# Astrometric Redshifts of Supernovae

Jaemyoung (Jason) Lee Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, PA 19104, USA  Masao Sako Department of Physics
and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
Richard Kessler Department of Astronomy and Astrophysics, University of
Chicago, Chicago, IL 60637, USA  Alex I. Malz McWilliams Center for Cosmology,
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
The LSST Dark Energy Science Collaboration

###### Abstract

Differential Chromatic Refraction (DCR) is caused by the wavelength dependence
of our atmosphere’s refractive index, which shifts the apparent positions of
stars and galaxies and distorts their shapes depending on their spectral
energy distributions (SEDs). While this effect is typically mitigated and
corrected for in imaging observations, we investigate how DCR can instead be
used to our advantage to infer the redshifts of supernovae from multi-band,
time-series imaging data. We simulate Type Ia supernovae (SNe Ia) in the
proposed Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) Deep
Drilling Field (DDF), and evaluate astrometric redshifts. We find that the
redshift accuracy improves dramatically with the statistical quality of the
astrometric measurements as well as with the accuracy of the astrometric
solution. For a conservative choice of a 5-mas systematic uncertainty floor,
we find that our redshift estimation is accurate at $z<0.6$. We then combine
our astrometric redshifts with both host galaxy photometric redshifts and
supernovae photometric (light-curve) redshifts and show that this considerably
improves the overall redshift estimates. These astrometric redshifts will be
valuable especially since Rubin will discover a vast number of supernovae for
which we will not be able to obtain spectroscopic redshifts.

Type Ia Supernovae — Differential Chromatic Refraction — Astrometry —
Redshifts

††thanks<EMAIL_ADDRESS>
##  1 Introduction

Observations of Type Ia supernovae (SNe) resulted in the unexpected but
groundbreaking discovery of the accelerating universe (Riess et al., 1998;
Perlmutter et al., 1999). While this discovery with only tens of SNe Ia is an
extraordinary feat, subsequent efforts have allowed for increasingly precise
constraints on the dark energy equation of state parameter $w$ and the matter
density $\Omega_{m}$. Indeed, recent measurements from Pantheon+ (Brout et
al., 2022) and the Dark Energy Survey (DES) (Abbott et al., 2024) now give
roughly $0.15$ uncertainty on the equation of state parameter $w_{0}$ for a
flat $w_{0}$CDM model from over 1000 SNe each. Combining supernovae
measurements with Cosmic Microwave Background (CMB) and Baryon Acoustic
Oscillations (BAO) measurements, which are largely independent from SNe Ia
measurements, significantly improves the cosmological constraints.

Accurate redshifts are crucial for SN Ia cosmology. Up to this date, SN
redshifts have relied on spectroscopic redshifts, either of the SN itself or
from the host galaxy (Howell et al., 2005; Sako et al., 2018; Lidman et al.,
2020; Smith et al., 2020). However, the Vera C. Rubin Observatory Legacy
Survey of Space and Time (LSST, Ivezić et al., 2019) will observe orders of
magnitude more SNe than all of the SNe detected so far, making it no longer
viable to perform spectroscopic follow-up for all of the detected SNe and
their hosts.

Consequently, there have been numerous efforts to utilize SN photometry to
accurately measure redshifts. Some of these efforts include using the peak
flux in the observed bands as inputs for an empirical, analytic redshift
estimator (Wang, 2006; Wang et al., 2007, 2015), adding the redshift as an
additional light-curve fitting parameter, also known as `LCFIT+z` (Kessler et
al., 2010; Palanque-Delabrouille et al., 2010), and using a machine learning
approach to learn the redshifts from light-curves (Qu & Sako, 2023). Many
studies have sought to improve SN photometric redshift measurements by adding
a host-galaxy photometric redshift (photo-$z$) prior, including three of the
aforementioned works, as well as Mitra et al. (2023), where the resulting
improvement in cosmological constraints were explored extensively.

While the impact of photometric redshift uncertainties on cosmology may not be
severe (Dai et al., 2018; Chen et al., 2022; Mitra et al., 2023), better
redshift measurements will lead to more constraining power in the cosmological
parameter estimation both by allowing for the inclusion of more SNe in the
analysis, and by reducing the uncertainties in the redshifts used for
cosmology. Here, we introduce another method to obtain redshift measurements
using imaging data. This method utilizes the fact that sources at different
redshifts (and hence different colors) are refracted by different amounts in
our atmosphere. As illustrated in Figure 1, shorter wavelength light is
refracted more in our atmosphere than longer wavelength light. This effect,
known as Differential Chromatic Refraction (DCR, Filippenko, 1982), is a
result of the refractive index of our atmosphere being wavelength dependent.

Normally DCR is an effect cosmologists want to mitigate for precision
cosmology in both weak lensing (Plazas & Bernstein, 2012; Meyers & Burchat,
2015; Carlsten et al., 2018) and SN Ia cosmology (Lee & Acevedo et al., 2023).
However, Kaczmarczik et al. (2009) found that treating the observed DCR shifts
in the Sloan Digital Sky Survey (SDSS) u and g bands as additional colors in
photometric redshifts improves quasar photometric redshifts, with the fraction
of objects within $\Delta z=\pm 0.1$ of the spectroscopic redshifts increasing
by 9% and catastrophic outliers reduced. As such, astrometric redshifts have
the potential to strengthen photo-$z$ constraints when combined with
photo-$z$’s. They also mention the possibility of applying these astrometric
redshifts (astro-$z$’s) to other sources with distinct spectral features such
as Type Ia and Type II supernovae.

Figure 1: A schematic diagram showing how DCR affects light in the Earth’s
atmosphere; shorter wavelengths are refracted more by the atmosphere, which
means that to an observer on Earth, a blue star appears to be higher up in the
sky than a red star located at the same position in space. $R_{450\rm{nm}}$
and $R_{650\rm{nm}}$ denote the amount of refraction by our atmosphere at an
air mass (AM) of 1.4 for 450 nm and 650 nm light respectively.

In this work, we develop the methodology to obtain astrometric redshifts for
Type Ia supernovae. For this development, we added DCR effects to the
`SNANA`111https://github.com/RickKessler/SNANA (SuperNova ANAlysis, Kessler et
al., 2009) simulation, and analyzed simulated SN Ia samples corresponding to
the proposed Deep Drilling Fields (DDF) for LSST. We show astro-$z$
constraints from only the DCR effect, utilizing multi-band, multi-epoch
measurements, and we show how combining them with both host and SN photo-$z$’s
improves the overall redshift estimates.

The content of this work is as follows. In Section 2, we compute the amount of
DCR shift that occurs depending on the air mass, source flux and filter
response functions and describe how SNe at different redshifts are affected.
Next, we describe the simulated LSST dataset in Section 3. Then, we describe
our methods to obtain redshift estimates from the simulated astrometry for
each SN observation in Section 4. In Section 5, we show our results for LSST-
like simulations in hypothetical (ideal) and realistic cases, as well as the
improvement when combined with both host galaxy and SNe photometric redshifts,
and extensively discuss our results. We also show the impact of several
systematic effects. Lastly, we end with a discussion and conclusion in Section
6.

Figure 2: Example SN Ia SED at peak epoch overplotted on top of the LSST
g-band at $z=0$ (top) and $z=0.6$ (bottom). As shown in Equation 1, the
resulting DCR shifts at the two redshifts are different since the overlap
between the filter and the SED changes significantly. Because the product of
the filter function and the SED, $F(\lambda)S(\lambda)$, is smaller for the
$z=0.6$ SN, the (absolute) DCR shift is also smaller, as shown in Figure 3.

##  2 Differential Chromatic Refraction

DCR is due to the dependency of our atmosphere’s refractive index on the
source wavelength. The centroid shift caused by DCR can be calculated as in
Plazas & Bernstein (2012):

$\bar{R}=\frac{\int d\lambda R(\lambda,z_{a})F(\lambda)S(\lambda)}{\int
d\lambda F(\lambda)S(\lambda)}$ (1)

where $\bar{R}$ is the altitude shift of the source towards the zenith,
$R(\lambda,z_{a})$ is the shift of each photon towards the zenith, $z_{a}$ is
the zenith angle, $F(\lambda)$ is the filter response function, and
$S(\lambda)$ is the source spectral energy distribution (SED).
$R(\lambda,z_{a})$ can be calculated using:

$R(\lambda,z_{a})\approx R_{0}\tan(z_{a}),\quad
R_{0}=\frac{n^{2}(\lambda)-1}{2n^{2}(\lambda)}$ (2)

with the air mass (AM), or the amount of air along the line of sight, being
$\mathrm{AM}=\sec z_{a}$ for $z_{a}<80^{\circ}$. For an explicit expression of
$n(\lambda)$, see Filippenko (1982). As apparent in Equation 1, the DCR
altitude shift depends on the filter transmission function. It also depends on
the source SED, which depends on both redshift and the epoch of the SN. The
dependency on $R(\lambda,z_{a})$ means that the shift becomes larger at higher
AM.

Figure 3: The resulting g-band DCR altitude shifts with respect to a K5V
reference star for a $z=0$ SN (blue) and $z=0.6$ SN (orange) at different air
masses (AMs), but both at peak epoch. As we expect, the shifts at the two
redshifts are different and we see more shift at higher AM. Comparison with
Figure 2 implies that the $z=0$ SN is “bluer” (in the g-band at least) than
the reference star while the $z=0.6$ SN is “redder.”

Figure 2 shows that as the SN redshift increases, the overlap of its SED with
the filter function changes, so the DCR altitude shift changes with redshift.
Although the amount of refraction that occurs due to DCR is usually between
43$\arcsec$ to 46$\arcsec$ for a given source viewed through the LSST ugriz
bands at an AM of 1.4, the observed shifts are much smaller because DCR also
affects the reference stars that are used to determine the astrometric
solution. Hence, we measure the shift with respect to the average color star
in a given CCD, which we take in this work to be a K5V star, adopted from the
DES-SN5YR analysis (Lee & Acevedo et al., 2023). For LSST, we do not know
beforehand the spectrum of the average color star, but it is straightforward
to use another spectral type as the reference star. Figure 3 shows the g-band
altitude shift vs. AM for a SN (with the light-curve stretch and color being
0) at rest-frame epoch $t_{\rm{rest}}=0$ for $z=0$ and $z=0.6$ where
$t_{\rm{rest}}$ is the number of days from the time of peak brightness in the
g-band. Compared to the average color star, the $z=0$ SN is DCR shifted
upwards (towards zenith) while the $z=0.6$ SN is shifted downwards (away from
zenith) at $\rm{AM}>1.0$. We also show the expected increase in the DCR shift
at higher AM.

##  3 Dataset and Simulations

We perform our analysis on a simulated data sample generated assuming LSST
survey characteristics. LSST is a ground-based survey using the 8.4 m Simonyi
Survey Telescope in Chile with expected first light in 2024. Over the period
of a decade, LSST will observe 18,000 $\rm{deg}^{2}$ of the southern sky with
its 3.2-gigapixel camera and a very wide 9.6 $\rm{deg}^{2}$ field of view in
six filters ugrizY covering $320$ to $1050$ nm. We only use the Deep Drilling
Fields (DDF) subsample (Scolnic et al., 2018) obtained by coadding exposures
from each of the bands taken within the same night.

We simulate LSST-like SNe Ia observations in the ugrizY bands using `SNANA`,
which uses LSST-like SN Ia observing characteristics following the Extended
LSST Astronomical Time-series Classification Challenge (ELAsTiCC) dataset
(Narayan & ELAsTiCC Team, 2023), which is a sequel to the earlier PLAsTiCC
dataset (Allam Jr et al., 2018; Kessler et al., 2019b). The simulation
includes realistic SN observations with a source model that generates SEDs for
each epoch, galactic extinction that impact the SED, and a noise model that
accounts for the PSF, sky noise and zeropoint as highlighted in Kessler et al.
(2019a).

For our SED model in the simulation and analysis, we use the SALT3 model
(Kenworthy et al., 2021) with wavelengths extended to the near-infrared
(coverage up to to 2,000 nm) following (Pierel et al., 2022). Host-galaxy
properties are modeled as described in Lokken et al. (2023).

Figure 4: LSST ugriz filters (from left to right) according to their relative
transmission and wavelength range. We do not show the Y band as it is not used
for the DCR calculations. Multiple detections with different bands are crucial
for accurate astrometric redshifts. Figure 5: Redshift and air mass (AM)
distributions for the LSST DDF simulations in SNANA. The peak redshift is
around $z=0.9$ for No selection cut and Epoch cut, but around $0.6<z<0.9$ for
the Epoch + S/N cut, while the AM peaks between 1.05 and 1.10 for all three
selection cuts. Higher AM observations lead to better astro-$z$ estimates.

