# Atacama Large Aperture Submillimeter Telescope (AtLAST) Science: Resolving
the Hot and Ionized Universe through the Sunyaev-Zeldovich effect

Luca Di Mascolo  Laboratoire Lagrange, Université Côte d’Azur, Observatoire de
la Côte d’Azur, CNRS, Blvd de l’Observatoire, CS 34229, 06304 Nice cedex 4,
France  Astronomy Unit, Department of Physics, University of Trieste, via
Tiepolo 11, Trieste 34131, Italy  INAF – Osservatorio Astronomico di Trieste,
via Tiepolo 11, Trieste 34131, Italy  IFPU – Institute for Fundamental Physics
of the Universe, Via Beirut 2, 34014 Trieste, Italy  Yvette Perrott  Victoria
University of Wellington, Wellington, New Zealand  Tony Mroczkowski  European
Southern Observatory (ESO), Karl-Schwarzschild-Strasse 2, Garching 85748,
Germany  Stefano Andreon  INAF – Osservatorio Astronomico di Brera, via Brera
28, 20121, Milano, Italy  Stefano Ettori  INAF – Osservatorio di Astrofisica e
Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy  INFN –
Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy  Aurora
Simionescu  SRON, Netherlands Institute for Space Research, Niels Bohrweg 4,
NL-2333 CA Leiden, the Netherlands  Leiden Observatory, Leiden University, PO
Box 9513, NL-2300 RA Leiden, the Netherlands  Kavli Institute for the Physics
and Mathematics of the Universe, The University of Tokyo, Kashiwa, Chiba
277-8583, Japan  Srinivasan Raghunathan  Center for AstroPhysical Surveys,
National Center for Supercomputing Applications, Urbana, IL 61801, USA
Joshiwa van Marrewijk  European Southern Observatory (ESO), Karl-
Schwarzschild-Strasse 2, Garching 85748, Germany  Claudia Cicone  Institute of
Theoretical Astrophysics, University of Oslo, P.O. Box 1029, Blindern, 0315
Oslo, Norway  Minju Lee  Cosmic Dawn Center (DAWN), Denmark  DTU-Space,
Technical University of Denmark, Elektrovej 327, DK2800 Kgs. Lyngby, Denmark
Dylan Nelson  Universität Heidelberg, Zentrum für Astronomie, Institut für
Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany
Laura Sommovigo  Center for Computational Astrophysics, Flatiron Institute,
162 5th Avenue, New York, NY 10010, USA  Scuola Normale Superiore, Piazza dei
Cavalieri 7, I-56126 Pisa, Italy  Mark Booth  UK Astronomy Technology Centre,
Royal Observatory Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK  Pamela
Klaassen  UK Astronomy Technology Centre, Royal Observatory Edinburgh,
Blackford Hill, Edinburgh EH9 3HJ, UK  Paola Andreani  European Southern
Observatory (ESO), Karl-Schwarzschild-Strasse 2, Garching 85748, Germany
Martin A. Cordiner  Astrochemistry Laboratory, Code 691, NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA.  Doug Johnstone  NRC Herzberg
Astronomy and Astrophysics, 5071 West Saanich Rd, Victoria, BC, V9E 2E7,
Canada  Department of Physics and Astronomy, University of Victoria, Victoria,
BC, V8P 5C2, Canada  Eelco van Kampen  European Southern Observatory (ESO),
Karl-Schwarzschild-Strasse 2, Garching 85748, Germany  Daizhong Liu  Max-
Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1 Garching,
Bayern, D-85748, Germany  Purple Mountain Observatory, Chinese Academy of
Sciences, 10 Yuanhua Road, Nanjing 210008, China  Thomas J. Maccarone
Department of Physics & Astronomy, Texas Tech University, Box 41051, Lubbock
TX, 79409-1051, USA  Thomas W. Morris  Brookhaven National Laboratory, Upton,
NY 11973, USA  Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Amélie Saintonge  Department of Physics and Astronomy, University College
London, Gower Street, London WC1E 6BT, UK  Max-Planck-Institut für
Radioastronomie (MPIfR), Auf dem Hügel 69, D-53121 Bonn, Germany  Matthew
Smith  School of Physics & Astronomy, Cardiff University, The Parade, Cardiff
CF24 3AA, UK  Alexander E. Thelen  Division of Geological and Planetary
Sciences, California Institute of Technology, Pasadena, CA 91125, USA.  Sven
Wedemeyer  Institute of Theoretical Astrophysics, University of Oslo, P.O. Box
1029, Blindern, 0315 Oslo, Norway  Rosseland Centre for Solar Physics,
University of Oslo, Postboks 1029 Blindern, N-0315 Oslo, Norway

###### Abstract

An omnipresent feature of the multi-phase “cosmic web” — the large-scale
filamentary backbone of the Universe — is that warm/hot ($\gtrsim
10^{5}\leavevmode\nobreak\ \mathrm{K}$) ionized gas pervades it. This gas
constitutes a relevant contribution to the overall universal matter budget
across multiple scales, from the several tens of Mpc-scale intergalactic
filaments, to the Mpc intracluster medium (ICM), all the way down to the
circumgalactic medium (CGM) surrounding individual galaxies, on scales from
$\sim 1$ kpc up to their respective virial radii ($\sim 100$ kpc). The study
of the hot baryonic component of cosmic matter density represents a powerful
means for constraining the intertwined evolution of galactic populations and
large-scale cosmological structures, for tracing the matter assembly in the
Universe and its thermal history. To this end, the Sunyaev-Zeldovich (SZ)
effect provides the ideal observational tool for measurements out to the
beginnings of structure formation. The SZ effect is caused by the scattering
of the photons from the cosmic microwave background off the hot electrons
embedded within cosmic structures, and provides a redshift-independent
perspective on the thermal and kinematic properties of the warm/hot gas.
Still, current and next-generation (sub)millimeter facilities have been
providing only a partial view of the SZ Universe due to any combination of:
limited angular resolution, spectral coverage, field of view, spatial dynamic
range, sensitivity, or all of the above. In this paper, we motivate the
development of a wide-field, broad-band, multi-chroic continuum instrument for
the Atacama Large Aperture Submillimeter Telescope (AtLAST) by identifying the
scientific drivers that will deepen our understanding of the complex thermal
evolution of cosmic structures. On a technical side, this will necessarily
require efficient multi-wavelength mapping of the SZ signal with an
unprecedented spatial dynamic range (from arcsecond to degree scales) and we
employ detailed theoretical forecasts to determine the key instrumental
constraints for achieving our goals.

## Keywords

galaxies: clusters: intracluster medium, intergalactic medium, galaxies:
halos, cosmic background radiation, submillimeter: general

## Plain language summary

The matter content of the Universe is organized along a large-scale
filamentary “cosmic web” of galaxies, gas, and an unseen “dark matter”
component. Most of the ordinary matter exists as a diffuse plasma, with
temperatures of $10-100$ million degrees. The largest concentrations of such
gas are found within gigantic galaxy clusters at the intersections of the
cosmic web, weighing as much as 100 trillion Suns. As such, their masses,
number and distribution across cosmic time probe the evolution and composition
of the Universe itself. Also, the thermodynamics of the warm/hot gas provides
an archeological record of those mechanisms that, over time, influence the
large-scale structures — from the energy deposition by supermassive black
holes to the collisions of massive clusters.

The best option for studying the cosmic warm/hot gas, especially at the
beginnings of their formation, is provided by the so-called Sunyaev-Zeldovich
(SZ) effect — a faint distortion of the Cosmic Microwave Background (CMB)
observable at (sub)millimeter wavelengths. Using the SZ effect to study cosmic
thermal history however requires technical advances not met by state-of-the-
art (sub)millimeter telescopes. In fact, many key questions on the co-
evolution of the warm/hot gas, the embedded galaxies, and the cosmic web
remain unanswered.

With these motivations in mind, we discuss the development of the Atacama
Large Aperture Submillimeter Telescope (AtLAST). Thanks to its unprecedented
combination of a 50-meter aperture and wide $2^{\circ}$ field of view
(4$\times$ wider than the full Moon), AtLAST will map vast sky areas and
detect extremely faint signals across multiple wavelengths. Overall, AtLAST
will push these studies beyond the legacy of the many (sub)millimeter
facilities that, during the 50 years since the theoretical foundations of the
SZ effect, have pioneered the exploration of the warm/hot Universe through the
SZ effect.

##  Introduction: clusters and the evolution of large scale structure

Clusters of galaxies, groups, and massive galaxies trace the large scale
structure of the Universe, and have therefore, since their discovery, served
as probes of cosmology (Kravtsov & Borgani 2012). For example, clusters
provided the first tentative hints of dark matter (Zwicky 1933) as well as
early evidence that we live in a universe with a low matter density
${}_{M}\sim 0.2-0.3$ (Bahcall & Cen 1992, White et al. 1993). While the large
catalogs compiled by cluster and large scale structure surveys have offered
the testbeds of, for example, the growth of structure and cosmic shear, these
tests are limited by systematics originating primarily from astrophysical
effects — shocks, feedback, non-thermal pressure, and the objects’ dynamical
and virialization states, to name a few — as well as from contamination due to
interlopers and sources within the systems that can bias our measurements and
any resulting cosmologically relevant observable (e.g., the cluster mass, a
key proxy of structure evolution).

The same sources that can contaminate measurements, primarily radio-loud
active galactic nuclei (AGN) and star forming galaxies, or cause departures
from thermal equilibrium, such as shocks, are also the main drivers of the
physical and thermodynamic evolution of the intracluster medium (ICM). The
ICM, in turn, is the large scale environment within which a large fraction of
galaxies reside, so the feedback and interactions between the two are
important to both cosmology as well as cluster and galaxy evolution. The study
of galaxy clusters is thus complex and multifaceted, but the reward to
understanding the nature of clusters and large scale structure is that it
allows us to test the properties of the ubiquitous dark matter, and to peer
into the dark universe itself.

Figure 1: Expanded diagram highlighting some of the aspects of galaxy clusters
and large-scale structures that will be studied through the Sunyaev-Zeldovich
(SZ) effect using AtLAST. The SZ effect is caused by the interaction of
photons from the cosmic microwave background (CMB) with reservoirs of
energetic electrons within cosmic large-scale structures. Thanks to AtLAST’s
unparalleled capabilities, it will be possible to fully exploit multiple
aspects of the SZ effect to characterize the impact of active galactic nuclei
(AGN) on the circumgalactic medium (CGM), and the multi-scale properties of
the intracluster medium (ICM) of the large-scale filaments of intergalactic
medium (IGM). The figure is an adaptation of the SZ schematic in Mroczkowski
et al. (2019), which was based on that from L. van Speybroeck as adapted by J.
E. Carlstrom.

Some of the dominant astrophysical processes occurring in clusters are shown
in the cartoon depiction in Fig. 1. Studying in greater detail the multi-scale
physical processes taking place within the most massive structures in the
Universe will ultimately allow us to build a more complete understanding of
the thermal history of our universe, how structure grew and evolved, or
fundamentally how the Universe came to be the way it is.

Here we seek to motivate deep, multi-band or multi-chroic high resolution and
wide field observations with the Atacama Large Aperture Submillimeter
Telescope111http://atlast-telescope.org/ (AtLAST; Klaassen et al. 2020,
Ramasawmy et al. 2022, Mroczkowski et al. 2023, 2024) that will address
questions of cluster astrophysics as well as the contamination that could
potentially plague cluster cosmology done at arcminute resolutions. At the
same time, the observations discussed here are not simply to aid cosmological
studies, but can probe interesting astrophysics and solve important questions
about astrophysics in their own right — with the unique potential of providing
a link between galaxy evolution, large-scale structure, and cosmological
studies. Our primary tool here is the Sunyaev-Zeldovich effect, described
below.

##  The multi-faceted Sunyaev-Zeldovich effect

The thermal Sunyaev-Zeldovich (SZ) effect (Sunyaev & Zeldovich 1970, 1972) —
an up-scattering to higher energies of photons from the cosmic microwave
background (CMB) by hot electrons — was proposed theoretically as an
alternative way to probe the pressure along the line of sight (l.o.s.) of the
hot gas in galaxy clusters. A schematic picture for the scattering due to the
thermal SZ effect, which scales as the Compton $y$ parameter ($y\propto
P_{\mathrm{e}}dl$ for electron pressure $P_{\mathrm{e}}$ and l.o.s. $l$), is
shown in Figure 1. Shortly after the theoretical foundations of the thermal SZ
effect, the kinetic SZ effect ($\propto n_{\mathrm{e}}v_{z}dl$ for electron
density $n_{\mathrm{e}}$ l.o.s. peculiar velocity $v_{z}$; see Sunyaev &
Zeldovich 1980) was proposed as a way to measure gas momentum with respect to
the CMB, our ultimate and most universal reference frame. The following decade
saw developments in the theory regarding relativistic corrections to the
thermal SZ and kinetic SZ effects as well as anticipating more exotic SZ
effects from non-thermal and ultra-relativistic electron populations (e.g.,
Itoh et al. 1998, Nozawa et al. 1998, Enßlin & Kaiser 2000). Observations of
the SZ effects, however, took longer to come to the fore, beginning with
pioneering measurements such as Birkinshaw et al. (1984) and culminating more
recently in several thousand measurements or detections from both low-
resolution (1-10′) SZ surveys (e.g., Swetz et al. 2011, Carlstrom et al. 2011)
and dedicated observations, often at higher (subarcminute) resolution (e.g.,
Mason et al. 2010, Plagge et al. 2013, Adam et al. 2014, Kitayama et al.
2016). For more comprehensive reviews of the various aspects of the SZ effect,
see, e.g., Birkinshaw (1999), Carlstrom et al. (2002), Kitayama (2014), and
Mroczkowski et al. (2019).

Figure 2: Various SZ spectra versus transmission in the top quartile (lighter
gray) and median (darker gray) atmospheric transmission conditions available
at the Chajnantor Plateau ($\approx 5000$ meters above sea level). The left
y-axis corresponds to transmission, and the right y-axis is appropriate for
the SZ intensity for a cluster with $y=10^{-4}$. The kinetic SZ values assume
a line of sight velocity component $v_{z}=-1000$ km/s (i.e. toward the
observer, implying a net blueshift in the CMB toward the cluster) and an
electron opacity $\tau=0.01$. Here we used SZpack to solve for the SZ spectral
distortions (Chluba et al. 2012), and the am code for atmospheric transmission
(Paine 2019). The optimal continuum bands of the proposed AtLAST SZ
observations, reported in Table 1, are shown as background shaded regions.

The proposed millimeter/submillimeter facility, AtLAST, presents novel, unique
capabilities that will revolutionize both deep targeted observations aiming
for detailed astrophysical studies, as well as wide-field surveys aiming to
push SZ observations to much lower mass limits and higher redshifts. Since the
epoch of reionization, the majority of baryons have been making their way up
to high enough temperatures ($>10^{5}\leavevmode\nobreak\ \mathrm{K}$) that
their emission is nearly completely undetectable at optical wavelengths
(visible and near-IR band), where the majority of telescopes operate. Such a
hot phase is an omnipresent feature of the multi-phase cosmic web,
representing a relevant contribution to the volume-filling matter budget on
multiple scales — from Mpc-scale filaments of intergalactic medium, to the
intracluster medium (ICM), and down to the circumgalactic medium (CGM)
surrounding individual galaxies up to their virial radius (up to few 100s of
kpc). Through the SZ effect, the millimeter/submillimeter wavelength regime
offers a view of this important component of galaxies and their surrounding
environments (clusters, groups, filaments) — components that are largely
invisible to all but X-ray and SZ instruments.

