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11institutetext: Laboratoire de Physique de Clermont Auvergne, IN2P3/CNRS,
F-63000 Clermont-Ferrand, France 22institutetext: Leinweber Center for
Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor,
MI 48109, USA 33institutetext: High Energy Physics Division, Argonne National
Laboratory, Lemont, IL 60439, USA 44institutetext: Fermi National Accelerator
Laboratory, USA 55institutetext: Department of Physics and Astronomy, Rutgers,
The State University of New Jersey, Piscataway, NJ 08854, USA

# Proposal for DESC cohesive Deep Drilling Programs
for LSST

Philippe Gris 11 Humna Awan 22 Matthew R. Becker 33 Huan Lin 44 Eric Gawiser
55 Saurabh W. Jha 55
the LSST Dark Energy Science Collaboration

# A Cohesive Deep Drilling Field Strategy for LSST Cosmology

Philippe Gris 11 Humna Awan 22 Matthew R. Becker 33 Huan Lin 44 Eric Gawiser
55 Saurabh W. Jha 55
the LSST Dark Energy Science Collaboration

The Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) will
image billions of astronomical objects in the wide-fast-deep primary survey
and in a set of minisurveys including intensive observations of a group of
deep drilling fields (DDFs). The DDFs are a critical piece of three key
aspects of the LSST Dark Energy Science Collaboration (DESC) cosmological
measurements: they provide a required calibration for photometric redshifts
and weak gravitational lensing measurements and they directly contribute to
cosmological constraints from the most distant type Ia supernovae. We present
a set of cohesive DDF strategies fulfilling science requirements relevant to
DESC and following the guidelines of the Survey Cadence Optimization
Committee. We propose a method to estimate the observing strategy parameters
and we perform simulations of the corresponding surveys. We define a set of
metrics for each of the science case to assess the performance of the proposed
observing strategies. We show that the most promising results are achieved
with deep rolling surveys characterized by two sets of fields: ultradeep
fields ($z\leavevmode\nobreak\ \lesssim\leavevmode\nobreak\ $1.1) observed at
a high cadence with a large number of visits over a limited number of seasons;
deep fields ($z\leavevmode\nobreak\ \lesssim\leavevmode\nobreak\ $0.7),
observed with a cadence of $\sim$3 nights for ten years. These encouraging
results should be confirmed with realistic simulations using the LSST
scheduler. A DDF budget of $\sim$8.5$\%$ is required to design observing
strategies satisfying all the cosmological requirements. A lower DDF budget
lead to surveys that either do not fulfill photo-$z$/WL requirements or are
not optimal for SNe Ia cosmology.

##  1 Introduction

The Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et
al. (2019)) will image billions of objects in six bands for ten years. The
observing time will be shared between the wide-fast-deep (WFD) primary survey,
which will cover half of the sky ($\sim$20,000 deg2), and a set of
minisurveys, including intensive observation of a set of deep drilling fields
(DDFs). In the WFD survey, fields are observed with a similar cadence and
pattern, the cadence beeing defined as the median internight gap in any filter
(high cadences correspond to small internight gaps). The DDF mini-survey is
characterized by a high cadence of observation, a high number of visits per
observing night, and a limited survey area (60-70 deg2).

The DDF mini-survey is vital for achieving the core science the LSST Dark
Energy Science Collaboration (DESC; Abate (2012)) and it has three main
drivers: cosmology, calibration of primary systematic uncertainties for
several cosmological measurements, and synergy with other surveys. DDFs are
essential for DESC to reach its Stage IV dark energy goals, that is to
multiply by $\sim$ 10 the Figure of Merit (FoM) of stage II surveys (Albrecht
et al., 2006).

DESC has long emphasized the importance of the observing strategy for science
probes (or key input to science probes) used to measure cosmological
parameters (dark energy equation of state) with high accuracy. Many studies
have already been achieved (Awan et al. (2016), Lochner et al. (2018), Scolnic
et al. (2018), Almoubayyed et al. (2020), Lochner et al. (2021a), Lochner et
al. (2021b), Gris et al. (2023), Alves et al. (2023)) and the work presented
in this paper is a continuation of many years of efforts aiming to define the
optimal observing strategy to accomplish the scientific objectives of the
Rubin Observatory. In the following we focus on the design of the DDF mini-
survey which is critical for three DESC cosmological measurements: photometric
redshifts, weak gravitational lensing, and Type Ia Supernovae, as outlined in
the text that follows.

LSST is expected to provide useful shape measurements in six-band photometry
for about 4 billion of galaxies (Abell et al., 2009). It is infeasible to
measure redshifts via spectroscopy for such a large number of galaxies,
extending to faint magnitudes, and widely distributed. LSST will thus
primarily rely on photometric redshifts (photo-$z$) based only on imaging
information. A key ingredient for all LSST photometric measurement techniques
is a training set of galaxies with highly reliable redshifts. DDFs in LSST
lead to much deeper coadded observations with respect to the WFD survey and
overlap with deep spectroscopic redshifts (Le Fèvre et al., 2013) and deep
photometric redshift samples (Weaver et al., 2022). These samples are
essential for redshift calibration purposes.

Weak gravitational lensing (WL), the deflection of light from distant massive
structures, is currently one of the most promising probes to constrain the
growth of the cosmic structure and unveil the nature of dark energy
(Mandelbaum (2018) and references therein). The LSST survey will dramatically
increase the statistical power of WL measurements by observing billions of
galaxies. Two key parameters are to be extracted from LSST data: the shear
from galaxy shapes, and the photometric redshift estimates. Shear signals are
to be measured with systematic errors lower than $\sim 10^{-3}$ to minimize
the degradation in cosmological parameter accuracies (Huterer et al., 2006).
The shear calibration bias, one of the largest systematic error sources, may
be reduced using self-calibration techniques such as metacalibration (Huff &
Mandelbaum (2017), Sheldon & Huff (2017)) where real images are used to
simulate the effect of a known shear. Using the metacalibration method in
images collected in the WFD survey would lead to a $\sim 20$$\%$ loss of
precision in the statistical shear measurements (Zhang et al., 2023). Reducing
this loss (pixel noise) to $\lesssim$5$\%$ in metacalibration estimators of
weak lensing signals requires using the DDFs and the deep-field
metacalibration technique (Zhang et al., 2023).

Type Ia Supernovae (SNe Ia) are the results of powerful and luminous
explosions of white dwarfs. SNe Ia can be considered as standardizable candles
to estimate distances used to measure cosmological parameters (equation of
state of dark energy) in a Hubble diagram (Betoule et al. (2014), Brout et al.
(2022)). Cosmological distances are estimated from SNe Ia parameters extracted
from photometric observations (SNe Ia light curves) generated by the observing
strategy. Hence measuring cosmological parameters with a high accuracy
requires optimizing the observing strategy to collect high quality light
curves (in terms of sampling and signal-to-noise ratio). LSST will observe a
large number of SNe Ia at low redshifts ($z\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ $0.4) in the WFD survey (Lochner et al., 2021a).
Measuring $w_{0}$ and $w_{a}$ parameters of the equation of state of dark
energy requires to observe (necessary condition) a large sample of SNe Ia at
higher redshifts (up to $z\leavevmode\nobreak\ \sim$ 1.1) and this is only
feasible with DDFs (Gris et al., 2023). DDFs are thus critical if LSST is
expected to address dark energy physics and cosmology with SNe Ia.

The purpose of this paper is to design a DESC cohesive DDF program that would
fulfill calibration requirements (photo-$z$ and WL) and cosmological
constraints (SNe Ia) whilst meeting the Survey Cadence Optimization Committee
(SCOC; Bianco et al. (2021)) phase 2 guidelines (The Rubin Observatory Survey
Cadence Optimization Committee, 2022). This article is divided in four
sections. The design requirements of the survey are detailed in §2 with a
summary of the SCOC phase 2 recommendations (§2.1) and a detailed explanation
of the requirements from cosmology measurements (§2.2). A set of metrics to
assess observing strategy and to quantify to which extent science requirements
are met are defined in §3. These metrics are based on statistical optimization
and do not include systematic uncertainties. A proposed cohesive DDF strategy
is presented in §4 along with the methodology used to define observing
strategy parameters leading to DDF surveys compliant with science
requirements. These strategies are simulated and assessed in §5 using the
metrics defined in §3.

##  2 Design requirements

###  2.1 SCOC phase 2 recommendations

The SCOC phase 2 recommendations are summarized in The Rubin Observatory
Survey Cadence Optimization Committee (2022). The main points concerning the
Deep Drilling program are:

  * •

budget (i.e. fraction of visits corresponding to DD observations) in the range
5$\%$-7$\%$;

  * •

5 DDFs to be observed for ten years. 4 fields (COSMOS, CDF-S, ELAIS-S1, XMM-
LSS) were selected in 2012 (https://ls.st/bki). A 5th field, the Euclid Deep
Field South (EDFS), is added, with a footprint roughly twice as large as the
extent of the other DDFs. This field will be observed at half the exposure
time of the other DDFs;

  * •

COSMOS should be prioritized with additionnal survey time to reach 10-year DDF
depth within the first three years of LSST. COSMOS will continue to be
observed thereafter at the same rate as other DDFs. There are two main reasons
for starting the DDF observations early: the DESC DDF photo-$z$ calibration
will be improved by the earlier information; at least one DDF should be
completed early for low-surface-brightness science and calibration purposes
(Galaxies Science Collaboration).

Two critical aspects are still to be defined, the cadence of observation, and
the filter allocation (i.e. the number of visits per band and per observing
night). The goal of this paper is to propose a set of values for those
parameters whilst taking into account the recommendations of the SCOC. The
resulting observing strategies should also meet Photo-z, Weak Lensing, and SNe
Ia requirements detailed in the following.

