contents=  Submitted to the Proceedings of the US Community Study on the
Future of Particle Physics (Snowmass 2021) ,
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# Snowmass2021 Cosmic Frontier White Paper: Enabling Flagship Dark Energy
Experiments to Reach their Full Potential

Jonathan A. Blazek  Department of Physics, Northeastern University, Boston,
MA, 02115, USA  Doug Clowe  Ohio University, Department of Physics and
Astronomy, Clippinger Labs 251B, Athens, OH 45701, USA  Thomas E. Collett
Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth
PO1 3FX, UK  Ian P. Dell’Antonio  Physics Department, Brown University,
Providence, RI, 02912, USA  Mark Dickinson  NSF’s National Optical-Infrared
Astronomy Research Laboratory, Tucson, AZ, USA  Lluís Galbany  Institute of
Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, E-08193
Barcelona, Spain  Institut d’Estudis Espacials de Catalunya (IEEC), E-08034
Barcelona, Spain  Eric Gawiser  Department of Physics and Astronomy, Rutgers,
the State University of New Jersey, Piscataway, NJ 08854, USA  Katrin Heitmann
Argonne National Laboratory, 9700 S. Cass Ave, Lemont, IL 60439, USA  Renée
Hložek  David A. Dunlap Department of Astronomy and Astrophysics & Dunlap
Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George
Street, M5S 3H4, Canada  Mustapha Ishak  Department of Physics, The University
of Texas at Dallas, Richardson, Texas 75080, USA  Saurabh W. Jha  Department
of Physics and Astronomy, Rutgers, the State University of New Jersey,
Piscataway, NJ 08854, USA  Alex G. Kim  Lawrence Berkeley National Laboratory,
1 Cyclotron Road, Berkeley, CA 94720, USA  C. Danielle Leonard  School of
Mathematics, Statistics and Physics, Herschel Building, Newcastle University,
Newcastle-upon-Tyne, NE1 7RU, United Kingdom  Anja von der Linden  Department
of Physics and Astronomy, Stony Brook University, Stony Brook University,
Stony Brook, NY 11794, USA  Michelle Lochner  Department of Physics and
Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South
Africa  South African Radio Astronomy Observatory (SARAO), The Park, Park
Road, Pinelands, Cape Town 7405, South Africa  Rachel Mandelbaum  McWilliams
Center for Cosmology, Department of Physics, Carnegie Mellon University,
Pittsburgh, PA 15213, USA  Peter Melchior  Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 08540, USA  Joel Meyers
Department of Physics, Southern Methodist University, Dallas, TX 75275, USA
Jeffrey A. Newman  Department of Physics and Astronomy and PITT PACC,
University of Pittsburgh, Pittsburgh, PA 15260, USA  Peter Nugent  Lawrence
Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
Department of Astronomy, University of California at Berkeley, New Campbell
Hall, Berkeley, CA 94720, USA  Saul Perlmutter  Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA  Department of Physics,
University of California Berkeley, 366 LeConte Hall MC 7300, Berkeley, CA,
94720, USA  Daniel J. Perrefort  Department of Physics and Astronomy and PITT
PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA  Center for Research
Computing, University of Pittsburgh, Pittsburgh, PA 15260, USA  Javier Sánchez
Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA
Samuel J. Schmidt  Department of Physics, University of California, Davis, One
Shields Avenue, Davis, CA 95616, USA  Sukhdeep Singh  McWilliams Center for
Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA
15213, USA  Mark Sullivan  School of Physics and Astronomy, Univeristy of
Southampton, Southampton, SO17 1BJ, UK  Aprajita Verma  Sub-department of
Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road,
Oxford OX1 3RH, UK  Rongpu Zhou  Lawrence Berkeley National Laboratory, 1
Cyclotron Road, Berkeley, CA 94720, USA

###### Abstract

A new generation of powerful dark energy experiments will open new vistas for
cosmology in the next decade. However, these projects cannot reach their
utmost potential without data from other telescopes. This white paper focuses
in particular on the compelling benefits of ground-based spectroscopic and
photometric observations to complement the Vera C. Rubin Observatory, as well
as smaller programs in aid of a DESI-2 experiment and CMB-S4. These additional
data sets will both improve dark energy constraints from these flagship
projects beyond what would possible on their own and open completely new
windows into fundamental physics. For example, additional photometry and
single-object spectroscopy will provide necessary follow-up information for
supernova and strong lensing cosmology, while highly-multiplexed spectroscopy
both from smaller facilities over wide fields and from larger facilities over
narrower regions of sky will yield more accurate photometric redshift
estimates for weak lensing and galaxy clustering measurements from the Rubin
Observatory, provide critical spectroscopic host galaxy redshifts for
supernova Hubble diagrams, provide improved understanding of limiting
astrophysical systematic effects, and enable new measurements that probe the
nature of gravity. A common thread is that access to complementary data from a
range of telescopes/instruments would have a substantial impact on the rate of
advance of dark energy science in the coming years.

##  1 Introduction

The newest generation of “Stage IV” Dark Energy experiments such as the Vera
C. Rubin Observatory Legacy Survey of Space and Time (LSST) and the Dark
Energy Spectroscopic Instrument (DESI) will play a major role in improving our
knowledge of cosmology in the coming years. However, additional data from
other ground-based facilities can enable these facilities to constrain models
of cosmic acceleration more strongly and reach their utmost potential.

The gains from additional data can arise in several ways. Strong lensing and
supernova cosmology constraints from LSST can be greatly strengthened by
making use of additional telescopes and instruments that will enable detailed
follow-up of individual objects. These facilities can provide single-object
imaging either in more bands, with higher spatial resolution, or with higher
cadence than the LSST observations provide, and can provide spectroscopic
measurements for rare objects, which are difficult to target efficiently for
multi-object spectroscopy. Spectroscopy of individual strong lens systems will
constrain lens properties and measure redshifts for both the lenses and
background objects; spectroscopy can also classify individual supernovae in
“live” follow up observations.

Other opportunities to enhance cosmological measurements from LSST would be
enabled by access to deep ($i\sim 25$), highly-multiplexed optical and near-
infrared multi-object spectroscopy on 8–40 m telescopes. Such capabilities
would enable improved photo-$z$ estimates and reduce systematic uncertainties
[1]. Every major cosmological probe based upon LSST data would benefit. As an
example, photometric redshift (photo-$z$) estimates depend on deep
spectroscopic datasets for optimization of algorithms; every major
cosmological probe will rely on photo-$z$’s directly or indirectly. However,
focused spectroscopic campaigns can also improve weak lensing cosmology by
constraining the intrinsic alignments between the orientations of galaxies;
contribute to galaxy cluster studies by measuring kinematics of galaxies in
clusters and testing photo-$z$ performance in regions of high density; and aid
strong lensing measurements by determining the distribution of foreground mass
along the lines of sight to systems.

A third area of gains to LSST would come from access to wide-field ($>20$ deg2
total survey area), highly-multiplexed optical and near-infrared multi-object
spectroscopy (MOS) on 4–15 m telescopes. LSST photo-$z$’s can be calibrated
with high precision via cross-correlations against large-area spectroscopic
samples. Spectroscopy with wide-field instruments can also constrain galaxy
intrinsic alignments using cross-correlations with shallower spectroscopic
samples; aid strong lensing measurements by providing a first characterization
of redshifts of candidate strong lens systems; provide tests of modified
gravity via a variety of cross-correlation techniques; and contribute to
supernova cosmology by providing redshifts for host galaxies of supernovae
that lack direct spectroscopy.

However, the Vera Rubin Observatory will not be the only beneficiary from
additional data obtained with other facilities. As an example, future surveys
with the DESI instrument can more efficiently target objects of interest if
deep, multiband imaging (such as that which LSST will produce) is available in
regions north of the coverage of the main LSST “Wide, Fast, Deep” survey.
Additionally, cosmology with galaxy clusters from CMB-S4 will benefit from
deep multiplexed spectroscopy, overlapping with the needs for LSST cluster
science. Additional discussions of the benefits to cosmological studies from
combining datasets from multiple surveys are provided in two other Snowmass
white papers [2, 3].

In the remainder of this white paper, we describe the benefits that additional
data can bring to upcoming Dark Energy experiments, considering the gains for
each science case separately. We concentrate this discussion on upcoming
experiments that are primarily focused on constraining the nature of Dark
Energy, but note that other cosmology experiments should also benefit from
access to the additional capabilities we outline here.

##  2 Enabling Measurements of the Supernova Distance – Redshift Relation with
LSST

###  2.1 Photometric follow-up

For supernova cosmology, we require measurements of supernova light curves
(brightness as a function of time) in multiple bands with dense enough time
sampling to accurately determine the peak brightness of the supernova and the
rate at which it declines. For some supernovae observed by LSST, the standard
“Wide, Fast, Deep” cadence will not be optimized for these measurements, and
additional photometry from other facilities can enhance supernova cosmology in
the coming decade.

There are two major ways in which additional photometry in the LSST fields can
benefit supernova cosmology. First, the cadence of the LSST will not be
optimal for all transients of interest. Many LSST science cases are focused on
high-precision photometry and astrometry spanning long periods of time for
transients which are brighter than cosmological supernovae (e.g., variable
stars, high-proper motion stars, etc.), while on the fainter end the most
common transients of interest have light curves that evolve only slowly (e.g.,
high-redshift Type Ia supernovae for cosmology). The need to serve these
science cases results in the standard LSST cadence being less optimal for some
classes of supernovae. Higher-cadence observations can complement LSST by
providing higher-quality light curves for nearby supernovae, improving
cosmological measurements.

Additionally, it is easier to obtain follow-up data (e.g., spectroscopy) for
more nearby, brighter objects, as then smaller and less expensive telescopes
can be put to use. Obtaining a timely spectrum is a key need for much
supernova science (as described below). However, much of the very nearby
universe is inaccessible to the Rubin Observatory as bright objects will
saturate its detectors. As a result a high-cadence, shallow survey is an
optimal complement to Rubin and the LSST for nearby supernova science such as
the measurement of peculiar velocities.

