by the $\text{SNN}_{+Z}$ classifier given the wrong host redshift
($P_{\mathrm{Ia,wronghost}}$) as opposed to the correct host redshift
($P_{\mathrm{Ia,correcthost}}$). Wrong redshifts indeed cause SNN to produce
incorrect predictions for both Ia and non-Ia SNe, leading to the observed drop
in accuracy with wrong redshifts. SCONE and $\text{SNN}_{NoZ}$ are oblivious
to any host matching changes and produce the same predictions for both sets of
simulations, as expected.

Figure 11 shows the biases on the binned Hubble diagram for each of the
photometric classifiers with $\Delta\mu$ defined as in Equation 11. Overall,
the classifiers perform quite similarly and exhibit very small differences in
$\Delta\mu$ over the full redshift range. As expected from the small $\Delta
w$ values, the $\Delta\mu$ curves for the Ia+CC simulations exhibit a slight
redshift-dependent bias, though still mostly consistent with 0 up to high
redshifts. Further validating the observed $\Delta w$ values for each
classifier, we see that the two classifiers with most similar $w$ shifts, the
perfect classifier (shown in orange) and $\text{SNN}_{NoZ}$ (green), have the
most similar $\Delta\mu$ values. $\text{SNN}_{+Z}$ (shown in blue), which has
the largest $w$ shift, also consistently appears furthest from $\Delta\mu=0$
across all redshift bins.

The binned Hubble residuals of SNe Ia in the Ia+CC simulations as predicted by
the 4 photometric classifiers are shown in Figure 12. This plot shows
$\Delta\mu=\mu_{\mathrm{match}}-\mu_{\mathrm{model}}$, as opposed to Figure
11, which shows $\mu_{\mathrm{match}}-\mu_{\mathrm{truehost}}$. This allows us
to compare residuals from the subpopulations of SNe with wrong and correct
host match with respect to a fiducial cosmology. The biases on Hubble
residuals from SNe Ia with wrong hosts (shown in teal diamonds on the top row)
appear similar between the four classifiers, reflecting the small recovered
$\Delta w$ values shown in Table 4. We also observe that the bias from wrong
hosts is much more pronounced in SNe Ia misclassified as CC (shown in green
triangles on the bottom row), particularly in the $\text{SNN}_{+Z}$ panel,
indicating that $\text{SNN}_{+Z}$ was able to identify severe redshift
outliers and rejected them from the SN Ia sample.

###  7.3 Robustness of Cosmological Biases

####  7.3.1 Impact of CMB Prior

The $w-\Omega_{m}$ contour estimated from measurements of the CMB exhibits a
nearly orthogonal direction of degeneracy to the SN-only contour for a flat
$w\text{CDM}$ model, providing strong constraints and drastically reducing the
impact of systematics that act along the SN degeneracy direction. All $\Delta
w$ values reported in Sections 7.1 and 7.2 were calculated with a CMB prior.
In this section, we evaluate the impact of the CMB prior on cosmological
biases.

The cosmological biases on both $w$ and $\Omega_{m}$ with and without the CMB
prior are shown in Table 5 for the SNIa-only sample. The associated
cosmological contours are shown in Figure 13. Both $\Delta w$ and
$\Delta\Omega_{m}$ are significantly inflated without the CMB prior, though
still within their uncertainties (right column). The larger shift in $w$ and
$\Omega_{m}$ are visible when comparing the contours in Figure 13. The
contours computed with a CMB prior (top panel) are very nearly identical,
whereas with a flat $\Omega_{m}$ prior, the contour with matched hosts is
visibly shifted from the true hosts contour. However, these results are still
consistent with 0 for the analysis we performed for the DES data, but should
be studied further in future surveys.

Parameter | with CMB prior | no CMB prior
---|---|---
$\Delta w$ | $0.0013\pm 0.0026$ | $-0.062\pm 0.072$
$\Delta\Omega_{m}$ | $0.0014\pm 0.0017$ | $0.028\pm 0.029$
Table 5: $\Delta w$ and $\Delta\Omega_{m}$ values for the Ia-only SN
population with and without a CMB prior. The values in the $\Delta w$ with CMB
prior cell are reproduced from Table 4. Figure 13: Cosmological contours
computed with one realization of simulated DES Y5 SNe Ia with (red) and
without host galaxy mismatch (gray). (top) SN+CMB contours, in which an
approximate CMB prior $\Omega_{m}=0.311\pm 0.010$ prior is applied (see
Section 6.3 for details). This results in good agreement between the two
contours. (bottom) SN-only contours showing a substantial shift in best fit
$w,\Omega_{m}$ between the two contours.

