11institutetext:  Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53,
avenue des Martyrs, 38000 Grenoble, France 22institutetext: Departamento de
Física Teórica and CIAFF, Módulo 8, Facultad de Ciencias, Universidad Autónoma
de Madrid, 28049 Madrid, Spain 33institutetext: I.N.A.F. - Osservatorio
Astronomico di Roma, Via Frascati 33, 00040 Monteporzio Catone, Roma, Italy

#  the three hundred : contrasting clusters galaxy density in hydrodynamical
and dark matter simulations

A. Jiménez Muñoz the three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulationsthe three hundred : contrasting
clusters galaxy density in hydrodynamical and dark matter simulationsthe three
hundred : contrasting clusters galaxy density in hydrodynamical and dark
matter simulationsthe three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulations J. F. Macías-Pérez the three
hundred : contrasting clusters galaxy density in hydrodynamical and dark
matter simulationsthe three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulations G. Yepes the three hundred :
contrasting clusters galaxy density in hydrodynamical and dark matter
simulationsthe three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulations M. De Petris the three hundred :
contrasting clusters galaxy density in hydrodynamical and dark matter
simulationsthe three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulations A. Ferragamo the three hundred :
contrasting clusters galaxy density in hydrodynamical and dark matter
simulationsthe three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulations W. Cui the three hundred :
contrasting clusters galaxy density in hydrodynamical and dark matter
simulationsthe three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulations J.S. Gómez the three hundred :
contrasting clusters galaxy density in hydrodynamical and dark matter
simulationsthe three hundred : contrasting clusters galaxy density in
hydrodynamical and dark matter simulations

Cluster number counts at visible and IR wavelengths will be a key cosmological
probe in the next decade thanks to the Euclid satellite mission. For this
purpose, cluster detection algorithm performance, which at these wavelengths
are sensitive to the spatial distribution of the cluster galaxy members and
their luminosity function, need to be accurately characterized. Using The
Three Hundred hydrodynamical and dark matter only simulations we study a
complete sample of massive clusters beyond 7 (5) $\times$ 1014 M⊙ at redshift
0 (1) on a $(1.48\ \mathrm{Gpc})^{3}$ volume. We find that the mass resolution
of the current hydrodynamical simulations (1.5 $\times$ 109 M⊙) is not enough
to characterize the luminosity function of the sample in the perspective of
Euclid data. Nevertheless, these simulations are still useful to characterize
the spatial distribution of the cluster substructures assuming a common
relative mass threshold for the different flavours and resolutions. By
comparing with the dark matter only version of these simulations, we
demonstrate that baryonic physics preserves significantly low mass subhalos
(galaxies) as have also been observed in previous studies with less
statistics. Furthermore, by comparing the hydro simulations with higher
resolution dark matter only simulations of the same objects and taking the
same limit in subhalo mass we find significantly more cuspy galaxy density
profiles towards the center of the clusters, where the low mass substructures
would tend to concentrate. We conclude that using dark matter only simulation
may lead to some biases on the spatial distribution and density of galaxy
cluster members. Based on the preliminary analysis of few high resolution
hydro simulations we conclude that a mass resolution of 1.8 $\times$ 108 h-1
M⊙ will be needed for The Three Hundred simulations to approach the expected
magnitude limits for the Euclid survey. These simulations are currently under
way.

###### Key Words.:

galaxy: clusters: general, methods: numerical, cosmology: large scale
structure of Universe, galaxies: luminosity function, mass function

##  1 Introduction

The abundance of clusters of galaxies (Press & Schechter 1974) constitutes a
major cosmological probe (Allen et al. 2011) for the next generation of large-
scale structure surveys like the one expected from the Euclid satellite
(Laureijs et al. 2011). The number of clusters per unit of mass and redshift
is driven by cosmological parameters as the dark matter and dark energy
densities, $\Omega_{m}$ and $\Omega_{\Lambda}$ as well as the rms of the
linear matter fluctuations at 8 Mpc scales, $\sigma_{8}$, via the halo mass
function, which can be computed from numerical simulations (e.g. Tinker et al.
2008). A large number of studies have been performed with multi-wavelengths
observations of galaxy clusters confirming their potential as cosmological
probes in X-ray (Liu et al. 2022; Pacaud et al. 2018; Adami et al. 2018;
Böhringer et al. 2017), in the optical (Abbott et al. 2020; Drlica-Wagner et
al. 2018; Maturi et al. 2019), and via the thermal Sunyaev-Zeldovich effect
(tSZ, Hilton et al. 2021; Bocquet et al. 2019; de Haan et al. 2016; Bleem et
al. 2015; Planck Collaboration et al. 2016, 2014). Nevertheless, to date
cluster cosmological constraints seem to be dominated by systematic effects
related to the observational characterization of their mass (see summary in
Pratt et al. 2019) and redshift (e.g. Benitez 2000, and references therin).

