# The Impact of Observing Strategy on Cosmological Constraints with LSST

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  Dan Scolnic  Department of Physics, Duke University, 120 Science
Drive, Durham, NC, 27708, USA  Husni Almoubayyed  McWilliams Center for
Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA
15213, USA  Timo Anguita  Departamento de Ciencias Fisicas, Universidad Andres
Bello, Fernandez Concha 700, Las Condes, Santiago, Chile  Millennium Institute
of Astrophysics, Nuncio Monsenor Sotero Sanz 100, Of. 104, Providencia,
Santiago, Chile  Humna Awan  Leinweber Center for Theoretical Physics,
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
Department of Physics and Astronomy, Rutgers, the State University,
Piscataway, NJ 08854, USA  Eric Gawiser  Department of Physics and Astronomy,
Rutgers, the State University, Piscataway, NJ 08854, USA  Satya Gontcho A
Gontcho  Department of Physics & Astronomy, University of Rochester, 500
Wilson Blvd., Rochester, NY 14627  Philippe Gris  Laboratoire de Physique de
Clermont,IN2P3/CNRS, Université Clermont Auvergne, F-63000 Clermont-Ferrand,
France  Simon Huber  Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-
Str. 1, 85748 Garching, Germany  Physik-Department, Technische Universität
München, James-Franck-Straße 1, 85748 Garching, Germany  Saurabh W. Jha
Department of Physics and Astronomy, Rutgers, the State University,
Piscataway, NJ 08854, USA  R. Lynne Jones  Department of Astronomy, University
of Washington, Seattle, WA 98195, USA  DiRAC Institute, University of
Washington, Seattle, WA 98195, USA  Alex G. Kim  Lawrence Berkeley National
Laboratory, 1 Cyclotron Rd., Berkeley, CA, 94720, USA  Rachel Mandelbaum
McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon
University, Pittsburgh, PA 15213, USA  Phil Marshall  Kavli Institute for
Particle Astrophysics and Cosmology, P.O. Box 20450, MS29, Stanford, CA 94309,
U.S.A.  Tanja Petrushevska  Centre for Astrophysics and Cosmology, University
of Nova Gorica, Vipavska 11c, 5270 Ajdovšc̆ina, Slovenia  Nicolas Regnault
Laboratoire de Physique Nucleaire et de Hautes-Energies, Barre 12-22 1er
etage, 4 place Jussieu, 75005 Paris, France  Christian N. Setzer  The Oskar
Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm
University, AlbaNova, Stockholm, SE-106 91, Sweden  Sherry H. Suyu  Max-
Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching,
Germany  Physik-Department, Technische Universität München, James-Franck-
Straße 1, 85748 Garching, Germany  Academia Sinica Institute of Astronomy and
Astrophysics (ASIAA), 11F of ASMAB, No.1, Section 4, Roosevelt Road, Taipei
10617, Taiwan  Peter Yoachim  Department of Astronomy, University of
Washington, Seattle, WA 98195, USA  Rahul Biswas  The Oskar Klein Centre for
Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova,
Stockholm, SE-106 91, Sweden  Tristan Blaineau  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  Isobel Hook  Department of Physics,
Lancaster University, Lancaster, Lancs LA1 4YB, U.K.  Marc Moniez  Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Eric Neilsen  Fermi
National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA  Hiranya
Peiris  The Oskar Klein Centre for Cosmoparticle Physics, Department of
Physics, Stockholm University, AlbaNova, Stockholm, SE-106 91, Sweden
Department of Physics & Astronomy, University College London, Gower Street,
London WC1E 6BT, UK  Daniel Rothchild  Department of Electrical Engineering
and Computer Sciences, UC Berkeley, 253 Cory Hall, Berkeley, CA 94720-1770
Christopher Stubbs  Department of Physics, Harvard University, 17 Oxford St,
Cambridge MA 02138  Department of Astronomy, Harvard University, 60 Garden St,
Cambridge MA 02138  the LSST Dark Energy Science Collaboration

###### Abstract

The generation-defining Vera C. Rubin Observatory will make state-of-the-art
measurements of both the static and transient universe through its Legacy
Survey for Space and Time (LSST). With such capabilities, it is immensely
challenging to optimize the LSST observing strategy across the survey’s wide
range of science drivers. Many aspects of the LSST observing strategy relevant
to the LSST Dark Energy Science Collaboration, such as survey footprint
definition, single visit exposure time and the cadence of repeat visits in
different filters, are yet to be finalized. Here, we present metrics used to
assess the impact of observing strategy on the cosmological probes considered
most sensitive to survey design; these are large-scale structure, weak
lensing, type Ia supernovae, kilonovae and strong lens systems (as well as
photometric redshifts, which enable many of these probes). We evaluate these
metrics for over 100 different simulated potential survey designs. Our results
show that multiple observing strategy decisions can profoundly impact
cosmological constraints with LSST; these include adjusting the survey
footprint, ensuring repeat nightly visits are taken in different filters and
enforcing regular cadence. We provide public code for our metrics, which makes
them readily available for evaluating further modifications to the survey
design. We conclude with a set of recommendations and highlight observing
strategy factors that require further research.

tablenum SIXtablenum

##  1 Introduction

The Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST), with its
ability to make rapid, deep observations over a wide sky area, will deliver
unprecedented advances in a diverse set of science cases (Abell et al., 2009,
hereafter The LSST Science Book).

LSST has an ambitious range of science goals that span the Universe: solar
system studies, mapping the Milky Way, astrophysical transients, and
cosmology; these are all to be achieved with a single ten-year survey. Around
80% of LSST’s observing time will be dedicated to the main or WFD survey,
which will cover at least 18,000 deg2. The remainder of the time will be
dedicated to “mini-surveys” (for instance, a dedicated Galactic plane survey),
“deep drilling fields” (DDFs) and potentially, “targets of opportunity”
(ToOs).

Because LSST has such broad science goals, the choice of observing strategy is
a difficult but critical problem. Important early groundwork was laid in the
community-driven paper on LSST observing strategy optimization (LSST Science
Collaborations et al., 2017, hereafter COSEP). To further address this
challenge, in 2018, the LSST Project Science Team and the LSST Science
Advisory Committee released a call for community white papers proposing
observing strategies for the LSST WFD survey, as well as the DDFs and mini-
surveys (Ivezić et al., 2018). In response to this call, the DESC Observing
Strategy Working Group (hereafter DESC OSWG) performed a detailed
investigation of the impact of observing strategy on cosmology with LSST. The
DESC OSWG made an initial set of recommendations for both the WFD (Lochner et
al., 2018) and DDF (Scolnic et al., 2018b) surveys, with proposals for an
observing strategy that will optimize cosmological measurements with LSST.
Following this call, many new survey strategies have been simulated to answer
the ideas in various white papers submitted; these strategies are discussed in
Jones et al. (2021). Furthermore, a Survey Cadence Optimization Committee
(SCOC) was formed111https://www.lsst.org/content/charge-survey-cadence-
optimization-committee-scoc with the charge of guiding the convergence of
survey strategy decisions across the multiple LSST collaborations. The SCOC
released a series of top-level survey strategy
questions222https://docushare.lsst.org/docushare/dsweb/Get/Document-36755,
where answers can be supported using analyses of the simulations in Jones et
al. (2021). In this paper, we evaluate a number of simulated LSST observing
strategies to support decisions on the survey strategy.

A review of the dark energy analyses planned by the DESC (which is a subset of
the fundamental cosmological physics that will be probed by LSST) is given in
the DESC Science Requirements Document (The LSST Dark Energy Science
Collaboration et al., 2018; hereafter DESC SRD). Each analysis working group
(weak lensing, large-scale structure, galaxy clusters, type Ia supernovae and
strong lensing) within DESC provided a forecast of the constraints on dark
energy expected from their probe, given a baseline observing strategy. As a
metric, the DESC SRD used the Dark Energy Task Force Figure of Merit (DETF
FoM), defined as the reciprocal of the area of the contour enclosing 68% of
the credible interval constraining the dark energy parameters, $w_{0}$ and
$w_{a}$, after marginalizing over other parameters (Albrecht et al., 2006).
Once statistical constraints were quantified, each group determined the
control of systematic uncertainties needed to reach the goals for a Stage IV
dark energy mission.

Three ways to increase the likelihood of achieving the goals set out in the
DESC SRD are to (a) improve the statistical precision of each probe, (b)
reduce each probe’s sensitivity to systematic uncertainties, or (c) reduce the
total uncertainty when combining multiple probes. Changes in observing
strategy have the potential to affect each of these. For instance, observing
strategies that yield more uniform coverage across the survey footprint, or
strategies with improved cadence, can have a strong impact on both the
statistical precision and the systematic control.

DESC encompasses multiple cosmological probes and it is the ultimate goal of
the DESC OSWG to be able to compute the combined DETF FoM using all probes, as
well as other combined metrics, for each proposed observing strategy. However,
at the level of LSST precision, careful treatment of systematic effects is
required, and work is still ongoing to include these in the forecasting
analysis tools. In addition, a full cosmological analysis is computationally
intensive and not feasible to test on hundreds of simulated LSST observing
strategies. Thus, while we include an emulated DETF FoM for certain dark-
energy probes, we also introduce a suite of metrics that are anticipated to
correlate with cosmological constraints but that are faster to run and simpler
to interpret. Most of the metrics in this paper are focused on the WFD survey,
but we make use of many of the same metrics (particularly for supernova
cosmology) for the DDFs. It should be noted that one of the cosmological
probes mentioned, galaxy clusters, is not explicitly included in our analysis.
This is because it is expected that clusters will have identical requirements
to large-scale structure and so should already be accommodated.

While the metrics we have developed focus entirely on the extragalactic part
of the WFD survey, there is one cosmological probe that relates to
observations near the Galactic plane: the study of dark matter with
microlensing. Microlensing is the light magnification due to the transit of a
massive compact object (lens) close enough to the line of sight of a
background star (Paczynski, 1986). The search for the dark matter component of
intermediate mass black holes within the Milky Way through microlensing
involves several months timescale events, which can be efficiently detected
only with long time-scale surveys such as LSST (Mirhosseini, A. & Moniez, M.,
2018). This search will not be sensitive to the details of observing strategy,
as long as gaps larger than a few months are avoided. Thus, for this work we
focus only on extragalactic probes.

Although all the metrics described in this paper are useful for understanding
the impact of observing strategy on cosmological measurements with LSST, some
are more closely related to the primary cosmology goals as outlined in the
DESC SRD than others. The 3x2pt correlation function, which measures structure
growth, and supernovae, which probe the expansion history of the Universe,
have the most constraining power. However, novel probes such as strong lensing
and kilonovae can be complementary and offer unique tests of cosmology beyond
the DETF FoM. Our recommendations and conclusions are generally guided by the
priorities outlined in the DESC SRD, but we attempt to quantify performance of
observing strategies in terms of the scientific opportunities offered by more
novel probes as well.

We structure the paper as follows: Section 2 outlines the factors that affect
LSST observing strategy, the simulator used and resulting sets of simulations
of different observing strategies. We split the metrics descriptions as
follows: general static science metrics (Section 3); static science-driven
metrics (Section 4), which include WL, LSS and photometric redshifts; and
transient science-driven metrics (Section 5), which include supernovae, strong
lensing of supernovae/quasars, and kilonovae. We draw together the results of
the analysis of our metrics on various simulated observing strategies in
Section 6 and Section 7. In addition to describing the analysis supporting the
proposal for various observing strategy choices, we provide metrics,
recommendations and conclusions in this paper that are meant to be of more
general use to future large-scale cosmology surveys.

##  2 LSST Observing Strategy

In this section, we describe the Rubin Observatory, the baseline LSST
observing strategy, the software used to generate realistic LSST observing
schedules that we make use of in this work, and the metrics framework used to
evaluate different strategies.

###  2.1 LSST Overview

An overview of the Vera C. Rubin Observatory telescope specifications can be
found in Ivezić & the LSST Science Collaboration (2013); we summarize here the
specifications that impact observing strategy. The Rubin Observatory is under
construction in the Southern Hemisphere, in Cerro Pachón in Northern Chile,
and will undertake the Legacy Survey of Space and Time (LSST); a 10-year
survey expected to start in 2023. The system has an 8.4m (6.7m effective)
diameter primary mirror, a 9.6 deg2 field of view, and a 3.2 Gigapixel camera.
The integrated filter exchange system can hold up five filters at a time, and
there are six filters available: $ugrizy$ that cover a wavelength range of
320–1050 nm. Typical $5\sigma$ depth (i.e., the apparent brightness in
magnitudes at which a point source is detected at $5\sigma$ significance) of
30 s exposures in $ugrizy$ are $23.9,25.0,24.7,24.0,23.3,22.1$ mag and co-
added over the full survey will reach approximately
$26.1,27.4,27.5,26.8,26.1,24.9$ mag. Several performance specifications
influence the survey cadence333The cadence is defined as the median inter-
night gap over a season.: filter change time (120 s), closed optics loop delay
or slews where altitude changes by more than 9 degrees (36 s), read time (2
s), shutter time (1 s), median slew time between fields (4.94 s)444These times
are estimated from the expected performance from the various components of the
telescope and camera and are what is used in the scheduling simulators. The
estimated fraction of photometric time is 53% of the 10-year survey. Standard
‘visit’ exposures are typically 30 seconds long (referred to as $1\times 30$
s). An alternative exposure strategy, to mitigate cosmic-ray and satellite
trail artifacts, are two successive short exposures called “snaps” (this is
referred to as $2\times 15$ s). The decision between these exposure strategies
has not yet been made. Throughout this paper, all simulations use $1\times 30$
s exposures, however we will return to this point in Section 6 to explicitly
examine the impact of using $2\times 15$ s exposures instead.

###  2.2 LSST Observing Strategy Requirements

The observing strategy of LSST is impacted by several factors, and its
optimization is a complex challenge. The LSST Science Requirements Document
(Ivezić & the LSST Science Collaboration, 2013, hereafter LSST SRD) defines
top-level specifications for the survey such that:

  * •

The sky area uniformly covered by the main survey will include at least 15,000
deg2, and 18,000 deg2 by design.

  * •

The sum over all bands of the median number of visits in each band across the
sky area will not be smaller than 750, with a design goal of 825.

  * •

At least 1,000 deg2 of the sky, with a design goal of 2,000 deg2, will have
multiple observations separated by nearly uniformly sampled time scales
ranging from 40 sec to 30 min.

