11institutetext: Max-Planck Institute for Astrophysics, Karl-Schwarzschild-
Str. 1, D-85741 Garching, Germany
11email<EMAIL_ADDRESS>
# FLows Around Galaxies (FLAG) I – The dependency of galaxy connectivity on
cosmic environments and effects on the star-formation rate

Daniela Galárraga-Espinosa  Enrico Garaldi  Guinevere Kauffmann

(Received XXX; accepted YYY)

With the aim of bringing substantial insight to the fundamental question of
how galaxies acquire their material for star-formation, we present in this
work the first comprehensive characterisation of the galaxy connectivity (i.e.
the number of filamentary streams connected to a galaxy) in relation with the
cosmic environment, and a statistical exploration of the impact of
connectivity on the star-formation rate at $z=2$. We detect kpc-scale
filaments directly connected to galaxies by applying the DisPerSE filament
finder to the dark matter (DM) density around 2977 massive central galaxies
($M_{*}>10^{8}$ $\mathrm{M}_{\odot}/h$) of the TNG50-1 simulation. Our results
demonstrate that more than half of the galaxies are connected to two or three
streams. About $15\%$ of them are connected to zero streams, i.e. these
galaxies are completely disconnected from the local web. We examine a variety
of factors that could influence the connectivity finding out that this
parameter increases with mass, decreases with local density for low mass
galaxies, and does not depend on local environment, estimated by the Delaunay
tessellation, for high mass galaxies. In order to better understand the
relation between the small-scale streams and the large-scale environment,
galaxies are further classified according to their location in five different
cosmic web environments. These are galaxy clusters, cluster outskirts, cosmic
filaments, filament outskirts, and walls and voids. On average, galaxies are
connected to two streams in filaments, to three in cluster cores, but are not
connected to streams in cluster outskirts. These differences reflect the
strength of the cosmic tides, which can prevent the formation of coherent
streams feeding the galaxies, or even disconnect the galaxy from its local
web. Finally, we show that, at fixed local density and cosmic web environment,
the sSFR of low mass galaxies is up to $7\,\sigma$ enhanced due to
connectivity. A milder impact is found for high mass galaxies, hinting at
different relative efficiencies of matter inflow via filamentary streams in
galaxies of different masses.

###### Key Words.:

(cosmology:) large-scale structure of Universe, galaxies: evolution ,
galaxies: star formation, galaxies: statistics, methods: numerical, methods:
statistical

##  1 Introduction

Under the action of gravity, matter on large scales in the Universe is
assembled to form a gigantic network composed of nodes, filaments, walls and
voids, called the cosmic web (de Lapparent et al. 1986; Bond et al. 1996).
Emerging from the initial density fluctuations (Zel’dovich 1970), this cosmic
skeleton is mainly composed of and ruled by the dynamics of dark matter (DM).
Driven by gravity, baryonic matter falls into the DM potential wells. The
structure of the cosmic web is highly multi-scale (Aragón-Calvo et al. 2010).
While the nodes of the web, hosting the most massive galaxy clusters, are
connected to large-scale cosmic filaments of widths of several Mpc (e.g. Gouin
et al. 2021, 2022; Galárraga-Espinosa et al. 2022), small haloes are also
attached to the web via smaller-scale filaments, characterised by widths of
tens of kpc (e.g. Ramsøy et al. 2021). These small-scale filaments, or
streams, are expected to have a great influence on the evolution and
properties of galaxies residing at the centre of those haloes.

Indeed, galaxies are thought to be formed at the intersection of these small-
scale filamentary streams which, in theory, feed the galaxies with the cold
and dense material necessary for star-formation (e.g. Birnboim & Dekel 2003;
Kereš et al. 2005; Ocvirk et al. 2008; Dekel et al. 2009; Pichon et al. 2011;
Faucher-Giguère & Kereš 2011; Faucher-Giguère et al. 2011; Danovich et al.
2012). The theoretical prediction is thus that these filaments act as highways
of matter, from the large-scale reservoirs down to the halo centres. This
picture is supported by studies in observations such as Bauermeister et al.
(2010), and more recently Prescott et al. (2015) and Zabl et al. (2019), who
have clearly demonstrated the need of gas replenishment from external
reservoirs. Nevertheless, other processes can also participate in the fuelling
of the galaxy with the material for star formation. These are, for example,
the precipitation of hot gas in virial equilibrium with the dark matter halo
(Kereš et al. 2005), the recycling of gas from the circum-galactic medium
(CGM), or even galaxy-galaxy mergers which drive gas from the outskirts of
galaxies into their centres where they form stars very rapidly, in a so-called
starburst. Interestingly, Nelson et al. (2013), showed that the numerical
scheme that is used in the hydrodynamical simulation strongly alters the
relative importance of gas accretion via cold streams and via cooling of
shock-heated gas. Thus, the question of how galaxies acquire the material for
star-formation and the relative efficiency of the processes involved is yet to
be understood.

Another active topic of investigation is the question of why galaxies stop
forming stars. The current picture involves a complex variety of feedback and
environmental processes that regulate the balance between gas inflows and
outflows around galaxies, and whose relative impact strongly depends on other
parameters such as galaxy mass and environment (e.g. Kauffmann et al. 2004;
Baldry et al. 2004; Bamford et al. 2009; Peng et al. 2010; Moutard et al.
2018). Star-formation could be suppressed either by internal mechanisms such
as energetic feedback from supernovae or accreting black holes, or by
environmental effects such as ram-pressure stripping or tidal interactions.
The latter are external processes that, according to Aragon Calvo et al.
(2019), are fundamentally linked with the disconnection (or detachment) of the
galaxy from its filamentary streams. This engenders a mechanical starvation
either by removing gas reservoirs or by preventing gas from reaching galaxies.

In this context, it is crucial to re-evaluate the relative effect of
filamentary streams on galaxy evolution in a cosmological context, i.e. taking
into account the environment where galaxies form and evolve. Indeed, a study
in a cosmological context is crucial because it is now well established, both
in observations and simulations, that beyond the trends with mass and local
environment, galaxy properties also vary as a function of their location in
the structures of the cosmic web. For example, galaxies located in cluster
environments are more massive, less star forming, redder, and with more
elliptical morphologies than those in less dense regions (see e.g. the reviews
of Dressler 1980; Boselli & Gavazzi 2006, 2014), and similar trends are found
in the cores of cosmic filaments with respect to regions further away from the
spines (e.g. Alpaslan et al. (2016); Chen et al. (2017); Malavasi et al.
(2017); Laigle et al. (2018); Bonjean et al. (2020); Rost et al. (2020);
Welker et al. (2020); Gouin et al. (2020); Winkel et al. (2021) in
observations and Kraljic et al. (2018, 2019); Ganeshaiah Veena et al. (2018,
2021); Malavasi et al. (2022) in simulations).

