# Cosmology Requirements on Supernova Photometric Redshift Systematics for
Rubin LSST and Roman Space Telescope

Ayan Mitra<EMAIL_ADDRESS>School of Engineering and Digital Sciences,
Nazarbayev University,
Nur-Sultan 010000, Kazakhstan  Eric V. Linder  Berkeley Center for
Cosmological Physics & Berkeley Lab, University of California, Berkeley, CA
94720, USA

Energetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan 010000,
Kazakhstan

###### Abstract

Some million Type Ia supernovae (SN) will be discovered and monitored during
upcoming wide area time domain surveys such as the Vera C. Rubin Observatory
Legacy Survey of Space and Time (LSST). For cosmological use, accurate
redshifts are needed among other characteristics; however the vast majority of
the SN will not have spectroscopic redshifts, even for their host galaxies,
only photometric redshifts. We assess the redshift systematic control
necessary for robust cosmology. Based on the photometric vs true redshift
relation generated by machine learning applied to a simulation of 500,000
galaxies as observed with LSST quality, we quantify requirements on
systematics in the mean relation and in the outlier fraction and deviance so
as not to bias dark energy cosmological inference. Certain redshift ranges are
particularly sensitive, motivating spectroscopic followup of SN at $z\lesssim
0.2$ and around $z\approx 0.5$–0.6. Including Nancy Grace Roman Space
Telescope near infrared bands in the simulation, we reanalyze the constraints,
finding improvements at high redshift but little at the low redshifts where
systematics lead to strong cosmology bias. We identify a complete
spectroscopic survey of SN host galaxies for $z\lesssim 0.2$ as a highly
favored element for robust SN cosmology.

##  I Introduction

Type Ia supernovae (SN) are incisive probes of the cosmic expansion history,
giving the tightest constraints on dark energy properties of any probe for a
given distance precision. Wide field surveys can multiplex the observations,
i.e. measure many SN at once in a given survey field, and time domain surveys
that revisit fields on a cadence compatible with SN rise and fall times can
monitor them throughout their lightcurve. Upcoming surveys such as ZTF ztf
and LSST lsst1 ; lsst2  are time domain surveys that are wide field (as well
as deep, measuring SN over a significant range of redshifts) and with multiple
wavelength bands, and thus will produce thousands to of order a million SN
lightcurves.

For best use as a cosmological probe, these sources need not only to be
observed in multiple bands with good photometry for a significant part of
their lightcurve (for accurate fitting and color corrections), as these
surveys will provide, but also be classified as Type Ia supernovae to ensure a
pure sample, and ideally subtyped as normal Type Ia. The last two
characteristics are most robustly established through spectroscopy. The SN
redshift must also be determined, e.g. through the redshift of the host
galaxy, either through spectroscopy or photometry.

This final step of the redshift determination for the distance-redshift
relation to measure the cosmic expansion is what we focus on here,
specifically the requirement on photometric redshift (‘photo-z’) accuracy.
Statistical uncertainties in the redshift will propagate through to increasing
the cosmological parameter uncertainties (see m0  for detailed calculations),
but systematic errors will bias the cosmology. We extend the analysis of f1 ,
which employed analytic toy models for redshift systematic uncertainties, to
use data simulated to reflect LSST observing characteristics, i.e. filters,
exposure depths, etc.

In Sec. II we review the methodology for propagating systematic redshift
errors into cosmological parameter bias. The simulated data and the derived
photo-z vs true redshift mapping is discussed in Sec. III, with the results
presented in Sec. IV, along with the requirements necessary to avoid
significant bias. We explore the effects of external near infrared data (NIR)
from the Roman Telescope in Sec. V, and conclude in Sec. VI.

##  II Method

To propagate an observational systematic to bias in cosmology inference, a
well used and convenient technique is the Fisher information bias formalism
9805012 ; 0604280 . Specifically we follow the approach in f1 , with a
parameter set including the supernova absolute magnitude parameter
$\mathcal{M}$, the present matter density $\Omega_{m}$ as a fraction of the
critical density, and the dark energy equation of state parameters $w_{0}$ and
$w_{a}$ giving its present value and a measure of its time variation. We
include the effects of a misestimated redshift $z$ on both the distances and
the lightcurve width-luminosity relation (but not extinction, which is
expected to be a small effect).

