11institutetext: Dipartimento di Fisica, Sapienza Università di Roma, Piazzale
Aldo Moro 5, I-00185 Rome, Italy
11email<EMAIL_ADDRESS>22institutetext: Departamento de Física
Teórica, Facultad de Ciencias, Universidad Autónoma de Madrid, Modulo 8,
E-28049 Madrid, Spain 33institutetext: Escuela de Doctorado UAM, Centro de
Estudios de Posgrado, Universidad Autónoma de Madrid, E-28049, Madrid, Spain
44institutetext: Centro de Investigación Avanzada en Física Fundamental
(CIAFF), Facultad de Ciencias, Universidad Autónoma de Madrid, E-28049,
Madrid, Spain 55institutetext: School of Physics and Astronomy, University of
Nottingham, Nottingham NG7 2RD, UK 66institutetext: Institute for Astronomy,
University of Edinburgh, Edinburgh EH9 3HJ, UK 77institutetext: INAF,
Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3,
I-40129 Bologna, Italy 88institutetext: INFN, Sezione di Bologna, viale Berti
Pichat 6/2, I-40127 Bologna, Italy

# The Three Hundred project: Estimating the dependence of gas filaments on the
mass of galaxy clusters

Sara Santoni The Three Hundred project: Estimating the dependence of gas
filaments on the mass of galaxy clustersThe Three Hundred project: Estimating
the dependence of gas filaments on the mass of galaxy clustersThe Three
Hundred project: Estimating the dependence of gas filaments on the mass of
galaxy clustersThe Three Hundred project: Estimating the dependence of gas
filaments on the mass of galaxy clustersThe Three Hundred project: Estimating
the dependence of gas filaments on the mass of galaxy clustersThe Three
Hundred project: Estimating the dependence of gas filaments on the mass of
galaxy clusters Marco De Petris The Three Hundred project: Estimating the
dependence of gas filaments on the mass of galaxy clustersThe Three Hundred
project: Estimating the dependence of gas filaments on the mass of galaxy
clusters Gustavo Yepes The Three Hundred project: Estimating the dependence of
gas filaments on the mass of galaxy clustersThe Three Hundred project:
Estimating the dependence of gas filaments on the mass of galaxy clustersThe
Three Hundred project: Estimating the dependence of gas filaments on the mass
of galaxy clustersThe Three Hundred project: Estimating the dependence of gas
filaments on the mass of galaxy clusters Antonio Ferragamo The Three Hundred
project: Estimating the dependence of gas filaments on the mass of galaxy
clustersThe Three Hundred project: Estimating the dependence of gas filaments
on the mass of galaxy clusters Matteo Bianconi The Three Hundred project:
Estimating the dependence of gas filaments on the mass of galaxy clustersThe
Three Hundred project: Estimating the dependence of gas filaments on the mass
of galaxy clusters Meghan E. Gray The Three Hundred project: Estimating the
dependence of gas filaments on the mass of galaxy clustersThe Three Hundred
project: Estimating the dependence of gas filaments on the mass of galaxy
clusters Ulrike Kuchner The Three Hundred project: Estimating the dependence
of gas filaments on the mass of galaxy clustersThe Three Hundred project:
Estimating the dependence of gas filaments on the mass of galaxy clusters
Frazer R. Pearce The Three Hundred project: Estimating the dependence of gas
filaments on the mass of galaxy clustersThe Three Hundred project: Estimating
the dependence of gas filaments on the mass of galaxy clusters Weiguang Cui
The Three Hundred project: Estimating the dependence of gas filaments on the
mass of galaxy clustersThe Three Hundred project: Estimating the dependence of
gas filaments on the mass of galaxy clustersThe Three Hundred project:
Estimating the dependence of gas filaments on the mass of galaxy clustersThe
Three Hundred project: Estimating the dependence of gas filaments on the mass
of galaxy clustersThe Three Hundred project: Estimating the dependence of gas
filaments on the mass of galaxy clustersThe Three Hundred project: Estimating
the dependence of gas filaments on the mass of galaxy clusters Stefano Ettori
The Three Hundred project: Estimating the dependence of gas filaments on the
mass of galaxy clustersThe Three Hundred project: Estimating the dependence of
gas filaments on the mass of galaxy clustersThe Three Hundred project:
Estimating the dependence of gas filaments on the mass of galaxy clustersThe
Three Hundred project: Estimating the dependence of gas filaments on the mass
of galaxy clusters

(Received …; accepted …)

###### Abstract

Context. Galaxy clusters are located in the densest areas of the universe and
are intricately connected to larger structures through the filamentary network
of the Cosmic Web. In this scenario, matter flows from areas of lower density
to higher density. As a result, the properties of galaxy clusters are deeply
influenced by the filaments that are attached to them, which are quantified by
a parameter known as connectivity.

Aims. We explore the dependence of gas-traced filaments connected to galaxy
clusters on the mass and dynamical state of the cluster. Moreover, we evaluate
the effectiveness of the cosmic web extraction procedure from the gas density
maps of simulated cluster regions.

Methods. Using the DisPerSE cosmic web finder, we identify filamentary
structures from 3D gas particle distribution in 324 simulated regions of
$30\,h^{-1}$ Mpc side from The Three Hundred hydrodynamical simulation at
redshifts z=0, 1, and 2. We estimate the connectivity at various apertures for
$\sim 3000$ groups and clusters spanning a mass range from
$10^{13}\,h^{-1}\,M_{\odot}$ to $10^{15}\,h^{-1}\,M_{\odot}$. Relationships
between connectivity and cluster properties like radius, mass, dynamical state
and hydrostatic mass bias are explored.

Results. We show that the connectivity is strongly correlated with the mass of
galaxy clusters, with more massive clusters being on average more connected.
This finding aligns with previous studies in literature, both from
observational and simulated data sets. Additionally, we observe a dependence
of the connectivity on the aperture at which it is estimated. We find that
connectivity decreases with cosmic time, while no dependencies on the
dynamical state and hydrostatic mass bias of the cluster are found. Lastly, we
observe a significant agreement between the connectivity measured from gas-
traced and mock-galaxies-traced filaments in the simulation.

###### Key Words.:

large-scale structure of Universe – Galaxies: clusters: general – Methods:
numerical – Methods: statistical

##  1 Introduction

Clusters of galaxies, considered the largest gravitationally-bound structures
in the universe, are located at the nodes of an intricate web-like structure
known as the Cosmic Web (Bond et al., 1996). This cosmic pattern of dark
matter (DM) and baryons and the formation of large-scale structures are
predicted as a consequence of the non-linear growth of primordial density
perturbations (Zel’dovich, 1970; Peebles, 1980). In this framework, matter
flows from underdense regions to overdense structures, resulting in compact
galaxy clusters at the nodes of elongated filamentary structures and filaments
at the intersection of sheet-like formations, known as walls, surrounded by
vast void regions.

The cosmic web has been observed by several galaxy surveys, such as the Sloan
Digital Sky Survey (SDSS, York et al., 2000) and the 2dF Galaxy Redshift
Survey (2dFGRS, Colless et al., 2001). Furthermore, the properties of the
cosmic web and its components have been identified through several large-scale
numerical simulations, both N-body and hydrodynamical, such as the Millennium
simulation (Springel, 2005) and the MilleniumTNG project (Hernández-Aguayo et
al., 2023), The Illustris project (Vogelsberger et al., 2014) and IllustrisTNG
(Nelson et al., 2019), the EAGLE project (Schaye et al., 2015) and the SIMBA
project (Davé et al., 2019).

Depending on the scientific goal of the study, different features of the
cosmic web can be examined, often without relying on unequivocal definitions.
Consequently, several algorithms have been developed to identify the cosmic
web, some tailored for simulations and others for observational data, as
illustrated in the exhaustive review by Libeskind et al. (2018). The different
algorithms generally converge on the identification and characterization of
specific features, but they reflect the complexity involved in the study and
definitions of these structures.