There are other additional requirements on PSF ellipticity correlations and
parallax constraints, the former of which will be indirectly analyzed in this
paper.

Given that these requirements only constrain a few aspects of observing
strategy, many remaining factors can still be optimized to maximize scientific
return.

###  2.3 Baseline Strategy

Here we summarize the baseline observing strategy (Jones et al., 2021), which
has evolved significantly over the years. The strategy described here is
considered the current nominal observing strategy plan pending further
modifications:

  * •

Visits are always $1\times 30$ s long (not $2\times 15$ s)555We note that the
choice between $1\times 30$ s and $2\times 15$ s exposures will not be
finalized until the commissioning phase of the LSST survey.. The baseline
simulation achieves about 2.2M visits over 10 years.

  * •

Pairs of visits in each night are in two filters as follows: $u-g$, $u-r$,
$g-r$, $r-i$, $i-z$, $z-y$ or $y-y$. Pairs are scheduled for approximately 22
minutes separation. Almost every visit in $gr$ or $i$ has another visit within
50 minutes. These visit pairs assist with asteroid identification.

  * •

The survey footprint is the standard baseline footprint, with 18,000 deg2 in
the WFD survey spanning declinations from -62 to +2 degrees (excluding the
Galactic equator), and additional coverage for the North Ecliptic Spur (NES),
the Galactic Plane (GP) and South Celestial Pole (SCP). The baseline footprint
includes WFD, NES ($griz$ only), GP, and SCP. WFD is $\sim 82\%$ of the total
time.

  * •

Five DDFs are included666Details of the four selected DDFs can be found here:
https://www.lsst.org/scientists/survey-design/ddf, with the fifth field being
composed of two pointings covering the Euclid Deep Field - South
(EDF-S)777This field has not been officially confirmed as part of the LSST
survey. Information on the Euclid field can be found here:
https://www.cosmos.esa.int/web/euclid/euclid-survey, devoting 5% of the total
survey time to DD fields.

  * •

The standard balance of visits between filters is 6% in $u$, 9% in $g$, 22% in
$r$, 22% in $i$, 20% in $z$, and 21% in $y$.

  * •

Owing to the limitation of five installed filters in the camera filter
exchange system, if at the start of the night the moon is 40% illuminated or
more (corresponding to approximately full moon +/- 6 nights), the $z$-band
filter is installed; otherwise the $u$-band filter is installed.

  * •

The camera is rotationally dithered nightly between -80 and 80 degrees. At the
beginning of each night, the rotation is randomly selected. The camera is
rotated to cancel field rotation during an exposure then reverts back to the
chosen rotation angle for the next exposure.

  * •

Twilight observations are done in $r$,$i$,$z$,$y$, and are determined by a
‘greedy’ algorithm which takes the maximum of the reward function at each
observation.

  * •

Non-rolling cadence: the nominal baseline strategy observes the entire
footprint each observing season. A _rolling cadence_ would prioritize sections
of the footprint at different times (e.g., observing half the footprint for
one year and changing to the other half the next) to improve cadence in that
section.

###  2.4 Survey Simulators

The simulations analyzed here are created using the Feature-Based Scheduler
(FBS, Naghib et al., 2018), which uses a modified Markov Decision Process to
decide the next observing direction and filter selection, allowing a flexible
approach to scheduling, including the ability to compute a detailed reward
function throughout a night. FBS is the new default scheduler for the LSST,
replacing the LSST Operations Simulator (OpSim Ridgway et al., 2010; Delgado
et al., 2014).

We note that, besides the LSST default schedulers (OpSim and FBS), there is an
alternate scheduler, AltSched, presented in Rothchild et al. (2019). AltSched
is a simple, deterministic scheduler, which ensures that telescope
observations take place as close to the meridian as possible, alternating
between sky regions North and South of the observatory latitude on alternate
nights, while cycling through the filter set and changing filters after
observing blocks. We do not include simulations from this scheduler; however
we note that its approach does produce encouraging results.

###  2.5 Observing Strategy Simulations

Sets of simulations have been periodically released for use by the community.
Table 1, we summarize the families of simulations used, number of simulations
in each family, and versions888Simulations can be downloaded at http://astro-
lsst-01.astro.washington.edu:8081/. New versions of simulated strategies are
released regularly with improvements to the scheduler, weather simulation and
changes to the baseline strategy. In this paper, we mostly focus on version
1.5 simulations, but select v1.6 and v1.7 simulations are included in certain
plots (see Jones et al., 2021, for details of the simulations). Certain
simulations are excluded from specific plots because they are unrealistic or
differ significantly from the baseline (for instance, with a dramatically
different footprint or visit allocation in WFD). It is very important to note
that for each version, the baseline changes somewhat. In particular, the
default choice of exposure strategy has changed from $2\times 15$ s in older
versions, to $1\times 30$ s in v1.5 and v1.6 and back to $2\times 15$ s in
v1.7, which has a large impact on overall efficiency and hence metric
performance. All figures in this paper are for relative improvements compared
to the baseline strategy corresponding to that simulation’s version.
LABEL:sec:simulations_appendix captures in detail exactly which simulations
are used for which plot and the corresponding baseline simulation. Section 2.6
is a lookup table for the short, simpler names for the simulations used in
some figures in this paper.

Table 1: Description of FBS simulation families used including a descriptive name of each family, the FBS version, the number of simulations, and a brief explanation of the family. Name | FBS Version | # of simulations |  Description
---|---|---|---
Baseline | 1.4/1.5/1.6 | 3 |  Baseline as described, with choice of 2 versus 1 snap, and mixed filters or not
U_pairs | 1.4 | 12 |  Varies how $u$-band visits are paired with other filters, how many visits occur in $u$-band and when $u$-band is loaded in or out of filter
Third_obs | 1.5 | 6 |  Adds third visit to some fields night, where total amount of time dedicated to these visits is 15–120 minutes per night
Wfd_depth | 1.5 | 16 |  Amount spent on WFD compared to other areas changes from 60–99%
Footprints | 1.5 | 12 |  Changes in WFD footprint, in North/South, Galactic coverage, Large/Small Magellanic Clouds
Bulge | 1.5 | 6 |  Different strategies for observing the Galactic bulge
Filter_dist | 1.5 | 8 |  Varying the ratios of time spent in $u,g,r,i,z,y$ filters
Alt_roll_dust | 1.5 | 3 |  Dust-limited WFD footprint with the alt-scheduler scheduling algorithm, with and without rolling, where a rolling strategy only observes a set fraction of the survey footprint each year
DDFs | 1.5 | 3 |  Different strategies for the DDFs, ranging from 3–5.5% of the total survey time
Goodseeing | 1.5 | 5 |  Aims to acquire at least one good-seeing visit at each field each year, varies which filters it’s needed for
Twilight_neo | 1.5 | 4 |  Adds a mini-survey during twilight to search for Near-Earth Objects (NEOs)
Short exposures | 1.5 | 5 |  Adds short exposures in all filters, from 1–5 seconds, 2–5 exposures per year
U60 | 1.5 | 1 |  Swaps 60 second $u$-band visit instead of 30 s
Var_expt | 1.5 | 1 |  Changes exposure time so that the single image depth is constant
DCR | 1.5 | 6 |  Adds high airmass observations in different combinations of filters 1 or 2 times per year
Even_filters | 1.6 | 4 |  Bluer filters are observed in moon bright time
Greedy footprint | 1.5 | 1 |  A greedy survey not run on ecliptic, where a portion of the sky that has the highest reward function is observed two times over a given time span (about 20–40 minutes).
Potential Schedulers | 1.6 | 17 |  Multiple variations at once for a particular science goal.

###  2.6 Proxy Metrics and Metrics Analysis Framework

As stated in Section 1, the ultimate goal of the DESC OSWG is to compute
cosmology figures of merit to evaluate observing strategies. However, this is
difficult and computationally intensive, making such an approach impractical
for evaluating many simulations. We thus largely focus on proxy metrics, which
can be quickly computed on any simulation. We make use of and incorporate our
metrics into the Metrics Analysis Framework (MAF, Jones et al., 2014). MAF is
a python framework designed to easily evaluate aspects of the simulated
strategies. It can compute simple quantities, such as the total co-added depth
or number of visits, but it can also be used to construct more complex metrics
that can evaluate the expected performance of a simulation for a given science
case. Here, we use a combination of independent codes (which are too slow to
run as part of MAF) and custom MAF metrics to evaluate the simulated observing
strategies. MAF metrics and external metrics created for this paper are linked
when described. Unless otherwise specified, each metric is run on a simulation
of the full ten year survey.

We note that in order to compare metrics directly, they must be transformed to
be able to be interpreted as “larger is better” and placed on the same scale.
Appendix A includes a table that describes how all metrics are transformed. In
all plots, to put the metrics on the same scale, they are normalized using
their values for the baseline simulation (which is different for each FBS
version) as:

$x_{\rm{normed}}=\frac{x-x_{\rm{baseline}}}{x_{\rm{baseline}}},$ (1)

where $x$ is the metric in question, $x_{\rm{baseline}}$ is the value of that
metric for the corresponding baseline simulation and $x_{\rm{normed}}$ is the
normalized metric.

A final point to note before introducing the metrics is that the focus of this
paper is the optimization of the WFD survey; thus all metrics are evaluated
only on the WFD observations of each simulation. However, we include some
discussion of DDF optimisation in Section 7.2, which is particularly important
for supernovae and also photometric redshift calibration.

Table 2: A lookup table for the shorter, simpler names used in the figures in
Section 4 and Section 5.

Simulation name | Short name
---|---
baseline_v1.5_10yrs.db | Baseline
baseline_2snapsv1.5_10yrs.db | 2x15s exposures
bulges_bs_v1.5_10yrs.db | More visits in Galactic bulge
footprint_big_sky_dustv1.5_10yrs.db | Larger extragalactic footprint
footprint_bluer_footprintv1.5_10yrs.db | Bluer filter distribution
footprint_newAv1.5_10yrs.db | Galactic plane focus
var_expt_v1.5_10yrs.db | Variable exposure
wfd_depth_scale0.65_noddf_v1.5_10yrs.db | 65% of visits in WFD
wfd_depth_scale0.99_noddf_v1.5_10yrs.db | 99% of visits in WFD
ss_heavy_v1.6_10yrs.db | Solar system focus
combo_dust_v1.6_10yrs.db | Combo dust

##  3 General Static Science Metrics

In this section, we introduce metrics relevant to static science topics that
will be useful to multiple cosmological probes. This includes metrics related
to general WFD characteristics as well as to photometric redshift (photo-$z$)
characteristics from the WFD galaxy sample.

###  3.1 WFD Depth, Uniformity and Area

In this subsection, we introduce three sets of metrics, where each set
includes information after years Y1, Y3, Y6 and Y10 respectively (where Y1
refers to the data collected after the first year, Y3 after the first three
years, etc.). Figure 1 shows the results from these metrics.

  1. 1.

Y1/Y3/Y6/Y10 area for static science (deg2): The effective survey area for
static science after Y1/Y3/Y6/Y10; this area is limited by depth and
extinction, and requires coverage in all 6 bands, and is described in more
detail below. Note that this area is also referred to as “the extragalactic
area” in later discussion.

  2. 2.

Y1/Y3/Y6/Y10 median $i$-band coadded depth: The median $i$-band 5$\sigma$
coadded depth in the effective survey area for static science after
Y1/Y3/Y6/Y10.

  3. 3.

Y1/Y3/Y6/Y10 $i$-band coadded depth stddev: The standard deviation of the
$i$-band 5$\sigma$ coadded depth distribution in the effective survey area for
static science after Y1/Y3/Y6/Y10, quantifying the non-uniformity in depth
coverage; smaller values of this metric indicate a more uniform survey.

Figure 1: Static science metrics as a function of selected observing
strategies. Section 2.6 contains the exact simulation names corresponding to
the short names used here. Note that as for all plots in this paper, metrics
are transformed using the equations in LABEL:tab:metric_transforms and are
taken relative to their values at baseline in order to be directly comparable,
with larger values always being better.

We follow the LSST Science Book in using $i$-band to track the brightness
limit of galaxies that can be detected in the survey, motivated by the fact
that almost all galaxies are brighter in $i$ than in $g$ or $r$, while the co-
added depths are similar for these three filters. We note that this could be
misleading when comparing observing strategies that vary the relative time
spent observing in $i$ versus other bands by a significant factor.

The metrics above are calculated using
HEALPix999http://healpix.sourceforge.net/ (Górski et al., 2005) maps, with
pixel resolution of 13.74 arcmin (achieved using the HEALPix resolution
parameter $N_{\mathrm{side}}$ = 256). For extragalactic static science using
high S/N measurements, we must restrict our analysis to a footprint that
provides the deep, high S/N galaxy samples needed for our science. To achieve
this, we implement an extinction cut and a depth cut, retaining only the area
with E(B-V)$<$0.2 (where E(B-V) is the dust reddening in magnitudes) with
limiting $i$-band coadded 5$\sigma$ point source detection depths of 24.65 for
Y1, 25.25 for Y3, 25.62 for Y6, and 25.9 for Y10; the E(B-V) cut ensures that
we consider the area with small dust uncertainties101010As noted by e.g.,
Leike & Enßlin (2019), the behavior of Galactic dust becomes more uncertain as
the amount of dust increases, and Schlafly & Finkbeiner (2011) identify
E(B-V)=0.2 as a threshold where the dust properties change. while the depth
cut ensures that we have high S/N galaxies, with the Y10 cut fixed by the LSST
SRD goal of yielding a “gold sample” of galaxies with $i$-band coadded
5$\sigma$ (extended source detection depth), $i<25.3$ after Y10. This is
achieved using a the MAF Metric object, egFootprintMetric111111
https://github.com/humnaawan/sims_maf_contrib/blob/master/mafContrib/lssmetrics/egFootprintMetric.py.

