# Designing an Optimal LSST Deep Drilling Program for Cosmology with Type Ia
Supernovae

Philippe Gris  Laboratoire de Physique de Clermont, IN2P3/CNRS, F-63000
Clermont-Ferrand, France  Nicolas Regnault  Laboratoire de Physique Nucléaire
et des Hautes Energies, IN2P3/CNRS, France  Humna Awan  Leinweber Center for
Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor,
MI 48109, USA  Isobel Hook  Physics Department, Lancaster University,
Lancaster, United Kingdom LA1 4YB, United Kingdom  Saurabh W. Jha  Department
of Physics and Astronomy, Rutgers, the State University of New Jersey,
Piscataway, NJ 08854, USA  Michelle Lochner  Department of Physics and
Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South
Africa  South African Radio Astronomy Observatory (SARAO), The Park, Park
Road, Pinelands, Cape Town 7405, South Africa  Bruno Sanchez  Department of
Physics, Duke University, Durham, NC 27708, USA  Dan Scolnic  Department of
Physics, Duke University, Durham, NC 27708, USA  Mark Sullivan  School of
Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
Peter Yoachim  University of Washington, 4333 Brooklyn Ave NE, Seattle, WA
98105, USA  the LSST Dark Energy Science Collaboration

###### Abstract

The Vera C. Rubin Observatory’s Legacy Survey of Space and Time is forecast to
collect a large sample of Type Ia supernovae (SNe Ia) that could be
instrumental in unveiling the nature of Dark Energy. The feat, however,
requires measuring the two components of the Hubble diagram - distance modulus
and redshift - with a high degree of accuracy. Distance is estimated from SNe
Ia parameters extracted from light curve fits, where the average quality of
light curves is primarily driven by survey parameters such as the cadence and
the number of visits per band. An optimal observing strategy is thus critical
for measuring cosmological parameters with high accuracy. We present in this
paper a three-stage analysis aiming at quantifying the impact of the Deep
Drilling (DD) strategy parameters (number of fields, cadence, number of
seasons of observation, number of visits per band, time budget) on three
critical aspects of the survey: the redshift completeness (originating from
the Malmquist cosmological bias), the number of well-measured SNe Ia, and the
cosmological measurements. Analyzing the current LSST survey simulations in a
first stage, we demonstrate that the current DD survey plans are characterized
by a low completeness (limited to $z\leavevmode\nobreak\ \sim$ 0.55-0.65), and
irregular and low cadences (few days) that dramatically decrease the size of
the well-measured SNe Ia sample (by about 30$\%$). We propose in a second
stage a modus operandi that provides the number of visits (per band) required
to reach higher redshifts. The results of this approach are used to design a
set of optimized DD surveys for SNe Ia cosmology in a third stage. We show
that most accurate cosmological measurements are achieved with Deep Rolling
surveys characterized by a high cadence (one day), a rolling strategy (each
field observed at least two seasons), and two sets of fields: ultra-deep
($z\gtrsim 0.8$) and deep ($z\gtrsim 0.6$) fields. We also demonstrate that a
deterministic scheduler including a gap recovery mechanism is critical to
achieve a high quality DD survey required for SNe Ia cosmology.

##  1 Introduction

Type Ia supernovae (SNe Ia) are transient astronomical events resulting from a
powerful and luminous explosion of a white dwarf. They display a
characteristic brightness evolution, with a luminosity peak about 15 days
after explosion, and a slow decrease lasting up to few months. SNe Ia can be
used as standardisable candles to determine cosmological distances. The Hubble
diagram of SNe Ia is the most statistically efficient approach to constrain
the dark energy equation of state (Betoule et al. 2014; Scolnic et al. 2018a).

The Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST Ivezić
et al. 2019) will discover few millions of supernovae during ten years of
operations (Abell et al. 2009). This number is quite impressive but in a sense
misleading. If the survey is not optimized, a large fraction of these SNe Ia
will be useless for cosmological measurements because of large luminosity
distance errors. An optimized survey aims at observing a large sample (few
thousands spanning a broad range of redshifts to be limited by systematic
uncertainties) of well-measured SNe Ia with distance measurements accurate to
better than 2-3$\%$.

The ten-year LSST will image billions of objects in six bands. 80-90$\%$ of
the observing time will be dedicated to Wide Fast Deep (WFD) primary survey,
which will cover half of the sky ($\sim$ 18000 deg2) at a universal111Fields
are observed with a similar cadence and pattern. cadence222The cadence is
defined as the median inter-night gap in any filter.High cadences are
characterized by low inter-night gaps.. The remaining observing time will be
shared among other programs (mini-surveys) including intensive scanning of a
set of Deep Drilling (DD) fields. It is not clear yet what fraction of SNe Ia
observed in the WFD (DD) survey will be confirmed from spectral features. But
spectroscopically confirmed SNe Ia will certainly represent a small part of
the SNe Ia sample. Accurate SNe Ia parameters will thus be estimated from
well-measured light curves characterized by a sampling of few days and high
Signal-to-Noise Ratio per band (SNRb). Obtaining these high quality light
curves is therefore a key design point of the SN survey: the average quality
of the light curves depends primarily on the observing strategy.

In a recent paper (Lochner et al. 2021a), the LSST Dark Energy Science
Collaboration (DESC) has presented an analysis of the WFD survey of observing
strategies simulated by the LSST project333
https://community.lsst.org/t/community-survey-strategy-highlights.. The
conclusion is that an unprecedented number of high quality SNe Ia will be
observed in the WFD survey (between 120k and 170k) up to redshifts
$z\leavevmode\nobreak\ \sim$ 0.3. The DD mini-survey of LSST is critical for
observing a sample of high-redshift and well measured SNe Ia so as to achieve
Stage IV dark energy goals (Albrecht et al. 2006). Optimizing the LSST DD
survey so as to collect a large sample of well-measured SNe Ia up to high-
redshift fulfilling this requirement while taking into account survey
constraints (budget) is one of the main purpose of this paper. The work
presented is a further step of a process started few years ago (Scolnic et al.
(2018b), Lochner et al. (2021b)).

There are critical LSST survey parameters, such as scanning strategy, cadence,
or filter allocation, that are not defined yet. Ongoing efforts are being made
to define the requirements to accomplish the four primary science objectives
of Rubin Observatory: dark energy and dark matter, inventory of the Solar
System, transient optical sky exploration and mapping the Milky Way. As of
2020, the Survey Cadence Optimization Committee (SCOC Bianco et al. (2021))
was charged to make specific recommendations for the survey parameter choices
for LSST (initial survey strategy for ten years) based on input from the
science community. The studies presented in this paper are part of the global
effort aiming to define optimal strategy parameters to accomplish the
scientific objectives of the Rubin Observatory.

This paper deals with the interplay between the DD strategy and the SNe Ia
sample collected by the survey. We perform a detailed study of the impact of
the strategy parameters (number of fields to observe, number of seasons,
season lengths, number of visits per night and per field) on the SNe Ia sample
quality to assess whether observing supernovae up to $z\leavevmode\nobreak\
\simeq$ 0.8-0.9 is achievable given design constraints, including in
particular the number of visits alloted to DDFs. This article is subdivided
into eight sections. The requirements for supernovae and the design
constraints of the DD program are presented in Section 2 and Section 3. The
metrics used to assess observing strategies are defined in Section 4 and used
in a detailed analysis of LSST simulations in Section 5. One of the
conclusions of this analysis is that the samples collected with the proposed
strategies could be too shallow and we propose in Section 6 a method aiming at
increasing the depth of the DD survey. We use the results of this method to
design optimized DD scenarios that would achieve the goal of observing high
quality SNe Ia up to higher redshifts (Section 7 to Section 9).

##  2 Requirements for supernovae

SNe Ia have been demonstrated as precise and reliable distance indicators in
the last few decades. Distances are derived from SNe Ia parameters that are
infered from photometric light curves. The accuracy of the distance estimation
reflects the precision of the photometric measurements. We discuss in the
following (subsection 2.1) the light curve quality criteria that are required
to obtain accurate distance measurements.

Supernovae surveys are magnitude limited and gather samples affected by a
selection effect called the Malmquist bias (Malmquist 1922, Teerikorpi 2015,
and references therein): brighter SNe Ia are preferentially discovered at the
faint limits of the survey. This redshift-varying bias has an impact on the
measurement of the cosmological parameters and is to be taken into account in
the design of the survey (subsection 2.2).

###  2.1 Distance measurement and well-measured supernovae

The diversity of SNe Ia light curves is usually parametrized by three
parameters: an amplitude (brightness), a color and a light curve width
(shape). The Tripp estimator (Tripp & Branch 1999 and references therein) uses
the B-band absolute magnitude, the (B-V) color, and the rate of decline during
the first 15 days after maximum, $\Delta m_{15}$, to standardize the SNe Ia
brightness and estimate the distance. In the SALT2 light curve model (Guy et
al. 2007, 2010), the distance modulus, $\mu$, is defined for each SN Ia by:

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

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 a restframe color equal to $B$-$V$ at peak brightness. 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 $B$ rest-frame magnitude. $\alpha$ and
$\beta$ are global nuisance parameters quantifying the correlation between
brightness with $x_{1}$ and $c$, respectively. The three parameters
$\alpha,\beta,M$ are fitted along with cosmological parameters by minimizing
the distance scatter. Accurate luminosity distances (i.e. accurate estimation
of the SNe Ia standardisation parameters $m_{B},x_{1},c$) are thus critical to
constrain cosmological parameters with better precision.

Figure 1: Contributions to the uncertainty of the distance modulus
$\sigma_{\mu}$ as a function of the redshift. Three components are
represented: the color component ($\beta\sigma_{c}$), the stretch component
($\alpha\sigma_{x_{1}}$) and the amplitude component ($\sigma_{m_{b}}$). The
color component starts to contribute significantly to the distance uncertainty
above $\beta\sigma_{c}\leavevmode\nobreak\ \sim$ 0.12 (or equivalently
$\sigma_{c}\leavevmode\nobreak\ \sim$ 0.04). This threshold corresponds to a
redshift limit value (here 0.71) for the observation of well-measured SNe Ia.
These results were obtained from full simulation of SNe Ia light curves
(regular cadence of one day). SNe Ia parameters were estimated from a SALT2
fit (see Section 4 for more details).

