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: High Density Galaxy Clustering in
the Regime of Cosmic Acceleration

Kyle Dawson  Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112, USA  Andrew Hearin  High Energy Physics Division, Argonne
National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA  Katrin
Heitmann  High Energy Physics Division, Argonne National Laboratory, 9700
South Cass Avenue, Lemont, IL 60439, USA  Mustapha Ishak  Department of
Physics, The University of Texas at Dallas, Richardson, TX 75080, USA
Johannes Ulf Lange  Kavli Institute for Particle Astrophysics and Cosmology
and Department of Physics, Stanford University, CA 94305, USA  Department of
Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High
Street, Santa Cruz, CA 95064, USA  Martin White  Berkeley Center for
Cosmological Physics, Department of Physics, University of California,
Berkeley, CA 94720, USA  Lawrence Berkeley National Laboratory, 1 Cyclotron
Road, Berkeley, CA 94720, USA  Rongpu Zhou  Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
for the Snowmass 2021 Cosmic Frontier 4 Topical Group

###### Abstract

Joint studies of imaging and spectroscopic samples, informed by theory and
simulations, offer the potential for comprehensive tests of the cosmological
model over redshifts $z<1.5$. Spectroscopic galaxy samples at these redshifts
can be increased beyond the planned DESI program by at least an order of
magnitude, thus offering significantly more constraining power for these joint
studies. Spectroscopic observations of these galaxies in the latter half of
the 2020’s and beyond would leverage the theory and simulation effort in this
regime. In turn, these high density observations will allow enhanced tests of
dark energy, physics beyond the standard model, and neutrino masses that will
greatly exceed what is currently possible. Here, we present a coordinated
program of simulations, theoretical modeling, and future spectroscopy that
would enable precise cosmological studies in the accelerating epoch where the
effects of dark energy are most apparent.

##  1 Introduction

Surveys to measure fluctuations in the cosmic microwave background (CMB)
radiation and large-scale structure (LSS) through galaxies and quasars provide
our best windows into the fundamental physics of the cosmos. The tightest
constraints on dark energy, modifications to General Relativity, mass limits
on light dark matter particles, and neutrino masses all come from one or both
of these measurements. Within the next few years, we expect to see a dramatic
leap in our ability to probe fundamental physics through cosmology with the
Dark Energy Spectroscopic Instrument [DESI; 1, 2], Rubin Observatory’s Legacy
Survey of Space and Time (LSST), and CMB-S4. DESI has already begun, and is
quickly mapping the three dimensional distribution of galaxies and the
intergalactic medium. The LSST on Rubin will begin in a few years, providing
deep imaging over unprecedented areas of the sky. With CMB-S4 closing out the
decade, our view of the cosmic microwave background will advance every bit as
much as in the optical.

With five years of spectroscopy from DESI, the expansion history of the
universe will be well mapped. However, growth of structure, the full shape of
the power spectrum, higher order clustering statistics, and cosmology from
joint lensing and spectroscopic surveys will be less well constrained. The
five-year DESI samples will allow initial investigations using these
techniques, but advances in theory, simulations, and spectroscopic samples
will allow significant advances.

In the latter half of this decade, DESI will remain the most competitive
instrument in the world for large-scale spectroscopic surveys. New
spectroscopic facilities have been proposed with even faster survey speeds,
offering the potential to increase the number density of DESI galaxies by an
order of magnitude or more. Long-term investments in spectroscopic surveys
will allow enhanced tests of dark energy, physics beyond the standard model,
and neutrino masses that will greatly exceed what is currently possible. A new
sample at a much higher number number density that covers the full epoch of
cosmic acceleration back to the matter dominated era will enable a unique
spectroscopic probe of the following cosmological questions:

  1. 1.

Under General Relativity, structure grows at a rate that can be predicted at
any time given the Hubble parameter, $H(z)$. New growth measurements over a
range of redshifts will thus allow improved constraints on the expansion
history and the equation of state for dark energy.

  2. 2.

By comparing the expansion history derived from growth measurements to that
from geometric measurements, we can test the self-consistency of General
Relativity or whether additional physics is required.

  3. 3.

The long redshift baseline afforded by growth measurements to $z<1.5$ will
further break degeneracies of standard parameters with common dark energy
models, and hence improve constraints on the dark energy equation of state.

  4. 4.

The amplitude of clustering with redshift depends on the sum of neutrino mass.
A survey of structure growth over a wide redshift range would tighten
constraints on the neutrino masses. If cosmology results favor the lowest
allowed neutrino masses, then these measurements will also help resolve the
question of mass hierarchy.

  5. 5.

Precise measurements of the power spectrum afforded by a large number of
tracers would provide tight limits on dark matter interactions, light relic
particles, early dark energy, and potential features in the primordial power
spectrum that appear on small scales.

  6. 6.

Current lensing measurements of the amplitude of clustering indicate potential
tension with predictions from the CMB under a $\Lambda$CDM model. As
highlighted in the report from the Astro2020 Decadal Survey, resolving this
tension with higher precision growth measurements could reinforce the
$\Lambda$CDM model or reveal extensions in the dark sector.

Order of magnitude increases over the initial DESI samples of $z<1.5$ galaxies
are within reach of several proposed spectroscopic programs. A survey of
galaxies at a high number density would allow high precision measurements down
to small scales, with the statistical power to constrain the clustering
amplitude at better than 0.1% precision on scales comparable to a typical dark
matter halo. With a large sample of spectroscopically-confirmed galaxies,
multiple tracers can be identified for clustering and cross-correlation
studies, thus alleviating shot noise and offering robust checks on systematic
errors. Vastly increased sample sizes will also facilitate studies of higher
order statistics that remain in the early stages of development. Finally, by
probing the same volume with spectroscopy and imaging data, such a survey will
reap the full benefits of lensing and three-dimensional clustering.