One key aspect for cluster cosmology is the determination of the so called
Selection Function (SF) of the survey, which gives the probability of finding
a cluster at a certain mass and redshift (see for example Adam et al. 2019,
and references therin). The SF is an intrinsic characteristic of the cluster
survey and depends on the cluster finder algorithm, the observational and
quality cuts, the characteristics of the survey, and the chosen observables to
estimate the cluster mass (X-ray emission, tSZ effect, richness, lensing,
velocity dispersion) and redshift (photometric or spectroscopic). The SF can
be estimated from simulations either from full mock galaxy catalogues
including clusters and field galaxies, or from individual simulated clusters
that are injected in the observed galaxy survey (e.g. Sartoris et al. 2016;
Maturi et al. 2019; Adam et al. 2019; Rykoff et al. 2014). In both cases
realistic physical properties of the clusters are needed over large ranges in
mass and redshift and for a large variety of physical conditions.

For cosmological studies large sky volumes are needed. However, it is very
difficult to produce full hydrodynamic simulations with sufficient resolution
for such large volumes. A possible solution to this problem is the ‘zoom-in’
technique, as adopted by The Three Hundred collaboration (Cui et al. 2018).
For this project, a large cosmological volume is simulated by N-body dark-
matter-only simulation, and only in the regions where a galaxy cluster is
found, full-physics simulations are performed. For having enough statistics it
is necessary to run a large number of independent simulations (324 regions for
the The Three Hundred). From these simulations clusters properties can be
derived and used to complement large volume dark matter only simulations (de
Andres et al. 2023). Alternatively, the individual cluster simulations and
derived cluster properties can also be used to create synthetic clusters,
which can be injected in real datasets.

Within the perspective of cluster cosmology with the next generation of
optical and infrared large scale structure surveys, and in particular of the
Euclid satellite mission, we concentrate in this paper in the study of cluster
member galaxies in the The Three Hundred. We will consider their luminosity
function and spatial distribution, which are expected to drive cluster
detection algorithms Adam et al. (2019). In Section 2 we describe the The
Three Hundred data used. Sect. 3 discusses resolution effects in the
determination of the luminosity function. In Section 4 we present results in
the spatial distribution of cluster galaxy members. We finally conclude in
Section 5.

##  2  The Three Hundred data

###  2.1  The Three Hundred simulations

The The Three Hundred simulations were derived from the MDPL2 MultiDark
Simulations (Klypin et al. 2016). The latter consists on a 1 h-1 Gpc cube
containing 38403 dark matter (DM) particles with a mass of 1.5 $\times$ 109
h-1M⊙ each. Once the dark-matter-only simulations are performed, a cluster
finder algorithm is ran. In this case the ROCKSTAR halo finder (Behroozi et
al. 2012), which will look for dark matter halos. A total of 324 spherical
regions were extracted from the halo finder results, selecting as center for
these regions the position of the most massive halo at redshift $z=0$. The
radius of each spherical region is 15 h-1 Mpc that is much larger than the
virial radius of the central cluster, which is the radius that encloses the
mass that corresponds to approximately 98 times of the critical density of the
Universe (at $z=0$), as given by the Spherical Collapse model. The phase space
initial conditions for the 324 selected regions are used to perform the ‘zoom’
re-simulations. For the study presented in this paper, the 300th collaboration
has ran these simulations in four different flavors based on the GIZMO-SIMBA
(Cui et al. 2022) code, which can generate both dark matter only or
hydrodynamical simulations. The GIZMO-SIMBA code is based on the Meshless
Finite Mass (MFM) method. The MFM is a Langrangian method for hydrodynamics
based on a kernel discretization of the volume coupled to a high-order matrix
gradient estimator and a Riemann solver acting over the volume ‘overlap’,
first proposed by Hopkins (2015). In the case of hydrodynamical simulations
the GIZMO-SIMBA code applies the following baryonic physics models:

  * •

The gas treatment consists of an homogeneous UV background (Haardt & Madau
2012) accounting for self-shielding on the fly (Rahmati et al. 2013) and gas
metal dependent cooling (Smith et al. 2017).

  * •

Star formation and stellar feedback are included using a stellar model by Davé
et al. (2016) and galactic stellar and substellar initial mass function by
Chabrier (2003). Decoupled two-phase winds with mass loading factor scaling
with stellar mass and wind velocity limit in jet mode of 7000 km/s are also
implemented.

  * •

Finally, GIZMO-SIMBA includes black hole seeding and growth and active
galactic nuclei (AGN) feedback (Davé et al. 2019).

###  2.2 Dataset

In this paper we have used four different flavours of The Three Hundred
simulations depending mainly on the resolution and on the physics used to re-
simulate the cluster regions:

  1. 1.

lr_dmonly Dark matter only simulations with a dark matter particle resolution
of 1.5$\times$109 h-1M⊙.

  2. 2.

hr_dmonly Dark matter only simulations at high resolution. They have twice
more particles per dimension than the lr_dmonly for a total of 76803
particles, and consequently eight times less mass per particle i.e.,
1.8$\times$ 108 h-1 M⊙ each.

  3. 3.

lr_hydro Full-physics hydrodynamics simulations at the resolution of the
lr_dmonly.

  4. 4.

hr_hydro Full-physics hydrodynamics simulations at the resolution of
hr_dmonly. For these simulations we only have 5 regions, due to the high
computational cost.