####  7.3.2 Bias Correction Simulations with Mismatch

In the earlier sections, distance moduli from both $S_{\mathrm{match}}$ and
$S_{\mathrm{truehost}}$ are bias corrected using a large simulation of SNe Ia
generated with the same parameters as $S_{\mathrm{truehost}}$, i.e. each SN is
matched to its true host. We define this set of perfectly matched bias
correction simulations as $B_{\mathrm{truehost}}$. An alternative bias
correction strategy is to “correct like with like”, i.e. generating two
separate bias correction simulations, one identical to $S_{\mathrm{match}}$ as
well as the existing one identical to $S_{\mathrm{truehost}}$. We define this
new set of DLR-matched bias correction simulations as $B_{\mathrm{match}}$.
$B_{\mathrm{match}}$ models the biases arising from mismatched hosts in the
bias correction simulations in order to correct for these biases in the
simulated data. The reference $w$ value, $w_{\mathrm{truehost}}$ is still
computed from $S_{\mathrm{truehost}}$ corrected with the baseline bias
correction simulations with correct hosts only ($B_{\mathrm{truehost}}$). In
this scheme, we define $w(S,B)$ as the best-fit $w$ value computed from
simulations $S$ corrected with bias correction simulations $B$. The resulting
$\Delta w$ equation then becomes

$\Delta w,\mathrm{bcor+mismatch}=\\\ \langle
w(S_{\mathrm{match}},B_{\mathrm{match}})-w(S_{\mathrm{truehost}},B_{\mathrm{truehost}})\rangle_{\mathrm{(25\;realizations)}}.$
(12)

The $\Delta w$ values following this approach are shown in Table 6. While the
$\Delta w$ values using this bias correction scheme are still consistent with
zero, the uncertainties are larger than those using bias correction
simulations with perfect host matching; this may arise from comparing $w$
values corrected with two statistically independent sets of bias correction
simulations, i.e. $S_{\mathrm{match}}$ is corrected with $B_{\mathrm{match}}$,
whereas $S_{\mathrm{truehost}}$ is corrected with $B_{\mathrm{truehost}}$.
Further investigation of the interplay of bias correction simulations with
different sources of bias, including incorrect redshifts from host mismatch,
will be addressed in a future work.

Table 6: $\Delta w$ values for the Ia-only and Ia+CC simulated SN populations with an alternative bias correction scheme (“bcor+mismatch”) that includes mismatched hosts in the bias correction simulations. The values in the $\Delta w$, baseline column are reproduced from Table 4. Classifier | SNe |  $\Delta w$, baseline |  $\Delta w$, bcor+mismatch
---|---|---|---
Perfect | Ia only | $0.0013\pm 0.0026$ | $-0.0094\pm 0.0099$
Perfect | Ia+CC | $0.0011\pm 0.0027$ | $-0.0110\pm 0.0120$
$\text{SNN}_{+Z}$ | Ia+CC | $0.0032\pm 0.0040$ | $-0.0081\pm 0.0086$
$\text{SNN}_{NoZ}$ | Ia+CC | $0.0009\pm 0.0028$ | $-0.0100\pm 0.0012$
SCONE | Ia+CC | $0.0016\pm 0.0032$ | $-0.0099\pm 0.0100$

####  7.3.3 Results from Varying Sérsic Scale

We vary the scaling of the fitted Sérsic $a$ and $b$ parameters, which
describe each galaxy’s semi-major and semi-minor axes, respectively. Simulated
SNe are placed according to the intrinsic light profile described by these
parameters. The scaled parameters $a^{\prime}$ and $b^{\prime}$ are calculated
as $a^{\prime}=ka,b^{\prime}=kb$, where $k$ is the scaling parameter we vary.
We tested $k\in[0.5,1.2]$ with an interval of 0.1 and evaluated $\chi^{2}$
values on the histograms in Figure 5 to find the best match between data and
simulations. We found that $k=0.8$ minimized the $\chi^{2}$ and was thus
chosen for the main analysis, as described in Section 3.2.3.

