11institutetext: IRAP, Université de Toulouse, CNRS, CNES, UT3-PS, Toulouse,
France
11email<EMAIL_ADDRESS>22institutetext: Department of Astronomy,
University of Geneva, Ch. d’Ecogia 16, CH-1290 Versoix, Switzerland
33institutetext: Department of Astrophysical Sciences, Princeton University, 4
Ivy Lane, Princeton, NJ 08544-1001, USA 44institutetext: Dipartimento di
Fisica e Astronomia, Universitá di Bologna, Via Gobetti 93/2, 40122, Bologna,
Italy 55institutetext: Center for Astrophysics $|$ Harvard $\&$ Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA 66institutetext: Université Paris-
Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France
77institutetext: INAF, Osservatorio di Trieste, v. Tiepolo 11, 34131, Trieste,
Italy 88institutetext: IFPU, Institute for Fundamental Physics of the
Universe, Via Beirut 2, 34014 Trieste, Italy 99institutetext: Istituto
Nazionale di Astrofisica, IASF-Milano, via A. Corti 12, 20133, Milano, Italy
1010institutetext: Hamburger Sternwarte, University of Hamburg, Gojenbergsweg
112, 21029 Hamburg, Germany 1111institutetext: Istituto di Radioastronomia,
INAF, Via Gobetti 101, 40122, Bologna, Italy 1212institutetext: INAF,
Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3,
40129 Bologna, Italy 1313institutetext: DiSAT, Università degli Studi
dell’Insubria, via Valleggio 11, I-22100 Como, Italy 1414institutetext:
Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5,
I-00185 Roma, Italy 1515institutetext: Università degli studi di Roma ‘Tor
Vergata’, Via della ricerca scientifica, 1, 00133 Roma,
Italy1616institutetext: INFN, Sezione di Roma 2, Università degli studi di
Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, Italy

# CHEX-MATE : turbulence in the ICM from X-ray surface brightness fluctuations

S. Dupourqué CHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuations N. Clerc CHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuations E. Pointecouteau CHEX-MATE : turbulence in the ICM
from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in the ICM
from X-ray surface brightness fluctuations D. Eckert CHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuations M. Gaspari CHEX-MATE :
turbulence in the ICM from X-ray surface brightness fluctuationsCHEX-MATE :
turbulence in the ICM from X-ray surface brightness fluctuations L. Lovisari
CHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuations G.W. Pratt CHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuations E. Rasia CHEX-MATE : turbulence in the ICM from X-ray
surface brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray
surface brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray
surface brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray
surface brightness fluctuations M. Rossetti CHEX-MATE : turbulence in the ICM
from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in the ICM
from X-ray surface brightness fluctuations F. Vazza CHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuations M. Balboni CHEX-MATE :
turbulence in the ICM from X-ray surface brightness fluctuationsCHEX-MATE :
turbulence in the ICM from X-ray surface brightness fluctuations I. Bartalucci
CHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuations H. Bourdin CHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuations F. De Luca CHEX-MATE : turbulence in the ICM from
X-ray surface brightness fluctuationsCHEX-MATE : turbulence in the ICM from
X-ray surface brightness fluctuationsCHEX-MATE : turbulence in the ICM from
X-ray surface brightness fluctuationsCHEX-MATE : turbulence in the ICM from
X-ray surface brightness fluctuations M. De Petris CHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuationsCHEX-MATE : turbulence in
the ICM from X-ray surface brightness fluctuations S. Ettori CHEX-MATE :
turbulence in the ICM from X-ray surface brightness fluctuationsCHEX-MATE :
turbulence in the ICM from X-ray surface brightness fluctuations S. Ghizzardi
CHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface brightness
fluctuations P. Mazzotta CHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuationsCHEX-MATE : turbulence in the ICM from X-ray surface
brightness fluctuations

(Received ???; accepted ???)

The intra-cluster medium is prone to turbulent motion that will contribute to
the non-thermal heating of the gas, complicating the use of galaxy clusters as
cosmological probes. Indirect approaches can estimate the intensity and
structure of turbulent motions by studying the associated fluctuations in gas
density and X-ray surface brightness. In this work, we want to constrain the
gas density fluctuations at work in the CHEX-MATE sample to obtain a detailed
view of their properties in a large population of clusters. To do so, we use a
simulation-based approach to constrain the parameters of the power spectrum of
density fluctuations, assuming a Kolmogorov-like spectrum and including the
stochastic nature of the fluctuation-related observables in the error budget.
Using a machine learning approach, we learn an approximate likelihood for each
cluster. This method requires clusters to be not too disturbed, as
fluctuations can originate from dynamic processes such as merging.
Accordingly, we remove the less relaxed clusters (centroid shift $w>0.02)$
from our sample, resulting in a sample of 64 clusters. We define different
subsets of CHEX-MATE to determine properties of density fluctuations as a
function of dynamical state, mass and redshift, and investigate the
correlation with the presence or not of a radio halo. We found a positive
correlation between the dynamical state and density fluctuation variance, a
non-trivial behavior with mass and no specific trend with redshift or the
presence/absence of a radio halo. The injection scale is mostly constrained by
the core region. The slope in the inertial range is consistent with Kolmogorov
theory. When interpreted as originating from turbulent motion, the density
fluctuations in $R_{500}$ yield an average Mach number of
$\mathcal{M}_{3D}\simeq 0.4\pm 0.2$, an associated non-thermal pressure
support of $P_{\text{turb}}/P_{\text{tot}}\simeq(9\pm 6)\%$ or a hydrostatic
mass bias $b_{\text{turb}}\simeq 0.09\pm 0.06$, which is in line with what is
expected from the literature.

###### Key Words.:

X-rays: galaxies: clusters, galaxies: clusters: intracluster medium,
turbulence

##  1 Introduction

Clusters of galaxies, which are the largest gravitationally bound structures
in the Universe, house the majority of their baryonic content within the
intracluster medium (ICM). This hot gas ($\sim 10^{7}-10^{8}$ K) quickly
develops turbulent cascades as a result of the various perturbations it
undergoes. Different phenomena will dictate the injection of kinetic energy
into the ICM, depending on the location within the cluster. The central parts
are dominated by feedback from the Active Galactic Nucleus (AGN, McNamara &
Nulsen, 2012; Gaspari et al., 2014a; Voit et al., 2017) while the outer parts
are prone to accretion from the cosmic web and merging with groups or clusters
of galaxies which will induce shocks and therefore turbulence (Nelson et al.,
2012). These perturbations and their associated turbulence significantly
contribute to the non-gravitational heating of the ICM, thereby influencing
the hydrostatic mass bias of galaxy clusters (Piffaretti & Valdarnini, 2008;
Lau et al., 2009; Rasia et al., 2012; Nelson et al., 2014; Biffi et al., 2016;
Pratt et al., 2019; Gianfagna et al., 2023).

Generally speaking, turbulence is characterized by chaotic, irregular motion
in a fluid, resulting from the injection of high-scale kinetic energy and high
Reynolds numbers. It involves the transfer of energy from large scales to
smaller scales within the fluid, leading to complex, unpredictable patterns of
motion. Turbulence in galaxy clusters is extensively modelled through
numerical simulations (e.g. Vazza et al., 2012; Schmidt et al., 2016; ZuHone
et al., 2018; Mohapatra & Sharma, 2019; Ayromlou et al., 2023). Studying
turbulence in galaxy clusters can help in characterising the non-thermal
pressure support of the ICM, as this should be one of its key components
(Vazza et al., 2018; Angelinelli et al., 2020), together with magnetic fields
and, to a lesser extent, cosmic rays.

Constraining this process is relatively straightforward with highly resolved
X-ray information in the spectral domain, as the motion of the gas is mirrored
in the observed spectrum through centroid shifts and broadening of the
emission lines. Direct measurements of the line centroid shift were performed
with the XIS instrument onboard Suzaku, deriving upper limits as a subsonic
line of sight motions in local clusters of various dynamical states (Ota et
al., 2007; Sugawara et al., 2009; Tamura et al., 2011, 2014; Ota & Yoshida,
2016). More recently, by calibrating the EPIC-pn instrument of XMM-Newton
using the instrumental fluorescence lines, measurements of velocity maps with
a precision of $\sim 200-500$ km s-1 were performed for local clusters, and
the first centroid shift structure function from direct observations was
derived (Gatuzz et al., 2022b, a, 2023b, 2023a). Broadening measurements were
performed using the RGS instrument of XMM-Newton (Sanders et al., 2011;
Sanders & Fabian, 2013; Pinto et al., 2015; Ogorzalek et al., 2017), providing
overall upper limits on subsonic motions. A comprehensive review of the study
of the ICM motion is proposed by Simionescu et al. (2019). Calibration
uncertainties and limited spatial resolution are the main showstoppers when
studying the ICM motions in the X-ray domain. The emergence of X-ray integral
field units will enable high-precision direct measurements of such turbulent
motions (e.g. Roncarelli et al., 2018; Zhang et al., 2023a). The Hitomi
Collaboration (2016) determined a line-of-sight velocity dispersion of $\sim
160\text{ km s}^{-1}$ in the core of Perseus cluster. Similar measurements
will be accessible with the recent launch of XRISM/Resolve (XRISM Science
Team, 2020), the Line Emission Mapper (LEM, Kraft et al., 2022) and
Athena/X-IFU in the long term (Barret et al., 2020). Indirect measurements can
be achieved by searching for fluctuations in the thermodynamical properties of
the ICM that can be due to turbulent processes. For instance, X-ray surface
brightness (X-SB) fluctuations point to a pure hydrodynamical flow in the ICM
(Schuecker et al., 2004). Such methods have been applied to the Coma (Churazov
et al., 2012; Gaspari & Churazov, 2013) and Perseus cluster (Zhuravleva et
al., 2015). Zhuravleva et al. (2018) used a small sample of clusters to
investigate the turbulent motions occuring in their cool-cores. Hofmann et al.
(2016) investigated the fluctuations of several thermodynamical properties of
the ICM. Khatri & Gaspari (2016) and Romero et al. (2023) showed a promising
approach, joining the X-ray images to the Sunyaev-Zel’dovich effect (SZE).