##  Proposed science goals

Here we provide a summary of the main applications in the context of SZ
studies enabled by AtLAST that will allow us to develop a more profound and
complete understanding of the thermal history of the Universe, ultimately
transforming our understanding of the numerous processes involved in structure
formation, evolution, feedback, and the quenching of star formation in
overdense environments. We refer to Lee et al. (in prep.) and van Kampen et
al. (in prep.) for companion AtLAST case studies focused on emission line
probes of the cold circumgalactic medium (CGM) of galaxies and on providing a
comprehensive survey of high-$z$ galaxies and protoclusters, respectively.
Common to all the specific science cases discussed below is the need for a
wide field, high angular resolution facility able to optimally probe the full
SZ spectrum (Figure 2). We refer to Sec. 4 for a more extended discussion of
the technical requirements for the proposed science goals.

###  Thermodynamic properties of the ICM: radial profiles and small-scale
perturbations

The morphological and thermodynamic properties of the ICM represent key
records of the many physical processes shaping the evolution of galaxy
clusters and groups. Non-gravitational processes — e.g., cooling, AGN
feedback, different dynamical states and accretion modes (Battaglia et al.
2012, Ghirardini et al. 2019) — are expected to leave their imprint on the
pressure distribution of the ICM in the form of deviations from the radial
models derived under universal and self-similar assumptions for structure
formation (see, e.g., Nagai et al. 2007, Arnaud et al. 2010, Sayers et al.
2023). On cluster scales, shock fronts induced by cluster mergers as well as
cosmological accretion deposit their kinetic energy into the ICM, contributing
to its overall thermalization (Markevitch & Vikhlinin 2007, Ha et al. 2018).
On smaller scales, turbulent motion (Schuecker et al. 2004, Khatri & Gaspari
2016, Romero et al. 2023) can induce significant non-thermal contributions to
the ICM pressure support, in turn hampering the validity of the hydrostatic
equilibrium assumption. We thus need robust constraints on the level of
turbulence affecting the energy budget of the ICM along with an independent
census of the “hydrostatic mass bias” (e.g., Biffi et al. 2016) via a
combination of fluctuations and resolved hydrostatic mass information. This
will be crucial for inferring corrections to the hydrostatic mass due to the
non-thermalized gas (see, e.g., Angelinelli et al. 2020, Ettori & Eckert 2022)
and therefore strengthening the role of thermodynamic quantities for
cosmological purposes (Pratt et al. 2019).

Figure 3: Observed Compton $y$ profiles when considering different fiducial
models for the ICM pressure distribution (top panels; Arnaud et al. 2010,
McDonald et al. 2014, Le Brun et al. 2015, Gupta et al. 2017, Sayers et al.
2023, Melin & Pratt 2023) and respective variations (bottom panels) with
respect to the universal pressure profile from Arnaud et al. (2010), commonly
adopted as reference model for the inference of cluster masses. In this plot,
we consider a cluster with fixed mass
($M_{\mathrm{500}}=10^{14}\leavevmode\nobreak\ \mathrm{M_{\odot}}$) for an
arbitrary set of redshifts ($z=\\{0.05,1.50,3.00\\}$). The different profiles
are computed on heterogeneous samples in terms of mass and redshift ranges,
and thus encode different biases associated to the intrinsic scatter of
pressure profiles, deviations from self-similar evolution and hydrostatic
equilibrium. Thanks to AtLAST’s sensitivity to Compton $y$ levels $\lesssim
10^{-7}$, it will be possible to characterize such effects, while providing a
model for the evolution of ICM pressure across cosmic history. As reference,
we report as dashed vertical lines the virial radius of the model clusters and
the instantaneous field of view expected for AtLAST (see Sec. 4). We further
denote as dotted horizontal lines the largest and smallest angular resolution
i achievable with AtLAST, respectively obtained in the proposed Band 2 and
Band 10 (see Sec. 4.1 and Table 1 below).

The thermal SZ effect provides a direct proxy for the (thermal) pressure due
to the free electrons in the ICM and, as such, the optimal tool for gaining a
direct calorimetric view of the gas thermal properties. In fact, observational
models for a statistically relevant sample of clusters are currently limited
to the indirect determination of resolved pressure models for clusters up to
$z\lesssim 1$ (Arnaud et al. 2010, McDonald et al. 2014, Sayers et al. 2023).
Direct constraints of the properties of the ICM within protoclusters and
clusters early in their formation have been obtained for only a handful of
extreme systems ($z>1$; Tozzi et al. 2015, Brodwin et al. 2016, Gobat et al.
2019, Di Mascolo et al. 2023, Andreon et al. 2021, 2023, van Marrewijk et al.
2023) or limited samples (e.g., Ghirardini et al. 2021b). Despite the
significant time investment with the Atacama Large Millimeter/Submillimeter
Array (ALMA; Wootten & Thompson 2009) or the 100-meter Green Bank Telescope
(GBT; White et al. 2022), these observations only allow one to perform a
characterization of the physical and thermodynamic state of these early
systems for a few select systems. Still, these have generally required the
combination with ancillary X-ray observations, due to observational
limitations including poor signal to noise or the data being limited to fewer
than 5 bands.

In order to gain a radially resolved view of pressure profiles and of their
small-scale perturbations for a large variety of clusters (in terms of
dynamical state, mass, and redshift; see Fig. 3), it is key to have
simultaneous access to enhanced sensitivity, high angular resolution, and wide
spectral coverage across the millimeter/submillimeter spectrum. These
observations are important, as from hydrodynamical simulations the pressure
distribution of high-$z$ galaxy clusters are predicted to diverge from the
universal pressure models (Battaglia et al. 2012, Gupta et al. 2017), leading
to a systematic offset between the mass-to-SZ observable scaling relation for
high-$z$ haloes with respect to local ones (Yu et al. 2015). Constraining such
deviations is crucial as they carry fundamental information on the complex
interplay between all those multi-scale processes — e.g., merger and accretion
events, AGN and stellar feedback, turbulent motion —- at epochs ($z>1$) when
their impact from galactic to cluster scales are expected to be the strongest.
At the same time, tracing the pressure profiles out to the cluster outskirts
will be key to pinpoint and characterize virial and accretion shocks (Hurier
et al. 2019, Anbajagane et al. 2022, 2024), whose existence is a fundamental
prediction of the current paradigm of large-scale structure formation (e.g.,
Ryu et al. 2003, Zhang et al. 2021). In particular, the location and
properties of their SZ features can be exploited to study the mass assembly of
galaxy clusters and to place direct constraints on their mass accretion rate
(a quantity otherwise difficult to infer observationally; see, e.g., Molnar et
al. 2009, Lau et al. 2015, Baxter et al. 2021, Towler et al. 2023, Baxter et
al. 2024). And finally, pressure perturbations due to the turbulent motion
within the ICM have been measured to result in fluctuations of the Compton $y$
signal with fractional amplitude $\lesssim 10^{-1}$ compared to the underlying
bulk SZ signal (Khatri & Gaspari 2016, Romero et al. 2023). The enhanced
sensitivity and calibration stability that will be achieved by AtLAST will
allow it to easily probe this level of fluctuations, providing important
albeit indirect information on the level of non-thermal pressure support in
the ICM. More in general, it is only with the unique technical prospects
offered by AtLAST that we will be able to probe to thermal SZ signal down to
the levels Compton $y\approx 10^{-7}$ (Sec. 4.2) required to probe the full
extent of the ICM pressure distribution (Fig. 3), unparalleled by any of the
current or forthcoming submillimeter facilities.

###  Measuring the ICM temperature via relativistic SZ effect

The classical formulation of the thermal SZ effect relies on a non-
relativistic assumption for the velocity distribution of the electron
populations responsible for the SZ signal. These can however manifest
velocities of the order of a few tens percent of the speed of light.
Accounting for any associated special-relativistic effects introduce a
temperature-dependent distortion of the SZ spectral model (Fig. 2). The
resulting relativistic SZ effect thus offers a valuable (yet largely
unexplored) opportunity to directly measure the temperature of ICM electrons.
This represents a key ingredient for enhancing our physical models of galaxy
clusters and improving their utility as cosmological probes via more accurate
tuning of mass calibrations and scaling relations (e.g., Lee et al. 2020,
Remazeilles & Chluba 2020). At the same time, having simultaneous access to
the full ICM thermodynamics (via temperature $T_{\mathrm{e}}$, as well as
pressure $P_{\mathrm{e}}$ and density $n_{\mathrm{e}}$ measurements via the
combination of the relativistic and purely thermal SZ effects) offers the key
chance of building a temporal census of the ICM entropy distribution ($\propto
T_{\mathrm{e}}\leavevmode\nobreak\ n^{-2/3}_{\mathrm{e}}$, or $\propto
T^{5/3}_{\mathrm{e}}P^{-2/3}_{\mathrm{e}}$ when considering thermodynamic
quantities directly probed by the SZ effect; Voit 2005). The many processes
affecting cluster evolution — e.g., AGN and stellar feedback, injection of
kinetic energy due to merger activity — are observed to modify the entropy
profiles throughout the cluster volumes (e.g., Pratt et al. 2010, Walker et
al. 2012, Ghirardini et al. 2017), compared to a baseline model that includes
only the non-radiative sedimentation of low-entropy gas driven by gravity
(Tozzi & Norman 2001, Voit et al. 2005). As such, the spatially resolved study
of the ICM entropy distribution provides a fundamental proxy of the thermal
evolution of cosmic structures as well as the specific dynamical state of
galaxy clusters.

Currently, estimates of the relativistic corrections to the thermal SZ effect
are limited to a few pioneering studies targeting individual systems (Hansen
et al. 2002, Prokhorov & Colafrancesco 2012) or focusing on stacking analyses
(Hurier 2016, Erler et al. 2018). Still, even in the case of individual
clusters with extremely rich observational spectral coverage (see, e.g.,
Zemcov et al. 2012 and Butler et al. 2022, focusing on the well-known cluster
RX J1347.5-1154), SZ-based inferences of the ICM temperature have commonly
resulted in constraints with limited significance. Higher angular resolutions,
such as those offered by AtLAST, will be an asset for constraining SZ
temperatures. First, the higher angular resolution allows spatially-distinct
foregrounds such as radio sources, dusty galaxies and the Galactic dust
foreground to be accurately modelled and removed. Second, the extraction of
resolved pressure and temperature profiles provides the unique opportunity of
performing the physical modeling of the ICM relying solely on the SZ effect.
This represents a key advantage. Although electron temperatures can be
measured using X-ray data, these are roughly density-square-weighted estimates
(e.g., Mazzotta et al. 2004) and therefore subject to biases due to clumping
(e.g., Simionescu et al. 2011). Further, observations can become prohibitive
at large cluster radii, due to the low X-ray emissivity, and at high redshift,
due to cosmological dimming222Churazov et al. (2015) showed that the self-
similar evolution of galaxy clusters would introduce a near independence of
redshift of the X-ray luminosity at fixed cluster mass — when this is defined
as the mass enclosed in the radius within which the average matter density
equals some fiducial cosmic overdensity value (e.g. $500\times{}_{\rm crit}$).
Nevertheless, we note that these considerations are valid only under the
assumption that the local mass-observable scaling relations are applicable at
high redshift. At the same time, both the resolved SZ signal and the
respective cluster-integrated flux would still be $(1+z)^{3/2}$ larger than
the X-ray emission from the same system at a given redshift $z$.. On the other
hand, since the SZ effect is characterized by a surface brightness that is
inherently independent of redshift, ICM temperature constraints can in
principle be derived without specific limits on the distance of the target
systems. And lastly, the temperatures inferred using data on the same clusters
but taken using different X-ray observatories may suffer large systematic
variations due to inherent calibration differences (Schellenberger et al.
2015, Migkas et al. 2024). In contrast, the SZ temperature estimate is
pressure-weighted and is therefore predicted to be less biased by emission
while being easier to constrain at large cluster radius due to the linear
(instead of squared) dependence on density. And even in the case of low-mass
(i.e., low-SZ surface brightness; see also Sec. 3.4) clusters for which it
will not be possible to extract resolved SZ-based temperature information, the
availability of deep, high angular resolution SZ observations for a large
sample of systems will still allow for matching resolution with X-ray
observations and to extract resolved full thermodynamic properties of the ICM
(see Sec. 3.1 and Sec. 5.2.4).

An exploratory study of AtLAST’s expected capabilities to measure temperature
via the relativistic SZ effect is presented in Sec. 4.3. We refer to this for
more details on the impact of the specific spectral setup on the
reconstruction of the relativistic SZ effect and on the technical requirements
for extending such measurements over broad ranges of cluster masses and
redshifts.

###  Kinematic perspective on large scale structures

The kinetic component of the SZ effect represents a valuable tool for
revealing the peculiar motion of cosmic structures. Nevertheless, its
properties – namely its shape, the fact that the kinetic SZ signal is
generally weaker than the thermal SZ effect (Fig. 2), and that it traces the
integrated line of sight momentum – make the kinetic SZ effect somewhat
elusive to measure and interpret. Further, the kinetic SZ spectral distortion
is consistent with a Doppler shift of the CMB photons, making it spectrally
indistinguishable from small-scale primordial CMB anisotropies.

Figure 4: Thermal (left) and kinetic (center) SZ effects, and X-ray luminosity
(right) from a simulated massive cluster undergoing a major merger
($M_{200}\simeq 1.5\times 10^{15}\leavevmode\nobreak\ \mathrm{M_{\odot}}$,
$z=0$) extracted from the TNG-Cluster simulation (Nelson et al. 2023). The
contours in both panels trace a Compton $y$ level of $2\times 10^{-7}$,
roughly corresponding to the reference SZ depth for a deep AtLAST survey (Sec.
4.2). As a reference, we mark with a circle the virial radius of the galaxy
cluster. This implies that AtLAST will be able to efficiently trace the SZ
signal out to the low-density outskirts of clusters.

Past targetted kinetic SZ studies (e.g., Mroczkowski et al. 2012, Sayers et
al. 2013, Adam et al. 2017, Sayers et al. 2019, Silich et al. 2023) have
already reported direct measurements of the kinetic SZ signal due to the
large-scale gas flows associated with merger events. All of these works
focused on individual, relatively extreme clusters (either in terms of overall
mass, dynamical state, or orientation of the merger direction with respect to
the line-of-sight). The broad spectral coverage and the expected sensitivity
of AtLAST, in combination with its capability of probing a high dynamic range
of angular scales, will instead allow for systematically including the kinetic
SZ information in the reconstruction of the thermodynamic characterization of
large statistical samples of galaxy clusters and groups.