ZTF, while focused in the north, is one such survey[4]. In the south, there is
the La Silla Schmidt Southern Survey [LS4; 5]. LS4 is a 5-year public, wide-
field, optical survey using the upgraded 20 square degree QUEST Camera on the
ESO Schmidt Telescope at the La Silla Observatory in Chile. It uses LBNL deep-
depletion CCDs to maximize the sensitivity in the optical up to 1 micron. As
outlined above, it complements the LSST by providing a higher cadence over
several thousand square degrees of sky in the $g-$ and $z$-bands, allowing a
more accurate characterization of brighter and faster evolving transients down
to 21st magnitude. It also opens up a new phase-space for discovery when
coupled with the LSST by probing the sky between 12–16th magnitude – a region
where the Rubin Observatory saturates. As such, it will greatly improve both
the identification and characterization of nearby Type Ia supernova for
several of the cosmology studies outlined below.

###  2.2 Single-object spectroscopic observations of active SNe

As described in subsection 2.3, the largest sets of redshift measurements for
LSST supernovae should come from targeting SNe and their host galaxies via
wide-field multi-object spectroscopy, either using a small fraction of fibers
in surveys that span the LSST footprint or via targeted surveys in the LSST
Deep Drilling Fields (DDFs). However, higher signal-to-noise observations will
be needed for a significant sample of low-$z$ SNe to characterize the range of
intrinsic spectra; and to compare this to the high-$z$ LSST Type Ia SN ($z\sim
0.5-1.2$) will require targeted long-exposure follow-up while the high-$z$ SNe
are still bright.

Typing to eliminate non-SNe Ia contamination and to validate photometric
classification at high (and low) redshift: Much work has gone into techniques
for determining supernova types and redshifts from photometry alone, but this
is still a difficult technique when applied over a large redshift range due to
changing filter coverage and unknown drifts in the spectroscopic properties of
the SNe. Current photometric identification techniques require
spectroscopically classified training samples, ideally over the full range of
redshifts under consideration. Fully characterizing some classes of objects
from LSST at the faintest magnitudes [e.g., SLSN and Type II SN, see 6, 7]
will require spectroscopic observations on 25–40 m class telescopes.
Furthermore, measuring the redshifts via spectra of “live” supernovae
eliminates type uncertainty, yielding high-purity samples with limited
systematics. Ultimately, detailed spectrophotometric studies of low-$z$ Type
Ia SN will be the best source of high-signal-to-noise information about the
variations over a large range of Type Ia SN subtypes, making spectroscopy on
smaller telescopes valuable as well. An example of such a study can be found
in Ref. [8].

Systematic error mitigation: Detailed understanding of a variety of potential
sources of systematic error will be crucial for supernova science with LSST,
including evolution of the Type Ia SN population with redshift; errors in the
specification of the spectral energy distribution (SED) in different
redshift/rest-frame wavelength ranges that can affect training of light curve
models; and mis-classifying fine sub-types (“twins”) of SNe Ia. Finally, we do
not know if the current catalog of spectral properties is exhaustive, and, if
not, the mismatch between assumed spectral model templates and actual spectra
will introduce systematic shifts in cosmological distance measurements. While
deep follow-up of a variety of LSST transients will be needed to understand
their physics and evolution, requirements for the use of SNe Ia as cosmic
probes of distance are particularly stringent, and hence detailed spectral
studies are needed.

Very high data quality is needed if spectra of “live” supernovae will be used
to detect the subtle differences between sub-types of SNe Ia, to study
evolutionary population drift, and to match highly similar supernovae (“twins”
as described in [9]) to reduce the dispersion of measured distances. Such
purposes require high signal-to-noise observations to track subtle
spectroscopic features and spectrophotometric data to allow a clean
subtraction of the host galaxy light. For systematics constraints on
measurements of high-redshift Type Ia SN to be commensurate with the precise
statistical uncertainties resulting from large numbers of LSST-discovered SNe,
new instrumentation for ground-based telescopes will be needed (e.g., IFU or
high-throughput spectrographs now in the planning stages for 2–4 m telescopes
and IFU reformatters on existing spectrographs, with or without AO, on 8–40 m
telescopes). The development of space-based observational avenues such as
coordinated observations with Euclid, WFIRST or other space missions now
planning spectrophotometric instrumentation could also be helpful.

###  2.3 Multi-object spectroscopic observations of supernovae and their host
galaxies

SNe Ia provide a mature and well-exploited probe of the accelerating universe
(e.g., [10]), and their use as standardizable candles is an immediate route to
measuring the equation of state of dark energy. LSST, for example, could
assemble around 100,000 SNe Ia to $z=1$, giving unprecedented insight into the
expansion history of the universe. A major systematic uncertainty will be the
photometric classification and redshift measurement of the supernova
detections. Wide-field spectroscopy can exploit the fact that wherever a
follow-up facility points in the extragalactic sky, there will be known time-
variable sources, including both recently discovered transients and older,
now-faded events.

Spectroscopy serves two main goals over both the typical LSST Wide, Fast, Deep
(WFD) footprint and the more frequently-observed Deep Drilling Fields (DDFs).
The first is the classification of live SNe and the construction of optimized
training samples for photometric classifiers to assemble the next generation
of SN Ia cosmological samples. Even the most advanced machine-learning
classification techniques are fundamentally dependent on large, homogeneous
and representative training sets [11]. The second goal is to obtain
spectroscopic redshifts (spec-$z$’s) for host galaxies of SNe that have faded
away. While conventional SN Ia cosmology analyses rely on spectroscopic
follow-up of all the SNe, new analyses [e.g. 12, 13, 14] show the possibility
of taking advantage of even larger samples of SNe after obtaining spec-$z$’s
of their host galaxies. Targeted campaigns of multi-object spectroscopy of SNe
and their hosts in LSST DDFs, combined with the assignment of fibers to such
objects in larger surveys covering the wide LSST footprint, can provide the
redshifts needed for most LSST SN Ia cosmology studies. More detailed
investigation of a smaller set of SNe will remain valuable, however, as
described in section 2.

####  2.3.1 Supernovae and their hosts in LSST Deep Drilling Fields (DDFs)

The best-characterized and deepest LSST SN samples will come from the higher-
quality light curves provided by more frequent and deeper photometry of the
DDFs. Because of the comparatively small area covered by these fields ($\sim
50$ deg2 total) and the low surface density of SNe, it is feasible to obtain
long exposures on all DDF live SNe or their hosts using wide-field
spectrographs (e.g., DESI or Subaru/PFS in the North and 4MOST in the South);
c.f. [15]. This approach has proven very successful in the OzDES survey [16],
and will be employed by the 4MOST/TiDES survey for LSST SNe [17]. In addition
to the cosmological measurements enabled by DDF SNe and host redshifts, the
resulting dataset should be an extremely valuable source of high signal-to-
noise templates for light curves to be used in cosmology for SNe that have
LSST data but no spectroscopy, and will also provide a wealth of training data
for photometric classification of transients.

Such surveys of supernova hosts can be performed efficiently at a variety of
current (e.g., DESI) or proposed facilities. Table 1 describes the total
amount of time it would take with different instruments and telescopes to
perform annual spectroscopy of 100 new $r<24$ galaxy hosts of supernovae per
square degree spanning five LSST deep drilling fields, each 10 square degrees
in area; characteristics of relevant facilities are summarized in [18]. We
scale the exposure time for each facility to be equivalent to 8 hours per
pointing on a 4m telescope, which should yield sufficient signal-to-noise to
obtain secure redshifts for at least 75% of such targets (Supernova hosts that
fail to yield redshifts could be retargeted in successive years to further
improve S/N). Over the course of the 10-year LSST survey, such follow-up would
provide redshifts for most of the $\sim 50,000$ best-characterized LSST SNe
Ia. Note that the time costs can be mitigated by incorporating SN host targets
into a denser MOS observing program serving multiple science needs. Indeed
OzDES survey exposure times simultaneously served SN host-galaxy redshifting,
photo-z calibration, active-transient classification, and QSO monitoring.

Details of the assumptions made about how instruments may be compared in these
calculations are given in [19]. Results are given in “dark-years”: one dark-
year consists of 365 8-hour-long dark nights, of which one-third are lost to
weather and instrumental overheads.

Figure 1: Left panel: The redshift distributions of high-quality supernovae that should be observed in the wide (WFD) and deep (DDF) regions of the LSST survey. LSST will vastly increase SN samples compared to current surveys; spec-$z$’s will greatly increase the value of the SNe for cosmological analyses. Right panel: The fraction of SNe Ia expected to have secure redshift measurements from 4MOST/TiDES observations of $i<22$ SN hosts in the WFD region (orange curve), as well as expectations for an $i<22.9$ Subaru/PFS sample in the DDFs (blue curve); the existence of such data is assumed in Ref. [1], and will be vital for SN Ia cosmology with LSST.  Table 1: Time required per epoch of SN host spectroscopy in LSST deep fields  Instrument / Telescope | Collecting | Field area | Multiplex | Total time
---|---|---|---|---
| Area (sq. m) | (sq. arcmin) |  | (dark-years)
4MOST | 10.7 | 14,400 | 1,400 | 0.05
Mayall 4m / DESI | 11.4 | 25,500 | 5,000 | 0.03
WHT / WEAVE | 13.0 | 11,300 | 1,000 | 0.06
Megamapper (Magellan-like) | 28.0 | 25,416 | 20,000 | 0.01
Subaru / PFS | 53.0 | 4,500 | 2,400 | 0.04
VLT / MOONS | 58.2 | 500 | 500 | 0.29
Keck / DEIMOS | 76.0 | 54 | 150 | 2.04
Keck / FOBOS | 76.0 | 314 | 1,800 | 0.35
ESO SpecTel | 87.9 | 17,676 | 15,000 | 0.01
MSE | 97.6 | 6,359 | 3,249 | 0.01
GMT/MANIFEST + GMACS | 368.0 | 314 | 420 | 0.07
TMT / WFOS | 655.0 | 25 | 100 | 0.51
E-ELT / Mosaic Optical | 978.0 | 39 | 200 | 0.2211For E-ELT, observations in both the optical and near-IR settings are required to achieve the required wavelength coverage, increasing total time required.
E-ELT / MOSAIC NIR | 978.0 | 46 | 100 | 0.19

####  2.3.2 Supernovae and their hosts in the Wide, Fast, Deep (WFD) Survey

While the smaller area of the deep fields ensures that host redshifts can be
obtained for a higher fraction of all SN hosts than in the WFD, the wide field
will yield a much larger number of SNe in total. This sample of hundreds of
thousands of SNe will revolutionize cosmological analyses and enable extensive
studies of systematics. The improvement is illustrated in the left panel of
Figure 1, which shows redshift histograms for the best-characterized SNe Ia in
the DDF and WFD, as compared to the current sample of known SNe. The LSST WFD
survey will increase the number of SNe dramatically in the intermediate
redshift range, complementing the deep sample from the DDF. The 4MOST/TIDES
program will target SNe and their hosts for spectroscopy across the WFD
footprint, but will run only for the first half of the LSST survey [17].

