# Evaluating Cosmological Biases using Photometric Redshifts for Type Ia
Supernova Cosmology with the Dark Energy Survey Supernova Program

R. Chen,1 D. Scolnic,1 M. Vincenzi,1,2,3 E. S. Rykoff,4,5 J. Myles,6 R.
Kessler,7,8 B. Popovic,1,9 M. Sako,10 M. Smith,11 P. Armstrong,12 D.
Brout,13,14 T. M. Davis,15 L. Galbany,16,17 J. Lee,10 C. Lidman,18,12 A.
Möller,19 B. O. Sánchez,20,1 M. Sullivan,11 H. Qu,10 P. Wiseman,11 T. M. C.
Abbott,21 M. Aguena,22 S. Allam,23 O. Alves,24 F. Andrade-Oliveira,25 J.
Annis,23 D. Bacon,26 D. Brooks,27 A. Carnero Rosell,28,22 J. Carretero,29 A.
Choi,30 C. Conselice,31,32 L. N. da Costa,22 M. E. S. Pereira,33 H. T.
Diehl,23 P. Doel,27 S. Everett,34 I. Ferrero,35 B. Flaugher,23 J. Frieman,23,8
J. García-Bellido,36 M. Gatti,10 E. Gaztanaga,16,26,17 G. Giannini,29,8 D.
Gruen,37 R. A. Gruendl,38,39 G. Gutierrez,23 K. Herner,23 S. R. Hinton,15 D.
L. Hollowood,40 K. Honscheid,41,42 D. Huterer,24 D. J. James,43 K. Kuehn,44,45
M. Lima,46,22 J. L. Marshall,47 J. Mena-Fernández,48 F. Menanteau,38,39 R.
Miquel,49,29 R. L. C. Ogando,50 A. Palmese,51 A. Pieres,22,50 A. A. Plazas
Malagón,4,5 A. Roodman,4,5 S. Samuroff,52 E. Sanchez,53 D. Sanchez Cid,53 I.
Sevilla-Noarbe,53 E. Suchyta,54 M. E. C. Swanson,38 G. Tarle,24 C. To,41 D. L.
Tucker,23 V. Vikram,10 N. Weaverdyck,55,56 and J. Weller57,58 (DES
Collaboration)
Affiliations shown at end of paper

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Cosmological analyses with Type Ia Supernovae (SNe Ia) have traditionally been
reliant on spectroscopy for both classifying the type of supernova and
obtaining reliable redshifts to measure the distance-redshift relation. While
obtaining a host-galaxy spectroscopic redshift for most SNe is feasible for
small-area transient surveys, it will be too resource intensive for upcoming
large-area surveys such as the Vera Rubin Observatory Legacy Survey of Space
and Time, which will observe on the order of millions of SNe. Here we use data
from the Dark Energy Survey (DES) to address this problem with photometric
redshifts (photo-$z$) inferred directly from the SN light-curve in combination
with Gaussian and full $p(z)$ priors from host-galaxy photo-$z$ estimates.
Using the DES 5-year photometrically-classified SN sample, we consider several
photo-$z$ algorithms as host-galaxy photo-$z$ priors, including the Self-
Organizing Map redshifts (SOMPZ), Bayesian Photometric Redshifts (BPZ), and
Directional-Neighbourhood Fitting (DNF) redshift estimates employed in the DES
3x2 point analyses. With detailed catalog-level simulations of the DES 5-year
sample, we find that the simulated $w$ can be recovered within $\pm 0.02$ when
using SN+SOMPZ or DNF prior photo-$z$, smaller than the average statistical
uncertainty for these samples of 0.03. With data, we obtain biases in $w$
consistent with simulations within $\sim 1\sigma$ for three of the five
photo-$z$ variants. We further evaluate how photo-$z$ systematics interplay
with photometric classification and find classification introduces a
subdominant systematic component. This work lays the foundation for next-
generation fully photometric SNe Ia cosmological analyses.

###### keywords:

cosmology:dark energy - supernovae

††pubyear: 2024††pagerange: Evaluating Cosmological Biases using Photometric
Redshifts for Type Ia Supernova Cosmology with the Dark Energy Survey
Supernova Program–B

##  1 Introduction

Type Ia Supernovae (SNe Ia) are standardizable candles used as a key
cosmological probe to measure the distance-redshift relation and understand
the nature of dark energy. In order to precisely constrain the dark energy
equation-of-state parameter w, the largest SN surveys and compilations to date
such as Pantheon+ (Brout et al., 2022) have relied on access to spectroscopy
of the live SN to confirm the type of the SN, as well as of the host galaxy to
obtain an accurate redshift. Recent cosmological analyses such as the Dark
Energy Survey 5-year SN analysis (DES-SN5YR; Vincenzi et al., 2024, DES
Collaboration, 2024) have shown that a transition to photometric
classification immensely boosts our statistical constraining power without
introducing significant systematic uncertainty. Upcoming surveys such as the
Vera Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et al.,
2019) and the Nancy Grace Roman Space Telescope High-Latitude Time-Domain
Survey (henceforth Roman; Spergel et al., 2015, Hounsell et al., 2018, Rose et
al., 2021) are poised to observe multiple orders of magnitude more supernovae
than our current largest compilations. As it becomes impossible to
spectroscopically follow-up every SN or its host galaxy, photometric
classification will increase the number of cosmologically suitable SNe.
However, cosmological analyses with these next-generation datasets will still
be limited if they rely on spectroscopic redshifts (spec-$z$) from host
galaxies. Therefore, efforts to develop the usability of photometric redshifts
(photo-$z$) for Ia cosmology, either from the SN, the host galaxy, or a
combination of the two, will be critical to optimize the potential of Stage IV
Dark Energy experiments.

Photometric redshift estimation is an area of active research, as other key
cosmological probe analyses already require redshifts for hundreds of millions
of galaxies for which it is impossible to obtain spectroscopic redshifts. In
weak gravitational lensing, these efforts are focused on characterizing the
redshift distributions of source and lens galaxy samples in several
tomographic bins, i.e. $n(z)$, to extreme precision, for example using Self-
Organizing Map photo-$z$ (SOMPZ; Buchs et al., 2019, Myles et al., 2021).
Other photometric redshift estimates are primarily concerned with the
performance of individual galaxy redshift estimates. Several data-driven
approaches make use of machine-learning methods to learn the mapping between
colors and redshift (e.g. Carrasco Kind & Brunner, 2013, Sadeh et al., 2016,
De Vicente et al., 2016), while others rely on templates of galaxy SEDs to
find the best fit redshift (e.g. Benítez, 2000, Feldmann et al., 2006, Brammer
et al., 2008). Each method is sensitive to different systematics that affect
the precision and accuracy of the final redshift estimates for different
galaxy populations to varying degrees. Each photo-$z$ estimate can be given as
a fully descriptive probability distribution function (PDF) but are frequently
simplified to a point estimate (mean or mode) and a Gaussian uncertainty.

Previous analyses have illustrated the potential for using photometric
redshifts for Type Ia Supernova cosmology. Chen et al. (2022) performed an
analysis using a subset of the DES-SN5YR sample containing SNe with redMaGiC
(a selection of Luminous Red Galaxies; Rozo et al., 2016) host galaxies. These
galaxies have particularly well-constrained photometric redshifts, with
$\frac{\sigma_{z}}{1+z}<0.02$, but comprise only $\sim$6% of the DES-SN5YR
sample. Using this sample of $\sim$125 SNe, they found that using the redMaGiC
host galaxy photo-$z$ directly in place of the spectroscopic redshift results
in a $w$-shift of $\sim 0.005$, and this result was validated with consistent
results from detailed catalog-level simulations. However, this analysis did
not model the effects of core-collapse contamination. redMaGiC host galaxies
are expected to have low to zero rates of core-collapse supernovae, as they
are old, passive galaxies, compared to the star-forming galaxies that core-
collapse progenitors occur in (Irani et al., 2022). This analysis was further
simplified by restricting the sample host galaxies to a single type, reducing
the need for modeling dependencies between the SN and host-galaxy properties.
Ruhlmann-Kleider et al. (2022) present an analysis using a combined sample of
the spectroscopic Joint Light-curve Analysis (JLA) SN Ia sample and the 3-year
photometric SuperNova Legacy Survey (SNLS) SN Ia sample. Using a mixture of
spectroscopic and photometric redshifts, they found that while a naive
analysis with photometric redshifts and contamination leads to biased matter
density ($\Omega_{\textrm{M}}$) for a Flat-$\Lambda$CDM model, the bias can be
corrected with an appropriately modeled magnitude bias correction computed for
selection effects.

An alternative to directly using the host-galaxy photo-$z$ is to infer the
photo-$z$ from the supernova light-curve. This can be done by extending the
SALT (Spectral Adaptive Light-curve Template; Guy et al., 2010) SN light-curve
model fitting framework such that the SN redshift is floated, rather than
fixed in the fit. These estimates are greatly improved by optionally including
a prior on the redshift from a host-galaxy photo-$z$ estimate (SN+host
photo-$z$: Kessler et al., 2010; Palanque-Delabrouille et al., 2010; Dai et
al., 2018). This photo-$z$ approach has two unique benefits: i) it combines
information from two independent photo-$z$ estimates, whereas other probes
rely on a single galaxy photo-$z$ and are thus more vulnerable to catastrophic
outliers, and ii) the redshift covariance is propagated to the other SALT
fitted parameters. This inferred photo-$z$ method was applied to the extended
DES-SN5YR sample classified without redshifts from Möller et al. (2024). They
used host photo-$z$ priors only when available and evaluated the biases of
this mixed sample. Using simulations they found structured offsets in redshift
and SALT2 parameter estimation that can be minimized by binning. However, they
did not evaluate the impact on cosmology. Mitra et al. (2023) presented an
analysis using a set of mock-data simulated LSST SNe from the Deep Drilling
Fields and found that supplementing the subset of SNe with spec-$z$ with
SN+host photo-$z$ increases the $w_{0}w_{a}$ figure-of-merit by 50%. In
particular, this helps extend the upper limit of the redshift range of usable
SNe, as obtaining spectroscopy becomes less feasible for fainter sources.
While this LSST analysis included a statistical+systematic covariance matrix
including calibration uncertainties, it did not include non-Ia contaminants or
intrinsic scatter systematics.