The final relation, as in f1 , for the apparent magnitude offset $\Delta m$ is

$\Delta m=\frac{5}{\ln 10}\ln{\left[\frac{D_{L}(z+\delta
z)}{D_{L}(z)}\right]}+1.4\,\frac{\delta z}{1+z}\ ,$ (1)

where $\delta z$ is the redshift systematic at redshift $z$ and $D_{L}$ is the
luminosity distance. This misestimation $\Delta m$ then biases the cosmology
parameters. Imposing constraints on the degree of cosmology bias in turn
propagates back to requirements on redshift systematics, as a function of
type, degree, and redshift at which they occur. We will present the confidence
contours in the dark energy $w_{0}$–$w_{a}$ plane (marginalized over the other
parameters), and require that the bias shift the cosmology by less than
$1\sigma$, i.e. staying within the 68.3% joint confidence level contour:
$\Delta\chi^{2}<2.3$.

In f1  we focused on toy models for additive and multiplicative systematics,
and outliers. Here we use the photometric vs true redshift mapping derived
from simulations, as described in the next section.

##  III Data

To obtain the redshift of a SN, the most common method is to measure the host
galaxy redshift. This either exists in previous catalogs or can be determined
by the survey itself, possibly after the SN has faded. Less common is getting
an estimate of the redshift from the SN colors (flux differences between
wavelength bands) kessler ; it is also possible that this could be useful in
avoiding catastrophic outliers in the galaxy photometric redshift kessler2 ;
dai ; zbeams . However this has not been fully tested for side effects and
confirmed, and here we consider only host galaxy redshifts.

###  III.1 Photo-z Catalog

The leading current simulations of LSST galaxy photometric redshift
distributions use color matched nearest neighbors (CMNN) estimators m1 ; m2 .
This incorporates the expected photometric data quality and survey
characteristics. The photo-z estimator is based on the location of a test
galaxy in color space, identifying it to the nearest color matched galaxy from
the spectroscopic training data set. Minimization on the distance metric is
done via $\chi^{2}$ to estimate the photo-z of the test galaxy. The $\chi^{2}$
(Mahalanobis) distance $D$ is computed as

$D=\sum_{1}^{N}\frac{(c_{\rm train}-c_{\rm test})^{2}}{(\delta c)^{2}}\,,$ (2)

where $c$ is the color, $N$ is the total number of colors, and $\delta c$
quantifies the measurement error in color. A test galaxy needs to be detected
with at least $N=3$ colors to be assigned a photo-z.

The training set is analogous to the spectroscopic galaxy sample while the
test set is composed of galaxies with simulated colors, for which the photo-z
are estimated. For training the CMNN estimator we have simulated a larger data
set than in m2 , of 500,000 galaxies and for testing a sample size of 90,000.
As both are simulated, the true redshifts are known and the distribution of
the derived photo-z relative to the true redshift can be mapped.

This estimator is designed such that the accuracy and the precision of the
photo-z are directly related to the precision of the survey’s photometry. The
training and testing sets were drawn entirely from the simulated photometry
data catalog, with both having the same distribution of redshift and magnitude
(flux). The simulated galaxy catalog is based on the Millennium simulation mil
and uses realistic photometric characteristics. Details on the galaxy catalog
construction are described in gonz ; mers .

The CMNN photo-z estimator was designed to model the optical ($ugrizy$) and
NIR ($YJHK$) properties of galaxies. For the NIR, note that from the figures
in m1  the NIR bands from the Euclid satellite euclid  do not significantly
impact the photo-z systematic uncertainties at redshifts $z\lesssim 1$ where
we observe SN with LSST (note this refers to the much tighter photo-z
requirements for supernovae; Euclid will be quite valuable for weak lensing
photo-z constraints), though those from the Nancy Grace Roman Space Telescope
ngrst  could. For the purposes of this work we use only the $10$ year LSST
projections, with the optical filters’ $5\sigma$ detection limits as tabulated
in Table 1, as the main input. In Section V we extend this to include Roman
$YJH$ bands.

$u$ | $g$ | $r$ | $i$ | $z$ | $y$
---|---|---|---|---|---
26.1 | 27.4 | 27.5 | 26.8 | 26.1 | 24.9
Table 1: Summary of the $5\sigma$ limiting magnitudes for each filter for a
LSST $10$ year forecast, based on the LSST simulation software package
connolly .

