# Forecasting the success of the WEAVE Wide-Field Cluster Survey on the
extraction of the cosmic web filaments around galaxy clusters

Daniel J. Cornwell1, Ulrike Kuchner1, Alfonso Aragón-Salamanca1, Meghan E.
Gray1, Frazer R. Pearce1, J. Alfonso L. Aguerri2,3, Weiguang Cui4,5, J.
Méndez-Abreu2,3, Luis Peralta de Arriba6,7, Scott C. Trager8
1School of Physics and Astronomy, University of Nottingham, Nottingham NG7
2RD, UK 2Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, E-38205
La Laguna, Tenerife, Spain 3 Departamento de Astrofísica, Universidad de La
Laguna, E-38200 La Laguna, Tenerife, Spain 4Institute for Astronomy,
University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom
5Departamento de Física Teórica, Módulo 15, Facultad de Ciencias, Universidad
Autónoma de Madrid, 28049 Madrid, Spain 6Institute of Astronomy, University of
Cambridge, Madingley Road, Cambridge CB3 0HA, UK 7 Centro de Astrobiología
(CAB, CSIC-INTA), ESAC Campus, 28692 Villanueva de la Cañada, Madrid, Spain 8
Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV
Groningen, The Netherlands E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Next-generation wide-field spectroscopic surveys will observe the infall
regions around large numbers of galaxy clusters with high sampling rates for
the first time. Here we assess the feasibility of extracting the large-scale
cosmic web around clusters using forthcoming observations, given realistic
observational constraints. We use a sample of 324 hydrodynamic zoom-in
simulations of massive galaxy clusters from TheThreeHundred project to create
a mock-observational catalogue spanning $5R_{200}$ around 160 analogue
clusters. These analogues are matched in mass to the 16 clusters targetted by
the forthcoming WEAVE Wide-Field Cluster Survey (WWFCS). We consider the
effects of the fibre allocation algorithm on our sampling completeness and
find that we successfully allocate targets to 81.7 $\%\pm$ 1.3 of the members
in the cluster outskirts. We next test the robustness of the filament
extraction algorithm by using a metric, $D_{\text{skel}}$, which quantifies
the distance to the filament spine. We find that the median positional offset
between reference and recovered filament networks is $D_{\text{skel}}=0.13\pm
0.02$ Mpc, much smaller than the typical filament radius of $\sim$ 1 Mpc.
Cluster connectivity of the recovered network is not substantially affected.
Our findings give confidence that the WWFCS will be able to reliably trace
cosmic web filaments in the vicinity around massive clusters, forming the
basis of environmental studies into the effects of pre-processing on galaxy
evolution.

###### keywords:

large-scale structure of Universe – galaxies: clusters: general – galaxies:
haloes – techniques: spectroscopic – methods: numerical – methods: data
analysis

††pubyear: 2022††pagerange: Forecasting the success of the WEAVE Wide-Field
Cluster Survey on the extraction of the cosmic web filaments around galaxy
clusters–C

##  1 Introduction

Galaxies spend a significant amount of time in cosmic web filaments. They are
the backbone of a complex network of geometrical structures that indicate how
matter is distributed in the Universe: as filaments, sheets, walls, knots and
voids (Aragón-Calvo et al., 2010), described by the Zel’dovich formalism
(Zel’dovich, 1970). The present day cosmic web has evolved from small over-
and underdensities already existent in the early Universe, and still grows and
changes today as a consequence of gravitational collapse. Since $z\approx 2$,
about half of the mass budget of the Universe is located within filaments,
whilst only contributing to $6\%$ of its volume. By contrast, only about
$10\%$ of the mass is located in galaxy clusters (Cautun et al., 2014).

However, clusters stand out as recognizable, bright peaks in the density field
and have been the focus of observational galaxy evolution studies for decades
(Gray et al., 2009; Balogh et al., 2017). They have uncovered a now
well–supported relation that finds higher fractions of quiescent and early
type galaxies in clusters compared to outside of clusters (Morphology-Density
relation, Dressler et al. 1997). Typically, this is explained through
astrophysical effects that quench and transform galaxies as they encounter the
extremely dense intra-cluster medium of cluster cores during their infall. Ram
pressure stripping is one of several possible mechanisms quenching galaxies
infalling onto a cluster (Zinger et al., 2018; Arthur et al., 2019). However,
the majority of the gas lies outside the boundaries where the clusters are
virialized, and in the intergalactic medium within filaments (Walker & Lau,
2022; Galárraga-Espinosa et al., 2022; Gouin et al., 2022).

Galaxy clusters are therefore not isolated islands, but assemble, replenish
and grow via ongoing mergers with smaller clusters, groups and clumps of gas,
as well as through a constant flow of gas and galaxies from filaments. The
most prominent of these filaments have hot gas temperatures and dense cores
that have the possibility to strip the gas from galaxies, but also to
replenish galaxies with pre-enriched filamentary gas (Vulcani et al., 2019),
impacting their mass assembly and star formation histories in very different
ways (Laigle et al., 2017; Song et al., 2020). It is clear that the challenge
of understanding galaxy evolution must include the impact of the large-scale
geometry and flows of the cosmic web, and that galaxy transformation begins
well before the galaxies fall into the cluster ("pre-processing", Zabludoff &
Mulchaey 1998). Physical processes in the outskirts of galaxy clusters are
therefore fundamentally different from cluster cores, and thus important areas
for the study of cluster assembly and their connection to the filaments of the
cosmic web (Sarron et al., 2019; Salerno et al., 2020; Malavasi, Nicola et
al., 2022; Gouin et al., 2020). However, they are challenging to capture.

Whilst filaments can be identified by mapping the gas distribution of galaxy
clusters in simulations (Kuchner et al., 2020; Vallés-Pérez et al., 2020;
Gouin et al., 2022), galaxies tend to trace these features of the cosmic web
and can therefore be used to detect filaments observationally (Einasto et al.,
2020; Malavasi et al., 2020). To correctly identify filaments that feed
clusters, we require a large area, high sampling density and depth to cover a
sufficient number of galaxies over a broad range of masses. Large-area surveys
such as the Sloan Digital Sky Survey (SDSS, York et al. 2000), the Galaxy and
Mass Assembly survey (GAMA, Liske et al. 2015), the Two-Degree Field Galaxy
Redshift Survey (2dFGRS, Colless et al. 2001a), the VIMOS Public Extragalactic
Redshift Survey (VIPERS, Colless et al. 2001b) and the Dark Energy
Spectroscopic Instrument (DESI, Secco et al. 2022) all probe the distribution
of galaxies over large redshift ranges, providing strong observational
evidence for the presence of a cosmic web. However, they either lack
statistically significant samples of galaxy clusters, or the necessary
sampling or detail required for an investigation on pre-processing by
filaments feeding clusters. Targeted spectroscopic studies that focus on
clusters may provide the required sampling, but they are either only available
as case studies of stand-out targets such as Virgo (Castignani et al., 2021,
2022), or do not extend far enough to bridge cluster infall regions to the
large-scale cosmic web filaments (e.g., GOGREEN Balogh et al. 2017 or
OmegaWINGS Gullieuszik et al. 2015; Moretti, A. et al. 2017).

To address the need for observing programmes that combine high sampling and
statistical power, we look towards next generation wide-field, multi-object
spectroscopic (MOS) surveys of galaxy clusters as they will enable detailed
study into the far-reaching lower-density cluster outskirts. They are designed
to reveal the complex interplay between the properties of galaxies and their
position in the cosmic web filaments that feed the clusters. Examples of next
generation MOS surveys are the upcoming WEAVE Wide Field Cluster Survey
(WWFCS; Kuchner et al. in prep) and the 4MOST CHileAN Cluster galaxy Evolution
Survey (CHANCES, Haines et al. in prep). We motivate our studies with the
WWFCS, which will cover 20 low redshift (z$\sim 0.05$) galaxy clusters in a
mass range of $\log(M_{\rm cl}/M_{\odot})=13.8-15.5$ out to and beyond
$5R_{200}$. For each cluster, thousands of new spectra will be obtained with a
galaxy stellar mass limit of $10^{9}M_{\odot}$, extending our current
understanding of these systems to include the infall regions and low-mass
galaxies.

Given the challenging task of accurately mapping filaments in the vicinity of
massive clusters (both in terms of extreme contrasts of interlaced high and
low density regions, 2D projections, and complications due to the Finger of
God effect Kuchner et al. 2021a), careful preparation is required. We have
planned, tested, and fine-tuned our steps to map and characterise the infall
regions of these clusters with a large statistical sample of simulated
clusters from TheThreeHundred survey, and investigated strategies for doing so
in redshift space (Kuchner et al., 2020, 2021a, 2021b; Rost et al., 2021). To
confidently carry out the observational programme with the 1000-fiber fed MOS
WEAVE at the Wiliam Herschel Telescope (Dalton et al., 2014), we now take the
final step from simulations to fully-informative mock observations to
understand the success and limitations of identifying filaments around
clusters.