###  2.2 Requirements from cosmological measurements

####  2.2.1 Photo-$z$

Photometric redshifts in LSST are estimated from the Spectral Energy Density
(SED) of distant galaxies, integrated over 6 bands. The type and redshift
measured from the flux observed in multi-band is frequently not uniquely
defined: two different rest-frame SEDs observed at two different redshifts can
be difficult to distinguish. The type/redshift degeneracy is one of the main
causes of uncertainty in photometric redshift calibration.

It may be possible to break degeneracies by combining wide-field, multi-band
measurements with accurate redshift galaxies. In deep-fields, precise
photometry of multi-band flux can be used to group galaxies into fine-grained
set of phenotypes (Sánchez & Bernstein, 2018), which are types of galaxy based
on observed flux rather than on rest-frame properties. The multi-band deep
fields provide greater opportunities to assess the impact of selection
effects.The set of galaxy photometry and a set of galaxies with confidently
known $z$ (obtained using many-band photometry observations or spectroscopy)
can be used as input to a photometric method such as SOMPZ which is based upon
the self-organizing map algorithm (Buchs et al. (2019), Myles et al. (2021)).

The coadded DDFs go much deeper than those of the WFD area and overlap deep
spectroscopic redshift surveys (e.g., VIMOS VLT Deep Survey (VVDS), Euclid
Complete Calibration of the Color-Redshift Relation (C3R2), Le Fèvre et al.,
2013; Stanford et al., 2021), deep photometric redshift samples (Weaver et
al., 2022, COSMOS2020,), Spitzer/IRAC data (Mauduit et al., 2012, SERVS,), and
deep NIR catalogs (McCracken, H. J. et al., 2012, UltraVISTA,; Jarvis et al.,
2012, VIDEO,). The deep $ugrizy$ photometry from the DDF overlaps of these
deep spec-z and photo-$z$ samples are essential for enabling calibrations of
the galaxy redshift distributions needed for LSST science. We therefore define
photo-$z$ requirements for the $ugrizy$ depths in the DDF areas based on
redshift calibration considerations, in particular in the following two cases.

Year 1 (Y1): We first set requirements at the end of LSST Y1, in order to
enable the use of the C3R2 and VVDS deep spectroscopic redshift samples for
redshift calibrations. We define the requirements based on measuring good
photometry for 95% of the C3R2 and VVDS samples, with good photometry defined
as measuring 2-arcsec diameter aperture magnitudes at 10-$\sigma$ for $griz$
and 5-$\sigma$ for $uy$. We measure these magnitude limits using the magnitude
histograms of C3R2 and VVDS galaxy redshift samples overlapping the DES Year 3
Deep Field photometry catalog (Hartley et al., 2021). These aperture magnitude
limits are translated to 5-$\sigma$ point source limits, assuming a Gaussian
point spread function (PSF) and typical seeing FWHM of 0.8 arcsec, resulting
in LSST Y1 5-$\sigma$ point source depths for the $ugrizy$ bands (Table 1).
These depths are meant to apply to the 5th percentile values of the
Coaddm5Metric (Equation 4) evaluated for healpixels within a radius of 1.6
degrees from the center of each DDF. The 5th percentile depth accounts for
significant spatial non-uniformities in depth in early years and is a
compromise between minimum (too stringent) and median (too relaxed) values.

Table 1: PZ requirements. 5-$\sigma$ point source depths for year 1 and for year 10. Depths for year 2-9 will be scaled from year 10 requirements. band | Y1 | Y10
---|---|---
u | 26.7 | 27.8
g | 27.0 | 28.1
r | 26.2 | 27.8
i | 25.8 | 27.6
z | 25.6 | 27.2
y | 24.7 | 26.5

Years 2-10 (Y10): We set requirements scaling with time so that at the end of
LSST Y10, the ugrizy magnitude limits match those of the COSMOS2020 imaging.
The flux limit scales with the square root of the exposure time and decreases
from Y1 to Y10. This would enable the deep DDF photometry, apart from their
overlaps with external spec-z and photo-$z$ samples, to be used for redshift
calibration, e.g., via self organizing map (SOM) methods as used in the Dark
Energy Survey (Myles et al., 2021). We translate the quoted COSMOS2020
3-$\sigma$ 2-arcsec diameter aperture magnitude depths (Weaver et al., 2022)
to LSST 5-$\sigma$ point source depths for the $ugrizy$ bands (Table 1).

####  2.2.2 Weak Lensing

The weak lensing requirements are driven by two primary uses of the DDFs.
First, DDF imaging that overlaps or will overlap with high-resolution space-
based imaging (e.g., COSMOS or Euclid) is essential for generating realistic
simulations for testing the calibration of weak lensing shape estimation
methods (e.g., MacCrann et al., 2022). Second, the DDFs are used to calibrate
certain weak lensing shear estimators, including the Bayesian Fourier Domain
(BFD) technique (Bernstein & Armstrong, 2014; Bernstein et al., 2016) and the
deep-field metacalibration technique (Zhang et al., 2023).

The requirements for both of these use cases can be phrased in terms of the
relative number of visits between the DDFs and the WFD. First, for the image
simulation use case, previous studies have found that image simulations for
weak lensing shear calibration need to have galaxies down to $\approx$1.5
magnitudes fainter than the limiting magnitude being calibrated in order to
limit calibration errors to $\lesssim 0.1\%$ (Hoekstra et al., 2015, 2017).
Assuming the sources are background-dominated, this requirement implies that
the DDFs used to derive input sources for WL shear calibration simulations
have $\gtrsim 16\times$ as many visits as the WFD. This number is derived from
Equation 4 below, assuming a target increased coadded depth of 1.5 magnitudes.
Importantly, this requirement would only apply to regions with imaging that
overlaps with a space-based survey. Further, a joint analysis of the ground-
and space-based imaging will greatly benefit from both this increased depth
and from the much smaller effective PSF of the space-based data. At minimum,
this would include COSMOS data but might also include Euclid data as well.

Second, the primary requirement for BFD and deep-field metacalibration is that
the pixel noise in the DDF coadd images be sufficiently smaller than the
typical pixel noise in WFD coadd images. Working in the background-dominated
limit and assuming a simple mean coaddition strategy, the relative pixel noise
in the mean coadd image between the DDF and WFD data will decrease as the
square root of the ratio of the number of DDF to WFD visits. Both techniques
require this ratio to be $\gtrsim 10$ for essentially the same reason:
limiting the DDF contribution to the variance in the weak lensing shape
estimator to $\lesssim 10\%$.

We finally end up with two WL requirements related to the ratio of the number
of DDF visits to the number of WFD visits $r=\frac{N_{v}^{DDF}}{N_{v}^{WFD}}$:
$r\gtrsim 16$ (WL shear calibration for e.g. COSMOS) and $r\gtrsim 10$ (pixel
noise). We thus assume a requirement of at least $10\times$ as many visits in
the DDFs as the WFD. We do not impose to have $r\gtrsim 16$ for e.g. COSMOS
but we will check whether this requirement is fulfilled in the strategies
proposed in this paper.

The WL requirements are summarized in Table 2. We have specified the required
number of visits for Y1 and Y10. However, note that the above WL requirements
must be satisfied at any data release in the survey that we wish to use for
cosmological analysis. These data releases will include Y1 and Y10, and will
also include intermediate data releases e.g., Y4 and Y7.

Table 2: Number of visits (per band) required to fulfill photo-$z$ and WL requirements. |  | photo-$z$ requirements | WL requirements | WL+photo-$z$ (WZ) requirements
---|---|---|---|---
band | $m_{5}^{single}$ | Nvisit | Nvisit | Nvisit
|  | season 1 | season 2-10 | season 1 | season 2-10 | season 1 | season 2-10
u | 23.53 | 343 | 2601 | 48 | 432 | 343 | 2601
g | 24.23 | 165 | 1252 | 72 | 648 | 165 | 1252
r | 23.70 | 100 | 1905 | 184 | 1656 | 184 | 1905
i | 23.33 | 94 | 2592 | 184 | 1656 | 184 | 2592
z | 22.95 | 131 | 2493 | 152 | 1368 | 152 | 2493
y | 22.17 | 105 | 2884 | 160 | 1440 | 160 | 2884

####  2.2.3 SNe Ia

The estimation of the cosmological parameters with SNe Ia is achieved by
fitting the distance modulus (indirect measure of distance using brightness)
versus redshift relation. The distance modulus depends on SNe Ia parameters
which are estimated from photometric observations (light curve) emanating from
the survey (observing strategy). Measuring cosmological parameters with a high
degree of accuracy requires minimizing SNe Ia parameter errors, that is to
maximize the light curve signal-to-noise ratio (SNR) per band $b$, SNRb,
defined by:

SNRb
$\displaystyle=\sqrt{\sum_{i=1}^{n^{b}}{\left(\frac{f_{i}^{b}}{\sigma_{i}^{b}}\right)^{2}}}$
(1)

where $f^{b}$, and $\sigma^{b}$ are the fluxes and flux uncertainties. The sum
runs over observations within a time-window around the SN Ia luminosity peak
MJD (see §3.3). The main strategy parameters impacting SNRb are the cadence of
observation (SNRb increases with an increase of the cadence) and the number of
visits per observing night and per band N${}_{visits}^{b}$. The light curve
flux errors $\sigma_{i}^{b}$ decreases with an increase of N${}_{visits}^{b}$.
This can be explained as follows. In the background-dominated regime one has
$\sigma_{i}^{b}\simeq\sigma_{5}^{b}$ where $\sigma_{5}^{b}$ is equal by
definition to

$\displaystyle\sigma_{5}^{b}$ $\displaystyle=\frac{f_{5}^{b}}{5}$ (2)

where $f_{5}^{b}$ is the 5-$\sigma$ flux related to the 5-$\sigma$ magnitude
$m_{5}^{b}$. The coadded 5-$\sigma$ depth increases (and $\sigma_{i}^{b}$
decreases) with the number of visits (see Equation 4). Nightly DDF coadded
observations lead to an increase of SNRb values which translates into a
significant increase of well-observed SNe Ia at higher redshift.