In this work, we present an evaluation of cosmological biases for the
Flat-$w$CDM model using data from the DES-SN5YR photometrically classified SN
sample and SN+host galaxy photometric redshifts. We also present an
implementation of SN+host photo-$z$ that utilizes a full photo-$z$ estimate
PDF from the host galaxy rather than a point estimate with Gaussian
uncertainty. To better evaluate our methodology, we focus solely on the subset
of DES-SN5YR SNe with host spectroscopic redshifts and thus do not evaluate
the statistical impact of a 5YR sample that includes events without a
spectroscopic redshift. For the anchoring low-$z$ sample we use spectroscopic
redshifts only.

The outline of the paper is as follows. In Section 2, we detail the DES-SN5YR
photometric supernova sample, the anchoring low-$z$ sample, and the associated
host-galaxy photometric redshift estimates and properties used for the
analysis. In Section 3, we detail the catalog-level simulations used to
validate the analysis and compute bias corrections for cosmology. In Section
4, we describe the analysis framework and cosmological parameter inference
formalism. In Section 5, we present the results of the analysis for
simulations and data. Finally, we discuss future prospects in Section 6 and
conclude in Section 7.

##  2 Data

###  2.1 The Dark Energy Survey

We use data from the Dark Energy Survey Supernova Program, which observed for
five seasons with the Dark Energy Camera (DECam; Flaugher et al., 2015) on the
4 meter Blanco telescope at Cerro Tololo Inter-American Observatory in Chile.
DES consists of two programs: a wide field survey optimized for weak
gravitational lensing, galaxy clustering, and galaxy cluster cosmology, and a
time-domain survey primarily for SN cosmology. The ten SN fields lie within
the footprint of the wide field but were observed with a much higher cadence
to detect transients. For both programs, raw images were pre-processed by the
DES Data Management team (Morganson et al., 2018) to produce calibrated images
and catalogs.

####  2.1.1 DES SN Data

The DES SN fields consist of ten 2.7 square degree fields, which were observed
in the griz filters at a cadence of $\sim 7$ days. Of these ten fields, two
‘deep’ fields (X3,C3) were observed to a single-visit depth of 24.5 mag in
each band, and eight ‘shallow’ fields (X1,X2,C1,C2,E1,E2,S1,S2) were observed
to a depth of 23.5 mag. To identify transient candidates, images were
processed through DIFFIMG (Kessler et al., 2015), the DES difference imaging
pipeline. Imaging artefacts were further removed using the AUTOSCAN algorithm
(Goldstein et al., 2015). A candidate light-curve was stored and updated if
there were at least two detections within 1 arcsec, separated by at least 1
but not more than 30 days (Kessler et al., 2015).

While DIFFIMG delivered real-time SN photometry with $\sim 2\%$ precision, our
DES-SN5YR cosmology analysis includes a more precise Scene Modeling Photometry
(SMP; Holtzman et al., 2008, Astier et al., 2013, Brout et al., 2019),
simultaneously modeling the time-varying SN flux and the static host-galaxy
flux. With SMP, 19,706 transient candidates were analyzed. Sánchez et al.
(2024) provides further details on the DES-SN5YR photometry. Corrections for
Differential Chromatic Refraction (DCR; Filippenko, 1982), which causes
wavelength-dependent effects on the SN PSF shapes and positions, were computed
for the DES-SN5YR sample (Lee et al., 2023) but are not included here, as they
are a subdominant effect on distance measurements.

Host galaxies are assigned based on the Directional Light Radius method (DLR;
Sullivan et al., 2006, Gupta et al., 2016) using a deep host galaxy library
built from coadds (Wiseman et al., 2020; Qu et al., 2024). While the DES-SN3YR
cosmological analysis (DES Collaboration, 2019) relied on a subset of
spectroscopically typed SNe ($\sim 200$), the DES-SN5YR cosmological analysis
(DES Collaboration, 2024) uses a sample of photometrically classified SNe
($\sim 1600$). These SNe comprise the DES-SN5YR cosmology sample and are
selected using only their light-curves and host-galaxy spectroscopic redshifts
using SuperNNova (Möller & de Boissière, 2020; Möller et al., 2022). For host
galaxy spectroscopic redshifts, DES relied on a partner program (OzDES) on the
Anglo-American Telescope (AAT; Lidman et al., 2020) using the AAOmega
spectrograph. Details of the spectroscopic follow-up program are provided in
Smith et al. (2020).

To evaluate cosmological biases, we start with the cosmological sample from
Vincenzi et al. (2024) and restrict the sample for this analysis to SNe whose
host galaxies: i) have available spectroscopic redshifts (as a ‘truth’ point
of comparison) and ii) have available photometric redshifts (described in
Section 2.3).

###  2.2 Low-$z$ SN Samples

To measure cosmological parameters, we anchor the Hubble diagram with external
low-$z$ SN samples. We use 76 SNe from several smaller samples (CSP;
Krisciunas et al., 2017, CfA3; Hicken et al., 2009, CfA4; Hicken et al., 2012)
and 123 SNe from the Foundation sample (Foley et al., 2018), totaling 199. For
each of these low-$z$ samples, we use the spec-$z$ rather than a photo-$z$, as
spectroscopic redshifts are expected to be readily available for such samples
even for next generation surveys.

Figure 1: Top panels: host-galaxy photo-$z$ vs. spectroscopic redshifts for
the $\sim 14,000$ SN candidate host galaxies identified in Y3GOLD. Bottom
panels: binned redshift bias ($\overline{\Delta z}$) in dotted pink and
redshift scatter in dashed-dotted blue for the three photo-$z$ estimates used
as host-galaxy priors for the photo-$z$ in this analysis: SOMPZ (left), DNF
(middle), BPZ (right).

###  2.3 Host Galaxy Photometric Redshifts

To estimate host galaxy photo-$z$, we use $griz$ photometry from the wide-
field program rather than the deeper SN-field photometry. Additional near
infrared (NIR) photometry ($JHKs$) has been used in a small subset of fields
for photo-$z$ training (Hartley et al., 2022), but we do not use the NIR
photometry in our analysis. While SN contamination can potentially bias the
photometry, the $\sim$ month long transient duration is small compared to the
30 months of co-added images.

We use host galaxy photo-$z$ estimates from three different methods to capture
some of the diversity of photo-$z$ algorithms, as each approach is sensitive
to different systematics. We focus on i) Self Organizing Map $p(z)$ (SOMPZ;
Buchs et al. 2019, Myles et al. 2021), ii) Bayesian photo-$z$ (BPZ; Benítez
2000), and iii) Directional Neighbourhood Fitting (DNF; De Vicente et al.
2016). For SOMPZ, we consider both point estimates with Gaussian uncertainty,
as well as the full PDF ($p(z)$). For BPZ and DNF, we consider only the point
estimates with Gaussian uncertainty.

Because of efforts from wide field static science probes, these host-galaxy
photometric redshifts are available from the Y3GOLD ‘value-added’ catalog
(Sevilla-Noarbe et al., 2021) and for the simulated galaxy catalogs we use to
build our host-galaxy library (see Section 3.2).

To evaluate the performance of photometric redshift point estimates, we
consider the following standard metrics:

  * •

photo-$z$ bias: $\Delta z\equiv z_{\textrm{photo}}-z_{\textrm{spec}}$

  * •

photo-$z$ scatter (Normalized Mean Absolute Deviation):

$\hskip 45.0pt\sigma_{\rm NMAD}\equiv
1.48\times\textrm{Median}\left(\frac{\left|\Delta z-\overline{\Delta
z}\right|}{1+z_{\textrm{spec}}}\right)$ (1)

  * •

fraction of outliers $\eta$ with $\left|\Delta z\right|>0.1$

####  2.3.1 Self-Organizing Map $p(z)$ (SOMPZ)

SOMPZ is one of three independent photo-$z$ methods used as part of the
overall DESY3 3x2 pt analysis redshift scheme, along with clustering redshifts
(Gatti et al., 2022) and shear ratios (Sánchez et al., 2022). The SOMPZ method
leverages information from subsamples of galaxies with deep 8-band photometry
(from the DES deep fields; Hartley et al., 2022) and secure redshifts (from
public and private spectroscopic and many-band photometric catalogs) to
determine the $n(z)$ of a broader source galaxy sample with only wide field
photometry in (g)riz. A self-organizing map is constructed for both the wide
field photometry and deep field photometry; i.e. each galaxy is assigned a
phenotype based on its colors. A synthetic source injection software (BALROG;
Everett et al., 2022) is used to inject simulated deep field galaxies into
real wide field images. These wide-deep pairings are then used to weight the
secure redshift information and constrain the $n(z)$ to 0.01 on the mean
redshift in four tomographic bins. Further details of the method are provided
in Myles et al. (2021). This method is designed specifically to precisely
calibrate the overall redshift distribution of a galaxy sample to weak lensing
cosmology requirements, rather than to constrain an individual galaxy point
estimate or $p(z)$ PDF. As an example, the DESY3 SOMPZ analysis was restricted
to riz fluxes due to constraints on using $g$-band data specific to the DES
lensing analysis.

For each individual galaxy, we assign the weighted wide field $p(z)$ for the
assigned phenotype, i.e. galaxies in the same SOM cell will have the same
$p(z)$. We use the SOMPZ estimates for the DESY3 weak lensing source galaxy
catalog. Both the SOMPZ $p(z)$ and point estimates are considered in our
analysis, but performance metrics are given only for point estimates. In the
left panel of Figure 1 we show the redshift bias and scatter for the SOMPZ
point estimates on the subset of Y3GOLD galaxies also detected as SN candidate
hosts. The median photo-$z$ bias across redshift bins is 0.023 and the median
photo-$z$ scatter is 0.071. The outlier fraction is 0.337.