The distribution of SN host redshift measurements will depend on the
allocation of spectroscopic resources from ground-based telescopes. We have
tested the impact on the Dark Energy Task Force Figure of Merit (FoM) from
altering the fiducial redshift efficiencies from [1], which are shown in
Figure 1,either by changing the total number of hours and simply rescaling the
amplitude of the selection function, or by modifying the distribution to
disfavor fainter hosts at higher $z$. From these tests we find that the DETF
FoM is most sensitive to the number of SNe at high redshift, and hence changes
in the time allocated to the DDFs have the greatest effect. High redshift
completeness can be achieved across a broad range of galaxy populations for
low-$z$ SNe in WFD, but that will likely be impossible for the higher-$z$ DDF
sample. This will make wide-area spectroscopy vital for investigating
systematic uncertainties related to host galaxy properties, which may be a
limiting factor in LSST SN cosmology.

##  3 Transient Strong Lensing

One of the most striking consequences of general relativity is that light from
distant sources is deflected by the gravitational field of massive objects
near the line of sight. When the deflection produces multiple images, the
phenomenon is known as “strong gravitational lensing.” The multiple images of
strongly lensed sources arrive at different times because they travel
different paths and through different gravitational potentials to reach us
[20, 21]. When a strongly lensed source is time variable, the arrival time
delays can be measured and combined with a mass model to yield cosmological
constraints, particularly on the Hubble constant $\mathrm{H}_{0}$. Strongly
lensed QSOs, compound lens systems, and strongly lensed supernovae each
provide opportunities for measurement of cosmological parameters with LSST,
and the LSST DESC plans to exploit them all [1].

###  3.1 Precision Cosmology with Strongly Lensed Supernova Time Delays

The prospect of cosmological constraints from strongly lensed supernovae was
illustrated with the ground-breaking Hubble Space Telescope observations of
the first multiply-imaged lensed supernova, SN Refsdal [22]. Ground-based
time-domain optical imaging surveys similar in spirit to LSST, such as the
intermediate Palomar Transient Factory (iPTF), played a key role in the
discovery and follow-up observations of another strongly lensed supernova with
resolved multiple images [23, see Figure 2].

Figure 2: iPTF16geu, a Type Ia supernova at $z=0.4$ strongly lensed by an
elliptical galaxy at $z=0.2$ (data from HST and Keck). This transient—the only
known lensed Type Ia SN with resolved multiple images—was discovered by a
predecessor of LSST, the intermediate Palomar Transient Factory (iPTF). The
multiple images of the supernova are indicated with circles. Due to the small
image separation ($\sim$0.3′′), high-resolution imaging was required to
confirm that it was a strongly-lensed supernova. Figure reproduced from [23].

Time delays from lensed supernovae present opportunities to observe the
earliest phases of supernova explosions, to infer cosmological parameters, and
to map substructure in lens galaxies, but many more systems are needed to
achieve these goals. Thanks to novel lensed-supernova hunting techniques, a
new generation of alert-based wide-field imaging surveys that began in
mid–2018 with the Zwicky Transient Facility [ZTF; 4] and will continue into
the 2030s with LSST will increase the size of this sample by orders of
magnitude. ZTF is currently expected to yield $\sim$20 lensed supernovae over
the course of its 3-year campaign, followed by thousands from LSST and WFIRST
[24]. This vast increase in sample size will enable groundbreaking new
measurements with the potential to rapidly deliver precision constraints on
the Hubble constant ($\mathrm{H}_{0}$) and dark energy. The Hubble constant
can be determined to exquisite precision with measurements of the nearby
universe using the distance ladder [25, $\mathrm{H}_{0}=73.24\pm
1.74\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$]. It can also be inferred with
measurements of the primordial universe using the cosmic microwave background
(CMB), assuming a $\Lambda$CDM cosmology [26, $\mathrm{H}_{0}=66.93\pm
0.62\,\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$]. The tension between these local
and distant measurements is palpable: they currently disagree by $\sim
4\sigma.$ It is potentially a sign of new fundamental physics, such as sterile
neutrinos or “phantom” dark energy [e.g., 27, 28, 29], but could also be a
sign of systematics in the measurements [e.g., 30, 31]. Time delays between
the multiple images of strongly lensed time-variable sources depend primarily
on $\mathrm{H}_{0}$, and the mass distribution along the line of sight [32,
33, 34, 35]. Refsdal [36] first suggested using time delays from strongly
lensed supernovae to measure $\mathrm{H}_{0}$, but today, more than 50 years
later, this measurement has yet to be made with precision. Time delays from
lensed supernovae have many advantages over time delays from strongly lensed
quasars, which have been used to measure $\mathrm{H}_{0}$ to 3.4% in a
$\Lambda$CDM cosmology Lensed supernovae require 100 times less monitoring
(i.e., a few weeks rather than decades) and are less sensitive to microlensing
[24], mass modeling systematics [37], and selection bias [38]. Consequently,
they provide the most direct and rapid route to sub-percent constraints on
$\mathrm{H}_{0}$ with strong lensing time delays.

Figure 3:  Examples of measurements of $\mathrm{H}_{0}$ since the year 2000.
Strong lensing time delays from systems discovered by LSST and observed with
supplemental monitoring and follow-up spectroscopy as recommended in this
white paper (black diamond) will resolve or increase the current tension in
$\mathrm{H}_{0}$. Figure adapted from [28].

We forecast that $\mathrm{H}_{0}$ can be measured to sub-percent precision
within $\Lambda$CDM cosmological models with time delays from systems
discovered in the first year of the LSST survey. The total number of supernova
time delay systems over the LSST survey is expected to be $\sim 100$. Our
projected constraint on $\mathrm{H}_{0}$ as shown in Figure 3 is comparable in
precision to the leading current measurement from the combination of Planck
and BOSS data [39, 40], and it is almost ten times better than the current
state-of-the-art constraints from quasar time delays [41]. In addition to
constraining $\mathrm{H}_{0}$, time delays from lensed supernovae are
sensitive to dark energy in a completely different way than cosmological
probes based on distances and volumes [e.g., the CMB, the Type Ia SN distance-
redshift relation, BAO, and galaxy clusters; 42, 43], making time-delay
measurements highly complementary. Adding in lensed supernovae from the first
year of LSST increases the Dark Energy Task Force [44] figure of merit by a
factor of 3 over an Type Ia SN-based constraint alone—a major gain.

Complementary methods: The use of gravitational lens time delays as “standard
clocks” to determine cosmological distances is complementary to the method of
“standard sirens” using the gravitational wave signal emitted by merging
neutron stars and black holes [45]. That method requires redshift measurements
for the host galaxies of the merger events. For binary neutron star mergers
with kilonova couterparts its precision can be improved with spectroscopic
observations of the optical-infrared (OIR) transient to constrain the merger
orientation and reduce the orientation–distance degeneracy. Planned upgrades
to gravitational wave detectors should greatly increase the samples for
standard siren measurements, enabling 1% measurement of $H_{0}$, but the more
distant OIR transients will be exceedingly faint and may require 25–40 m-class
telescopes for spectroscopic follow-up. Traditional “standard candle” distance
measurements using the infrared tip of the red giant branch (IR-TRGB) method
can also be extended to large samples of galaxies out to $\sim 100$ Mpc
distances, well into the Hubble Flow, using deep, high angular resolution
observations from 25–40 m-class telescopes, providing a third independent and
complementary method for precision measurement of $H_{0}$ [46].

###  3.2 Photometric and Spectroscopic Follow-up of QSO Lens Systems

LSST is expected to deliver hundreds of cosmologically useful lensed
supernovae [47] and thousands of cosmologically useful lensed QSOs [48].
Initial observations of strong lenses piggybacking on wide-field multi-object
spectroscopic surveys can help to characterize lensed QSO systems and identify
the most useful ones. However, measuring cosmological parameters to percent-
level accuracy with strong-lens time delays requires two main ingredients once
the ideal systems are identified. First, one must measure the time delays
between the multiple images of a source, and second, the lensing potential
must be inferred to convert the observed time delays into a time-delay
distance.

To measure time delays, high-cadence, high-resolution, multi-filter imaging of
the resolved lensed images is required. In general, the LSST cadence may be
insufficient for this purpose; as a result, dedicated follow-up imaging of
lens systems with more frequent visits will be needed, but this can often be
performed on smaller telescopes (2–4 m). To model the lens systems, both
source and lens redshifts are required, as well as high-resolution imaging of
the lens galaxy and lensed host galaxy to measure apparent positions with high
precision. Kinematic velocity dispersions derived from lens galaxy spectra can
also improve the models. Adaptive optics (AO) IFU spectroscopy on 8–40 m-class
telescopes (with aperture required depending on brightness) can satisfy all of
these needs simultaneously, but the combination of AO imaging with slit
spectroscopy would also suffice. Additionally, multi-object spectroscopy of
faint galaxies near the line of sight can help to resolve mass-sheet
degeneracies and improve lens modeling.