The redshift distribution of the sample of well-measured SNe Ia is critical to
measure dark energy parameters. The WFD survey will provide a low-$z$ sample
(0.01 $\gtrsim\leavevmode\nobreak\ z\leavevmode\nobreak\
\gtrsim\leavevmode\nobreak\ 0.4-0.6$) which will be combined with a high-$z$
(0.3 $\gtrsim\leavevmode\nobreak\ z\leavevmode\nobreak\
\gtrsim\leavevmode\nobreak\ 1.1$) DDF SNe Ia sample necessary to measure
($w_{0},w_{a}$) parameters with high accuracy (Gris et al., 2023). It is
possible to quantify the depth of the survey using the redshift completeness
$z_{\mathrm{complete}}$ (see the definition in Appendix A), a parameter
correlated with SNRb (Gris et al., 2023). Increasing $z_{\mathrm{complete}}$
leads to an increase of the number of well-sampled SNe Ia at higher redshifts
and requires to boost SNRb values with a survey strategy characterised by a
high cadence of observation (1-2 nights), a large number of visits per
observing night (typically more than hundred for a cadence of 1 night), and a
filter allocation optimized to observe at higher redshifts (Table 3).

Table 3: Number of visits per observing night and filter allocation required to reach $z_{\mathrm{complete}}$ of 0.80, 0.75, and 0.70. The cadence of observation is of one night and the season length of 180 days. These results are extracted from Gris et al. (2023). $z_{\mathrm{complete}}$ | Nvisit/obs. night | budget
---|---|---
| total | $g/r/i/z/y$ | per season
0.80 | 149 | 2/9/62/56/20 | 1.3$\%$
0.75 | 108 | 2/9/44/38/15 | 0.9$\%$
0.70 | 81 | 2/9/33/31/6 | 0.7$\%$

The DDF budget is limited to 5-7$\%$ (The Rubin Observatory Survey Cadence
Optimization Committee, 2022). It is thus impossible, according to the numbers
in Table 3 to observe a full sample of SNe Ia up to high redshift completeness
($z_{\mathrm{complete}}$ $\gtrsim$ 0.75) with a survey scanning 5 fields for
ten years. One way to maximize the number of SNe Ia at higher redshifts is
then to define two types of DDFs: Deep Fields (DFs) with a redshift
completeness lower than $\sim$ 0.60; Ultra-Deep Fields (UDF) with observing
strategy parameters corresponding to Table 3 for a limited number of seasons
(typically between 2 to 4); UDFs are observed with the same strategy as DFs
for the remaining seasons. Optimized scenarios include the observation of two
UDFs for two ($z_{\mathrm{complete}}$$\leavevmode\nobreak\
\sim\leavevmode\nobreak\ $0.80), three
($z_{\mathrm{complete}}$$\leavevmode\nobreak\ \sim\leavevmode\nobreak\ $0.75),
or four ($z_{\mathrm{complete}}$$\leavevmode\nobreak\ \sim\leavevmode\nobreak\
$0.70) seasons.

##  3 Metrics to assess observing strategies

###  3.1 Photo-$z$

Photo-$z$ calibration requirements can be translated to 5-$\sigma$ depth point
source constraints (subsubsection 2.2.1). We define the photo-$z$ metric as
the (coadded per band) 5-$\sigma$ depth difference between observing strategy
observations and the requirements defined in Table 2:

$\mbox{$\Delta
m_{5}^{b}$}=\mbox{$m_{5}^{b,OS}$}-\mbox{$m_{5}^{b,PZ\leavevmode\nobreak\
req.}$}$ (3)

where $b$ is the band and $m_{5}^{b,OS}$ is the coadded 5-$\sigma$ depth:

$\mbox{$m_{5}^{b,OS}$}=1.25\log_{10}\sum_{i=1}^{N_{v}^{b}}(10^{0.8\leavevmode\nobreak\
\mbox{$m_{5}^{b,i}$}})$ (4)

Equation 4 equates to adding signal-to-noise ratio for a background-limited
object in quadrature and may be written as:

$\mbox{$m_{5}^{b,OS}$}=1.25\log_{10}N_{v}^{b}+<\mbox{$m_{5}^{b,i}$}>+\mbox{$\Delta
m_{5}^{b,i}$}$ (5)

where $N_{v}^{b}$ is the number of visits in the $b$ band and ¡$m_{5}^{b,i}$¿
is the 5-$\sigma$ depth mean value. $\Delta m_{5}^{b,i}$ depends on the
standard deviation of the $m_{5}^{b,i}$ values. From the LSST simulation
baseline_v3.0_10yrs we obtain ¡$\Delta m_{5}^{b,i}$¿$\sim\mathcal{O}(0.01)$
for the COSMOS field. In this paper we have used $\Delta m_{5}^{b,i}$=0 since
we have considered median observing conditions per season (§5).

Photo-$z$ requirements are fulfilled if $\Delta m_{5}^{b}$ $\geq$ 0.

###  3.2 Weak Lensing

Following the WL requirements (subsubsection 2.2.2) we define the photo-$z$
metric as:

$\mbox{$r^{WL}$}=\mbox{$\frac{N_{visits}^{OS}}{N_{visits}^{WL\leavevmode\nobreak\
req}}$}$ (6)

where $N_{visits}^{OS}$ is the number of visits (per band) of the proposed
strategies and $N_{visits}^{WL\leavevmode\nobreak\ req}$ the number of visits
corresponding to WL requirements (Table 2). WL requirements are fulfilled if
$r^{WL}$ $\geq$ 1.

###  3.3 SNe Ia

The metric used to assess the strategies proposed in this paper is based on
the Dark Energy Task Force (DETF) Figure of merit (Albrecht et al., 2006). It
is estimated from the result of a cosmological fit using SNe Ia and is defined
as:

$\mathrm{Supernova\leavevmode\nobreak\ Metric\leavevmode\nobreak\
of\leavevmode\nobreak\ Merit}\leavevmode\nobreak\
(\mbox{SMoM})\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{\pi}{A}$ (7)

where $A$ is the area of the confidence ellipse defined by:

$A=\pi\Delta\chi^{2}\sigma_{w_{0}}\sigma_{w_{a}}\sqrt{1-\rho^{2}}$ (8)

with $\Delta\chi^{2}$ = 6.17 (95.4$\%$ C.L. for 2 degrees of freedom). $\rho$
is the correlation factor equal to
Cov($w_{0},w_{a}$)/($\sigma_{w_{0}}\sigma_{w_{a}})$. $w_{0}$ and $w_{a}$ are
the parameters of the Chevallier-Polarski-Linder (CPL) model of the dark
energy equation of state (Chevallier & Polarski, 2001; Linder, 2003):

$w=w_{0}+w_{a}{{z}\over{1+z}}$ (9)

The cosmological parameters are estimated by minimizing:

$-ln\mathcal{L}=\sum_{i=1}^{N_{SN}}\frac{(\mu_{i}-\mbox{$\mu_{th}(z_{i},\Omega_{m},w_{0},w_{a})$})^{2}}{\sigma_{\mu_{i}}^{2}+\mbox{$\sigma_{int}^{2}$}}\\\
$ (10)

where $N_{SN}$ is the number of well-sampled SNe Ia (definition below),
$\sigma_{\mu}$ the distance modulus error, and $\sigma_{int}$ the intrinsic
dispersion of SNe Ia ($\sigma_{int}$ $\sim$ 0.12 mag).

We simulate SNe Ia distance moduli using:

$\mu_{i}(z_{i})\sim\mathcal{N}(\mbox{$\mu_{th}(z_{i},\Omega_{m},w_{0},w_{a})$},\sigma^{2}=\sigma_{\mu_{i}}^{2}+\mbox{$\sigma_{int}^{2}$})$
(11)

Distance moduli errors, $\sigma_{\mu_{i}}$, are estimated from SNe Ia
simulations. We use the SALT3 model (Kenworthy et al., 2021) to simulate and
fit SNe Ia light curves. In this model, the distance modulus, $\mu$, is
defined for each SN Ia by:

$\mu=m_{B}+\alpha x_{1}-\beta c-M$ (12)

where $m_{B}=-2.5\leavevmode\nobreak\ \mathrm{log_{10}}(x_{0})+10.635$, where
$x_{0}$ is the overall flux normalization, $x_{1}$ describes the width of the
light curve, $c$ is equal to a color offset (with respect to the average) at
the date of peak brightness in $B$-band, $c=(B-V)_{MAX}-<B-V>$. For each SN
Ia, $m_{B},\leavevmode\nobreak\ x_{1},\leavevmode\nobreak\ c$ parameters are
estimated from a fit of a SN Ia model to the measurements of a multicolor
light curve. $\alpha$, $\beta$ and $M$ are global parameters estimated from
the data. $M$ is the absolute magnitude in rest-frame $B$-band for a median SN
Ia with ($x_{1}$,$c$) = (0.0, 0.0). $\alpha$ and $\beta$ are global nuisance
parameters quantifying the correlation between brightness with $x_{1}$ and
$c$, respectively. The three parameters $\alpha,\leavevmode\nobreak\
\beta,\leavevmode\nobreak\ M$ are usually fitted along with cosmological
parameters by minimizing Equation 10. In this study we set
$\alpha,\leavevmode\nobreak\ \beta$, and $M$ to their simulated values so as
to fit only cosmological parameters.

In the SALT3 model a SNe Ia is described by five parameters: $x_{0}$, $x_{1}$,
$c$, $z$ (redshift), and $T_{0}$ (time of maximum luminosity). For each of the
proposed strategies we have simulated a sample of SNe Ia corresponding to 30
times the number of expected SNe Ia (we use the rate Hounsell et al. (2018))
in the full redshift range [0.01,1.1]. Concerning the ($x_{1}$,$c$)
distribution of SNe Ia, we use the G10 intrinsic scatter model (Scolnic &
Kessler, 2016) where ($x_{1}$,$c$) distributions are described by asymmetric
gaussian distributions with three parameters. We choose random $T_{0}$ values
spanning over the season duration of a group of observations.