However, these studies do not account for the stochastic nature of the surface
brightness fluctuations. Indeed, when observing a random field with a finite
size sample and trying to assess its properties, an additional source of
irreducible variance is added to the total error budget. As shown in Clerc et
al. (2019) and in Cucchetti et al. (2019), the so-called sample variance may
be dominant at any spatial scale achievable by XMM-Newton and its prevalence
grows with scale up to an order of magnitude with respect to other sources. In
Dupourqué et al. (2023), we performed an X-ray surface brightness fluctuation
analysis on the X-COP cluster sample (Eckert et al., 2017a), with a forward
model approach to explicitly account for the sample variance. In this paper,
we apply this approach to the CHEX-MATE cluster sample, which is an order of
magnitude higher in terms of statistics, and contains a more diverse
population of clusters.

In Sect. 2, we describe the methodology to compute the density fluctuation
parameters. In Sect. 3, we present the joint constraints on density
fluctuations, applied to various subsamples of CHEX-MATE, show the effect of
excluding the central part of clusters in our analysis, and investigate the
correlation with radio data using the LOw Frequency ARray (LOFAR) Two Meter
Sky Survey second data release (LoTSS DR2, Shimwell et al., 2022; Botteon et
al., 2022). In Sect. 4, we discuss the Mach number obtained when assuming that
the density fluctuations originate from turbulent processes, and the resulting
turbulent mass bias. Throughout this paper, we assume a flat $\Lambda$CDM
cosmology with $H_{0}=70\text{ km s}^{-1}$ and
$\Omega_{m}=1-\Omega_{\Lambda}=0.3$. Scale radii are defined according to the
critical density of the Universe at the corresponding redshift. The $R_{500}$
and $M_{500}$ values are obtained from the MMF3 detection method (Melin et
al., 2006) as highlighted in Appendix A of The CHEX-MATE Collaboration (2021).
The Fourier transform conventions are highlighted in Appendix A.

##  2 Data and Method

###  2.1 The CHEX-MATE Sample

The CHEX-MATE program111http://xmm-heritage.oas.inaf.it/, detailed in The
CHEX-MATE Collaboration (2021) paper, is a three mega-second, multi-year XMM-
Newton Heritage Programme aimed at obtaining X-ray observations of 118
minimally biased, signal-to-noise-limited galaxy clusters detected by Planck
through SZE (The Planck Collaboration, 2016). The project’s objectives are to
accurately understand the statistical properties of the cluster population,
examine how gas properties are influenced by dark matter halo collapse, reveal
non-gravitational heating origins, and address major uncertainties in mass
determination that restrict the use of clusters for cosmological parameter
estimation.

To achieve these goals, a sample of 118 Planck clusters was chosen based on
their SZE signal (S/N ¿ 6.5) and categorized into two subsamples: Tier 1
contains 61 low-redshift objects in the northern sky that provide an unbiased
view of the most recent cluster population ($0.05<z<0.2$ and
$2<\frac{M_{500}}{10^{14}M_{\odot}}$ ¡ 9); Tier 2 is representative of the
most massive systems formed in the Universe’s history ($z<0.6$ and
$\frac{M_{500}}{10^{14}M_{\odot}}>7.25$). These subsamples share four common
clusters. The XMM-Newton observations are characterized by an exposure time
ensuring an S/N of 150 within $R_{500}$ in the [0.3-2.0] keV band. This
requirement was chosen to satisfy three conditions: i) to calculate the
temperature profile up to $R_{500}$ (with an accuracy of $\pm 15\%$ in the
[0.8–1.2] $R_{500}$ region), ii) to measure the mass obtained from the $Y_{X}$
mass proxy (Kravtsov et al., 2006), in which $Y_{X}$ = $M_{g,500}T_{X}$
($M_{g,500}$ represents the gas mass within $R_{500}$ and $T_{X}$ refers to
the spectroscopic temperature estimated in the [0.15-0.75] $R_{500}$ range),
with an uncertainty of $\pm 2\%$, and iii) to derive the mass derived from
hydrostatic equilibrium (HE) at $R_{500}$ with a precision of approximately
$15-20\%$.

For further information on the sample, scientific objectives, and observation
strategies in X-ray and other wavebands, readers should consult the
presentation paper: The CHEX-MATE Collaboration (2021).

###  2.2 Data preparation

The images in this study were generated using the pipeline established for the
X-COP sample (Eckert et al., 2017a; Ghirardini et al., 2019) and adopted by
the CHEX-MATE collaboration. Specifically, XMM-Newton data were processed with
SAS software (version 16.1.0) and the Extended Source Analysis Software
package (ESAS, Snowden et al., 2008). The choice of SAS version 16.1 is
motivated in Rossetti et al. 2023 (submitted) to which we refer for more
details. In brief, we aim to avoid the presence of unresolved bugs in
intermediate versions of SAS and a major refactoring effort for version 21.0,
which makes existing reduction pipelines incompatible for the time being.
However, we ensured that the latest calibration available in January 2021 was
applied, as specified by Bartalucci et al. (2023). Count images, exposure
maps, and particle background maps were extracted in the narrow [0.7-1.2] keV
band to maximize the source-to-background emission ratio and minimize
systematics related to EPIC background subtraction. A detailed account of the
procedure is presented in Bartalucci et al. (2023), and a complete gallery of
images is displayed in Fig. 6 in The CHEX-MATE Collaboration (2021). Point
sources are extracted using the ewavelet routine in the two energy bands
[0.3-2] and [2-7] keV. Every source with flux smaller than the maximum of the
logN-logS distribution determined from the point source extraction is masked.

###  2.3 Bayesian inference for density fluctuations

The methodology described below was first developed and discussed in detail in
Dupourqué et al. (2023). This is a 3-step procedure that consists of
determining an unperturbed surface brightness model, defining the associated
surface brightness fluctuations and deriving a meaningful observable, and
finally constraining parameters of the 3D power spectrum, including the whole
error budget. We summarise it briefly in the following sections but refer
interested readers to the aforementioned publication for further details.

####  2.3.1 X-SB unperturbed model

In this study, we aim to model the unperturbed surface brightness $S_{X}$ of a
galaxy cluster to define fluctuations. Using $\@vec{r}=(x,y,\ell)$ as the 3D
position parametrization, and $\@vec{\rho}=(x,y)$ as the 2D equivalent, the
surface brightness can be expressed as:

$S_{X}(\vec{\rho})=\int^{+\infty}_{-\infty}\Psi(\@vec{r})\>n_{e}^{2}(\@vec{r})\>\mathop{}\\!\mathrm{d}\ell+B,$

where $n_{e}(\vec{r})$ is the electronic density and $\Psi(\@vec{r})$
represents a combination of various factors such as the cooling function,
cosmological dimming, Galactic absorption, and convolution with the XMM
response functions. $B$ denotes a constant surface brightness sky background.
This radial model is folded in a triaxial shape for the cluster, with no
scaling along the line of sight. The centre position and ellipticity are left
free, and the centre is fitted with a normal prior centred on the X-ray peak
as defined by Bartalucci et al. (2023), with a 50% relative scatter. A
modified Vikhlinin model (with $\alpha=0$ and $\gamma=3$, see Vikhlinin et
al., 2006; Shi et al., 2016) is used to parametrize the density profile:

$n_{e}(r)=n_{e,0}\frac{(1+r^{2}/r_{c}^{2})^{-3\beta/2}}{(1+r^{3}/r_{s}^{3})^{\epsilon/6}}.$

The $\Psi(\@vec{r})$ function is determined using an analytical functional
form that we fit on a count-rate grid estimated using XSPEC 12.11.1. To do so,
we compute the expected count-rate of a PhAbs*APEC model with the same
instrumental setup and redshift for each cluster in the CHEX-MATE sample, more
details are available in App. C of Dupourqué et al. (2023). The metallicity
was set to $0.3\leavevmode\nobreak\ Z_{\odot}$ and the abundances fixed
according to Anders & Grevesse (1989). Galactic absorption is accounted for
using the $N_{H}$ data from the HI4PI survey (HI4PI Collaboration, 2016). The
count image is rebinned using Voronoi tessellation (Cappellari & Copin, 2003),
with approximately 100 counts per bin, to increase the speed of the inference
given the computationally intense surface brightness model that we consider.
This binning introduces a median bias of $\sim 0.5\%$ which peaks at 4 $\%$ in
the outermost (and low significance) region, which is still lower than the
expected $10\%$ systematic error due to Poisson noise, see App. D in Dupourqué
et al. (2023) for more details. The number of counts expected in each bin is
forward modelled using the aforementioned surface brightness model. The model
parameters are determined using Bayesian inference with a Poisson likelihood
in each bin. The posterior distributions are sampled using the No U-Turn
Sampler (Hoffman & Gelman, 2014) as implemented in the numpyro library
(Bingham et al., 2019; Phan et al., 2019).

