# Mapping the Intracluster Medium in the Era of High-resolution X-ray
Spectroscopy

Congyao Zhang,1 Irina Zhuravleva,1 Maxim Markevitch,2 John ZuHone,3 François
Mernier,2,4 Veronica Biffi,5 Ákos Bogdán,3 Priyanka Chakraborty,3 Eugene
Churazov,6,7 Klaus Dolag,8,6 Stefano Ettori,9,10 William R. Forman,3 Christine
Jones,3 Ildar Khabibullin,8,6,7 Caroline Kilbourne,2 Ralph Kraft,3 Erwin T.
Lau,3 Sheng-Chieh Lin,11 Daisuke Nagai,12 Dylan Nelson,13 Anna Ogorzałek,2,4
Elena Rasia,5,14 Arnab Sarkar,15 Aurora Simionescu,16,17,18 Yuanyuan Su,11
Mark Vogelsberger,15 and Stephen Walker19

1 Department of Astronomy and Astrophysics, The University of Chicago,
Chicago, IL 60637, USA
2 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
3 Center for Astrophysics ❘ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
4 Department of Astronomy, University of Maryland, College Park, MD 20742, USA
5 INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, I34131 Trieste,
Italy
6 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741
Garching, Germany
7 Space Research Institute (IKI), Profsoyuznaya 84/32, Moscow 117997, Russia
8 University Observatory Munich, Scheinerstr 1, D-81679 Munich, Germany
9 INAF, Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti
93/3, 40129 Bologna, Italy
10 INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
11 University of Kentucky, 505 Rose street, Lexington, KY 40506, USA
12 Physics Department, Yale University, New Haven, CT 06511, USA
13 Universitat Heidelberg, Zentrum für Astronomie, Institut für theoretische
Astrophysik, Albert-Ueberle-Str. 2,
69120 Heidelberg, Germany
14 Institute for Fundamental Physics of the Universe, via Beirut 2, 34151,
Trieste, Italy
15 Department of Physics, Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
16 SRON Netherlands Institute for Space Research, Niels Bohrweg 4, NL-2333 CA
Leiden, the Netherlands
17 Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, the
Netherlands
18 Kavli Institute for the Physics and Mathematics of the Universe, University
of Tokyo, Kashiwa 277-8583, Japan
19 University of Alabama in Huntsville, Department of Physics and Astronomy,
Huntsville, AL 35899 E-mail<EMAIL_ADDRESS>
###### Abstract

High-resolution spectroscopy in soft X-rays will open a new window to map
multiphase gas in galaxy clusters and probe physics of the intracluster medium
(ICM), including chemical enrichment histories, circulation of matter and
energy during large-scale structure evolution, stellar and black hole
feedback, halo virialization, and gas mixing processes. An eV-level spectral
resolution, large field-of-view, and effective area are essential to separate
cluster emissions from the Galactic foreground and efficiently map the cluster
outskirts. Several mission concepts that meet these criteria have been
proposed recently, e.g., LEM, HUBS, and Super DIOS. This theoretical study
explores what information on ICM physics could be recovered with such missions
and the associated challenges. We emphasize the need for a comprehensive
comparison between simulations and observations to interpret the high-
resolution spectroscopic observations correctly. Using Line Emission Mapper
(LEM) characteristics as an example, we demonstrate that it enables the use of
soft X-ray emission lines (e.g., O VII/VIII and Fe-L complex) from the cluster
outskirts to measure the thermodynamic, chemical, and kinematic properties of
the gas up to $r_{200}$ and beyond. By generating mock observations with full
backgrounds, analysing their images/spectra with observational approaches, and
comparing the recovered characteristics with true ones from simulations, we
develop six key science drivers for future missions, including the exploration
of multiphase gas in galaxy clusters (e.g., temperature fluctuations, phase-
space distributions), metallicity, ICM gas bulk motions and turbulence power
spectra, ICM-cosmic filament interactions, and advances for cluster cosmology.

###### keywords:

galaxies: clusters: intracluster medium – methods: numerical – techniques:
imaging spectroscopy – X-rays: galaxies: clusters

††pubyear: 2023††pagerange: Mapping the Intracluster Medium in the Era of
High-resolution X-ray Spectroscopy–References

##  1 Introduction

The intracluster medium (ICM) is a gaseous component of galaxy clusters that
fills the space between galaxies, contributing the largest fraction of the
cluster’s baryonic mass. It is predominantly composed of low-density but
remarkably hot ($\sim 10^{7}-10^{8}$ K) weakly-magnetized plasma, emitting
X-rays via bremsstrahlung and lines of heavy elements (see, e.g., Kravtsov &
Borgani 2012; Vikhlinin et al. 2014; Walker et al. 2019 for reviews).

A number of processes that drive cluster growth (e.g., smooth accretion,
mergers) and evolution of the galaxies (e.g., feedback, gas stripping, and
mixing) are imprinted on the ICM, allowing detailed studies with X-ray and
sub-mm/Sunyaev-Zel’dovich (SZ) observations. In recent years, the gas
thermodynamic and chemical properties of many clusters have been measured in
the past years from core regions (e.g., by Chandra, XMM-Newton) up to virial
radii (e.g., Suzaku, SRG/eROSITA, and Planck). However, low spectral
resolution of current CCD-type X-ray detectors prevents direct gas velocity
measurements through the broadening and shift of emission lines.111With the
Reflection Grating Spectrometers (RGS) on XMM-Newton, it is possible to probe
gas dynamics within the compact innermost regions of some galaxy clusters,
group and massive galaxies (e.g., Sanders et al., 2010; de Plaa et al., 2012;
Pinto et al., 2015; Ogorzalek et al., 2017). Only in some special cases, CCD-
type detectors could be pushed to their limit to provide tentative detections
of the ICM bulk motions (e.g., Tamura et al., 2014; Liu et al., 2015; Sanders
et al., 2020; Gatuzz et al., 2022). The lack of sufficient energy resolution
also prevents robust measurements of different gas phases in the ICM and their
coupling, and abundances of various elements beyond Fe in most cases (see
Mernier et al. 2018 for a review).

Only recently, the Hitomi satellite, with an X-ray microcalorimeter on board,
mapped the velocity field and measured metal abundances in the central region
of the Perseus cluster with an unprecedented spectral resolution of $\simeq
5{\rm\,eV}$ (Hitomi Collaboration, 2016, 2017). As its successor, the
recently-launched XRISM observatory will soon observe many extended X-ray
sources with similar spectral capabilities (XRISM Science Team, 2020; Ezoe,
Ohashi, & Mitsuda, 2021), resolving individual lines in spectra of galaxy
clusters, providing kinematic properties of the ICM and allowing multi-
component gas decomposition (both in temperature and velocity). Given a modest
effective area of the telescope, XRISM will mainly focus on the brightest
cluster regions within $r_{2500}$222The $r_{\Delta}$ represents the radius
within which an average mass density is $\Delta$ (e.g., $2500,\ 500,\ 200$)
times the critical density of the Universe at the cluster redshift. with
$\simeq 1^{\prime}$ angular resolution.

Resolving X-ray spectra beyond the central cluster regions ($\gtrsim
r_{2500}$) is crucial for understanding chemical enrichment histories, global
energy transfer during large-scale structure evolution, how material is
brought from the intergalactic medium (IGM) into the ICM and gets mixed,
gaseous boundaries and halo virialization, and cluster cosmology. From a
theoretical point of view, understanding ICM physics in the cluster outskirts,
especially the IGM-ICM transition regions, is a complex multi-scale problem,
spanning $3-4$ orders of magnitude, from Mpc to sub-kpc (if not smaller,
considering the effects of magnetic fields). These outer regions are poorly
resolved in current cosmological simulations, given the extremely low gas
number density ($\lesssim 10^{-4}\,{\rm cm}^{-3}$). Therefore, properly
capturing non-linear processes, e.g., small-scale turbulence, multiphase gas
interactions, and discontinuities, is non-trivial.

Figure 1:  High-resolution X-ray spectroscopy is crucial for separating
diffuse components of the Milky Way and the ICM emission from the galaxy
cluster outskirts. The black and red curves show spectra of a cluster in the
annulus 1 - 1.1$r_{500}$ (ICM temperature $\simeq 1.5{\rm\,keV}$) at $z=0.08$,
including all background components and the ICM-only component, respectively,
simulated with the $2{\rm\,eV}$ resolution. Multiple strong emission lines
from the cluster (vertical dashes) stand out above the background. For
comparison, the blue curve shows the total spectrum with optimistic CCD
resolution ($50{\rm\,eV}$), for which separating the ICM from the background
is impossible (see Section 1).

Detailed observations of the ICM in cluster outskirts through mapping
thermodynamic, chemical, and kinematic properties will shed light on the
missing elements of the current theoretical picture and stimulate the
development of more physically motivated models. For this goal, one needs a
telescope with a large grasp - the product of the effective area and field-of-
view (FOV) of a telescope and high spectral resolution. A large effective area
is required to collect sufficient photons with reasonable exposure times and a
large FOV to map extended outer regions efficiently.

Intermediate and low-mass clusters ($\lesssim 5\times 10^{14}{\,M_{\odot}}$)
dominate the cluster mass function. Their typical gas temperatures are $\simeq
1-2{\rm\,keV}$ beyond $r_{500}$, corresponding to the peak of their X-ray
emission in a soft X-ray band, at energies below $\simeq 2{\rm\,keV}$.
However, at these low energies, the X-ray emission is dominated by a signal
from the Milky Way that shows a forest of emission lines, with the strongest
ones O VII and O VIII at the rest energies $\simeq 0.57{\rm\,keV}$ and
$0.65{\rm\,keV}$, respectively. In order to detect emission lines from the ICM
distinguished clearly from the Galactic foreground, the targets have to be
selected within specific redshift ranges, ideally $z\simeq
0.06-0.1$333Corresponding to $\simeq 0.24{\rm\,Gpc}^{3}$ comoving volume,
where it is expected to find a few thousand clusters with
$M_{200}>10^{14}{\,M_{\odot}}$ (e.g., Bocquet et al., 2016)., such as Abell
3112 and 2597 (e.g., Nevalainen et al., 2003; Tremblay et al., 2018), and the
X-ray spectral resolution of the instrument should be at least $\simeq
1-2{\rm\,eV}$ for the sake of accurate line profile measurements. This is
illustrated in Fig. 1. The redshifted strong emission lines from the ICM
(e.g., Fe XVII and O VIII) lie prominently above the background. They provide
valuable information on gas temperature, metal abundances, velocity
dispersion, and bulk velocity through line ratios, broadening, and shift
measurements. Emission lines in soft X-rays are extremely sensitive to gas
temperatures below $\simeq 2{\rm\,keV}$, see an illustration in Fig. 2 based
on the Astrophysical Plasma Emission Code (APEC) model (Smith et al., 2001).
Such features provide a unique opportunity to separate multiple gas phases,
particularly important in the cluster outskirts, where projection effects mix
the hot, volume-filling ICM, infalling substructures and their stripped
medium, and/or penetrating filaments.

Figure 2: Spectral template as a function of gas temperature $T_{\rm gas}$
(colours) from the APEC model, assuming solar metallicity. In the soft X-ray
band, emission lines are extremely sensitive to various temperatures below
$\simeq 2{\rm\,keV}$, providing a direct opportunity to investigate multiphase
gas in cluster outskirts (see Section 1).

Recently, several mission concepts that meet all of the above criteria have
been proposed, including Line Emission Mapper (LEM; see Kraft et al. 2022),
Hot Universe Baryon Surveyor (HUBS; see Cui et al. 2020; Bregman et al. 2023),
and Super Diffuse Intergalactic Oxygen Surveyor (Super DIOS; see Yamada et al.
2018; Ohashi et al. 2018) - all are soft X-ray missions with $1-2{\rm\,eV}$
spectral resolution.444Athena/NewAthena is another X-ray mission concept that
covers a broader energy band ($0.2-12{\rm\,keV}$) and enables spatially
resolved high-resolution X-ray spectroscopy with its X-IFU (Barret et al.,
2020). It shows strong capabilities to map ICM velocities in massive clusters
through the bright $6.7{\rm\,keV}$ Fe XXV K$\alpha$ line (e.g., Biffi, Dolag,
& Böhringer 2013; Roncarelli et al. 2018; see also Cucchetti et al. 2018;
Mernier et al. 2020 for the measurement of metal abundance within $r_{500}$).
However, limited by the small grasp of the X-IFU, observations would require a
prohibitive time investment to cover the entire cluster outskirts. It is
important to explore quantitatively what opportunities such missions provide
in the studies of ICM, in particular, what can be learnt about gas kinematics
and thermodynamics, chemical composition, and gas multiphaseness, especially
in complex and dynamic cluster outer regions. This is the primary scope of
this paper, i.e., to provide a theoretical guide on developing ICM/IGM science
drivers for future missions. Without loss of generality, we will focus on LEM
characteristics – an X-ray Probe Mission Concept for the 2030s, equipped with
a large microcalorimeter array covering a $30^{\prime}\times 30^{\prime}$ FOV
with $15^{\prime\prime}$ pixel size,555We used a $32^{\prime}\times
32^{\prime}$ LEM FOV in all our calculations, which was recently updated
during the manuscript preparation. This update does not affect any of our
conclusions. $1600\,{\rm cm}^{2}$ effective area at $0.5{\rm\,keV}$, and
$2{\rm\,eV}$ energy resolution and a central sub-array ($5^{\prime}\times
5^{\prime}$) with $0.9{\rm\,eV}$ resolution (Kraft et al., 2022), when
producing mock data based on cosmological simulations. In addition to
exploring what information could be recovered and the challenges, a
comprehensive comparison between simulations and observations will be carried
out to highlight the complexities of interpreting real data and understanding
their underlying physics. As shown below, this is a crucial step for correctly
bridging simulations and observations. In general, our methodology and most of
the conclusions are applicable to any of the telescopes mentioned above.