The relative contribution of the SN Ia parameter errors to the uncertainty on
the distance modulus $\sigma_{\mu}$ is driven by the values of the nuisance
parameters $\alpha$ and $\beta$. Recent measurements (Scolnic et al. 2018a,
Abbott et al. 2019) confirm that $\beta$ is larger than 3, that $\alpha$ is
around 0.16 and that measured values of $x_{1}$ and $c$ lie in limited ranges,
$[-3.0,3.0]$ and $[-0.3,0.3]$, respectively. The consequence is that the color
term $\beta\sigma_{c}$ is dominant in the $\sigma_{\mu}$ budget as illustrated
in Fig. 1. The dispersion of Hubble residuals due to the intrinsic scatter of
standardized SNe Ia brightness (Brout & Scolnic 2021) is of 0.12-0.14 mag
(Betoule et al. 2014, Scolnic et al. 2018a). The measurement uncertainties on
the color above $\sim$ 0.04 will thus make a significant contribution to the
distance modulus errors. The requirement $\sigma_{C}\leavevmode\nobreak\
\lesssim$ 0.04 is one of the main criteria (see Section 4) that designates a
well-measured SN Ia. It implicitly defines a redshift limit $z_{\mathrm{lim}}$
(Fig. 1) above which SNe Ia light curve measurements lead to inaccurate
distance estimation and:

$\displaystyle\mbox{$\sigma_{c}$}\leq 0.04$ $\displaystyle\Longrightarrow$
$\displaystyle\mbox{$z_{\mathrm{lim}}$}$ (2)

Figure 2: Color uncertainty as a function of the Signal-to-Noise Ratio per
band. Requiring $\sigma_{c}$$\leq$ 0.04 is equivalent to applying the
following selections: $SNR^{i}\leavevmode\nobreak\ \geq$ 59 and
$SNR^{z}\leavevmode\nobreak\ \geq$ 39 and $SNR^{y}\leavevmode\nobreak\ \geq$
20\. These results were obtained from full simulation of SNe Ia light curves
for a medium SN Ia (regular cadence of one day). SNe Ia parameters were
estimated from a SALT2 fit (see Section 4 for more details)

The uncertainty on $m_{b}$, $x_{1}$, and $c$ is driven by the quality of the
collected light curves which is defined by the sampling frequency of the
measurements (cadence of observation) and by the light curve points
uncertainties (observing conditions). $\sigma_{c}$ estimation is a function of
the Signal-to-Noise Ratio (SNR) per band b, SNRb, defined by:

SNRb
$\displaystyle=\sqrt{\sum_{i=1}^{n^{b}}{\left(\frac{f_{i}^{b}}{\sigma_{i}^{b}}\right)^{2}}}$
(3)

where $f^{b}$, and $\sigma^{b}$ are the fluxes and flux uncertainties. The
summation runs over the number of light curve points. Requiring $\sigma_{c}$
$\leq$ 0.04 is equivalent to requiring a minimal SNR per band (Fig. 2) and the
link between $z_{\mathrm{lim}}$ and SNRb may be written:

$\displaystyle\land(\mbox{SNR${}^{b}$}\geq\mbox{SNR${}^{b}_{\mathrm{min}}$})\Longrightarrow\mbox{$\sigma_{c}$}\leq
0.04$ $\displaystyle\Longrightarrow\mbox{$z_{\mathrm{lim}}$}$ (4)

where the logical symbol $\land$ means that the requirement
$\mbox{SNR${}^{b}$}\geq\mbox{SNR${}^{b}_{\mathrm{min}}$}$ is to be fulfilled
for all the considered bands.

###  2.2 Redshift completeness

As with all flux-limited surveys, a larger fraction of bright SNe Ia of the DD
survey will systematically be observed at high-redshift. SNe Ia observed at
the fainter ends of the luminosity distribution are characterized by a mean
intrinsic peak brightness higher than the mean of the whole sample. This bias
increases with redshift and affects the distance estimation: the effective
luminosity is biased towards brighter values. This leads to shorter distance
measurements.

It is possible to estimate distance biases using simulation of the unobserved
events (Kessler et al. 2013, Scolnic & Kessler 2016). Recent cosmological
analysis (Scolnic et al. 2018a, Riess et al. 2019) use the BEAM with Bias
Corrections (BBC) framework (Kessler & Scolnic 2017) which includes
corrections dependent on $\alpha$ and $\beta$ in the SNe Ia cosmology
likelihood. With this method, the distance bias corrections have a clear
impact on the cosmological measurements (the shift of the dark energy
parameter $w$ decreases from 7$\%$ to 1$\%$) but the systematic uncertainty
related to the selection bias still accounts for more than 20$\%$ of the total
systematic error budget (Scolnic et al. 2018a). An incomplete high redshift
sample is thus affected by a systematic uncertainty (due to selection bias)
that could be dominant in high-redshift, magnitude-limited surveys like the
LSST DD survey. More importantly, the Malmquist bias leads to a decrease of
the number of SNe Ia for $z\leavevmode\nobreak\ \geq\leavevmode\nobreak\
\mbox{$z_{\mathrm{complete}}$}$: the fraction of higher redshift SNe Ia
decreases with $z_{\mathrm{complete}}$. The redshift completeness value has
thus an impact on SNe Ia cosmology since accurate cosmological measurements
with a Hubble diagram heavily rely on the distribution of the number of well-
measured SNe Ia as a function of the redshift, N($z$). We will quantify this
impact in Section 7.

There are two ways to optimize the cosmological constraints from SNe Ia in a
budget-limited survey. In a first scenario, the total number of well-sampled
SNe Ia can be maximized by observing all the DD fields for ten years. The
second approach consists of maximizing the redshift completeness by observing
the DD fields a limited number of years. We will study these two types of
scenarios, (high NSN, low $z_{\mathrm{complete}}$) and (low NSN, high
$z_{\mathrm{complete}}$), in the following (Section 7).

##  3 Observing strategy constraints

The design parameters are the number of fields to be observed, the number of
seasons of observation and the season length, the cadence of observation, the
filter allocation, and the total observing time budget.

The Rubin Observatory defined in 2012444http://ls.st/bki. four extragalactic
Deep Drilling Fields: COSMOS, ELAIS-S1, XMM-LSS, CDF-S (Tab. 1). More
recently, the DESC collaboration has supported the LSST DDF coverage of the
southern deep fields area (Lochner et al. 2021b) to ensure contemporaneous
observations with Euclid (Laureijs et al. 2011; Amendola et al. 2013) and
Roman Space Telescope (Spergel et al. 2015), at the begining and at mid-term
of the LSST survey, respectively.

Table 1: Location of the DD fields considered in this study. AKARI Deep Fields (ADF-A and ADF-B) are examples of southern fields in the Euclid/Roman area simulated in LSST observing strategy. Field | Central RA | Central Dec
---|---|---
Name | (J2000) | (J2000)
ELAIS-S1 | 00:37:48 | -44:01:30
XMM-LSS | 02:22:18 | -04:49:00
CDF-S | 03:31:55 | -28:07:00
COSMOS | 10:00:26 | +02:14:01
ADF-A | 04:51:00 | -52:55:00
ADF-B | 04:35:00 | -54:40:00

The number of observed supernovae is proportional to the number of seasons of
observation and to the season duration (Perrett et al. 2012). The season
length of a field is equal to the period of observability555An astronomical
target is said to be observable if it is visible (i.e. for Rubin Observatory
with altitude 20° $\leq$ alt $\leq$ 86.5°and airmass $\leq$ 1.5) for a minimal
amount of time. which depends on its location w.r.t. the Vera C. Rubin
Observatory. It is driven by the nightly observable time that can be converted
to a number of visits of 30 s per observing night ($\mathrm{N_{visits}}$). The
estimation of the season length as a function of the total number of visits
for the fields defined in Tab. 1 (Fig. 3) suggests a decrease from 275-200 to
150-100 days when $\mathrm{N_{visits}}$ increases from 1 to 400. Season
lengths of at least six months are required to collect at least 80$\%$ of SNe
Ia of the northernmost fields. Maximizing season length is particularly
important in the DDFs because of time dilation. High-$z$ SNe Ia light curves
last longer than low-$z$ ones. SNe Ia collected at the beginning and at the
end of the season are characterized by poorly reconstructed light curves
leading to inaccurate distance measurements. Time dilation effects may be
quantified as an effective season length decreasing with $z$.

Figure 3: Maximal season length as a function of the nightly observable time
(in hour) (lower $x$-axis) or the number of visits of 30$s$ (upper $x$-axis)
per observing night. Fields are observable if the following requirements are
met: 20°$\leq$ altitude $\leq$ 86.5°, airmass $\leq$ 1.5. This plot was made
using scripts and tools of the LSST scheduler. The black line corresponds to a
season length of 180 days.

A regular cadence of observation ($\sim$ 3 days max) is required to collect
well-sampled light curves (LC). The number of large gaps ($>$ 10 days) between
visits degrades the measurements of luminosity distances, and potentially
result in rejecting large sets of light curves of poor quality.

Measuring cosmological parameters with high accuracy requires observing SNe Ia
over a wide redshift range $z\in[0.01,1.1]$. Five filters of the VRO have thus
to be used: $g,r,i,z,y$ with a number of visits per band and per night
depending on the redshift completeness of the survey (Section 6).

It is expected that 5-15$\%$ of the total number of LSST visits will be
alloted to the DD program and shared among science topics interested by DD
observations (such as AGN, supernovae, photo-$z$ training, …). The DD budget
is defined as the fraction of observing time alloted to DDFs during the
survey. For the sake of simplicity we will assume that the exposure time of
observation does not change during the survey. In that case the budget is
defined by:

$\displaystyle\mathrm{DD_{budget}=N_{visits}^{DD}/(N_{visits}^{DD}+N_{visits}^{non-
DD})}$ (5)

where $N_{visits}^{DD}$ is the total number of visits (for the 10 years of
Rubin Observatory operation) allocated to DDFs and is defined by:

$\displaystyle\mathrm{N_{visits}^{DD}}=\sum_{i=1}^{\mathrm{N_{fields}}}\sum_{j=1}^{\mathrm{N_{season}^{i}}}\mathrm{N_{visits,night}^{ij}\times
seaslen^{ij}/cad^{ij}}$ (6)

where $\mathrm{N_{fields}}$ is the number of DD fields, $\mathrm{N_{season}}$
the number of season of observations per field, seaslen the season length (in
days), cad the cadence of observation, and $\mathrm{N_{visits,night}^{ij}}$
the total number of visits per observing night, per field, and per season. The
total number of visits corresponding to all fields but the DDFs,
$N_{visits}^{non-DD}$, was estimated from a sample of LSST simulations and set
to 2122176 visits (10 years of survey). The budget is fairly strongly
dependent on the number of visits per observing night and on the season length
(Fig. 4): the total number of visits is multiplied by 5 if the budget
increases from 3% to 15%.

Figure 4: Budget contours in the plane ($\mathrm{N_{visits}}$, season length)
for a configuration of 5 fields with two (black) or ten (red) observing
seasons per field and a cadence of one day. The number of visits dramatically
decreases with the number of seasons of observation when the time budget is
limited. For a typical season length of 180 days and a budget extending from
3$\%$ (minimal) to 15$\%$ (maximal), the number of visits ranges from 7 to 42
for 10 seasons (dashed red lines). Decreasing the number of seasons to 2 lead
to an increase of the possible number of visits, from 36 to 208 (solid black
lines).

##  4 Metrics to assess observing strategies

The metrics used to assess observing strategies are estimated from full
simulation and fit of light curves. We have used
SNCosmo666https://sncosmo.readthedocs.io/en/latest/index.html(Barbary et al.
2016), a Python library synthesising supernova spectra and photometry from SN
models. It includes a lot of supernova models (SALT2, MLCS2k2, Hsiao, Nugent,
PSNID, SNANA and Whalen models), as well as a variety of built-in bandpasses
and magnitude systems. It includes functions for fitting and sampling SN model
parameters given photometric light curve data. We have used the SALT2 model
(Guy et al. 2007, 2010) where a SNe Ia is described by five parameters:
$x_{0}$, $x_{1}$, $c$, $z$, and $T_{0}$, the time of maximum luminosity. A
flat-$\Lambda$CDM model was used to estimate cosmological distances, with
$H_{0}$ = 70 km s-1, $\Omega_{m}$ = 0.3 and $\Omega_{\Lambda}$ = 0.7.

In SALT2, model uncertainties of $g$ and $r$ bands (rest-frame UV) light
curves fluxes are large (Guy et al. 2007), and $g$ and $r$ observations with
relative error model larger than 5$\%$ have not been considered in this study.
This requirement implies that the list of filters useful to measure
photometric light curves (observer-frame) is redshift-dependent (Tab. 2).

