# Reconsidering the dynamical states of galaxy clusters using PCA and UMAP

Roan Haggar,1,2 Federico De Luca,3,4 Marco De Petris,5 Elizaveta Sazonova,1,2
James E. Taylor,1,2 Alexander Knebe,6,7,8 Meghan E. Gray,9 Frazer R. Pearce,9
Ana Contreras-Santos,6 Weiguang Cui,6,7,10 Ulrike Kuchner,9 Robert A.
Mostoghiu Paun,11,12 Chris Power8,13
1Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
2Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario
N2L 3G1, Canada
3Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca
Scientifica 1, I-00133 Roma, Italy
4INFN, Sezione di Roma 2, Università degli studi di Roma Tor Vergata, Via
della Ricerca Scientifica, 1, Roma, Italy
5Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5,
I-00185 Roma, Italy
6Departamento de Física Teórica, Módulo 15 Universidad Autónoma de Madrid,
28049 Madrid, Spain
7Centro de Investigación Avanzada en Física Fundamental (CIAFF), Universidad
Autónoma de Madrid, 28049 Madrid, Spain
8International Centre for Radio Astronomy Research, University of Western
Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
9School and Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD,
UK
10Institute for Astronomy, University of Edinburgh, Royal Observatory,
Edinburgh EH9 3HJ, UK
11Centre for Astrophysics & Supercomputing, Swinburne University of
Technology, 1 Alfred St, Hawthorn, VIC 3122, Australia
12ARC Centre of Excellence for Dark Matter Particle Physics (CDM), Australia
13ARC Centre of Excellence for All-Sky Astrophysics in 3 Dimensions (ASTRO
3D), Australia E-mail<EMAIL_ADDRESS>
(Accepted 2024 June 21. Received 2024 June 20; in original form 2023 September
30)

###### Abstract

Numerous metrics exist to quantify the dynamical state of galaxy clusters,
both observationally and within simulations. Many of these correlate strongly
with one another, but it is not clear whether all of these measures probe the
same intrinsic properties. In this work, we use two different statistical
approaches – principal component analysis (PCA) and uniform manifold
approximation and projection (UMAP) – to investigate which dynamical
properties of a cluster are in fact the best descriptors of its dynamical
state. We use measurements taken directly from The Three Hundred suite of
galaxy cluster simulations, as well as morphological properties calculated
using mock X-ray and SZ maps of the same simulated clusters. We find that four
descriptions of dynamical state naturally arise, and although correlations
exist between these, a given cluster can be “dynamically relaxed” according to
all, none, or some of these four descriptions. These results demonstrate that
it is highly important for future observational and theoretical studies to
consider in which sense clusters are dynamically relaxed. Cluster dynamical
states are complex and multi-dimensional, and so it is not meaningful to
classify them simply as “relaxed” and “unrelaxed” based on a single linear
scale.

###### keywords:

galaxies: clusters: general – methods: numerical – dark matter – galaxies:
kinematics and dynamics – X-rays: galaxies: clusters

††pubyear: 2024††pagerange: Reconsidering the dynamical states of galaxy
clusters using PCA and UMAP–B

##  1 Introduction

The structure of a galaxy cluster consists of a large dark matter halo,
typically with a mass between $10^{14}-10^{15}\ M_{\odot}$, filled with hot
intracluster gas, and populated with hundreds or thousands of galaxies. A wide
range of physical processes take place within clusters, meaning they play a
crucial role in many areas of astrophysics and cosmology.

The evolution of galaxies is strongly dependent on their cosmic environment,
and clusters represent one of the most extreme environments for a galaxy.
Strong tidal forces due to a cluster’s gravitational potential can disrupt the
structure of a galaxy, dramatically changing its morphology (Moore et al.,
1996; Mihos et al., 2017; Knebe et al., 2020; Haggar et al., 2023).
Additionally, processes such as ram-pressure stripping (Gunn & Gott, 1972) by
the intracluster gas can remove both the cold gas in a galactic disk and the
surrounding halo gas, quenching star formation in galaxies (Larson et al.,
1980; Zabel et al., 2019). Galaxy clusters are also an important tool in
constraining cosmology – for example, studying the shapes of cluster halo
density profiles can provide information about the nature of dark matter
(Eckert et al., 2022; Limousin et al., 2022). Galaxy clusters can also be used
as a proxy for measuring cosmological parameters, either through cluster
number counts from large surveys (Evrard, 1989; de Haan et al., 2016; Abdullah
et al., 2020, for example), or by using cluster properties. For example,
Amoura et al. (2021) measure the formation times of clusters in a suite of
simulations with varying values of $\Omega_{\rm{M}}$, the matter density
parameter, and $\sigma_{8}$, the amplitude of the matter power spectrum at a
scale of $8\ h^{-1}\rm{Mpc}$, and show that the formation times of galaxy
clusters depend on the values of $\Omega_{\rm{M}}$ and $\sigma_{8}$.

One of the core properties of a galaxy cluster is its dynamical state – that
is, whether or not it is dynamically relaxed. Numerous areas of galaxy cluster
physics, including many of those described above, are dependent on an
understanding of cluster dynamical states. It has been shown that various
phenomena in the intra-cluster medium (ICM), such as turbulence, differ
between relaxed and unrelaxed clusters (Lau et al., 2009; Vallés-Pérez et al.,
2021; Simonte et al., 2022). Furthermore, Nagai & Lau (2011) and Vazza et al.
(2013) both show that, in dynamically disturbed clusters, the hot X-ray-
emitting ICM contains dense, cooler clumps of gas, associated with infalling
galaxies and galaxy groups. The intra-cluster medium drives galaxy evolution
mechanisms such as ram-pressure stripping; this has also been shown to be
enhanced in dynamically disturbed groups and clusters (Mauduit & Mamon, 2007;
Roberts & Parker, 2017; Lourenço et al., 2023).

Additionally, many of the astronomical and cosmological measurements described
above are indirectly related to the cluster dynamical states, as they rely on
accurate measurements of the masses of clusters, which are in turn dependent
on a their dynamical state. Cluster masses are often calculated under the
assumption that a cluster is in hydrostatic equilibrium. As a result of this,
clusters that are dynamically unrelaxed (and so are not in hydrostatic
equilibrium) can have their masses underestimated by up to $20\%$ (Nagai et
al., 2007; Kravtsov & Borgani, 2012). Relaxed clusters can also have their
masses underestimated, albeit to a lesser extent (Lau et al., 2013; Gianfagna
et al., 2021), and the strength of this bias is dependent on the redshift of
the cluster in question (Bennett & Sijacki, 2022). Cluster scaling relations
also differ between clusters that are relaxed, and those that are rapidly
accreting material (Planelles & Quilis, 2009; Lau et al., 2015; Chen et al.,
2019). Additionally, properties such as location of a cluster’s splashback
radius (Adhikari et al., 2014; More et al., 2015) are dependent on its
dynamical state. This is the radius within which the cluster material
dominates over the surrounding infalling material, and so this implies that
the region in which a galaxy is impacted by a cluster is also dependent on
cluster dynamical state. Because of this, a thorough understanding of cluster
dynamical states is vital if we are to use clusters as an astronomical and
cosmological tool.

In its simplest form, a system of collisionless particles can be described as
dynamically relaxed and virialised once the velocities of particles in the
system are uncorrelated with their initial velocities. Equivalently, this
means the average magnitude of the velocity of each particle, $\bm{v}$, is
equal to the change in velocity of the particle due to interactions with other
particles, $\delta{\bm{v}}$. The typical time required for a system to reach
this stage is given by the relaxation time, $t_{\rm{relax}}$,

$t_{\rm{relax}}\approx\frac{0.1N}{{\rm{ln}}(N)}t_{\rm{cross}}\,,$ (1)

where $N$ is the number of particles in a system, and $t_{\rm{cross}}$ is the
average crossing time for a particle in the system (Binney & Tremaine, 1987).
For a galaxy cluster, the number of particles (galaxies) is $\sim 10^{3}$, and
the crossing time is $\sim 1\ \rm{Gyr}$, leading to a typical relaxation time
of $\sim 10\ \rm{Gyr}$, comparable to the age of the Universe. However, in the
context of galaxy clusters, this definition of dynamical state is not
particularly useful for several reasons. Firstly, clusters are not
collisionless systems. Galaxies in clusters frequently experience near-misses
or tidal interactions (Knebe et al., 2006; Muldrew et al., 2011; Bahé et al.,
2019), and the intra-cluster gas, which makes up a significant portion of a
cluster’s mass, is not a collisionless fluid. Secondly, this definition
describes a closed system, which galaxy clusters are not; clusters are
continuously accreting material from their surrounding environment. As such,
the effects from their “boundary” need to be included to quantify their
dynamical state. Finally, according to this definition, only material that has
been in a cluster for greater than $10\ \rm{Gyr}$ can be dynamically relaxed.
This means that a $z=0$ cluster can only be truly relaxed if it accreted all,
or most, of its material before $z=1.5$. While this is technically possible,
it is an exceptionally rare scenario in a typical $\Lambda$CDM cosmology.

Consequently, throughout the literature, numerous properties of clusters are
treated as indicators of “dynamical state”, each of which is used to quantify
how relaxed is a galaxy cluster. Many of these are theory-based metrics, taken
from simulations of galaxy groups or clusters. In Cui et al. (2017), the
dynamical state of a cluster is described by three properties: dynamically
unrelaxed clusters are those with a centre of mass that is offset from the
cluster halo density peak, those with large amounts of substructure, and those
that are not virialised.

This combination of observables has been widely used for some time. For
example, Neto et al. (2007) classify dark matter haloes based on these three
parameters, and place a limit on each of these, describing relaxed haloes as
those that satisfy all three of these conditions, and unrelaxed haloes as
those that do not. Cui et al. (2018) place similar constraints on simulated
galaxy clusters, and use this to classify clusters as relaxed and unrelaxed.
Haggar et al. (2020), by contrast, combine the three into a single parameter,
$\chi_{\rm{DS}}$, giving a single continuous parameter describing how relaxed
is a cluster. Other work also uses these measures, such as Wen & Han (2013),
who use the amount of substructure as an indicator of dynamical state. They
describe how a large amount of structure is indicative of a recent merger
event. The results of Kimmig et al. (2023) also demonstrate this – they show
that the amount of substructure, and in particular the size of the eighth most
massive substructure, are indicative of the merger history of a cluster over
the last two gigayears. Further works have shown that numerous other cluster
properties correlate with recent merger activity, such as the virialisation
and centre of mass offset of a cluster (Power et al., 2012), its concentration
(Wang et al., 2020), or similarly its sparsity (Richardson & Corasaniti,
2022). Contreras-Santos et al. (2022) also use the time since a major merger
as an measure of cluster relaxation, and show how this correlates with the
$\chi_{\rm{DS}}$ measure from Haggar et al. (2020).