###  III.2 Photo-z Systematics

Given the catalog of photo-z’s and true redshifts, one can carry out various
statistical analyses to assess robustness of the distribution. Ref. m2
presented different statistical measures of the photo-z quality based on the
results from the CMNN estimators on the mock galaxy catalog. In this analysis,
we will follow these definitions, with alterations as described in Sec. IV.
From these we will derive the quantity needed for the cosmology requirements
on systematics, $\delta z(z)$ in Eq. (1).

To model the LSST-like uncertainty, Gaussian random photometric scatter was
added to the simulated observed apparent magnitudes from the true catalog with
a standard deviation equal to the predicted magnitude error for each galaxy.
The magnitude error was modeled as for LSST, based on the description in
ivezic . For computing the cosmology bias we use two types of photo-z
systematic offsets for $\delta z$, the bias in the core of the photo-z
distribution (referred to in m2  as “robust bias”) and in the outlier
distribution.

####  III.2.1 Robust Bias

If the mean photo-z relation not only scatters about the true redshift but is
biased from it, generally by different amounts at different redshifts, this
will lead to a cosmology bias. This core offset, or robust bias, is defined as
the mean bias in the inter quartile range (IQR; defined as including 50% of
the galaxies) of the galaxy photo-z error m2 . For a photo-z error defined as

$\Delta z_{(1+z)}=\left(\frac{z_{\rm true}-z_{\rm phot}}{1+z_{\rm
phot}}\right)\ ,$ (3)

where $z_{\rm true}$ is the true or spectroscopic catalog redshift and $z_{\rm
phot}$ is the estimated photometric redshift, the robust bias is taken to be
the mean over the inter quartile range

${\rm Robust\ Bias}=\overline{\Delta z_{(1+z),{\rm IQR}}}\ .$ (4)

We refer to this as $\delta z_{\rm core}$.

####  III.2.2 Outlier Bias

Photo-z errors sometimes have large departures from the true values, lying
outside the core. There are two types of these: outliers and catastrophic
outliers. Here we need to know not only the degree of offset but the fraction
of photo-z’s that are outliers.

Catastrophic outliers are defined as $|z_{\rm true}-z_{\rm phot}|>1.5$ and it
is highly unlikely that a SN with such a mistaken redshift would go
unrecognized and be placed on a Hubble diagram (its peak magnitude, lightcurve
width, etc. would lie well off expectation for a reasonable variation of
cosmology). Therefore we do not consider catastrophic outliers.

We classify outliers as photo-z’s that lie outside the core, defined as
$\Delta z_{(1+z)}>3\sigma(z)$ where $\sigma(z)$ is the standard deviation for
the IQR galaxies. Furthermore we require $\Delta z_{(1+z)}>0.06$.

####  III.2.3 Sample Selection

To select our data set for the analysis, we modified and reran the CMNN
simulation with two alterations relative to m2 :

  * •

$0<z_{\rm true}\leq 1.2$, as the LSST supernova survey is expected to cover
this range, i.e. SN at higher true redshifts would likely be too faint for the
survey magnitude limits. This restricts the redshift range on the high end
relative to m2 , but expands it on the low end since m2  did not identify
outliers for $z\leq 0.2$.

  * •

$|\delta z|<1.5$, i.e. we distinguish between outliers and catastrophic
outliers.

Figure 1 shows the result of the updated simulation analysis. Core points are
in green, outliers are in blue if they fall within the $z_{\rm true}$ cut and
red if they are at higher true redshift, while gold points are catastrophic
outliers and not included in the cosmology analysis. Green core points out to
$z_{\rm true}\leq 1.2$ determine $\delta z_{\rm core}(z_{\rm true})$ while
blue points determine $\delta z_{\rm out}(z_{\rm true})$ and the fraction of
outliers $f_{\rm out}(z_{\rm true})$, equaling the ratio of the number of blue
points to blue+green points at that redshift. We use a binning on the
statistics of $\Delta z=0.05$, much finer than the original $\Delta z=0.3$. To
train the CMNN estimator, we simulated 500,000 galaxies for training and
90,000 galaxies for testing (out to $z_{\rm true}=3$).