Statistical measurements of the kinetic SZ effect in disturbed and merging
systems represent a crucial ingredient for cosmological studies via direct
measurements of the amplitude and the growth rate of cosmological density
perturbations (e.g., Bhattacharya & Kosowsky 2007, Soergel et al. 2018). They
can also be used to distinguish CDM from alternative cosmologies with modified
gravitational forces (Kosowsky & Bhattacharya 2009, Mueller et al. 2015,
Bianchini & Silvestri 2016). Further, correlating the velocity structure with
information from facilities at other wavelengths on the baryonic and dark
matter content of merging systems will represent a preferential probe of the
collisional nature of dark matter (Silich et al. 2023). In the case of
relatively relaxed systems (i.e., with velocity fields not manifesting complex
morphologies), the joint analysis of the thermal and kinematic SZ effects
would naturally complement the inference of the ICM pressure and temperature
distributions with information on the bulk peculiar velocity of galaxy
clusters and tighter constraints on the ICM density (Mroczkowski et al. 2019).

The detailed spatial mapping of the kinetic SZ effect could also be used to
characterize turbulent motions and to identify their driving dissipation
scales which are relevant for feedback mechanisms. This can be done, in
particular, by computing the velocity structure function (VSF), defined as the
average absolute value of the line of sight velocity differences as a function
of projected scale separation. The VSF is an effective way of characterizing
turbulent motions and identifying their driving and dissipation scales (see,
e.g., Li et al. 2020, Ganguly et al. 2023, Gatuzz et al. 2023, Ayromlou et al.
2023). Determining the driving scale of turbulence would constrain the
relative importance of gas motions driven by AGN feedback on small scales and
mergers on large scales, while the dissipation scale is sensitive to the
microphysics of the ICM, such as its effective viscosity (Zhuravleva et al.
2019). In general, constraints on the small-scale properties of the velocity
field associated with turbulent motion (Sunyaev et al. 2003, Nagai et al.
2003), coherent rotation of gas within their host dark matter haloes (Cooray &
Chen 2002, Baldi et al. 2018, Baxter et al. 2019, Altamura et al. 2023,
Bartalesi et al. 2024), or merger-induced perturbations (Biffi et al. 2022)
can complement the reconstruction of ICM thermodynamic fluctuations (Khatri &
Gaspari 2016, Romero et al. 2023) and the potential mitigation of biases due
to non-thermal pressure support (e.g., Shi et al. 2016, Angelinelli et al.
2020, Ansarifard et al. 2020, Ettori & Eckert 2022) discussed in Sec. 3.1.
Perturbations in the kinetic SZ distribution will result in small-scale
kinetic SZ fluctuations more than an order of magnitude smaller than the
corresponding thermal SZ component (Sunyaev et al. 2003, Mroczkowski et al.
2019, Biffi et al. 2022) even for massive systems. The clear requirement of
extremely demanding observations (along with the difficulty in spectrally
disentangling the kinetic SZ effect from the underlying CMB signal;
Mroczkowski et al. 2019) have so far limited the possibility of directly
measuring any small-scale kinetic SZ feature. However, AtLAST will be able to
efficiently measure percent-level deviations from the dominant thermal SZ
effect (see, e.g., Sec. 4.3 below for a discussion in the context of
relativistic SZ corrections) and to swiftly survey wide sky areas at $\sim
1.5-35$ arcsec resolution, thus opening a novel observational window on ICM
velocity substructures.

Finally, AtLAST’s simultaneous sensitivity to both small and large spatial
scales facilitates studies of the distortions in the CMB across a broad range
of spatial scales ($300\lesssim\ell\lesssim 20000$). Existing and forthcoming
CMB experiments cannot probe beyond $\ell\sim 4000$, whose power spectrum is
dominated by both regular CMB anisotropies and CMB lensing effects. However,
at 220 GHz around $\ell\approx 7000$, the kinetic SZ effect becomes the
dominant contributor to the angular power spectra (Smith & Ferraro 2016), thus
enabling studies on the kinetic SZ imprint from the epoch of reionization,
originating from relativistic electrons within expanding ionizing bubbles
(Ferraro & Smith 2018) – the so-called “patchy kinetic SZ” signal.

###  Overcoming cluster selection biases

It is becoming generally appreciated that X-ray selected clusters offer a
biased view of the cluster population (Pacaud et al. 2007, Stanek et al. 2006,
Eckert et al. 2011, Planck Collaboration 2011, 2012, Maughan et al. 2012,
Andreon et al. 2017, 2019). This is because, in a given sample, bright
clusters are over-represented (see, e.g., Mantz et al. 2010 for discussion of
Malmquist and Eddington biases), whereas those systems fainter-than-average
for their mass are underrepresented, if not missing altogether. This bias is
difficult to correct because the correction depends on assumptions about the
unseen population (Vikhlinin et al. 2009, Andreon et al. 2017). On the other
hand, SZ-selected cluster samples are generally thought to offer a less biased
view and indeed show a larger variety (e.g., in gas content) than X-ray
selected samples (e.g., Planck Collaboration 2011, 2012). Comparisons of the
X-ray properties of SZ-selected systems (see, e.g., CHEX-MATE Collaboration
2021) have highlighted the fact that ICM-based selection biases can depend on
the specific morphology (Campitiello et al. 2022) or the presence of a
dynamically relaxed cool core (the so-called “cool-core bias”; Rossetti et al.
2017).

However, the selection of clusters via their galaxies (i.e., based on the
identification of cluster members) or via gravitational lensing (i.e., based
on the effect of the cluster potential on the images of background sources)
can provide an observational perspective that is potentially unbiased with
respect to the thermodynamic state of the ICM. Although methods based on
galaxies can still suffer from significant biases due to contamination and
projection effects (e.g., Donahue et al. 2002, Willis et al. 2021), the fact
that they are not dependent on the ICM-specific biases have granted the
possibility of unveiling the existence of a variety of clusters at a given
mass larger than X-ray or current SZ-based approaches. In particular, the low-
surface brightness end of the unveiled new population of clusters is changing
our view of galaxy clusters. These are found to introduce significant scatter
in many ICM-based mass-observable scaling relations (Andreon et al. 2022), at
very the heart of our understanding of cluster physics and broadly used in the
context of cluster cosmology. Characterizing such a population of low surface
brightness clusters will necessarily require a major leap in the SZ
sensitivity with respect to state-of-the-art facilities.

The possibility of performing deep, high angular resolution mapping over wide
sky areas offered by AtLAST will allow observers to efficiently detect those
clusters that are presently underrepresented in, or entirely missing from,
catalogs due to an SZ or X-ray signal inherently fainter than expected from
their mass. Indeed, clusters with low X-ray surface brightness tend to have
low central values of Compton $y$, of the order of few $10^{-6}$ (based on
Andreon et al. 2022), at the very limit of long pointed observations with
current single-dish telescopes, when not beyond their effective detection
capabilities. In combination with X-ray, strong and weak-lensing data, this
will allow for a thorough characterization of their physical and thermodynamic
state, and for discriminating between any variation in the inherent properties
of the intracluster gas and observational biases induced by any astrophysical
processes more or less associated with the specific evolution and physics of
the target clusters — e.g., energetic AGN feedback, recent merger events, low
gas fraction, enhanced clustering of millimeter-bright galaxies.

Figure 5: Mass vs. redshift detection forecast for AtLAST assuming different
survey strategies (covering $1000\leavevmode\nobreak\ \mathrm{deg^{2}}$,
$4000\leavevmode\nobreak\ \mathrm{deg^{2}}$, and $20000\leavevmode\nobreak\
\mathrm{deg^{2}}$, respectively, for a fixed survey time of 5 years) in
comparison with next-generation wide-field millimeter surveys (Abazajian et
al. 2016, Sehgal et al. 2019) and the eROSITA all-sky (X-ray) survey (Bulbul
et al. 2024). Poisson realizations of the thermal SZ confusion were simulated
in the AtLAST survey case, while the other resolution surveys used Gaussian
realizations appropriate in the case where the lower resolution and
sensitivity limit the ability to surpass the thermal SZ confusion limit. We
refer to Raghunathan (2022) for a more general discussion of the different
treatments of the SZ confusion noise. For comparison, we report as green
points the clusters from the available SZ survey samples (Planck Collaboration
2016, Bleem et al. 2020, Hilton et al. 2021, Bleem et al. 2024), as well as
relevant high-$z$ clusters from the literature: XDCP J0044-2033 (Tozzi et al.
2015), IDCS J1426.5+3508 (Brodwin et al. 2016), JKCS 041 (Andreon et al.
2023), XLSSC 122 (Mantz et al. 2020, van Marrewijk et al. 2023), CL J1449+0856
(Gobat et al. 2019), and the Spiderweb protocluster (Di Mascolo et al. 2023).
This figure is adapted from Raghunathan et al. (2022).

###  Identification and thermodynamic characterization of high-$z$ clusters
and protocluster

Next generation SZ facilities like Simons Observatory (SO; Simons Observatory
Collaboration 2019) and CMB-S4 (Abazajian et al. 2016) will extend our
observational window into the high-$z$ and low-mass realm (see, e.g.,
Raghunathan et al. 2022 and Fig. 5) of galaxy clusters and protoclusters.
Tracing the earliest phases of their evolution will be crucial for
constraining the physical origin of the thermal properties of the large-scale
structures observed in the nearby Universe.

Nevertheless, current forecasts estimate that next-generation wide-field
surveys (Gardner et al. 2024) will detect less than $20\%$ of the most massive
(proto)clusters ($M_{200}\lesssim 10^{14}\leavevmode\nobreak\
\mathrm{M_{\odot}}$, $z>2$). This is mostly a consequence of the competing
impact of inherently low SZ amplitudes (due to low mass, disturbed state, and
severe deviations from full gas thermalization and virialization; Sereno et
al. 2021, Bennett & Sijacki 2022, Li et al. 2023), the low angular resolution
of the facilities, and of the increasing contamination level due to, e.g.,
enhanced star formation and AGN activity, or possibly due to massive CGM gas
and dust reservoirs at high redshift (Lee et al. in prep.). And as already
broadly discussed in Sec. 3.1, extreme limitations are also faced in the case
of high angular resolution measurements. Clearly, having access to deep, high
angular resolution and multi-band SZ observations will allow observers to
simultaneously tackle all such issues, making AtLAST the optimal telescope
that will definitively shape our perspective on high-$z$ (proto)clusters.

Figure 6: Composite Hubble Space Telescope (HST) image based on ACS/WFC F475W
and F814W data of the Spiderweb protocluster field. Overlaid (orange) is the
thermal SZ signal from the ICM assembling within the protocluster complex as
observed by ALMA over a total of more than $12\leavevmode\nobreak\ \mathrm{h}$
of on-source integration time. In a similar amount of time and with the same
spectral tuning, AtLAST will achieve a depth comparable to ALMA, however
providing a dramatic improvement of $\sim 10^{3}$ in field of view and, thus,
in overall mapping speed. At the same time, AtLAST will provide a novel
perspective on the reservoirs of cold gas (light blue overlay; Emonts et al.
2016) coexisting with the warm/hot phase within protocluster cores (see the
CGM science case study by Lee et al. in prep. for a discussion). For
comparison, we also include the bright jet of radio emission output from the
central galaxy as observed by VLA (the linear east-west feature, shown in red;
Carilli et al. 2022). The present figure is adapted from Di Mascolo et al.
(2023) and the corresponding ESO Press Release eso2304.

The correlation of such constraints with the properties of the galactic
populations observed within the (proto)cluster complexes will further allow
for directly linking the evolution of the forming intracluster gas to the
multi-phase protocluster environment and its only partially understood impact
on galaxy formation and evolution (Fig. 6). Current multi-wavelength
observations have highlighted that the environmental effects might act on Mpc
scales and well beyond the more or less virialized regions within these
protocluster galaxy overdensities (Alberts & Noble 2022). These studies
however rely on the characterization of environmental processing solely from
the perspective of protocluster galaxies (Overzier 2016). On the other hand,
the wide field, the extreme sensitivity and the capability of AtLAST to trace
low density regions thanks to the SZ effect will allow for an efficient
imaging of the complex galaxy-environment puzzle with a comprehensive glance
of the multi-scale and multi-phase nature of high-$z$ (proto)cluster systems.

###  Impact of AGN feedback and halo heating

Acting in practice as a calorimeter of astrophysical electron populations, the
thermal SZ effect can shed light on the interplay of feedback processes and
heating of large-scale halos from galactic to cluster scales. This is
particularly relevant in the context of AGN studies, in relation to the
specific impact of feedback and AGN-driven outflows in contributing to the
heating of cosmic haloes (Fabian 2012). In fact, despite the importance of
supermassive black holes (SMBH) in driving the evolution of cosmic structures,
we still have a limited understanding of the complex connection between multi-
scale physical properties of SMBH and their host galaxies (Gaspari et al.
2020).

Current multi-wavelength observations support a rough duality in the feedback
framework (Padovani et al. 2017), with the level of radiative efficiency
depending on the specific scenario regulating SMBH accretion (Husemann &
Harrison 2018, Hlavacek-Larrondo et al. 2022). From the perspective of the
observational properties of the hot ICM/CGM phase, different feedback models
would naturally result in different levels of energy injection and, thus, in
deviations from the halo thermal budget expected from virial considerations.
At the same time, the strong interaction of winds and jets with the
surrounding medium introduces a significant amount of non-thermal support to
the overall pressure content — in the form, e.g., of turbulent motion,
buoyantly rising bubbles of extremely hot plasma ($\gtrsim
100\leavevmode\nobreak\ \mathrm{keV}$) and associated shock-heated gas cocoons
(Pfrommer et al. 2005, Abdulla et al. 2019, Ehlert et al. 2019, Marchegiani
2022, Orlowski-Scherer et al. 2022). All this implies, however, that gaining a
detailed view of the thermodynamic properties of the circumgalactic haloes
would allow us to obtain better insights into the AGN energetics and improve
our feedback models.

Measurements of the integrated thermal SZ signal have already been broadly
demonstrated to provide an efficient means for probing the evolution of the
imprint of feedback on the thermal energy of cosmic structures (Crichton et
al. 2016, Hall et al. 2019, Yang et al. 2022). These are however limited
mostly to stacking measurements of arcminute-resolution SZ data, and are thus
hampered by the low angular resolution of the wide-field survey data employed.
On the other hand, targeted observations at higher angular resolution
currently comprise an extremely small set of high-$z$ quasars (Lacy et al.
2019, Brownson et al. 2019, Jones et al. 2023). The overall limited
sensitivity as well as interferometric effects such as poor $uv$-coverage and
the filtering of large scales, however, resulted only in what appear to be low
significance detections of the SZ signal in the direction of these systems.
While these works have been pioneering for high resolution studies, they so
far provide little constraining power on the AGN energetics and feedback
scenarios.