Once the regions are re-simulated, they are analysed by the Amiga’s Halo
Finder (AHF; Knollmann & Knebe 2009), producing a catalogue with the halos
found within the different regions. In the case of hydrodynamical simulations,
for each halo different properties are computed, such as its radius
$R_{200}$111Radius at which the mean spherical density of the cluster is 200
times the critical density of the Universe at the cluster redshift, mass
$M_{200}$, density profile, galaxy luminosities for several spectral bands
covering from far-UV to radio. The galaxy luminosities are computed from the
identified stars of the AHF finder using the STARDUST code (Devriendt et al.
1999). The spectral energy distribution (SED) of each galaxy is convolved with
the bandpass of each photometric filter to compute the galaxy luminosity. In
the case of dark matter only simulations we have the same properties except
those related to baryon physics (e.g., stellar mass and luminosities). Each
halo can have smaller halos gravitationally bounded to it, which we will call
subhalos, with their own properties. The more massive and central halo is
known as the host halo. A low mass threshold of $8\times
10^{14}\mathrm{h}^{-1}$ M⊙ at $z=0$ is imposed for the central halo. For each
simulation flavours the final data consists of a halos and substructure
catalogue per region.

Figure 1: Luminosity function in the H-band for the 324 lr_hydro regions
(black), for five hr_hydro regions (red) and for the same five regions for
lr_hydro (blue) at z = 1. The dots and the associated uncertainties are
computed from the mean and dispersion in the bins in magnitude using all
available clusters. The shaded areas are the 1$\sigma$ and 2$\sigma$
uncertainties for the best Schechter model fit, respectively. The vertical
dashed line represents the observational limit for Euclid.

(a) z=0.0

(b) z=0.3

(c) z=0.5

(d) z=0.8

(e) z=1.0

(f) z=1.4

Figure 2: Mass distribution of the selected clusters for the different
snapshots in redshifts for the lr_dmonly (blue), hr_dmonly (orange) and
lr_hydro (green) datasets described in the text.

###  2.3 Identifying Galaxy Cluster Members.

To define a galaxy, first we consider a mass threshold, for the subhalo mass,
for each of four simulations. This translates into a mass resolution limit.
Notice that we consider the particle mass instead of the number of particles
as our threshold because for the different simulations the particle mass is
different. Thus, the same number of particles does not translate into the same
subhalo mass. Taking into account the particle mass and the simulation
resolution we adopt the following cuts:

  * •

For lr_dmonly we consider 3 x 1010 h-1M⊙ as the lower limit. This is
equivalent to considering that the substructure is formed by, at least, 20
dark matter particles.

  * •

For hr_dmonly considering the same mass threshold as for the low resolution
ones, it leads to be at least 160 dark matter particles, because the
resolution is 8 times higher. Since the resolution is significantly higher, we
can vary this threshold with respect to the low resolution ones. So to check
resolution effects, we choose a mass threshold of 9 x 109 h-1M⊙, corresponding
to 50 dark matter particles.

  * •

lr_hydro shares resolution with the lr_dmonly case. However the particle mass
is different because in this case we have dark matter particles and gas
particles. This means that the same mass threshold between lr_dmonly and
lr_hydro does not translate into the same number of particles. A
hydrodynamical simulated structure can have the same mass as a dark-matter-
only one, but without dark matter particles. We have to ensure that a galaxy
has dark matter particles. For this reason we apply, on top of the mass
threshold, a number of particles threshold. Thus we consider a mass threshold
of 3 x 1010 h-1M⊙ (as for the lr_dmonly case) and a minimum number of
particles threshold of 10. In this way, we assure the presence of dark matter
components in the baryonic structures, and also a minimum mass for having
enough resolution.

  * •

hr_hydro shares resolution with the hr_dmonly case. So as in the latter, we
choose a mass threshold of 9 x 109 h-1M⊙. However, following the reasoning of
the lr_hydro case, instead of having a minimum of 50 dark matter particles as
for hr_dmonly we choose 30 dark matter particles.

The next step is to identify for each re-simulated region the substructures
(galaxies) associated to the halos (clusters). The output of the AHF algorithm
is a list of structures with their associated physical properties like for
example the total, gas, and stellar mass, as well as the halo to which is
gravitationally bounded. For each cluster we define as galaxy members those
structures bounded to it. After this process, our dataset is formed by a list
of clusters with galaxies bounded to them. In the case of hydrodynamical
simulations, we also need to check the mass ratio between the dark matter and
the stellar content of the galaxy. The mass of a real galaxy is mainly coming
from the dark matter halo surrounding the stars and gas, so we impose that the
stellar mass component is not higher than $30\%$ of the total mass of the
galaxy. We finally get rid of contaminated low resolution particles that
initially were outside the region of interest in order to maintain only well
resolved structures. From now in this paper, we use subhalos or substructures,
interchangeably, to refer to galaxies. The same applies to halos or clusters,
to refer to galaxy clusters.

(a) MB1

(b) MB2

(c) MB3

(d) MB4

Figure 3: Cumulative galaxy mass function for the hr_hydro, hr_dmonly,
lr_hydro and lr_dmonly simulations at redshift 0 for the four bins in mass
considered. The shaded regions correspond to the standard deviation across
clusters. Resolution effects are clearly visible for the low mass region. The
vertical lines represent the minimum relative mass necessary to avoid
resolution effects.

(a) lr_hydro

(b) lr_dmonly

(c) hr_dmonly

Figure 4: Redshift evolution of the cumulative galaxy mass function for the
lr_hydro (left), lr_dmonly (center) and hr_dmonly (right) simulations for the
four bin mass defined in Table 1 (from top to bottom). The shaded regions are
obtained from the mean and standard deviation across clusters as described in
the text.