This paper is the first of a series providing an updated picture of the impact
of filamentary flows on galaxy evolution. We use the recent TNG50 simulation
(Pillepich et al. 2019; Nelson et al. 2019a) to perform a statistical analysis
of the number of (kpc scale) streams connected to galaxies, hereafter referred
to as the galaxy connectivity, as a function of the environment of the galaxy
in the cosmic web (defined at Mpc scales). While potential inflows and
outflows of baryons along these streams will be studied in the second part of
this project, we provide in this paper a first exploration of the impact of
galaxy connectivity on the specific star-formation rate (sSFR), defined as the
star formation rate normalised by galaxy stellar mass. We emphasise that the
multi-scale analysis performed in this work is different from previous
studies, which have rather focused on how large-scale structures, such as
groups or clusters, are connected to large-scale cosmic filaments on Mpc
scales (Kraljic et al. 2020; Gouin et al. 2021, 2022), yielding relevant
conclusions on the properties of the cosmic environments where galaxies live,
but not on the properties of galaxies themselves. Moreover, we note that this
type of study has only recently been enabled thanks to the advent of large-
scale hydro-dynamical simulations with more robust baryonic models and
increasing resolution (e.g. Tremmel et al. 2017; Pillepich et al. 2019; Dubois
et al. 2021), and is crucial in order to interpret future observations.

This paper is organised as follows. Section 2 introduces the TNG50 simulation
and the dataset of galaxies. We present the detection of the small-scale
filamentary streams as well as the large-scale cosmic web in Sect. 3. Results
about galaxy connectivity are first introduced in Sect. 4, and the impact of
the large-scale environments is discussed in Sect. 5. Finally, the relation
between connectivity and star-formation rate is explored in Sect. 6, and we
summarise our conclusions in Sect. 7. Throughout this paper, we adopt the
values of the cosmological parameters given by Planck Collaboration et al.
(2016), i.e. $\Omega_{\Lambda,0}=0.6911$, $\Omega_{m,0}=0.3089$,
$\Omega_{b,0}=0.0486$, $\sigma_{8}=0.8159$, $n_{s}=0.9667$ and $h=0.6774$. The
error-bars correspond to the errors on the mean values, derived from bootstrap
resampling.

##  2 Data

###  2.1 The TNG50-1 simulation

The analysis presented in this work uses the outputs of the TNG50-1
simulation, which is the highest resolution box of the gravito-
magnetohydrodynamical simulation suite, IllustrisTNG111https://www.tng-
project.org (Pillepich et al. 2018; Nelson et al. 2019b; Pillepich et al.
2019). With a mass resolution of $m_{\mathrm{DM}}=3.07\times
10^{5}\,\mathrm{M}_{\odot}/h$ and a volume of $(35\,\mathrm{cMpc/h})^{3}$,
this box is adapted to study the small-scale (kpc) filamentary streams in a
statistical way. We note that the IllustrisTNG project was run with the
moving-mesh code Arepo (Springel 2010), and the baryonic models and
prescriptions were specifically calibrated on observational data to match the
observed galaxy properties and statistics (Pillepich et al. 2018; Nelson et
al. 2019b). All the following results are derived from the TNG50-1 snapshot at
redshift $z=2$. This redshift typically corresponds to the so-called ‘cosmic
noon’, the epoch where galaxies were the most actively star-forming and is
therefore the ideal time to examine galaxy connectivity and its influence on
star formation.

typically in cosmic noon epoch

###  2.2 Resolution considerations

Recent studies (Ramsøy et al. 2021) have argued that the small-scale
filamentary streams, already present at $z=4$, need to be resolved with a
minimum of $\sim 2$ kpc in order to fully capture their physical properties
(e.g. shocks in the temperature profiles). These constraints were obtained by
performing convergence tests using simulations of different resolutions run
with the RAMSES adaptive mesh refinement (AMR) code. In order to check that
the TNG50-1 is a good simulation for studying these filamentary structures, we
compare the resolution scale cited above with the average diameter,
$d_{\mathrm{cell}}$, of the TNG50-1 Arepo cells at the typical density of the
filaments detected in Ramsøy et al. (2021). Assuming that cells can be
approximated by spheres, the cell diameter is given by
$d_{\mathrm{cell}}=2\times\left(3\,V_{\mathrm{cell}}\,/\,(4\,\pi)\right)^{1/3}$,
where $V_{\mathrm{cell}}$ is the cell volume. The latter is given by the ratio
between the TNG50-1 target baryon mass, $m_{\mathrm{target}}=8.5\times
10^{4}\,\mathrm{M}_{\odot}$ (Pillepich et al. 2019), and the typical filament
density found in Ramsøy et al. (2021), which is
$\rho=10^{-26.5}\,\mathrm{g}.\mathrm{cm}^{-3}=4.7\times
10^{4}\,\mathrm{M}_{\odot}.\mathrm{kpc}^{-3}$. The resulting cell diameter is
thus $d_{\mathrm{cell}}=1.51$ kpc, which is comparable with the minimum
resolution scale of $\sim 2$ kpc. Thus, the TNG50-1 simulation allows the
study of the small-scale filamentary streams at an optimal spatial resolution.

###  2.3 Galaxy selection

From the subhalo catalogue of the TNG50-1 simulation at $z=2$ (produced using
the Subfind code Springel (2005)), we selected the central objects with
stellar masses larger than $M_{*}>10^{8}$ $\mathrm{M}_{\odot}/h$. The maximum
subhalo stellar mass is $4\times 10^{11}\,\mathrm{M}_{\odot}/h$. This
selection in mass chooses subhaloes at $z=2$ that will most likely become
systems of typical mass $10^{9}-10^{12}$ $\mathrm{M}_{\odot}/h$ at $z=0$
(Brinchmann et al. 2004; Taylor et al. 2011). Importantly, we emphasise that
in this work we focus on central subhaloes only, defined as the objects at the
centre of their corresponding friends-of-fiends (FoF) halo. In the following,
we refer to them as galaxies. Finally, in order to ease the procedure of
extraction of the filamentary streams (see next section), we conservatively
choose to discard all the galaxies located at distances lower than 1.5 cMpc/h
from the edges of the full simulation box. Given the selections presented
above, the total number of galaxies analysed in this work is 2977. We finally
note that all these galaxies are star-forming, as shown by their main sequence
in the $M_{*}-\mathrm{SFR}$ plane presented in Appendix A.

##  3 Finding small and large-scale filaments

In this section we explain the procedure adopted to extract both the small-
scale streams connected to galaxies, and the large-scale (Mpc) cosmic web
skeleton. To detect these multi-scale structures with an optimal resolution we
employed the filament finder DisPerSE (Sect. 3.1) to adapted regions of the DM
density field. In a nutshell, the small-scale streams were detected from
selected sub-boxes centred on the position of individual galaxies (see Sect.
3.2), while the entire simulation box was used to find the large-scale
filamentary skeleton, as explained in Sect. 3.3.