On the other hand, the SZ signal from AGN-inflated bubbles has been robustly
detected in one, extreme case (MS 0735.6+7421; Abdulla et al. 2019, Orlowski-
Scherer et al. 2022). Still, the observations required 10s hours with the
current-generation MUSTANG-2 instrument (Dicker et al. 2014), and 100s of
hours with the previous-generation CARMA interferometer (Woody et al. 2004),
and were limited to single frequency observations. Since the SZ signal scales
as the amount of energy displaced, future observations with current
instruments to observe additional, less energetic AGN outbursts could require
much more time on the source. As such, this singular example serves largely as
a proof-of-principle for further, future resolved studies. We note that some
progress will be made in this decade with, e.g., TolTEC (Bryan et al. 2018),
though the Large Millimeter Telescope Alfonso Serran (LMT; Hughes et al. 2010)
was designed to achieve a surface accuracy of $\sim 50\leavevmode\nobreak\
\mu$m (2.5$\times$ worse than AtLAST), and regardless will be limited by the
atmospheric transmission to $\nu\lesssim 350\leavevmode\nobreak\ \mathrm{GHz}$
in all but the most exceptional weather (see, e.g., the site comparison in
Klaassen et al. 2020). Other single dish facilities delivering similarly high
resolution will be limited to even lower frequencies (e.g. Nobeyama, Green
Bank Telescope, Sardinia Radio Telescope), while ALMA has difficulty
recovering scales larger than $1^{\prime}$ in all but its lowest bands (see
Section 4).

Recently, multiple studies (e.g., Moser et al. 2022, Chakraborty et al. 2023,
Grayson et al. 2023) showed that obtaining high angular resolution
observations of the thermal SZ effect (in combination with X-ray observations)
would allow for constraining the distinct contribution from different feedback
models. First observational studies based on the cross-correlation of the
thermal and kinetic SZ signals (e.g., Amodeo et al. 2021, Schaan et al. 2021,
Vavagiakis et al. 2021, Das et al. 2023) already showed independent and
competitive constraints. Recently, Coulton et al. (2024) demonstrated that the
so-called “patchy screening” can provide an alternative and highly
complementary perspective on feedback mechanisms. Still, the low angular
resolution of such measurements is not sufficient to spatially separate first
and higher-order halo terms, and are thus hampered by respective systematics.
On the other hand, based on numerical predictions for different feedback
models (Yang et al. 2022), extending our observational constraints to include
a broad range of masses and redshift and distinguishing between different
feedback models will be highly impractical with current high angular
resolution facilities. Further, it is worth noting that strongly asymmetric
outflows from quasars, as well as gas inflows, would result in small-scale
distortions of the overall SZ signal due to the localized thermal, kinetic and
relativistic SZ contributions (see, e.g., Bennett et al. 2024). Similarly, the
inflation of cavities by large-scale jets and the consequent generation of
shock fronts and turbulent motion would imprint observable deviations in the
global SZ signal in the direction of AGN hosts (Ehlert et al. 2019). Having
access to sensitive, multi-frequency observations as provided by AtLAST would
thus be crucial, on the one hand, for reducing any biases associated with the
missing decomposition of the different SZ components to the measured signal as
well as any contamination (due to, e.g., millimeter/submillimeter bright
emission from the AGN within the studied haloes). On the other hand, it will
allow for cleanly dissecting the spectral and morphological features
characteristic of the different feedback scenarios.

Concerning the reconstruction of the thermal properties of the CGM, this will
have an impact even beyond the context of the evolution of the physical
processes driving the heating of cosmic haloes. In fact, it will be possible
to swiftly build a multi-phase picture of the CGM333For a broader discussion
of the importance of multi-phase CGM studies in the context of galaxy
formation and evolution, we refer to the companion AtLAST CGM science case
study by Lee et al. in prep. by concurrently tracing its cold phase along with
direct constraints on the otherwise elusive warm/hot constituent — comprising
$\approx 80\%$ of the total baryonic material in the CGM overall (e.g.,
Schimek et al. 2024). This is an unparalleled feature of (sub)millimeter
measurements, that necessarily require a combination of high spectral and
angular resolution, along with the capability of mapping large-scale diffuse
signals. Clearly, AtLAST will be the optimal facility for such a task.

###  Galaxy cluster outskirts and intercluster structures

Figure 7: Comparison of the Compton-$y$ maps produced with Planck alone (left,
10′ resolution) and ACT (right, 1.7′ resolution), adapted from Hincks et al.
(2022). AtLAST will deliver an 8.3$\times$ improvement in resolution, and
69.4$\times$ the instantaneous sensitivity per beam, with respect to that of
ACT (and other 6-m CMB experiments) for the same observing frequencies,
allowing one to image substructures in intercluster bridges and directly
identify and remove source contamination.

A significant portion of the baryonic content of the Universe at $z\lesssim 3$
is expected to lie well beyond the virial boundaries of cosmic structures (Cen
& Ostriker 1999). This diffuse “warm-hot intergalactic medium” (WHIM) is
expected to have temperatures $T_{\mathrm{e}}\approx
10^{5}-10^{7}\leavevmode\nobreak\ \mathrm{K}$, largely invisible at optical
wavelengths and generally too low in temperature for all but the deepest X-ray
observations, often being limited to line of sight absorption studies
(Nicastro et al. 2018). Obtaining a detailed view of the large-scale WHIM is
however crucial. Accurately constraining the actual amount of matter
constituting the WHIM will provide fundamental information on the “missing
baryons” budget associated to this specific phase of the filamentary
intergalactic medium (e.g., Shull et al. 2012). This will be connected to the
specific mechanisms driving the heating of large-scale structure on
cosmological scales: on the one hand, matter inflows and mergers along large-
scale filaments driving strong accretion and virialization shocks (Baxter et
al. 2021, Anbajagane et al. 2022, 2024; see also Sec. 3.1); on the other hand,
the impact of feedback processes and of the environmental pre-processing of
galaxies (e.g., Fujita 2004, Alberts & Noble 2022).

To date, the identification and characterization of the physical properties of
the filamentary WHIM has been performed mostly through stacked SZ and/or X-ray
measurements (e.g., de Graaff et al. 2019, Tanimura et al. 2019, 2020, 2022),
and is often dominated by the hottest extremes of the range of temperatures
expected for the WHIM (see Lokken et al. 2023 for discussion). Recently,
direct SZ imaging of a nearby intercluster bridge was presented in Hincks et
al. (2022), which used the combination of ACT+Planck data to reveal details at
a much higher spatial dynamic range than the previous results using Planck
alone. The results are shown in Fig. 7. This work, while serving as a valuable
pathfinder, highlighting the SZ substructures possible to image at even
modestly higher ($\sim 6\times$) resolution, was still limited to nearby
($z\approx 0.05$) massive clusters. Deep maps with AtLAST will allow improved
spatial dynamic range and higher fidelity, enabling such studies for many more
clusters going to both higher redshifts and lower mass regimes.

Thanks to its sensitivity and to the possibility of recovering large scales
over extremely wide fields, AtLAST will provide the ideal tool for searching
for the presence of the SZ effect in accreting and unbound intergalactic gas
surrounding the virialized volume of clusters and groups. In particular, this
will make it possible to routinely explore intercluster structures in a large
number of cluster pairs without the need for time demanding observations. For
instance, it will be possible to achieve the same Compton $y$ (or surface
brightness) sensitivity as in the observation of the A399-A401 observations by
Hincks et al. (2022, see also Fig. 7) in less than $\sim
10\leavevmode\nobreak\ \mathrm{h}$ of integration, but with better spectral
coverage and an order of magnitude improvement in the angular resolution. On
the other hand, we can consider as a rough lower limit of the expected
amplitude for large-scale filaments the results from previous stacking
experiments on intergalactic gas. For instance, de Graaff et al. (2019)
provide estimates of the average SZ signal to have amplitudes in Compton $y$
unit of $\lesssim 10^{-8}$, corresponding to a maximum amplitude of the
thermal SZ signal of $\gtrsim-64\leavevmode\nobreak\
\mathrm{nJy\leavevmode\nobreak\ beam^{-1}}$ for the decrement, and $\lesssim
9.5\leavevmode\nobreak\ \mathrm{nJy\leavevmode\nobreak\ beam^{-1}}$ for the
increment. Although impractical for performing any direct imaging of WHIM
between and around individual galaxies, the extreme observing speed of AtLAST
will allow for extending the stacking constraints to higher redshift and
resolutions, providing a resolved evolutionary perspective on the hot phase of
the cosmic web and the processes driving their thermal properties. Similarly,
the broad spectral coverage will allow for reducing contamination from inter-
filamentary structures, while providing the means for directly inferring
robust temperature constraints (currently representing the main limitation for
using the SZ effect for determining the overall contribution of WHIM to the
missing baryon budget).

##  Technical Justification

The field of view of a (sub)millimeter telescope represents a key parameter in
the context of SZ science. Current high resolution instruments on large single
dish telescopes — e.g., MUSTANG-2 (Dicker et al. 2014), NIKA2 (Adam et al.
2018), TolTEC (Bryan et al. 2018) — lose signals on scales larger than their
instantaneous fields of view ($\approx 4-6^{\prime}$; Romero et al. 2020),
where much of the most interesting, faint target SZ signals exist.444We note
that continuum observations using the 12-meter antennas in the ALMA Total
Power Array (TPA; Iguchi et al. 2009) suffer even more egregiously from being
unable to remove atmospheric contamination due to their limited fields of
view. They also suffer from the poor mapping speeds associated with single
beam observations, and from relatively small collecting areas. This issue is
arguably more restrictive in the case of interferometric observations, which
generally feature maximum recoverable scales that fall within the sub-
arcminute regime (e.g., ALMA Bands 4-10)555For details, see the ALMA Technical
Handbook at https://almascience.eso.org/proposing/technical-handbook. So far,
instruments with much larger instantaneous fields of view, which are therefore
better able to recover larger scales, have been employed in the context of
CMB/SZ survey experiments like ACT (Swetz et al. 2011), SPT (Carlstrom et al.
2011), SO (Simons Observatory Collaboration 2019), CMB-S4 (Abazajian et al.
2016), CCAT-prime (Fred Young Submillimeter Telescope or FYST; CCAT-Prime
Collaboration 2023). Still, these feature small apertures ($\leq 15$-m). This
results in poor source sensitivity due to their limited collecting areas and
in arcminute-level angular resolution, making these telescopes not suitable
for imaging the small-scale morphologies of galaxy clusters and protoclusters
(except for a few systems in the nearby Universe).

In general, larger scales are difficult to recover due to the large, and
largely common mode, atmospheric signal which dominates. A field of view of
reduced size requires a tailored observational strategy and data reduction
pipeline to mitigate signal loss at large scales. Nevertheless, even in such a
case, the recovery of astrophysical information beyond the maximum recoverable
scales of such facilities would still be severely hampered. This would
critically affect many of the proposed science goals, particularly those
requiring both wide field of views and extended recoverable scales.
Intergalactic filaments are in fact expected to extend over tens of Mpc (e.g.,
Galárraga-Espinosa et al. 2020) and, thus, extending over degree-scales in the
case of nearby superclusters (Ghirardini et al. 2021a). Similar physical
extents are observed also in the case of high-$z$ protocluster complexes
($\lesssim 20$ arcmin; see, e.g., Matsuda et al. 2005, Cantalupo et al. 2019,
Hill et al. 2020, Jin et al. 2021). And as shown in Figs. 3 and 4, effectively
probing the distribution of the ICM thermodynamic properties out to the
cluster outskirts requires mapping the SZ signal beyond $\sim
1\leavevmode\nobreak\ \mathrm{deg}$ in clustercentric distance. Therefore, the
capability of gaining instantaneous observations of structures extending from
few arcminutes up to degree scales will represent a crucial benefit of AtLAST
compared to state-of-the-art and future telescopes covering the same
observational windows.

Figure 8: A simulated nearby galaxy clusters ($M_{\mathrm{500}}=1.28\times
10^{15}\leavevmode\nobreak\ \mathrm{M_{\odot}}$, $z=0.0688$; left) as observed
MUSTANG-2 (bottom center), and ALMA+ACA in Band 3 (top center), and by AtLAST
in Band 3 (right). The respective beams are shown in the bottom right corner
of each panel. The input simulation is extracted from the Dianoga cosmological
simulation suite (Rasia et al. 2015, Bassini et al. 2020). Overlaid as dashed
white circles are the ACA+ALMA and MUSTANG-2 footprints. We note that the
respective panels on the central column are scaled up arbitrarily with the
goal of highlight any observed features, and do not reflect the relative
angular sizes of the fields. For all cases, we consider an on-source time of 8
hours. The mock AtLAST and MUSTANG-2 observations are generated using the
maria simulation tool (see van Marrewijk et al. 2024 for details), assuming an
AtLAST setup with the minimal detector counts of 50,000 (Sec. 4.2). For
ACA+ALMA, we employ the simobserve task part of the Common Astronomy Software
Applications (CASA; CASA Team 2022).

Multi-band observations are also critical to suppress/mitigate non-SZ signals
below the detection threshold, making wide spectral coverage essential for
many of the science goals detailed above. Current high resolution facilities
on large telescopes have at most three bands, and are limited to relatively
low-frequency observations — e.g. $\leq 350$ GHz for the LMT (Hughes et al.
2010), $\leq 270$ GHz for the 30-meter Institute for Millimetric Radio
Astronomy (IRAM), and $\leq 115$ GHz for the 100-meter GBT (White et al.
2022), the 64-meter Sardinia Radio Telescope (SRT; Prandoni et al. 2017), or
any potential single dish component of the ngVLA (Selina et al. 2018). This
implies that any current or next-generation facilities will provide limited
constraining power for the relativistic and kinetic SZ, as well as
contamination from the cosmic infrared background or diffuse dust
contamination.

Most foregrounds should be spatially distinguishable from the SZ signal.
However there may be a spatially coincident large-scale dust component
originating from within clusters themselves (e.g., Erler et al. 2018) which
makes at least two bands in the range $400-900\leavevmode\nobreak\
\mathrm{GHz}$ indispensable to trace the Rayleigh-Jeans tail of the dust
spectral energy distribution and to mitigate biases in the SZ spectral
modeling. An additional band at $\approx$ 1200 GHz would be even more helpful
to resolve degeneracies between dust and SZ signals, but this is precluded by
the severe reduction in atmospheric transmission (see Figure 2).

To meet the observational requirements for pursuing the proposed scientific
goals (Sec. 2), we perceive the most salient instrumentation requirements to
be the ability to achieve high continuum mapping speeds over large areas and
in multiple bands. This would convert into the key demand of densely filling
the telescope focal plane with a large count of multi-frequency detectors. In
this regard, current state-of-the-art continuum cameras (e.g., transition edge
sensor bolometers or kinetic inductance detectors) have already demonstrated a
high technical readiness level, providing background-limited performance in
the (sub)submillimeter the possibility of being read out in large numbers
(tens-to-hundreds of thousands, as noted in Klaassen et al. 2020) through
frequency multiplexing, allowing the construction of large imaging arrays. We
further refer to the AtLAST Memo 4 for details on the expected instrumental
specifications for AtLAST.