Some survey characteristics particularly important for this work include the
filter response
functions222https://github.com/lsst/throughputs/tree/1.9/baseline shown in
Figure 4, and the simulated air mass as well as redshift distributions, shown
in Figure 5. Because the DCR shifts are typically the largest in the u and
g-bands, observations in those bands will be especially helpful in measuring
SN redshifts. The redshift distribution peaks around $z=0.9$ without selection
requirements (cuts; see end of Section 3), but at $0.6<z<0.9$ with cuts. Most
of the observations are at $1.05<\rm{AM}<1.25$, regardless of the selection
cut, meaning that the vast majority of the SNe observations will result in
some sort of shift away from the reference star. Roughly 23% of the
observations are at high AM ($>1.4$) where we see the most extreme shifts.

To simulate realistic astrometry, we added new functionalities in the `SNANA`
simulations. We simulate true DCR shifts in the Right Ascension (RA, $\alpha$)
and Declination (DEC, $\delta$) directions by first calculating the DCR
altitude shifts for the SN ($\bar{R}_{\rm{ALT,SN}}$) and K5V star
($\bar{R}_{\rm{ALT,K5V}}$) by performing the integral given in Equation 1 then
adding $\Delta_{\rm{DCR}}\alpha$ and $\Delta_{\rm{DCR}}\delta$ to the
reference star (K5V) RA and DEC after DCR where

$\begin{split}\Delta_{\rm{DCR}}\alpha&=(\bar{R}_{\rm{ALT,SN}}-\bar{R}_{\rm{ALT,K5V}})\frac{\sin
q}{\cos\delta}\\\
\Delta_{\rm{DCR}}\delta&=(\bar{R}_{\rm{ALT,SN}}-\bar{R}_{\rm{ALT,K5V}})\cos{q}\end{split}$
(3)

and $q$ is the parallactic angle. While Equation 3 assumes flat Euclidean
geometry, the true shift calculated using spherical trigonometry results in a
very similar answer. This is because the shifts considered are smaller than
$1\arcsec$. We show in Appendix A a diagram of the celestial sphere for an
observer in the Southern Hemisphere as well as detailed calculations of the
shifts.

Next, because real observations include Poisson noise, the SN centroid
accuracy depends on the S/N of the SN measurement. Thus, we add random
Gaussian smearing as a function of ($\rm{FWHM}_{\rm{PSF}}$/[S/N]) for our
statistical uncertainties. To parametrize the angular resolution, we use the
GalSIM333https://galsim-developers.github.io/GalSim/_build/html/index.html
(Rowe et al., 2015) package to simulate images with Poisson noise, calculate
the standard deviation of RA and DEC shifts depending on the
$\rm{FWHM}_{\rm{PSF}}$ and S/N at multiple AMs, and fit a 2nd-order polynomial
of ($\rm{FWHM}_{\rm{PSF}}$($\arcsec$)/[S/N]). While it is possible to simulate
each SN observation using GalSIM including Poisson noise as well as DCR
effects and measure the centroid shift for each observation, this is
computationally expensive. Therefore, we chose to simulate the shift arising
from Poisson noise and DCR instead of simulating each image. Our fitted
polynomial function for the standard deviation with a systematic floor applied
gives the total uncertainties,
$\sigma_{\mathrm{tot}}^{2}=\sigma_{\mathrm{syst}}^{2}+\sigma_{\mathrm{stat}}^{2}$,
where

$\sigma_{\mathrm{stat}}=0.66\left(\frac{\rm{FWHM}_{\rm{PSF}}(\arcsec)}{\rm{[S/N]}}\right)+\frac{0.1}{1\arcsec}\left(\frac{\rm{FWHM}_{\rm{PSF}}(\arcsec)}{\rm{[S/N]}}\right)^{2}$
(4)

with $\sigma_{\rm{syst}}=5$ mas. The simulated RA and DEC shifts due to the
total uncertainties are:

$\Delta_{\sigma,\rm{tot}}\alpha=\sigma_{\rm{tot}}\frac{\mathcal{N}_{\alpha}(0,1)}{\cos{\delta}},\quad\Delta_{\sigma,\rm{tot}}\delta=\sigma_{\rm{tot}}\mathcal{N}_{\delta}(0,1)$
(5)

where $\mathcal{N}_{\alpha}(0,1)$ and $\mathcal{N}_{\delta}(0,1)$ are random
Gaussian deviates each with a mean of zero and standard deviation of 1.
Combining everything, we have:

$\alpha_{\rm{obs}}=\alpha_{\rm{true}}+\Delta_{\rm{DCR}}\alpha+\Delta_{\sigma,\rm{tot}}\alpha$
(6)

and likewise for $\delta_{\rm{obs}}$. When the S/N is too low,
$\sigma_{\rm{stat}}$ becomes larger than the largest DCR shifts we observe, or
around $0.1\arcsec$ to $0.2\arcsec$, which typically occur at high AM
observations in the ug-bands at $z<0.1$. Thus, we only use DCR shifts when the
S/N is larger than 3 ($\sigma_{\rm{stat}}\approx 0.23\arcsec$ for a $1\arcsec$
PSF when S/N = 3). It is possible in theory to use all observations including
when $\mathrm{S/N}<3$, but this is likely not feasible for real observations
due to the difficulty of accurately fitting for centroids at such low S/N.
Because Poisson noise causes the centroids to shift in random directions,
using only $\mathrm{S/N}>3$ SNe for our analysis will not result in a bias.

To compare the simulated DCR measurement and the model DCR shift on a given
epoch, we subtract the peak MJD of the SN from its observed date and take this
value to be the observed epoch $T_{\rm{obs}}$. To estimate peak MJDs, we use
the ‘Fmax-clump’ method as described in the `SNANA` manual, where peak MJD is
defined to be the epoch of maximum flux in the sliding 50-day window
containing the most detections. This method is robust because there is no
fitting, and it ignores pathological fluxes that are far away from peak
brightness. Next, we use the light-curve fitting module `LCFIT+z` (Kessler et
al., 2010) with a weak cosmology prior as described in Appendix C, only for
the purpose of obtaining light-curve fit peak MJD values that are input to the
astro-$z$ pipeline. Since we use the very high-cadence DDF only, the ‘Fmax-
clump’ method is adequate. However, for surveys with poorer cadence (e.g.,
WFD), light-curve fit may be needed to get a better estimate of peak MJDs.
Because the light-curve fitting is not always successful, this process leaves
us with 14,170 out of 20,000 candidates, or 7,864 out of the 8,622 candidates
that pass the selection cuts as described later in this section. The primary
reason for unsuccessful light-curve fitting is not having observations pre-
and post-peak epoch, while having low S/N is another reason. Thus, we combine
the astro-$z$ measurements obtained using `LCFIT+z` peak MJDs (when available)
with ‘Fmax-clump’ peak MJDs (when `LCFIT+z` peak MJDs are not available) when
showing our default results in Section 5.2. We also show some performance
metrics i) with only ‘Fmax-clump’ peak MJDs and ii) assuming we know precisely
the peak MJDs in Table 4.

Our simulation does not account for other wavelength dependent effects in the
telescope optics and detectors affecting SN centroid measurements. Some of
these effects were investigated in detail for DECam in Bernstein et al. (2017,
2018). The largest optics effect comes from refraction in the lenses, shifting
sources in different bands by different amounts particularly near the
boundaries of the focal plane. Such color-dependent radial displacements (also
known as lateral color) are shown to be around $0.050\arcsec$/mag and
$0.005\arcsec$/mag in the g and r bands in $g-i$ magnitudes respectively near
the edges of the focal plane for DES. While Lee & Acevedo et al. (2023)
mentions that these shifts are at worst, similar to the DCR positional effects
for DES, it is plausible to expect that at least some of these effects will be
corrected for in LSST. For DECam, the uncertainty in the astrometric solution
due to telescope optics is about 3-6 mas on the focal plane (Bernstein et al.,
2017), so our assumption of a $\sigma_{\mathrm{syst}}=5$ mas systematic floor
is plausible. Therefore, the uncertainties we model in this study will account
for the bulk of the uncertainties we will face in real observations.

Because we advocate the combination of photo-$z$’s with our astro-$z$’s, we
apply selection cuts typically applied to photo-$z$’s to our simulated samples
when showing results for realistic simulations. Following Kessler et al.
(2010), we first require that there are observations before $T_{\rm{rest}}=-3$
days and after $T_{\rm{rest}}=10$ days when showing the results for all types
of simulations. This means that for each candidate we use, there are
observations pre- and post-peak epoch. Since we do not know $T_{\rm{rest}}$
beforehand, we relax the epoch cuts to $T_{\rm{rest}}/(1+z_{\rm{max}})$ as in
Kessler et al. (2010), or observations before $T_{\rm{obs}}=-3/2.2$ days and
after $T_{\rm{obs}}=10/2.2$ since our $z_{\rm{max}}=1.2$. For the same reason,
we use peak MJDs from the ‘Fmax-clump’ method to impose epoch cuts. Second, we
require that there are at least 3 bands with detections of S/N $>$ 8 for
realistic simulations. The number of candidates that remain after selection
cuts for the astro-$z$ only results are shown in Table 1. After imposing the
epoch selection cut, we are left with 13,821 or 69.1% of the SNe Ia, while
also imposing the S/N cut leaves us with 8,622 SNe Ia out of the simulated
20,000, or about 43.1%.

Lastly, when we combine SN photo-$z$’s with the other two redshift estimates,
we take the fit probability (FITPROB), calculated by integrating the tail of
the $\chi^{2}$ distribution from the $\chi^{2}$ of the light-curve fit and the
number of degrees of freedom for each of the candidates (p-value), and require
that $\mathrm{FITPROB}>0.01$ when combining with SN photo-$z$’s. The number of
candidates removed by this selection cut is less than 10% for all combinations
as shown in Table 6, but noticeably improves the performance metrics.

Type | Number of Candidates
---|---
No selection cut | 20,000
Epoch cut | 13,821
Epoch + S/N cut (default) | 8,622
Epoch + S/N cut + AM cut | 7,590
Table 1: Number of candidates that pass the given selection cut. Out of the
20,000 simulated candidates, the ones that pass the Epoch cut are used to show
the results for Perfect and $\sigma_{\mathrm{syst}}$ only simulations, while
the default for realistic simulations is Epoch + S/N cut. The AM cut
additionally requires that at least one observation for a candidate has an
$\rm{AM}>1.4$.

##  4 Analysis Methodology

To measure the astro-$z$ for a given SN, we minimize the $\chi^{2}$ between
the model and `SNANA` simulated DCR shifts. As introduced in Section 3, we use
the extended SALT3 model to calculate the model DCR altitude shifts (MODEL) on
a grid of (FILTER, $z$, EPOCH, AM, $x_{1}$, $c$) with $x_{1}$ and $c$ being
the SALT3 light-curve parameters for stretch and color. The grids for each of
the parameters are:

$\begin{split}z:\text{ }&0.00\text{ to }1.20\text{, increments of }\Delta
z=0.01\\\ \rm{EPOCH}:\text{ }&-18\text{ to }50\text{, increments of 1 day}\\\
\text{AM}:\text{ }&1.00\text{ to }3.00\text{, increments of 0.01}\\\
x_{1}:\text{ }&-3.0\text{ to }2.0\text{, increments of 0.5}\\\ c:\text{
}&-0.30\text{ to }0.50\text{, increments of 0.05}\end{split}$ (7)

When calculating the DCR model shifts, we force the SED to be zero whenever
the computed SALT3 fluxes are below zero. These negative fluxes sometimes
occur at early phases ($T_{\mathrm{rest}}\leq-15$) in the ultraviolet, mostly
at extreme color ($|c|\geq 0.2$) and stretch ($|x_{1}|\geq 2$). This can
sometimes cause an issue, because the DCR shifts are infinite when the
denominator of Equation 1 is zero. In such cases, we extrapolate the DCR
shifts along the epoch axis using `scipy.interpolate`.