###  3.3 Wide-field spectroscopy for lens systems

LSST will discover $\sim 100,000$ strong gravitational lenses [49], 100 times
more than are currently known. This drastic change opens new opportunities to
precisely constrain dark energy with lensed quasars [41], lensed supernovae
[50], lenses with multiple background sources [51], and lenses with a
spectroscopic velocity dispersion [52], as well as by using strong lensing as
a calibration tool for weak lensing shear [53]. Each of these science cases
requires redshifts for lens and source, both to confirm candidates as lenses
and to convert lensing-derived quantities into cosmologically meaningful
constraints. The LSST lenses will be uniformly distributed across the LSST
footprint, with typical $r$-band magnitudes of $20.1\pm 2.4$ for the lens and
$23.7\pm 0.7$ for the source. As a result, many lenses are bright enough for
redshift measurements via targeted fibers within very wide-area surveys,
enabling identification of the systems best suited for follow-up as described
above.

##  4 Peculiar Velocities Measured From Supernovae

In the upcoming decade cadenced wide-field imaging surveys will increase the
number of identified Type Ia supernovae (SNe Ia) at $z<0.2$ from hundreds up
to one hundred thousand. The increase in the number density and solid-angle
coverage of SNe Ia, in parallel with improvements in the standardization of
their absolute magnitudes, now make them competitive probes of the growth of
structure and hence of gravity. Each SN Ia can have a peculiar velocity
$S/N\sim 1$ with the distance probative power of $30-40$ galaxies, so a sample
of 100,000 SNe Ia is equivalent to a survey of 3 million galaxies, in a
redshift range where we currently have $\lesssim$100,000 galaxies. The
peculiar velocity power spectrum is sensitive to the effect of gravity on the
linear growth of structure. Measurements with 4–5$\sigma$ confidence will be
possible in the coming decade by exploiting the large numbers of $z<0.09$ SNe
Ia now and soon to be discovered [54, 55, 56] together with even more out to
$z=0.2$ and beyond to come from the Vera C. Rubin Observatory LSST [57]. These
constraints have the same statistical significance as those from DESI or
Euclid, but are measured essentially at a given cosmic time rather than
averaged over a broad range $z\sim$1–2, and at low redshift where
modifications of gravity may be most apparent. Together, SNe Ia and high-
redshift peculiar velocity tracers are sensitive to physics beyond the
standard $\Lambda$CDM cosmology.

Given the anticipated stream of fresh SNe, a coordinated network of follow-up
facilities is then required to determine their peculiar velocities. The
signals the network must provide are deviations from the average Hubble
expansion:

Early-Phase Screening: Cadenced wide-field imaging surveys discover SNe Ia
together with a background of other transient events. Though not strictly
required, screening enables the efficient use of limited telescope resources
by identifying a subset of likely active SNe Ia targets for follow-up at peak
brightness. Screening is primarily based on the temporal and color evolution
during the early post-discovery phases of transient light curves.

SN Ia Classification: Pure SN Ia samples have lower absolute magnitude
dispersion, and hence better velocity precision, relative to samples that
suffer from non-Ia contamination. SNe Ia are defined based on their
spectroscopic features, so spectroscopic coverage of the relevant features
ensures the purity of the analysis sample. Photometric classification dilutes
the signal in the scatter of the Hubble diagram, while selection from only
early-type hosts has core-collapse contamination and significantly reduced
sample size.

Host-Galaxy Precision Redshifts: Peculiar velocities come from the difference
between host-galaxy and cosmological redshifts. Fractional host-galaxy
redshift uncertainties from moderate-resolution spectroscopy ($R>100$) do not
contribute significantly to the error budget. In many cases the redshifts
could come naturally from the galaxy signal in the spectroscopic transient
classification. Alternatively, moderate-resolution bright-galaxy redshift
surveys can provide multiplexed observations of a significant fraction of
nearby SN hosts before discovery or after transient light has faded away.

Precise SN Ia Absolute Magnitudes: SNe Ia are standardizable candles, in that
their absolute magnitudes can be inferred from intrinsic observables such as
multi-band light-curve shapes, spectroscopic features, and host-galaxy
properties. For the same supernova, the absolute magnitude uncertainty derived
from a dataset with sparse light curves, a small number of filters, limited
wavelength coverage, and lacking spectral-feature measurements is larger than
the dispersion derived from a broader dataset with dense light curves, a large
number of filters, broad wavelength coverage, and spectral-feature
measurements. Surveys with three well-sampled rest-frame optical light curves
obtain $\sim 7$%[58] distance uncertainties; standardization studies show that
supplemental infrared data [59, 60] or spectrophotometry at peak brightness
[61, 62] can yield distances as good as $\lesssim 4$%. Follow-up measurements
that give this better precision yield SNe with 3$\times$ the peculiar-velocity
probative power compared to those with only optical light-curves.

SN Ia Observed Magnitudes: The difference between observed and absolute
magnitudes gives the distance to a supernova. Generally the observed
magnitudes come from the amplitudes of the same multi-band light curves whose
shapes and colors are used to infer absolute magnitudes.

Two reference surveys are introduced to assess the supplemental resources
needed for a complete SN peculiar velocity program. The first “pathfinder”
survey corresponds to a sample of $\sim 5000$, $g<19$ mag, $z<0.09$ SN Ia
discoveries as would be discovered by ASAS-SN+ATLAS, ZTF-II during a three-
year survey. The second “legacy” survey is composed of a sample of
$\sim$150,000, $r<20.5$ mag, $z<0.2$ SN Ia as will be discovered by LSST over
the course of ten years.

Nominal follow-up networks that collect the requisite peculiar-velocity data
for the above reference surveys are now described. The follow-up strategies
presented have the advantage of being based on successful precedents, but do
not represent the only viable strategy.

Wide-field ($\gtrsim 1$ sq. deg.) Imaging Survey on a $\lesssim 2$m Telescope:
Early-phase screening ideally would come from this search. The observing
strategy of ZTF (and presumably ZTF-II) is highly cadenced and so provides
good early photometric classification[63]. However, the original LSST Wide
Fast Deep (WFD) survey produces sparse per-band light curves with little
early-time data and poor distance determinations [64]. The ultimate LSST WFD
strategy is currently unspecified; hence its capability for early typing
remains unknown. The Rubin Observatory has convened the Survey Cadence
Optimization Committee, which is currently slated to decide on the observing
strategy by late 2022111See https://www.lsst.org/content/charge-survey-
cadence-optimization-committee-scoc. This source of risk is mitigated by
planning for a wide-field imaging survey on a different telescope designed to
1) fill in the temporal gaps of the WFD survey to enable early classification;
2) implement an observing strategy optimized for low redshift peculiar
velocities, e.g. favor larger solid angle and shallower depth than envisioned
for WFD. The LS4 project[5], a next-generation transient search on the La
Silla Schmidt Telescope with an upgraded camera, satisfies these needs. Other
options include the VST/OmegaCam[65] and DECam.

Targeted (IFU) Spectroscopy of Active SNe: Spectroscopic follow-up with
moderate spectral resolution of likely SNe Ia provides 1) classification; 2)
host-galaxy redshifts; 3) precision SN Ia absolute magnitudes. The SNIFS
Integral Field Unit (IFU) spectrograph [66] mounted on the UH-88” is a proof
of concept, having generated a time series of $\sim 300$ SNe Ia[67] at
$z<0.08$. (Other IFU designs[68, 69] are also viable.) The IFU allows for the
subtraction of a host-galaxy reference yielding a spectrophotometric SN-only
spectrum. For the “pathfinder survey”, a 2.2 m telescope requires an average
20 minute exposure to measure the spectral features of one SN Ia near maximum
light, which translates to $\sim 160$ clear nights per year for three years to
get the full sample. Total clock time includes overheads for poor weather and
non-Ia contamination. In addition, a comparable amount of non-time-critical
reference spectra must be obtained at least a year after the SN light has
faded. The “legacy” survey of 150,000 SNe Ia requires added 4m-class
telescopes with 7$\times$ the open-shutter time per year (3 telescope years
per year plus weather overhead) for ten years plus additional time for
references.

Targeted Spectroscopy of Host Galaxies: The redshifts of a subset of hosts
will not be accessible from the IFU data. In these cases redshifts of host
galaxies can be efficiently obtained using wide-field multi-object
spectroscopy with DESI[70] or 4MOST[71] even after the supernova light has
faded.

There are other approaches for designing the network that can be evaluated
during the Snowmass Process. One option is to focus on NIR photometry to
improve SN Ia distances, e.g. at UKIRT. Another is to decouple the
classification and redshifting instruments by obtaining (sub)classifications
using $R<100$ spectroscopy that cannot deliver a precision redshift, e.g., SED
Machine[72], and getting redshifts from dedicated host-galaxy spectroscopy;
indeed this approach is being applied to the shallow discoveries of ZTF[73].
Another option is to use photometry only, without supplemental spectroscopy,
to calibrate SN absolute magnitudes; the probative power of each SN drops
significantly in this case, but still remains much better than competing
Fundamental Plane and Tully-Fisher distances. Similarly, follow-up of a
reduced subset of SNe with scaled-back follow-up resources, while not taking
full advantage of all the discoveries, can still outperform galaxy-derived
distance surveys.

##  5 Photometric redshift training and calibration

Photometric redshifts – i.e., redshift estimates obtained from imaging data
alone – will be critical for all LSST probes of cosmology. The cosmological
tests enabled by LSST all rely on determining the behavior of some quantity
with redshift, $z$, but we cannot measure spectroscopic redshifts (spec-$z$’s)
for the large numbers of objects detected in the LSST imaging data. Even in
cases where follow-up spectroscopy of individual objects will be needed (e.g.,
strong lens systems and some supernova studies), photo-$z$’s are used to
identify targets of interest. However, if photometric redshift estimates are
systematically biased, dark energy inference can be catastrophically biased as
well (see, e.g., [74]); as a result photo-$z$’s are both a critical tool and a
major source of concern affecting all cosmological analyses. The great depth
of LSST data is a compounding factor, as it makes the challenges associated
with obtaining follow-up spectroscopy to the depths of the samples used for
LSST more difficult.

