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Future of Particle Physics (Snowmass 2021) ,
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# Snowmass2021 Cosmic Frontier White Paper:
Prospects for obtaining Dark Matter Constraints with DESI

Monica Valluri  University of Michigan, Ann Arbor, MI 48109, USA  Solene
Chabanier  Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Vid
Iršič  Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK  Cavendish Laboratory, University of Cambridge, 19 J. J.
Thomson Ave., Cambridge CB3 0HE, UK  Eric Armengaud  IRFU, CEA, Université
Paris-Saclay, F-91191 Gif-sur-Yvette, France  Michael Walther  University
Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr.
1, 81677 Munich, Germany  Excellence Cluster ORIGINS, Boltzmannstr. 2, 85748
Garching, Germany  Connie Rockosi  University of California, Santa Cruz, CA
95064, USA  Miguel A. Sánchez-Conde  Instituto de Física Teórica (IFT) UAM-
CSIC and Departamento de Física Teórica, Universidad Autónoma de Madrid, Calle
Nicolás Cabrera, 13-15, 28049 Madrid, Spain  Leandro Beraldo e Silva
University of Michigan, Ann Arbor, MI 48109, USA  Andrew P. Cooper  Department
of Physics, National Tsing Hua University, Taiwan  Elise Darragh-Ford  Kavli
Institute for Particle Astrophysics and Cosmology and Department of Physics,
Stanford University, CA 94305, USA; SLAC National Accelerator Laboratory,
Menlo Park, CA 94025  Kyle Dawson  University of Utah, Department of Physics
and Astronomy, 115 S 1400 E, Salt Lake City, UT 84112, USA  Alis J. Deason
Institute for Computational Cosmology, Durham University, Durham, DH1 3LE, UK
Simone Ferraro  Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Jaime E. Forero-Romero  Departamento de Física, Universidad de los Andes, Cra.
1 No. 18A-10 Edificio Ip, CP 111711, Bogotá, Colombia  Antonella Garzilli
EPFL Laboratoire d’astrophysique, BSP 320 – Cubotron, CH – 1015 Lausanne,
Switzerland  Ting Li  David A. Dunlap, Dept of Astronomy and Astrophysics,
University of Toronto, Ontario, M5S 3H4 Canada  Zarija Lukić  Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Christopher J. Manser
Astrophysics Group, Department of Physics, Imperial College London, Prince
Consort Rd, London, SW7 2AZ, UK  Nathalie Palanque-Delabrouille  Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  IRFU, CEA, Université
Paris-Saclay, F-91191 Gif-sur-Yvette, France  Corentin Ravoux  IRFU, CEA,
Université Paris-Saclay, F-91191 Gif-sur-Yvette, France  Ting Tan  Sorbonne
Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes
Energies, LPNHE, 4 Place Jussieu, F-75252 Paris, France  Wenting Wang
Shanghai Jiaotong University, Shanghai, China  Risa H. Wechsler  Kavli
Institute for Particle Astrophysics and Cosmology and Department of Physics,
Stanford University, CA 94305, USA; SLAC National Accelerator Laboratory,
Menlo Park, CA 94025  Andreia Carrillo  Institute for Computational Cosmology,
Durham University, Durham, DH1 3LE, UK  Arjun Dey  NSF’s National Optical-
Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719
Sergey E. Koposov  Institute for Astronomy, University of Edinburgh, Royal
Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK  Yao-Yuan Mao  Department
of Physics and Astronomy, Rutgers, The State University of New Jersey,
Piscataway, NJ 08854, USA  Paulo Montero-Camacho  Department of Astronomy,
Tsinghua University, China  Ekta Patel  Department of Astronomy, University of
California, Berkeley CA 94720, USA  Miller Institute for Basic Research in
Science, University of California, Berkeley CA 94720, USA  Graziano Rossi
Sejong University, Department of Physics and Astronomy, Seoul, 143-747,
Republic of Korea  L. Arturo Ureña-López  Universidad de Guanajuato, México
Octavio Valenzuela  Universidad Nacional Autónoma de México, Instituto de
Astronomía

Executive Summary

For the past several decades efforts to directly detect dark matter (DM), the
particle thought to constitute $\sim 24\%$ of the matter–energy density of the
universe have largely focused on cold dark matter. Despite heroic and
sustained efforts, highly sensitive direct-detection experiments have not yet
led to the discovery of the WIMP (although the allowed mass/energy scale and
interaction cross-section have been significantly constrained; see, e.g., [1]
for a recent review). This has led to increasing interest in alternatives to
the $\Lambda$CDM paradigm and a growing impetus to explore alternative DM
scenarios (including fuzzy DM, warm DM, self-interacting DM, etc.). In many of
these scenarios, particles cannot be detected directly and constraints on the
properties of these DM particles can ONLY be arrived at using astrophysical
observations.

The Dark Energy Spectroscopic Instrument (DESI), which recently began its
regular operations, is currently one of the most powerful instruments for
wide-field surveys. The synergy of DESI with current (e.g. ESA’s Gaia
satellite) and future observing facilities including the James Webb Space
Telescope (JWST), the Vera Rubin Observatory’s Legacy Survey of Space and Time
(LSST), and the Nancy Grace Roman Space Telescope’s High Latitude Survey (HLS)
will yield datasets of unprecedented size and coverage that will enable
constraints on DM over a wide range of physical and mass scales and across
redshifts.

DESI will obtain spectra of the Lyman-$\alpha$ (Ly-$\alpha$) forest out to
$z\simeq 5$ and radial velocities of $\simeq$ 10 million stars in the Milky
Way (MW) and Local Group satellites. DESI will detect four times as many
Ly-$\alpha$ QSOs than were observed in the largest previous survey, yielding
about 1 million spectra, which will put constraints on clustering of the low-
density intergalactic gas. Radial velocities from DESI in conjunction with
astrometry from Gaia (and eventually Roman’s HLS) will enable us to constrain
the global distribution of DM within the MW and in dwarf satellites, and the
mass function of dark subhalos (the DM distribution on small scales). This
paper describes how DESI will enable tighter constraints on the nature of DM
if the survey is supported by a large simulation and modeling effort.

The paradigm that the formation of cosmological structure can be described as
a solution to Boltzmann equations has been extensively tested with simulations
in many different environments — from high redshifts and using the gas between
galaxies to low redshifts and dwarf galaxies, all the way to kinematics within
our Galaxy. The majority of these simulations to date have focused on
collisionless CDM. Simulations with alternatives to CDM have recently been
gaining ground but are still in their infancy. Specifically, while there are
numerous publicly available large-box and zoom-in simulations in the
$\Lambda$CDM framework, there are no comparable publicly available WDM, SIDM,
FDM simulations. DOE should fund such a public simulation suite to enable a
more cohesive community effort to compare observations from DESI (and other
surveys) with numerical predictions. To aid such comparisons it is also
necessary to support the creation of suites of queryable mock datasets from
simulations which account for selection effects and survey volume. Machine
learning (deep learning) tools should be supported with the goal of assessing
(from blinded mock datasets) the likelihood that a given set of observational
constraints on DM is consistent with simulation predictions within a specific
DM scenario. Such an effort will greatly enhance the impact of DESI on DM
science.

##  1 Introduction

Much of what we know about dark matter (DM) on galactic scales comes from
comparing the predictions of cosmological simulations with the observed
properties of galaxies on these scales. Cosmological simulations make several
key predictions about the DM halos of galaxies that can be validated using the
phase-space coordinates of stars in the Milky Way and in Local Group dwarf
galaxies as well as with the structure of the Lyman-$\alpha$ forest at high
redshifts. These predictions differ for different types of DM in ways that
reflect both the effect of the DM particle on primordial structure formation
as well as possible interactions between the DM particles themselves and
interactions between the DM particles and baryonic matter.