We measure the DCR shift for simulated data similar to how we would for real
data. First, we use the “observed” RA and DEC to compute the altitude of the
SN at the telescope site, and subtract the altitude of the
($\sigma_{\rm{tot}}^{2}$ weighted) Y-band average RA and DEC for each
candidate. This subtraction is needed because we cannot determine the exact
coordinates of the K5V reference star as all the astronomical objects in a
given exposure are shifted by DCR as well. Hence, we take the Y-band average
coordinates of the SN to be a proxy for the reference star coordinates, as the
DCR shifts are smallest in the Y-band. As Poisson noise still affects the
Y-band positions the same way it affects other bands, the Y-band positions
will have some scatter, but the average Y-band coordinates are reasonable
approximations for the reference star coordinates especially if we have
multiple Y-band observations or a few high S/N Y-band observations since the
scatter will average out. As only about 12,700 candidates have Y-band
observations, we use the z-band average coordinates instead when we do not
have Y-band observations for a given SN candidate. When we use the z-band
average coordinates as the K5V reference star coordinates, we do not use the
z-band DCR shifts when constructing our probability distribution functions
since the z-band average coordinates are sensitive to the DCR shifts in the
z-band. Using a combination of both Y-band and z-band average coordinates
result in a slight, but noticeable improvement in terms of the outlier
fractions (defined in Section 5) particularly for the Perfect and systematics-
only simulations from when using just z-band average coordinates especially
because we can utilize z-band shifts when using the Y-band average coordinates
as the reference star coordinates.

It is also possible to use the i-band average coordinates as the reference
star coordinates, which could be a better estimate than the z or Y-band
average coordinates since the S/N tends to be much higher in the i-band.
However, we also need to consider the fact that DCR shifts tend to be larger
in the i-band compared to the zY-bands, sometimes to a non-negligible degree.
If a high S/N i-band observation happens to have a large DCR shift, the
average i-band coordinates for that particular candidate will no longer be a
good proxy for the reference star coordinates. Furthermore, using the i-band
average coordinates could bias the iz-band DCR shift measurements and using
only the ugr-band DCR shifts reduces constraining power. We leave to future
works to investigate in more detail which is most appropriate to use as the
reference star coordinates.

The $\chi^{2}$ between the model and the observed altitude shifts are defined
as:

$\chi^{2}=\sum_{i}\frac{(\Delta_{\rm{OBS},i}-\Delta_{\rm{MODEL},i})^{2}}{\sigma_{i,\rm{total}}^{2}}$
(8)

where OBS indicates a real or simulated observation, $i$ denotes each of the
observations for a given SN, and
$\sigma_{i,\rm{total}}^{2}=\sigma_{i,\rm{stat}}^{2}+\sigma_{i,\rm{syst}}^{2}$
with $\sigma_{i,\rm{stat}}$ given in Equation 4 and $\sigma_{i,\rm{syst}}=5$
mas as stated in Section 3. After the $\chi^{2}$ is calculated for a given
candidate over the 3-dimensional grid of FILTER, $x_{1}$, $c$, we calculate
the posterior:

$\mathcal{P}(z)=\int_{-0.3}^{0.5}\int_{-3.0}^{2.0}P(z,x_{1},c)P(x_{1})P(c)dx_{1}dc$
(9)

where $P(z,x_{1},c)\propto e^{\frac{-\chi^{2}}{2}}$ with the $\chi^{2}$ being
the sum over the ugriz band-$\chi^{2}$ (ugri when Y-band observations are
absent) and $P(x_{1})$ and $P(c)$ are priors for the light-curve stretch and
color. $P(x_{1})$ and $P(c)$ are obtained by binning asymmetric Gaussian
distributions with the binning shown in Equation 7. The value with highest
probability, low-sided and high-sided Gaussian widths are taken to be
($\bar{x}_{1}$, $\sigma_{x_{1},-}$, $\sigma_{x_{1},+}$) = ($-0.054$, $0.043$,
$0.101$) and ($\bar{c}$, $\sigma_{c,-}$, $\sigma_{c,+}$) = ($0.973$, $1.472$,
$0.222$) as in Scolnic & Kessler (2016), which are also used as the `SNANA`
input. We do not need to normalize $\mathcal{P}(z)$ since we only need to
compute the relative probabilities at each redshift.

Figure 6: The relative posterior probabilities, $z_{\rm{true}}$, CDF50, and
PDFPeak estimates for three cases highlighting the advantages and
disadvantages for the two methods: (Left) CDF50 gives a better estimate
(multimodal with similar peak amplitudes), (Middle) PDFPeak gives a better
estimate (distinct peak but not large in amplitude), and (Right) both CDF50
and PDFPeak estimate the correct redshift.

We use $\mathcal{P}(z)$ to estimate the redshift in two different ways: (1)
taking the redshift at the 50th percentile of the cumulative distribution
function (hereafter, CDF50) and defining the 68% confidence interval as 16th
to 84th percentiles, and (2) taking the redshift at the peak of the
probability distribution function (hereafter PDFPeak, equivalent to the
maximum likelihood) with uncertainties defined as the $\pm 34$ percentiles
around the peak.

There are advantages and disadvantages for either method. CDF50 gives better
results when the posterior probabilities are multimodal without a distinct
peak and the error bars are always within the range of redshifts we explore.
However, when the $\chi^{2}$ is unconstraining, the redshift is measured to be
near the midpoint of the redshift range, meaning that the results are biased.
On the other hand, PDFPeak can result in catastrophic outliers if the
posterior has multimodal peaks or has similar values throughout all redshifts.
Figure 6 shows examples for which (i) CDF50 gives a better redshift estimate
(multimodal with similar probabilities), (ii) PDFPeak gives a better estimate
(distinct peak but not large in amplitude), and (iii) both CDF50 and PDFPeak
are able to determine the redshift very well (well-defined peak).

When computing the posterior PDFs for Perfect
($\sigma_{\rm{syst}}=\sigma_{\rm{stat}}=0$) simulations with or without
marginalization of $c$ or $x_{1}$ (Section 5.3), the denominator of Equation
(9) is zero in theory, and $\chi^{2}$ becomes infinite while
$\mathcal{P}(z)=0$. To avoid this numerical artifact, we calculate
$\mathcal{P}(z)$ with $\sigma_{\rm{tot}}=1.0$ mas. PDFPeak estimates are
independent of the $\sigma_{\rm{tot}}$ value. We note that the CDF50 estimates
are sensitive to our choice of $\sigma_{\rm{tot}}$, with a smaller
$\sigma_{\rm{tot}}$ resulting in the CDF50 value becoming closer to the
PDFPeak value. This is because an overall multiplicative factor ($>1$) in the
$\chi^{2}$, corresponding to smaller $\sigma_{\rm{tot}}$, translates to the
highest amplitude peak in $\mathcal{P}(z)$ becoming more distinct. We
confirmed that as $\sigma_{\rm{tot}}$ decreases, the CDF50 estimates
increasingly match that of PDFPeak. Additionally, the uncertainties converge
to zero as $\sigma_{\rm{tot}}$ decreases. Therefore, we show the same
(PDFPeak) results for CDF50 and PDFPeak for the Perfect simulations without
uncertainties where $\sigma_{\rm{tot}}$ should be as close to zero as
possible. While taking $\sigma_{\rm{tot}}=1.0$ mas renders $\mathcal{P}(z)=0$
for all redshifts for a small fraction of candidates when marginalization is
included, the percentages of this occurring is at most 0.5% of the candidates
that pass the epoch cut.

Symbol | Measurement Method
---|---
$z_{\rm{DCR}}$ | from astrometry using DCR effect
$z_{\rm{Host}}$ | host photo-$z$ PDF
$z_{\rm{DCR+Host}}$ | combined PDF from $z_{\rm{DCR}}$ and $z_{\rm{Host}}$
$z_{\rm{SN}}$ | from `LCFIT+z`, using flat redshift prior
$z_{\rm{SN+Host}}$ | from `LCFIT+z`, using $z_{\rm{Host}}$ PDF as prior
$z_{\rm{SN+DCR}}$ | from `LCFIT+z`, using $z_{\rm{DCR}}$ PDF as prior
$z_{\rm{SN+DCR+Host}}$ | from `LCFIT+z`, using $z_{\rm{DCR+Host}}$ PDF as prior
Table 2: How we denote our redshift estimates using various methods and their
combinations.

In addition to our astro-$z$ results, we combine our astrometric redshifts
with host galaxy photometric redshifts (Host photo-$z$’s) as well as SN Ia
photometric redshifts (SN photo-$z$’s) in Section 5. Moving forward, we denote
the redshift measurements using the three methods as well as their
combinations as shown in Table 2. $z_{\rm{Host}}$ are derived from ELAsTiCC
host photometry by using the `pzflow`
package444https://github.com/jfcrenshaw/pzflow (Crenshaw, 2021) as a
generative model, trained on the CosmoDC2 simulations (Korytov et al., 2019),
and are stored as quantiles corresponding to integrated (CDF) probabilities of
[$0\%$, $10\%$, … , $100\%$]. The $z_{\rm{SN}}$’s are obtained from `LCFIT+z`
(Kessler et al., 2010) using the ugrizY bands as with our $z_{\rm{DCR}}$
simulations, where the redshift is fit simultaneously along with the light-
curve parameters, i.e., time of maximum brightness $t_{0}$, color $c$, stretch
$x_{1}$, and flux normalization $x_{0}$.

When combining $z_{\rm{DCR}}$ with $z_{\rm{Host}}$, we use the `qp`
package555https://github.com/LSSTDESC/qp (Malz et al., 2018) to reconstruct
PDFs from the $z_{\rm{Host}}$ quantiles before multiplying the $z_{\rm{DCR}}$
and $z_{\rm{Host}}$ PDFs together. To combine either $z_{\rm{DCR}}$ or
$z_{\rm{Host}}$ (or both) with $z_{\rm{SN}}$, we provide the respective
quantiles as priors in `LCFIT+z`. We do not determine $P(z_{\rm{DCR}})$,
$P(z_{\rm{Host}})$, and $P(z_{\rm{SN}})$ independently and combine the three
PDFs for a joint PDF, because the SALT3 parameters such as $x_{0}$, $x_{1}$,
and $c$ and their uncertainties would be incorrectly associated with
$P(z_{\rm{SN}})$ only. Thus the correct method is to combine
$P(z_{\rm{Host}})$ and $P(z_{\rm{DCR}})$, and use this combined prior in
`LCFIT+z` to not only obtain the combined redshift estimates, but also the
SALT3 parameters with correct uncertainties. When showing combined results, we
impose the default selection cut described in Section 3 on the candidates with
successful light-curve fits with $z_{\rm{SN}}$ and just the default selection
cut when $z_{\rm{SN}}$ are not used.

Sometimes, the $z_{\rm{DCR}}$ and $z_{\rm{Host}}$ estimates cannot be
determined (e.g. when the SN candidate has no z or Y band observations for
$z_{\rm{DCR}}$). In this case, we replace the PDF with a flat prior; if there
is no $z_{\rm{DCR}}$ estimate for a candidate, only the $z_{\rm{Host}}$ prior
is used and vice versa.

When combining two or more PDFs, sometimes the PDFs have negligible overlap,
meaning that it is meaningless to combine them. In particular,
$z_{\rm{DCR+Host}}$ perform worse than $z_{\rm{DCR}}$ by itself at $z<0.4$
where $z_{\rm{DCR}}$ perform better than $z_{\rm{Host}}$ when $z_{\rm{DCR}}$
and $z_{\rm{Host}}$ are combined without considering the compatibility between
$z_{\rm{DCR}}$ and $z_{\rm{Host}}$ PDFs. Hence, we impose an additional
selection cut for $z_{\rm{DCR+Host}}$ when $z_{\rm{DCR}}$ and $z_{\rm{Host}}$
PDFs are incompatible. We assess the compatibility between the two PDFs using
the Overlapping Index (OVL) between two PDFs, which is defined in the
literature such as Inman & Bradley Jr (1989), but originally postulated in a
related form in Pearson (1894):

$\mathrm{OVL}(p_{0},p_{1})=\int\min(p_{0}(x),p_{1}(x))dx$ (10)

where $p_{0}(x)$ and $p_{1}(x)$ are PDFs. In Appendix B, we discuss the
overlapping index in more detail and in Figure 24, we show $\mathrm{OVL}$
values depending on the number of standard deviations the means of two
normalized Gaussian distributions are apart from one another. Based on the
performance metrics (discussed in the beginning of Section 5) for
$z_{\rm{DCR+Host}}$ compared to $z_{\rm{DCR}}$ and $z_{\rm{Host}}$ by
themselves, we remove candidates for $z_{\rm{DCR+Host}}$ when
$\mathrm{OVL}<0.0344$, or when two normalized Gaussian distributions with the
same standard deviations ($\sigma$) have means that are $3\sqrt{2}\sigma$
apart from each other. This choice eliminates only about 2.5% of the
candidates that pass the Default selection cut, mostly at the lowest redshifts
where incorrect $z_{\rm{Host}}$ PDFs can degrade $z_{\rm{DCR+Host}}$
considerably, but significantly improves performance metrics at those
redshifts.