Lacking a comprehensive knowledge of galaxy evolution, the only way in which
photo-$z$ errors can be reduced and biases characterized is via robust
spectroscopic redshift measurements of a set of galaxies. We follow [19] in
dividing the uses of spec-$z$’s into two classes, “training” and
“calibration.” Training is the use of samples with known $z$ to develop or
refine algorithms, and hence to reduce random errors on individual objects’
photo-$z$’s: i.e., to improve the performance of algorithms at estimating the
redshift of a galaxy. Photometric redshifts that are trained from larger and
more complete spectroscopic samples and hence have smaller uncertainties could
greatly improve the constraining power of LSST, for example by providing
sharper maps of the large-scale structure and improved clustering statistics,
providing better photometric classifications for supernovae, enabling
identification of lower-mass galaxy clusters at higher confidence, and
yielding better intrinsic alignment mitigation for weak lensing measurements.
If photo-$z$ accuracies are limited only by photometric errors (as would be
true if a perfect – i.e., very large and totally unbiased – training set could
be obtained), it has been found in simulations that LSST can deliver photo-$z$
estimates with sub-$2\%$ uncertainties ($\sigma_{z}<0.02(1+z)$). However,
errors in redshifts from real data sets at LSST depth with our current
knowledge of galaxy spectral energy distributions are closer to 5%. Achieving
the ideal performance by having a large training sample spanning the
properties of objects used for cosmology would improve the Dark Energy Task
Force Figure of Merit from LSST lensing and Baryon Acoustic Oscillations alone
by $\sim 40\%$ [75, 76].

Calibration is the problem of determining the true overall $z$ distribution of
a sample of objects; i.e., the characterization of redshift distributions that
will be used in analyses. Miscalibration will lead to systematic errors in
photo-$z$’s and hence downstream analyses [77, 76, 78, 79, 74]. Photo-$z$
calibration requirements for LSST are extremely stringent [1]. If photo-$z$’s
are systematically biased in an unknown way or if their errors are poorly
understood, dark energy inference will be biased as well (e.g., because we are
effectively measuring the distance to a different redshift than is assumed in
calculations).

For both training and calibration, we need sets of galaxies for which the true
$z$ is securely known; this generally requires medium or high-resolution
spectroscopy (low-resolution spectroscopy or many-band imaging yield less-
secure redshifts with high incorrect-redshift rates, which can have a
catastrophic impact on the training of algorithms and the characterization of
redshift distributions). If spec-$z$’s could be obtained for a large, unbiased
sample of objects down to the faint limit of the samples to be used for LSST
cosmology, both needs could be fulfilled using the same data; we describe a
sample which could optimistically serve this dual purpose in subsection 5.1.
However, many faint galaxies in existing spectroscopic samples fail to yield
secure redshift measurements. Photometric redshift performance will still
improve if only incomplete spectroscopic samples are obtained, but the
accuracy of the characterization of redshift distributions would be
compromised. In that case, other methods may be needed for the calibration of
redshift distributions. Complementary spectroscopic datasets that can serve
this purpose are described in subsection 5.3.

###  5.1 Deep spectroscopy for improving photometric redshift performance

In a white paper developed for the previous Snowmass process [19], it was
concluded that an effective training set of photometric redshifts for the LSST
weak lensing sample would require highly-multiplexed medium-resolution ($R\sim
4000$) spectroscopy covering as much of the optical/infrared window as
possible with very long exposure times on large telescopes. To enable
photo-$z$ direct calibration errors to be subdominant to other uncertainties
if the training set were used for that purpose, the spec-$z$ sample must
comprise at least 20,000 galaxies spanning the full color and magnitude range
used for cosmological studies, reaching $i=25.3$ for LSST. As can be seen in
Fig. 4, improvements in photo-$z$ errors and outlier rates are slow beyond
this point, so larger samples have limited utility for improving the
performance of algorithms as well as for calibration.

Figure 4: Orange points show photometric redshift errors and outlier rates
versus the number of galaxies in the training set for galaxies with simulated
LSST photometric errors. Photo-$z$’s were calculated using a random forest
regression algorithm. The left panel shows the photo-$z$ error, quantified by
the normalized median absolute deviation (NMAD) in
$(z_{\text{phot}}-z_{\text{spec}})/(1+z_{\text{spec}})$, as a function of
training set size; similarly, the right panel shows the fraction of 10%
outliers, i.e. objects with
$|z_{\text{phot}}-z_{\text{spec}}|/(1+z_{\text{spec}})>0.1$. A vertical dashed
line shows the sample size for the baseline training survey from [19]. The
blue curves represent simple fits to the measurements as a function of the
training set size, $N$. This analysis uses a set of simulated galaxies from
Ref. [80] that spans the redshift range of $0<z<4$, using a randomly-selected
testing set of $10^{5}$ galaxies for estimating errors and outlier rates;
these catalogs are based upon simulations from Refs. [81],[82], and [83].

A critical issue for machine learning-based photometric redshift algorithms
[e.g., 84] is that sample/cosmic variance in the regions with spectroscopy can
imprint on the inferred redshift distribution of objects over the whole sky,
biasing photo-$z$ results. To both quantify and mitigate this effect, the
survey strategy described in Ref. [19] seeks to obtain spectroscopy spanning
at least 15 widely-separated fields a minimum of 20′ in diameter. Such a
survey has comparable sample/cosmic variance to the Euclid C3R2 strategy of
six 1 sq. deg. fields [85], but requires only $\sim$ 22% as much sky area to
be covered; by spanning more fields than C3R2 it also allows more robust
identification of regions that are overdense or underdense at a given $z$.

We have calculated the amount of dark time required for a survey for training
LSST photometric redshifts for a variety of instruments and telescopes in
Table 2, updated from the results presented in Ref. [19]. In this table we
calculate the total time required to obtain spectroscopy of objects down to
magnitude $i=25.3$ with the same signal-to-noise that would be achieved with
the Keck/DEIMOS instrument for objects with $i=22.5$, for a total sample of at
least 20,000 objects distributed across 15 fields of at least 20 arcminutes in
diameter, 6 fields of at least 1 degree in diameter, or 4 fields of at least 2
degrees in diameter (in order to mitigate the effects of sample/cosmic
variance). Only instruments or combinations of instruments with the capability
of obtaining spectra over the full optical window with sufficient spectral
resolution to resolve the [OII] 3727 angstrom doublet are considered; with
lower resolution, it is infeasible to obtain highly-secure redshifts for most
galaxies at $z>1$. Results are given in units of “dark-years”: one dark-year
consists of 365 8-hour-long dark nights, of which one-third are lost to
weather and instrumental overheads.

We note that such a survey has strong synergies with studies of galaxy
evolution, offering the possibility of co-funding that extends beyond HEP. It
would determine the range of galaxy SEDs (and hence star formation histories)
as a function of their local environment across the redshift and magnitude
range covered by LSST cosmology samples (reaching down to $\sim L_{*}$ at
$z=2$, and including brighter galaxies to $z=3$). Improved photo-$z$’s will
also enhance a variety of galaxy evolution science from LSST, making the
results useful to broader communities.

Table 2: Time required for photometric redshift training spectroscopy Instrument / Telescope | Collecting | Field area | Multiplex | Total time
---|---|---|---|---
| Area (sq. m) | (sq. arcmin) |  | (dark-years)
4MOST | 10.7 | 14,400 | 1,400 | 1.4
Mayall 4m / DESI | 11.4 | 25,500 | 5,000 | 1.4
WHT / WEAVE | 13.0 | 11,300 | 1,000 | 1.6
Megamapper (Magellan-like) | 28.0 | 25,416 | 20,000 | 0.6
Subaru / PFS | 53.0 | 4,500 | 2,400 | 0.4
VLT / MOONS | 58.2 | 500 | 500 | 2.7
Keck / DEIMOS | 76.0 | 54 | 150 | 6.8
Keck / FOBOS | 76.0 | 314 | 1,800 | 0.8
ESO SpecTel | 87.9 | 17,676 | 15,000 | 0.2
MSE | 97.6 | 6,359 | 3,249 | 0.2
GMT/MANIFEST + GMACS | 368.0 | 314 | 420 | 0.5
TMT / WFOS | 655.0 | 25 | 100 | 1.2
E-ELT / Mosaic Optical | 978.0 | 39 | 200 | 0.511For E-ELT, observations in both the optical and near-IR settings are required to achieve the required wavelength coverage, increasing total time required.
E-ELT / MOSAIC NIR | 978.0 | 46 | 100 | 0.8

###  5.2 Deep spectroscopy for testing the impact of blending on photometric
redshifts

Due to its unprecedented depth and sensitivity to the low-surface-brightness
outer regions of galaxies, the probability of two or more sources overlapping
in LSST data is high. Approximately 63% of LSST sources will have at least 2%
of their flux coming from other objects, in contrast to $\sim$30% of sources
in Dark Energy Survey (DES) data [86, 87]. These overlaps complicate the
measurement of galaxy fluxes and shapes, requiring accurate deblending [88,
89] or statistical correction techniques. Any residual light contamination
will result in biases to individual-object photo-$z$’s or potentially even in
estimates of overall redshift distributions [90].

Deep multi-object, medium-resolution spectroscopy can detect the presence of
blends and the redshifts of each component by identifying superimposed
features in a spectrum, providing new training samples for deblenders and
constraints on statistical corrections for deblending effects. While many
blends can be detected as having multiple components in space-based data, some
are so close that only spectroscopy can provide a definitive indication of a
problem. As a result, the proposed photometric redshift training spectroscopy
should also greatly enhance studies of deblending down to the depth of the
LSST Gold sample, $i_{\text{lim}}=25.3$ [1].