###  3.1 The filament extractor code DisPerSE

DisPerSE (Sousbie 2011; Sousbie et al. 2011) is a publicly available code
which detects the cosmic skeleton from the topology of the density field (e.g.
the DM density), using the Discrete Morse theory and the theory of persistence
(see Sousbie 2011, and references therein). This algorithm identifies the
critical points of the field, i.e the points with a vanishing density
gradient. Filaments are defined as the ridges of the density field connecting
maximum density critical points (hereafter CPmax) to saddles222For an
illustration, we refer the reader to Fig. 2 of Galárraga-Espinosa et al.
(2020). Importantly, the minimum significance of the detected filaments with
respect to the noise can be set by fixing the persistence threshold of the
corresponding pairs of CPmax-saddle critical points. In the case of density
fields computed on regular grids (such as in this work), the persistence
threshold needs to be set via the cut parameter. The value of this parameter
should correspond to the amplitude of the noise of the input density grid, so
that any CPmax-saddle critical pair with density difference lower than the
adopted threshold is rejected. For further details, we refer the reader to the
DisPerSE presentation papers (Sousbie 2011; Sousbie et al. 2011) and
website333http://www2.iap.fr/users/sousbie/web/html/indexd41d.html.

###  3.2 Extracting the small-scale streams

Figure 1: Examples of 2D projected DM density fields. For each sub-box of side
3 cMpc/h, the red star corresponds to the central galaxies of mass
$M_{*}>10^{8}\,\mathrm{M}_{\odot}$. The small white stars and white dots
indicate respectively other centrals and satellite galaxies located in the
sub-box. The orange line at the bottom left part of the panels indicates the
resolution scale of the grid (20 ckpc/h).

Figure 2: Examples of the resulting small-scale filamentary streams detected
with DisPerSE (blue lines), for the same galaxies as in Fig. 1. For
illustration, the black points represent a random sub-sample (1/1000) of the
DM particle distribution in the sub-box, but the filaments were detected from
3D grids of the DM density field, as described in Sect. 3.2.

We detected the small-scale (kpc) filamentary streams connected to galaxies by
applying DisPerSE to the local DM density field. For each individual galaxy,
we selected the DM particles in sub-boxes of side $L=3$ cMpc/h centred on the
position of the galaxy. This value was chosen in order to capture the galaxy’s
environment beyond the typical scales of the CGM. The DM density field was
then computed by projecting the particles inside the sub-box onto a regular
grid of $N_{\mathrm{pix}}=150$ pixels per side. We applied a Gaussian filter
of standard deviation equal to the size of a pixel (i.e. $3/150=20$ ckpc/h) to
the grid values, and we rescaled the resulting pixel values by the standard
deviation. These steps enable the application of DisPerSE with the same
parameterisation to all the 2977 density grids. Figure 1 presents some
examples of DM density grids (projected along the z-axis). For each panel, the
analysed galaxy (red star) is at the centre of the sub-box. Other centrals and
satellites located in the same sub-box are shown by white stars and dots,
respectively.

DisPerSE was then applied to each one of the 2977 regular grids, so that each
galaxy possesses its own set of small-scale filaments. The persistence
threshold, which acts as a filter of the features that are likely to have been
generated by noise, was determined by exploring a broad range of values of the
DisPerSE cut parameter. Acceptable parameterisations were defined as those
whose skeletons successfully identified the central galaxies as CPmax, and
with streams in good visual agreement with the underlying density field. These
criteria were satisfied by skeletons extracted with cut values between 20 and
35. We then performed the statistical analyses presented in Appendix B which
demonstrated that, at fixed mass and local density, the relative difference of
the galaxy connectivity between skeletons extracted with cut=20 and cut=35 was
always less than $\sim 1.5\%$. Finally, we chose to fix the persistence
threshold to cut=30 after visual inspection of several random galaxies.
Indeed, the skeletons extracted with this value of the persistence threshold
had the lowest number of spurious secondary branches with respect to the
outputs obtained with lower values. Also, this threshold kept some filament
portions that were visually in good agreement with the underlying DM density
field, but absent in the cut=35 outputs. The positions of the resulting
streams were then smoothed using the DisPerSE skelconv function. By
straightening the skeleton segments and smoothing sharp and possibly non-
physical edges between them, this final step alleviates the effect of shot
noise on the geometry of the filaments. Note that this procedure does not
affect the topology of the density field (Codis et al. 2018), and thus keeps
the connectivity unchanged. Figure 2 presents some examples of the resulting
streams in 3D boxes. These correspond to the same galaxies as in Fig. 1.

It is worth noting that we explicitly chose not to identify the filaments from
the DM particle distribution in order to avoid the inevitable contamination
from small clumps (of scales $<10$ kpc) and, most of all, from the high shot
noise levels provoked by the great number of DM particles. Indeed, we found
that skeletons detected in the particle distribution were extremely sensitive
to the slightest changes of persistence threshold, making filaments appear and
disappear, and provoking radical changes in the position of even the most
prominent structures. Running DisPerSE on a DM density grid is thus a more
stable method, but has the draw-back of setting an intrinsic resolution scale,
$L/N_{\mathrm{pix}}=20$ ckpc/h (see tiny orange squares in Fig. 1
corresponding to the size of a pixel). This means that the positions of the
filament spines are determined with a precision of $\pm 10$ ckpc/h. While this
precision limit could potentially compromise the accuracy of radial density
profiles (because of the uncertainty on the exact position of the filament
cores), we emphasise that it does not undermine the results on connectivity
presented in this work.

###  3.3 Extracting the large-scale cosmic skeleton

Figure 3: Cosmic filaments (black lines) of the full TNG50-1 simulation. The
red dots show the position of clusters of galaxies, i.e. the FoF haloes of
masses $M_{200}>10^{12}\,\mathrm{M}_{\odot}/h$. The blue stars correspond to
the 2977 central galaxies analysed in this work. The detection of the cosmic
skeleton is presented in Sect. 3.3.

With a similar method as for the small-scale streams, the large-scale (Mpc)
cosmic skeleton was detected by projecting the full TNG50-1 DM particle
distribution onto a regular grid of 150 pixels per side, yielding an intrinsic
resolution scale of $35/150=0.23$ cMpc/h for these large-scale cosmic
filaments. The persistence threshold was set after analysing the outputs
obtained with different values of the cut parameter. In the case of the LSS, a
physical criterion to determine the robustness of the skeleton is that the
DisPerSE CPmax points match with the positions of the most massive haloes,
like those of groups and clusters of galaxies. The results of this matching is
presented in Appendix C, in which the choice running of DisPerSE with a
persistence threshold of 6 is also justified. Figure 3 shows the resulting
cosmic filaments in the 3D box of the TNG50-1 simulation. We remind the reader
that the identification of the large-scale cosmic skeleton in this work is
done solely with the aim of classifying the galaxies into different cosmic
environments, as will be shown in Sect. 5.