In this white paper, we describe a program to use full-scale clustering (where
both non-linear and linear scales are jointly modeled) to test the
cosmological model over the redshift interval $0<z<1.5$. In Section 2, we
provide a brief overview of current Stage-IV Dark Energy programs and
expectations for constraints on fundamental physics. In Section 3, we describe
the effort in theory and simulations that is required to reach the precision
allowed by these new techniques. Investment in that theory and simulation work
should begin now for future spectroscopic observations to be most effective.
Finally, in Section 4, we describe the science drivers and new spectroscopic
samples that are possible with existing and proposed facilities.

##  2 Stage-IV Dark Energy Surveys

Question |  Optimized approach
---|---
What is the physical origin of cosmic acceleration at late times (Dark energy)? |  $z<1.5$ spectroscopy at high number density $+$ Rubin lensing measurements
What is the sum and hierarchy of the neutrino masses? |  Full-shape power spectrum measurements from $z<1.5$ spectroscopy at high density $+$ characterization of underlying dark matter distribution through Rubin lensing measurements
What is the particle nature of dark matter? |  three-dimensional clustering on small scales with $z<1.5$ spectroscopy at high number density $+$ galaxy-galaxy lensing from Rubin, Roman, & Euclid
Does the rate of cosmic expansion and the growth of structure indicate new particle or field content? |  All of the above
Table 1: P5 advances enabled by efforts at $z<1.5$ with theory, simulations,
Rubin imaging, and DESI spectroscopy. Spectroscopic sample sizes significantly
larger than those planned with DESI are possible in the near future, allowing
tighter constraints on each of these science drivers.

Large-scale cosmology surveys spanning imaging, spectroscopy, and lensing will
independently advance four of the five key science drivers identified by the
2014 Particle Physics Project Prioritization Panel (P5) [3]. Analyses that
jointly utilize imaging and spectroscopy will substantially enhance the
science returns that can be reaped from the data, as shown in Table 1.

Stage-IV Dark Energy surveys will pursue these science drivers in three
approximate redshift intervals using complementary techniques. At the lowest
redshifts ($z<1$), Rubin will assess the distribution of dark matter on all
scales through weak lensing, while DESI will assess the three dimensional
clustering of galaxies with precise spectroscopy. Over moderate redshifts
($1<z<2$), Rubin will measure the distribution of galaxies through photometric
redshifts, while DESI will provide three-dimensional positions with fiber-
based spectroscopy. Finally, at the highest redshifts ($z>2$), Rubin will
measure the distribution of galaxies through photometric redshifts, and DESI
will map the three-dimensional distribution of matter through quasar and
Lyman-$\alpha$ forest spectroscopy. In each of these redshift regimes, imaging
and spectroscopy play critical, complementary roles. DESI in particular will
provide essential fiber-based, optical spectroscopy over the full redshift
range $0<z<4$.

###  2.1 DESI at $z<1.5$

The currently planned DESI program will obtain a spectroscopic sample of
roughly 13 million bright galaxies (BGS), seven million luminous red galaxies
(LRG) at redshifts $z<1$, and 16 million emission line galaxies at redshifts
$0.6<z<1.6$. Covering roughly 2/3 of the observable extragalactic sky, the
baryon acoustic oscillation (BAO) distance measurements from DESI over
redshifts $z<1.0$ will be near the sample-variance limit. Increasing the
footprint or adding new $z<1$ targets will only have marginal returns for
these BAO measurements. The DESI BAO distance measurements over $1<z<1.5$ will
be within a factor of two of the sample-variance limit for this same survey
footprint, lending potential for future surveys of new areas and higher target
densities.

These spectroscopic samples will also enable measurements of the growth of
structure through redshift space distortions (RSD). RSD result from the
imprint of gravitational infall on the measured redshifts of galaxies. These
peculiar velocities record the history of structure growth which can be
modelled with clustering measurements performed along and transverse to the
line-of-sight. RSD introduce anisotropy to this three-dimensional clustering,
typically parameterized as $f\sigma_{8}=\frac{\partial\sigma_{8}}{\partial\ln
a}$, where $a=(1+z)^{-1}$ is the dimensionless cosmic expansion factor.
Evaluating the amplitude of clustering at redshift $z=0$ under an assumption
of a flat $\Lambda$CDM cosmology, $\sigma_{8}$ has been measured to a
precision of 3.5% [4] using BAO and RSD in the Stage-III Baryon Oscillation
Spectroscopic Survey [BOSS; 5] and its successor, eBOSS [6]. Under the same
$\Lambda$CDM model, with the same priors on the baryon density from Big Bang
Nucleosynthesis (BBN), and the sample priors on the spectral index of the
primordial power spectrum, the precision on $\sigma_{8}$ is expected to
improve to 0.5% in DESI when using similar algorithms and scales to
$0.20\,\,h{\rm Mpc}^{-1}$ to perform the RSD measurements. While a significant
improvement over Stage-III, theoretical and modeling work will be needed to
reach this goal. Additional data will be needed to reach the precision of 0.2%
identified as a goal in the Astro2020 Decadal Survey to test consistency with
the Planck [7] $\sigma_{8}=0.811\pm 0.006$ constraint.

###  2.2 Rubin observatory at $z<2$

The Vera C. Rubin Legacy Survey of Space and Time (LSST) uses the Simonyi
Survey Telescope, which is designed to observe a 18,000 deg2 region of the sky
in six optical bands [8]. LSST will run for a decade allowing the detection of
$\sim$20 billion galaxies with photometric redshifts extending well-above
unity. It will measure shapes for $\sim$2 billion galaxies allowing lensing
tomography to constrain the growth rate of structure and the equation of state
of dark energy to reach the requirements for a Stage IV dark energy
experiments as defined in the Dark Energy Task Force report [9]. It will also
measure clustering of galaxies and their cross correlation with lensing
measurement in the so-called $3\times 2$pt analysis. LSST will also discover
and measure at least 500 Supernovae Type Ia (SNe Ia) per season which will
provide tens of thousands of well-measured SNe Ia light curves up to $z\sim 1$
during the survey. LSST will discover $\sim$100,000 clusters of galaxies.
Finally, LSST will measure strong gravitational lensing and time delay for the
multiple images providing a sample of $\sim$2600 time-delayed lensing systems.
A comprehensive analysis of the expected dark energy constraining power
combining the probes can be found in Ref. [10].