In particular, the most used extraction methods can be grouped as graph and
percolation techniques (e.g MST, Alpaslan et al. (2014) and T-Rex, Bonnaire et
al. (2020)), stochastic methods (e.g. Bisous model, Tempel et al. (2016)),
methods based on the Hessian of the density, tidal or velocity fields
(e.g.T-web, Forero-Romero et al. (2009), V-web, Hoffman et al. (2012)) and,
among these, those that follow a multi-scale approach (e.g. MMF, Aragón-Calvo
et al. (2007), NEXUS and NEXUS+, Cautun et al. (2013). Moreover, the cosmic
web can be identified with topological methods (e.g. Spineweb, Aragón-Calvo et
al. (2010a) and DisPerSE, Sousbie (2011); Sousbie et al. (2011)) and, lastly,
phase-space methods (e.g. ORIGAMI, Falck et al. (2012).

In addition to the galaxy clusters, filaments constitute the densest regions
of the cosmic web, containing approximately $50\%$ of the matter in the
Universe (Cautun et al., 2014). Filaments serve an important function in the
accretion phase of galaxy clusters, acting as pathways for matter
transportation towards the centre of the node to which they are connected. The
connectivity of the cosmic web, which measures the number of filaments
connected to a cluster, offers a method to quantify the interaction between
clusters and the filaments surrounding them, enabling the assessment of
structural evolution (Codis et al., 2018). Several works in literature have
been investigating the correlation of connectivity and the cluster’s
properties, such as the mass and the dynamical state (see e.g. Darragh Ford et
al., 2019; Sarron et al., 2019; Malavasi et al., 2020b; Gouin et al., 2021;
Galárraga-Espinosa et al., 2024).

Furthermore, several studies proved a significant role of filaments in the so-
called ’pre-processing’ of the galaxies that reside within the filamentary
structures. Filaments affect galaxy evolution and galaxy properties, before
entering the cluster, through several processes as ram-pressure, tidal effects
and mergers (see the reviews of Boselli & Gavazzi (2006, 2014) and more recent
works e.g. Kraljic et al. (2018); Winkel et al. (2021); Hellwing et al.
(2021); Song et al. (2021); Donnan et al. (2022); Kotecha et al. (2022);
Kuchner et al. (2022); Malavasi et al. (2022); Galárraga-Espinosa et al.
(2024); Bulichi et al. (2024)). Therefore, the outskirts of clusters provide a
distinctive environment for examining matter accretion dynamics and galaxy
properties.

Unlike the galaxy content of filaments, which has been extensively identified
with galaxy surveys, the detection of the gas component has proved to be more
challenging. Hydrodynamical simulations (see e.g. Bond et al., 1996; Davé et
al., 2001; Shull et al., 2012; Tuominen et al., 2021) predict that the
filamentary structures of the cosmic web are hosting a large fraction of the
missing baryons 111When compared to the concordance cosmology model, we
observe a $\approx 30\%$ deficit in the census of observed baryons from low-
redshift sources, such as galaxies, groups and clusters, inter-galactic and
circum-galactic medium. in a low-density and warm-hot state known as Warm-Hot
Intergalactic Medium (WHIM), with a density $\delta_{b}\approx 10-100$ and a
temperature in the range of $10^{5}-10^{7}$ K.

The hot phase of WHIM, with a $\log T(K)\approx 5.5-7$, is not observable
through absorption, being fully ionised, contrary to the low-temperature
phase, with a $\log T(K)\approx 5-5.5$, which has been detected several times
through metal absorption lines (Danforth et al., 2016). Despite having a faint
emission, several attempts have been made to detect the warmest phase of WHIM
through direct X-ray observations (see e.g. Eckert et al., 2015; Nicastro et
al., 2018) and the Sunyaev-Zel’dovich effect, employing stacking methods
(Tanimura et al., 2019, 2020, 2022) or looking at the densest regions between
two close galaxy clusters (Bonjean et al., 2018; Hincks et al., 2022).

In this work, we use the outputs of The Three Hundred (Cui et al., 2018a)
hydrodynamical simulations to extract the cosmic web in the surrounding
volumes of galaxy clusters, with a specific focus on the filamentary
structures, from the gas particle distribution using the cosmic web finder
DisPerSE. By analysing the connectivity estimates of simulated galaxy
clusters, this investigation explores the influence of filaments on the main
properties of galaxy clusters, such as their mass and dynamical state, and
assess the cosmic web extraction from diffuse gas particles within the
simulated The Three Hundred regions.

This paper is structured as follows: Sect. 2 provides a detailed overview of
the The Three Hundred simulation and the theoretical framework behind the
cosmic web finder DisPerSE. The process of creating the gas density maps and
calibrating DisPerSE is discussed in Sect. 3. In Sect. 4 we present the
results of the connectivity analysis, while in Sect. 5 we investigate the
effect of cosmic web tracers by comparing the gas-traced filaments to mock-
galaxy-traced filaments. Finally, Sect. 6 summarizes the main results of our
study.

##  2 Simulation and methods

###  2.1 The Three Hundred project

The data set analysed in this work is extracted from The Three Hundred
project222https://the300-project.org (Cui et al., 2018a). The Three Hundred
hydrodynamical simulation consists in 324 zoom-in re-simulated regions
centered on the most massive haloes selected at $z=0$ from the MultiDark
Planck 2 (hereafter as MDPL2) simulation (Klypin et al., 2016). MDPL2 is a DM-
only cosmological simulation, consisting of a periodic cube with a side length
of $1\,h^{-1}$ Gpc with $3840^{3}$ DM particles, each with a mass of
$1.5\times 10^{9}\,h^{-1}\,M_{\sun}$. The cosmological parameters used in
MDPL2 and The Three Hundred are those reported in the 2015 Planck data release
(Planck Collaboration XIII, 2016) ($h=0.678$, $n_{s}=0.96$,
$\sigma_{8}=0.823$, $\Omega_{\Lambda}=0.693$, $\Omega_{\textrm{M}}=0.307$, and
$\Omega_{\textrm{B}}=0.048$).

The 324 spherical regions of The Three Hundred, centered on the most massive
haloes ($M_{vir}>8\times 10^{14}\,h^{-1}\,M_{\sun}$), have a radius of
$15\,h^{-1}$ Mpc. The DM particles within these regions are split into
separate dark matter and gas particles, according to the cosmological baryonic
mass fraction, with masses $m_{\textrm{DM}}=12.7\times
10^{8}\,h^{-1}\,M_{\sun}$ and $m_{\textrm{gas}}=2.36\times
10^{8}\,h^{-1}\,M_{\sun}$.

In this analysis, we use the regions re-simulated with the smoothed particle
hydrodynamics (SPH) code Gadget-X (Springel, 2005; Beck et al., 2016) \- a
modified version of the Gadget3 code - which implements radiative cooling and
star formation (Springel & Hernquist, 2003), chemical stellar evolution
(Tornatore et al., 2007), super massive black hole (BH) accretion and active
galactic nuclei (AGN) and Supernovae feedbacks (Steinborn et al., 2015). The
haloes and subhaloes within each region are identified with the publicly
available333http://popia.ft.uam.es/AHF/ Amiga Halo Finder (AHF; Knollmann &
Knebe, 2011), which includes DM, gas, stars and BHs in the density peak
finding process and estimates the coordinates and properties of the clusters,
such as their mass, radius and luminosities at various overdensities. The
simulation runs are stored in 129 snapshots, from redshift $z=17$ to $z=0$.

In this work, we analyse the galaxy clusters’ catalogues at redshift $z=0$,
$z=1.03$ and $z=2.02$, which span, at $z=0$, a mass range of $10^{13}\leq
M_{200}\,h^{-1}\,M_{\sun}\leq 2.62\times 10^{15}$.

###  2.2 DisPerSE

The cosmic web in the simulated regions is extracted using the publicly
available444https://www2.iap.fr/users/sousbie/web/html/indexd41d.html Discrete
Persistent Structure Extractor (DisPerSE; Sousbie, 2011; Sousbie et al.,
2011). DisPerSE is a topological structure finder, designed to identify multi-
scale structures. It relies on the Discrete Morse theory (Forman, 1998;
Gyulassy et al., 2008) to identify the critical points of density fields. The
noise introduced into the detection by the finite sampling of the density
field is reduced with the approach of the Persistent homology theory
(Edelsbrunner et al., 2002) and topological simplification (Edelsbrunner et
al., 2002; Gyulassy et al., 2008).