The design of DESI was optimized for cosmological galaxy redshift surveys,
since the primary objective of the DESI survey is to probe the origin of
cosmic acceleration by employing the baryon acoustic oscillation technique,
with redshifts out to $z=3.7$. However, DESI will also obtain spectra of the
Ly-$\alpha$ forest out to $z\approx 5$ using a sample of QSOs $\sim 4$ times
as large as the previous comparable spectroscopic survey. During bright time
(unsuitable for cosmological targets), the project will measure the radial
velocities of $\approx 10$ million stars in the Milky Way (MW) and in Local
Group dwarfs. DESI data will enable models of the DM distribution both at high
redshifts that contribute to the Ly-$\alpha$ forest and in the Milky Way and
its vicinity.

##### The DESI Instrument and Survey:

The Dark Energy Spectroscopic Instrument (DESI) is currently the premier
multi-object spectrograph for wide-field surveys [2, 3]. DESI deploys 5000
fibers over a $3.1^{\circ}$ diameter field of view at the prime focus of the
Mayall 4m telescope at Kitt Peak National Observatory. The fibers feed ten
identical 3-arm spectrographs, each spanning 3600-9824 Å at a FWHM resolution
of about 1.8 Å. Each fiber can be positioned individually by a robotic
actuator within a radius of $1.48$ arcmin, with a small overlap between the
patrol regions of adjacent fibers. The full set of fiber positioners can be
reconfigured in 3-5 minutes. This design allows large areas of sky to be
surveyed very rapidly at a density of $\sim 600$ targets per square degree.

##### The DESI Milky Way Survey:

The DESI Milky Way Survey (MWS) will be carried out alongside its primary
5-year cosmological program. The main DESI-MWS is leveraging the already
considerable impact of the European Space Agency’s Gaia mission by targeting
stars between $16<G\leq 20.5$. Gaia has revolutionized our understanding of
the Milky Way by delivering precision astrometric parallaxes, proper motions
and photometry for more than a billion stars to $G\simeq 20-21$ mag. Gaia’s
proper motion accuracy is $\simeq 0.25$ mas/yr at the faint limit,
corresponding to a velocity accuracy of tens of km/s at characteristic halo
distances of 50 kpc. The Gaia Radial Velocity Spectrograph will also obtain
radial velocities for the brighter stars ($\sim 33$ million stars with $G<14$
in Gaia DR3). In contrast DESI-MWS will obtain radial velocities for stars
down to $r=19.5$ (which is approximately $G=19.5$ for redder stars) and with
velocity errors of $\lesssim 10\,\mathrm{km\ s}^{-1}$. DESI which has
outstanding survey speed due to its sensitive spectrographs and robotically
controlled fiber positioners, is able to achieve this radial velocity accuracy
with $\sim 300$ seconds of effective exposure time or less.

The MWS is primarily aimed at obtaining kinematics of the stellar halo and
thick disk and will also study the stellar populations of the Milky Way by
yielding stellar parameters ($T_{\rm eff}$, $\log{g}$, [Fe/H], [$\alpha/$Fe],
and elemental abundances for a few additional elements). MWS will have greater
depth, greater homogeneity, and will result in an order-of-magnitude increase
in sample size compared to existing stellar surveys with similar spectral
resolution. The main MWS sample will focus on large-scale spatial and
kinematic structure up to 100 kpc from the Sun. Radial velocities from MWS
combined with Gaia astrometry will constrain the 3D distribution of dark
matter in the Galaxy. Radial velocities of stars in thin, dynamically cold
stellar tidal streams will enable us to set constraints on the population of
dark subhalos in the Galaxy that could also be used as a powerful tool to
discriminate between different cosmological scenarios. DESI’s large field of
view makes it an efficient instrument for obtaining radial velocities of stars
in dwarf satellites in the Local Group, especially in the outer parts of these
galaxies where evidence of tidal disruption is easiest to observe. Such radial
velocities, in combination with proper motions (0.25$\mu$as yr-1) from the
Roman Space telescope will enable the tightest constraints to date on the
matter density profiles of dwarf galaxies. This, in turn, will allow us to
obtain more accurate predictions of dark matter-induced photon fluxes from
these objects. Such predictions are critical to setting more reliable and
robust constraints on the properties of the dark matter particle. Finally, the
DESI Backup Program (a program which will operate during very bright and poor
seeing conditions when the main program cannot be successfully executed) will
obtain spectra of $\simeq 8$ million stars including those closer to the MW
disk plane enabling constraints on the local DM density - important for direct
DM detection experiments.

##### The DESI Ly-$\alpha$ Survey:

The Ly-$\alpha$ forest is detected as the series of absorption lines in the
quasar spectra, caused by the Lyman-alpha transitions of neutral hydrogen in
the low-density, high-redshift intergalactic medium (IGM). It is a biased
continuous tracer of the quasi-linear matter density field, and the auto
(cross) correlation function of the forests (with quasars) have been used to
detect the Baryon Acoustic Osillations (BAO) signal[4]. Furthermore, dark
matter models can be tested using the 1D power spectrum $P_{\rm 1d}(k)$ of the
Ly-$\alpha$ forest – measuring the clustering down to small ($\sim$ Mpc)
scales along the radial direction while averaging over all lines-of-sight –
allowing constraints to be placed on the underlying dark matter distributions
from large scales to small scales.

DESI is designed to obtain about a million Ly-$\alpha$ quasars, as compared to
the $\sim 200,000$ Ly-$\alpha$ quasar data set of the Sloan Digital Sky
Survey’s (SDSS) extended Baryon Oscillation Spectroscopic Survey (BOSS)[4],
the largest existing data set available. The observations are conducted with a
310 deg-2 target density, and the combination of the larger mirror and
improved target selection allow a fainter limiting magnitude of
$\textit{r}\sim 23$ compared to previous experiments. This enables DESI to
detect quasars with a target density of 200 deg-2, of which more than 60
quasars deg-2 probe $z>2.1$ and thus contain an observable part of the
Ly-$\alpha$ forest. It also allows one to probe higher redshifts $z>4.6$,
where the impact of exotic DM models on matter distribution is less affected
by non-linearities, with greater statistical power than previously possible.

The spectrographs used by DESI have a resolution between R=2000 and R=5100, up
to a factor of 2 better than SDSS, and the survey is the first to apply the
spectro-perfectionism approach [5] to optimize the data reduction accuracy.
The increased spectral resolution enables structures of smaller physical size
to be probed with unprecedented precision: previous samples of similar or
better resolution have been constrained to $\lesssim 1000$ objects in total
[6, 7, 8]. DESI uses a Random Forest machine learning algorithm for the quasar
target selection, and a combination of a spectral fitting classifier
(Redrock[9]) and a deep convolutional neural network classifier
(QuasarNET[10]) to distinguish between quasars, galaxies and stars, measure
the objects’ redshifts and generate a quasar catalog. Using this catalog, a
software package (picca111https://github.com/igmhub/picca) is used to
statistically determine the quasar continuum, create a Ly-$\alpha$ absorption
catalog, which contains the transmission fields as the fraction of observed
flux over the mean absorbed continuum, and calculate the correlation functions
and power spectra.

##  2 Observational Signatures of Dark Matter Physics

The standard cold dark matter with dark energy model ($\Lambda$ CDM) has been
extremely successful at matching theoretical predictions from numerical
simulations with the observed distributions of galaxies on large scales.
However inconsistencies persist when the predictions from simulations are
compared to observational data on small scale. This is the so-called “small-
scale crisis” of CDM. This and some of the other key predictions of
$\Lambda$CDM that can be probed by DESI and could yield insights into the
nature of dark matter are described below.

##### The “cusp-core” problem” and the “diversity of rotation curves”:

Simulations carried out with CDM predict universal dark matter density
profiles with steeply rising central density cusps [11]. However models of the
central stellar and gas kinematics in dwarf spheroidal galaxies, low surface
brightness disks and gas-rich dwarf galaxies overwhelmingly favor cored
density profiles, although exceptions exist. This is referred to as the
“cusp/core problem” [for a review see, 12].