We performed cosmological parameter estimation using simulations with and
without Sérsic scaling ($k=0.8$ and $k=1$) and found a modest benefit of using
the $k=0.8$ scaling for redshift-independent photometric classifiers as well
as perfectly classified SNe Ia, but a noticeable improvement for SNe
classified using the redshift-dependent classifier, $\text{SNN}_{+Z}$. This
seems to indicate that the Sérsic scaling improves the host matching
efficiency significantly, since $\text{SNN}_{+Z}$ is most affected by
incorrect redshifts. This is notably the same scaling factor found to best
match the DES3YR data when comparing distributions of host galaxy surface
brightness at the SN position (see Figure 6 of Kessler et al., 2019). The
recovered biases on $w$ from mismatched hosts ($\Delta w$) with and without
Sérsic scaling for all simulations and classifiers are shown in Table 7.

Table 7: $\Delta w$ values for the Ia-only and Ia+CC simulated SN populations with and without Sérsic scaling ($k=0.8$ and $k=1$). The primary results presented in this work (Sections 7.1 and 7.2) use simulations scaled with $k=0.8$. The $\text{SNN}_{+Z}$ results with $k=1$ have an inflated $\Delta w$ as well as $\sigma_{\Delta w}$ due to a realization with poor $\chi^{2}$ fit from wfit. Classifier | SNe | $\Delta w(k=0.8)$ | $\Delta w(k=1)$
---|---|---|---
Perfect | Ia only | $0.0013\pm 0.0026$ | $0.0030\pm 0.0042$
Perfect | Ia+CC | $0.0011\pm 0.0027$ | $0.0016\pm 0.0026$
$\text{SNN}_{+Z}$ | Ia+CC | $0.0032\pm 0.0040$ | $0.0120\pm 0.0640$
$\text{SNN}_{NoZ}$ | Ia+CC | $0.0009\pm 0.0028$ | $0.0021\pm 0.0031$
SCONE | Ia+CC | $0.0016\pm 0.0032$ | $0.0025\pm 0.0034$

####  7.3.4 Host Confusion Parameter

Equation 3 in Gupta et al. (2016) defines a quantity characterizing the
likelihood of a wrong match: the host confusion parameter, or $HC$. This
parameter is a function of the $d_{\mathrm{DLR}}$ values of each galaxy in the
search radius around a SN, and aims to distinguish situations with a clear
correct host from those without. In this analysis, we attempt to remove wrong
hosts by cutting out SNe with high host confusion ($HC\geq-2.5$). The
distribution of $HC$ values for wrong and correct hosts in our simulations is
shown in Figure 14. The $-2.5$ threshold was chosen by visual inspection of
this distribution. The $HC$ distributions for both populations look quite
similar, but 72% of SNe with an incorrect host match are removed by the
$HC\geq-2.5$ cut, while 12% of SNe with correct host matches are removed.

We performed a full cosmology analysis with perfectly classified Ia-only
simulations and find $\Delta w=-0.0043\pm 0.0047$ by comparing Ia populations
with and without wrong hosts, both subject to the $HC<-2.5$ selection
requirement. In this cosmology analysis, we use bias correction simulations
with host matching (i.e. $B_{\mathrm{match}}$ from Section 7.3.2) and select
only the subset of the bias correction simulations with the same $HC<-2.5$
requirement for consistency. Both simulations (with perfectly matched hosts
and with wrong hosts) are bias corrected with $B_{\mathrm{match}}$ in order to
apply the $HC$ selection requirement globally, but it may not be suitable to
correct simulations without wrong hosts in this way. The magnitude of this
$\Delta w$ value is larger than that of the baseline (i.e. no $HC$ selection
requirement), likely due to our inclusion of wrong hosts in the bias
correction simulations, but is still consistent with zero.

Figure 14: Distributions of $HC$ values for SNe with wrong and correct host
matches. The dotted line is drawn at our chosen threshold of $HC=-2.5$. Figure
15: $\Delta w$ values from all tested variants of simulated DES SNe with CMB
prior. Note that the total uncertainty on $w$ from the DES3YR analysis (Abbott
et al., 2019), $\delta w=0.059$, is much larger than the bounds of this plot.
Details about each variant can be found in Section 7.

##  8 Conclusions

Matching SNe to their host galaxies is a non-trivial problem when using
2-dimensional images of 3-dimensional space. Accurate host galaxy matches are
important for cosmology because host galaxies are used to measure the vast
majority of SN redshifts. Identifying incorrect host galaxies alters the shape
of the Hubble diagram and can impact the resulting fitted cosmological
parameters.

In this work, we investigated the impact of mismatched host galaxies for the
DES Y5 SN cosmology analysis. To this end, we created a galaxy catalog from
DES deep field images that was sufficiently deep and dense, and calculated
photometric redshifts, galaxy parameters, as well as Sérsic parameter fits for
these 4 million galaxies. Simulations using this galaxy catalog were verified
to be sufficiently similar to the DES Y5 photometric SN Ia sample using
distributions such as Figure 5. This simulation enabled us to predict the
prevalence of mismatched host galaxies in DES data and, in turn, characterize
the effect of host mismatches on cosmological parameter estimates.