When assessing the compatibility between $z_{\rm{SN}}$ PDFs and the two other
estimates $z_{\rm{DCR}}$ and $z_{\rm{Host}}$, we cannot use the full
$z_{\rm{SN}}$ PDFs to compute the overlapping index with the other estimates
since `LCFIT+z` returns only the estimated redshift and the $1\sigma$ error
bars. While it is possible to compute the overlapping index assuming that
$z_{\rm{SN}}$ PDFs are Gaussian (although this is not true as shown in Sako et
al. (2018)), we found that the OVL values between $z_{\rm{SN}}$ and
$z_{\rm{DCR}}$ or $z_{\rm{Host}}$ are very small, due to the $z_{\rm{SN}}$
$1\sigma$ error bars from `LCFIT+z` being small, with a mean of $0.017$ and
median of $0.011$. The `LCFIT+z` $z_{\rm{SN}}$ error bars are computed using
the default MINUIT (James & Roos, 1975) error option (MIGRAD, ‘Migration by
Gradientes’). Better photo-$z$ error estimates can be obtained, such as by
using the MINOS (‘Minimization by Orthogonal Simplex’) option in MINUIT, or
MCMC fitting, but the fitting speed would be significantly slower. The
$\mathrm{FITPROB}>0.01$ cut mentioned in Section 3 resolves some of the
incompatibility issues and we believe that our treatment is sufficient for
this initial investigation of combining SN Ia astrometric redshifts with
photometric redshifts.

Figure 7:  $z_{\rm{DCR}}$ vs. $z_{\rm{true}}$ in the Perfect case for PDFPeak. As pointed out in Section 4, CDF50 results are equivalent to PDFPeak results for this case and the $1\sigma$ uncertainties are zero. The right panels are the same as the left, but zoomed to $z<0.6$ and the outlier fraction $\eta_{0.10}$ is defined as the fraction where $|\Delta z|>0.10$, with $\Delta z\equiv\frac{z_{\rm{estimate}}-z_{\rm{true}}}{1+z_{\rm{true}}}$. The imperfect $z_{\rm{DCR}}$ estimates are due to degeneracies in the DCR shifts at several redshifts, which is most apparent at $z_{\rm{true}}>0.8$. Type | Bias (CDF50) | Bias (PDFPeak) |  $\eta_{0.10}$ (CDF50) |  $\eta_{0.10}$ (PDFPeak) |  $\sigma_{\rm{MAD}}$ (CDF50) |  $\sigma_{\rm{MAD}}$ (PDFPeak)
---|---|---|---|---|---|---
Perfect | -0.003$\pm$0.000 | -0.003$\pm$0.000 | 0.016 | 0.016 | 0.008 | 0.008
Realistic | -0.049$\pm$0.001 | -0.027$\pm$0.002 | 0.342 | 0.331 | 0.088 | 0.077
Table 3: Bias and error on the mean, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$
for Perfect and Realistic simulations. We show in Section 5.3 that most of the
degradation shown here for the Realistic case is caused by statistical
uncertainties.

Additionally, for $z_{\rm{SN+DCR+Host}}$, we use either $z_{\rm{DCR}}$ alone
or $z_{\rm{Host}}$ alone as priors when $z_{\rm{DCR}}$ and $z_{\rm{Host}}$ are
incompatible with each other, depending on whether the overlapping index
between $z_{\rm{SN}}$ and $z_{\rm{DCR}}$ or $z_{\rm{Host}}$ is higher. This
choice further improves the performance metrics at low-$z$, as with
$z_{\rm{DCR+Host}}$ discussed earlier.

##  5 Results

In this section, we show the results of our $z_{\rm{DCR}}$ estimates for LSST-
like simulations for three cases: (i) Perfect ($\sigma_{\rm{stat}}=0$ and
$\sigma_{\rm{syst}}=0$) (ii) Realistic (with non-zero $\sigma_{\rm{stat}}$ and
$\sigma_{\rm{syst}}$), and (iii) Systematic Effects only. For the Realistic
case, we also show the results combined with $z_{\rm{Host}}$ as well as with
$z_{\rm{SN}}$ obtained from `LCFIT+z`. Along with the comparison between the
estimated redshifts and true redshifts, we also show three different metrics
depending on the case and the combination of PDFs: the mean bias (or binned
residuals), outlier fraction, and the MAD (Median Absolute Deviation)
deviation.

As is customary in photo-$z$ literature (Kessler et al., 2010; Pasquet et al.,
2019; Qu & Sako, 2023), we define the residuals as:

$\Delta z\equiv\frac{z_{\rm{estimate}}-z_{\rm{true}}}{1+z_{\rm{true}}}$ (11)

where $z_{\rm{estimate}}$ is the estimated redshift and the bias is
$\langle\Delta z\rangle$. The uncertainty on $\langle\Delta z\rangle$ is
defined as $\frac{\mathrm{RMS}(\Delta z-\langle\Delta
z\rangle)}{\sqrt{N_{\mathrm{SN}}}}$ where RMS is the root-mean-square and
$N_{\mathrm{SN}}$ is the number of SN events in the sample or redshift bin.
The outlier fraction $\eta_{x}$ is defined as the fraction of candidates where
$|\Delta z|>x$: $x=0.10$ is our default. Lastly, the precision metric MAD
deviation is defined as $\sigma_{\rm{MAD}}=1.4826\times\rm{Median}|\Delta
z-\rm{Median}(\Delta z)|$.

###  5.1 Perfect Simulations

We first show our results in Figure 7 for the Perfect case without any
statistical or systematic uncertainties, assuming that our peak MJD
measurements have no error, and that we know both $x_{1}$ and $c$. This test
validates the analysis and demonstrates the ideal performance of
$z_{\rm{DCR}}$ for the LSST DDF SNe Ia. Only the epoch selection cut is
applied, since we are essentially assuming infinite S/N. As highlighted in
Section 4, we set $\sigma_{\mathrm{syst}}=1$ mas in the $\chi^{2}$ denominator
to avoid infinite $\chi^{2}$ (Equation 8), while $\sigma_{\mathrm{syst}}=0$ in
the simulation. The results we show here are equivalent for PDFPeak and CDF50.

In this perfect case, $z_{\rm{DCR}}$ give very precise estimates of the SN Ia
redshifts for all redshifts, but especially for $z<0.6$ where $z_{\rm{DCR}}$
almost always coincides with $z_{\rm{true}}$. When all redshifts are included,
the outlier fraction $\eta_{0.10}$ is 1.58%, $\sigma_{\rm{MAD}}$ is 0.008, and
bias is $-0.003\pm 0.000$ as shown in Table 3.

Figure 8: Binned residuals (bias), error on the mean (bold error bars) and the
size of $\sigma_{\mathrm{MAD}}$ (light error bars) as a function of
$z_{\rm{true}}$ for Perfect and Realistic simulations for CDF50 (left) and
PDFPeak (right). CDF50 and PDFPeak values are equivalent for Perfect
simulations. For clarity, Realistic points (blue) are plotted at the midpoints
of each redshift bin ($0.05,0.15,0.25,...$), while Perfect are displaced 0.02
to the right. The bias for the Realistic case noticeably deviates away from
zero starting at $z_{\rm{true}}=0.7$, especially for CDF50, where
$z_{\rm{DCR}}\approx 0.6$ for the unconstrained candidates. PDFPeak, on the
other hand, displays significantly larger $\sigma_{\rm{MAD}}$ at higher
$z_{\rm{true}}$. This is due to $z_{\rm{DCR}}$ being centered at $0.6$ with
little spread for CDF50, meaning that the redshift estimates are precise, but
not accurate. Figure 9: Binned outlier fractions as a function of
$z_{\rm{true}}$ for Perfect and Realistic simulations for CDF50 (left) and
PDFPeak (right). The Perfect simulations show $<5\%$ outlier fractions at all
redshifts. For the Realistic simulations, the outlier fractions remain below
40% for both CDF50 and PDFPeak until $z_{\rm{true}}=0.7$. At
$z_{\rm{true}}\geq 0.8$, CDF50 estimates are mostly incorrect as the PDFs are
largely flat and unconstrained. Note that the outlier fractions for PDFPeak
decreasing at the highest redshifts for the Realistic case is due to the
possible estimated redshift range being limited from 0.0 to 1.2. Figure 10:
Similar to Figure 7, but for the Realistic case; $z_{\rm{DCR}}$ vs.
$z_{\rm{true}}$ and the $1\sigma$ error bars for the Realistic case for CDF50
(top panels) and PDFPeak (bottom panels). We see noticeable degradation in the
$z_{\rm{DCR}}$ estimates, especially at $z_{\rm{true}}>0.8$. The
$z_{\rm{true}}<0.6$ panels still show good agreement.

In Figure 8 and Figure 9, the green X symbols show the binned bias,
$\sigma_{\mathrm{MAD}}$, and the outlier fractions versus $z_{\mathrm{true}}$.
Again, the values shown for CDF50 and PDFPeak are equivalent. For the Perfect
case, the bias is below 0.01 and $\eta_{0.10}$ is also below 5% in all
redshift bins up to $z_{\rm{true}}=1.2$.

Although the results shown here are for the Perfect case, there are some
degeneracies and catastrophic outliers in Figure 7 (clumps or streaks)
especially at the high redshifts. These artifacts are caused by the DCR shifts
being degenerate at multiple redshifts at a given AM and observed epoch; fewer
observations result in a higher probability that these degeneracies remain.
Additionally, at higher redshifts, we typically do not have detections in the
ug bands due to low S/N (we only use detections where the coadded
$\mathrm{S/N}>3$ as mentioned in Section 3) where the DCR shifts are largest
and therefore most informative.

Figure 11: Histograms of the residuals $\Delta z$ for the Realistic case for
CDF50 (left) and PDFPeak (right). While the overall bias is significantly
larger than the Perfect case, we see little bias at $z_{\rm{true}}<0.6$.

###  5.2 Realistic Simulations

In Figures 10 and 11, we present our results for the realistic case with both
$\sigma_{\rm{stat}}$, $\sigma_{\rm{syst}}$, and realistic (mostly light-curve
fit) peak MJD measurements. Additionally, we marginalize over $x_{1}$ and $c$.
After imposing the S/N selection cut on top of the epoch selection cut, we are
left with about 43.1% of the simulated candidates (see Section 3).

Type | Bias (CDF50) | Bias (PDFPeak) |  $\eta_{0.10}$ (CDF50) |  $\eta_{0.10}$ (PDFPeak) |  $\sigma_{\rm{MAD}}$ (CDF50) |  $\sigma_{\rm{MAD}}$ (PDFPeak)
---|---|---|---|---|---|---
No selection cut | -0.103$\pm$0.001 | -0.065$\pm$0.001 | 0.579 | 0.443 | 0.138 | 0.122
Default | -0.049$\pm$0.001 | -0.027$\pm$0.002 | 0.342 | 0.331 | 0.088 | 0.077
Default + AM cut | -0.043$\pm$0.001 | -0.022$\pm$0.002 | 0.315 | 0.311 | 0.081 | 0.071
Fmax-clump | -0.050$\pm$0.001 | -0.028$\pm$0.002 | 0.353 | 0.346 | 0.091 | 0.082
Ideal $T_{\mathrm{obs}}$ | -0.050$\pm$0.001 | -0.028$\pm$0.002 | 0.333 | 0.322 | 0.085 | 0.073
Table 4: Bias and error on the mean, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$
for the analysis methods indicated in the first column. With stricter
selection cuts, we see improved metrics. Using peak MJDs from LCFIT+z (which
is mostly the case for ‘Default’) improves performance compared to using
‘Fmax-clump.’

Here, we find that the $z_{\rm{DCR}}$ estimates have degraded compared to the
Perfect case, especially at high redshifts ($z_{\rm{true}}>0.7$). CDF50
estimates very few SNe to be above $z=0.8$ because the PDFs are nearly flat
and the 50th percentile is clustered near the middle of the redshift range as
shown in Figure 10 and reflected in the histograms of the residuals (Figure
11). PDFPeak shows slightly better performance when all redshifts are
included, both in terms of $\eta_{0.10}$ (33.1% compared to 34.2% for CDF50)
and bias ($-0.027\pm 0.002$ compared to $-0.049\pm 0.001$ for CDF50). This
suggests that even at $z_{\rm{true}}>0.8$, the high redshift PDF peaks still
provide some information, although by a modest amount.