##  4 Galaxy connectivity

After detecting the small-scale filamentary streams, in this section we
present a statistical analysis of the galaxy connectivity, i.e. the number of
streams to which each galaxy is connected. Sect. 4.1 presents general results
for all the galaxies, and secondary dependencies with galaxy mass and local
environment are respectively analysed in Sect. 4.2 and Sect. 4.3.

###  4.1 General results

Figure 4: Histogram of the number of streams, $N_{\mathrm{streams}}$,
connected to each galaxy of the full galaxy catalogue. The vertical dotted and
dashed lines show respectively the mean and median values. The number of
streams to which a galaxy is connected defines the galaxy connectivity.

Figure 4 shows the distribution of the number of streams to which a galaxy is
connected, for all the 2977 galaxies of the dataset. This number was simply
obtained by counting the filaments that are directly connected to the CPmax at
the position of the central galaxy. From this figure we can see that the
galaxy connectivity spans a broad range (from 0 to 9), and follows an
irregular and asymmetric distribution. There is a long tail towards high
connectivity values, indicating that high connectivity is possible, but occurs
quite rarely. The highest peaks are seen for $N_{\mathrm{streams}}=2$ and 3,
with $31.9\%$ and $25.3\%$ of galaxies connected to two and three streams,
respectively. Moreover, $14.5\%$ of galaxies have exactly zero streams. For
reference the mean and median values of the distribution are respectively
$2.3$ and 2.

The irregular shape and broad range of the distribution presented in Fig. 4
hints that there may be additional factors that might have an effect on the
galaxy connectivity. In order to interpret the results presented in this
section, we thus need to disentangle possible secondary dependencies such as
galaxy mass and local environment. This will be addressed in the next section.

###  4.2 Trends with galaxy mass

Figure 5: Connectivity distribution by bins of galaxy mass (from yellow to
purple in increasing mass). The mean value of each distribution is 2.06, 2.45,
2.59, 2.92, and 3.00, respectively from the lowest to the highest masses.

In this section we investigate the effects of the galaxy mass on its
connectivity. Figure 5 shows the $N_{\mathrm{streams}}$ distribution for
galaxies separated in five different bins of stellar mass. A clear trend
emerges, as we see that the distributions of the highest mass bins (e.g.
purple) are shifted towards higher connectivity values with respect to those
of the lowest mass bins (e.g. yellow). More massive galaxies are therefore
more connected than lower mass ones. This result provides an extension to
lower masses of the trend well established in galaxy clusters, on Mpc scales
(Aragón-Calvo et al. 2010; Codis et al. 2018; Darragh Ford et al. 2019; Sarron
et al. 2019; Malavasi et al. 2020; Kraljic et al. 2020; Gouin et al. 2021).
The mean and median values of the distributions of Fig. 5 also reflect the
described trend. Respectively from the lowest to the highest masses, they mean
connectivity values are 2.06, 2.45, 2.59, 2.92, and 3.00. According to Codis
et al. (2018), a higher connectivity is predicted for high density peaks
(massive galaxies in the present context) because all the eigenvalues of the
Hessian matrix (i.e. the matrix of the second derivatives of the density
field) are equal at the vicinity of these peaks, thus describing a situation
of local isotropy where all incoming directions become possible (see also
Pichon & Bernardeau 1999).

We note that the separation in different mass bins allows us to better
understand the irregular and asymmetric shape of the total
$N_{\mathrm{streams}}$ distribution presented in Fig. 4. It is clearly
observed that the peak at $N_{\mathrm{streams}}$$=0$ is associated to the
lowest mass galaxies, whose distribution in Fig. 5 is clearly bi-modal. With
increasing mass, the distributions become more regular and symmetric and, for
highest mass bin, reach a unimodal Gaussian-like shape centred at the mean
value.

Figure 6: Relation between galaxy mass and mean connectivity,
$\overline{N}_{\mathrm{streams}}$ (blue curve). The red dashed line shows the
resulting fit-curve as presented in Eq. 1.

To go further in the quantitative analysis, we present in Fig. 6 the relation
between the mean connectivity, $\overline{N}_{\mathrm{streams}}$, and galaxy
mass. A simple logarithmic model is used to fit this relation, and the best-
fit result is shown by the red dashed diagonal. With a reduced chi-squared of
$\chi^{2}_{\nu}=0.90$, the resulting parameters are given by

$\overline{N}_{\mathrm{streams}}=(0.49\pm
0.03)\times\log(M_{*}\,\,[\mathrm{M}_{\odot}/h])-(1.96\pm 0.31).$ (1)

We have checked that the $\sim 0.5$ slope is independent of the number and
limits of the mass bins. These results show that the trends of galaxy
connectivity with mass can be captured quite well by a simple relation in the
$\overline{N}_{\mathrm{streams}}-\log(M_{*})$ plane. This relation echoes the
theoretical results of Codis et al. (2018), using peak theory.

In this section we have shown that the number of streams connected to a galaxy
depends on galaxy mass. We found the clear trend that more massive galaxies
are on average more connected than less massive ones. Let us now explore any
dependencies with the local environment of the galaxy, quantified by the local
density.

###  4.3 Trends with local density

We estimate the local density of a galaxy using the Delaunay Tessellation
Field Estimator (DTFE, Schaap & van de Weygaert 2000; van de Weygaert & Schaap
2009) on the distribution of galaxies. Each galaxy of our dataset is assigned
with a value of local over-density, $1+\delta_{\mathrm{DTFE}}$, which is
defined as:

$1+\delta_{\mathrm{DTFE}}=\frac{\rho_{\mathrm{DTFE}}}{\langle\rho_{\mathrm{DTFE}}\rangle}.$
(2)

In this equation $\rho_{\mathrm{DTFE}}$ is the resulting Delaunay density at
the position of the galaxy, and $\langle\rho_{\mathrm{DTFE}}\rangle$ is the
average density of all the galaxies. Physically, this quantity can be
interpreted as a proxy for the crowding of the local environment of the
galaxy. Galaxies in crowded regions (i.e. with many other neighbouring
galaxies) are associated with high values of local over-density, whereas low
local over-density values pertain to galaxies living in more locally empty,
less crowded spaces.

Figure 7: Mean connectivity $\overline{N}_{\mathrm{streams}}$ in the galaxy
mass vs local over-density plane. The pixel colours represent the means of the
number of streams for a given mass and $1+\delta_{\mathrm{DTFE}}$ bin. The
gray dashed lines show the limits of the four different galaxy populations
studied in Sect. 6.