####  2.3.2 Surface brightness & density fluctuations

Surface brightness emission from galaxy clusters in the soft X-ray band
originates mainly from the thermal bremsstrahlung occurring in the ICM plasma.
The intensity of this radiation is directly proportional to the squared
electronic density. Hence, over-densities and under-densities in the gas
distribution can be found as fluctuations in the surface brightness images,
and dominate the fluctuations when compared to other sources (Churazov et al.,
2012). To constrain the properties of the density fluctuations, we choose to
model it as a Gaussian random field, which can be described using only a
second order moment such as the power spectrum or the structure function. As
we expect a strong link between density fluctuations and turbulence (Gaspari &
Churazov, 2013; Gaspari et al., 2014b; Zhuravleva et al., 2014b; Simonte et
al., 2022), we model the 3D power spectrum of the random field using a
Kolmogorov-like functional form (e.g. ZuHone et al., 2016; Dupourqué et al.,
2023) since turbulence in clusters is compatible with the purely
hydrodynamical case (Schuecker et al., 2004):

$\mathcal{P}_{3D,\delta}(k)=\sigma_{\delta}^{2}\frac{e^{-\left(k/k_{\text{inj}}\right)^{2}}e^{-\left(k_{\text{dis}}/k\right)^{2}}k^{-\alpha}}{\int
4\pi
k^{2}\,e^{-\left(k/k_{\text{inj}}\right)^{2}}e^{-\left(k_{\text{dis}}/k\right)^{2}}k^{-\alpha}\mathop{}\\!\mathrm{d}k},$
(1)

where $k_{\text{inj}}$ and $k_{\text{dis}}$ are, respectively, the Fourier
frequencies corresponding to the injection scale $\ell_{\text{inj}}$ and
dissipation scale $\ell_{\text{dis}}$, $\alpha$ is the inertial range spectral
index (shortened to “slope”), and $\sigma_{\delta}^{2}$ is the variance of
fluctuations. We fix the dissipation scale for our entire sample at
$10^{-3}\leavevmode\nobreak\ R_{500}$ (e.g. Brüggen & Vazza, 2015), which is
significantly smaller than the spatial scales accessible with CHEX-MATE
($\gtrsim 10^{-2}\leavevmode\nobreak\ R_{500}$. In any case, the arbitrary
choice of dissipation scale has little to no impact on the normalization of
the power spectrum.

####  2.3.3 X-SB fluctuations and power spectrum

In this analysis, the X-SB fluctuations are assumed to originate exclusively
from intrinsic density fluctuations (see Churazov et al., 2012, for other
sources of X-SB fluctuations). The squared density profile can be split into a
rest component $n_{e,0}^{2}$ and fluctuations $\delta<<1$, such that the
density is given by
$n_{e}(\@vec{r})=n_{e,0}(\@vec{r})\times(1+\delta(\@vec{r}))$. Using a first
order expansion of the density, the raw surface brightness image
$S_{X}(\vec{\rho})$ can be decomposed as the sum of an unperturbed image
$S_{X,0}(\vec{\rho})$ and a surface brightness fluctuation map. We define the
fluctuation map as follows:

$\Delta(\vec{\rho})\overset{\operatorname{def}}{=}\frac{S_{X}(\vec{\rho})-S_{X,0}(\vec{\rho})}{2}\simeq\int^{+\infty}_{-\infty}\epsilon_{0}(\vec{r})\delta(\vec{r})\mathop{}\\!\mathrm{d}\ell,$
(2)

where $S_{X,0}$ can be obtained using the best-fit X-SB model and
$\epsilon_{0}(\vec{r})=\Psi(\vec{r})\leavevmode\nobreak\ n_{e,0}^{2}(\vec{r})$
is the associated best-fit emissivity. The factor two arises when considering
the square dependency of $\delta$ with the surface brightness. To study the
contribution of each spatial scale to surface brightness fluctuations, an
analysis in Fourier space is performed. We note the Fourier 3D frequencies as
$\vec{k}=(k_{x},k_{y},k_{\ell})$, and associated 2D frequencies as
$\vec{\kappa}=(k_{x},k_{y})=(\kappa,\varphi_{k})$ in cylindrical coordinates.
The power spectrum of the fluctuations, $\mathcal{P}_{2D,\Delta}$, is defined
as:

$\mathcal{P}_{2D,\Delta}(\kappa)=\frac{1}{2\pi}\int|\hat{\Delta}(\vec{\kappa})|^{2}\mathop{}\\!\mathrm{d}\varphi_{k},$

where $\hat{\Delta}$ is the Fourier transform of the two-dimensional (2D) map
$\Delta$ and $\varphi$ is the direction angle in Fourier space. The numerical
evaluation of $P_{2D,\Delta}$ is performed using the method described by
Arévalo et al. (2012), which computes the variance of images filtered by
Mexican hats on a characteristic scale to estimate the power spectrum,
implicitly handling irregular masks such as the excluded point sources or XMM-
Newton mosaics. In practice, we divide the X-SB fluctuation power spectrum
$\mathcal{P}_{\text{2D, }\Delta}$ by the power spectrum of the best-fit X-SB
model with Poisson noise $\mathcal{N}_{\text{2D, }\Delta}$, to isolate
exclusively the fluctuations due to density variations. This observable is
defined as a Signal-to-Noise ratio (S/N) as follows:

$\text{S/N}_{\Delta}\overset{\operatorname{def}}{=}\frac{\mathcal{P}_{\text{2D,
}\Delta}}{\mathcal{N}_{\text{2D, }\Delta}}.$ (3)

We compute this spectrum on scales defined by the size of the cluster in
$R_{500}$ units. The observational strategy detailed in The CHEX-MATE
Collaboration (2021) and Rossetti et al. 2023 (submitted) ensures that the
thermodynamic properties can be reconstructed up to $R_{500}$, so we use
photons inside this radius. Since the CHEX-MATE sample has a wide range of
redshifts, the lowest scale we can investigate is greater for distant clusters
than for nearby clusters. We choose a conservative, redshift-dependent scale
that allows us to study all clusters respecting the Nyquist-Shannon criterion
and minimizing the lost information, in the form of an affine function:

$\rho_{\text{low}}\simeq 0.123\times z+0.023\leavevmode\nobreak\
\left[R_{500}\right].$ (4)

The lowest scale for each cluster in the sample as a function of $z$ is
displayed in Fig. 1, where Eq. 4 is shown with a dotted line. The effect of
this redshift-dependent scale will be to reduce the importance of distant
clusters compared with closer ones, since the information linked to them is
intrinsically less constraining.

Figure 1: Lowest scale accessible as a function of the redshift, and best
conservative affine relationship used in this paper as defined in Eq. 4, which
is defined as an affine function with the same slope as the points, but raised
to be above them.

####  2.3.4 Simulation-based inference

To link our observable quantities to the parameters of the density
fluctuations, we need to establish the formal link between these two
quantities, which is encapsulated by the likelihood function in a Bayesian
framework. However, we cannot derive an analytical formula or closed form for
this likelihood in this case. It is however possible to generate mock
observables, by generating a Gaussian random field with known parameters and
injecting it into our best-fit emissivity model before projection along the
line of sight. By including all our sources of uncertainty in this forward
modelling, it is possible to assess the dispersion associated with our
parameters and our observable due to the sample variance and Poisson noise.
This procedure is repeated for 300,000 simulations per cluster to build the
training sample, whose generating parameters are randomly drawn from the
following priors: $\sigma_{\delta}\sim 10^{\mathcal{U}\left(-2,0\right)}$,
$\ell_{\text{inj}}\sim 10^{\mathcal{U}\left(-2,0.3\right)}$ and
$\alpha\sim\mathcal{U}\left(2,5\right)$. We showed in Dupourqué et al. (2023)
that this number of simulations is sufficient to build a likelihood estimator
enabling reconstruction of the expected parameters for mock observations. We
train a neural network on these simulations to estimate an approximation of
the likelihood of our problem, using a masked autoregressive flow
(Papamakarios et al., 2017, 2019) as implemented in the sbi package (Tejero-
Cantero et al., 2020). In a nutshell, a masked autoregressive flow is a
density estimator that is adjusted to estimate any well-behaved distribution
(in the mathematical sense). In this problem, the likelihood of a specific
observation is learned from the 300,000 joint samples of parameters and mock
observations. Once it is trained, the model can return an approximation of the
likelihood of any observation. This approximate likelihood is sampled and
inverted into a posterior distribution using classical MCMC methods such as
NUTS (Hoffman & Gelman, 2014).

###  2.4 Splitting CHEX-MATE into subsamples

Subsample | Definition | Size
---|---|---
State (I) | $0.001<w<0.005$ | 22
State (II) | $0.005<w<0.011$ | 21
State (III) | $0.011<w<0.019$ | 21
Mass (I) | $2.2<M_{500}<3.77$ | 11
Mass (II) | $3.77<M_{500}<4.65$ | 10
Mass (III) | $4.65<M_{500}<8.77$ | 10
Redshift (I) | $0.09<z<0.26$ | 12
Redshift (II) | $0.26<z<0.42$ | 11
Redshift (III) | $0.42<z<0.6$ | 11
Radio halo | Halo in LoTSS DR2 | 8
No radio halo | No halo in LoTSS DR2 | 13
Table 1: Definition of the subsamples used in the analysis and relative size
of each sample. The total number of clusters used in the analysis is 64.
Masses are in $10^{14}M_{\odot}$ unit.

The presented methodology allows the estimation of the likelihood of
individual observations of each cluster. By joining them, one can estimate the
parameters of the density fluctuations on selected subsets of the CHEX-MATE
sample. For this study, we exclude some clusters for quality or modelling
reasons. Our modelling is effective on regular clusters. Disturbed clusters,
which have strong departures from spherical symmetry, produce very
inhomogeneous density fluctuations due to structural residues. It is important
to emphasize that the descriptive power of the Gaussian field model decreases
in clusters featuring structures resulting from its dynamic assembly, such as
sloshing spirals or cool cores. To mitigate this effect, we want to
systematically exclude clusters for which this approach is not suited, and use
morphological indicators to do so. Campitiello et al. (2021) extracted
morphological parameters of the CHEX-MATE sample and compared it to a visual
classification in three classes: i) relaxed, ii) mixed, iii) disturbed. The
centroid shift $w$, which relates to the variation of the distance of the peak
of luminosity and the centroid of the emission for a varying aperture up to
$R_{500}$, was computed for every cluster in the CHEX-MATE sample, and shows
good separation of the three states. This quantity is dimensionless as it is
scaled to $R_{500}$. We choose to exclude clusters such as $w>0.02$, which is
comparable to the threshold proposed by Lovisari et al. (2017) to classify
disturbed clusters, and correspond to 43 objects in the whole CHEX-MATE
sample. Campitiello et al. (2021) also derived a combined $M$ parameter (see
also Rasia et al., 2013; De Luca et al., 2021), whose purpose was to estimate
the disturbance of clusters on a continuum using a combination of multiple
morphological indicators. Using $w$ or $M$ yields similar results in this
study, as both can define a threshold to exclude the most visually disturbed
clusters. However, using $w$ instead of $M$ removes more mixed clusters from
the sample with the same threshold definition, which is arguably a more
conservative choice. Finally, $w$ correlates better with the variance of
density fluctuations than $M$ in the whole sample, with a Pearson $R$ of
respectively 0.5 and 0.36, showing that $w$ is, in this case, more suited to
track the surface brightness disturbances. It is worth noting that despite
this threshold, some disturbed clusters remain in our sample, such as
PSZ2G266.04-21.25, the famous “Bullet Cluster”, using either criteria to
perform the selection. Along with this disturbance threshold, we have
systematically excluded double clusters, as well as clusters that are
contaminated by the galactic emission (in particular CIZA clusters, see
Ebeling et al., 2000), Virgo emission, or by clusters in the foreground or
background.