Figure 5: Redshift and mass evolution of the parameters from the power-law fit
of the 3D cumulative galaxy mass function. The right column represents the
$\alpha$ parameter and the right column the normalization $N_{-4}$, for $z=0$
up to $z=1.0$. From top to bottom we present results for the different cluster
mass bins of Table 1.

Figure 6: 3D cumulative galaxy density radial profile for the lr_hydro (left
column) and hr_dmonly (right column) simulations at redshift zero. From top to
bottom we show results for the four bin in mass in Table 1. The black dots and
error bars represent the stacked profile in the bin and uncertainties. The red
lines are the best-fit Einasto models to the stacked profiles following a MCMC
approach as described in text. The dark and light red shaded areas are the
1$\sigma$ and 2$\sigma$ uncertainties for the best Einasto model fit,
respectively.

##  3 Resolution Effects

###  3.1 Luminosity Function

As discussed above a key property for the detection of clusters of galaxies in
large scale structure surveys is their Luminosity Function (LF), defined as
the projected density of galaxies per unit of magnitude (see Adam et al. 2019,
for discussion). We have therefore computed the LF for the lr_hydro
simulations as shown in Figure 1 (black dots and associated uncertainties).
For simplicity we have considered in the figure only the clusters at $z=1$. To
ease the comparison to the results on the Mock Euclid catalogue presented in
Adam et al. (2019) we compute the LF for apparent magnitudes in the H-band.
First of all, we observe a clear deficit of galaxies at faint magnitudes with
a clear drop at magnitude 21. This is even more obvious when trying to fit a
Schechter model to the data via a MCMC analysis following Adam et al. (2019).
The dark and light gray shaded areas in the figure correspond to the 1$\sigma$
and 2$\sigma$ uncertainties over the best-fit Schechter model, respectively.

The observed drop in the number of faint galaxies is most probably due to a
lack of resolution in the lr_hydro simulations. To test this hypothesis, the
The Three Hundred collaboration has produced five hr_hydro simulations as
presented in Sect. 2. We present in red dots in the Figure 1 the LF for
several clusters in the hr_hydro simulations. We compute the LF as for the
lr_hydro simulations and also fit it to a Schechter model. We observe that the
LF reaches a maximum magnitude of 24 which corresponds to the observational
limit expected for Euclid (green vertical line in Figure 1). We also observe
larger uncertainties as only five regions are considered but no drop in the
number of galaxies. Furthermore the Schechter model seems to be a good fit to
the data as expected. We observe fainter galaxies for the lr_hydro simulations
most probably due to the lack of statistics in the hr_hydro simulations. To
check this, we present in blue dots in Figure 1 the LF for the lr_hydro
simulations for the same five regions as the hr_hydro simulations. Indeed, we
observe that the fainter part of the LF agrees between both resolutions. From
these results we conclude that the resolution of the lr_hydro simulations is
not enough for studying the properties of the LF of clusters for the next
generation of large scale structure surveys as Euclid. Producing hr_hydro is
computationally very expensive and so for this paper we only dispose of five
regions. In the following we will use them for qualitative comparison and
limit our quantitative analysis to the lr_hydro, lr_dmonly and hr_dmonly
simulations.

###  3.2 Subhalo Mass Function

The subhalo mass function gives the number of halo member galaxies (subhalos)
of a given mass, relative to the mass of the halo. The final mass resolution
of the simulation will imprint in the subhalo mass function as a deficit of
low mass subhalos.

Mass Bin | Cluster Mass [1014 h-1M⊙]
---|---
MB1 | 1.0 $\leq$ $M_{200}$ ¡ 3.5
MB2 | 3.5 $\leq$ $M_{200}$ ¡ 7.0
MB3 | 7.0 $\leq$ $M_{200}$ ¡ 10
MB4 | 10 $\leq$ $M_{200}$ $\leq$ 100
Table 1: Definition of the cluster mass bins used for the analyses presented
in the main text. We give the interval in mass considered in units of 1014h-1
M⊙.

To compute the subhalo mass function we first divide our sample in four bins
in mass as described by Table 1. These were defined at redshift zero so that
we can have bins as narrow as possible in mass (to ensure equivalent
properties for the clusters in the bin) while preserving sufficient statistics
per bin. We show in Fig. 2 the mass distribution of the selected clusters for
the 6 redshift snapshots considered in this paper. The mass histograms for the
lr_hydro, lr_dmonly and hr_dmonly simulations are displayed in green, blue and
orange, respectively. On the one hand, we find that the mass distribution at
each redshift is very similar for the three simulation flavors. On the other
hand, we observe that at high redshift the number of massive clusters
decreases significantly as one would expect. This will increase the number of
clusters for the lowest mass bins.

The resolution effect in the subhalo (galaxy) mass function can be clearly
observed in Figure 3. For the four bin in mass and at redshift zero we present
the cumulative galaxy mass function for the four simulation flavors. We use
all the available regions for each of the simulation flavors. The shaded
regions displayed in the figure are obtained for each flavor from the mean and
dispersion of the galaxy mass function of individual clusters in the mass bin.
We plot the total number of substructures (galaxies or subhalos) per halo
(cluster) with relative mass, $M_{\rm{substruct}}/M_{\rm{halo}}$, large than a
certain threshold. For the second mass bin the hr_hydro simulations has no
cluster because of the low statistics (only five regions available).