####  3.1.1 Uniformity and dithering

Survey uniformity, as measured by our $i$-band coadded depth stddev metric, is
critical for all static science probes. Non-uniformity can be introduced by
spending more observing time or having better atmospheric conditions in
certain parts of the sky, or when a survey is tiled and the overlaps in fields
introduce an artificial structure to the survey. The latter effect can be
effectively mitigated using dithering: small offsets in the pointing of the
telescope when it returns to a field (see, e.g., Awan et al., 2016). Dithering
can be translational or rotational, both of which are useful for reducing
different types of systematics. Figure 1 shows that the stddev metric varies
by less than 5% across the simulations, meaning that the current observing
strategy proposals are implementing effective translational dithering
strategies. While most metrics in this paper focus on the performance of the
full ten-year LSST survey, we note that it is important that survey uniformity
is achieved at specific release intervals, such as Y1, Y3, Y6 and Y10, in
order to enable periodic analyses of datasets suitable for cosmology. The
current baseline achieves this by default, but it will be important to
consider if a strategy is chosen whereby only parts of the footprint are
observed each season (so-called rolling cadence).

####  3.1.2 General Conclusions from Static Science

Static science systematics can be reduced by increasing survey uniformity via
frequent translational and rotational dithers; the impacts of these can be
probe-specific, as noted below. We note that Y1 is especially sensitive to the
specific cadence, and while the different kinds of cadences/footprints
converge for Y3-Y10 area, very few simulations yield close to the desired
18,000 deg2 WFD area for extragalactic science.

We also note that spectroscopic observations in the extragalactic part of the
survey will be essential to calibrate LSST photometric redshifts. For this
purpose, overlap with upcoming spectroscopic surveys is quite critical and is
further discussed in Section 6.

###  3.2 Photometric Redshifts

While photometric redshifts impact multiple probes, including transients such
as supernovae, they are in turn not affected by time-sensitive aspects of
observing strategy. We thus generally include photo-$z$ metrics with the
static science metrics.

We introduce four metrics for the quality of photo-$z$ determination in the
WFD. A summary of the results of these metrics can be seen in Figure 2.

  1. 1.

Photo-$z$ standard deviation at high (1.8–2.2) and at low (0.6–1.0) redshift.

  2. 2.

Outlier fraction at high and at low redshift.

We evaluate the relative quality of simulated photometric redshift estimates
for each simulation by determining the average coadded depth in extragalactic
fields, and using those depths to simulate observed apparent magnitudes and
photo-$z$ for a mock galaxy catalog using the color-matched nearest neighbors
(CMNN) photometric redshift estimator Graham et al. (2018)121212A
demonstration of the CMNN photo-$z$ estimator is available on GitHub at
https://github.com/dirac-institute/CMNN_Photoz_Estimator.. The CMNN estimator
does not produce the “best” or “official” LSST photo-$z$, but does produce
results for which the relative quality of the photo-$z$ is directly related to
the input photometric quality, and thus is an appropriate photo-$z$ estimator
for evaluating the bulk impact on photo-$z$ results due to changes in the
photometric depth of a survey.

First, we determine the 5$\sigma$ point-source limiting magnitudes of the
10-year coadded images from the WFD program in sky regions ($\sim$220 arcmin
wide) for all simulations. We consider extragalactic fields as appropriate for
cosmological studies if their Galactic dust extinction is E(B-V)$<$0.2 mag,
and if they receive at least 5 visits per year in all six filters (i.e., the 5
visits define a minimum coadded depth). The median 10-year 6-filter depths
across all such appropriate extragalactic fields for each simulation are then
input to the CMNN photo-$z$ estimator. The CMNN estimator uses the depths to
to synthesize apparent observed magnitudes and errors for a simulated galaxy
catalog; then it splits the catalog into test and training sets, and uses the
training set to estimate photo-$z$ for the test set. The 5$\sigma$ depths are
the only input; thus, only observing strategies which result in photometric
depths that differ significantly from the baseline cadence will result in
significantly different photometric redshift results.

We used the same mock galaxy catalog as described in Graham et al. (2018) and
Graham et al. (2020), which is based on the Millennium simulation (Springel et
al., 2005) and the galaxy formation models of Gonzalez-Perez et al. (2014),
and was fabricated using the lightcone construction techniques described by
Merson et al. (2013)131313Documentation for this catalog can be found at
http://galaxy-catalogue.dur.ac.uk. To both the test and training sets we apply
cuts on the observed apparent magnitudes of 25.0, 26.0, 26.0, 25.0, 24.8, and
24.0 mag in filters ugrizy, and enforce that all galaxies are detected in all
six filters. These cuts are all about half a magnitude brighter than the
brightest 5$\sigma$ depth of any given simulation we considered. This cut is
applied because it imposes a galaxy detection limit across all simulations
that is independent of the depth (i.e., independent of the photometric
quality). If such a cut is not imposed, the default setting is for the CMNN
estimator to apply a cut equal to the 5$\sigma$ limiting magnitude. This
default setting results in more fainter galaxies being included in the test
and training sets for simulations with deeper coadds. Although this default
setting is realistic – fainter galaxies are included in real galaxy catalogs
made from deeper coadds – due to the fact that fainter galaxies generally have
poorer-quality photo-$z$ estimates, this also results in some simulations with
deeper coadded depths appearing to produce worse photo-$z$ estimates. These
bright magnitude cuts ensure an “apples-to-apples” comparison across all
simulations.

All other CMNN estimator input parameters are left at their default values,
except for the number of test (50,000) and training (200,000) set galaxies,
and the minimum number of colors which is set to 5 (from a default of 3) to
only include galaxies that are detected in all six filters. The other CMNN
parameters141414Other CMNN parameters include, e.g., the minimum number of
CMNN training-set galaxies, the mode of determining the photo-$z$ from the
CMNN subset, and the percent-point function value which is used to generate
the CMNN subset. impact the final absolute photo-$z$ quality, and so it is
important to keep in mind that the results of the CMNN estimator should not be
interpreted as absolute predictions for the photo-$z$ quality, but as relative
predictions for the photo-$z$ quality produced by different observing
strategies (i.e., different 10-year coadded depths). It is important to note
that, because the test and training sets are drawn from the same population,
they have the same apparent magnitude and redshift distributions. This
contrived scenario in which the training set is perfectly representative of
the test set does not produce photo-$z$ results with realistic systematic
effects or biases. Additionally, we use input parameters for the CMNN
estimator that produce photo-$z$s with a very good absolute quality. The
combination of nearly perfectly matched test and training sets, optimized
input parameters, and bright magnitude cuts all result in very small bias
values (where bias is the average of $|z_{true}-z_{phot}|$ over all test
galaxies) for our simulations, which is why the photo-$z$ bias is not being
used as one of the metrics (described below) for evaluating the simulations.
In future photo-$z$ simulations, variations in the test and training sets
could be established which correspond to different simulations (e.g., building
a deep training set from the DDFs) – but we consider this out of the scope of
the present analysis.

We evaluate the photo-$z$ results with two statistics: precision and inlier
fraction. To calculate the precision, we first reject catastrophic outliers
with $|z_{\rm true}-z_{\rm phot}|>1.5$ (this is a non-standard definition,
chosen for this simulated data set, and used also in Graham et al. 2020). Then
we calculate the robust standard deviation in the photo-$z$ error, $\Delta
z_{1+z}=(z_{\rm true}-z_{\rm phot})/(1+z_{\rm phot})$, as the full-width at
half maximum (FWHM) of the interquartile range (IQR) divided by $1.349$, and
use this as the precision. The inlier fraction is one minus the outlier
fraction, which is the number of galaxies with a photo-$z$ error greater than
three times the standard deviation or greater than three times 0.06, whichever
is larger (this definition matches that used for photo-$z$ outliers by the
LSST SRD. We calculate these two statistics for a low-$z_{\rm phot}$ bin
(0.6-1.0) and a high-$z_{\rm phot}$ bin (1.8-2.2), for each of the
simulations.

The results are shown in Figure 2. As mentioned above, the results of the CMNN
estimator should be considered as relative predictions for the photo-$z$
quality. Thus in Figure 2 we show the results as fractional changes from the
results for the baseline simulation.

Figure 2: Photometric redshift metrics as a function of selected observing
strategies. Section 2.6 contains the exact simulation names corresponding to
the short names used here. As described in Section 3.2, the low $z$ bin is
(0.6–1.0) and the high $z$ bin is (1.8–2.2).

####  3.2.1 General Conclusions from Photometric Redshifts

We find that the photo-$z$ quality is optimized by observing strategies that
lead to deeper limiting magnitudes, which is as expected. As there is a trade-
off in any survey between depth and area, and because areal coverage is
required by a variety of LSST science goals, we recognize that the observing
strategies which optimize photo-$z$ quality might not be optimal for
cosmological studies. Different science cases have different photo-$z$ needs,
and a single photo-$z$ metric that captures the science impact is very
challenging to define. Full end-to-end simulations of scientific results that
incorporate photo-$z$ quality would be the correct approach, but are beyond
the scope of this work.

##  4 Weak Lensing and Large-Scale Structure

###  4.1 Weak Lensing

Weak gravitational lensing (WL) is the deflection of light from distant
sources due to the gravitational influence of matter along the line-of-sight.
In practice, the coherent distortions of background galaxy shapes, or ‘shear’
(measured in different redshift ranges), reveal the clustering of matter as a
function of time, including both luminous and dark matter. The evolution of
matter clustering is affected by the expansion history of the Universe, which
means that WL is also sensitive to the accelerated expansion rate of the
Universe caused by dark energy (for a review, see Kilbinger, 2015).

We introduce two new metrics associated with WL. A summary of the results from
the WL metrics can be found in Figure 3.

  1. 1.

WL+LSS Figure of Merit (Section 4.1.1)- DETF Figure of Merit for cosmological
WL and large-scale structure measurement; larger numbers correspond to larger
statistical constraining power.

  2. 2.

WL Average visits (Section 4.1.2)- Average visits metric in $r$, $i$, and
$z$-bands; higher numbers are better for WL shear systematics mitigation.

Figure 3: WL metrics as a function of selected observing strategies. Section
2.6 contains the exact simulation names corresponding to the short names used
here. As shown, the metrics that describe statistical constraining power (‘WL
+ LSS DETF Figure of Merit’) and reduction in systematic uncertainties in WL
shear (‘WL Average visits’) do not necessarily correlate, as the former tends
to prefer increased area while the latter tends to prefer an increased number
of visits at each point in the WFD survey on average. Thus, observing strategy
changes such as increasing the fraction of time in the WFD survey can increase
both metrics, while observing strategy changes that increase area at the
expense of decreasing the average number of visits across the survey area will
lead to opposing changes in the metrics.

####  4.1.1 WL+LSS Figure of Merit

The 3x2pt (DETF) Figure of Merit (FoM) is the inverse of the area of the 68%
confidence interval in the space of the dark energy equation of state
parameters $w_{0},w_{a}$, and is an indicator of the statistical constraining
power of the survey’s static probes. In this section, we follow the convention
taken by ongoing WL surveys that the canonical analysis is a joint measurement
of WL and large-scale structure. Since forecasting the cosmological
constraining power for such a measurement is quite resource-intensive, our
approach is to carry out the calculation for a limited number of survey
strategies, and use that to build an emulator of the 3x2pt FoM for arbitrary
scenarios based on interpolation. This section briefly describes the emulation
process for the 3x2pt FoM; for more detail, see Eifler & Motka in
prep.151515https://github.com/hsnee/sims_maf/blob/master/python/lsst/sims/maf/metrics/summaryMetrics.py

The emulator is defined in a six-dimensional parameter space, where the
dimensions are: effective survey area, survey median depth in $i$-band,
systematic uncertainty in the WL shear calibration, photometric redshift
scatter, and the size of priors on photometric redshift bias and scatter. A
total of 36 points in this parameter space were selected to build the
emulator, following a Latin hypercube design (often used for efficient
sampling of high-dimensional parameter spaces in cosmological emulators -
e.g., Mead et al. 2021). For each selected point in this parameter space, the
galaxy redshift distributions, and the observable quantities for joint WL and
large-scale structure measurement, and their covariance matrix are calculated
using CosmoLike (Krause & Eifler, 2017). This joint WL and large-scale
structure measurement is often referred to as ‘3x2pt’, as it involves the
combination of three two-point correlation functions: shear-shear, galaxy-
shear, and galaxy-galaxy correlations. The 3x2pt FoM is then calculated using
MCMC constraints on cosmological parameters (marginalizing over key sources of
astrophysical systematic uncertainty, such as galaxy bias, intrinsic
alignments, and baryonic physics) in a simulated likelihood analysis based on
the observables and their covariances. The reason to marginalize over those
systematic uncertainties when computing the FoM is that this is the process
used in a real WL analysis to propagate the aforementioned systematic
uncertainties into uncertainties on cosmological parameters. The emulator was
then built from those 36 FoM values in the six-dimensional space using a
Gaussian process regression.

Area and median depth are calculated after making an extinction cut of
$E(B-V)<0.2$, to exclude high extinction areas, along with minimum depth cuts
of 24.5 and 25.9 for Y1 and Y10 respectively, to ensure that the survey depth
is relatively homogeneous; as well as a cut that guarantees at least some
coverage in all 6 bands to ensure photo-$z$ quality. These cuts are consistent
with the extragalactic cuts applied in Section 3.1. See the DESC SRD for more
details about sample definition and methods of estimating redshift
distributions and number counts, though the figures of merit in that document
were calculated with slightly different choices of redshift binning and
modeling of systematic uncertainty.

Note that the plots in this paper only use the area and depth dimensions of
the emulator, with the remaining parameters fixed to their fiducial values
from the DESC SRD. We used the emulator with marginalization over default
values for the photometric redshift systematics parameters; therefore the only
effects it captures are the varying area and median depth of the different
survey strategies. The effect of area and median depth changes are the
dominant factors for the emulator, with changes in photometric redshift
systematics being subdominant. For example, a 15% change in photometric
redshift variance (maximum change for the strategies considered in this work)
would cause a $\sim$ 2% change in the 3x2pt FoM (Eifler & Motka in prep),
while a similar change in area or median depth changes the 3x2pt FoM by $\sim$
10%. Changes due to the priors on the photometric redshift variance and bias
have negligible effects for the type of strategies considered (specifically,
the considered changes of overlap with spectroscopic surveys used for
calibrating photo-$z$ errors), and are likewise not included in this work. As
a rule, the 3x2pt FoM prefers greater median depths and larger survey areas.
The 3x2pt FoM metric described here is an improved version of the associated
metric presented in Lochner et al. (2018); while the improved version includes
more sources of systematic uncertainty, its trends with survey area and depth
are similar to those in Lochner et al. (2018).