To illustrate the observational capabilities of the proposed AtLAST continuum
setup, we generate mock observations using the maria simulation library (van
Marrewijk et al. 2024) and consider a simulated galaxy cluster extracted from
the Dianoga hydrodynamical cosmological simulations (Rasia et al. 2015,
Bassini et al. 2020) as input. The results are shown in Fig. 8. For
comparison, we include simulated observations performed with MUSTANG-2 and
jointly with ALMA and the 7-m Atacama Compact Array (ACA; Iguchi et al. 2009).
The clear result is the superior capability of AtLAST in recovering spatial
features over a broad range of scales at high significance, while MUSTANG-2
and ACA+ALMA suffer from limited sensitivity and significant large-scale
filtering, respectively. We note that, in this test, we are considering only
single-band observations at the same frequency to facilitate the comparison.
Although ALMA Band 1 offers an improved sensitivity, spatial dynamic range,
and field of view compared to Band 3, it still provides a limited sampling of
large-scale SZ structures (with a maximum recoverable scale
$\mathrm{MRS}\lesssim 1.20^{\prime}$ when ALMA is in its most compact
configuration).

###  Optimizing the spectral setup

As mentioned broadly in Sec. 2 and discussed in the introduction to this
section, among the critical aspects for performing a robust reconstruction of
the SZ effect is the requirement of cleanly separating the multiple spectral
components of the SZ signal from contaminating sources. From a technical point
of view, this converts to maximizing the spectral coverage while requesting
maximum sensitivity (i.e., lowest noise root-mean-square) for each of the
bands. Given the deteriorating atmospheric transmission when moving to higher
frequencies, this is not obtained by trivially expanding the effective
bandwidth arbitrarily. At the same time, we would like to consider a minimum
setup in order not to result in an over-sampling of the target spectral range.

Figure 9: The high-frequency bands (Band 8-10) will be crucial for optimally
sampling the peak of the dust continuum emission from individual high-$z$
background galaxies. Along with inferring the physical properties of their
dust content, this will be crucial for minimizing the contamination of the SZ
signal due to cospatial dusty components (see also Sec. 4.3). As a reference,
we show here model emission for star-forming galaxies at varying redshift.
Here, the dust contribution is based on the $z$-dependent dust temperature
model from Sommovigo et al. (2022) for a dust mass
$M_{\mathrm{d}}=10^{8}\leavevmode\nobreak\ \mathrm{M_{\odot}}$ (consistent
with the galaxy REBELS sample; Bouwens et al. 2022). The low-frequency radio
component reproduces the radio model from Delvecchio et al. (2021), assuming
an infrared-to-radio luminosity ratio of $q_{IR}=2.646$. The dashed line
denotes the lowest redshift at which the dust emission peak falls within the
AtLAST spectral range ($z\simeq 3.60$).

A summary of the selected bands, specifically optimized to minimize the output
noise root-mean-square level per given integration time, is provided in Table
1. Our low-frequency set ($\lesssim 300GHz$) extend upon the multi-band set-up
proposed for CMB-S4 (Abazajian et al. 2016), shown in forecasts to provide an
optimal suppression of the contribution from astrophysical foregrounds and
backgrounds (Abazajian et al. 2019). Nevertheless, motivated by the expected
coverage of the $\lesssim 30\leavevmode\nobreak\ \mathrm{GHz}$ range by future
radio facilities (e.g. SKA, ngVLA), we decide do not include the synchrotron-
specific $20\leavevmode\nobreak\ \mathrm{GHz}$ band. On the other hand, given
the centrality of the high-frequency ($\gtrsim 500\leavevmode\nobreak\
\mathrm{GHz}$) for maximizing AtLAST’s capability of separating different SZ
components and the signal from contaminating sources (see Sec. 4.3), we extend
the overall spectral coverage beyond $300\leavevmode\nobreak\ \mathrm{GHz}$ to
include four additional bands up to $900\leavevmode\nobreak\ \mathrm{GHz}$.
Compared to FYST’s choice of survey bands (CCAT-Prime Collaboration 2023), our
choice will allow one to better sample the high-frequency end of AtLAST’s
spectral range and, in turn, to gain a better handle on the relativistic SZ
effect and on the dust contamination (Fig. 9; we further refer to van Kampen
et al. in prep. for a direct comparison of the large-scale distribution of
submillimeter bright sources observed by the arcminute-resolution ACT and
AtLAST). Currently, we are investigating the possibility of integrating an
additional band covering the $\sim 500\leavevmode\nobreak\ \mathrm{GHz}$
atmospheric window, but the low transmission and limited fractional bandwidth
are expected to limit the effectiveness of such an addition in terms of an
increase of the overall SZ sensitivity. However, we emphasize that AtLAST
coverage of the ALMA Band 8 frequencies up to $\nu=492\leavevmode\nobreak\
$GHz would be fundamental for other application in the context of AtLAST
science (we refer the interested readers to the companion AtLAST case studies
by Lee et al. in prep. and Liu et al. in prep.).

band ref. frequency bandwidth band edges beam sensitivity survey noise — [GHz]
[GHz] [GHz] [arcsec] [$\mathrm{\mu Jy\leavevmode\nobreak\
beam^{-1}\leavevmode\nobreak\ h^{1/2}}$] [$\mathrm{\mu
K_{cmb}-arcmin\leavevmode\nobreak\ h^{1/2}}$] 2 42.0 24 30–54 35.34 6.60 2.40
3 91.5 51 66–117 16.22 6.46 1.27 4 151.0 62 120–182 9.83 7.14 1.21 5 217.5 69
183–252 6.82 9.22 1.86 6 288.5 73 252–325 5.14 11.91 3.71 7 350.0 50 325–375
4.24 23.59 12.26 8 403.0 38 384–422 3.68 39.98 34.70 9 654.0 118 595–713 2.27
98.86 $1.67\times 10^{3}$ 10 845.5 119 786–905 1.76 162.51 $3.70\times 10^{4}$

Table 1: Frequencies, sensitivities and beam sizes for an AtLAST type of SZ
experiment. The sensitivity levels are computed assuming the standard values
for weather condition (2th octile) and elevation ($\alpha=45\deg$), but
consider the broad-band re-implementation of the AtLAST sensitivity
calculator. The specific frequencies of the band edges correspond to the ones
that minimizes the output noise RMS level in the corresponding band per given
integration time.

###  Survey strategy and detector requirements

To obtain a straightforward estimate of the instrumental performance expected
when adopting the proposed spectral setup, we extend the analysis performed by
Raghunathan (2022) to simulate an AtLAST-like facility (we refer to the
aforementioned paper for technical details). As broadly discussed in the
previous sections, performing a clean and robust separation of the multiple
spectral components determining the millimeter/submillimeter sky will
represent the major observational challenge to the achievement of the proposed
SZ science goals. This will inherently result in more or less severe residual
noise, as a combination of any contributions from instrumental noise, galactic
foregrounds and extragalactic backgrounds are not properly accounted for the
separation. As such, it represents a limiting factor in the detectability of
any SZ signal and could be interpreted as the final SZ depth of the proposed
observations.

Figure 10: Residual Compton-$y$ noise power spectra as a function of the sky
coverage ($f_{\mathrm{sky}}$) in the case of a wide-field AtLAST survey
(adapted from Raghunathan et al. 2022, which we refer to for details). We
include as a reference the residual noise curve expected in the case of the
CMB-HD survey (Sehgal et al. 2019). The shaded band denotes the power spectrum
for a fiducial thermal SZ sky as extracted from the bahamas simulations
(McCarthy et al. 2017), with $1\sigma$ and $2\sigma$ credible intervals based
on George et al. (2015).

In Fig. 10 we present the result of an optimal internal linear combination of
simulated multi-frequency AtLAST maps when adopting a wide-field survey
strategy over a period of 5 years. The proposed spectral configuration will in
particular allow for reaching a lower mass limit almost a factor of $2\times$
lower than achievable with CMB-HD (Sehgal et al. 2019), a reference next-
generation CMB facility in terms of proposed survey depth, and with better
angular resolution. This implies that AtLAST will be able to probe the SZ
signal to Compton $y$ levels $\lesssim 5\times 10^{-7}$ over an extreme
dynamic range of spatial scales when considering a deep survey approach
($<4000\leavevmode\nobreak\ \mathrm{deg^{2}}$). Targeted observations will
allow to reach a beam-level Compton $y$ depth of $\sim 2\times 10^{-6}$ per
hour of integration time. Previous studies (e.g., Dolag et al. 2016,
Raghunathan 2022) have predicted a Compton $y$ confusion floor of $2-5\times
10^{-7}$ from $<10^{13}\leavevmode\nobreak\ \mathrm{M_{\odot}}$ haloes
(corresponding to predicted detection threshold for AtLAST). As such, the
estimated sensitivity imply that AtLAST will obtain SZ confusion limited
observations in $\sim 100\leavevmode\nobreak\ \mathrm{h}$ with single-pointing
strategy. Nevertheless, we note that this sensitivity estimate corresponds to
the residual Compton $y$ root-mean-square noise obtained when applying a
constrained internal linear combination procedure to a simple set of mock
AtLAST observations666We generate flat-sky sky realizations at the AtLAST
bands including Galactic foregrounds and extragalactic background. The
foreground model is based on the pysm3 models (Thorne et al. 2017), but the
code was ported and adapted to extend the stochastic components down to
arcsecond scales. The output mock realization comprises the dust (d11 model),
AME (default), free-free (default) and synchrotron (s6) from the Milky Way.
The background signal is composed by a random CMB realization, as well as
infrared and radio background from unresolved sources as extracted from the
SIDES (Béthermin et al. 2017) and RadioWebSky simulations (Li et al. 2022),
respectively. To reproduce a clean subtraction of the point source
contamination, we excluded all the radio and infrared components with fluxes
in at least two bands larger than $3\times$ the corresponding noise root-mean-
square. under the assumption of a clean subtraction of any contaminating
compact source. As such, it should be considered as a rough ground reference
for the actual depth achievable with future AtLAST measurements. Future
forecasting studies will particularly investigate how different observation
strategies, source subtraction, and modeling techniques will affect the
contamination mitigation and the effective SZ sensitivity.

Still, achieving such frontier capabilities will necessarily demand a
considerable mapping speed and, thus, a crucial effort in the optimization of
the detector array. To estimate a minimal detector count for filling the focal
plane, we consider the sensitivity estimates reported in Table 1 as target
depths for surveys with varying observing period and sky coverage (see Fig.
5). The results are reported in Fig. 11. A detector count
$n_{\mathrm{det}}\simeq 5\times 10^{4}$ is sufficient for achieving the
sensitivity goal in the case of a narrow survey configuration
($1000\leavevmode\nobreak\ \mathrm{deg^{2}}$) both in Band 2 and Band 3, key
for tracing the decrement regime of the thermal SZ signal. For the same bands,
the same $n_{\mathrm{det}}$ constraints would allow to achieve a similar
survey sensitivity also in the intermediate $4000\leavevmode\nobreak\
\mathrm{deg^{2}}$ case over $\sim 4-5\leavevmode\nobreak\ \mathrm{years}$.
Nevertheless, extending these considerations to other bands or a wide-field
scenario would require a significant increase in $n_{\mathrm{det}}$. For
instance, in the case of Band 5 — crucial for constraining the departures from
the thermal SZ effect due to kinetic and relativistic contributions — such a
boost would range over almost an order of magnitude.

Figure 11: Required number of detectors to reach the target sensitivity
estimates listed in Table 1 for different bands and considering different
survey strategies. For comparison, the squares on the ordinate marks the
detector counts required in each band for fully covering a
$1\leavevmode\nobreak\ \mathrm{deg^{2}}$ field of view. The horizontal line
traces the minimal number of detectors (50,000) identified for reaching the
target depth in Band 2 and 3 in the case of $1000\leavevmode\nobreak\
\mathrm{deg^{2}}$ and $4000\leavevmode\nobreak\ \mathrm{deg^{2}}$ surveys. We
note that this is consistent with the estimated specifications reported in the
AtLAST Memo 4 for the 1st generation instruments.

As broadly highlighted in Sec. 2, constraining the small-scale fluctuations in
the thermal and kinetic SZ effects, while tracing the temperature-dependent
relativistic SZ corrections would imply measuring deviations from the global
SZ distribution order of magnitudes smaller than the bulk, non-relativistic
thermal SZ signal. This would in turn require a significant reduction of any
systematic effects hampering the overall calibration accuracy. In this regard,
an interesting technical aspect of AtLAST is the plan for closed-loop
metrology for tracking the alignment of the primary mirror panels (see, e.g.,
Mroczkowski et al. 2024, Reichert et al. in prep). By using active, closed-
loop metrology such as the laser system currently being developed for the
Sardinia Radio Telescope (Attoli et al. 2023) or the wavefront sensing system
being developed on the Nobeyama Radio Observatory 45-m (Tamura et al. 2020,
Nakano et al. 2022), the errors in the beam can be kept down to sub-percent
levels, meaning the beam will be diffraction limited and stable throughout
observations. This in turn will improve the calibration accuracy and reduce
systematics (see, e.g., Naess et al. 2020 for discussion of the diurnal
effects on the ACT beams) that have been shown to impact CMB and SZ results at
the several percent level in the case of passive optics ($3-5\%$; Hasselfield
et al. 2013, Lungu et al. 2022), with the result that the daytime data have
generally been excluded from cosmological analyses. Future dedicated forecasts
will analyze the benefits of metrology for secondary CMB measurements using
AtLAST, including improvements to the calibration, reduction of systematics,
and the ability to recover larger angular scales on sky. However, the salient
takeaway message is that uncertainties in the beam should no longer be a
leading source of systematic error.

###  Mock reconstruction of the relativistic SZ effect

The relative amplitude of the relativistic component compared to the thermal
and kinetic SZ effects makes this modeling task highly challenging. To test
the prospects of using AtLAST measurements for performing a spectral
separation and analysis of the SZ effect, we thus perform a mock
reconstruction of the intracluster temperature using the relativistic SZ
effect.

####  SZ-only reconstruction

As a test case, we consider a galaxy cluster777We note that, despite
representing relatively extreme (but realistic) values, the setup
$(T_{\mathrm{SZ}},y)=(10\leavevmode\nobreak\ \mathrm{keV},10^{-4})$ is chosen
to facilitate this first study of the AtLAST capabilities of providing
spectral constraints on temperature-dependent distortions of the thermal SZ
effect. A broader exploration of the parameter space will be presented in Sec.
4.3.3. with temperature $T_{\mathrm{SZ}}=10\leavevmode\nobreak\ \mathrm{keV}$
and Compton $y=10^{-4}$. The amplitude of the SZ signal at each of the
selected bands in the minimal spectral set is obtained by integrating the
relativistically-corrected thermal SZ (rtSZ) spectrum across each band
assuming flat bandpasses. We then obtained estimates for the corresponding
uncertainties based on the sensitivity estimates from the AtLAST sensitivity
calculator. First, we compute the integration time required to achieve a
signal-to-noise (SNR) of 50 in Band 8, arbitrarily chosen among the two
spectral windows closest to the peak in the rtSZ effect (Figs. 2 & 12). The
resulting noise root-mean-square (RMS) is defined as the corresponding
uncertainty. The uncertainties $\delta I$ for each of the remaining bands are
thus computed assuming the same integration time as for the Band 8 estimation
above, but taking into account both the differing point-source sensitivities
and beam sizes across frequency bands,

$\frac{\delta I(n)}{\delta
I(\mathrm{8})}=\frac{\sigma(n)}{\sigma(\mathrm{8})}\times\frac{\Omega(\mathrm{8})}{\Omega(n)}.$
(1)

Here, $n$ denotes the band index ($n=\\{2,...,10\\}$), while and $\Omega(\nu)$
are the flux density RMS and the beam size at a given frequency ,
respectively. The resulting simulated measurements are shown in Fig. 12.