Figure 7 presents the variation of the mean connectivity (see pixel colours)
in the mass vs local density parameter space. Given that mass and local
density are intrinsically correlated (e.g. Aragón-Calvo et al. 2010), it is
crucial to analyse these two parameters together in order to simultaneously
capture their influence on galaxy connectivity. For reference, the number of
galaxies contributing to each pixel of this 2D plane is shown in Fig. 16 of
Appendix D.

In addition to the already described trends with galaxy mass, Fig. 7 shows
interesting trends with $1+\delta_{\mathrm{DTFE}}$. For low mass galaxies
(with stellar masses lower than $\sim 10^{9.5}$ $\mathrm{M}_{\odot}/h$), the
mean number of streams strongly increases with decreasing local over-density.
The least connected galaxies ($\overline{N}_{\mathrm{streams}}\sim 0$) are
galaxies living in the highest local density environments, i.e. very locally
crowded regions (see yellow region at the top left of the plot). Galaxies in
these regions are subject to stronger (local) tidal effects (e.g. Hahn et al.
2009), which increase the probability of strong interactions with the
environment (e.g. by mergers) with respect to galaxies in lower density
environments. Aragon Calvo et al. (2019) has shown that these interactions can
lead to the disconnection of galaxies from their filamentary web, thus leading
to very low connectivity values. In line with these interpretations, we see in
Fig. 7 that low mass galaxies embedded in less crowded regions
($\log(1+\delta_{\mathrm{DTFE}})<-1.5$) are more connected to the web, with
connectivity averages of two to four streams.

In stark contrast with low mass galaxies, high mass galaxies ($M_{*}>10^{9.5}$
$\mathrm{M}_{\odot}/h$) do not show any significant trend with local over-
density. Their mean connectivity varies between two and five (with few
exceptions) regardless of specific values of mass and density. We note that
the tail at the highest $M_{*}$ and $1+\delta_{\mathrm{DTFE}}$ values (top
right corner) is due to the intrinsic correlation between mass and local
environment. High mass galaxies are less sensitive to the tides driven by the
local density, so their high connectivity is thus probably dominated by the
trends with mass discussed in Sect. 4.2.

Local density gives a first-order description of the environment of a galaxy,
but does not encode information on the location of such galaxy in the large-
scale environment, set by the different structures of the cosmic web. Knowing
the position of a galaxy in the large-scale structures is crucial to fully
understand the the results presented in this work. This will be shown in the
next section.

##  5 Connectivity in different cosmic web environments

Figure 8: 2D illustration of the five different cosmic environments considered in this work. These are clusters of galaxies (red), cluster outskirts (purple), cosmic filaments (green), filament outskirts (orange) and ‘other’ environments (blue). The exact definitions and corresponding number of galaxies belonging to each environment are presented in the main text. Cosmic structure | Number of galaxies
---|---
Voids + Walls | 1212
Filament outskirts | 457
Filaments | 1224
Cluster outskirts | 31
Galaxy clusters | 53
Table 1: Number of galaxies in each of the five different cosmic environments
considered in this work.

In this section we explore the effect of large-scale cosmic environment on
galaxy connectivity. It is important to extend the study of environment beyond
the first order analysis of local densities because of the well established
influence of large-scale cosmic tides on matter assembly (Hahn et al. 2009;
Musso et al. 2018; Paranjape et al. 2018). Before presenting our results, we
remind the reader that information about the local over-density of a galaxy
does not allow us to unambiguously determine the position of this object in
the cosmic web. This is due to the degeneracies between local and global
(cosmic) environments (e.g. Cautun et al. 2014). Figure 13 of their paper
clearly illustrates this point, as the $1+\delta$ distributions of matter in
the cosmic environments of nodes, filaments, walls, and voids largely overlap.

Figure 9: Connectivity distribution as a function of cosmic environments. The
five cosmic environments studied in this work are defined in the main text.
The vertical lines in each panel correspond to the mean values of the galaxy
connectivity, which are from left to right 2.75, 1.96, 1.95, 0.06 and 3.23,
respectively. Figure 10: Mean galaxy connectivity (shown by the pixel colours)
in the mass vs $1+\delta_{\mathrm{DTFE}}$ plane as a function of cosmic
environments (from left to right panels). The gray dashed lines show the
limits of the four different galaxy populations studied in Sect. 6.

We associate galaxies to one of the five different cosmic environments
presented in the illustration of Fig. 8. The five cosmic environments are
defined as follows and the number of galaxies in each is reported in Table 1.

  * •

Galaxy clusters are the spheres of radius $R_{200}$ centred on the positions
of the friends-of-friends (FoF) haloes of masses
$M_{200}>10^{12}\,\mathrm{M}_{\odot}/h$.

  * •

Cluster outskirts are defined as the spherical shells of inner and outer
radius 1 and $3\times R_{200}$, centred on the positions of clusters of
galaxies.

  * •

Cosmic filaments are the cylinders aligned with the spine of the (large-scale)
skeleton detected in Sect. 3.3, and of radius 1 cMpc/h. This value was chosen
in order to select the regions associated to the cores of cosmic filaments
(Galárraga-Espinosa et al. 2022).

  * •

Filament outskirts are the regions between 1 and 2 cMpc/h from the axis of
cosmic filaments (filament cores excluded).

  * •

Void and wall environments are all the other regions which do not belong to
one of four described above. We note that in this work we analyse together
galaxies in voids and walls. This is due to the little current information
regarding physical properties of these cosmic structures (e.g. wall average
thickness, void size), which are needed in order to associate galaxies with
the structures of the cosmic web described by DisPerSE.

Figure 9 presents the connectivity distribution of galaxies split according to
the five cosmic environments defined above. These distributions are very
different, showing a clear dependency of connectivity on the location of the
galaxy in the cosmic web. In order to better understand these distributions,
we study in Fig. 10 the corresponding connectivities in the mass vs
$1+\delta_{\mathrm{DTFE}}$ plane. For completeness, the number of galaxies in
each bin of this 2D parameter space is shown in the 2D histogram of Fig. 17.