####  4.1.2 WL Average visits and WL Shear Systematics

The statistical constraining power for WL was covered in the previous
subsection. For this reason, the text below summarizes observing strategy
considerations related to WL shear systematics in the WFD survey, for which a
metric has been introduced in
MAF161616https://github.com/lsst/sims_maf/blob/master/python/lsst/sims/maf/metrics/weakLensingSystematicsMetric.py.
See Almoubayyed et al. (2020) for further detail.

WL analysis typically involves measuring coherent patterns in galaxy shapes
due to WL shear. For this reason, any effects that are not associated with WL
but that cause apparent galaxy shape distortions with any spatial coherence
must be well understood and controlled to avoid the measurement being
systematics-dominated. The LSST provides a new opportunity to control WL
systematics using the observing strategy. This opportunity was not as feasible
in previous surveys because the LSST will be the first survey to dither at
large scales (relative to the field of view) with a very large number of
exposures. This large number of exposures means that a source of systematic
error with a particular spatial direction in one exposure may contribute with
a different direction in other exposures for a given object, thus reducing the
amount of systematic error that must be controlled in the image analysis
process. Similar studies for systematics associated with differential
chromatic refraction and CCD fixed frame distortions were conducted in the
COSEP. and were found to be minimized for a uniform distribution of
parallactic angles and the position angle of the LSST camera over all visits.

Additive shear systematics, such as those induced by errors in modeling the
point-spread function (PSF) and errors arising from the CCD charge transfer,
often have a coherent spatial pattern in single exposures. This type of
systematic can potentially be mitigated and averaged down in coadded images,
depending on the details of the dithering and observing strategy

In Almoubayyed et al. (2020), we developed a physically-motivated analysis
related to the impact of observing strategy WL shear systematics, then used
our findings with it to design a simpler proxy metric. Here, we describe the
physically-motivated analysis in four steps: (a) We select a large number
(e.g., 100,000) of random points at which the PSF is to be sampled,
distributed uniformly in the WFD area of each survey, with cuts based on the
co-added depth and dust extinction as explained in Section 3.1. (b) We create
a toy model for the PSF modeling errors as a function of position in each
exposure as a radial error in the outer 20% of field of view (and no error in
the inner 80%), and for modeling CCD charge transfer errors and the brighter-
fatter effect, we use a horizontal (CCD readout direction) error over the
stars in the entire field of view. This model is motivated by observed spatial
patterns in PSF model errors in ongoing surveys (Bosch et al., 2018; Jarvis et
al., 2016). (c) To approximate the effect of coaddition, we average down the
modeling errors across exposures via their second moments, since the
coaddition process is linear in the image intensity and therefore in the
(unweighted) moments. (d) We propagate the systematic errors for the PSF in
the coadded image into the bias on the cosmic shear using the
$\rho$-statistics formulation (Rowe, 2010; Jarvis et al., 2016).

To create a proxy metric that connects more directly with survey parameters
and is more practical to run for every simulation, we note that given a chosen
dithering pattern (e.g., a random translational dither per visit with random
rotational dithering at every filter change), the reduction in systematic
errors is directly related to the number of visits that are used in WL
analysis. This is due to the fact that more visits leads to a better sampling
of a rotationally uniform distribution around the coherent direction
associated with the additive shear systematic. We, therefore, use the average
number of $r$, $i$, and $z$-band visits for a large number of objects – for
practical reasons, picked as the centers of cells in a HEALPix grid (Górski et
al., 2005). The higher this number, the better a survey strategy performs.
Increasing the number of visits need not be done at fixed exposure time, so it
is not necessarily the case that increasing the number of visits requires a
decrease in survey area; rather, there are a variety of area-depth tradeoffs
possible for scenarios with increased numbers of visits. Even a decrease to 20
s exposures in some bands can be impactful for this metric. Note that the
bands to be used for WL shear estimation have not been decided yet; however,
$riz$ are likely to dominate due to their higher S/N for WL-selected samples,
which is why we choose to focus on them here.

Due to uncertainties in the level of detector effects and other sources of
additive shear systematics, and in the performance of instrument signature
removal methods (which determine how sensor effects may contaminate the PSF
estimated from bright stars), this metric has an arbitrary normalization and
can only differentiate between the relative improvement between different
strategies. Existence of on-sky LSSTCam data will provide a direct estimate of
the level of additive shear systematics that need to be mitigated via
observing strategy. Therefore, the impact of this metric cannot currently be
directly compared with that of the 3x2pt FoM defined in Section 4.1.1. There
is a potential trade-off between improving on 3x2pt FoM and mitigating WL
shear systematics, as the 3x2pt FoM prefers an increase in area, while WL
shear systematics are mitigated with a larger number of well-dithered visits;
the relative importance of these metrics for science will likely only be clear
at the time of commissioning. Strategies that increase the usable area for WL
and decrease exposure time can lead to improvement in both metrics
simultaneously.

###  4.2 Large-Scale Structure

Large-scale structure (LSS) constrains cosmological parameters via
observations of galaxy clustering. LSS is a more localized tracer of the
matter distribution, rather than an integral along a line of sight like WL. As
a result, the constraining power of LSS is more sensitive to bias, scatter,
and catastrophic errors in photometric redshift estimation, as these determine
how much the clustering signal is degraded by projection along the line-of-
sight (Chaves-Montero et al., 2018). Artificial modulations in the observed
galaxy number density caused by depth variations and observing conditions
(e.g., sky brightness, seeing, clouds) provide key systematic errors in
measuring galaxy clustering (Awan et al., 2016). Additionally, Galactic dust
impacts the brightness and color of each galaxy (e.g., Li et al., 2017), and
correcting for these effects becomes more difficult in regions with high
levels of Galactic dust reddening.

For the large-scale structure probe, we introduce 2 new metrics in detail
below. A summary of the results of these metrics can be seen in Figure 4.

  1. 1.

Y1/Y10 $N_{\mathrm{gal}}$ at $0.66<z<1.0$ (Section 4.2.1): Estimated number of
galaxies at $0.66<z<1.0$ based on Y1/Y10 $i$-band coadded depth in the
effective survey area.

  2. 2.

LSS systematics FoM for Y1/Y10 (Section 4.2.2)): LSS systematics diagnostic
FoM for Y1/Y10, comparing the uncertainty added by Y1/Y10 survey non-
uniformity vs. that achieved for the baseline strategy using the Y10 gold
sample (as defined in Section 3.1).

Figure 4: Large-scale structure metrics as a function of selected observing
strategies. Section 2.6 contains the exact simulation names corresponding to
the short names used here. As noted in the text, all of the strategies
generate galaxy samples for which shot noise is small, making even 10% changes
in galaxy number unimportant for cosmological constraints and placing more
emphasis on the LSS systematics FoM and the WL+LSS 3x2pt FoM of Figure 3.

####  4.2.1 Galaxy Counts

In order to get cosmological constraints from $n$-point statistics, e.g.,
2-point correlation functions or the 2-point power spectra, LSST will offer an
unprecedentedly large and deep galaxy sample, allowing us to carry out
analyses in a regime where statistical uncertainties will be subdominant to
systematic ones. To estimate the number of galaxies, we follow Awan et al.
(2016) and propagate the simulated 5$\sigma$ coadded depth to the number of
galaxies using light-cone mock catalogs.

Using the MAF object from Awan et al. (2016), we create a new MAF object,
depthLimitedNumGalMetric171717
https://github.com/humnaawan/sims_maf_contrib/blob/master/mafContrib/lssmetrics/depthLimitedNumGalMetric.py,
that calculates the number of galaxies in the extragalactic footprint (as
defined in Section 3.1).

While we find small variations in the total galaxy counts across the observing
strategies, these variations are not critical for LSS since all strategies
lead to samples comprising billions of galaxies that easily beat the shot
noise limit and hence offer similar contributions to the 3x2pt FoM in Section
4.1.1.

####  4.2.2 Systematics Introduced by the Observing Strategy

Spatial fluctuations in galaxy counts represent large-scale structure and
hence are of interest for dark energy science. As discussed in Awan et al.
(2016), artificial structure induced by the observing strategy leads to
systematic uncertainties for LSS studies. Specifically, while a systematic
bias induced in the measured large-scale structure can be corrected, the
uncertainty in our knowledge of this bias leads to uncertainties that affect
our measurements. In order to quantify the effectiveness of each cadence in
minimizing the uncertainties in the artificial structure that is induced by
the observing strategy, we update the LSS FoM given in Equation 9.4 in the
COSEP. Specifically, we have:

$\mathrm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}LSS\
FoM}}\equiv\sqrt{\frac{\sum\limits_{\ell}{\left({\sqrt{\frac{2}{f_{\mathrm{sky,Y10baseline}}(2\ell+1)}}C_{\ell,\mathrm{LSS,Y10baseline}}}\right)}^{2}}{\sum\limits_{\ell}\left[{\left({\sqrt{\frac{2}{f_{\mathrm{sky}}(2\ell+1)}}\left\\{{C_{\ell,\rm{LSS}}+\frac{1}{\bar{\eta}}}\right\\}}\right)^{2}+\sigma_{C_{\ell,\mathrm{OS}}}^{2}}\right]}}$
(2)

where the numerator differs from that in the COSEP as we now compare the
uncertainty for each year vs. that achieved for the baseline strategy using
the Y10 gold sample; briefly $f_{\mathrm{{sky}}}$ is the fraction of the sky
used for analysis, while $C_{\ell}$ denotes the angular power spectrum and
$\bar{\eta}$ is the surface number density of the galaxies in units of
steradians-1; we refer the reader to COSEP for further details. The first two
terms in the denominator represent the standard sample variance and shot
noise, and their combination adds to the final term giving variance caused by
the observing strategy (OS); the final term is calculated, as in Awan et al.
(2016), as the standard deviation of $C_{\ell,\mathrm{OS}}$ across the $ugri$
bands, to model uncertainties due to detecting galaxy catalogs in different
bands. Note that this FoM approaches 1 if the observing strategy and shot
noise contributions are negligible and the statistical power matches the Y10
baseline strategy. It can be greater than 1 by Y10 for an observing strategy
that covers more area than the baseline, but in that case, this improvement
will duplicate that seen in the static science FoM. Improvements in survey
uniformity, however, will affect this LSS FoM and not the static science FoM.

For $f_{\rm{fsky,baseline}}$, we use the Y10 sky coverage from the baseline
for a given FBS simulation (i.e., we use baseline_v1.5_10yrs for v1.5 sims;
baseline_nexp2_v1.6_10yrs for v1.6 sims that implement the $2\times 15$ s
exposure (identified by the nexp2 tag) and baseline_nexp1_v1.6_10yrs for the
rest; baseline_nexp2_v1.7_10yrs for v1.7 sims); the footprint is the
extragalactic footprint (as defined in Section 3.1), designed to achieve the
target Y10 gold sample of galaxies.

###  4.3 General Conclusions from Weak Lensing and Large-Scale Structure

For both WL and large-scale structure, statistical constraining power and
observational systematics are both impacted by choices in observing strategy.
As seen in Figure 3, changes in observing strategy that lead to more visits in
$r$, $i$, or $z$ bands are preferred for the WL systematics metric. The 3x2pt
FoM benefits from any of the following: larger survey area at low dust
extinction, greater median depth, or improved photometric redshifts. While the
latter two both favor depth versus area, within the variations available from
simulated surveys, the 3x2pt FoM shows a greater improvement for simulations
that maximize the area at low dust extinction. The LSS metrics in Figure 4
follow the general trend of the 3x2pt DETF FoM in favoring a larger effective
survey area despite the corresponding modest loss of median depth. The LSS FoM
metric prefers both increased area and greater survey uniformity; the latter
responds favorably to nightly translational dithers (as shown in Awan et al.
2016, COSEP), which have now been implemented as a default in the FBS
simulations.  On a higher level, we find that the statistical power for
combined WL and LSS prefers more area, as do the observational systematics for
LSS, while observational systematics for WL prefer more visits. In the end,
for the majority of the static science metrics explored in this section, the
gain from larger area is greater than that from more visits.

To illustrate the tension specifically, for example,
footprint_bluer_footprintv1.5_10yrs and wfd_depth_scale0.65_noddf_v1.5_10yrs
both reduce exposure in the extragalactic area (or in bands that are used for
WL shear estimation) resulting in worse performance for both the 3x2pt FoM and
the systematics metric; while the opposite is true for
wfd_depth_scale0.99_noddf_v1.5_10yrs. For bulges_bs_v1.5_10yrs and
footprint_newAv1.5_10yrs, we see a trade-off between the two metrics, due to
the fact that these simulations generally increase the area of the survey
while decreasing the average number of visits in this area. The FBS simulation
footprint_big_sky_dustv1.5_10yrs is beneficial to the 3x2pt FoM due to the
increase in area without harming the WL systematics metric due to reducing the
area that is effectively ignored by the metric.

##  5 Transient Science

###  5.1 Supernovae

As of today, the Hubble diagram of type Ia supernovae (SNe Ia) contains of the
order of $\mathcal{O}(10^{3})$ supernovae (SN) (Betoule et al. 2014; Scolnic
et al. 2018). LSST will discover an unprecedented number of SNe Ia –
$\mathcal{O}(10^{6})$. The key requirements to turn a significant fraction of
these discoveries ($\sim$10%) into distance indicators useful for cosmology
are (1) a regular sampling of the SNe Ia light curve in several rest-frame
bands, (2) a relative photometric calibration (band-to-band) at the $10^{-3}$
level, (3) a good understanding of the SNe Ia astrophysical environment, (4) a
good estimate of the survey selection function, and (5) a precise measurement
of the redshift host galaxy based on LSST photo-$z$ estimators. The first
point is crucial. It determines how well we can extract the light-curve
observables used by current and future standardization techniques (stretch,
rest-frame color(s), rise time). It also determines how well photometric
identification techniques are going to perform, as live spectroscopic follow-
up will only be possible for 10% of SNe Ia.

The average quality of SNe Ia light curves depends primarily on the observing
strategy through five key facets: (1) a high observing cadence (2 to 3 days
between visits) delivers well sampled light curves, which is key to distance
determination and photometric identification; (2) a regular cadence allows
minimizing the number of large gaps ($>$10 days) between visits, which
degrades the determination of luminosity distances, and potentially result in
rejecting large batches of light curves of poor quality; (3) a filter
allocation ensuring the use at least 3 bands (rest-frame) to select high-
quality supernovae; (4) the season length determines the number of SNe Ia with
observations before and after peak; due to time dilation maximizing season
length is particularly important in the DDFs; (5) finally, the integrated
signal-to-noise ratio over the SNe Ia full light curve determines the
contribution of measurement noise to the distance measurement. It is a
function of the visit depth and the number of visits in a given band.