The CDM density profiles are almost identical to the profiles predicted by
warm dark matter [WDM, e.g. a sterile neutrino, 13], except for halo masses
below $\sim 10^{10}$ $\mathrm{M}_{\odot}$ [14], but quite different from the
cored density profiles that result from simulations with self-interacting dark
matter (SIDM) [15, 16] and fuzzy dark matter (FDM) [17, 18, 19]. The central
density cores observed in low surface brightness and dwarf galaxies are more
consistent with the central density profiles of halos produced in SIDM
simulations [15] than with the cuspy halos predicted by CDM and WDM
simulations. Also SIDM more naturally produces the observed diversity of
rotation curves [20].

The “cusp/core” problem is significantly alleviated by considering feedback
from bursty star formation [21] which is able to produce cores in subhalos of
mass $\lesssim 10^{10}$$\mathrm{M}_{\odot}$ [for an alternative view see 22].
However, classical and ultra faint dSphs ($M_{*}/M_{\mathrm{halo}}\leq
10^{-3}$) have formed so few stars over their lifetimes, that it is
challenging to invoke baryonic feedback (e.g. stellar winds and supernovae) as
the main mechanism responsible for drastically transforming their central dark
matter distributions [23]. Cosmological hydrodynamical zoom-in simulations
with different types of dark matter (CDM, WDM, SIDM and mixed DM, e.g. WDM
with self-interaction, [24]) show that the addition of baryons substantially
decreases differences between the simulations with different types of dark
matter. Small differences do exist however, e.g. baryons decrease the sizes of
cores in SIDM and WDM+SIDM subhalos compared to SIDM-only simulations, but
SIDM halos have lower central densities than CDM halos with baryons. It has
also been found [24] that baryons in the SIDM and WDM+SIDM simulations result
in a relation between circular velocity at the stellar half-mass radius
$V_{1/2}$ and half mass radius $r_{1/2}$ that is broadly consistent with
observations. One of the greatest successes of the SIDM+baryons model is
probably its ability to account for both the diversity of rotation curves
[25], where the DM profile closely follows that of the baryons [26, 27] and
produce ab initio the radial acceleration relation [28] \- the relation
between total acceleration and the acceleration due to baryons. DESI’s large
field of view, high sensitivity and excellent radial velocity precision will
enable us to obtain radial velocities for significant samples of stars in
dwarf spheroidal and ultra faint satellites of the MW enabling us to set
better constraints on their density profiles (see § 3.1.4 for details.)

##### The total mass and number of satellites of the Milky Way:

Substantially fewer satellites are found around MW-like galaxies than
predicted by dark matter-only $\Lambda$CDM simulations [the “missing
satellites problem”, e.g. 29, 30]. On the other hand, WDM, and a variety of
other alternatives to CDM, result in a suppression of small-scale power at
early times and consequently significantly supresses the number of subhalos of
mass $\leq 10^{8}$$\mathrm{M}_{\odot}$[e.g. 14, 31], and similarly for the
case of FDM [32, 33]. The subhalo mass function in SIDM simulations is largely
indistinguishable from CDM simulations and the only models that lead to a
difference relative to CDM have velocity-independent cross-section of
$\sigma/m\gtrsim 10\,$cm2g-1, which has already been ruled out by observations
of clusters of galaxies, like the Bullet Cluster [34]. However, baryonic
feedback from energetic stellar winds and supernovae can also decrease the
number of observed dwarf satellites by blowing out the baryons and hence
quenching subsequent star formation in smaller subhalos. This could result in
a population of (nearly or completely) dark subhalos whose presence could be
inferred by their gravitational impact on visible structures in the Galaxy
(such as stellar streams) or gamma ray emission if composed of WIMPs. In
external galaxies strong lensing can also be used to infer the presence of
dark subhalos [35]. $\Lambda$CDM simulations also predict that the most
massive satellites of Milky Way mass galaxies are more massive and denser on
average than the observed satellites [referred to as the “too-big-to-fail
problem”, 36].

Uncertainty in the total (virial) mass of the Milky Way greatly affects our
ability to assess whether or not the observed number of satellites and their
masses are consistent with predictions from $\Lambda$CDM. Although the mass of
the Milky Way is a fundamental quantity of interest for comparisons with
predictions from cosmological simulations, it is surprisingly poorly
constrained. The availability of distances, radial velocities and even proper
motions for huge numbers of individual stars, globular clusters and satellite
galaxies has led to numerous efforts to determine the Milky Way halo
parameters. Recent reviews give estimates of the mass of the Milky Way in the
range 0.55-2.62$\times 10^{12}\mbox{$\mathrm{M}_{\odot}$}$ [e.g. 37, 38].
Methods using tidal streams, give a similarly large range of masses. The most
recent measurements (based on Gaia and/or HST data) demonstrate that despite
the availability of high quality data, the mass of the Milky Way is still
uncertain by a factor of about two. These uncertainties affect our ability to
determine how severe the “missing satellites problem” and “too-big-to-fail
problem” really are. DESI’s radial velocity data will enable us to decrease
the uncertainty in the mass from its current value of 200% to $\lesssim 25$%
using tracers out to at least 100 kpc, which will result in improved
constraints on a range of DM models (see § 3.1.1 for details).

##### The shapes of dark matter halos:

Dark matter halos in CDM-only simulations have overwhelmingly triaxial shapes
[e.g. 39, 40, 41, 42] with flattening (short/long axis ratios $c/a\sim 0.6$
and intermediate/long axis ratios $b/a\sim 0.8$). Triaxiality correlates with
halo mass, with triaxiality decreasing with decreasing halo mass [43]. Dark
matter-only simulations with SIDM also predict triaxial dark matter halos, but
the degree of triaxiality depends on the self-interaction cross section
($\sigma/m$) [44]. WDM halos (made e.g. of sterile neutrinos) resemble CDM
halos on Milky Way mass scales but are rounder on smaller scales [14]. Baryons
tend to make the distinctions between different types of dark matter less
clear. In CDM-only simulations, baryons make halos more oblate-axisymmetric
within the inner third of the virial radius, becoming more triaxial at large
radii [45, 46, 47]. Recent SIDM simulations of Milky Way mass halos with
different interaction cross-sections show that the addition of baryons results
in halo shapes that are similar to CDM halos [48]. However the radius at which
the shape of the stellar halo deviates from the shape of the total matter
distribution is a sensitive indicator of the SIDM self-interaction cross-
section $\sigma/m$. By obtaining radial velocities for Blue Horizontal Branch
(BHB) stars, RR Lyrae, and luminous red giants out to $\sim 100$ kpc, DESI
will be able to measure the radial change in shape of both the DM halo and
stellar halo, providing new constraints on the nature of the DM particle (see
§ 3.1.1 for details.)

##### Angular momentum content of dark matter halos:

Halos acquire a net angular momentum through tidal torques during growth in
the linear regime [49, 50, 51]. This angular momentum can subsequently be
modified and redistributed by mergers. It is now common to measure of halo
angular momentum by defining a dimensionless spin parameter
$\lambda=J/(\sqrt{2}Mr_{\mathrm{v}ir}v_{\mathrm{v}ir})$, where $J$ is the
total angular momentum of a galaxy, $M$ is the total mass, and
$r_{\mathrm{v}ir}$ and $v_{\mathrm{v}ir}$ are the virial radius and velocity,
respectively [52]. In cosmological simulations it has been found that the
angular momentum of the dark matter halo is correlated with the angular
momentum of stellar disk as well as the angular momentum of the stellar halo
[53], enabling dark matter spin to be measured via stellar kinematics. The
angular momentum of merger remnants can be either in the form of streaming
motions of individual particles or figure rotation (or tumbling). It has been
long known from CDM cosmological $N$-body simulations that $\sim$90% of dark
matter halos are triaxial and can tumble with very slow pattern speeds [54,
55, 56]. Unlike several other properties of DM halos, both $\lambda$ and
pattern speed are independent of mass. The measurement of these quantities in
the MW will allow us to distinguish between particle models of dark matter and
non-gravitational and non-Newtonian theories [55]. DESI’s measurement of
radial velocities and precision spectrophotometric distances for halo stars,
in conjunction with proper motions from Gaia (and Roman’s HLS) will enable
some of the first measurements of the angular momentum content (halo spin and
pattern speed) of the MW halo (see § 3.1.1 for details.)