###  2.3 New Approaches with Stage-IV Dark Energy Surveys

Whereas the expansion history will be well mapped by DESI at redshifts
$z<1.5$, new techniques will be developed to improve clustering measurements
at smaller scales. As a step toward this goal, several projects within DESI
have begun to characterize DESI galaxies through galaxy-galaxy lensing from
public weak lensing surveys (KiDS, DES, and HSC) and CMB lensing in order to
deduce cosmological constraints from full-scale clustering studies. Such
cross-correlations often rely on a much higher number density than the BAO
measurements, but may not need a spectrum of every object. Ongoing analyses
will help establish the balance between photometric and spectroscopic sample
sizes. Additional projects are also underway to compute higher order
clustering statistics to better characterize scale-dependent bias that is
degenerate with cosmological information at small scales. These studies will
also help establish the relationship between number density and cosmological
constraints.

##  3 Immediate Analysis Needs

Theoretical forecasts of the cosmological constraining power of the nonlinear
regime of structure formation now date back many years [e.g., 11, 12, 13, 14],
and suggest that an improved ability to use smaller-scale information can
result in factors of 2-4 improvement on dark energy constraints. The natural
question arises as to whether these gains can be realized in practice, or
whether the need to marginalize over nuisance parameters capturing systematic
uncertainty results in an excessive loss of constraining power. Recent work
analyzing BOSS galaxy samples has shown that these potential gains could
indeed be a reality [15, 16, 17]; by including measurements from nonlinear
scales, these recent analyses have achieved a full factor of 2 improvement
beyond previous BOSS analyses [e.g. 18] that restricted attention to the
quasi-linear regime.

In addition to pushing the boundaries of a single survey by improving analysis
capabilities, cross-correlations of data from different wavebands will enhance
the science returns from ongoing and upcoming surveys (see Ref. [19] for a
detailed analysis). Therefore, a modest investment into an overarching
simulation and modeling program that will enable the exploitation of small
scales and cross-correlations has the potential to substantially extend the
scientific reach of ongoing and upcoming surveys (see also Ref. [20]).

###  3.1 High-Fidelity Simulations

Simulations play an important role in fully exploiting the information
available from cosmological surveys, in particular on smaller length scales.
The simulation needs for ongoing and upcoming surveys span a wide range, from
large numbers of realizations for covariance estimates, to detailed
simulations that allow the creation of high-quality synthetic sky catalogs to
enable tests of analysis pipelines and evaluation of systematic effects, to
simulations that allow the exploration of physics beyond the confines of
$\Lambda$CDM. Simulations beyond $\Lambda$CDM are important for many reasons.
They provide predictions across cosmological parameters for a range of
cosmological observables and enable the exploration of signatures of new
physics beyond the current cosmological paradigm for both dark energy and dark
matter.

The importance of simulation campaigns has been recognized widely in the
community. Critical simulation campaigns have been carried out for
contemporary surveys, including the MICE [21] and Buzzard simulations [22] for
DES, the DC2 simulated catalogs for LSST DESC [23, 24], and the AbacusSummit
suite for DESI [25]. A related White Paper [26] provides details regarding the
roles of simulations in Cosmic Frontier science.

Conventional N-body simulations are now extremely mature tools; over the
years, many code comparison and verification projects have been carried out,
e.g., Refs. [27, 28, 29, 30, 31, 32]. These studies have shown that
contemporary N-body codes with appropriate settings for initial conditions,
force and mass resolution and coverage of large enough volumes, agree at the
1% level out to scales of $k\sim 1h^{-1}$Mpc, beyond which baryonic effects
need to be taken into account. The statistical power of future spectroscopic
surveys can be be 0.1% or better at these scales, offering substantial power
to constrain the assumptions in N-body simulations. Simulation development
over the course of the next decade is required to keep pace with the
statistical precision allowed by these potential surveys.

As a very recent example, in Ref. [32], several N-body simulation codes have
been considered within DESI to test theoretical models and systematic errors
in two-point clustering and other measurement techniques. These simulations
must accurately predict the growth of the dark matter halos that host the
galaxies that are used as tracers of the underlying dark matter distribution.
The simulations codes under consideration make different approximations to
gravitational interactions between particles and therefore slightly different
predictions for the halo mass distribution and its clustering properties. At
scales out to $k_{\rm max}=0.20h\,\mathrm{Mpc}^{-1}$, where previous RSD
measurements have been performed, there is almost perfect agreement between
the high resolution SWIFT simulations and the ABACUS simulations (Figure 1).
However, even for these two simulations, the agreement degrades to the 0.2%
level around $k=1h\,\mathrm{Mpc}^{-1}$. While these halos have not yet been
populated with galaxies, the exercise demonstrates that a theoretical floor of
at least 0.2% precision currently exists in the first step of modeling halo
clustering at small scales.

Figure 1: The real-space power spectra of halos from the N-body simulations
using GADGET-2 [33] and SWIFT [34] relative to the ABACUS [35] code. Figure
reproduced from “The DESI $N$-body Simulation Project I: Testing the
Robustness of Simulations for the DESI Dark Time Survey” [32].

###  3.2 Synthetic Skies from High-Fidelity Simulations

Carrying out sufficiently accurate simulations is only the first step,
however, establishing robust connections to the observable universe is
essential for making useful predictions for galaxy clustering on nonlinear
scales – the process of galaxy formation is not directly modeled in N-body
simulations. It is important, therefore, to develop improved techniques for
relating the galaxy population to the underlying N-body simulations, a
quantitative improvement in the so-called ‘galaxy-halo’ connection.