We use SNCosmo (Barbary et al., 2016), a Python library synthesizing SN
spectra and photometry from SN models, to simulate and fit SNe Ia light curves
using the SALT3 model. The telescope throughput curves (detector, lenses,
mirrors, filters) used to estimated SNe Ia fluxes are consistent with the LSST
simulations considered in this paper (v1.5 of
https://github.com/lsst/throughputs). The set of well-sampled SNe Ia used to
estimate SMoM is composed of SNe Ia fulfilling the requirements detailed in
Table 4.

Table 4: Selection criteria leading to a sample of well-sampled SNe Ia based on the available phases p=$\frac{t-T_{0}}{1+z}$ of photometric observations at time $t$. (a) and (b) correspond to Guy et al. (2010) and Betoule et al. (2014), respectively. Selection | Rationale
---|---
light curve (LC) |
$N_{epochs}$($p\leq-10$) $\geq$ 1 | low phase LC
$N_{epochs}$($p\geq+20$) $\geq$ 1 | high phase LC
$N_{epochs}$($-10\leq p\leq 35$) $\geq$ 4 | SN parameters (a)
$N_{epochs}$($-10\leq p\leq 5$) $\geq$ 1 | SN parameters (a)
$N_{epochs}$($+5\leq p\leq 20$) $\geq$ 1 | LC shape (a)
$N_{epochs}$($-8\leq p\leq 10$) $\geq$ 2 | SN peak luminosity
(two bands) | and color (a)
SN parameters |
$\sigma_{T_{0}}\leavevmode\nobreak\ \leq$ 2 | LC sampling (b)
$\sigma_{x_{1}}\leavevmode\nobreak\ \leq$ 1 | LC sampling (b)

One may notice that the SMoM metric is estimated by combining SNe Ia samples
observed in the WFD (low-$z$) and DDF (high-$z$) surveys. It can not be used
to check directly whether SNe Ia specifications (subsubsection 2.2.3) related
to DDFs are fulfilled. But it has been shown that UDF scenarios with
$z_{\mathrm{complete}}^{\mathrm{UD}}$ corresponding to SNe Ia requirements
tend to maximize SMoM values (Gris et al., 2023).

##  4 Designing a cohesive DDF strategy

As shown in §2, measuring cosmological parameters (dark energy equation of
state) with high accuracy with SNe Ia requires to observe (necessary
condition) a large sample of SNe Ia at higher redshifts ($z\geq$0.8). This can
be achieved with a high cadence of observation ($\leq$ 2 night) and a large
number of visits per observing night (more than 150 visits for a 2 night
cadence) with a specific filter allocation defined by the $z$-depth to be
reached (see Gris et al. (2023) for all the details). Nonetheless, this
corresponds to a total number of visits (per season) quite costly in terms of
budget: observing a SNe Ia sample up to a redshift completeness of $\sim$ 0.75
leads to a fraction of DD visits of about 0.9% per season (Table 3). Since the
DDF budget is limited to 5-7%, it is not realistic to design a uniform DD
survey (leading to accurate cosmological measurements) where all the fields
would be observed at the same cadence, with the same number of visits per
observing night, and the same filter allocation, for ten years.

Table 5: List of parameters defining DDF scenarios. Constraints are taken from the SCOC phase 2 recommendations, photo-$z$/WL calibration constraints, and SNe Ia cosmological requirements. parameter | definition | constraint | source
---|---|---|---
budget${}^{\text{DD}}$ | DD budget | 5-7% | SCOC phase 2 recommendations
N${}_{\text{v}}^{\text{LSST}}$ | total number of LSST visits | 2.1 million | Survey simulation
N${}_{\text{f}}^{\text{Y1}}$ | Number of fields observed in season 1 | 5 | SCOC phase 2 recommendations
N${}_{\text{v}}^{\text{Y1}}$ | Number of visits per field in season 1 | 1187 | photo-$z$/WL req.
N${}_{\text{f}}^{\text{UDF}}$ | Number of UD fields | 2 | SNe Ia req.
N${}_{\text{s}}^{\text{UDF}}$ | Number of seasons for UD fields | 2-4 | SNe Ia req.
N${}_{\text{v}}^{\text{UDF}}$ | Number of visits (per season) per UD field | 14580-26820 | SNe Ia req.
N${}_{\text{s}}^{\text{UDF,DF}}$ | Number of seasons of UD fields = Deep fields | 9-N${}_{\text{s}}^{\text{UDF}}$ | SCOC phase 2 recommendations
N${}_{\text{f}}^{\text{DF}}$ | Number of Deep fields | 5-N${}_{\text{f}}^{\text{UDF}}$ | SCOC phase 2 recommendations
N${}_{\text{s}}^{\text{DF}}$ | Number of seasons for Deep fields | 9 | SCOC phase 2 recommendations
N${}_{\text{v}}^{\text{DF}}$ | Number of visits (per season) per Deep field | 1525 | photo-$z$,WL req.

One possible solution is to design a DDF survey with two types of fields:
ultra-deep fields leading to a high number of well-measured SNe Ia used to
achieve accurate cosmological measurements; deep fields satisfying the
5-$\sigma$ point source criteria from photo-$z$ and WL. The depth (coadded
5-$\sigma$ depth for photo-$z$/WL or $z_{\mathrm{complete}}$ for SNe Ia) is
primarily driven by survey parameters such as the cadence and the number of
visits per band and per observing night.

The method to design DESC cohesive DDF strategies proceeds in two steps. The
first stage is to define a set of scenarios in a 4d parameter space: number of
UD fields (and season of observation), number of visits per season of
observation (for UDFs and DFs). Survey parameters such as cadence and season
length of observation, filter allocation per observing night are to be defined
in a second phase to build observing strategies from these scenarios. These
parameters are chosen so that requirements from cosmological measurements are
fulfilled (§2.2).

Figure 1: Graphical estimation of the parameters of DESC cohesive DDF
strategies in the (N${}_{\text{v}}^{\text{UDF}}$/observing night,
N${}_{\text{v}}^{\text{DF}}$/season) parameter space. Black lines correspond
to three configurations
(N${}_{\text{f}}^{\text{UDF}}$,N${}_{\text{s}}^{\text{UDF}}$) for a budget of
7$\%$, a number of visits of 2.1 million, a cadence of observation of 2 days
and a season length of 180 days for UDFs. The
(N${}_{\text{v}}^{\text{UDF}}$/observing night,
N${}_{\text{v}}^{\text{DDF}}$/season) parameter space corresponding to
photo-$z$+WL (calibration) requirements (red dotted vertical lines) and SNe Ia
cosmological constraints (red dotted horizontal lines) is used to define the
strategies (intersection points with the lines corresponding to the three
(N${}_{\text{f}}^{\text{UDF}}$,N${}_{\text{s}}^{\text{UDF}}$) configurations.
On this plot are also reported the results corresponding to the SCOC phase 2
recommendations (SCOC_p2) and to a universal survey. Yellow areas around
photo-$z$+WL and SNe Ia requirements correspond to $\Delta
m_{5}^{b}$=$\pm$0.05 and $\Delta$$z_{\mathrm{complete}}$=$\pm$0.01,
respectively. Markers correspond to strategies optimal for SNe Ia (squares),
fulfilling photo-$z$/WL requirements (stars), or corresponding to compromises
between photo-$z$/WL and SNe Ia constraints (circles).

The total number of DDF visits N${}_{\text{v}}^{\text{DDF}}$ can be expressed
as a function of eleven configuration parameters (see definitions in Table 5)
characterizing a DDF scenario (in the following: f=field; s=season, v=visit):

N${}_{\text{v}}^{\text{DDF}}$
$\displaystyle=\mbox{N${}_{\text{v}}^{\text{LSST}}$}\times\mbox{budget${}^{\text{DD}}$}$
(13)
$\begin{split}\mbox{N${}_{\text{v}}^{\text{DDF}}$}&=\mbox{N${}_{\text{f}}^{\text{DF}}$}\times\mbox{N${}_{\text{s}}^{\text{DF}}$}\times\mbox{N${}_{\text{v}}^{\text{DF}}$}+\mbox{N${}_{\text{f}}^{\text{UDF}}$}\times\mbox{N${}_{\text{s}}^{\text{UDF}}$}\times\mbox{N${}_{\text{v}}^{\text{UDF}}$}\\\
&+\mbox{N${}_{\text{f}}^{\text{UDF}}$}\times(9-\mbox{N${}_{\text{s}}^{\text{UDF}}$})\times\mbox{N${}_{\text{v}}^{\text{DF}}$}+\mbox{N${}_{\text{f}}^{\text{Y1}}$}\times\mbox{N${}_{\text{v}}^{\text{Y1}}$}\end{split}$
(14)

Combining Eqs (13) and (14) leads to:

$\begin{split}\mbox{A}&=[\mbox{N${}_{\text{f}}^{\text{UDF}}$}\times(9-\mbox{N${}_{\text{s}}^{\text{UDF}}$})+\mbox{N${}_{\text{f}}^{\text{DF}}$}\times\mbox{N${}_{\text{s}}^{\text{DF}}$}]\times\mbox{N${}_{\text{v}}^{\text{DF}}$}\\\
\mbox{N${}_{\text{v}}^{\text{UDF}}$}&=\dfrac{\mbox{budget${}^{\text{DD}}$}\times\mbox{N${}_{\text{v}}^{\text{LSST}}$}-\mbox{N${}_{\text{f}}^{\text{Y1}}$}\times\mbox{N${}_{\text{v}}^{\text{Y1}}$}-\mbox{A}}{\mbox{N${}_{\text{f}}^{\text{UDF}}$}\times\mbox{N${}_{\text{s}}^{\text{UDF}}$}}\end{split}$
(15)

Photo-$z$/WL requirements for Y1 are different from the ones concerning Y2-Y10
(Table 2). This is why we chose to have a uniform DDF cadence in Y1 with
observing strategy parameters (cadence of observation, season length, filter
allocation) leading to surveys fulfilling photo-$z$/WL requirements. The
observing strategy Y1 is thus the same for all the strategies proposed in this
paper.