Already, it is apparent that different measures of a cluster’s dynamical state
are probing different core properties. Measurements such as the substructure
fraction and centre of mass offset are quantities that can be measured
observationally at a single epoch, although in observations they are projected
into two dimensions. This is in contrast to the time since the last major
merger, which is a property of the history of a cluster. Other studies also
take this approach to dynamical state, connecting it to the total history of a
cluster. For example, Diemer & Kravtsov (2014) introduced the accretion rate
proxy $\Gamma$, variants of which have since become widely used as measures of
halo relaxation (e.g. Vallés-Pérez et al., 2020). An alternative approach is
to use a redshift-dependent definition of dynamical state, based on the fact
that different cluster properties evolve over different timescales (e.g.
Mendoza et al., 2023; Vallés-Pérez et al., 2023).

Gouin et al. (2021) probe the dynamical state of clusters from the
IllustrisTNG simulation (Nelson et al., 2019) using three separate measures of
cluster growth history: the $z=0$ halo growth rate, the $z=0.5$ mass accretion
rate, and the cluster formation time ($z_{0.5}$, see Section 2.2.1). They also
show that dynamically disturbed clusters are more strongly connected to the
cosmic web – that is, they have more cosmic filaments attached to the cluster
(see Santoni et al., 2024, for a similar study with The300 clusters).
Similarly, Darragh Ford et al. (2019) find that simulated clusters that have
recently experienced a major merger have a higher connectivity. This is yet
another interpretation of dynamical state, as a property of the surrounding
region of the Universe. This cosmic environment can also impact global
properties of the cluster and its dark matter halo, such as its shape (Gouin
et al., 2021; Smith et al., 2023) and concentration (Neto et al., 2007).

Additional complexity comes from the large number of cluster dynamical state
metrics and morphological parameters that are used in observational astronomy.
For example, cluster shapes can be mapped using X-ray and Sunyaev-Zel’dovich
(SZ) effect observations. From these, many quantifiable properties can be
measured, such as the amount of substructure (Ge et al., 2016), offset of the
brightest cluster galaxy (BCG) from the X-ray/SZ peak (Zenteno et al., 2020)
or the power spectrum of the hot gas distribution (Cerini et al., 2023).
Combinations of multiple X-ray or SZ morphological parameters have been shown
to provide even more robust measures of dynamical state than these individual
parameters (Parekh et al. (2015); Yuan & Han (2020); Campitiello et al.
(2022); Yuan et al. (2022); see Rasia et al. (2013) and Zhang et al. (2024)
for more comprehensive discussions). Other observational studies include Wen &
Han (2013), who quantify cluster substructure using optical data from the
Sloan Digital Sky Survey (SDSS; Aihara et al., 2011).

The result of having so many different properties in common use is that the
“dynamical state” of a cluster is not clearly or consistently defined in the
literature. Consequently, when the dynamical states of clusters are inferred
from observations, it is not entirely apparent which fundamental property of a
cluster these are probing. It is also not clear if they are all probing the
same intrinsic cluster property, or if the “dynamical state” of a cluster is
actually made up of several properties.

In this study, we aim to gain a deeper understanding of cluster dynamical
state measurements by investigating the connections between theory-based and
observable properties of dynamical state, the degeneracy between different
measures of dynamical state, and the core, intrinsic properties of clusters
that these measurements are actually probing. Our primary approach to this is
through dimensionality reduction – reducing a large number of dynamical state
metrics to a smaller set of variables will make the nature of dynamical state
clearer and easier to interpret. Previous studies have taken a similar
approach to ours, although mostly to consider the accretion histories of dark
matter haloes, rather than the dynamical states of galaxy clusters. One
notable example is Wong & Taylor (2012), who use principal component analysis
(PCA; Jolliffe & Cadima, 2016) to determine the principal components of 10
input properties of dark matter haloes, and study the mass accretion histories
of these haloes. They show that splitting these clusters by their first and
second principal components naturally displays two separate modes of accretion
history: the halo formation time, and the acceleration/deceleration of a
halo’s accretion. For further discussion, we also refer the reader to Jeeson-
Daniel et al. (2011) and Skibba & Macciò (2011), who perform similar analyses
on simulations of dark matter haloes.

In this work we develop the methods used in previous studies, applying
principal component analysis to 17 indicators of cluster dynamical state,
based on 3D data (as opposed to directly observable quantities) from
simulations of $z=0$ galaxy clusters. This analysis uses The Three Hundred
project, a suite of hydrodynamical zoom simulations of massive galaxy
clusters, taken from a large $1\ h^{-1}\rm{Gpc}$ cosmological volume. We
physically interpret the principal components that naturally arise from this
analysis, and show how they correspond to the mass accretion histories of
these clusters. Despite this work focusing on a theoretical approach to
dynamical states, we also compare our PCA to X-ray and SZ properties measured
from mock observations of the same simulated clusters, originally calculated
in De Luca et al. (2021). Finally, we expand further on previous studies in
this area by analysing the same simulated clusters using uniform manifold
approximation and projection (UMAP; McInnes et al., 2018), an alternative
approach to dimensionality reduction. While we do not develop a quantitative
means to classify clusters, we demonstrate qualitatively the different
dynamical states of clusters that exist in our simulations.

We interpret our results as showing that the dynamical state of a cluster is
not a single property. Instead, a single cluster has multiple “dynamical
states”, and can be relaxed in all of these dimensions, or none of them, or
some of them.

The paper is structured as follows: in Section 2 we introduce the simulation
data used throughout this work, as well as the dynamical state indicators we
use directly from the simulations (Section 2.2.1) and from mock X-ray and SZ-
effect maps generated from the simulation data (Section 2.2.2). In Section 3
we study these parameters using both PCA and UMAP, and discuss how the results
connect to the mock observations of clusters, and their mass accretion
histories. Finally, in Section 4 we summarise our findings, and the
implications this has for how the dynamical state of galaxy clusters should be
interpreted.

##  2 Simulations and methodology

This work utilises simulation data from The Three Hundred project. We study
these galaxy clusters in the final snapshot of the simulations ($z=0$). In
Section 2.1 we provide an overview of the simulation data, but for a detailed
description we refer the reader to Cui et al. (2018).

###  2.1 Simulation data

The Three Hundred project (hereafter The300) is a suite of hydrodynamical
resimulations of large galaxy clusters. The simulations are based on the MDPL2
MultiDark simulation (Klypin et al., 2016)111The MultiDark simulations are
publicly available from the CosmoSim database, https://www.cosmosim.org..
MDPL2 is a large dark matter-only simulation, with a comoving box size of $1\
h^{-1}\rm{Gpc}$, which uses Planck cosmology ($\Omega_{\rm{M}}=0.307$,
$\Omega_{\rm{B}}=0.048$, $\Omega_{\Lambda}=0.693$, $h=0.678$,
$\sigma_{8}=0.823$, $n_{\rm{s}}=0.96$) (Planck Collaboration et al.,
2016)222The reduced Hubble constant, $h$, is defined such that the Hubble
constant, $H_{\rm{0}}=h\times 100\ \rm{km}\ \rm{s}^{-1}\ \rm{Mpc}^{-1}$..

From this simulation, the 324 most massive dark matter haloes were selected,
and resimulated from their initial conditions (at an initial redshift of
$z_{\rm{init}}=120$) with baryonic physics included. For each cluster, the $1\
h^{-1}\rm{Gpc}$ dark matter-only simulation box was re-centred on the cluster,
and the particles within $15\ h^{-1}\rm{Mpc}$ of the cluster centre at $z=0$
were traced back to their initial conditions. Each of these particles was then
split into a dark matter particle and a gas particle, with masses of
$m_{\rm{DM}}=1.27\times 10^{9}\ h^{-1}M_{\odot}$ and $m_{\rm{gas}}=2.36\times
10^{8}\ h^{-1}M_{\odot}$ respectively, set according to the cosmic baryon
fraction. The resolution of dark matter particles outside of this radius was
degraded, allowing the large-scale tidal forces acting on the cluster to be
maintained with a reduced computational cost.

In this work, we use clusters from The300 dataset simulated using the GadgetX
code. GadgetX is an updated version of the Gadget3 code (Springel, 2005; Beck
et al., 2016; Sembolini et al., 2016), and uses a smoothed-particle
hydrodynamics scheme to simultaneously evolve the baryonic and dark matter
components of a simulation. As well as gas, the stochastic star formation in
GadgetX produces stellar particles of varying mass, typically around
$m_{\rm{star}}\sim 4\times 10^{7}\ h^{-1}M_{\odot}$ (Tornatore et al., 2007;
Murante et al., 2010; Rasia et al., 2015). Type II supernova feedback and AGN
feedback are included, as described in Springel & Hernquist (2003) and
Steinborn et al. (2015) respectively. The final dataset consists of 324 galaxy
clusters, with masses ranging from $M_{200}=5\times 10^{14}\ h^{-1}M_{\odot}$
to $M_{200}=2.6\times 10^{15}\ h^{-1}M_{\odot}$. Here, $M_{200}$ is the mass
enclosed within a sphere of radius $R_{200}$, defined such that the average
density within this sphere is equal to 200 times the critical density of the
Universe at that redshift. For the cluster masses in The300, the $15\
h^{-1}\rm{Mpc}$ high-resolution region corresponds approximately to
$7-10R_{200}$. As well as the extensive description in Cui et al. (2018),
further description of The300 dataset is available in other previous studies
that have used these data. We particularly refer the reader to studies that
have investigated cluster dynamical states using The300 (Haggar et al., 2020;
Capalbo et al., 2021; De Luca et al., 2021; Contreras-Santos et al., 2022; Li
et al., 2022, for example).

In this work, we specifically use the data from the final snapshot of The300
simulations ($z=0$), as the focus of this work is on present-day galaxy
clusters. However, some of the cluster properties we calculate also rely on
cluster properties at $z>0$ (see Section 2.2.1 for details). This information
requires the construction of halo merger trees, which are described in the
following section.