Many contemporary efforts to derive cosmological constraints from nonlinear
scales are built upon simplistic empirical models such as the Halo Occupation
Distribution (HOD). Due to the very formulation of HOD-type models,
incorporating new constraints from multiple redshifts and more than a single
tracer galaxy population requires a significant expansion of the parameter
space, and/or reliance upon plausibly-violated assumptions about the galaxy-
halo connection. Thus conventional halo occupation models actually penalize
attempts to incorporate new constraining data. Moreover, this older generation
of models was devised at a time when consensus in the field had not yet been
reached on the reliability of cosmological simulations to resolve halo
substructure, and so the HOD and related models are founded merely upon host
halos identified at a particular simulated snapshot. But the field of
computational cosmology has seen dramatic progress in the quality of simulated
data products over the last fifteen years: subhalo catalogs with merger trees
are becoming widely available for high-resolution, survey-scale simulations
[36, 37, 38, 39]; thus the continued use of HOD-type models as the basis of
our cosmological predictions fails to capitalize upon the now well-established
ability of contemporary codes to track the evolution of halos and their
substructure across cosmic time. These core limitations of traditional halo
occupation models highlight how they bear the mark of the era in which they
were developed. Hydrodynamical simulations or Semi-Analytic Models (SAMs) have
also been used to generate predictions across redshift in a physically
motivated way. These models remain irreplaceable in the effort to understand
the detailed physics of galaxies and clusters, but the full scientific
potential of data in the 2020s can only be delivered with expansive
explorations of parameter space based on high-resolution, Gpc-scale
simulations, and so new techniques beyond the traditional implementations of
these models are needed.

Considerable recent progress has been made by a promising new generation of
empirical models that bridge the gap between the level of complexity achieved
by SAMs and the computational efficiency of empirical models, e.g.,
UniverseMachine [40] and EMERGE [41]. The ability of these models to make
predictions for multiple tracers across redshift is promising, but further
advances are needed on both the modeling and computation side for this new
approach to conduct cosmological inference with survey-scale simulations. To
meet the predictive needs and to maximize the scientific returns of the
upcoming surveys, we consider it critical to invest in the development of a
new generation of modeling approaches that address the limitations of
contemporary techniques.

###  3.3 Lack of Inter-collaboration structures

The general requirements on simulations to analyze surveys are similar for the
different surveys. While some surveys need larger volumes, other surveys
possibly require higher mass resolution. However, developing an overarching
simulation program that allows surveys to share simulations and derived data
products such as synthetic catalogs would be very valuable. In Ref. [42] a
strong case was made for a joint simulation program across the Vera Rubin
Observatory (Rubin) Dark Energy Science Collaboration (DESC), the Nancy Grace
Roman Space Telescope, and Euclid. This case clearly carries over to DESI,
CMB-S4, and other future surveys as well.

##  4 Future Survey Prospects

Advances in simulations and the theory connecting galaxies to dark matter will
allow more accurate modeling of the cosmological signal. Meanwhile, future
spectroscopic programs have the potential to significantly improve the
statistical precision allowed in these analyses. After DESI completes, there
will still be a very large number of spectroscopic modes available to probe
clustering at small scales at $z<1.5$. New spectroscopic galaxy samples at
these redshifts will allow precise measurements of structure growth and the
full-shape of the power spectrum during the critical time when the Universe
transitioned from a matter-dominated expansion to a dark energy-dominated
expansion.

Figure 2: Number of spectroscopic tracers required to reach $\bar{n}P(k)=1$ as
a function of redshift for scales $k_{\rm{max}}=$0.2, 0.5, and 1.0
$h\,\rm{Mpc}^{-1}$. The DESI BGS, LRG, and ELG number densities are as
expected based on current survey operations. Left: Number density as a
function redshift. Right: Cumulative surface density as a function of
redshift. The dependence of large-scale bias on number density is based on
extrapolations of the HOD models [43] derived from clustering in the
photometric catalogs of the DESI LRG sample [44], an assumed halo mass
function [45], and an assumed halo mass-halo bias relation [46]. The large-
scale bias decreases by a factor of roughly 1.7 between the number densities
reported for $k_{\rm{max}}=$0.2 $h\,\rm{Mpc}^{-1}$ and those reported for
$k_{\rm{max}}=$1.0 $h\,\rm{Mpc}^{-1}$.

Number density, redshift range, spectroscopic completeness, and volume for a
future spectroscopic program will need to be matched to analysis techniques
and a systematic error budget that will steadily improve over the next decade.
In the absence of a specific systematic error budget or well-accepted metrics
to optimize area and number density for cross-correlation studies or higher
order statistics, we refer to the product $\bar{n}P(k)=1$ for characterizing
the potential of future samples. Here, $\bar{n}$ is the average number density
of the sample and $P(k)$ is the amplitude of the observed power spectrum at a
scale $k$. This parameterization reflects a fairly optimal balance between
area and number density for two-point statistics in a sample of a fixed size
[for further discussion, see 47]. As shown in the left panel of Figure 2, the
DESI luminous red galaxy (LRG) and emission line galaxy (ELG) samples will
achieve number densities that exceed the limit $\bar{n}P(k)=1$ for scales
$k_{\rm{max}}=0.20\,h\,\rm{Mpc}^{-1}$.

The bright galaxy sample (BGS) will reach $\bar{n}P(k)=1$ for scales
$k_{\rm{max}}=1\,h\,\rm{Mpc}^{-1}$ at redshifts $z<0.3$. A scale of
$k_{\rm{max}}=1\,h\,\rm{Mpc}^{-1}$ corresponds roughly to the transition
between sampling the halo population (two halo term) and sampling the galaxies
within a halo (one halo term). It is this region where the nuisance terms
resulting from connecting galaxies to halos are best constrained, thus
breaking degeneracies with the cosmological information.