Our goal for this paper is to quantify what effect the physical constraints of
assigning fibers to targets—a necessary and important step in the design of a
MOS survey—has on filament finding. Our previous investigations assumed that
all theoretical cluster structure members are targeted and return spectra,
thus featuring in the mapping and subsequent analysis. However, in reality,
instruments only have a finite capacity to place fibers on targets, and
physical restrictions imposed by the geometry and size of the instrumental
components require us to make decisions that will ultimately influence the
success of finding filaments. In addition, limitations of a realistic target
selection may lead to losing valuable fibers to background galaxies. Fiber
collisions in dense regions like groups and substructures in the outskirts, as
well as decisions on which galaxies should receive higher priority than
others, directly link to the input for filament identification algorithms and
thus could impact the analysis of pre-processing in infall regions.

In this paper, we therefore close the circle of comparing simulations to
observations—from a theoretical 3D volume to a fully configured 2D projection.
We design a framework for determining the feasibility of reliably
characterizing the large scale structure from galaxies that can be observed in
current wide-field cluster surveys, using concrete constraints that are
matched to the WEAVE instrument and the WWFCS.

The paper is structured in the following way: Section 2 describes the data we
have used. This includes the spectroscopic survey inspiring this paper and the
numerical simulations used to create mock observations. Section 3 reports our
generation of mock observations and a summary on how we extract the cosmic
web. Section 4 displays the results and discussion, we describe the accuracy
in which we trace the large-scale structure surrounding our simulated clusters
and explain the importance of our results in probing the success of next-
generation spectroscopic surveys. We present our conclusions on the likely
success of filament retrieval from WWFCS in Section 5.

##  2 Galaxy cluster information

The framework presented in this paper is designed with the WWFCS in mind, but
should also work for similar surveys making the adjustments required by the
specific observational strategy and constraints. We will use simulation
outputs in tandem with algorithms underpinning the observational processes of
next generation wide-field cluster surveys. This section describes the planned
observations and simulated data relevant to this work.

###  2.1 The WEAVE Wide-Field Cluster Survey

WEAVE (William Herschel Telescope Enhanced Area Velocity Explorer) is a next
generation MOS. The spectrograph makes use of $\sim 1000$ individual fibres
deployable over a 2-degree-diameter field-of-view. The instrument also
includes 20 small deployable integral field units (mini-IFUs), as well as one
large IFU. In this paper we are only concerned with the (MOS) observing mode.
Low and high spectral resolution observing modes are available. Further
details on the instrument can be found in Balcells et al. 2010; Dalton et al.
2014; Dalton 2016 and Jin et al. (in prep).

A number of surveys will be carried out with WEAVE in the next few years, one
of which is the WEAVE Wide-field Cluster Survey (WWFCS; Kuchner et al. in
prep). The WWFCS will utilize the 1000 fibre-fed MOS to study the infall
regions of galaxy clusters in unprecedented detail. The WWFCS will observe up
to 20 clusters at low redshift ($0.04<z<0.07$) and will return spectra for
thousands of cluster members for each cluster, out to several virial radii.
The sample consists of galaxy clusters previously observed in the WINGS
(Fasano, G. et al., 2006) and OmegaWINGS (Moretti, A. et al., 2017) surveys.
The WINGS sample covers a wide range of cluster masses ($\sigma=500-1200\text{
kms}^{-1}$; $\log{L_{\text{X}}}=43.3-45\text{ ergs}^{-1}$; virial masses
$\log(M_{\rm cl}/M_{\odot})=13.8-15.5$). The WWFCS selected clusters have
velocity dispersions and X-ray luminosities that are statistically
indistinguishable from the parent sample and are therefore, unbiased in terms
of their mass distribution (Kuchner et al. in prep). The WWFCS will use the
low spectral resolution ($R\sim 5000$) mode and obtain optical spectra in the
366 nm $<\lambda<$ 959 nm range. These spectra will yield accurate redshifts,
velocity dispersions as well as quantitative information on the star formation
histories of the different galaxy populations.

The WWFCS observing strategy is illustrated by Figure 1. We show a simulated
galaxy cluster (cf. Section 2.2) overlaid with WEAVE 2-degree diameter MOS
fields (white circles). The inner yellow dashed circle corresponds to the
cluster’s $R_{200}$, the radius at which the density is equal to two hundred
times the critical density of the Universe. The outer yellow dot-dashed circle
corresponds to $5R_{200}$. Note the large over- and under-dense regions
reaching far out from the very dense cluster core. The large field of view we
will be able to cover with WEAVE will allow us to explore and map these
environments – including filaments – in great detail, reaching much larger
cluster-centric distances than hitherto possible (beyond $5R_{200}$), and also
study the properties of the galaxies that inhabit them.

Figure 1: Example simulated cluster from TheThreeHundred with a similar mass
and redshift to the cluster Abell 602, one of the WWFCS targets. The projected
dark matter density distribution is shown, derived using a Kernel Density
Estimation (KDE) with a $500\,$kpc smoothing scale. This box has a depth of 10
Mpc. The green dots indicate the positions of galaxy-mass halos. Each white
circle encloses a WEAVE field with a 2 degree diameter. The central yellow
dashed circle corresponds to $R_{200}$ and the larger dot-dashed yellow circle
to $5R_{200}$.

###  2.2 TheThreeHundred galaxy cluster simulations

TheThreeHundred111https://the300-project.org/ project (Cui et al., 2018) is a
set of zoom-in resimulations of the Multidark Dark Matter only (MDPL2)
cosmological simulations (Klypin et al., 2016). MDPL2 is a periodic cube of
comoving length $1\,h^{-1}\,$Gpc containing $3840^{3}$ dark matter particles,
each with mass $1.5\times 10^{9}\,h^{-1}{M_{\odot}}$. MDPL2 uses Planck
cosmology
($\Omega_{\text{M}}=0.307,\Omega_{\text{B}}=0.048,\Omega_{\Lambda}=0.693,h=0.678,\sigma_{8}=0.823,n_{s}=0.96$).

This simulation suite extracts the 324 most massive individual regions
($M_{\text{vir}}>8\times 10^{14}h^{-1}M_{\odot}$), follows them back to their
initial conditions and resimulates the hydrodynamics of the volume surrounding
a 15 $h^{-1}$ Mpc radius sphere enclosing the cluster and its environment at a
higher resolution. Outside of this high resolution region are a set of
consecutive shells, hosting lower mass resolution particles that reproduce the
tidal fields of the large-scale structure at a reduced computational cost. The
highest resolution dark matter particles are divided into dark matter and gas,
following the cosmological baryonic fraction using the Planck 2015 cosmology:
${\Omega_{\text{b}}}/{\Omega_{\text{M}}}\approx 0.16$. This gives a combined
mass resolution of $m_{\text{DM}}+m_{\text{gas}}=1.5\times
10^{9}h^{-1}M_{\odot}$. There are 128 individual time snapshots for all 324
zoomed-in Lagrangian regions, ranging from $z=17$ to $z=0$. Throughout this
work, we use the last snapshot at $z=0$ given the low redshift of the WWFCS
sample.

These zoom-in re-simulations have been run with the smoothed-particle
hydrodynamics codes (SPH): Gadget-MUSIC (Sembolini et al., 2012), Gadget-X
(Beck et al., 2015; Rasia et al., 2015) and mesh-less code GIZMO-SIMBA (Davé
et al., 2019; Cui et al., 2022). We only focus on Gadget-X which incorporates
full-physics galaxy formation, star formation and feedback from both SNe and
AGN. The work in this paper utilizes the AMIGA Halo Finder (Gill et al., 2004;
Knebe et al., 2011) to determine the halo properties.

TheThreeHundred simulations provide a useful testbed to develop the
observational strategy and forecast the performance of the WWFCS. First, the
large volume of the parent dark-matter simulation (MDPL2) ensures a high
number of massive clusters are available for statistical purposes. Second, the
high-resolution re-simulations reach out as far as $15h^{-1}\,$Mpc from each
cluster centre, comparable to the area that the WEAVE observations will cover.
The extensive information available from the cluster centre all the way to
beyond $5R_{200}$ allows us to study all the environments present – from
individual galaxy halos to filaments, groups, and the cluster core.