In the $z_{\rm{true}}<0.6$ panels of Figure 10, the accuracy is still
relatively high compared to $z_{\rm{true}}>0.6$, but CDF50 shows better
performance. This is because CDF50 estimates the redshift to be $z\approx 0.6$
when the PDF is unconstraining, reducing catastrophic outliers when
$z_{\rm{true}}<0.6$. In Figure 8 and Figure 9, the biases are close to zero at
$z_{\mathrm{true}}<0.7$ and the outlier fractions are below 20% at
$z_{\mathrm{true}}<0.6$ for both CDF50 and PDFPeak. The bias however, deviates
from zero at $z_{\mathrm{true}}>0.7$ for both CDF50 and PDFPeak, but to a
greater extent for CDF50 since most of the estimated redshifts are around
$0.6$. This is also the reason why the $\sigma_{\rm{MAD}}$ error bars are much
larger for PDFPeak compared to CDF50; the CDF50 estimates are precise, but not
accurate at high-$z$. The diagonal streak (Figure 10) spanning
$(z_{\rm{true}},z_{\rm{pred}})$ = $(0.3,0.5)$ to $(0.5,0.3)$ could be due to
the g-band DCR shifts typically being close to zero throughout all epochs
around these redshifts, while there likely are not many u-band observations,
increasing room for degeneracies.

Figure 12: $z_{\rm{Host}}$ vs. $z_{\rm{true}}$ and the $1\sigma$ error bars
for CDF50 (top panels) and PDFPeak (bottom panels). For around 5% of the SNe
that passes the Default selection cut, there are no $z_{\rm{Host}}$ estimates,
so we assume flat PDFs as discussed in Section 4. While we do not show these
candidates as data points and error bars in this figure for clarity, they are
always included for the performance metrics. We see that the $z_{\rm{Host}}$
is more accurate at higher redshifts contrary to $z_{\rm{DCR}}$. We also note
that CDF50 performs better here, although this could be the result of the PDFs
being constructed from quantiles with 11 bins.

We show in Table 4 the bias, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$ depending
on the selection criteria: no selection cuts, default selection cut (43.1% of
total candidates) and default + AM selection cut (38.0% of total candidates)
where the AM cut requires at least one observation at $\rm{AM}>1.4$ for a
given candidate. As expected, the performance of $z_{\rm{DCR}}$ improves with
more stringent cuts. We also see larger bias in CDF50 (compared to PDFPeak)
for all three types of cuts, which is expected given that CDF50 estimates most
$z_{\mathrm{true}}>0.7$ SNe to be at $z_{\rm{DCR}}\approx 0.6$. Consequently,
the outlier fractions are also larger for CDF50 than for PDFPeak, but this is
most apparent for ‘No selection cut.’ Additionally requiring the AM cut
improves the estimates, but not significantly. While Kaczmarczik et al. (2009)
suggests that a number of moderately high AM observations will result in
considerably more accurate redshift estimates for quasars, we note that for
the LSST DDF SNe Ia simulations, most of the candidates that pass the default
selection cut also pass the AM cut as there are multiple (usually over 5)
observations per SN. We also show the metrics depending on the choice of
$T_{\rm{obs}}$, and as expected, using peak MJDs from `LCFIT+z` results in
some improvement from the ‘Fmax-clump’ method, but less than the Ideal
$T_{\rm{obs}}$ case, where we use the exact peak MJDs.

####  5.2.1 Combining with $z_{\rm{Host}}$

In this section, we discuss the performance of host galaxy photometric
redshifts ($z_{\rm{Host}}$) combined with astrometric redshifts. Since we
estimate $z_{\rm{Host}}$ using PDFs constructed from quantiles, we apply the
CDF50 and PDFPeak methods the same way as for $z_{\rm{DCR}}$. First, in Figure
12, we show $z_{\rm{Host}}$ as a function of $z_{\rm{true}}$ for CDF50 and
PDFPeak, with the same default selection cut imposed for the $z_{\rm{DCR}}$
only simulations. Compared to $z_{\rm{DCR}}$, the $z_{\rm{Host}}$ are much
more accurate at higher redshifts, but less accurate at lower redshifts. This
is due to the distinct 4000Å break for galaxies being in the u-band at low
redshifts, where the sensitivity is typically poor. Also, CDF50 gives more
accurate estimates than PDFPeak with $\eta_{0.10}=25.2\%$ for CDF50 and
$\eta_{0.10}=32.4\%$ for PDFPeak. As shown in Table 5, CDF50 performs better
than PDFPeak in terms of bias and $\sigma_{\rm{MAD}}$ as well.

Type | Bias (CDF50) | Bias (PDFPeak) |  $\eta_{0.10}$ (CDF50) |  $\eta_{0.10}$ (PDFPeak) |  $\sigma_{\rm{MAD}}$ (CDF50) |  $\sigma_{\rm{MAD}}$ (PDFPeak)
---|---|---|---|---|---|---
$z_{\rm{DCR}}$ | -0.049$\pm$0.001 | -0.027$\pm$0.002 | 0.342 | 0.331 | 0.088 | 0.077
$z_{\rm{Host}}$ | -0.008$\pm$0.001 | -0.029$\pm$0.002 | 0.252 | 0.324 | 0.067 | 0.083
$z_{\rm{DCR+Host}}$ | -0.016$\pm$0.001 | -0.014$\pm$0.001 | 0.154 | 0.190 | 0.047 | 0.051
Table 5: Bias and error on the mean, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$
for $z_{\rm{DCR}}$ only, $z_{\rm{Host}}$ only, and $z_{\rm{DCR+Host}}$.
Combining the two reduces the bias (PDFPeak) and $\sigma_{\rm{MAD}}$, while
the outlier fractions are reduced by at least 10%. Figure 13:
$z_{\rm{DCR+Host}}$ vs. $z_{\rm{true}}$ and the $1\sigma$ error bars for CDF50
(top panels) and PDFPeak (bottom panels). We see noticeable improvement from
Figure 12, especially at $z_{\rm{true}}<0.6$ where the $z_{\rm{Host}}$ are not
very constraining. Figure 14: Binned residuals (bias), error on the mean (bold
error bars) and the size of $\sigma_{\mathrm{MAD}}$ (light error bars) as a
function of $z_{\rm{true}}$ for $z_{\rm{DCR}}$, $z_{\rm{Host}}$, and
$z_{\rm{DCR+Host}}$ for CDF50 (left) and PDFPeak (right). Again,
$z_{\rm{DCR}}$ is placed at the midpoints of each redshift bin, while
$z_{\rm{Host}}$ and $z_{\rm{DCR+Host}}$ are displaced to the left and right
respectively. Contrary to $z_{\rm{DCR}}$, $z_{\rm{Host}}$ display more bias
and larger error bars at lower redshifts, but combining with $z_{\rm{DCR}}$
alleviates this. Figure 15: Binned outlier fractions as a function of
$z_{\rm{true}}$ for $z_{\rm{DCR}}$, $z_{\rm{Host}}$, and $z_{\rm{DCR+Host}}$
for CDF50 (left) and PDFPeak (right). Combining with $z_{\rm{DCR}}$
significantly reduces the outlier fractions of $z_{\rm{Host}}$ at
$z_{\rm{true}}<0.6$ for both CDF50 and PDFPeak, while the high-$z$ outlier
fractions are not affected by much.

In Figure 13, we show the combined results, $z_{\rm{DCR+Host}}$ vs.
$z_{\rm{true}}$. The estimated redshifts more tightly follow the true
redshifts with $\eta_{0.10}$ being 15.4% for CDF50 and 19.0% for PDFPeak,
meaning that the outlier fractions are reduced by at least 10% for both CDF50
and PDFPeak compared to $z_{\rm{Host}}$. The improvement is particularly
striking at $z_{\mathrm{true}}<0.6$, where the $z_{\rm{Host}}$ are not very
constraining.

In Table 5, we also highlight the bias and $\sigma_{\rm{MAD}}$ for
$z_{\rm{DCR}}$, $z_{\rm{Host}}$, and $z_{\rm{DCR+Host}}$. We find that
combining the two removes most of the bias for PDFPeak and $\sigma_{\rm{MAD}}$
is considerably reduced for both CDF50 and PDFPeak.

Figures 14-15 show the mean binned residuals vs $z_{\mathrm{true}}$. We see
large bias for $z_{\rm{Host}}$ at low redshifts for both CDF50 and PDFPeak,
little bias at the mid redshifts, and some bias at the higher redshifts.
Combining with $z_{\rm{DCR}}$ reduces the bias, especially at the low
redshifts. We note a similar trend for the outlier fractions as well; the
$z_{\rm{Host}}$ outlier fractions are reduced significantly when combined with
$z_{\rm{DCR}}$ at low redshifts.

####  5.2.2 Combining with $z_{\rm{SN}}$

| Number of LCFIT+z events with:
---|---
Type | No cuts | Default cuts | Default + FITPROB cuts
$z_{\rm{SN}}$ | 12,589 | 7,295 | 6,657
$z_{\rm{SN+Host}}$ | 12,324 | 7,176 | 6,615
$z_{\rm{SN+DCR}}$ | 12,246 | 7,660 | 7,255
$z_{\rm{SN+DCR+Host}}$ | 12,779 | 7,568 | 7,215
Table 6: Number of candidates with converged LCFIT+z light-curves (No cuts),
with the default selection cut imposed on the LCFIT+z converged candidates
(Default cuts), and with both the default selection cut and FITPROB $>0.01$
cut imposed (Default + FITPROB cuts). Combining $z_{\rm{DCR}}$ recovers about
600 more candidates (9%) where LCFIT+z for photo-$z$’s only do not converge.

In this section, we show the redshift estimates for supernovae photometric
redshifts ($z_{\rm{SN}}$) using `LCFIT+z` as well as $z_{\rm{SN}}$ combined
with $z_{\rm{Host}}$ and $z_{\rm{DCR}}$. Assessing the improvement resulting
from combining the $z_{\rm{DCR}}$ with $z_{\rm{SN+Host}}$ is crucial to
understanding how $z_{\rm{DCR}}$ can be used for upcoming surveys such as
Rubin where the number of supernovae detected will be far too large for
spectroscopic redshift measurements.

We show in Table 6, the number of candidates remaining from the original
20,000 after requiring `LCFIT+z` convergence which is about 60 to 65% of the
candidates, and then the default and FITPROB selection cuts, leaving 33% to
37% of the candidates. In Figure 16, we show $z_{\rm{SN}}$ vs.
$z_{\mathrm{true}}$ (top panels), $z_{\rm{SN+Host}}$ vs. $z_{\mathrm{true}}$
(second panels), $z_{\rm{SN+DCR}}$ vs. $z_{\mathrm{true}}$ (third panels), and
$z_{\rm{SN+DCR+Host}}$ vs. $z_{\mathrm{true}}$ (bottom panels) with the
`LCFIT+z` \+ Default selection cut. We would like to emphasize that including
$z_{\mathrm{DCR}}$ to the photo-$z$’s results in a 9% increase in the number
of candidates that pass the Default + FITPROB cuts.

As with $z_{\rm{DCR}}$, $z_{\rm{SN}}$ have streaks of degeneracies at several
locations and is more accurate at lower redshifts, although the performance is
generally better with $\eta_{0.10}$ being 20.5% for all redshifts and 9.4%
when limited to $z_{\rm{true}}<0.6$. Combining with $z_{\rm{Host}}$
significantly improves the estimates at all redshifts, but the improvement is
more pronounced at $z_{\mathrm{true}}>0.6$, with $\eta_{0.10}$ reduced by
about 14% for the all-$z$ sample. This improvement is expected since
$z_{\rm{SN}}$ and $z_{\rm{Host}}$ are independent estimates (as are
$z_{\rm{DCR}}$ with $z_{\rm{Host}}$ as discussed in Section 5.2.1). The third
panels show that adding $z_{\rm{DCR}}$ to $z_{\rm{SN}}$ considerably improves
the estimates with $\eta_{0.10}$ reduced by about 6% for the all-$z$ sample
compared to $z_{\rm{SN}}$ only. Contrary to combining $z_{\rm{Host}}$, the
$z_{\mathrm{SN+DCR}}$ improvement is more prominent at lower redshifts with
$\eta_{0.10}$ reduced by 7% when limited to $z_{\rm{true}}<0.6$. Lastly,
combining $z_{\rm{SN}}$ with $z_{\rm{Host}}$ and $z_{\rm{DCR}}$ shows the full
potential of non-spectroscopic redshift estimates, with $\eta_{0.10}$
decreasing to 6.1% for all redshifts and 1.8% when limited to
$z_{\mathrm{true}}<0.6$.

Table 7 shows the bias, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$ for the four
combinations of redshift estimation methods involving $z_{\rm{SN}}$. The
measurements involving $z_{\rm{SN}}$ are unbiased, with the exception of
$z_{\mathrm{SN+DCR}}$. Combining $z_{\rm{Host}}$ with $z_{\rm{SN}}$ reduces
$\sigma_{\rm{MAD}}$ to about two thirds of the $z_{\rm{SN}}$-only value, while
combining $z_{\rm{DCR}}$ with $z_{\rm{SN}}$ shows a similar impact. When all
three methods are combined, the bias is nearly zero and $\sigma_{\rm{MAD}}$ is
about 0.014.