Figure 2: Correlation between the mass $M_{500}$, the redshift $z$ and the
centroid shift $w$ in the CHEX-MATE sample without disturbed or contaminated
clusters, including the clusters in LoTSS DR2 with and without radio halos.

Applying this selection leads to a set of 64 clusters, which is $\sim 60\%$ of
the total cluster number in CHEX-MATE. This set is divided into several
subsamples, which are defined in Tab. 1 and displayed in Fig. 4(a). The
symbols (I), (II), (III) for each subsample denote an increase in the quantity
used to define it, for instance, the State (I) contains the most relaxed
clusters while the State (III) contains the most disturbed. In particular, we
are interested in the dependence of the density fluctuation parameters on the
mass of the clusters, their redshift, and their dynamical state. The sample is
divided into bins of almost equal sizes to have comparable statistics. The
CHEX-MATE sample is built on two subsamples, Tier 1 and Tier 2, which are
respectively made up of close clusters of varying mass, and high mass clusters
of varying redshifts. Hence, we perform mass separation using Tier 1 and
redshift separation using Tier 2. The dynamical state separation is performed
on the whole sample based on the parameter $w$. The CHEX-MATE sample is also
partially observed by LOFAR. We use the radio halo data and power from the
LoTSS-DR2 (Botteon et al., 2022). This survey covers 40 clusters from the
CHEX-MATE sample, in which 18 admit detected radio haloes. We apply the same
filtering procedure and exclude the most disturbed clusters as done
previously, lowering the number of clusters to 8 with a halo and 13 without a
halo. To investigate the correlation between X-SB fluctuations and radio data,
we further define two subsamples of clusters containing or not a radio halo.
The good mixing between mass, redshift, dynamical state, and the presence of
radio halo is shown in Fig. 2, guaranteeing that we are not introducing any
selection bias with our subsample definition.

##  3 Results

###  3.1 Joint constraints on the sample

The constraints on surface brightness fluctuations can be significantly
affected by sample variance when each cluster is studied individually. To
mitigate this effect and increase the overall signal, we combine the analysis
of density fluctuation properties across multiple clusters by summing their
log-likelihoods. We calculate the X-SB fluctuation spectra using information
within the $R_{500}$ radius. As cluster cores are often contaminated by cool-
cores and structural residuals, we investigate the impact of excluding the
central part up to 0.15 $R_{500}$ from our analysis. The joint parameters of
the density fluctuations and their associated power spectra, estimated across
the CHEX-MATE sample without disturbed clusters (64 objects), are shown in
Fig. 3 and in Tab. 2, Tab. 3, both with and without core exclusion. These are
compared with the previous results on the X-COP sample, including the
constraints from Dupourqué et al. (2023) for the two rings between 0.1-0.25
and 0.25-0.5 $R_{500}$ as well as the entire $R_{500}$ area. The results
obtained for the fluctuation variance $\sigma_{\delta}$ and the slope $\alpha$
of CHEX-MATE are structurally compatible with the X-COP 0.1-0.25 and 0.25-0.5
$R_{500}$ rings, except for the injection scale $\ell_{\text{inj}}$. This is
further discussed in Sec. 3.2. The slope of the CHEX-MATE analysis including
the core, as in X-COP, is fully compatible with Kolmogorov (1941, K41) theory.

Figure 3: (left) Joint posterior distributions of the standard deviation,
$\sigma_{\delta}$, the injection scale, $\ell_{\text{inj}}$, and the spectral
index $\alpha$ of the density fluctuation power spectrum parameters, estimated
on the cleaned CHEX-MATE sample and with the X-COP parameters in the 0.1-0.25
and 0.25-0.5 $R_{500}$ rings, as computed in Dupourqué et al. (2023). The
black dashed line represents the expected 11/3 index from Kolmogorov theory.
(right) Associated 3D power spectra as defined from Eq. 1. The dashed slope
indicates the expected slope from Kolmogorov. The scales are in units of
$R_{500}$.

The posterior distributions obtained for the joint fit across the state, mass,
and redshift subsamples, as defined in Tab. 1, are illustrated in Fig. 4(b)
when the core is included and in Fig. 4(c) where the core is excluded. The
separation of the dynamical states (Fig. 4(d) and Fig. 4(g)) shows an increase
in the variance of the fluctuations with the centroid shift $w$. These results
can also be compared with those found for the X-COP sample (Dupourqué et al.,
2023), which showed a positive correlation between dynamical indicators that
increase with cluster perturbation, specifically the centroid shift $w$ and
the Zernike moments. Despite such differences, a Kolmogorov/hydrodynamical
cascade (with -11/3 slopes) appears to be a good approximation for the type of
turbulence developing in such systems, as the slope is constrained to
$\alpha=3.58\pm 0.11$ including the core and $\alpha=3.28\pm 0.08$ excluding
it. The flatter slope in the outer regions might be correlated with a more
predominant magnetic field (Vestuto et al., 2003).

(a) Subsamples in $M_{500}-z$ plane

(b) Full sample with core

(c) Full sample without core

(d) Splitting by states (with core)

(e) Splitting by masses (with core)

(f) Splitting by redshifts (with core)

(g) Splitting by state (without core)

(h) Splitting by mass (without core)

(i) Splitting by redshift (without core)

Figure 4: (a) Distribution of the subsamples in the mass-redshift plane. The
shaded areas indicate the mass and redshift subsamples extracted from Tiers 1
and 2 respectively. The colour of the dots reflects the dynamical state of the
clusters as measured by the centroid shift $w$. (b) Joint posterior
distributions of the standard deviation, $\sigma_{\delta}$, the injection
scale, $\ell_{\text{inj}}$, and the spectral index $\alpha$ of the density
fluctuation power spectrum parameters, evaluated on the full sample including
the core $<0.15R_{500}$ region. The black dashed line represents the expected
11/3 index from Kolmogorov theory. (d), (e) and (f) are the same plot using
the subsamples associated respectively with the state, mass, and redshift as
defined in Tab. 1, including the core region. (g), (h), and (i) use the same
subsamples without the core regions.

The separation of the Tier 1 sample into several mass bins (Fig. 4(e) and Fig.
4(h)) shows a non-trivial correlation between mass and variance of the
fluctuations. Regarding previous observational constraints, Hofmann et al.
(2016) find a mild anti-correlation between relative density fluctuations (or
Mach number) and masses, decreasing by a factor of 1.5$\times$ from poor to
massive clusters. This is in line with our highest and lowest mass bins but
not with the intermediate mass bin which shows much lower normalisations. Note
that the low-mass bin does not contain any bin State (I) objects, i.e. the
most relaxed clusters in this sample, the high level of normalisation measured
can therefore also be linked to the dynamical state within the bin. As for the
injection scale, clusters are expected to virialize self-similarly, however,
meso- and micro-scale astrophysics introduces key deviations depending on the
hydrodynamical process at play (e.g., feedback, condensation, sloshing). For
instance, the injection scale tends to reside near the $R_{500}$ region, but
lower mass systems are capable to reach smaller injection scales toward the
core region, likely due to the relatively stronger AGN feedback impact (e.g.
Gaspari et al. 2014a). The slope is instead roughly invariant with mass, again
suggesting a similar type of turbulence (hydrodynamical). The Mass (III)
sample is the most comparable to the previous study on X-COP, and shows
comparable variance and slope, but increased injection, which is interpreted
as a spatial resolution effect in Sec. 3.2.

The last corner plots (Fig. 4(f) and Fig. 4(i)) show the separation of the
sample over the Tier 2 redshift range. At high redshifts $(z\gtrsim 1)$,
galaxy clusters are expected to experience more frequent mergers (e.g. Huško
et al., 2022), which can induce stronger turbulence and density fluctuations
in the ICM. With increasing redshift, we observe a mild increase in density
variance (hence turbulence) and injection scales. However, our redshifts reach
at best up to $z\sim 0.5-0.6$, far from the early universe formation. Thus,
over the redshift range studied via Tier 2, most turbulence parameters appear
to be fairly similar.

###  3.2 Effect of core exclusion & spatial resolution

The first differences between the analyses with and without the cluster core
involve an increase in fluctuation variance, especially in the Mass (III),
State (III) and global results, which might suggest that clusters are more
disturbed on average in the outer regions. There is also a decrease in the
global slope, which could indicate either that the outer regions are noisier
due to a lower signal-to-noise ratio or that turbulent processes undergo
reinjection at multiple scales, thereby flattening the observed spectrum.
About the injection scale, energy injection processes in the cluster cores,
such as AGN feedback and sloshing, have a smaller characteristic scale than
the outer regions, which are dominated by accretion and merger events. Our
constraints on the injection scale average the low injections in the centre
and the high injections from the outer parts. This might explain the reason
for which we constrain $\ell_{\text{inj}}$ to approximately $R_{500}$ when
including the core region but reduce it to an upper limit of roughly
$2R_{500}$—the largest scale that can be measured in our field—when the core
is excluded. Using CHEX-MATE, the low angular size of clusters prevents us
from assessing properly the low injection scale in the central region, due to
the low number of pixels in this region. As an illustration, there is a median
number of $\sim 200$ pixels below 0.15 $R_{500}$ in CHEX-MATE, compared to
$\sim 10000$ pixels below 0.15 $R_{500}$ in X-COP. However, such behaviour has
already been observed in the radial study of X-COP.