We observe, in terms of resolution, for very low mass subhalos we find
significant differences, with high resolution simulations showing more
subhalos at low mass than their respective low resolution simulations.
However, the subhalo mass function for the four simulation flavours are very
similar at intermediate masses and at high mass they converge into very
similar distribution within the variance. We stress that the hr_hydro
simulations are shown only for qualitative comparison because of the lack of
statistics. These two effects can be explained by the fact that low mass
substructures can not be formed unless sufficient resolution is attained.

We also observe some physical effects. For the lr_hydro simulation we can see
for all mass bins more galaxies than for the lr_dmonly ones. This is probably
due to baryonic physics, which diminishes the ripping out of the objects
because of cooling down processes of the gas, and so permits keeping more less
massive galaxies. The increase of resolution in the hr_dmonly simulations
tends to increase also the number of low mass substructures (galaxies).
However, we observe how the total number of substructures is the same between
the hr_dmonly simulations and the lr_hydro ones.

###  3.3 Redshift evolution of the subhalo mass function

| $\left<\alpha\right>_{z}$
---|---
Mass Bin | lr_dmonly | lr_hydro | hr_dmonly
MB1 | $-0.746\pm 0.011$ | $-0.921\pm 0.018$ | $-0.819\pm 0.010$
MB2 | $-0.813\pm 0.031$ | $-0.943\pm 0.016$ | $-0.838\pm 0.029$
MB3 | $-0.860\pm 0.032$ | $-0.986\pm 0.058$ | $-0.886\pm 0.024$
MB4 | $-0.838\pm 0.032$ | $-0.939\pm 0.044$ | $-0.842\pm 0.038$
Table 2: Redshift averaged and uncertainties of the slope, $\alpha$, of the subhalo mass function for the four bins in mass and for the different simulation flavors considered. | $M_{\rm{substruc}}/M_{\rm{halo}}$
---|---
Mass Bin | z=0.0 | z=0.3 | z=0.5 | z=0.8 | z=1.0 | z=1.4
MB1 | 0.000177 | 0.000177 | 0.000177 | 0.000177 | 0.000177 | 0.000205
MB2 | 0.000074 | 0.000074 | 0.000074 | 0.000074 | 0.000085 | 0.000074
MB3 | 0.000041 | 0.000048 | 0.000048 | 0.000048 | 0.000055 | 0.000074
MB4 | 0.000027 | 0.000031 | 0.000031 | 0.000036 | 0.000041 | 0.000074
Table 3: Relative galaxy to cluster mass, $M_{\rm{substruc}}/M_{\rm{halo}}$,
cut applied for comparing the lr_hydro, lr_dmonly and hr_dmonly simulations
for the six different redshift snapshots.

We show in Fig 4 the cumulative subhalo mass function for the lr_hydro (left
column), lr_dmonly (center column) and hr_dmonly (right column) simulations
for the six redshift snapshots considered. They have been computed as for Fig.
3 – the shaded region being obtained from the mean and dispersion across
clusters. From top to bottom the rows correspond to the bins in mass defined
in Table 1. These subhalo mass functions can be compared to those presented in
(Dolag et al. 2009). With respect to the latter we have increased in this work
significantly the statistics as well as extended the cluster mass and redshift
ranges considered. Furthermore, we dispose of simulations at different
resolutions.

These cumulative galaxy mass functions can be well approximated by a power-law
function as in Dolag et al. (2009),

$N_{m}=N_{-4}\left(\frac{m_{sub}/M_{vir}}{M_{\odot}}\right)^{\alpha},$ (1)

where N-4 is a normalization, $\alpha$ the slope and
M${}_{substruct}/$Mcluster is the ratio between the virial mass of the
substructures (galaxies) and that of their host halo. For obtaining the best-
fit parameters we have performed a least square fits of the mean value
accounting for the uncertainties computed from the dispersion across clusters.
In addition, we have also performed a fit of the cumulative galaxy mass
function per cluster and we compute the dispersion of the best-fit parameters
across clusters for each bin in mass and for each redshift slice.

We present in Figure 5 the evolution with redshift for both, the normalization
(left column) and the slope (right column) best-fit parameters for the four
cluster mass bins discussed above (top to bottom). The color dots represent
the best fit parameters for the four simulation flavors: lr_hydro (orange),
lr_dmonly (blue) and hr_dmonly (green). The uncertainties are computed from
the dispersion of the power-law fit of the 3D cumulative galaxy mass function
distribution per cluster, and from the intrinsic uncertainty on the fit of the
mean cumulative galaxy mass function as discussed before. We observe no
evolution with redshift for both parameters as was already the case in Dolag
et al. (2009). However, we observe in average a slight reddening of the slope
with cluster mass. This is more obvious on Table 2 where we present the
redshift averaged slope and uncertainties for the four bins in mass.

Figure 7: Redshift evolution of the 2D probability distribution of the
$\alpha$ and r0 parameters of the Einasto model for the lr_hydro (left column)
and hr_dmonly (right column) simulations. From top to bottom we give results
for the four bin in mass in Table 1. The inner and outer contours correspond
to the 68 % and 95.4 % C.L., respectively.