Table 2: List of filters useful to measure photometric light curves (observer-frame) as a function of the redshift. $z$ | [0.01,0.1] | [0.1,0.35] | [0.35,0.65] | [0.65,1.1]
---|---|---|---|---
bands |  $g$$r$$i$ |  $g$$r$$i$$z$ |  $r$$i$$z$$y$ |  $i$$z$$y$

Following the requirements for supernovae (Section 2), we rely on two metrics
to assess observing strategies: the redshift completeness
$z_{\mathrm{complete}}$, and the number of well-measured SNe Ia,
N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$. A well-measured SNe Ia is
defined by the following tight selection criteria:

  * •

only light curve points with SNR $\geq$ 1 are considered;

  * •

at least four (ten) epochs before (after) maximum luminosity are required, as
well as at least one point with a phase777The phase of a LC point at time t is
equal to $\frac{t-T_{0}}{1+z}$. lower (higher) than -10 (20);

  * •

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

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 group of observations. $z_{\mathrm{complete}}$
is defined as the 95th percentile of the
$z_{\mathrm{lim,faint}}^{\mathrm{SN}}$ cumulative distribution.

The metrics are measured in HEALPix (Gorski et al. (2005)) pixels of size 0.21
deg2 over the region of the DDFs. For each pixel in the sky light curves are
generated from observations of the simulated survey. Flux errors are estimated
from the 5-$\sigma$ point source limiting magnitude (5-$\sigma$ depth). Light
curves are fitted (using the SALT2 model implemented in SNCosmo) to estimate
SNe Ia parameters.

##  5 Analysis of LSST simulations

LSST project has periodically released sets of simulations containing a large
number of survey strategies. The simulations analyzed in this section were
performed with the Feature-Based Scheduler (FBS Naghib et al. 2019) based on a
modified Markov Decision Process that maximizes the scientific outcome of the
Vera C. Rubin Observatory during its ten-year survey. It allows a flexible
approach for scheduling. The sequential decisions of which filter and which
pointing to select are estimated from features (weather conditions/image
depth, slew time, footprint) to optimize observing strategy. The output of the
simulations is composed by a set of observing parameters888The list of
parameters is available at
https://github.com/lsst/sims_featureScheduler. estimated at the center of the
field of view of the telescope. These parameters serve as input for the
generation of SNe Ia light curves (Section 4).

The diversity of DD scenarios proposed in LSST simulations is rather limited
and we have chosen to analyze a representative set of DD surveys on the basis
of the following criteria: number of visits (and filter allocation) per
observing night, cadence of observation, dithering, and budget. The list of
LSST simulated observing strategies considered in this study is given in Tab.
3.

Table 3: Survey parameters for the list of observing strategies analyzed in
this paper. For the cadence and season length, the numbers correspond to
ADFS1/ADFS2/CDFS/COSMOS/ELAIS/XMM-LSS fields, respectivelly. The numbers
following the filter allocation ($\mathrm{N_{visits}}$ column) are the minimum
and maximum mean fraction of visits (per field over seasons) in the filter
distribution. Only filter combinations with a contribution higher than 0.01
have been considered.

Observing | cadence | $\mathrm{N_{visits}}$ | season length | area | DD budget | family
---|---|---|---|---|---|---
Strategy | [days] | u/g/r/i/z/y | [days] | [deg2] | (%) |
agnddf_v1.5_10yrs | 2.0/2.0/2.0/2.0/2.0/2.0 | -/1/1/3/5/4 [0.99-1.] | 164/165/235/189/171/177 | 112.9 | 3.4 | agn
baseline_v1.5_10yrs | 4.5/4.5/10.0/4.0/4.5/5.0 | -/10/20/20/26/20 [0.28-0.43] | 131/131/200/164/150/152 | 109.7 | 4.6 | baseline
|  | 8/10/20/20/-/20 [0.56-0.71] |  |  |  |
daily_ddf_v1.5_10yrs | 2.0/2.0/2.0/2.0/2.0/2.0 | -/1/1/2/2/2 [0.60-0.61] | 161/161/236/188/171/178 | 113.5 | 5.5 | daily
|  | 1/1/1/2/-/2 [0.38-0.39] |  |  |  |
ddf_heavy_v1.6_10yrs | 2.0/2.0/2.0/2.0/2.0/2.0 | -/10/20/20/26/20 [0.26-0.39] | 116/116/201/167/152/150 | 110.6 | 13.4 | baseline
|  | 8/10/20/20/-/20 [0.60-0.72] |  |  |  |
|  | -/2/4/8/-/- [0.37-0.5] |  |  |  |
descddf_v1.5_10yrs | 2.0/2.0/3.0/2.0/2.0/2.5 | -/-/-/-/25/4 [0.30-0.38] | 147/146/228/178/165/171 | 112.5 | 4.6 | desc
|  | -/-/-/-/-/4 [0.19-0.25] |  |  |  |
dm_heavy_v1.6_10yrs | 7.5/6.0/14.0/8.5/8.0/7.0 | -/10/20/20/26/20 [0.31-0.45] | 119/119/195/142/139/138 | 188.6 | 4.6 | baseline
|  | 8/10/20/20/-/20 [0.54-0.68] |  |  |  |
ddf_dither0.00_v1.7_10yrs | 4.0/4.0/6.0/2.0/3.0/3.0 | -/10/20/20/26/20 [0.17-0.43] | 121/123/204/165/153/159 | 69.2 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.56-0.81] | 121/123/204/165/153/159 | 69.2 | 4.6 |
ddf_dither0.05_v1.7_10yrs | 4.0/4.0/6.0/2.0/3.0/3.0 | -/10/20/20/26/20 [0.16-0.42] | 116/116/218/168/153/161 | 71.8 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.57-0.83] |  |  |  |
ddf_dither0.10_v1.7_10yrs | 4.0/4.0/6.0/2.0/3.0/3.0 | -/10/20/20/26/20[0.19-0.43] | 120/120/220/165/150/165 | 74.7 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.57-0.81] |  |  |  |
ddf_dither0.30_v1.7_10yrs | 4.0/4.0/6.5/3.0/3.0/3.0 | -/10/20/20/26/20 [0.21-0.45] | 118/118/201/167/146/146 | 83.5 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.54-0.78] |  |  |  |
ddf_dither0.70_v1.7_10yrs | 4.5/4.5/9.0/4.0/4.0/4.25 | -/10/20/20/26/20 [0.19-0.43] | 123/137/201/163/146/146 | 104.5 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.57-0.79] |  |  |  |
ddf_dither1.00_v1.7_10yrs | 4.0/4.0/14.0/5.5/5.0/5.0 | -/10/20/20/26/20 [0.23-0.43] | 113/118/198/153/143/143 | 124.4 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.56-0.77] |  |  |  |
ddf_dither1.50_v1.7_10yrs | 4.5/4.5/16.5/8.5/6.75/6.0 | -/10/20/20/26/20 [0.20-0.42] | 121/121/196/145/135/139 | 159.3 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.57-0.79] |  |  |  |
ddf_dither2.00_v1.7_10yrs | 4.0/4.0/19.0/12.0/9.5/9.0 | -/10/20/20/26/20 [27-44] | 112/111/193/137/118/133 | 199.3 | 4.6 | baseline
|  | 16/10/20/20/-/20 [0.57-0.79] |  |  |  |

Four sets of observing strategies can be defined from Tab. 3 according to the
filter allocation per night, the parameter that has the most significant
impact on $z_{\mathrm{complete}}$ value : baseline-like (11 observing
strategies), agn, daily, and desc family. Estimation of the pair metric
(N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$, $z_{\mathrm{complete}}$)
(defined in Section 4) for these families shows (Fig. 5) that higher redshift
limits are reached for the baseline-like family. Most (10/11) of these
observing strategies reach $z_{\mathrm{complete}}$ $\sim$ 0.65. ddf_heavy, the
strategy with the largest DD budget, reaches $z_{\mathrm{complete}}$ $\sim$
0.72 and also collects the larger number of well-sampled SNe Ia. daily and
desc are characterized by a lower depth but by a significant number of well-
measured SNe Ia.

Figure 5: N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$ vs
$z_{\mathrm{complete}}$ for the LSST simulated observing strategies considered
in this paper.

The metric output (N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$,
$z_{\mathrm{complete}}$) is driven by the probability of a SNe Ia light curve
to fulfill the requirements defined in Section 4 (see Appendix A for more
details). This observing efficiency depends on the signal-to-noise ratio per
band which is defined by the sampling frequency of the light curve points. The
number of well-sampled SNe Ia is thus strongly dependent on the cadence of
observation, as illustrated in Fig. 6 (top): as expected, higher cadences lead
to higher (N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$,
$z_{\mathrm{complete}}$).

Figure 6: Median number of well-measured supernovae N${}_{\mathrm{SN}}^{z\leq
z_{\mathrm{complete}}}$ as a function of the cadence of observation (top) and
median cadence as a function of maximal inter-night gap (bottom) for a set of
LSST simulated strategies studied in this paper. Yellow areas correspond to
observing strategy parameters (cadence and max inter-nigh gaps) leading to a
high quality SNe Ia sample.

Observing strategies studied in this paper are characterised by a wide range
of cadences (Fig. 6, top). Only two surveys, ddf_heavy and daily, have more
than 50% of their observations with a 1-day cadence (Tab. 4). baseline and
dither_00, two of the strategies with the lowest N${}_{\mathrm{SN}}^{z\leq
z_{\mathrm{complete}}}$, present a large fraction of observations with a
cadence of at least 3 days. These cadence distributions can be explained by
some period with no observations. Two sources of gaps can be identified. One
is the telescope downtime due to clouds and/or telescope maintenance. The
other is the scanning strategy, when choices have to be made on which fields
are to be observed on a given night. Inter-night gaps arising from telescope
downtime lead to about 16-20$\%$ of nights without observation per season and
are not expected to exceed few days (except for longer maintenance periods
that could last up to 16 days see Section 9). But the cadence may
significantly increase for gaps higher than $\sim$ 10 days. Large gaps of few
tens of days lead to a dramatic decrease of the cadence of observation, as
illustrated in Fig. 6 (bottom).

Table 4: Cadence distribution for a set of strategies studied in this paper. The cadence is estimated from nightly visits corresponding to all the filters $grizy$ for all the strategies but descddf which is characterized by observations related to $gri$ and $zy$ filters nightly interleaved (i.e. $gri$ visits one night and $zy$ visits the night after). Strategy cadence | 1-d | 2-d | 3-d | 4-d |  $\geq$ 5-d
---|---|---|---|---|---
agnddf | 37.6% | 56.5% | 5.0% | 0.7% | 0.1%
baseline | 0.0% | 28.8% | 44.7% | 11.9% | 14.6%
daily | 56.4% | 21.2% | 11.2% | 5.6% | 5.6%
dither0.00 | 0.0% | 36.1% | 43.7% | 14.4% | 5.8%
ddf_heavy | 62.4% | 22.8% | 10.5% | 3.2% | 1.1%
descddf | 9.9% | 58.9% | 16.4% | 7.0% | 7.8%
dm_heavy | 0.0% | 31.0% | 39.3% | 18.4% | 11.2%

The Rubin Observatory will provide a combination of large-scale dithers at
each point of observation. Dithering patterns are composed of translational
and rotational dithers (Awan et al., 2016). The former corresponds to offsets
of the telescope pointings, the latter to offsets of the camera rotational
angles. We have studied the impact of the translational dithering on the
metrics. It is expected to affect both the number of well-sampled supernovae
and the redshift completeness for each of the pixels of the field. With no
dithering, N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$ and
$z_{\mathrm{complete}}$ distributions are uniform across the whole field area.
The translational dithering has an impact on edge pixels which are thus
characterized by a lower cadence w.r.t. central pixels. A decrease of both
N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$ and $z_{\mathrm{complete}}$
(per pixel) is then observed for edge pixels. $z_{\mathrm{complete}}$ tends to
decrease with an increase of the translational dither offset (tdo), with a
greater effect for high cadences. The number of supernovae is the result of
the trade-off between two effects (Fig. 7): an increase of the survey area
(which increases with tdo) and a decrease of the cadence ( which decreases
with tdo). The increase of the survey area lead to an increase of the number
of supernovae for high cadences and low tdo values.