Furthermore, as CHEX-MATE clusters are, on average, smaller in the sky than
X-COP clusters, the lowest spatial limit that can be exploited on X-COP is 50%
smaller than that on CHEX-MATE, which implies a better sampling of the
inertial range. This is illustrated in Fig. 1, where the conservative lowest
scale we defined is shown along the individual scales that could be achieved
for each cluster, and the clusters common with X-COP are on the lower left.
This definition for the lowest scale allows consistency of analysis over the
whole sample but results in less sensitivity to small scales which are found
in the core regions of clusters, skewing the results toward larger injection
scales. In Fig.3, the constraints from X-COP align with those from CHEX-MATE
when we compare the outer rings, which exclude the core region driving the low
injection. Systematically excluding the core from this kind of analysis will
reduce the contamination from substructures such as the sloshing spirals, the
cool core or cold fronts from older merger, which further increase the
correlation between measured density fluctuations and turbulent process, see
Sec. 4.2 for further discussions.

###  3.3 Correlation with radio data

The presence of turbulence in the ICM is expected to induce particle
acceleration through second order Fermi mechanisms in turbulent medium (e.g.
Brunetti & Lazarian, 2016) which could be responsible for the presence of
radio haloes in galaxy clusters. Eckert et al. (2017b) showed a bimodality in
the distribution of the amplitude of X-SB fluctuations at 150 kpc, separating
clusters with and without radio haloes and the correlation between the radio
halo power at 1.4 GHz and the one dimensional Mach number $\mathcal{M}_{1D}$,
using a sample of 51 clusters. More recently, Zhang et al. (2023b) performed a
similar analysis on a smaller sample of 36 halos, detected at lower
frequencies with LOFAR. Confirmation of the bimodality was not achievable due
to the limited sample size. Additionally, no clear indications of a
correlation between the velocity dispersion and the radio power at 150 MHz
were found, possibly due to the LoTSS DR2 survey containing more objects with
an ultra-steep spectrum. These objects experienced less energetic mergers when
compared to clusters in Eckert et al. (2017b) (see Brunetti & Jones (2014) for
a review). In this section, we use the CHEX-MATE, which is partially covered
by the LoTSS (Shimwell et al., 2019), searching for a trend in our density
fluctuation model regarding the presence of radio halo, using the whole
available spatial scales.

Figure 5: (left) Visualization of the clusters in LoTSS DR2 in the mass-
redshift plane, with estimated halo size and power at 150 MHz. (right) Density
fluctuation parameters estimated when using subsamples with confirmed or
excluded radio halo.

In Fig. 5, we plot the contour obtained for the analysis split into the common
clusters with LoTSS DR2, with and without radio halo. We note that, on
average, the parameters we examine align well. We do not observe any
significant difference between the two subsamples. It is important to note
that the exclusion of the most perturbed clusters leads us to study a
population of radio-halo with a homogeneous distribution regarding their
dynamical state, as illustrated in Fig. 2. Hence, this population is very
different from the populations previously studied in Eckert et al. (2017b) and
Zhang et al. (2023b), which may justify the absence of a clear trend in our
results. The limited sample size hinders the determination of trends between
fluctuations and radio halos in this study, but the upcoming radio coverage of
the CHEX-MATE sample (82 clusters in total) will allow for the confirmation or
refutation of this trend.

##  4 Discussions

###  4.1 Interpretation as turbulent motion

Assuming turbulence in the ICM as the primary source of relative density
fluctuations, precursory hydrodynamical and theoretical studies (Gaspari &
Churazov 2013; Gaspari et al. 2014b; Zhuravleva et al. 2014b) demonstrated
that the characteristic amplitude of turbulent velocity dispersion – which
defines the Mach number $\mathcal{M}=\sigma_{v}/c_{\rm s}$ (with $\sigma_{v}$
the 3D velocity dispersion and $c_{\rm s}$ the sound speed) – is linearly tied
to the characteristic amplitude of density fluctuations, unlike in non-
stratified fluids, where a quadratic trend is expected. This relation has also
been studied in the context of turbulence in a box and corroborated for major
stratification levels (Mohapatra & Sharma, 2019; Shi & Zhang, 2019; Mohapatra
et al., 2020, 2021). Gaspari & Churazov (2013) initially found the linear
scaling relation between the density fluctuations at different scales and the
associated Mach number, by testing varying plasma physics (turbulence, thermal
conduction, electron-ion equilibration) in high-resolution hydrodynamical
simulations of galaxy clusters. As thermal conduction is expected to be highly
suppressed in the observed ICM, an argument supported by our nearly Kolmogorov
slopes and by theoretical work (e.g. ZuHone & Roediger, 2016), the relation
from Gaspari & Churazov (2013), involving the peak of the amplitude spectrum,
can be simplified to :

$\mathcal{M}_{3D}\approx
2\sigma_{\delta}\left(\frac{l_{\text{inj}}}{L_{\text{500}}}\right)^{-1/5},$
(5)

where $L_{\text{500}}$ is an injection scale of 500 kpc. Accounting for
different levels of turbulence and related spectral slopes, this adds a
scatter of $\sim$ 0.1 dex.

Figure 6: Mach number $\mathcal{M}_{3D}$ and turbulent mass bias
$b_{\text{turb}}$ as obtained using the relationship from Eq. 7 and Eq. 8
respectively, applied to the dynamical state, the mass, the redshift and the
radio halo subsamples. The blue and green bands represent the average value
over the whole sample, respectively, with and without the core. (I) to (III)
relates to the increasing value which has been used to define the subsample,
i.e. increasing disturbance for the state, and increasing mass and redshift.
See Tab. 1 for proper definition of each subsample.

Subsequently, Zhuravleva et al. (2014a) and Gaspari et al. (2014b) retrieved
an analogous linear relationship with the one-dimensional Mach number
($\mathcal{M}_{1D}=\mathcal{M}_{3D}/\sqrt{3}$):

$\mathcal{M}_{1D}\approx(1\pm 0.3)\,\sigma_{\delta}.$ (6)

Overall, the above-quoted studies converge toward a linear relation with
roughly unitary normalization with $\mathcal{M}_{1D}$. Recently, this was also
validated in a set of cosmological simulations with a sample of 80 clusters
(Zhuravleva et al. 2023), in particular for non-merging systems, which is also
the focus of our work. Eq. 5 has a low dependency on the injection scale, and
is comparable to the scaling from Eq. 6. To account for the dispersion between
the two relations, we choose to convert the density fluctuations in Mach
number using a unity scaling with a dispersion that accounts for the
dispersion between Eq. 5 and Eq. 6:

$\mathcal{M}_{3D}\approx\sqrt{3}\times(1\pm 0.4)\,\sigma_{\delta}.$ (7)

Further, we can estimate the contribution of such processes to the non-thermal
pressure support globally occurring in our sample. The ratio between the non-
thermal pressure and the total pressure can be expressed as a function of the
3D turbulent Mach number $\mathcal{M}_{3D}$ (Eckert et al., 2019) if the non-
thermal contribution exclusively comes from turbulence:

$\frac{P_{\text{turb}}}{P_{\text{tot}}}=\frac{\mathcal{M}_{3D}^{2}\gamma}{\mathcal{M}_{3D}^{2}\gamma+3},$
(8)

where $\gamma=5/3$ is the polytropic index. The ratio from Eq. 8 can directly
be interpreted as a turbulent mass bias $b_{\text{turb}}$ if $\gamma=5/3$. The
Mach numbers and turbulent mass bias $b_{\text{turb}}$ deduced in $R_{500}$
are shown in Fig. 6 and Tab. 2 and Tab. 3 for each subsamples, using the
scaling from Eq. 7, with a $\mathcal{M}_{3D}\simeq 0.41\pm 0.17$ when
evaluated on the whole sample without the core. Consistently with previous
studies (e.g., Hofmann et al. 2016) and physical expectations, 3D Mach numbers
remain in the subsonic regime, typically in the 0.3 - 0.5 range. Direct
comparison with existing works is not straightforward, as the value of the
$\mathcal{M}_{3D}$ derived in this work requires comparison to the 1D velocity
broadening measurements. Moreover, the analysis regions we used cover a much
larger fraction of the cluster volume than the core region that is probed in
most other works. In any case, our results are compatible with subsonic
motions as implied by Sanders et al. (2011); Sanders & Fabian (2013); Pinto et
al. (2015) The measured $\mathcal{M}_{3D}$ number value translates into
turbulent pressure support of $P_{\text{turb}}/P_{\text{tot}}\simeq(9\pm
6)\%$, which is in line with the typical value retrieved in cosmological
simulations (e.g., Vazza et al. 2018), and previous observational works on the
X-COP sample (Eckert et al., 2019; Dupourqué et al., 2023). The bias of
$b=0.09\pm 0.06$ the entire considered sample is compatible with the value
obtained in Lovisari et al. (2023) when accounting for the fluctuations at all
scales, and previous observational works in X-ray (e.g. Eckert et al., 2019;
Ettori & Eckert, 2022) or using lensing (e.g. Mahdavi et al., 2008; Zhang et
al., 2010). As noted in Sec. 3.1, Fig. 6 also shows a clear increase in
turbulent Mach numbers toward more disturbed systems, as well as toward lower
cluster masses. The redshift trend is instead minor, as expected by our
limited range $z<0.6$. A complementary approach, proposed by Lovisari et al.
(2023), determines the Mach number and the non-thermal contribution from
temperature fluctuations measured on a CHEX-MATE subsample. By performing our
joint analysis on the same subsample, we obtain a Mach number of the order of
$\mathcal{M}_{3D}\simeq 0.41\pm 0.18$ which is compatible with
$\mathcal{M}_{3D}\simeq 0.36\pm 0.10$ as obtained from Lovisari et al. (2023).
We note that the resolution of the X-SB maps is better than the resolution of
the temperature maps, so the most sensitive scales of turbulence might be
different. We stress that these results are derived under the assumption of
homogeneous and isotropic turbulence, which consequently is described by a
Gaussian random field. The stratification of the turbulence, its multiphase
aspect, and its general anisotropies constitute departures from the assumption
of Gaussianity. In particular, not taking stratification into account
(Mohapatra et al., 2021) and not considering accreted clumps or structural
merger remnants may lead to an overestimate of our turbulent Mach number.
Accounting for the stochasticity of this process via the sample variance
constitutes a non-negligible barrier when it comes to accurately constrain
more complex models. Nevertheless, our use of a sample of clusters and the
exclusion of central regions limits contamination by clumps, and our use of
scaling relations derived from cosmological simulations (Gaspari & Churazov,
2013; Simonte et al., 2022; Zhuravleva et al., 2023) partially considers of
the effect of anisotropies into our final Mach number calculation.