In addition to the trends with stellar mass and local density already analysed
in Sect. 4, the following trends with cosmic environment are detected. First,
in cosmic filaments and outskirts of filaments and clusters, we see that low
mass galaxies in high density regions (top left corner) are very poorly
connected, with a large number of mean connectivity values close to zero.
Nevertheless, the same galaxies belonging to voids and walls have a mean
connectivity of $\overline{N}_{\mathrm{streams}}\sim 2-3$ instead. This can be
explained by the different strengths of the tidal flow in the different cosmic
environments (Kraljic et al. 2019). For example, due to the stronger
gravitational pull, galaxies belonging to filaments are subject to stronger
tides than galaxies in the less violent environments of walls and voids. As
argued by Hahn et al. (2009), strong tides can prevent the convergence of
matter flows onto galaxies and hence the formation of coherent streams, thus
probably leading to the low connectivity values in cosmic filaments (where the
tides are the strongest). Of course, this tidal effect could also be combined
with potential strong interactions of the galaxy with its local environment,
leading to a reduction of the number of streams (Aragon Calvo et al. 2019) as
explained in Sect. 4.3. The interpretation of these results is less
straightforward in the case of cluster outskirts. Indeed, given that they are
at the intersection between cosmic filaments and clusters, cluster outskirts
are unique environments where galaxies having different histories and
properties co-exist (e.g. galaxies falling through filaments, splash-back
galaxies, galaxies in groups, e.g. Kuchner et al. 2022; Borrow et al. 2022).
In addition, the question of how galaxies are accreted into cluster cores and
the physical processes they undergo during their infall is currently under
active investigation (e.g. Gouin et al. 2022; Kotecha et al. 2022, and
references therein). Thus, at this stage we can only argue that the very low
connectivity values found for galaxies in cluster outskirts are probably a
combined effect of galaxy diversity and interactions in this unique
environment.

In stark contrast with the previously studied cosmic environments, clusters of
galaxies host systems with the highest connectivity values among all. This is
shown by the last panel of both Figs. 9 and 10. Given that these cosmic
structures dominate the local gravitational field, they are rather insensitive
to the cosmic tidal flows. The great number of streams of galaxies in clusters
is therefore driven by the high galaxy masses found in these cosmic
structures, following the trends presented in Sect. 4.2.

The results of this section echo the analysis in the zoom-in simulations of
Borzyszkowski et al. (2017); Romano-Díaz et al. (2017); Garaldi et al. (2018).
In these papers, the authors focus on a few selected haloes, and separate the
accreting from stalled ones, finding that their different mass assembly
histories are explained by the location of the halo in the cosmic web (see
e.g. Fig. 10 of Borzyszkowski et al. 2017). While a careful study of accretion
rates and outflows along the galactic streams found in the present work will
be done in a follow-up project, from Figs. 9 and 10 one can already hint that
accreting haloes might be highly connected objects residing in cosmic
environments where the tidal field is relatively weak, whereas the stalled
ones might rather be disconnected from their matter supply and embedded in
structures where the cosmic flow is strong (e.g. in filaments).

##  6 Impact on star-formation

Figure 11: Influence of galaxy connectivity on star-formation. The curves show the mean galaxy sSFR as a function of connectivity, for different galaxy populations (from A to D, see Fig. 7) and cosmic environments. The horizontal lines show the average sSFR of a given galaxy population and cosmic environment, regardless of the connectivity value. We note that the y-axis is in logarithmic scale. | A | B | C | D
---|---|---|---|---
Voids + Walls | 981 | 172 | 45 | 13
Filament outskirts | 281 | 138 | 17 | 23
Filaments | 489 | 501 | 89 | 145
Cluster outskirts | 0 | 29 | 0 | 2
Galaxy Clusters | 0 | 1 | 5 | 47
Table 2: Number of galaxies in the four different galaxy populations (from A to D, see text in Sect. 6) and cosmic environments considered in this work. | A | B | C | D
---|---|---|---|---
Voids + Walls | $4.9\,\sigma$ | $2.9\,\sigma$ | $1.5\,\sigma$ | -
Filament outskirts | $3.1\,\sigma$ | $4.3\,\sigma$ | - | -
Filaments | $3.4\,\sigma$ | $6.7\,\sigma$ | $-1.4\,\sigma$ | $1.5\,\sigma$
Table 3: Significance $\Delta_{\mathrm{max-min}}$ (see Eq. 3) of the
enhancement of sSFR due to connectivity, for different galaxy populations and
cosmic environments (see left panels of Fig. 11).

After studying the connectivity of galaxies and understanding its dependencies
on mass and environment, we present in this section a first exploration of the
impact of galaxy connectivity on star-formation. This is a crucial analysis
given that the material for star formation (cold and dense gas) is predicted
to be accreted to the galaxy via the small-scale streams (e.g. Kereš et al.
2005; Ocvirk et al. 2008; Dekel et al. 2009), such as those detected and
studied in this work. While a more comprehensive analysis including studies of
mass accretion rates and gas properties of the filamentary streams is left for
a follow-up project, at this stage we can already try to identify any possible
effects solely driven by topology, i.e. by the number of connections of the
galaxy to filamentary streams.

In order to break the well-known degeneracies between star-formation, galaxy
mass and local density, and probe the specific effects of connectivity, we
separate galaxies in the four different populations presented in the mass-
overdensity plane of Fig. 7 (see gray dashed lines). From A to D, galaxies
increase in local density and mass. The limits between populations are
$M_{*}=10^{9.5}\,\mathrm{M}_{\odot}/h$ and
$1+\delta_{\mathrm{DTFE}}=10^{-1.5}$. Moreover, since it is also established
that galaxy properties can be impacted by the large-scale environment via the
cosmic tidal field (e.g. Hahn et al. 2007b, a; Laigle et al. 2015;
Borzyszkowski et al. 2017; Musso et al. 2018; Paranjape et al. 2018; Malavasi
et al. 2022, and references therein) we further differentiate galaxies with
respect to their location in the cosmic web. For reference, the number of
galaxies belonging to a given population (from a A to D) and cosmic
environment is presented in Table 2.

The panels of Fig. 11 present the variation of the mean specific star-
formation rate (sSFR) of the four considered galaxy populations as a function
of the galaxy connectivity, respectively in voids and walls, filament
outskirts, and filaments. We refrain from performing this study in clusters
and cluster outskirts due to the very low number of galaxies in these
structures. Also, we precise that, in order to have statistical results, bins
of $N_{\mathrm{streams}}$ with less than ten galaxies have been removed (these
usually correspond to the highest connectivity values). For reference, the
average sSFR of the entire galaxy population in its corresponding cosmic
environment is marked by the dotted horizontal lines.

These panels show a positive correlation between star-formation and
connectivity. Galaxies with a higher number of connections are associated to
higher values of star-formation with respect to less connected galaxies, and
the lowest sSFR values are always seen for galaxies with zero streams.
Interestingly, we find that the strength of this relation depends on galaxy
population and cosmic environment, as shown by Table 3. The values of this
table are computed by

$\Delta_{\mathrm{max-
min}}=\frac{\mathrm{sSFR}_{\mathrm{max}}-\mathrm{sSFR}_{\mathrm{min}}}{\sqrt{\sigma(\mathrm{sSFR}_{\mathrm{max}})^{2}+\sigma(\mathrm{sSFR}_{\mathrm{min}})^{2}}},$
(3)

where, for a given galaxy population and cosmic environment,
$\mathrm{sSFR}_{\mathrm{max}}$ and $\mathrm{sSFR}_{\mathrm{min}}$ are the
maximum and minimum values of the corresponding curves, and
$\sigma(\mathrm{sSFR}_{\mathrm{max}})$ and
$\sigma(\mathrm{sSFR}_{\mathrm{min}})$ represent their respective bootstrap
errors. These values quantify the enhancement of star-formation driven by
connectivity, and we comment them in what follows.