Figure 7: Ratio of the number of supernovae $N_{SN}/N_{SN}^{nodither}$ (top)
and $z_{\mathrm{complete}}$ difference (bottom) as a function of translational
dither offset. The simulations labelled as ‘Fakes’ (dotted lines) correspond
to regular cadences (1, 2, 3, 5 days) with median observing conditions
(5-$\sigma$ depth single exposure: 24.13/23.84/23.45/22.74/22.10 for
$g/r/i/y/z$ bands, respectively.)

In summary, the LSST simulated strategies lead to the observation of a sample
of well-measured SNe Ia with a rather low $z_{\mathrm{complete}}$, the
redshift limit of the complete cosmology-grade SNe Ia sample. It will be shown
in the following (Section 7) that reaching $z_{\mathrm{complete}}$ $\sim$ 0.6
is not sufficient to achieve a measurement of $w$ with a high degree of
precision. $z_{\mathrm{complete}}$ is mainly driven by SNRb values (Eq. 3)
which depends on the number of visits in the corresponding band
N${}_{\mathrm{visits}}^{b}$ (Section 6). The number of visits per band has
thus to be increased to reach higher $z_{\mathrm{complete}}$. We propose in
Section 6 a method for assessing the relationship between
N${}_{\mathrm{visits}}^{b}$ and $z_{\mathrm{complete}}$. The second conclusion
of these studies is that the cadence is a key parameter to collect a large
sample of well-measured SNe Ia. High cadences are favored to maximize SNRb.
Large inter-night gaps are harmful as they lead to a decrease of the number of
well-measured supernovae. It is critical to reduce the inter-night gaps
originating from the survey strategy so as to maximize the size and depth of
the well-measured SNe Ia sample. Finally, larger translational dithers reduce
the DDF area with high cadence and lead to a dramatic decrease of the number
of well-sampled SNe Ia for low cadence ($\gtrsim$ 3 days) strategies.

##  6 Optimization of the number of visits

The analysis of LSST simulations has shown (see Section 5) that it seems
difficult to collect complete samples of SNe Ia with redshift higher than
$z_{\mathrm{complete}}$ $\sim$ 0.55-0.65. The proposed cadences of
observation, filter allocation and season lengths do not allow to reach higher
redshifts for a DD budget of $\sim$ 5$\%$. According to Eq. 4, reaching higher
$z_{\mathrm{complete}}$ requires increasing SNRb.

The signal-to-noise ratio per band is the complex result of the combination of
the SN Ia flux distribution ($z$-dependent), the number of visits per band,
the cadence of observation, and observing conditions (5-$\sigma$ depth). It is
thus not possible to estimate the observing strategy parameters required to
reach higher redshifts from the results of Section 5 (by a simple rescaling
for instance). This is why we present in this section a study to assess the
relationship between the redshift completeness and the number of visits per
band and per observing night (for a defined cadence). The optimized number of
visits per band required to reach higher redshifts estimated with this
approach is a key parameter to build DD scenarios consistent with the list of
constraints presented in Section 2 and in Section 3.

As described in Eq. 5 the DD budget depends primarily on 5 parameters: the
number of fields to observe, the season length (per field and per season), the
number of seasons of observation (per field), the cadence of observation (per
field and per season), and the number of visits N${}_{\mathrm{visits}}^{b}$
per filter and per observing night. N${}_{\mathrm{visits}}^{b}$ is related to
SNRb through the flux measurement uncertainties $\sigma_{i}^{b}$. In the
background-dominated regime one has $\sigma_{i}^{b}\simeq\sigma_{5}^{b}$ where
$\sigma_{5}^{b}$ is equal by definition to

$\displaystyle\sigma_{5}^{b}$ $\displaystyle=\frac{f_{5}^{b}}{5}$ (7)

where $f_{5}^{b}$ is the 5-$\sigma$ flux related to the 5-$\sigma$ depth
magnitude $m_{5}^{b}$ through:

$\displaystyle m_{5}^{b}$ $\displaystyle=-2.5\log f_{5}^{b}+zp^{b}$ (8)

where $zp^{b}$ is the zero point of the considered filter. $m_{5}^{b}$ is
related to N${}_{\mathrm{visits}}^{b}$through:

$\displaystyle m_{5}^{b}-m_{5}^{b,\mathrm{single}}$ $\displaystyle\approx
1.25\log(N_{visits}^{b})$ (9)

where $m_{5}^{b,\mathrm{single}}$ is the 5-$\sigma$ depth corresponding to a
single visit, a parameter depending on observing conditions. These equations
(7)-(9) describe the relationship between SNRb and
N${}_{\mathrm{visits}}^{b}$. The requirement
$\mbox{SNR${}^{b}$}\leavevmode\nobreak\ \geq\leavevmode\nobreak\
\mbox{SNR${}^{b}_{\mathrm{min}}$}$ is equivalent to
$\mbox{N${}_{\mathrm{visits}}^{b}$}\leavevmode\nobreak\
\geq\leavevmode\nobreak\ \mbox{$\mathrm{N_{visits,min}^{b}}$}$ and Eq. (4) may
be written:

$\displaystyle\land(\mbox{N${}_{\mathrm{visits}}^{b}$}\leavevmode\nobreak\
\geq\leavevmode\nobreak\
\mbox{$\mathrm{N_{visits,min}^{b}}$})\Longrightarrow\mbox{$\sigma_{c}$}\leq
0.04\Longrightarrow\mbox{$z_{\mathrm{lim}}$}$ (10)

where the logical symbol $\land$ means that the requirement
$\mbox{N${}_{\mathrm{visits}}^{b}$}\leavevmode\nobreak\
\geq\leavevmode\nobreak\ \mbox{$\mathrm{N_{visits,min}^{b}}$}$ is to be
fulfilled for all the considered bands. The relations (4) and (10) are not
univocal. Many SNRb (N${}_{\mathrm{visits}}^{b}$) combinations lead in fact to
the same result and constraints have to be applied to choose optimal
configurations.

We have used the following method to estimate N${}_{\mathrm{visits}}^{b}(z)$.
A systematic scan of the SNR parameter space (SNRg, SNRr, SNRi, SNRz, SNRy) is
performed. Median observing conditions estimated from DD simulations are used,
namely m${}_{5}^{g,\mathrm{single}}$ = 24.48, m${}_{5}^{r,\mathrm{single}}$ =
23.60, m${}_{5}^{i,\mathrm{single}}$ = 24.03, m${}_{5}^{z,\mathrm{single}}$ =
22.97, m${}_{5}^{y,\mathrm{single}}$ = 22.14. For each SNR combination and for
a set of cadences (1 day to 4 days), light curves of an intrinsically faint
SNe Ia (i.e. with $x_{1}$ = -2.0, $c$ = 0.2) in the redshift range [0.01,1.0]
are simulated using templates and SNe Ia parameter errors ($\sigma_{c}$,
$\sigma_{x_{1}}$) are estimated using the Fisher Matrix formalism. This
approach considerably reduces the processing time (compared to full
simulation+fit) while ensuring the highest degree of accuracy of the LC points
(fluxes and flux errors) and of the supernovae parameter errors. The light
curves fulfilling the requirements defined in Section 4 are used to define the
SNR parameter space, or equivalently the $\mathrm{N_{visits}}$ parameter space
(N${}_{\mathrm{visits}}^{g}$, N${}_{\mathrm{visits}}^{r}$,
N${}_{\mathrm{visits}}^{i}$, N${}_{\mathrm{visits}}^{z}$,
N${}_{\mathrm{visits}}^{y}$) according to Eq. 10, corresponding to well-
measured SNe Ia. Optimal combination are selected by minimizing the total
number of visits per observing night and by requiring a maximum number of
$y$-band visits. This selection aims at reducing the (potentially severe)
systematic effects affecting the $y$-band measurements (High et al., 2010).
The result is displayed in Fig. 8 (top) for a 1 day cadence. The number of
visits strongly increases with the redshift completeness for
$z_{\mathrm{complete}}$ $\gtrsim\leavevmode\nobreak\ 0.7$ where only three
bands $i$$z$$y$ can be used to construct SNe Ia light curves. About 130 visits
(1 hour and 5 minutes of observation) are required to reach
$z_{\mathrm{complete}}$ $\sim$ 0.8 for a one day cadence. Since the number of
visits required to reach a given $z_{\mathrm{complete}}$ value increases
linearly (as a first approximation) with the cadence, this corresponds to
$\sim$ 3.25 hours of exposure time per night for a 3-day cadence.

Figure 8: Top: number of visits per observing night as a function of the
redshift completeness. 131 visits with the following filter allocation
(N${}_{\mathrm{visits}}^{g}$, N${}_{\mathrm{visits}}^{r}$,
N${}_{\mathrm{visits}}^{i}$, N${}_{\mathrm{visits}}^{z}$,
N${}_{\mathrm{visits}}^{y}$)=(2, 9, 45, 64, 11) are required per observing
night to reach $z_{\mathrm{complete}}$$\sim$0.8 for a cadence of one day.
Bottom: budget per field and per season of observation as a function of
$z_{\mathrm{complete}}$. A 1-day cadence and a season length of 180 days have
been assumed. Observing a field every night for 180 days with a redshift
completeness of 0.80 corresponds to a budget of 1.1$\%$.

It is known that the restframe UV region is subject to large fluctuations
between SNe Ia in the SALT2 light curve model (see Section 4). One of the
consequences is that only three (two) bands, $i,z,y$ ($z,y$), may be used to
reconstruct light curves for redshifts higher than $\sim$ 0.7 ($\sim$ 1.1).
The SNe Ia parameter errors depend on the SNR values, but also on the shape of
the light curves per band. The contribution of SNRy to the SNe Ia parameter
errors tends to increase with $z_{\mathrm{complete}}$ if the total number of
visits remains relatively constant. A high number of
N${}_{\mathrm{visits}}^{y}$ for low $z_{\mathrm{complete}}$($\sim$ 0.7) leads
to a net loss of well-sampled SNe Ia because of bad color measurements. The
optimal N${}_{\mathrm{visits}}^{y}$ as a function of $z_{\mathrm{complete}}$
has been estimated by computing the redshift limit of a medium SNe Ia in
configurations with 5$\leavevmode\nobreak\ \leq\leavevmode\nobreak\
$N${}_{\mathrm{visits}}^{y}$$\leavevmode\nobreak\ \leq\leavevmode\nobreak\
$80\. Requiring a $z_{\mathrm{lim}}$ variation lower than 0.01 leads to the
results of Tab. 5.

Table 5: A set of optimal numbers of $y$-band visits as a function of $z_{\mathrm{complete}}$. $z_{\mathrm{complete}}$ | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90
---|---|---|---|---|---|---
N${}_{\mathrm{visits}}^{y}$ | 3 | 7 | 16 | 21 | 30 | 38

It is possible to estimate the budget per field and per season of observation
as a function of $z_{\mathrm{complete}}$ by using Eq. 5 and the results of
Fig. 8 (top). As expected (Fig. 8, bottom) a significant increase is observed
for higher redshifts and the observation of a field for 180 days with
$z_{\mathrm{complete}}$$\leavevmode\nobreak\ \sim\leavevmode\nobreak\ $0.9
requires a budget of 2$\%$.

The optimized number of visits required to reach higher redshift completeness
is the last piece of the puzzle to be included in the budget estimator (Eq.
5). We have now the tools to design realistic and optimal DD scenarios.

##  7 Optimal LSST DD surveys for cosmology with SNe Ia

In the following, we examine three key points of the surveys (redshift
completeness, cadence of observation, and cosmological measurements) before
presenting a set of optimized scenario.