The DESI BGS sample has such a high number density in part to serve as a
testbed for new cosmological analysis techniques. Future programs have the
potential to expand these techniques to higher redshifts and larger volumes.
As shown in the right panel of Figure 2, a program that achieves
$\bar{n}P(k)=1$ at $k_{\rm{max}}=1\,h\,\rm{Mpc}^{-1}$ to a redshift $z=1$
would require roughly 4000 galaxies deg-2. A program that extends a clustering
sample of this scope to $z=1.5$ would require roughly 10,000 galaxies deg-2.
With samples of this size extending to higher redshift, the techniques
developed on the BGS sample can eventually be applied over a redshift range
that includes the matter dominated era and the transition to the accelerating
epoch.

###  4.1 Science Drivers for Low-z clustering with future surveys

Current lensing measurements indicate tension with predictions from the CMB
under a $\Lambda$CDM model (see, e.g., Ref. [48] for recent results). These
tensions may be explained by an additional field or interaction that distorts
the shape of the power spectrum, modifications to General Relativity, other
extensions to the Standard Model, a chance statistical realization of the
analyses, or systematic errors in the data analysis. Three-dimensional
clustering data from DESI and lensing from Rubin observatory will allow us to
improve the precision of these measurements. Advances in simulations will
allow us to improve the theoretical modeling of these measurements. New
spectroscopic data samples that sample clustering at the densities presented
in Figure 2 over a wide redshift range can advance our understanding of cosmic
acceleration as follows:

  1. 1.

Growth of structure: probe origin of cosmic acceleration at late times

  2. 2.

Growth of structure: constrain neutrino mass

  3. 3.

Full-shape power spectrum measurements: provide limits on models for new
physics that distort the shape of the power spectrum at scales
$k<1\,h\,\rm{Mpc}^{-1}$

  4. 4.

Calibrating samples of voids, clusters, peculiar velocities, and other
techniques: perform comprehensive tests of the cosmologial model

  5. 5.

Higher-order statistics: resolving degeneracies or identifying cosmological
signatures not present in the power spectrum

  6. 6.

Multiple probes: test for new particle or field contents in the cosmological
model

The techniques to perform these cosmological constraints will likely arise
from a combination of three-dimensional clustering at small scales, higher
order clustering statistics, and cross-correlations with lensing surveys.

###  4.2 Clustering Samples from DESI with Extended LRG Selections

Figure 3: Expected redshift distribution for the DESI LRG samples compared to
various extended selection algorithms. Left: A broad extension to the DESI
algorithm that excludes criteria intended to reduce the number of low redshift
galaxies. Right: An extended selection that incorporates criteria intended to
reduce the number of $z<0.3$ galaxies. In both cases, the redshifts are from
COSMOS photo-z’s [49] for the $z_{\rm fiber}$-limited samples and DESI
spectroscopic redshifts for DESI LRGs. Pixel-level simulations indicate that
the samples described by a selection to $z_{\rm fiber}<21.6$ can be
spectroscopically classified in the same exposure time as planned for the
ongoing DESI program. Simulations indicate that galaxies with $21.6<z_{\rm
fiber}<22.0$ require exposures twice as long while galaxies with $22.0<z_{\rm
fiber}<22.4$ require exposures four times as long.

The reddest, most luminous LRGs exhibit relatively high large-scale-structure
bias, which enhances the amplitude of their clustering. Because of their
strong 4000 Å break and their well-behaved, red spectral energy distributions,
LRGs can be selected from imaging catalogs and spectrally classified with high
efficiency. Pixel-level simulations indicate that more than 95% of these
targets can be assigned reliable redshifts using a selection to $z_{\rm
fiber}<21.6$, where $z_{\rm fiber}$ is the $z$-band magnitude in an aperture
matched to the DESI fiber diameter.

The currently planned DESI program will obtain an LRG spectroscopic sample of
roughly six million galaxies at redshifts $z<1$. Beyond this approximate
redshift, the strong 4000 Å break and characteristic spectral features move
into a region of the spectrum with strong sky line contamination, thus
complicating redshift classification. As shown in Figure 3, the DESI LRG
sample will cover the redshift range $0.3<z<1$ with a surface density of
approximately 600 deg-2. These observations will be obtained over five years,
using a subset of the 50M fibers that will be available over all pointing
centers.

In the latter half of this decade, DESI will remain the most competitive
instrument for large-scale spectroscopic surveys. DESI will be able to
maintain a high redshift efficiency for targets identified in extensions to
the LRG selections that include bluer galaxies with lower bias. Potential
selections have been explored using infrared and optical photometry, with
simulations revealing consistently high redshift efficiency at the same
exposure times planned for the current DESI survey. Two such selections are
presented in orange in Figure 3. At 6600 deg-2, the first selection (left
panel) includes a high fraction of low redshift galaxies and very high
completeness as a function of stellar mass. The second selection (right panel)
is tuned for minimal redshift overlap with the BGS sample, leading to a
density of 3000 deg-2. This second selection exceeds a number density of
$1\times 10^{-3}\,h^{3}\,Mpc^{-3}$ out to $z<0.9$. As demonstrated in Figure
2, this number density would be sufficient to reach the threshold
$\bar{n}P(k)=1$ at $k_{\rm{max}}=1\,h\,\rm{Mpc}^{-1}$. The galaxies in these
two selections are sufficiently bright that spectra can be classified at high
efficiency using the same exposure times as planned for the current LRG
sample. At this exposure time, roughly 10M spectra can be obtained per year.

Longer exposure times could be used to include even fainter galaxies, as shown
in the green and red curves of Figure 3. Extending the magnitude limit to
$z_{\rm fiber}<22.0$ (green) would require a factor of two increase in
exposure time for the additional 1700 deg-2 galaxies included in the
selection. Doing so would extend the redshift coverage to $z<1$ where the
density allows sampling of clustering beyond the shotnoise limit. A factor of
four increase in exposure time (red) would allow observations of a sample to
$z_{\rm fiber}<22.4$, thus vastly exceeding the required number densities over
$0.3<z<1$ and extending coverage to $z<1.1$.