###  4.2 Discussing the correlation between $\sigma_{\delta}$ and
$\mathcal{M}_{3D}$

The linear link between density fluctuations and turbulent velocities can be
analytically derived via simplified hydrodynamics equations for a stratified
medium like the ICM (Gaspari & Churazov, 2013; Zhuravleva et al., 2014b). This
correlation has been investigated and tested using hydrodynamical simulations
(Gaspari & Churazov, 2013; Zhuravleva et al., 2014b; Gaspari et al., 2014b;
Simonte et al., 2022; Zhuravleva et al., 2023). At variance with the above
works, Simonte et al. (2022) applied small-scale filtering techniques to
disentangle the spurious contribution of bulk flows and extract the isotropic
part of turbulent motions, and excised gas clumps in the simulated ICM.
Following this approach, they reported a substantially lower normalization
(factor of 2-to-3) and larger scatter in the above relation, also with
significant differences in the slope of the relation, depending on the
dynamical state of clusters. They concluded that, in general, the presence of
residual gas density fluctuations not directly linked to turbulence, but to
the crossing of self-gravitating clumps and substructures in the ICM, is hard
to excise. It introduces a large scatter in the observational link between
projected density fluctuations and the true turbulent budget of the ICM. On
the other hand, without distinguishing between turbulent and bulk gas motions
in the ICM, cosmological runs by Zhuravleva et al. (2023) show such
mild/strong correlation and normalization. To disentangle both sources of
fluctuations, a clump filtering method has to be applied to simulated and
observed datasets. Typically, this involves identifying the high-density tail
of the gas density distribution. For example, Simonte et al. (2022) cut the
top 5% tail, while Zhuravleva et al. (2023) more conservatively cut less than
0.05% of the values (3.5 $\sigma$). The former reports an applied threshold
$\sigma_{\delta}^{2}=0.22$, hence potentially filtering motions
$\mathcal{M}_{1D}\geq 0.47$, which may account for the difference among the
various theoretical predictions.

From an observational perspective, it is non-trivial to understand the nature
of gas clumping in clusters. Besides filtering obvious mergers, the results of
our analysis can be biased toward some residual density structures, which may
have varying effects on the parameters depending on their distance from the
centre of the cluster, its mass and its dynamical state. Gas clumps in the
outer parts of the cluster will induce high $\sigma_{\delta}$ as they
represent a large relative fluctuation, which could contribute to the
correlation between dynamical states and fluctuation normalizations as
disturbed clusters contain more substructures. In the central parts, gas
clumps are understood as highly significant fluctuations in a very dense
region, which will drive the overall spectrum to lower values of
$\sigma_{\delta}$. The highly structured fluctuations associated with clumps
can also bring high slopes and constraints on the injection scale that are
close to the clump size. This highlights the challenge of disentangling
density fluctuations emerging from turbulent motions and substructures/clumps,
such as those also observed in the central parts of X-COP clusters (Dupourqué
et al., 2023) and in the CHEX-MATE sample, as illustrated in Fig. 7. One
approach to estimating the contamination due to these substructures could be
to use simulations involving contamination by modelled cool-cores or sloshing
spirals, and to estimate the biases induced on the reconstruction of the
parameters of the fluctuation field as well as the robustness of this approach
to the individual deviations of each cluster. This exercise would require the
exploration of a large space of contaminant parameters, and will be the
subject of a future study. Nevertheless, the results obtained by excluding the
central regions of each cluster should be more robust to the presence of
artifacts, as shown by the exclusion of the bullet of PSZ2G266.04-21.25, and
other clumps in Fig. 7.

##  5 Conclusions

We have applied the methodology introduced in Dupourqué et al. (2023) to the
CHEX-MATE sample. We focused on a sample of 64 objects, excluding the most
disturbed clusters, as our analysis is not suitable for strongly perturbed
surface brightness images. We used this subsample to characterise the gas
density fluctuations through simulation-based inference applied to the power
spectrum of surface brightness fluctuations, increasing the statistic by a
factor of $\sim 5$ when compared to X-COP.

  * •

We derived the parameters of the density fluctuation spectrum for different
CHEX-MATE subsamples. The high sample statistic allows us to define 3 classes
in terms of dynamical state, mass, and redshift. The fluctuations show a
normalisation that increases with the dynamical state, which relates to the
fact that disturbed clusters tend to have more density fluctuations. We
observe an anticorrelation between mass and normalisation of the fluctuations.
The redshift subsamples show invariant behaviour, which is consistent with the
fact that the turbulence within the ICM is not expected to change drastically
over the studied redshift range ($z<0.6$).

  * •

We assessed the effects of cluster core on the overall results by comparing
the analysis in $R_{500}$ to the analysis excluding the central $0.15R_{500}$
in each cluster. It can be seen that, in particular, the injection scale is no
longer constrained when the centre is excluded, which means that the injection
we measure is mainly driven by the high signal and low scale processes
occurring in the central region, while the signal outside $0.15R_{500}$ only
allows us to establish lower limits. The core exclusion reduces the
contamination from central structures such as the cool-cores or sloshing
spiral, making a more reliable measurement for the statistics of fluctuations.

  * •

We conducted a correlation analysis between our measurements and radio data
from the LoTSS-DR2, investigating potential relationships between density
fluctuation parameters and the presence of radio halos. No specific trends
emerged when comparing individual parameters of clusters with associated radio
halo parameters, which is in line with expectations from the population of
haloes we studied, which are low-frequency radio haloes in relaxed clusters.

  * •

We interpreted the density fluctuations as resulting from turbulent processes.
Using two scaling relations, we derived 3D Mach numbers for various subsamples
defined using the CHEX-MATE sample. We obtained an average
$\mathcal{M}_{3D}\simeq 0.4\pm 0.2$ and a corresponding non-thermal pressure
support of $P_{\text{turb}}/P_{\text{tot}}\simeq 9\pm 6\%$ or
$b_{\text{turb}}\simeq 0.09\pm 0.06$ in $R_{500}$, which is consistent with
what can be found in the literature.

More advanced constraints could be obtained in the future by crossing the
fluctuations of the different ICM observables, such as the fluctuations in the
SZE, which carry information at further distances from the centre thanks to
dependency over $n_{e}$ instead of $n_{e}^{2}$ for the X-Ray emissivity.
Pioneering work such as Khatri & Gaspari (2016) and Romero et al. (2023)
demonstrated the feasibility of this approach, yet it remains very challenging
even using high resolution SZ instruments and still has to integrate the
sample variance. In addition, direct and unambiguous measurement of gas
motions could be possible in the coming year with XRISM/Resolve (Terada et
al., 2021) and in the next decade with the Line Emission Mapper (LEM Kraft et
al., 2022) and Athena/X-IFU (Barret et al., 2020), through spatially resolved
observations of spectral lines centroid shift and broadening.

###### Acknowledgements.

EP, NC, SD, and GWP acknowledge the support of CNRS/INSU and CNES. LL
acknowledges the financial contribution from the INAF grant 1.05.12.04.01. MDP
acknowledges support from Sapienza Università di Roma thanks to Progetti di
Ricerca Medi 2020, RM120172B32D5BE2 and Medi 2021, RM12117A51D5269B. FV
acknowledges the financial support from the Cariplo ”BREAKTHRU” funds Rif:
2022-2088 CUP J33C22004310003. MR, IB, SE, SG acknowledge the financial
contribution from the contracts Prin-MUR 2022, supported by Next Generation EU
(n.20227RNLY3 The concordance cosmological model: stress-tests with galaxy
clusters), ASI-INAF Athena 2019-27-HH.0,“Attività di Studio per la comunità
scientifica di Astrofisica delle Alte Energie e Fisica Astroparticellare”
(Accordo Attuativo ASI-INAF n. 2017-14-H.0), and from the European Union’s
Horizon 2020 Programme under the AHEAD2020 project (grant agreement n.
871158). This work was granted access to the HPC resources of CALMIP
supercomputing center under the allocation 2022-22052. LOFAR data products
were provided by the LOFAR Surveys Key Science project (LSKSP; https://lofar-
surveys.org/) and were derived from observations with the International LOFAR
Telescope (ILT). LOFAR (Haarlem et al., 2013) is the Low Frequency Array
designed and constructed by ASTRON. It has observing, data processing, and
data storage facilities in several countries, that are owned by various
parties (each with their own funding sources), and that are collectively
operated by the ILT foundation under a joint scientific policy. The efforts of
the LSKSP have benefited from funding from the European Research Council,
NOVA, NWO, CNRS-INSU, the SURF Co-operative, the UK Science and Technology
Funding Council and the Jülich Supercomputing Centre. This research was
supported by the International Space Science Institute (ISSI) in Bern, through
ISSI International Team project #565 (Multi-Wavelength Studies of the
Culmination of Structure Formation in the Universe). This work made use of
various open-source packages such as matplotlib (Hunter, 2007), astropy
(Robitaille et al., 2013; The Astropy Collaboration et al., 2018),
ChainConsumer (Hinton, 2016), cmasher (Velden, 2020), sbi (Tejero-Cantero et
al., 2020), pyro (Bingham et al., 2019), jax (Bradbury et al., 2018), haiku
(Hennigan et al., 2020), numpyro (Bingham et al., 2019; Phan et al., 2019)

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##  Appendix A Fourier transform convention

In this paper, we define the Fourier transform with the classical signal
processing convention, namely $(0,-2\pi)$, see Weisstein (1995). This pairs
results in the forward transform highlighted in Eq. 9 and 10. We use $\hat{f}$
and $\tilde{f}$ to refer respectively to the 2D and 3D Fourier transform of a
function $f$.