This dataset has already been used previously to generate theoretical
expectations with the WWFCS in mind. For instance, Kuchner et al. (2021a)
quantified the impact of redshift space distortions (’Fingers of God’) on the
identification of cosmic filaments. They found that trying to correct for this
effect statistically in the virialized regions of clusters and groups does not
lead to a more reliable extraction of the ‘true’ filamentary networks. For
this reason, Kuchner et al. (2021a) forecast that the identification of the
cosmic web in the regions surrounding massive clusters using spectroscopic
surveys should rely primarily on the 2D positions of the galaxies on the sky.
However, they also point out that accurate spectroscopic redshifts are crucial
in defining and isolating the cluster volume from which these galaxies should
be selected.

##  3 Generation of mock observations

This section describes the framework we have developed to create mock
observations and their optimization using the simulated clusters. We also
discuss how we use WEAVE’s fibre allocation algorithm to generate realistic
WWFCS-like ‘simul-observed’ galaxy samples from the simulations, and the
method for identifying the cosmic web using these samples. In other words, we
describe the steps we take to go from simulations to observations.

###  3.1 Optimizing the WWFCS field positions

It is important to optimize the observational strategy of upcoming wide-field
spectroscopic surveys to improve the reliability of the filament extraction
process while maximising the observational efficiency. The WWFCS can place MOS
fibres on targets over a 2-degree-diameter field (Figure 1). In order to map
the filamentary structures that feed clusters it is necessary to maximise the
spatial coverage, reaching out to and beyond $5R_{200}$. Such radial coverage
is a good compromise between the available observing time and the need to
cover as far as possible into the infall regions around the clusters (Kuchner
et al. , in prep.). We therefore need to design a tiling strategy to cover a
circular region around the clusters that reaches $5R_{200}$ using the minimum
number of WEAVE fields (or pointings). The tiling strategy we have used to
find the optimal position of the WEAVE pointings for each WWFCS cluster is
described in detail in Appendix A (see examples in Figures 1, 3, and 4). The
‘simul-observations’ described below follow the same strategy. We note that by
applying this optimisation process we have been able to reduce the required
number of pointings (and thus the required observing time) by $\sim 15\%$ from
our initial estimate, allowing us to increase the number of clusters we will
be able to observe in the available time from $\sim 16$ to $\sim 17$–$19$
without compromising the accuracy of our filament mapping.

###  3.2 Deriving WWFCS cluster properties

In order to develop mock-observations from the simulations, we need to
determine the properties of the clusters selected for the WWFCS. We firstly
calculate $R_{200}$ and $M_{200}$ (the radius and mass where the density is
200 times the critical density of the Universe) of the clusters using their
spectroscopic redshift $z$ and velocity dispersion $\sigma$ from the WINGS
survey (Moretti, A. et al., 2017) using the following equation (Poggianti et
al., 2010; Finn et al., 2005):

$R_{200}=\frac{1.73\sigma}{(\Omega_{\Lambda}+\Omega_{\text{M}}(1+z)^{3})^{1/2}}.$
(1)

The cluster mass inside $R_{200}$ ($M_{200}$) can then be estimated using this
value and the critical density of the universe. The complete list of the WWFCS
targets and their properties can be found in Kuchner et al. (in prep.). The
bottom panel of Figure 2 shows the mass distributions of the WWFCS cluster
sample (blue) and those from TheThreeHundred simulations (orange). The mass
distribution of the simulated clusters is skewed towards higher masses than
those of the clusters selected for the WWFCS. This is to be expected since
TheThreeHundred resimulates the most massive haloes in a large cosmological
volume, making it possible to find the rarest objects. By contrast, the
clusters selected for the WWFCS are more representative of clusters at low
redshift and deliberately span a large range in X-ray luminosity (Section
2.1). We address this mismatch in next section.

###  3.3 Generating the simulated cluster and galaxy samples

We discuss now the generation of the clusters and galaxies that will be
included in our mock observations using the halo data from TheThreeHundred.
The main aim is to create mock sample analogues to the ones we expect from the
WWFCS.

First, we impose some quality constraints on the cluster halos in the
simulations so that we only select the highest quality data. We confine our
study to the high resolution region of the cluster re-simulation, a spherical
region with a radius of $15\,h^{-1}\,$Mpc centered on the cluster centre. We
then require that the mass fraction in high resolution particles for the zoom
simulation needs to be greater than 0.99 ($fM_{\text{hires}}>0.99$). This
criterion rejects low-resolution dark matter particles that may have travelled
inwards into the high resolution region during the simulation of the clusters
evolution. Finally, we only accept halos with a mass greater than the
simulation resolution ($3\times 10^{10}M_{\odot}$), corresponding to 20 dark
matter particles, as explained in Kuchner et al. 2020.

####  3.3.1 Creating mass-matched cluster samples

In order to make a meaningful comparison between the clusters from
TheThreeHundred simulations and the WWFCS clusters, we need to create a sample
of simulated clusters whose masses match those of the observational sample.
Past studies have shown that for a flat Universe, on scales large enough to
neglect baryonic physics, dark matter halos evolve self similarly (Kaiser,
1986; Mostoghiu et al., 2018). Self-similarity implies that the dark matter
distribution (and hence the location of dark-matter halos) in less massive
galaxy clusters is well represented by that of more massive clusters that have
been scaled down appropriately taking into account their mass ratio.

To have reasonable statistics, our goal is to create a mass-matched sample of
simulated WWFCS cluster analogues containing 10 simulated clusters for each
WWFCS cluster. Because dark matter halos evolve self similarly on large
scales, we are able to do so using the large sample of simulated clusters from
TheThreeHundred project.

We describe in detail the methodology behind the mass-scaling of the simulated
clusters in Appendix B. In short, when the mass of a WWFCS cluster is too
small to be able to find similar mass clusters in TheThreeHundred simulations,
we use a mass scaling factor $M_{\text{F}}$ to reduce the mass of all the
simulated clusters to approximately match the mass of the WWFCS cluster. The
mass of each individual dark matter halo in the relevant cluster simulation is
also scaled down by the same factor, and included in the halo sample if its
mass is above a mass threshold. These mass thresholds are chosen to ensure
that the number of galaxy-mass halos in each simulated cluster approximately
matches that expected in the the observed cluster. For each WWFCS cluster, ten
analogue clusters with similar masses are randomly selected from the scaled
clusters.

Figure 2 demonstrates that three values of the mass-scaling factor
$M_{\text{F}}$ (1, 2 and 5) are sufficient to provide enough mass-matched
simulated clusters. Of course, $M_{\text{F}}=1$ implies no scaling is applied.
The bottom panel shows the mass distribution of the WWFCS clusters (black)
together with that of the un-scaled and scaled TheThreeHundred clusters. The
orange, green and red histograms represent the mass distributions of all 324
simulated clusters with $M_{{F}}=1,2,$ and $5$ respectively.

Figure 2: Bottom panel: normalized cluster count as a function of cluster mass
showing the mass distribution of the WWFCS clusters (solid blue line) and
TheThreeHundred un-scaled clusters ($M_{\text{F}}=1$, solid orange line), and
scaled by $M_{\text{F}}=2$ (green dashed line) and $M_{\text{F}}=5$ (red dot-
dashed line). Three scaling values ($M_{\text{F}}=1,2,$ and $5$) are
sufficient to fully cover the mass range of the WWFCS cluster sample. Top
panel: the number of galaxy halos contained within $5R_{200,\text{scaled}}$
for each simulated cluster as a function of cluster mass is displayed by the
small dots and the coloured density distribution. Red crosses correspond to
the mass-matched simulated clusters (ten per WWFCS cluster). This gives an
indication of the approximate number of galaxies that can be ‘observed’ in
each cluster simul-observation.

####  3.3.2 Scaling cluster properties

Having built a mass-matched sample of simulated clusters, we now describe how
the spatial properties of the clusters and subhaloes (linear and angular size,
halo positions) are likewise scaled.

The radius of the mass-scaled clusters $R_{200,\text{scaled}}$ can be derived
from the relationship between $R_{200}$ and $M_{200}$, adapted from Poggianti
et al. 2010,

$R_{200}^{3}=\frac{M_{200}}{K\times
h^{2}(\Omega_{\Lambda}+(1+z)^{3}\Omega_{0})},$ (2)

where $K=2.32\times 10^{14}\,M_{\odot}$Mpc-3, making
$M_{200}=M_{200,\text{scaled}}$. The spatial coordinates $x$, $y$ and $z$ of
all the halos in the cluster (and thus their clustercentric distances) are
therefore multiplied by a factor
$R_{200,\text{scaled}}/R_{200,\text{unscaled}}$.

For each WWFCS cluster the angular diameter distance is calculated using their
redshifts (Moretti, A. et al., 2017) with the adopted cosmology. Next, for
each of the corresponding ten analogue simulated clusters, we convert 3D
positions to angular distances between the haloes and the centre of their
clusters, and thus their relative positions on the simulated sky.