In short, from a given 2D or 3D distribution of tracers, DisPerSE provides the
position of the extreme points – where the gradient of the field is null – and
the integral lines that connect them, which form the filaments. The starting
point of DisPerSE can either be a catalogue of the tracers’ coordinates (e.g.
galaxies or haloes) or a pixelated density map, as used in this work. In the
first case, the density field is extracted from the Delaunay tessellation of
the distribution with the Delaunay Tessellation Field Estimator (DTFE; Schaap
& van de Weygaert, 2000; van de Weygaert & Schaap, 2009).

From the density field, the identified extreme points are classified as
maxima, minima, saddle and bifurcation points, i.e. where a filament splits in
two. The ridges are presented as a set of segments that connect two extreme
points – usually a maximum and a saddle point – following the density field.
The robustness of filaments, i.e. their reliability, is quantified through a
threshold parameter called persistence cut, defined as the absolute difference
of the density value of the two critical points at the filament’s extremes.
The choice of the persistence allows for low-significance and noisy structures
to be filtered out.

##  3 Cosmic web extraction

###  3.1 DisPerSE calibration

In order to extract a significant filamentary skeleton from the density
distribution, a combination of parameters – the Gaussian smoothing of the
input density map and the persistence cut of DisPerSE – needs to be tuned. The
first step of the cosmic web extraction is the smoothed gas density field
reconstructed from the regions of the hydrodynamical simulation. The Three
Hundred gas-traced cosmic web has been widely analysed in previous works,
namely to study the properties of filaments (Rost et al., 2021, 2024) and as a
benchmark for the identification of galaxy-traced filaments (Kuchner et al.,
2020, 2021). In this work, we choose to analyse the filamentary structures
traced from the gas particle distribution because we plan, in future studies,
to use 3D gas filaments and their projection as a benchmark for the detection
of filaments from mock Compton-y maps generated in The Three Hundred
simulation, as described in Cui et al. (2018a).

The gas particles of each simulated region were first binned in a 3D grid of
$30\,h^{-1}$ Mpc per side, with a pixel resolution of $150\,h^{-1}$ kpc. Then,
to avoid sharp variations from one pixel to another, we applied a Gaussian
kernel to smooth the density field. As the smoothing kernel of the map
increases, only the more distinct structures will be identified by the
filament finder, leading to a network that only features robust critical
points and filaments. For this analysis, we applied a Gaussian kernel with a
full width at half-maximum (FWHM) of 4 pixels, equivalent to $600\,h^{-1}$
kpc. We opted for this specific kernel size, ensuring it remains smaller than
the diameter of the galaxy clusters that we analysed, to preserve the
resolution of these structures, as shown in Fig. 1, where we compare the
chosen value and two larger smoothing kernels to the distribution of the
diameters of clusters at $z=0$.

In addition to the smoothing level of the density map, the robustness of the
extracted network depends also on the DisPerSE parameter persistence. The
effect of the persistence cut on the network is similar to the Gaussian
smoothing, with higher values of persistence leading to less critical point
pairs and therefore to the removal of less significant filaments. The optimal
value of persistence cut for our data set is 0.5, which identifies the
threshold between significant topological features and noisy ones as seen in
the DisPerSE Persistence Diagram Viewer (pdview), accessible with the option
-interactive of the function mse. The density maps are first analysed by
DisPerSE with the function mse, with the options -forceLoops, -periodicity
000, -cut 0.5 and -upSkl, to compute the Morse-smale complexes. During this
step, the filaments are extracted and the critical points are identified.
Then, the networks are post-treated with the function skelconv, which we
implemented with the following options: -smooth 1, -breakdown, -rmOutside and
-rmBoundary. Here, the option -breakdown merges the infinitely close pieces of
arcs and adds the bifurcations points, while the -smooth option determines the
straightness of the filament’s trace, while keeping the critical points
coordinates fixed. We note that the -smooth function does not affect the
extraction and positions of filamentary structures and, although having a
similar name, is different from the Gaussian smoothing kernel applied on the
density maps, which operates on the density field prior to the cosmic web
extraction procedure on DisPerSE.

Figure 1: Diameter $2R_{200}$ distribution of galaxy clusters at $z=0$. The
vertical lines represent the smoothing kernel FWHM at 4, 6, and 8 pixels,
respectively in yellow (solid), orange (dot-dashed) and red (dashed).

###  3.2 Simulated cosmic web network

In the DisPerSE output networks, the galaxy groups and clusters are defined
only by the coordinates of the maxima (the nodes), while the filaments are
defined only by the position of their spines. To associate the extracted
skeleton to the density distribution of the simulation, one needs to match the
nodes to The Three Hundred galaxy clusters, identified with AHF. While there
are several ways to associate the nodes to clusters (for a complete study we
refer the reader to Cohn, 2022), in this analysis we require the distance
between the coordinates of the node and the cluster to be less than the
smoothing kernel of $4$ pixels. Where multiple pairs that satisfy this
condition, we match the DisPerSE node to the nearest AHF cluster.

To evaluate the accuracy of the DisPerSE catalogue and of the matches, we can
compute the completeness of the sample, defined as the number of groups and
clusters matched to nodes, normalized by the total number of groups, as in
Cornwell et al. (2024). The completeness depends on the smoothing scale of the
density maps, with a larger and less filtering smoothing kernel resulting in
lower completeness of the sample, as shown in Fig. 2. Here, we display the
completeness of the sample as a function of the Gaussian smoothing FWHM,
averaged over the 324 simulated regions. To test only the effect of the
smoothing scale on the completeness of the sample, we use the same persistence
cut for all the smoothing values, which alters the skeleton in a similar way
of the smoothing. This result agrees with Cohn (2022) and Galárraga-Espinosa
et al. (2024), where they studied the effect of the smoothing on the
completeness respectively on skeletons extracted from dark matter and galaxies
distributions.

The sample analysed in this work has an average completeness value of $89\%$,
indicating that almost every group or cluster identified in the simulated
regions has a matched DisPerSE node. It is worth noticing that since the
completeness is lower than the ideal value of $100\%$, some clusters remain
unmatched by any DisPerSE nodes. This discrepancy might be attributed to their
coordinates’ distance exceeding the threshold we set. This phenomenon could be
explained by considering that DisPerSE node coordinates are determined solely
from the gas-only density distribution, unlike cluster coordinates, which also
incorporate data from dark matter particles. Therefore, the different spatial
distributions of dark matter and gas particles near the centre of the regions
could affect the peak density finding process (Cui et al., 2018b; Lokken et
al., 2023). Conversely, there are DisPerSE nodes without any corresponding
clusters. This population of nodes likely represents local maximum points of
the density field not classified as haloes by AHF or nodes associated to sub-
haloes. To avoid possible contamination by low-resolution DM particles near
the borders, we consider only the matched nodes inside a sphere of
$13\,h^{-1}$ Mpc radius from the centre of each region. Lastly, we note that
the galaxy groups and clusters analysed in this work - except the 324 most
massive ones - reside within the $15\,h^{-1}$ Mpc environment of very massive
galaxy clusters ($M_{vir}>8\times 10^{14}\,h^{-1}\,M_{\odot}$).

The final sample used for this analysis, including all the 324 simulated
regions, amounts to 3412 galaxy groups and clusters and $12\,643$ filaments at
$z=0$. The filament sample includes filaments going from a maximum point to
another – with a saddle point in the middle –, filaments from one maximum
point to a bifurcation point and filaments with two bifurcation points at the
extremes. The filament length distribution, computed as the sum of the lengths
of all segments belonging to each filament, is shown in Fig. 3, along with the
median values of the distribution. We plot the length distribution of the
whole sample in black, in red the length distribution of maximum-maximum
filaments and in grey the length of bifurcation-maximum and bifurcation-
bifurcation filaments. The mean length of the maximum-maximum filaments is
$6.67\,h^{-1}$ Mpc, comparable to the length of filaments extracted in Rost et
al. (2021) from the same gas particle distribution but with different levels
of smoothing of the density map and persistence. Due to the exploration of the
limited volume of each region, compared to different studies in literature,
the extracted filaments predominantly belong to the short filaments
population, as shown in previous studies (Bond et al., 2010a; Galárraga-
Espinosa et al., 2020).