##### The small-scale Ly-$\alpha$ forest power spectrum

The primordial power spectrum is ideally probed by the Ly-$\alpha$ forest on
scales ranging roughly an order-of-magnitude around $k\sim 1\,h^{-1}$ Mpc. The
density fluctuations as probed by the Ly-$\alpha$ forest are still in a mildly
non-linear regime since the forest signal is generated by the IGM in
relatively empty regions of the cosmic web, and it is measured at redshifts
$z>2$. This makes the Ly-$\alpha$ forest a unique probe of primordial small-
scale density fluctuations where signatures of alternative DM models arise. A
small-scale cut-off in the early matter power spectrum is predicted in several
DM scenarios beyond CDM. In particular, it is a consequence of the velocity
dispersion of non-thermal WDM particles, e.g. sterile neutrinos produced by
resonant transitions [57, 58, 59, 60, 61, 62] or parent particle decays [63,
64, 65, 66, 67, 68]; or of the quantum pressure produced by FDM particles,
e.g. axion-like particles, with extremely small masses $m\sim
10^{-22}\text{eV}$ [69, 70, 71, 72] (for an early discussion of the
cosmological consequences of ultra-light boson particles for dark matter see
also [73, 74, 75, 76]) ; or of DM interactions described by e.g. the
Lagrangian-based Effective theory of structure formation (ETHOS) [77, 78]; or
oscillations induced by interactions in the dark sector [79, 72, 80, 81]. The
various beyond-CDM scenarios predict different shapes for the power spectrum
suppressions: for example FDM and ETHOS predict oscillations at very small
scales (for FDM see for instance [82, 83]), contrarily to WDM. However, the
Ly-$\alpha$ forest is not sensitive in general to these detailled features,
and a general $3$-parameter formula can be used to approximate the transfer
function of those DM scenarios [79], with a precision adapted to Ly-$\alpha$
observations. Besides, using high-resolution quasar spectra, different studies
have detected a cut-off in the small-scale power spectrum [84, 85, 86]
coherent with the signature of alternative DM models such as WDM or FDM,
preventing the growth of small-scale structures. However, a similar cut-off in
the flux power spectrum is produced by well established astrophysical effects,
such as the thermal Doppler broadening due to the instantaneous temperature of
the IGM, and the smoothing of the gas distribution due to the photo-heating of
the gas by the first sources of reionization and the ongoing emission from
galaxies and quasars [87, 88, 89]. To break this degeneracy one can exploit
the different redshift evolution of those effects, which requires particularly
accurate modeling of IGM physics in hydrodynamical cosmological simulations,
the only way to provide predictions that match the scales and precision
achieved by the data (see § 4.2). Due to the strong improvement in resolution
and the observation of fainter quasars (see § 1), DESI will be able to measure
the Ly-$\alpha$ power spectrum at scales as small as 5 Mpc.h-1 at redshifts as
high as $z$=5 enabling significantly tightened constraints on alternative DM
models.

##  3 Dark Matter Constraints from DESI

###  3.1 Constraints from the Milky Way Survey

Our unique vantage point within the Milky Way enables us to directly test
several different predictions of cosmological simulations described in Section
2. DESI in concert with Gaia and future proper motion surveys can create the
large, homogeneous datasets required for testing various dark matter
properties. These data will enable us to set competitive constraints on both
the global and small-scale properties of the dark matter distribution within
the Milky Way and its satellites, thereby allowing us to place the Milky Way’s
measured properties and its evolutionary history in a cosmological context.
Suites of high spatial and mass resolution (zoom-in) cosmological simulations
with baryons run with different dark matter scenarios, will be needed to
assess the likelihood of obtaining the observed global properties of the Milky
Way halo and the ensemble properties of its subhalo population with each type
of dark matter particle. The global and small-scale constraints on dark matter
that DESI will contribute to are described below.

####  3.1.1 Global halo constraints

DESI will yield an unprecedented sample of radial velocities for halo giants
(hundreds of thousands to millions, depending on the density confirmed by DESI
spectroscopy), as well as stellar standard candles such as RR Lyrae and BHB
stars, out to $\sim 100$ kpc. These radial velocities with Gaia DR5 proper
motions, and spectrophotometric distances from DESI (with $\sim 10$%
precision) will yield the 6D phase-space coordinates for field halo stars that
are needed to determine the cumulative mass of the Galaxy out to 100 kpc with
an accuracy of a few percent [90, 91] to 25% [92], significantly reducing the
current uncertainty of 200%. In addition, accurate measurement of the
velocities of halo stars in the wake produced by the LMC [93, 94] will allow
us to determine the masses of the LMC and the Milky Way and to correct for the
effects of disequilibrium caused by the LMC’s tidal field [95, 96].

These 6D phase-space coordinates for field halo stars will also enable the
measurement of the 3D shape of the dark matter halo and its variation with
radius, in order to enable comparisons with cosmological simulations with
different types of dark matter. Sophisticated stellar distribution function
modeling methods have recently been developed and validated with mock data for
this purpose [97]. The tests with mock Gaia DR2 data generated from Milky Way
like galaxies in cosmological hydrodynamical simulations show that the
flattening (short axis to long ratio $c/a$) can be measured to a precision of
$\Delta(c/a)\pm 0.15$ even without radial velocities. Adding radial velocities
decreases the uncertainty to $\Delta(c/a)\pm 0.1$. The current measurement of
$c/a=0.96$ obtained with Gaia RR Lyrae stars [98, 97] are already mildly in
tension with $\Lambda$CDM simulations which predict $c/a=0.8\pm 0.1$ [47].
DESI will make it possible, for the first time, to measure the variation in
shape of both the dark matter and stellar halos, enabling comparisons between
the MW and cosmological simulations which predict differences between SIDM and
CDM [48].

Stellar tidal streams have also been used to derive the shape of the Milky
Way’s dark matter halo. Since tidal streams deviate only slightly from the
orbits of their progenitors, they provide an important way to constrain the
orbits of objects in the global potential of the Galaxy. Previous models of
the Sagittarius stream [99, 100, 101] have yielded highly discrepant results,
in large part due to the absence of precision kinematic data (proper motions
and line of sight velocities) along the entire stream. Although Gaia provides
proper motions along the entire stream, the proper motion uncertainty is
$\gtrsim 10-20$ $\,\mathrm{km\ s}^{-1}$at $50-100$ kpc where much of the
stream lies. The best recent model of the Sagittarius stream [102] used Gaia
DR2 proper motions and limited radial velocity data to quantify the effects of
the LMC on the MW and the Sagittarius stream but did not attempt to constrain
the shape of the halo. Since the Sagittarius stream occupies a significant
portion of the sky, the large area and depth of DESI will, over the course of
the 5-year survey, significantly increase the sample of stars with radial
velocities with precision of $\lesssim 10\,\mathrm{km\ s}^{-1}$ out to 100 kpc
along very large contiguous lengths of the Sagittarius stream, enabling us to
set stronger constraints on the global variation in halo shape. Constraining
the orbits of multiple stellar streams simultaneously can serve to set
stronger constraints on the global mass distribution (the shape and density
profile of the halo) [103]. It has been shown that the availability of full 6D
phase-space coordinates for about a dozen stream can yield constraints of a
few percent on the total halo mass (within the radius occupied by the streams)
[90, 91]. DESI is the only current US survey with the field of view large
enough be able to efficiently map the radial velocities of stars along the
long streams (tens of degrees across the sky) that are most useful for global
halo modeling.