####  2.3.2 Directional Neighborhood Fitting photo-$z$ (DNF)

To estimate photometric redshifts, the DNF machine-learning algorithm (De
Vicente et al., 2016) combines a neighbourhood fitting (NF) estimator with a
modified neighborhood metric (Directional Neighborhood). In brief, the NF
approach allows the algorithm to not only identify galaxies with similar
observables (i.e. magnitudes) as neighbors, but also galaxies with similar
relative observables (i.e. colors). The Directional Neighborhood is defined as
the product of the Euclidean and Angular Neighborhoods, i.e. the Euclidean
distance and angle between vectors in multi-magnitude space. DNF combines
these concepts and constructs a best-fit hyperplane to the Directional
Neighborhood of a given galaxy to produce a photo-$z$ $p(z)$. The redshift
bias and scatter are shown in the center panel of Figure 1 and the median bias
and scatter across redshift are 0.000 and 0.020 respectively. The outlier
fraction is 0.106. By these established metrics, DNF outperforms both SOMPZ
and BPZ, although works such as Schmidt et al. (2020) emphasize that photo-$z$
metrics should be specific to a given science case. We note that the training
sample used for the DES DNF estimates include OzDES spec-$z$, which were
primarily obtained for SN candidate host galaxies. The negligible bias
reported here is therefore likely attributable to the overlap between our SN
host galaxies and the DNF training sample, rather than a direct measure of the
algorithm’s performance.

####  2.3.3 Bayesian Photometric Redshifts (BPZ)

BPZ is a template-fitting photo-$z$ estimator that produces point estimates
and Bayesian posteriors (Benítez, 2000). The redshift posterior is obtained by
calculating the $\chi^{2}$ likelihood of a set of galaxy photometry given a
galaxy template at a given redshift and marginalizing over six base galaxy
model templates. A magnitude-dependent luminosity prior is applied for each
galaxy based on the redshift-evolving luminosity functions for elliptical,
spiral, and star-burst galaxies. The estimates used for this analysis are
taken from the DES Y3GOLD catalog, and further details on the DES-specific
code and calibration are found in Hoyle et al., 2018. The redshift bias and
scatter are shown in the right panel of Figure 1 and the median bias and
scatter across redshift are -0.002 and 0.041 respectively. The outlier
fraction is 0.191.

##  3 Simulations

To validate our analysis and to calculate bias corrections for known selection
effects in our sample, we use catalog-level simulations generated with the
SuperNova ANAlysis Software (SNANA; Kessler et al., 2009) and orchestrated
with the Pippin (Hinton & Brout, 2020) software. As we include a full
treatment of core-collapse contamination in the analysis, the simulations are
also used to train the photometric classifier, as well as to model the core-
collapse likelihood (see Section 4.3).

###  3.1 Simulation Overview

We generate a modified version of the simulations built for the main DES-SN5YR
analysis (Kessler et al., 2019; Vincenzi et al., 2021; DES Collaboration,
2024). An overview of the simulation process is as follows: first, a time-
varying source SED is generated based on the SALT3 (Guy et al., 2007, 2010;
Kenworthy et al., 2021) spectro-photometric light-curve model, and
cosmological and astrophysical effects are applied (redshift, cosmological
dimming, lensing, extinction, peculiar velocities). The top-of-atmosphere SEDs
are integrated across survey filters to obtain the observed fluxes. Second,
measurement noise is added according to the survey observing conditions (PSF,
sky noise, zeropoints). Lastly, the detection trigger is applied based on
survey characteristics such as detection efficiency vs. SNR.

To model the parent populations (SN stretch and color) as a function of host-
galaxy stellar mass, we use the parameters from the main DES-SN5YR analysis
(Vincenzi et al., 2024), which are obtained following the method described in
Popovic et al. (2021). To model the intrinsic scatter $\sigma_{\rm int}$, i.e.
the post-standardization scatter in Hubble residuals, we use a dust-based
model (Brout & Scolnic, 2021) with parameters constrained with the Dust2Dust
software (Popovic et al., 2023). This results in fitted distributions of
intrinsic color $c_{\rm int}$, intrinsic color-luminosity relation $\beta_{\rm
int}$, and extinction model parameters ($R_{V}$ and $E(B-V)$) for high and low
mass galaxies. The modeling is done separately for both the low-$z$ and DES
samples.

For non-Ia SN populations, we use templates, rates, and luminosity functions
for simulations as detailed in Vincenzi et al. (2021, 2023). These include
peculiar SN Ia populations (91bg-like, Iax; Kessler et al., 2019 and
references therein) and core-collapse SNe (stripped-envelope, hydrogen-rich;
Vincenzi et al., 2019).

SN host galaxies are simulated and associated using a galaxy catalog (or host-
galaxy library; HOSTLIB) which contains at minimum a galaxy redshift and is
detailed in the following section.

###  3.2 Host Galaxy Library

The nominal DES-SN5YR analysis uses a HOSTLIB built with data (Qu et al.,
2024) which contains all the galaxies detected using deep coadds (Wiseman et
al., 2020). In this work we instead use a simulation-based HOSTLIB which
allows us to consider multiple photo-$z$ algorithms without having to rerun
photo-$z$ estimation codes. To accurately capture the multi-dimensional
relationship between galaxy colors, magnitudes, and photo-$z$ performance, we
rely on the Buzzard suite of N-body simulations (DeRose et al., 2019, 2022) to
generate our host-galaxy library. This set of N-body simulations was designed
to mock the DESY3 source and lens galaxy samples and SOMPZ, DNF, and BPZ have
previously been run on the resulting catalog photometry. As a result, we do
not need to analytically model the photo-$z$ performance ourselves. We use the
v0.25 SOMPZ run on Buzzard v2.0.0 and use one million random galaxies and
their properties from the catalog for the simulation HOSTLIB. We do not
include host galaxy flux contribution to the SN Poisson noise.

To model SN dependencies on host galaxy properties, we focus on the host
galaxy stellar mass. We use the Buzzard host galaxy photometry and follow the
Spectral Energy Distribution fitting method from Sullivan et al. (2010),
following templates and assumptions made as in DES Collaboration (2024), to
fit masses for the simulation HOSTLIB. We also associate SNe with their host
galaxies as a function of host galaxy stellar mass based on the weighting map
from Vincenzi et al. (2021) and Wiseman et al. (2021).

###  3.3 Redshift Efficiency

Because we require both a host spec-$z$ to evaluate biases as well as a host
photo-$z$, we pay careful attention to model the efficiency of obtaining both
redshifts. Normally the redshift efficiency would be more lax (i.e. be
efficient to fainter magnitude) for photo-$z$ than for spec-$z$. However,
because we require a photo-$z$ from the wide-field Y3GOLD catalog, which is
shallower than the DES-SN survey, our photo-$z$ efficiency is actually
stricter (i.e. drops off at brighter magnitude) than the spec-$z$ efficiency.
While the spec-$z$ efficiency is typically modeled with a “host efficiency”
map (Vincenzi et al., 2021), here the photo-$z$ efficiency is determined by
the magnitude distribution of simulated Buzzard galaxies combined with the
depth of the Y3GOLD catalog. In other words, the photo-$z$ efficiency is built
into the simulations by the availability of photo-$z$ for galaxies in Buzzard
and we do not explicitly model a spec-$z$ efficiency.

In Figure 2 we show the $r$-band magnitude distribution for the host galaxies
in the DES-SN5YR sample (black histogram) compared to the magnitude
distribution for DES-SN5YR hosts which are also detected in Y3GOLD (red
histogram). This illustrates how the sample used for this analysis is skewed
brighter than the nominal DES-SN5YR SN sample due to the requirement of wide
field detection for photo-$z$.

Figure 2: r-band magnitude distributions for host galaxies of the final DES-
SN5YR SN sample (black; DES Collaboration, 2024) and the DES-SN5YR host
galaxies which are detected in the Y3GOLD catalog (red).

##  4 Analysis

Here we detail the analysis pipeline from light-curve fitting (Section 4.1) to
final cosmology constraints (Section 4.8).

Figure 3: Top panels: SN+host photo-$z$ estimates vs. true spectroscopic
redshifts. A one-to-one relation is given in blue dashed line. Bottom panels:
the dotted pink lines show the binned redshift bias ($\overline{\Delta z}$)
and the dashed-dotted blue lines show the redshift scatter (defined in Sec.
2.3) across redshift bins for each SN+host photo-$z$ variant. Figure 1 shows
the equivalent for host only photo-$z$. The host only redshift scatter from
Figure 1 is plotted in dashed-dotted orange lines for comparison.

###  4.1 Light-curve Fitting

To measure SN distances, we use the SALT3 light-curve model framework
(Kenworthy et al., 2021), based on SALT2 (Guy et al., 2010), to fit for
standardization parameters. The model is parametrized by the following
parameters: $z$ (redshift), $x_{0}$ or $m_{x}$ (overall amplitude, with
$m_{x}=-2.5\log_{10}(x_{0})$), $t_{0}$ (time of peak brightness), $x_{1}$
(stretch), and $c$ (color). To obtain the distance modulus $\mu$, we use the
Tripp estimator (Tripp, 1998):

$\mu=m_{x}+\alpha x_{1}-\beta
c-M-\delta\mu_{\textrm{bias}}+\delta\mu_{\textrm{host}}$ (2)

where $M$ is the absolute magnitude of a SN Ia with $c=0$, $x_{1}=0$,
$\alpha$, $\beta$ are coefficients parametrizing the stretch-luminosity and
color-luminosity relations, and $\delta\mu_{\textrm{bias}}$ is the bias
correction applied to distances (see Section 4.6). The last term,
$\delta\mu_{\textrm{host}}$, is a correction for additional host-galaxy
property and SN Ia property dependencies, where here we use host-galaxy
stellar mass $M_{*}$. This correction is defined as a step function:

$\delta\mu_{\rm host}=\begin{cases}+\gamma/2&\quad\text{if
}M_{*}>10^{10}M_{\odot},\\\ -\gamma/2&\quad\text{otherwise,}\end{cases}$ (3)

where $\gamma$ is the size of the ‘mass step,’ and $10^{10}M_{\odot}$ is the
location of the ‘mass step’ (Kelly et al., 2010; Lampeitl et al., 2010;
Sullivan et al., 2010). For light-curve fitting, we use the implementation in
SNANA, which uses a $\chi^{2}$ minimization to obtain best-fit parameters and
uncertainties.

