# Indirect Measurements of Gas Velocities in Galaxy Clusters: Effects of
Ellipticity and Cluster Dynamic State

Irina Zhuravleva,1 Mandy C. Chen,1 Eugene Churazov,2,3 Alexander A.
Schekochihin,4,5 Congyao Zhang,1 Daisuke Nagai6
1 Department of Astronomy and Astrophysics, The University of Chicago,
Chicago, IL 60637, USA
2 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741
Garching, Germany
3 Space Research Institute (IKI), Profsoyuznaya 84/32, Moscow 117997, Russia
4 Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK
5 Merton College, Oxford OX1 4JD, UK
6 Department of Physics, Yale University, New Haven, CT 06520, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

While awaiting direct velocity measurement of gas motions in the hot
intracluster medium, we rely on indirect probes, including gas perturbations
in galaxy clusters. Using a sample of $\sim 80$ clusters in different dynamic
states from Omega500 cosmological simulations, we examine scaling relations
between the fluctuation amplitudes of gas density, $\delta\rho/\rho$,
pressure, $\delta P/P$, X-ray surface brightness, Sunyaev-Zel’dovich (SZ)
y-parameter, and the characteristic Mach number of gas motions, $M_{\rm 1d}$.
In relaxed clusters, accounting for halo ellipticities reduces
$\delta\rho/\rho$ or $\delta P/P$ by a factor of up to 2 within $r_{500c}$. We
confirm a strong linear correlation between $\delta\rho/\rho$ (or $\delta
P/P$) and $M_{\rm 1d}$ in relaxed clusters, with the proportionality
coefficient $\eta\approx 1$. For unrelaxed clusters, the correlation is less
strong and has a larger $\eta\approx 1.3\pm 0.5$ ($1.5\pm 0.5$) for
$\delta\rho/\rho$ ($\delta P/P$). Examination of the power-law scaling of
$M_{\rm 1d}$ with $\delta\rho/\rho$ shows that it is almost linear for relaxed
clusters, while for the unrelaxed ones, it is closer to
$\delta\rho/\rho\propto M_{\rm 1d}^{2}$, supporting an increasing role of non-
linear terms and compressive modes. In agreement with previous studies, we
observe a strong correlation of $M_{\rm 1d}$ with radius. Correcting for these
correlations leaves a residual scatter in $M_{\rm 1d}$ of $\sim 4(7)$ per cent
for relaxed (perturbed) clusters. Hydrostatic mass bias correlates with
$M_{\rm 1d}$ as strongly as with $\delta\rho/\rho$ in relaxed clusters. The
residual scatters after correcting for derived trends is $\sim 6-7$ per cent.
These predictions can be verified with existing X-ray and SZ observations of
galaxy clusters combined with forthcoming velocity measurements with X-ray
microcalorimeters.

###### keywords:

galaxies: clusters: intracluster medium – turbulence – methods: numerical –
methods: data analysis hydrodynamics

††pubyear: 2022††pagerange: Indirect Measurements of Gas Velocities in Galaxy
Clusters: Effects of Ellipticity and Cluster Dynamic State–A

##  1 Introduction

The intracluster medium (ICM), filled with hot ($T\sim 10^{7}-10^{8}$ K),
X-ray-emitting gas, is continuously perturbed by matter accretion along cosmic
filaments, mergers, feedback processes, motions of galaxies, etc. These
processes generate gas motions inside the cluster potential well that
contribute significantly to the energy and pressure budget of the ICM and
prevent clusters from reaching a full hydrostatic equilibrium (e.g., Rasia et
al., 2006; Nagai et al., 2007a; Iapichino & Niemeyer, 2008; Lau et al., 2009;
Vazza et al., 2009; Nelson et al., 2012; Battaglia et al., 2012; Zhuravleva et
al., 2013; Nelson et al., 2014a; Shi et al., 2016; Biffi et al., 2016;
Angelinelli et al., 2020; Barnes et al., 2021). Gas motions transfer energy
from large to small scales and eventually dissipate into heat, regulating
cooling and star-forming processes, re-accelerate relativistic particles,
reorder and amplify magnetic fields in the ICM (e.g., Schekochihin & Cowley,
2006; Brunetti & Lazarian, 2007; Vazza et al., 2009; Zhuravleva et al., 2014a;
Miniati & Beresnyak, 2015; Shi et al., 2018, 2020). With a few exceptions
(e.g., Sanders et al., 2010; Tamura et al., 2014; Hitomi Collaboration et al.,
2018a; Sanders et al., 2020), direct velocity measurements await future high-
resolution X-ray spectrometers, such as Resolve on XRISM (launch 2023, XRISM
Science Team 2020) and projected missions like Athena (Nandra et al., 2013),
LEM 111http://lem.cfa.harvard.edu and $Lynx$
222https://www.lynxobservatory.com. Currently, we rely on indirect probes of
gas motions, including resonant scattering (e.g., Gilfanov et al., 1987;
Churazov et al., 2004; Ogorzalek et al., 2017; Hitomi Collaboration et al.,
2018b), an impact of gas motions on metal abundance profiles (Rebusco et al.,
2005), widths of shocks (Nulsen et al., 2013), and gas density and pressure
fluctuations (e.g., Schuecker et al., 2004; Churazov et al., 2012; Zhuravleva
et al., 2014a; Zhuravleva et al., 2018; Walker et al., 2015; Khatri & Gaspari,
2016).

The latter method is particularly attractive as it (1) provides a way of
measuring not only velocity amplitudes but also their length scales and power
spectra - characteristics that are otherwise difficult to extract from
observations; and (2) utilizes high-resolution X-ray imaging data that has
been extensively collected with Chandra and XMM-Newton over the past 20+
years. It is also possible to use sub-mm/SZ images of galaxy clusters to
measure velocities through imprinted pressure fluctuations (see Mroczkowski et
al., 2019, for a review). However, with the limited spatial resolution of
Planck data, only large-scale fluctuations could be constrained (Khatri &
Gaspari, 2016).

The idea of the method, employing that density fluctuations (revealed by X-ray
images) can be linked to slow gas motions, rests on the consideration of a
weakly perturbed stably stratified atmosphere. Namely, it has been shown that
in atmospheres of relaxed galaxy clusters with predominantly subsonic motions
driven on large, buoyancy-dominated scales (Nagai et al., 2013; Shi et al.,
2018; Shi & Zhang, 2019; Shi et al., 2020), there is a linear relation between
the amplitude of density fluctuations, $(\delta\rho/\rho)_{\rm k}$, measured
at a wavenumber333Here, we adopt a wavenumber $k$ that is related to a length
scale $l$ without a factor $2\pi$. $k=1/l$, and a one-component velocity of
gas motions, $v_{\rm 1d,k}$, namely, $(\delta\rho/\rho)^{2}_{\rm
k}\approx\eta_{\rm k}^{2}(v^{2}_{\rm 1d,k}/c^{2}_{\rm s})$, where $c_{\rm s}$
is the sound speed within the gas and $\eta_{\rm k}$ is the proportionality
coefficient (Zhuravleva et al., 2014b). In essence, it is the radial entropy
gradient of the atmosphere that sources density variations of displaced gas
lumps. In the same paper, the proportionality coefficient was calibrated using
a small sample of relaxed and quasi-spherical galaxy clusters from Omega500
cosmological simulations (Nagai et al., 2007a, b; Nelson et al., 2014a),
giving $\eta_{\rm k}\approx 1.0\pm 0.3$. Performing high-resolution 3D
hydrodynamic simulations of idealized turbulence in a stratified cluster
atmosphere, Gaspari et al. (2014) confirmed this scaling and additionally
showed that (1) the scaling may break on small scales in the presence of
strong thermal conduction, providing a promising method to constrain
conduction in the bulk ICM, and (2) pressure perturbations become substantial
when turbulence Mach number $\gtrsim 0.5$, providing an additional source of
correlation between velocity and pressure/density fluctuations.

In soft X-rays (e.g., $0.5-3$ keV band for $T>3$ keV gas), the X-ray
emissivity per unit volume is proportional to squared gas density in galaxy
clusters. Hence, power spectra of density fluctuations can be directly
measured from X-ray images. Pressure fluctuation could be probed using harder
X-ray images (e.g., Forman et al., 2017, see their fig.2) or SZ maps.
Measuring velocity power spectra through density fluctuations in a sample of
cool cores in relaxed clusters inferred typical velocities between $\sim
100-150$ km/s on scales $<50$ kpc, and up to $\sim 300$ km/s on larger scales
$\sim 100$ kpc (Zhuravleva et al., 2018). These velocities are consistent with
those measured with Hitomi in the core of the Perseus cluster given the
dominant scale of motions in this central region is $<100$ kpc (confirmed with
observations, Hitomi Collaboration et al. 2018a). The same (or similar) ideas
have been extended outside relaxed cool cores, to unrelaxed clusters
(including rather extreme cases) and on large spatial scales up to $500-700$
kpc (e.g., Churazov et al., 2012; Gu et al., 2009; Hofmann et al., 2016;
Eckert et al., 2017; Bonafede et al., 2018).