Finally, although it has been long known from CDM cosmological simulations
that $\sim$90% of dark matter halos exhibit steady figure rotation (tumbling)
[54, 55, 56], it has only recently been shown that it is possible to measure
(or at least set an upper limit) on the pattern speed of figure rotation of
the Milky Way halo using the phase-space distributions of stars in long halo
tidal streams like the Sagittarius stream [104]. The smallness of the pattern
speed predicted ($0.15h-0.8h\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}\sim
9-45^{\circ}\mathrm{Gyr}^{-1}$) implies the need for precise 6D phase-space
data over a significant portion of the stream, implying that such a
measurement is not possible in any external galaxy. While there are currently
no published studies of figure rotation of dark matter halos arising from
different types of dark matter particles, even the detection of figure
rotation will allow us to rule out [55] non-standard gravitational theories
such as MOND [105, 106, 107]. The first measurement of the spin parameter
$\lambda$ of the Milky Way halo has also been made recently from Gaia
measurements of the kinematics of Milky Way halo stars [53]. This measurement
is based on correlations between the angular momentum content of the stellar
halos and dark matter halo spin in $\Lambda$CDM hydrodynamical simulations.
DESI will greatly increase the sample of stars and the fraction of the volume
of the halo from which the spin of the stellar halo can be measured, enabling
us to improve the measurement of halo angular momentum.

####  3.1.2 Constraints on dark matter subhalos from stellar tidal streams

In addition to being powerful probes of the global dark matter distribution in
the Milky Way, stellar streams originating from globular clusters (GC) are
powerful probes for the dark matter distribution on smaller scales. GCs have
small velocity dispersions ($\lesssim 3-5\,\mathrm{km\ s}^{-1}$), consequently
tidal streams from GC progenitors are dynamically “cold” and narrow. This
makes them sensitive probes of perturbations due to gravitational interactions
with small-scale perturbers such as dark subhalos [108, 109, 110], since such
interactions produce gaps, density irregularities and off-stream structures
that are easier to observe in the otherwise thin streams. In particular, since
subhalos of mass less than $10^{8}$$\mathrm{M}_{\odot}$ are predicted to exist
in Milky Way like halos in the standard $\Lambda$CDM cosmological scenario,
but are hard to detect due to their lack of baryons, observations of such gaps
provide a crucial test of the clustering of dark matter at mass scales below
the threshold of galaxy formation. Gaps have already been observed in streams
like the Palomar 5 stream [111] and the GD1 stream [112]. Models for the
formation of the gap in the GD1 stream for instance require a compact object
of $\sim 10^{7}$$\mathrm{M}_{\odot}$ with much higher density than expected
for $\Lambda$CDM subhalos [113]. The measurement of the velocities of stars in
the vicinity of gaps can lead to stronger constraints on the orbit, mass, and
velocity of a possible perturber [110]. More precise predictions on the
locations of the perturbers based on DESI data could also lead to the
discovery of brand new targets for gamma-ray dark matter searches [114]. It
has been shown that analyzing the power spectrum of density fluctuations in a
stream can enable constraints on the mass function of dark subhalos, since
subhalos of different masses affect the density structure of streams on
different scales [115]. Recent observations of gaps in the GD1 stream have
been used to infer the total abundance of dark subhalos normalized to standard
CDM [116], and can possibly enable constraints to be set on the nature of the
dark matter particle [117].

At least half of the Milky Way’s GCs are thought to have formed in external
dwarf galaxies as massive star clusters that were subsequently accreted into
the Galaxy. Thin and dynamically cold GC streams have recently been shown to
be sensitive probes of the central dark matter distributions in the dwarf
galaxies that accreted GCs were born in. GCs accreted within cosmologically
motivated subhalos (hosting dwarf galaxies) experience a phase of “pre-
accretion tidal evolution” within their parent subhalo [118, 119, 120]. After
this pre-accretion phase, the GC and its host dwarf galaxy are accreted by the
Milky Way and the dwarf galaxy is tidally disrupted, while the GC stream
continues to evolve within the Milky Way potential. This two-phase evolution
produces tidal streams that are more highly structured than GC tidal streams
that evolve in the MW alone. Accreted GC streams comprise a thin, linear
component plus a broader “cocoon” component as well as off-stream structures
(“spurs”) and can have numerous gaps and branches. Cocoons, spurs and gaps
have now been detected with Gaia photometry and proper-motions in several GC
streams [121, 122]. It has been shown recently that quantities that measure
the “hottness” of accreted GC tidal streams (physical width, line-of-sight
velocity dispersion, tangential velocity dispersion, and angular momentum
spread) are sensitive to whether the parent satellite from which the GC was
accreted had a cuspy or cored dark matter halo [119, 123]. These simulations
have found that stellar streams formed in cored DM halos are systematically
narrower (in physical width) and have lower velocity dispersion (both line-of-
sight and tangential) than those formed in cuspy halos [119, 123]. A
comparison of simulations with existing Gaia proper motions in about 7 streams
already imply that many observed GC streams are too cold to be consistent with
formation in cuspy CDM halos of mass $\gtrsim 10^{9}$$\mathrm{M}_{\odot}$.
DESI will enable the measurement of line-of-sight velocity dispersions in many
more streams, including those too distant for making a measurement with Gaia
proper motions.

This novel probe of the central density profiles of dwarf galaxies is a
powerful complement to density profile measurements made with stellar
kinematics in dwarf galaxies (see Section 3.1.4). Gaia has discovered dozens
of Milky Way GC streams and measured the proper motions of their stars [124,
125]. Many of these lie within the DESI footprint and DESI’s large field-of-
view and high survey speed will enable the radial velocity and stellar
abundance measurements needed to obtain constraints on the density profiles of
the parent halos of accreted GCs. Although impacts on GC streams from dark
subhalos and pre-accretion tidal evolution of accreted GC streams can both
produce gaps and off-stream features like those observed in the GD1 stream,
kinematics of stars along substantial sections of tidal streams can
distinguish between these two scenarios.

Radial velocities are a crucual complement to proper motion surveys. DESI,
with its large field-of-view and survey area, is currently the most efficient
way to obtain radial velocities and determine stream membership for a
significant number of stars in stellar streams. In other words, a
spectroscopic survey like DESI is crucial for leveraging the power of GC
streams to probe the nature of dark matter.

####  3.1.3 Constraints on disk dark matter from galactoseismology

Direct dark matter detection experiments rely on precise knowledge of the dark
matter density in the solar neighborhood. Traditionally this has been
estimated by using the stellar number density and velocity distributions
(typically perpendicular to the disk) or modeling the rotation curve [for a
recent review see, 126]. Some cosmologically motivated simulations have
predicted that a dark disk, co-planar with the stellar disk, could exist in
the Milky Way, formed either from the accretion of satellites [127, 128] or
from a subdominant species of dark matter particle with much stronger
interactions than the collisionless WIMP. Such a particle could interact
dissipatively forming a rotationally supported disk [129, 130]. While searches
for such a “dark disk” have been largely unsuccessful [131], this lack of
success could in part be due to the assumption that the disk is in steady
state [132].

Spectroscopic surveys of the stellar disk from SEGUE, RAVE, LAMOST revealed
that the local stellar disk of the Milky Way exhibits bending and breathing
modes perpendicular to the midplane within a few kpc from the sun [for a
review see 133]. Recently 6D phase-space data from the Gaia satellite has led
to the discovery of even more complex kinematic substructures in the stellar
disk in form of “phase-space spirals” and ridges [134]. The presence of the
phase-space spiral in stars of all stellar ages [135, 136], and the detection
of the spiral at different radii and azimuthal angles in the disk [137]
suggests that these features are a large scale (disk-wide) response that is
possibly the consequence of the passage of the Sagittarius dwarf through the
stellar disk.

It has recently been shown that the phase-space spiral, an inherently non-
equilibrium feature, especially its morphological structure (its width,
flattening and degree of winding), can be used to set extremely tight
constraints on the potential of the Galaxy, especially the potential
associated with a thin dark disk [132, 138]. These authors used Gaia EDR3
kinematics to constrain the surface density of a thin dark disk with a scale
height of less then 50 pc to be 5$\mathrm{M}_{\odot}$pc-2 much tighter than
the constraints placed by previous steady-state models that also used Gaia
data [139], essentially ruling out the existence of a “dark disk” and
therefore more dissipative species of dark matter particles.