Figure 16: $z_{\rm{SN}}$ vs. $z_{\mathrm{true}}$ (top panels), $z_{\rm{SN+Host}}$ vs. $z_{\mathrm{true}}$ (second panels), $z_{\rm{SN+DCR}}$ vs. $z_{\mathrm{true}}$ (third panels), and $z_{\rm{SN+DCR+Host}}$ vs. $z_{\mathrm{true}}$ (bottom panels). Combining $z_{\rm{SN}}$ with either $z_{\rm{Host}}$ or $z_{\rm{DCR}}$ significantly improves the estimates, more so at $z_{\rm{true}}>0.6$ for $z_{\rm{Host}}$ and $z_{\rm{true}}<0.6$ for $z_{\rm{DCR}}$. Combining all three shows much better overall (throughout all redshifts) accuracy than any two combinations. Type | Bias | $\eta_{0.10}$ | $\sigma_{\rm{MAD}}$
---|---|---|---
$z_{\rm{SN}}$ | 0.006$\pm$0.001 | 0.205 | 0.025
$z_{\rm{SN+Host}}$ | -0.001$\pm$0.001 | 0.068 | 0.016
$z_{\rm{SN+DCR}}$ | -0.021$\pm$0.001 | 0.147 | 0.017
$z_{\rm{SN+DCR+Host}}$ | -0.004$\pm$0.001 | 0.061 | 0.014
Table 7: Bias and error on the mean, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$
for $z_{\rm{SN}}$, $z_{\rm{SN+Host}}$, $z_{\rm{SN+DCR}}$, and
$z_{\rm{SN+DCR+Host}}$. Combining all three methods significantly lowers the
outlier fraction compared to $z_{\rm{SN}}$ only.

There are a few points worth noting. First, while combining $z_{\rm{SN}}$ with
$z_{\rm{Host}}$ reduces the outlier fraction by 14% as opposed to 6% with the
addition of $z_{\rm{DCR}}$, only about 37% of the true redshifts are below
0.6, where the $z_{\rm{DCR}}$ are more informative. Because the $z_{\rm{DCR}}$
and $z_{\rm{Host}}$ each perform better at different redshifts, we show in
Table 8 the bias, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$ for the different
combinations at high-$z$ ($z_{\mathrm{true}}>0.6$) and low-$z$
($z_{\mathrm{true}}<0.6$) separately. For $z_{\rm{DCR}}$, $z_{\rm{Host}}$, and
$z_{\rm{DCR+Host}}$, we show the PDFPeak values. Second, it may be counter-
intuitive that combining $z_{\rm{SN}}$ with $z_{\rm{DCR}}$ results in any
noticeable improvement, since the information for both redshift estimates is
based on the same SN Ia SED time series. We evaluate the independence of these
two redshift estimates by computing the Pearson correlation coefficient
between $z_{\rm{SN}}-z_{\rm{true}}$ and $z_{\rm{DCR}}-z_{\rm{true}}$, which we
find to be 0.151. Such a low value shows that the two measurements are largely
independent. This is because $z_{\rm{DCR}}$ relies on the product of the
astrometric shifts due to DCR and the SED within a band (Equation 1) while
$z_{\rm{SN}}$ relies on only the SED integrated within a band. It is therefore
reasonable that combining these two measurements results in a better redshift
constraint.

In Table 8 (where we show the bias, $\eta_{0.10}$ and $\sigma_{\rm{MAD}}$ at
low-$z$ and high-$z$ separately for PDFPeak when not combining with
$z_{\rm{SN}}$), both the bias and outlier fractions are much lower at low-$z$
than high-$z$ for $z_{\rm{DCR}}$, while the outlier fractions are much lower
at high-$z$ for $z_{\rm{Host}}$ as seen earlier. $z_{\rm{SN}}$ show larger
bias at low-$z$ but a much smaller outlier fraction (9.4%) than at high-$z$
(26.5%) as with $z_{\rm{DCR}}$. Combining $z_{\rm{SN}}$ with $z_{\rm{Host}}$
reduces both the bias and outlier fractions significantly at both low-$z$ and
high-$z$. While it is no surprise that the outlier fraction at high-$z$ is
about a third of that for $z_{\rm{SN}}$ (8.9%), it is intriguing that the
outlier fraction at low-$z$ is also 6% lower, at 3.2% given that the
$z_{\rm{Host}}$-only outlier fraction at low-$z$ is 42.1%. Combining
$z_{\rm{SN}}$ with $z_{\rm{DCR}}$ greatly reduces the bias and outlier
fraction at low-$z$, with the outlier fraction being 2.3%, but the bias at
high-$z$ is only a little smaller than $z_{\rm{DCR}}$ at high-$z$, and the
outlier fraction at high-$z$ is reduced by a small amount (22.4%) as one would
expect. When all three methods are combined, the bias at both low-$z$ and
high-$z$ are close to zero, and the outlier fractions are similarly low at
1.8% and 8.7% for low-$z$ and high-$z$ respectively, although using
$z_{\rm{SN+DCR}}$ at low-$z$ and $z_{\rm{SN+Host}}$ at high-$z$ shows similar
performance. A similar trend can be seen with $\sigma_{\rm{MAD}}$ as well;
$z_{\rm{DCR}}$ has a much lower value at low-$z$ while $z_{\rm{Host}}$ has a
lower value at high-$z$, and hence combining each with $z_{\rm{SN}}$ displays
lower (or at least similar) values at low-$z$ and high-$z$ respectively.
$z_{\rm{SN+DCR+Host}}$ has the lowest $\sigma_{\rm{MAD}}$ values for both
low-$z$ and high-$z$.

Type | Bias (low-z) | Bias (high-z) |  $\eta_{0.10}$ (low-z) |  $\eta_{0.10}$ (high-z) |  $\sigma_{\rm{MAD}}$ (low-z) |  $\sigma_{\rm{MAD}}$ (high-z)
---|---|---|---|---|---|---
$z_{\rm{DCR}}$ | 0.006$\pm$0.002 | -0.047$\pm$0.002 | 0.144 | 0.439 | 0.031 | 0.121
$z_{\rm{Host}}$ | 0.012$\pm$0.003 | -0.052$\pm$0.002 | 0.421 | 0.268 | 0.123 | 0.066
$z_{\rm{DCR+Host}}$ | 0.005$\pm$0.001 | -0.025$\pm$0.002 | 0.151 | 0.212 | 0.039 | 0.057
$z_{\rm{SN}}$ | 0.014$\pm$0.001 | 0.002$\pm$0.001 | 0.094 | 0.265 | 0.007 | 0.064
$z_{\rm{SN+Host}}$ | 0.005$\pm$0.001 | -0.004$\pm$0.001 | 0.032 | 0.089 | 0.008 | 0.024
$z_{\rm{SN+DCR}}$ | 0.003$\pm$0.001 | -0.036$\pm$0.001 | 0.023 | 0.224 | 0.007 | 0.044
$z_{\rm{SN+DCR+Host}}$ | 0.003$\pm$0.001 | -0.008$\pm$0.001 | 0.018 | 0.087 | 0.007 | 0.023
Table 8: Bias and error on the mean, $\eta_{0.10}$, and $\sigma_{\rm{MAD}}$
for low-$z$ ($z_{\mathrm{true}}<0.6$) and high-$z$ ($z_{\mathrm{true}}>0.6$)
for various combinations of the three methods discussed in this work. For
$z_{\rm{DCR}}$, $z_{\rm{Host}}$, and $z_{\rm{DCR+Host}}$, we show the PDFPeak
values. $z_{\rm{DCR}}$ shows good performance at low-$z$ while $z_{\rm{Host}}$
shows good performance at high-$z$, which leads to $z_{\rm{SN+DCR}}$ being
significantly improved at low-$z$ compared to $z_{\rm{SN}}$ and
$z_{\rm{SN+Host}}$ being significantly improved at high-$z$.
$z_{\rm{SN+DCR+Host}}$ show similar performance to $z_{\rm{SN+DCR}}$ and
$z_{\rm{SN+Host}}$ at low-$z$ and high-$z$ respectively. Figure 17: Binned
residuals (bias), error on the mean (bold error bars) and the size of
$\sigma_{\mathrm{MAD}}$ (light error bars) as a function of $z_{\rm{true}}$
for $z_{\rm{SN}}$, $z_{\rm{SN+Host}}$ , $z_{\rm{SN+DCR}}$, and
$z_{\rm{SN+DCR+Host}}$. Combining all three reduces the bias throughout all
redshifts.

Lastly, in Figure 17 and Figure 18, we show the mean binned residuals and its
$\sigma_{\rm{MAD}}$ error bars, as well as the outlier fractions by true
redshift. The smallest bias occurs at $z_{\mathrm{true}}<0.2$ for
$z_{\rm{SN+DCR}}$, $0.2<z_{\mathrm{true}}<0.8$ for $z_{\rm{SN+DCR+Host}}$, and
$z_{\mathrm{true}}>0.8$ for $z_{\rm{SN+Host}}$. Similarly, the lowest outlier
fractions occur at $z_{\mathrm{true}}<0.8$ for $z_{\rm{SN+DCR+Host}}$ and
$z_{\mathrm{true}}>0.8$ for $z_{\rm{SN+Host}}$. For $z_{\rm{SN+DCR+Host}}$,
the outlier fractions are lowest at $0.0<z_{\mathrm{true}}<0.2$ (close to 0)
and highest at $1.1<z_{\mathrm{true}}<1.2$ (still less than 30%). Contrary to
when $z_{\rm{DCR}}$ are combined with $z_{\rm{Host}}$, combining
$z_{\rm{DCR}}$ with $z_{\rm{SN}}$ noticeably degrades the performance at
$z_{\mathrm{true}}>0.8$.

Figure 18: Binned $\eta_{0.10}$ as a function of $z_{\rm{true}}$ for
$z_{\rm{SN}}$, $z_{\rm{SN+Host}}$ , $z_{\rm{SN+DCR}}$, and
$z_{\rm{SN+DCR+Host}}$. Combining all three lowers the outlier fractions
throughout all redshifts, although $z_{\rm{SN+Host}}$ show slightly lower
outlier fractions at $z_{\rm{true}}>0.8$. Figure 19: Diagnostics for
$z_{\mathrm{SN+DCR+Host}}$. The blue points show the distributions for
outliers, while the red histograms show the relative number of candidates for
non-outliers. The red bars have been scaled such that the area under the
$\eta_{0.10}$ histograms and Not $\eta_{0.10}$ histograms are equal. We show
the error bars for the $\eta_{0.10}$ points only because Not $\eta_{0.10}$ has
many more candidates than $\eta_{0.10}$ and hence smaller error bars. As
expected, outliers have lower FITPROB than non-outliers as well as slightly
higher prior-$\chi^{2}$. Additionally, outliers are more likely to occur in
lower mass host galaxies. Figure 20: $z_{\rm{SN}}$ vs. $z_{\rm{true}}$ with a
weak cosmology prior for LCFIT+z. Compared to the top panels of Figure 16
where there is no cosmology prior, the estimates are closer to the true values
throughout all redshifts and the outlier fractions are lower.

####  5.2.3 Diagnostics and Cosmology Prior

As discussed at the end of Section 3, we impose a $\mathrm{FITPROB}>0.01$
selection cut when combining with $z_{\mathrm{SN}}$, which addresses some of
the compatibility issues between the different redshift estimation methods.
While this is sufficient for our initial analysis with simulations, we dive
deeper and present some diagnostics as well as how much improvement can result
from using a cosmology prior for `LCFIT+z`.

In Figure 19, we show some diagnostics histograms for
$z_{\mathrm{SN+DCR+Host}}$ (with the $\mathrm{FITPROB}>0.01$ cut and the
compatibility cut for the $z_{\mathrm{DCR+Host}}$ prior) for candidates that
are within $\eta_{0.10}$ and outside $\eta_{0.10}$: FITPROB, prior-$\chi^{2}$,
and host galaxy $\log$ mass. For each of the panels, the blue points show the
number of candidates within the $x$-axis bins for outliers, while the red
histogram shows the distribution of non-outlier (Not $\eta_{0.10}$)
candidates, scaled to have the same integral as the outlier distribution.