All the studies presented in this section on light-curve simulations of SNe
Ia. We have used the SALT2 model (Guy et al. 2007, 2010) where a type Ia
supernova is described by five parameters: $x_{0}$, the normalization of the
spectral energy distribution (SED) sequence; $x_{1}$, the stretch; $c$, the
color; $T_{0}$, the day of maximum luninosity; and $z$, the redshift. The
time-distribution and the photometric errors of the light-curve points are
estimated from observing conditions (cadence, 5$\sigma$-depth, season length)
given by the scheduler. We consider two types of SNe Ia defined by
($x_{1}$,$c$) parameters to estimate the metrics: (intrinsically) faint
supernovae, defined by ($x_{1}$,$c$)= (-2.0,0.2), and medium supernovae,
defined by ($x_{1}$,$c$)= (0.0,0.0). ($z_{\mathrm{faint}}$,
$N_{z<z_{\mathrm{faint}}}\ $) gives an assessment of the size and depth of the
redshift limited sample (i.e. the sample of supernovae usable for cosmology)
with selection function having minimal dependence on understanding the noise
properties. ($z_{\mathrm{med}}$, $N_{z<z_{\mathrm{med}}}\ $) gives an
assessment of the size and depth of the sample of SNe Ia with precise
distances. We will get higher statistics with the medium sample, but need also
a better understanding of noise to determine the selection function.

All the metrics described below are estimated from a sample of well-measured
SNe Ia that passed the following light curve (LC) requirements: visits with
SNR$>$ 10 in at least 3 bands; 5 visits before and 10 visits after peak,
within [-10;+30] days (rest-frame); $\sigma_{C}<$ 0.04 where $\sigma_{C}$ is
the SALT2 color uncertainty; all observations satisfying 380 nm
$~{}<~{}\bar{\lambda}_{obs}/(1+z)~{}<$ 700 nm.

For the SNe Ia probe, we introduce  7 new metrics. A summary of the results
from the metrics can be seen in Figure 5.

  1. 1.

Faint SNe Ia redshift limit - Redshift limit corresponding to a complete SNe
Ia sample (z${}_{lim}^{faint}$) (Section 5.1.1)

  2. 2.

Medium SNe Ia redshift limit - Redshift limit corresponding to a complete SNe
Ia sample (z${}_{lim}^{medium}$) (Section 5.1.1)

  3. 3.

Number of SNe Ia with z$\leq$zlim(faint) - Number of well-sampled SNe Ia with
z$\leq$z${}_{lim}^{faint}$ (Section 5.1.1)

  4. 4.

Number of SNe Ia with z$\leq$ zlim(med) - Number of well-sampled SNe Ia with
z$\leq$z${}_{lim}^{medium}$ (Section 5.1.1)

  5. 5.

SNe Ia $r$-band Signal-to-Noise Ratio - Fraction of faint SNe Ia with a
$r$-band Signal-to-Noise Ratio (SNR) higher than a reference SNR corresponding
to a regular cadence (Section 5.1.2).

  6. 6.

SNe Ia $r$-band redshift limit - $r$-band redshift limit of faint SNe Ia
(Section 5.1.2)

  7. 7.

Peculiar velocities - SNe Ia host galaxy velocities (Section 5.1.3)

Figure 5: Supernova metrics as a function of selected observing strategies.
Section 2.6 contains the exact simulation names corresponding to the short
names used here. As described in Section 5.1.1, $z_{\mathrm{lim}}$ corresponds
to the redshift beyond which supernovae no longer pass light curve
requirements. “med” and “faint” refer to a sample of typical and faint
supernovae respectively.

####  5.1.1 Number of well-measured type Ia supernovae/Survey completeness

We use, as our primary metric, the size and depth of a subset of well-sampled
SNe Ia using the redshift limit $z_{\mathrm{lim}}$ and the number of well-
sampled supernovae $N_{z<z_{\mathrm{lim}}}$below this redshift.
$z_{\mathrm{lim}}$ corresponds to the redshift beyond which supernovae no
longer pass light curve requirements.

The WFD footprint is quite large (at least 18000 deg2) forbidding the use of
full (time-consuming) LC simulations. We have opted for a slightly different
approach. The celestial sphere is pixellized in HEALPIX superpixels
($N_{\mathrm{side}}$=64, which corresponds to 0.8 deg2 per pixel). The
directions (i.e. (RA,Dec) positions)/ healpixel affected by a Galactic
extinction $E(B-V)$ larger than 0.25 are masked (to minimize reddening
effects) and not included in our assessment. We consider only the $griz$
observations which are the ones that matter to derive SN luminosity distances.

We process observing strategies using a simple model of the LSST focal plane
and estimate:

$\displaystyle z_{\mathrm{lim}}$ $\displaystyle=$
$\displaystyle\mathrm{max}\left(z|\mathrm{LC(z)\ fulfill\
requirements}\right)$ (3) $\displaystyle N_{z<z_{\mathrm{lim}}}$
$\displaystyle=$
$\displaystyle\delta\Omega_{\mathrm{pix}}\int_{0}^{z_{\mathrm{lim}}}\frac{\Delta
T_{\mathrm{step}}}{1+z}\ {\mathcal{R}}(z)\ dV(z)$ (4)

where $\delta\Omega_{\mathrm{pix}}$ is the solid angle subtended by one pixel;
$dV$ is the (differential) co-moving volume; $\Delta T_{\mathrm{step}}$ is the
time interval for supernovae simulations (in observer frame days), that is
only supernovae with a peak luminosity during this time range are simulated;
$\mathcal{R}(z)$ is the SN Ia volumetric rate (Perrett et al., 2012). We also
compute the average cadence (in day-1), i.e. the number of $g,r,i$ or $z$
visits in a fiducial restframe interval. The quantities above are determined
for each pixel and each night (identified by its Modified Julian Date MJD) and
may be used to build full sky maps giving, as a function of the position on
the sky (1) the density of supernovae, (2) the median maximum redshift (over
the observed area), and (3) the median cadence.

This metric is the most precise to assess observing strategies, but also the
most intricate to implement. A lot of effort has been put to design algorithms
combining speed, reliability, and accuracy. The codebase is accessible through
Github181818https://github.com/LSST-
nonproject/sims_maf_contrib/blob/master/science/Transients/SN_NSN_zlim.ipynb
in the Metric Analysis Framework.

We note that while we expect the quality cuts described here will ensure
accurate classification of SNe Ia and separate them from other classes of
transients, we do not yet have a transient metric to ensure this is the case
and leave this important step to future work (see Section 7.2 for a detailed
discussion).

####  5.1.2 $SNR_{\mathrm{rate}}$ and redshift limit

While the metric in Section 5.1.1 is our most accurate metric, we developed
two proxy metrics that have also been incorporated in MAF where
straightforward, fast and easy-to-run metrics are preferred. These two metrics
are quite simple (they do not require the use of a light curve fitter) and
just need templates of supernova light curves as input. They are estimated for
each band thus providing tools for further comparison of observing strategy
performance. They are sensitive to two key points of observing strategies: the
median cadence, and the inter-night gap variations. The codebase is accessible
through Github191919https://github.com/LSST-
nonproject/sims_maf_contrib/blob/master/science/Transients/SNSNR.ipynb.

These two metrics rely on Signal-to-Noise Ratio (SNR) of the light curves (per
band) that may be written as:

$SNR_{b}=\sqrt{\sum_{i}{\left({{f_{i}^{b}}\over{\sigma_{i}^{b}}}\right)^{2}}}$
(5)

where $b$ is the band, $f^{b}$ are the fluxes and $\sigma^{b}$ the flux errors
(summation over LC points). In the background-dominated regime, flux errors
may be expressed as a function of the $5-\sigma$ limiting flux of each visit
$f_{i|5}$. We may rewrite Equation 5 by defining $\delta_{b}$ as the number of
observations per bin in time $\Delta T$ and per band $b$:

$f_{|5}<\delta_{b}>^{-1/2}=\frac{5\sqrt{\Delta
T}\sqrt{\sum_{i}(f_{i}^{b})^{2}}}{SNR_{b}}$ (6)

We describe below two metrics that may be extracted from Equation 5 and
Equation 6: the $SNR_{\mathrm{rate}}$ and the redshift limit metrics.

$SNR_{b}$ (Equation 5) is the result of the combination of observing strategy
features (5$\sigma$-depth and cadence) and supernovae  parameters. We fix some
of these parameters ( we considered faint supernovae with $z$=0.3, where the
sample is not affected by the Malmquist bias) so as to estimate $SNR_{b}(t)$
for a supernova with $T_{0}$=t-10. We also evaluate $SNR_{b}^{regular}(t)$
using the same supernova parameters but opting for median values for
5-$\sigma$ depth and cadence. The $SNR_{rate}$ metric is then defined as the
fraction of time (in a season) when the requirement $SNR_{b}(t)\geq
SNR_{b}^{regular}(t)$ is fullfilled.

The two above-mentioned contributions to $SNR_{b}$ are clearly visible in
Equation 6, with observing conditions on the left side (5-$\sigma$ limiting
flux times cadence), and flux (supernovae properties) on the right side. We
fix some of the supernova parameters ( faint supernovae with $T_{0}$=0.) and
we use median values of 5-$\sigma$ depth and cadences to estimate redshift
values defining the second metric, dubbed as $z_{\mathrm{lim}}$. We used
Equation 6 with the following SNR (ANDed) requirements:
$SNR_{g}>30,SNR_{r}>40,SNR_{i}>30,SNR_{z}>20$. Combining these selections is
equivalent to requesting $\sigma_{c}\leq$ 0.04 and will ensure observation of
well-measured SNe Ia.

####  5.1.3 Peculiar Velocities

The goal of the peculiar velocities metric is to study modified gravity
through its effects on the overdensities and velocities of type Ia supernova
host galaxies. Gravitational models are efficiently parameterized by the
growth index, $\gamma$, which influences the linear growth rate as
$f=\Omega_{M}^{\gamma}$ where
$\Omega_{M}=\frac{\Omega_{M_{0}}}{\Omega_{M_{0}}+(1-\Omega_{M_{0}})a^{3}}$ and
$\Omega_{M_{0}}$ is the mass density today. The parameter dependence enters
through $fD$, where $D$ is the spatially-independent “growth factor” in the
linear evolution of density perturbations and $f\equiv\frac{d\ln{D}}{d\ln{a}}$
is the linear growth rate where $a$ is the scale factor (Hui & Greene, 2006;
Davis et al., 2011). Two surveys with the same fractional precision in fD will
have different precision in $\gamma$, with the one at lower redshift providing
the tighter constraint. We thus use the uncertainty in the growth index,
$\sigma_{\gamma}$, as the peculiar velocity metric.

For a parameterized survey and redshift-independent $\gamma$,
$\sigma_{\gamma}$ is bounded in Kim et al. (2019) by using the Fisher matrix

$\displaystyle F_{ij}$ $\displaystyle=\frac{\Omega}{8\pi^{2}}\int_{r_{\rm
min}}^{r_{\rm max}}\int_{k_{\rm min}}^{k_{\rm
max}}\int_{-1}^{1}r^{2}k^{2}\text{Tr}\left[C^{-1}\frac{\partial
C}{\partial\lambda_{i}}C^{-1}\frac{\partial
C}{\partial\lambda_{j}}\right]d\mu\,dk\,dr$ (7)

where $\Omega$ is the sky coverage of the survey, $r_{max}$ ($r_{min}$) are
the comoving distances corresponding to the upper (lower) redshift limits of
each redshift bin, and we set $k_{max}$ = 0.2h Mpc-1 and
$k_{min}=2\pi/r_{max}$. $\mu$ is the cosine of the angle $\phi$ between the
k-vector and the observer’s line-of-sight. The covariance matrix $C$ is
defined by:

$C(k,\mu)=\begin{bmatrix}P_{\delta\delta}(k,\mu)+\frac{1}{n}&P_{v\delta}(k,\mu)\\\
P_{v\delta}(k,\mu)&P_{vv}(k,\mu)+\frac{\sigma^{2}}{n}\end{bmatrix}$ (8)

and the parameters considered are $\lambda\in\\{\gamma,bD,\Omega_{M_{0}}\\}$.

The SNe Ia host-galaxy radial peculiar velocity power spectrum is
$P_{vv}\propto(fD\mu)^{2}$, the count overdensity power spectrum is
$P_{\delta\delta}\propto(bD+fD\mu^{2})^{2}$, the overdensity-velocity cross-
correlation is $P_{vg}\propto(bD+fD\mu^{2})fD$, where $b$ is the galaxy bias
and $\mu\equiv\cos{(\hat{k}\cdot\hat{r})}$ where $\hat{r}$ is the direction of
the line of sight and $n$ is $\epsilon\phi t$ where $\epsilon$ is the sample
selection efficiency, $\phi$ is the observer-frame SN Ia rate and $t$ is the
duration of the survey. While the $bD$ term does contain information on
$\gamma$, its constraining power is not used here. The variance in $\gamma$ is
$\left(F^{-1}\right)_{\gamma\gamma}$. Our figure of merit is the inverse
variance, so that a larger value corresponds to a more precise measurement and
hence a better survey strategy.

The parameters in Equation 7 that are primarily affected by survey strategy
are the survey solid angle $\Omega$ and the  number density $n$ of well-
measured SNe Ia. The other parameters related to the follow-up strategy of
these SN discoveries are the survey depth $r_{\rm max}$ and the intrinsic
velocity dispersion $\sigma$, which is related to the intrinsic  magnitude
dispersion of well-measured SNe Ia. The estimate of $\sigma_{\gamma}$ is
sensitive to both the sample variance $P_{vv}$ and shot noise
$\frac{\sigma^{2}}{n}$ in the range of proposed surveys, meaning that its
accurate determination cannot be taken in either the sample- or shot-noise
limit. A follow-up strategy must also be specified for the calculation of
$\sigma_{\gamma}$ since Rubin/LSST will not generate all the information
needed for this measurement, e.g. redshift, SN classification. Here, we adopt
a maximum survey redshift of $z=0.2$ and follow-up that gives 0.08 mag
magnitude dispersion per SN. The minimum redshift is $z=0.01$, number
densities are based on 65% of the SNe Ia rates of Dilday et al. (2010), and
$k_{\rm max}=0.1h$ Mpc-1, $b=1.2$, $\Lambda$CDM cosmology with
$\Omega_{M_{0}}=0.3$, and overdensity power spectra for the given cosmology as
calculated by CAMB (Lewis & Bridle, 2002).