While these results are encouraging, the velocity - density (or pressure)
fluctuations scaling has not been verified for unrelaxed clusters and beyond
the inner regions, where the amplitudes of perturbations become large
invalidating the assumption of a nearly hydrostatic atmosphere. To explore
this problem, a large sample of unrelaxed clusters is required as the expected
scatter could be substantially larger than for relaxed systems. This brings us
to the main scope of this work to calibrate further the velocity-fluctuations
relation, accounting for different cluster dynamic states, halo ellipticities,
and using a large sample of clusters from cosmological simulations. We will
consider both density and pressure fluctuations, as well as their observable
(“projected”) characteristics, namely X-ray surface brightness, $I_{\rm X}$,
and SZ $Y_{\rm SZ}$ parameter. A closely related question, namely, whether
this link between velocity and the amplitude of fluctuations is more
fundamental (i.e., causal) compared to the one arising from radial trends of
increasing levels of perturbations and velocities of gas motions found in
cosmological simulations (e.g., Lau et al., 2009; Nelson et al., 2012, 2014b;
Biffi et al., 2016; Vazza et al., 2017), will also be examined.

In the past couple of years, there has been an interesting theoretical
development in the field. Conducting high-resolution hydrodynamic simulations
of subsonic turbulence with different levels of stratification characterized
through the Richardson number, $Ri$, Mohapatra et al. (2020) showed that the
amplitude of density fluctuations, characterized through the standard
deviation of the logarithmic density fluctuations, increases with increasing
$Ri$ (i.e., with the level of stratification). They further verified the
relation between the amplitude of density fluctuations and velocity, showing
that it was reaching earlier predictions by Zhuravleva et al. (2014a) in the
limit of low Mach number and $Ri\gtrsim 10$. They also showed that pressure
fluctuations are independent of the level of stratification and only depend on
the Mach number of turbulence (see also Mohapatra et al., 2021). Another
recent study by Simonte et al. (2022) explored the density fluctuations-
velocity scaling using a sample of 20 clusters (in various dynamic states)
from the Itasca cluster sample from cosmological simulations, confirming a
linear relation between the amplitude of density fluctuations (namely, the
root mean square of density fluctuations) and velocity. They found that
relaxed objects show a slightly steeper slope $\eta=1.1\pm 0.06$ compared to
the earlier predictions, which is consistent with $1\pm 0.3$ given a large
scatter. Perturbed clusters followed a flatter and weaker relation. In
contrast to Mohapatra et al. (2020), they did not find a strong correlation
between the logarithmic density fluctuations and the level of stratification.

Our study further explores the scaling relations between various amplitudes of
fluctuations and velocities in clusters in different dynamic states,
accounting for radial variations of these characteristics and halo
ellipticity. Besides a fundamental interest in gas dynamics in the ICM,
measuring velocities of gas motions and associated non-thermal pressure is
important for precise cluster mass measurements through hydrostatic
equilibrium for cosmology (e.g., Pratt et al., 2019, for a recent review).
Given a link between the amplitudes of fluctuations and velocities, we examine
these amplitudes as potential proxies for the cluster mass bias.

The structure of the paper is as follows. Simulations used for this work,
sample selection, and classification of clusters are described in Section 2.
Methodology, in particular, the ellipticity-measurement algorithm,
characterization of fluctuations and gas velocity field are summarized in
Section 3. Section 4 shows our main results on fluctuation amplitudes,
correlations between the amplitudes and velocities, and calibration of the
proportionally coefficient $\eta$ in various cases. A possible connection of
fluctuations to cluster mass bias is discussed in Section 5. The main
conclusions are summarized in Section 6.

Figure 1: Illustration of the clump removal procedure used to prepare data for
the fluctuation analysis. The top and bottom rows show two different clusters
(relaxed and unrelaxed, respectively). The left panels show initial maps of
projected gas density, while the corresponding middle and right panels show
the same projected density after applying the clump removal algorithm with
$f_{\rm cut}=3.5$ (middle) and $f_{\rm cut}=2.5$ (right). The box size used
for these projected maps is $\sim 5$ Mpc on each side. The color scales of all
maps are the same. The images are lightly smoothed for display purposes. The
clumps are effectively removed regardless of their size, density, and location
within the cluster.

##  2 Sample of galaxy clusters

Our sample includes 78 galaxy clusters from the non-radiative hydrodynamic
cosmological simulations Omega500 (Nagai et al., 2007a, b; Nelson et al.,
2014a). The input cosmology corresponds to a flat $\Lambda$CDM model with
$\Omega_{m}=0.27$, $\Omega_{b}=0.0469$, $h=0.7$ and $\sigma_{8}=0.82$. These
simulations use an Adaptive Refinement Tree (ART) hydrodynamic solver
(Kravtsov & Klypin, 1999; Rudd et al., 2008), which is particularly good at
capturing turbulence, shocks, and sharp contact discontinuities. The default
simulation volume is resolved using six levels of mesh refinement, implying a
maximum comoving spatial resolution of $\sim 15$ $h^{-1}$ kpc. We checked that
a finer refinement of the data does not affect our results. The total masses
of clusters in our sample range between $M_{\rm 200c}=2\cdot
10^{14}h^{-1}M_{\odot}$ and $1.6\cdot 10^{15}h^{-1}M_{\odot}$ with the median
value $\sim 5\cdot 10^{14}h^{-1}M_{\odot}$ at redshift $z=0$. The center of
each cluster corresponds to the location of the dark matter particle with the
most bounded gravitational energy, which translates to the densest peak of the
most massive cluster in cases of merging clusters.

Dense clumps of gas that could be associated with infalling structures or
formed in simulations due to incomplete (sub-grid) physics prescription or
insufficient resolution could bias mean thermodynamic characteristics of the
hot gas in clusters (e.g., Nagai & Lau, 2011; Rasia et al., 2014; Avestruz et
al., 2014), affect ellipticity measurements (Section 3.1), and the amplitude
of projected fluctuations and velocity in the bulk gas (Section 4.1). It is
crucial to remove them carefully before analyzing fluctuations in the bulk
gas444In observations, such dense clumps are usually identified in X-ray
images and removed from the analysis of gas fluctuations.. Following the
method proposed by Zhuravleva et al. (2013), we identified the clumps through
high-density tails of the probability density distribution of gas density
within considered regions (see their Fig. 2), using $f_{\rm cut}=3.5$. Namely,
in each region, all cells with a density larger than the median density by 3.5
$\sigma$, where $\sigma$ is the log10-based standard deviation of density
distributions, are associated with the dense clumps. These identified clumps
were then removed from the data, leaving the bulk gas component intact. To
illustrate the procedure, Fig. 1 shows two examples of clusters (a relaxed one
on top and unrelaxed on the bottom) with prominent clumps. Their initial
projected density with all the clumps is shown on the left, while the same
projected density with removed clumps is shown in the middle ($f_{\rm
cut}=3.5$, default) and right ($f_{\rm cut}=2.5$) panels. The procedure works
well and the difference between the two choices of $f_{\rm cut}$ is minor. We
checked that varying $f_{\rm cut}$ within a reasonable range, by $\pm 30$ per
cent, does not affect our main conclusions.

Figure 2: Projected gas density distribution ($\propto\int\rho dl$) of a
typical relaxed (left), unrelaxed (right), and in-between (middle) cluster in
our sample. White/purple color traces the highest/lowest projected density
values. Only data within the radius $\sim 2r_{\rm 500c}$ is used for these
images.

Visually inspecting all clusters in our sample, we classified them into three
groups: relaxed (smooth morphology), unrelaxed (very perturbed), and in-
between (see also Shi et al., 2016; Chen et al., 2019, for other
classifications). For the classification, we projected the 3D density
distributions (with removed high-density clumps) along three axes and ranked
each projection based on how easily the cluster center could be identified
(e.g., a few central densest pixels vs. a more diffuse region), the symmetry
of gas distribution, the presence of merger-driven structures (e.g.,
filaments, large-scale contact discontinuities), and how substantial is the
clump removal. Fig. 2 shows representative examples of clusters from each
group. The final subsamples of relaxed, unrelaxed, and in-between clusters
include 19, 27, and 32 objects, respectively.

Our mass-limited sample is cosmologically representative in terms of cluster
dynamical state. This means that the split between relaxed, in-between, and
unrelaxed clusters should be roughly similar to samples of nearby clusters.
While the visual classification of clusters in these three categories is
subjective (although it could be implemented with modern machine-learning
techniques), the most important are the fractions, namely 0.24, 0.35, and 0.41
for relaxed, in-between and unrelaxed clusters, respectively. When using other
criteria for apparent deviations from the relaxed state, one can hope that
choosing $\sim$25 per cent of the most relaxed clusters would approximately
match our relaxed sample.

##  3 Methodology

###  3.1 Ellipticity measurements

Gas distribution in the ICM often deviates from perfect spherical symmetry.
Cosmological simulations show that the average ellipticity of the hot gas
varies depending on the distance from the cluster center and the physics
involved in the simulations, reflecting elongation of the underlying
gravitational potential dominated by dark matter (e.g., Lau et al., 2012; Chen
et al., 2019; Harvey et al., 2021). Using spherical shells to characterize the
amplitude of fluctuations in elongated clusters could bias the amplitude,
effectively increasing it on large scales. Therefore, we performed the
analysis in both spherical and elliptical shells.