Figure 1:  Photo-z’s vs true redshifts resulting from the CMNN galaxy
simulation m2 , with modified selection criteria. Galaxies are classified as
lying in the core of the distribution (green), outliers (blue if $z_{\rm
true}\leq 1.2$, red if $z_{\rm true}>1.2$), or catastrophic outliers (gold).
The grey horizontal dashed line shows the $z_{\rm true}\leq 1.2$ cut and the
black diagonal line shows a perfect survey, with $z_{\rm phot}=z_{\rm true}$.
Any deviation in the mean from the diagonal gives a bias.

##  IV Analysis

We propagate the quantities $\delta z_{\rm core}$, $\delta z_{\rm out}$, and
$f_{\rm out}$ derived from analysis of Fig. 1 into the cosmology analysis,
i.e. Eq. (1) and then the Fisher information analysis. The quantity $f_{\rm
out}$ enters there since only a fraction $f_{\rm out}$ of the SN have the
ensuing outlier bias $\Delta m$. Thus, both $\delta z_{\rm out}$ and $f_{\rm
out}$ are important: if the redshift offset $\delta z_{\rm out}$ is high, but
happens only rarely (low $f_{\rm out}$), this will give a small cosmology
bias, as will a large outlier fraction $f_{\rm out}$ but with only a small
offset $\delta z_{\rm out}$.

For the cosmology calculation we follow the analysis described in f1 , with SN
distributed over the range $z=[0-1.2]$. We evaluate $\delta z$ in bins of
width $\Delta z=0.05$ (see discussion below). We treat the core bias and
outlier bias separately, for clarity.

Figure 2 shows the bias in cosmology as a result of the core bias redshift
systematic. The input cosmology has $(w_{0},w_{a})=(-1,0)$ and the ellipse
shows the $1\sigma$ (68.3%) joint confidence contour, marginalized over the
other parameters (matter density and SN absolute magnitude ${\mathcal{M}}$),
for a rough approximation of the LSST SN sample plus a Planck CMB prior on the
distance to last scattering.

Applying the $\delta z_{\rm core}(z)$ systematic to the redshift bins one by
one, the corresponding shift in the cosmology is shown by the individual
square boxes, beginning with the orange box ($z=[0,0.05]$), and ending with
the blue box ($z=[1.15,1.2]$). The red line connecting the boxes traces the
locus of the cosmology bias with increasing bin redshift. The green arrow
shows the net effect (vector sum) for the systematic present on all the
redshift bins. Note the direction of bias from some redshifts is such that it
can diminish bias from another redshift. Cosmology is most sensitive to the
low redshift systematics, with systematics in the three lowest bins biasing
cosmology outside the $1\sigma$ contour. This can be a hopeful sign in that
these can be the most easily addressed with supplementary observations.

Figure 2:  Cosmology bias in the dark energy $w_{0}$–$w_{a}$ plane coming from
core bias redshift systematics is shown by the red curve tracing the effect
from one redshift bin systematic $\delta z_{\rm core}(z)$ at a time, marked by
squares from the lowest (orange) to highest redshift (blue). The green arrow
gives the vector sum over all redshifts, with the inset showing the full
extent of the bias. For reference, the statistical uncertainty is shown as the
$1\sigma$ (68.3%) joint confidence contour (blue ellipse).

Next we turn to the outlier bias. For this we find the fine redshift bins of
width $\Delta z=0.05$ to be important to treat properly the sharp outlier
features, especially at low redshift. Quantitatively, the cosmology parameter
bias is misestimated by using bins of $\Delta z=0.1$ rather than 0.05 by $\sim
0.03$ in $w_{0}$ and 0.13 in $w_{a}$ only at $z<0.2$; above this the maximum
errors are $\lesssim 0.015$ and 0.06 respectively, and generally much less.
Recall that for redshift systematics due to outliers we care about both the
offset and the fraction of galaxies exhibiting the systematics. Except at low
redshift (where $dm/dz$ is steep), the cosmology bias is basically
proportional to their product, $f_{\rm out}\,\delta z$.

Figure 3 shows the simulation results for $\delta z(z)$, $f_{\rm out}(z)$, and
their product. Although the product looks quite small, one must propagate it
to the cosmology bias to determine its impact. Even a small systematic can
have a significant effect on a high precision survey such as LSST.

Figure 3: Simulation output for outlier redshift systematics magnitude $\delta
z_{\rm out}(z)$, fraction of outliers $f_{\rm out}(z)$, and their product.
Sharp features in the outlier systematics require fine binning in redshift.