Figure 12:  Predicted rtSZ measurements for a cluster with a temperature of
$10\leavevmode\nobreak\ \mathrm{keV}$ and Compton $y$ of $10^{-4}$, assuming
flat bandpasses in the Bands 2–10 (denoted as gray bands; see also Table 1).
We assume the same exposure time in each band and account for flux sensitivity
and beam size differences, tuned to achieve a $\mathrm{SNR}=50$ in Band 8. For
comparison, the non-relativistic approximation is also shown. The bottom axis
shows the difference between the relativistic and non-relativistic spectra.

The derived SZ measurements can then be used to perform a simple joint
inference of the Compton $y$ and electron temperature for the target case. If
only lower-frequency data points ($\lesssim 200\leavevmode\nobreak\
\mathrm{GHz}$) are measured, then there is a complete degeneracy between
$T_{\mathrm{e}}$ and the Compton $y$ parameter. In this spectral range, in
fact, an increase in the electron temperature reduces the signal in a similar
manner as decreasing the overall Compton $y$ amplitude. When higher-frequency
points are included, the degeneracy can be minimized as shown in Fig. 13. By
dropping one band at a time from the fit, we find that Band 6 ($\approx$ 240
GHz) has the greatest influence in breaking the degeneracy since the signal-
to-noise on the difference between nearly-degenerate models is greatest in
this band.

Figure 13:  Fits to the simulated measurements shown in Fig. 12, using all
bands (grey) and all except Band 6 (red). Excluding Band 6 increases the
$y$-$T_{\mathrm{e}}$ degeneracy substantially. The black diamond denotes the
input parameters.

####  SZ+dust reconstruction

So far, we have assumed that the only signal present is the SZ signal.
However, in reality there will of course be other astrophysical foregrounds
and backgrounds present along the line of sight, resulting in non-negligible
contamination of the overall SZ signal observed in the direction of a galaxy
cluster. In this test, we however assume that signals that are not spatially
correlated to the SZ effect can be removed by means of component separation
methods (see Sec. 4.2) or targeted forward modeling procedures in the case of
unresolved compact sources (e.g., Ruppin et al. 2017, Di Mascolo et al. 2019a,
Kéruzoré et al. 2020, Kitayama et al. 2020, Andreon et al. 2021). However,
previous studies (e.g. Erler et al. 2018) showed that there is a spatially
correlated signal within clusters associated with the diffuse dust emission.
To understand the impact on the capability of AtLAST in constraining any rtSZ
deviation, we repeat the above test by adding an additional dust-like spectral
component (Fig. 14). In particular, we assume a modified black body signal
given by (Erler et al. 2018)

$I_{\mathrm{Dust}}(\nu)=A_{\mathrm{dust}}^{857}\left(\frac{\nu}{{}_{0}}\right)^{{}_{\mathrm{Dust}}+3}\frac{\exp\left[h{}_{0}/(k_{B}T_{\mathrm{Dust}})\right]-1}{\exp\left[h\nu/(k_{B}T_{\mathrm{Dust}})\right]-1},$
(2)

where ${}_{0}=857$ GHz is chosen as the reference frequency, and
$A_{\mathrm{dust}}^{857}$ is the amplitude at this frequency. We use the
(Erler et al. 2018) parameter fits for $A_{\mathrm{dust}}^{857}$, Dust and
$T_{\mathrm{Dust}}$ to generate a dust signal, and add them as free parameters
to our fit with uniform priors on all parameters. The uncertainties on the
measurements in each band are the same as in the SZ-only fit (Sec. 4.3.1)

Figure 14: Same as Fig. 12, but with the addition of a modified black body
dust component based on the model from Erler et al. 2018. The red dotted line
shows the dust signal. The red solid and dashed lines show the total signal
from the dust and non-relativistic and relativistic signals respectively.

In this case, more bands become necessary to correctly constrain the rtSZ
temperature and disentangle the rtSZ and dust spectral components. The best
minimal combination comprises Bands 2, 4, 6, 8 and 10, that provide almost
identical constraints to the full set of bands on the rtSZ parameters, while
achieving a lower precision on the dust parameters (as shown in Fig. 15). Most
importantly, it is important to note that the broad spectral coverage offered
by the proposed setup allows a clean separation of the rtSZ and dust signals
with only a marginal impact on the rtSZ constraints compared to the SZ-only
case (Sec. 4.3.1).

Figure 15:  Fits to the simulated measurements shown in Fig. 14, using all
bands (grey) and Bands 2, 4, 6, 8 and 10 only (blue). With this optimal set of
five bands, the constraints on the rtSZ parameters are almost equivalent to
the constraints with all bands, but we obtain a slightly reduced constraining
power on the dust parameters. The dotted diamonds denote the input parameters.

####  Required sensitivity and time forecasts

The reference SNR of 50 employed above was mainly intended to achieve a
general perspective on the spectral constraining power of the proposed setup
without being limited by the inherent significance of the test SZ signal.
Thus, we now investigate what SNR is required to achieve good temperature
constraints from rtSZ measurements. In particular, we run similar fits for
different values of the reference SNR and different temperatures.

The impact of the varying SNR on the temperature reconstruction is summarized
in Fig. 16. If we require, for example, an accuracy of $1\leavevmode\nobreak\
\mathrm{keV}$ in temperature, a reference SNR of 40 is sufficient for all the
temperatures tested. A reference SNR of 30 is instead sufficient for all
except the very hottest clusters when dust is included.

Figure 16: Precision achieved on the temperature reconstruction as a function
of the SNR in the reference band (384–422 GHz; Band 8) and for a range of ICM
temperatures. The horizontal line denotes the target temperature
$T_{\mathrm{SZ}}=1\leavevmode\nobreak\ \mathrm{keV}$ discussed in the text.

Although AtLAST will be able to observe clusters spanning a broad range in
mass and redshift (and, hence, temperature), the example analysis presented in
the previous section is aimed only at forecasting AtLAST capabilities of
measuring relativistic deviations from the standard thermal SZ and not at
testing the expected detection threshold as a function of cluster properties.
To take into account the evolution of the rtSZ effect with the mass and
redshift of a galaxy cluster, we aim here at estimating the required observing
time to reach a target SNR in Band 8, our reference spectral window (Sec.
4.3.1).

To do so, we construct cluster signal maps for a range of masses and
redshifts, using the physical model given in Olamaie et al. (2012) and Javid
et al. (2019). Assuming hydrostatic equilibrium, this model gives us
physically consistent pressure and temperature profiles which we use with the
szpack (Chluba et al. 2012, Chluba et al. 2013) temperature-moment method to
predict relativistic SZ effect signal maps, taking into account the spatial
variation of the temperature. The resulting observing time predictions for a
reference SNR of 30 are shown in Fig. 17. For reasonable observing times
($<16\leavevmode\nobreak\ \mathrm{hours}$) we can get average temperature
constraints for most clusters at redshifts up to $z\approx 0.1$, and high-mass
clusters ($M_{200}\gtrsim 4\times 10^{14}\mathrm{M}_{\odot}$) up to
arbitrarily high redshift. It is important to note that the enhanced angular
resolution of AtLAST could easily allow one to obtain spatially resolved
information on the temperature distribution, once the SNR requirements are
satisfied for each spatial element considered for the analysis — e.g., radial
bins or spectrally homogeneous regions as generally considered in high-
resolution X-ray studies (e.g., Sanders 2006).

Figure 17:  Beam-level noise root-mean-square and observing times as a
function of cluster redshift and mass $M_{200}$ required to reach an SNR of 30
in the reference spectral band (384–422 GHz; Band 8), allowing average SZ
temperatures to be well constrained.

##  AtLAST SZ studies in a multi-probe context

AtLAST will provide an unprecedented speed and spectral grasp across the
(sub)millimeter spectrum. This will make AtLAST inherently relevant beyond
just SZ science, and will open up possibilities for fundamental synergies in
the a multi-wavelength and multi-probe exploration of the Universe.

###  AtLAST scientific cross-synergies

Thanks to the novel multi-instrument design (Mroczkowski et al. 2024), AtLAST
will be aimed at representing a high-impact (sub)millimeter facility with a
broad and varied scientific reach. As such, this will set the ground for a
natural cross-synergy across the different scientific applications identified
as part of the AtLAST Science Development effort.

In the case of a wide-field continuum survey discussed Sec. 4.2, the multi-
band coverage and the extended temporal span will make the SZ-driven
observations extremely valuable for temporally-dependent studies as for
transient surveys (Orlowski-Scherer et al. in prep.). Similarly, the multiple
bands and likely polarization sensitivity will be useful in the study of
Galactic dust and molecular clouds (Klaassen et al. in prep.), building on the
lower resolution results with, for example, the Simons Observatory (Hensley et
al. 2022).

Related to the science goals proposed in this work, the availability of a
wide-field spectroscopic survey of the distant Universe (van Kampen et al. in
prep.) will immediately enhance the validity of the SZ identification and
study of high-$z$ clusters and protoclusters by providing accurate redshift
information. The broad spectral coverage achieved thanks to the proposed
multi-band setup will actually play a crucial role in maximizing the redshift
domain. At the same time, as already mentioned in previous sections, having
access simultaneously to constraints on the physical properties of large-scale
environments via the SZ effect (Sec. 3.5) and on the associated galaxy
populations (via the spectral characterization of their cold molecular gas and
the inference of their dust content; see Fig. 9) will represent an
unprecedented opportunity in the context of galaxy-environment co-evolution
studies.

Similarly, the information on the warm/hot component of galactic haloes will
be essential for building a comprehensive picture of the diffuse and multi-
phase CGM (Lee et al.. A combination of the novel perspective offered by
AtLAST on the cold contribution with the tight measurements of the thermal and
kinetic properties of such elusive haloes (Sec. 3.6) will represent the only
way for shedding light on the many potential evolutionary routes of the
elusive large-scale CGM.

###  Synergies with other state-of-the-art and forthcoming facilities

In the context of large scale structures, AtLAST’s constraints on the multi-
faceted SZ effect will be highly complementary to multi-wavelength information
on the galaxy motions and distribution, the gravitational potentials of the
systems, the X-ray emission, magnetic field structure, and the highly non-
thermal and relativistic emission traced by radio emission. Below, we
highlight some of the key facilities and experiments that provide the most
synergy with AtLAST.

####  Radio

The radio waveband offers information that can complement and enhance many of
the science cases outlined above, and next-generation instruments such as the
Square Kilometer Array (SKA; Huynh & Lazio 2013) and the next-generation Very
Large Array (ngVLA; Selina et al. 2018) will have the angular resolution and
sensitivity required to provide it. On the one hand, getting a clear view of
the resolved SZ effect and searching for intrinsic scatter and surface
brightness fluctuations requires sensitive detection and removal of
contaminating radio sources (e.g., Dicker et al. 2021). While this will
already be possible with AtLAST’s data itself thanks to its spectral coverage
and $\approx 5^{\prime\prime}\lambda\leavevmode\nobreak\ \rm mm^{-1}$
resolution (i.e. 10′′ at 2 mm), interferometric observations at lower
frequency (SKA-MID) will aid in pinpointing the location and morphology of the
radio sources, while also being more sensitive to fainter sources as most will
be brighter at lower frequency.

On the other hand, radio information provides a powerful complementary probe
of the astrophysics in the cluster, being sensitive to magnetic fields and
populations of non-thermal electrons. The reference surveys proposed for the
SKA (Prandoni & Seymour 2015) predict the detection of $\sim$1000s of radio
halos with SKA1-LOW, out to redshifts of at least $0.6$ and masses
$M_{500}>10^{14}\leavevmode\nobreak\ \mathrm{M_{\odot}}$ (Cassano et al. 2015,
Ferrari et al. 2015; see also, e.g., Knowles et al. 2021, 2022, Duchesne et
al. 2024 for preliminary results from SKA precursors), along with potential
first detections of the polarization of radio halos (Govoni et al. 2015). This
offers the opportunity not only to compare the detailed astrophysics of the
thermal and non-thermal components of clusters, shedding light on the
turbulent properties currently limiting the accuracy of mass estimation (see
Section 3.1), but also to potentially discover new populations of clusters via
their radio signals. When also observed by AtLAST, these populations will
offer insight into the variation in cluster properties when selecting by
different methods (see also Section 3.4). Faraday Rotation Measure
observations of polarized sources behind galaxy clusters as well as studies of
tailed radio galaxies _in_ clusters will enable the study of cluster magnetic
fields in unprecedented detail (Bonafede et al. 2015, Johnston-Hollitt et al.
2015a, b), contributing to our astrophysical understanding and ability to make
the realistic simulations crucial for interpreting observations.

Ultimately, the direct correlation of the SZ information with the spatial,
spectral, and polarimetric properties of the multitude of radio structures
observed in the direction of galaxy clusters will be essential for
constraining the detailed mechanisms governing particle (re)acceleration
within the ICM (van Weeren et al. 2019). Specifically, there is mounting
evidence that the non-thermal plasma observed in the form of (multi-scale)
radio halos (e.g., Gitti et al. 2015, Cuciti et al. 2022) as well as
intercluster bridges (e.g., Botteon et al. 2020b, Bonafede et al. 2022,
Radiconi et al. 2022) originates due to turbulent (re)acceleration (Brunetti &
Lazarian 2007, Brunetti & Jones 2014, Eckert et al. 2017, Cassano et al.
2023). On the other hand, radio relics are connected to (re)acceleration at
shock fronts (Akamatsu & Kawahara 2013, van Weeren et al. 2017, Botteon et al.
2020a). While it is clear that cluster mergers are driving both processes
(turbulence and shocks), our understanding of the physics of (re)acceleration
in clusters is limited by two factors: (i) information about the distribution
of gas motions in the ICM is currently sparse and usually inferred via
indirect methods (for a review, see Simionescu et al. 2019), and (ii)
characterizing shocks in the low-density cluster outskirts, where radio relics
are usually found, is very challenging. Detailed mapping of the thermal
(sensitive to shocks) and kinetic (sensitive to gas motions) SZ signals
throughout the volume of a large sample of galaxy clusters (potentially
extending out into the cosmic web), and how these signals relate to features
observed in the radio band, will be invaluable towards painting a clear
picture of the connection between large-scale structure assembly, magnetic
field amplification, and cosmic ray acceleration.

Understanding the impact of AGN feedback, on the other hand, (Section 3.6)
requires complementary observations of the AGN themselves. Gitti et al. (2015)
finds that even with early SKA1 (50% sensitivity), all AGN with luminosity
$>10^{23}\leavevmode\nobreak\ \mathrm{W\leavevmode\nobreak\ Hz^{-1}}$ can be
detected up to $z\leq 1$ with subarcsecond resolution, and the radio lobes
thought to be responsible for carving out the X-ray cavities should be
detectable in any medium – large mass cluster at any redshift in the SKA1-MID
deep tier surveys. Moreover, SKA1-MID is predicted to detect intercluster
filaments at around $2.5$ – $6\sigma$ (Giovannini et al. 2015), providing
information on their magnetic fields as a complement to the SZ information on
their thermodynamic properties (Section 3.7).