This paper is organized as follows. Section 2 describes the cluster samples
used in this study and their gas properties directly related to the X-ray
spectra. In Section 3, we present the main method of generating and analysing
mock observational spectra. In Section 4, we elucidate six key scientific
objectives in detail for the mission and their implications for understanding
cluster physics. In Section 5, we summarize our conclusions.

##  2 TNG300 Cluster Sample and ICM Properties

Table 1: Properties of clusters selected from the TNG300-1 sample for detailed studies in this work (see Section 2).  Name666The cluster’s name used in the paper. | TNG-ID777The ID of the cluster given in TNG300-1’s group catalog. | $M_{200}\ (10^{\rm 14}{\,M_{\odot}})$ | $r_{500}\ (\rm Mpc)$ | $r_{200}\ (\rm Mpc)$ | Status |  $z$888The assumed redshift of the cluster for LEM mock data, though our clusters are all from the TNG300-1 simulation’s $z=0$ snapshot. It is selected so that the major emission lines of the ICM can be well separated from the Galactic foreground and the virial region of the cluster ($\leq r_{200}$) can be covered by one LEM FOV.
---|---|---|---|---|---|---
CL-RS | 446262 | $0.95$ | $0.64$ | $0.97$ | fully-relaxed | 0.06
CL-RM | 180645 | $2.8$ | $0.92$ | $1.38$ | fully-relaxed | 0.08
CL-RL | 55060 | $6.3$ | $1.19$ | $1.81$ | relaxed | 0.12
CL-U | 150265 | $3.3$ | $0.95$ | $1.46$ | unrelaxed | 0.08

In this section, we briefly describe the simulated clusters used in this study
and characterize their physical properties directly relevant to the high-
resolution X-ray spectra.

Our cluster sample comes from the TNG300-1 cosmological simulation, a
simulation set with the largest box size ($\simeq 300{\rm\,Mpc}$) in the
IllustrisTNG project (Pillepich et al., 2018; Springel et al., 2018; Naiman et
al., 2018; Marinacci et al., 2018; Nelson et al., 2018, 2019), including
$\simeq 340$ galaxy clusters (total bound mass $\geq 10^{14}{\,M_{\odot}}$) at
redshift zero. Among these clusters, we select $30$ most relaxed ones based on
the following criteria: the clusters show visually smooth and spherical/near
spherical gas distributions in their cores ($\lesssim r_{500}$) and no
prominent substructures within $r_{200}$ based on their projected gas
distributions. The statistical properties of the full and relaxed cluster
samples are compared in this section.

For the spectral analysis, we mostly focus on four clusters from the TNG300
sample to demonstrate the capability and robustness of high-resolution X-ray
spectroscopy to understand the ICM physics (see Table 1): three relaxed ones,
Virgo- to Perseus-like clusters in terms of their total masses, and one
unrelaxed system (CL-U) with deeply penetrated filaments. The cluster CL-RM
with intermediate mass is one of the most relaxed clusters in the TNG
simulations, representing an ideal target for investigating cluster
hydrostatic mass bias. CL-RS, also fully relaxed, is similar to a fossil
group/cluster (e.g., Jones et al., 2003; Milosavljević et al., 2006), isolated
from prominent large-scale structures (e.g., halos, filaments; see Fig. 26) on
the $\sim 10{\rm\,Mpc}$ scale. CL-RL is a relatively perturbed cluster in our
relaxed cluster sample, revealing velocity substructures within $r_{500}$.

Current X-ray observations suggest a flat metallicity radial profile of the
ICM at $\gtrsim 0.5r_{500}$ (e.g., Urban et al., 2017; Ghizzardi et al.,
2021). There are, however, still large uncertainties in the cosmological
simulation predictions, sensitive to the sub-grid models and their parameters.
For the feasibility study, we assume that our clusters have uniform
metallicity in units of angr solar abundances ($Z=0.2$; see Anders & Grevesse
1989), throughout their volumes, unless otherwise stated. It provides a
simple, physically-motivated baseline model to characterize plausible errors
and biases in our mock observational measurements (see more discussions in
Section 4.2).

###  2.1 Gas radial velocity histograms

In this study, we focus on two types of cluster: (1) relaxed ones and (2)
perturbed ones with deeply penetrated filaments largely driven by mergers.
They represent two key states of clusters in the hierarchical structure
formation scenario. Analysing histograms of gas radial velocities in radial
shells provides a robust and convenient way to identify them from a large
cluster sample. This is illustrated in Fig. 3. Each curve in the right panels
shows a volume-weighted ($m_{\rm gas}/\rho_{\rm gas}$) histogram of gas radial
velocity at a given radius scaled by $u_{200}=\sqrt{GM_{200}/r_{200}}$, where
$G$ is the gravitational constant, $m_{\rm gas}$ and $\rho_{\rm gas}$ are mass
and density of gas cells. The ripples and bumps emerging in the distribution
highlight visually infalling substructures (e.g., small halos, filaments) and
merger-driven outflows.

Figure 3: Examples of a relaxed (the top panels) and a perturbed with deeply-
penetrated filaments (the bottom panels) clusters (see Fig. 26 for the other
two relaxed clusters listed in Table 1). The left and middle panels show dark
matter surface density and broad-band X-ray surface brightness
($0.5-2{\rm\,keV}$), including all dark matter particles/gas cells within
$4r_{200}$. The white dashed circles indicate cluster $r_{200}$. The black
dotted line marks a Mpc-scale bow shock identified in CL-U. The right panels
show histograms of gas radial velocities in narrow radial shells, providing a
convenient way to identify infalling substructures (e.g., subhalos,
filaments). The green dashed line illustrates the free-fall velocity profile
of the system, i.e., $u_{\rm r}/u_{200}=-\sqrt{2r_{200}/r}$. Penetrating
filaments appear as ripples and bumps on the negative side (towards the
cluster centre) of the velocity distributions (see Section 2.1).

In the top panels, we show the relaxed cluster CL-RM. Its radial velocity
distribution is smooth and narrow within $\simeq 1.5r_{200}\,(\simeq
r_{100}$), demonstrating the well-relaxed state of the cluster. At larger
radii, substructures start to appear (e.g., ripples and asymmetries). Most of
them have negative radial velocities, indicating that they are falling towards
the cluster centre. While we see subhalos in the dark matter surface density
and broad-band999Note that, we use the energy range $0.5-2{\rm\,keV}$ in our
X-ray predictions throughout this paper (referred to as ‘broad-band’
hereafter), including both surface brightness and X-ray-weighted quantities.
We have tested a wider range $0.2-2{\rm\,keV}$, which does not affect any of
our conclusions. X-ray surface brightness within $r_{200}$, they are mostly
projected from larger radii.

The bottom panels of Fig. 3 show the unrelaxed CL-U. It is one of the TNG300
clusters we identified that has large-scale filaments deeply penetrated into
the ICM, similar to those discussed by Zinger et al. (2016). In both dark
matter and X-ray maps, one can see the primary filament along the $y$-axis.
The cluster is also elongated along the same direction in projection. The
filament gas inflows all the way down to $0.5r_{200}$ and approximately
follows the free-fall velocity profile $u\propto\sqrt{1/r}$ assuming a simple
point-mass gravitational potential (green dashed line in the right panel). In
3D, the filament has a $\simeq 30-45^{\circ}$ inclination angle from the line-
of-sight (LOS; i.e., $z$-direction in the simulation). In Section 4.6, we will
see that such a geometric configuration is ideal for detecting large-scale
cosmic flows inside the ICM driven by filaments in observations.

It is worth noting that CL-U is a merging system. An infalling subcluster has
just passed the core and is located near the inner end of the filament,
leading the filament penetration into the ICM. The bow shock driven by the
subcluster is marked by the black dotted lines. The associated gas radial
inflow shown in the bottom right panel is, therefore, a mix of the merger-
driven wake and filament. Since most of the mergers occur along the filaments,
it is non-trivial to separate the diffuse gas of the filament and embedded
subhalos within the filament. Therefore, in this study, we refer to these two
components collectively as a ‘filament’. In the TNG300 cluster sample, we
identified $\sim 10$ clusters (out of $\simeq 340$) with clear large-scale
radial inflows reaching at least $r_{200}$, similar to CL-U. They all show
signs of ongoing mergers (e.g., multiple subhalos near the cluster core, bow
shocks) taking place close to the filamentary directions. Mergers likely play
an important role in enabling filaments to penetrate deeply into the ICM.
However, more quantitative studies on this topic are beyond the scope of this
paper. We postpone them to our future work.

###  2.2 Gas velocity dispersion

Fig. 4 shows radial profiles of characteristic gas 1-dimensional (1D) velocity
dispersion $\sigma_{\rm 1D}(r)$ normalized by the local sound speed $c_{\rm
s}(r)$ of our cluster sample,

$\sigma_{\rm 1D}(r)=\sqrt{\frac{1}{3}\sum_{i=x,y,z}{\sigma_{i}(r)^{2}}},$ (1)

where $\sigma_{i}$ is the mass-weighted velocity dispersion along $x$, $y$,
and $z$-directions at the given radius. The shaded regions indicate the 10th
and 90th percentile range of $\sigma_{\rm 1D}$ over the sample. Note that
large-scale bulk motions are not filtered out in the estimation (c.f., Vazza
et al., 2018) so that we can directly compare these results with observations.
Mergers and their associated infalling substructures may significantly boost
the velocity dispersion. To illustrate such an effect, we compare the velocity
dispersion profiles over the full TNG300-cluster sample (grey) and the
subgroup of relaxed clusters (green). Relaxed clusters show smaller
$\sigma_{\rm 1D}$ scatter by a factor of $\sim 2$ at all radii. The average
Mach number of their gas motions ranges from $\simeq 0.1$ in the central
region to $\simeq 0.4$ near $r_{200}$. The latter corresponds to a non-thermal
pressure fraction $\sim 20\%$, generally consistent with those reported in the
literature (e.g., Nelson, Lau, & Nagai, 2014; Biffi et al., 2016; Zhuravleva
et al., 2023). Interestingly, two samples show comparable lower boundaries
(10th percentile) of the velocity dispersion, reflecting the level of gas
random motions in the most quiescent regions of a cluster. Velocity dispersion
profiles of the four selected clusters are overlaid in the top panel (solid
lines). CL-RM and CL-RS are clearly among the most relaxed clusters in our
sample.

The bottom panel of Fig. 4 compares $\sigma_{\rm 1D}$ of our relaxed clusters
with a broad-band X-ray-weighted LOS velocity dispersion $\sigma_{\rm los}$
that includes all gas cells within $4r_{200}$. Cold clumps101010In the paper,
we use the term ‘clumps’ to refer generally to cold, dense, small gaseous
structures inside a cluster or along its LOS, which may or may not associate
with any galaxies or subhalos. make $\sigma_{\rm los}$ significantly deviate
from $\sigma_{\rm 1D}$ even within $r_{500}$ due to the projection effect.
Taking CL-RM as an example, a large difference between $\sigma_{\rm los}$ and
$\sigma_{\rm 1D}$ starts from $\sim 0.5r_{200}$ (see the red dotted and solid
lines), although the cluster is well relaxed and lacks prominent substructures
within $r_{200}$ (see Fig. 3). To exclude cold clumps, we take into account
only the hot gas with $T_{\rm gas}>1{\rm\,keV}$, which excludes $\lesssim 30$
per cent gas mass near $r_{200}$ but $\simeq 70$ per cent of X-ray flux along
the LOS – a simple and robust way to remove the bulk clumps’ contributions
(see Fig. 25 and Appendix A). It corresponds to the red dashed line, appearing
close to $\sigma_{\rm 1D}$. The deviation near $r_{200}$ is mainly caused by a
large substructure in the east of the cluster (see Fig. 3). This suggests that
the observable $\sigma_{\rm los}$ is a robust tracer of the true 1D velocity
dispersion (in 3D) of the bulk gas component in relaxed clusters if it is
possible to separate spatially and/or spectroscopically the cold gas
contribution from the hot ICM. In Section 4.4, we demonstrate the capability
of high-resolution X-ray spectroscopy to recover $\sigma_{\rm 1D}$ in relaxed
clusters.