On the left panel, we see that $\eta_{0.10}$ has more candidates at lower
FITPROB while Not $\eta_{0.10}$ display the opposite, as expected. Adding
useful information from FITPROB into a more complex machine learning algorithm
could therefore improve the selection of events. In the middle, we see that
the prior-$\chi^{2}$ histograms are similar for $\eta_{0.10}$ and Not
$\eta_{0.10}$, although $\eta_{0.10}$ is skewed towards slightly higher
prior-$\chi^{2}$ values. On the right panel, $\eta_{0.10}$ is shown to have
more low host mass contribution compared to Not $\eta_{0.10}$. This could be
due to low-mass host galaxies having lower S/N at high redshifts, degrading
all three redshift estimates.

Figure 21: Binned residuals (bias), error on the mean (bold error bars) and
the size of $\sigma_{\mathrm{MAD}}$ (light error bars) as a function of
$z_{\rm{true}}$ for $z_{\rm{SN}}$, with and without cosmology priors. Using a
cosmology prior generally tends to reduce the bias and $\sigma_{\mathrm{MAD}}$
except at $0.6<z_{\rm{true}}<0.8$. Figure 22: $\eta_{0.10}$ as a function of
$z_{\rm{true}}$ for $z_{\rm{SN}}$, with and without cosmology priors. Similar
to the bias and $\sigma_{\mathrm{MAD}}$, a cosmology prior generally reduces
$\eta_{0.10}$.

For all of the aforementioned results, we did not impose any cosmology prior
apart from determining $T_{\mathrm{obs}}$ using `LCFIT+z` for the Realistic
case (Section 3). Here, we investigate potential improvement from using a weak
cosmology prior for `LCFIT+z`. More detail on how the prior is applied is
given in Appendix C. In Figures 20 \- 22, we show the performance of
$z_{\rm{SN}}$ only with a weak cosmology prior of $w=-1.0\pm 0.1$ and
$\Omega_{m}=0.3\pm 0.03$. In Figure 20, we see that compared to the top panel
of Figure 16, the outlier fractions have noticeably decreased, $\eta_{0.10}$
is 0.205 and 0.186 without and with the cosmology prior respectively for all
redshifts, while for $z<0.6$ only, $\eta_{0.10}$ is 0.094 and 0.052
respectively. Note that the number of candidates that pass the selection cuts
(`LCFIT+z` convergence + epoch and S/N cut + $\mathrm{FITPROB}>0.01$ cut) have
also increased by 9% compared to the case without the cosmology prior. In
Figures 21 and 22, we show that the bias, $\eta_{0.10}$, as well as
$\sigma_{\mathrm{MAD}}$ are generally slightly reduced with the exception of
some bins between $0.6<z_{\mathrm{true}}<0.9$. This additional analysis shows
that in practice, we can utilize a weak cosmology prior to obtain better
estimates for both photometric and astrometric redshifts.

###  5.3 Systematic Effects

Type | Bias (CDF50) | Bias (PDFPeak) |  $\eta_{0.10}$ (CDF50) |  $\eta_{0.10}$ (PDFPeak) |  $\sigma_{\rm{MAD}}$ (CDF50) |  $\sigma_{\rm{MAD}}$ (PDFPeak)
---|---|---|---|---|---|---
Perfect | -0.003$\pm$0.000 | -0.003$\pm$0.000 | 0.016 | 0.016 | 0.008 | 0.008
$\sigma_{\rm{syst}}=2$ mas | -0.007$\pm$0.000 | -0.003$\pm$0.000 | 0.015 | 0.016 | 0.009 | 0.009
$\sigma_{\rm{syst}}=5$ mas | -0.010$\pm$0.000 | -0.004$\pm$0.000 | 0.028 | 0.024 | 0.011 | 0.011
$\sigma_{\rm{syst}}=10$ mas | -0.019$\pm$0.000 | -0.007$\pm$0.000 | 0.065 | 0.046 | 0.019 | 0.015
$c$ marginalized | -0.004$\pm$0.000 | -0.004$\pm$0.000 | 0.018 | 0.018 | 0.007 | 0.007
$x_{1}$ marginalized | -0.003$\pm$0.000 | -0.003$\pm$0.000 | 0.015 | 0.015 | 0.008 | 0.008
realistic $T_{\rm{obs}}$ | -0.002$\pm$0.000 | -0.002$\pm$0.000 | 0.041 | 0.041 | 0.010 | 0.010
All syst. for realistic | -0.010$\pm$0.001 | -0.004$\pm$0.001 | 0.064 | 0.073 | 0.016 | 0.014
Table 9: Bias, outlier fractions, and $\sigma_{\rm{MAD}}$ for various
systematics. The impact of $\sigma_{\rm{syst}}=2$ or $5$ mas, marginalization
over $c$ and $x_{1}$ is small, while using realistic $T_{\rm{obs}}$ results in
more degradation. Larger $\sigma_{\rm{syst}}$ results in substantially larger
degradation compared to other systematics. ‘All syst. for realistic’ means
using realistic $T_{\rm{obs}}$, marginalizing over $c$ and $x_{1}$, and
assuming $\sigma_{\rm{syst}}=5$ mas.

Since we have assessed the Perfect and Realistic cases, here we show how
sensitive our $z_{\rm{DCR}}$ results are to various levels of systematic
effects. In Table 9, we show again the bias, $\eta_{0.10}$, and
$\sigma_{\rm{MAD}}$ for the Perfect case as we saw in Section 5.1 as
reference. We also show the metrics for a Perfect simulation with the
following systematic variations: $\sigma_{\rm{syst}}=2$ mas,
$\sigma_{\rm{syst}}=5$ mas, $\sigma_{\rm{syst}}=10$ mas, $c$ marginalized,
$x_{1}$ marginalized, and realistic $T_{\rm{obs}}$. For each systematic we
tested, all the other conditions were taken to be the same as the Perfect
case. As described in Section 4, CDF50 and PDFPeak results are the same when
$\sigma_{\rm{tot}}$ should be zero; so for Perfect, $c$ marginalized, $x_{1}$
marginalized, and realistic $T_{\mathrm{obs}}$.

First, we show that $\sigma_{\rm{syst}}=5$ mas, which we took to be our
default systematic floor throughout Section 5.2, only causes a slight
degradation compared to the Perfect case with $\eta_{0.10}$ being around 2.5%
and $\sigma_{\rm{MAD}}$ being around 0.011 for both CDF50 and PDFPeak. CDF50
and PDFPeak show similar performance in terms of $\eta_{0.10}$ and
$\sigma_{\rm{MAD}}$, which indicates that the degradation primarily occurs at
the highest redshifts where the PDFs tend to be flatter than at lower
redshifts, meaning that systematic effects can smear them out enough to result
in incorrect redshift estimates. Bias values are farther away from zero for
CDF50 than PDFPeak, closely resembling the Realistic case.

Marginalizing over $x_{1}$ and $c$, as well as taking $\sigma_{\rm{syst}}=2$
mas does not cause much degradation, and sometimes the metrics slightly
improve. This slight improvement is due to several factors that prevents the
Perfect case from being truly ideal. First, DCR shifts are sensitive to color
and stretch of the light-curve, meaning that the $c$ and $x_{1}$ bins shown in
Equation 7 may be too coarse for the ideal case, although a finer grid is
unrealistic since the mean $c$ and $x_{1}$ errors are around 0.05 and 0.7
respectively. Indeed, for some of the candidates that are outliers for the
Perfect case but not for $c$ marginalized, taking the $c$ for the MODEL to be
$\pm 0.05$ results in an accurate estimate. Second, in many cases the
degeneracies that result in incorrect redshift estimates at the high redshifts
as shown in Figure 7 are caused by several peaks in $\mathcal{P}(z)$ that are
similar in amplitude. Marginalizing over $c$, $x_{1}$, or taking
$\sigma_{\mathrm{syst}}=2$ mas smears out these peaks by a small amount, which
sometimes leads to more accurate redshift estimates.

Using realistic $T_{\rm{obs}}$ (mostly using `LCFIT+z`, with 11,828 candidates
remaining after `LCFIT+z` out of the 13,821 that pass the Epoch cut) causes a
larger degradation, with $\eta_{0.10}$ at about 4.1%, or around two to three
times that of the Perfect case while $\sigma_{\rm{MAD}}$ also increases but to
a smaller extent. The bias values are similar to the Perfect case, but have
larger error bars. We find that the $z_{\rm{DCR}}$ performance with all of the
systematics included in the Realistic case (using realistic $T_{\rm{obs}}$,
marginalizing over $c$ and $x_{1}$, and assuming $\sigma_{\rm{syst}}=5$ mas),
but with $\sigma_{\rm{syst}}=0$, is considerably better than the Realistic
case with $\eta_{0.10}=7.3\%$ and $\sigma_{\rm{MAD}}=0.014$ for PDFPeak
(similar values for CDF50) and the bias being $-0.010\pm 0.001$ and $-0.004\pm
0.001$ for CDF50 and PDFPeak respectively, meaning that our $z_{\rm{DCR}}$
estimates are mostly limited by S/N.

As mentioned in Section 3, we note that our assumption of a
$\sigma_{\rm{syst}}=5$ mas floor is reasonable and possible to achieve for
LSST given the DECam value of 3–6 mas (Bernstein et al., 2017). Additionally,
we consider the $z_{\rm{DCR}}$ performance when $\sigma_{\rm{syst}}$ is 2 mas
and 10 mas. In Table 9, we show that $\sigma_{\rm{syst}}=2$ mas results in
$\eta_{0.10}$ being about half that of $\sigma_{\rm{syst}}=5$ mas or similar
to the Perfect case while the bias and $\sigma_{\rm{MAD}}$ similarly improve
slightly. $\sigma_{\rm{syst}}=10$ mas results in $\eta_{0.10}$ being about 3
to 4 times larger than the Perfect case at 6.5% and 4.6% for CDF50 and PDFPeak
and the bias and $\sigma_{\rm{MAD}}$ also degrade substantially more compared
to the other systematics considered here. The larger discrepancy in the
metrics between CDF50 and PDFPeak for $\sigma_{\rm{syst}}=10$ mas suggests
that the high-$z$ estimates are affected disproportionately. The metrics we
present here for larger $\sigma_{\rm{syst}}$ shows that limiting
$\sigma_{\rm{syst}}$ to 5 mas or lower will be crucial when implementing
$z_{\rm{DCR}}$ for LSST.

##  6 Discussion and Conclusion

In this paper, we outlined a method to obtain astrometric redshifts of SNe Ia
from multi-band, multi-epoch observations using Differential Chromatic
Refraction or DCR, which is caused by the wavelength dependence of our
atmosphere’s refractive index. Because the DCR shifts depend on the source
SED, it can in principle be used to infer its redshift if sufficient number of
observations are available at moderately high air masses.

We evaluate our method on LSST-like simulations using an updated version of
`SNANA` incorporating statistical and systematic uncertainties and detailed
calculations of the DCR effect. We provide two types of point estimates, one
where we take the 50th percentile of the cumulative distribution function of
the posterior to be the estimated redshift (CDF50), and one where we take the
peak of the posterior to be the estimated redshift (PDFPeak). We find that in
the hypothetical cases of Perfect (no noise) simulations, $z_{\rm{DCR}}$
perform very well, although the estimate is not always accurate due to
degenaracies. For the Realistic case, we find that our estimates at
$z_{\rm{true}}>0.7$ are substantially degraded, but the $z_{\rm{DCR}}$
estimates are still quite reasonable at $z_{\rm{true}}<0.6$.

We then combine $z_{\rm{DCR}}$ with host galaxy photometric redshifts as well
as SN Photometric redshifts using `LCFIT+z`. As $z_{\rm{Host}}$ are completely
independent from $z_{\rm{SN}}$ and $z_{\rm{DCR}}$, and $z_{\rm{DCR}}$ are
close to independent from $z_{\rm{SN}}$, which we confirmed by calculating the
Pearson correlation coefficients between them, combining the different methods
results in substantial improvement in the redshift estimates. Combining all
three methods results in reduced bias, outlier fractions, and MAD deviation.
The outlier fractions for $z_{\rm{SN}}$ is $\eta_{0.10}=20.5\%$, while
combining with both $z_{\rm{Host}}$ and $z_{\rm{DCR}}$ reduces this to about
6.1% for the all-redshifts sample.

We believe that our analysis is important in several ways. First, this is the
first demonstration of astrometric redshift measurements for SNe Ia. Contrary
to Kaczmarczik et al. (2009), where the DCR shifts in the u and g-bands were
treated as colors in the photo-$z$ algorithm, we constructed and exploited the
full PDFs. Astrometric redshifts are especially useful since they are
independent of host galaxy photometric redshifts, and nearly independent from
SN photometric redshifts, allowing the combination to result in more accurate
redshift measurements. $z_{\rm{DCR}}$ perform better at lower redshifts,
complementing $z_{\rm{Host}}$, which perform better at higher redshifts. This
will also allow us to use more SNe Ia for cosmology compared to using just SN
+ Host photometric redshifts, as well as reduce uncertainties.