Although Gaia has been crucial for the discovery of the phase-space spiral and
the models arising from its discovery, Gaia’s Radial Velocity Spectrograph
will only obtain radial velocities for stars brighter than $G=16$. Gaia’s end
of mission radial velocity precision for stars at $G=16$ is expected be 10-20
km s-1 (depending on stellar type). Furthermore the Gaia database will not
include stellar abundances. While the DESI MWS main survey is primarily
targeting stars at $|b|>30^{\circ}$, the Backup Program (conducted during poor
seeing and very bright conditions) will get much closer to the disk plane
targeting stars with $16<r<18$ and $|b|>7^{\circ}$. The current performance of
DESI during the Backup Program suggests that we will be able to obtain $\sim
1-10$km s-1 precision for $r=16-18$ respectively. This will enable constraints
to be set on the dark matter density in the disk out to much larger distances,
thereby setting stronger constraints on exotic dark matter particle scenarios.

####  3.1.4 Constraints on dark matter distributions of Local Group dwarf
galaxies

Dwarf spheroidal galaxies and ultra faint dwarf galaxies are amongst the most
dark matter dominated, most metal-poor and oldest stellar systems in the
Universe representing the low end of the galaxy luminosity function [140]. The
fact that they contain very few stars may imply that these systems are some of
the only places in the Universe where one can directly probe the distribution
of dark matter on small scales. While a few dwarf spheroidal galaxies have
been found to possess cuspy DM profiles e.g., Draco [141, 142] and Sculptor
[143], the majority of galaxies modeled to date strongly favor DM
distributions with shallower, almost cored, inner density profiles. Cored
central density distributions have now been measured in low surface brightness
disks [144, 145, 146], and various dwarf satellites of the Milky Way, e.g.
Ursa Minor [147], Fornax [148, 149, 150, 151], NGC 6822 [152] and Eridanus II
[153]. However, despite the strong claims of cores in the literature, the
current measurements of central density profiles are generally based on very
small numbers (hundreds to a few thousand) of line-of-sight velocities in each
galaxy and strong assumptions about the internal velocity anisotropy. The
velocity dispersion in these galaxies is of order 2-5 $\,\mathrm{km\
s}^{-1}$and the precision of the line-of-sight velocity measurements is of the
same order resulting in fairly large uncertainties in the central mass
profiles and velocity anisotropy distributions [154, 155]. Furthermore,
although dSphs are generally modeled as spherical, they are observed to be
elongated (prolate or triaxial) and this results in significant uncertainties
in both the density and anisotropy profiles [156, 157]. While proper motions
can help to reduce the uncertainties [158, 155] the precision of proper-
motions from the Gaia satellite are currently inadequate to distinguish
between a cusp and a core in dwarf galaxies like Sculptor [159]. For many dSph
galaxies, the 2$\,\mathrm{km\ s}^{-1}$ tangential velocity precision needed to
obtain tight constraints on the central density profiles will become available
following several years of observations with the Roman Space Telescope.

In addition to the primary DESI-MWS, DESI will also conduct a secondary
program to observe all the dwarf spheroidal galaxies and ultra-faint dwarf
galaxies within DESI footprint. Most dwarf spheroidal and ultra faint dwarf
satellites of the Milky Way comfortably fit within DESI’s 3∘ field of view.
The large numbers of fibers is ideally suited to obtaining radial velocities
for a significant fraction of stars in these systems. The candidate member
stars are selected using the proper motion and parallax information from Gaia
EDR3, together with the photometry from the DESI Legacy Survey DR9 [160], and
will be observed during the bright time survey for stars at $16<r<20$ and
during the dark time survey for stars at $19<r<21$. We expect that with
multiple passes surveying the sky in the 5 year DESI survey, we will observe
most of the member stars in the ultra-faint dwarf galaxies, and a significant
number of stars (a few thousand stars per galaxy) in the dwarf spheroidals.

Radial velocities of stars in the cores of dwarf galaxies in conjunction with
$\sim 25~{}\mu$arcsec proper motions (expected from the Roman Space Telescope)
will provide precision measurements of their central density profiles,
addressing the cusp/core problem and enabling indirect dark matter searches to
place more robust and stringent constraints, not only on CDM WIMPs using
gamma-ray data [161, 162, 163, 164, 165] but also on WDM particle candidates –
such as sterile neutrinos or gravitinos – from observations in the X-ray and
gamma-ray energy bands [166, 167, 168], axion dark matter with radio data
[169], interacting and self-interacting dark matter models [170, 171] and
possibly other, even more exotic scenarios [e.g. 172].

Despite substantial efforts to obtain spectra of stars in the centers of these
systems, very few spectroscopic observations have been taken in their
outskirts, mainly due to the limited field of view of existing spectroscopic
facilities. Recent work suggests at least some of these galaxies may be
surrounded by significant low surface brightness structures. Such structures
have been predicted by simulations for certain types of dark matter even for
galaxies that are not presently losing significant mass through tidal
stripping [e.g. 173]. The primary DESI MWS will provide a wide but sparse
sampling of several members in the outskirts of these galaxies. Dedicated
follow up in secondary programs will enable the study of large-scale
metallicity gradients and tidal effects. With both proper motions and radial
velocities dynamical models may be able to set constraints on the 3D shapes of
dwarf satellite galaxies [174] enabling comparisons with simulations.

###  3.2 Dark Matter Constraints from the Ly-$\alpha$ Forest

Until now, the major statistical tool to constrain DM scenarios with
Ly-$\alpha$ has been the one-dimensional Ly-$\alpha$ transmission power
spectrum, $P_{\rm 1d}(k)$, introduced in [175] with first applications to
constrain DM properties in [176]. This statistical property of the Ly-$\alpha$
forest is obtained by computing the power spectrum of Ly-$\alpha$ forest
transmission in individual quasar lines of sight and then averaging over a
quasar sample, ignoring 3D correlations. This allows the structure of matter
fluctuations on small scales to be probed without being limited by the
sampling of objects on the sky. The smallest scales that can be accurately
probed are instead determined by the resolution of the spectrograph, the S/N
of the dataset as well as astrophysical contaminants e.g. in the form of metal
absorption systems.

Using large scale structure surveys, $P_{\rm 1d}(k)$ was measured for scales
$0.001<k<0.02\,\rm s/km$ and redshifts $2.1<z<4.7$ from SDSS [177], BOSS
[178], and eBOSS [179] with the latest measurement being based on $>45,000$
quasar spectra and reaching $\sim 1.5\%$ precision for some redshifts. These
measurements are complemented by smaller, high-resolution, high S/N [85, 180,
181, 8] $P_{\rm 1D}$ measurements from 8m-class telescopes, probing both
smaller scales (higher k) and extending the redshift range to $1.8<z<5.4$,
albeit at lower precision. Going to smaller scales and higher redshifts
significantly improves constraints since alternative DM models have more
impact at small scales.

The sensitivity of the high redshift Ly-$\alpha$ forest to DM is due to
differences in the redshift evolution of typical physical scales shaping the
signal. The DM models imprint a typical scale on the cosmic web at a fixed
redshift (e.g. at decoupling) that is frozen in comoving distance units. At
the same time, the physics of the intergalactic medium imprints two scales:
the hydrodynamical response to reionization heating that increases with time
elapsed since the heating occurred; and random motions due to finite
temperature of the gas that is frozen in velocity units, resulting in
increasing distance scale towards lower redshifts due to the Universe’s
expansion. Relative to the imprint of the baryonic physics, the scales
dominating the dark matter sector become increasingly more important towards
higher redshifts, see [182, 88] for an in-depth discussion.

Additionally, due to the strong increase in quasar density, DESI for the first
time will be able to measure small-scale Ly-$\alpha$ fluctuations both along
and across the line-of-sight (LOS) to quasars, i.e. the 3D Ly-$\alpha$ power
spectrum, $P_{\rm 3d}(k)$ [183]. $P_{\rm 3d}(k)$ measurements will be a cosmic
gold mine for Ly$\alpha$ DM studies for two reasons. First, it will add
valuable constraining power at small scales, where we look for deviations from
the $\Lambda$CDM model. Second, it will allow us to break degeneracies between
the IGM temperature that smooths absorption along the LOS, and dark matter
mass that smooths absorptions in all directions (but note that pressure
broadening of the gas distribution due to past photo-heating is also a 3D
effect and thus still degenerate with the WDM feature).