Host matching was performed using the directional light radius (DLR) method, a
technique that identifies the host galaxy by minimizing SN-galaxy distance
normalized by galaxy radius. This analysis aims to characterize the systematic
error due to host misidentification by the DLR method for the DES Y5 cosmology
analysis.

The main findings of this work are summarized in Figure 15, which shows the
observed shifts in the dark energy equation of state parameter $w$ as a result
of host galaxy mismatches with different approaches to the analysis. We
defined $\Delta w$ by producing two identical sets of simulations: one with
perfect host matching and one with host matches from the DLR method. We also
probed the interplay between host galaxy mismatches and photometric
classification, as classifiers such as SuperNNova use SN redshift estimates
for more accurate SN type predictions. Finally, we explored the impact of
variations to our analysis, such as the choice of $\Omega_{m}$ prior, bias
correction simulations, Sérsic scaling factor, and additional selection cuts
to remove incorrect host matches.

We found that the baseline $w$ shift with perfectly classified type Ia SNe is
$\Delta w=0.0013\pm 0.0026$, and changes in certain properties can increase
this to $|\Delta w|\sim 0.004$. We also find that the choice of photometric
classifier makes an impact on the $w$ shift: classifiers requiring redshift
estimates for prediction tend to misclassify SNe with the wrong redshift,
leading to a larger $w$-bias. When the CMB prior is replaced with a flat
$\Omega_{m}$ prior, the $\Delta w$ value changes to $-0.062\pm 0.072$. Though
the $\Delta w$ uncertainty is larger with the flat $\Omega_{m}$ prior, the
shift is still consistent with 0 given the associated inflation in
uncertainty.

In conclusion, we find that our current estimate on the systematic error
associated with host galaxy mismatch is substantially smaller than the
statistical error for DES Y5, but as future surveys continue to discover more
SNe Ia, we encourage continued study and improvement of the accuracy of host
galaxy matching and prevalence of catastrophic errors in redshift.

## Acknowledgements

H.Q., J.L., and M.S. were supported by DOE grant DE-FOA-0002424 and NSF grant
AST-2108094. L.G. acknowledges financial support from the Spanish Ministerio
de Ciencia e Innovación (MCIN), the Agencia Estatal de Investigación (AEI)
10.13039/501100011033, and the European Social Fund (ESF) “Investing in your
future” under the 2019 Ramón y Cajal program RYC2019-027683-I and the
PID2020-115253GA-I00 HOSTFLOWS project, from Centro Superior de
Investigaciones Científicas (CSIC) under the PIE project 20215AT016, and the
program Unidad de Excelencia María de Maeztu CEX2020-001058-M. PW acknowledges
support from the Science and Technology Facilities Council (STFC) grant
ST/R000506/1. L.K. thanks the UKRI Future Leaders Fellowship for support
through the grant MR/T01881X/1.

This work was completed in part with resources provided by the University of
Chicago’s Research Computing Center, as well as resources of the National
Energy Research Scientific Computing Center (NERSC), 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.

Funding for the DES Projects has been provided by the U.S. Department of
Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology Facilities Council of the
United Kingdom, the Higher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of Illinois at
Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the
Ministério da Ciência, Tecnologia e Inovação, the Deutsche
Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy
Survey.

The Collaborating Institutions are Argonne National Laboratory, the University
of California at Santa Cruz, the University of Cambridge, Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the
University of Chicago, University College London, the DES-Brazil Consortium,
the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH)
Zürich, Fermi National Accelerator Laboratory, the University of Illinois at
Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the
Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence
Cluster Universe, the University of Michigan, NSF’s NOIRLab, the University of
Nottingham, The Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, Texas A&M University, and the OzDES
Membership Consortium.

Based in part on observations at Cerro Tololo Inter-American Observatory at
NSF’s NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J. Frieman), which is managed
by the Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation.

The DES data management system is supported by the National Science Foundation
under Grant Numbers AST-1138766 and AST-1536171. The DES participants from
Spanish institutions are partially supported by MICINN under grants
ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-2016-0597,
and MDM-2015-0509, some of which include ERDF funds from the European Union.
IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
Research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Program
(FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We
acknowledge support from the Brazilian Instituto Nacional de Ciência e
Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2).

This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics.

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