Figure 8: Redshift evolution of the 2D probability distribution of the r0 and
n0 parameters of the Einasto model for the lr_hydro (left column) and
hr_dmonly (right column) simulations. From top to bottom we give results for
the four bin in mass in Table 1. The inner and outer contours correspond to
the 68 % and 95.4 % C.L., respectively.

Figure 9: 3D radial number density of galaxies for the four bins in mass
defined in Table 1 (from top to bottom) at redshift zero. The left column
gives the best-fit Einasto model and uncertainties for the lr_hydro (blue) and
hr_dmonly (red) simulations. In the middle and right column we display the
radial galaxy density distribution per galaxy-cluster relative mass for the
lr_hydro and hr_dmonly simulations, respectively. The colorbars the number
density of galaxies.

###  3.4 Resolution Cuts on the Galaxy Mass Function

As discussed above the subhalo mass function is very similar for the four
simulations flavors apart from resolution effects (see Fig 3). The latter
affect more severely to the lr_dmonly simulations than to the lr_hydro ones.
Comparing these two flavors in the following would require to apply a cut on
the minimum mass of the galaxies considered to avoid resolution effects on the
lr_dmonly simulations. A better alternative is to compare hr_dmonly and
lr_hydro simulations by applying a cut derived from the latter.

For this we have computed for the lr_hydro simulations the minimum
$\left(M_{\rm{substruc}}/M_{\rm{halo}}\right)$ value for which resolution
effects are not significant:

$\|N^{\rm{LR\\_HYDRO}}/\max{\left(N^{\rm{LR\\_HYDRO}}\right)}-1\|>0.1$

where $N^{\rm{LR\\_HYDRO}}$ is the subhalo mass function for the lr_hydro
simulations. The mass cut found are given in Table 3 for each bin in mass and
for each redshift slice. These mass cuts will be applied in the rest of this
paper when comparing the properties of the lr_hydro and hr_dmonly simulations.

##  4 Galaxy Density Distribution

We study now the distribution of cluster galaxy members. For this, we
concentrate on the lr_hydro and hr_dmonly simulations to minimize resolution
effects while preserving sufficient statistics (number of cluster regions). We
apply the galaxy relative mass cuts presented in Table 3. This should ensure
fair comparison both in terms of statistics and resolution. We stress that in
most cases dark matter only simulations can be used to assess cluster
detection algorithm performance (see for example Adam et al. 2019).

###  4.1 Galaxy number density radial profiles

To assess the distribution of cluster galaxy members we compute galaxy density
radial profiles for each of the identified clusters. We use equally spaced
logarithmic radial bins in $R_{200}$ units so that we can easily compare and
combine clusters of different masses and redshifts. We calculate the
cumulative cluster galaxy member density from the center to the outskirt of
the cluster. For a spherical shell at a distance $R/R_{200}$ from the cluster
center, we count all the galaxies that are inside this sphere, and compute the
associated spherical volume. We have also produced non cumulative radial
profiles but found them less stable in terms of model fitting and will not be
further discussed here.

We present in Figure 6 the cumulative galaxy number density profiles for the
lr_hydro (left column) and hr_dmonly (right column) simulations at redshift
zero. From top to bottom we show results for the four bins in mass presented
in Table 1. The black dots correspond to stacked radial profiles. The stacked
profiles are computed by counting all galaxies for all the clusters in the
given radial bin and dividing by the volume and the number of clusters
considered. Uncertainties are obtained assuming a Poisson distribution for the
number of galaxies. We find that for the most inner radius the profiles are
badly reconstructed due to a lack of statistics. This effect leads to a clear
bias of the stacked galaxy density radial profiles as discussed in Appendix A
(Figure 10). In the following we will restrict the analysis to the external
radial region as represented by the stacked profiles in Figure 6.

###  4.2 Modelling and Fitting

It is difficult to extract physical information from a direct comparison of
the galaxy density profiles. Better insights can be obtained from well known
analytical models. At this respect, we have fitted the galaxy density profiles
to an Einasto model (Einasto 1965; Navarro et al. 2004) defined as

$\rho(r)=n_{0}\exp{\left(\frac{-2}{\alpha}\left[\left(\frac{r}{r_{0}}\right)^{\alpha}-1\right]\right)},$
(2)

where n0, r0 and $\alpha$ are free parameters. We summarize here the physical
interpretation of the parameters of the model:

  * •

n0 gives the normalization of the galaxy density.

  * •

$\alpha$ is related to the slope of the galaxy density distribution. When
$\alpha$ decreases, the slope increases.

  * •

r0 is a kind of characteristic radius. For a radius larger than r0 the galaxy
density profile drops rapidly.

We have fitted the stacked cumulative galaxy density profiles to the Einasto
model. Notice that the latter has been integrated to obtain a cumulative
radial profile matching the procedure performed on the simulation data. For
finding the best-fit parameters we have considered a MCMC approach based on
the emcee python package (Foreman-Mackey et al. 2013).

We present in Figure 6 the results of the fit for the lr_hydro and hr_dmonly
simulations at redshift zero and for the four bin in mass. Overall the best-
fit models are good fits to the data within the region of interest. The red
lines correspond to the best-fit model for the stacked profiles in the case of
the MCMC algorithms. We observe that the best-fit model represent a good fit
to the data in the region of interest. The MCMC results also allowed us to
handle the uncertainties on the parameters.