Figure 4 shows the resulting cosmology bias from the outlier systematics.
While the effect on the first redshift bin is less than from the core bias,
systematics from the remainder of the redshift range have comparable,
significant effects. Of course both types of systematics will be present, and
their induced bias goes in the same direction for distorting the cosmology
inference.

Figure 4: As Figure 2, but for outlier bias. Again, low redshift bins
$z\lesssim 0.2$ are particularly affected.

Since the cosmology bias from the redshift core systematics and the outlier
systematics, and their sum, are well beyond the desired $1\sigma$ statistical
confidence contour, it must be reduced for useful cosmology estimation from
LSST photometric supernovae. We present three possible strategies for
amelioration: 1) modeling, 2) select spectroscopy, 3) external imaging data.

The redshift systematics found by the simulations is a raw systematic, without
remediation. One could attempt to model the systematics and correct for them,
up to the fidelity of the modeling process, leaving a smaller, residual
systematic. Figure 5 shows that if the residual core bias is scaled down by a
factor of 20, i.e. leaving only 5% of the simulation systematic, then the
cosmology bias lies within the $1\sigma$ confidence contour. (In fact, one
should not take the factor 20 too precisely: the Fisher bias formalism is
valid for small changes in the observable, i.e. $\Delta m$, so the effect of
large changes at the lowest redshifts is possibly exaggerated.) The inset
shows the case for the systematics reduced all the way to the bootstrap
sampling error111The bootstrap error in the core bias is derived by randomly
drawing galaxy subsets with replacement and recalculating the statistics 1000
times, and then using the standard deviation of all recalculations as the
error. m2 . A similar process could potentially be applied to the outlier
bias.

Figure 5:  As Figure 2, but with the core bias now scaled down by a factor 20,
mimicking modeling of the systematic to leave a small residual. At this level
of control, the net bias is contained within the $1\sigma$ statistical
confidence contour. The inset shows the result if the systematics were modeled
down to the bootstrap sampling error from 1000 simulations.

The second strategy involves targeting the systematics in particular redshift
ranges. We have seen that the low redshift systematics produce the highest
cosmology bias. This is expected as the SN apparent magnitude $m$ on the
Hubble diagram starts off steeply varying with redshift, roughly $\sim 1/z$,
and then flattens at higher redshift. So a small $\delta z$ at low redshift
has a significant effect. This is fortunate in that the low redshift region is
most amenable to use of spectroscopy to determine the SN redshift (e.g.
through targeting its host galaxy).

Figure 6 shows the application of this approach to the case of outlier
systematics. The three arrows correspond to the total cosmology bias that
ensues from SN over the full range $z=[0,1.2]$ if those (and only those) SN at
$z<z_{\star}$ get spectroscopic redshifts, and so there are no outliers there:
$f_{\rm out}(z<z_{\star})=0$. Spectroscopic redshifts for $z\lesssim 0.2$ SN
bring the bias under control. One possibility for carrying this out is through
the secondary target program of the DESI Bright Galaxy Survey bgs , or other
multifiber spectrographs mos ; elt . Such a low redshift spectroscopic SN
sample is quite interesting scientifically as it also has power as a cosmic
probe of gravity through peculiar velocities pecvel ; peculiar .

Figure 6: As Figure 4, but with spectroscopic redshifts, and hence $f_{\rm
out}=0$, for $z<z_{\star}$, for three different values of $z_{\star}$.

We consider the third strategy of utilization of external imaging data to
improve redshift systematics in the next section.

##  V LSST + Roman Space Telescope

During the LSST, supplementary imaging data that can help constrain the
photometric redshift estimation will be available, notably from the Euclid
satellite and the Nancy Grace Roman Space Telescope, both with near infrared
wavelength measurements. As discussed in Section III.1, Euclid’s leverage
comes at redshifts higher than those whose photometric systematics most
significantly impact the supernova cosmology. Roman, however, extends to lower
redshifts and we consider the benefit from adding YJH222The K filter, really
F184, is useful at much higher redshift than we consider here. See Fig. 10 and
Sec. 4 in m2  for discussion. imaging data with exposure depths comparable to
those from Roman to the photometric redshift determination (26.7, 26.9, 26.0
mag respectively for $5\sigma$ limiting depths). We follow m2  for the joint
analysis photo-z catalogs. While Roman will of course have its own highly
incisive sample of spectroscopic SN, here we consider only its effect on LSST
photo-z.