$\mathcal{FT}_{2D}\left\\{f\right\\}\equiv\int\mathop{}\\!\mathrm{d}^{2}\vec{\rho}\,f(\vec{\rho})e^{-2i\pi\vec{k_{\rho}}.\vec{\rho}}=\hat{f}(\@vec{k}_{\rho})$
(9)

$\mathcal{FT}_{3D}\left\\{f\right\\}\equiv\int\mathop{}\\!\mathrm{d}^{3}\vec{r}\,f(\@vec{r})e^{-2i\pi\vec{k_{r}}.\vec{r}}=\tilde{f}(\@vec{k}_{r})$
(10)

##  Appendix B Table with values

The results obtained for the variance of density fluctuations
$\sigma_{\delta}$, the injection scale $\ell_{\text{inj}}$, the spectral index
$\alpha$, the 3D Mach number $\mathcal{M}_{3D}$ and the turbulent mass bias
$b_{\text{turb}}$ are displayed in Tab. 2 with the core region is included and
in Tab. 3 with the 0.15 $R_{500}$ inner region excluded.

Subsample | $\sigma_{\delta}$ | $\ell_{\text{inj}}$ | $\alpha$ | $\mathcal{M}_{3D}$ | $b_{\text{turb}}$
---|---|---|---|---|---
State (I) | $0.16\pm 0.01$ | $1.03\pm 0.21$ | $3.34\pm 0.15$ | $0.27\pm 0.11$ | $0.04\pm 0.03$
State (II) | $0.18\pm 0.02$ | $1.45\pm 0.23$ | $3.24\pm 0.12$ | $0.31\pm 0.13$ | $0.05\pm 0.04$
State (III) | $0.24\pm 0.02$ | $0.87\pm 0.24$ | $3.71\pm 0.29$ | $0.42\pm 0.17$ | $0.09\pm 0.07$
Mass (I) | $0.53\pm 0.06$ | $1.62\pm 0.27$ | $2.64\pm 0.09$ | $0.91\pm 0.39$ | $0.32\pm 0.16$
Mass (II) | $0.14\pm 0.02$ | $0.60\pm 0.20$ | $3.30\pm 0.25$ | $0.24\pm 0.10$ | $0.03\pm 0.03$
Mass (III) | $0.17\pm 0.01$ | $0.73\pm 0.16$ | $3.85\pm 0.18$ | $0.29\pm 0.12$ | $0.05\pm 0.04$
Redshift (I) | $0.18\pm 0.02$ | $0.94\pm 0.25$ | $3.94\pm 0.23$ | $0.32\pm 0.13$ | $0.05\pm 0.04$
Redshift (II) | $0.19\pm 0.02$ | $1.30\pm 0.31$ | $3.56\pm 0.27$ | $0.33\pm 0.14$ | $0.06\pm 0.05$
Redshift (III) | $0.20\pm 0.03$ | $1.12\pm 0.27$ | $3.57\pm 0.29$ | $0.35\pm 0.15$ | $0.06\pm 0.05$
Radio Halo | $0.24\pm 0.04$ | $1.48\pm 0.53$ | $3.47\pm 0.47$ | $0.40\pm 0.18$ | $0.08\pm 0.07$
No Radio Halo | $0.20\pm 0.02$ | $0.84\pm 0.24$ | $3.25\pm 0.19$ | $0.34\pm 0.14$ | $0.06\pm 0.05$
Joint | $0.19\pm 0.01$ | $1.04\pm 0.14$ | $3.58\pm 0.11$ | $0.34\pm 0.14$ | $0.06\pm 0.04$
Table 2: Joint marginalized constraints on the parameters for each of the subsamples defined in Tab. 1, uncertainties correspond to the 68% confidence ranges. Subsample | $\sigma_{\delta}$ | $\ell_{\text{inj}}$ | $\alpha$ | $\mathcal{M}_{3D}$ | $b_{\text{turb}}$
---|---|---|---|---|---
State (I) | $0.16\pm 0.02$ | $1.83\pm 0.14$ | $3.46\pm 0.16$ | $0.28\pm 0.12$ | $0.04\pm 0.04$
State (II) | $0.22\pm 0.02$ | $1.60\pm 0.21$ | $3.41\pm 0.18$ | $0.37\pm 0.16$ | $0.07\pm 0.06$
State (III) | $0.33\pm 0.03$ | $1.84\pm 0.12$ | $3.19\pm 0.10$ | $0.57\pm 0.23$ | $0.15\pm 0.10$
Mass (I) | $0.45\pm 0.06$ | $1.42\pm 0.24$ | $2.88\pm 0.13$ | $0.77\pm 0.34$ | $0.25\pm 0.14$
Mass (II) | $0.15\pm 0.03$ | $1.75\pm 0.19$ | $3.16\pm 0.23$ | $0.25\pm 0.12$ | $0.03\pm 0.04$
Mass (III) | $0.28\pm 0.03$ | $1.61\pm 0.21$ | $3.07\pm 0.11$ | $0.49\pm 0.20$ | $0.12\pm 0.08$
Redshift (I) | $0.22\pm 0.02$ | $1.71\pm 0.26$ | $4.00\pm 0.31$ | $0.37\pm 0.16$ | $0.07\pm 0.05$
Redshift (II) | $0.19\pm 0.02$ | $1.50\pm 0.24$ | $4.05\pm 0.25$ | $0.34\pm 0.14$ | $0.06\pm 0.05$
Redshift (III) | $0.22\pm 0.03$ | $1.47\pm 0.30$ | $3.70\pm 0.35$ | $0.39\pm 0.17$ | $0.08\pm 0.06$
Radio Halo | $0.20\pm 0.03$ | $1.55\pm 0.28$ | $3.64\pm 0.31$ | $0.35\pm 0.15$ | $0.06\pm 0.05$
No Radio Halo | $0.21\pm 0.02$ | $1.72\pm 0.20$ | $3.41\pm 0.19$ | $0.37\pm 0.16$ | $0.07\pm 0.05$
Joint | $0.24\pm 0.01$ | $1.93\pm 0.07$ | $3.29\pm 0.08$ | $0.41\pm 0.17$ | $0.09\pm 0.06$
Table 3: Joint marginalized constraints on the parameters for each of the
subsamples defined in Tab. 1, after exclusion of the central 0.15 R500 region.
Uncertainties correspond to the 68% confidence ranges.

##  Appendix C Fluctuation gallery

The gallery of surface brightness fluctuations on the CHEX-MATE subsample used
for this analysis is shown in Fig. 7, sorted with increasing values of the
centroid shift $w$.

Figure 7: Surface brightness fluctuation gallery with masked point sources, as
defined in Eq. 2 for the 63 clusters used in the analysis (see Fig. 4(a)), in
ascending order of $w$. The two rings denote $0.15$ and $1R_{500}$.


# Progress Metrics in DAG-based Consensus

Quan Nguyen  FANTOM  James Henderson  FANTOM  Egor Lysenko  FANTOM

###### Abstract

Lachesis protocol [2] leverages a DAG of events to allow nodes to reach fast
consensus of events. This work introduces DAG progress metrics to drive the
nodes to emit new events more effectively. With these metrics, nodes can
select event timing and can choose previous events as parents for their own
new events. Our results show that our event timing and parent selection
methods can help reaching consensus quicker and thus can reduce lower time to
finality significantly.

_K_ eywords event timing, parent selection, OPERA, directed-acyclic-graph, DAG
progress, metrics, Lachesis, consensus protocol

##  1 Introduction

Lachesis protocol [2, 1] is a BFT consensus protocol that reduces
communications overhead as events can be reused in the form of event DAG.
Nodes participating in Lachesis consensus produce events asynchronously. Nodes
generate event blocks to form a DAG of events, each of which contains
transactions. Each event block has references to parent event blocks. For an
event block $e$ and it’s parents $e_{p}$, there is no restriction about the
creator nodes of $e$ and that of $e_{p}$. Each node computes Roots to find
Atroposes in order to reach consensus on finalized event blocks.

Each node can generate new own event block and can choose which previous
events as the parents for this new event. Throughout this paper, the terms
‘event block‘ and ‘event‘ are used interchangeably. Event generation involves
two steps _Event Timing_ and _Parent Selection_.

  * •

Event Timing: node can decide by itself when it can generate a new own event
block.

  * •

Parent Selection: this is the process of select a suitable set of previous
event blocks and use them as parents (aka. $e_{p}$) for the new event.

For the network to perform efficiently, nodes need a set of criteria so as to
produce a new event in an effective manner. If nodes produced too many events,
it could increase communication overheads as peers will need to handle those
events. At the same time, many of such events are possibly abundant in the
sense that they do not necessarily contribute to reach a new root. Thus, it is
important to know whether a node should emit a new event and whether they
should choose certain events as the parents for this new event.