After removing high-density clumps, we replaced the removed substructure with
the median values of considered characteristics at that cluster-centric radius
and estimated the ellipticity of ICM at each radius using an iterative method
described in Zemp et al. (2011). Summarizing the method, we first calculate
the shape tensor in each spherical shell as

$S_{ij}=\frac{\sum_{k}\rho_{k}(\mathbf{r}_{k})_{i}(\mathbf{r}_{k})_{j}}{\sum_{k}\rho_{k}}\,,$
(1)

where $i$, $j$ indices represent the $x$, $y$ or $z$ axes, $({\bf r}_{k})_{j}$
is the $j-$component of the position vector of the $k$th cell, and $\rho_{k}$
is the gas density in the $k$th cell. The eigenvalue of $S_{ij}$ estimates the
axis ratio of the ellipse in the $ij$ plane, and the eigenvector represents
the orientation of the ellipse. We then update the $S_{ij}$ tensor with a new
shell definition that takes into account the ellipticity and orientation of
the ellipsoid and repeat the calculation until it converges. For the
ellipticity measurements of gas pressure distribution, we follow the same
procedure, substituting density with the gas thermal pressure in relation (1).

Note that for an ellipsoidal shell, the definition of the radius is given by

$r_{\mathrm{ell}}=\sqrt{x_{\mathrm{ell}}^{2}+\frac{y_{\mathrm{ell}}^{2}}{(b/a)^{2}}+\frac{z_{\mathrm{ell}}^{2}}{(c/a)^{2}}}\,,$
(2)

where $x_{\mathrm{ell}}$, $y_{\mathrm{ell}}$ and $z_{\mathrm{ell}}$ are the
coordinates along the eigenvectors of the shape tensor $S_{ij}$ after
convergence, $a$, $b$ and $c$ are the semi-principal axes with $a\geq b\geq
c$. Below, we denote radius as $r$, which corresponds to the semi-major axis
$a$ of the ellipsoid surface in elliptical shells.

Fig. 3 shows radial profiles of axis ratios averaged over our samples of
relaxed, unrelaxed, and in-between clusters. One can see that relaxed clusters
are more spherical compared to unrelaxed ones, especially at $r<r_{500c}$.
Ellipticities calculated from density and pressure perturbations are very
similar for the relaxed and unrelaxed clusters, however, there are some
differences in the in-between group. In all cases, the effects of ellipticity
are the strongest within the inner $r_{500c}$ region. Note that if gaps from
the high-density clumps are smoothly filled with the density values that
separate clumps from the bulk gas (i.e., there are no sharp edges associated
with removed clumps), the ellipticity changes maximum by 5 and 7 per cent in
relaxed in unrelaxed clusters, respectively. This difference is noticeable
beyond $1.5r_{500c}$.

Figure 3: Radial profiles of axis ratios averaged over our subsamples of
relaxed, unrelaxed, and in-between clusters. $a,b$ and $c$ are semi-principal
axes with $a\geq b\geq c$. The ratios are shown for density and pressure
distributions with removed high-density clumps assuming $f_{\rm cut}=3.5$.

Besides the 3D characteristics, we also consider observational (“projected”)
characteristics such as the X-ray surface brightness $I_{\rm X}\propto\int
n_{\rm e}^{2}\Lambda(T)dl$ and SZ $y$-parameter $Y_{\rm SZ}\propto\int n_{\rm
e}Tdl$. Here, $n_{\rm e}$ is the electron number density, $\Lambda(T)$ is the
X-ray emissivity calculated within the $0.5-2.$ keV band, $T$ is the electron
temperature, and the integration is along a line of sight $l$ (the length of
the line of sight is $\sim 5.5$ Mpc in our analysis). Elliptical annuli for
these observational characteristics are calculated similarly to the 3D case.
We calculate the shape tensor (1), using $I_{\rm X}$ or $Y_{\rm SZ}$ instead
of density and $i,j$ representing only $x$ and $y$ axes. The eigenvalue and
eigenvector of the shape tensor estimate the ellipticity and orientation of
the best-fitting ellipse at each distance from the cluster center. Based on
the calculated ellipticity, we construct the elliptical projected radial grid
used in this work.

###  3.2 Characterizing gas fluctuations

We calculated the amplitude of the bulk density, pressure, X-ray surface
brightness, and SZ $y$-parameter fluctuations (hereafter, we refer to any of
these amplitudes as $\delta\xi/\xi$) following the procedure proposed by
Zhuravleva et al. (2013). Consistent with earlier studies, we confirmed that
the probability density distributions of these characteristics in each
considered region follow log-normal distributions (Kawahara et al., 2007). In
each shell/annulus (spherical or elliptical), we measured the width of the
probability density distribution of $\xi$ as
$\delta\xi/\xi=\log_{10}({\xi_{1}/\xi_{2}})$, where $\xi_{1}$ is the 12th-
quantile of the characteristic’s distribution within the shell/annulus, and
$\xi_{2}$ is the 88th-quantile. If the distribution is approximated as a log-
normal distribution with a standard deviation (natural log based) $\sigma$,
then $\displaystyle{\frac{\delta\xi}{\xi}=\frac{2\sqrt{2\ln 2}}{\ln
10}\sigma\simeq 1.02\sigma}$. Note that this definition for the total
amplitude (i.e., measured on all scales together) of density or pressure
fluctuations is not sensitive to the presence of high-density clumps or the
procedure used to identify and remove them from the data.

For density fluctuations, we also considered an average amplitude of
fluctuations as a function of an isotropic wavenumber
$k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. This scale-dependent amplitude of
density fluctuations has been measured in several observed galaxy clusters
using X-ray imaging data (e.g., Churazov et al., 2012; Walker et al., 2015;
Arévalo et al., 2016; Zhuravleva et al., 2018). Following the observational
procedure, we calculated a radial profile of X-ray surface brightness and
approximated it with a $\beta-$model for each cluster. Dividing the gas
density distribution by the corresponding best-fitting $\beta-$model, we
obtained a data cube of density fluctuations. We then calculated a power
spectrum of density fluctuations, $P(k)$, using the modified $\Delta-$variance
method (Arévalo et al., 2012; Ossenkopf et al., 2008). Note that this method
of calculating power spectra is not sensitive to gaps in the data (caused by
the removal of high-density clumps) and non-periodic boundaries of the data
cube. Finally, we calculated the amplitude of density fluctuations,
$(\delta\rho/\rho)_{k}$, at each $k$ as $(\delta\rho/\rho)_{k}=\sqrt{P(k)4\pi
k^{3}}$. Velocity power spectra were calculated for each velocity component
using the same $\Delta-$variance method.

###  3.3 Characterizing gas velocity field

After removing the high-density clumps from our data cubes, we calculated the
characteristic RMS velocity amplitude of the bulk-gas component in each
shell/annulus as

$V_{\rm rms}=\sqrt{\langle(V_{\rm x}-\langle V_{\rm x}\rangle)^{2}+(V_{\rm
y}-\langle V_{\rm y}\rangle)^{2}+(V_{\rm z}-\langle V_{\rm
z}\rangle)^{2}\rangle}\,,$ (3)

where $\langle\rangle$ denotes averaging over all particles within a
shell/annulus, and $\langle V_{\rm x}\rangle$, $\langle V_{\rm y}\rangle$,
$\langle V_{\rm z}\rangle$ are the components of mean velocities within each
region (the reference velocity). Our experiments with different choices of the
reference velocity (e.g., averaging within some central regions instead)
showed no significant differences in the $V_{\rm rms}$ averaged over the
subsamples of clusters. Consistent with previous studies, we saw that the RMS
velocity of the bulk component had very regular behavior with radius in
contrast to the velocity of the high-density clumps.

It is important to emphasize that we do not decompose gas motions into bulk,
laminar motions and genuine turbulence, and consider the whole velocity field
in our study. This is different from the other recent study by Simonte et al.
(2022). Such velocity decomposition relies on an assumption about the
filtering scale that is difficult to define unambiguously given the variety of
velocity driver scales in the ICM (e.g., Vazza et al., 2017). Moreover, the
Reynolds number in cosmological simulations is relatively small, typically
below 100. Therefore, the filtered small-scale motions do not necessarily
correspond to fully-developed turbulence. Given these difficulties, we choose
a conservative approach to explore the whole velocity field. Idealized
numerical simulations of merging clusters and turbulence in stratified
atmospheres are better suited for addressing the relation between fluctuations
and turbulent motions of the ICM gas (e.g., Gaspari et al., 2014; Mohapatra et
al., 2020).

We define the characteristic Mach number of gas motions as $M=V_{\rm
rms}/c_{\rm s}$, where the sound speed of the gas is $c_{\rm s}=\sqrt{\gamma
k_{\rm B}T/\mu m_{\mathrm{p}}}$, $\gamma=5/3$ is the adiabatic index for ideal
monatomic gas, $k_{\rm B}$ is the Boltzmann constant, $\mu=0.588$ is the mean
atomic weight and $m_{\mathrm{p}}$ is the proton mass. One-component Mach
number is formally defined as $M_{\rm 1d}=M/\sqrt{3}$. When exploring
projected fluctuations ($y-$parameter or X-ray surface brightness) and their
relation to the Mach number of gas motions, we used velocities and sound
speeds averaged within shells even if the fluctuations are probed within
geometrically-different regions (along the line of sight at a given distance
of each annulus from the cluster center). This choice is motivated by our
desire to extract radial velocity information in observed clusters. In other
words, we are trying to link the potential observables at a given projected
distance from the cluster center to gas properties at a similar distance in
3D.