##### Redshift completeness

Spectroscopic datasets from cosmological endeavors overlapping with LSST in
area and timing are essential for SNe Ia cosmology. They provide enormous
added benefits through (a) the follow-up of a subset of the full sample of
well-measured supernovae (to improve the models used to build training sample
for photometric classification), and (b) the measurement of host-galaxy
redshifts with high accuracy. Three of the spectroscopic resources
contemporaneous with LSST, Dark Energy Spectroscopic Instrument (DESI
Aghamousa et al. 2016), Primary Focus Spectrograph (PFS/Subaru Tamura et al.
2016) and 4MOST (de Jong et al. 2019), will provide vital live spectra and
host spectroscopic redshifts (Mandelbaum et al. 2019).

DESI is a ground-based dark energy experiment used to conduct a five-year
survey that will cover 14,000 deg2. More than 30 million galaxy and quasar
redshifts will be measured to study baryon acoustic oscillation and the growth
of structure through redshift-space distorsions. The DESI survey will overlap
with at least 4000 deg2 of the LSST footprint in the northern hemisphere.

The PFS/Subaru spectroscopic follow-up survey is designed to observe two of
LSST DDFs accessible from the Subaru telescope: COSMOS and XMM-LSS. About 2000
spectra of live supernovae and 20,000 host galaxy redshifts up to $z\sim 0.8$
will be collected after 10 years.

The 4MOST Time-Domain Extragalactic Survey (TiDES Swann et al. 2019) is
dedicated to the spectroscopic follow-up of extragalactic optical transients
and variable sources selected from e.g. LSST. The goal is to collect spectra
for up to 30,000 live transients to $z\sim 0.5$ and to measure up to 50,000
host galaxy redshifts up to $z\sim 1$. This corresponds to both the DD and the
WFD fields.

Two sets of LSST fields may then be defined to fully benefit from the synergy
with DESI, PFS/Subaru and 4MOST. DESI and PFS/Subaru will provide live-spectra
and spectroscopic redshifts for the northernmost fields, COSMOS and XMM-LSS,
over a broad range in redshift. Southern fields, Euclid/Roman, CDF-S,
ELAIS-S1, will take advantage from 4MOST/TiDES measurements.

##### Cadence of observation

Few arguments point in favour of high cadences: the total number of visits per
night, the season length, the translational dithering, and inter-night gaps.

More than 240 visits are required to reach $z_{\mathrm{complete}}$ $\geq$ 0.9
for a 1-day cadence (Fig. 8, top). The same depth is obtained for a 3-day
cadence with more than 720 visits, that is about 6 hours of observation.
Reaching higher $z_{\mathrm{complete}}$ with low cadence observing strategies
is thus not realistic: it would potentially jeopardize the uniformity of the
WFD survey.

Because of their northernmost positions, COSMOS and XMM-LSS are characterized
by shorter season lengths w.r.t. other fields (for the same observing time per
night) (Fig. 3). Requiring more than 480 visits per night (to reach
$z_{\mathrm{complete}}$ $\geq$ 0.9 with a 2d cadence) would dramatically
degrade the season lengths (to less than 90 days for these two fields) and
would drastically reduce the size of the well-measured supernovae sample (by
more than 50 $\%$).

As shown on Fig. 7 the translational dithering has a limited impact on the
number of well-measured SNe Ia and on $z_{\mathrm{complete}}$ up to tdo$\sim$1
degree for high cadences. The number of well-measured SNe Ia is falling
rapidly with tdo for cadences lower than 3 days.

One of the main conclusions of the analysis of the proposed LSST simulations
(Section 5) is that large inter-night gaps have harmful effects on the
sampling and on the quality of SNe Ia light curves. They can be reduced to a
minimum (i.e. to unavoidable gaps related to telescope maintenance or bad
weather conditions) by either observing DDFs at high cadences or including a
mechanism in the scheduler that would ensure to keep a high observing rate of
SNe Ia (see Section 9 for suggestions).

##### Cosmological metric

The most accurate way to measure cosmological parameters from a sample of
well-measured SNe Ia is to perform a maximum likelihood analysis by
minimizing:

$-ln\mathcal{L}=(\mu-\mbox{$\mu_{th}(z,\Omega_{m},w)$})^{\intercal}C(\mu-\mbox{$\mu_{th}(z,\Omega_{m},w)$})$
(11)

where $C$ is the covariance matrix and $\mu$ is the distance modulus (Eq. 1).
$\mu_{th}(z,\Omega_{m},w)$ is defined by:

$\mbox{$\mu_{th}(z,\Omega_{m},w)$}=5\log_{10}[d_{L}(Mpc)]+25$ (12)

In a flat universe, the luminosity distance $d_{L}$ is defined by:

$d_{L}(z,\Omega_{m},w)={\frac{c(z+1)}{H_{0}}}\int_{0}^{z}{\frac{dz^{{}^{\prime}}}{\sqrt{(1-\Omega_{m})+\Omega_{m}(1+z^{{}^{\prime}})^{3\mbox{$w$}}}}}$
(13)

where $\Omega_{m}$ the dark matter density parameter, and $w$ is the parameter
of the dark energy equation of state. Five parameters are to be estimated: the
cosmological parameters ($\Omega_{m},\mbox{$w$}$) and the nuisance parameters
($M,\alpha,\beta$) (Eq. 1).

The goal of this section is to study a large set of surveys by varying the
number of fields to be observed, the redshift completeness (i.e. having
samples with redshift completeness field-dependent) and the number of seasons.
Using the above-mentioned method for each scenario would require to produce a
lot of (time-consuming) simulations and fits of SNe Ia light curves to
estimate cosmological parameters. We have thus chosen to work with distance
moduli of supernovae simulated using:

$\mu_{i}(z_{i})\sim\mathcal{N}(\mbox{$\mu_{th}(z_{i},\Omega_{m},w)$},\sigma^{2}=\sigma_{\mu_{i}}^{2}+\mbox{$\sigma_{int}^{2}$}+\mbox{$\sigma^{2}_{syst_{i}}$})$
(14)

where $\sigma_{int}$ is the intrinsic dispersion of supernovae
($\mbox{$\sigma_{int}$}\sim$ 0.12) and $\sigma_{syst}$ accounts for systematic
uncertainties. $\sigma_{\mu}(z)$ is the distance modulus error for each SNe Ia
of redshift $z_{i}$. It has been estimated from a complete simulation of DD
surveys with varying $z_{\mathrm{complete}}$ using the method developed in
Section 4.

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)$})^{2}}{\sigma_{\mu_{i}}^{2}+\mbox{$\sigma_{int}^{2}$}+\mbox{$\sigma^{2}_{syst_{i}}$}}$
(15)

where $N_{SN}$ is the number of well-measured SNe Ia used to perform the fit.
The realistic simulations used to estimate $\sigma_{\mu}$ and $N_{SN}$ take
into account redshift bias (see Appendix B for more details). For each DD
scenario considered in the following, a sample of about 100,000 low-$z$ SNe Ia
(10000 per year) has been added up to $\mbox{$z_{\mathrm{complete}}$}\sim$0.2
(Lochner et al., 2021a).

The following sections describe a set of optimized scenarios based on two
different approaches. One is optimizing the number of well-sampled SNe Ia
collected by the survey. The other aims at probing high redshift completeness
domains. A survey is characterized by three parameters, the redshift
completeness, the number of DDFs, and the cadence of observation. Three
metrics are presented to assess the proposed scenario: the DD budget, the
cosmological metric (we have chosen the error on the $w$ parameter,
$\sigma_{\mbox{$w$}}$), and the total number of well-sampled SNe Ia.

###  7.1 Deep Universal Surveys

In the Deep Universal (DU) survey, all the DDFs are observed for ten years
with a similar cadence, season length and $z_{\mathrm{complete}}$ (i.e. the
same number of visits per band and per observing night). The budget is the
major factor limiting the redshift completeness of the survey (Fig. 9, top). A
budget higher than 15$\%$ is required to reach $z_{\mathrm{complete}}$
$\lesssim$ 0.65 with 5 DDFs (and 2 pointings for Euclid/Roman). This
configuration leads to a high number of well-sampled SNe Ia ($\sim$ 14,000)
and to the best cosmological constraints ($\sigma_{\mbox{$w$}}$ $\sim$ 1$\%$).
$z_{\mathrm{complete}}$ and NSN dramatically decrease for a budget lower than
5$\%$ and the cosmological measurements get significantly worse
($\sigma_{\mbox{$w$}}$ $\geq$ 2$\%$).

Figure 9: NSN (red dashed lines), DD budget (black solid lines) and
$\sigma_{\mbox{$w$}}$ (blue dotted lines, left plot) for three types of
surveys: Deep Universal (top), Deep Rolling 10 years (middle) and Intensive
Deep Rolling (bottom). Subscripts correspond to the number of seasons of
observation (6 months season length) and superscripts to the redshift
completeness. The number of $y$-band visits is less than 20 (per observing
night). Only statistical uncertainties are included.

##### Cadence of observation

Reaching $z_{\mathrm{complete}}$$\sim$0.65 can be achieved with 41 visits per
night of observation with a 1-day cadence (Section 6). Up to 4 fields (because
of its northernmost location, COSMOS is visible independently of the other
DDFs) may have to be scanned for some nights, which corresponds to 1 hour and
42 minutes of observation. Moving to a 3-day cadence would require 123 visits
per field and per night of observation. In that case the scanning of 4 fields
(5 pointings) has to be spread over 3 nights with at least two fields observed
per night. This corresponds to about 2 hours of observation.

###  7.2 Deep Rolling Surveys

Observing 5 fields for ten years up to $z_{\mathrm{complete}}$ $\sim$ 0.9
would certainly give access to a large sample of well-measured SNe Ia (around
19k) but also to an unrealistic scenario
(N${}_{visits}^{DD}\leavevmode\nobreak\ \gtrsim\leavevmode\nobreak\
\mathrm{N}_{visits}^{WFD}$). The only way to reach higher
$z_{\mathrm{complete}}$ while remaining within a reasonable budgetary envelope
is to reduce the number of fields to be observed and/or the number of seasons
of observation. We propose the Deep Rolling (DR) strategy characterized by a
limited number of seasons of observation per field (at least 2) and a large
number of visits per observing night (more than 130 for higher
$z_{\mathrm{complete}}$).

A realistic large scale high-$z$ DR survey is characterized by: (a) a high
cadence of observation (one day), (b) a rolling strategy (with a minimal of
two seasons of observation per field), and (c) two sets of fields, ultra-deep
(COSMOS, XMM-LSS, with $z_{\mathrm{complete}}$ $\gtrsim$ 0.8) and deep
(Euclid/Roman, CDF-S, ELAIS-S1, with $z_{\mathrm{complete}}$ $\gtrsim$ 0.7)
fields. We study two DR scenarios in the following, characterized by a minimal
number of seasons of observation per field or by a minimal number of fields to
be observed.

####  7.2.1 Deep Rolling 10 Years

In this scenario all the DDFs are observed for two seasons. The results of the
triplet (budget, $\sigma_{\mbox{$w$}}$, NSN) as a function of the number of
DDFs and redshift completeness are given on Fig. 9 (middle). It seems
difficult, with a budget lower than 5$\%$, to perform cosmological
measurements of $w$ with an accuracy better than 1.5$\%$. A sample of $\sim$
4000-4500 well-measured SNe Ia with a reasonable budget of $\sim$ 8$\%$ would
lead to $\sigma_{\mbox{$w$}}$ $\sim$ 1.1$\%$.

This scenario appears to have two essential weaknesses. The number of seasons
of observation per field is low. Periods of bad weather could affect the
progress of the survey and the quality of the SNe Ia sample. What is more,
this survey requires a precise timeline that may be difficult to tune (see an
example below).