An example of a simulated region at $z=0$ is shown in Fig. 4(a). Here, we
display the whole simulated skeleton in grey, while the critical points are
represented in the following way: the nodes as red stars, the saddle points as
blue dots and the bifurcation points as yellow triangles. Moreover, we plot
the $R_{200}$ spheres of the matched central cluster and less massive groups
in the region. A zoom-in of the central cluster and an example of bifurcation
points inside the $R_{200}$ sphere are displayed in the bottom panels of the
figure, respectively in Figs. 4(b) and 4(c).

Figure 2: Completeness of the sample. Each dot refers to the completeness of
the sample, averaged over the 324 simulated regions, extracted with different
smoothing kernels . The error bars represent the standard deviation error.
Figure 3: Filaments length distribution. In the black solid line we plot the
distribution of length for the whole sample of filaments. In the red dashed
line, the length distribution of the filaments with two maxima at the extremes
and in the grey dot-dashed line the length of bifurcation-maximum and
bifurcation-bifurcation filaments. The vertical lines are the median values of
each distribution.

(a) Filaments in region 1 at $z=0$

(b) Zoom-in of a central cluster

(c) Example of bifurcation points inside the cluster

Figure 4: Examples of DisPerSE simulated networks. The top panel (4(a)) shows
the full filament network of region 1 in grey. The nodes are shown as red
stars, the saddle points as blue dots and the bifurcation points as yellow
triangles. The light blue spheres are the $R_{200}$ of matched clusters. The
bottom left panel (4(b)) is a zoom-in of the central cluster of region 6. The
bottom right panel (4(c)) is an example of two bifurcation points inside the
$R_{200}$ sphere of the central cluster of region 2.

##  4 Results

To investigate the dependence of cosmic web filaments on the properties of
galaxy clusters, we estimate the connectivity of clusters, denoted with k. The
connectivity of a galaxy cluster, as defined by Codis et al. (2018), is the
number of filaments globally connected with the cluster. More specifically,
the connectivity is the number of filaments connected to the galaxy cluster at
a specific aperture, or in other words, the number of filaments intersecting
the surface of a sphere with a certain radius $R_{\Delta}$555The subscript
$\Delta$ represents the overdensity, i.e. value of the ratio between the
density of the cluster inside that radius and the critical density of the
Universe $\rho_{c}=3H^{2}/(8\pi G)$ at the cluster’s redshift..

For the complete data set, we estimate the connectivity of groups and clusters
at $R_{500}$, $R_{200}$ and $R_{vir}$, denoted respectively as $k_{500}$,
$k_{200}$ and $k_{vir}$, where the subscript $vir$ refers to the virial
radius666The virial radius refers to the aperture where the halo satisfies the
conditions of the virial theorem.. With this definition, the connectivity
value takes into account filaments that start from a node or a bifurcation
point within the aperture considered. An example of this phenomenon is
displayed in the bottom right panel of Fig 4(c), where the cluster has a
connectivity $k_{200}$ equal to 7, considering the filaments originating from
the bifurcation points inside $R_{200}$. We observe that, for the sake of
redundancy, we omit from the connectivity symbol the subscripts denoting the
cosmic web tracer and dimensionality, which in our case should be $gas$ and
$3D$, although these aspects should always be specified when considering the
cosmic web identification.

###  4.1 Connectivity and galaxy cluster mass

The number of filaments connected to a galaxy cluster – the connectivity – is
expected to scale with the cluster’s mass. As filaments feed the cluster
centre with dark matter, gas and galaxies, more connected clusters are on
average more massive.

In The Three Hundred data set, we estimate the connectivity for 3412 groups
and clusters at $z=0$, with a total mass range of $10^{13}\leq
M_{200}\,h^{-1}\,M_{\sun}\leq 2.62\times 10^{15}$. Our results of the
connectivity as a function of the cluster’s mass $M_{200}$ at $z=0$ are
presented in Fig. 5. We find the expected correlation between the mass and k
at all the apertures explored – $R_{500}$, $R_{200}$ and $R_{vir}$ – whose
mean values and errors on the mean, computed with the bootstrap method 777For
the bootstrap re-sampling, we draw 1000 new samples with replacement, i.e. the
same data point can be selected several times, each sample with the same
length as the original sample. The error is the standard deviation of the new
distribution., are plotted respectively in red, pink and blue. We performed
linear fittings, plotted as dashed lines in Fig. 5 and whose parameters are
shown in Table 1, as the following: $\log k_{\Delta}=A\cdot\log M_{200}+B$.

Table 1: The fitting parameters for the $\log k_{\Delta}-\log M_{200}$ relation Aperture | A | B
---|---|---
$R_{500}$ | 0.295 $\pm$ 0.016 |  $-3.69$ $\pm$ 0.24
$R_{200}$ | 0.308 $\pm$ 0.019 |  $-3.86$ $\pm$ 0.27
$R_{vir}$ | 0.324 $\pm$ 0.021 |  $-4.09$ $\pm$ 0.30

We observe a flattening in the growing trend among low-massive clusters,
specifically in the range $13\leq\log[M_{200}/h^{-1}\,M_{\sun}]\leq 14.5$, not
present for the massive and central clusters. This phenomenon is likely
attributed to the intrinsic limitation on the minimum connectivity value,
introduced by our sample construction, which can not be less than one. The
clusters for which we compute the connectivity are, in fact, matched to
DisPerSE critical nodes, which are, by definition, all linked to at least one
filament.

A direct computational consequence of including other critical points, besides
the node matched to the cluster, inside the aperture in the estimate of the
connectivity is that the connectivity depends on the aperture chosen, with
$k_{vir}$ always equal to or larger than $k_{200}$ and $k_{500}$. This result
is also visible in Fig. 5. We note that this effect is more evident for more
massive clusters – the clusters at the centre of the simulated boxes – which
lie in a denser environment and can have several substructures, leading to
more nodes identified by DisPerSE. Further studies are needed to investigate
other physical explanations of this phenomenon, such as the diffusion of
filaments inside the apertures considered. This result is consistent at larger
redshifts, namely for $z=1$ and $z=2$, as shown in Appendix A.

Finally, it’s worth noting that the linear fits and the findings shown in Fig.
5 are computed based on a constant mass, $M_{200}$, for both the dataset at
$R_{500}$ and $R_{Vir}$. This decision was made specifically to emphasize how
the connectivity of a fixed cluster mass varies with the aperture.
Nevertheless, it’s important to mention that the positive correlation between
the connectivity and mass would persist even if we were to consider the
general relationship between $k_{\Delta}$ \- $M_{\Delta}$.

Figure 5: The mean connectivity of The Three Hundred groups and clusters at
$z=0$ as a function of the mass $M_{200}$. The connectivity is estimated at
different over-densities, respectively $k_{500}$ in red, $k_{200}$ in pink and
$k_{vir}$ in blue. The dashed lines represent the linear fitting. The error
bars are the errors on the mean values, computed with the bootstrap method,
multiplied by 2 for visual clarity. The mean values are shifted in mass for
visual clarity.

Our measurements of the connectivity show a strong agreement with other
connectivity values from literature, over a wider mass range. This result is
shown in Fig. 6, where we plot The Three Hundred data compared to other
connectivity samples from literature, estimated from simulations and
observations. The data of Aragón-Calvo et al. (2010b), plotted as blue
triangles, are derived from a $\Lambda$ cold dark matter N-body simulation,
where a multiscale morphological filter (MMF; Aragón-Calvo et al. (2007)) was
used to extract cosmological structures as walls, clusters and filaments. The
values of Gouin et al. (2021) and Galárraga-Espinosa et al. (2024) are
estimated from hydrodynamical simulations, respectively, the IllustrisTNG and
the MilleniumTNG and are plotted as grey dots (relaxed groups) and crosses
(unrelaxed groups) and as pink triangles. The connectivity value of the
simulated Coma cluster of Malavasi et al. (2023), plotted as violet diamond,
is estimated from a dark matter-only N-body simulation. The observed samples
are the AMASCFI clusters sample, plotted as green stars, extracted from the
CFHTLS photometric redshift survey of Sarron et al. (2019) and the COSMOS2015
photometric redshift catalogue of Darragh Ford et al. (2019), plotted as
yellow squares. Lastly, the observed connectivity value of the Coma cluster in
Malavasi et al. (2020b) is computed from the cosmic web extracted from the
Main Galaxy Sample of the Sloan Digital Sky Survey (SDSS) (Malavasi et al.,
2020a) and plotted as a red diamond.