##  4 Results

###  4.1 Amplitude of gas fluctuations in relaxed and dynamically perturbed
clusters

Figure 4: Radial profiles of the amplitudes of density and pressure
fluctuations (top two rows) and fluctuations of X-ray surface brightness and
SZ $y$-parameter (bottom two rows) averaged over the subsamples of relaxed
(left), in-between (middle), and unrelaxed (right) clusters. Light blue and
solid curves/regions show results in spherical shells or annuli, while navy
dashed/hatched curves/regions - are in elliptical regions. Gray regions show
results for relaxed clusters in elliptical shells/annuli and are plotted for
visual comparison with clusters in other dynamic states. Dashed red curves
show results for $f_{\rm cut}=2.5$ (cf. the navy dashed curves). Figure 5:
Correlations between the amplitudes of gas fluctuations (from top to bottom:
density, pressure, X-ray surface brightness, SZ $y-$parameter) and one-
component Mach number of gas motions. Color represents a radial distance from
the cluster center. At each radius, fluctuations are calculated within a
corresponding elliptical shell/annulus. Each track of points corresponds to an
individual cluster. Only measurements within $0.2r_{500c}<r<1.7r_{500c}$ are
used. All distributions are fitted with a linear function (blue), $M_{\rm
1d}=\alpha+\gamma(\delta\xi/\xi)$, as well linear function between the log-10
of these characteristics (red), i.e., $M_{\rm
1d}=10^{\alpha_{1}}(\delta\xi/\xi)^{\gamma}=\alpha(\delta\xi/\xi)^{\gamma}$.
Corresponding Pearson’s ranks, $R_{\rm p}$, and the best-fitting parameters
are shown in legends. The dashed gray lines indicate a one-to-one correlation
for visual guidance. See Section 4.2 for details. Figure 6: Radial profiles of
the proportionality coefficient $\eta$ between the amplitude of density (top)
or pressure (bottom) fluctuations and one-component Mach number averaged over
a sample of relaxed (left), in-between (middle), and unrelaxed (right)
clusters. Calculations in spherical shells are shown with light blue, solid
curves/regions, while results in elliptical shells are plotted with navy,
dashed/hatched curves/regions. All panels show sample-averaged mean values and
scatter. The averaged values within the entire region ($r<2r_{500c}$,
accounting for the ellipticity) are written in the top-right or bottom-right
corners.  Figure 7: Proportionally coefficient between a power spectrum of
density fluctuations and one-component velocity spectrum as a function of a
wavenumber $k$ averaged over a sample of relaxed (left), in-between (middle
left), unrelaxed (middle right), and all (right) clusters. The mean
values/scatters are shown with solid curves/regions. The region between the
dotted lines corresponds to earlier predictions based on a small sample of the
most relaxed clusters, $\eta_{\rho,k}=1\pm 0.3$ (Zhuravleva et al., 2014b). We
only show wavenumbers not affected by the resolution of simulations
(correspond to scales $\sim 60-300$ kpc). Figure 8: Proportionality
coefficient $\eta$ between the amplitude of X-ray surface brightness (left,
middle) or $y-$parameter (right) fluctuations and one-component Mach number
averaged over a subsample of relaxed (left, right) or in-between (middle)
clusters. Notations are the same as in Fig. 6.

Fig. 4 shows the widths of density, pressure, X-ray surface brightness, and
$y$-parameter distributions (relative to the median value) as functions of the
radius or projected radius. These widths characterize the total (i.e.,
integrated over all scales within a given shell or annulus) amplitude of
fluctuations of the corresponding characteristics. The calculations are done
in spherical (solid, light blue) and elliptical (hatched/dashed, navy) shells
or annuli and averaged over the groups of relaxed (left), in-between (middle),
and unrelaxed (right) clusters. Accounting for ellipticity reduces the mean 3D
amplitudes for relaxed and in-between clusters by a maximum factor of $\sim
1.9$ and $\sim 1.4$, respectively, and $\sim 1.3$ for unrelaxed ones. When
measured in elliptical shells, the scatter around the mean value increases for
unrelaxed clusters. This is mainly caused by the strongly asymmetric
morphology of some unrelaxed clusters and large offsets between the chosen
cluster center (the location of the dark matter particle with the most bounded
gravitational energy) and the peak of considered gas characteristics. The
average amplitudes are below 60 (70) per cent in elliptical shells for relaxed
(in-between) clusters and could be closer to 90 per cent for unrelaxed
clusters. The small amplitude of density fluctuations, $\lesssim 25$ per cent,
in the innermost regions is consistent with the observed amplitudes measured
through X-ray surface brightness fluctuations in nearby clusters (e.g.,
Churazov et al., 2012; Walker et al., 2015; Zhuravleva et al., 2018) 555We
checked that using emissivity-weighted density, which is closer to the
observational characteristic, does not affect the amplitude of density
fluctuations in relaxed clusters and may only change the average amplitude in
unrelaxed clusters by a small factor of $\sim 1.2$.. The larger values of
$\delta P/P$ in the outer regions are also consistent with the pressure
fluctuations measured with Planck (Khatri & Gaspari, 2016).

While the ratio of pressure and density fluctuations amplitudes is about $\sim
1.1-1.5$ for clusters in any dynamic state, the ratio between the projected
characteristics, the X-ray surface brightness, and the $y-$parameter
fluctuations, is larger, a factor of $\sim 1.6-2.4$. This is not surprising
given that both observational characteristics scale differently with the
density. $\delta I_{\rm X}/I_{\rm X}$ and $\delta Y_{\rm SZ}/I_{\rm SZ}$ are
both below $40$ per cent within $r_{500}$ for relaxed clusters and
significantly larger (up to $100$ per cent in some cases) for unrelaxed
systems. These conclusions are independent of the choice of projection axis.
Overall, $\delta\rho/\rho$ has the smallest scatter for clusters in all
dynamic states. $\delta I_{\rm X}/I_{\rm X}$, in contrast, has the largest
scatter compared to fluctuations of other characteristics. This is not
surprising given that $\delta I_{\rm X}/I_{\rm X}$ is most sensitive to the
presence of extended structures around the high-density clumps. This is
illustrated with the red dashed curves calculated using $f_{\rm cut}=2.5$ (cf.
the dashed navy curves). Exclusion of more substructure around the high-
density clumps affects $\delta I_{\rm X}/I_{\rm X}$ only mildly and at $r_{\rm
proj}>r_{500c}$, however, suppresses the mean value of fluctuation in
unrelaxed clusters by $\sim 20-30$ per cent. Note that $\delta Y_{\rm
SZ}/Y_{\rm SZ}$ is not sensitive to these extended clumping structures.

One can use the amplitude of fluctuations as a proxy for the dynamical state
of a cluster when classifying clusters into relaxed or unrelaxed groups.
Several methods for such classification have been proposed and used in
observations and simulations, including asymmetry of the X-ray surface
brightness, X-ray light concentration ratio, centroid shifts, power ratios,
third-order power ratio, the cross-correlation of the X-ray surface brightness
and mass distribution (see e.g., Buote & Tsai, 1995; Mohr et al., 1995; Santos
et al., 2008; Weißmann et al., 2013; Nurgaliev et al., 2013; Rasia et al.,
2013; Mantz et al., 2015; Lovisari et al., 2017; Yuan & Han, 2020; De Luca et
al., 2021; Cerini et al., 2022, and references therein). Fig. 4 (gray vs. navy
curves/regions) clearly shows the difference between fluctuations in relaxed
and dynamically active clusters. At each radius, we performed a Kolmogorov-
Smirnov test to assess whether the samples of $\delta\rho/\rho$ or $\delta
P/P$ values for relaxed clusters belong to a different population compared to
the population of unrelaxed clusters. By generating 100 bootstrap samples, we
also estimated uncertainties on the p-values (if the p-value is below 0.05
then the null hypothesis that the two samples were drawn from the same
distribution is rejected). The typical p-value is smaller than $10^{-6}$ for
$\delta\rho/\rho$ and $10^{-3}$ for $\delta P/P$. The lowest p-value is
reached at $r=(0.5-0.7)r_{500c}$ in both elliptical and spherical shells,
meaning that the pressure and density fluctuations are statistically very well
separated between relaxed and unrelaxed clusters. Similar conclusions are
valid when comparing relaxed and in-between clusters, however, the p-values
are slightly higher. This classification method is particularly convenient for
large samples of simulated galaxy clusters as the widths of density and
pressure distributions can be measured robustly (e.g., not sensitive to the
presence of high-density clumps) and straightforwardly. We also checked
projected characteristics, finding that, overall, the p-value is $\lesssim$
0.01 at $r<r_{500c}$ and could be higher at $r>(1-1.5)r_{500c}$. Therefore,
using 3D fluctuations (density, pressure) to classify clusters based on their
dynamic states is more robust compared to projected characteristics.