The code used for the calculations are available in
Github202020https://github.com/LSSTDESC/SNPeculiarVelocity/blob/master/doc/src/partials.py.

####  5.1.4 General Conclusions from Supernovae

Collecting a large sample of well-measured type Ia supernovae is a
prerequisite to measure cosmological parameters with high accuracy. Our
analysis (Section 5.1.1 to Section 5.1.3 and Figure 5) has shown that the key
parameter to achieve this goal is the effective cadence delivered by the
survey. For the WFD survey, a regular cadence in the $g$, $r$ and $i$ bands is
essential to (1) secure a high-efficiency photometric identification of the
detected SNe Ia and (2) secure precise standardized SNe Ia distances, by
optimizing the integrated Signal-to-Noise Ratio along the SNe Ia light curves.
Gaps of more than $\sim$ 10 days in the cadence have a harmful impact on the
size and depth of the SNe Ia sample. The cadence of observations is by far the
most important parameter for SNe Ia science, before observing conditions: on
the basis of studies conducted, it is preferable to have pointings with sub-
optimal observing conditions rather than no observation at all.

Three main sources of gaps have been identified: telescope down time (clouds,
maintenance), filter allocation, and scanning strategy (ie the criteria used
to move from one pointing to another). While we are aware that it is difficult
to minimize the impact of down time, there is still room for improvement on
filter allocation and scanning strategy. Significant efforts have been made to
make sure that nightly revisits of the same field are performed in different
bands. Relaxing the veto on bluer bands around full moon, or increasing the
density of visits (ie the number of visits per square degree) during a night
of observation (by decreasing the observed area for instance) will help to
achieve an optimal cadence for supernovae of 2 to 3 days in the $g$, $r$ and
$i$ bands.

###  5.2 Strong Lensing

The Hubble constant $H_{0}$ is one of the key parameters to describe the
universe. Current observations of the CMB (cosmic microwave background)
assuming a flat $\Lambda$CDM cosmology and the standard model of particle
physics yield $H_{0}=67.36\pm 0.54\,{\rm km\,s^{-1}\,Mpc^{-1}}$ (Planck
Collaboration, 2020), which is in tension with $H_{0}=74.03\pm 1.42\,{\rm
km\,s^{-1}\,Mpc^{-1}}$ from the local Cepheid distance ladder (Riess et al.,
2016, 2018, 2019), although is statistically consistent with the $H_{0}$
derived using the tip of the red giant branch in the distance ladder by
Freedman et al. (2019). In order to verify or refute this $4.4\sigma$ tension,
further independent methods are needed.

One such method is lensing time delay cosmography which can determine $H_{0}$
in a single step. The basic idea is to measure the time delays between
multiple images of a strongly lensed variable source (Refsdal, 1964). This
time delay, in combination with mass profile reconstruction of the lens and
line-of-sight mass structure, yields directly a “time-delay distance” that is
inversely proportional to the Hubble constant ($t\propto D_{\Delta t}\propto
H_{0}^{-1}$). Applying this method to six lensed quasar systems, the H0LiCOW
collaboration (Suyu et al., 2017) together with the COSMOGRAIL collaboration
(e.g. Eigenbrod et al., 2005; Tewes et al., 2013b; Courbin et al., 2017)
measured $H_{0}=73.3^{+1.7}_{-1.8}\,{\rm km\,s^{-1}\,Mpc^{-1}}$ (Wong et al.,
2020) in flat $\Lambda$CDM, which is in agreement with the local distance
ladder but higher than CMB measurements. Including the distance measurement to
another lensed quasar system from the STRIDES collaboration (Shajib et al.,
2020), the newly formed TDCOSMO organisation has further investigated
potential residual systematic effects (e.g., Millon et al., 2020; Gilman et
al., 2020; Birrer et al., 2020). Another promising approach goes back to the
initial idea of Refsdal (1964) using lensed supernovae (LSNe) instead of
quasars for time-delay cosmography (e.g., Grillo et al., 2020; Mörtsell et
al., 2020; Suyu et al., 2020). In terms of discovering strong lens systems
from the static LSST images for cosmological studies, having $g$-band
observations with comparable seeing as in the $r$\- and $i$-band would
facilitate the detection of strong lens systems (Verma et al., 2019).

In this section, we investigate the prospects of using LSST for measuring time
delays of both lensed supernovae and lensed quasars. In particular, we focus
on the number of lens systems that we would detect for the various observing
strategies as our metrics. From the investigation of LSNe by galaxies, we
define a metric for the number of LSNe Ia with good time delay measurement.
For lensed quasars, we have additional metrics defining how well we can
measure the time-delay distances.

For the Strong Lensing (SL) probe, we introduce 4 new metrics. A summary of
the results from these metrics can be seen in Figure 6

  1. 1.

Number of SNe Ia lensed by galaxies - Number of SNe Ia strongly lensed by
galaxies with accurate and precise time delays between the multiple SN images
(Section 5.2.1)

  2. 2.

Number of SNe Ia lensed by clusters - Number of strongly lensed SNe Ia in the
multiply-imaged galaxies behind well-studied galaxy clusters (Section 5.2.2)

  3. 3.

Number of lensed quasars - Number of strongly lensed quasars with accurate and
precise time delays between the multiple quasar images (Section 5.2.3)

Figure 6: Strong lensing metrics as a function of selected observing
strategies. Section 2.6 contains the exact simulation names corresponding to
the short names used here. In general, a long cumulative season length and
high sampling frequency at sufficient depth are preferred for strong lensing
in terms of maximizing the number of lens systems, particularly SNe and
quasars lensed by galaxies (blue and orange). For the specific case of LSNe Ia
in the five cluster fields, the number of LSNe Ia is sensitive to number of
visits to the cluster fields, the image depth, the gap of these observations,
and the filter used. For the observing strategies plotted here, the depth and
the mean gap between the observations of the considered clusters remain
roughly the same, while the main difference is the number of observations
(lower than the baseline), which negatively affects the number of expected
cluster-lensed SNe Ia. Furthermore, the number of observations to the clusters
in WFD 65% strategy is higher than the WFD 99% one, which is also reflected in
the expected cluster-lensed SNe Ia.

####  5.2.1 Number of supernovae lensed by galaxies

For constraining cosmological parameters with supernovae lensed by galaxies as
well as possible, ideally we would like to maximize the number of accurate and
precise time-delay distance measurements. Currently there are only two known
lensed SN systems, namely SN “Refsdal” (Kelly et al., 2016a, b) and iPTF16geu
(Goobar et al., 2017), but LSST will play a key role in detecting many more
LSNe (Oguri & Marshall, 2010; Goldstein et al., 2018; Wojtak et al., 2019). A
measurement of a time-delay distance from a strongly lensed SN system requires
(1) the detection of the system, (2) the measurement of time delays between
the multiple SN images from their observed light curves, and (3) the lens mass
modeling of the system to infer the distance from the time delays. LSST’s
observing strategies affect both (1) and (2), and the uncertainties in the
time delays from (2) enter directly into the uncertainties on the time-delay
distances. Therefore, we use as a metric the number of lensed supernovae
systems that could yield time-delay measurements with precision better than 5%
and accuracy better than 1%, in order to achieve $H_{0}$ measurement that has
better than 1% accuracy from a sample of lensed SNe. We refer to time delays
that satisfy these requirements as having “good” delays.

Huber et al. (2019) have presented a detailed study to obtain the number of
lensed SNe, and we summarize the results here. To simulate LSST observations
of a lensed SN system, Huber et al. (2019) have used 202 mock LSNe Ia from the
OM10 catalog (Oguri & Marshall, 2010), and produced synthetic light curves for
the mock SN images via ARTIS (Applied Radiative Transfer In Supernovae)
(Kromer & Sim, 2009) for the spherically symmetric SN Ia W7 model (Nomoto et
al., 1984) in combination with magnification maps from GERLUMPH (Vernardos et
al., 2015) to include the effect of microlensing similar to Goldstein et al.
(2018). Huber et al. (2019) have then simulated data points for the light
curves, following the observation pattern from different observing strategies
from the FBS scheduler212121https://cadence-
hackathon.readthedocs.io/en/latest/current_runs.html and uncertainties are
calculated according to The LSST Science Book. To measure the time delay from
the simulated observation, Huber et al. (2019) have used the free knot spline
optimizer from PyCS (Python Curve Shifting) (Tewes et al., 2013a; Bonvin et
al., 2016). For each mock lens system from OM10, there are 100 random starting
configurations, where a starting configuration corresponds to a random
position in the microlensing map, a random field on the sky where the system
is located and a random time of explosion in the observing seasons such that
the detection requirement from OM10 is fulfilled. For each starting
configuration, there are 1000 noise realisations. Through these realistic
simulations, Huber et al. (2019) have then quantified the precision and
accuracy of the measured time delays for each mock system. Taking into account
the different areas and cumulative seasons lengths of the different observing
strategies, Huber et al. (2019) have then estimated the number of lensed SNe
that would be detected with LSST and would have “good” delays.

Huber et al. (2019) have considered two scenarios: (1) LSST data only for both
detection and time-delay measurements, and (2) LSST data for detection with
additional follow-up observations with a more rapid cadence than LSST. Follow-
up observations are assumed to take place every second night in three filters
(g, r, and i) going to a baseline-like 5$\sigma$ point-source depth of 24.6,
24.2, and 23.7, respectively. For our science case of measuring time delays
from as many lensed SNe as possible (Huber et al., 2019), it would be more
effective to use LSST as a discovering machine with additional follow-up,
instead of relying on LSST completely for the delay measurements. Based on the
investigations of Huber et al. (2019), long cumulative seasonal lengths and a
more frequent sampling are important to increase the number of LSNe Ia with
well measured time delays. Specifically, we request 10 seasons with a season
length of 170 days or longer. Rolling cadences are clearly disfavored, because
their shortened cumulative season lengths (only 5 instead of 10 seasons for
two declination bands) lead to overall a more negative impact on the number of
LSNe Ia with delays, compared to the gain from the more rapid sampling
frequency. To improve the sampling, single visits per night, meaning that a
given LSST field will be only observed once a night, would be helpful. Since
this will make the science case of fast-moving transients impossible, we
suggest doing one revisit within a night but in a different band than the
first visit. Further improvements are the replacement of $2\times 15$ s
exposure by $1\times 30$ s for an increased efficiency and redistributing some
of the visits in $y$-band to $g$, $r$, $i$ and $z$.

We note that Goldstein et al. (2019) performed detailed simulations of the LSN
population using a completely independent technique and pipeline, and reached
similar conclusions to the ones presented here: rolling cadences are strongly
disfavored, and wide-area, long-season surveys with well sampled light curves
are optimal.

To evaluate further observing strategies, we have defined a metric based on
the investigations of Huber et al. (2019). The number of LSNe Ia with well
measured time delays using LSST and follow-up observations for a given
observing strategy can be approximated as

$\displaystyle N_{\mathrm{LSNeIa,\,good\,\,delay}}\approx
45.7\frac{\Omega_{\mathrm{WFD}}}{$20\,000\text{\,}\mathrm{{}^{2}}\deg$}\frac{t_{\mathrm{csl}}}{2.5\,\mathrm{yr}}\cdot\frac{1}{2.15}\mathrm{exp}(-0.37t_{\mathrm{gap}})$
(9)

The first part of the metric (separated by a dot from the second part) is the
rescaling of the value predicted in OM10 (Oguri & Marshall, 2010) by taking
into account the survey area of the WFD ($\Omega_{\mathrm{WFD}}$) and the mean
of the cumulative season length ($t_{\mathrm{csl}}$; summed over all season
lengths) for a given observing strategy. The second part contains a fit based
on the numbers of LSNe Ia with well measured time delay, presented in Huber et
al. (2019), in comparison with the total number of LSNe Ia which will be
detected (first part of Equation 9). The fit function depends on the median
inter-night gap between any filters $t_{\mathrm{gap}}$ and is measured in
days, which is an important parameter because we assume a detection of the
LSNe Ia in Huber et al. (2019) after the third data point exceeds the
5$\sigma$ point-source depth in any filter. The inter-night gap
($t_{\mathrm{gap}}$), survey area $\Omega_{\mathrm{WFD}}$, and cumulative
season length $t_{\mathrm{csl}}$ can be calculated via
MAF222222https://me.lsst.eu/gris/ and a Python script
232323https://github.com/shuber891/LSST-metric-for-LSNe-Ia/ where we only take
observations into account, with a 5$\sigma$ point-source depth greater than
22.7, 24.1, 23.7, 23.1, 22.2, and 21.4 for the filters $u$, $g$, $r$, $i$,
$z$, and $y$. These cuts are motivated by Huber et al. (2019) and are
important to restrict visits to the WFD, where the metric from Equation 9 is
valid The results are summarized in Figure 6, where we see that $1\times 30$ s
exposures are favored over $2\times 15$ s exposures and more time on the WFD
is preferred. Our conclusions including results from Huber et al. (2019) are
summarized in Section 5.2.4.

####  5.2.2 Number of supernovae lensed by galaxy clusters

Here, we focus on prospects of observing LSNe which are strongly lensed by
galaxy clusters which have well-studied lens models. High-z galaxies that
appear as multiple images in the cluster field can host SN explosions. The
first discovery of this kind was SN Refsdal, which was classified as a core-
collapse (CC) explosion (Kelly et al., 2015, 2016a). Several teams predicted
the re-appearance of SN Refsdal almost a year later, which allowed to test
their lens models (e.g., Grillo et al., 2016; Kelly et al., 2016a). By
measuring the time delays of SN Refsdal and having a high-quality strong
lensing model of the galaxy cluster, it was shown that it is possible to
measure _H 0_ with 6% total uncertainty (Grillo et al., 2018, 2020). Dedicated
ground-based searches for lensed SNe behind galaxy clusters have been
performed using near-infrared instruments at the Very Large Telescope
(Stanishev et al., 2009; Goobar et al., 2009; Petrushevska et al., 2016,
2018b). Most notably, they reported the discovery of one of the most distant
CC SN ever found, at redshift $z=1.703$ with a lensing magnification factor of
$4.3\pm 0.3$ (Amanullah et al., 2011). Furthermore, thanks to the power of the
lensing cluster, it was possible to estimate the volumetric CC SN rates for
$0.4\leq z<2.9$, and compare it with the predictions from cosmic star
formation history (Petrushevska et al., 2016). Knowing the absolute brightness
of SNe Ia allows estimating the absolute magnification of SNe Ia, therefore to
break the so-called mass-sheet degeneracy of gravitational lenses (Holz,
2001). Thus, LSNe Ia could be used to put constraints on the lensing
potential, if the background cosmology is assumed to be known (see e.g., Patel
et al., 2014; Nordin et al., 2014; Rodney et al., 2015).