It is possible from Equation 15 to estimate the parameter space
(N${}_{\text{v}}^{\text{DF}}$, N${}_{\text{v}}^{\text{UDF}}$) for a choice of
(N${}_{\text{f}}^{\text{UDF}}$,N${}_{\text{s}}^{\text{UDF}}$) configurations
if all other parameters are assigned the values of Table 5. Since SNe Ia
requirements are related to the number of visits per observing night
(subsubsection 2.2.3), we define a DDF strategy by the four parameters
(N${}_{\text{v}}^{\text{UDF}}$/obs. night, N${}_{\text{v}}^{\text{DF}}$,
N${}_{\text{f}}^{\text{UDF}}$, N${}_{\text{s}}^{\text{UDF}}$). This requires
to choose a cadence of observation (cad${}^{\text{UDF}}$) and season length
(sl${}^{\text{UDF}}$) for UD fields. We have used cad${}^{\text{UDF}}$=2
nights and sl${}^{\text{UDF}}$=180 days in this paper.

Cohesive DDF strategies have to fulfill requirements from photo-$z$, WL and
SNe Ia. We use a graphical method to define these strategies as described in
the following. From Equation 15 we estimate the
(N${}_{\text{v}}^{\text{UDF}}$/obs. night, N${}_{\text{v}}^{\text{DF}}$)
parameter space (sets of lines) for three UD configurations
(N${}_{\text{f}}^{\text{UDF}}$, N${}_{\text{s}}^{\text{UDF}}$)=[(2, 2),(2,
3),(2, 4)]. We superimpose to these results the parameter space corresponding
to photo-$z$, WL and SNe Ia requirements (§2) which correspond to a set of
vertical (photo-$z$+WL) and horizontal (SNe Ia) lines. DESC cohesive
strategies parameters are defined by the intersection of these two parameter
spaces. This graphical method (Figure 1) leads to the definition of 9 cohesive
DDF strategies defined in Table 6. We observe that it is not possible to
design scenarios fulfilling SNe Ia and photo-$z$/WL requirements with a DDF
budget of 7$\%$. Fulfilling all the requirements would require a DDF budget of
$\sim$8.5$\%$.

We have added two strategies on Figure 1. In the Deep Universal scenario all
the DDFs are observed for ten years with the same cadence and the same depth
(same number of visits per observing night). This scenario does not count any
UD field. The SCOC_p2 scenario is designed from the SCOC phase 2
recommendations stipulating that the COSMOS field should have additionnal
survey time to reach a depth corresponding to 10-year DDF within three years
of LSST (§2.1).

Table 6: List of DESC cohesive scenarios. Requirements are fulfilled if $\Delta$Nvisits $\geq$0 (photo-$z$/WL) or $\Delta z_{\mathrm{complete}}^{\mathrm{UD}}$ $\geq$0 (SNe Ia).  | Requirements |  |
---|---|---|---
Strategy | $\Delta z_{\mathrm{complete}}^{\mathrm{UD}}$ | $\Delta$Nvisits | (N${}_{\text{f}}^{\text{UDF}}$,N${}_{\text{s}}^{\text{UDF}}$) | Note
DDF_DESC_0.80_SN |  | -701 | (2,2) | optimal
DDF_DESC_0.75_SN | 0 | -899 | (2,3) | for
DDF_DESC_0.70_SN |  | -865 | (2,4) | SN
DDF_DESC_0.80_WZ |  |  | (2,2) | optimal
DDF_DESC_0.75_WZ | -0.05 | 0 | (2,3) | for
DDF_DESC_0.70_WZ |  |  | (2,4) | PZ/WL
DDF_DESC_0.80_co |  | -351 | (2,2) | compromise
DDF_DESC_0.75_co | -0.03 | -450 | (2,3) | between
DDF_DESC_0.70_co |  | -433 | (2,4) | PZ/WL and SN
Figure 2: Photo-$z$ metric $\Delta m_{5}$ for the proposed strategies, for
each DDF and each band. Photo-$z$ requirements are fulfilled if $\Delta m_{5}$
$\geq$ 0\. The dashed line corresponds to $\Delta m_{5}$=0.

Completing the definition of the DESC cohesive DDF strategies requires to
choose three additional parameters: season length, cadence of observation, and
filter allocation (number of visits per band and per observing night). For
UDFs, these parameters are coming from depth requirements related to the
redshift completeness of the SNe Ia survey (§2.2 and Table 5). For DFs, we
have chosen a season length of 180 days (to maximize the number of observed
SNe Ia ) and a cadence of three days (median cadence observed in the
simulations). The number of visits per band and per observing night (filter
allocation) for the DFs is estimated from the total number of visits per
season N${}_{\text{v}}^{\text{DF}}$/season (Figure 1).
N${}_{\text{v}}^{\text{DF}}$/season is distributed among filters and observing
nights in such a way that photo-$z$ and WL requirements are fulfilled
(subsubsection 2.2.1 and subsubsection 2.2.2). If this is not possible (which
is the case for all the proposed scenarios except the ones optimal for
photo-$z$/WL) the number of visits is split in such a way that the relative
number of visits per band is equal to the one estimated from photo-$z$ and WL
requirements.

Figure 3: WL metric
$r^{WL}=\frac{N_{visits}^{OS}}{N_{visits}^{WL\leavevmode\nobreak\ req}}$ for
the proposed strategies and for each field/band. The dashed (dotted) line
correspond to $r^{WL}$=1 (1.6). WL requirements are fulfilled if $r^{WL}$
$\geq$ 1.

The LSST camera has 6 filters ($ugrizy$) and only five can be loaded in the
filter exchange carousel. The current filter load strategy is to replace
during daytime one of the $z$ or $y$ by the $u$ band at the start of dark
time, when the lunar phase is lower than 20%. The process is reversed when the
lunar phase is above 20%. We have performed a study using SNe Ia to assess the
impact of the swap $z\leftrightarrow u$ or $y\longleftrightarrow u$ on two
metrics ($z_{\mathrm{complete}}$ and N${}_{\mathrm{SN}}^{z\leq
z_{\mathrm{complete}}}$). The results are detailed in Appendix A . The
conclusion is that the minimal loss is observed for the $y\leftrightarrow u$
swap. This can be explained by the fact that the number of visits in the $y$
band is lower than the number of visits in the $z$ band. The decrease of the
light curve signal-to-noise ratio due to the swap is lower if $y$ and $u$
bands are exchanged. Following the conclusion of this study we have swapped
the $y$ and $u$ bands when the lunar phase is low in the design of the
proposed strategies. The total number of $u$ ($y$) visits is distributed among
the nights with a Moon phase lower (higher) than 20%.

The observing strategy parameters for the proposed DESC cohesive scenarios are
summarized on Figures 8-9. Exposure times are given in Tables 10-9. The
coadded 5-$\sigma$ depth and the total number of visits per field/band/season
are given in Table 11. As expected, the number of visits per observing night
increases with the depth of observation ($z_{\mathrm{complete}}$) and higher
values (about 300 visits that is 2.5 hours of observing time) are reached for
$z_{\mathrm{complete}}$$\sim$ 0.80 in the DDF_DESC_0.80_SN scenario. As
expected, a large fraction of the DDF budget is exhausted after few seasons
for strategies with two UDFs (i.e. DDF_DESC_* scenarios - Figure 10).

##  5 Assessment of the proposed DESC cohesive DDF strategies

###  5.1 Simulation of the proposed strategies

Assessing the DESC cohesive strategies (§4) using the metrics defined in §3
requires to have simulations of the proposed scenarios. We have performed
these simulations using the parameters defined in Figure 8 and Figure 9. We
have used median observing conditions per season for the only observing
parameter considered, the 5-$\sigma$ depth parameter. We are aware of the
imprecise nature of these simulations. But they can be used to provide some
insight about the relative performance of the proposed scenarios. As will be
stresses below, the preliminary findings obtained in this study need to be
checked with more accurate simulations using the LSST scheduler.

Figure 4: SNe Ia metric SMoM as a function of the season for all the
strategies designed in this paper. A prior on $\Omega_{m}$ is added to
Equation 10 ($\sigma_{\Omega_{m}}=0.0073$ Aghanim et al. (2020)) to achieve
these results. The observed sample is composed exclusively of SNe Ia with
spectroscopic host galaxy redshift.

###  5.2 Photo-$z$ metric

The photo-$z$ metric defined in Sec. 3, $\Delta m_{5}\leavevmode\nobreak\
=\leavevmode\nobreak\ m_{5}^{OS}-m_{5}^{PZ\leavevmode\nobreak\ req}$, is
estimated using the above-described simulations. Photo-$z$ requirements are
fulfilled if $\Delta m_{5}\leavevmode\nobreak\ \geq\leavevmode\nobreak\ 0$.

Results for season 1 (Table 12) show that two bands, $u$ and $g$, do not meet
photo-$z$ requirements for the COSMOS field, with $\Delta m_{5}$ $\sim$ -0.08
and -0.11, respectively. These differences correspond to a one observing night
variation in the survey. Most of the bands for EDFS-a and EDFS-b do not
fulfill photo-$z$ requirements. This can be explained by the fact that the
total number of visits for these two fields is twice lower (SCOC phase 2
recommendations). $\Delta m_{5}$ is positive for $r$ and $i$ bands because the
number of visits for season 1 is driven by WL requirements (Tab. 2).