####  2.1.1 Galaxy identification and tree-building

In each snapshot of The300 data, the haloes and subhaloes are identified using
the Amiga Halo Finder, ahf333http://popia.ft.uam.es/AHF (see Gill et al.,
2004; Knollmann & Knebe, 2009, for a detailed description of ahf). ahf is a
density peak halo finder, and is used to identify the particles in the main
cluster halo, subhaloes of the cluster, and haloes in the surrounding region.
For each of these (sub)haloes, properties such as the position, velocity and
mass are given as outputs. The halo catalogues in each snapshot were linked
together using the mergertree tree-builder, which is part of the ahf package.
This tree-builder uses a merit function, $M_{\rm{i}}$, given in Table B1 of
Knebe et al. (2013), to identify the main progenitor of each halo by searching
for particles that are common between the two haloes. mergertree is also able
to “skip” snapshots, meaning that if ahf is unable to resolve a halo in one
snapshot, the tree-builder can instead find an appropriate progenitor in an
earlier snapshot (see Knebe et al., 2011; Srisawat et al., 2013, for
additional details on mergertree).

For most of this work, we include all of these subhaloes (and thus all of the
particles) in the calculations of dynamical state parameters. For those
measures that rely on galaxy properties (e.g. cluster richness, $N_{200}$, and
magnitude difference between galaxies, $m_{12}$, detailed in Section 2.2.1),
we include all galaxies with a total mass greater than $10^{10.5}\
h^{-1}M_{\odot}$, and a stellar mass of greater than $10^{9.5}\
h^{-1}M_{\odot}$. These limits have an extremely minor impact on our results,
as most dynamical state indicators based on galaxy properties are dependent on
the largest, brightest galaxies in a cluster.

###  2.2 Measures of dynamical state

####  2.2.1 3D measures

From the simulations, we utilise 17 different properties, each of which is
associated with cluster dynamical state. These are detailed below, along with
relevant information on how they were calculated in our simulations. Many of
these quantities are calculated using $R_{200}$ as an outer boundary, as this
is the characteristic radius used by ahf in calculating halo properties; we
state explicitly when this is not the case. These properties are also
calculated using all particles within the relevant radius (typically $R_{200}$
unless stated otherwise), apart from $c_{200}$ which only uses dark matter
particles.

  * •

$f_{\rm{s}}$: The fraction of cluster mass inside a given radius that is
contained within subhaloes. Two values of this are used, $f_{\rm{s}}(R_{200})$
and $f_{\rm{s}}(R_{500})$, equal to the substructure mass fraction inside the
radii $R_{200}$ and $R_{500}$ respectively. Using these two radii allows us to
probe both the outer and inner regions of a cluster: as Cui et al. (2017)
show, dynamical state indicators are dependent on cluster-centric distance.

  * •

$\Delta_{\rm{r}}$: The offset of the centre of mass of the cluster from the
density peak of the cluster halo, as a fraction of the cluster radius
$R_{200}$. Similarly to $f_{\rm{s}}$, $\Delta_{\rm{r}}(R_{200})$ and
$\Delta_{\rm{r}}(R_{500})$ are both used, which are each calculated using the
centre of mass of all material inside the radii $R_{200}$ and $R_{500}$
respectively.

  * •

$\eta$: The virial ratio, a measure of how well a cluster obeys the virial
theorem, based on its total kinetic energy, $T$, its total potential energy,
$W$, and its energy from surface pressure, $E_{\rm{s}}$. It is typically
defined in the literature as $\tilde{\eta}=(2T-E_{\rm{s}})/|W|$, so that
$\tilde{\eta}=1$ for virialised haloes. However, PCA is designed to capture
linear, monotonic relationships between variables. This variable, where the
“extreme” cases (most virialised haloes) correspond to an intermediate value
($\tilde{\eta}=1$) is therefore not well-suited to PCA. Consequently, we
perform a transformation, defining a new $\eta$ such that:

$\eta=\left|\frac{\left(2T-E_{\rm{s}}\right)}{|W|}-1\right|\,.$ (2)

Increasing values of this quantity correspond to a greater deviation from
virialisation, and thus to less virialised haloes, making this quantity better
suited for use in PCA. Note also that this virial ratio differs from the
classic definition of virialisation, due to the additional surface pressure
term which accounts for clusters’ ongoing accretion of material (Poole et al.,
2006; Shaw et al., 2006). The surface pressure is calculated as the energy
from surface pressure integrated over the halo boundary – a detailed
mathematical description is given in Section 3 of Cui et al. (2017). The
virial ratio is also calculated twice, for all material inside $R_{200}$ and
$R_{500}$.

  * •

$z_{0.5}$: The formation time of a cluster: the redshift at which the cluster
mass, $M_{200}$, is equal to half its value at $z=0$.

  * •

$\lambda$: A dimensionless spin parameter, used to describe the bulk rotation
of a cluster. It is defined in the same way as Bullock et al. (2001),

$\lambda=\frac{J_{200}}{\sqrt{GM_{200}^{3}R_{200}}}\,,$ (3)

where $J_{200}$ is the total angular momentum of material inside $R_{200}$. We
note that other definitions of the cluster spin also exist (e.g. Peebles,
1969), but we use this value as it is calculated using only mass within a
well-defined radius, $R_{200}$.

  * •

$c/a$: Sphericity, the ratio of the minor and major axes of the cluster halo’s
moment of inertia tensor. As this calculation of the moment of inertia
includes all particles (dark matter, gas, and stars), it accounts for these
particles’ varying masses accordingly. We note that several alternative
measures of cluster shape are also used throughout the literature, including
triaxiality, ellipticity, and prolaticity (Lau et al., 2021), but we only use
sphericity in this work.

  * •

$c_{\rm{200}}$: The concentration of the dark matter halo. This is equal to
the ratio between $R_{200}$ and $R_{\rm{s}}$, the scale radius of a halo, as
defined by an NFW profile (Navarro et al., 1996). Here, $\rho(r)$ is the dark
matter density of a halo as a function of halo-centric distance, and
$\rho_{0}$ is some characteristic density:

$\rho(r)=\frac{\rho_{0}}{\left(\frac{r}{R_{\rm{s}}}\right)\left(1+\frac{r}{R_{\rm{s}}}\right)^{2}}\,.$
(4)

This is an output from ahf, which selects cluster centres based on a density
peak finder. The concentration of a halo is not calculated directly, but is
instead a numerical solution to Equation 9 in Prada et al. (2012), based on
the maximum circular velocity of a cluster.

  * •

$z_{\rm{merge,50}}$: The redshift at which a cluster last experienced a merger
that increased its mass by more than $50\%$. These are equal to the values of
$z_{\rm{start}}$ calculated in Contreras-Santos et al. (2022), which describe
the jump in the mass accretion history of the cluster and therefore the onset
of a merger phase.

  * •

$\gamma$: The mass accretion rate of the cluster, defined as the fractional
increase in $M_{200}$ within the last dynamical time, $t_{\rm{d}}$. We use the
crossing time of a cluster as its dynamical time (see Binney & Tremaine, 1987;
Contreras-Santos et al., 2022, for example), equal to:

$t_{\rm{d}}\simeq\frac{R_{200}}{v_{\rm{circ}}}=\sqrt{\frac{R_{200}^{3}}{GM_{200}}}\,.$
(5)

Using the definition of $M_{200}$, and the fact that the critical density
$\rho_{\rm{crit}}=3H^{2}/8\pi G$, we find that the dynamical time at $z=0$ is
given by $t_{\rm{d}}\simeq 1/(10H_{0})$. For the cosmology used in The300
simulations, this dynamical time is equal to approximately $1.4$ Gyr, or a
redshift of $z=0.1$. Hence, the value of $\gamma$ is the fractional increase
in mass between $z=0.1$ and $z=0$.

  * •

$N_{\rm{fil}}$: The number of filaments, or connectivity, of a cluster. The
connectivity of each cluster was calculated using the DIScrete PERsistent
Structure Extractor (DisPerSE) filament finding algorithm (Sousbie, 2011). In
our specific case, the cosmic filaments were identified based on the number
density of gas particles around a cluster. Multiple definitions for the
connectivity of clusters exist – we use it to refer to the number of these
filaments beginning at the cluster centre (node) and passing through a sphere
of radius $R_{200}$ surrounding the cluster. Cosmic filaments around the
clusters in The300 have been studied extensively in other works via galaxies
(Cornwell et al., 2022; Kuchner et al., 2022) and their gas component (Santoni
et al., 2024).

  * •

$D_{n,f}$: This environment parameter is defined as the distance to the $n$th
nearest halo whose mass is greater than $fM_{200}$, in units of $R_{200}$. In
our case, we use $n=1$, $f=0.1$, and so $D_{1,0.1}$ is the distance to the
nearest halo whose mass is greater than one tenth of the cluster’s mass (see
also Jeeson-Daniel et al., 2011; Wong & Taylor, 2012). This is consequently a
measure of how isolated a galaxy cluster is from other clusters with masses of
the same order of magnitude.

  * •

$N_{200}$: Richness, the number of cluster members whose absolute magnitude is
between $m_{3}$ and $m_{3}+2$, where $m_{3}$ is the magnitude of the third-
brightest cluster member (this definition is given by Abell, 1958, and while
multiple definitions of cluster richness exist, this is the one we use
throughout this work). We use $R$-band luminosities, calculated using the
stellar population synthesis code STARDUST (see Devriendt et al., 1999).

  * •

$m_{12}$: Difference in magnitude between the brightest and second-brightest
cluster member galaxies, or “magnitude gap”. Multiple names are used
throughout the literature to refer to this property, or similar properties
(e.g. the “fossil parameter” in Ragagnin et al., 2019, also used as a measure
of dynamical state). As in our calculations of cluster richness, we calculate
this using $R$-band magnitudes.

  * •

$\sigma_{\rm{BCG}}$: The velocity dispersion of stars in the brightest cluster
galaxy (BCG) of our simulated clusters. This was calculated using the stellar
particles located within a spherical aperture of radius $200\
h^{-1}{\rm{kpc}}$ around the centre of the cluster halo, a radius which should
include most of the material associated with the BCG (Lin & Mohr, 2004;
Contreras-Santos et al., 2022).

It should be noted that the order in which we present these 17 dynamical state
indicators is arbitrary. In principle, one could group these parameters
together – for example, $z_{0.5}$, $z_{\rm{merge,50}}$ and $\gamma$ all depend
on temporal information about a cluster. However, we have not chosen to do
this here, as these properties are taken by our analysis in a random order,
without any further information relating them to one another. Later, we will
deliberately reorder these 17 parameters based on our principal component
analysis (see Section 3.1).