The top panel of Figure 2 displays the number of galaxy halos contained within
$5R_{200,\text{scaled}}$ from their cluster centers as a function of cluster
mass (small dots and density distribution). The red crosses correspond to the
clusters in the mass-matched sample (ten per WWFCS cluster). Note that the
planned WWFCS observations will generally cover well beyond $\sim 5R_{200}$
(cf. Kuchner et al., in prep.; see also Figure 1 and Appendix A), and
therefore the number of potential targets for each cluster shown in the figure
is a conservative lower limit.

####  3.3.3 Defining galaxy sample and properties

To bring our mock ‘observational’ sample closer to the real observations, each
of the WWFCS clusters’ mass-matched simulated analogues are placed at the
appropriate redshift and sky location. We then allocate WEAVE pointings using
the field positions determined in Appendix A. Only halos covered by these
pointings will be considered as possible spectroscopic targets.

Galaxy-size dark-matter halos in each simulated cluster are then given in-
fibre magnitudes in the SDSS $r$-band (similar to the ones that will be used
the observational target selection) following a simple procedure that ensures
the target galaxies have comparable numbers and magnitude ranges to the
planned observations. The actual galaxy magnitudes have no impact on the
findings of this paper, but the fibre allocation program Configure (Terrett et
al., 2014) requires them as input. Explicitly, the total $r$-band magnitude of
a galaxy is estimated from the mass of the simulated halo using the equation

$r_{\text{total}}=W-2.5\log_{10}({M_{\text{halo}}/M_{\odot}}),$ (3)

where $W$ is a constant that is calculated by mapping the least massive halos
in each simulated cluster (Appendix B) to the planned $r$-band limit of the
WWFCS spectroscopic observation ($r_{\text{total}}<19.75$, corresponding to an
approximate galaxy stellar mass limit of $\sim 10^{9}\,M_{\odot}$, Kuchner et
al., in prep). An average offset between total and in-fibre magnitudes of
$1.75\,$mag, estimated through a least-square fit to the appropriate SDSS
magnitudes, is then applied. The in-fibre magnitude limit of the WWFCS galaxy
sample is therefore $r_{\text{fibre}}<21.50$, which sets the planned exposure
times of $\sim 1\,$hour. This exposure time is expected to yield reasonable
signal-to-noise ($S/N>5$ per Å, for all the spectra up to this magnitude
limit), and we therefore expect close to 100% redshift completeness for the
observed (and thus mock) galaxies (see Kuchner et al. in prep.).We use a
simple procedure to allocate magnitudes to the galaxy-sized dark matter halos
here, as the results in this paper only require accurate spatial distributions
of mock galaxies and their expected number densities. Our simple approach
ensures this without relying on uncertain model galaxy properties. As Cui et
al. (2018) show (see, e.g., their Figure 8), large uncertainties still remain
in the model observed magnitudes and colours, and the results depend strongly
on the specific baryonic model used, particularly at low galaxy masses. While
the simulations have appropriate resolution to yield reliable masses and
locations for the dark matter haloes, the additional step of predicting
observable properties through the available hydrodynamic or semi-analytic
models would require making uncertain choices which are not necessary for our
purposes.

At this point we have created a set of 160 simulated galaxy clusters (10 per
WWFCS target cluster), populated them with mock galaxies, placed them at the
appropriate redshift and sky position, and covered them with WEAVE pointings
exactly as those planned in the observations.

###  3.4 Allocating spectroscopic fibres to mock galaxies using Configure

An integral part of creating mock observations is to carry out a realistic
allocation of spectroscopic fibres to the mock galaxies since, this process
can potentially distort and limit the spatial information that can be derived
from the real observations. Geometric and mechanical constraints (such as
fibre collisions and overlap) mean that it is not possible to assign fibres to
all the galaxies on a pointing.

Optimising fibre allocation is not a trivial task, and sophisticated software
is generally used to reduce costly human intervention. Configure is the
program that WEAVE will use to find an optimal set of assignments of fibres to
positions on the sky (Terrett et al., 2014). Each field (or pointing) will
contain not only science targets, but also a set of calibration objects and
guide stars. Configure uses a probabilistic technique named ‘simulated
annealing’ (Kirkpatrick S., 1983) to simulate the thermal motion of a system
to be optimized. The ‘energy’ of each fibre with a target assigned to it is
given by $(1.0+s)/p$, where $s$ is a measure of how straight the fibre is and
$p$, the target priority, is an integer value between 0 and 10 that is used to
prioritize objects on the fields. In our case, we assign a maximum priority of
10 to all of the cluster mock galaxy members, and lower values to other
targets (Table 1). The algorithm then optimises the fibre allocation by
finding the configuration with the lowest ‘energy’ by swapping the position of
fibres until the minimum is found.

This process determines the galaxies that will be allocated a fibre and
therefore decides which galaxies will have spectroscopic information. It may
play a crucial role in determining the accuracy in extracting cosmic-web
information from the WWFCS observations, and its effect will be thoroughly
tested in this paper.

Figure 3: A demonstration of the process of ‘configuration’, the allocation of
spectroscopic fibres to targets, on a simulated cluster mass-matched to WWFCS
cluster RX0058 ($M\sim 4.3\times 10^{14}M_{\odot},R_{200}\sim 1.54\,$Mpc), an
average mass WWFCS cluster. Left panel: mock ‘observation’ of the simulated
cluster containing 8 2-degree diameter WEAVE fields. Blue points show the
positions on the sky of the simulated galaxies. Middle panel: zoom-in on the
central field from the plot on the left, with the simulated target galaxies
before fibre configuration shown as red points. There are 667 simulated
cluster members that are potentially observable in this field. Right panel:
the same central field after configuration, where 567 cluster members have
been assigned a fibre (blue dots), while galaxies without a fibre are shown in
red. Figure 4: Results for the full configuration process of one simulated
analogue to cluster RX0058. We plan to observe this cluster using 8 WEAVE
individual pointings (black circles). The central pointing will be observed
twice given the high density of targets in the cluster core. For this cluster,
$74\%$ of its simulated galaxies have at least one fibre allocated and $69\%$
background objects also have allocated fibres. Top left: all cluster members
(blue dots) that could be assigned a fibre. Top middle-left: galaxies with one
fibre assigned (green dots). Top middle-right: galaxies with two fibres
assigned (red dots). Top right: combined sample of target galaxies with one or
two fibres (purple dots) Bottom left: galaxies with no fibre allocated (grey
crosses). Bottom middle-left: background objects allocated one fibre (orange
dots) and blank sky positions assigned one fibre (salmon dots). Bottom right:
fractional target completeness (i.e., fraction of cluster galaxy targets with
one or two fibres; see Section 4.1) in clustercentric distance bins. $R_{200}$
and the radius of the central WEAVE field are plotted for reference. The bin
width is 1/6 of of the radius of one WEAVE pointing (i.e., 1/6 of one degree).

To make a more realistic analogue to the planned observations, before running
Configure we pollute our target catalogues with background objects. We
randomly place 1400 background objects on each WEAVE field, corresponding to a
number density $\sim 450\,$deg-2. This is somewhat larger than the galaxy
number density corresponding to an in-fibre magnitude limit
$r_{\text{fibre}}<21.5$, the planned WWFCS limit, but over-populating the
background is not a problem because these objects will be assigned a low
priority of $1$ (see Table 1). We note that the background objects are not
designed to be representative of the larger-scale cosmic web, but are
implemented to test the usage of free fibers (i.e., fibers that haven’t been
assigned to cluster members according to their photometric redshifts). This is
especially relevant further away from the cluster center as fibers are then
free to be placed on background targets if they are brighter than the
magnitude limit. In this paper, we stress-test this assumption by putting a
slightly exaggerated number of background targets in the catalog to compete
with higher priority cluster targets. If their spectroscopic redshifts from
the WWFCS reject them from cluster membership, they will not feature in the
filament finding, as described in Section 3.5.1. For this paper, we assume
that the vast majority of our WEAVE target selection—based on magnitude,
colour and precise photometric redshifts—correctly rejects galaxies that lie
outside the volume in space that corresponds to TheThreeHundered volume. This
is supported by our tests using available observational data of the cluster
centers. While we will only know the exact number after analysing early WEAVE
observations, we can expect that not every background galaxy will be
identified correctly. In practice, we will therefore use a range of priorities
for background objects. However, given the high quality of the J-PLUS
photometric redshifts (Cenarro et al., 2019), in combination with conservative
colour and magnitude cuts, we expect the percentage of interlopers to be small
and that the majority of background objects can be accurately de-prioritised.
We keep them in the target catalog for the sole reason that unallocated fibres
can be used and they do not feature in our filament finding, as described in
Section 3.5.1.