At the top of the SKA frequency range, it will be possible to directly access
thermal SZ information. Future extensions to the SKA-MID Phase 1 setup (with
the integration of the high-frequency Band 6; we refer to the SKA Memo 20-01
for details) and the ones envisioned for SKA Phase 2
888https://www.skao.int/en/science-users/118/ska-telescope-specifications
(2030+) will allow SKA to probe the low-frequency ($\lesssim
24\leavevmode\nobreak\ \mathrm{GHz}$) domain of the SZ spectrum, less affected
by kinetic and relativistic deviations than the range probed by AtLAST. In
fact, Grainge et al. (2015) find that 1 hour of integration is sufficient for
obtaining a $14\sigma$ detection of the SZ effect from a $M_{200}=4\times
10^{14}\leavevmode\nobreak\ \mathrm{M_{\odot}}$ cluster at $z>1$. On the other
hand, the clean perspective offered by AtLAST on the multiple SZ components
will provide the means, e.g., for cleanly disentangling the SZ footprint of
galaxy clusters from the faint, diffuse signal from mini- to cluster-scale
radio haloes, large-scale relics, and back-/foreground and intracluster radio
galaxies, enhancing their joint study.

####  Millimeter/Submillimeter

Millimetric/submillimetric survey experiments like SO (Simons Observatory
Collaboration 2019) and its upgrades, CCAT-prime/FYST (CCAT-Prime
Collaboration 2023), upgrades to the South Pole Telescope (Anderson et al.
2022), and ultimately CMB-S4 (Abazajian et al. 2016) will cover roughly half
the sky over the next few years to a decade, predominantly in the Southern
sky. Along with past and current facilities, these however have $\sim$ arcmin
resolution, well-matched to the typical angular size of clusters in order to
optimize their detection but not optimal for peering inside clusters to
explore astrophysical effects (aside from few nearby exceptional clusters).
Nevertheless, while limited to resolutions approximately $8.33\times$ lower
than AtLAST at the same frequencies, their data will provide robust
constraints at large scales, lending itself naturally to joint map-making and
data combination, as well as valuable source finders for deep AtLAST follow-
up. On the other hand, AtLAST will be able to resolve any structures probed by
these wide-field surveys, defining a natural and intrinsic synergy.

At still higher resolutions, ALMA will undergo a number of upgrades improving
its bandwidth and sensitivity over the next decade. These upgrades are called
the Wideband Sensitivity Upgrade (WSU; Carpenter et al. 2023), which in the
context of SZ science could deliver $2-4\times$ ALMA’s current bandwidth. Wide
field mapping capabilities are however not part of the key goals for the WSU,
and it is unlikely ALMA will ever map more than a few tens of square
arcminutes. Nevertheless, the improved sensitivity and bandwidth could allow
for exploiting ALMA to complement AtLAST observations with a high spatial
resolution view of astrophysics through detailed follow-up studies. At the
same time, such observations will require AtLAST to recover more extended
scales (see Section 4). The necessity of such a combination will however allow
to fully leverage the synergistic strengths of single-dish and interferometric
facilities for gaining an unprecedented view of the hot baryonic content of
the Universe, along with its multi-phase counterparts.

####  Optical/infrared

The Euclid mission (Laureijs et al. 2011, Euclid Collaboration et al. 2022)
has recently started surveying the optical/infrared sky, and is expected to
result in the identification of $\gtrsim 10^{5}$ galaxy clusters and
protoclusters across the entire cluster era ($0\lesssim z\lesssim 2$; Euclid
Collaboration 2019). Complemented with data from the Legacy Survey of Space
and Time (LSST) survey by the forthcoming Vera C. Rubin Observatory (Ivezić et
al. 2019), these will represent a wealth of complementary constraints on the
cluster and protocluster populations that will be essential for enhancing the
scientific throughput of AtLAST in the context of SZ studies. The
characterization of the weak lensing footprint of galaxy clusters and groups
jointly with resolved information on the thermodynamics of their ICM will
enable a thorough exploration of the many processes biasing our cluster mass
estimates (Sec. 3.1). At the same time, the detailed characterization of the
SZ signal from the vast number of weak-lensing selected systems (and, thus,
with different selection effects than surveys relying on ICM properties) will
allow for studying in detail the origin of the under-luminous clusters (Sec.
3.4). Similarly, AtLAST will provide an unprecedented view on the ICM forming
within the wealth of high-$z$ galaxy overdensities that will be identified by
Euclid/LSST, in turn providing an unbiased means for constraining the physical
processes driving the thermalization of protoclusters complexes into the
massive clusters we observe at $z\lesssim 2$.

More in general, the access to a rich set of imaging and spectroscopic
measurements by wide-field surveys — Euclid, Rubin Observatory, and the next
generation Nancy Grace Roman Space Telescope999https://roman.gsfc.nasa.gov/
(Spergel et al. 2015), SPHEREx (Doré et al. 2014) — along with deep, targeted
observation from high-resolution facilities — e.g., JWST (Gardner et al.
2006), or the upcoming Extremely Large Telescope101010https://elt.eso.org/ —
will be greatly complemented by the resolved, wide perspective of AtLAST on
the SZ Universe. Tracing the faint warm/hot backbone of large-scale structure
(Sec. 3.7), as well as tightly correlating resolved thermodynamic constraints
for the large-scale cluster environment with the physical properties of the
galaxies embedded within them (Alberts & Noble 2022, Boselli et al. 2022) and
the distribution of the more elusive intracluster light (Contini et al. 2014),
will be essential to shed light on their complex and dynamical co-evolution.

In addition, to facilitate precision cosmology studies with Euclid/LSST, it is
imperative to gain a better understanding of the impact of galactic processes
on the redistribution of baryons over large scales. Different prescriptions of
feedback employed in various cosmological simulations alter the predicted
amplitude and scale dependence of the matter power spectra at separations
under 10 Mpc on a level that is considerably larger than the statistical
uncertainty expected from upcoming cosmology experiments (Chisari et al. 2019,
van Daalen et al. 2020). Mapping the gaseous contents in the low-density
outskirts of galaxy groups and WHIM filaments through sensitive AtLAST
measurements will thus provide invaluable observational priors necessary to
model baryonic feedback for survey cosmology.

####  X-ray

On the X-ray side, Chandra111111https://chandra.harvard.edu/ and XMM-
Newton121212https://www.cosmos.esa.int/web/xmm-newton, launched in 1999 with
CCDs capable of $0.5-5^{\prime\prime}$ spatial resolution, are still providing
a reasonable thermodynamic mapping of the brightest regions of the collapsed
structures up to redshift $\sim 1.2$. The
eROSITA131313https://www.mpe.mpg.de/eROSITA telescope (launched in 2019) has
recently delivered the first release of its X-ray all-sky surveys (Merloni et
al. 2024) and new catalogs of cluster candidates up to redshift $z\simeq 1.3$
(Bulbul et al. 2024). Still, the large point spread function ($\sim
15^{\prime\prime}$) and the limited sensitivity does not allow for resolving
the temperature structure of the ICM and any derived quantities (e.g.,
pressure, entropy, mass) with the exception of nearby and bright galaxy
clusters (e.g., Iljenkarevic et al. 2022, Sanders et al. 2022, Whelan et al.
2022, Liu et al. 2023).

The $5\leavevmode\nobreak\ \mathrm{eV}$ spectral resolution of the Resolve
microcalorimeter onboard XRISM141414https://xrism.isas.jaxa.jp/en/, launched
in September 2023, will soon enable the first systematic investigation of the
gas kinematics in hot, X-ray bright galaxy clusters. However, these studies
will be limited by the low ($\sim 1.3^{\prime}$) angular resolution, small
field of view and effective area, especially at soft X-ray energies. This
particularly hinders the study of less massive haloes (galaxy groups and CGM)
and the mapping of extended cluster outskirts and WHIM filaments. Forthcoming
space missions are expected to improve all these performances through the
development of next-generation instruments with higher both spectral and
spatial resolutions over a wider field of view and with a larger collecting
area: Athena151515https://www.the-athena-x-ray-observatory.eu/en (expected to
be adopted as a L-mission by ESA in 2027 for a launch in 2037) will outperform
the current satellites thanks to the larger effective area by an order of
magnitude with a spatial resolution better than 10 arcsecs;
LEM161616https://www.lem-observatory.org/ (a proposed US Probe mission;
Patnaude et al. 2023) is designed to effectively map the thermodynamics and
kinematics of the low-density CGM and WHIM using spectral imaging of soft
X-ray line emission; AXIS171717https://blog.umd.edu/axis/ (another US Probe
proposal) will extend and enhance the science of sensitive, high angular
resolution X-ray imaging.

The complementarity of SZ and X-ray measurements of the warm/hot content of
cosmic large-scale structures has long represented a valuable asset for a
cross-enhancement of the respective astrophysical information. The different
dependence of these tracers on the physical properties of the ionized gas has
been broadly exploited — from, e.g. obtaining tighter constraints on the
thermodynamics of the hot gas in distant clusters and cluster outskirts (e.g.,
Ghirardini et al. 2019, Castagna & Andreon 2020, Ghirardini et al. 2021b,
Andreon et al. 2021, Ruppin et al. 2021, Lepore et al. 2023) and large-scale
filaments (e.g., Planck Collaboration 2013, Akamatsu et al. 2017, Hincks et
al. 2022), to studying local deviations from particle and thermal equilibrium
(e.g., Basu et al. 2016, Di Mascolo et al. 2019b, Sayers et al. 2021),
deriving detailed morphological models of the three-dimensional distribution
of ionized gas (e.g., De Filippis et al. 2005, Limousin et al. 2013, Umetsu et
al. 2015, Sereno et al. 2018, Kim et al. 2023), or obtaining measurements of
the Hubble–Lemaître parameter independently of more standard probes (e.g.,
Bonamente et al. 2006, Kozmanyan et al. 2019, Wan et al. 2021).

The enhanced sensitivity, spatial resolution, and mapping speed of AtLAST for
various flavors of the SZ effect, combined with the capabilities of next-
generation X-ray facilities, will undoubtedly take these already existing
synergies one leap further. On the other hand, the novel high spectral
resolution imaging capabilities in the soft X-ray band (expected to become
available in the next decade with e.g. LEM and Athena) will give rise to new
opportunities for complementary measurements with the SZ band. Namely, X-ray
observations are primarily expected to map the line emission or absorption
signals from metals in the diffuse, warm-hot gas permeating large-scale
structure WHIM filaments or the low-mass haloes of individual L* galaxies. The
X-ray continuum emission from these targets will be swamped by the foreground
continuum from our own Milky Way, and extremely difficult to probe (see, for
instance, Kraft et al. 2022). The ideal path to obtaining a full picture of
the physical properties of this diffuse gas component of the cosmic web,
therefore, is to combine diagnostics about the metal content (from X-ray line
intensities), metal dynamics (from X-ray line widths and shifts), temperature
(from X-ray line ratios and relativistic SZ terms) with the gas pressure
cleanly measured through the thermal SZ signal. We can then solve for the gas
density (knowing the pressure and temperature), and the gas metallicity
(knowing the metal content and gas density). Taking this one step even
further, by detecting the kinetic SZ signal from the same gas, it will be
possible to compare the velocities of metal-poor (primordial) gas from the
kinetic SZ measurements which may be different than the velocities of metals
probed from the X-ray lines. This will provide truly groundbreaking
information about the circulation of gas and metals in and out of galaxies, by
offering the opportunity to map, for instance, metal-rich outflows driven by
feedback, and metal-poor inflows driven by accretion from the cosmic web,
leading to a revolution in our understanding of galaxy evolution.

##  Summary and conclusions

AtLAST will provide a transformational perspective on the SZ effect from the
warm/hot gas in the Universe. The high angular resolution enabled by the
50-meter aperture, the extensive spectral coverage, and the extreme
sensitivity swiftly achievable over wide areas of the (sub)millimeter sky will
provide the unprecedented opportunity to measure the SZ signal over an
instantaneous high dynamic range of spatial scales (from few arcsecond to
degree scales) and with an enhanced sensitivity ($\lesssim 5\times 10^{-7}$
Compton $y$).

Such a combination of technical advances will allow us to constrain
simultaneously the thermal, kinematic, and relativistic contribution to the SZ
effect for a vast number of individual systems, ultimately opening a novel
perspective on the evolution and thermodynamics of cosmic structures. Such an
unmatched capability will provide the means for exploring key astrophysical
issues in the context of cluster and galaxy evolution.

  * •

By resolving the multi-faceted SZ footprint of galaxy clusters, low-mass
groups, and protoclusters, it will be possible to trace the temporal evolution
of their thermodynamic properties across (and beyond) the entire cluster era
($z\lesssim 2$), over an unprecedented range in mass. The complementary
information on the full spectrum of small-scale ICM perturbations that will be
accessed thanks to AtLAST’s superior resolution and sensitivity will thus
allow to build a complete picture of the many intertwined processes that make
galaxy clusters deviate from the otherwise hydrostatic equilibrium and self-
similar evolution. At the same time, we will be able to get a complete census
of the cluster population, circumventing the inherent biases associated with
current cluster selection strategies. Overall, such studies will allow AtLAST
to be pivotal in firming the role of galaxy clusters as key cosmological
probes.

  * •

The possibility offered by AtLAST of accessing the low-surface brightness
regime will open an SZ window on the low-density warm/hot gas within the
cosmic large-scale structure — ranging from the characterization of the mostly
unexplored properties of the assembling ICM seeds within protocluster
overdensities to the barely bound outskirts of galaxy clusters. These
represent the environments where the same process of virialization begins. As
such, they are ideal for studying how deviations from thermalization, gas
accretion, and strong dynamical processes impact the thermal history of galaxy
clusters.

  * •

By tracing the imprint on the thermodynamics properties of circumgalactic
medium surrounding galaxies and of the cluster cores, AtLAST will allow to
constraint energetics and physical details of AGN feedback. This will provide
the means for moving a fundamental step forward in our understanding of the
crucial impact of AGN on the evolution of the warm/hot component of cosmic
structures over a wide range of spatial scales and across cosmic history.

To achieve these ambitious goals, it will essential to satisfy the following
technical requirements:

  * •

Degree-scale field of view. The superior angular resolution achievable thanks
to the 50-meter aperture planned for AtLAST will need to be complemented by
the capability of effectively recovering degree-level large scales. Such a
requirement is motivated by the aim of mapping the SZ signal from at low or
intermediate redshift astrophysical sources that are inherently extended on
large scales (e.g., intercluster filaments) and with diffuse signals (e.g.,
protocluster overdensities). At the same time, we aim at performing a deep
($\sim 10^{-7}$ Compton $y$) and wide-field ($>1000\leavevmode\nobreak\
\mathrm{deg^{2}}$) SZ survey, key for effectively probing a varied sample of
SZ sources. In turn, our requirement consists of an instantaneous field of
view covering $>1\leavevmode\nobreak\ \mathrm{deg^{2}}$. Clearly, combining
wide-field capabilities with enhanced sensitivity will be highly demanding in
terms of minimal detector counts. To reach the target sensitivities reported
in Table 1, we forecast that the focal plane array should be filled by
$\gtrsim 50,000$ detectors.