Figure 4: Top panel: radial profiles of gas 1D-velocity dispersion
$\sigma_{\rm 1D}(r)$ in units of the local sound speed. The bands indicate the
10th and 90th percentiles of the dispersion over the whole cluster sample
(grey) and the relaxed cluster subsample (green). The solid lines show the
profiles of four individual clusters listed in Table 1. The vertical dashed
line marks $r_{500}$. Bottom panel: a comparison between $\sigma_{\rm 1D}$
(green band and red solid line, same as in the top panel but not normalized to
the sound speed) and broad-band X-ray-weighted gas LOS velocity dispersion
$\sigma_{\rm los}$ (pink band, and red dashed and dotted lines) of the relaxed
clusters. In particular, the lines compare the results for CL-RM. The dotted
and dashed lines take into account all gas cells and those that are only
hotter than $1{\rm\,keV}$, respectively. The cold clumps and small
substructures make $\sigma_{\rm los}$ strongly deviate from $\sigma_{\rm 1D}$
even within $r_{500}$ due to the projection effect (see Section 2.2).

###  2.3 Temperature distribution

Gas atmospheres in galaxy clusters are multi-temperature environments. Central
regions are often prone to rapid radiative cooling, leading to the presence of
cold gas phases mixed with the warm and hot components (e.g., Fabian et al.,
2003; Churazov et al., 2003; Sanders et al., 2016; Pinto et al., 2016; Hitomi
Collaboration, 2018), while mergers and accretion of matter continuously
supply outer cluster regions with multiphase gaseous structures (e.g.,
Roncarelli et al., 2013; Angelinelli et al., 2021; Gouin et al., 2023),
driving shocks and turbulence. Additionally, 3D temperature gradients,
including rapid temperature drop beyond $0.1-0.2r_{200}$ (e.g., Markevitch et
al., 1998; Vikhlinin et al., 2006), will be imprinted on the observed 2D
temperature. All of these will lead to broad temperature distributions at a
given radius and, if not taken into account in the spectral modelling, may
bias the recovered gas characteristics, especially in the cluster outskirts.
Similar biases on metal abundances and spectroscopic-like temperatures have
been discussed in the literature (e.g., Buote, 2000; Rasia et al., 2008;
Simionescu et al., 2009; ZuHone et al., 2023a).

Figure 5: Sample median PDFs of the ICM temperature in radial shells weighted
by gas mass (top) and in 2D annuli along the LOS weighted by broad-band X-ray
emissivity (bottom). The distance is shown in colour, $T_{0}$ is the median
temperature in radial shells (mass-weighted) or annuli (X-ray emissivity
weighted; see Section 2.3). Figure 6: PDF of temperature fluctuations at four
different radii shown in Fig. 5. Solid lines compare their median profiles in
radial shells (blue) and in 2D annuli (purple). The corresponding bands
indicate the 10th and 90th percentiles, illustrating the level of scatters
over the cluster sample. The black dotted and dashed lines show our best-fit
log-normal and Gaussian distributions for the profiles, respectively (see
Section 2.3).

In Fig. 5, we characterize ICM temperature fluctuations in our entire
TNG300-cluster sample relative to the median temperature. The top panel shows
sample median profiles of the probability distribution function (PDF) for
temperature fluctuation in narrow radial shells with bin width $0.05r_{200}$
weighted by gas mass. The PDFs can be well described by log-normal
distributions (Kawahara et al., 2007; Zhuravleva et al., 2013; Khedekar et
al., 2013), being broader at larger cluster radii (see also Fig. 6). The PDF
distribution can be understood as arising from the cumulative effects of
multiple processes (e.g., turbulence, shocks, merging structures) driving
temperature fluctuations. We approximate the radial dependence of the PDF’s
variance within the radial range $\leq 1.5r_{200}$ by

$\sigma_{\rm logt}(r)=\sigma_{0}(e^{\eta r/r_{200}}-1)+\sigma_{\rm logt}(0),$
(2)

where three best-fit parameters are $\sigma_{0}=1.0$, $\sigma_{\rm
logt}(0)=0.08$, and $\eta=0.2$, respectively. We note that different numerical
codes and sub-grid models used in the simulations may affect these parameters,
but the log-normal shape of the temperature distribution is unlikely changed
(Rasia et al., 2014). Moreover, the PDF of an individual cluster is not
smooth. There are variations/scatters caused by (resolved) individual
substructures, e.g., clumps and shocks, especially in the cluster outskirts.
In Fig. 5, we show only median profiles, where most scatters have been
cancelled. Our relaxed cluster-only sample gives almost the same results (not
shown in the paper). These profiles are referred to as the intrinsic
temperature distribution of the ICM. In Section 3.3, we will demonstrate that
it is crucial to take this intrinsic temperature distribution into account to
prevent systematic bias while fitting ICM model parameters (e.g.,
metallicity).

In the bottom panel of Fig. 5, we show the temperature PDF again but in 2D
annuli of projected temperature along the LOS, weighted by broad-band X-ray
emissivity. They are directly connected with X-ray spectra and show asymmetric
profiles due to the projection effect. The low-temperature tail of the
distribution is mostly contributed by the low-temperature gas at a larger
radius in the LOS (compare the top and bottom panels).

A direct comparison between PDF in 3D and in projection is shown in Fig. 6.
The latter is no longer log-normal but closer to a Gaussian distribution
because it is a sum of many log-normal distributions along the LOS. The best-
fit Gaussian variance ranges from $\simeq 0.2\ (0.1r_{200})$ to $0.3$
($r_{200}$; see also the blue line in Fig. 12). We have noticed that a skew-
normal distribution fits the PDF in the projection better, especially at the
high temperature end. However, as we discuss in Section 4.1, the skewness can
hardly be recovered when fitting mock X-ray spectra. Therefore, we no longer
consider it in this work.

##  3 Mock X-ray Images and Spectra

To illustrate quantitatively the unique opportunities of studying galaxy
clusters with soft X-ray calorimeters, we mainly focus on the LEM mission
concept. It provides a representative example of several perspective X-ray
spectroscopy missions, taking advantage of the large grasp and high-spectral
resolution, ideal for probing weak X-ray signals of cosmic structures.
However, our main results are general and applicable to other similar missions
(e.g., HUBS, Super DIOS; see Cui et al. 2020; Yamada et al. 2018).

###  3.1 Mock X-ray observation

We generate LEM mock cluster event files using pyXSIM111111https://hea-
www.cfa.harvard.edu/~jzuhone/pyxsim and SOXS121212https://hea-
www.cfa.harvard.edu/soxs Python packages, which simulate observations of X-ray
sources with current and future X-ray telescopes. Specifically, pyXSIM samples
synthetic photons according to gas distributions in TNG clusters with a Monte
Carlo approach based on the PHOX algorithm (Biffi et al., 2012; Biffi, Dolag,
& Böhringer, 2013), assuming the APEC model. Resonant scattering,
photoionization, and non-equilibrium ionization are ignored and likely
subdominant compared to other uncertainties. Then, SOXS convolves the sky-
projected event positions and energies with an instrumental response
($2{\rm\,eV}$ energy resolution unless stated otherwise). To properly capture
projection effects, we include all gas cells with zero star formation rate,
gas temperature $T_{\rm gas}>10^{5.5}{\rm\,K}$, and density $\rho_{\rm
gas}<0.5\,{\rm cm^{-3}}$ (close to the star formation density threshold in the
TNG simulations)131313These conditions are used to exclude a small number of
unphysically overdense, isolated, and X-ray bright gas cells in the simulation
that would not be expected to exist in real clusters but are a numerical
artifact. They represent a very small fraction of the mass and X-ray emission
of the cluster. Performing such cuts is a common procedure in similar works
(Rasia et al., 2013; Barnes et al., 2021; Pop et al., 2022; ZuHone et al.,
2023b) within at least $4r_{200}$ of a cluster when generating the mock.

Figure 7: Sky background model. The spectrum in grey indicates the Galactic
foreground. Four solid curves show $100$, $50$, $10$, and $1$ per cent of the
CXB described by an absorbed power law model. LEM sky background is adopted as
the sum of unresolved CXB ($50$ per cent) and Galactic radiation (black
spectrum). For comparison, broad-band X-ray surface brightness $I_{\rm x}(r)$
vs. X-ray temperature $T_{\rm x}(r)$ of three relaxed galaxy clusters listed
in Table 1 and one of the most massive TNG300 clusters (ID 11748) are overlaid
as the colour curves with points. The colour encodes the cluster radius. The
pairs of big triangles indicate $r_{500}$ and $r_{200}$, respectively. The ICM
X-ray signal from the cluster outskirts is weak and has a soft spectrum
(depending on the halo mass). The large grasp and high spectral resolution are
vital to measuring their signals (see Section 3.1).  Figure 8: Mock LEM
spectra from a TNG300 cluster (CL-RM) at redshift $0.08$. The left two panels
show distributions of X-ray-weighted temperature and X-ray surface brightness
of the cluster. Two white circles indicate $r_{500}$ and $r_{200}$. The right
panels show spectra extracted from two annuli marked in the bottom left panel
(darker regions). The green, yellow, and purple curves represent the
background (including both sky and instrumental contributions), ICM emission,
and the total spectrum, respectively. This figure illustrates the key role of
high spectral resolution in detecting a signal (i.e., several strongest
individual lines) from ICM, allowing unambiguous separation of the cluster’s
and Milky Way’s signals, even in low surface brightness, outer regions of
clusters (see Section 3.1).

Besides cluster emission, the background is also a major component of the
X-ray signal. The background includes Galactic foreground, cosmic X-ray
background (CXB), and detector background. Fig. 7 shows spectra of the (1)
Galactic foreground that includes both local hot bubble and gas halo (McCammon
et al., 2002) and (2) CXB approximated as an absorbed power-law model with
energy index $\Gamma=1.47$ (Hickox & Markevitch, 2006). For LEM, we expect
that $\sim 50$ per cent of CXB will be resolved, i.e., $\simeq 50-75$
brightest point sources per FOV, corresponding to a loss of $\simeq 1$ per
cent detector area while excising them. The rest of the (unresolved) CXB,
along with the Galactic emission, will be the sky background of LEM (black
solid curve), which will be taken into account when fitting the spectra. For
comparison, we show radial profiles of X-ray surface brightness
($0.5-2{\rm\,keV}$) of four individual relaxed clusters as a function of their
X-ray-weighted temperature $T_{\rm X}$ (colour lines with points). Two large
triangles mark $r_{500}$ and $r_{200}$, respectively. The curves are overlaid
in the figure by assuming that $k_{\rm B}T_{\rm X}$ corresponds to the
dominant X-ray photon energy in the emission (approximately peak position of
the bremsstrahlung continuum), where $k_{B}$ is the Boltzmann constant. This
figure illustrates that ICM X-ray emissions from the cluster outskirts
($r>r_{500}$) are generally soft and weak except for very massive halos (e.g.,
$\gtrsim 10^{15}{\,M_{\odot}}$). Below $2{\rm\,keV}$, it is non-trivial to
measure the continuum outside $r_{500}$. High-spectral resolution along with a
large grasp are essential to measure the ICM there via the detection of
individual emission lines. Finally, for the detector background, we simply
assume a flat spectrum $1\,{\rm cnt\,s^{-1}{\rm\,keV}^{-1}\,FOV^{-1}}$, based
on conservative scaling of the Athena X-IFU estimates (Barret et al., 2020),
which corresponds to $\simeq 10^{-10}\,{\rm
cnt\,s^{-1}\,cm^{-2}\,arcsec^{-2}{\rm\,keV}^{-1}}$ around $0.5-1{\rm\,keV}$.

Examples of our mock LEM spectra are shown in Fig. 8, extracted from two
annuli marked in the bottom left panel. Within $r_{500}$, the cluster’s signal
(yellow line) dominates the spectrum. Both continuum and emission lines (e.g.,
O VIII and Fe-L complex) are well resolved. However, this is not the case
outside $r_{500}$. Near $r_{200}$, the ICM continuum is more than 10 times
weaker than the background (see the bottom right panel). Thanks to the
cluster’s proper redshift, multiple strong lines from the cluster (e.g., O
VII, O VIII, and Fe XVII) are redshifted and separate unambiguously from the
Milky Way’s signals, providing an observational window to map the ICM in the
cluster outskirts.