Finally, the input catalogues fed to Configure contains also blank sky
positions for sky-subtraction purposes. These correspond to real celestial
positions devoid of objects visible in SDSS images of the target clusters.
They are given a priority of 1. Although in all cases we have more suitable
sky positions, Configure is set to allocate a maximum of 50 sky fibres per
field, as per the observational strategy described in Kuchner et al.

Therefore, each WEAVE field target list consists of $N_{\text{gal}}$ cluster
galaxies, determined by the simulated cluster galaxy sample (see Section 3.3.3
and Figure 2), $N_{\text{sky}}$ sky positions, and
$N_{\text{back}}=1400-N_{\text{gal}}-N_{\text{sky}}$ background targets. As
mentioned above, the exact number of background objects does not matter, and
we limit $N_{\text{back}}$ in this way to keep the size of the target
catalogues small enough to keep the Configure running time manageable.

To fully ‘configure’ a cluster’s mock-observation we need to apply the
Configure software to each individual WEAVE field (or pointing) sequentially,
taking into account that these pointing overlap (Figures 3 and 4) and that the
central pointing will be observed twice in order to deal with the high density
of targets in the cluster core (Kuchner et al., in prep.). The aim of the
process is to maximise the number of target galaxies with at least one fibre
allocated. Maximizing the number of galaxies with measured spectroscopic
redshifts, particularly in the cluster outskirts and infall regions, is a key
goal of the observational strategy that will enable a more accurate mapping of
the cosmic web.

Each field intersects a minimum of 3 other fields, meaning that target objects
in the overlap region have multiple chances of having a fibre allocated. To
obtain information on data quality and repeatability, it is desirable to have
some repeated observations, but we do not want these to have a significant
impact on the final sample of galaxies with redshifts. We therefore aim at no
more than $\sim 10$–$20\%$ of the cluster galaxies to be observed twice and we
chose to not artificially select an upper limit on the number of galaxies
observed twice as we want to minimize the number of empty fibers. We thus
allow target galaxies to have at most two fibres allocated (in separate
pointings), but sky positions and background galaxies are only allocated one
fibre at most. This process is controlled by the Configure targeting
priorities (Table 1)222In this exercise we do not include the flux calibration
targets and guide stars since given their small numbers they have a negligible
effect in our results..

For all simulated clusters we sequentially apply Configure to each WEAVE field
(see Figure 4). We start with the central one, which we configure twice, and
then continue with the outer fields. After each step we update the priorities
for all objects in the target list taking into account whether an object
(cluster galaxy, background galaxy or sky position) has been allocated a fibre
in a previous iteration. If a cluster galaxy has already been allocated one
fibre, its priority is reduced to 1. If it has already been allocated 2
fibres, its priority goes to 0. Background galaxies and sky positions with
fibres allocated previously get also a priority of 0 (Table 1). The process is
illustrated with one example for the cluster RX0058 in Figures 3 and 4.
Obviously we are not able to allocate a fibre to each target cluster galaxy,
In the typical cluster shown in these figures, $74\%$ of the cluster galaxies
have at least one fibre assigned, with little radial variation beyond the
radius of the inner WEAVE field. The success rate there is higher despite the
higher density because this field is observed twice. Beyond that, no strong
spatial biases are apparent, but we will analyse quantitatively the effect the
configuration process has on our ability to map the large-scale structure and
filaments around the clusters next.

Table 1: Target priorities used in Configure.  Object type | Target priority $p$
---|---
Cluster galaxy | 10
Background galaxy | 1
Sky position | 1
Cluster galaxy with one fibre already allocated | 1
Background galaxy with one fibre already allocated | 0
Sky position with one fibre already allocated | 0

###  3.5 Cosmic web extraction method

The rationale of this paper is to assess the ability of upcoming spectroscopic
surveys such as the WWFCS to accurately map and characterise the cosmic web.
We describe in this section the techniques we use for that purpose.

####  3.5.1 DisPerSE

To map the large-scale structure around clusters and extract filaments, we
utilize the topological structures extractor DisPerSE (Sousbie, 2011; Sousbie
et al., 2011), which uses the concept of Morse theory (Stein et al., 1963)
applied to matter distributions. It identifies structures, such as nodes,
walls, knots, and voids as components of the Morse-Smale complex – the set of
all ascending manifolds of the input function – and is able to classify
regions using critical points and integral lines (nulls of the density field
and tangents to the critical points). DisPerSE identifies topologically
significant regions in the Delaunay-2D/3D tessellation density field by taking
a set of discrete 2D or 3D positions (positions of haloes, mock galaxies, gas
particles, etc.). It computes a set of segments representing the skeleton of
the filamentary network.

In what follows, DisPerSE is run in 2D on the sky positions of the simulated
cluster galaxies to mimic the observations since, uncertainties in the radial
position of the galaxies due to peculiar velocities mean than filament
extraction in 2D is preferable when redshifts (and not true distances) are
available (Kuchner et al., 2020, 2021a).

Figure 5: Example filament network extracted with different persistence
thresholds. Top row: $\text{persistence}=2\sigma$. Bottom row:
$\text{persistence}=3\sigma$. All plots correspond to the same cluster. Left
column: KDE smoothed matter density distribution with the extracted filament
network overlaid. Right column: filament network only.

DisPerSE analyses the topology of the pre-defined region and finds the saddle
points that join together the main nodes set by a pre-defined ‘persistence’
threshold. The theory of persistence allows a user to account for uncertainty
and Poisson noise in data sets (in terms of $\sigma$) and is analogous to
signal-to-noise ratio for observations. When extracting filaments with
DisPerSE, setting a higher persistence threshold returns only the most robust,
topologically significant, large scale structure. Lower persistence values
enable the detection of smaller tendrils. Therefore, there is a trade-off
between the number of filaments that are extracted and their astrophysical
significance. Figure 5 demonstrates the implementation of different
persistence thresholds. Both filamentary networks shown are derived by
DisPerSE with the same input galaxy positions but using two different
persistence values, $2\sigma$ and $3\sigma$. It is evident that the least
robust filament segments are not present in the network associated with higher
persistence. Therefore, persistence strongly dictates the level of structure
that is extracted, and it is important the its value is optimised for the
scientific application intended. In the analysis that follows we will use a
persistence of $2.5\sigma$ for the simulated reference network and $2.1\sigma$
for the network obtained from the analogue observations (after Configure is
applied). We use different persistence values as the underlying density field
will change upon target selection for each cluster. These choices are
justified in Appendix C. There is also a smoothing parameter that is input
into the Disperse runs that influences the rigidity of the identified filament
paths. We chose a smoothing parameter of 5, as used in Kuchner et al. (2020),
although using values between 1 and 5 has virtually no impact our results.

####  3.5.2 Dealing with boundary conditions

The complex 2D geometrical shape defined by the positions of the WEAVE
pointings that will tile each cluster and its environment (Appendix A)
influences the features that are detected by DisPerSE, particularly near the
boundaries. If the shape of the field tiling is not properly accounted for,
artificial nodes are detected that trace the outer boundary of the sky region
covered by the WEAVE fields. To avoid that, we populate the region outside the
boundaries of the area covered by the planned WEAVE pontings with a random
uniform distribution of artificial galaxies that will act as ‘guard particles’
(cf. Sousbie, 2011) to prevent the appearance of these artificial nodes and
their associated filaments. The surface density of the artificial galaxies is
chosen to be similar to that of cluster galaxies in the outer regions. In
practice, the number density of galaxies that lie beyond $2R_{200}$ is
computed for each cluster, and a random uniform distribution of ‘guard
particles’ with this number density is added outside the outer boundary of the
‘observed’ fields, reaching $7.5R_{200}$. After testing different values for
this radius, we find that the recovered networks are very similar when one
increases the guard particle boundary beyond $7.5R_{200}$. This is
sufficiently far away from the cluster centre to prevent the true filamentary
network being distorted by the irregular boundaries.

The positions of cluster members and ’guard particles’ are fed into DisPerSE
and, once the filament network is computed by the filament finder, we truncate
the network outside of the ‘observed’ region, keeping only the filament
segments inside. This procedure works remarkably well, and visual inspection
indicates that spurious nodes and filaments associated with the boundaries are
eliminated without affecting the filament network inside the observed fields.

##  4 Results and discussion

With all the necessary elements in place, in what follows we will compare the
filamentary networks that are extracted using DisPerSE before and after
applying the MOS ‘fibre configuration’ process. In other words, we will
quantify the difference between the filaments extracted when all simulated
galaxies are fed into DisPerSE with those we obtained if we only use the
‘mock-observed’ galaxy sample, where some galaxies are lost due to fibre-
positioning constraints. This will allow us to forecast the impact that
realistic observational constraint will have on the information we can derive
about the filamentary networks around clusters from spectroscopic survey like
the WWFCS.