###  4.2 Linear relation between the amplitude of gas perturbations and
velocity Mach number

First, we check how strong the linear correlation is between the Mach number
of gas motions and all types of fluctuations in the ICM. Fig. 5 shows a one-
component Mach number vs. all the amplitudes $\delta\xi/\xi$. All
characteristics are measured as a function of the elliptical radius (color-
coded) within a $r\lesssim 2r_{500c}$ region (the central $0.2r_{500c}$ are
excluded from the analysis as these regions are dominated by radiative
physics) in each cluster. We fit all the distributions with a linear function,
$M_{\rm 1d}=\alpha+\gamma(\delta\xi/\xi)$, and check the strength of each
correlation through a standard Pearson’s rank correlation coefficient $R_{\rm
p}$. The best-fitting functions and all the parameters are plotted in blue.
Relaxed clusters show a strong correlation ($R_{\rm p}\approx 0.8-0.9$)
between $M_{\rm 1d}$ and density or pressure fluctuations (top two rows in
Fig. 5, left panels). The best-fitting line (blue curve), in this case,
closely follows a one-to-one correlation (dashed gray lines) obtained earlier
based on a small sample of relaxed, quasi-spherical simulated clusters
(Zhuravleva et al., 2014b). For in-between clusters, these correlations are
weaker but still relatively strong ($R_{\rm p}\approx 0.6$). Unrelaxed
clusters show a weaker correlation between $\delta\rho/\rho$ and $M_{\rm 1d}$
($R_{\rm p}\sim 0.5$), while the correlation between $\delta P/P$ and $M_{\rm
1d}$ is still moderately strong ($R_{\rm p}\sim 0.7$). This trend for
unrelaxed clusters is not surprising since gas motions are stronger with
$M\gtrsim 0.5$ and, hence, the role of compressive modes is progressively
increasing. “Projected” (i.e., potentially observable) amplitudes, $\delta
I_{\rm X}/I_{\rm X}$ and $\delta Y_{\rm SZ}/Y_{\rm SZ}$, show significantly
weaker correlations ($R_{\rm p}<0.5$) with $M_{\rm 1d}$ in half of the cases
except for relaxed clusters ($R_{\rm p}\sim 0.5-0.7$) and, in the case of
density fluctuations, for in-between ones ($R_{\rm p}\sim 0.5$). While
specific values for $R_{\rm p}$ could vary slightly, our conclusions for
projected characteristics qualitatively are essentially independent of the
choice of projection axis and the $f_{\rm cut}$ parameter.

Calibration of the statistical relations between the amplitudes of
fluctuations and velocity at different radii, namely,
$\delta\xi/\xi=\eta_{\xi}M_{\rm 1d}$, is shown in Fig. 6. The top panels show
$\eta_{\rho}=(\delta\rho/\rho)/M_{\rm 1d}$, while the bottom ones - $\eta_{\rm
P}=(\delta P/P)/M_{\rm 1d}$. All $\eta$ are averaged over subsamples of
clusters at each radius (curves and regions) as well as averaged within the
entire $r<2r_{500c}$ region (labels in the top- or bottom-right corners). One
can see that accounting for ellipticity reduces $\eta$ in relaxed and in-
between clusters, bringing it closer to the earlier-predicted value $1\pm 0.3$
(Zhuravleva et al., 2014b). $\eta$ measured in elliptical shells in unrelaxed
clusters has a larger scatter compared to the same calculations in radial
shells, while the mean value remains almost unchanged. The dynamic state of
clusters is reflected in the average scatter. Within $r<2r_{500c}$, it is the
smallest for relaxed clusters ($\sim 20-30$ per cent) and could be up to $\sim
50$ per cent for unrelaxed ones. Therefore, when using $\eta$ to infer
velocities from observed amplitudes of density or pressure fluctuations, it is
important to take into account the ellipticity of gas distribution in relaxed
and in-between clusters (affects the mean, does not change the scatter), while
using spherical shells for the unrelaxed ones (does not matter for the mean
$\eta$ yet the scatter is lower).

Besides the global scalings, it is interesting to check their scale-by-scale
versions as they are most relevant to recently-measured velocity power spectra
from the observed power spectra of density fluctuations (e.g., Churazov et
al., 2012; Walker et al., 2015; Arévalo et al., 2016; Zhuravleva et al.,
2018). We checked the scaling within the central $r<0.5r_{500}$ region, where
the adaptive mesh resolution of the simulations is the highest, and our
resolution study showed convergence on a range of relevant wavenumbers.
Following the observational procedure (Arévalo et al., 2012; Churazov et al.,
2012), we calculated radial profiles of X-ray surface brightness for each
cluster and approximated them with a $\beta-$model. Dividing the gas density
by the corresponding best-fitting model, we obtained a data cube of relative
density fluctuations. We then calculated the power spectra of density
fluctuations and RMS velocity and took the ratio of both to get
$\eta_{\rho,k}$. The averaged results for all types of clusters are shown in
Fig. 7. We only focus on scales from $\sim 60-300$ kpc, where the power
spectra are not affected by the resolution of the simulations. Overall, the
mean value of $\eta_{\rho,k}$ is consistent with global (integrated over all
scales) results shown in Fig. 6 for all types of clusters.

In observations, when reprojecting 2D (i.e., $I_{\rm X}$, $Y_{\rm SZ}$)
amplitudes of fluctuations to the 3D ones (i.e., $\rho$, $P$), one relies on
the assumption of many independent fluctuations of a given scale along the
line of sight (see Section 3 in Churazov et al., 2012). Therefore, if
projected amplitudes could be used to measure the velocities of gas motions,
it would significantly simplify the observational procedure. Fig. 8 shows
$\eta$ from projected amplitudes that show strong correlations with Mach
number in Fig. 5. Interestingly, $\eta_{\rm X}$ for relaxed and in-between
clusters has a scatter that is a factor of $\sim 2$ larger than for the 3D
density amplitudes, while $\eta_{\rm SZ}$ for relaxed clusters has the same
scatter of $\sim 20-30$ per cent as in the 3D density or pressure case.
Therefore, for the relaxed clusters, all types of fluctuations, except for
$\delta I_{\rm X}/I_{\rm X}$, can be used interchangeably to probe velocities
of gas motions. These predictions are only slightly affected by the choice of
projection axis and $f_{\rm cut}$ parameter. Depending on the choice, the mean
value of $\eta$ may change by $\pm 0.1$ \- still consistent with the default
results within the scatter.

To summarize, the 3D amplitude of density or pressure fluctuations is a
preferred proxy for velocity amplitude measurements in clusters in any dynamic
state. For relaxed and in-between clusters, using elliptical shells instead of
spherical ones improves the correlations between the amplitudes and Mach
number. Instead, for unrelaxed clusters, using the amplitudes in spherical
shells should give the tightest constraints on $M_{\rm 1d}$. The $y-$parameter
fluctuations are also as robust proxies for velocities as any of the 3D
amplitudes in relaxed clusters. We also checked that using a finer resolution
of cosmological simulations negligibly affects $\eta$ from 3D characteristics.
The effect on mean $\eta$ from projected amplitudes within $r_{500c}$ is
slightly stronger yet consistent with the fiducial case within the
uncertainties.

###  4.3 Improved proxies for Mach number

As is clear from Figs. 4 and 9, the fluctuation amplitudes of all quantities
and velocity Mach number show strong radial trends. These trends reflect a
typical state of the ICM in clusters that keep growing by accretion. In other
words, the gas becomes progressively more and more perturbed with the
increasing radius, and the level of perturbations depends on the recent
accretion history. Several interesting questions arise naturally. Are the
correlations seen in Fig. 5 driven purely by these radial dependencies? Is the
correlation between $M_{\rm 1d}$ and, e.g., density perturbations, which in
stratified atmospheres can be established by processes outlined above,
stronger or weaker than the typical radial trend? To what accuracy can $M_{\rm
1d}$ be predicted using a pure radial dependence or taking into account other
proxies too?