###  3.2 Spatially identifying cold clumps

Figure 9: Top panel: modelled X-ray surface brightness from emission lines
between $510$ and $540{\rm\,eV}$ (observer frame) of the cluster CL-RM at
redshift $z=0.08$ (no continuum and instrumental response). In the cluster
frame, this energy interval includes the O VII lines near $570{\rm\,eV}$. Two
white circles indicate $r_{500}$ and $r_{200}$, respectively. Cold-gas clumps
from stripped infalling galaxies and/or groups are clearly seen in this
narrow-band image due to their strong O VII emissions. However, the central
extended bright halo is largely contributed by Fe XXIV line emissions from the
hot ICM. Bottom panel: $2{\rm\,Ms}$ mock LEM image – a difference between two
narrow energy bands to remove the continuum (see Section 3.2 for more
details). It reproduces well the simulation shown in the top panel,
demonstrating the robustness of identifying cold clumps with a LEM narrow-band
image. The red circles mark visually-identified clumps, which would be masked
when extracting spectra as one of our strategies in the study (see Section
3.3).  Figure 10: Similar to Fig. 9 but for the line Fe XXIII near
$975{\rm\,eV}$, tracing the hot ICM of the cluster (see Section 3.2).

The ICM is multiphase, especially in the cluster outskirts, as discussed in
Sections 2.2 and 2.3. It is possible to trace different gas phases through
narrow-band X-ray imaging (see Fig. 2). For instance, relatively cold and
dense clumps could be seen through the mapping of O VII triplet lines near
$570{\rm\,eV}$ that are largely populated by $T_{\rm gas}\simeq
0.1-0.5{\rm\,keV}$ gas, Fe XVII ($826{\rm\,eV}$) by $0.3-1{\rm\,keV}$ gas, and
Fe XXIII ($1050{\rm\,eV}$) by $\gtrsim 1{\rm\,keV}$ gas.

Fig. 9 shows narrow-band images of the CL-RM cluster around the O VII line.
The top panel shows a simulated prediction of the surface brightness from
emission lines only between $510$ and $540{\rm\,eV}$ in the rest frame of the
observer. In the cluster frame, this band contains O VII lines. The continuum
and instrumental response are not included on the map. Many cold gas clumps
are clearly seen through their strong O VII emissions. Most of them are
stripped material from infalling galaxies and/or groups. The biggest
substructure within the FOV is in the east of the cluster, visible even in the
broad-band X-ray and temperature distributions (see Figs. 8 and 3). Note that
the extended bright halo in the central region is not from O VII but weak Fe
XXIV lines emitted from the hot ICM ($T_{\rm gas}\gtrsim 1{\rm\,keV}$).

For comparison, the bottom panel of Fig. 9 shows a $2{\rm\,Ms}$ mock LEM
surface brightness $2I_{510-540}-I_{495-555}$, where $I_{510-540}$
($I_{495-555}$) is the surface brightness of the ICM in the energy band
$510-540{\rm\,eV}$ (495-555 eV). Subtraction of these two narrow bands removes
the continuum. For simplicity, we did not include any background in the image
since the background is assumed to be uniform spatially in our model, and
there are no strong Galactic emission lines in the adopted energy bands. The
background will only contribute Poisson noise to our results. The mock image
captures all the major features of the simulated image, demonstrating the
robustness of applying LEM narrow-band images to resolve cold clumps in the
ICM up to $r_{200}$ and even beyond. The red circles in the figure mark the
visually-identified clumps, which will be masked when extracting the spectra
as one of our strategies in the study (see Section 3.3). The mask used here is
rather aggressive – $\simeq 1/3-1/2$ of the region outside $r_{500}$ is
excluded.

Fig. 10 shows similar images but for Fe XXIII over a narrow energy band
$972-982{\rm\,eV}$ (i.e., $2I_{972-982}-I_{967-987}$), revealing distribution
of the hot gas phase of the cluster up to $r_{200}$, distinguishable from the
cold clumps’ distribution in Fig. 9.

###  3.3 Spectral analysis and Xspec model

We extract ICM X-ray spectra from mock event files and generate the background
component by using the fakeit command in the X-ray spectral fitting package
Xspec (Arnaud, 1996) while including the Galactic foreground, 50 per cent CXB
(as 50 per cent is removed as discrete, detected sources), and detector
background as described in Section 3.1 (see examples in Fig. 8).

We fit the total spectra to a combined model including both cluster and
background components within the energy range $0.4-1.5{\rm\,keV}$, assuming
Cash statistics (Cash, 1979). Fitting a wider energy band does not provide
much additional information since there are no strong emission lines. Fitting
errors are quoted at the $90$ per cent confidence level unless otherwise
noted. For simplicity, we fix all the parameters involved in the background so
that the background contributes merely statistical uncertainties to the
fitting results. This is a reasonable assumption for the purpose of this
study, given that the majority of strong lines from the cluster have been
redshifted away from the Galactic foreground lines. In reality, however,
temporal and spatial variations of the background could be taken into account
by treating some (if not all) background parameters free. It is expected that
our knowledge of the Milky Way foreground will soon be advanced through the
ongoing SRG/eROSITA and XRISM missions (Predehl et al., 2021; XRISM Science
Team, 2020).

We have shown in Section 2.3 that the distribution of the X-ray-emissivity-
weighted ICM temperature along the LOS can be approximated as a Gaussian
distribution. To account for this, we construct an Xspec model gaussbvapec. It
is similar to the built-in model vgadem with only two differences: (1)
velocity broadening is included as in the bvapec model and (2) standard
deviation of the Gaussian temperature distribution normalized by the mean
temperature is a free parameter ($\sigma_{\rm temp}$) of the model. In Section
4.2, we demonstrate that such an extended temperature distribution plays an
important role in suppressing bias in metal abundances based on the single-
temperature bvapec model.

Besides the intrinsic temperature spread of the ICM, there are also resolved
cold gas clumps that can significantly impact our fitting results. We apply
two strategies to deal with them and compare their pros and cons:

  * •

We have demonstrated in Section 3.2 that one can apply a mask to exclude the
majority of the cold clump contributions from the extracted spectra (see,
e.g., Fig. 9). The remaining (unresolved) cold gas makes a minimal
contribution to the spectra. We then need only a hot gas component in the
fitting model, either gaussbvapec (1g for short hereafter) or bvapec (1b), to
fit the ICM spectrum. The latter is, however, used only for comparison.

  * •

High-resolution X-ray spectroscopy provides the capability of resolving cold
clumps spectroscopically. It requires a major component and $N$ additional
colder components to fit the hot ICM and cold clumps, respectively. We apply a
gaussbvapec model for the former and place a prior constraint on its mean
temperature parameter (i.e., $0.5-10{\rm\,keV}$). We start our fitting from
$N=0$. In the following iterations, we carry out a new fitting each time by
adding one more clump component and check if it makes a non-negligible
contribution to the cluster signals, i.e., its normalization is larger than 1
per cent of the major component. This way is more robust than checking only
$\chi^{2}$, since background always dominates the spectrum in the cluster
outskirts. For simplicity, we model each clump component by only a bapec model
with a prior temperature constraint as $0.05-1{\rm\,keV}$. We stop the
iteration and take the previous best-fit results if the newly added cold
component is insignificant (weaker than 1 per cent of the major component). In
our experiment, $N\leq 2$ always provides sufficiently good fittings for our
sample. We refer to this model as 1g2b in the following discussions (see Table
2 for a summary of the models used in the study).

In general, we find that (1) with a mask, we are able to constrain metal
abundances of the hot ICM better (see Section 4.2); (2) without a mask, the
1g2b model accurately captures the multiphase gas in galaxy clusters,
providing a better constraint on the velocity dispersion especially in the
cluster outer regions since more photons are used in the fitting (see Section
4.4). The spectroscopically resolved cold clumps provide important phase-space
information of the infalling substructures in galaxy clusters (see Section
4.3).

Table 2: A summary of our Xspec models used to fit the LEM mock spectra (see Section 3.3).  Model name |  With mask | Hot comp. | Additional cold comp.
---|---|---|---
(1st) | (2nd | 3rd)
1g | Yes | gaussbvapec | - | -
1b | Yes | bvapec | - | -
1g1b141414This model is only used in Section 4.6. | No | gaussbvapec | bapec | -
1g2b | No | gaussbvapec | bapec | bapec

##  4 Science Drivers

In this section, we discuss six key science drivers for future LEM-type X-ray
missions on the ICM science and highlight the capabilities of high-resolution
X-ray spectroscopy in pushing ICM measurements and understanding their
underlying physics to a new level.

Based on our TNG300 cluster sample, we present (1) measurements of gas
thermodynamic (Section 4.1), metal abundance (Section 4.2), and non-thermal
pressure (Section 4.4) profiles up to $r_{200}$, (2) reconstruction of gas
phase-space distributions (Section 4.3), and (3) mapping of gas LOS velocity
distributions to constrain ICM turbulence properties (Section 4.5) and detect
ICM-filament interactions (Section 4.6). We provide strategies to optimize
data analysis and interpretation. Unless otherwise stated, we assume that the
observation duration of each cluster is $t_{\rm exp}=2{\rm\,Ms}$. X-ray
photons are radially binned to measure the ICM radial profiles. The minimum
number of counts from the cluster in each bin and the minimum radial size of
the bin are $10^{5}{\rm\,Ms}^{-1}$ and $1^{\prime}$, respectively.

###  4.1 Gas thermodynamics of the ICM

Fig. 11 shows the LEM best-fit gas density radial profile of the cluster CL-RM
and compares it with the true profile in simulation (green line). The
deprojected radial profiles are reconstructed from the best-fit normalization
of the hot ICM component using the standard onion-peeling approach (Kriss et
al., 1983). The statistical uncertainties are small ($\lesssim 5$ per cent
near $r_{200}$) and barely visible in the plot. The simulation curve is
estimated by excluding all gas cells colder than $1{\rm\,keV}$ to remove cold
clumps that create strong spikes on top of the global radial profile. The LEM
measurements recover the density radial profile of the ICM. Different fitting
approaches show almost identical results. Determining slopes of gas density
profiles outside $r_{500}$ is vital to understanding the evolution of the
runaway merger shocks and how they re-shape matter distributions in the
cluster outskirts (Zhang et al., 2019, 2020). Cosmological simulations suggest
a steep gas density profile near the virial radius ($\propto r^{-3}$ or
steeper; see, e.g., Roncarelli et al. 2006; Lau et al. 2015), implying the
existence of a “habitable zone” for runaway shocks that can power radio relics
in the peripheries of the cluster (e.g., Lyskova et al., 2019). However,
current X-ray measurements are often contaminated by the complexity of
clumpiness (e.g., Nagai & Lau, 2011). In our experiment, we do not see any
significant gas density overestimation caused by such an issue, thanks to
LEM’s capability of separating the hot ICM and cold clumps spectroscopically.
Fig. 11 shows that high-resolution X-ray spectroscopy accurately recovers the
normalization and slope of the ICM density radial profile up to $r_{200}$ and
beyond, which will allow distinguishing different slopes of gas density
profiles along filamentary and non-filamentary directions (see, e.g., Vurm et
al. 2023 and also Section 4.6).

Figure 11: A comparison of the LEM best-fit deprojected density radial profile
of the hot ICM (points with error bars) with the simulation prediction (green
line) for the cluster CL-RM. The model includes only gas cells with $T_{\rm
gas}\geq 1{\rm\,keV}$ to exclude contributions from cold clumps. The vertical
dashed line marks $r_{500}$. The LEM measurements accurately recover the
simulation, independent of the strategies for resolving cold clumps (see
Section 4.1).

In Fig. 12, we show the best-fit ICM projected temperature profile (the top
panel). The error bars represent the intrinsic temperature spread of the ICM
$T_{\rm gas}\times\sigma_{\rm temp}$ rather than the uncertainty of the
temperature statistics (smaller than $\simeq 3$ per cent at $r_{200}$). For
comparison, we estimate the broad-band X-ray-weighted temperature histogram as
a function of the projected radius shown as the grey band. We use the same
mask as in the mock LEM analysis to exclude clumps (see Fig. 9). The upper and
lower boundaries indicate $\pm 1\sigma$ of the distribution, and the middle
line shows the mean temperature. Our mock measurements agree with the
underlying model. They slightly overestimate the temperature near $r_{200}$ by
$\lesssim 10$ per cent due to the complex temperature structure in the cluster
outskirts (e.g., deviation from Gaussian; see Fig. 15). Our 1g and 1g2b models
show almost identical results.

Figure 12: Top panel: LEM measurements of the projected temperature profile of
the ICM in CL-RM (points with error bars). The vertical error bars show the
intrinsic temperature spread of the ICM $T_{\rm gas}\times\sigma_{\rm temp}$
rather than the statistical uncertainties, which are very small ($\sim$ a few
per cent). The grey band shows the simulated broad-band X-ray-weighted
temperature distribution. The upper and lower boundaries indicate $\pm
1\sigma$ of the distribution, and the middle line shows the mean. Cold clumps
are excluded using the same mask as in the mock LEM analysis. The green lines
show the modelled X-ray-weighted temperature profiles without any mask, while
the solid line includes only gas cells with $T_{\rm gas}>1{\rm\,keV}$. Bottom
panel: Best-fit standard deviation $\sigma_{\rm temp}$ of the ICM temperature
distribution normalized by the mean gas temperature in the gaussbvapec model
in each annulus. The green and blue lines show the predicted profile of CL-RM
and the average over the entire TNG300-cluster sample, respectively. This
figure shows that, in addition to the mean temperature, LEM can recover
$\sigma_{\rm temp}$, providing essential information about the ICM temperature
structure (see Section 4.1).