###  4.2 SN+host Photometric Redshifts

For this analysis, we do not use the host galaxy photo-$z$ directly in place
of the spec-$z$ but rather to help inform a photo-$z$ fitted from the SN
light-curve. The SALT light-curve model framework can be extended to include a
fitted photometric redshift from the SN photometry; i.e. by floating the
redshift rather than fixing it in the fit (Kessler et al., 2010). A host
galaxy redshift prior can also be provided to improve the redshift estimates.
Previous studies (Mitra et al., 2023) have included Gaussian host galaxy
redshift priors (Graham et al., 2018), with the median or average of the PDF
as the mean of the Gaussian, and the RMS as the width of the Gaussian. Here,
we also implement in the SNANA framework the ability to use a full host galaxy
photo-$z$ PDF as a prior, which should provide a more principled estimate of
uncertainties and better encapsulate the degeneracies endemic to photo-$z$
estimation. These PDFs are read and stored in the format of 11 quantiles in
10% probability bins, to optimally preserve information and minimize storage
space (Malz et al., 2018). We consider the $p(z)$ only for SOMPZ redshift
estimates, as they were previously saved in the Y3 processing and did not need
to be recomputed. We include a photo-$z$ variant where the SOMPZ $p(z)$ is
approximated by a Gaussian to study the impact on the final SN+host photo-$z$
estimate. For BPZ and DNF, we consider only point estimates with Gaussian
uncertainties and refer to the final photo-$z$ estimates as SN+BPZ (Gaussian)
prior and SN+DNF (Gaussian) prior.

We note a subtlety related to the implementation of this photo-$z$ fitting in
SNANA; if the SED model range does not extend sufficiently into bluer
wavelengths, this may cause filter dropouts at high redshift. This can result
in artificially small $\chi^{2}$ values that cause low redshift SNe to be
pathologically fit with high redshift values. To avoid this problem, we use
the SALT3 extended wavelength model with fit range 2000-13000 $\AA$ and linear
extrapolation to zero flux at $\lambda=500\AA$.

In Figure 3 we show the SN+host photo-$z$ estimates plotted against true
spectroscopic redshifts. We show the redshift bias and scatter for each
SN+host photo-$z$ variant in magenta and blue in the bottom panels. We also
show the binned redshift scatter for each host-galaxy photo-$z$ prior (without
SN information) in orange in the bottom panels. Except for the case of SN+DNF
prior, the redshift scatter obtained from host galaxy photometry only is
reduced by $\sim 50\%$ when adding SN information. This indicates that adding
SN data significantly improves the photo-$z$ estimate. In the SN+DNF case, the
redshift scatter is not improved from the DNF host galaxy photo-$z$ because
the host photo-$z$ information is much more precise compared to the SN.

The best performing variant by the photo-$z$ bias and scatter metrics is
SN+DNF prior with the median redshift bias and scatter across redshift bins
being -0.003 and 0.019 respectively. We also find that the final SN+host
photo-$z$ is not significantly improved by using the full SOMPZ $p(z)$ prior
compared to a Gaussian approximation. We discuss other potential uses for the
$p(z)$ information in Section 6. In particular, we note that the SN+SOMPZ
$p(z)$ prior, SN+SOMPZ Gaussian prior, and SN $z$ fit with flat prior show a
redshift-dependent bias at $z\gtrsim 0.8$, which may be caused by asymmetric
migration due to the redshift cut at $z=1.2$ such that events with
$z_{\textrm{photo}}<z_{\textrm{spec}}$ remain in the sample but events with
$z_{\textrm{photo}}>z_{\textrm{spec}}$ are rejected.

###  4.3 Sample Selection

To select our sample, we apply standard cosmological quality cuts as follows:

  * •

fitted color $\left|c\right|<0.3$

  * •

fitted stretch $\left|x_{1}\right|<3.0$

  * •

fitted stretch uncertainty $\sigma_{x_{1}}<1.0$

  * •

fitted $t_{0}$ uncertainty $\sigma_{t_{0}}<2.0$ days

In Table 1 we show the number of SNe remaining after sequentially applying
each cut and for different host-galaxy photo-$z$ priors. In brackets, we show
the percentage of SNe lost from each cut. We note that the spec-$z$ sample is
not exactly identical to the main DES-SN5YR sample, due to several differences
including i) requiring host photo-$z$, ii) requiring photo-$z$ fit convergence
and iii) using an extended SALT3 model for light-curve fitting (see Section
4.2) to be consistent with the photo-$z$ cases.

|  $\\#$ SNe for Redshift Method [$\%$ cut]
---|---
Cut | spec-$z$ |  SN+SOMPZ
$p(z)$ prior |  SN+SOMPZ
Gaussian prior |  SN+DNF
Gaussian prior |  SN+flat prior |  SN+BPZ
Gaussian prior
Host-galaxy photo-$z$ available | 13,793 |  13,793 |  13,793 |  13,793 |  13,793 |  13,793
SALT3 fit converged | 2377 |  2427 |  2392 |  2392 |  2400 |  2392
SN color $\left|c\right|<0.3$ | 2087 [13.9%] |  2131 [13.9%] |  2114 [13.2%] |  2114 [13.2%] |  2132 [12.6%] |  2114 [13.2%]
SN stretch $\left|x_{1}\right|<3.0$ | 1902 [9.7%] |  1817 [17.3%] |  1857 [13.8%] |  1857 [13.8%] |  1777 [20.0%] |  1857 [13.8%]
SN stretch uncertainty $\sigma_{x_{1}}<1.0$ | 1623 [17.2%] |  1547 [17.5%] |  1562 [18.9%] |  1562 [18.9%] |  1491 [19.2%] |  1562 [18.9%]
SN $t_{0}$ uncertainty $\sigma_{t_{0}}<2.0$ | 1614 [0.6%] |  1537 [0.7%] |  1552 [0.6%] |  1552 [0.6%] |  1484 [0.5%] |  1552 [0.6%]
Valid bias correction | 1577 [2.3%] |  1481 [3.8%] |  1505 [3.1%] |  1506 [3.1%] |  1420 [4.5%] |  1474 [5.3%]
Final sample size | 1577 |  1481 |  1505 |  1506 |  1420 |  1474
Table 1: Summary Table of SN sample cuts (DES-only) Figure 4: Distributions of
spec-$z$, fitted SN light-curve parameters using spec-$z$ ($c$, $x_{1}$,
$m_{B}$), and their errors for data (black points) and simulations (blue
histogram). The simulation histograms are normalized to the number of data
points. Figure 5: Distributions of SN host-galaxy properties for data (points)
and simulations (blue histogram).

###  4.4 Data and Simulation Comparison

In Figure 4 we show the distributions comparing simulations and data for
fitted SALT3 light-curve parameters and their uncertainties, normalized to the
number of SNe in the data. We also compare the distributions for simulations
and data for host-galaxy properties in Figure 5. The agreement in host
magnitude in particular illustrates that the “photo-$z$ selection function” is
sufficiently reproduced, although the fainter tails for the simulation
redshift and $m_{x}$ distributions also indicate that this modeling can be
improved. We emphasize that correctly modeling and propagating the
relationship between galaxy properties and redshift biases is a crucial
component of the analysis. In Appendix A, we show the equivalent light-curve
parameter comparison plots for when the redshift is floated simultaneously
with varying host-galaxy priors. We manually scale down the simulated total
number of SNe, as well as the rate of core-collapse contamination, to better
match the data, as we are unable to replicate the same values from first
principles alone.

###  4.5 SN Classification

To classify our supernova sample, we use the photometric classifier SuperNNova
(SNN, Möller & de Boissière 2020), a recurrent neural network trained on
simulated light-curves. Previous analyses have rigorously evaluated the
performance of SNN for the DES-SN5YR analysis, with a prediction of less than
1.5 percent contamination in the final sample (Vincenzi et al., 2021; Möller
et al., 2022). We train a model of SNN on our nominal Ia and non-Ia simulated
light-curves as detailed in Section 3.1.

###  4.6 BEAMS with Bias Corrections

We use an extension of the Bayesian Estimation Applied to Multiple Species
(BEAMS; Kunz et al., 2007), BEAMS with Bias Corrections (BBC; Kessler &
Scolnic, 2017) to address core-collapse contaminants and to correct for known
selection effects and Malmquist bias for our SN sample. The BEAMS method
incorporates probabilistic information from a photometric classifier to
marginalize over core-collapse contamination. In the BEAMS framework, the
likelihood is modeled with two terms, one for the SN Ia population (modeled
analytically), and one for core-collapse contaminants (empirically modeled
using simulations; see Section 3.1). Using SNN, each SN in the sample is
assigned a probability of being a Ia, with zero being non-Ia and one being
likely-Ia. These probabilities are then used to weight contributions to the
Hubble diagram.

BBC extends the BEAMS framework and incorporates the method from Marriner et
al. (2011) to determine standardization nuisance parameters in a cosmology-
independent way. The BBC fit determines global nuisance parameters ($\alpha$,
$\beta$, $\gamma$, $\sigma_{\rm int}$) and also determines bias-corrected
distances in redshift bins to create a binned Hubble diagram. When including
systematic uncertainties, an unbinned Hubble diagram approach results in
smaller uncertainties than a binned approach (Brout et al., 2022; Kessler et
al., 2023), but for a statistical uncertainty only approach, there is no
difference. As we do not include a full systematic covariance matrix in this
analysis, we use the binned BBC approach rather than unbinned or re-binned.

For bias corrections, BBC applies a distance correction based on the true and
measured distances for a large set of survey-specific simulations
($\delta\mu_{\rm bias}$ in Equation 2). We use the BBC-4D framework (Popovic
et al., 2021), which computes the bias corrections in bins of
{$z,x_{1},c,\log{M_{*}}$} to be compatible with dust-based models. In our
analysis, we use the respective measured redshift in each case (e.g. spec-$z$,
SN+SOMPZ Gaussian prior, SN+DNF Gaussian prior, etc.) to compute the
corrections. The bias corrections for the low-$z$ sample are computed
separately following Scolnic et al. (2018), Kessler et al. (2019), and DES
Collaboration (2024). We also compute the bias correction $\sigma_{\rm int}$
separately by sample to prevent an overinflated $\sigma_{\rm int}$ for the
low-$z$ and Foundation samples, which use spec-$z$.