Although the connectivity values span almost the same ranges, it is important
to note the differences in how measurements were taken. This difference may
include variations in how the connectivity is defined and at which aperture is
estimated, the type of tracers used to extract the cosmic web and the filament
finder employed. In particular, the connectivity can be computed at a fixed
aperture for all the clusters, as in Aragón-Calvo et al. (2010b), where they
adopt a fixed aperture of $3\,h^{-1}$ Mpc, and in Sarron et al. (2019), with a
fixed aperture of $1.5$ cMpc, or it can be estimated at a set overdensity, as
in the case of Darragh Ford et al. (2019) and Malavasi et al. (2020b, 2023),
whose connectivity is at $1.5\,R_{vir}$ or as Gouin et al. (2021) and
Galárraga-Espinosa et al. (2024), with an aperture of $1.5\,R_{200}$.

The different web finder employed can also affect the recovery of the
filamentary structures – and therefore the estimate of the connectivity –
since they rely upon different approaches (we refer the reader to Libeskind et
al. (2018) for a review of the most common web finders). The most used
filament finder is DisPerSE, employed in this work as well as in Sarron et al.
(2019); Darragh Ford et al. (2019); Malavasi et al. (2020b, 2023) and
Galárraga-Espinosa et al. (2024), although with different parameters tuning.
The cosmic web in Gouin et al. (2021) is extracted with the graph-based
algorithm T-ReX (Bonnaire et al., 2020).

The final distinction to consider is the type of tracers used to extract the
web. In this work, the filamentary structures are traced by the gas particle
distribution, in Aragón-Calvo et al. (2007) by dark matter particles, while
the remaining studies make use of observed or simulated galaxy catalogues.

All the previous studies confirm a positive correlation between the number of
filaments connected to a cluster and its mass, thus affirming the significant
role filaments play in the accretion of matter towards the clusters. However,
it is important to take the possible biases into account, summarized in Table
2, when referring to the connectivity.

Figure 6: Connectivity of The Three Hundred clusters compared to literature
values (see the legend for details). The black dots are the mean connectivity
values $k_{200}$ of The Three Hundred groups and clusters. The error bars of
The Three Hundred points are the errors on the mean, estimated with the
bootstrap method, while the shaded grey area is the standard deviation error.
For the literature values, the error bars represent the error on the mean,
except for the data set of Aragón-Calvo et al. (2010b); Sarron et al. (2019);
Malavasi et al. (2020b) and Malavasi et al. (2023). For details and
differences of each sample see the main text and Table 2.  Table 2: A summary
of the parameters of previous literature studies compared to this work.

Sample | Data | $k_{R}$ | Mass | CW extraction | Web finder | References
---|---|---|---|---|---|---
The300 | Hydro-sim | $R_{200}$ | $M_{200}$ | 3D gas particles | DisPerSE | this work
N-body sim | DM-sim |  $3\,h^{-1}$ Mpc | $\leavevmode\nobreak\ M_{vir}$ | 3D DM particles | MMF | 1
AMASCFI | Observations |  $1.5\,h^{-1}$ Mpc | $M_{200}$ | 2D galaxies | DisPerSE | 2
COSMOS2015 | Observations | $1.5\,R_{vir}$ | $M_{200}$ | 2D galaxies | DisPerSE | 3
Coma obs. | Observations | $1.5\,R_{vir}$ | $M_{200}$ | 2D galaxies | DisPerSE | 4,5
Coma sim. | DM-sim | $1.5\,R_{vir}$ | $M_{200}$ | 2D galaxies | DisPerSE | 6
IllustrisTNG | Hydro-sim | $1.5\,R_{200}$ | $M_{200}$ | 3D galaxies | T-ReX | 7
MilleniumTNG | Hydro-sim | $1.5\,R_{200}$ | $M_{200}$ | 3D galaxies | DisPerSE | 8

(1) Aragón-Calvo et al. (2010b); (2) Sarron et al. (2019); (3) Darragh Ford et
al. (2019); (4) Malavasi et al. (2020b) (5) Malavasi et al. (2020a); (6)
Malavasi et al. (2023); (7) Gouin et al. (2021); (8) Galárraga-Espinosa et al.
(2024).

###  4.2 Connectivity evolution with redshift

According to the standard theory of anisotropic collapse (Zel’dovich, 1970),
matter in the universe is predicted to flow from less dense to denser regions
over cosmic time, progressing from walls to filaments and eventually to
clusters. As a consequence, in recent times the universe is predominantly
composed of large under-dense regions – the voids – in terms of volume, while
clusters and filaments dominate in terms of mass. Moreover, it has been shown
(see e.g. Cautun et al. (2014)) that, at high redshift, the filaments of the
cosmic web are dominated by thin structures, which are then prevailed by more
significant ones at later times. In this framework, we investigate the
evolution of cluster connectivity with redshift, which is expected to decrease
with cosmic time, as shown in Codis et al. (2018) and Galárraga-Espinosa et
al. (2024).

To detect the cosmic web skeletons at higher redshifts, we create the density
maps from the gas particle distributions at redshift $z=1.03$ and $z=2.02$,
with the same parameters used for the $z=0$ sample, with a pixel resolution of
$150\,h^{-1}$ kpc and a Gaussian smoothing with a FWHM of 4 pixels. We use the
same DisPerSE functions and persistence cut of $0.5$, in order to extract
filamentary structures with the same robustness and significance. The two
final samples have a completeness value of $82\%$ and $83\%$, respectively at
redshift $z=1$ and $z=2$, indicating that almost every cluster is matched by a
node, similar to the sample at $z=0$.

We compare the connectivity mean values estimated at $R_{200}$ of clusters in
the mass range of $10^{13}\leq M_{200}\,h^{-1}\,M_{\odot}\leq 10^{14}$,
populated at all the redshifts explored. In Fig. 7 we plot the connectivity
values at $z=0$ in red, $z=1$ in purple and $z=2$ in yellow. We recover the
expected trend of the connectivity decrease from high redshift to present
times, in agreement with what has been found both in theoretical predictions
from 2D and 3D estimates (Codis et al., 2018) and, more recently, in
hydrodynamical simulations (Galárraga-Espinosa et al., 2024). We observe that,
for masses below $2.5\times 10^{13}\,h^{-1}\,M_{\odot}$, the connectivity at
$z=0$ shows a flattening trend and remains comparable to the values of higher
redshifts. A possible reason for this behaviour can be found in the intrinsic
minimum connectivity value introduced by the sample construction, as explained
in detail in Subsect. 4.1.

The decrease of connectivity with cosmic time is both a consequence of the
anisotropic collapse of the cosmic web structures, with two filaments that
merge into one, or a result of their stretch and disconnection caused by the
cosmological expansion of voids regions in later times due to the dark energy.
However, further research is required to explore the influence of matter
inflow from walls to filaments on filamentary evolution with cosmic time. We
note that future studies of the evolution of connectivity with redshift with
different dark energy models (see e.g. Codis et al., 2018), and critical
events, such as filaments mergers (see e.g. Cadiou et al., 2020), could offer
an interesting probe for dark energy models.

Figure 7: Evolution with redshift of the connectivity-mass relation. In
yellow, the mean values of connectivity $k_{200}$ extracted from the gas
skeleton at $z=2$, in purple the values at $z=1$, in red the values at $z=0$.
The error bars represent the errors on the mean, computed with the bootstrap
method.