The top panel of Fig. 12 shows a direct view of the intrinsic temperature
distribution of the ICM. It cannot be characterized merely by a single
temperature at a given radius. The green lines show the modelled broad-band
X-ray-weighted temperature profiles without any mask. In particular, the solid
line includes only the gas cells hotter than $1{\rm\,keV}$. The lines show up
to $\sim 40$ per cent deviations around $r_{200}$ from each other (also
compared to the grey line). It is non-trivial to unambiguously define the
X-ray temperature of the ICM in the cluster outskirts, even in original
simulations; it is sensitive to the chosen criteria of separating the hot ICM
and cold gas components. For this reason, a two-parameter model (i.e.,
temperature peak and spread) provides a more comprehensive characterization of
the ICM temperature structure.

In the bottom panel of Fig. 12, we show a direct comparison of our best-fit
$\sigma_{\rm temp}$ with the simulation. The green and blue lines show a
modelled profile of the cluster CL-RM and an average over our TNG300 cluster
sample (see Fig. 6). The recovered $\sigma_{\rm temp}$ is encouragingly close
to the prediction and is not sensitive to the scheme used to exclude the
resolved cold clumps, given the complexity of resolving the ICM temperature
structure. The well-relaxed CL-RM shows a slightly smaller $\sigma_{\rm temp}$
than the average but a similar radial profile shape. This result demonstrates
high-resolution spectroscopy’s capability of measuring the intrinsic gas
temperature distribution, which could be used to explore turbulence, mixing,
and gas transport properties of the ICM. Besides, the mild rise of the
$\sigma_{\rm temp}$ profile in the innermost region of the cluster ($\lesssim
0.2r_{200}$) likely reflects the active galactic nucleus (AGN) activity, which
can be used to constrain the impact of feedback on the ICM.

###  4.2 Metal abundances

When and how the hot and/or warm gas phase of the intergalactic medium
(including the ICM) was enriched with metals is an important open question in
large-scale structure evolution. Metal-abundance radial profiles in cluster
outskirts provide a clean diagnostic for different enrichment scenarios for
several reasons: (1) the cluster outer regions ($\gtrsim r_{500}$) are not
affected strongly by the activity of the cluster’s central supermassive black
hole; (2) metal distributions in the cluster outskirts reflect the enrichment
of gas being accreted recently from the cosmic web; and (3) early AGN feedback
($z>2-3$) that stirs and mixes metals outside of galaxies considerably impacts
the slopes of abundance profiles outside $\sim 0.5r_{200}$ (see, e.g., Biffi
et al. 2018 and also Mernier et al. 2018 for a review).

Figure 13: Best-fit metal abundances of the cluster CL-RM while assuming a
uniform metallicity distribution ($Z=0.2$). The panels show the abundance
parameters fitted by our model: oxygen-to-iron ratio, iron abundance, and
abundance of other elements (from the top to bottom). The horizontal dotted
lines mark the input value. The 1g model recovers the parameters the best,
demonstrating that modelling a temperature distribution (e.g., a Gaussian
distribution) is vital for correct measurements of abundance profiles (see
Section 4.2).

Suzaku measured the radial profiles of Fe abundance in massive clusters (e.g.,
Perseus) up to $r_{200}$ based on Fe-K lines (e.g., Werner et al., 2013; Urban
et al., 2017), suggesting a flat iron distribution extending from $\simeq
0.3r_{200}$ to at least $r_{200}$ ($\simeq 0.3$ solar). However, it is still
under debate whether such a feature is universal from massive clusters to
galaxy groups and if other, lighter elements (e.g., $\alpha$-elements) follow
the same pattern (e.g., Simionescu et al., 2015; Sarkar et al., 2022). Future
high-energy resolution X-ray missions will have the capability to address
these questions, as discussed below. In this study, we focus on recovering
iron and oxygen abundances that emit the strongest lines in soft X-rays with
high-resolution X-ray spectroscopy. While fitting the spectra, we use three
parameters in the model: $Z_{\rm Fe}$, $Z_{\rm O}/Z_{\rm Fe}$, and $Z_{\rm
others}$, representing iron abundance, oxygen-to-iron ratio, and the abundance
of other elements, all with respect to solar.

Fig. 13 shows the results based on our baseline cluster model - with uniform
metallicity ($Z=0.2$) throughout the cluster, where we can unambiguously
characterize the biases and uncertainties in the measurements. We compare our
best-fits from different fitting schemes. The horizontal dotted lines mark the
input abundance value. The 1g model correctly recovers all parameters up to
$r_{200}$. In contrast, the 1b model reveals prominent biases outside
$0.3r_{200}$, $Z_{\rm Fe}$ in particular. The deviation of $Z_{\rm O}/Z_{\rm
Fe}$ from unity is largely caused by the bias in Fe abundance. Masks are
applied in both cases to spatially exclude the cold clumps’ contribution (see
Fig. 9). The strong dependence of the Fe-L complex on gas temperature (see
Fig. 2 and also Gastaldello et al. 2021) largely explains why the single-
temperature assumption leads to large biases in our fitting. It highlights the
importance of modelling the ICM temperature structure (a Gaussian
distribution) to accurately measure metal distributions. In Fig. 13, the 1g
and 1g2b models show similar constraints on $Z_{\rm Fe}$ and $Z_{\rm others}$.
The latter reveals a larger scatter in $Z_{\rm O}/Z_{\rm Fe}$, because cold
clumps generate both O VII and O VIII, complicating the spectral modelling and
interpretation in the 1g2b model.

Figure 14: Similar to Fig. 13 but assuming a non-uniform cluster metallicity
directly from the cosmological simulation. The yellow and orange lines show
the simulated Fe-abundance radial profile in 3D and the broad-band X-ray-
weighted profile in projection where only gas cells with $T_{\rm
gas}>1{\rm\,keV}$ are included. The points with error bars show our best-fit
parameters $Z_{\rm Fe}$ (green) and $Z_{\rm O}/Z_{\rm Fe}$ (blue) from the 1g
model, fully consistent with the simulation input. The non-uniform metallicity
distribution does not make the abundance recovery more difficult (see Section
4.2).

In Fig. 14, we explore a more realistic situation, where the metallicity
varies spatially (i.e., from the simulation directly). It is known that the
ICM metallicity in TNG300-1 does not converge to those in TNG100-1 due to the
dependence of stellar mass on numerical resolution (Vogelsberger et al.,
2018). We include a factor of $1.6$ to adjust TNG300-1 to TNG100-1 as
suggested by Vogelsberger et al. (2018) and still assume solar abundance
ratios for all the elements, e.g., $Z_{\rm Fe}/Z=1$ (Simionescu et al., 2015;
Mernier et al., 2017). The yellow and orange lines show the predicted Fe-
abundance radial profile in 3D and projection (including only $T_{\rm
gas}>1{\rm\,keV}$ gas and weighted by broad-band X-ray emissivity). The
projection effect shows only a mild effect on the curve. The green and blue
points show our best-fit $Z_{\rm Fe}$ and $Z_{\rm O}/Z_{\rm Fe}$ based on the
1g model, fully consistent with the predictions, similar to those shown in
Fig. 13, although with larger error bars due to lower metallicity in this
model (e.g., $Z\simeq 0.1$ near $r_{200}$).

###  4.3 Multiphase gas in the outskirts

Although cold clumps complicate the X-ray spectra and recovery of the ICM
temperature and metal abundances, they provide valuable information on the
assembly history of galaxy clusters and star formation quenching (e.g., merger
rate, gas stripping) and transport properties of the ICM (e.g., viscosity,
thermal conduction). High-resolution spectroscopy enables cold clumps to be
separated from the hot ICM spatially and spectroscopically, allowing direct
measurements of the clump velocities and other physical properties.

Figure 15: Broad-band X-ray-weighted gas temperature PDF (top) and LOS
velocity PDF (bottom) of the cluster CL-RM in projection. The line colour
indicates the annular radius. The corresponding shaded regions show the PDFs
after excluding clumps using the same mask as in the mock LEM analysis. The
thin dotted lines in the top panel show the best-fit Gaussian distributions
for the shaded temperature PDFs. This figure illustrates the complexity of gas
structures in an individual cluster that encodes valuable information in the
high-resolution X-ray spectra (see Section 4.3).

Fig. 15 illustrates the complexity of the temperature and velocity
distributions of the ICM (including all gas phases and substructures) in an
individual cluster (cf., Fig. 6). The top panel shows the temperature PDF. It
follows approximately a Gaussian distribution in the inner region, where the
contribution of small substructures to X-ray emissivity is negligible. At
larger radii, the distribution becomes more asymmetric and reveals a broad
low-temperature tail with $T_{\rm gas}\sim 0.1-1{\rm\,keV}$. Outside
$0.7r_{200}$, one can see a secondary temperature peak caused by the largest
subhalo in the east of the cluster (see Fig. 9). We emphasize that this
distribution is a mix of the intrinsic ICM temperature distribution (see
Section 2.3) and cold clumps in the cluster outskirts along the LOS. The
corresponding shaded regions show the same PDFs after excluding resolved cold
clumps using the same mask as in the LEM mock analysis (see Fig. 9),
highlighting the intrinsic ICM temperature distribution. The thin dotted lines
show their best-fit Gaussian distributions, which capture the majority of the
PDFs but not the remaining low-temperature tails. These tails contribute a
second-order effect to our fitting results (e.g., intrinsic temperature spread
of the ICM), which require a more quantitative characterization in future
work. The bottom panel of Fig. 15 shows the corresponding LOS velocity PDF.
The central peak represents the hot ICM component with a peculiar velocity
$\simeq-200{\rm\,km\,s^{-1}}$, which tends to be slightly broader at a larger
radius, reflecting the velocity dispersion of the system. The bumps/peaks on
the side show substructures of the cluster. Most of them are infalling towards
the cluster centre, which can be clearly seen in Fig. 16.

Figure 16: Top panel: phase-space distribution of gas cells in CL-RM – radial
position vs. radial velocity. The gas cells are randomly selected (1 out of
50) from the simulation, whose colour encodes the temperature. The cyan line
shows the (inverse) escape velocity profile. Bottom panel: phase-space
distribution in projection, i.e., projected radius vs. LOS velocity, using the
same set of gas cells. The cyan dashed lines show the maximum escape velocity
projected along the LOS (see more details in Section 4.3). Our mock LEM
measurements are overlaid as the points with error bars. The full bar sizes
(vertical) reflect the FWHM of the velocity fields ($\propto\sigma_{\rm
los}$). The white points show the primary component, the hot ICM. The green
and yellow points show the second and third cold gas components. We slightly
shift the green and yellow points horizontally for a clear view. The points
overlap consistently with the distribution of the hot and cold gas (see also
Fig. 17), demonstrating the capability of LEM to resolve the multiphase gas in
the cluster outskirts clearly and to determine its kinematics (see Section
4.3).

The top panel of Fig. 16 shows the phase-space distribution of gas cells in
the cluster, i.e., radial position vs. radial velocity (see also Fig. 3). The
colour encodes the gas temperature. The cyan line shows the (inverse) escape
velocity profile $v_{\rm esc}(r)=-\sqrt{2\Phi(r)}$ as a reference, where
$\Phi(r)$ is the gravitational potential of the cluster. Inside $r_{500}$
(even $r_{200}$), we barely see any velocity substructures. Most of the cold
gas (e.g., $T_{\rm gas}<0.5{\rm\,keV}$) lies outside $\simeq 1.2r_{200}$ and
has a negative radial velocity close to $v_{\rm esc}$. In observations,
instead of gas radial velocities, LOS velocities are measured. The cluster’s
phase-space distribution in projection is shown in the bottom panel of Fig.
16, i.e., gas projected radial position vs. LOS velocity, using the same set
of gas cells as in the top panel. Unlike the top panel, we see a fair amount
of cold gas inside $r_{200}$ (even within $r_{500}$). This is a pure
projection effect, implying that a non-negligible cold-gas component should
always be expected in the X-ray spectra, even extracted from an inner region
of a well-relaxed cluster. On the other hand, the same projection effect
allows us to probe gas dynamics outside $r_{200}$ in 3D but at projected radii
below $r_{200}$, opening a window to gas dynamics in non-virialized regions.
The two cyan dashed lines show the maximum escape velocity projected along the
LOS, i.e., $v_{\rm esc,max}(r_{\rm proj})=\max{\Big{[}v_{\rm
esc}(r)\sqrt{1-(r_{\rm proj}/r)^{2}}\Big{]}}$, where $r\in[r_{\rm
proj},\infty)$. They are shifted by $-200{\rm\,km\,s^{-1}}$ by taking the
peculiar velocity of the cluster into account, enveloping most of the gas
points.