We show in Figures 7 and 8 the MCMC 2D probability distribution for the best-
fit parameters of the stacked profiles in Figure 10. For both figures we plot
the 68 % and 95.4 % C.L. contours for lr_hydro (left column) and hr_dmonly
(right column) simulations. We give results for the four bins in mass in Table
1 (top to bottom) and for six slice in redshift from 0 to 1.4 (color coded).
In some cases and for the highest redshift the statistic is not sufficient to
obtain reliable fits.

We observe in Fig. 7 a clear anti-correlation between the $\alpha$ and $r_{0}$
parameters. For both simulations the $r_{0}$ parameter decreases (smaller
clusters) with increasing redshift as one would expect. Overall we observe
larger values of the $\alpha$ and $r_{0}$ for the hr_dmonly simulations with
respect to the lr_hydro ones. This would indicate that the former are more
concentrated and show steeper density profiles. Furthermore, for the hr_dmonly
simulations there seems to be a hint of redshift evolution of the $\alpha$
parameter with larger values at higher redshift. By contrast, for the lr_hydro
simulations the $\alpha$ parameter shows very little evolution with redshift
within the uncertainties. Nevertheless, it is difficult to give general
conclusions on redshift evolution for any of the simulations as for a given
bin in mass the distribution of cluster masses is very different across
redshifts. Nevertheless, we conclude that as observed in Figures 6 and 10
there are more galaxy in the inner cluster region for the lr_hydro simulations
than for the hr_dmonly ones.

The latter conclusions seem to be confirmed by the results shown in Figure 8
where we present the MCMC 2D probability distribution for the $r_{0}$ and
$n_{0}$ parameters. We observe that the normalization parameter is larger for
the lr_hydro simulations with respect to the hr_dmonly ones. We also find that
overall at lower redshift $n_{0}$ and $r_{0}$ are smaller and larger,
respectively. This would indicate clusters are more extended at lower redshift
as expected. As for the results in Fig. 7 the differences in the cluster mass
distribution across redshift may explain some variations with respect to this
general pattern.

###  4.3 Discussion

From the previous results we can conclude that the density radial profile for
lr_hydro simulations are more concentrated and cuspier, than those of the
hr_dmonly simulations. This can be clearly observed on the left column of
Figure 9 where we present the best-fit Einasto galaxy density models and
uncertainties (shaded region) for the lr_hydro (blue) and hr_dmonly (red)
simulations at redshift zero and for the four mass bins in Table 1 (top to
bottom). The lr_hydro clusters present a clear excess of galaxies in the
central region and a drop in the profile for smaller radius than the hr_dmonly
ones. In addition, we have seen from the comparison of the subhalo mass
function in Section 3.2 that the lr_hydro (hr_hydro) simulations have more low
mass galaxies than the lr_dmonly (hr_dmonly) ones. We interpret these results
as the fact that baryons physics preserve low mass substructures that tend to
locate in the center of the cluster.

We have checked the latter statement by studying the galaxy density as a
function of the relative galaxy-cluster mass,
$M_{\rm{substruc}}/M_{\rm{halo}}$. We show the results on the middle and right
columns of Figure 9 at redshift zero for the lr_hydro and hr_dmonly
simulations, respectively. We represent the radial galaxy density (color
coded) for different bins in $M_{\rm{substruc}}/M_{\rm{halo}}$. We confirm a
clear excess of low mass galaxy in the inner cluster region for the lr_hydro
simulations with respect to the hr_dmonly ones. We also find that the low mass
galaxies are present at all radius and are the main component in number
density for both types of simulations.

##  5 Conclusions

In this paper we have studied the properties of galaxies in the The Three
Hundred simulations in terms of luminosity function and radial distribution in
the perspective of future large scale structure surveys at optical and
infrared wavelengths. For this we have used full physics hydrodynamical
simulations at at mass resolution of 1.5$\times$109 h-1 M⊙. We have completed
these with equivalent resolution and eight times more resolved dark matter
only simulations to disentangle possible effects from baryonic physics.

In terms of the luminosity function we have found that the current mass
resolution of the The Three Hundred hydrodynamical simulations is not
sufficient leading to artificial cut above magnitudes of about 21 when future
surveys like Euclid is expected to go as deep as magnitudes of 24. These
results have motivated the The Three Hundred collaboration to produce five
high resolution full-physics hydrodynamical regions (eight times more resolved
in mass). We have proved that for those the reconstruction of the luminosity
function goes up to magnitudes of 24. An effort to produce more high
resolution hydro simulations is under way.

We have computed subhalo (galaxies) mass function for the four flavor of The
Three Hundred simulations (hydro low and high resolution and dark matter high
and low resolution) for four bins in mass and for six snapshots in redshift.
We find that the lack of resolution leads to significantly less low mass
galaxies both for the hydro and dark matter only simulations. We also find
that the overall number of galaxies is the same between the low resolution
hydro simulations and the high resolution dark matter only ones. We observe
that overall the mass function are quite similar in the high mass regions
where there are no resolution effects. We also find that baryonic physics
tends to preserve significantly more low mass galaxies. We have approached the
different subhalo mass functions by a power law and have consistent slopes
across simulations flavors and no evolution with redshift. We conclude that
the main difference between hydrodynamical and dark matter only simulations is
the increased in the number of low mass galaxies.