Continuing DESI observations in the area that overlaps the Vera Rubin
Observatory survey footprint would allow joint studies of a high density
spectroscopic sample and high signal-to-noise lensing sample. In Table 2, we
demonstrate several potential survey strategies that can be completed with the
DESI instrument in five years or less. With only an investment in DESI
operations, such a survey would allow enhanced tests of dark energy, physics
beyond the standard model, and neutrino masses using very high density probes
of large-scale structure out to an approximate redshift of $z=1$.

Survey Boundaries | Area | Survey Time (years) | Sample Size
---|---|---|---
Selection at 6600 deg-2 without low redshift cut ($z_{\rm fiber}<21.6$)
$0<\delta<12$ | 3372 deg2 | 2 years | 22M
$-10<\delta<12$ | 6024 deg2 | 4 years | 40M
Selection at 3000 deg-2 with low redshift cut ($z_{\rm fiber}<21.6$)
$0<\delta<12$ | 3372 deg2 | 1 year | 10M
$-10<\delta<12$ | 6024 deg2 | 2 years | 18M
Selection at 4700 deg-2 with low redshift cut ($z_{\rm fiber}<22.0$)
$0<\delta<12$ | 3372 deg2 | 2 years | 16M
$-10<\delta<12$ | 6024 deg2 | 4 years | 28M
Selection at 7100 deg-2 with low redshift cut ($z_{\rm fiber}<22.4$)
$0<\delta<12$ | 3372 deg2 | 5 years | 24M
Table 2: Potential survey footprints for four different selection algorithms
using DESI as shown in Figure 3. The reported area corresponds to the overlap
with Vera Rubin imaging in regions at galactic latitudes that avoid excessive
Galactic extinction and stellar contamination.

###  4.3 Future Clustering Samples at $1<z<1.5$

As shown in Figure 2, roughly 6000 galaxies deg-2 are required to reach a
threshold of $\bar{n}P(k)=1$ at $k_{\rm{max}}=1\,h\,\rm{Mpc}^{-1}$ over the
redshift range $1<z<1.5$. At these redshifts, the absorption features that
allow robust classification of passive, LRG spectra become harder to detect
due to increased sky background and decreased continuum signal. For this
reason, the DESI survey relies on the detection of [O ii] line flux for galaxy
redshift classification in this redshift range. At a wavelength of 3727 Å, the
[O ii] emission line that is often associated with star formation is visible
with Silicon detectors to approximately $z<1.6$. However, selecting these
targets from imaging data to have both significant line strength and redshifts
in the range $1.0<z<1.5$ is far more challenging than the selection of LRG
targets described in Section 4.2. As can be seen in the right panel of Figure
2, the DESI ELG selection produces only around 400 deg-2 robust redshift
classifications in this redshift range. These redshifts are obtained from a
spectroscopic sample of roughly 1400 deg-2, some of which are at lower
redshift and some of which are not classified with high confidence. More than
an order of magnitude increase in number density is required to reach the
shot-noise limited densities over this redshift range.

If a selection algorithm with new imaging data is identified to select strong
[O ii] emitting galaxies over $1<z<1.5$ with high efficiency, then it will be
possible to improve clustering statistics with new DESI observations. However,
it is most likely that new selections will experience similar contamination
from low redshift galaxies and require longer exposure times to reach to
fainter [O ii] line fluxes. While such an investigation remains to be done,
one can assume similar rates of redshift success. The [O ii] luminosity
distribution is also unknown, but for illustrative purposes, we assume that an
order of magnitude increase in sample size will produce limiting [O ii] line
strengths that are between a factor of one and two smaller than in the DESI
sample. Such a sample would require between 21,000 and 84,000 fiber-exposures
deg-2 at equivalent DESI exposure times. A pilot survey with DESI could
complete between 120 and 500 deg2 in one year under these assumptions.

Observations at this high density are more efficiently pursued with a future
spectroscopic facility with a significantly higher survey speed. Three
concepts for future spectroscopic programs were described in white papers
submitted to the Astro2020 Decadal Survey: MegaMapper [50], Mauna Kea
Spectroscopic Explorer (MSE) [51], and SpecTel [52]. Each of these offers the
potential for even larger sample sizes than would be obtained with DESI. Both
the proposed fiber positioner designs and the collecting areas are yet to be
finalized for these three facilities. Accepting these uncertainties, we
approximate the collecting area of the 6.5-meter MegaMapper telescope to be
three times that of DESI and the collecting area of both the 12-meter class
MSE and SpecTel telescopes to be ten times that of DESI. We assume a similar
focal plane of 20,000 fibers for all three designs, and assume that sufficient
targets are available to populate the fibers regardless of field of view. The
survey areas available to each facility simply scales with the product of
fiber number and collecting area.

Under these idealized assumptions, we find that in order to reach
$\bar{n}P(k)=1$ at $k_{\rm{max}}=1\,h\,\rm{Mpc}^{-1}$ over the redshift range
$1<z<1.5$:

  * •

MegaMapper could complete between 1,500 and 6,000 deg2 per year;

  * •

MSE could complete at least 5000 deg2 per year; and

  * •

SpecTel could complete at least 5000 deg2 per year.