Results for seasons 2-10 are given on Figure 2. UDFs tend to meet photo-$z$
requirements for all the bands but the $u$ one. The larger number of visits in
$grizy$ bands for UDFs explains this result. As expected $u$-band photo-$z$
requirements are fulfilled for scenarios corresponding to an optimization of
photo-$z$ and WL criteria. A variation of $\Delta m_{5}$ $\sim$ -0.5 is
observed for strategies built to meet SNe Ia requirements. This $u$-band
result for UDFs is to be extended to all the bands for CDF-S and ELAIS-S1
where lower values of $\Delta m_{5}$ $\sim$ -0.5 are reached for optimal SN
strategies. Lower $\Delta m_{5}$ values of $\sim$-1 mag are reached by $z$
band measurements for EDFS-a and EDFS-b. As for season 1, these results are to
be explained by the lower depth of these fields. One may observe that the
scenario DDF_Univ_SN is far from meeting the minimum requirements for $u$ and
$y$ bands.

###  5.3 WL metric

The WL metric defined in §3,
$r^{WL}=\frac{N_{visits}^{OS}}{N_{visits}^{WL\leavevmode\nobreak\ req}}$, is
estimated using the above-described simulations. WL requirements are fulfilled
if $r^{WL}$ $\geq$ 1.

Season 1 results (Tab. 13) show that WL requirements are met for all fields
except for ECDFS-a and ECDFS-b in the southern region. The number of visits in
$r$,$i$,$z$,$y$ for season 1 is driven by WL requirements (Tab. 2). $r^{WL}$
is thus close to one for these bands and for all fields but EDFS-a and EDFS-b.
For these two fields the number of visits is a factor of 2 lower compared to
the other DDFs (SCOC phase 2 recommendations) and the ratio is close to $\sim$
0.5. For this season the requirement $r^{WL}$$\gtrsim$16 (subsubsection 2.2.2)
is fullfilled for all the fields but EDFS-a and EDFS-b and only for $u$ and
$g$ bands.

WL requirements are exceeded for UDFs observations during seasons 2 to 10 for
all the strategies but DDF_Univ_SN. $r^{WL}$ lies in the range [3-10] with
higher values corresponding to $z$ and $y$ bands. Results for CDF-S and
ELAIS-S1 are much more mitigated. Except the $u$-band, WL requirements are
fulfilled for all scenarios but the ones corresponding to strategies optimal
for SN. The situation is worse for southernmost fields EDFS-a and EDFS-b where
WL requirements are not met (except the $u$-band) for all strategies with
UDFs.

###  5.4 SNe Ia metric

The SMoM metric is estimated from a Hubble diagram cosmological fit and
requires to have a sample of SNe Ia in the redshift range [0.01,1.1]. We thus
generate a SNe Ia sample observed in a WFD survey using the
baseline_v3.0_10yrs simulation (see Appendix C). This sample is then combined
to each of the proposed DDF strategies to estimate SMoM values.

We assume that the SNe Ia used to estimate SMoM have host galaxy redshifts
measured from spectroscopic observations. We (conservatively) consider two
spectroscopic resources contemporaneous with LSST to provide vital spectra and
host redshifts, PFS/Subaru (Tamura et al., 2016) and 4MOST/TiDES (de Jong et
al. (2019); Swann et al. (2019)). The PFS/Subaru spectroscopic follow-up
survey is designed to observe two of the northernmost LSST DDFs, COSMOS and
XMM-LSS. About 20,000 host galaxy redshifts up to $z\leavevmode\nobreak\ \sim$
0.8 will be measured after ten years. The 4MOST/TiDES survey is dedicated to
the spectroscopic follow-up of extragalactic optical transients. About 50,000
host galaxy redshifts up to $z\leavevmode\nobreak\ \sim$ 1 will be collected
for 5 years. This corresponds both to the WFD and the DD fields. The fraction
of SNe Ia expected to have secure redshift measurements is taken from
Mandelbaum et al. (2019) (Figure 1, right plot).

Table 7: SMoM values (ascending order) after a 10-season survey. The standard deviation corresponding to SMoM distribution resulting from the fit of 50 random surveys is also quoted. Observing Strategy | SMoM
---|---
DDF_Univ_WZ | 155 $\pm$ 1
DDF_SCOC_p2 | 173 $\pm$ 2
DDF_Univ_SN | 179 $\pm$ 2
DDF_DESC_0.80_WZ | 187 $\pm$ 3
DDF_DESC_0.80_co | 190 $\pm$ 2
DDF_DESC_0.80_SN | 191 $\pm$ 3
DDF_DESC_0.75_WZ | 195 $\pm$ 2
DDF_DESC_0.70_WZ | 196 $\pm$ 1
DDF_DESC_0.70_co | 199 $\pm$ 3
DDF_DESC_0.75_co | 201 $\pm$ 2
DDF_DESC_0.75_SN | 202 $\pm$ 4
DDF_DESC_0.70_SN | 206 $\pm$ 3

SMoM values are displayed on Figure 4. As expected an increase of SMoM is
observed up to season 6 ($\sim$7,000 SNe Ia from the WFD survey are added each
season) when 4MOST/TiDES is expected to end its 5-year survey. After season 6,
only SNe Ia observed in DDFs are added to the SNe Ia sample used to estimate
SMoM. A saturation of SMoM values is observed in season 6 for all the
strategies having two UDFs. Final SMoM values (full survey) are given in Table
7.

Measuring cosmological parameters ($w_{0},w_{a}$) with high accuracy (i.e.
maximizing SMoM) requires to observe (necessary condition) a large sample of
well-measured SNe Ia at higher redshifts (Gris et al., 2023). Scenarios with
UDFs lead (by design) to samples with $\sim$1000, $\sim$1500, $\sim$1800 SNe
Ia with $z\geq 0.8$ for DDF_DESC_0.80*, DDF_DESC_0.75*, DDF_DESC_0.70*
observing strategies, respectively (Figure 12). Furthermore, a large fraction
of SNe Ia in these samples (65-80$\%$, 40-60$\%$, 20-40$\%$, respectively)
lead to accurate distance measurements ($\sigma_{\mu}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \sigma_{int}$).

We have estimated SMoM for surveys composed of spectroscopic and photometric
($\mathcal{O}$(10,000) per year for ten years) host galaxy redshifts. These
surveys lead to higher SMoM values (as expected) but also to biased fits.
Studying the origin of the biases requires additional work (beyond the scope
of this paper). At this stage the SNe Ia metric proposed in this paper, SMoM,
should only be used with surveys with spectroscopic host galaxy redshifts.

###  5.5 Combining metric results

A quick analysis of the metric results (Figs 2-4) tend to show that the
strategies leading to highest SMoM values do not all fulfill photo-$z$ and WL
requirements. It may be interested to study in more details these strategies
both to quantify to which extent the requirements are not met and propose
adjustments of the strategies designed in this paper.

Table 8: Mean $\Delta m_{5}^{b}$ values for field/band that do not fulfill photo-$z$ requirements for the DDF_DESC_*_co strategies. Field | band | $\Delta m_{5}^{b}$
---|---|---
COSMOS | u | -0.26
XMM-LSS | u | -0.16
| r | -0.05
CDF-S | u | -0.10
| y | -0.23
| z | -0.26

The list of strategies that do not fulfill photo-$z$ or WL requirements is
given on Figure 5 along with photo-$z$ and WL metric values. The fields EDFS-a
and EDFS-b are not included in this results (photo-$z$ and WL requirements are
difficult to meet for these fields which are observed at a depth twice lower
than the other fields). WL requirements are fulfilled by DDF_DESC_0.80_co and
DDF_DESC_0.70_co. For DDF_DESC_0.75_co requirements are not met for the
$r$-band and two fields (ELAIS-S1 and CDF-S). This could be solved by adding
one $r$-band visit per observing night. It would correspond to a (very modest)
increase of the DD budget of $\sim$0.05$\%$.

Figure 5: Photo-$z$ (top) and WL (bottom) metric values for
field/band/strategy that do not fulfill photo-$z$ or WL requirements. The
dotted lines correspond to $\Delta m_{5}^{b}$ = 0 (photo-$z$ requirement, top)
and $r^{WL}$ = 1 (WL requirement, bottom). Southernmost fields (EDFS-a and
EDFS-b) were not included.

Observations used for photo-$z$ calibration (reference catalogs listed in
subsubsection 2.2.1) overlap with COSMOS, XMM-LSS and CDF-S. Mean $\Delta
m_{5}^{b}$ values for these three fields are given in Table 8. A lower depth
with respect to reference observations would not be optimal for photo-$z$
calibration (a fraction of COSMOS2020 galaxy sample of 89$\%$ and 74$\%$ would
be used for $\Delta m_{5}^{b}$ -0.2 and -0.5 mag, respectively) but the final
impact has to be estimated using more realistic simulations of the strategies
proposed in this paper.

In summary, the most promising strategies are the ones optimal for
photo-$z$/WL requirements (DESC_DDF*_WZ). These surveys lead to highest SMoM
values whilst fulfilling photo-$z$/WL requirements. Observing strategies
corresponding to a compromise between the constraints from photo-$z$, WL and
SNe Ia (DESC_DDF*_co) lead to high SMoM, fulfill WL requirements, but are
close (i.e. showing the highest $\Delta m_{5}^{b}$ $<$ 0) to meet photo-$z$
constraints. Further studies are needed to quantify the impact of these
strategies on photo-$z$ calibration. Strategies optimal for SNe Ia do not
fulfill photo-$z$ and WL requirements. Fulfilling photo-$z$/WL constraints for
these strategies requires to increase the DDF budget to 8.5$\%$.

##  6 Conclusion

Cosmological measurements with LSST rely heavily on the DDFs for photometric
redshift training, weak gravitational lensing, and particularly to provide a
deep sample of high-redshift SNe Ia. In this paper we have proposed a method
to design a set of DESC cohesive DDF strategies fulfilling both constraints
from cosmological measurements (calibration requirements for photo-$z$ and WL,
precision cosmology for SNe Ia) and recommendations from the SCOC. We have
simulated these strategies and we have defined a set of metrics (for each
science case) to evaluate their performance.