The sequence of observations (field/night) of the DR survey must fulfill a
couple of constraints. LSST observation of Euclid/Roman has to be
contemporaneous with Euclid (years 2 and 3) and with Roman Space Telescope
(years 5 and 6). Observing multiple fields per night is not optimal if the
number of visits per field is high. It may jeopardize the uniformity of the
WFD survey (if the total number of DD visits is too high) and have a negative
impact on the regularity of the DD cadence (if a choice has to be made among
the DDFs). Overlap of field observations should thus be minimized. This means
that the DR survey should be deterministic with a timely sequence defined in
advance. An example of the progress of a DR survey is given in Fig. 10 for a
configuration of 5 fields and a survey complete up to $z\lesssim$ 0.8 and
$z\lesssim$ 0.7 for ultra-deep and deep fields, respectively.

Figure 10:  Cumulative sum of the number of nights (per field and per season)
as a function of the time since survey start (assumed to be late 2023). The
following sequence of observations is considered: COSMOS, Euclid/Roman (x2),
XMM-LSS, COSMOS, Euclid/Roman (x2), XMM-LSS, ELAIS-S1 (x2), CDF-S (x2) , with
a maximum season length of 180 days for the deep fields, a cadence of one day,
and ensuring only one field is observed per night. The fields are required to
be observable (airmass $\leq$ 1.5 and 20° $\leq$ altitude $\leq$ 86.5°) for at
least 1 hour and 5 minutes (131 visits) for the ultra-deep fields and 34
minutes (68 visits) for the deep fields. The overlap, defined as the fraction
of nights with more than one field observed during a night, is
$\sim\leavevmode\nobreak\ 4\%$

####  7.2.2 Intensive Deep Rolling

In this scenario a minimal number of fields are observed and the goal is to
maximize the redshift completeness of the survey999It is defined as the median
redshift completeness of the observed fields.. The choice of the fields may be
motivated by the following considerations: the need to explore high redshift
completeness domains, and the synergy with surveys contemporaneous with Rubin
Observatory operations. Fulfilling these requirements leads to the choice of
three fields: COSMOS, XMM-LSS and Euclid/Roman. In this scenario Euclid/Roman
is observed for 4 seasons up to $z_{\mathrm{complete}}$$\sim$ 0.7 and COSMOS
and XMM-LSS are observed every year with a redshift completeness of at least
0.7. Reaching higher $z_{\mathrm{complete}}$ domains requires to increase the
number of visits per observing night ($\sim$ 130 visits to reach
$z_{\mathrm{complete}}$ $\sim$ 0.8 for a 1-day cadence). The DD budget may
then be used up in few years and this is why this scenario may be dubbed as
“intensive”. The results of the triplet (budget, $\sigma_{\mbox{$w$}}$, NSN)
as a function of the number of seasons of observation and redshift
completeness of the two ultra-deep fields are given on Fig. 9 (bottom). About
3500-4000 well-sampled SNe Ia are collected after 3 to 4 years of observation
of COSMOS and XMM-LSS up to $z_{\mathrm{complete}}$ $\lesssim$ 0.8. The
corresponding budget is 8$\%$ and $\sigma_{\mbox{$w$}}$$\sim$1.1-1.2$\%$.

###  7.3 Conclusion: DR surveys yield more accurate cosmological measurements

Few conclusions can be drawn from a comparison of the surveys presented in
subsection 7.1 and 7.2:

  * •

redshift completeness: it is impossible to reach $z_{\mathrm{complete}}$$\geq$
0.6-0.65 for DU${}^{\leavevmode\nobreak\ }$surveys. This is to be explained by
the DD budget envelope leading to a limited number of visits per observing
night if a large number of fields are observed all seasons for ten years. This
results in a low number of visits in the redder bands ($z$ and $y$) imposing a
limit on $z_{\mathrm{complete}}$.

  * •

accuracy of cosmological measurements: under the assumption of an identical
budget, IDR surveys lead to more accurate cosmological measurements. With a DD
budget of 5$\%$, the $w$ parameter can be measured with $\sigma_{\mbox{$w$}}$
$\sim$ 1.3-1.4$\%$ for IDR scenarios and to $\sigma_{\mbox{$w$}}$ $\geq$ 2$\%$
for DU${}^{\leavevmode\nobreak\ }$surveys. This result is mainly due to the
fact that the distribution of the number of SNe Ia N($z$) depends on the
redshift completeness value of the survey (see Fig. 18 in Appendix B). IDR
surveys present a higher fraction of SNe Ia at higher redshift compared to
DU${}^{\leavevmode\nobreak\ }$surveys (Fig. 11) and lead to more accurate
cosmological measurements.

  * •

DD Budget impact: moving from a budget of 5$\%$ to 8$\%$ would lead to a
relative decrease of $\sigma_{\mbox{$w$}}$ of 20-25$\%$, depending on the
scenarios. Measuring $w$ with a precision on $\leavevmode\nobreak\
\sim\leavevmode\nobreak\ $1.1$\%$ requires a minimal budget of 8$\%$ and
15$\%$ for DR and DU scenarios, respectively.

Figure 11: Fraction of SNe Ia ($\geq\leavevmode\nobreak\ z$) as a function of
the redshift ($z$) for two scenarios: IDR${}_{0.80}^{0.60}$ (full red line)
and DU0.65 (dotted blue line). The fraction of SNe Ia with
$z\leavevmode\nobreak\ \geq\leavevmode\nobreak\ $0.8 is of 45$\%$ and 22$\%$
for IDR${}_{0.80}^{0.60}$ and DU0.65 surveys, respectively.

##  8 Realistic surveys - impact of host-galaxy redshifts

The goal of this section is draw a comparison of the performance of realistic
surveys using the triplet (budget, $\sigma_{\mbox{$w$}}$, NSN). Effects of two
critical aspects have to be included to get a more accurate comparison of the
proposed surveys: the Malmquist bias correction and the SNe Ia host galaxy
redshift estimation.

Systematic uncertainties related to observational selection effects will
probably account for a major part of the error budget in the era of LSST. We
have considered two components related to the selection bias : a statistical
contribution, related to the limited number of SNe Ia per redshift bin ; a
bias contribution, due to the (limited) knowledge of the ($x_{1}$,$c$)
distribution of the selected SNe Ia. We have used the G10 intrinsic scatter
model (Scolnic & Kessler 2016) where ($x_{1}$,$c$) distributions are described
by asymmetric gaussian distributions with three parameters and their
uncertainties $\sigma$. We have performed simulations by individually varying
each parameter of $\pm$1$\sigma$. The differences of distance modulus values
w.r.t. the nominal configuration were added quadratically to provide
systematic uncertainties.

Measuring cosmological parameters with a high degree of accuracy requires to
minimize uncertainties of the two components of the SNe Ia Hubble diagram: the
distance modulus and the redshift. Collecting a large sample of well-measured
SNe Ia leading to accurate distance measurements is a guiding thread of this
paper. It is achievable by optimizing the cadence of the survey, by adapting
the number of visits per observing night, and by imposing selection criteria
on photometric light curves. Redshifts can be derived either from the host
galaxy (spectrum and/or photometric measurements) or from the spectrum of the
supernova itself. Only host galaxy spectroscopic redshifts are considered in
this study. We assume that 4MOST/TiDES will provide $\sim$ 2500101010This
number corresponds to $\sim$ 5$\%$of the host galaxy redshifts measured by
4MOST/TiDES. The actual number is not known yet. host galaxy redshifts (after
5 years) for the DD fields (CDF-S, ELAIS-S1, Euclid/Roman) and that equatorial
fields will benefit from PFS/Subaru measurements ($\sim$ 20,000 spectra after
ten years). The current PFS/Subaru strategy is to cover 5 deg2 (4 PFS FoV) for
equatorial fields to accumulate $\sim\leavevmode\nobreak\ $12,000 host galaxy
spectroscopic redshifts over 10 years. The remaining DD area would be observed
once the LSST survey is completed. The fraction of SNe Ia expected to have
secure redshift measurements is taken from Fig. 1 of Mandelbaum et al. 2019.

Figure 12: DD budget as a function of the total number of well-sampled SNe Ia
(left) and time budget (right) for a set of Intensive Deep Rolling (IDR), Deep
Rolling (DR) and Deep Universal (DU) surveys. Subscripts (superscripts)
correspond to the redshift completeness of ultra-deep (deep) fields. Black
dotted lines correspond to DD budgets of 5$\%$ and8$\%$.

We have chosen a large set of surveys among the possible configurations
presented above (observing strategy parameters are listed in Tab. 6):

  * •

Deep Universal survey (DU): 5 DD fields - COSMOS, XMM-LSS, CDF-S, ELAIS-S1,
Euclid/Roman - are observed every season with $z_{\mathrm{complete}}$
$\in\leavevmode\nobreak\ $[0.60,0.80] ($\Delta z\leavevmode\nobreak\
=\leavevmode\nobreak\ 0.05)$

  * •

Deep Rolling 10 years (DR): 5 DD fields - COSMOS, XMM-LSS, CDF-S, ELAIS-S1,
Euclid/Roman - are observed for two seasons each according to the timeline
defined in Fig. 10. The redshift completeness ranges are [0.60,0.80] and
[0.50,0.75] ($\Delta z\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.05)$ for
ultra-deep and deep fields, respectively.

  * •

Intensive Deep Rolling (IDR): 3 DD fields are considered: two ultra-deep
fields (COSMOS,XMM-LSS) with $z_{\mathrm{complete}}$ $\in\leavevmode\nobreak\
$[0.70,0.75,0.80] and one deep field (Euclid/Roman) with
$z_{\mathrm{complete}}$ $\in\leavevmode\nobreak\ $[0.50,0.70] ($\Delta
z\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.05)$ . Ultra-deep fields are
observed every year and Euclid/Roman during four seasons.

Figure 13: Number of well-measured SNe Ia observed in the DD survey (top: SNe
Ia observed in ultra-deep fields, bottom: SNe Ia observed in deep fields) as a
function of time (year of survey) for deep fields (CDF-S, ELAIS-S1,
Euclid/Roman) (left) and ultra-deep fields (COSMOS, XMM-LSS) (right). The
black circles (squares) correspond to a budget of 5$\%$ (8$\%$).

The budget as a function of the total number of well-sampled SNe Ia and the
time budget are presented in Fig. 12. As expected the larger number of SNe Ia
is provided by DU surveys and the minimal number by IDR scenarios with a ratio
$N_{SN}^{DU}/N_{SN}^{IDR_{0.80}}\simeq$ 1.8. The lowest time budget is
obtained with an IDR scenario with a high $z_{\mathrm{complete}}$ (0.8) for
ultra-deep fields: the 5$\%$ budget limit is reached after $\sim$ 2.6 years.

The number of well-measured SNe Ia observed in the DD survey is given on Fig.
13 for deep (CDF-S, ELAIS-S1, Euclid/Roman) and ultra-deep (COSMOS, XMM-LSS)
fields. IDR scenarios lead to samples mostly composed of SNe Ia observed in
equatorial fields (the total number of SNe Ia provided by non-equatorial
fields is of $\sim\leavevmode\nobreak\ $100). Increasing the DD budget from
5$\%$ to 8$\%$ lead to an increase of the the SNe Ia sample by a factor of
$\sim\leavevmode\nobreak\ $1.6.