In this work, we introduce two new metrics, which measure the DAG progress of
an event and can be used to optimize the way that a node emits a new event.
The two metrics are _Quorum Indexer_ (QI) metric and _Root Knowledge_ (RK)
metric. The metrics enable nodes to determine whether a proposed new event
block will have a positive contribution towards reaching a new root.

Based on the two metrics, we have defined new criteria for event timing and
parent selection for nodes. In event timing, node can decide itself whether
creating a new event will be beneficial. Leveraging the metrics ensures that
nodes attempt to create events only if they can produce substantial DAG
progress toward finality. The metrics can also improve parent selection, in
which nodes can select previous events as candidate parents for its new event,
and thus it can improve the contribution of each event toward reaching
finality.

We compare the two metrics in a set of experiments. Our experimental results
show that RK metric outperformed QI metric. Our results also show that event
timing and parent selection based on the RK metric have significantly reduced
the node overheads (i.e. avoid generate excessive events and handle them) and
time-to-finality (TTF). Efficiency is achieved as nodes will create events
when they are able to make substantial DAG progress, reducing the
computational time and resources used for processing and transmitting events.

##  2 DAG Progress Metrics

This section describes two new DAG progress metrics. These metrics are used in
event timing and parent selection.

###  2.1 Quorum Indexer Metric

Quorum Indexer (QI) is a measure of DAG progress. QI metric is based on the
sequence numbers of HighestBefore events.

Let $v$ be a node. For a local DAG of $v$, we determine each highest (latest)
event $e_{i}$ for each node $i$. HighestBefore calculations are performed for
each node $v$ as follows.

  * •

Median: Each node $v$ computes the highest events $e_{i}$ for each node $i$ in
a local DAG of $v$. For each $e_{i}$, it then finds the sequence number of
$e_{i}$ in the subgraph. Then it calculates the median sequence number
weighted by the stake of each highest event creator’s stake. Intuitively, this
gives an estimate of the median highest event created by $i$ in all nodes.

  * •

Current Self: A node $v$ finds the sequence number of highest event created by
$v$ in the subgraph of its own.

  * •

New Self: A node pre-computes the sequence number of highest event created by
$v$ that would be in the subgraph if it created such a new event.

The above three sub-metrics for node $v$ are combined into a single QI metric
of DAG progress produced by a new event. To do so, the three metrics are
compared and transformed using a piecewise linear function into a metric in
the range $[0,w_{i}/W]$, where $w_{i}$ is the stake of node $v$ and $W$ is the
total stake of all nodes.

The QI metric $h^{(e)}$ is defined as summation across all nodes $i$ to give a
final metric of DAG progress for a new event $e$. The metric is in the range
[0,1], with 0 indicating little DAG progress, and 1 indicating significant DAG
progress.

###  2.2 Root Knowledge Metric

Progress in Lachesis consensus is achieved via the production of new roots and
frames. Root Knowledge (RK) metric is a more effective metric as it is built
on the knowledge of previous roots.

An event $e$ is a root when in $e$’s subgraph quorum roots of the previous
frame forkless-cause $e$. This means that within $e$’s subgraph quorum roots
of the previous frame are each in the subgraph of events created by quorum
nodes, potentially with different sets of quorum nodes for each previous root.
Further, there can be no forks detected for any of the nodes involved in
confirming the forkless-cause condition.

####  2.2.1 Notations

The progress of an event $e$ toward meeting the required conditions for the
next root can be described via a root knowledge matrix $\mathbf{K}^{(e)}$ (or
simply $\mathbf{K}$).

We write knowledge of previous roots among $n$ nodes in an event $e$’s
subgraph as an $n\times n$ matrix $\mathbf{K}$, with entries

$\mathbf{K}^{(e)}_{ij}=\begin{cases}1,&\text{if the subgraph of node $i$
contains a root created by node $j$ and no forks are observed for $i$ or
$j$.}\\\ 0,&\text{otherwise.}\end{cases}$ (1)

Let $k^{(e)}$ be the sum of all elements in matrix $\mathbf{K}^{(e)}$. The
metric $k^{(e)}$ is in the range $[0,1]$.

$k^{(e)}=\frac{1}{n^{2}}\sum_{i}\sum_{j}\mathbf{K}^{(e)}_{ij}.$ (2)

where the numerator is a count of the number of nodes observing each known
root in the subgraph. The denominator $n^{2}$ normalises $k^{(e)}$ to the
range [0,1], where $k^{(e)}=1$ if all nodes know all roots.

We define the progress of an event $e$ as a scalar metric of DAG progress,
$k^{(e)}$. This scalar metric will enable a comparison the progress of two
events. Figures 1(a) and 1(b) show $\mathbf{K}$ and $k$ examples for simple
DAGs.

(a)  Root knowledge of event $e$ on the left is $k^{(e)}$= 1/9.

(b)  Root knowledge of event $e$ on the left is $k^{(e)}$= 5/9.

Figure 1: Examples of root knowledge matrix $\mathbf{K}^{(e)}$ and root
knowledge metric $k^{(e)}$. The network consists of three nodes which index
the matrix

A new root will typically be produced before $k^{(e)}$ because producing a new
root only requires quorum prior roots to forkless cause the new root.

####  2.2.2 Pseudo-code

The rootProgress function in calculates the root knowledge $k^{(e)}$ of an
input event $e$. The function rootProgress does not explicitly construct the
$n\times n$ matrix $\mathbf{K}^{(e)}$ in memory. Instead, $k^{(e)}$ can be
calculated by looping over the indexes of $\mathbf{K}^{(e)}$ without storing
the matrix’s elements in memory. The implementation uses function
ForklessCauseProgress to calculate each column of $\mathbf{K}^{(e)}$.

Algorithm 1 rootProgress $k^{(e)}$ calculation

function rootProgress($e$)

R $\leftarrow$ roots in $e$’s subgraph

s $\leftarrow$ 0

for r in range(R) do

$c$ = ForklessCauseProgress(e, r)

$s$ = $s$ \+ sum($c$)

return s

In Alg. 1, ForklessCauseProgress returns a column of $\mathbf{K}^{(e)}$,
corresponding to the number of nodes that have an event whose subgraph
contains root, without forks.

##  3 Parent Selection and Event Timing

Based on the metrics, a node can rank candidate events for its new event. For
each QI metric and RK metric, a parent selection method is defined to rank
candidate parents into a sorted list. Parents are selected from the list if
they provide the greatest progress toward producing the next root.

Based on the two metrics, we also define two different event timing
strategies. We define new event timing metrics $t$ for a node $v$ based on DAG
progress metric (e.g., either QI metric $h^{(e)}$ or RK metric $k^{(e)}$).

We describe the timing metric $t$ based on RK metric $k^{(e)}$ in more details
as follows. The timing metric $t$ can be defined base on QI metric $h^{(e)}$
in a similar way.

Let $v$ be a node, and let $e_{v}$ be the latest event of $v$. Within node
$v$’s local DAG, for a node $i$, let $e_{i}$ be the latest event of $i$. The
new metric $t$ counts the number of nodes $i$, whose highest event has metric
$k^{(e_{i})}$ that exceeds that of $e_{v}$.
$t=\sum_{i}H(k^{(e_{i})}-k^{(e_{v})})s_{i}$, where $s_{i}$ is the stake of
node $i$ and $H(x)$ is the step function. $H(x)$= 1 if $x>0$; 0, otherwise.

The event timing metric is used to order nodes and is a direct measure of the
progress a node has made toward producing a new root and progressing toward
the next frame. Nodes emit an event when they fall below a threshold level in
an ordering of nodes, and new emission is to help increase their DAG progress
metric.

##  4 Simulation Results

Figures 2(a), 2(b), and 2(c) compare the performance in simulation between
parent selection and event timing methods using QI metric and RK metric.

(a)  Comparing ’Frames produced Per Event’

(b)  Comparing ’Frames produced Per Second’

(c)  Comparing ’Frame Rate’

Figure 2: Comparing mean Frame Rate between RK metric and QI metric parent
selection and event timing. Larger is better.

The experiments were made on a simulation of a network consisting of 30 nodes,
with the stake of each node randomly sampled. A dataset of real world internet
latencies between cities is used to model latencies between pairs of nodes. It
randomly allocated each node to a city in the dataset. Each of the 100
simulations shown are simulations of 100 seconds of network activity.

On average, the new RK metric is more efficient, producing each frame using
fewer events compared to the QI metric. The new RK metric produces more frames
per second on average compared to QI metric, and thus, the new methods will
lower TTF.

Figure 3 compares frame rate. The experiments were made with a simulation of
40 nodes, each event has 3 parents. Latency was set around 100ms.

Figure 3: Comparison of frame rate for different combination of event timing
and parent selection

##  5 Conclusion

In this work, we introduce two new metrics to measure the DAG progress of a
new event. By comparing between different possible new event blocks it can
generate, a node can determine which new event block is the most effective to
be made in order to progress toward achieving a new root.

Based on the metrics, we presented new methods for event timing and parent
selection. These methods will give several advantages. First, they can reduce
the number of generated events and can avoid generating many excessive events
and hence can reduce communication overheads. Second, using these methods, it
can achieve faster finality as it can compute roots and Atroposes quicker.
Third, it can reduce events per frame and increase frame rate, as more events
are generated only if they help achieve roots quicker. As such, it improves
the number of frames per second (frame rate).

##  6 Reference

##

  * [1] Q. Nguyen, A. Cronje, M. Kong, A. Kampa, and G. Samman.  Stairdag: Cross-dag validation for scalable BFT consensus.  CoRR, abs/1908.11810, 2019.
  * [2] Q. H. Nguyen, A. Cronje, M. Kong, E. Lysenko, and A. Guzev.  Lachesis: Scalable asynchronous BFT on DAG streams.  CoRR, abs/2108.01900, 2021.