###  4.7 Distance Uncertainties

The individual distance modulus uncertainty for a SN in BBC (with the $i$
index ignored for clarity) is given by Kessler & Scolnic (2017); Brout et al.
(2022):

$\displaystyle\sigma_{\mu}^{2}=$ $\displaystyle f(z,c,M_{*})\sigma_{\rm
SALT3}^{2}+(\sigma_{\mu}^{\textrm{vpec}})^{2}+(\sigma_{\mu}^{z})^{2}+\sigma_{\rm
lens}^{2}$ (4)

The light-curve fit parameter uncertainty $\sigma_{\rm SALT3}^{2}$ is given
by:

$\displaystyle\sigma_{\rm
SALT3}^{2}=C_{m_{B},m_{B}}+\alpha^{2}C_{x1,x1}+\beta^{2}C_{c,c}+2\alpha
C_{m_{B},x1}$ (5) $\displaystyle-2\beta C_{m_{B},c}-2\alpha\beta C_{x1,c},$

where $C$ is the fitted covariance matrix for the light-curve parameters.
$\sigma_{\mu}^{\textrm{vpec}}$ is the contribution from the peculiar velocity
uncertainty $\sigma_{\textrm{vpec}}$ given by:

$\sigma_{\mu}^{\textrm{vpec}}=\left(\frac{5}{\ln(10)}\right)\frac{1+z}{z(1+z/2)}\left(\frac{\sigma_{\textrm{vpec}}}{c}\right),$
(6)

as explained in Davis et al. (2011), and $\sigma_{\mu}^{z}$ is the
contribution from the redshift uncertainty $\sigma_{z}$. We set
$\sigma_{\mu}^{z}=0$, as $\sigma_{z}$ already inflates uncertainties on
$m_{x},x_{1}$, and $c$ when redshift is floated in the light-curve fit;
therefore its inclusion causes an overestimated uncertainty due to the
correlated color error (see Chen et al., 2022 Appendix A.1 for details). The
$f(z,c,M_{*})$ error scaling factor is introduced following the original BBC
and determined from simulations such that rms($(\mu-\mu_{\rm
true})/\sigma_{\mu})=1$ in {$z,c,M_{*}$} bins. The $f(z,c,M_{*})$ scaling is
needed because the naively computed uncertainty does not account for Malmquist
bias that removes faint events and thus reduces the scatter (and uncertainty)
at higher redshifts.

We do not include an error floor term $\sigma_{\rm floor}^{2}(z,c,M_{*})$ that
was included in the main DES-SN5YR analysis. We discuss the subtleties related
to this distance uncertainty error scaling when using photo-$z$ in Section 5.
Details concerning the justification and implementation of the error
corrections can be found in Kessler & Scolnic (2017) and Vincenzi et al.
(2024).

###  4.8 Cosmological Parameter Inference

To constrain cosmological parameters, the $\chi^{2}$ of the SN likelihood is
given as

$\chi^{2}=\Delta\mu^{T}\cdot C_{\rm stat}^{-1}\cdot\Delta\mu,$ (7)

where $\Delta\mu$ is the data vector of distance measurements
$\\{\mu_{\textrm{obs,}i}-\mu_{\textrm{theory,}i}(\Omega_{M},w)\\}_{i=1,...,N_{\textrm{SNe}}}$
and $C_{\rm stat}$ is the statistical-only uncertainty covariance matrix as
defined in DES Collaboration (2024).

We use the “wfit” $\chi^{2}$ minimization program as implemented in SNANA with
a prior from the Cosmic Microwave Background (CMB) R-shift parameter, which is
derived from simulations with the same underlying cosmology as our simulated
samples. We fix the R-shift parameter uncertainty to $\sigma_{R}=0.006$
following the constraining power of Planck Collaboration et al. (2020).

##  5 Results

We perform an identical analysis following the framework presented in Section
4 using spectroscopic redshifts and using photometric redshifts, where the
photo-$z$ variations are:

  * •

SN + SOMPZ $p(z)$ prior

  * •

SN + SOMPZ Gaussian prior

  * •

SN + DNF Gaussian prior

  * •

SN + BPZ Gaussian prior

  * •

SN + flat prior

To isolate the biases that result from using photo-$z$, we consider the
difference in constraints on $w$ when using photo-$z$ and spec-$z$ defined as:

$\Delta w\equiv w_{\textrm{photo-$z$}}-w_{\textrm{spec-$z$}},$ (8)

where $w_{\textrm{spec-$z$}}$ is the constraint obtained using spectroscopic
redshifts and $w_{\textrm{photo-$z$}}$ is the constraint obtained using a
given photometric redshift variant.

We note that the set of SNe used to obtain $w_{\textrm{spec-$z$}}$ and
$w_{\textrm{photo-$z$}}$ are not identical (see Table 1). We choose not to use
a uniform SN sample across photo-$z$ variations, as in the future we will not
have spec-$z$ for all SNe and therefore will not be able to approximate a
selection based on which SNe would pass cuts when using spec-$z$. Instead, we
take a realistic approach and allow each redshift case to be selected by our
conventional cuts (see Table 1). In other words, when calculating $\Delta w$
values the spec-$z$ sample size is fixed to the larger value while the
photo-$z$ sample sizes vary.

To validate our analysis methods, we verify with simulations that when using
spec-$z$ with identically generated bias corrections, we recover the true
cosmology of $w=-1$ to within 0.01. In Section 5.1 we discuss the impact of
using photo-$z$ on the SN light-curve parameters and BBC nuisance parameters.
In Section 5.2 we present results for simulations and in Section 5.3 we
present results for data. In Section 5.4 we describe subtleties related to
error scaling for distance modulus uncertainties.

| Data | Simulations Ia+CC
---|---|---
Redshift | $\alpha$ | $\beta$ | $\gamma$ | $\sigma_{\rm int}$ | $\alpha$ | $\beta$ | $\gamma$ | $\sigma_{\rm int}$
spec-$z$ | 0.159$\pm$0.004 | 3.274$\pm$0.034 | 0.026$\pm$0.009 | 0.102 | 0.140$\pm$0.003 | 3.087$\pm$0.046 | -0.015$\pm$0.009 | 0.078
SN+SOMPZ $p(z)$ prior | 0.138$\pm$0.000 | 2.821$\pm$0.002 | 0.037$\pm$0.010 | 0.162 | 0.142$\pm$0.005 | 3.089$\pm$0.063 | -0.014$\pm$0.015 | 0.157
SN+SOMPZ Gaussian prior | 0.137$\pm$0.004 | 2.885$\pm$0.004 | 0.033$\pm$0.010 | 0.153 | 0.142$\pm$0.006 | 3.097$\pm$0.061 | -0.015$\pm$0.016 | 0.150
SN+DNF Gaussian prior | 0.146$\pm$0.000 | 3.027$\pm$0.000 | 0.004$\pm$0.009 | 0.131 | 0.142$\pm$0.004 | 3.085$\pm$0.051 | -0.012$\pm$0.013 | 0.118
SN+flat prior | 0.147$\pm$0.003 | 2.972$\pm$0.031 | 0.009$\pm$0.010 | 0.175 | 0.142$\pm$0.005 | 3.066$\pm$0.055 | -0.018$\pm$0.013 | 0.169
SN+BPZ Gaussian prior | 0.156$\pm$0.004 | 3.076$\pm$0.044 | -0.019$\pm$0.010 | 0.198 | 0.143$\pm$0.005 | 3.065$\pm$0.039 | -0.017$\pm$0.012 | 0.202
Table 2: BBC nuisance parameter values for data and simulations

###  5.1 Impact on light-curve and nuisance parameters

In Figure 6 we show how fitted light-curve parameters, their errors, and their
covariances, are changed as a function of the photo-$z$ bias for the entire
data sample. In Table 2, we show the fitted BBC nuisance parameters
$\alpha,\beta,\gamma$, and $\sigma_{\rm int}$ for each redshift case, for data
and simulations. The errors for data are given from the BBC fit, while the
errors for simulations are given as the standard deviation of the 25
simulation instances. The true input $\alpha$ and $\gamma$ are 0.145 and 0
respectively, while the input $\beta$ is not comparable with the fitted
$\beta$ (see Popovic et al., 2023 for further explanation on the difference
between $\beta_{SN}$ and $\beta_{\rm SALT}$).

Most notably, biases in redshift ($\Delta z$) are strongly correlated with
biases in SN color ($\Delta c$). As a result, the color-luminosity relation
$\beta$ is also biased with respect to the values obtained using spec-$z$. As
explained in Chen et al. (2022), these effects do not cause a bias in
cosmology because the $\Delta\mu$ calculated from $\Lambda$CDM for small
$\Delta z$ values is roughly equal to $-\beta\Delta c$ such that SNe are self-
corrected toward the fiducial cosmology (see also Sec 5.3 of Mitra et al.,
2023). However, this effect is not propagated to uncertainty in nuisance
parameters and may lead to a drastically underestimated error on $\beta$ as is
seen for the data. Further, the recovered $\beta$ is consistent across each
redshift case in the simulations but fluctuates in the data.

We recover a mass step $\gamma$ ($\sim 0.03$) in the data nearly consistently
across redshift cases, whereas in the simulations, we recover a mass step
consistent with zero for each redshift. This is consistent with what is seen
in the DES-SN5YR analysis, indicating that a dust-based intrinsic scatter
model alone is not sufficient to completely explain the mass step (see also
Wiseman et al., 2022; Kelsey et al., 2023). We also note that $\alpha$ is
consistent between redshift cases in the simulations, but biased by $\sim
0.02$ when using photo-$z$ for the data.

For both data and simulations, the intrinsic scatter is notably increased when
using photo-$z$. While this is an indication that our method of accounting for
redshift uncertainty contribution to distance uncertainties could be improved,
we leave this investigation to future work.