Our conclusion is that it seems possible to design DDF surveys meeting the
specifications. The most promising results are obtained with deep rolling
surveys composed of deep fields (cadence of observation: $\sim$ 3 nights;
30-40 visits per observing night) and of ultradeep fields ( cadence of
observation: $\lesssim$ 2 nights; 80-100 visits per observing night for 3 to 4
seasons; same cadence as deep fields thereafter). These surveys (like
DESC_DDF_*_WZ) lead to the most accurate cosmological measurements for time
domain science whilst fulfilling requirements for static science for
extragalactic deep fields.

The method used in this paper to design such cohesive strategies is based on
the depth required to perform accurate calibration of primary systematic
uncertainties (photo-$z$, WL) and precise cosmological measurements (SNe Ia).
It could be used to design other large surveys.

All the strategies proposed in this paper correspond to a DDF budget of
$\sim$7$\%$, the maximal budget recommended by the SCOC. We have shown that it
was not possible in this budget to design cohesive strategies fulfilling all
requirements (photo-$z$/WL and SNe Ia). A DDF budget of $\sim$8.5$\%$ is
required to design observing strategies satisfying all the cosmological
requirements. This 1.5$\%$ DDF budget increase would lead to an increase of up
to 5$\%$ of SMoM values.

SNe Ia results have highlighted the necessity to observe a large sample of SNe
Ia in the WFD survey leading to accurate distance measurements ($\sigma_{\mu}$
$\lesssim\leavevmode\nobreak\ \mbox{$\sigma_{int}$}$). It was also shown that
the filter load strategy consisting of swapping $u$ and $y$ bands according to
the Moon phase leads to higher number of SNe Ia at higher redshifts. It should
be the (default) filter load strategy used in LSST.

These results were achieved with a relatively basic simulation as far as
observing parameters (skybrightness, clouds, dithering) are concerned. A
validation with realistic simulations using the LSST scheduler is required.
These simulations would help to estimate realistic metric values and may lead
to a tuning of the survey parameters of the cohesive DDF strategies proposed
in this paper.

### Acknowledgments

The DESC acknowledges ongoing support from the Institut National de Physique
Nucléaire et de Physique des Particules in France; the Science & Technology
Facilities Council in the United Kingdom; and the Department of Energy, the
National Science Foundation, and the LSST Corporation in the United States.
DESC uses resources of the IN2P3 Computing Center (CC-IN2P3–Lyon/Villeurbanne
- France) funded by the Centre National de la Recherche Scientifique; the
National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded
by UK BIS National E-infrastructure capital grants; and the UK particle
physics grid, supported by the GridPP Collaboration. H. Awan acknowledges
support from Leinweber Postdoctoral Research Fellowship and DOE grant DE-
SC009193.

This paper has undergone internal review in the LSST Dark Energy Science
Collaboration. The authors would like to thank René Hlozek, Eric Gawiser, and
Saurabh Jha for their helpful comments and reviews.

This research has made use of the following Python software packages: NumPy
(van der Walt et al., 2011), SciPy (Virtanen et al., 2020), Matplotlib (et al,
2007), Astropy (Price-Whelan et al., 2018), SNCosmo (Barbary et al., 2016),
Pandas (McKinney, 2011), sn_pipe (https://github.com/lsstdesc/sn_pipe).

Author contributions are listed below.
Ph. Gris: initiated, set up and coordinated the analysis as DESC Observing
Strategy/Survey Coordination Working Group co-convener; conceptualization,
methodology (sections 3-5) software, analysis (metrics), writing (original
draft; review & editing); formal analysis of the SNe Ia parts;
H. Awan: initiated, coordinated, and contributed to the paper as the DESC
Observing Strategy Working Group co-convener with Ph. Gris, building on years
of engagement as an active OSWG member;
M. Becker: developed the weak lensing DDF requirements and provided extensive
feedback on the text of the document.
H. Lin: developed photo-z DDF requirements and provided related text.

##  Appendix A Impact of the filter load strategy on SNe Ia metrics

To study the impact of the filter load strategy on SNe Ia metrics we generate
a set of strategies where the filter $u$ is exchanged with one of the other
filters ($rizy$) with varying Moon phases. For each of these strategies we
simulate and fit SNe Ia light curves to extract SNe Ia parameters used to
estimate the redshift completeness $z_{\mathrm{complete}}$ and the number of
well-sampled SNe Ia, NSN.

Observing strategies are generated with a regular cadence (two nights) and a
number of visits per observing night equal to 10/2/9/50/81/28 with a single
visit 5$\sigma$-depth equal to 23.48/24.26/23.87/23.46/22.76/22.02 for
$u/g/r/i/y/z$ bands, respectively. The lunar phase value to exchange the $u$
band ranges from 0% to 40%. We generate one season with a length of 180 days.

Figure 6: The number of well-sampled SNe Ia with $z\leavevmode\nobreak\
\geq\leavevmode\nobreak\ 0.8$, NSN, normalized to the configuration where the
Moon phase is equal to 0 (i.e. no swap), as a function of the Moon phase
threshold corresponding to the swap of the $u$ filter. Figure 7:
$z_{\mathrm{complete}}$ variation, normalized to the configuration where the
Moon phase is equal to 0 (i.e. no swap), as a function of the Moon phase
threshold corresponding to the swap of the $u$ filter.

We use these strategies to simulate and fit SNe Ia light curves using SALT3
with:

  * •

$T_{0}$ random in [$\mathrm{MJD_{min}}$,$\mathrm{MJD_{max}}$] where
$\mathrm{MJD_{min}}$ and $\mathrm{MJD_{max}}$ are the min and max MJD of the
season, respectively.

  * •

$z$ random in [0.01,1.1]

  * •

($x_{1}$,$c$)=(-2.0,0.2) to estimate $z_{\mathrm{complete}}$

  * •

($x_{1}$,$c$) randomly distributed using the G10 intrinsic scatter model
(Scolnic & Kessler (2016)) to estimate NSN.

The following tight selection criteria are applied:

  * •

only light-curve points with S/N $\geq$ 1 are considered;

  * •

at least 4 epochs before the maximum of luminosity and 10 epochs after the
maximum luminosity are required;

  * •

at least one point with a phase lower than -10 and one point with a phase
higher than 20 are required; and

  * •

$\sigma_{c}$$\leq$ 0.04 is required to ensure accurate distance measurement.

By definition, for this study, a well-sampled (or well-observed) SN Ia
fulfills these criteria. The redshift limit is defined as the maximum redshift
of supernovae passing these selection criteria. The redshift of a complete
sample, $z_{\mathrm{complete}}$, is estimated from the redshift limit
distribution, $z_{\mathrm{lim,faint}}^{\mathrm{SN}}$, of a simulated set of
intrinsically faint supernovae (i.e. with ($x_{1},\leavevmode\nobreak\ c$) =
(-2.0, 0.2)) with $T_{0}$ values spanning over the season duration of a set of
observations. $z_{\mathrm{complete}}$ is defined as the 95th percentile of the
$z_{\mathrm{lim,faint}}^{\mathrm{SN}}$ cumulative distribution.

The number of well-sampled SNe Ia , NSN, and the $z_{\mathrm{complete}}$
variation, both normalized to the configuration where the Moon phase is equal
to 0, as a function of the Moon phase are given in Figure 6 and Figure 7 for
the following filter load configuration: $u\leavevmode\nobreak\
\leftrightarrow\leavevmode\nobreak\ r$, $u\leavevmode\nobreak\
\leftrightarrow\leavevmode\nobreak\ i$, $u\leavevmode\nobreak\
\leftrightarrow\leavevmode\nobreak\ z$, $u\leavevmode\nobreak\
\leftrightarrow\leavevmode\nobreak\ y$. As expected the swap
$u\leavevmode\nobreak\ \leftrightarrow\leavevmode\nobreak\ r$ has a limited
effect since the SNe Ia flux is mainly collected with the $izy$ bands in the
considered redshift range. The lowest impact is observed for the
$u\leavevmode\nobreak\ \leftrightarrow\leavevmode\nobreak\ y$ configuration.

##  Appendix B DESC cohesive DDF strategies

A definition of the observing strategies proposed in this paper is provided in
Figure 8 and Figure 9. The number of visits (per observing night) as a
function of the season is given for each of the fields considered. The filter
allocation (i.e. the number of visits per band and per observing night) and
the cadence of observation (per season) are also mentioned. A summary of
exposure times is given in Table 9 (season 1) and Table 10 (seasons 2-10). The
coadded 5-$\sigma$ depth and the total number of visits per field/band/season
are given on Table 11.