Figure 9: Radial profiles of one-component Mach number of gas motions averaged over the subsamples of relaxed, in-between, and unrelaxed clusters. Only the bulk component of the gas is considered (i.e., the high-density clumps are removed). The color coding is the same as in Fig. 4. Elliptical shells are taken from density distributions. | $\delta\xi/\xi$ |  | $M_{\rm 1d}=\alpha+\beta(r/r_{500c})+\gamma(\delta\xi/\xi)$
---|---|---|---
Sample | $\xi=$ | $\alpha$ | RMSi |  | $\alpha$ | $\beta$ | Rank | RMS |  | $\alpha$ | $\gamma$ | Rank | RMS |  | $\alpha$ | $\beta$ | $\gamma$ | RMS
(a) | (b) | (c) | (d) |  | (e) | (f) | (g) | (h) |  | (i) | (j) | (k) | (l) |  | (m) | (n) | (o) | (p)
Relaxed | $\rho$ | 0.25 | 0.09 |  | 0.08 | 0.19 | 0.87 | 0.04 |  | 0.04 | 0.91 | 0.83 | 0.05 |  | 0.04 | 0.13 | 0.41 | 0.04
| $P$ | 0.25 | 0.09 |  | 0.08 | 0.19 | 0.88 | 0.04 |  | 0.08 | 0.67 | 0.86 | 0.05 |  | 0.06 | 0.11 | 0.35 | 0.04
| $I_{\rm X}$ | 0.25 | 0.09 |  | 0.08 | 0.19 | 0.87 | 0.04 |  | 0.16 | 0.32 | 0.49 | 0.08 |  | 0.06 | 0.18 | 0.09 | 0.04
| $Y_{\rm SZ}$ | 0.25 | 0.09 |  | 0.08 | 0.19 | 0.88 | 0.04 |  | 0.13 | 0.79 | 0.66 | 0.07 |  | 0.06 | 0.16 | 0.26 | 0.04
In-betw. | $\rho$ | 0.33 | 0.11 |  | 0.19 | 0.15 | 0.59 | 0.09 |  | 0.15 | 0.55 | 0.59 | 0.09 |  | 0.11 | 0.1 | 0.36 | 0.08
| $P$ | 0.33 | 0.10 |  | 0.19 | 0.15 | 0.62 | 0.08 |  | 0.19 | 0.36 | 0.58 | 0.08 |  | 0.13 | 0.11 | 0.24 | 0.07
| $I_{\rm X}$ | 0.33 | 0.11 |  | 0.19 | 0.15 | 0.59 | 0.09 |  | 0.21 | 0.26 | 0.52 | 0.09 |  | 0.14 | 0.12 | 0.17 | 0.08
| $Y_{\rm SZ}$ | 0.33 | 0.10 |  | 0.19 | 0.15 | 0.62 | 0.08 |  | 0.25 | 0.34 | 0.38 | 0.10 |  | 0.16 | 0.14 | 0.17 | 0.08
Unrel. | $\rho$ | 0.43 | 0.11 |  | 0.26 | 0.19 | 0.70 | 0.08 |  | 0.29 | 0.25 | 0.48 | 0.10 |  | 0.18 | 0.17 | 0.16 | 0.07
| $P$ | 0.43 | 0.12 |  | 0.25 | 0.20 | 0.71 | 0.08 |  | 0.28 | 0.23 | 0.67 | 0.09 |  | 0.21 | 0.14 | 0.14 | 0.07
| $I_{\rm X}$ | 0.43 | 0.11 |  | 0.26 | 0.19 | 0.70 | 0.08 |  | 0.38 | 0.07 | 0.27 | 0.11 |  | 0.23 | 0.18 | 0.04 | 0.08
| $Y_{\rm SZ}$ | 0.43 | 0.12 |  | 0.25 | 0.20 | 0.71 | 0.08 |  | 0.35 | 0.19 | 0.42 | 0.11 |  | 0.22 | 0.18 | 0.10 | 0.08
| $\delta\xi/\xi$ |  | $M_{\rm 1d}=\alpha(r/r_{500c})^{\beta}(\delta\xi/\xi)^{\gamma}$
Sample | $\xi=$ | $\alpha$ | RMSi |  | $\alpha$ | $\beta$ | Rank | RMS |  | $\alpha$ | $\gamma$ | Rank | RMS |  | $\alpha$ | $\beta$ | $\gamma$ | RMS
(a) | (b) | (c) | (d) |  | (e) | (f) | (g) | (h) |  | (i) | (j) | (k) | (l) |  | (m) | (n) | (o) | (p)
Relaxed | $\rho$ | 0.25 | 0.09 |  | 0.27 | 0.63 | 0.89 | 0.04 |  | 0.83 | 0.83 | 0.81 | 0.05 |  | 0.4 | 0.47 | 0.29 | 0.04
| $P$ | 0.25 | 0.09 |  | 0.27 | 0.63 | 0.89 | 0.04 |  | 0.65 | 0.68 | 0.85 | 0.04 |  | 0.4 | 0.42 | 0.29 | 0.04
| $I_{\rm X}$ | 0.25 | 0.09 |  | 0.27 | 0.63 | 0.89 | 0.04 |  | 0.41 | 0.42 | 0.48 | 0.08 |  | 0.3 | 0.6 | 0.1 | 0.04
| $Y_{\rm SZ}$ | 0.25 | 0.09 |  | 0.27 | 0.63 | 0.89 | 0.04 |  | 0.63 | 0.50 | 0.64 | 0.07 |  | 0.33 | 0.56 | 0.11 | 0.04
In-betw. | $\rho$ | 0.33 | 0.11 |  | 0.34 | 0.37 | 0.65 | 0.09 |  | 0.63 | 0.60 | 0.67 | 0.08 |  | 0.51 | 0.22 | 0.39 | 0.08
| $P$ | 0.33 | 0.10 |  | 0.34 | 0.38 | 0.67 | 0.08 |  | 0.50 | 0.45 | 0.65 | 0.08 |  | 0.45 | 0.26 | 0.28 | 0.07
| $I_{\rm X}$ | 0.33 | 0.11 |  | 0.34 | 0.37 | 0.65 | 0.09 |  | 0.43 | 0.35 | 0.52 | 0.09 |  | 0.4 | 0.30 | 0.19 | 0.08
| $Y_{\rm SZ}$ | 0.33 | 0.10 |  | 0.34 | 0.38 | 0.67 | 0.08 |  | 0.49 | 0.28 | 0.46 | 0.09 |  | 0.42 | 0.33 | 0.14 | 0.08
Unrel. | $\rho$ | 0.43 | 0.11 |  | 0.45 | 0.37 | 0.70 | 0.08 |  | 0.55 | 0.42 | 0.58 | 0.10 |  | 0.52 | 0.30 | 0.25 | 0.08
| $P$ | 0.43 | 0.12 |  | 0.45 | 0.38 | 0.70 | 0.08 |  | 0.52 | 0.41 | 0.74 | 0.08 |  | 0.50 | 0.21 | 0.27 | 0.07
| $I_{\rm X}$ | 0.43 | 0.11 |  | 0.45 | 0.37 | 0.70 | 0.08 |  | 0.44 | 0.18 | 0.36 | 0.11 |  | 0.46 | 0.35 | 0.09 | 0.08
| $Y_{\rm SZ}$ | 0.43 | 0.12 |  | 0.45 | 0.38 | 0.70 | 0.08 |  | 0.52 | 0.21 | 0.48 | 0.10 |  | 0.49 | 0.33 | 0.09 | 0.08
Table 1: Summary of correlations between the distributions of Mach number and
the amplitude of fluctuations, and Mach and radius. (a): subsample used for
calculations; (b): fluctuating characteristic; (c-d): the mean value of Mach
number and RMS of Mach number around the mean value calculated from the
initial $M_{\rm 1d}-\delta\xi/\xi$ correlation (see Fig. 5); (e) - (h): the
best-fitting parameters of the correlation between $M_{\rm 1d}$ and radius
$r/r_{500c}$, Pearson’s rank, and the RMS of Mach number after subtracting
this best-fitting model; (i)-(l): the same but for the correlation between
$M_{\rm 1d}$ and $\delta\xi/\xi$, see also Fig. 5; (m) - (p): the best fitting
parameters of the correlation between $M_{\rm 1d}$, $r/r_{500c}$ and
$\delta\xi/\xi$, and the RMS of Mach number once this best-fitting model is
subtracted from the correlation. Note that for each subsample, the values
(e-h) are almost the same for all types of fluctuations because the
ellipticities of these characteristics are very similar. The top half of the
table shows results for linear relations, while the bottom half summarizes
correlations between log-10s of the considered characteristics (i.e.,
exploring a power law correlation).

To answer these questions, we checked all the correlations, summarizing the
results in Table 1. For all three subsamples, we calculated the mean and RMS
of the Mach number relative to the mean value from the initial $M_{\rm
1d}-\delta\xi/\xi$ correlations [columns (c-d)]. Not surprisingly, the mean
$M_{\rm 1d}$ is increasing with the dynamic state of a cluster, from $\sim
0.25$ for relaxed ones and reaching $\sim 0.43$, on average, for unrelaxed
ones. It is remarkable, though, that the scatter around the mean [column (d)]
is relatively small, $\sim 0.1$, and is essentially the same for all groups.

Columns (e)-(h) show results for the $M_{\rm 1d}-r/r_{500c}$ correlation,
namely, the best fitting intercepts and slopes, Pearson’s rank, and the RMS of
Mach number after this best-fitting model is removed from the correlation.
Relaxed clusters show a strong correlation of $M_{\rm 1d}$ with radius,
$R_{\rm p}\sim 0.9$. For perturbed clusters, the correlation is weaker but
still within the strong range, $R_{\rm p}\sim 0.6-0.7$. Once the best-fitting
models are subtracted from the correlations, the Mach number RMS reduces by a
factor of $\sim 2/1.3/1.5$ for relaxed/in-between/unrelaxed clusters,
respectively.

As for the $M_{\rm 1d}-\delta\xi/\xi$ [columns (i)-(l), the distributions are
shown in Fig. 5], one can see that the correlations with 3D amplitudes
(density or pressure) are essentially as strong as with radius in relaxed
clusters, $R_{\rm p}\sim 0.8-0.9$. This means that both correlations could be
used interchangeably. When fitting $M_{\rm 1d}$ with two parameters ($r$ and
$\delta\xi/\xi$), it does not reduce the RMS further. “Projected” amplitudes
correlate weaker with $M_{\rm 1d}$, especially $\delta I_{\rm X}/I_{\rm X}$
(as we saw in Fig. 5), and only reduce the initial RMS by 0.01-0.02. For in-
between clusters, the correlation with 3D amplitudes is almost as strong as
with $r/r_{500c}$, and combining all three characteristics reduced the RMS the
most, by a factor of $\sim 1.6$. In contrast to the relaxed group, the
correlation with $\delta I_{\rm X}/I_{\rm X}$ is significantly stronger than
with $\delta Y_{\rm SZ}/Y_{\rm SZ}$. Unrelaxed clusters show the strongest
correlation with the amplitude of pressure fluctuations, comparable with the
correlation strength with the radius. When radial correlations are combined
with the 3D amplitudes, the initial RMS is reduced the most, by a factor of
$\sim 1.6-1.7$.