Figure 6: Bias in fitted light-curve parameters ($m_{x},x_{1},c$), their errors ($m_{x}$ERR, $x_{1}$ERR, $c$ERR), and their covariances ($C_{x1,x0},C_{x1,c},C_{c,x0}$) as a function of $z$ bias in the data using SN+DNF Gaussian prior photo-$z$. | Data | Simulations Ia+CC | Simulations Ia Only
---|---|---|---
Redshift | $\Delta w$ | $\sigma_{w}$ | $\Delta w$ |  $\Delta w$ Error | $\sigma_{w}$ |  $\Delta w$ RSD | $\Delta w$ |  $\Delta w$ Error | $\sigma_{w}$ |  $\Delta w$ RSD
spec-$z$ | 0.000 | 0.023 | 0.000 | 0.000 | 0.020 | 0.000 | 0.000 | 0.020 | 0.000 | 0.000
SN+SOMPZ $p(z)$ prior | -0.019 | 0.026 | -0.007 | 0.003 | 0.024 | 0.014 | -0.005 | 0.024 | 0.003 | 0.014
SN+SOMPZ Gaussian prior | -0.002 | 0.025 | -0.006 | 0.002 | 0.024 | 0.012 | -0.009 | 0.024 | 0.002 | 0.010
SN+DNF Gaussian prior | -0.017 | 0.025 | -0.003 | 0.002 | 0.022 | 0.009 | -0.002 | 0.022 | 0.002 | 0.009
SN+flat prior | -0.035 | 0.027 | 0.004 | 0.002 | 0.034 | 0.011 | 0.013 | 0.034 | 0.003 | 0.015
SN+BPZ Gaussian prior | -0.038 | 0.029 | 0.007 | 0.003 | 0.025 | 0.013 | 0.006 | 0.025 | 0.002 | 0.012
Table 3: Difference in $w$ ($\Delta w$) values between spec-$z$ and each
photo-$z$ case Figure 7: Hubble diagram from analyzing simulated data (upper
panels) and Hubble residuals (lower panels; w.r.t a flat $\Lambda$CDM
cosmology) using spec-$z$ (left) and SN+DNF prior photo-$z$ (right). DES SNe
are shown in blue, and low-$z$ SNe are shown in red. The Hubble residual means
in redshift bins are overplotted in the lower panels as triangles. The DES-SN
sample is color coded by probability of being an SN Ia as given by SNN. Figure
8: $\Delta w=w_{\textrm{photo-$z$}}-w_{\textrm{spec-$z$}}$ values for
simulations with Ia+CC SNe (black circle) and Ia Only (magenta triangle) with
the standard error on $\Delta w$. $\Delta w$ values for data are shown as blue
X’s with standard deviation (RSD in Table 3) from 25 instances of simulations.
The greyed area indicates photo-$z$ cases which have significantly discrepant
results between the simulations and data.

###  5.2 Cosmology Results from Simulations

For the spec-$z$ and each photo-$z$ case, we generate 25 statistically
independent Hubble diagrams with bias corrected distances, with $\sim$ 1450
SNe in the simulated samples. In the top panels of Figure 7, we show one of
the 25 Hubble diagrams when using spec-$z$ and the SN+DNF prior photo-$z$,
which has the lowest redshift scatter. In the bottom panel we plot the binned
and unbinned Hubble residuals, the difference between our measured distances
and distances from the best fit $w$CDM cosmology, as a function of redshift.
The spec-$z$ case has Hubble scatter, i.e. the rms scatter of the Hubble
residuals, of 0.152 while the SN+DNF prior photo-$z$ has Hubble scatter 0.176.
The Hubble diagrams for the remaining photo-$z$ cases are shown in Appendix B,
along with the Hubble scatter in each panel. We find that the Hubble scatter
is larger than the spec-$z$ case across each photo-$z$ method as expected.

We provide a summary of the resulting $\Delta w$ values (with respect to
spec-$z$) in Table 3 for simulations including non-Ia SNe (Ia+CC, middle
section) and for Ia-only simulations (right section). Each $\Delta w$ value is
averaged over the 25 statistically independent realizations of the simulation.
The robust standard deviation (RSD; $1.48\times\rm Median(\left|\Delta
w\right|)$) and standard error (RSD/$\sqrt{25}$) of the $\Delta w$ values are
also given for each case. We also report the average uncertainty on $w$,
$\sigma_{w}$, as a point of comparison for the significance of the given
$\Delta w$ values. We find $\Delta w$ values $<|0.01|$ for each photo-$z$
method. In particular, SN+DNF prior, SN+SOMPZ $p(z)$ and SN+SOMPZ Gaussian
prior are able to recover the true $w$ to $-0.003\pm 0.002$, $-0.007\pm
0.003$, and $-0.006\pm 0.002$ respectively. We further discuss in Section 5.3
why SN+flat prior and SN+BPZ prior are disfavored despite their small $\Delta
w$ values.

Given that the SN+SOMPZ $p(z)$ and SN+SOMPZ Gaussian prior estimates have
higher redshift scatter as seen in Figure 3, it is somewhat unexpected that
their use results in a small bias. SOMPZ estimates are also calibrated
specifically to provide unbiased redshift distributions for wide tomographic
bins (and as such the $p(z)$ for individual galaxies are principled but wide).
Our ability to infer unbiased cosmology using SOMPZ estimates in our current
framework, which requires a point estimate redshift, is of note. We discuss
further potential applications for $p(z)$ in Section 6. We also note that
using SN+SOMPZ $p(z)$ and SN+SOMPZ Gaussian prior result in significantly more
lost events compared to the other photo-$z$ cases.

####  5.2.1 Photometric Classification

To understand the interacting effects of photometric classification with
photo-$z$, we consider simulations in which each SN is classified perfectly,
i.e. cosmology is constrained only using SNe Ia. For simulations containing
only SNe Ia, the $\Delta w$ values shift only marginally compared to
simulations with both Ia+CC, on average across cases by $-0.0008$. While the
$\Delta w$ values for Ia+CC sims are consistently larger than for Ia only
sims, the shifts are small and comparable to those reported in DES
Collaboration (2024). In Figure 8, we show the $\Delta w$ values and their
standard errors for the Ia+CC sims and Ia only sims for each photo-$z$
variant. In each case, except for SN+flat prior, the change in $\Delta w$ with
and without CC SNe are within $1\sigma$ of each other, showing that core-
collapse contamination is a minor systematic, even when combined with
photometric redshifts.

###  5.3 Cosmology Results from Data

Using the same analysis pipeline as the simulations, we generate Hubble
diagrams and fit cosmology using the dataset described in Section 2. We show
the Hubble diagrams for spec-$z$ and SN+DNF photo-$z$ in the top panels of
Figure 9 and Hubble residuals as a function of redshift in the bottom panels.
Hubble diagrams for the remaining photo-$z$ cases are given in Appendix B. As
we do not have truth values for classification in the data, the Ia-only
subsample is defined as SNe having SNN probability of being a Ia $>0.5$.

The Hubble scatter across each photo-$z$ case is larger than the spec-$z$ case
as expected from simulations. However, the Hubble scatter for the spec-$z$
case is 0.167, $0.015$ larger than the sims (0.152) and $0.017$ larger for the
Ia-only scatter (0.160 vs. 0.143). This trend in Hubble scatter is not seen
coherently for each photo-$z$ case, where the scatter is larger in only two
out of five cases in the data compared to the sims.

We find $\Delta w$ values of $-0.019$ and $-0.002$ for SN+SOMPZ $p(z)$ prior
and SN+SOMPZ Gaussian prior respectively and $-0.017$ for SN+DNF prior. In
Figure 8, we show the data $\Delta w$ values with an uncertainty which is
given by the standard deviation of 25 $\Delta w$ values from simulations. The
data $\Delta w$ values are consistent with simulations to $\sim 1\sigma$ for
these three redshift cases and subdominant to the overall uncertainty on $w$.

However, our data results for SN+flat prior and SN+BPZ prior are significantly
discrepant with our simulation results. We are unable to attribute this to any
particular cause but note that i) $\sigma_{\rm int}$ is higher for these two
cases across data and simulations than the others and ii) the redshift scatter
and outlier rate are higher in the simulations than the data. The largest
differences between each of these photo-$z$ cases are in the Hubble residuals
for redshift bins at $z<0.3$, but there are no clear trends in either the
simulations or data that explain the final $\Delta w$ discrepancies. We also
note that unlike in the simulations, using SN+SOMPZ $p(z)$ prior and SN+SOMPZ
Gaussian prior estimates do not result in a significantly smaller final
sample.

###  5.4 Error scaling for Distance Modulus Uncertainties

In Section 4.7 we describe the multiplicative error scaling term introduced in
the original BBC framework to address naively overestimated distance
uncertainties. Recent analyses such as DES Collaboration (2024) and Brout et
al. (2022) have also included an additive (in quadrature) term $\sigma_{\rm
floor}^{2}(z,c,M_{*})$ in $\sigma_{\mu}^{2}$, which is intended to allow for
additional scatter beyond those computed in $\sigma_{\textrm{SALT3}}$.
However, we find that the inclusion of this additive term results in a BBC
reduced $\chi^{2}$ significantly $<1$. The inclusion of the error floor
results in overestimated uncertainties, which we hypothesize could be due to
the fact that bias corrections are computed at the measured redshift rather
than the true redshift, which may cause redshift bin migrations and therefore
mismeasured error scaling factors from BBC. As we are able to recover the
input cosmology with reduced $\chi^{2}\sim 1$ with only the multiplicative
error scaling and without the additive error floor, we leave investigation of
this subtlety to future work.

##  6 Discussion

While there are clear benefits to expanding the size of our usable SN samples
using photometric redshifts for SN Ia cosmology, this approach also presents
new systematics to understand and quantify. Here we have developed efforts
showing that true cosmological parameters can be estimated using
photometrically classified SNe with photo-$z$ for a cosmological analysis. We
use realistic simulated photo-$z$, photometric classification to address non-
Ia contaminants, and a dust-based intrinsic scatter model. In this section we
detail further work that is needed to rigorously measure unbiased cosmological
parameters in a completely photometric analysis.

###  6.1 Systematic Uncertainties

In this work we have focused primarily on the recovery of unbiased
cosmological parameters relative to those inferred with spec-$z$ by
considering $\Delta w$ values. However, future cosmological analyses will
require a full accounting of systematics (calibration, intrinsic scatter
model, etc.) using a statistical+systematic covariance matrix (Conley et al.,
2011) as in DES Collaboration (2024).