###  4.1 Recovery of cluster galaxies after Configure

Physical constraints from the fiber positioner imply that we will never have a
$100\%$ completeness of cluster galaxies. Some galaxies won’t be targeted as
they will appear too close together as well as the constraint of fiber
overlap. Therefore, the first test to quantify the success of the WEAVE-like
MOS fibre configuration is to estimate the fraction of simulated cluster
galaxies with at least one fibre assigned. A high fraction – particularly
outside the cluster core – will help us achieve our science goals. The overall
average fraction of galaxies covered by the WEAVE pointings with at least one
fibre allocated (overall target completeness) is $72.7\%\pm 1.7\%$, where the
errors denote the scatter of the values for the 160 simulated clusters. If we
restrict our calculation to the regions outside the central pointing, which is
dominated by the cluster core, the corresponding fraction (outer target
completeness) is $81.7\%\pm 1.3\%$. We argue that this value is more relevant
than the overall one when dealing with the characterisation of the filament
network since the whole cluster core will behave just as a single node
(Kuchner et al., 2020, 2021a).

We have checked whether the fraction of galaxies selected by configure depends
on galaxy mass. If we divide the galaxy sample at the median mass into two
equal subsamples, we find that the fraction of high-mass galaxies that are
“configured” is $\sim 77$%, while the corresponding fraction of low-mass
galaxies is $\sim 69$%. This is due to the fact that the central regions of
the cluster, which contain a higher fraction of massive galaxies – high-mass
galaxies cluster more strongly than low-mass ones – are observed twice. Beyond
$\sim 2R_{200}$ the fractions are approximately equal.

We find that, whilst the overall target completeness stays relatively constant
as a function of cluster mass, the outer target completeness decreases
slightly for higher cluster masses. This not surprising since the more massive
clusters will have larger cores and a higher surface density of galaxies at
all radii. In any case, the sample size reduction induced by the observational
constraints seems moderate at all radii.

Note also that the very high number of WEAVE fibres will allow us to observe
thousands of background objects per cluster (Figure 4), providing a thorough
test of the accuracy of our photometric redshifts and the quantification of
any the possible biases their inclusion in the target selection may introduce
(see Kuchner et al., in prep.).

###  4.2 Filament network comparison metrics

We have found that the completeness rates we find are encouragingly high,
suggesting that the sample size statistics will not be very severely impacted
by the observational constraints. We now need to check whether this sample
reduction introduces any biases or changes in the properties of the recovered
filamentary networks.

####  4.2.1 Skeleton distance

A useful metric designed to quantify the accuracy of filament extraction is
the ‘skeleton distance’ $D_{\text{skel}}$ (Laigle et al., 2017; Malavasi et
al., 2016; Sarron et al., 2019). After running the cosmic web extractor
software (DisPerSE in our case), we obtain a series of segments that delineate
the cosmic filamentary structure. When comparing two different networks
derived in the same region of space, $D_{\text{skel}}$ measures the distance
between the start of a segment in the reference network and the nearest one in
the other network. The segments that are found are much smaller than the
length of a typical filament, allowing us to use the position of the start of
a segment as a proxy of a segments position. This is illustrated in Figure 6.
The left panel shows two filamentary networks, the reference one derived from
the full simulated cluster galaxy sample in red, and the ‘configured’ network
recovered from the mock-observed galaxy sample in green. The middle panel
shows an enlarged version of the pink-boxed region of the left panel, where
the differences between the red and green networks are largest, showing the
individual segments. The right panel illustrates how we calculate cluster
connectivity and is discussed in 4.2.2. The $D_{\text{skel}}$ values are
calculated for each segment in the reference network by finding the distance
to the nearest segment in the ‘configured’ network. Note that the calculation
can also be done in the opposite direction, starting from the segments in the
‘configured’ network instead, and the distribution of $D_{\text{skel}}$ values
will not be necessarily the same (see below). In both cases, the distribution
of $D_{\text{skel}}$ values quantifies how well both filament networks match
each other.

Figure 7 shows the reference network (left panel) and the ‘configured’ network
(middle panel) for one of the simulated clusters mass-matched to one of the
WWFCS target clusters, RX0058. The right panel shows the normalised
probability distribution function for $D_{\text{skel}}$, calculated going from
the reference network to the ‘configured’ one (R2C, in green) and vice-versa
(C2R, in red). The median values are indicated. Both medians are much smaller
than the typical radius of filaments ($\sim 1\,$Mpc; Kuchner et al. 2020). A
large proportion of $D_{\text{skel}}>$ 1 Mpc would indicate that a filament in
this cluster has no counterpart in the corresponding mock-observational
cluster. Note that, typically, the median $D_{\text{skel,R2C}}$ is smaller
than the median $D_{\text{skel,C2R}}$ because there are generally more
segments in the reference network than in the ‘configured’ one, and thus the
likelihood of finding a nearer corresponding segment is higher in the R2C
direction. If both networks are very similar, both $D_{\text{skel}}$ median
values will not only be very small, but also very similar to each other.
Therefore, the median values of $D_{\text{skel}}$ and their ratio can be used
to quantify the accuracy of filament network reconstruction and also to derive
the optimal parameters used by DisPerSE (see Appendix C).

Another measure of the similarity between the reference and ‘configured’
filament networks is provided by the fraction of $D_{\text{skel}}$ values that
are larger than $\sim 1\,$Mpc (the typical radius of filaments). The right
panel of Figure 7 shows that this fraction is also reassuringly small ($\sim
10\%$) in both cases.

Figure 6: Illustration of the methods used to calculate $D_{\text{skel}}$ and
cosmic connectivity. Left panel: the reference filament network (red) and
‘configured’ network (green) are plotted on top of the KDE-smoothed halo
density distribution of a simulated cluster, an analogue to WWFCS cluster
RX0058. A filament is the amalgamation of many discrete segments, as clearly
seen in the middle panel. Middle panel: zoom-in on the region shown by the
pink box on the left panel, where the two networks show large differences to
demonstrate how $D_{\text{skel}}$ is calculated (see Section 4.2.1). Right
panel: zoom-in on the cluster core (yellow box in the left panel), only
plotting the reference network for illustrative purposes. The circle
corresponds to $R_{200}$ and is used to calculate the connectivity as the
number of filaments that stem from the main node and cross the $R_{200}$
circle, (see Section 4.2.2). For this cluster, the connectivity has a value of
three. Figure 7: Illustration of the recovery of the filament network around a
simulated analogue of the RX0058 cluster, the same cluster as Figure 6. Left
panel: KDE-smoothed density distribution of the simulated cluster galaxies
with reference filament network in red. Middle panel: as the left panel, but
showing the density distribution and filament network (green line) recovered
from the ‘configured’ (mock-observed) galaxy sample. Right panel:
$D_{\text{skel}}$ distribution function obtained going from the reference
network to the ‘configured’ network (R2C) and vice-versa (C2R), as described
in the text. The thick vertical line represents the medians of each
distribution, while the shaded yellow region correspond to the 25th and 75th
percentiles. The dot-dashed black line represents the typical radius of a
filament ($\sim 1\,$Mpc). The values of the medians and the percentage of
segments with $D_{\text{skel}}>1\,$Mpc are shown in the legend. We normalized
the PDF’s to have a common peak value.

####  4.2.2 Cluster connectivity

Another useful parameter to quantify the accuracy of the filament network
derived from the mock observations is the cluster connectivity $C$. We define
connectivity as the number of filaments that stem from the main node (cluster
core) and terminate beyond $R_{200}$ away from the cluster centre. This
definition is slightly different from that of Laigle et al. (2017), where the
authors use the cluster virial radius instead of $R_{200}$. The last panel in
Figure 6 gives an example of how $C$ is calculated – there are three filaments
stemming from the main node of the network (cluster core) that cross the
circle with $R_{200}$ radius, resulting in a cluster connectivity of three. A
weak positive correlation between cluster connectivity and cluster mass has
been reported in the literature (Sarron et al., 2019; Gouin et al., 2021;
Darragh Ford et al., 2019; Kraljic et al., 2019), albeit with considerable
scatter. Our simulated clusters show a similar correlation. If the recovered
filament network is similar to the reference one, their connectivity should be
the same. Therefore, comparing network connectivities will also allow us to
assess the accuracy of the recovered filaments.