We carry out the joint analysis in two distinct ways. First, we consider the
effect of the added Roman information on the outlier systematics analysis of
Section IV, comparing LSST alone with LSST+Roman. However, since the addition
of Roman data can decrease the photo-z standard deviation, the outlier
fraction can actually increase since it is defined in terms of galaxy
photo-z’s deviating from $z_{\rm true}$ by $3\sigma$. (Also, LSST catastrophic
outliers can become LSST+Roman regular outliers.) Therefore we also discuss
the full systematics – from outlier galaxies (outside $3\sigma$) and inlier
galaxies (within $3\sigma$) – in a second cosmology analysis.

###  V.1 LSST+Roman Outlier Analysis

The color matching nearest neighbors (CMNN) algorithm of m2  works somewhat
differently when combining external data with LSST, so the interested reader
should consult that paper for a full discussion. The main point to note here
is that because including new filters not only adds information but also
degrees of freedom, i.e. fit parameters, in the algorithm, if the additional
NIR filter does not carry clear photo-z information (as can occur at, say, low
redshift) then the combination can actually give worse results than LSST
alone. Future work could consider how to treat this, either by cutting or
tapering multiband information in such cases, or adjusting the CMNN algorithm.

Figure 7 compares outlier systematics in terms of $\delta z_{\rm out}$ and
$f_{\rm out}$ for LSST and LSST+Roman. We see that Roman does help to tame the
excess systematic in the lowest redshift bin, although it actually has a
higher fraction of outliers there. Over the range $z=[0.2,0.6]$ there is
little impact on $\delta z_{\rm out}$ from the NIR bands, but Roman data
improves the photo-z significantly for $z\gtrsim 0.6$. For $z\gtrsim 1$, the
outlier fraction when including Roman is strongly reduced, so what deviations
$\delta z_{\rm out}$ do exist only affect a small number of galaxies.

Figure 7: Comparison between LSST+Roman and LSST simulation output for outlier
redshift systematics magnitude $\delta z_{\rm out}(z)$ and the corresponding
fraction of outliers $f_{\rm out}(z)$.

The outlier systematics at $z\lesssim 0.2$ and around $z\approx 0.5$ still
remain, so the cosmology bias issue is not solved. Figure 8 shows the
cosmological effects of the photo-z systematic. By comparing to Figure 4 we
see qualitatively similar behavior, and the quantitative aspects are also not
very different. The improvement in photo-z at high redshifts from Roman does
not result in a large impact because high redshifts give a fairly modest
contribution to the cosmology bias, which is much more sensitive to low
redshifts. Thus, the need for low redshift spectroscopy to remove the photo-z
outlier systematic remains strong.

Figure 8: As Figure 4, but cosmology bias from the outlier systematics in the
LSST+Roman case.

As mentioned, adding Roman NIR data to LSST changes which galaxies are
considered outliers, and so the ameliorating effects of the extra data are
somewhat obscured. We can consider only those galaxies classified as outliers
using just LSST, and then add Roman data to those alone and examine the
properties of those that remain outliers: this is a “like to like” comparison.
Figure 9 shows the cosmology bias for this case, and we see that indeed
significant improvement is evident.

Figure 9: As Figure 8, but for a like to like treatment where the LSST+Roman
outliers are restricted to galaxies that were also LSST alone outliers.
Cosmology bias is reduced from LSST alone (Figure 4) and from LSST+Roman
general outliers (Figure 8).

###  V.2 LSST+Roman All Galaxy Analysis

Adding NIR not only affects the photo-z outliers, but changes the size of the
core and hence the dividing line between outliers and inliers. To take into
account all the effects on the photo-z systematics of adding NIR data, we
consider here all the galaxies together, outliers and inliers, and compute the
total cosmology bias.

Figures 10 and 11 show the total cosmology bias from all galaxy photo-z
systematics for LSST alone and for LSST+Roman. Again, qualitatively they are
similar and quantitatively there is not a large difference. While cosmology
bias is significantly reduced around $z\approx 0.5$ (by at least $\delta
w_{a}>0.15$), due to a combined reduction in photo-z outlier and inlier
systematics there, and photo-z systematics is improved at $z\gtrsim 1$ – but
systematics there causes relatively little cosmology bias – the low redshift
systematics remains, and this substantially drives the cosmology bias.