###  5.3 Wide-area spectroscopy for characterizing photometric redshift
distributions

Given the level of accuracy with which the redshift distributions of LSST
photometric samples must be known, if we wish to characterize them with a
representative sample of galaxies such as would be obtained from the
spectroscopy described in subsection 5.1, that spectroscopy would need to have
an extremely high ($>99\%$) spectroscopic redshift success rate (i.e., the
fraction of spectroscopic targets for which a secure redshift is obtained) and
an extremely low ($<1\%$) incorrect-redshift rates (i.e., the fraction of
putatively secure redshift measurements which are not in fact correct).
Existing deep spectroscopic samples have fallen well short of these goals,
meeting the incorrect-redshift requirement only for the most robust (and hence
most incomplete) subsets of redshift measurements [91, 92]. Based on the
systematic incompleteness of existing deep surveys, we therefore proceed under
the assumption [e.g., as applied in 19] that direct calibration via a large,
representative spec-$z$ sample will not be possible at the depth of LSST weak
lensing samples.

However, methods based upon cross-correlating the locations on the sky of LSST
galaxies with the positions of spec-$z$ samples as a function of the
spectroscopic $z$ (sometimes referred to as “clustering redshifts”) can
provide an accurate photo-$z$ calibration for dark energy applications [93].
These techniques exploit the fact that all galaxies cluster together on large
scales, so those objects with spec-$z$’s intrinsically cluster on the sky only
with the subset of galaxies in other samples that are at nearly the same
redshift, but not with closer or more distant objects. We can use this to
determine the true redshift distribution of objects in an unknown sample
[e.g., 94, 95] as input to dark energy measurements. Precise calibration only
requires that the spectroscopic samples span the redshift range of the LSST
objects and cover a wide area of sky.

A spectroscopic sample containing of order $\sim$$10^{5}$ objects spanning
thousands of square degrees and the full redshift range of LSST photometric
samples and covering a fair sample of observing conditions, extinction, etc.
should be sufficient for calibrating LSST photometric redshifts [19]. The DESI
survey [70] which began in 2021 should meet the sample size and redshift range
requirements [19] for this application: it will overlap with at least 4000
deg2 of the LSST footprint, obtaining more than 10 million redshifts of
galaxies from $0<z<1.6$ and quasars at $z<4$ over that area. However, because
DESI is located at the northern end of the LSST footprint, there is a risk of
the LSST photometry – and hence the photo-$z$’s to be calibrated – being
systematically different in this region (e.g., due to difficulties in
correcting for the atypical distribution of airmasses). As a result, greater
southern coverage is extremely desirable. The 4MOST survey, which will use a
dedicated wide-field multi-object spectrograph on the VISTA telescope at
Paranal Observatory [96], plans to obtain samples with DESI-like target
selection (but lower number density) over $\sim$1000 deg2 for Emission Line
Galaxies (ELGs) and $\sim$7500 deg2 for Luminous Red Galaxies (LRGs) and
quasars; these areas may overlap with DESI. It would thus be valuable to
enlarge the samples available for cross-correlation calibration in the
southern part of the LSST footprint; this would require new survey data in
regions of sky not covered by the DESI survey. A follow-on survey with the
DESI instrument that covers southern regions targeted by LSST but not the DESI
Legacy Survey imaging could potentially serve this purpose, or alternatively
samples from a high-multiplex instrument located further to the south.

##  6 Intrinsic Alignments

Intrinsic correlations between shapes of galaxies that are physically near
each other (“intrinsic alignments” or IA) are a known contaminant to precision
weak gravitational lensing (WL) measurements – see, e.g., the reviews [97, 98]
and references therein. If not accurately accounted for, their presence can
generate significant biases in cosmological analyses [99, 100, 101, 102]. More
recently, it has also been recognized that through selection effects, IA can
contaminate redshift-space distortion (RSD) analyses in DESI and other future
spectroscopic galaxy clustering projects [103].

There are two types of IA correlations. The first happen when considering
pairs of galaxies that are both within the same large-scale structure, thus
directly inducing intrinsic shape correlations known as “II” type. The second
type is due to the fact that a given matter structure will both align
physically nearby galaxies and shear background galaxies. The resulting
correlations are known as “GI” type. The latter is particularly problematic as
it is present even between galaxy bins significantly separated along the line
of sight and thus cannot be easily removed, even with precise redshift
information. It is this latter effect that can also impact RSD studies, since
the apparent impact on galaxies will depend on the angle with the observer’s
line of sight.

Mitigation of IA typically requires jointly modeling the effect along with the
cosmological lensing signal as well as other astrophysical and observational
effects.222“IA self-calibration” [e.g. 102] is an alternative approach that
removes IA without explicitly marginalizing over model parameters. However,
constructing the relevant data vectors still requires an accurate
understanding of IA correlations. Improved understanding of the underlying IA
behavior will allow us to robustly perform this modeling without significantly
degrading the overall constraining power of the analysis. Indeed, to optimize
final statistical uncertainties, IA modeling should have as few parameters as
possible, and these parameters will ideally have strong priors from targeted
observational studies. Moreover, as our understanding of IA improves, we will
not only be able to effectively remove them as a contaminant, but also extract
additional cosmological information from the shape correlations (e.g. [104]).

Despite this clear need for observational IA constraints, to date it has been
challenging to obtain measurements with sufficient signal to noise, especially
for the samples most relevant to future lensing projects such as LSST. Because
IA and weak lensing can mimic each other, and both are weak signals compared
to the intrinsic “shape noise” of galaxies, IA measurements are most effective
when precise 3D locations of galaxies are known, i.e. when spectroscopic-
quality redshifts have been obtained. Below, we briefly describe how both
deep/narrow and wide/shallow MOS data in the future can enable significantly
improved constraints on Dark Energy, and cosmology more broadly, in the
presence of IA.

###  6.1 Deep spectroscopy for direct tests of intrinsic alignments

By enabling the 3D localization of galaxies, a deep MOS campaign would provide
greatly-improved direct constraints on intrinsic alignments for typical weak
lensing sources, rather than only for the bright and nearby objects which
current datasets constrain [105, 86, 106]. Such data would extend our
knowledge of IA to unexplored regimes in galaxy parameter space, resolve the
current inconsistencies between predictions of different hydrodynamical
simulations [107, 108, 104], and allow better priors to be placed on IA model
parameters. All of these advances will improve the cosmological constraining
power of future lensing studies, including LSST [100, 102, 109].

Currently, the faintest magnitude-limited sample with a significant IA
detection has $i_{\text{lim}}<19.8$ [110]. Forecasts based on recent
measurements [111, 110] indicate that meaningful IA constraints require
$\gtrsim 10^{5}$ galaxies with measured shapes and spectra. Obtaining this
many spectra for a representative sample at the magnitude limit of the LSST
Gold sample ($i_{\text{lim}}=25.3$) would require a significant expansion of
the photo-$z$ training program described above, and may be infeasible with
currently planned facilities. However, even if we cannot reach magnitudes as
faint as $i_{\rm lim}=25.3$, extending direct IA tests closer to the LSST
limits would be very valuable. Most of the IA signal will come from scales
where shape noise dominates, and thus total IA constraining power (for a fixed
number of galaxies) is maximized if the surface density of targets is high.
Such a dense sample can be obtained by switching out targets to other,
brighter galaxies during the photo-$z$ training survey once secure redshifts
are obtained; in this way, $\gtrsim 10^{5}$ spec-$z$’s for objects with
$i\lesssim 24$ should be obtainable during the training survey (only 9% as
much observing time is needed to obtain the same S/N at $i=24$ as at
$i=25.3$). Additional information will come from cross-correlations with
shallower, wider-area surveys, as described below.

###  6.2 Wide-area spectroscopy for cross-correlation studies of intrinsic
alignments

Wide-field MOS will enable measuring the cross-correlation between positions
of bright galaxies with spectroscopy and intrinsic shapes of fainter galaxies
used for lensing analyses, constraining IA models with greater precision than
possible with photometric data alone. Such measurements can place upper limits
on IA or constrain parameters associated with IA models [e.g., 112, 111, 110].
Wide MOS over a contiguous area would also enable cosmic web reconstruction
[e.g., as in 113], which could be helpful in testing some of the predictions
of cosmological simulations [114]. Furthermore, cross-correlations using the
spec-$z$ sample will allow us to break degeneracies between IA parameters and
photo-$z$ uncertainties. Current measurements from spectroscopic samples [111,
110] have shown that wide-field samples of $\sim$ 100k–500k objects with both
shapes and spec-$z$ measurements can place useful constraints on IA amplitude
and scale dependence. The LSST-DESI overlap should enable measurements which
greatly surpass what is currently possible, especially if northern coverage by
LSST is increased (as proposed in [115]). 4MOST galaxy samples will further
enhance these analyses. However, larger samples over wider areas would yield
better constraints on IA models, improving S/N in IA measurements by up to
$\sim 2\times$ if samples could be expanded to cover the full LSST
extragalactic footprint. This would likely require a new high-multiplex
facility in the Southern hemisphere. Smaller-field surveys of fainter
galaxies, described above, will provide complementary constraints on IA.

Intrinsic alignments have also been proposed as a probe of cosmology and novel
physics in their own right [116, 104], and a wide MOS campaign would improve
their performance for exploring primordial non-Gaussianity, potential new
interactions in the dark sector, and as a multi-tracer probe of large-scale
structure.

##  7 Galaxy cluster studies

The observed abundance of galaxy clusters as a function of their mass provides
a sensitive probe of the growth of structure and, to a lesser degree, the
expansion history of the universe. However, galaxy cluster abundance
measurements from LSST will require additional data to mitigate systematic
effects; as an example, the accurate calibration of the relationship between
observable cluster properties and their total mass must be very well-
understood.

###  7.1 Deep spectroscopy in galaxy cluster fields

Deep multi-object spectroscopy of galaxies in a set of fields containing
galaxy clusters will improve LSST cluster cosmology in a number of ways, while
simultaneously resolving open questions about galaxy evolution by determining
differences between SEDs of galaxies in clusters versus the field, potentially
leading to interest from outside HEP in joint projects.