###  4.3 Quantifying the quality of the recovered filament networks

Figure 8: $D_{\text{skel}}$ distributions for all the simulated WWFCS cluster
analogues. Each panel shows the individual cluster comparison (thin lines) and
the average for the 10 simulated cluster mass-matched to each WWFCS cluster
(thick lines). The format of each panel follows that of the right-hand panel
of Figure 7. There is little variation in $D_{\text{skel}}$ over different
WWFCS analogue clusters and the $D_{\text{skel}}$ median is always much less
than a typical filament radius of 1Mpc. Figure 9: Stacked probability density
function (top panel) and cumulative density function (lower panel) of the
$D_{\text{skel}}$ distributions corresponding to all the simulated clusters
shown in Figure 8. The positional difference in the networks is minimal – our
mock observations of filaments around galaxy clusters successfully recreate
the ’true’ simulated filament network.

We are now in a position to use the $D_{\text{skel}}$ and connectivity metrics
to assess quantitatively the impact of the WWFCS observational strategy and
constraints on the recovery of the filament networks surrounding galaxy
clusters.

As described above, Figure 7 illustrates the filament network comparison
process for a single simulated cluster, mass-matched to RX0058. Visually,
there is remarkable similarity in the reference and ‘configured’ filament
networks. For this particular cluster, $71.8\%$ of the simulated cluster
galaxies have at least one MOS fibre allocated, which is close to the average
for the whole sample. The median values of $D_{\text{skel}}$ are $0.12\,$Mpc
going in the R2C direction and $0.16\,$Mpc going in the C2R direction (cf.
Section 4.2.1). These values are much smaller than $\sim 1\,$Mpc, the typical
radius of filaments. Moreover, only $8\%$ and $10\%$ of the filamentary
segments lie at a distances greater than $\sim 1\,$Mpc.

The cluster connectivity of the reference network is 3, while one of the
filaments in the ‘configured’ network bifurcates inside $R_{200}$, increasing
the connectivity to 4. Changes in connectivity of $\pm 1$ are not uncommon,
indicating that the recovery is not perfect. However, larger changes in
connectivity are rare (see below).

These results, if replicated for the whole cluster sample, are very
encouraging, suggesting that the data provided by the WWFCS will allow the
reliable recovery of the filamentary structures around clusters since the
impact of the observational constraints will be moderate.

Figure 8 confirms that the $D_{\text{skel}}$ results shown for the RX0058
analogue are indeed typical of the whole sample. We can therefore stack the
$D_{\text{skel}}$ distributions for the 160 simulated clusters (Figure 9) and
derive representative average quantities for the whole sample. On average, the
median values of $D_{\text{skel,R2C}}$ and $D_{\text{skel,C2R}}$ are $0.13\pm
0.02\,$Mpc. The values are not only reassuringly small, but also almost
exactly the same when going in both directions, strongly suggesting the
compared filament networks are very similar. Furthermore, typically only
$11$–$13\%$ of the corresponding filamentary segments are more that $1\,$Mpc
away from each other.

Figure 10: Comparison between the cluster connectivity $C$ of the reference
and ‘configured’ filament networks. The 1-to-1 line is shown in black. Each
point corresponds to the mean connectivity of the 10 mass-matched analogues of
each WWFCS cluster, and the error bars show the $\pm 1\sigma$ scatter.

Figure 10 shows a comparison between the connectivity of the reference
filament networks $C_{\text{reference}}$ and that of the ‘configured’ ones
$C_{\text{configured}}$. The mean values corresponding to each WWFCS match
very well, with no significant bias, and the small scatter indicates changes
in $C$ are generally kept within the $\pm 1$ range.

In summary, the quantitative tests we have carried out for the whole cluster
sample confirm our initial assessment that the impact of the observational
constraints imposed by the WWFCS on the recovery of the filament networks
around galaxy clusters will be very moderate.

##  5 Summary and Conclusions

The outskirts and infall regions of galaxy clusters act as the points of
contact linking the large-scale structure of the Universe to the highly dense
virialized cores of the clusters themselves, containing some of the key
environments affecting galaxy evolution. Since next generation spectroscopic
surveys such as the Weave Wide Field Cluster Survey (WWFCS333Although in this
paper we have focused on the WWFCS, our methodology could easily be adapted
and applied to other wide-field spectroscopic surveys.) will explore and map
in detail these complex regions, in this paper we forecast how successful such
surveys will be at identifying the filaments that link together the ‘nodes’ in
the large-scale structure – clusters and groups – and channel galaxies into
them.

We aim at quantifying the impact the observational limitations will have on
our ability to detect the filamentary structures that feed the clusters in the
WWFCS. To achieve that aim we have used a large sample of simulated massive
galaxy clusters from TheThreeHundred project (Cui et al., 2018) and created a
set of simulated cluster galaxy samples closely matching the selection and
observational constraints imposed by the WWFCS (Kuchner et al., in prep.). For
each one of the 16 WWFCS target clusters we have extracted 10 mass-matched
analogue clusters from the simulations and built mock-observed galaxy samples
reaching beyond $\sim 5R_{200}$, where cosmic filaments trace and connect
galaxy clusters to the cosmic web. We summarise our main results below.

  1. 1.

We have then followed closely the strategy, selection, and observational
constraints of the WWFCS. Applying the same MOS fibre configuration tool that
the WEAVE spectrograph will use, we find that, on average, we are able to
allocate fibres to $72.7\%\pm 1.7\%$ of all the target galaxies. More
importantly, outside the cluster core – in the outer regions that are crucial
for filament identification – the success rate increases to $81.7\%\pm 1.3\%$.
The number of cluster galaxies that are targeted ranges from 1284 – 4062. The
high completeness that the WEAVE observations will allow, together with the
large field coverage, are key to the success of the survey.

  2. 2.

In each of the simulated cluster regions we have used the filament finder
DisPerse (Sousbie, 2011) to trace the cosmic-web filament skeleton before and
after the observational constraints (including MOS fibre positioning) are
imposed on the galaxy samples. We then compared quantitatively the resulting
filament networks and find that we are able to recover the original network
remarkably well. Specifically, we find that the median distance between
corresponding filament segments $D_{\text{skel}}$ in the reference and
recovered networks is only $0.13\pm 0.02\,$Mpc on average, an much smaller
than the typical filament radius of $\sim 1\,$Mpc. Furthermore, only $\sim
11-13\%$ of all recovered filament segments lie at a distance larger than
$1\,$Mpc away from their corresponding reference segment.

  3. 3.

As a further test on the integrity of the recovered filament networks we
computed their connectivity, the number of filaments that stem from the
cluster core and terminate beyond $R_{200}$ away from the cluster centre. We
find that the connectivities of the reference and recovered networks match
very well, without any significant bias, indicating that their global
properties are also recovered well.

These findings make us confident that the WWFCS will be able to reliably trace
cosmic-web filaments in the vicinity of massive galaxy clusters. The next
step, when we start receiving data from WEAVE, will be to identify the
galaxies that belong to these filaments, and compare their properties (e.g.,
mass, metallicity, star formation, stellar populations) to those of galaxies
inhabiting other environments such as groups, the clusters cores, and the
general field. With the combination of a statistical sample of clusters
together with high target sampling rate, the WWFCS will provide a detailed
look at the influence of all environments in the cluster infall regions on
galaxy evolution.

## Acknowledgements

Funding for the WEAVE facility has been provided by UKRI STFC, the University
of Oxford, NOVA, NWO, Instituto de Astrofísica de Canarias (IAC), the Isaac
Newton Group partners (STFC, NWO, and Spain, led by the IAC), INAF, CNRS-INSU,
the Observatoire de Paris, Région Île-de-France, CONCYT through INAOE, Konkoly
Observatory of the Hungarian Academy of Sciences, Max-Planck-Institut für
Astronomie (MPIA Heidelberg), Lund University, the Leibniz Institute for
Astrophysics Potsdam (AIP), the Swedish Research Council, the European
Commission, and the University of Pennsylvania. The WEAVE Survey Consortium
consists of the ING, its three partners, represented by UKRI STFC, NWO, and
the IAC, NOVA, INAF, GEPI, INAOE, and individual WEAVE Participants. Please
see the relevant footnotes for the WEAVE
website444https://ingconfluence.ing.iac.es/confluence//display/WEAV/The+WEAVE+Project
and for the full list of granting agencies and grants supporting
WEAVE555https://ingconfluence.ing.iac.es/confluence/display/WEAV/WEAVE+Acknowledgements.

For the purpose of open access, the authors have applied a creative commons
attribution (CC BY) to any journal-accepted manuscript.

This work has been made possible by TheThreeHundred collaboration, which
benefits from financial support of the European Union’s Horizon 2020 Research
and Innovation programme under the Marie Skłodowskaw-Curie grant agreement
number 734374, i.e. the LACEGAL project. TheThreeHundred 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.