Low mass galaxies (populations A and B) show the strongest enhancement of
their sSFR with connectivity. Indeed, the significance of this enhancement is
always higher than $2.9\sigma$, and reaches up to $6.7\sigma$ for galaxies in
population B (i.e. with low masses and high local over-densities) sitting in
cosmic filaments. This strong sSFR enhancement for low mass galaxies is in
line with the so-called ‘cold’ accretion mode introduced in (Kereš et al.
2005). Namely, the haloes hosting these galaxies may not be massive enough to
support shocks, enabling the cold gas flowing along the filamentary streams to
reach down to the centre of the halo, thus feeding the central galaxy with
material for star-formation. Consequently, the greatest the number of streams,
the highest the sSFR enhancement.

On the other hand, high mass galaxies (populations C and D) show a much milder
increase of sSFR with connectivity. In this case, the significance is roughly
a factor of two lower than for low mass galaxies, and always lower than
$1.5\sigma$. We even detect a $1.4\sigma$ decrease of sSFR with connectivity
for galaxies in population C embedded in filaments. These results indicate
that star-formation in massive galaxies is less dependent on the number of
connections of the galaxy to the matter reservoirs outside the halo. In line
with the so-called ‘hot’ accretion mode (Kereš et al. 2005), this hints that
in massive systems star-formation might have little to do with potential
inflows of cold gas via the filamentary streams, and might rather be regulated
by internal processes, such as the recycling of gas within the halo, or the
cooling of gas that has been shock-heated by accretion into the halo. In this
scenario, a more important parameter to understand star-formation in massive
galaxies could be the cooling rate of gas, rather than the galaxy
connectivity.

##  7 Summary and conclusions

The question of how galaxies acquire the material from the cosmic web to fuel
star formation is fundamental to galaxy evolution. In this work we presented
the first comprehensive characterisation of the galaxy connectivity (i.e. the
number of filamentary streams attached to a galaxy) in relation with the
cosmic environment. We also showed the first steps towards assessing the
impact of this topological property on the galaxy star-formation rate. The
main results derived in this paper are summarised in the following:

  * •

(i) We have found that most of the galaxies ($>50\%$) are connected to two or
three streams, $\sim 15\%$ of them have exactly zero streams, and less than
$5\%$ of galaxies are connected to five streams or more. This is shown by the
total connectivity distribution of Fig. 4.

  * •

(ii) Galaxy connectivity strongly depends on galaxy mass. We have found that
low mass galaxies are on average less connected than high mass galaxies, and
that the connectivity distributions of galaxies with increasing masses go from
bi-modal to unimodal (Fig. 5). Empirically, we have established (in Fig. 6)
the following simple relation between mean connectivity and galaxy mass:
$\overline{N}_{\mathrm{streams}}\propto
0.5\,\,\log(M_{*}\,[\mathrm{M}_{\odot}/h])$.

  * •

(iii) Galaxy connectivity also depends on local environment, with differences
between low and high mass galaxies (Fig. 7). We have found that low mass
galaxies (with stellar masses lower than $\sim 10^{9.5}$
$\mathrm{M}_{\odot}/h$) sitting in high local over-density environments are
connected to significantly smaller numbers of streams with respect to galaxies
of the same mass located in lower over-dense regions. This trend with local
environment was interpreted by the influence of the stronger tidal forces felt
by low mass galaxies in high over-density environments (Hahn et al. 2009;
Aragon Calvo et al. 2019). Concerning high mass galaxies, we have showed that
their connectivity is independent of local over-density, and that their
greater number of connected streams is probably driven by their mass.

  * •

(iv) By further disentangling galaxies in different cosmic environments, we
have found that the average galaxy connectivity decreases from cosmic voids
and walls to filament outskirts, from the latter to filament cores, and is the
lowest among all in cluster outskirts (Fig. 10 and 9). This decrease might be
due to the increasing strength of cosmic tides respectively in these cosmic
environments (e.g. Musso et al. 2018; Paranjape et al. 2018; Kraljic et al.
2019). On the other hand, we have shown that the average galaxy connectivity
is the highest of all in galaxy clusters, where the most massive galaxies
reside.

  * •

(v) We have found that galaxy connectivity has an impact on star-formation
(Fig. 11). At fixed mass, local density, and cosmic environment, the overall
galaxy star-formation increases with the number of streams connected to the
galaxy. Interestingly, in the case of low mass galaxies, we have measured an
enhancement of galaxy sSFR due to connectivity of about $3-7\,\sigma$. This
enhancement was found to be at least twice as low for high mass galaxies.

These results draw a picture in which star-formation is linked to an external
parameter describing topology, the galaxy connectivity. Within this picture,
high number of connected streams might favour the accretion of cold material
from the large-scales and thus boost the galaxy star-formation, especially in
the case of low mass galaxies. As mentioned in the main body of the paper, it
remains to be investigated whether galaxy connectivity is a fundamental
parameter, or rather a proxy for e.g. gas accretion rates. For example, are
all the DM streams actively transporting matter towards the galaxy? What is
the fraction of gas accreted via the streams with respect to an isotropic
accretion? And more fundamentally, is mass the result of connectivity (because
of an efficient accretion of matter through the streams) or is the
connectivity driven by mass? These questions will be tackled in the second
part of this series of papers, where we will also investigate the gas
properties of the streams.

Moreover, throughout this paper we have seen that cosmic filaments host
galaxies with the most diverse ranges of masses, local densities, and
connectivity values (see for example the middle panel of Fig. 10). Different
galaxy populations therefore co-exist in these cosmic environments, which are
also are less extreme than those of clusters of galaxies, and present a rich
diversity in terms of gas density and temperature (e.g. Galárraga-Espinosa et
al. 2021, 2022). This diversity makes cosmic filaments an interesting
playground for galaxies, where it can be possible to study the evolution of
different populations of galaxies in the broader cosmological picture.

###### Acknowledgements.

DGE would like to thank Raúl Angulo, Rüdiger Pakmor, and Nir Mandelker for
useful and insightful discussions. We thank Adam L. Schaefer and Céline Gouin
for providing comments on the final version of the draft. We also thank the
IllustrisTNG team for making their data publicly available, and for creating a
user-friendly and complete website.

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##  Appendix A Galaxies in the $M_{*}-\mathrm{SFR}$ plane

The relation between stellar mass and star-formation rate of the TNG50-1
galaxies studied in this work is presented in Fig. 12. This figure shows that
almost all of these $z=2$ galaxies are in the main sequence. A quenched
population of galaxies is absent.