Figure 17: Same data as in the bottom panel of Fig. 16 but showing the gas
temperature-LOS velocity diagram. The background points include only the gas
cells within $r_{\rm proj}<1.1r_{200}$ (approximately LEM FOV). The colour
encodes the projected radius. The cold gas within $(0.7-1)r_{200}$ is
recovered as two components with the temperature $0.2-0.3$ and
$0.5-0.6{\rm\,keV}$, respectively (see Section 4.3).

The overlaid points with error bars in Fig. 16 show our mock LEM measurements
based on the 1g2b model. The bars indicate the best-fit velocity dispersion
$\kappa\sigma_{\rm los}$, where $\kappa\ (=1.2)$ is a factor included so that
the full bar size reflects the full width at half maximum (FWHM) of the
velocity field if Gaussian. The white points show the hot ICM component. The
cluster’s peculiar velocity is accurately recovered. The best-fit FWHM (i.e.,
the length of the bars) is $\simeq 300-400{\rm\,km\,s^{-1}}$, consistent with
the distributions shown in Fig. 15. The green and yellow points indicate the
second and third components in our fitting, modelling the cold gas, ordered by
their normalization parameter. To exhibit only the most prominent cold clumps
resolved in spectra, we show the data points with normalization greater than 5
per cent of the hot component. Their corresponding temperature-LOS velocity
diagram is shown in Fig. 17. Most of these points have positive LOS
velocities, in line with the asymmetric velocity PDF shown in Fig. 15 (solid
lines), largely contributed by the prominent substructure in the east of the
cluster. Between $0.7-1r_{200}$, the cold clumps appear as two temperature
components in our model: the first one with $T_{\rm gas}\simeq
0.2-0.3{\rm\,keV}$ and the other with $T_{\rm gas}\simeq 0.5-0.6{\rm\,keV}$,
reflecting the structures of the temperature PDF in Fig. 15. The colder
component tends to have larger temperature uncertainties in the fitting. The
points showing large velocity dispersions reflect overlapping clumps moving in
opposite directions. Note that our simplified model fitting could be improved,
especially for resolving multiple cold gas clumps along the LOS, i.e., the
points shown in the figure may be further split into more gas components in a
more detailed analysis. Additional multi-wavelength observations (e.g.,
optical, infrared) may also provide prior constraints on the model’s
parameters, helping to interpret the spectra.

Overall, our mock measurements capture the phase-space distribution of the
cold gas and the temperature distribution, demonstrating the capability of LEM
to resolve cold clumps spectroscopically and recover their physical
properties. The determined $u_{\rm los}$ and $\sigma_{\rm los}$ can provide an
approximate radial position of the corresponding cold gas based on the $v_{\rm
esc}-v_{\rm esc,max}$ relation. Two important questions to answer for missions
like LEM in the future are: what is the minimum radius that the falling cold
gas ($\sim 0.1-1{\rm\,keV}$) can reach in relaxed clusters? And how does this
gas co-exist and co-move (if there is any) with its host galaxies and the hot
gas phase?

###  4.4 Velocity dispersion and non-thermal pressure

Velocity dispersion driven by ICM turbulence can be measured directly through
the broadening of X-ray emission lines. In clusters, turbulence dominantly
contributes to non-thermal pressure support for atmospheres (Lau, Kravtsov, &
Nagai, 2009; Lau, Nagai, & Nelson, 2013), balancing part of the gravitational
attraction. For this reason, the hydrostatic mass usually underestimates the
true cluster mass (e.g., Evrard, 1990; Kay et al., 2004; Rasia et al., 2006;
Nagai, Vikhlinin, & Kravtsov, 2007) and biases the cluster mass function – the
basis of the X-ray cluster cosmology. Direct measurements of the non-thermal
pressure are essential to address this issue, correcting cluster mass biases.
Hitomi mapped the inner region of the Perseus cluster and found an approximate
uniform velocity dispersion distribution $\sigma_{\rm los}\simeq
150-200{\rm\,km\,s^{-1}}$ (Hitomi Collaboration, 2016). XRISM will soon
observe more clusters, focusing mainly on their brightest central regions. It
is, however, non-trivial to extrapolate measurements of $\sigma_{\rm los}$ in
the core to the outskirts due to the fact that: (1) AGN feedback and merger-
driven sloshing generate strong bulk flows that often dominate the velocity
field; and (2) cosmological simulations suggest that the non-thermal pressure
fraction is an increasing function of cluster radius, expected to be low near
the cluster centre (e.g., Nelson, Lau, & Nagai, 2014; Shi et al., 2015). The
outskirts of relaxed clusters, alternatively, provide a unique opportunity to
measure non-thermal pressure, overcoming both aforementioned issues. This is
practically doable only if the instrument simultaneously has both a large
grasp and high-spectral resolution.

Figure 18: Best-fit velocity dispersion profiles of the cluster CL-RM. The
grey line and shaded region show the simulated profile of $\sigma_{\rm 1D}$
and its scatter across three independent directions (see Eq. 1 and also Fig.
4). The black dashed line shows the broad-band X-ray-weighted velocity
dispersion along the LOS, taking into account only gas cells with $T_{\rm
gas}>1{\rm\,keV}$. The three panels compare our mock LEM measurements
depending on different fitting models (top), exposures (middle), and energy
resolutions (bottom). All the models show consistent results, in line with the
simulated expectation. Their errors scale with the exposure $t_{\rm exp}$ as
expected. A sub-eV spectral resolution can dramatically improve the velocity
dispersion measurements (see Section 4.4).

Fig. 18 shows our experiments on how well LEM can measure $\sigma_{\rm los}$
in the bulk gas component up to $r_{200}$ based on the cluster CL-RM, one of
the most relaxed clusters in the TNG simulations. The points show the best-fit
velocity dispersion profiles obtained through different fitting models using
different exposures and energy resolutions. The grey line and the shaded
region show the simulated prediction of $\sigma_{\rm 1D}$ (see Eq. 1) and its
scatter in three independent directions ($x,\ y$, and $z$), which implies the
small velocity dispersion anisotropy in the cluster. The black dashed line
shows the broad-band X-ray-weighted velocity dispersion along the LOS,
including only the gas cells hotter than $1{\rm\,keV}$ (see also Fig. 4). Our
fitting schemes/models all provide consistent results, in line with the
simulated prediction, shown in the top panel. The 1g2b model gives smaller
error bars since there are more photons in the data without a mask. Similar
results from the 1g and 1b models imply that the measurement of the velocity
dispersion is not very sensitive to the intrinsic temperature structure of the
ICM. In the middle panel, we check the dependence of our velocity measurements
on the exposure time, $t_{\rm exp}$. We find that the uncertainties in
$\sigma_{\rm los}$ scale with an exposure time as $\varpropto 1/\sqrt{t_{\rm
exp}}$ and $t_{\rm exp}\gtrsim 1{\rm\,Ms}$ is necessary to achieve
$50{\rm\,km\,s^{-1}}$ accuracy in the cluster outskirts ($\geq r_{500}$)
depending on the radial binning schemes. In the bottom panel, we experiment
with $0.9{\rm\,eV}$ energy resolution while keeping all other instrumental
settings the same. The sub-eV resolution turns out to be a game changer. With
only half of the exposure ($1{\rm\,Ms}$), the error bars are reduced by a
factor of $\simeq 2$, equivalent to $\sim 8$ times more efficient in
determining the velocity dispersion than in $2{\rm\,eV}$ resolution. LEM will
have a $0.9{\rm\,eV}$ sub-array in the middle of its detector, which would be
useful to advance the measurements, especially in cluster outskirts (e.g.,
covering a larger region using the same exposure and achieving the same level
of velocity precision with a multi-pointing strategy).

Figure 19: Non-thermal pressure fraction profile of the cluster CL-RM. The
grey line shows the simulated prediction. The points show the mock LEM
measurements, estimated based on the recovered gas density, temperature, and
velocity dispersion profiles. They are consistent with the theoretical curve,
showing $\sim 30$ per cent uncertainties near $r_{200}$. This figure
illustrates the LEM’s capability of distinguishing low ($\simeq 0.1-0.2$) and
high ($\gtrsim 0.4$) non-thermal pressure fraction within $\simeq
r_{500}-r_{200}$ by measuring the most relaxed clusters (see Section 4.4).

Fig. 19 shows the non-thermal pressure fraction profile of the cluster CL-RM,
defined as

$f_{\rm nth}(r)=\frac{P_{\rm nth}(r)}{P_{\rm gas}(r)+P_{\rm nth}(r)},$ (3)

where $P_{\rm gas}(r)$ and $P_{\rm nth}(r)\equiv\rho_{\rm gas}(r)\sigma_{\rm
1D}(r)^{2}$ are the gas thermal and non-thermal pressure profiles,
respectively. The grey line shows the simulated prediction, derived from the
ICM density, temperature, and velocity dispersion profiles shown in Figs. 11,
12, and 18. The data points show our mock LEM measurements, estimated based on
the recovered gas profiles. Their error bars take into account only the
velocity dispersion errors, since the statistical uncertainties on recovered
density and temperature are negligible by comparison. The $\sigma_{\rm 1D}$ in
Eq. 3 is replaced by $\sigma_{\rm los}$, valid for well-relaxed clusters. Both
1g and 1g2b models recover the prediction, with $\simeq 30$ per cent
uncertainties near $r_{200}$. The measured $f_{\rm nth}$ can be converted to
cluster hydrostatic mass bias ($1-b$) based on cosmological simulations (see,
e.g., fig. B1 in Shi et al. 2016 for a conversion at $r_{500}$) and/or
analytical models (e.g., Shi & Komatsu, 2014). The precision of our
measurements is sufficient to distinguish low-level $f_{\rm nth}\simeq
0.1-0.2$ within $\simeq r_{500}-r_{200}$, corresponding to the mass bias
$1-b\gtrsim 0.8$ suggested by the current cosmological simulations (e.g., Lau,
Kravtsov, & Nagai, 2009; Shi et al., 2015), and high-level $f_{\rm nth}\gtrsim
0.3$ corresponding to $1-b\lesssim 0.7$, considered as a plausible solution
for the tension between the cluster X-ray/SZ constraints on cosmological
parameters ($\Omega_{\rm m},\sigma_{8}$) and those from the primary cosmic
microwave background anisotropies (Planck Collaboration, 2014, 2016). We note
that future weak lensing observations show great potential for calibrating
cluster mass scaling relations. However, their mass measurements can still
suffer from various systematic effects (see, e.g., Umetsu 2020 for a review).
Direct measurements of cluster hydrostatic masses will be an essential
complementary for precision cluster cosmology, as they come with different
systematics.

Figure 20: Similar to Fig. 18 but for the clusters CL-RL (top) and CL-RS
(bottom; see Section 4.4).  Figure 21: Gas LOS velocity distribution of the
cluster CL-RM. Three panels from left to right show the simulation (broad-band
X-ray weighted), LEM mock measurement, and measurement errors, respectively.
In the prediction, we exclude gas cells with $T_{\rm gas}<1{\rm\,keV}$ to
remove contributions from the cold clumps. Three white dashed circles indicate
characteristic radii. LEM accurately recovers the velocity distribution except
for some regions overlapping prominent clumps (e.g., the reddish regions in
the middle panel). Within $r_{500}$, we get $\lesssim 30$ per cent
uncertainties in most of the region (see Section 4.5).

In Fig. 20, we show the velocity dispersion measurements for two other
clusters: CL-RL (top), a massive cluster with perturbed gas velocity field at
redshift $z=0.12$, and CL-RS (bottom), a quiescent fossil group/cluster at
$z=0.06$ (see Table 1). In the latter, we obtain a good recovery of
$\sigma_{\rm 1D}$, similar to Fig. 18, with even smaller uncertainties
everywhere. Because of the absence of prominent substructures within the whole
FOV, the spectra from CL-RS can be fitted well with a single gas component.
This is confirmed by our 1g model, showing similar results with and without a
mask to exclude small clumps. The temperature of the cluster lies below
$1{\rm\,keV}$ outside $r_{500}$, emitting bright O VII and Fe XVII lines that
provide additional constraints on the velocity dispersion of the hot
atmosphere compared to more massive clusters like CL-RM. This explains the
small velocity uncertainties in our measurement ($\lesssim 20$ per cent near
$r_{200}$), suggesting that quiescent fossil groups/clusters are ideal targets
for future LEM-type missions to determine the lower limit of $f_{\rm nth}$ in
galaxy clusters. The cluster CL-RL, on the contrary, is massive and not fully
relaxed. Nevertheless, our measurements are still able to recover $\sigma_{\rm
los}$. However, in this cluster, the X-ray-weighted $\sigma_{\rm los}$ profile
(dashed curve) noticeably deviates from $\sigma_{\rm 1D}$ (grey region) due to
the complexity of the gas substructures (see Fig. 26). The measured
$\sigma_{\rm los}$ does not fully reflect the $f_{\rm nth}$ profile, which
implies that future observations should focus on well-relaxed clusters to
investigate $f_{\rm nth}$.