DC acknowledges support from STFC through a studentship. He thanks Kenneth
Duncan, Sarah Hughes and Gavin Dalton for helping setup and manipulating the
software Configure. He also thanks Benedetta Vulcani and Daniela Bettoni for
useful feedback on the contents of the paper. He thanks Tomas Hough for
providing the SAG galaxy property files. UK acknowledges support from the
Science and Technology Facilities Council through grant number RA27PN. 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
further acknowledges the science research grants from the China Manned Space
Project with NO. CMS-CSST-2021-A01 and CMS-CSST-2021-B01. LPdA acknowledges
financial support from grant PGC2018-094671-B-I00 funded by
MCIN/AEI/10.13039/501100011033 and by the European Union
NextGenerationEU/PRTR. The authors contributed to this paper in the following
ways: DC, UK, AAS, MEG and FRP formed the core team. DC analysed the data,
produced the plots and wrote the paper along with UK, AAS and MEG.

## Data Availability

Data available on request to TheThreeHundred collaboration:
https://www.the300-project.org.

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##  Appendix A Process of optimizing the WWFCS field positions

As mentioned in section 3.1, the WWFCS performs observations by arranging 2
degree fields into a mosaic pattern, covering the cluster core, infall region
and outskirts (Figure 1). To optimize the observational strategy, we aim to
design the field positions in a way that maximizes the cluster coverage.
Firstly, we place two fields at the core of each cluster, the region of the
highest number density. This is so we can maximise the number of targeted
cluster members over the total field of view. To optimize the tiling for each
cluster, we adopt the following regime, such that if

$\frac{\text{Area within }5R_{200}\text{ for N - 1 fields}}{\text{Area within
}5R_{200}\text{ for N fields}}>97\%,$ (4)

Figure 11: An example field layout of cluster A2572a before this work (top)
and after this work (bottom). The red circles are individual 2-degree diameter
WEAVE fields whilst the black outer circle represents the angular diameter
corresponding to $5R_{200}$ of this cluster. The numbers displayed are the
total number of fields required to cover this cluster $N$, the cluster
redshift and $R_{200}$ taken from (Moretti, A. et al., 2017). Figure 12: WEAVE
field pattern for each WWFCS cluster. The caption in each panel states the
name of the cluster, $5R_{200}$ in degrees and the number of fields $N$ used
for each observation.

then we can remove one field, (we use 97% to ensure that we are still covering
a significant area within $5R_{200}$). We iterate through this process by
removing fields in the outer region of the cluster, manually inspecting each
time one is removed, until the 97% threshold is exceeded. Starting from a
’naive’ geometric tiling pattern (illustrated in Figure 11, top), the total
number of WEAVE fields required to cover the 16 clusters was 155, adding up to
$147\,250$ fibre hours. Using the new optimised tiling method (illustrated in
Figure 11, bottom) the total number of fiber hours is reduced to $130\,390$.
For the example shown in Figure 11, even though we have removed four fields,
we are still covering out to $5R_{200}$. Of the clusters, 12 out of 16 have
full coverage out to $5R_{200}$, whilst overall we have lost a total of
$0.06\%$ area within $5R_{200}$.

##  Appendix B Galaxy cluster scaling

This sections details our method for scaling down the mass of the clusters
from TheThreeHundred to match the WWFCS selected clusters, as mentioned in
Section 3.3.1.

Figure 13: Cumulative sum of individual halo masses of simulated galaxies for
one galaxy cluster in TheThreeHundred that has been modified using the three
$M_{F}$. All haloes follow the criterion set for selecting ’high–quality data’
from the simulations, as described in Section 3.3. Different colours represent
different scale factors that have been applied to the simulated catalogues.
The vertical dotted lines represent different mass resolutions.

Firstly, we arbitrarily chose three mass-scaling factors: $M_{F}=1,2$ and 5,
which the simulated cluster mass is divided by. We chose the most-massive halo
in the corresponding cluster catalogue to act as a proxy for the cluster.
Also, we increase our cluster catalogue sample size by a factor of 3 by
including each 2D-plane, (xy, xz, yz). The resulting mass distributions of the
clusters that have been scaled down by $M_{F}$ are in Figure 2, where we have
demonstrated that by choosing these mass factors, we have covered the entire
WEAVE mass range.

For each scaling factor $M_{F}$, we divide the mass range into 20 mass bins.To
create a statistically significant sample, we draw from these bins with the
aim of identifying 10 mass-matched simulated analogue clusters for each of the
16 WEAVE clusters. Our total sample of analogue clusters is thus 160. The
presence of companion clusters within $5<R<15$ Mpc of the WWFCS clusters (the
radius of the simulation volume) is unknown, therefore we do not exclude
analogue clusters with secondary clusters within this distance. Within the
sample of 160 analogue clusters there are six such configurations.

In scaling the mass of the clusters from the simulations to match WEAVE, we
have to also individually scale the mass of all the associated halos for each
cluster. Figure 13 displays the cumulative number of simulated halos that lie
above a mass interval for one cluster. We see that for higher mass scaling
factors ($M_{F}$) we are shifting the masses of all the haloes associated with
the cluster to lower values.

We are limited in our recovery of haloes by two thresholds: The ’scaled’
simulation mass resolution and the observational mass resolution. In the case
where we don’t scale the simulations, ($M_{F}=1$), the mass resolution limit
is that of the dark matter particles in the simulations (Kuchner et al.,
2021a), given by the dotted grey line in Figure 4. For $M_{F}>1$, we reduce
the simulation mass resolution by dividing it by the mass scaling factor
$M_{F}$. Whilst we can change the simulated mass resolution threshold, there
is a hard limit on the observational mass limit. For $F>3$, the mass
resolution stays at the observational limit $M_{\text{obs}}=10\times
10^{10}M_{\odot}$, (corresponding to the r band limit used for WEAVE:
$r_{\text{total}}<19.75$ which is equivalent to a stellar mass of
$10^{9}M_{\odot}$, (Kuchner et al., in prep)). At $M_{F}>3$, we artificially
lose halos that are massive enough to be simulated, but are too small to meet
the WWFCS observational criteria. However, as shown by the top panel in Figure
2, we are still obtaining thousands of cluster galaxies per cluster. This
reduction in the cluster member population mimics the expected relation of
lower mass cluster’s hosting fewer subhaloes (see for example Poggianti et al.
2010).

##  Appendix C DisPerSE input parameters

Figure 14: This figure illustrates how the change in the density field when
‘configuring’ a cluster makes a change in the persistence parameter necessary
when extracting the filament network. Left panel: KDE-smoothed density field
of a simulated cluster analogue to WWFCS target RX0058, with the filaments
identified by the topological structures extractor DisPerSE traced on top
(using our choice of persistence for the ‘pre-configured’ model, $2.5\sigma$.
Middle panel: the ‘mock-observational’ cluster with filaments extracted using
the same persistence, $2.5\sigma$. Right panel: the same but with our choice
of persistence for the ‘configured’ model, $2.1\sigma$. Lowering the
persistence yields a more accurate reconstruction of the cosmic web around
galaxy clusters.

As mentioned in Section 3.4, to extract the filament networks with DisPerSE we
need to set a persistence threshold. We introduce a metric $\Psi$ with 3 key
parameters:
1) $D_{\text{skel}}$ median (the positional difference in the reference
network spine and the configured network spine),
2) $D_{\text{skel}}$ ratio: the ratio of the two methods of calculating
$D_{\text{skel}}$,
3) Cluster connectivity of ’configured’ network ($C$, number of filaments that
stem from the main node and terminate outside $R_{200}$) comparison to
reference network.

All of the above parameters are normalized against their maximum output given
a persistence and are equally weighted. We compute this metric in a suitable
range of persistences $2<\sigma<3$, in 0.1$\sigma$ intervals, and the
persistence that minimizes $\Psi$ returns the most accurate reconstruction of
the filamentary network. A $\Psi$ of three corresponds to the worst possible
reconstruction of the network, whilst $\Psi$ values close to zero represent
the most accurate filamentary mapping. We have automated a process in
determining the best persistence given a reference skeleton. Our scientific
rationale requires a high completeness, therefore, we explore low persistence
values that not only map out the most robust structure, but also filaments
that connect nodes with smaller persistence ratios. After analysing $\Psi$ for
different networks for different clusters, we selected a persistence of
$\sigma=2.5$ for the reference network and $\sigma=2.1$ for the configured
network. In the process of ‘configuring’ a cluster, we are effectively
altering the underlying density field and therefore, it is necessary to change
the input persistence, as demonstrated in Figure 14. Although varying the
persistence cluster-by-cluster can change $D_{\text{skel}}$, the median and
PDFs of $D_{\text{skel}}$ do not vary significantly with changes in
persistence when averaged over all 160 clusters.