## References

  * [1] DESI Collaboration, Amir Aghamousa, et al.  The DESI Experiment Part I: Science,Targeting, and Survey Design.  arXiv e-prints, page arXiv:1611.00036, October 2016.
  * [2] DESI Collaboration, Amir Aghamousa, et al.  The DESI Experiment Part II: Instrument Design.  arXiv e-prints, page arXiv:1611.00037, October 2016.
  * [3] Steve Ritz et al.  Building for Discovery: Strategic Plan for U.S. Particle Physics in the Global Context.  5 2014.
  * [4] eBOSS Collaboration et al.  Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory.  Phys. Rev. D., 103(8):083533, April 2021.
  * [5] Kyle S. Dawson et al.  The Baryon Oscillation Spectroscopic Survey of SDSS-III.  AJ, 145(1):10, January 2013.
  * [6] Kyle S. Dawson et al.  The SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Overview and Early Data.  AJ, 151(2):44, February 2016.
  * [7] Planck Collaboration et al.  Planck 2018 results. VI. Cosmological parameters.  A&A, 641:A6, September 2020.
  * [8] Željko Ivezić et al.  LSST: From Science Drivers to Reference Design and Anticipated Data Products.  Astrophys. J., 873(2):111, March 2019.
  * [9] Andreas Albrecht, Gary Bernstein, Robert Cahn, Wendy L. Freedman, Jacqueline Hewitt, Wayne Hu, John Huth, Marc Kamionkowski, Edward W. Kolb, Lloyd Knox, John C. Mather, Suzanne Staggs, and Nicholas B. Suntzeff.  Report of the Dark Energy Task Force.  arXiv e-prints, pages astro–ph/0609591, September 2006.
  * [10] The LSST Dark Energy Science Collaboration, Rachel Mandelbaum, et al.  The LSST Dark Energy Science Collaboration (DESC) Science Requirements Document.  arXiv e-prints, page arXiv:1809.01669, September 2018.
  * [11] A. R. Zentner, E. Semboloni, S. Dodelson, T. Eifler, E. Krause, and A. P. Hearin.  Accounting for baryons in cosmological constraints from cosmic shear.  Phys. Rev. D, 87(4):043509, February 2013.
  * [12] Beth A. Reid, Hee-Jong Seo, Alexie Leauthaud, Jeremy L. Tinker, and Martin White.  A 2.5 per cent measurement of the growth rate from small-scale redshift space clustering of SDSS-III CMASS galaxies.  MNRAS, 444(1):476–502, October 2014.
  * [13] E. Krause, T. F. Eifler, and the DES Collaboration.  Dark Energy Survey Year 1 Results: Multi-Probe Methodology and Simulated Likelihood Analyses.  arXiv:1706.09359, June 2017.
  * [14] Andrés N. Salcedo, Benjamin D. Wibking, David H. Weinberg, Hao-Yi Wu, Douglas Ferrer, Daniel Eisenstein, and Philip Pinto.  Cosmology with stacked cluster weak lensing and cluster-galaxy cross-correlations.  MNRAS, 491(3):3061–3081, January 2020.
  * [15] Benjamin D. Wibking, David H. Weinberg, Andrés N. Salcedo, Hao-Yi Wu, Sukhdeep Singh, Sergio Rodríguez-Torres, Lehman H. Garrison, and Daniel J. Eisenstein.  Cosmology with galaxy-galaxy lensing on non-perturbative scales: emulation method and application to BOSS LOWZ.  MNRAS, 492(2):2872–2896, February 2020.
  * [16] Johannes U. Lange, Andrew P. Hearin, Alexie Leauthaud, Frank C. van den Bosch, Hong Guo, and Joseph DeRose.  Five-percent measurements of the growth rate from simulation-based modelling of redshift-space clustering in BOSS LOWZ.  MNRAS, submitted, page arXiv:2101.12261, January 2021.
  * [17] Michael J. Chapman et al.  The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the growth rate of structure from the small-scale clustering of the luminous red galaxy sample.  arXiv:2106.14961, page arXiv:2106.14961, June 2021.
  * [18] BOSS Collaboration et al.  The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample.  MNRAS, 470(3):2617–2652, September 2017.
  * [19] Authors et al.  Snowmass2021 Cosmic Frontier White Paper: Cosmology from Combined Probes.  Contribution to Snowmass 2021, 2022.
  * [20] Kyle Dawson, Josh Frieman, Katrin Heitmann, Bhuvnesh Jain, Steve Kahn, Rachel Mandelbaum, Saul Perlmutter, and Anže Slosar.  Cosmic Visions Dark Energy: Small Projects Portfolio.  arXiv e-prints, page arXiv:1802.07216, February 2018.
  * [21] P. Fosalba, M. Crocce, E. Gaztañaga, and F. J. Castander.  The MICE grand challenge lightcone simulation - I. Dark matter clustering.  MNRAS, 448(4):2987–3000, April 2015.
  * [22] Joseph DeRose et al.  The Buzzard Flock: Dark Energy Survey Synthetic Sky Catalogs.  arXiv e-prints, page arXiv:1901.02401, January 2019.
  * [23] The LSST Dark Energy Science Collaboration, Danila Korytov, et al.  CosmoDC2: A Synthetic Sky Catalog for Dark Energy Science with LSST.  Astrophys. J.S., 245(2):26, December 2019.
  * [24] LSST Dark Energy Science Collaboration (LSST DESC).  The LSST DESC DC2 Simulated Sky Survey.  Astrophys. J.S., 253(1):31, March 2021.
  * [25] Nina A. Maksimova, Lehman H. Garrison, Daniel J. Eisenstein, Boryana Hadzhiyska, Sownak Bose, and Thomas P. Satterthwaite.  ABACUSSUMMIT: a massive set of high-accuracy, high-resolution N-body simulations.  MNRAS, 508(3):4017–4037, December 2021.
  * [26] Simon Birrer, Salman Habib, Katrin Heitmann, Zarija Lukić, , Yuuki Omori, and Yi-Min Zhong.  Snowmass2021 Computational Frontier White Paper: Cosmological Simulations and Modeling.  Contribution to Snowmass 2021, 2022.
  * [27] Katrin Heitmann, Paul M. Ricker, Michael S. Warren, and Salman Habib.  Robustness of Cosmological Simulations. I. Large-Scale Structure.  ApJS, 160(1):28–58, September 2005.