A good performance of the simple (linear) radial model is encouraging, given
its simplicity. Verifying these predictions with X-ray and sub-mm observations
via measuring X-ray surface brightness and/or projected pressure fluctuations
and converting them to 3D proxies would be very interesting. Since the $M_{\rm
1d}-r$ correlation is driven by the history of structure formation while
$M_{\rm 1d}-\delta\xi/\xi$ by intrinsic fluid properties within the stratified
ICM, measuring Mach numbers through both proxies and comparing them with
direct velocity measurements could provide important constraints for
cosmological models of galaxy clusters.

Now that we found the best-fitting correlations of $M_{\rm 1d}$ and
$\delta\xi/\xi$ with the radius, it is interesting to check a residual $M_{\rm
1d}-\delta\xi/\delta\xi$ correlation, i.e., the correlation between these
characteristics after the corresponding radial trends are subtracted. This is
shown in Fig. 11 in Appendix A. One can see that the correlations with
$\delta\rho/\rho$ and $\delta P/P$ remain moderately strong for relaxed
clusters ($R_{\rm p}\sim 0.5-0.6$), while become weaker for in-between and
unrelaxed clusters (yet, at a moderate level, $R_{\rm p}\sim 0.4-0.5$). This
exercise confirms that, at least for weakly perturbed clusters, the amplitude
of density/pressure perturbations bears additional information on the gas
velocities that are not captured by the pure radial trends.

As is seen from Table 1, the remaining scatter of the Mach number around the
model that uses two-parameters fits increases from $\sim 0.04$ for the relaxed
clusters to $\sim 0.08$ for the in-between and unrelaxed sub-samples. If the
sample-averaged value is of interest, this factor of 2 increase in the RMS
could be compensated by a factor of 4 larger samples, so that the factor
$1/\sqrt{N}$, where $N$ is the number of objects in the sample, compensates
for the increased RMS. Since the relaxed sample makes up about a quarter of
all clusters, there is a certain value in dealing with the entire sample.
However, in practice, working with a cleaner and smaller sample has many other
benefits and appears as a viable option.

Besides the linear correlations, it is interesting to check a linear scaling
between the logarithms of the characteristics or, equivalently, a power-law
scaling between the initial quantities. Red lines and parameters in Fig. 5
show the best-fitting results between the log10 characteristics. The second
half of Table 1 also summarizes all considered cases. For the correlations
with radius and $\delta\xi/\xi$, both linear and power-law scalings provide
consistent results: the same cases of strong correlations, RMS is reduced by
similar factors. The amplitude $\delta\rho/\rho\sim M_{\rm 1d}^{1.2}$ for
relaxed clusters, i.e., close to the confirmed linear scaling, while for
unrelaxed clusters, the scaling is closer to $\sim M_{\rm 1d}^{2}$ (namely,
$\delta\rho/\rho$ and $\delta P/P$ are $\sim M_{\rm 1d}^{2.4}$). Idealized
hydrodynamic simulations of turbulence in stratified cluster atmospheres in a
static gravitational potential with pure solenoidal driving (Gaspari et al.,
2014; Mohapatra et al., 2020) already showed that such scaling is expected at
$M_{\rm 1d}>0.3$. It is interesting that despite all the complexity of
structure formation, clusters in cosmological simulations confirm this result.

##  5 Discussion

###  5.1 Proxies for hydrostatic mass bias

Figure 10: Correlations between the mass bias, $M_{\rm hse}/M_{\rm tot}$, and
radius (left), Mach number of gas motions (middle left), the amplitude of
density (middle right) or pressure (right) fluctuations in a sample of relaxed
clusters. The amplitudes and Mach numbers are calculated in elliptical shells.
The color indicates the amplitude of density fluctuations in the left and
radius in all other panels. Each track of points corresponds to an individual
cluster in the sample. Red lines are the best fitting linear functions,
$M_{\rm hse}/M_{\rm tot}=\alpha+\beta X$, where $X$ is the corresponding
characteristic on the X-axis. The best-fitting $\alpha,\beta$ and Pearson’s
ranks, $R_{\rm p}$, are shown in legends.

Measuring velocities of gas motions in the ICM is important for precise
cluster mass measurements through their X-ray or SZ observations under the
assumption of hydrostatic equilibrium between the thermal pressure of the gas
and gravity. Gas motions may provide significant non-thermal pressure support,
up to 10-20 per cent at $r_{500}$ (e.g., Lau et al., 2009; Nelson et al.,
2012; Nelson et al., 2014a; Nelson et al., 2014b), even in relaxed clusters,
and should be taken into account for the mass measurements suitable for high-
precision cosmology (Pratt et al., 2019, for a recent review).

In the context of this work, it is interesting to examine various approximate
proxies for mass bias, namely, the Mach number of gas motions (that will be
observed soon, e.g., with XRISM) and various amplitudes of fluctuations (can
be probed with current X-ray and SZ imaging data), and compare them with
predictions from the radial trends of mass bias in cosmological simulations.
We define the mass bias as $M_{\rm bias}=M_{\rm hse}/M_{\rm tot}$, where
$M_{\rm hse}$ is the mass derived from the equation of hydrostatic
equilibrium, namely, $M_{\rm hse}(<r)=-\frac{r^{2}}{G\rho(r)}\frac{dP_{\rm
th}(r)}{dr}$, where $\rho(r)$ and $P_{\rm th}(r)$ are the radial profiles of
gas density and thermal pressure, respectively, and $M_{\rm tot}$ is the total
mass enclosed within a sphere of radius $r$. Fig. 10 shows the correlations of
the mass bias with radius, Mach number, and the amplitudes of density and
pressure fluctuations. As before, we measured these characteristics in each
cluster at each radius. Only relaxed clusters are considered, and calculations
are done within $\approx 2r_{500c}$, excluding the central $0.2r_{500c}$. In
all four cases, the correlations are moderately strong, with Pearson’s rank
$R_{\rm p}\sim 0.5$. Not surprisingly, the best correlation of $M_{\rm bias}$
is found with the Mach number, although a pure radial trend performs almost
equally well. Among all the considered amplitudes of fluctuations,
$\delta\rho/\rho$ shows the strongest correlation with $M_{\rm bias}$, $R_{\rm
p}\approx 0.53$, while “projected” amplitudes correlate very weakly (not shown
in the Figure). We fit all correlations with a linear function, $M_{\rm
bias}=\alpha+\beta X$, where $X$ is one of the considered characteristics on
the X-axes. The best-fitting parameters and functions are shown in red. One
can see that the global trends of $M_{\rm bias}$ could be easily removed using
any of the four characteristics. As for the scatter (RMS of $M_{\rm bias}$
around the mean value), it is reduced from the initial value $\sim 8$ per cent
to $\sim 6$ per cent when removing the correlation with the Mach number. For
the radius, $\delta\rho/\rho$ and $\delta P/P$, the residual RMS in $M_{\rm
bias}$ amounts to $\sim 7$ per cent.

This result shows that, on average, it is possible to correct hydrostatic
masses for mass bias for a sample of relaxed galaxy clusters using any of the
discussed observables (Mach number, $\delta\rho/\rho$, $\delta P/P$). The
corrections based on 3D amplitudes work almost as well as the $M_{\rm 1d}$
correction, i.e., the remaining scatter after any of these corrections is
about the same $\sim 6-7$ per cent. We also checked if combining multiple
proxies for the mass bias may further reduce the scatter (i.e., performing
two-parameter fits to $M_{\rm bias}$), however, the scatter always remained at
the same level. The origin of this residual scatter requires further studies
and is beyond the scope of this study. Note that measuring radial information
(radial profile of velocity or the amplitude of fluctuations) is crucial since
local (i.e., within smaller radial regions) correlations are significantly
weaker. The success of the radial trends of $M_{\rm bias}$ (Fig. 10, left) is
encouraging. Indeed, if confirmed and calibrated with the X-ray and SZ imaging
and XRISM measurements for $M_{\rm 1d}$, it will represent the easiest way of
making the first-order hydrostatic bias corrections to derived masses.

###  5.2 Caveats and future studies

Our results are based on a sample of clusters simulated with minimal (non-
radiative) physics. Any additional physics, e.g., gas cooling, feedback
process, MHD, may affect gas clumpiness, the amplitude of fluctuations, and
considered correlations. Moreover, various prescriptions for these additional
physics in cosmological simulations may also have an effect. Therefore, it is
important to extend the analysis to other cosmological simulations in the
future.