Training and Testing Photometric Redshifts in Cluster Fields: Photometric
redshifts are a critical tool for calibrating the masses of galaxy clusters
via weak gravitational lensing. Photo-$z$’s are already a leading source of
systematic uncertainty in current work [117, 118], but the requirements on
photo-$z$ accuracy for LSST will be even more stringent[1]. Photo-$z$’s are
vital for distinguishing the lensed background galaxy population from the
unlensed foreground and cluster population; unaccounted-for contamination of
the background will lead to underestimation of cluster masses. However,
photo-$z$ performance may degrade in higher-density regions due to the
differing galaxy populations of clusters vs. the field, magnification and
reddening of background sources, and severe blending due to cluster galaxies
[e.g. 119].

Photo-$z$ algorithms are generally trained and evaluated on fields selected
not to contain massive structures. However, there is reason to expect that
photo-z algorithms may not perform as well in areas of the sky that host
clusters as in the field. The incidence of blending is worse in such areas due
to the denser environment and contamination by intra-cluster light, causing
larger errors in the photometry. Lensing magnification changes the observed
brightness and size of background galaxies while the mix of early- vs. late-
type galaxies is different in clusters than in the field; these effects may
lead to mismatches with photo-$z$ priors. Furthermore, due to the profound
impact of the cluster environment on galaxy evolution, clusters might host
galaxy types that are very rare in the field. They also may host dust which
would cause reddening of background galaxies.

As a result, it will be important to quantify the performance of photo-z
algorithms in cluster fields specifically. Doing so requires comprehensive
multi-object spectroscopy down to weak lensing depths ($i\sim 25.3$) for a
sample of $\sim 20$ clusters spanning a range of redshifts. This is best
achieved with high-throughput, high-multiplex spectrographs with FOVs of $\sim
10^{\prime}$ (wide-format IFUs may be suitable in cluster cores) on the
largest telescopes available; the multi-object spectrographs planned for the
US-ELT facilities would be well-suited for this work.

Such a program would be able to characterize the performance of photo-$z$
probability distributions as a function of magnitude, redshift, and cluster
properties. A sample of 1000 objects per cluster would enable a crude binning
into 3 magnitude and 3 redshift bins with $\sim$100 galaxies per bin, enough
for a statistically meaningful evaluation of photo-$z$ performance at the
percent level. To cover a range of cluster redshifts, masses, and dynamical
states, $>20$ clusters should be surveyed. It is critical to achieve near-
complete redshift success rates to avoid biases from target populations for
which no redshift is measured in a first attempt. Such data will also yield
very valuable insight into LSST deblending performance in general.

Measurements of Cluster Kinematics and Infall Velocities: Deep multi-object
spectroscopy in fields hosting clusters will also enable direct mass estimates
for galaxy clusters via the infall method [120, 121, 122, 123], providing an
additional calibration of the mass-richness relation [124] “for free” from the
photometric redshift training/test spectroscopy. We hope to obtain $\sim 200$
redshifts at projected separation $<5$ Mpc for each cluster targeted for
photo-$z$ studies, providing purely kinematic mass estimates. In conjunction
with weak-lensing measurements, spectroscopy of halo infall regions also
enables sensitive comparison between the dynamical and the lensing potential
of gravity, which can differ from each other at measurable levels in several
modified gravity theories that seek to provide an alternate explanation for
cosmic acceleration [e.g., 125].

###  7.2 Additional constraints from Sunyaev-Zel’dovich measurements

Planned CMB surveys such as CMB-S4 [126, 127], will detect a very large number
of galaxy clusters through the thermal Sunyaev-Zel’dovich (tSZ) effect [128].
These tSZ-identified clusters will be detected to high redshift, and will
provide complementary information to optical measurements of the systems
detected in both wavelength bands. The amplitude of the tSZ effect for a given
cluster provides a low-scatter proxy for the cluster mass, but the mapping of
tSZ decrement to mass must be calibrated by additional observations. The
synergy between CMB-S4 and LSST will be important for fully exploring the
potential of tSZ-selected clusters as cosmological probes and extracting
maximal information from both datasets. For example, LSST weak lensing
measurements of tSZ-identified galaxy clusters can provide a cluster mass
calibration. They also will give an important sanity check on internal CMB-
based lensing mass measurements, since the systematics affecting the two
experiments are nearly independent. Conversely, tSZ measurements and CMB
lensing will provide additional methods for establishing masses for LSST-
selected clusters.

Furthermore, since the tSZ effect is redshift-independent, SZ-selected
clusters will span a large range in $z$. Cross-matching the cluster catalogs
from CMB-S4 and LSST will provide high-precision photometric redshift
estimates for tSZ clusters at $z\lesssim 1.5$ and a lower redshift bound for
even more distant tSZ-selected clusters. Photometric redshift distributions in
the directions of the tSZ-selected clusters may be sufficient to determine the
redshifts for even higher-$z$ objects.

A mass- and redshift-calibrated cluster catalog produced by the combination of
LSST and CMB-S4 will serve as a powerful probe of the growth of structure
across cosmological times, and thus of the properties of dark energy and the
sum of neutrino masses [129]. The significant overlap between the survey
regions of LSST and CMB-S4 ensures that this synergy will be possible across a
large fraction of the sky.

The combination of LSST and CMB-S4 data will also enable new analyses that
will constrain the nature of dark energy using probes beyond galaxy clusters.
A variety of other cross-correlation based measurements, which can exploit
information from CMB lensing and the kinetic and polarized Sunyaev-Zel’dovich
effects as well as Rubin Observatory lensing and large-scale-structure
measurements, will be enabled by the two experiments. A separate white paper
[2] describes these opportunities in more detail.

##  8 Testing the Nature of Gravity with Wide-Area Spectroscopy

Galaxies and weak lensing maps trace the same underlying matter field.
Galaxies are biased tracers of matter but with spec-$z$’s can provide full
three-dimensional information, while lensing maps are unbiased tracers that
mainly provide information about projected density. Cross-correlating galaxies
with lensing maps can provide very fine tomographic information about matter
clustering independent of galaxy bias [see, e.g., 130, 131].

As a result, combining cross-correlations between galaxy density and lensing
with measurements of redshift-space distortions in galaxy clustering allows
for tests of gravity on cosmological scales. Galaxy positions and velocities
probe the behavior of non-relativistic matter under gravitation, while lensing
is sensitive to the paths of purely relativistic particles (photons). For
instance, previous work has defined a statistic $E_{G}$ which combines
information from these quantities and is sensitive to deviations from General
Relativity [132, 133]. In order for the dependence on linear galaxy bias to
cancel from $E_{G}$ (or similar statistics), redshift-space distortion
measurements (which require spectroscopic redshift measurements) must be made
on the same galaxy sample as that used to measure clustering and as lenses in
galaxy-galaxy lensing analyses. Therefore, a sample of galaxies with
spec-$z$’s that overlaps the LSST WL sample is required to measure $E_{G}$.

DESI and 4MOST should provide such overlapping samples via both LRGs with
$\bar{z}\approx 0.77$ and ELGs with $\bar{z}\approx 1.0$. The combined samples
of DESI and 4MOST ELGs and LRGs should each enable a determination of $E_{G}$
to $\sim 0.004$, roughly 10 times more precise than current constraints.
Enlarging the northern LSST footprint (as proposed in Ref. [115] and discussed
in the next section) would reduce errors by $\sim 25\%$ more; greater
improvements are possible with enlarged spectroscopic samples in the southern
LSST footprint.

##  9 Rubin Observatory Imaging to Enable a DESI-2 Survey

At the conclusion of the planned DESI survey (c. 2026), the instrument should
still be a nearly-uniquely capable facility for wide-area spectroscopic
surveys; it is unlikely to be surpassed by other projects until 2030 or later.
The strategy for a follow-on, DESI-2 survey is still yet to be determined;
however, the availability of $ugrizy$ imaging from Rubin Observatory should
enable much more powerful target selection methods to be applied than were
possible for the current DESI program.

A limiting factor, however, is that from its location in the Northern
hemisphere, much of the LSST sky area can be observed only at high airmass
(requiring long observation times) or not at all. Many probes of cosmology,
including Baryon Acoustic Oscillation measurements and tests for primordial
non-Gaussianity from large-scale clustering, will be steadily more sensitive
the wider area of sky that is covered. As a result, we would wish to maximize
the sky area that both has LSST-like $ugrizy$ imaging and is accessible to
DESI.

The current LSST “Wide, Fast, Deep” main survey design covers the Northern sky
up to declination of +12 degrees; most of the LSST footprint is far to the
south. However, due to its location at latitude +32∘ N, DESI can observe the
far-Southern sky only inefficiently. If DESI is restricted to declination
$>-20^{\circ}$, the region where its survey speed would be degraded by at most
$\sim 2\times$ due to having to point near the horizon, the DESI-LSST overlap
would consist of roughly 8700 square degrees of sky. This may be compared to
the 14,000 square degrees which the current DESI survey is expected to cover.

In Ref. [134] a mini-survey with LSST was previously recommended which would
enable this. They proposed performing shallower observations with LSST
covering declinations up to +30∘, primarily to improve photometric redshifts
for the Euclid satellite. Such data would also be sufficient to enable a
DESI-2 survey of maximal sky area, if the mini-survey observations are
performed early in the LSST survey. Such front-loading is required because
target selection for a second-generation DESI survey will need to be completed
around 2026. If such data could be obtained, it could greatly improve the
targeting efficiency of a DESI-2 program, and hence its constraining power.
Extending the survey area to cover declinations from $-$20∘ to +30∘ would
increase the DESI-2/LSST overlap to 13,300 square degrees, only a little short
of the sky coverage of the current DESI survey.