Table 9: Exposure times for season 1. | nightly | season
---|---|---
| [sec] | [h]
Moon phase $>$ 20% | 480 | 28.3
Moon phase $\leq$ 20% | 930 | 24.3

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: Number of visits as a function of the season for COSMOS, XMM-LSS,
CDF-S, ELAIS-S1, EDFS-a, EDFS-b fields for several considered DESC cohesive
DDF strategies. Red points correspond to nights with a lunar phase lower than
20%. Blue crosses correspond to nights with a lunar phase higher than 20%. The
filter allocation (per observing night) and the cadence of observation (per
season) are specified for each season (magenta).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: Number of visits as a function of the season for COSMOS,XMM-LSS, CDF-S, ELAIS-S1, EDFS-a, EDFS-b fields for the remaining DESC cohesive DDF strategies. Red points correspond to nights with a lunar phase lower than 20%. Blue crosses correspond to nights with a lunar phase higher than 20%. The filter allocation (per observing night) and the cadence of observation (per season) are specified for each season (magenta). Table 10: Exposure times (in hours) for seasons 2-10. | nightly |
---|---|---
Strategy | UDF | DF | survey
DDF_DESC_0.80_WZ | 1.8 | 0.2 | 1195.2
DDF_DESC_0.80_SN | 2.5 | 0.1 | 1153.9
DDF_DESC_0.80_co | 2.2 | 0.2 | 1191.9
DDF_DESC_0.75_WZ | 1.3 | 0.2 | 1213.9
DDF_DESC_0.75_SN | 1.8 | 0.1 | 1150.7
DDF_DESC_0.75_co | 1.5 | 0.1 | 1152.2
DDF_DESC_0.70_WZ | 1.0 | 0.2 | 1215.5
DDF_DESC_0.70_SN | 1.4 | 0.1 | 1181.7
DDF_DESC_0.70_co | 1.2 | 0.1 | 1175.8
DDF_SCOC_p2 | 0.8 | 0.3 | 1212.0
DDF_Univ_SN | 0.4 | 0.4 | 1193.4
DDF_Univ_WZ | 0.4 | 0.4 | 1196.0
Table 11: Coadded 5-$\sigma$ depth and total number of visits $N_{v}$ per band. Strategy | Field | season | $m_{5}$ | $N_{v}$
---|---|---|---|---
|  |  | u/g/r/i/z/y | u/g/r/i/z/y
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.80_WZ |  | 2-10 | 27.7/28.3/28.6/28.9/28.1/26.8 | 2643/1582/4984/18151/16331/6158
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.9/28.2/28.1/27.9/27.2/26.5 | 2640/1098/2196/2745/2745/3072
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.80_SN |  | 2-10 | 27.4/28.2/28.5/29.0/28.2/26.8 | 1476/1155/4130/23849/21238/6204
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.6/27.8/27.8/27.6/26.7/26.1 | 1485/549/1098/1647/1098/1536
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.80_co |  | 2-10 | 27.6/28.3/28.5/29.0/28.2/26.8 | 1968/1582/4557/21000/18998/6181
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.7/28.2/28.0/27.8/27.0/26.3 | 1980/1098/1647/2196/2196/2304
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.75_WZ |  | 2-10 | 27.7/28.4/28.7/28.9/28.1/26.5 | 2651/1824/6378/18483/17664/3968
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.9/28.2/28.1/27.9/27.2/26.5 | 2640/1098/2196/2745/2745/3072
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.75_SN |  | 2-10 | 27.2/28.3/28.6/29.0/28.3/26.8 | 1094/1458/5280/24756/21480/6528
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.5/27.8/27.4/27.4/26.7/26.0 | 1155/549/549/1098/1098/1152
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.75_co |  | 2-10 | 27.5/28.4/28.6/28.9/28.2/26.6 | 1858/1824/5646/21027/19935/4928
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.7/28.2/27.8/27.6/26.9/26.2 | 1815/1098/1098/1647/1647/1920
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.70_WZ |  | 2-10 | 27.7/28.4/28.8/28.8/28.1/26.4 | 2644/2066/7772/17541/17541/3504
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.9/28.2/28.1/27.9/27.2/26.5 | 2640/1098/2196/2745/2745/3072
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.70_SN |  | 2-10 | 27.3/28.4/28.7/29.0/28.2/26.5 | 1177/1761/7162/24634/23178/3714
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.5/27.8/27.8/27.4/26.7/26.0 | 1155/549/1098/1098/1098/1152
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_DESC_0.70_co |  | 2-10 | 27.5/28.4/28.8/28.9/28.2/26.4 | 1865/2066/7467/20935/19843/3630
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 27.7/28.2/28.0/27.6/26.9/26.2 | 1815/1098/1647/1647/1647/1920
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_SCOC_p2 |  | 2-10 | 28.0/28.5/28.7/28.7/28.0/26.5 | 4729/2556/7110/13122/13029/4544
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 28.2/28.6/28.3/28.2/27.4/26.8 | 4785/2196/3294/4941/4392/5376
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_Univ_SN |  | 2-10 | 26.2/28.4/28.8/28.3/27.7/25.2 | 165/2196/9882/7686/8235/384
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 26.4/28.6/28.9/28.4/27.8/25.4 | 165/2196/9882/7686/8235/384
| UDF | 1 | 26.6/27.0/26.7/26.3/25.5/24.7 | 342/183/183/183/183/172
DDF_Univ_WZ |  | 2-10 | 28.1/28.5/28.3/28.1/27.4/26.7 | 5445/2745/3843/5490/4941/6144
| DF | 1 | 27.0/27.4/26.9/26.5/25.7/25.0 | 342/183/183/183/183/172
|  | 2-10 | 28.3/28.7/28.4/28.3/27.5/26.9 | 5445/2745/3843/5490/4941/6144

Figure 10: Cumulative DDF budget as a function of the season. A large fraction
of the budget is consumed in a few seasons for strategies with two UDFs (i.e.
DDF_DESC* scenarios).

##  Appendix C Simulation results

The proposed DDF strategies have been simulated using simplified observing
conditions (median 5-$\sigma$ depth per field/season) and metrics have been
processed using these strategies as input.

The cumulative DD budget as a function of the season is given on Figure 10. As
expected (by design) strategies with two UDFs use a large part of the budget
in a few seasons (more than 80$\%$ in 4 seasons for DDF_DESC_0.70* surveys).

Results of the PZ metrics are given on Table 12 for the first year of the
survey. PZ requirements are fulfilled for all the fields/band but EDFS-a
andEDFS-b (differences for $u$ and $z$ bands for COSMOS are due to a one night
difference in the simulations w.r.t. the requirements). Differences observed
for EDFS-a and EDFS-b are explained by the fact that these fields are not as
deep as other DDFs (SCOC phase 2 recommendations).

Results of the WL metrics are given on Table 13 for the first year of the
survey. WL requirements are met for all fields/bands.

Table 12: $\Delta m_{5}\leavevmode\nobreak\ =\leavevmode\nobreak\
m_{5}^{OS}-m_{5}^{PZ\leavevmode\nobreak\ req}$ for year 1. photo-$z$
requirements are fulfilled if $\Delta m_{5}\leavevmode\nobreak\
\geq\leavevmode\nobreak\ 0$.

.  season band $\Delta m_{5}\leavevmode\nobreak\ =\leavevmode\nobreak\
m_{5}^{OS}-m_{5}^{PZ\leavevmode\nobreak\ req}$ u -0.08/0.18 g 0.01/0.50
COSMOS/ r 0.49/0.85 XMM-LSS i 0.52/0.83 z -0.11/0.38 y 0.04/0.43 u 0.30/0.21 g
0.36/0.28 CDF-S/ r 0.67/0.66 ELAIS-S1 i 0.72/0.85 z 0.09/0.35 y 0.29/0.40 u
-0.10/-0.07 g -0.06/-0.08 EDFS-a/ r 0.23/0.30 EDFS-b i 0.19/0.36 z -0.30/-0.24
y -0.13/-0.16

Table 13: $\frac{N_{visits}^{OS}}{N_{visits}^{WL\leavevmode\nobreak\ req}}$
for year 1. ELAIS-S1, XMM-LSS, and CDF-S fields have similar results to
COSMOS, and EDFS-b to EDFS-a. WL requirements are fulfilled if
$\frac{N_{visits}^{OS}}{N_{visits}^{WL\leavevmode\nobreak\
req}}\leavevmode\nobreak\ \geq\leavevmode\nobreak\ 1$.

.  season band $\frac{N_{visits}^{OS}}{N_{visits}^{WL\leavevmode\nobreak\
req}}$ u 7.12/4.35 g 2.54/1.29 COSMOS/ r 0.99/0.51 EDFS-a i 0.99/0.51 z
1.20/0.61 y 1.08/0.50

We have performed a simulation of the WFD survey using the LSST simulation
baseline_v3.0_10yrs. We have simulated a sample of SNe Ia corresponding to 10
times the number of expected SNe Ia (we use the rate from Hounsell et al.
(2018)) in the full redshift range [0.01,0.7]. We use the G10 intrinsic
scatter model (Scolnic & Kessler (2016)) where ($x_{1}$,$c$) distributions are
described by asymmetric gaussian distributions with three parameters. We
choose random $T_{0}$ values spanning over the season duration of a group of
observations.

Figure 11: Number of SNe Ia after ten years for the WFD survey of
baseline_v3.0_10yrs.

SNe Ia fulfilling the selection criteria defined in Table 4 make-up the sample
of well-measured SNe Ia used for cosmology measurements. About $\sim$ 900,000
SNe Ia are expected to be observed after ten years (Figure 11) with a rate
between 70,000 and 120,000 SNe Ia per year. About 10-15$\%$ of the observed
SNe Ia lead to accurate distance measurements
($\sigma_{\mu}\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \sigma_{int}$).

##  Appendix D Metric results

The number of well-measured SNe Ia with $z\geq 0.8$ used to estimate SMoM
(Figure 4) as a function of the season is given on Figure 12 for the
strategies proposed in this paper. Scenarios with UDFs lead clearly to a
higher number of SNe Ia at higher redshifts. Highest numbers are reached with
DDF_DESC_0.70* strategies. This is expected, as UDFs, by design, provide a
large part (more than 80$\%$ for DDF_DESC_0.70_co) of the SNe Ia sample for
$z\geq 0.8$.

Figure 12: Number of well-sampled SNe Ia with $z\geq 0.8$.

SMoM values displayed in Figure 4 are estimated using a random distribution
($z\in[0.01,0.4]$) for the redshifts of the SNe Ia observed in the WFD sample.
This choice reflects the way TiDES will observe the sky ((Swann et al., 2019))
but it leads to samples with a low number of SNe Ia at low redshifts
($z\lesssim 0.1-0.2$). We have processed the SMoM metric on samples maximizing
the number of low$z$ SNe Ia of the WFD survey. An increase of SMoM is of
$\sim$12-13$\%$ is observed.

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