These measurements are all dependent on information that is only available
from the full 3D version of our simulations. While the strength of this
dependence varies, it means that none of these are directly measurable from
observations. Some of these properties can be measured quite well in
observations, such as the velocity dispersion of the BCG, $\sigma_{\rm{BCG}}$,
which can be measured using spectroscopy, and the magnitude difference between
the two brightest galaxies, $m_{12}$. Conversely, some are much more sensitive
to 2D projection effects. For example, the centre of mass offset,
$\Delta_{\rm{r}}$, is strongly dependent on the viewing angle, particularly if
the offset is caused by a single major merger event. If the merger takes place
along the line of sight, the apparent offset will be minimal. If instead it
occurs in the plane of the sky, the calculated offset of the centre of mass
will be apparent and measurable (see Zenteno et al., 2020). A similar property
is the sphericity, $c/a$ – very different values of this will be measured
depending on whether a cluster’s major axis is aligned along the line of sight
or not. Other properties such as the substructure fraction, $f_{\rm{s}}$, and
cluster richness, $N_{200}$, are likely to be measured quite well in
observations, but will still suffer somewhat due to the presence of interloper
galaxies along the line of sight. Additionally, some of these cluster
properties cannot be measured at all. The cluster formation time, $z_{0.5}$,
requires knowledge of the growth histories of a cluster over several
gigayears, which can be inferred from other properties but not measured
directly.

The fact that some of these properties are difficult, or impossible, to
measure observationally is a topic we plan to address in a future study (see
also Section 4). However, in this work our focus is on the actual properties
of a cluster, rather than the limitations of what can be measured. The
exception is the mock X-ray and SZ data that we use, which we describe in the
following subsection.

####  2.2.2 X-ray and SZ morphological parameters

In De Luca et al. (2021), mock observations were created for all the clusters
of The300 sample. Maps were generated for each of these clusters in X-rays,
and also as they would be seen through the SZ effect, in terms of the Compton
parameter. From these maps, six morphological parameters are calculated, as
described below. More rigorous descriptions of these parameters, as well as
details on the production of the mock maps and a more thorough review of the
literature, can be found in De Luca et al. (2021).

  * •

$A$: Asymmetry (Schade et al., 1995), the normalised difference in flux
between the original map, and a rotated map. The value of $A$ chosen is the
maximum calculated from four different rotations/reflections ($90\degree$,
$180\degree$, and reflection along the main cluster axes).

  * •

$K$: Light concentration ratio (Santos et al., 2008), the ratio of surface
brightness computed within two concentric apertures. For the X-ray maps these
are $0.025R_{500}$ and $0.25R_{500}$, and for the SZ maps these radii are
$0.05R_{500}$ and $0.25R_{500}$.

  * •

$W$: Centroid shift (Mohr et al., 1993), a measure of how much the centroid of
a map shifts as different apertures are used to calculate the centre.

  * •

$P$: Power ratio (Buote & Tsai, 1995), is based on a multipole decomposition
applied to the maps of the ICM. Specifically, $P$ is the third order power
ratio.

  * •

$G$: Gaussianity (Cialone et al., 2018), the ratio of the two values for
standard deviation required to describe a 2D Gaussian fit to the map. This can
distinguish elongated and circular clusters, and so is analogous to the
cluster sphericity, $c/a$.

  * •

$S$: Strip variation (Cialone et al., 2018), the normalised difference between
four light profiles, inclined by $45\degree$ to one another, passing through
the centroid.

Additionally, De Luca et al. (2021) calculate $M$, a normalised, linear sum of
these six morphological parameters, each weighted such that the difference in
$M$ between relaxed and unrelaxed clusters is maximised. Each of these seven
total parameters (six parameters, plus the combined measure) is calculated for
both X-ray and SZ-effect maps, and we use subscripts to distinguish between
these. For an additional study using similar methods, we also refer the reader
to Campitiello et al. (2022).

##  3 Results

###  3.1 PCA of dynamical state indicators

Principal component analysis is a commonly used dimensionality reduction
technique, which defines new variables (“principal components”) in a multi-
dimensional parameter space. These principal components are linear sums of the
input parameters, defined by the eigenvectors of the covariance matrix of the
data, meaning they are orthogonal and uncorrelated to one another. The
principal components can also be ordered based on the variance in the data for
which they account, thus allowing one to only consider the “most important”
components. These components can be interpreted physically, and can be used to
identify correlations, trends and degeneracies in high-dimensional data. PCA
requires the input data to be standardised; we have applied this to each of
our 17 parameters, and also for our subsequent UMAP analysis (Section 3.4).
Some discussion of the non-standardised distributions can be found in Appendix
B.

Fig. 1 shows the contributions of each of the 17 dynamical state indicators to
the four major principal components, PC1, PC2, PC3 and PC4, as determined by
principal component analysis. For each dynamical state indicator, its
contribution (i.e. the coordinate value in that dimension) to each of these
four components is shown by the height of the bars, coloured based on the
component number. Horizontal lines are marked at $\pm 0.24$. We take component
coordinates that have an absolute value of greater than 0.24 to be “important”
contributing parameters to a principal component; as these principal
components are normalised, the root-mean-square contribution is
$N_{\rm{par}}^{-0.5}\approx 0.24$, where $N_{\rm{par}}$ is the dimensionality
of the data (17 in our case). While this distinction is still somewhat
arbitrary, it is in line with previous similar work, such as Wong & Taylor
(2012), who use a boundary of 0.3 for $N_{\rm{par}}=13$. These data are also
shown in Table 1, and component coordinates greater than 0.24 are shown in
bold.

Figure 1: Contribution of each of the 17 dynamical state parameters to the four main principal components, as determined by principal component analysis. Dynamical state indicators are listed horizontally, and the value of the bars show the coordinate value of that dimension for PC1, PC2, PC3 and PC4, coloured based on the component number. Horizontal lines are marked at $\pm 0.24$, the root-mean-square contribution of any parameter to a given principal component – this is the boundary at which we consider a parameter to be an “important” contributor to that principal component. This data is tabulated in Table 1. Table 1: Coordinate values for the four major dynamical state principal components (this data is also shown in Fig. 1). Component coordinates with an absolute value greater than 0.24 are highlighted in bold. Parameter | PC1 | PC2 | PC3 | PC4
---|---|---|---|---
$f_{\rm{s}}(R_{\rm{200}})$ | -0.36 | -0.12 | 0.03 | 0.15
$f_{\rm{s}}(R_{\rm{500}})$ | -0.32 | 0.09 | 0.13 | 0.09
$\Delta_{\rm{r}}(R_{\rm{200}})$ | -0.32 | -0.13 | 0.03 | -0.02
$\Delta_{\rm{r}}(R_{\rm{500}})$ | -0.33 | 0.04 | 0.08 | -0.08
$\eta(R_{\rm{200}})$ | -0.16 | 0.39 | -0.11 | 0.05
$\eta(R_{\rm{500}})$ | -0.09 | 0.37 | -0.23 | 0.11
$z_{\rm{0.5}}$ | 0.35 | -0.01 | -0.11 | 0.13
$\lambda$ | -0.18 | -0.12 | -0.32 | -0.18
$c/a$ | 0.18 | 0.16 | -0.13 | -0.56
$c_{\rm{200}}$ | 0.10 | 0.52 | -0.07 | -0.11
$z_{\rm{merge,50}}$ | 0.19 | -0.07 | -0.22 | 0.65
$\gamma$ | -0.27 | 0.28 | -0.12 | -0.12
$N_{\rm{fil}}$ | 0.02 | 0.18 | 0.77 | 0.09
$D_{1,0.1}$ | 0.16 | 0.19 | 0.34 | -0.04
$N_{200}$ | 0.26 | 0.11 | 0.01 | 0.04
$m_{12}$ | 0.35 | 0.10 | -0.05 | -0.07
$\sigma_{\rm{BCG}}$ | -0.11 | 0.43 | -0.09 | 0.36

Throughout the remainder of this work, we consider only these four dominant
principal components. Between them, these four explain $64\%$ of the variance
of the 17 dynamical state indicators ($38\%$, $14\%$, $6\%$ and $6\%$ for
PC1-PC4 respectively), as shown in Fig. 5. We choose to consider only four
parameters as this is the minimum number such that every dynamical state
indicator contributes strongly to at least one principal component, and one of
the aims of this work is to group these indicators in as simple a way as
possible. While there is not a sharp decrease in the importance of components
after PC4, we do not believe that including these additional components allows
for significantly more scientific interpretation; this is discussed further in
Appendix A.

The major contributors to each principal component can be summarised as:

  * •

PC1: Substructure fraction, centre of mass offset, formation time, accretion
rate, cluster richness, and dominance of BCG.

  * •

PC2: Virial ratio, concentration, accretion rate, and BCG velocity dispersion.

  * •

PC3: Cluster spin, connectivity, and distance to nearest large halo.

  * •

PC4: Sphericity, time since last major merger, and BCG velocity dispersion.

Throughout the remainder of this work, we have reordered our 17 indicators
from the order in which they were presented in Section 2.2.1, based on the
principal component to which they strongly contribute (see Fig. 2, for
example).

We can interpret these principal components physically, as four different
forms of dynamical state.

PC1 describes the formation time of a cluster, and the properties of galaxies
in this cluster – that is, whether a cluster is a recently-forming rich
cluster with many bright galaxies, or an old, poor cluster dominated by a
single BCG. This component primarily describes the history of substructure
accretion by this cluster, and is the “most important” principal component,
explaining more than one-third of the total variance in the dataset.

PC2 describes the relaxation and virialisation of the dark matter halo of the
cluster. It is dependent on whether the halo is virialised or not, and how
concentrated it is. In turn, the BCG velocity dispersion is also included in
this. Highly concentrated haloes have a greater BCG velocity dispersion as
their central dark matter density is greater, while low-concentration haloes
(those with more of a central core) have a lower BCG velocity dispersion.

PC3 represents the local environment of a cluster: how connected it is to
cosmic filaments, and whether it is in an isolated region of the Universe. The
local environment will impact the shear forces on a cluster, potentially
explaining the inclusion of spin in this component.