  * •

Wide frequency coverage. To perform a spectral inference of the multiple SZ
components, along with their clean separation from foreground and background
astrophysical contamination, it will be crucial to probe the spectral regime
from $30\leavevmode\nobreak\ \mathrm{GHz}$ up to $905\leavevmode\nobreak\
\mathrm{GHz}$ with multi-band continuum observations. We specifically identify
an overall set of nine spectral bands (centered at 42.0, 91.5, 151.0, 217.5,
288.5, 350.0, 403.0, 654.0, and 845.5 GHz), specifically selected to maximize
the in-band sensitivity at fixed integration time. By testing this spectral
configuration in the context of a mock spectral component separation, we
demonstrated that such a choice allows for achieving a clean separation of
multiple SZ components, as well as of the signal from dominant contamination
sources.

  * •

Sub-percent beam accuracy. An accurate calibration will be essential for
reducing potential systematics in the small-amplitude fluctuations of the SZ
signal associated with local pressure and velocity perturbations, or to
relativistic distortions. As such, we require a sub-percent level control of
the beam stability.

## Data and Software Availability

The calculations used to derive integration times for this paper were done
using the AtLAST sensitivity calculator, a deliverable of Horizon 2020
research project “Towards AtLAST”, and available from this link.

## Competing Interests

No competing interests were disclosed.

## Grant information

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No. 951815 (AtLAST).
L.D.M. is supported by the ERC-StG “ClustersXCosmo” grant agreement 716762.
L.D.M. further acknowledges financial contribution from the agreement ASI-INAF
n.2017-14-H.0. This work has been supported by the French government, through
the UCAJ.E.D.I. Investments in the Future project managed by the National
Research Agency (ANR) with the reference number ANR-15-IDEX-01.

Y.P. is supported by a Rutherford Discovery Fellowship (“Realising the
potential of galaxy clusters as cosmological probes”) and Marsden Fast Start
grant (“Turbulence in the Intracluster Medium: toward the robust extraction of
physical parameters”).

S.A. acknowledges INAF grant “Characterizing the newly discovered clusters of
low surface brightness” and PRIN-MIUR 20228B938N grant “Mass and selection
biases of galaxy clusters: a multi-probe approach”.

S.E. acknowledges the financial contribution from the contracts Prin-MUR 2022
supported by Next Generation EU (n.20227RNLY3 The concordance cosmological
model: stress-tests with galaxy clusters), ASI-INAF Athena 2019-27-HH.0,
“Attività di Studio per la comunità scientifica di Astrofisica delle Alte
Energie e Fisica Astroparticellare” (Accordo Attuativo ASI-INAF n.
2017-14-H.0), and from the European Union’s Horizon 2020 Programme under the
AHEAD2020 project (grant agreement n. 871158).

M.L. acknowledges support from the European Union’s Horizon Europe research
and innovation programme under the Marie Skłodowska-Curie grant agreement No
101107795.

D.N. acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG)
through an Emmy Noether Research Group (grant number NE 2441/1-1).

T.Mo. acknowledges the support of L. Page.

S.W. acknowledges support by the Research Council of Norway through the EMISSA
project (project number 286853) and the Centres of Excellence scheme, project
number 262622 (“Rosseland Centre for Solar Physics”).

## Acknowledgements

We thank William Coulton for useful insights on the CMB systematics and
recovery of larger scales, and Matthieu Béthermin for the support with the
analysis and processing of the SIDES simulations. We further thank Adam D.
Hincks for sharing the Planck and ACT SZ data for Abell 399–401.

## References

  * Abazajian et al. (2019) Abazajian, K., Addison, G., Adshead, P., et al. 2019, arXiv e-prints, arXiv:1907.04473
  * Abazajian et al. (2016) Abazajian, K. N., Adshead, P., Ahmed, Z., et al. 2016, arXiv e-prints, arXiv:1610.02743
  * Abdulla et al. (2019) Abdulla, Z., Carlstrom, J. E., Mantz, A. B., et al. 2019, ApJ, 871, 195
  * Adam et al. (2014) Adam, R., Comis, B., Macías-Pérez, J. F., et al. 2014, A&A, 569, A66
  * Adam et al. (2017) Adam, R., Bartalucci, I., Pratt, G. W., et al. 2017, A&A, 598, A115
  * Adam et al. (2018) Adam, R., Adane, A., Ade, P. A. R., et al. 2018, A&A, 609, A115
  * Akamatsu & Kawahara (2013) Akamatsu, H., & Kawahara, H. 2013, PASJ, 65, 16
  * Akamatsu et al. (2017) Akamatsu, H., Fujita, Y., Akahori, T., et al. 2017, A&A, 606, A1
  * Alberts & Noble (2022) Alberts, S., & Noble, A. 2022, Universe, 8, 554
  * Altamura et al. (2023) Altamura, E., Kay, S. T., Chluba, J., & Towler, I. 2023, MNRAS, 524, 2262
  * Amodeo et al. (2021) Amodeo, S., Battaglia, N., Schaan, E., et al. 2021, Phys. Rev. D, 103, 063514
  * Anbajagane et al. (2022) Anbajagane, D., Chang, C., Jain, B., et al. 2022, MNRAS, 514, 1645
  * Anbajagane et al. (2024) Anbajagane, D., Chang, C., Baxter, E. J., et al. 2024, MNRAS, 527, 9378
  * Anderson et al. (2022) Anderson, A. J., Barry, P., Bender, A. N., et al. 2022, in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy XI, 1219003
  * Andreon et al. (2019) Andreon, S., Moretti, A., Trinchieri, G., & Ishwara-Chandra, C. H. 2019, A&A, 630, A78
  * Andreon et al. (2022) Andreon, S., Trinchieri, G., & Moretti, A. 2022, MNRAS, 511, 4991
  * Andreon et al. (2017) Andreon, S., Wang, J., Trinchieri, G., Moretti, A., & Serra, A. L. 2017, A&A, 606, A24
  * Andreon et al. (2021) Andreon, S., Romero, C., Castagna, F., et al. 2021, MNRAS, 505, 5896
  * Andreon et al. (2023) Andreon, S., Romero, C., Aussel, H., et al. 2023, MNRAS, 522, 4301
  * Angelinelli et al. (2020) Angelinelli, M., Vazza, F., Giocoli, C., et al. 2020, MNRAS, 495, 864
  * Ansarifard et al. (2020) Ansarifard, S., Rasia, E., Biffi, V., et al. 2020, A&A, 634, A113
  * Arnaud et al. (2010) Arnaud, M., Pratt, G. W., Piffaretti, R., et al. 2010, A&A, 517, A92
  * Attoli et al. (2023) Attoli, A., Poppi, S., Buffa, F., et al. 2023, in 2023 XXXVth General Assembly and Scientific Symposium of the International Union of Radio Science (URSI GASS), 135
  * Ayromlou et al. (2023) Ayromlou, M., Nelson, D., Pillepich, A., et al. 2023, arXiv e-prints, arXiv:2311.06339
  * Bahcall & Cen (1992) Bahcall, N. A., & Cen, R. 1992, ApJ, 398, L81
  * Baldi et al. (2018) Baldi, A. S., De Petris, M., Sembolini, F., et al. 2018, MNRAS, 479, 4028
  * Bartalesi et al. (2024) Bartalesi, T., Ettori, S., & Nipoti, C. 2024, A&A, 682, A31
  * Bassini et al. (2020) Bassini, L., Rasia, E., Borgani, S., et al. 2020, A&A, 642, A37
  * Basu et al. (2016) Basu, K., Sommer, M., Erler, J., et al. 2016, ApJ, 829, L23
  * Battaglia et al. (2012) Battaglia, N., Bond, J. R., Pfrommer, C., & Sievers, J. L. 2012, ApJ, 758, 74
  * Baxter et al. (2021) Baxter, E. J., Adhikari, S., Vega-Ferrero, J., et al. 2021, MNRAS, 508, 1777
  * Baxter et al. (2024) Baxter, E. J., Pandey, S., Adhikari, S., et al. 2024, MNRAS, 527, 7847
  * Baxter et al. (2019) Baxter, E. J., Sherwin, B. D., & Raghunathan, S. 2019, J. Cosmology Astropart. Phys., 2019, 001
  * Bennett & Sijacki (2022) Bennett, J. S., & Sijacki, D. 2022, MNRAS, 514, 313
  * Bennett et al. (2024) Bennett, J. S., Sijacki, D., Costa, T., Laporte, N., & Witten, C. 2024, MNRAS, 527, 1033
  * Béthermin et al. (2017) Béthermin, M., Wu, H.-Y., Lagache, G., et al. 2017, A&A, 607, A89
  * Bhattacharya & Kosowsky (2007) Bhattacharya, S., & Kosowsky, A. 2007, ApJ, 659, L83
  * Bianchini & Silvestri (2016) Bianchini, F., & Silvestri, A. 2016, Phys. Rev. D, 93, 064026
  * Biffi et al. (2022) Biffi, V., ZuHone, J. A., Mroczkowski, T., Bulbul, E., & Forman, W. 2022, A&A, 663, A76
  * Biffi et al. (2016) Biffi, V., Borgani, S., Murante, G., et al. 2016, ApJ, 827, 112
  * Birkinshaw (1999) Birkinshaw, M. 1999, Phys. Rep, 310, 97
  * Birkinshaw et al. (1984) Birkinshaw, M., Gull, S. F., & Hardebeck, H. 1984, Nature, 309, 34
  * Bleem et al. (2024) Bleem, L., Klein, M., Abbott, T. M. C., et al. 2024, The Open Journal of Astrophysics, 7, arXiv:2311.07512
  * Bleem et al. (2020) Bleem, L. E., Bocquet, S., Stalder, B., et al. 2020, ApJS, 247, 25
  * Bonafede et al. (2015) Bonafede, A., Vazza, F., Brüggen, M., et al. 2015, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), 95
  * Bonafede et al. (2022) Bonafede, A., Brunetti, G., Rudnick, L., et al. 2022, ApJ, 933, 218
  * Bonamente et al. (2006) Bonamente, M., Joy, M. K., LaRoque, S. J., et al. 2006, ApJ, 647, 25
  * Boselli et al. (2022) Boselli, A., Fossati, M., & Sun, M. 2022, A&A Rev., 30, 3
  * Botteon et al. (2020a) Botteon, A., Brunetti, G., Ryu, D., & Roh, S. 2020a, A&A, 634, A64
  * Botteon et al. (2020b) Botteon, A., van Weeren, R. J., Brunetti, G., et al. 2020b, MNRAS, 499, L11
  * Bouwens et al. (2022) Bouwens, R. J., Smit, R., Schouws, S., et al. 2022, ApJ, 931, 160
  * Brodwin et al. (2016) Brodwin, M., McDonald, M., Gonzalez, A. H., et al. 2016, ApJ, 817, 122
  * Brownson et al. (2019) Brownson, S., Maiolino, R., Tazzari, M., Carniani, S., & Henden, N. 2019, MNRAS, 490, 5134
  * Brunetti & Jones (2014) Brunetti, G., & Jones, T. W. 2014, International Journal of Modern Physics D, 23, 1430007
  * Brunetti & Lazarian (2007) Brunetti, G., & Lazarian, A. 2007, MNRAS, 378, 245
  * Bryan et al. (2018) Bryan, S., Austermann, J., Ferrusca, D., et al. 2018, in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy IX, 107080J
  * Bulbul et al. (2024) Bulbul, E., Liu, A., Kluge, M., et al. 2024, The SRG/eROSITA All-Sky Survey: The first catalog of galaxy clusters and groups in the Western Galactic Hemisphere, arXiv:2402.08452
  * Butler et al. (2022) Butler, V. L., Feder, R. M., Daylan, T., et al. 2022, ApJ, 932, 55
  * Campitiello et al. (2022) Campitiello, M. G., Ettori, S., Lovisari, L., et al. 2022, A&A, 665, A117
  * Cantalupo et al. (2019) Cantalupo, S., Pezzulli, G., Lilly, S. J., et al. 2019, MNRAS, 483, 5188
  * Carilli et al. (2022) Carilli, C. L., Anderson, C. S., Tozzi, P., et al. 2022, ApJ, 928, 59
  * Carlstrom et al. (2002) Carlstrom, J. E., Holder, G. P., & Reese, E. D. 2002, ARA&A, 40, 643
  * Carlstrom et al. (2011) Carlstrom, J. E., Ade, P. A. R., Aird, K. A., et al. 2011, PASP, 123, 568
  * Carpenter et al. (2023) Carpenter, J., Brogan, C., Iono, D., & Mroczkowski, T. 2023, in Physics and Chemistry of Star Formation: The Dynamical ISM Across Time and Spatial Scales, 304
  * CASA Team (2022) CASA Team. 2022, PASP, 134, 114501
  * Cassano et al. (2015) Cassano, R., Bernardi, G., Brunetti, G., et al. 2015, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), 73
  * Cassano et al. (2023) Cassano, R., Cuciti, V., Brunetti, G., et al. 2023, A&A, 672, A43
  * Castagna & Andreon (2020) Castagna, F., & Andreon, S. 2020, A&A, 639, A73
  * CCAT-Prime Collaboration (2023) CCAT-Prime Collaboration. 2023, ApJS, 264, 7
  * Cen & Ostriker (1999) Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1
  * Chakraborty et al. (2023) Chakraborty, A., Chatterjee, S., Lacy, M., et al. 2023, ApJ, 954, 8
  * CHEX-MATE Collaboration (2021) CHEX-MATE Collaboration. 2021, A&A, 650, A104
  * Chisari et al. (2019) Chisari, N. E., Mead, A. J., Joudaki, S., et al. 2019, The Open Journal of Astrophysics, 2, 4
  * Chluba et al. (2012) Chluba, J., Nagai, D., Sazonov, S., & Nelson, K. 2012, MNRAS, 426, 510
  * Chluba et al. (2013) Chluba, J., Switzer, E., Nelson, K., & Nagai, D. 2013, MNRAS, 430, 3054
  * Churazov et al. (2015) Churazov, E., Vikhlinin, A., & Sunyaev, R. 2015, MNRAS, 450, 1984
  * Contini et al. (2014) Contini, E., De Lucia, G., Villalobos, Á., & Borgani, S. 2014, MNRAS, 437, 3787
  * Cooray & Chen (2002) Cooray, A., & Chen, X. 2002, ApJ, 573, 43
  * Coulton et al. (2024) Coulton, W. R., Schutt, T., Maniyar, A. S., et al. 2024, arXiv e-prints, arXiv:2401.13033
  * Crichton et al. (2016) Crichton, D., Gralla, M. B., Hall, K., et al. 2016, MNRAS, 458, 1478
  * Cuciti et al. (2022) Cuciti, V., de Gasperin, F., Brüggen, M., et al. 2022, Nature, 609, 911
  * Das et al. (2023) Das, S., Chiang, Y.-K., & Mathur, S. 2023, ApJ, 951, 125
  * De Filippis et al. (2005) De Filippis, E., Sereno, M., Bautz, M. W., & Longo, G. 2005, ApJ, 625, 108