Although we use the coadded DDF for LSST in this initial analysis, we expect
that astrometric redshifts will also be useful for the Wide-Fast-Deep (WFD)
survey where the chances of obtaining spectroscopic redshifts are much lower.
While astrometric redshifts are not expected to provide much information at
the higher redshifts, especially for the WFD due to low S/N and reduced
cadence, we believe they will be similarly constraining as shown in this work
for $z_{\mathrm{true}}<0.6$, which we have seen in preliminary analysis with
the Dark Energy Survey (DES) shallow fields. Although we found in Section 5.2
that restricting to candidates with at least one observation with
$\rm{AM}>1.4$ does not improve $z_{\rm{DCR}}$ significantly for the DDF, a few
intentional high AM observations might be beneficial for the Wide-Fast-Deep
(WFD) fields as the number of observations per candidate are expected to be
lower than the DDF on average.

We plan to implement astrometric redshifts of SNe Ia into our SNe Ia analysis
pipeline for LSST observations, although we will have to account for the
additional uncertainties and bias coming from the telescope instrumental
properties discussed in Section 3. In this work, we did not consider
atmospheric variability, although its effect on DCR shifts is likely to be
small and it is possible to incorporate the recorded atmospheric conditions
for each observation. We recommend that LSST record the quantities required to
calculate the atmospheric refraction index for each observation, such as air
temperature, air pressure, and water vapor pressure. Ideally, the index of
refraction would be calculated for each observation and stored for later
access. We also did not include uncertainties arising from host-galaxy
modeling, which will make analysis in real observations more difficult.
Additionally, we note that the performance of astrometric redshifts will
depend sensitively on the detailed survey strategy such as the cadence and the
air mass distribution of the observations, which have not yet been finalized
for LSST.

Some additional potential applications of this work include: utilizing the PSF
shape changes due to DCR and wavelength-dependent seeing to estimate redshifts
as the 2D shape changes tend to be larger than the 1D DCR shifts, and
extending astrometric redshifts to Type II supernovae. The former method is
encouraging because the shape change caused by wavelength-dependent effects
result in more apparent magnitude shifts than the positional shifts due to
DCR, but the shape changes will almost certainly be more difficult to measure
than 1D positional shifts. Type II supernovae, like Type Ia supernovae, also
display distinct emission lines which result in substantially different DCR
shifts depending on the redshift. Thus, we expect that astrometric redshifts
for Type II supernovae will also be useful as we have shown in this work for
Type Ia supernovae. Another potential application could be to find mismatched
host-galaxies using $z_{\rm{SN+DCR}}$. If the estimated $z_{\rm{SN+DCR}}$ is
very different from $z_{\rm{Host}}$, the host galaxy may not have not been
identified correctly. This could be useful whether we are using host-galaxies
for photometric redshifts or spectroscopic follow-up.

Acknowledgements. This paper has undergone internal review in the LSST Dark
Energy Science Collaboration. We thank the internal reviewers Gautham Narayan
and Robert Knop. LSST DESC acknowledges ongoing support from the Institut
National de Physique Nucléaire et de Physique des Particules in France; the
Science & Technology Facilities Council in the United Kingdom; and the
Department of Energy, the National Science Foundation, and the LSST
Corporation in the United States. LSST DESC uses the resources of the
IN2P3/CNRS Computing Center (CCIN2P3–Lyon/Villeurbanne—France) funded by the
Centre National de la Recherche Scientifique; the Univ. Savoie Mont
Blanc—CNRS/IN2P3 MUST computing center; the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility supported
by the Office of Science of the U.S. Department of Energy under contract No.
DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded by UK BIS National
E-infrastructure capital grants; and the UK particle physics grid, supported
by the GridPP Collaboration. This work was performed in part under DOE
contract DE-AC02-76SF00515. J.L. and M.S. were supported by DOE grant DE-
FOA-0002424 and NSF grant AST-2108094. The work of A.I.M. was supported by
Schmidt Sciences. Author Contributions are Listed Below. J. Lee: led main
analysis, wrote code and scripts for analysis, as well as the manuscript. M.
Sako: project initiator, supervision of lead graduate student, manuscript
editing, administration, and management. R. Kessler: updated SNANA simulation
to model atmospheric effects, light-curve fitting strategy, and manuscript
editing. A.I. Malz: generated the photometric redshift PDFs.

##  Appendix A DCR calculations

Figure 23: The celestial sphere for an observer in the Southern Hemisphere.

In this section, we describe more explicitly how we implemented the DCR RA
($\alpha$) and DEC ($\delta$) shifts into `SNANA` as mentioned in Section 3.
This is similar to the Appendix of Lee & Acevedo et al. (2023) but for an
observer in the Southern Hemisphere. Figure 23 is a diagram of the celestial
sphere for an observer (who would be in the center of the sphere) in the
Southern Hemisphere, where Z is the zenith, (S)P and (N)P (denoted as just ‘P’
from hereon) are the South and North celestial poles, and $\phi$ is the
latitude of the observatory. Note that $\phi<0$ for an observer in the
Southern Hemisphere. This means that the angle between the South Pole and the
observer’s South or the angle between the North Pole and the observer’s North
is $|\phi|$. For an object located at X, $a$ is the altitude, $\alpha$ and
$\delta$ are the RA and DEC, with $\delta<0$ in this diagram, and $\alpha$
increases in the counterclockwise direction as viewed from above the North
Pole. The angle XPZ or $H$, is the Hour Angle, and angle ZXP or $q$, is the
parallactic angle, which are both typically defined using the North Celestial
Pole.

In `SNANA`, we calculate the parallactic angles using the spherical sine and
cosine laws:

$\begin{split}\frac{\sin(q)}{\sin(90^{\circ}+|\phi|)}=\frac{\sin(H)}{\sin(90^{\circ}-a)}=\frac{\sin(360^{\circ}-A)}{\sin(90^{\circ}-\delta)}\rightarrow\sin(q)=\frac{\cos(\phi)\sin(H)}{\cos(a)}=\frac{-\cos(\phi)\sin(A)}{\cos(\delta)}\\\
\cos(90^{\circ}+|\phi|)=\cos(90^{\circ}-a)\cos(90^{\circ}+|\delta|)+\sin(90^{\circ}-a)\sin(90^{\circ}+|\delta|)\cos{q},\rightarrow\cos(q)=\frac{\sin(\phi)-\sin(a)\sin(\delta)}{\cos(a)\cos(\delta)}\end{split}$
(A1)

where we have used that $|\phi|=-\phi$ and $|\delta|=-\delta$ and $A$ is the
azimuth of the object.

Figure 24: Overlapping Index depending on number of standard deviations apart
for two identical normalized Gaussian distributions with different means.
Because the means of the two distributions are fixed, the larger the $\sigma$,
the higher the overlap. We take the case when the means of the two
distributions are $3\sqrt{2}$ apart or when $\mathrm{OVL}>0.0344$ as a
threshold when combining $z_{\mathrm{DCR}}$ and $z_{\mathrm{Host}}$.

Now, consider a DCR altitude shift towards the zenith, $\Delta a$. In flat
Euclidean geometry, we would calculate the $\alpha$ and $\delta$ shifts as
$\Delta\alpha=\Delta a\sin(180^{\circ}-q)=\Delta a\sin(q)$ and
$|\Delta\delta|=\Delta a\cos(180^{\circ}-q)=-\Delta a\cos(q)$. However, these
relations do not necessarily hold on a sphere. To calculate the $\alpha$ and
$\delta$ shifts properly, we can utilize Napier’s laws for right spherical
triangles:

$\begin{split}\sin(\Delta\alpha)=\sin(180^{\circ}-q)\sin(\Delta
a)=\sin(q)\sin(\Delta a)\\\ \sin(|\Delta\delta|)=\sin(x)\sin(\Delta
a)=\tan(\Delta\alpha)\cot(180^{\circ}-q)=-\tan(\Delta\alpha)\frac{\cos(q)}{\sin(q)}\end{split}$
(A2)

where the angle between $\Delta\alpha$ and $|\Delta\delta|$ is $90^{\circ}$,
$|\Delta\delta|$ and $\Delta a$ is $180^{\circ}-q$, and $\Delta a$ and
$\Delta\alpha$ is labeled as $x$. Since the shifts (altitude, RA, and DEC) we
are considering are well below $1^{\circ}$,
$\sin(\Delta\alpha)\approx\tan(\Delta\alpha)\approx\Delta\alpha$, and Equation
A2 can be approximated as:

$\Delta\alpha\approx\sin(q)\Delta
a,\quad|\Delta\delta|\approx-\Delta\alpha\frac{\cos(q)}{\sin(q)}=-\cos(q)\Delta
a$ (A3)

Lastly, when we calculate the projected DCR shifts, we must account for the
size of $1^{\circ}$ becoming smaller as we move farther away from the
celestial equator, and $\Delta\delta$ is clearly negative for an upward DCR
altitude shift, so we have $\Delta\alpha=\frac{\sin(q)}{\cos(\delta)}\Delta a$
and $\Delta\delta=\cos(q)\Delta a$ as in Equation 3.

##  Appendix B Overlapping Index

In Section 4, we defined the Overlapping Index (OVL) as:

$\mathrm{OVL}(p_{0},p_{1})=\int\min(p_{0}(x),p_{1}(x))dx$ (B1)

where $p_{0}(x)$ and $p_{1}(x)$ are PDFs. In Figure 24, we show two identical
normalized Gaussian distributions with means of 20 (red) and 26 (blue),
depending on their standard deviations ($\sigma$) and hence number of
$\sigma$’s the means of the two distributions are apart. When normalized as
shown here, we can say that the blue curve overlaps 3.44% with the red curve
when their means are $3\sqrt{2}\sigma$ apart. In the main text, we take
$\mathrm{OVL}>0.0344$ to be the threshold for two PDFs being compatible with
each other. As the means of the two distributions become farther away from
each other in terms of $\sigma$, the Overlapping Index decreases, and we see
smaller overlap.

##  Appendix C Cosmology Prior for Light-Curve Fitting

In Section 3, we use a weak cosmology prior for `LCFIT+z` to obtain
$T_{\rm{obs}}$ for Realistic $z_{\rm{DCR}}$. Additionally, we use a cosmology
prior in Section 5.2.3 to see improvement in the $z_{\rm{SN}}$ performance.
Here, following Kessler et al. (2010) and Möller et al. (2024), we describe
how the prior is applied.

We begin with the approximate fitted distance modulus, $\mu_{\rm{SALT3}}$:

$\mu_{\rm{SALT3}}=30.0-2.5\log_{10}{x_{0}}+\alpha x_{1}-\beta c$ (C1)

with $x_{0}$ (amplitude), $x_{1}$ (stretch), and $c$ (color) being SALT3
light-curve parameters. We use $\alpha=0.14$ and $\beta=3.2$ as our default.
The difference between the fitted and theoretical distance modulus,
$\mu_{\rm{DIFF}}$, is given by:

$\mu_{\rm{DIFF}}=\mu_{\rm{SALT3}}-\mu_{\rm{th}}(z_{\rm{SN}})$ (C2)

where $\mu_{\rm{th}}$ is the theoretical distance modulus. The distance
modulus uncertainty, $\sigma_{\mu}$, can be computed using:

$\sigma_{\mu}^{2}=\bigg{(}\frac{d\mu}{d\Omega_{m}}\Omega_{m,\rm{ERR}}\bigg{)}^{2}+\bigg{(}\frac{d\mu}{dw}w_{\rm{ERR}}\bigg{)}^{2}$
(C3)

where $\Omega_{m,\rm{ERR}}$ and $w_{\rm{ERR}}$ are uncertainties for
$\Omega_{m}$ and $w$ respectively. As mentioned in Section 5.2.3, we use
$\Omega_{m,\rm{ERR}}=0.03$ and $w_{\rm{ERR}}=0.1$. The reduced
$\chi_{\rm{SALT3}}^{2}$ is then given by:

$\chi_{\rm{SALT3}}^{2}=\bigg{(}\frac{\mu_{\rm{DIFF}}}{\sigma_{\mu}}P_{\mu\rm{ERRSCALE}}\bigg{)}^{2}.$
(C4)

$P_{\mu\rm{ERRSCALE}}$ is the distance modulus prior: a Gaussian profile with
a standard deviation of $2\sigma_{\mu}$ when obtaining $T_{\rm{obs}}$ in
Section 3, and $1\sigma_{\mu}$ when assessing improvement in the $z_{\rm{SN}}$
performance in Section 5.2.3.

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