More realistic redshift systematics tests and systematic variations for a
covariance matrix will rely on a systematic uncertainty quantification from a
given photo-$z$ algorithm. One such example is the rigorously quantified
sources of uncertainty from SOMPZ, which include contributions from shot noise
and sample variance, inherent SOMPZ method uncertainty, redshift sample
uncertainty, photometric calibration uncertainty, and synthetic source
injection method uncertainty (Table 2, Myles et al., 2021). As the
uncertainties are given on the mean of each of four tomographic bins, one can
interpolate across the sample redshift range to consider a redshift-dependent
bias for individual host galaxies. However, other photo-$z$ algorithms tend to
include only statistical uncertainties with less obvious or explicit choices
for systematic variations. We emphasize that it is most important to
understand any redshift-dependent systematics, as they can be highly
degenerate with cosmological constraints. Future cosmological analyses will
require detailed studies of the modeling choices for systematics beyond a 1D
analytic description of photo-$z$ bias.

While here we study the primary effect of using photo-$z$ on the Hubble
diagram, there are several second order systematics that are also reliant on
redshifts and will require further study for an entirely photometric SN
cosmology analysis. Two of the leading systematic uncertainties in the main
DES-SN5YR analysis are those of SN intrinsic scatter and SN correlations with
their host galaxies. In this work we relied on only one realization of a dust-
based intrinsic scatter model with parameters constrained using spectroscopic
redshifts. As these parameters are constrained using SN color, Hubble
residuals, and Hubble scatter, using photo-$z$ for the SN light-curve fit may
cause biases which will need to be further studied. While here we have used
host galaxy stellar masses obtained using host galaxy photometry with
spectroscopic redshifts, future studies may require more detailed
consideration of how galaxy mass estimates may be biased by using photo-$z$
instead, as these effects also propagate to the SN intrinsic scatter model.

Figure 9: Same as Figure 7 but for data.

###  6.2 Future Work

As described in Section 4.6, we use a binned Hubble diagram for this analysis,
i.e. bias-corrected distances are determined in redshift bins. Brout et al.
(2021) and Kessler et al. (2023) have shown that an unbinned Hubble diagram
and covariance matrix (for spectroscopically confirmed samples) or re-binned
Hubble diagram (for photometrically confirmed samples) result in a $1.5\times$
smaller systematic uncertainty compared to a binned Hubble diagram. However,
previous work (Mitra et al., 2023) has shown that using an unbinned Hubble
diagram with photometric redshifts results in smaller uncertainties but
unresolved cosmological biases. This outstanding issue must be addressed in
order to take advantage of the so-called data “self-calibration” which allows
for the most robust results with a given dataset.

While analyses with spectroscopic redshifts have treated redshift
uncertainties as negligible, a crucial component that has not yet been fully
developed is the treatment of redshift uncertainties when they are non-
negligible, as is the case for photometric redshifts. In this analysis we
discuss the use of host galaxy photo-$z$ PDFs as priors in the SN+host
redshift fit, but each SN is ultimately placed on the Hubble diagram with a
single redshift without its (non-Gaussian) redshift uncertainties accounted
for in the cosmology fit. Roberts et al. (2017) presents a potential
resolution to this using a hierarchical Bayesian framework inspired by the
BEAMS formalism that marginalizes over redshift uncertainties to
simultaneously address photometric classification, host misassociation, and
photometric redshifts.

Ruhlmann-Kleider et al. (2022) investigate two other potential methods for
propagating redshift uncertainties to cosmology: i) refitting individual
photometric redshifts simultaneously with cosmology and ii) sampling from the
redshift probability distribution and propagating to the cosmology fit
$\chi^{2}$ values. For a flat $\Lambda$CDM model, they find that sampling the
redshift resolution function reduces bias in $\Omega_{M}$ to $0.3\sigma$ as
long as the sample is bias corrected accounting for photo-$z$ bias and core-
collapse contamination. They also conclude that propagating the redshift
resolution to the cosmological likelihood is likely a method of secondary
importance for reducing biases compared to i) containing CC contamination to
below the few percent level, ii) relying on spectroscopic redshifts at
$z<0.5$, as redshift biases lead to stronger shifts in luminosity distances at
low-$z$, and iii) ensuring bias corrections include photo-$z$ modeling.

As shown in Section 4.2, the use of the full $p(z)$ as a prior for a SN+host
photo-$z$ was not more informative than a prior approximated as a Gaussian.
Future analyses could consider developing the cosmological framework required
to use a $p(z)$ as the final redshift information on the Hubble diagram,
rather than solely as a prior. This could involve computing a weighted
distance contribution at various sampled redshifts from a PDF, similar to
Ruhlmann-Kleider et al. (2022), but would also likely require the development
of an emulator or rapid light-curve fitting for computational feasibility in
order to calculate distances at multiple redshifts for thousands of
supernovae.

In this study, we have limited our sample to SNe with spec-$z$ available; if
this requirement is lifted, a larger cosmological DES-SN sample can be built
using photo-$z$ only, as investigated in Möller et al. (2024). We estimate
this could comprise a sample size of at least $\sim 2200$ SNe after
cosmological cuts, compared to the $\sim 1600$ in the DES-SN5YR photometric
sample. While here we have used all photo-$z$s available without having to
rerun algorithms, an alternative approach to using photo-$z$ for SN cosmology
could be to subselect a sample with higher precision photo-$z$ only, such as
by introducing a cut on the photo-$z$ uncertainty. Subsamples of SNe may also
have overlapping information from many-band photometric surveys with high-
quality, many-band photo-$z$ for a subsample of SNe. Recent and upcoming
surveys with Euclid and Roman will provide additional information in NIR
bands, which can also be used to drastically improve photo-$z$ estimates. The
performance of other photometric redshift algorithms beyond those considered
in this work should also be investigated, e.g. from Qu & Sako (2023), which
uses a convolutional neural network to estimate photo-$z$ PDFs from SN light-
curves without host information.

Here we have focused on a cosmological SN sample which is entirely reliant on
photometric redshifts, but in the future it will likely be more realistic and
ideal to use a combination of spec-$z$ when available and photo-$z$ when not
(at higher redshifts). This will require careful attention to the modeling of
spectroscopic and photometric redshift efficiencies and study of the impact of
the redshift cutoff at which spec-$z$ are available.

##  7 Conclusion

In this work, we have shown that SN+host photo-$z$ are sufficient for an
unbiased Type Ia Supernova cosmology analysis. Using 1577 SNe from the DES-
SN5YR photometrically classified sample and fitting photo-$z$ from the SN
light-curve with host-galaxy priors, we obtain $\pm\Delta w$ values as low as
$0.01$ and consistently subdominant to the overall $w$ uncertainty. With
SN+host photo-$z$ estimates using SOMPZ and DNF priors we are able to obtain
$w$ estimates consistent with those obtained using spec-$z$ within
uncertainties. We also show, using simulations, that our analysis methods are
robust. We present the first efforts toward understanding host-galaxy
photo-$z$ modeling and algorithm related systematics in preparation for future
SN Ia cosmology analyses to be done with photometric redshifts and highlight
remaining open questions to be addressed in upcoming works.

## Acknowledgements

Author contributions: RC developed and performed the analysis and wrote the
manuscript. DS, MV, ER, JM, RK, and BP assisted in developing the project and
provided useful discussion and insight. MSa and MSm served as internal
reviewers. All authors contributed to this paper and/or carried out
infrastructure work that made this analysis possible. Additional highlighted
contributions include: Construction and validation of the DES-SN5YR data and
Hubble diagram: AM, BP, BS, CL, DB, DS, GQ, JL, LG, MSa, MSm, MSu, PA, PW, MV.
Contributed to the internal review process: AM, CL, HQ, JL, LG, MSa, MSm, MSu,
PW, TD. Development and/or maintenance of software: PA, RK. The remaining
authors have made contributions to this paper that include, but are not
limited to, the construction of DECam and other aspects of collecting the
data; data processing and calibration; developing broadly used methods, codes,
and simulations; running the pipelines and validation tests; and promoting the
science analysis.

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Workforce Development for Teachers and
Scientists, Office of Science Graduate Student Research (SCGSR) program. The
SCGSR program is administered by the Oak Ridge Institute for Science and
Education for the DOE under contract number DE-SC0014664.

DS is supported by Department of Energy grant DESC0010007, the David and
Lucile Packard Foundation, the Templeton Foundation and Sloan Foundation. LG
acknowledges financial support from the Spanish Ministerio de Ciencia e
Innovación (MCIN) and the Agencia Estatal de Investigación (AEI)
10.13039/501100011033 under 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, and from the Departament de Recerca i Universitats de la
Generalitat de Catalunya through the 2021-SGR-01270 grant.

This paper has gone through internal review by the DES collaboration. 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, NFS’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.

Based in part on data acquired at the Anglo-Australian Telescope, under
program A/2013B/012. We acknowledge the traditional owners of the land on
which the AAT stands, the Gamilaraay people, and pay our respects to elders
past and present.

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 work was completed in part with resources provided by the University of
Chicago’s Research Computing Center.

## Data Availability

Data used in this article is publicly available from the DES-SN5YR Data
Release (Sánchez et al., 2024) on Github (“github.com/des-science/DES-SN5YR”).

## Affiliations

1 Department of Physics, Duke University Durham, NC 27708, USA
2 NASA Einstein Fellow
3 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble
Road, Oxford OX1 3RH, United Kingdom
4 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA
5 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
6 Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA
7 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL
60637, USA
8 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL
60637, USA
9 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS, IP2I Lyon / IN2P3, IMR 5822,
F-69622, Villeurbanne, France
10 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
11 School of Physics and Astronomy, University of Southampton, Southampton,
SO17 1BJ, UK
12 The Research School of Astronomy and Astrophysics, Australian National
University, ACT 2601, Australia
13 Department of Astronomy, Boston University, 725 Commonwealth Ave., Boston,
MA 02215, USA
14 Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA
02215, USA
15 School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia
16 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
17 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
18 Centre for Gravitational Astrophysics, College of Science, The Australian
National University, ACT 2601, Australia
19 Centre for Astrophysics & Supercomputing, Swinburne University of
Technology, Victoria 3122, Australia
20 Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
21 Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile
22 Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
23 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,
USA
24 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
25 Physik-Institute, University of Zurich, Winterthurerstrasse 190, 8057
Zurich, Switzerland
26 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, UK
27 Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
28 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain
29 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
30 NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771,
USA
31 Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK
32 University of Nottingham, School of Physics and Astronomy, Nottingham NG7
2RD, UK
33 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029
Hamburg, Germany
34 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Dr., Pasadena, CA 91109, USA
35 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
36 Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049
Madrid, Spain
37 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81679 Munich, Germany
38 Center for Astrophysical Surveys, National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA
39 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002
W. Green Street, Urbana, IL 61801, USA
40 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA
41 Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA
42 Department of Physics, The Ohio State University, Columbus, OH 43210, USA
43 Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
44 Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113,
Australia
45 Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA
46 Departamento de Física Matemática, Instituto de Física, Universidade de São
Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
47 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
48 LPSC Grenoble - 53, Avenue des Martyrs 38026 Grenoble, France
49 Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
50 Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil
51 Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania
15312, USA
52 Department of Physics, Northeastern University, Boston, MA 02115, USA
53 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
54 Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831
55 Department of Astronomy, University of California, Berkeley, 501 Campbell
Hall, Berkeley, CA 94720, USA
56 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA
94720, USA
57 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse,
85748 Garching, Germany
58 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians
Universität München, Scheinerstr. 1, 81679 München, Germany

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##  Appendix A Simulation and data comparisons for photo-$z$

Following Section 4.4 we show comparison plots of light-curve parameters
between the simulations and data when redshift is fit jointly using different
host-galaxy photo-$z$ priors.