Table 6: Observing strategy parameters for a representative set of optimized surveys. The cadence of observation is of one day and the season length of 180 days (max). Observing Strategy |  |  |  |
---|---|---|---|---
type | name | Fields | $z_{\mathrm{complete}}$ | Nseasons/field | $\mathrm{N_{visits}}$
|  |  |  |  | $g/r/i/z/y$
Deep Universal | DU0.65 | COSMOS, XMM-LSS | 0.65 | 10 | 2/9/10/15/3
ELAIS-S1, CDF-S, Euclid/Roman
Deep Rolling | DR${}_{0.75}^{0.65}$ | COSMOS, XMM-LSS | 0.75 | 2 | 2/9/26/35/16
ELAIS-S1, CDF-S, Euclid/Roman | 0.65 | 2/9/10/15/3
Intensive Deep Rolling | IDR${}_{0.70}^{0.60}$ | COSMOS, XMM-LSS | 0.70 |  $\geq$2 | 2/9/20/29/7
Euclid/Roman | 0.60 | 4 | 2/9/1/1/
IDR${}_{0.75}^{0.60}$ | COSMOS, XMM-LSS | 0.75 |  $\geq$2 | 2/9/26/35/16
Euclid/Roman | 0.60 | 4 | 2/9/1/1/
IDR${}_{0.80}^{0.60}$ | COSMOS, XMM-LSS | 0.80 |  $\geq$2 | 2/9/37/52/21
Euclid/Roman | 0.60 | 4 | 2/9/1/1/

Since only SNe Ia with host spectroscopic redshifts are considered, the
cosmological metric $\sigma_{\mbox{$w$}}$ values depend on the spectroscopic
scenario (i.e. the number of host spectroscopic measured per year for
PFS/Subaru and 4MOST/TiDES) and on the budget. The variation of
$\sigma_{\mbox{$w$}}$ as a function of time (year of survey) is given in Fig.
14 (top left) assuming the current PFS/Subaru strategy described above. IDR
scenarios tend to lead to more accurate cosmological metric estimation for the
same budget. Taking into account the whole set of spectroscopic redshifts
provided by PFS/Subaru ($\sim\leavevmode\nobreak\ $20,000) lead to the results
of Fig. 14 (top right). Significant differences are observed between IDR and
DU scenarios and relative decrease of $\sigma_{\mbox{$w$}}$ of about 30$\%$
for a budget of 5$\%$ can be observed.

Figure 14: Left: $\sigma_{\mbox{$w$}}$ (top) and DETF FoM (bottom) a function
of time (year of survey) for scenarios with a DD budget lower than 10$\%$. The
black circles (squares) correspond to a budget of 5$\%$ (8$\%$). Right:
$\sigma_{\mbox{$w$}}$ (top) and DETF FoM (right) after ten years for DD
strategies considered in this paper and and for DD budgets of 5$\%$ (solid
line) and 8$\%$ (dashed line).

Measuring cosmological parameters with a high degree of accuracy with
supernovae requires to observe a large sample of well-measured SNe Ia in the
full redshift range [0.01,1.1]. The results of Figs 14 (top) indicate that the
shape of the NSN($z$) distribution is critical to achieve low
$\sigma_{\mbox{$w$}}$ values. The profile of NSN($z$) is affected by the
Malmquist bias for $z\leavevmode\nobreak\ \geq\leavevmode\nobreak\
$$z_{\mathrm{complete}}$ and the fraction of well-measured SNe Ia at high
redshift depends on the detection threshold $z_{\mathrm{complete}}$ (Fig. 11).
Between two surveys collecting the same number of well-measured SNe Ia, the
most accurate cosmological measurements are achieved by surveys characterized
by the highest $z_{\mathrm{complete}}$ (Fig. 14, top right). Low
$z_{\mathrm{complete}}$ surveys have to collect a higher number of well-
measured SNe Ia to achieve the same accuracy.

For each of the scenarios considered in this section we have estimated the
figure of merit defined by the Dask Energy Task Force (Albrecht et al., 2006):

$\mathrm{DETF\leavevmode\nobreak\ FoM}=\frac{\pi}{A}$ (16)

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

$A=\pi\Delta\chi^{2}\sigma_{w_{0}}\sigma_{w_{a}}\sqrt{1-\rho^{2}}$ (17)

with $\Delta\chi^{2}$ = 6.17 (95.4$\%$ C.L.). $\rho$ 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):

$w=w_{0}+w_{a}{{z}\over{1+z}}$ (18)

The cosmological parameters ($\Omega_{m}$,$w_{0}$,$w_{a}$) have been estimated
from the minimization of Eq. 15 (modified to account for the definition of $w$
in Eq. 18) and a prior has been added on the $\Omega_{m}$ parameter (with
$\sigma_{\Omega_{m}}\leavevmode\nobreak\ =\leavevmode\nobreak\ $0.0073 Aghanim
et al. 2020).

The $2-\sigma$ figure of merit of the scenarios considered in this section are
given on Fig. 14 (bottom left and right). The conclusion is the same as above:
IDR scenarios lead to the highest FoM in comparison with DU surveys (+25$\%$
between IDR0.80 and DU0.65 for a DD budget of 5$\%$). This result reflects the
above-mentioned dependence of the shape of NSN($z$) on the redshift
completeness of the survey.

One of the main conclusions of the studies presented in this section is that
the design of an optimal LSST DD mini-survey for cosmology with SNe Ia has to
include external critical datasets such as precise host redshift measurements
to be realistic. Further studies are needed to assess the impact of
spectroscopic datasets from PFS/Subaru, 4MOST/TiDES, and of synergy with
Euclid and Roman Space Telescope, on the scenarios proposed in this paper.
Optimizing the use of limited spectroscopic resources requires to select DD
surveys that deliver a sample containing a low number of well-measured SNe Ia
while leading to accurate cosmological measurements. IDR (high
$z_{\mathrm{complete}}$) surveys fulfill both criteria.

The large number of surveys presented in this section were achieved with
optimal observing conditions (regular cadence, median 5-$\sigma$ depth).
Additional simulations are required to assess the effect of realistic
observing conditions such as variations of 5-$\sigma$ depth values or of the
Moon brightness. The results of such studies might require to tune the
parameters of the surveys proposed in this paper.

##  9 Gap/budget recovery

All the results presented above assume a regular cadence with median observing
conditions (i.e. median $m_{5}$) and no translational dithering. But it is
known that the LSST survey will be affected by gaps originating from telescope
downtimes related to maintenance periods and to poor observing conditions
(clouds). The probability to have low gaps (few nights) is high (more than
80$\%$ for gaps lower than 3 nights) and mainly due to dome closed periods
(Fig. 15). Larger gaps are explained by telescope maintenance times and are
not exceptional: the probability to have a gap of 14 nights is of about 25
$\%$. It has been shown (Section 5) that gaps affect the redshift completeness
and the total number of well-measured SNe Ia, but also the DD budget if the
season length is fixed.

Figure 15: Probability to have gaps (full line, left $y$-axis) and mean number
of gaps (dotted line, right axis) as a function of the night gap estimated
with a sliding window of 180 days using the LSST baseline simulation dubbed
baseline_nexp1_v1.7_10yrs. Figure 16: Gap tracker metric as a function of MJD.
The full black lines correspond to SNRz and the blue points to observations
($m_{5}$). Four cases are represented: 3-day cadence surveys with no gaps (a),
with gaps (b), with gaps and a recovery method based on the number of visits
(c), with gaps and a recovery method based on the gap tracker metric (d).

The metrics used up to this point were estimated using the whole set of data.
The impact of the gaps is estimated at the end of the survey. It may be
interesting to define a metric that would trace gap effects on a nightly
basis. We propose as a gap tracker metric the $z$-band SNR (SNRz) of a medium
SNe Ia (i.e. with ($x_{1}$,$c$)=(0.0,0.0)) at a redshift $z\sim$0.6. This SNe
Ia is characterized by $T_{0}$=$\mathrm{MJD_{night}}$ where
$\mathrm{MJD_{night}}$ is the Modified Julian Date of a given night. SNRz is
estimated from the rising part of the SNe Ia light curve. The value of SNRz is
constant for regular cadences (Fig. 16a) and decreases with gaps (Fig. 16b).

Table 7: Variations of budget, $z_{\mathrm{complete}}$, NSN, and season length(sl) for a set of surveys with gaps and three recovery methods. The reference values are taken from a survey with a regular cadence of three days (no gaps) and a 6 months season length. Survey |  $\Delta$Budget |  $\Delta$$z_{\mathrm{complete}}$ |  $\Delta$NSN |  $\Delta$sl
---|---|---|---|---
| $\%$ |  | $\%$ | days
gaps | -25 | -0.03 | -18 | 0
season length extens. | 0 | -0.02 | +11 | +45
nightly $\Delta N_{visits}$ | 0 | -0.01 | -6 | 0
gap tracker thresh. | -8 | -0.02 | -9 | 0

A survey with the gap distribution of Fig. 16b lead to a decrease of
$z_{\mathrm{complete}}$ of about 0.03 and to a loss of about 18$\%$ of well-
measured SNe Ia. The budget is also affected (-25$\%$). Three recovery methods
can be used to get ($z_{\mathrm{complete}}$, NSN, budget) values close to a
survey with a regular cadence (as in Fig. 16a). The first approach would
consist in recovering the initial budget (i.e. the total number of observing
night) by adding, at the end of each season, a number of observing nights
corresponding to the number of downtime nights. This method has limited
effects on $z_{\mathrm{complete}}$ but lead to an increase of the size of the
well-measured SNe Ia due to the increase of the season length (Tab. 7). The
second approach relies on the comparison, at the beginning of a night, between
the number of visits $N_{obs}$ and the number of expected visits $N_{exp}$
corresponding to a survey without gaps. Observations are added while $\Delta
N_{visits}=N_{obs}-N_{exp}<0$ (Fig. 16c). This method leads to a complete
recovery of the budget and to a minimal loss of well-measured SNe Ia (Tab. 7).
The last approach exploits the fact that the gap tracker metric strongly
decreases with gap widths. The recovery is made by adding observations if SNRz
values are lower than a threshold defined from the survey without gaps (Fig.
16a). A partial recovery can be obtained from this method (Fig. 16d and Tab.
7) which requires some tuning of the threshold value.

A closer look at the results of Tab. 7 indicates that one of the best ways to
recover from gap effects is to extend the season length of observations. This
method could nonetheless not be applicable for all the DDFs considered in this
paper. Because of their northernmost positions, COSMOS and XMM-LSS have the
lowest season lengths (Fig. 3) and extensions may not be possible (depending
on the number of visits). In that case the second ($\Delta N_{visits}$
comparison) or third method (gap tracker) should be preferred.

##  10 Conclusions

In this paper we have presented a three-phase study to assess the impact of
the LSST Deep-Drilling mini-survey on the size and depth of a sample of well-
measured SNe Ia: (a) thorough analysis of DD strategy proposed by LSST, (b)
development of a method to probe higher redshift completeness domains and (c)
proposal of a set of optimized DD surveys.

  1. (a)

A comprehensive analysis of LSST simulations has been achieved on a subset of
representative DD scenarios. We have studied the impact of cadence, gaps, and
translational dithering using the metric ($z_{\mathrm{complete}}$, NSN). It
was shown that reaching a redshift completeness higher than 0.55-0.65 is
difficult with a reasonable ($\sim$ 5$\%$) budget allocation.

  2. (b)

Reaching higher redshift completeness requires increasing the signal-to-noise
ratio of the photometric light curves, while considering band-flux
distribution ($z$-dependent), cadence, and observing conditions. We have
proposed a method providing the relationship between the optimized number of
visits per band and the redshift completeness. We have used this result to
design a set of realistic strategies.

  3. (c)

Two classes of optimized surveys have been studied. In the Deep Universal
strategy all the DDFs are observed with the same cadence of observation,
season length and number of visits per observing night (i.e. same
$z_{\mathrm{complete}}$). The Deep Rolling strategy, where two classes of
fields are defined (deep and ultra-deep), aims at probing higher
$z_{\mathrm{complete}}$ domains. Host galaxy spectroscopic measurements from
surveys contemporaneous with LSST have been included to design realistic
optimized surveys.