PC4 includes the information on whether this cluster is a recent merger.
Currently-merging clusters are unlikely to be spherical as they will consist
of a superposition of two approximately spherical haloes. Moreover, Contreras-
Santos et al. (2022) use The300 data to show that BCG properties are strongly
impacted by major mergers, explaining the inclusion of BCG velocity dispersion
in this component.

Interestingly, we note that PC4 actually consists of a positive
$z_{\rm{merge,50}}$ component and a negative $c/a$ component, implying that
the value of this component is greater for elongated clusters, and those that
last merged long ago. While this is not necessarily expected, we believe that
this behaviour is a consequence of the definition of $z_{\rm{merge,50}}$,
which only includes mergers that have already finished – ongoing mergers will
not be counted as a “recent merger”, even though their measure of $c/a$ will
be strongly impacted (this is the same as the “reduced” merger sample in
Contreras-Santos et al., 2022). Additionally, we note that both of these
parameters also appear in PC1 (albeit with lower contributions than in PC4)
and vary in the same direction. PC1 explains more of the total variance than
PC4, and so some of the merger behaviour is also encompassed within PC1, as
well as in PC4. This is supported by the fact that there is no significant
overall correlation between $z_{\rm{merge,50}}$ and $c/a$ (Spearman’s rank,
$\rho_{\rm{s}}=0.06$, $p=0.30$, as shown later in Fig. 3).

Similarly, we note that PC2 includes positive contributions from $\eta$,
$c_{200}$, $\gamma$ and $\sigma_{\rm{BCG}}$, indicating that a greater value
of this component corresponds to clusters that are non-virialised, rapidly
accreting, have a disturbed BCG, and are highly concentrated. Mergers are
generally known to make cluster haloes less concentrated, and so this result
is also unexpected. It can potentially be explained by the fact that some of
the components of PC2 (such as $\gamma$) also contribute to PC1, similarly to
how some elements of PC4 contribute to PC1. Also, as shown in Wang et al.
(2020), the relationship between mergers and halo concentration can be
somewhat complex and non-monotonic; we discuss this further in Section 3.4.

Each of these four principal components are, by definition, orthogonal and
uncorrelated, and they are each driven by specific properties of a cluster.
However, it is important to note that these four different forms of the
dynamical state of a galaxy cluster are not independent of one another;
multiple dynamical state indicators contribute in a non-negligible way to
several principal components. We do not perform a detailed, qualitative
analysis of the relationship between these different forms in this work, but
instead focus on exploring these four different dimensions of dynamical state.

###  3.2 X-ray and SZ morphological indicators

This paper is primarily a theoretical study, focusing on properties that are
taken directly from simulations, rather than from mock observations of
simulated clusters. In this section we draw some connections to observational
astronomy, by comparing these parameters to mock X-ray and SZ observations of
the same galaxy clusters.

To test which of the different interpretations of “dynamical state” described
in Section 3.1 are actually probed by X-ray and SZ morphological parameters,
in Fig. 2 we show the Spearman’s rank correlation coefficient between each of
these 12 observables (plus the composite parameters, $M_{\rm{X}}$ and
$M_{\rm{SZ}}$) and the 17 dynamical state indicators that we use. Here we only
show the magnitude of the Spearman’s rank – that is, we do not distinguish
between positive and negative correlations, we are only studying the strength
of the correlation.

Figure 2: Spearman’s rank correlation coefficient, $\rho_{\rm{s}}$, between
the 17 dynamical state parameters described in Section 2.2.1, and the seven
observable morphological parameters described in Section 2.2.2, for both X-ray
and SZ mock observations (separated by a horizontal black line, with X-ray
measurements in the top half). The colour of each cell represents the
(absolute) value of the Spearman’s rank; lighter colours represent a stronger
correlation. The value of the Spearman’s rank is also written in each cell.
Note that the 17 dynamical state indicators have been reordered, and grouped
together based on the principal component to which they contribute (see
Section 3.1).

Overall, the morphological parameters correlate well with dynamical state
indicators that contribute strongly to the primary principal component, PC1,
with a median correlation coefficient of 0.53 (median of the absolute values
of the Spearman’s rank coefficients). The exception to this is the Gaussianity
of the cluster, which does not correlate as well with the PC1 parameters,
regardless of whether it is measured from X-ray or SZ maps. This result is not
unexpected – previous studies have also found that Gaussianity is generally a
less informative parameter than other X-ray/SZ morphological parameters. For
example, De Luca et al. (2021) show that $G$ is weighted three to four times
weaker than the other parameters in the calculation of the composite parameter
$M$. They attribute this to several factors: Gaussianity is mostly dependent
on the global shape of a cluster, not on cluster substructure, and is also
highly dependent on projection effects. Furthermore, even virialised dark
matter haloes can have an ellipsoidal (non-spherical) shape, and so it is not
necessarily a good metric for distinguishing relaxed and unrelaxed clusters.
Similarly, Cialone et al. (2018) show that Gaussianity calculated from SZ maps
is less effective at separating clusters by dynamical state, and give similar
reasons for this finding.

In both of these sets of mock observations, the weighted sums of morphological
parameters ($M_{\rm{X}}$ and $M_{\rm{SZ}}$) correlate most strongly with the
elements of PC1. This is predictable, given that $M$ is defined in such a way
that the difference in $M$ between relaxed and unrelaxed clusters is
maximised. These results are in general agreement with De Luca et al. (2021).
It is important to note that the apertures for calculating these morphological
parameters were chosen such that clusters could be separated based on
$f_{\rm{s}}(R_{500})$ and $\Delta_{\rm{r}}(R_{500})$. However, the
correlations of the morphological parameters with $f_{\rm{s}}(R_{200})$ and
$\Delta_{\rm{r}}(R_{200})$ are similarly strong, and so we do not consider
this to be an issue in our analysis.

The X-ray and SZ morphological parameters correlate far less well with the
parameters that contribute to PC2, PC3 and PC4 – the median (absolute)
Spearman’s rank coefficient is equal to 0.24 for these combinations. The
exception to this is, once again, the cluster Gaussianity, particularly in
X-rays. For parameters in PC1, the correlation coefficients associated with
$G_{\rm{X}}$ were lower than those for the other morphological indicators, but
for several parameters in PC2, PC3 and PC4, $G_{\rm{X}}$ is comparable to the
other morphological indicators. For example, $G_{\rm{X}}$ has a moderate
correlation with the cluster sphericity $c/a$ ($\rho_{\rm{s}}=0.23$,
$p=2\times 10^{-5}$), due to the fact that $G_{\rm{X}}$ is a direct measure of
a cluster’s shape.

These results are summarised in Table 2, which shows how each of these 14
morphological measurements of each cluster correlate with the four main
principal component values of that cluster. Indeed, we see that (apart from
the Gaussianity), each of these X-ray/SZ measures correlates well with PC1,
which we interpret as a measure of the time since much of the cluster’s galaxy
population was accreted and built up (see Section 3.1). However, the halo
virialisation (PC2), local environment (PC3) and recent merger history (PC4)
do not correlate as well with any of the morphological parameters.

Table 2: Spearman’s rank correlation coefficient between X-ray and SZ morphological parameters, and each of the four principal components. This data is similar to that shown in Fig. 2, but showing the correlation coefficient for each full principal component, not for their contributing dynamical state indicators. Observable | PC1 | PC2 | PC3 | PC4
---|---|---|---|---
$A_{\rm{X}}$ | -0.65 | -0.11 | -0.08 | 0.08
$K_{\rm{X}}$ | 0.72 | 0.13 | -0.07 | 0.04
$W_{\rm{X}}$ | -0.71 | -0.17 | 0.04 | 0.06
$P_{\rm{X}}$ | -0.70 | -0.03 | 0.01 | 0.02
$G_{\rm{X}}$ | 0.25 | 0.11 | 0.02 | -0.07
$S_{\rm{X}}$ | -0.57 | -0.12 | -0.04 | 0.06
$M_{\rm{X}}$ | -0.78 | -0.13 | 0.01 | 0.05
$A_{\rm{SZ}}$ | -0.64 | -0.11 | -0.04 | 0.11
$K_{\rm{SZ}}$ | 0.75 | 0.23 | -0.11 | 0.02
$W_{\rm{SZ}}$ | -0.73 | -0.11 | -0.01 | 0.07
$P_{\rm{SZ}}$ | -0.60 | -0.01 | 0.02 | 0.03
$G_{\rm{SZ}}$ | 0.18 | -0.05 | -0.05 | -0.05
$S_{\rm{SZ}}$ | -0.63 | -0.06 | -0.08 | 0.12
$M_{\rm{SZ}}$ | -0.82 | -0.14 | 0.02 | 0.06

Despite the fact that the observable morphological properties of a cluster
only strongly correlate with the dynamical state indicators that make up PC1,
there are still correlations between other indicators. Fig. 3 shows the
Spearman’s rank correlation coefficient between all 17 of these 3D (non-
observational) parameters. For example, the time since the last major merger,
$z_{\rm{merge,50}}$, and the cluster formation time, $z_{\rm{0.5}}$, have a
positive correlation ($\rho_{\rm{s}}=0.50$). This result is quite intuitive,
as clusters that have recently merged will not have a high formation redshift.
Other dynamical state indicators also correlate well – for instance, all five
of the parameters that contribute to PC2 show weak to moderate correlations
with each other ($|\rho_{\rm{s}}|\geq 0.20$, $p\leq 2\times 10^{-4}$). These
correlations are weaker for the dynamical state indicators in PC3 and PC4, but
this is not unexpected given that these components explain less of the
variance of the total dataset (see Fig. 5).

Figure 3: Spearman’s rank correlation coefficient, $\rho_{\rm{s}}$, between
all 3D dynamical state parameters. Darker coloured cells represent stronger
correlations; the value of the Spearman’s rank is also written in each cell. A
corner plot showing the distributions of each of these parameters can be found
in Appendix B. The principal component to which each dynamical state indicator
contributes is indicated at the top of the figure, and by the coloured bars
along the axes.

This figure also displays some counter-intuitive results. For example, one
might consider clusters with lots of substructure to be “rich”, but there is
actually a negative correlation between $f_{\rm{s}}$ and $N_{200}$ (for
$f_{\rm{s}}(R_{200})$, $\rho_{\rm{s}}=-0.63$, and for $f_{\rm{s}}(R_{500})$,
$\rho_{\rm{s}}=-0.46$). This is due to the fact that $N_{200}$ is defined by
the number of galaxies of similar magnitude to the third brightest, and
clusters with lots of substructure are more likely to have several bright
galaxies, meaning that the threshold for galaxies to be counted in $N_{200}$
is higher. Nevertheless, $N_{200}$ is a useful dynamical state indicator,
although using the term “richness” for this is somewhat ambiguous. While we do
not explicitly show the correlations between these parameters in Fig. 3 (for
the sake of clarity), we have included a corner plot showing these
correlations in Appendix B.