###  4.3 Connectivity and galaxy cluster dynamical state

To investigate if the filaments are related to the dynamical state of galaxy
clusters or if the connectivity can be used as a tracer of the dynamical
state, we study the correlation between the connectivity and the dynamical
state of each cluster. In this work, we assess the dynamical state of a
cluster following the definition of De Luca et al. (2021), based on 3D
dynamical parameters estimated from the physical properties of each particle
in the simulation. To evaluate the dynamical state in regions different from
the virial radius volumes we consider two indicators of the cluster’s
dynamical state, the total sub-halo mass fraction $f_{s}$ and the center-of-
mass offset $\Delta_{r}$. The total sub-halo mass fraction $f_{s}$ is defined
as the sum of all the sub-halo masses $M_{i}$, identified by AHF within a
specific aperture $R_{\Delta}$, normalized by the total cluster mass
$M_{\Delta}$:

$f_{s}=\frac{\sum_{i}M_{i}}{M_{\Delta}}$ (1)

The center-of-mass offset $\Delta_{r}$ is defined as:

$\Delta_{r}=\frac{|\textbf{r}_{cm}-\textbf{r}_{c}|}{R_{\Delta}}$ (2)

where $\textbf{r}_{cm}$ is the position of the centre-of-mass and
$\textbf{r}_{c}$ is the theoretical centre of the cluster, defined as the
highest density peak, which typically coincides with the brightest cluster
galaxy (BCG). The $\Delta_{r}$ parameter is used to describe the cluster’s
deviation from smoothness and spherical symmetry, as objects with high values
of $\Delta_{r}$ are considered morphologically disturbed.

Specifically, we classify the clusters as relaxed if both the conditions
$f_{s}<0.1$ and $\Delta_{r}<0.1$ are simultaneously satisfied. On the other
hand, a cluster is classified as dynamically disturbed if both $f_{s}>0.1$ and
$\Delta_{r}>0.1$ are true. The clusters whose indicators are not
simultaneously verified – therefore if $f_{s}>0.1$ and $\Delta_{r}<0.1$ or if
$f_{s}<0.1$ and $\Delta_{r}>0.1$ – are classified as hybrid.

We estimate the dynamical state of the clusters at redshift $z=0$ and compute
the degree of relaxation as in Eq. 3, considering the parameters at $R_{200}$.
In our sample, $66\%$ of the clusters are classified as relaxed, $20\%$ as
hybrid and $14\%$ as disturbed. In Fig. 8, we plot the connectivity-mass
relation, computed at $R_{200}$, for the three sub-samples of dynamical state,
respectively relaxed clusters in green, hybrid in yellow and unrelaxed
clusters in light blue. The three sub-samples overlap and are comparable to
each other, within the errors estimated with the bootstrap method, indicating
that the connectivity-mass relation is not affected by the dynamical state –
assessed in terms of the two indicators $f_{s}$ and $\Delta_{r}$ – of the
cluster. This result is not dependent on the redshift, as shown in Appendix B,
where we present this analysis at redshifts $z=1$ and $2$, nor on the
overdensity at which the connectivity and the dynamical state are estimated,
as shown in Appendix C, where we present the same analysis for $R_{500}$ and
$R_{vir}$.

Figure 8: Connectivity and mass relation for three dynamical state sub-samples
at redshift $z=0$. In green the connectivity mean values $k_{200}$ for the
dynamically relaxed clusters, in yellow for hybrid clusters, and in light blue
the values for dynamically disturbed clusters. The error bars represent the
errors of the mean, computed with the bootstrap method.

To further investigate the relation between the dynamical state of the cluster
and the filaments connected to it, we now focus on the continuous dynamical
state indicator $\chi$. The degree of relaxation combines the dynamical
parameters in a single measure of the cluster dynamical state and is defined
by Haggar et al. (2020) as:

$\chi=\left[\frac{\sum_{i}(\frac{x_{i}}{x_{0,i}})^{2}}{N}\right]^{-1/2}$ (3)

where $x_{i}$ is the general indicator of the dynamical state, $x_{0,i}$ is
the threshold between the relaxed and the disturbed regimes and $N$ is the
number of indicators used. As a result, larger values of $\chi$ (with
$\log\chi>0$) indicate a more relaxed dynamical state, while negative values
of $\log\chi$ represent a disturbed dynamical state. In literature, the
definition of the degree of relaxation can vary both in the choice of the
parameters used and in the selection of the threshold values (see e.g. Cui et
al., 2017). In this work, we estimate $\chi$ with $f_{s}$ and $\Delta_{r}$,
both considered with a threshold of $0.1$.

To possibly disentangle the impact of connectivity and mass on the dynamical
state, given their correlation, we divide the total data set in three bins of
connectivity and three bins of mass. This approach is necessary to study the
relationship between the degree of relaxation and the connectivity
independently of the mass, and vice versa.

The three connectivity bins divide the data set into $\sim$700 weakly
connected groups, with a $k_{200}<2$, $\sim$2000 average-connected clusters,
with $k_{200}\in\\{2,3\\}$, and $\sim$500 highly connected clusters with
$k_{200}>3$. These thresholds were chosen taking the mean connectivity value
of $k_{200}=2.42$ of the whole cluster population at redshift $z=0$ into
account.

In the left panel (a) of Fig. 9 we plot the mean degree of relaxation as a
function of the mass for the three connectivity sub-samples, respectively in
green the weakly-connected, in yellow the average-connected and in light blue
the highly connected clusters. At fixed mass, the three connectivity bins
overlap and are comparable to each other within the errors, confirming that
the connectivity does not influence the dynamical state of the cluster,
computed with the $f_{s}$ and $\Delta_{r}$, corroborating the result shown in
Fig. 8.

This outcome is not consistent with that of Gouin et al. (2021) who found a
correlation between the degree of relaxation and the connectivity values, with
more connected clusters being on average more dynamically unrelaxed. We note
that the disagreement could be due to the different definition of the degree
of relaxation used in Gouin et al. (2021), which differs from ours in the
dynamical indicators and thresholds used, as well as the differences in the
data set and the cosmic web extraction as illustrated in Table 2.

Focusing now on the mass bins, we divide the data set into low-mass groups,
with $M_{200}<5\times 10^{13}\,h^{-1}\,M_{\odot}$, medium groups and clusters,
with $10^{13}\leq M_{200}\,h^{-1}\,M_{\odot}<5.5\times 10^{14}$ and massive
clusters with $M_{200}\geq 5.5\times 10^{14}\,h^{-1}\,M_{\odot}$, which
correspond to the large galaxy clusters at the centre of each simulated
region. The three sub-samples are plotted in the right panel (b) of Fig. 9,
respectively in green, yellow and light blue. Contrary to the connectivity
sub-samples shown in Fig. 9(a), at fixed connectivity the three mass bins show
a distinct separation from one another. This behaviour suggests a correlation
between the degree of relaxation $\chi_{200}$ and the mass of the cluster,
independently of their connectivity value, with less massive groups appearing
to be more dynamically relaxed, and vice versa.

(a) Degree of relaxation for three connectivity bins

(b) Degree of relaxation for three mass bins

Figure 9: Dynamical state dependence on connectivity and cluster total mass at
redshift $z=0$. The left panel (a) shows the degree of relaxation $\chi_{200}$
as a function of the total mass $M_{200}$. Green, yellow and light blue dots
indicate respectively weakly connected ($k_{200}<2$), average-connected
($k_{200}\in\\{2,3\\}$) and highly connected ($k_{200}>3$) clusters. The right
panel (b) shows the degree of relaxation $\chi_{200}$ as a function of the
connectivity $k_{200}$. Green, yellow and light blue dots indicate
respectively low mass ($M_{200}<5\times 10^{13}\,h^{-1}\,M_{\odot}$), medium
($5\times 10^{13}\leq M_{200}\,h^{-1}\,M_{\odot}<5.5\times 10^{14}$) and
massive ($M_{200}\geq 5.5\times 10^{14}\,h^{-1}\,M_{\odot}$) clusters. The
shaded areas and the error bars indicate the error of the mean, computed with
the bootstrap method.