###  4.5 Line-of-sight bulk motions and velocity structure function

Many physical questions require statistical information about the velocity
field as a function of scale or spatial separation measured through, e.g.,
power spectrum or structure function. For instance, the slope of the velocity
amplitude measured as a function of scale could tell us how the kinetic energy
is transported from large to small scales and how it dissipates, providing a
window to constrain the microphysical properties of the ICM (e.g., Gaspari et
al., 2014; Zhuravleva et al., 2019). By mapping LOS velocities and estimating
their velocity structure function, one can constrain the power spectrum of ICM
motions, i.e.,

${\rm SF}(d)\propto\int_{0}^{\infty}{P_{\rm 3d}(k)k{\rm d}{k}},$ (4)

where ${\rm SF}(d)\equiv\langle|u_{\rm los}(\textbf{r}+\textbf{d})-u_{\rm
los}(\textbf{r})|^{2}\rangle_{\textbf{r}}$ is the second-order structure
function, $\langle\rangle_{\textbf{r}}$ indicates an average over all velocity
pairs on the sky with a spatial separation $d\ (=|\textbf{d}|)$, and $P_{\rm
3d}(k)$ is the 3D velocity power spectrum (see ZuHone, Markevitch, &
Zhuravleva 2016, their eqs. 8-12, and also Zhuravleva et al. 2012). The
injection scales ($\ell_{\rm inj}$) and dissipation ($\ell_{\rm diss}$)
scales, key parameters in $P_{\rm 3d}$, define the inertial range. The former
is closely related to velocity drivers in the system (i.e., mergers and
feedback), and the latter is sensitive to the microphysics of the ICM.

Fig. 21 shows our mock LEM results on measuring the LOS velocity distribution
of the relaxed cluster CL-RM (middle) and its comparison with the simulated
prediction (left). The right panel presents the corresponding measurement
error. In the analysis, we adaptively bin X-ray photons with a centroid
Voronoi tessellation approach (e.g., Cappellari & Copin, 2003) by limiting the
minimum spatial resolution $l_{\rm min}$ of the bins and the minimum photon
number $N_{\rm cnt}$ from the cluster in each bin. In this way, we get roughly
uniform grids in the cluster’s inner region (e.g., $\lesssim 0.5r_{500}$) and
a sufficient signal-to-noise ratio in the outskirts. Note that fixing the
signal-to-noise ratio over the entire FOV is difficult, given the steep radial
profile of the X-ray surface brightness in the cluster outskirts. In this
section, we use $l_{\rm min}=0.5^{\prime}\ (\simeq 45{\rm\,kpc})$ and $N_{\rm
cnt}=10^{4}{\,\rm cnt{\rm\,Ms}^{-1}}$, and fit the spectra with our standard
1g2b model. There are, in principle, a sufficient number of photons to achieve
a higher spatial resolution in the cluster’s inner regions. To better resolve
the hot ICM component, we slightly tighten our prior constraint on its mean
temperature parameter ($T_{\rm gas}\in[0.9,\,10{\rm\,keV}]$; see Section 3.3).
However, this affects only the regions dominated by cold clumps (reddish
regions in the middle panel).

Figure 22: Square root of the second-order velocity structure function of the
cluster CL-RM. The red line and the shaded region show our LEM mock
measurement using all velocity pairs within $0.5r_{500}$ and its
uncertainties. The red points with error bars indicate the contribution from
the velocity statistical errors (see Eq. 5). Yellow and purple lines show the
corresponding simulation results in 2D and 3D, where the solid and dashed
lines are estimated within $0.5r_{500}$ and $r_{500}$, respectively. This
figure demonstrates LEM’s capability of measuring the ICM velocity structure
function and using it to constrain the injection scale (see Section 4.5).

The middle panel of Fig. 21 shows our recovery of $u_{\rm los}$, accurately
matching the simulation, particularly within $r_{500}$. Some differences
appear in the regions that overlap with the resolved cold clumps (see Fig. 9).
In our simulation result (the left panel), we exclude gas cells with $T_{\rm
gas}<1{\rm\,keV}$ to diminish cold-clump effects. In the LEM mock data,
however, it is difficult to resolve hot ICM components from outer regions
dominated by clumps (e.g., the big one in the east), even with a tighter prior
temperature parameter constraint. We tend to see large velocity scatter and
uncertainties in the fitting there. Within $\simeq r_{500}$, no significant
clumps are identified. We are able to achieve $\lesssim 30\ (20)$ per cent
velocity errors (neglecting calibration and gain uncertainties) within
$r_{500}\ (0.5r_{500})$, which allows us to constrain the injection scale and
a slope of the velocity power spectrum, as discussed below.

Fig. 22 shows the square root of the second-order velocity structure function
${\rm SF}(d)$. The red line and the shaded region show the LEM mock
measurement and its uncertainties (90 per cent confidence level) using all
velocity pairs within $0.5r_{500}$. The uncertainties are modelled using a
Monte Carlo approach taking into account (1) the statistical velocity
measurement errors, assumed to follow Gaussian distributions (see the right
panel in Fig. 21) and (2) the finite spatial size of each bin and its
irregular shape. The latter is subdominant in our case, where all bins within
$\simeq 0.5r_{500}$ have high roundness. In observations, the measured
structure function can be written as

${\rm SF}_{\rm obs}(d)={\rm SF}_{\rm true}(d)+{\rm SF}_{\rm err}(d),$ (5)

a sum of the true function and the velocity error contribution, if the gas LOS
velocity $u_{\rm los}$ and its statistical uncertainty $\Delta u_{\rm los}$
are spatially uncorrelated (see appendix C in ZuHone, Markevitch, & Zhuravleva
2016). The term ${\rm SF}_{\rm err}(d)$ is simply flat if the velocity error
is normally distributed with zero mean. In reality, $\Delta u_{\rm los}$
inevitably depends on the radius of the cluster. We model ${\rm SF}_{\rm
err}(d)$ in our mock measurement, shown as red points with error bars. It
depends only mildly on the separation, explains the slight flattening, and
increases uncertainties of the observed structure function at small
separations.

The yellow lines in Fig. 22 show the projected structure function estimated
directly from the simulation data within $0.5r_{500}$ (solid) and $r_{500}$
(dashed). It is not surprising that the LEM result agrees with the simulation,
given the well-recovered $u_{\rm los}$ map (Fig. 21). For comparison, we also
show the velocity structure functions measured in 3D (purple lines) using the
3D velocity field information rather than the X-ray-weighted projection. The
2D functions reveal smaller amplitudes and steeper slopes due to the strong
projection effect. In our cluster sample, the patchiness of the velocity
becomes significant outside $\simeq 0.5r_{500}$, largely driven by
substructures moving in the ICM. Such bulk motions on $\sim 10^{2}{\rm\,kpc}$
length scales complicate the velocity structure function (compare the solid
and dashed lines). For the goal of characterizing turbulence (or random
motions) in the ICM, any significant bulk motions should be avoided if
possible, e.g., masking regions with prominent LOS bulk motion patterns.

Figure 23: Constraining the velocity power spectrum (injection scale and the
slope of the inertial range) with LEM’s measured structure function. The red
line with points shows our LEM mock measurement, the same as the one in Fig.
22. The colour bands show theoretical models with different underlying
injection scales ($\ell_{\rm inj}$) indicated by the corresponding vertical
arrows, based on Gaussian random velocity fields. The modelled structure
functions show a strong dependence on $\ell_{\rm inj}$ if $\ell_{\rm
inj}\lesssim 1{\rm\,Mpc}$ (approximately the measurement’s aperture size
$r_{500}$). Our LEM result favours $\ell_{\rm inj}\gtrsim 1{\rm\,Mpc}$,
demonstrating the robustness of using the velocity structure function to
constrain $\ell_{\rm inj}$ of the ICM (see Section 4.5).

To interpret the observed structure function, we generate Gaussian random
velocity fields with underlying energy power spectra $E(k)$ as baseline models
(Hosking & Schekochihin, 2022),

$E(k)=C_{0}\frac{(k\ell_{\rm inj})^{\alpha_{\rm e}}}{1+(k\ell_{\rm
inj})^{\alpha_{\rm e}-4}}e^{-k\ell_{\rm diss}},$ (6)

where $C_{0}$ is a normalization constant setting the standard deviation of
$u_{\rm los}$ as $70{\rm\,km\,s^{-1}}$ to match our mock LEM structure
function’s amplitude, $\alpha_{\rm e}=-5/3$ is the spectral index, $\ell_{\rm
inj}$ and $\ell_{\rm diss}$ are injection and dissipation scales,
respectively. We model the LOS velocity distribution weighted by X-rays as
$u_{\rm los}=\int{\varepsilon_{\rm x}u_{z}{\rm d}z}$, where $\varepsilon_{\rm
x}=\rho_{\rm gas}^{2}/\int{\rho_{\rm gas}^{2}{\rm d}z}$ is the approximate
X-ray emissivity. The gas radial density profile of the cluster CL-RM is
applied in the model (see Fig. 11). The velocity structure functions are shown
as the colour bands in Fig. 23 using the same aperture radius $0.5r_{500}$ as
in the LEM mock analysis (red points). The bandwidth shows a standard
deviation over 100 realizations, illustrating the level of cosmic variance. We
fix $\ell_{\rm diss}=1{\rm\,kpc}$ and find that the modelled structure
function exhibits a strong dependence on $\ell_{\rm inj}$ when $\ell_{\rm
inj}\lesssim 1{\rm\,Mpc}$. Our TNG300 cluster sample favours a large
(dominant) injection scale $\ell_{\rm inj}\gtrsim 1{\rm\,Mpc}$, and its
structure function slope is consistent with Kolmogorov’s $5/3$ power law. We
stress that flattening of the structure function occurs near $d\simeq\ell_{\rm
inj}/3-\ell_{\rm inj}/2$. The aperture size in the measurement (e.g.,
$r_{500}$ in our case) is thus approximately the upper limit of the injection
scale that can be constrained via the velocity structure function.
Furthermore, the apparent slope of the function on small scales is shallower
than the (asymptotic) theoretical expectation ($\propto d^{5/3}$; see eq. 12
in ZuHone, Markevitch, & Zhuravleva 2016), even at $d\simeq\ell_{\rm
inj}/10^{2}$, suggesting that it is important to fit the entire structure
function to constrain parameters (e.g., $\ell_{\rm inj}$ and $\alpha_{\rm e}$)
while taking into account projection effects, error contributions, and
flattening at large separation. We have further tested various $\ell_{\rm
diss}$ from $1{\rm\,kpc}$ to $\ell_{\rm inj}/10$ and found that the effect,
i.e., slight steepening of the curve, is weak, barely distinguished by LEM and
other similar telescopes.

###  4.6 Penetrating filaments in galaxy clusters

The measured gas LOS velocity is contributed by two sources in general: (1)
large-scale bulk motions driven by mergers and accretion and (2) volume-
filling random motions in the ICM. The latter often dominates the velocity
fields in relaxed clusters, as already exemplified in the previous section.
Mapping the LOS velocity of unrelaxed clusters, on the other hand, can provide
direct evidence of interaction between galaxy clusters and the surrounding
cosmic web.

The velocity structures of merging systems can be resolved by observing
cluster mergers with their merger axes away from the sky plane. Similar
strategies have been applied in the cluster kinematic SZ observations (e.g.,
Mroczkowski et al., 2012; Adam et al., 2017; Sayers et al., 2019). On the
other hand, filaments that penetrate into the ICM can also drive Mpc-scale
bulk motions (Zinger et al., 2016), playing an important role in thermalizing
the IGM. Our TNG300 cluster sample suggests that mergers and deep filament
penetration are likely to be often associated with each other. The radial
velocities they drive are up to $\sim 10^{3}{\rm\,km\,s^{-1}}$, comparable to
their free-fall velocities, corresponding to $u_{\rm los}\gtrsim
500{\rm\,km\,s^{-1}}$ with $\gtrsim 30^{\circ}$ inclination angles.

Figure 24: Gas LOS velocity distribution of the cluster CL-U – an unrelaxed
cluster with two deeply penetrating filaments indicated by white arrows. The
left panel shows the broad-band X-ray-weighted velocity distribution,
including all gas cells in the simulation. The overlaid black contours
represent the X-ray surface brightness (see also Fig. 3). Zoom-in images of
the region marked by the red box are highlighted in the four right panels – a
comparison between the simulations (the middle column) and LEM mock
measurements (the right column). The top (bottom) panels compare the LOS
velocity of the hot (cold) ICM component, respectively (see more details in
the text). Our LEM results well resolve two large-scale filaments associated
with the cluster. The north one appears in the hot gas phase, and the south
one in the cold phase (see Section 4.6).