To approximately estimate the robustness of our conclusions, we analyzed a
subsample of our clusters that were modeled in a cosmological context,
including additional physics (cooling, star formation, SN and AGN feedback, UV
background). While the mean amplitudes of fluctuation at specific radii may
change maximum by a factor of $\sim 1.5-2$, they are consistent with the NR
results within the scatter. Projected amplitudes show a slightly increased
scatter with the additional physics included, while the mean values remain
consistent. The main conclusions from the correlations with $M_{\rm 1d}$ are
qualitatively the same for both types of simulations, and the proportionality
coefficients $\eta$ agree well within the scatters. Overall, the implemented
additional physics in the Omega500 simulations does not seem to affect our
main conclusions.

It is important to mention that any comparisons of these types of studies have
to take into consideration that adopted procedures for velocity filtering
(e.g., decomposition of the velocity field into coherent and turbulent flows)
may lead to substantial differences in results. The results also depend quite
strongly on the choice of considered regions and ellipticity treatment in the
calculations.

Finally, it is important to mention that, to some extent, our results rely on
a somewhat subjective, visual classification of clusters based on their
dynamical state (relaxed, in-between, unrelaxed). Other classifications of the
same sample have been used in previous works (see e.g., Shi et al., 2016; Chen
et al., 2019). Comparison with the cluster classification based on the mass
ratio of mergers within the last 4.5 Gyr by Chen et al. (2019) shows that
while the boundary between unrelaxed and in-between clusters could be blurred,
the selection of relaxed clusters is quite robust. Indeed, about 84 per cent
of relaxed clusters, identified based on our visual classification, were
classified as smooth mergers (equivalent to relaxed) from the mass accretion
histories. Overall, the classification of more than 60 per cent of clusters in
our sample is weakly dependent on the chosen classification method. This means
that our main results on relaxed vs. perturbed clusters should be stable to
the choice of the classification method.

##  6 Conclusions

In this study, we explored gas density and pressure fluctuations in the ICM as
well as their directly observable characteristics, namely fluctuations of
X-ray surface brightness and SZ $y-$parameter, using a sample of 78 galaxy
clusters from cosmological simulations Omega500. We examined scaling relations
between the amplitudes of these fluctuations and velocities of gas motions in
clusters in different dynamic states (relaxed, in-between, and unrelaxed),
accounting for radial variations of these characteristics and halo
ellipticity. We further calibrated a commonly-used in observations statistical
relation between the amplitude of fluctuations and velocity Mach number in the
central and outer cluster regions. We studied whether the amplitude of density
fluctuations provides a much more accurate proxy for velocity amplitudes
compared to a mean radial dependence of amplitudes with radius. Finally, we
discussed whether these characteristics could be used as reliable proxies for
hydrostatic mass bias and the expected level of remaining scatter in the bias.
Our main findings are summarized below:

  * •

With a large sample of simulated galaxy clusters, we confirm that the averaged
over the subsamples of relaxed, in-between, and unrelaxed clusters amplitudes
of fluctuations increase with the distance from the cluster center. The
sample-averaged amplitude of density or pressure fluctuations in relaxed
clusters is below 60 per cent, while it could be up to 90 per cent in
unrelaxed ones. “Projected” amplitudes, $\delta I_{\rm X}/I_{\rm X}$ and
$\delta Y_{\rm SZ}/Y_{\rm SZ}$ integrated over all spatial scales, vary
between $\sim 0.2-0.5$ and $\sim 0.1-0.3$ at $r=(0.2-2)r_{500c}$ in relaxed
objects and between $\sim 0.6-1$ and $0.25-0.7$ in unrelaxed ones,
respectively. Accounting for halo ellipticity is important, especially for
relaxed clusters within the central $\sim r_{500c}$ regions, as it may reduce
the average amplitude by up to a factor of $\sim 2$ at some radii. For
unrelaxed clusters, halo ellipticity almost does not affect the mean
amplitude, while increasing the scatter (Fig. 4). These predictions could be
checked with current X-ray and SZ observations using cluster imaging data.

  * •

Among the considered amplitudes of fluctuations, the sample-averaged mean
$\delta\rho/\rho$ at a given $r$ has the lowest scatter in clusters in all
dynamic states. The scatter in $\delta I_{\rm X}/I_{\rm X}$ appears the
largest, however, it is most prone to the method and parameters used to remove
high-density clumps from the data.

  * •

One can use the amplitude of density or pressure fluctuations for classifying
galaxy clusters based on their dynamic state. Relaxed, in-between, and
unrelaxed clusters are statistically separated especially well at
$r\sim(0.5-0.7)r_{500c}$ in both spherical and elliptical shells. This
classification can be performed for both simulated and observed galaxy
clusters.

  * •

There is a strong linear, almost one-to-one, correlation between the
distributions of $\delta\rho/\rho$ (or $\delta P/P$) and the Mach number of
gas motions in relaxed galaxy clusters within the $2r_{500c}$ region. For
perturbed clusters, the correlation is weaker but still within the moderately
strong regime. This is encouraging as one can use the measured amplitudes as
robust proxies for velocities of gas motions in galaxy clusters in various
dynamic states. The average proportionality coefficient between the amplitude
of density (pressure) fluctuations and Mach number is $0.9\pm 0.2$ ($1\pm
0.2$), $1\pm 0.3$ ($1.2\pm 0.4$), and $1.3\pm 0.5$ ($1.5\pm 0.5$) for relaxed,
in-between and unrelaxed clusters, respectively. When using this method, it is
important to account for the ellipticity of gas distribution, especially for
relaxed and mildly perturbed clusters.

  * •

Linear correlations between “projected” amplitudes and Mach number are
significantly weaker (Pearson ranks $<0.5$) except for $\delta I_{\rm
X}/I_{\rm X}$ for in-between clusters and $\delta Y_{\rm SZ}/Y_{\rm SZ}$ for
relaxed ones. While these “projected” amplitudes could be directly measured
from the X-ray and SZ images, it is important to convert them to
$\delta\rho/\rho$ or $\delta P/P$ for the most robust constraints on $M_{\rm
1d}$. This conversion is non-trivial and requires further studies, however,
the idea could be similar to the one proposed for scale-dependent amplitudes
and applied to observed power spectra of X-ray surface brightness fluctuations
(see Churazov et al., 2012; Zhuravleva et al., 2015, for details).

  * •

The power-law scalings between $M_{\rm 1d}$, radius, and the amplitude of
fluctuations are almost as strong as the linear correlations. They also lead
to similar-level scatter in residual Machs as the linear models once removed
from the initial data. For relaxed clusters, a power-law scaling between
$\delta\rho/\rho$ and $M_{\rm 1d}$ is close to the linear one, while unrelaxed
clusters show the best-fitting scaling closer to $\delta\rho/\rho$ (the same
for $\delta P/P$) $\propto M_{\rm 1d}^{2}$, indicating an increased role of
compressive modes and/or strong variations of the perturbed gravitational
potential in unrelaxed systems.

  * •

The hydrostatic mass bias calculated in relaxed galaxy clusters at each radius
within the $r\sim 2r_{500c}$ region correlates moderately strongly with the
radius, Mach number, and the amplitudes of density or pressure fluctuations,
with the Pearson rank $\sim 0.5$. One can use the provided best-fitting
scalings to correct for the average mass bias in observations. The remaining
scatter is expected $\sim 6$ per cent when using the correlation with $M_{\rm
1d}$ and $\sim 7$ per cent if any of the other three correlations is used. If
the radial trend of the mass bias predicted in simulations is confirmed with
observations, it would provide the easiest way of making the first-order
hydrostatic mass bias correction to the observed cluster masses.

Measuring Mach numbers through various proxies could provide important
constraints for cosmological models since the $M_{\rm 1d}-r$ correlation is
driven by the history of structure formation while $M_{\rm 1d}-\delta\xi/\xi$
by intrinsic fluid properties within the ICM. Current X-ray and SZ
observations of galaxy clusters can verify our predictions on gas
fluctuations, while soon-launched XRISM will provide velocities necessary for
testing the remaining predictions.

## Acknowledgements

Part of the analyses 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.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A $M_{\rm 1d}-\delta\xi/\xi$ correlations after removing radial
variations

It is interesting to check a residual correlation between the Mach number of
gas motions and fluctuation amplitudes after removing the radial variations of
these characteristics. Fig. 11 shows the distributions after the best-fitting
radial trends were subtracted from the initial $M_{\rm 1d}$ (see Table 1 for
the best-fitting parameters) and $\delta\xi/\xi$. Both linear (navy) and
power-law (light blue) models for the radial trends were considered. One can
see that (1) the correlations with 3D amplitudes remain moderately strong with
$R_{\rm p}\sim 0.4-0.6$ for clusters in all dynamic states; (2) the scatter is
the smallest for relaxed clusters; (3) projected amplitudes show a weak
correlation with Mach ($R_{\rm p}<0.4$), especially in dynamically-perturbed
clusters. Despite smaller values of $R_{\rm p}$ compared to those in Fig. 5,
this result indicates the fundamental origin of the correlation between the
velocities of gas motions and gas fluctuations in the ICM.

Figure 11: The same correlations as in Fig. 5 with the radial variations of
amplitudes and Mach numbers being removed by subtracting corresponding best-
fitting linear (navy) or power-law (light blue) models. The corresponding
Pearson ranks are shown in legends. The dashed lines indicate a one-to-one
match for visual guidance.