Thus, the addition of NIR data does not obviate the need for spectroscopic
redshifts for SN host galaxies at $z\lesssim 0.2$, and further improvements
around $z\approx 0.5$ would be useful as well.

Figure 10: As Figure 2, but cosmology bias from the total outlier and inlier
photo-z systematics for LSST.  Figure 11: As Figure 10, but for LSST+Roman.

##  VI Conclusions

Supernovae continue to be one of the most incisive probes of cosmic
acceleration. With forthcoming surveys the data available will vastly
increase, limited however by spectroscopic followup. We consider the use of
Type Ia supernovae with only photometric redshifts, assessing the impact of
systematics in the redshift estimation through large simulations of host
galaxy colors. This builds on the analytic work of f1  by using mock data
meant to emulate LSST observations to derive the systematics requirements for
controlling bias in the cosmological parameter estimation, particularly for
dark energy.

Simulating half a million galaxies provides us with good statistics on the
redshift systematics, categorized into outlier and core systematics, as a
function of true redshift. We propagate this to cosmological parameter bias
for a LSST-like survey, showing the impact of each individual redshift range
as well as the full sum. Our results show that for both outliers and inliers,
the redshift systematic requirement is reduction by an order of magnitude –
principally in the lower redshift range – for the bias not to exceed the 68%
confidence statistical uncertainty in the $w_{0}$–$w_{a}$ plane.

The low redshift photometric systematics are the most dangerous. Fortunately,
they are also easiest to mitigate with further observations. The favored
situation would be to use the photometric supernova for cosmology only for
$z>0.3$, and obtain spectroscopic followup for all supernovae at $z\lesssim
0.2-0.3$ (see below).

We explore three potential methods for controlling systematics: modeling,
select spectroscopy, and external imaging data. Fully successful modeling,
i.e. a residual at only the level of the bootstrap uncertainty on a large
suite of simulations, would be ideal, while a residual an order of magnitude
below base is necessary. Improvement in understanding host galaxy properties
(for both the core bias and the outlier bias) is desirable, e.g. are the
spectroscopic catalogs used for training the CMNN estimators – and the
resulting outliers – representative, in particular of Type Ia SN host
galaxies. Machine learning algorithms to estimate the redshifts will likely
continue to become better; Ref. cmnn2  has summarized the accuracy for several
different algorithms, and while the CMNN estimator fares quite well in
comparison to most of the other machine learning estimators, the authors of m2
state that it is not optimized for the absolute best fit, but rather aims to
assess differences in survey strategy.

Select spectroscopy is an attractive and highly practical solution. This would
involve a multiobject spectrograph survey to obtain SN host galaxy redshifts
out to $z\lesssim 0.2$, where the systematics have the greatest impact on
cosmology bias. Such data has numerous other astrophysical applications,
including for probing gravity with peculiar velocities pecvel ; peculiar .

External imaging data will be available from NIR surveys such as the Nancy
Grace Roman Space Telescope and the Euclid satellite. We carry out an analysis
including Roman NIR filter constraints on galaxy photometric redshifts with
LSST mock data in the simulations. We consider the impact on systematics from
photo-z outliers, inliers (the complement of outliers), the total set, and a
special like to like comparison where we choose the same set of galaxies from
both LSST and LSST+Roman. Especially for the like to like comparison, Roman
helps in controlling the systematics. However, overall the main redshift
estimation improvement is at high redshifts where the cosmology bias is less.
The sensitive low redshifts show relatively little gain. So even in the era of
LSST, Euclid, and Roman, low redshift spectroscopy of SN host galaxies will be
quite important for enabling the full potential of dark energy constraints
from supernova cosmology through time domain surveys.

###### Acknowledgements.

We are very grateful to Melissa Graham for making her data and code public,
and many useful tips on how to adapt them. We also thank Rick Kessler, Alex
Kim, Gautham Narayan, and the LSST DESC Supernova and Photometric Redshift
working groups for comments and suggestions. AM acknowledges the support of
ORAU Grant No. 110119FD4534. EL is supported in part by the U.S. Department of
Energy, Office of Science, Office of High Energy Physics, under contract no.
DE-AC02-05CH11231, by NASA ROSES grant 12-EUCLID12-0004, and by the Energetic
Cosmos Laboratory.

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