###  3.3 Mass accretion histories of clusters

In order to try and learn more about the dynamical histories of these
clusters, we separate our sample of 324 clusters based on the coordinate
values of their principal components, and study their mass accretion
histories. A similar analysis was performed by Wong & Taylor (2012), who also
split simulated dark matter haloes into different classes based on PCA
analysis. They show that the mass accretion histories of their classes of
clusters differ – their first principal component separates the clusters into
early-forming and late-forming, and their second component describes whether a
halo’s growth is accelerating or decelerating.

For our similar analysis, we find the upper and lower quartiles of the values
of each of our principal components. For example, based on PC1, we have a
group of clusters that have low values of PC1, and a group with high values.
As the contribution of $z_{0.5}$ to PC1 is positive (0.35, see Table 1), this
also implies that selecting clusters with low values of PC1 is equivalent to
choosing a group with low values of $z_{0.5}$, making them a “late forming”
quartile. Similarly, we describe the group with high values of PC1 (and so
higher values of $z_{0.5}$) as the “early forming” quartile. We then look at
the median mass accretion history of the clusters in each of these two extreme
groups, ignoring the intermediate clusters. The median mass accretion
histories for each principal component are shown in Fig. 4.

Figure 4: Median mass accretion histories for clusters, split based on the
values of their principal components, with a different principal component
considered in each panel (PC1 in top-left, PC2 in top-right, PC3 in bottom-
left, PC4 in bottom-right). Upper and lower quartile of each principal
component are shown. Horizontal axis shows redshift (bottom axis), and
lookback time in units of the dynamical time at $z=0$ (top axis), equal to
approximately 1.4 Gyr as defined in Eq. 5. Vertical axis shows the fraction of
the $z=0$ mass that has been built up by a given time. Faint lines show the
mass accretion histories of individual galaxy clusters. Some of these have
temporary decreases in mass, which can be explained by several well-
established mechanisms, such as the decrease in $M_{200}$ as a halo virialises
after a period of rapid growth (Tormen et al., 1997, e.g.), or through
mismatches in halo tracking (Fakhouri & Ma, 2009; Behroozi et al., 2015). The
thick lines show the median accretion history for that group, the light shaded
regions and thin lines show the $1\sigma$ (16-84 percentile) spread in the
data, and the dark shaded regions show the uncertainty in the median.

When splitting clusters by their PC1 coordinate values, we see a clear
difference between the mass accretion histories. As expected, the mass of the
early forming clusters is built up far earlier. On average, the early forming
clusters have built up half of their mass by a redshift of 0.7, while the late
forming clusters do not do this until a redshift of 0.2. This is very similar
to the first principal component found by Wong & Taylor (2012), which also
splits their haloes into early-forming and late-forming.

We interpret the PC2 coordinate values as separating the clusters based on how
virialised are their dark matter haloes. Separating the clusters based on
their PC2 coordinate values, we find that the difference in their mass
accretion histories is less pronounced than for PC1, and that there is a
significant overlap in the spread of the data. However, by “bootstrapping” the
data, we can study the uncertainty in the median mass accretion histories.
From this, we find that there is a significant difference between the clusters
split by PC2, particularly at low redshifts. The “more virialised” clusters
have experienced only a very small change in mass since $z=0.1$, having an
average mass at $z=0.1$ of $(0.97\pm 0.01)M_{200}(z=0)$ (median and
uncertainty). Meanwhile, the “less virialised” clusters have an average mass
of $(0.80^{+0.02}_{-0.07})M_{200}(z=0)$ at $z=0.1$. This indicates that this
“virialisation” component is strongly dependent on the very recent growth
history of a cluster, although we emphasise that there is a large overlap in
the spread of these data. We also note that the shapes of these mass accretion
histories are similar to the mass accretion histories of haloes that Wong &
Taylor (2012) separate by PC2. They describe these as accelerating or
decelerating growth rates, although their PC2 component is interpreted as
describing the shape and spin of their haloes, not the “virialisation” as we
find. Some slight variations between our results and those of Wong & Taylor
(2012) are to be expected, as their work uses dark matter-only simulations,
not hydrodynamical simulations. The baryonic effects in The300 are
particularly strong in the cluster centres (Haggar et al., 2021), a region on
which PC2 appears to be strongly dependent (see also Section 4).

The component values of the third principal component, PC3, do not appear to
have a strong impact on the mass accretion histories of clusters. Splitting
the clusters by their coordinate values in this component (into “symmetric
environments” with many filaments and a large distance to the nearest large
halo, and “asymmetric environments” with few filaments and a nearby large
halo) does not show a large difference in mass accretion histories. This
indicates that the present-day local environment of a cluster is not closely
tied to its mass accretion history, although Fig. 3 does show a weak
correlation between $z_{0.5}$ and $D_{1,0.1}$ ($\rho_{\rm{s}}=0.22$,
$p=6\times 10^{-5}$). We also note that a physical interpretation of this
component is challenging. Based on past work, one would expect that relaxed
clusters have both fewer filaments and a large distance to their nearest
neighbour, but splitting clusters based on their PC3 values puts clusters with
many filaments in a group with those that have a large distance to the nearest
neighbour. However, this is likely due to the weak correlations between the
components of PC3 that we find throughout this work, particularly
$N_{\rm{fil}}$ (also shown in Fig. 3). We discuss the strength of these
correlations further in Section 3.4.

For the fourth principal component, PC4, the accretion histories are similar,
although there is a slight difference in the shapes of the profiles; similarly
to in PC1, the “relaxed” (non-merging) clusters build up their mass at
slightly earlier redshifts. However, the main difference between the recent
mergers and non-mergers is that the spread in mass accretion histories is far
greater in recently merged clusters. For example, at $z=0.2$, the non-merging
clusters have an average mass of $0.77^{+0.17}_{-0.23}$ times their present
day mass (median and $1\sigma$ spread). In contrast, the recently merging
clusters have an average mass of $0.73^{+0.28}_{-0.32}$ times their present
day mass, corresponding to a $50\%$ greater spread. This increased spread in
accretion histories for recently merging clusters is likely due to the
stochastic nature of mergers – these objects will have experienced a large
jump in their mass at some recent time, but the exact time of this jump varies
between clusters. Again, this is similar to the findings of Wong & Taylor
(2012), who explain that mass accretion histories are not well modelled by
smooth curves due to the stochasticity of merger events.

###  3.4 Cluster dynamical state with UMAP

Uniform manifold approximation and projection (UMAP; McInnes et al., 2018) is
an alternative dimensionality reduction technique to PCA. UMAP is commonly
used as part of more complex machine learning studies, but is also a useful
data visualisation method in its own right. UMAP involves first constructing a
graph of datapoints (in our case, 324 galaxy clusters), each connected to
their nearest $N$ neighbours in a high-dimensional space (the 17 dimensions
defined by the dynamical state indicators). Some additional connections
between points are also added, with a decreasing likelihood for points
separated by greater distances. Next, a low-dimensional space, the “embedding
space”, is populated randomly with (324) corresponding datapoints. These are
iteratively repositioned according to some loss function. This function is
designed to match the pairwise distances between connected points in the new
low-dimensional space to the pairwise distances in the original high-
dimensional space.

This process preserves local structure in the dataset, meaning that objects
with similar properties remain close together in the low-dimensional
“embedding space”. It also preserves global properties of the dataset, so that
groups of objects that are far apart in the high-dimensional space remain
separated from one another. Similarly to PCA, UMAP is able to reduce a high-
dimensional datasets into a smaller number of parameters (in our case, two).
Unlike PCA however, the stochastic, iterative nature of UMAP allows for
complex, non-linear relationships between parameters to also be captured. This
means that the outputs of UMAP are harder to physically interpret – whereas
PCA produces mathematically well-defined axes, the output “axes” of UMAP are
determined numerically. Similarly, a simple measure of the fractional variance
that is captured by UMAP, analogous to the “explained variance” of PCA, does
not exist (McInnes et al., 2018). We use UMAP as a complementary method to
PCA, approaching the same problem but in a different way.

Fig. LABEL:fig:umap shows the results of our UMAP analysis for the 17
dynamical state indicators also used in the PCA. Each point represents one
galaxy cluster, with the horizontal and vertical axes showing the two combined
parameters produced by the two dimensional UMAP analysis, analogous to the
principal components that we summarised in Table 1. Due to the definition and
optimisation processes in UMAP, points that are close together on a UMAP plot
are close together in the 17-dimensional space that we started with.
Consequently, these are likely to be dynamically similar clusters. As the UMAP
axes are a dimensionless (due to the standardisation of the data), complex
combination of many parameters, these axes are left unlabelled for clarity.

Each panel in this figure shows data for one of our 17 3D dynamical state
indicators, and the colour of each point in this panel represents the value of
this dynamical state indicator for each cluster. Linear colour scales are used
for each, but the colour scale is flipped for some quantities, such that the
clusters we interpret as more dynamically relaxed (according to this
individual parameter) are shown by darker colours. This allows us to see which
regions of this new UMAP embedding space contain dynamically relaxed clusters,
according to each of the different definitions. Similarly to in Fig. 2, the
panels are grouped by the principal component to which they most strongly
contribute, according to our principal component analysis – otherwise, this
approach is fully independent of our PCA.