##  10 Summary of Opportunities

As described above, studies of dark energy via the Rubin Observatory can
benefit greatly from additional data to follow up supernovae and strong lens
systems, improve photometric redshifts, and constrain astrophysical
systematics. The facilities needed for such work can broadly be separated into
those which enable photometry or spectroscopy of individual objects; those
which provide wide-field, highly multiplexed spectroscopy; and those which
provide high-multiplex spectroscopy over smaller fields of view on telescopes
of much greater light-gathering power. We summarize the opportunities to make
gains via each of these classes of capabilities separately.

###  10.1 Single-target imaging and spectroscopy

Cosmological studies based on SNe and strongly lensed systems in the new era
of large-area, high-cadence optical surveys such as LSST will require single-
object follow-up data of several types:

  * •

Spectrophotometry of SNe Ia will enable direct cosmological measurements, the
development of spectral templates and the calibration of photometric
classification performance at high redshift (where it is the most uncertain),
as well as tests of systematic effects. Given the wide range of LSST SN
brightnesses, this work will be reliant on large amounts of low-to-medium
resolution spectroscopy on ground-based telescopes with apertures from 4 m to
$>20$ m (and, ideally, space-based telescopes as well).

  * •

Follow-up imaging with more frequent cadence than LSST will be valuable for
measuring strong lensing time delays; this can generally be conducted on 2–4 m
telescopes.

  * •

Adaptive optics IFU spectroscopy, or AO imaging plus slit spectroscopy, is
needed for precision image position measurements and lens system modeling;
this work will require access to both 8–10 m and $>20$ m telescopes with AO
capabilities.

The Kavli/NOAO/LSST report (summarized in Table 3) provided quantitative
estimates of the telescope time required to support cosmology measurements
with LSST supernovae and strong-lens systems [15].

Facility | Supernova single-object follow-up requirements
---|---
4 m spectroscopy | 60–180 nights total
8 m spectroscopy | 180–540 nights total
$>20$m spectroscopy | 180–540 nights total
Facility | Strong Lensing single-object follow-up requirements
2–4 m non-AO imaging |  $<8000$ hours total
$>8$m AO imaging |  $\sim 30$ hours, split amongst $>$8 m+ and $>20$ m telescopes
$>8$m spectroscopy |  $\sim 100$ hours, split amongst $>$8 m+ and $>20$ m telescopes
Table 3: Summaries of the required resources, as estimated in the
Kavli/NOAO/LSST study on Ground-based Optical/IR follow-up.

Although in many cases suitable instruments exist (e.g., Keck/OSIRIS) or are
being developed (e.g., Gemini/SCORPIO), cases remain where the telescopes best
positioned for this work have suboptimal instrumentation; as a result,
additional development, not just telescope time, may be valuable.

###  10.2 Wide-field spectroscopy

LSST cosmological constraints can be significantly improved in a number of
ways by incorporating additional wide-field, highly-multiplexed spectroscopy:

  * •

Spectroscopic follow-up for strong lens systems and SNe/hosts will still be
needed after 4MOST/TiDES observations are finished circa 2027 [17]. This could
be pursued via an extension of the 4MOST survey or by using PFS (for deep
drilling field surveys), DESI, or a future spectroscopic facility.

  * •

Increasing the area in the southern hemisphere with DESI-like target selection
of galaxies and quasars would improve cross-correlation calibration of
photo-$z$’s and other cross-correlation science. This could be achieved by
performing additional surveys with the 4MOST instrument covering thousands of
square degrees.

  * •

Denser and/or higher-redshift sampling of galaxies over thousands of square
degrees (e.g., in the Northern part of the LSST survey footprint) could
significantly improve constraints on theories of modified gravity and models
of intrinsic alignments for weak lensing studies. Such a project could be
pursued efficiently with the DESI spectrograph, PFS at Subaru, or a future
wide-field spectroscopic facility such as Megamapper [135], the Maunakea
Spectroscopic Explorer (MSE, [136]), or SpecTel [137].

The optimal solution for wide-field spectroscopy spanning the full LSST
footprint would be a highly-multiplexed, wide field-of-view instrument in the
South, ideally on an aperture at least as large as LSST’s; unfortunately no
such capability exists in the US OIR system. Such a facility could improve
LSST cosmology while simultaneously enabling new constraints of dark energy,
as described in other Snowmass white papers [138, 139]. The MegaMapper, MSE,
and SpecTel projects all would be well-suited for this role.

At the same time a facility of this type was targeting extragalactic objects
for LSST cosmology and other dark energy studies, it could also pursue other
projects with different fibers (e.g., wide-field surveys of stars in the Milky
Way halo, which can constrain dark matter models; cf. [140, 141]. This new
capability should also lead to advances across a broad range of astrophysics,
expanding the legacy of LSST and benefiting a diversity of communities. As a
result, it is likely that such a facility could attract funding from multiple
sources, building on the model of collaboration between HEP and other partners
in the Sloan Digital Sky Survey and its successor programs.

###  10.3 Deep but highly-multiplexed spectroscopy

Access to modestly-wide-field, highly-multiplexed, large aperture
spectroscopic facilities can lead to substantial gains in the constraining
power of LSST and CMB-S4 compared to what would be possible without additional
data:

  * •

Improvements to deep photometric redshift training sets will yield better
redshift estimates for individual objects, particularly improving constraints
from large-scale structure and galaxy clusters.

  * •

Tests of blending and intrinsic alignment effects will reduce the impact of
critical systematics on cosmological measurements.

  * •

Spectroscopy in a statistical sample of galaxy cluster fields will enable
calibration of mass-observable relations and reductions in systematics.

For developing photometric redshift training sets as well as testing models of
blending and intrinsic alignments, the key need is access to an instrument
with maximal multiplexing with a field of view of at least 20′ diameter on as
large a telescope aperture as possible. It is likely that all of these goals
can be pursued simultaneously with a common dataset. Subaru/PFS, Megamapper,
MSE, SpecTel, and GMT/GMACS with the MANIFEST fiber feed are all well-suited
for this work (wider-field fiber-fed spectrographs on other $>$6 m telescopes
could also be suitable), as summarized in Table 2. Personnel costs are likely
to be high on smaller telescopes because of the longer time required for deep
surveys on them, which may make instruments on 4 m telescopes such as DESI
less attractive for this work.

The telescope time required to obtain data for photo-$z$ training, blending,
and intrinsic alignment constraints is substantial, but it can be spread out
over the ten-year span of the LSST survey, reducing the impact in any one
year. Additionally, these studies are highly synergistic with both the
training needs for space-based experiments such as Euclid and the Nancy Grace
Roman Space Telescope and with studies of galaxy evolution, potentially
allowing combined surveys with a large impact on multiple fields of research
and funded by multiple sources.

For galaxy cluster studies with both Rubin Observatory and CMB-S4, targeting
more objects over smaller fields of view is desirable. For that work, suitable
options may include Gemini/GMACS, Keck/DEIMOS or LRIS, GMT/GMACS (in multislit
mode), TMT/WFOS, or a new, higher-multiplex spectrograph at one of these
observatories; wider-field facilities such as DESI, Subaru/PFS, Megamapper,
MSE or SpecTel would be less well suited due to the lower fiber density and
excessively large field of view.

In addition to the value that additional data can provide to LSST analyses,
the Vera C. Rubin Observatory itself is uniquely capable of increasing the
impact of a DESI-2 survey by providing modestly deep six-band imaging over a
large area of sky. A Northern mini-survey conducted in the first 1-2 years of
LSST would enable optimized target selection methods to be employed uniformly
over a DESI-2 footprint of more than 13,000 square degrees, which could
provide great improvements to the capabilities of DESI-2 at low cost.

Above, we have discussed a wide variety of ways that smaller investments in
other projects can enable improved dark energy constraints from upcoming,
flagship dark energy experiments, allowing us to obtain the maximum impact
from these flagship experiments. Obtaining this additional data in a timely
fashion will both reduce statistical uncertainties (via expanded strong lens
and supernova samples and photometric redshifts that extract the full
potential out of LSST data, or via improved target selection and enlarged
footprints for DESI-2 surveys) and can mitigate systematics which otherwise
are expected to be comparable to statistical errors from LSST [1]. Many of
these programs could be pursued at much smaller cost than a new Stage IV dark
energy experiment, fitting well below the P5 small projects threshold, but
offer similar improvements to constraining power as a separate new project.
For instance, comparatively modest amounts of funding could enable the
purchase of dedicated telescope time, in-kind contributions of new
instrumentation or components with guaranteed time provided in compensation,
or the collection and analysis of datasets obtained via telescope time
obtained via either proposal processes or international contributions.

In contrast, a new dedicated spectroscopic facility would require major
investment. In the short term, funding to enable instrument improvements and
develop the international collaborations needed to make such a project happen,
with a construction start before the end of this decade if possible, would be
valuable. Such a new capability would not only provide new constraints on
cosmology of its own as described in other white papers, but also would be
capable of improving the science yield from the LSST imaging survey, enabling
us to obtain the maximum yield from this unprecedented dataset. It may not be
possible to obtain the deep spectroscopic datasets that can enable improved
photometric redshift training and the characterization of blending and
intrinsic alignments before such a facility is available, unless dedicated
funding or in-kind contributions for such observations are made available. In
that case, a new spectroscopic telescope could enable improved constraints on
cosmology for the final LSST analyses in the middle of the next decade, as
well as pursuing a variety of explorations of the dark sector of its own that
would utilize LSST data for target selection.

Acknowledgements

Members of the LSST Dark Energy Science Collaboration acknowledge ongoing
support from the Institut National de Physique Nucléaire et de Physique des
Particules in France; the Science & Technology Facilities Council in the
United Kingdom; and the Department of Energy, the National Science Foundation,
and the LSST Corporation in the United States. DESC uses resources of the
IN2P3 Computing Center (CC-IN2P3–Lyon/Villeurbanne - France) funded by the
Centre National de la Recherche Scientifique; the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded by UK BIS National
E-infrastructure capital grants; and the UK particle physics grid, supported
by the GridPP Collaboration. This work was performed in part under DOE
Contract DE-AC02-76SF00515.

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