The results shown in this paper represent a first step towards the design of
an optimal LSST DD survey for cosmology with SNe Ia. Simulations of the DD
survey with the LSST scheduler would help in quantifying the impact of
realistic observing conditions (cadence, gaps, 5-$\sigma$ depth, Moon
brightness) on the proposed surveys. Additional studies are required to fully
benefit from spectroscopic resources (DESI, PFS/Subaru, 4MOST/TiDES) and to
optimize the synergy with Euclid and Roman Space Telescope.

The studies presented in this paper lead to following main conclusions:

  1. 1.

Simulated LSST surveys do not lead to precision cosmology
The DD surveys proposed by LSST lead to a sample of well-measured SNe Ia too
shallow to measure cosmological parameters with a high degree of accuracy. The
redshift completeness of the survey is too low. Large inter-night gaps lead to
low cadences of observation and to a dramatic decrease in the SNe Ia sample
size (up to 30$\%$). Having a deterministic scheduler would provide
significant improvements in the quality of the survey, in terms of cadence
regularity or season length. The scheduler could include metrics (gap recovery
mechanism) to monitor the DD survey on a nightly basis and to correct for gaps
so as to achieve a high quality observing strategy for SNe Ia cosmology with
LSST. Large translational dithers reduce the DDF area observed with high
cadence and lead to a dramatic decrease of the number of well-measured SNe Ia
for low cadence surveys.

  2. 2.

Intensive Deep Rolling for precision cosmology
Of the variety of optimized DD scenarios studied in this paper, Intensive Deep
Rolling (IDR) surveys lead to the most accurate cosmological measurements.
They are characterized by a minimal configuration of 3 fields, with two ultra-
deep (COSMOS and XMM-LSS up to $z_{\mathrm{complete}}$$\leavevmode\nobreak\
\sim\leavevmode\nobreak\ $0.8) and one deep (Euclid/Roman up to
$z_{\mathrm{complete}}$$\leavevmode\nobreak\ \sim\leavevmode\nobreak\ $0.6)
fields. These scenarios require a high cadence of observation (every night), a
large number of visits per night (about 130) and could be achieved in few
years.
Measuring cosmological parameters with a high degree of accuracy requires to
have a sample of well-measured SNe Ia with a significant fraction of
supernovae at higher redshifts. This fraction increases with the redshift
completeness of the sample. IDR are characterized by highest
$z_{\mathrm{complete}}$ and lead to the best figure of merit.

  3. 3.

Accurate $z_{host}$ critical for precision cosmology
The accuracy of the cosmological measurements with SNe Ia is crucially
dependent on the precision of the host galaxy redshifts ($x$-axis of the
Hubble Diagram). Spectroscopic resources will be critical for higher redshift
($z\leavevmode\nobreak\ \gtrsim\leavevmode\nobreak\ $0.7) SNe Ia that
significantly contribute to the measurement of cosmological parameters such as
$(w_{0},w_{a})$. PFS/Subaru is currently the only survey able to provide a
significant number of spectra (few thousands) in the range
$z\leavevmode\nobreak\ \in\leavevmode\nobreak\ $[0.7,1.1] during LSST era.
This explains why the best strategies proposed in this paper are based on the
intense observation of two equatorial fields (COSMOS and XMM-LSS). Only half
of the well-measured SNe Ia sample benefit from host galaxy spectroscopic
measurements with the current PFS/Subaru strategy. Final cosmological
measurements within three years of survey with the full SNe Ia sample would
involve the delivery of about 800 spectroscopic redshifts by PFS/Subaru per
year with 8 PFS FoV per equatorial field.
Additional studies are required for southern fields which benefit from
4MOST/TiDES host galaxy spectroscopic redshift measurements. The performance
of optimized strategies were obtained under the assumption that the fraction
of SNe Ia expected to have secure measurements in the deep fields is identical
for WFD and DD fields (this fraction is not known for DD fields yet). This is
probably pessimistic for DD fields and this fraction will probably be higher
at high-$z$ (it is of $\sim$ 20$\%$ for $z$ = 0.7 for WFD fields). The
accuracy of cosmological measurements would improve with an increase of the
fraction of SNe Ia with secure measurements of higher redshifts.

  4. 4.

Photometric redshifts, number of fields and budget
Using photometric redshifts to perform accurate cosmological measurements with
SNe Ia in a Hubble diagram is a challenging task. It requires measuring
photo-$z$ with a high degree of accuracy and controlling catastrophic outlier
redshifts to minimize photo-$z$ systematics. The current LSST minimal target
$\sigma_{z}$$\leavevmode\nobreak\ \sim\leavevmode\nobreak\ 0.02(1+z_{phot})$
(Graham et al. 2018) induces an error on the distance modulus higher than 0.10
mag (full redshift range). The impact of SNe Ia with photo-$z$ on the
measurement of cosmological parameters is in that case marginal. A lot of
efforts is made to develop techniques leading to lower $\sigma_{z}$ (Schmidt
et al. 2020). But it will be difficult to reach an accuracy similar to
spectroscopic measurements ($\sigma_{z}$ $\sim\leavevmode\nobreak\ 10^{-3}$)
and to have a set of photo-$z$ redshifts with a precision corresponding to the
stringent calibration requirements of LSST (The LSST Dark Energy Science
Collaboration et al. 2018).
It is thus difficult, at this stage, to design a set of DD surveys optimizing
cosmological measurements while collecting a large set of SNe Ia with
photometric redshifts. It would be more reasonable to opt for surveys
optimizing the use of spectroscopic resources (host galaxy redshifts). These
scenarios would guarantee a minimal figure of merit that could be improved by
complementary surveys with sets of SNe Ia with photo-$z$. All the DD fields
that Rubin guarantees to observe could be included in such strategies. For
example, if 8$\%$ of LSST time is spent on DDFs for supernovae, the survey
could be composed of an IDR${}_{0.80}^{0.60}$ strategy using $\sim$ 5$\%$ of
the DD budget (in $\sim$ 3 years) followed by a DU0.65 survey with the
remaining 4 fields observed two seasons each. In this scenario, a minimal DETF
FoM of 150 would be guaranteed and all the DD fields would be observed.

### 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. This work was performed
in part under DOE Contract DE-AC02-76SF00515. M. Lochner acknowledges support
from the South African Radio Astronomy Observatory and the National Research
Foundation (NRF) towards this research. Opinions expressed and conclusions
arrived at, are those of the authors and are not necessarily to be attributed
to the NRF. 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.
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).

Author contributions are listed below.
Philippe Gris: conceptualization, software, analysis, writing.
Nicolas Regnault: conceptualization, analysis, writing.
Humna Awan: participated in (and led some) discussions on DD survey
optimization, especially in regards to translational (and rotational)
dithering of observations. Provided detailed comments on the paper during CWR,
which hopefully strengthens the paper and makes its key takeaways clearer.
Contributed to the general optimization of survey sims via Builder work.
Isobel Hook: general paper discussion/suggestions. Acted as internal reviewer,
where comments led to several improvements. e.g. clarification of the
underlying metric of success used in this paper (section 7).
Saurabh Jha: builder, enabling contributions include helping to develop the
emphasis on ”well-measured” supernovae in section 2.1 and section 4 (per-
object uncertainty less than or on the order of the SN Ia intrinsic scatter);
suggested some survey strategies considered in Table 3; participated in (and
led some) discussion of paper content and helped oversee paper review as
Observing Strategy Working Group co-convener.
Michelle Lochner: general paper discussion/suggestions including y-band
optimization analysis and FoM implementation; edits of the draft; co-lead of
the Observing Strategy Working Group, which provided context for strategies
proposed in this paper, particularly the rolling strategies; builder.
Bruno Sanchez: general discussions, SNWG co-convener.
Dan Scolnic: general paper discussion/suggestions. Acted as internal reviewer,
with comments on earlier sections and framing of the metrics used. Also, co-
led Observing Strategy Working Group, which provided context for strategies
proposed in this paper, particularly the rolling strategies.
Mark Sullivan: general paper discussion/suggestions. Internal reviewer during
review process, providing significant/detailed comments which led to several
changes in the paper.
Peter Yoachim: observing strategy simulations.

##  Appendix A Metric (N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$,
$z_{\mathrm{complete}}$) estimation

The pair metric (N${}_{\mathrm{SN}}^{z\leq z_{\mathrm{complete}}}$,
$z_{\mathrm{complete}}$) is estimated from the combination of observing
efficiency curves and SNe Ia production rate. Observing efficiencies are
estimated from a set of simulated light curves of SNe Ia. A systematic scan of
the SNe Ia parameter space ($T_{0}$,$z$) is performed to estimate efficiency
curves with a high degree of accuracy: $z$ $\in[0.01,1.0]$ (step: 0.05) and
$T_{0}$$\in[MJD_{season}^{min}+15*(1+z),MJD_{season}^{max}-30*(1+z)]$ (step: 2
days) where MJD is the Modified Julian Date. The observing efficiency is
defined as the fraction of simulated light curves fulfilling the requirements
defined in Sec. 4, per redshift bin. Example curves are given in Fig. 17
(left). For a regular cadence, one would expect to have the following
(reference) shape: flat efficiency (close to 1) up to a completeness redshift
beyond which efficiency decreases down to 0. Fig. 17 (left) reveals that, in
practise, efficiency curves may significantly deviate from this reference
shape: while results achieved in season 1 are satisfactory, it appears that
observations collected in season 6 lead to poor efficiency curves (less that
50% at max). This difference is primarily explained by the cadence of
observation and the inter-night gaps that drive the sampling frequency of the
light curve measurements.

Figure 17: Left: Observing efficiencies (color curves) and SNe Ia rate (black
dashed curve) as a function of the redshift. Right: Normalised cumulative
distribution of the number of well-measured faint supernovae as a function of
redshift. The 95th percentile limit defines the $z_{\mathrm{complete}}$ value.
These plots correspond to one HEALPix pixel of the CDF-S field (with the
number 144428) of the daily strategy.

The number of well-measured SNe Ia is estimated from the combination of
efficiency curves and a SNe Ia production rate (Perrett et al., 2012):
N${}_{\mathrm{SN}}(z)$ = efficiency($z$)$\times$
Rate${}_{\mathrm{{SNe\leavevmode\nobreak\ Ia}}}(z)$. The normalised cumulative
sum of NSN is used to estimate to estimate the redshift completeness as
illustrated in Fig. 17 (right). The results are then used as input to
estimated the number of well-sampled SNe Ia up to
$z\leq\mbox{$z_{\mathrm{complete}}$}$, N${}_{\mathrm{SN}}^{z\leq
z_{\mathrm{complete}}}$.

##  Appendix B Realistic simulations using $\sigma_{\mu}$ and $N_{SN}$

The method used to estimate cosmological parameters (Section 6) relies on the
simulation of distance moduli and requires the knowledge of $\sigma_{\mu}$ and
$N_{SN}$ as a function of $z$. These two quantities were estimated from full
simulation and fit of SNe Ia light curves. The simulations were performed for
a set of redshift completeness values, namely
$z_{\mathrm{complete}}$$\in[0.50,0.90]$ (step: 0.05) so as to include
Malmquist bias effects as illustrated by Fig. 18: the dependence of
$\sigma_{\mu}$ on $z_{\mathrm{complete}}$ and the decrease of $N_{SN}$ for
$z\leavevmode\nobreak\ \geq\leavevmode\nobreak\ $$z_{\mathrm{complete}}$ is to
be explained by the redshift bias.

Figure 18: Left: mean values of $\sigma_{\mu}$ as a function of the redshift
for a set of surveys labelled by the redshift completeness. Right: number of
SNe Ia as a function of the redshift for a set of surveys labelled by the
redshift completeness for the CDF-S field (survey area of 9.6 deg2 and season
length of 180 days).

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