For each of these panels, we calculate the direction in which the $z$-axis
(i.e. the value of the dynamical state indicator) varies most quickly, and
thus the direction in this space in which the clusters become more dynamically
unrelaxed. We do so using the partial correlation coefficients (PCCs) approach
described in Lawrance (1976); see also Baker et al. (2022) for a clear
explanation of this method. The partial correlation coefficient between two
quantities, $A$ and $B$, whilst controlling for a third quantity, $C$, is
given by $\rho_{AB|C}$, and is calculated using the equation below:

$\rho_{AB|C}=\frac{\rho_{AB}-\rho_{AC}\rho_{BC}}{\sqrt{1-\rho_{AC}^{2}}\sqrt{1-\rho_{BC}^{2}}}\,.$
(6)

Here, $\rho_{XY}$ is the Spearman’s rank correlation coefficient between two
quantities, $X$ and $Y$. Eq. 6 allows the partial correlation coefficent
between the colour (the value of the dynamical state indicator) and the
vertical UMAP axis to be found, whilst controlling for the horizontal axis.
Similarly, we find the PCC between the colour and the horizontal UMAP axis,
whilst controlling for the vertical axis. The ratio of these two partial
correlation coefficients can then be used to calculate the maximal variation
direction of the colour in the UMAP embedding space (see also Bluck et al.,
2020, for further details of this method). We plot each of these directions of
maximal dynamical state variation in the top-left panel of Fig.
LABEL:fig:umap, coloured by the principal component to which they contribute
most strongly. The length of each arrow is equal to the Spearman’s rank
correlation coefficient between the dynamical state indicator (colour) and the
position of clusters along this direction in embedding space. Each arrow is
also shown on the bottom-left corner of the panel to which it corresponds.
This top-left panel therefore shows which direction in this embedding space
contains the unrelaxed clusters, according to each of the 3D dynamical state
measures.

Fig. LABEL:fig:umap shows that each of the parameters making up the first
principal component of dynamical state ($f_{\rm{s}}(R_{200})$,
$f_{\rm{s}}(R_{500})$, $\Delta_{\rm{r}}(R_{200})$, $\Delta_{\rm{r}}(R_{500})$,
$z_{0.5}$, $m_{12}$ and $N_{200}$) vary in approximately the same direction
across the UMAP plots, from bottom to top. According to these measures, the
relaxed clusters are generally found at the bottom of these plots, and the
unrelaxed clusters at the top. This indicates that the UMAP has also grouped
clusters based on this interpretation of dynamical state, separating clusters
that have formed and developed their substructure recently, from those that
formed long ago. The top-left panel of Fig. LABEL:fig:umap confirms that the
increasingly disturbed clusters, according to PC1 parameters, are at the top
of the UMAP plots.

However, the dynamical state indicators that contribute to the second, third
and fourth principal components do not all vary in the same way. The PC4
parameters ($c/a$ and $z_{\rm{merge,50}}$) do follow a similar trend to those
in PC1, with more dynamically disturbed clusters (low-$z_{\rm{merge,50}}$ and
low-$c/a$) being found near the top of the plot – this is consistent with our
interpretation of PC4 in Section 3.1, where we described how some of the
merger history behaviour is also included in PC1. The PC2 indicators,
representing the virialisation and concentration of the cluster haloes
($\eta(R_{200})$, $\eta(R_{500})$ and $c_{200}$) instead vary more strongly
from right to left, although not all in the same direction; $\gamma$ follows a
similar trend. The UMAP can also separate clusters by their environmental
dynamical state indicators (PC3: $\lambda$, $N_{\rm{fil}}$ and $D_{1,0.1}$),
but not very well, indicated by the weak trends in these panels. The
high-$\lambda$ and low-$D_{1,0.1}$ clusters, usually interpreted as unrelaxed,
are located towards the top of the plot – these trends are not strong, but are
significant, and are along the same axis as the PC1 components’ trends. For
spin, $\lambda$, $|\rho_{\rm{s}}|=0.44$. For the environment parameter,
$D_{1,0.1}$, $|\rho_{\rm{s}}|=0.24$ ($p=10^{-5}$). The number of filaments and
UMAP position correlate in the opposite direction along the same axis, but
this correlation is not significant ($|\rho_{\rm{s}}|=0.10$, $p=0.06$).

This UMAP analysis gives many subtle results that are not immediately
straightforward to interpret, but overall it confirms the indications of our
PCA – the different groupings of clusters in the plots correspond to different
classes of dynamically relaxed objects. For example, the early-forming, low-
substructure clusters (PC1) are found at the bottom of the UMAP plots. These
clusters have highly-concentrated haloes, as would typically be expected of a
relaxed cluster (Yuan & Han, 2020). At the top of the UMAP plots, we instead
find recently-formed clusters. However, there is also a third major population
of clusters in the top-left of the plots – these are dynamically unrelaxed
haloes, that have formed recently, have much substructure and have non-
virialised haloes, yet are highly-concentrated. This population have recently
experienced major mergers ($z_{\rm{merge,50}}$), and have very high accretion
rates, $\gamma$, indicating that they are currently accreting large amounts of
material. This is in contrast to the other clusters at the top of these plots,
which have recently merged but have not accreted lots of material in the last
dynamical time (since $z=0.1$). This difference is equivalent to the mass
accretion histories in the top-right panel of Fig. 4, where we group clusters
based on PC2 and show that their recent ($z<0.2$) growth histories are
different.

These three populations can be interpreted as follows: high-concentration
galaxy clusters are likely to have formed long ago, while low-concentration
clusters are likely recently formed, often after a major merger. However, if a
merger is still ongoing, or if large amounts of galaxies are pulled into the
cluster immediately afterwards (potentially accompanying the merging cluster),
this can make a cluster appear highly-concentrated, and therefore dynamically
relaxed. This explanation is strongly supported by Wang et al. (2020), who
show that major mergers can produce oscillations in concentration, driving
concentration up significantly when merging material reaches the first
pericentre of its orbit, before concentration quickly decreases again.
Alternatively, if a cluster is continuing to accrete large amounts of diffuse
material into its halo (rather than a single large object), this could also
result in a high mass accretion rate without a corresponding decrease in halo
concentration.

As a result of this, a simple measurement of the concentration of a cluster is
insufficient to draw conclusions about the virialisation of its halo, its
formation time, or its merger history. Additionally, the dynamical state of a
cluster’s BCG ($\sigma_{\rm{BCG}}$) also seems to only probe some of these
unrelaxed clusters. This complex behaviour cannot be fully accounted for by a
1D linear fit, which explains our previous counter-intuitive result that
highly-concentrated clusters appear to be less virialised and rapidly-
accreting (Fig. 1 and Fig. 3); simplifying this to a single correlation
coefficient does not capture the complete behaviour.

Additionally, the UMAP analysis shows that the early-forming clusters can be
split into two groups based on their merger histories. The group in the
bottom-left have not experienced major mergers for a very long time ($\sim 10$
Gyr), or not at all throughout their history. Meanwhile, those in the bottom-
right have experienced a major merger more recently, although still not for
several gigayears. There seems to be a (weak) correlation with the
connectivity of these clusters, $N_{\rm{fil}}$: early-forming clusters that
have still experienced a major merger are more strongly connected than early-
forming clusters that have never experienced a major merger. This may be
indicative of the extremely long timescales ($\sim 10$ Gyr) over which cosmic
filaments are persistent, but a more detailed analysis would be needed to
investigate this connection.

##  4 Discussion and Conclusions

Using hydrodynamical simulations of galaxy clusters, we show that the wide
variety of parameters used in the literature to quantify the “dynamical state”
of a cluster are actually probing multiple properties of a cluster.
Consequently, by applying PCA, we conclude that the dynamical state of a
cluster as described in the literature is actually made up of approximately
four different properties, summarised in Section 3.1. To use any measurement
(or set of measurements) to describe a cluster as simply “dynamically relaxed”
or “dynamically unrelaxed” is an incomplete description. Different classes of
dynamically relaxed clusters exist, and so instead one must specify in which
sense is a cluster relaxed.

The main component of a cluster’s dynamical state is its “formation dynamical
state”, describing the formation time of a cluster – that is, whether a
cluster built up most of its mass and galaxy population recently or long ago.
This formation state is indicated by galaxies and substructure within the
cluster. Recently forming clusters have more substructure and a greater offset
between their centre-of-mass and the position of their BCG. Additionally these
clusters are typically less dominated by a single, bright galaxy, and are
accreting material at a faster rate. This “formation state” is similar to the
dynamical state used in many previous studies (Cui et al., 2017; Gouin et al.,
2021, for example). It should be noted that several such studies include the
virial ratio, $\eta$, in this measure of dynamical state, making it analogous
to a combination of our PC1 and PC2, however it has been shown that the virial
ratio is not an important contributor to this measure of dynamical state
(Haggar et al., 2020; De Luca et al., 2021). As we see in Fig. 4, separating
clusters based on their first principal component from our PCA is equivalent
to separating them based on their formation times, in agreement with Wong &
Taylor (2012). The “late-forming” clusters have a median formation time of
$z_{0.5}=0.2\pm 0.1$, compared to $z_{0.5}=0.7\pm 0.2$ for the relaxed “early-
forming” clusters.

In addition to this dynamical description of the subhaloes and galaxies in a
cluster, the diffuse material in the cluster halo itself can be virialised and
dynamically relaxed. This “virialisation dynamical state” describes how well
the material in a cluster obeys the modified virial theorem (accounting for
surface pressure, see Poole et al., 2006; Shaw et al., 2006). It is
particularly well-described by the properties of the central regions of a
cluster – the NFW concentration of the halo, and the velocity dispersion of
the brightest cluster galaxy – as well as the present day accretion rate of
the cluster, $\gamma$. The UMAP analysis in Section 3.4 also showed that, in
this space, the “virialisation state” varies along a different axis to the
“formation state”, described by PC1. These complex relationships can be seen
through the apparent inconsistencies when only considering monotonic
relationships between quantities. For example, Fig. 3 shows that highly-
concentrated haloes (typically considered “relaxed”) correlate with having
less substructure (a sign of being relaxed), but also with a less virialised
halo according to $\eta$. The complex relationships between these dynamical
state indicators are much more apparent in Fig. LABEL:fig:umap.

Clusters can also be dynamically relaxed in terms of their local environment,
or in terms of their merger history. These components of dynamical state are
driven by global properties of a cluster, such as their shape and spin. These
components are typically noisier, and the cluster properties that constitute
these components do not correlate as well with one another. However, it is
still meaningful to separate clusters based on this – for instance, Fig. 4
shows an increased spread in the mass accretion histories of recently-merged
clusters.