The cluster CL-U (see Table 1) represents such systems, connected with two
remarkable filaments from the north and the south, respectively. Fig. 24 shows
its gas LOS velocity distribution. The left panel exhibits a broad-band X-ray-
weighted velocity distribution, including all gas cells in the cluster. The
overlaid contours reveal the X-ray surface brightness (see Fig. 3). Both
filaments extend beyond the scope of the image and show negative LOS
velocities along them (i.e., moving away from the observer). Our mock LEM
observations focus on the virial region marked by the red box. To bin X-ray
counts, we select $l_{\rm min}=1^{\prime}$ and $N_{\rm cnt}=2\times 10^{4}{\rm
cnt{\rm\,Ms}^{-1}}$ for our primary goal of identifying large velocity
structures in the cluster outskirts. We apply a 1g1b model to fit the spectra
from the cluster, i.e., one component for the hot gas phase and no more than
one for the cold ($N\leq 1$; see Section 3.3). No additional cold components
are included because of the low signal-to-noise ratios at large radii. It is
sometimes non-trivial to split cold gas into multiple motion components based
on several emission lines (see Section 4.3). We have compared our results
based on the 1g1b and 1g2b models. They show similar velocity distributions
for the hot ICM component. The 1g2b model, however, gives larger velocity
scatter and larger errors for the cold gas.

The right four panels in Fig. 24 show our best-fit LEM results (the right
column) and their comparisons with the simulations (the middle column). In the
model, we separate the contributions from the different gas phases by using
only gas cells with $T_{\rm gas}>1{\rm\,keV}$ (top) and $0.1<T_{\rm
gas}<1{\rm\,keV}$ (bottom). Note that the filament in the north appears mostly
in the bottom panels, implying its low characteristic temperature, i.e., $\sim
5-10$ times lower than the ambient ICM. The filament in the south is revealed
in both gas phases in the simulation. The cold phase has a higher LOS velocity
than its hot companion; the hot phase shows azimuthally more extended
distributions. However, our LEM mock results only recover the latter. In the
mock observational results, we compare the best-fit hot and cold components in
the fitting model. The top and bottom panels in the right column show the
hottest components with the best-fit temperature $T_{\rm gas}>1{\rm\,keV}$ and
$T_{\rm gas}<1{\rm\,keV}$, respectively. The latter represents the cold gas
phase, since the hot ICM cannot always be resolved from the cluster outskirts
(i.e., two components both capture the cold gas in the fitting). We present
only the bins with velocity errors $|\Delta u_{\rm
los}|<200{\rm\,km\,s^{-1}}$, i.e., non-empty regions in the figure. The LEM
images show similar velocity patterns as in the simulation. Two filaments are
well resolved not only in their LOS velocities but also their morphologies in
projection (e.g., gaseous boundaries, opening angles), which are important to
explore the co-evolution of multiphase gas and member galaxies inside the
filaments.

Filaments connected with a cluster can lead to anisotropic mass accretion and
reshape ICM boundaries (Zhang et al., 2021; Vurm et al., 2023). Their
interactions with the ICM are still poorly resolved in the current
cosmological simulations, e.g., the formation of boundary layers and shock
structures. This is why we focus only on detecting large-scale features of the
penetrating filaments in this section (e.g., gas velocity and geometry). In
reality, the complexity of soft X-ray lines can be vital diagnostics of the
instabilities and mixing formed near the filament-ICM interfaces. Filaments
inside clusters are typically cold (i.e., $0.1-1{\rm\,keV}$), unable to
produce Fe-K lines. LEM-type X-ray telescopes provide unique capabilities to
explore their co-evolution with the clusters.

##  5 Conclusions

High-resolution X-ray spectroscopy in soft X-rays will open a new window to
map multiphase gas in galaxy clusters, especially in the cluster outskirts,
allowing us to probe the transition between halos’ virialized and non-
virialized regions. Using LEM as an example, we illustrate the essential role
of eV-level spectral resolution in separating cluster emission from the
Galactic foreground. It enables us to probe the strongest lines (e.g., O VII,
O VIII, and Fe-L complex) from the ICM and cold substructures and to measure
gas velocities and metal abundances from line shifts, line broadening, and
line ratios. Prominent line features and their dependence on gas temperature
guarantee that they are sensitive tracers of gas in different phases. Cluster
targets must be within specific redshift ranges, ideally $z\simeq 0.06-0.1$.

By mocking LEM observations with full background information, analysing their
images/spectra, and comparing extensively with simulation predictions, we
demonstrate that, with $\simeq 1-2{\rm\,Ms}$ exposures, LEM can reliably probe
the gas thermodynamic, chemical and kinetic properties of the ICM from the
cluster core to the outskirts. We emphasize that our methodology and main
findings are applicable to other similar X-ray missions as well, e.g., HUBS
and Super DIOS, providing theoretical guides for their development of the
ICM/IGM science objectives. Our main conclusions are summarized below.

  * •

Gas density and temperature radial profiles of the bulk, volume-filling ICM
can be measured with high accuracy out to $r_{200}$ and beyond, as well as the
width of the Gaussian-shape intrinsic temperature distribution. They are not
prominently affected by gas clumpiness. Both the mean temperature and the
normalized variation ($\sigma_{\rm temp}\simeq 0.1-0.3$) are essential
parameters to describe the ICM temperature. The latter provides a unique
opportunity to constrain physical processes in galaxy clusters (e.g.,
turbulence, mixing, and feedback). Our measurements are robust to the methods
used to account for cold clumps when fitting spectra (see Section 4.1).

  * •

Iron and oxygen abundances of the ICM can be accurately recovered in our
experiments up to $r_{200}$. To avoid any biases in measuring these abundances
(Fe in particular), it is important to assume a temperature distribution of
the ICM with a specific non-zero width in the fitting model (see Section 4.2).

  * •

Multiple gas phases, particularly prominent in the cluster outskirts, can be
separated spectroscopically with a spectral resolution similar to that of LEM
by measuring the velocities of different gas components along the LOS. It
allows us to map the gas phase-space distribution, providing valuable
information on the assembly history of galaxy clusters, star formation
quenching, and transport properties of the ICM. In particular, thanks to
projection effects, cold gas measured within $r_{200}$ (even $r_{500}$)
provides a window to probe gas dynamics in non-virialized regions (see Section
4.3).

  * •

We measure gas velocity dispersion profiles with $\leq 50{\rm\,km\,s^{-1}}$
uncertainties in well-relaxed clusters and estimate the corresponding non-
thermal pressure fraction. We demonstrate LEM’s capability in distinguishing
low non-thermal pressure fraction $f_{\rm nth}\simeq 0.1-0.2$ suggested by the
current cosmological simulations and high $f_{\rm nth}\gtrsim 0.4$, considered
as a plausible solution for the ($\Omega_{\rm m},\ \sigma_{8}$) tension in
cluster cosmology (see Section 4.4).

  * •

We map the gas LOS velocity distribution of a well-relaxed cluster with
$\lesssim 30\,(20)$ per cent uncertainties within $r_{500}\,(0.5r_{500})$ and
use it to estimate the 2nd-order velocity structure function. The projected
structure function is steeper than the 3D one, whose shape is sensitive to the
dominant injection scale of the system if it is smaller than the aperture size
of the observation. Our TNG300 cluster shows consistency with Kolmogorov’s
$5/3$ turbulence power law in 3D and favours a large injection scale
$\ell_{\rm diss}\gtrsim 1{\rm\,Mpc}$, implying LEM’s feasibility to constrain
properties of ICM turbulence (see Section 4.5).

  * •

We identify two Mpc-scale penetrating filaments inside a TNG300 unrelaxed
cluster in our mock LEM observation. We resolve their spatial morphology,
e.g., gaseous boundaries in projection and opening angles, and LOS velocities
(both cold and hot components). Such observation will open a new windows in
the future to investigate ICM-filament interactions, e.g., boundary layers,
mixing, and stripping of the gas (see Section 4.6).

In this study, we provide an example of bridging numerical simulations and
X-ray observations self-consistently. Instrumental response, observational
background, and modelling mock data with observational approaches are among
the important steps to be taken for correct predictions from numerical
simulations testable with observations. On the one hand, cosmological
simulations provide insights into developing observational projects and
interpreting observational data. For example,

  * •

We show that the dynamical state of galaxy clusters has strong effects on
various X-ray observables. Only well-relaxed clusters can be used to constrain
the non-thermal pressure fraction of the ICM (see Section 2.2). In
simulations, we show that estimating the gas radial velocity histogram in
radial shells provides a robust way to classify clusters into different
dynamical states. The TNG300 simulation also suggests that deep filament
penetration into the ICM is often associated with the merger process, which
will guide the selection of cluster targets in future observations (see
Section 2.1).

  * •

We characterize the PDFs of temperature fluctuations of the ICM in TNG300
clusters. They can be well described by log-normal distributions in radial
shells and Gaussian distributions with some skewness in projected 2D annuli
(see Section 2.3). These results motivate our gaussbvapec model applied in the
mock observational analysis. Furthermore, we show that the resolved cold
clumps ($\simeq 0.1-1{\rm\,keV}$) are widely spread not only in the cluster
outskirts but also within $r_{500}$ even in well-relaxed clusters due to the
projection effect. These clumps must be resolved spatially (see Section 3.2)
and/or spectroscopically (see Section 3.3) to take their effects into account.

On the other hand, we emphasize that some physical processes might not be
properly resolved or might even be absent (e.g., small-scale turbulence,
mixing layers) in the current cosmological simulations, which limits our
discussion on their properties in this study. Future observations will open a
large discovery space in these areas and stimulate the development of more
physically-motivated theoretical models. Moreover, the present study assumes a
perfect understanding of the atomic database and sky background. However,
these are non-trivial in reality, especially for high-resolution X-ray
spectroscopy. In the coming decade, we will benefit from the missions
including XRISM and SRG/eROSITA in preparation for the era of LEM, HUBS, and
Super DIOS.

##  Appendix A Supplementary figures

Figure 25: Normalized cumulative distribution of the gas temperature in radial
shells weighted by gas mass (top) and in 2D annuli along the LOS weighted by
broad-band X-ray emissivity (bottom) for the cluster CL-RM. The colour shows
only the range of $0.01-0.99$. The white contours indicate 0.3, 0.5, 0.7, and
0.9. This figure illustrates the robustness of using a simple $T_{\rm
gas}>1{\rm\,keV}$ criterion to exclude cold clumps along the LOS within
$r_{200}$ (see Appendix A).  Figure 26: Similar to Fig. 3 but for the clusters
CL-RL (top) and CL-RS (bottom). CL-RL is a massive one and not yet fully
relaxed. Despite lacking prominent substructures, its radial velocity
histogram is noticeably asymmetric within $r_{200}$. CL-RS is a quiescent
fossil group/cluster, isolated from any other large-scale structures (e.g.,
filaments and halos). It provides an ideal opportunity to constrain the lower
limit of non-thermal pressure fraction in clusters (see Appendix A).

Fig. 25 shows the cluster CL-RM’s normalized cumulative distribution of the
gas temperature in radial shells weighted by gas mass (top) and in 2D annuli
weighted by broad-band X-ray emissivity (bottom). Near and outside $r_{500}$,
a large amount of X-ray flux is contributed by cold clumps and diffuse low-
temperature gas at large cluster radii along the LOS. In this study, we use a
uniform temperature criterion (i.e., $T_{\rm gas}>1{\rm\,keV}$) to exclude
contributions of cold clumps in the simulated predictions (see, e.g., Figs. 4,
11, and 12). Fig. 25 shows that $\lesssim 1\ (30)$ per cent gas mass is colder
than $1{\rm\,keV}$ at $r_{500}\ (r_{200})$. However, along the LOS, the gas
contributes $\simeq 20\ (70)$ per cent of the X-ray flux with $T_{\rm
gas}<1{\rm\,keV}$. Given the steep X-ray surface brightness radial profile of
the ICM, our approach is simple and robust. However, we note that, due to the
broad ICM-temperature distribution in the cluster outskirts, a small fraction
of the ICM with moderate temperature is inevitably excluded in our estimation.
This does not significantly affect our results except for the predicted X-ray-
weighted temperature (see Fig. 12).

Fig. 26 shows the dark matter, X-ray surface brightness, and gas radial
velocity distributions of the clusters CL-RL and CL-RS, similar to Fig. 3.

## Acknowledgments

The material is based upon work supported by NASA under award number
80GSFC21M0002. Part of the simulations presented in this paper were carried
out using the Midway computing cluster provided by the University of Chicago
Research Computing Center. IZ is partially supported by a Clare Boothe Luce
Professorship from the Henry Luce Foundation. SE acknowledges the financial
contribution from the contracts 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). D. Nelson acknowledges funding from the Deutsche
Forschungsgemeinschaft (DFG) through an Emmy Noether Research Group (grant
number NE 2441/1-1). D. Nagai is supported by NSF (AST-2206055 & 2307280) and
NASA (80NSSC22K0821 & TM3-24007X) grants. WF acknowledges support from the
Smithsonian Institution, the Chandra High Resolution Camera Project through
NASA contract NAS8-03060, and NASA Grants 80NSSC19K0116, GO1-22132X, and
GO9-20109X.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author. The IllustrisTNG simulations, including TNG300, are
publicly available at www.tng-project.org/data (Nelson et al., 2019). The LEM
instrumental responses are available at https://www.lem-observatory.org (Kraft
et al., 2022).

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