As a metric we use the expected number of LSNe in the selected galaxy cluster
fields in 10 years of LSST. For the details regarding the methods in this
section, we refer to Petrushevska (2020) based on the work in (Petrushevska et
al., 2016, 2018b, 2018a). Here, we present a short summary. We consider the
six Hubble Frontier Fields clusters (Lotz et al., 2017) and Abell 1689, given
in Table 3. These clusters have been extensively studied, and given the good
quality data, well-constrained magnification maps and time delays can be
obtained from the lensing models (Petrushevska et al., 2016, 2018b, 2018a). We
consider the multiply imaged galaxies in the cluster fields that have a
spectroscopic redshift. Given the redshift range of the multiply-imaged
galaxies considered here (see Table 3), the most important bands are $i$, $z$
and $y$ (see Figure 2 in Petrushevska (2020)). The observability of a SN in
the multiply-imaged galaxies is sensitive to the redshift, star formation rate
and magnification of the galaxy, but also on the observing strategy parameters
such as depth, separation between the two consecutive observations and the
filter. The expected number of LSNe Ia in the five cluster fields is
relatively low, mainly for two reasons. First, we have only considered 268
images of the background galaxies in the five cluster fields. Second, given
the redshift range of $0.73<z<5.75$ (see Table 3), the ground-based Rubin
Observatory filter set is not optimal for detecting SNe in these galaxies.
However, thanks to the magnification from the galaxy clusters, LSST is
sensitive of detecting LSNe Is to very high redshifts ($0.73<z<1.95$). We note
that the resulting expected number of LSNe in the selected galaxy cluster
fields are a lower limit, since we have only considered few clusters and the
multiply-imaged galaxies with spectroscopic redshift. Beyond the clusters that
we have considered here, LSST will observe $\sim 70$ galaxy clusters with
Einstein radii larger than $\theta_{E}>20"$ that have $\sim 1000$ multiply-
imaged background galaxies (The LSST Science Book). As the expectations of
strongly lensed SNe in cluster fields depend on several factors, including the
star formation rate and the stellar mass of the host galaxy, it is not
straightforward to make a reliable prediction and we leave the expectation in
all clusters visible to LSST for a future study.

The conclusions of all the metrics presented in the Section 5.2 are presented
in Section 5.2.4. The conclusions for the galaxy-lensed SNe and cluster-lensed
SNe are similar in general, with small differences (see Figure 6). Since here
we are focusing on particular cluster fields, what matters more is the number
of visits to the cluster fields rather than having generally large area.
Furthermore, in order to optimize the sensitivity to high-redshift SNe with
multiple images in galaxy cluster fields, deeper images in the reddest bands
($i$, $z$ and $y$) are preferred. This can be obtained by co-adding images
from visits closely separated in time. As mentioned in the previous section,
the LSST will serve to detect the LSNe, but additional follow-up by other
photometric and spectroscopic instruments will be needed to securely measure
the time delays.

Table 3: The galaxy clusters considered in this work. The number of unique galaxies behind the cluster is given in column 2 and the number of their multiple images of these galaxies in column 3. The redshift range of these galaxies is given in column 4. Cluster | $N_{systems}$ | $N_{images}$ | $z_{min}-z_{max}$
---|---|---|---
Abell 1689 | 18 | 51 | $1.15-3.4$
Abell 370 | 21 | 67 | $0.73-5.75$
Abell 2744 | 12 | 40 | $1.03-3.98$
Abell S1063 | 14 | 42 | $1.03-3.71$
MACS J0416.1-2403 | 23 | 68 | $1.01-3.87$
Total | 88 | 268 |

####  5.2.3 Number of “golden” lensed quasars

The goal of this section242424Summarized and updated version of (Cosmology
chapter of LSST Science Collaboration et al., 2017) is to evaluate the
precision we can achieve in measuring time delays in strongly lensed active
galactic nuclei (AGN), and as such, the precision on the measurement of the
Hubble constant from all systems with measured time delays.

Anticipating that the time-delay accuracy would depend on night-to-night
cadence, season length, and campaign length, we carried out a large scale
simulation and measurement program that coarsely sampled these schedule
properties. In Liao et al. (2015), we simulated 5 different light curve
datasets, each containing 1000 lenses, and presented them to the strong
lensing community in a “Time Delay Challenge” (TDC). These 5 challenge “runs”
differed by their schedule properties. Entries to the challenge consisted of
samples of measured time delays, the quality of which the challenge team then
measured via three primary diagnostic metrics: time delay accuracy ($|A|$),
time delay precision ($P$), and useable sample fraction ($f$). The accuracy of
a sample was defined to be the mean fractional offset between the estimated
and true time delays within the sample. The precision of a sample was defined
to be the mean reported fractional uncertainty within the sample.

Focusing on the best challenge submissions made by the community, we derived a
simple power law model for the variation of each of the time-delay accuracy,
time-delay precision, and useable sample fraction, with the schedule
properties cadence (cad), season length (sea) and campaign length (camp). They
are given by the following equations:

$\displaystyle|A|_{\rm model}$ $\displaystyle\approx 0.06\%\left(\frac{\rm
cad}{\rm 3days}\right)^{0.0}\left(\frac{\rm sea}{\rm
4months}\right)^{-1.0}\left(\frac{\rm camp}{\rm 5years}\right)^{-1.1}$ (10)
$\displaystyle P_{\rm model}$ $\displaystyle\approx 4.0\%\left(\frac{\rm
cad}{\rm 3days}\right)^{0.7}\left(\frac{\rm sea}{\rm
4months}\right)^{-0.3}\left(\frac{\rm camp}{\rm 5years}\right)^{-0.6}$ (11)
$\displaystyle f_{\rm model}$ $\displaystyle\approx 30\%\left(\frac{\rm
cad}{\rm 3days}\right)^{-0.4}\left(\frac{\rm sea}{\rm
4months}\right)^{0.8}\left(\frac{\rm camp}{\rm 5years}\right)^{-0.2}$ (12)

All three of these diagnostic metrics would, in an ideal world, be optimized:
this could be achieved by decreasing the night-to-night cadence (to better
sample the light curves), extending the observing season length (to maximize
the chances of capturing a strong variation and its echo), and extending the
campaign length (to increase the number of effective time delay measurements).

As accuracy and precision in time-delay measurements (assuming identical lens
“modeling uncertainty”) is roughly proportional to the statistical uncertainty
on the Hubble constant. Our analysis thus consists of selecting only the sky
survey area that allows time-delay measurements with accuracies of $<1\%$ and
precision $<5\%$. This high accuracy and precision area can be used to define
a “gold sample” of lenses. The TDC useable fraction averaged over this area
gives us the approximate size of this sample: we simply re-scale the 400
lenses predicted by Liao et al. (2015) by this fraction over the 30% found in
TDC. Note that naturally there is a strong dependence with footprint: assuming
the single visit depth is deep enough to detect the typical lensed quasar
image, the number of lenses will scale linearly with the survey area. The
uncertainty on the Hubble constant, will then finally scale as one over the
square root of the number of lenses. While these numbers are approximate, the
ratios between different observing and analysis strategies provide a useful
indication of relative merit.

Our calculations are performed using the full 10 years of LSST operations with
observations in all bands contributing equally (i.e. monochromatic intrinsic
variability). However, it is important to note that there is a rather large
caveat: Even when AGN variability can show almost negligible difference
between bands close in wavelength, the difference can be important between the
bluest and reddest LSST bands. As such, the numerical values have to
considered as an optimistic lower limit but, as mentioned before, the ratios
between observing strategies are an indication of relative merit.

####  5.2.4 General Conclusions from Strong Lensing

For our science case, a long cumulative season length and improved sampling
frequency (cadence) is important. The cumulative season length provided by the
baseline cadence is sufficient, but rolling cadences are disfavored as pointed
out by Huber et al. (2019), because of the reduced cumulative season length.
In terms of the cadence, Huber et al. (2019) showed that, revisits of the same
field within a single night should be done in different filters. Further
improvements could be achieved by doing single snaps instead of two snaps, as
shown in Figure 6. We also note that increasing the overall area would
naturally yield an improvement in the number of “golden” lensed quasars,
however, in this case it comes at the cost of having more visits in Galactic
plane and having fewer WFD visits which would be detrimental for the number of
lensed SNe with accurate and precise time delays.

###  5.3 Kilonovae

Within the next decade kilonovae (kNe) will mature as a probe of cosmological
physics. Gravitational-wave (GW) observatories will begin to run nearly
continuously and, with expected upgrades, become ever more sensitive to the
mergers of binary neutron stars (BNS) that produce kNe. Detecting the
electromagnetic (EM) counterpart to these sources, i.e., the kNe, will enable
improved constraints on the accelerating expansion of the universe, via
measurements of the Hubble constant (Holz & Hughes, 2005; Nissanke et al.,
2013), complementary to other probes of cosmic expansion (Nissanke et al.,
2010; Chen et al., 2018; Mortlock et al., 2018). This is predicated on the
detection of EM and gravitational waves from the same BNS mergers. The LSST
will be able to detect kNe at distances beyond the range that future GW
observatories operating in the 2020s will be sensitive (Scolnic et al., 2018a;
Setzer et al., 2019).

Combined with the large area and rapid cadence of LSST’s WFD survey this
offers the opportunity for the LSST to detect and identify kNe that will not
be detected with any other instrument. However, GW detectors do not need to
point in a given direction to make a measurement of a signal and are sensitive
to signals from the entire sky. In principle, if GW detectors are operating
coincidentally to a detection of a kN by the LSST, the signal of the BNS
merger will be in the GW data, but possibly below the significance threshold
used to claim a merger detection. The kN detection by the LSST can then be
used as prior information to reverse-trigger a search through GW data for an
accompanying merger signal (Kelley et al., 2013). With this information the
population of sources with both EM and GW signals, i.e., standard sirens, used
for studies of fundamental physics can be increased and studies can be made of
the underlying BNS population. This may be critical for studying the selection
effects of GW-detection of standard sirens.

For the kilonovae probe, we introduce 3 new metrics. A summary of the results
from this section can be seen in Figure 7.

  1. 1.

GW170817-like kNe Counts 10yrs - This metric represents the number of
GW170817/AT2017gfo-like (Abbott et al., 2017) kilonovae that are detected
according to a set of criteria over the full 10 year survey.

  2. 2.

kNe MAF Mean Counts - This metric represents the MAF implementation which
evaluates the number of GW170817/AT2017gfo-like kilonovae that are detected on
average per region of the sky assuming a kilonova is always “going-off” at a
fixed redshift of 0.075.

  3. 3.

kN Population Counts 10yrs - This metric represents the number of kilonovae
drawn from a population that are detected according to a set of criteria over
the full 10 year survey.

Figure 7: Kilonova metrics as a function of selected observing strategies.
Section 2.6 contains the exact simulation names corresponding to the short
names used here.

####  5.3.1 Serendipitous Kilonova Detections

To show the potential for this promising science case we use the number of
detected kNe as a metric to compare survey strategies. We here focus only on
serendipitous detection of kilonovae and not target-of-opportunity follow-up
of gravitational wave triggers which has different requirements. We introduce
three metrics: the expected kilonova population counts after 10 years, the
GW170817-like counts of kilonova after 10 years, and the MAF-Mean Counts of
kilonovae.

To classify a kNe as being detected we use the criteria from (Scolnic et al.,
2018a) and used again by (Setzer et al., 2019). These criteria are the
following:

  * •

Two LSST alerts separated by $\geq 30$ minutes.

  * •

Observations in at least two filters with $\mathrm{SNR}\geq 5$.

  * •

Observations with $\mathrm{SNR}\geq 5$ separated by maximum of 25 days (i.e.
no large gaps).

  * •

A minimum of one observation of the same location within 20 days before the
first $\mathrm{SNR}\geq 5$ observation.

  * •

A minimum of one observation of the same location within 20 days after the
last $\mathrm{SNR}\geq 5$ observation.

Several of these requirements were implemented to reject potential
contaminants such as asteroids, AGN and superluminous supernovae (Scolnic et
al., 2018; Setzer et al., 2019). However, they are insufficient to perfectly
separate kNe from other transient classes. While work is ongoing to implement
a transient classification metric (as discussed in Section 7.2) and we
anticipate such a metric to correlate with the same quantities that impact our
kNe metric, we acknowledge that this is an important step that must be studied
in future work.

As was concluded by Setzer et al. (2019), survey strategies which increase the
cadence of observing a region of the sky in multiple filters, e.g., obtaining
the nightly pair of observations in different filters, provide greater numbers
of detected kNe. We direct the reader to Setzer et al. (2019) for a full
discussion of the kN models and methodology used for this similar analysis.
Since that work, the default setting for new observing strategies has been to
implement the nightly pairs of observations in different filters. In this
analysis of additional survey strategies, we find several more features which
improve the numbers of detected kNe relative to the baseline strategy.

The first feature-change which benefits our science is when the number of
visits in the WFD survey is increased. Whether this is achieved through the
use of $1\times 30$ s exposures instead of $2\times 15$ s to increase survey
efficiency or directly increasing the time-allocation of the WFD, more extra-
galactic observations increases the number of detected kNe. This increase in
the number of observations, given a fixed survey length, effectively increases
the cadence of observations for any field.