  * [28] Katrin Heitmann, Zarija Lukić, Patricia Fasel, Salman Habib, Michael S Warren, Martin White, James Ahrens, Lee Ankeny, Ryan Armstrong, Brian O’Shea, et al.  The cosmic code comparison project.  Computational Science & Discovery, 1(1):015003, 2008.
  * [29] Katrin Heitmann, Martin White, Christian Wagner, Salman Habib, and David Higdon.  The Coyote Universe. I. Precision Determination of the Nonlinear Matter Power Spectrum.  ApJ, 715(1):104–121, May 2010.
  * [30] Aurel Schneider, Romain Teyssier, Doug Potter, Joachim Stadel, Julian Onions, Darren S. Reed, Robert E. Smith, Volker Springel, Frazer R. Pearce, and Roman Scoccimarro.  Matter power spectrum and the challenge of percent accuracy.  JCAP, 2016(4):047, April 2016.
  * [31] Lehman H. Garrison, Daniel J. Eisenstein, and Philip A. Pinto.  A high-fidelity realization of the euclid code comparison n-body simulation with abacus.  MNRAS, 485(3), 3 2019.
  * [32] Cameron Grove et al.  The DESI $N$-body Simulation Project I: Testing the Robustness of Simulations for the DESI Dark Time Survey.  arXiv e-prints, page arXiv:2112.09138, December 2021.
  * [33] Volker Springel.  The cosmological simulation code GADGET-2.  MNRAS, 364(4):1105–1134, December 2005.
  * [34] Matthieu Schaller, Pedro Gonnet, Aidan B. G. Chalk, and Peter W. Draper.  SWIFT: Using task-based parallelism, fully asynchronous communication, and graph partition-based domain decomposition for strong scaling on more than 100,000 cores.  arXiv e-prints, page arXiv:1606.02738, June 2016.
  * [35] Lehman H. Garrison, Daniel J. Eisenstein, Douglas Ferrer, Nina A. Maksimova, and Philip A. Pinto.  The ABACUS cosmological N-body code.  MNRAS, 508(1):575–596, November 2021.
  * [36] Chia-Hsun Chuang et al.  UNIT project: Universe N-body simulations for the Investigation of Theoretical models from galaxy surveys.  MNRAS, 487(1):48–59, July 2019.
  * [37] Katrin Heitmann, Nicholas Frontiere, Esteban Rangel, Patricia Larsen, Adrian Pope, Imran Sultan, Thomas Uram, Salman Habib, Hal Finkel, Danila Korytov, Eve Kovacs, Silvio Rizzi, Joe Insley, and Janet Y. K. Knowles.  The Last Journey. I. An Extreme-scale Simulation on the Mira Supercomputer.  ApJS, 252(2):19, February 2021.
  * [38] Tomoaki Ishiyama, Francisco Prada, Anatoly A. Klypin, Manodeep Sinha, R. Benton Metcalf, Eric Jullo, Bruno Altieri, Sofía A. Cora, Darren Croton, Sylvain de la Torre, David E. Millán-Calero, Taira Oogi, José Ruedas, and Cristian A. Vega-Martínez.  The Uchuu simulations: Data Release 1 and dark matter halo concentrations.  MNRAS, 506(3):4210–4231, September 2021.
  * [39] Sownak Bose, Daniel J. Eisenstein, Boryana Hadzhiyska, Lehman H. Garrison, and Sihan Yuan.  Constructing high-fidelity halo merger trees in AbacusSummit.  arXiv:2110.11409, page arXiv:2110.11409, October 2021.
  * [40] P. Behroozi, R. Wechsler, A. Hearin, and C. Conroy.  UniverseMachine: The Correlation between Galaxy Growth and Dark Matter Halo Assembly from z=0-10.  arXiv:1806.07893, June 2018.
  * [41] B. P. Moster, T. Naab, and S. D. M. White.  EMERGE - An empirical model for the formation of galaxies since $z\sim 10$.  ArXiv:1705.05373, May 2017.
  * [42] Nick Battaglia, Andrew Benson, Tim Eifler, Andrew Hearin, Katrin Heitmann, Shirley Ho, Alina Kiessling, Zarija Lukic, Michael Schneider, Elena Sellentin, and Joachim Stadel.  Report from the Tri-Agency Cosmological Simulation Task Force.  arXiv e-prints, page arXiv:2005.07281, May 2020.
  * [43] Zheng Zheng and David H. Weinberg.  Breaking the Degeneracies between Cosmology and Galaxy Bias.  ApJ, 659(1):1–28, April 2007.
  * [44] Rongpu Zhou et al.  The clustering of DESI-like luminous red galaxies using photometric redshifts.  MNRAS, 501(3):3309–3331, March 2021.
  * [45] Giulia Despali, Carlo Giocoli, Raul E. Angulo, Giuseppe Tormen, Ravi K. Sheth, Giacomo Baso, and Lauro Moscardini.  The universality of the virial halo mass function and models for non-universality of other halo definitions.  MNRAS, 456(3):2486–2504, March 2016.
  * [46] Jeremy L. Tinker, Brant E. Robertson, Andrey V. Kravtsov, Anatoly Klypin, Michael S. Warren, Gustavo Yepes, and Stefan Gottlöber.  The Large-scale Bias of Dark Matter Halos: Numerical Calibration and Model Tests.  ApJ, 724(2):878–886, December 2010.
  * [47] Andreu Font-Ribera, Patrick McDonald, Nick Mostek, Beth A. Reid, Hee-Jong Seo, and Anže Slosar.  DESI and other Dark Energy experiments in the era of neutrino mass measurements.  JCAP, 2014(5):023, May 2014.
  * [48] Catherine Heymans et al.  KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints.  A&A, 646:A140, February 2021.
  * [49] C. Laigle et al.  The COSMOS2015 Catalog: Exploring the 1.  The Astrophysical Journal Supplement Series, 224(2):24, June 2016\.
  * [50] David Schlegel, Juna A. Kollmeier, and Simone Ferraro.  The MegaMapper: a z¿2 spectroscopic instrument for the study of Inflation and Dark Energy.  In Bulletin of the American Astronomical Society, volume 51, page 229, September 2019.
  * [51] Jennifer Marshall et al.  The Maunakea Spectroscopic Explorer.  In Bulletin of the American Astronomical Society, volume 51, page 126, September 2019.
  * [52] Richard Ellis and Kyle Dawson.  SpecTel: A 10-12 meter class Spectroscopic Survey Telescope.  In Bulletin of the American Astronomical Society, volume 51, page 45, September 2019.