Of these four forms of dynamical state, X-ray and SZ measurements of
morphological properties of clusters, such as their asymmetry, concentration,
and centroid shift, overwhelmingly probe the formation dynamical state (PC1).
Generally, X-ray and SZ morphological parameters are very well-suited to
describing the formation of clusters and their accretion of substructures, but
not well-suited to describing the virialisation of cluster haloes, their
merger history, or their cosmic environment. This is not unexpected: as we
discussed in Section 3.4, mergers appear to have a somewhat chaotic effect on
other dynamical state parameters, and can lead to a cluster halo appearing
either relaxed or unrelaxed. Additionally, these morphological measurements,
particularly the X-ray measurements, are heavily weighted towards the central
regions of clusters ($R_{500}$ or smaller). We would therefore not necessarily
expect them to be good indicators of clusters’ larger-scale cosmic
environments. There are also some slight exceptions to this rule – for
example, the light concentration ratio calculated from the SZ-effect,
$K_{\rm{SZ}}$, does indeed correlate with the NFW concentration, $c_{200}$
($\rho_{\rm{s}}-0.37$).

As stated previously, this work is very much a theoretical, simulation-focused
study of dynamical state, and any further detailed analysis of observable
cluster properties is beyond the scope of this paper. However, numerous other
observable properties of clusters exist, and in future work we hope to examine
how these observable quantities correspond to dynamical state. For example,
although the mock X-ray observations used in this work are focused on the
cluster centres, X-rays can also be used to map out substructures in cluster
outskirts (Zhang et al., 2020), and even cosmic filaments (Walker et al.,
2019; Biffi et al., 2022). Other measures of cluster X-ray and SZ maps also
exist, such as decomposition into Zernike polynomials (Capalbo et al., 2021).
Beyond this, optical data can also be used to quantify the dynamical states of
clusters, by determining properties such as their substructure (Wen & Han,
2013), richness, and brightest galaxies magnitude difference, $m_{12}$
(Zhoolideh Haghighi et al., 2020). A future study (Cornwell et al., in prep.)
will use The300 simulations to investigate how cluster dynamical states can be
determined using spectroscopic measurements.

The results from this paper confirm that, although a description of dynamical
state (see Binney & Tremaine, 1987) is theoretically quite simple, the
dynamical state of galaxy clusters in practice is more complex. This has
implications for wider work on galaxy clusters. Many studies split clusters
into two samples, “relaxed” and “unrelaxed”, based on a small number of
properties, but this has the potential to combine clusters with very different
dynamical histories into a single group. Instead, it is important to describe
in which sense a cluster is known to be dynamically relaxed or unrelaxed – for
example, in terms of their substructure accretion history, their recent merger
history, or how virialised they are at the present day.

In future we plan to carry out a more quantifiable analysis of this multi-
dimensional dynamical state, rather than the qualitative description presented
in this work. Instead of a linear scale of dynamical state, it may be more
natural to quantify the dynamical state of clusters in two or more dimensions;
the UMAP analysis in this paper is similar to this, but it not simple to
interpret physically. Such a study could also be extended to study the
dynamical state in different “apertures”, looking at the kinematics of the
cluster centre, or of its outskirts. Additionally, we plan to investigate
whether clusters can be naturally separated into groups based on their
dynamical states. Previous work (e.g. Zhang et al., 2022) has investigated
defining a bimodal function to describe dynamical state, to allow clusters to
be split into “relaxed” and “unrelaxed” groups. In a future study, we will
investigate whether a multimodal description of dynamical state exists, that
would allow clusters to be split into more than two groups. This would provide
a means to compare the properties of clusters that do actually have similar
dynamical states, and to select samples of truly relaxed galaxy clusters in
order to reduces biases in cosmological studies.

## Acknowledgements

We would like to thank the anonymous referee for their comments on this work,
which have helped to improve the paper by adding robustness and clarity.

This work has been made possible by The Three Hundred
collaboration444https://www.the300-project.org. This work has received
financial support from the European Union’s Horizon 2020 Research and
Innovation programme under the Marie Skłodowskaw-Curie grant agreement number
734374, i.e. the LACEGAL
project555https://cordis.europa.eu/project/rcn/207630_en.html. The Three
Hundred simulations used in this paper have been performed in the MareNostrum
Supercomputer at the Barcelona Supercomputing Center, thanks to CPU time
granted by the Red Española de Supercomputación.

WC is supported by the STFC AGP Grant ST/V000594/1 and the Atracción de
Talento Contract no. 2020-T1/TIC-19882 granted by the Comunidad de Madrid in
Spain. He also thanks the Ministerio de Ciencia e Innovación (Spain) for
financial support under Project grant PID2021-122603NB-C21 and ERC: HORIZON-
TMA-MSCA-SE for supporting the LACEGAL-III project with grant number
101086388. M.D.P. acknowledges financial support from PRIN 2022 (Mass and
selection biases of galaxy clusters: a multi-probe approach - n. 20228B938N)
and from Sapienza Università di Roma - Progetti di Ricerca Medi 2022,
RM1221816758ED4E. JET acknowledges support from an NSERC Discovery Grant.

This work makes use of the SciPy (Virtanen et al., 2020), NumPy (van der Walt
et al., 2011), Matplotlib (Hunter, 2007), pandas (McKinney, 2010) and Scikit-
learn (Pedregosa et al., 2011) packages for Python666https://www.python.org.

The authors contributed to the paper in the following ways: RH carried out the
analysis, wrote the paper, and produced the figures. FDL and MDP calculated
the morphological parameters from the X-ray/SZ mock observations, which were
produced by WC. ES carried out the UMAP analysis. AK produced the simulation
halo catalogues and merger trees, from which several of the 3D dynamical state
measures were calculated. AK also assisted with the general direction of the
work, along with JET, MEG and FRP. ACS calculated the times since last major
merger. WC calculated the dynamical state indicators $f_{\rm{s}}$,
$\Delta_{\rm{r}}$ and $\eta$. UK calculated the cluster connectivities. RAMP
calculated the cluster formation times, $z_{0.5}$. CP calculated the BCG
velocity dispersions. All authors had the opportunity to comment on the paper.

## Data Availability

The simulations underlying this work have been provided by The Three Hundred
collaboration. The data may be shared on reasonable request to the
corresponding author, with the permission of the collaboration.

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##  Appendix A Explained variance of principal components

The four principal components selected and analysed throughout this work
collectively describe $64\%$ of the total variance of the 17 dynamical state
indicators described in Section 2.2.1. In Fig. 5, we show the proportion of
the variance explained by all 17 of the principal components, and the
cumulative explained variance.

Figure 5: Cumulative proportion of variance of the dynamical state indicators
dataset explained by the ordered principal components (black line and points),
and by the individual components (blue bars). Throughout our analysis, we only
consider the first four components, indicated by the vertical red dashed line.

Clearly, PC1 is the dominant component, explaining $38\%$ of the total
variance alone. Other than this, there is not a clearly visible “cut-off”
point, beyond which parameters are far less important. Consequently, the
decision to keep only the first four principal components is not obviously
mathematically justifiable. This choice was made in order to select the
minimum number of components such that each of the dynamical state indicators
contributed strongly to at least one of them, to provide an idea of the number
of dimensions along which dynamical state varies. For example, PC5 (which
explains $5\%$ of the total variance) is dominated by $\lambda$,
$N_{\rm{fil}}$ and $D_{1,0.1}$, which all also strongly contribute to PC3 (see
Table 3). We interpret this as PC5 providing a “second order” correction to
the environment of a cluster, as described by PC3; we believe that including
such a component would not meaningfully add to this work.

Table 3: Similar to Table 1, but showing coordinate values for PC5, which is excluded throughout this work. Component coordinates with an absolute value greater than 0.24 are highlighted in bold; the three parameters that contribute strongly to PC5 ($\lambda$, $N_{\rm{fil}}$ and $D_{1,0.1}$) also contribute strongly to PC3, albeit with different weights. Parameter | Contribution to PC5
---|---
$f_{\rm{s}}(R_{\rm{200}})$ | -0.04
$f_{\rm{s}}(R_{\rm{500}})$ | 0.05
$\Delta_{\rm{r}}(R_{\rm{200}})$ | -0.02
$\Delta_{\rm{r}}(R_{\rm{500}})$ | 0.02
$\eta(R_{\rm{200}})$ | -0.14
$\eta(R_{\rm{500}})$ | -0.13
$z_{\rm{0.5}}$ | -0.04
$\lambda$ | -0.40
$c/a$ | -0.23
$c_{\rm{200}}$ | 0.08
$z_{\rm{merge,50}}$ | -0.14
$\gamma$ | 0.09
$N_{\rm{fil}}$ | -0.55
$D_{1,0.1}$ | 0.59
$N_{200}$ | -0.24
$m_{12}$ | -0.05
$\sigma_{\rm{BCG}}$ | 0.05

##  Appendix B Correlations between dynamical state indicators

In Fig. 3 we show the Spearman’s rank correlation coefficient between each of
the 3D dynamical state indicators used in this study (described in Section
2.2.1). In Fig. 6 below, we explicitly show the correlations between these
measures, from which these correlation coefficients were calculated. As in
Fig. 3, we also group these parameters based on the principal component to
which they most strongly contribute. Finally, we also include the cluster mass
as an additional parameter here, to explicitly show the mass dependence of
these quantities.

Figure 6: Corner plot showing the correlations between the 17 dynamical state
indicators described in Section 2.2.1. The Spearman’s rank correlation
coefficients displayed in Fig. 3 are calculated from these scatter plots. This
figure also includes histograms showing the distribution of these parameters.
We also include the logarithmic mass of each galaxy cluster as an additional
quantity here, to show any mass dependence of these 17 parameters.

It should be noted that several of these parameters have a highly skewed
distribution (for example, $\eta$). These parameters were all standardised for
the PCA and UMAP analysis in this work to approximate a normal distribution
with a mean of zero and a unit standard deviation, but here we show the raw
distributions. Most of these parameters also display either no dependence, or
a weak dependence, on cluster mass ($|\rho_{\rm{s}}|\leq 0.15$, $p\geq
0.005$). The exception is the velocity dispersion of the BCG
($\rho_{\rm{s}}=0.74$), which has a fairly strong positive correlation with
the cluster mass. This result is expected, as numerous previous studies have
found that the BCG velocity dispersion scales with the mass of its host
cluster (e.g Sohn et al., 2020). One could account for this mass dependence by
normalising the BCG velocity dispersion by the maximum circular orbital speed
around the cluster, $v_{\rm{circ}}$. For The300, the Spearman’s rank
correlation coefficient between $M_{200}$ and
$\sigma_{\rm{BCG}}/v_{\rm{circ}}$ is far reduced ($\rho_{\rm{s}}=0.19$,
$p=6\times 10^{-4}$); although we have chosen to just use $\sigma_{\rm{BCG}}$
in this study, this ratio is a potential alternative choice that one could use
instead.