This result was also reported by Kuchner et al. (2020) – although with a
different definition of the degree of relaxation, both in the parameters and
the thresholds chosen – who suggest that the most massive clusters tend to be
in a more dynamically disturbed state. This phenomenon is likely attributed to
the fact that the most massive clusters are currently still in a growing-
process, resulting in larger centre-of-mass offsets and in the presence of
internal substructures (Power et al., 2012). Moreover, the high-density
environment in which they are situated can make them more likely to accrete
matter, resulting in less relaxed states (Kuchner et al., 2020).

###  4.4 Connectivity and mass bias

Classical approaches to infer the mass of galaxy clusters from SZ and X-rays
observations usually rely on the hydrostatic equilibrium (HE) assumption (for
a review, see Kravtsov & Borgani, 2012). This hypothesis assumes that the gas
thermal pressure is balanced by the gravitational forces, leading the cluster
in a state of equilibrium. Deviations from the HE can introduce errors on the
estimation of observable properties of clusters and their mass.

We investigate if the presence of filaments, quantified with the connectivity,
has an impact on the hydrostatic equilibrium of the galaxy clusters of The
Three Hundred simulation. For this purpose, we refer to the analysis of
Gianfagna et al. (2023), who explored the deviations of the HE mass $M_{HE}$
from the total mass $M_{true}$ of the 324 simulated central clusters,
quantified with the hydrostatic mass bias $b$. The $M_{true}$ refers to the
real mass, computed by summing all the DM, gas and stars particle masses
inside a radius. On the other hand, the hydrostatic equilibrium mass $M_{HE}$
is the mass inferred from the gas thermal pressure profile estimated from SZ
observations or from the temperature and electron density profiles from X-rays
observations, leading respectively to the mass biases $b_{SZ}$ and $b_{X}$,
defined as:

$b=\frac{M_{true}-M_{HE}}{M_{true}}$ (4)

We refer to Gianfagna et al. (2023) for the complete derivation of the
parameters. We note that a positive bias, if defined in this way, denote an
underestimation of the true mass.

We plot in Fig. 10 the median values of the hydrostatic mass biases $b_{SZ}$
and $b_{X}$, estimated at $R_{200}$ as a function of the connectivity
$k_{200}$ of the cluster. No correlation is found between the bias and the
connectivity, independently of observable property used to estimate it.
Moreover, we note that we do not observe any variation of the scatter with
connectivity, meaning that the dispersion of the bias is independent of the
number of filaments connected. On the other hand, the scatter of the
hydrostatic mass bias has been found to vary as a function of the dynamical
state, quantified with different indicators, with disturbed clusters showing a
wider dispersion (Piffaretti & Valdarnini, 2008; Rasia et al., 2012;
Ansarifard et al., 2020; Gianfagna et al., 2023). This result suggests that
the connectivity may not be used as a reliable indicator of the dynamical
state of a galaxy cluster, as already shown in Sec. 4.3. The same result is
observed for the connectivity and the bias estimated at $R_{500}$, as shown in
Appendix D.

Figure 10: Connectivity and hydrostatic mass bias relation at redshift $z=0$.
In green the mass bias median values $b_{X}$ at $R_{200}$ and in violet mass
bias median values $b_{SZ}$ at $R_{200}$. The shaded areas represent the 16th
and 84th percentiles.

##  5 Mock-galaxy filaments

In this section, we aim to investigate the effect of the chosen tracer on the
extraction of the cosmic web. In this context, we compare the connectivity
estimates from filaments extracted from gas particle distribution to those
identified from the mock-galaxy distribution in The Three Hundred regions.

The simulated cosmic web is traced by the mock-galaxies catalogue of the full
$1\,h^{-1}$ Gpc MDPL2 box at redshift $z=0$. Haloes are defined as a mock-
galaxy if they have a minimum stellar mass of $M_{*}\geq 3\times
10^{9}\,h^{-1}\,M_{\odot}$, which corresponds to haloes with $M_{halo}>3\times
10^{10}\,h^{-1}\,M_{\odot}$. The density is computed from the Delaunay
tessellation of the points with the DTFE. The tessellation is mass-weighted,
with a weight associated with each tessellation vertex corresponding to the
mass of the halo at this vertex, to improve the filament identification, as
explained in detail by Kuchner et al. (2020). The cosmic web skeleton is then
identified with DisPerSE with a persistence threshold of $6.5\sigma$ 888Here,
the persistence threshold is in terms of number of $\sigma$ of the
distribution and it is defined as the ratio – not the difference, as in the
persistence cut – of the values of the points in a critical pair. , which
provides the best agreement with the cosmic web extracted from gas particles,
as tested in Kuchner et al. (2020). A more detailed description of the
extraction procedure and in-depth analysis of filament properties from the
full MDPL2 box will be provided in a dedicated paper (Bianconi et al., in
prep).

We associate the DisPerSE nodes to simulated groups, considering as matched
those with a distance of less than 4 pixels, as explained in detail in Section
3.2. The final sample, for all the 324 regions, includes 836 galaxy groups and
clusters with a mass range of $10^{13}\leq M_{200}\,h^{-1}\,M_{\sun}\leq
2.69\times 10^{15}$. Lastly, we estimate the connectivity $k_{200}$ of the
galaxy clusters at $R_{200}$, as defined in Section 4.

The results of the connectivity $k_{200}$ estimated from mock-galaxies
filaments are plotted in Fig. 11 in red, as a function of the cluster’s mass
$M_{200}$. We plot as blue dots the connectivity values computed from gas
filaments, presented in Section 4.1. We performed the same linear fitting as
for the gas connectivity, whose parameters are shown in Table 3.

Table 3: The fitting parameters for the $\log k_{200}-\log M_{200}$ relation for gas-traced filaments and mock-galaxy-traced filaments.  Tracer | A | B
---|---|---
Mock-Galaxies | 0.291 $\pm$ 0.018 |  $-3.66$ $\pm$ 0.26
Gas | 0.308 $\pm$ 0.019 |  $-3.86$ $\pm$ 0.27
Figure 11: The mean connectivity of The Three Hundred groups and clusters at
$z=0$, estimated from gas and mock-galaxies filaments. The mean values of
connectivity $k_{200}$ from gas filaments are plotted in blue, the values of
$k_{200}$ from mock-galaxies filaments in red. The solid and dashed lines
represent the linear fitting, respectively for the gas and the mock-galaxies.
The error bars are the errors on the mean values, computed with the bootstrap
method.

The strong agreement between the two connectivities is an indication that the
number of filaments traced by gas or mock-galaxies is similar and therefore
that these are representative of the same structures. However, potential
discrepancies between the skeletons extracted from gas or DM particles and
haloes catalogues are expected, as noted in other studies (e.g. Bond et al.,
2010a, b; Laigle et al., 2018; Kuchner et al., 2020; Zakharova et al., 2023).
These discrepancies are likely to be attributed to the finer sampling of the
density field by gas particles, which outnumber the mock-galaxies, and
therefore could result in reduced noise during the cosmic web extraction
process. Moreover, employing different density field estimation methods - a
regular grid for the gas particles and the Delaunay tessellation for the mock-
galaxies - may introduce some variations in the extracted skeletons, such as a
preferred direction for the filamentary structures. The discrepancies between
the two tracers and the effects on the recovered cosmic web are investigated
more in detail by Bianconi et al. (in prep).

##  6 Conclusions

The outskirts of galaxy clusters can provide insightful information for the
study of large-scale structures. In particular, the presence of filamentary
structures, acting as pathways for matter transportation towards the centre of
clusters, and their impact on clusters’ properties can be investigated. The
aim of this study is to examine the correlation between the connectivity and
the main galaxy clusters properties.