By combining high- and moderate-resolution surveys, the $P_{\rm 1d}(k)$
measurement was used to constrain WDM mass (thermal relic, sterile neutrino)
[184, 84, 85, 185, 186, 187, 188, 189]. Most recent works agree on mass bounds
of at least $m_{X}\geq 3.5\rm keV$ when assuming a thermal relic type dark
matter (with the strongest bounds $m_{X}\geq 5.3\rm keV$ obtained when
combining Ly-$\alpha$ samples [190, 188]) placing the Ly-$\alpha$ forest among
the strongest probes of DM mass (see [191] for a recent comparison with other
probes). We note that these constraints require careful modeling of IGM
astrophysics in hydrodynamical simulations, which can relax constraints to
$m_{X}\geq 1.9\rm keV$ [189] when considering e.g. a particularly late
reionization scenario.

Other models such as fuzzy dark matter predict a similar impact to WDM on the
power spectrum of matter. The measurement of the $P_{\rm 1d}(k)$ also
constrained these fuzzy dark matter models [86, 192, 193]. In particular,
using high and moderate resolution spectra, the authors in [192] excluded the
mass range of $10^{-22}<m_{a}<2.9\times 10^{-21}$ eV at 95 % CL, although a
possible self-interaction parameter of the axion particles remains
unconstrained [194, 193].

DESI will significantly improve upon eBOSS measurements. On the one hand,
there will be a tremendous gain in statistical precision, going from tens of
thousands of spectra to hundreds of thousands of spectra, and DESI will have
one additional high redshift bin. On the other hand, DESI will double the
$k$-range due to its improved spectral resolution, 2 times better than SDSS
(see §1). As a result, DESI will be much less dependent on small high-
resolution data sets and will obtain the best Ly-$\alpha$ constraints on its
own. As a benchmark scenario, we consider a thermal relic WDM model, and
compute $P_{\rm 1d}(k)$ predictions within this scenario using the
hydrodynamical simulation grid of [195]. We define the sensitivity of a
measurement as the 95% CL bound on the WDM mass $m_{X}$, obtained on average
when fitting mock CDM data to the WDM model. Assuming $P_{\rm 1d}(k)$ is
measured from SDSS data only, with uncertainties identical to those of eBOSS
[179], the sensitivity to the mass of WDM is $m_{X}>2.3$ keV [TBC]. On the
other hand, we consider a DESI $P_{\rm 1d}(k)$ measurement with a factor of 4
increased statistics, and taking into account its improved resolution which
permits measurements up to $k=0.035$ s/km on average. In order to be
conservative, we assume systematic uncertainties are similar to those
estimated in eBOSS[179], except that the improved resolution is taken into
account. We then find that the DESI-only sensitivity is $m_{X}>3.6$ keV [TBC].
This represents an improvement by a factor of 1.6 per se compared to the
eBOSS-only result. Moreover, the constraining power on IGM thermal parameters
is improved by a factor of 2.6 [TBC] on average: this will prove crucial to
improving the robustness of constraints with respect to uncertainties in the
IGM thermal modeling.

##  4 Simulations for Dark Matter Science with DESI

The vast majority of constraints currently available on the nature of dark
matter come from comparisons between observations and cosmological simulations
with $\Lambda$CDM, although there have been significant advances in the past
decade on simulations with other types of DM (for a review see CF3-solicited
white paper by Banerjee, Boddy et al.). Although there are clearly
identifiable differences between the predictions from different types of dark
matter, it has become increasingly clear over the past decade that it is much
more difficult to distinguish between different DM scenarios when baryonic
effects are considered [196]. However, the next decade is expected to witness
a substantial increase in the spatial and mass resolution of cosmological
hydrodynamical simulations that will undoubtedly lead to significant
improvements in the representation of DM halos both at high redshift and in
the local Universe.

###  4.1 Simulations for Milky Way and Local Group Science

While modeling the data from DESI and other surveys will enable us to obtain
constraints on the DM distribution in the MW and Local Group, these
observationally derived parameters of the DM distribution need to be compared
with simulations in order to set robust constraints on the nature of DM.
Understanding the full context of the Milky Way itself requires comparing to
halos with similar accretion histories and satellite populations and
understanding how such halos connect to the full distribution of predicted CDM
systems. Example simulation suites that can enable such studies in
$\Lambda$CDM are the “Milky Way” simulation suite (first introduced in [197])
that consists of 45 resimulated host halos with $z=0$ virial masses of
$10^{12.1\pm 0.03}\mbox{$\mathrm{M}_{\odot}$}$, and a complementary set of 25
simulations (“Milky Way-est” suite) drawn from the same parent box but chosen
to have MW-like merger histories (e.g. a Gaia–Enceladus [198, 199] analog
merger at $z\sim 2$ then quiescent until the merger of an LMC analog with an
infall time of less than 2 Gyr from $z=0$; Buch et al. 2022 (in prep)). This
combined simulation suite has already been used to successfully model the
satellite populations observed by DES and Pan-STARRS [200, 201, 202, 203]. In
addition to the CDM simulations, three of the halos from the Milky Way mass
suite have been resimulated using SIDM models, including velocity-dependent
cross sections, from the same initial conditions [204, 205, 206]. Comparisons
between the SIDM and CDM simulations have already provided valuable insight on
the relative subhalo population between the two models in the presence of the
LMC [206] and on the abundance, radial distribution, and density profile of
SIDM subhalos [205]. These works provide simulation-based predictions from
different microphysical dark matter models for the internal subhalo dynamics
that DESI will be able probe. In addition to resimulating the rest of the
Milky Way-est suite using the SIDM prescription, future work will include
using UniverseMachine [207] to model star-formation histories for individual
subhalos as well as comprehensive particle tracking down to the resolution
limit of the simulations ($\sim 8\times 10^{7}\mbox{$\mathrm{M}_{\odot}$}$) to
make detailed measurements of the profiles of both intact and disrupted
subhalos in different dark matter models. There is also work being done to
resimulate individual halos including realistic disk potentials. Combining
these methods will allow for detailed predictions for dynamical clustering of
disrupted substructure for different dark matter microphysics.

In addition to these on-going DM-only simulations that are ideal for
investigating the subhalo populations, we need simulations that will improve
up on existing cosmological hydrodynamical simulations which are achieving
stellar mass resolution of $10^{3}-10^{4}\mbox{$\mathrm{M}_{\odot}$}$ [e.g.,
EAGLE, Auriga, APOSTLE, FIRE-2, 208, 209, 210, 211].

To meet the objectives laid out in this document we will need a multi-pronged
effort comprising the following:

  * •

A large scale cosmological hydrodynamical simulation effort designed to make
dozens ($\sim 30-50$) of Milky Way-like disk galaxies, possibly including a
nearby satellite as massive as the LMC and a massive companion similar to M31,
that possibly had a significant accretion event [like Gaia-Enceladus, 198,
199] in the past $\sim 8-10$ Gyr.

  * •

This large suite of simulations should also be run with different types of DM
and should start with identical (or as similar as possible) initial
conditions.

  * •

The simulations should include realistic feedback prescriptions (e.g. from
stellar winds, cosmic rays, and radiative mode feedback from AGN) designed to
produce stellar halos that resemble the observed stellar halos of nearby MW
mass disks. Currently most simulated stellar halos are too metal rich, too
massive and have too many stars that formed in situ (i.e. were not accreted)
[e.g. 212, 213, 214, 215, 216].

  * •

The simulations should produce stellar disks that resemble the global
properties of Milky Way more closely. Currently stellar disks in cosmological
simulations are too thick and too hot dynamically) [e.g. 217, 218, 219]; this
is crucial for setting realistic constraints on the DM density in the disk.

  * •

Controlled cosmologically motivated simulations that evolve individual
globular clusters (with realistic stellar populations) as well as populations
of globular clusters in MW-like halos to generate thin tidal streams [e.g.
220, 221, 222]. Such simulations should include impacts from cosmologically
motivated dark subhalos populations expected in different dark matter
scenarios.