Figure 12: Galaxies in the $M_{*}-\mathrm{SFR}$ plane. The total number of
galaxies is 2977.

##  Appendix B Setting the value of DisPerSE persistence threshold for
extraction of the small-scale filamentary streams

Figure 13: Effect of the DisPerSE persistence threshold (set by the value of
the cut parameter) on the resulting small-scale skeleton around three random
galaxies (from top to bottom). The black points in the background present a
random sub-sample 1/1000) of the underlying DM distribution, and the central
white star indicates the position of the galaxy. These boxes have a side
length of 3 cMpc/h. Figure 14: Effect of the DisPerSE persistence threshold on
the mean connectivity, as a function of galaxy mass and local density. This
figure is similar to Fig. 6, but shows the results for random 291 galaxies.
Figure 15: Relative difference of the mean connectivity. This figure compares
the outputs of extreme persistence thresholds between them (i.e. cut=20 vs
cut=35). The colors correspond to the residuals between the
$\overline{N}_{\mathrm{streams}}$ pixel values of the first and the last
panels of Fig. 14, divided by the mean galaxy connectivity of the cut=35
output, 2.55.

Figure 13 shows examples of the skeletons detected with values of the
persistence threshold between 20 and 35. By visual inspection, one can see
that different values of the cut parameter do not significantly change the
resulting skeleton. With increasing persistence thresholds, the number of
secondary branches decreases (compare e.g. the upper-left to the upper-right
boxes), and the galaxy connectivity of the top and bottom panels is unchanged
(with respectively two and three streams connected to those galaxies). The
middle panel shows an example where a spurious filament branch, directly
connected to the galaxy, is removed by high values of persistence threshold.
In this case, the connectivity decreases from six (cut=25) to five (cut=35).

A more quantitative comparison is shown in Fig. 14. This figure presents the
effect of the different persistence thresholds on the mean galaxy
connectivity, studied in the mass-overdensity plane (as in Fig. 7). We note
that, given the high numerical costs of detecting the streams with DisPerSE,
we limit this analysis to a random set of 291 galaxies. This number represents
$\sim 10\%$ of the total galaxy dataset. For each of these 291 galaxies,
different skeletons were identified, each with a value of the persistence
threshold between 20 and 35. As already pointed out by the example of Fig. 13,
overall higher connectivity values are detected in the outputs with the lowest
persistence thresholds. Nevertheless, the differences between these panels are
tiny. Indeed, as shown by Fig. 15, the relative difference between the two
extremes (i.e. cut=20 and cut=35) is always less than $\sim 1.5\%$ for
galaxies in the same range of mass and local-density. Moreover, the same
trends as in Fig. 7 are observed for all the cut values. These results thus
show that the statistical analysis performed in this work has a very weak
dependency on the exact value of the DisPerSE persistence threshold, and thus
any value of the cut parameter between 20 and 35 can be an adequate choice.

##  Appendix C Setting the value of DisPerSE persistence threshold for the
extraction of the cosmic skeleton

Testing the robustness of the positions of the cosmic filaments recovered by
DisPerSE is far from being trivial. Indeed, a ‘true’ physical reference can
not be easily determined in the simulations, and extracting the skeleton with
another detection technique (e.g. Cautun et al. 2013; Tempel et al. 2014;
Bonnaire et al. 2020) and comparing the outputs would rather be a
characterisation of the similarities and differences of the different methods
(already performed by Libeskind et al. 2018), and not a study of the detection
of the actual cosmic filaments.

Therefore, in order to check the robustness of the cosmic skeleton and fix the
value of the DisPerSE persistence threshold, we focus on the positions of the
ending points of the filaments, the CPmax, which by construction correspond to
the topological nodes of the skeleton. We compare the positions of the CPmax
to those of the most massive FoF haloes of the TNG50-1 simulation. The latter
were divided in the following mass bins, which were chosen in order to
encompass the main classes of objects:

  * •

$M_{200}\in[10^{11},5\times 10^{11}[\,\,\mathrm{M}_{\odot}/\mathrm{h}$ (small
haloes),

  * •

$M_{200}\in[5\times 10^{11},10^{12}[\,\,\mathrm{M}_{\odot}/\mathrm{h}$
(groups),

  * •

$M_{200}\geq 10^{12}\mathrm{M}_{\odot}/\mathrm{h}$ (clusters).

Table 4 presents the results of this comparison, for four skeletons detected
with different values of the DisPerSE persistence threshold (from 4 to 7).
This range encompasses adapted cut values given the pixel values of the input
DM density grid. The numbers of this table correspond to the fractions of FoF
haloes hosting a CPmax within their $R_{200}$ (first block) and, conversely,
the fractions of CPmax coinciding with the positions of the selected FoF
haloes (also within $R_{200}$), in the second block.

A robust skeleton is one which maximises the fraction of CPmax in the most
massive haloes (by reducing the number of critical points in less massive
objects or other less dense environments) while keeping $100\%$ of the most
massive haloes identified as CPmax. This is the case for the skeleton
extracted with a persistence threshold of 6, which is therefore adopted for
the classification of galaxies in different cosmic environments.

Table 4: Fraction of FoF haloes of the TNG50-1 simulation hosting DisPerSE CPmaxs, and conversely. | cut 4 | cut 5 | cut 6 | cut 7
---|---|---|---|---
Small haloes hosting CPmax | $26.7\%$ | $18.4\%$ | $13.3\%$ | $9.34\%$
Groups hosting CPmax | $91.1\%$ | $89.4\%$ | $88.6\%$ | $78.9\%$
Clusters hosting CPmax | $100\%$ | $100\%$ | $100\%$ | $98.4\%$
CPmax in small haloes | $52.9\%$ | $46.4\%$ | $39.0\%$ | $33.2\%$
CPmax in Groups | $24.4\%$ | $30.6\%$ | $35.2\%$ | $37.9\%$
CPmax in Clusters | $13.7\%$ | $17.4\%$ | $20.3\%$ | $24.2\%$
Total fraction of matched CPmax | $91.1\%$ | $94.4\%$ | $94.5\%$ | $95.3\%$

##  Appendix D Distribution of galaxies in the 2D parameter space of mass vs
local density

This Appendix presents the histograms of the distribution of galaxies in the
mass-$1+\delta_{\mathrm{DTFE}}$ plane, for all the galaxies (Fig. 16), and for
galaxies in the different cosmic environments (Fig. 17).

Figure 16: 2D histogram of the number of galaxies in the mass-overdensity
plane (see Sect. 4.3). Figure 17: 2D histogram of the number of galaxies in
the mass-overdensity plane, further segregated by cosmic environment (see
Sect. 5).