In this work, we have used the 324 regions with a $30\,h^{-1}$ Mpc side of The
Three Hundred hydrodynamical simulation to extract the filamentary structures
around galaxy clusters and groups. We ran the DisPerSE algorithm on the gas
particle density maps at redshifts $z=0,1$ and $2$. The full procedure of
generating the density maps has been described together with the setting of
relevant parameters of DisPerSE. We have estimated the connectivity at several
apertures, in particular at $R_{500}$, $R_{200}$ and $R_{vir}$ for $\sim
10^{3}$ groups and clusters with a total mass range of $10^{13}\leq
M_{200}\,h^{-1}\,M_{\sun}\leq 2.62\times 10^{15}$ at redshift $z=0$.
The main results of this work can be summarized as follows:

  1. (i)

The connectivity, computed as the number of gas filaments crossing the sphere
at a specific aperture, quantified in terms of overdensity, depends on the
chosen radius, with larger values of connectivity when estimated at a larger
radius. This is a consequence of including filaments that start from
substructures – maxima and bifurcation points – that lie within the radius
considered. Therefore, it is important to specify the aperture chosen when
analysing the connectivity of galaxy clusters.

  2. (ii)

Filaments have an impact on galaxy cluster mass accretion, transporting matter
from the outskirts to the centre. On average, more massive galaxy clusters are
also more connected. This result is in agreement with several literature
studies, both from observed and simulated data sets.

  3. (iii)

We observe a decrease in the connectivity mean values over cosmic time, with
clusters being more connected at higher redshifts. This result is likely to be
a consequence of the anisotropic collapse of the cosmic web structures.

  4. (iv)

No dependence of connectivity with the cluster’s dynamical state is found. The
connectivity-mass correlation does not have any significant difference when
computed for three sub-samples of clusters: dynamically relaxed, hybrid and
disturbed. Moreover, we do not observe any correlation between the
connectivity and the degree of relaxation parameter $\chi$. On the other hand,
we find that, independently of the connectivity, more massive clusters are on
average more dynamically disturbed.

  5. (v)

No correlation of the hydrostatic mass bias with the redshift is found for the
324 central clusters. This result is independent of the aperture at which the
quantities are computed and the observable properties used to infer the
hydrostatic mass bias. Moreover, we do not observe a variation in the scatter
of the bias with the connectivity, contrary to what is found with the
dynamical state of clusters. This suggests that the connectivity may not be an
indicator of the dynamical state.

  6. (vi)

We observe a strong agreement between the connectivity evaluated from gas-
traced filaments and mock-galaxies-traced filaments, extracted from the mock
haloes catalogue of the full MDPL2 box. This result suggests that the gas and
mock-galaxies are distinct and complementary tracers of the same structures.

This work underlines the dependence of filaments, traced by the gas particle
distribution, on the properties of galaxy clusters to which they are connected
and shows a correlation between the connectivity and the total mass of the
clusters. This result affirms the role that filaments have in the accretion of
matter towards the centre of the cluster.

The results presented in this work and the simulated cosmic web skeletons will
be used as a reference for cosmic web extraction from Compton-y simulated
maps, to provide insightful forecasts for observations. With further
investigations on filaments evolution with cosmic time, useful information on
the study of cosmology and structure evolution can be provided. Further work
is needed to assess the differences in the cosmic web extraction process from
distinct tracers and from different density field estimation methods. Future
analysis with different hydrodynamical codes, such as the Gizmo-Simba run of
The Three Hundred (Cui et al., 2022), will be necessary to investigate the
dependence of the results on different baryonic models.

###### Acknowledgements.

This work has been made possible by The Three Hundred collaboration. The
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. As part of The Three Hundred
project, this work has received financial support from the European Union’s
Horizon 2020 Research and Innovation programme under the Marie Skłodowska-
Curie grant agreement number 734374, the LACEGAL project. S.S. acknowledge
support from Sapienza Università di Roma thanks to ”Progetti per Avvio alla
Ricerca”, n. AR123188AF0D68BF, and ”Bando Mobilità Internazionale PhD”, D.R.
n. 1505/2023. M.D.P., S.S. and A.F. acknowledge financial support from PRIN
2022 (Mass and selection biases of galaxy clusters: a multi-probe approach -
n. 20228B938N) and from Sapienza Università di Roma, thanks to Progetti di
Ricerca Medi 2022, RM1221816758ED4E. G.Y., W.C. and S.S. would like to thank
also Ministerio de Ciencia e Innovación (Spain) for financial support under
project grant PID2021-122603NB-C21. W.C. is also supported by the Atracción de
Talento Contract no. 2020-T1/TIC-19882 granted by the Comunidad de Madrid in
Spain, and the science research grants were from the China Manned Space
Project.

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##  Appendix A Connectivity and cluster overdensity at higher redshift

The connectivity depends on the aperture at which it is estimated,
independently of the redshift. We plot the mean values of $k_{500}$, $k_{200}$
and $k_{vir}$ as a function of the cluster mass at redshift $z=1$ and $z=2$,
respectively in Figs. 12 and 13.

Figure 12: The mean connectivity of The Three Hundred groups and clusters at
$z=1$ as a function of the mass $M_{200}$. The connectivity is estimated at
different over-densities, respectively $k_{500}$ in red, $k_{200}$ in pink and
$k_{vir}$ in blue. The dashed lines represent the linear fitting. The error
bars are the errors on the mean values, computed with the bootstrap method.
The mean values are shifted in mass for visual clarity. Figure 13: The mean
connectivity of The Three Hundred groups and clusters at $z=2$ as a function
of the mass $M_{200}$. The connectivity is estimated at different over-
densities, respectively $k_{500}$ in red, $k_{200}$ in pink and $k_{vir}$ in
blue. The dashed lines represent the linear fitting. The error bars are the
errors on the mean values, computed with the bootstrap method. The mean values
are shifted in mass for visual clarity.

##  Appendix B Connectivity and dynamical state at higher redshift

The connectivity and mass correlation is not dependent on the dynamical state
of the cluster, regardless of the redshift. We plot in Figs. 14 and 15 the
connectivity mean values as a function of the mass for dynamically relaxed,
hybrid and disturbed clusters, respectively at redshift $z=1$ and $z=2$.

Figure 14: Connectivity and mass relation for three dynamical state sub-
samples at redshift $z=1$. In green the connectivity mean values $k_{200}$ for
the dynamically relaxed clusters, in yellow for hybrid clusters, and in light
blue the values for dynamically disturbed clusters. The error bars represent
the errors of the mean, computed with the bootstrap method.  Figure 15:
Connectivity and mass relation for three dynamical state sub-samples at
redshift $z=2$. In green the connectivity mean values $k_{200}$ for the
dynamically relaxed clusters, in yellow for hybrid clusters, and in light blue
the values for dynamically disturbed clusters. The error bars represent the
errors of the mean, computed with the bootstrap method.

##  Appendix C Connectivity and dynamical state at different overdensities

The connectivity and mass correlation is not dependent on the dynamical state
of the cluster, independently of the overdensity at which the connectivity and
the dynamical state are estimated. We plot in Figs. 16 and 17 the connectivity
mean values as a function of the mass for dynamically relaxed, hybrid and
disturbed clusters, estimated respectively at $R_{500}$ and $R_{vir}$.

Figure 16: Connectivity and mass relation for three dynamical state sub-
samples at redshift $z=0$, estimated at $R_{500}$. In green the connectivity
mean values $k_{500}$ for the dynamically relaxed clusters, in yellow for
hybrid clusters, and in light blue the values for dynamically disturbed
clusters. The error bars represent the errors of the mean, computed with the
bootstrap method. Figure 17: Connectivity and mass relation for three
dynamical state sub-samples at redshift $z=0$, estimated at $R_{vir}$. In
green the connectivity mean values $k_{vir}$ for the dynamically relaxed
clusters, in yellow for hybrid clusters, and in light blue the values for
dynamically disturbed clusters. The error bars represent the errors of the
mean, computed with the bootstrap method.

##  Appendix D Connectivity and HE bias at $R_{500}$

We plot a in Fig. 18 the median values of the hydrostatic mass biases $b_{X}$,
in green, and $b_{SZ}$, in violet, computed at $R_{500}$ as a function of the
connectivity $k_{500}$. A correlation between the connectivity and the
hydrostatic mass bias is not observed.

Figure 18: Connectivity and hydrostatic mass bias relation at redshift $z=0$.
In green the mass bias median values $b_{X}$ estimated at $R_{500}$, in violet
the mass bias median values $b_{SZ}$ at $R_{500}$. The shaded areas represent
the 16th and 84th percentiles.