  * •

Tools to generate mock stellar populations with realistic stellar velocities
properly accounting for observational selection functions of DESI and
astrometric surveys (like Gaia, LSST, Roman-HLS) need to be applied in a
consistent manner to all simulations and produce mock catalogs that are served
on ADQL servers that can be queried in similar manner to observational
catalogs, so that blinded studies can be easily performed.

  * •

Support for machine learning/deep learning methods that can be applied to DESI
data (and other datasets) as well as mock catalogs to identify multi-
dimensional correlations and patterns in the data. Given the numerous possible
constraints that can be set on DM distribution in the MW, in the event that no
one observational probe is found to provide “smoking-gun evidence” of a
specific DM particle, tools are needed to assess the likelihood that a given
set of observational constraints is statistically consistent with predictions
from simulations within a specific DM scenario.

  * •

Simulations of the formation and tidal evolution of dwarf galaxies (both
classical dwarf spheroidal galaxies and ultra faint dwarf galaxies), and their
tidal evolution within the MW halo and Local Group.

The CF3-solicited white paper by Banerjee, Boddy et al. provides a
comprehensive overview of the types of simulation effort needed for DM
searches as a whole.

Regarding the creation of mock datasets from simulations there have been
recent advances in this direction[e.g. Aurigaia [218], Ananke [223], COSANG
[224]]. These mock datasets yield realistic numbers of stars expected in
different magnitude limited surveys as well as realistic representations of
their phase-space distributions. However, the software to readily create mock
datasets for different observational data releases (i.e. with different survey
depths, data precision etc.) are publicly unavailable. In addition, not all
these methods for creating mock data from simulations retain the kinematical
properties of the constituent star particles and hence the “stars” they
produce cannot be used as dynamical tracers of the gravitational potential.
Tests to ensure that the mock datasets are able to recover the true properties
of the DM distribution, accounting for the effects of disequilibrium and
cosmic variance are essential.

###  4.2 Simulations for Ly-$\alpha$ science

The thermodynamical state of the intergalactic gas giving rise to the
Ly-$\alpha$ forest signal is the result of the complex interplay between
large-scale structure evolution driven by the dark-matter density
fluctuations, and small-scale baryonic physics [225]. Baryons in the
intergalactic medium (IGM) trace dark matter fluctuations on Mpc scales, while
on smaller ($\lesssim 100$ kpc scales) the $T\sim 10^{4}K$ gas is pressure
supported against gravitational collapse following the classic Jeans argument.
Baryonic fluctuations are therefore suppressed relative to the pressureless
dark matter and gas is pressure-smoothed or “filtered” on small scales. At any
given redshift, the pressure smoothing scale depends not only on the
prevailing pressure and temperature at that epoch, but also on the temperature
history of the IGM [226]. The density distribution of the baryons in the IGM
is sensitive to the nature of dark matter in a similar, but importantly not
identical way. The free-streaming horizon of dark matter particles lead to a
3D smoothing at high redshift (i.e. in the initial conditions, in case of a
simulation) that keeps decreasing as non-linearities increase. Therefore, the
small-scale suppression in non-CDM dark matter models moves from very large
scales to smaller scales at progressively lower redshifts. The major task of
simulations in the context of constraining dark matter properties is thus to
allow decoupling of reionization heating impact from the nature of dark matter
on the Ly-$\alpha$ forest statistics.

At first order, we thus require hydrodynamical simulations that accurately
model the gravitational collapse of baryons and dark matter, and physical
effects contributing to the gas pressure support, most notably radiative
cooling and photoionization heating from the UV background. Studies have shown
that to produce converged Ly-$\alpha$ 1-dimensional power spectra on the range
of scales and redshifts produced by DESI, we need simulations of at least
$\approx$120 Mpc with a $\approx$30 kpc resolution [227, 228, 229]. As the the
IGM primarily probes densities around the mean density, i.e.
$0.1\lesssim\Delta\lesssim 10$ depending on the redshift [227], the above
accuracy constraints translate into a minimal requirement of $4096^{3}$ cells
for unigrid codes (e.g. Nyx [230, 231], RAMSES [232]) or $8192^{3}$ particles
for Lagrangian codes (e.g. Gadget [233] and its variants). Larger numerical
requirement on Lagrangian codes are stemming from the fact that at higher
redshifts, $z>3$, Ly-$\alpha$ forest originates mostly from underdense regions
difficult to accurately sample with Lagrangian tracer particles (Chabanier et
al. 2022, in prep.).

Recent improvements in high-performance computing facilities as well as
cosmological hydrodynamical codes allow running simulations of this size. Note
that to allow cosmological analysis, we do not only need a single simulation,
but grids of simulations exploring the cosmological and astrophysical
parameters allowing accurate predictions of the Ly-$\alpha$ forest power
spectrum. Those grids are used to train a fast interpolation scheme (emulator)
for cosmological parameter inference [234, 235]. This is needed to bridge the
gap between the need of millions of model evaluations we need to get accurate
posterior probabilities on parameters of interest and the tens of simulations
we can actually run. In effect, the emulator is used as a fast interpolation
scheme between the actually simulated grid points in the final inference
[236]. We foresee requiring tens of hydrodynamical simulations to achieve
sufficient accuracy on this interpolation [237, 229].

While the IGM is very sensitive to the timing and heat injection of the Epoch
of Reionization [238, 239], complex and poorly understood physical processes
related to galaxy formation have often be thought to only play a relatively
minor role in determining its structure [240, 241], and are being neglected in
most existing studies. However, as the statistical errors of DESI will
significantly decrease compared to previous surveys, physical processes like
star formation and stellar and Active Galactic Nuclei (AGN) feedback, which
inject energy in the intergalactic medium, should be taken into consideration
[242, 243, 244]. However, modeling all those processes from “first principles”
and jointly exploring both the cosmological and astrophysical/nuisance
parameters arising from these models will require computational capabilities
that are not feasible in the near future, even using the largest high-
performance computing systems. We therefore foresee including the impact of
galaxy formation and reionization on Ly-$\alpha$ statistics at the post-
processing stage with analytical recipes established via auxiliary
hydrodynamical simulations [243].

To summarize, for the Ly-$\alpha$ forest analyzes the following simulations
are necessary:

  * •

Grids of $\sim 10$ hydrodynamical simulations with limited astrophysics (i.e.
no feedbacks, no/simplified star formation, uniform reionization/heating) to
model the dependencies of the overall IGM on cosmological parameters including
the nature of dark matter, but providing enough resolution to resolve low-
density IGM gas physics

  * •

Additional full galaxy formation and reionization simulations as auxilliary
inputs to derive analytical corrections for astrophysical effects

##  5 Conclusion and Future Prospects

The Milky Way and Ly-$\alpha$ constraints on the properties of dark matter
described here require wide field, deep spectroscopy to create large samples
of faint targets over a wide area of sky. In the latter part of this decade,
after the current DESI survey is finished, DESI will still have the highest
survey speed of all the dedicated facilities available. A next generation
spectroscopic survey with DESI is being designed. The science goals include a
high-density galaxy survey at $z<$1 to get improved precision on the dark
energy equation of state in the cosmic acceleration epoch, an emission-line
galaxy (ELG) survey at 1$<z<$1.6 for improved baryon acoustic oscillation
(BAO) and red-shift distortion (RSD) studies, and a quasar and galaxy survey
at $z>$2 that would give direct tracers for BAO and RSD as well as the
possibility of mapping the Ly-$\alpha$ forest. The parameters of these
surveys, which are designed to constrain the properties of dark energy, are
well-matched to the dark matter investigations described in this document. A
next-generation DESI survey could include programs to map stellar velocities
in the Milky Way halo and to acquire a larger quasar sample with higher
magnitude that would improve the statistical precision of Ly-$\alpha$
cosmology constraints at DESI’s resolution. Including these programs would
allow the DESI facility to address both dark matter and dark energy science
goals within the same next-generation survey program and operations plan. DOE
funding for dark matter related science with DESI will expand the scope and
impact of this unique and powerful observational facility.

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