This is expected, kNe are fast-evolving transients and their detection will be
sensitive to the cadence of observations. However, we find that rolling
observing strategies, which most directly increase the cadence of
observations, do not notably improve the number of kN-detections over the
baseline observing strategy. Kilonovae are not only fast-evolving, but they
are also rare, and thus our metric is sensitive to the amount of cosmological
volume, effectively sky-area, that is observed. Rolling-style observing
strategies fundamentally decrease the observed sky-area active within an
observing season by separating the sky into bands of declination. We find any
improvement that might be expected from the increased cadence is, at best,
negated by the substantial decrease in actively observed sky-area.

Conversely, increasing the total survey area with a fixed number of
observations will decrease the cadence of observations. We find if only the
observed sky-area is increased, there is not an improvement to our metric.
However, if the increase in sky-area is accompanied by the addition of more
observations, such as a reduction in total time allocated to mini-surveys,
e.g., the removal of uiy-band observations outside the WFD survey, we see a
modest increase in the number of detected kNe. We conclude increasing the
footprint area of the survey is beneficial for increasing the number of
detected kNe only if the number of observations per field is not significantly
decreased. Furthermore, a larger extra-galactic sky area can be achieved by
moving the observed sky-area to regions of low dust-extinction. We find,
additionally, the proposed survey strategies which avoid the Milky-way by a
dust or galactic-latitude cut are quite beneficial for detecting greater
numbers of kNe.

Lastly, we note that while kNe are expected to be quite red transients, we do
not find a redistribution of observations into redder filters to improve
detections. Of the proposed distributions of observations by filter, the
baseline filter-distribution performs best for both the population model of
kNe and the model based on GW170817. This can be understood by considering the
LSST bandpass efficiency, which is highest for the $g$-band and decreases for
the redder bands (LSST SRD). The best filter distribution with which to detect
kNe is hence dependent on both the underlying spectral energy distribution of
kNe and the bandpass efficiencies of LSST. It appears that the current
baseline filter allocation is close to optimal, at least for the chosen
population model.

While the full detection analysisis in general computationally prohibitive for
hundreds of proposed survey strategies, a simplified version of this is
implemented into the Metric Analysis Framework (MAF) that is used by the LSST
Project and is publicly available. The MAF implementation252525
https://github.com/LSST-
nonproject/sims_maf_contrib/blob/master/mafContrib/GW170817DetMetric.py,
labelled above descriptively as kN MAF Mean Counts, considers only
GW170817-like kNe. Like other MAF transient metrics, it considers that these
transients go off at a single, user-specified, cosmological redshift and occur
one after another at each point on the sky. From this, observations are made
of the light curves according to the chosen survey strategy and these are
checked whether they pass certain detection criteria. Of the criteria listed
above, we are limited to implementing only criteria two and three. From this
we obtain a number of detected kNe; however, as this is not a proper
cosmological transient distribution based on comoving volumetric rates this
metric should not be used to forecast detected counts of kNe per survey, as
was done in the full analysis. In this case, we instead compute the mean
number of detections, i.e., the number of detected kNe averaged over the
number of fields in the analyzed survey strategy, as our metric for
comparison. The use of these numbers should only be in comparison to other
survey strategies. Given the limitations in the MAF implementation, the MAF
metric does not exactly emulate all metric results we see from the full
analysis. However, the number of outliers is small and the overall trends are
reproduced.

####  5.3.2 General Conclusions from Kilonovae

We conclude the most important observing strategy feature for improving the
number of kilonovae detections is still obtaining a pair of observations in
different filters within a single night. Second to this, increasing the number
of the observations in the extragalactic WFD survey is very beneficial.
Lastly, we find observing strategies that increase the observed sky-area in
low dust-extinction regions, and do not substantially decrease the cadence of
observations to achieve this larger area, are also preferable.

##  6 Discussion

Combining the insights gained from our cosmology-related metrics is not a
trivial task. In an ideal world, a full DETF FoM could be calculated for each
simulation allowing the objective determination of the “optimal” observing
strategy for cosmology. However, as we expect systematic effects to play a
significant role in final cosmological constraints with LSST, we do not
currently have a realistic enough FoM to make final decisions based on it
alone. We make use of the 3x2pt FoM described in Section 4.1.1 to summarize
the impact of observing strategy on the main static science probes, but also
consider separately WL systematics and transient metrics. These simpler, more
interpretable metrics can assist in gaining deeper insight and making general
recommendations for observing strategy. We expect metrics such as the total
number of well-measured supernovae and survey uniformity to correlate strongly
with cosmological parameter constraining power once systematic effects are
taken into account. In this section, we use select metrics to investigate and
draw insight into various aspects of observing strategy.

###  6.1 Footprint, area and depth

In general, there is a trade-off between depth and area in any survey. Figure
8 shows several key metrics as a function of effective survey area (the survey
area meeting the cuts described in Section 3.1). The first thing to note is
that the 3x2pt FoM has a simple linear relationship with area and the larger
the survey, the better the 3x2pt FoM. Photometric redshifts however tend to
prioritize depth over area. There is also a trade-off between WL shear
systematics, which improve as more visits are taken (essentially ending up
with more depth) and the 3x2pt FoM, which prefers larger area. To fully
quantify this trade-off, the WL systematics would need to be included in the
full DETF FoM pipeline. Each of the transient metrics has a more complicated
relationship with area since they are impacted by other observing strategy
choices such as cadence and filter distribution.

Figure 8: Selected metrics, relative to their values at baseline, as a
function of effective area (i.e., the area that meets the cuts described in
Section 3.1) of different observing strategies. To improve readability, points
that are nearby in the $x$-axis are binned with only the mean and error bar
plotted for that bin. It can be seen that the 3x2pt FoM metric simply prefers
more area while the situation is more complex for time-dependent metrics. We
have highlighted specific simulations with numbered annotations. The
simulation at 1 reduces the overall area available for cosmology by reducing
visits in the crucial redder bands. Simulations 2 and 3 have larger area
footprints, but degrade transient metrics and photometric redshifts due to a
reduced number of visits in the extragalactic part of the WFD. Simulation 4
reverses this by completely prioritizing visits in areas of low dust
extinction, resulting in both large area and a large number of visits and the
best performance for all cosmology metrics. List of annotations:
1-footprint_bluer_footprintv1.5_10yrs, 2-footprint_newBv1.5_10yrs,
3-bulges_bs_v1.5_10yrs, 4-footprint_big_sky_dustv1.5_10yrs.

Of particular interest are the observing strategies with the largest area. It
can be seen that some strategies yield large area (around $16,500$ deg2) but
poor performance for the transient metrics and the WL systematics metric. This
is because these strategies have a large footprint but prioritize visits in
the Galactic bulge and plane, reducing cadence in the extragalactic area. The
largest area simulation, which gives simultaneous high performance for the
3x2pt FoM and the number of supernovae, is the `footprint_big_sky_dust`
simulation which uses an extinction-based cut to define the WFD footprint.
This allows both increased area and depth for extragalactic science. The
number of visits in WFD and the other surveys are the same as baseline,
however the number of visits in the Galactic plane is dramatically reduced.

Figure 9 shows a subset of metrics as a function of median co-added $5\sigma$
$i$-band depth. Here we see the lack of sensitivity of the 3x2pt FoM, as long
as sufficient depth is achieved. On the other hand, two transient metrics,
number of supernovae and number of kilonovae, improve by as much as 27% for a
deeper survey. This is really a consequence of an improved number of visits
and thus increased cadence. However, most of the simulations that are
significantly deeper than the baseline are not actually realistic, as they
place artificially large amounts of survey time into WFD. While these
experiments are useful to understand the behavior of the metric, they do not
represent viable observing strategies. However, it is entirely possible to
choose a footprint or strategy that reduces the overall depth of the survey,
to the detriment of the transient probes and photometric redshifts.

Within the constraints of the observing strategy requirements from the LSST
SRD, it appears to be quite challenging to achieve a depth much greater than
the current baseline. The ideal approach from the standpoint of cosmology is
to maintain a WFD footprint of 18,000 deg2 but prioritize regions with low
dust extinction, allowing for increased area while maintaining current depth.

Figure 9: Selected metrics, relative to their values at baseline, as a
function of $i$-band median co-added depth of different observing strategies.
To improve readability, points that are nearby in the $x$-axis are binned with
only the mean and error bar plotted for that bin. Here we note that the 3x2pt
FoM metric is generally indifferent to depth, implying that larger area is
more important (assuming the changes in depth remain small). Photometric
redshifts improve with more depth and the transient science probes,
particularly supernovae and kilonovae, are strongly affected since generally
greater depth corresponds to increased cadence. We have highlighted specific
simulations with numbered annotations. Simulation 1 has the poorest
performance because it removes visits from the extragalactic part of the
footprint. Simulation 2 is a large area footprint which improves the static
science metrics, but still reduces overall cadence and depth. Simulations 3
and 4 artificially perform well for the transient metrics because they
unrealistically ignore all mini-surveys and put all visits into WFD. List of
annotations: 1-footprint_newAv1.5_10yrs, 2-bulges_cadence_i_heavy_v1.5_10yrs,
3-wfd_depth_scale0.99_v1.5_10yrs, 4-wfd_depth_scale0.99_noddf_v1.5_10yrs.

###  6.2 A simple combined metric to analyze the depth-area trade-off

To better understand the trade-off between area and depth, we combine metrics
in a very simple way based on the DESC SRD. Because of the complexity involved
in combining the systematic related metrics such as the WL systematics and
photometric redshift metrics, we here only include the 3x2pt FoM and the
number of well-measured supernovae metric, in a 50–50 ratio after normalizing
each metric to represent the ratio in the amount of information provided by a
given simulation versus the baseline. We note that both these metrics
incorporate some (but not all) systematic effects. While we caution the reader
that this is a simple approximation of the much more complex full DETF FoM
that includes all systematics, it can be helpful in gaining insight. The
general trade-off between number of visits (and hence depth) and area can be
seen and, while it is difficult to improve over the current baseline, many
simulations are much worse. Our preferred footprint, indicated by “Larger
extragalactic footprint” that implements a dust extinction cut, has the best
overall performance for cosmology by increasing area without reducing depth.
One proposed simulation, indicated as “Combo dust” also has a larger area
footprint defined by dust extinction, but includes a significant number of
visits in the Galactic plane. This simulation is roughly equivalent to
baseline for this combined metric, as can be seen in Figure 10, because what
is gained in area is lost in depth.

It is important to note that key probes such as strong lensing and kilonovae
are not included in the combined metric because their constraining power is
not comparable to the other probes and they would be unfairly downweighted.
Although they do not contribute as significantly to the overall DETF FoM,
these probes are still essential for other aspects of cosmology (unravelling
the $H_{0}$ tension, for example). We thus consider them separately from this
combined metric.

Both kilonovae and strong lensing suffer if the area is increased so our
recommendation is that any increases in area should maintain the depth in WFD
as far as possible by redefining the footprint, using rolling cadence
(discussed in Section 7.2) and re-optimizing the number of visits and filter
distribution in the Galactic plane and other mini-surveys for those specific
science cases.

Figure 10: A simple attempt to combine multiple metrics to produce a combined
cosmology metric. We show the combined metric (using the usual (metric-
baseline)/baseline)) as a function of effective area (i.e., the area that
meets the cuts defined in Section 3.1) and median number of visits (which
correlates with depth for most simulations). Although the produced map is
complex, the trade-off between depth and area can be seen. Several specific
strategies are highlighted, the simulation names of which can be found in
Section 2.6. Unlike for the rest of the figures which focus on simulations
that more or less vary one aspect of observing strategy at a time, we have
highlighted here strategies from FBS v1.6 which are proposed observing
strategies that aim to satisfy multiple science goals. “Solar system focus”
and “Galactic plane focus” both take observations away from WFD to prioritize
other science cases. “Combo dust” is a proposed large area footprint which has
more area but less depth than “Baseline”, thus producing similar performance.
“Larger extragalactic footprint” is a larger area footprint that is defined by
dust extinction and gives improved cosmological constraints due to the area
gained by avoiding extinction. We caution the reader against assuming the
performance in this metric would correspond to the exact percentage
improvement/degradation in cosmological constraints. Only a full DETF FoM
including systematics can indicate the exact numerical impact of observing
strategy choice.

###  6.3 Overlap with other surveys

Data from other surveys is key to enhancing the cosmological analysis with
LSST data. Planned surveys using multi-object spectroscopy including the Dark
Energy Spectroscopic Instrument (DESI; DESI Collaboration et al., 2016) and
the 4-metre Multi-Object Spectroscopic Telescope (4MOST; de Jong et al., 2019)
will provide spectroscopic data for millions of objects. This can be used for
photometric redshift calibration and training, cross correlation to improve WL
and host galaxy redshift identification for transients (Mandelbaum et al.,
2019). The Time-Domain Extragalactic Survey (TiDES; Swann et al., 2019) on
4MOST will provide live spectra for up to 30,000 transients, including
supernovae. The Euclid telescope (Laureijs, 2011) will survey over 15,000
deg2, obtaining photometric and spectroscopic data in the infrared bands,
complementary to LSST and enabling better photometric redshift calibration and
training, improvements to WL and improved de-blending of galaxies (Capak et
al., 2019).

Section 6.3 shows how the overlap with DESI, 4MOST and Euclid can be improved
with a larger area footprint. Even better coverage, particularly for DESI, can
be achieved by extending the footprint further North, as described in Section
7.

Table 4: Area overlap for the 10-year LSST survey (extragalactic footprint)
with other surveys for two types of footprint: baseline and a larger area
footprint with a boundary defined by a dust extinction cut (see Section 2.6
for the precise simulation names). The larger footprint provides nearly 2000
deg2 additional overlap with DESI and over 1000 deg2 with 4MOST and Euclid.
Even greater gains can be made by extending the footprint further North.

Finally, we also note that external data in the DDFs will be crucial for
photometric redshift training and supernova spectroscopic follow-up (both live
and later host galaxy redshift determination). It is particularly important
and challenging to align DDF observations temporally with telescopes such as
Euclid, meaning that the optimal strategy will likely be to prioritize certain
DDFs in certain years. We leave it to future work to determine the optimal DDF
strategy for cosmological measurements with LSST.

###  6.4 Filter distribution

While the LSST SRD enforces a minimum number of visits in each filter, there
is still room to increase or decrease the number of visits (and hence the
cadence and depth) in any particular band. This choice of filter distribution
has a complex impact on our cosmology metrics, as seen in Figure 11. Here we
see some small tensions between probes: supernovae prefer a redder
distribution while any change from baseline is detrimental to the number of
kilonovae. The choice of filter distribution is especially complex for