Figure 10: Comparison of fitted SN light-curve parameters and their errors, as
in Fig. 4.

##  Appendix B Hubble diagrams

Here we present the Hubble diagrams for each photo-$z$ case for simulations
and data as detailed in Section 5.

Figure 11: Simulation Hubble diagrams for each redshift variant. Each point is
shaded by the probability of being a Ia as given by SNN.

Figure 12: Data Hubble diagrams for each redshift variant. Each point is
shaded by the probability of being a Ia as given by SNN.


[b]Alexander Studenikin

# A proposal for experiment with high-intensity tritium neutrino source in
Sarov: The search for coherent elastic neutrino-atom scattering and neutrino
magnetic moment

Matteo Cadeddu  Georgy Donchenko  Francesca Dordei  Carlo Giunti  Konstantin
Kouzakov  Bayarto Lubsandorzhiev  Vladimir Trofimov  Maxim Vyalkov  Arkady
Yukhimchuk

###### Abstract

A description of the current state of the project for the study of coherent
elastic neutrino-atom scattering using a tritium source and liquid helium
detector is given. The project was proposed in our paper in 2019 and its main
goal is to obtain a new record limit on the neutrino magnetic moment at a
level below $10^{-12}\mu_{B}$ using a tritium antineutrino source with an
intensity of 10 MCi or even 40 MCi.

## Introduction

The studies of neutrino electromagnetic properties can open a window to new
physics [1]. This is because the neutrino magnetic moments, the most well
understood and studied among neutrino electromagnetic characteristics, are
zero in the Standard Model with massless neutrinos. In a minimal extension of
the Standard Model the diagonal magnetic moment of a massive Dirac neutrino
$\nu_{i}$ is given [2] by
$\mu^{D}_{ii}=\frac{3eG_{F}m_{i}}{8\sqrt{2}\pi^{2}}\approx 3.2\times
10^{-19}\Big{(}\frac{m_{i}}{1\ \mathrm{eV}}\Big{)}\mu_{B}$, where $\mu_{B}$ is
the Bohr magneton. The strongest upper bounds for the neutrino magnetic
moments in laboratory experiments are obtained with reactor neutrinos:
$\mu_{\nu}\leq 2.9\times 10^{-11}\mu_{B}$ (GEMMA Collaboration [3]), and solar
neutrinos: ${\mu}_{\nu}\leq 2.8\times 10^{-11}\mu_{B}$ (Borexino Collaboration
[4]). An order of magnitude stronger constraints are provided by studies of
astrophysical neutrinos (see for example [1]). Thus, it is an important goal
of the present fundamental physics research to fill the gap between the
present experimental limits and the theoretical predictions for the neutrino
magnetic moments.

In our recent paper [5] we have proposed an experimental setup to observe
coherent elastic neutrino-atom scattering ($CE\nu AS$) using electron
antineutrinos from tritium decay and a supefluid 4He target. In this
scattering process with the whole atom, that has not been observed so far, the
electrons tend to screen the weak charge of the nucleus as seen by the
electron antineutrino probe.

Figure 1: Atomic recoil spectrum.

The differential number of neutrino-induced events as a function of the atomic
recoil energy $T_{R}$ in a logarithmic scale is shown in Fig. 1. The coherent
elastic neutrino nucleus scattering ($CE\nu NS$) differential number is shown
by the black solid line while the $CE\nu AS$ one is shown by the dashed red
line. The dashed-dotted blue line represents the additional term appearing in
the $CE\nu AS$ differential number of events assuming a neutrino magnetic
moment of $\mu_{\nu}=10^{-12}\mu_{B}$ while the dotted green line represents
the total differential number of $CE\nu AS$ for the same value of $\mu_{\nu}$.
We find that it is possible to set an upper limit that is about two orders of
magnitude smaller than the current experimental limits from GEMMA and
Borexino. A corresponding experiment involving an intense tritium neutrino
source is currently being prepared in the framework of the research program of
the National Center for Physics and Mathematics in Sarov (Russia).

##  1 Tritium source of neutrinos

The advantages of a tritium source (TS) as compared to solar, nuclear,
accelerator and other $\beta$-active sources may be summarized as follows [6]:
1) more intensive (anti)neutrino fluxes as compared with the reactor and
accelerator sources and strongly suppressed correlated background; 2) small
sizes that make it possible to locate an experiment in a moderate-size low-
background underground laboratories and flux modulation for the non-correlated
background deduction; 3) the (anti)neutrino spectrum defined with high
accuracy; 4) a quite low maximum of the neutrino energy spectrum ($E_{0}=18.6$
keV), thus the bremsstrahlung radiation does not penetrate beyond the limits
of the source and there is no necessity of a passive protection between the
source and detector.

To measure the magnetic moment of neutrinos, it was proposed in [6, 7] to use
a tritium source with an activity of 40 MCi (4 kg of tritium) (see Fig. 2).

Figure 2: Experimental setup: an assembly of semiconductor detectors is
located inside a spherical tritium source ($TiT_{2}$).

The tritium in the TS is in a chemically bound state with Ti, which provides
an adsorption capacity of tritium of 1700 ${\text{cm}^{3}}/{\text{cm}^{3}}$, a
low equilibrium pressure at room temperatures of $\sim 10^{-9}$ mbar, a high
decomposition temperature of 550-620 oC hydride and absence of radioactive
impurities of the uranium series.

Figure 3: The temperature distribution in the TS.

In the works [6, 7, 8], the possibility of a fundamental implementation of
such an experimental scheme was shown. However, the thermal calculation of the
TS, under conditions of natural storage, showed its significant heating. In
the center of the source, the temperature exceeded 309 oC (see Fig. 3), which
caused certain difficulties during transportation, storage and operation of
such a source.

In [9], the TS design was optimized. The following factors were taken into
account: the available amount of tritium; obtaining the maximum number of
neutrino interaction events with the detector material; feasibility of
manufacturing; safety at all stages of the TS life cycle; ease of installation
and disassembly.

Taking into account the above factors, Fig. 4 shows the evolution of views on
the design of TS. The design scheme of the source is a cylindrical ring
assembly of 80 tubular elements (see Fig. 5) filled with titanium tritide.
Detectors are placed inside the ring assembly. Advantages of such a design:
free access to a specific tubular element during its installation/disassembly;
easily removal of the heat from the decay of tritium; the transportation of
each of a single fuel element is carried out within the framework of existing
international and Russian standards.

Figure 4: The evolution of designs of the TS.

Figure 5: The tubular element.

##  2 Experimental scheme

We consider a detector setup such that the tritium source is surrounded with a
cylindrical superfluid-helium tank, as depicted in Fig. 6. This configuration
allows us to maximize the geometrical acceptance, while allowing us to have a
top flat surface where helium atoms could evaporate after a recoil.

Figure 6: Schematic representation of the detector proposed to observe
CE$\nu$AS processes. The recoil of a helium atom after the scattering with an
electron antineutrino coming from the tritium source produces phonons and
rotons which, upon arrival at the top surface, cause helium atoms to be
released by quantum evaporation. An array of bolometers on the top surface
detects the number of helium atoms evaporated.

##  3 Expected number of events

The source and tank dimensions depend on the tritium source activity. We
consider the tank’s volume of about 1 m3 and two values of the tritium source
activity: 10 MCi (minimally expected) and 40 MCi (maximally expected).

The expected number of the CE$\nu$AS events for 5 yr of data taking are
$N^{\rm CE\nu AS}=58.9$ and $195.2$ for the tritium source activity of 10 and
40 MCi, respectively. These numbers allow us to reach the sensitivity to
CE$\nu$AS of at least 5$\sigma$. A successful implementation of such scheme
will also allow us to test the neutrino magnetic moment at the level of
$3.8\cdot 10^{-13}\mu_{B}$ and $2.3\cdot 10^{-13}\mu_{B}$ at $90\%\ C.L.$ for
10 MCi and 40 MCi tritium source, respectively.

##  4 Prospects

The proposed experiment with a superfluid 4He target and a tritium neutrino
source will allow us to study for the first time the CE$\nu$AS process and to
test the neutrino magnetic moment and other neutrino electromagnetic
properties (millicharge and charge radius) at an unprecedented high level of
sensitivity. We plan to start the measurements in 2026 in a low-background
neutrino lab that will be established in Sarov for this purpose. We expect to
use the tritium source with the activity of at least 10 MCi (1 kg of tritium)
and the amount of tritium can be increased to reach 40 MCi (4 kg of tritium).

We also plan to develop and use a Si detector with a low-energy threshold for
testing the neutrino magnetic moment and other neutrino electromagnetic
properties by measuring elastic neutrino-electron scattering. The 20-kg Si
detector is expected to operate at temperatures of few tens of mK and to have
the energy threshold as low as $\sim 10$ eV or even $\sim 1$ eV due to the
Neganov-Trofimov-Luke effect [10]. With such a detector and the discussed
tritium source, we expect to achieve a sensitivity to the neutrino magnetic
moment at a level below $10^{-12}\mu_{B}$.

The work is supported by the Russian Science Foundation under grant
No.22-22-00384.

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