Finally, we have studied the radial distribution of cluster member galaxies.
For study possible baryonic physic effects we compare the low resolution
hydrodynamical simulations and the high resolution dark matter only resolution
after imposing a cut in the relative mass of the particle for each bin in mass
and for each redshift slice. Overall, we find that radial galaxy number
density profiles of the hydro simulations show an excess of galaxies in the
inner part of the cluster and are more concentrated than those of the dark
matter only simulations. Furthermore, we observe that for the hydro
simulations the low mass galaxies tend to concentrate on the inner part of the
cluster. We conclude that baryonic physics preserves significantly more low
mass substructures and in particular in the inner cluster region.

## References

  * Abbott et al. (2020) Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2020, Phys. Rev. D, 102, 023509
  * Adam et al. (2019) Adam, R., Vannier, M., Maurogordato, S., et al. 2019, Astronomy & Astrophysics, 627, A23
  * Adami et al. (2018) Adami, C., Giles, P., Koulouridis, E., et al. 2018, A&A, 620, A5
  * Allen et al. (2011) Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409
  * Behroozi et al. (2012) Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2012, The Astrophysical Journal, 762, 109
  * Benitez (2000) Benitez, N. 2000, The Astrophysical Journal, 536, 571
  * Bleem et al. (2015) Bleem, L. E., Stalder, B., de Haan, T., et al. 2015, ApJS, 216, 27
  * Bocquet et al. (2019) Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55
  * Böhringer et al. (2017) Böhringer, H., Chon, G., & Fukugita, M. 2017, A&A, 608, A65
  * Chabrier (2003) Chabrier, G. 2003, Publications of the Astronomical Society of the Pacific, 115, 763
  * Cui et al. (2022) Cui, W., Dave, R., Knebe, A., et al. 2022, MNRAS, 514, 977
  * Cui et al. (2018) Cui, W., Knebe, A., Yepes, G., et al. 2018, Monthly Notices of the Royal Astronomical Society, 480, 2898
  * Davé et al. (2019) Davé, R., Anglés-Alcázar, D., Narayanan, D., et al. 2019, MNRAS, 486, 2827
  * Davé et al. (2016) Davé, R., Thompson, R., & Hopkins, P. F. 2016, MNRAS, 462, 3265
  * de Andres et al. (2023) de Andres, D., Yepes, G., Sembolini, F., et al. 2023, MNRAS, 518, 111
  * de Haan et al. (2016) de Haan, T., Benson, B. A., Bleem, L. E., et al. 2016, ApJ, 832, 95
  * Devriendt et al. (1999) Devriendt, J. E. G., Guiderdoni, B., & Sadat, R. 1999, A&A, 350, 381
  * Dolag et al. (2009) Dolag, K., Borgani, S., Murante, G., & Springel, V. 2009, Monthly Notices of the Royal Astronomical Society, 399, 497
  * Drlica-Wagner et al. (2018) Drlica-Wagner, A., Sevilla-Noarbe, I., Rykoff, E. S., et al. 2018, ApJS, 235, 33
  * Einasto (1965) Einasto, J. 1965, Trudy Astrofizicheskogo Instituta Alma-Ata, 5, 87
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Haardt & Madau (2012) Haardt, F. & Madau, P. 2012, ApJ, 746, 125
  * Hilton et al. (2021) Hilton, M., Sifón, C., Naess, S., et al. 2021, ApJS, 253, 3
  * Hopkins (2015) Hopkins, P. F. 2015, MNRAS, 450, 53
  * Klypin et al. (2016) Klypin, A., Yepes, G., Gottlöber, S., Prada, F., & Heß, S. 2016, MNRAS, 457, 4340
  * Knollmann & Knebe (2009) Knollmann, S. R. & Knebe, A. 2009, The Astrophysical Journal Supplement Series, 182, 608
  * Laureijs et al. (2011) Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv e-prints, arXiv:1110.3193
  * Liu et al. (2022) Liu, A., Bulbul, E., Ghirardini, V., et al. 2022, A&A, 661, A2
  * Maturi et al. (2019) Maturi, M., Bellagamba, F., Radovich, M., et al. 2019, MNRAS, 485, 498
  * Navarro et al. (2004) Navarro, J. F., Hayashi, E., Power, C., et al. 2004, Monthly Notices of the Royal Astronomical Society, 349, 1039
  * Pacaud et al. (2018) Pacaud, F., Pierre, M., Melin, J. B., et al. 2018, A&A, 620, A10
  * Planck Collaboration et al. (2014) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, A20
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A24
  * Pratt et al. (2019) Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25
  * Press & Schechter (1974) Press, W. H. & Schechter, P. 1974, The Astrophysical Journal, 187, 425
  * Rahmati et al. (2013) Rahmati, A., Pawlik, A. H., Raičević, M., & Schaye, J. 2013, MNRAS, 430, 2427
  * Rykoff et al. (2014) Rykoff, E. S., Rozo, E., Busha, M. T., et al. 2014, ApJ, 785, 104
  * Sartoris et al. (2016) Sartoris, B., Biviano, A., Fedeli, C., et al. 2016, MNRAS, 459, 1764
  * Smith et al. (2017) Smith, B. D., Bryan, G. L., Glover, S. C. O., et al. 2017, MNRAS, 466, 2217
  * Tinker et al. (2008) Tinker, J., Kravtsov, A. V., Klypin, A., et al. 2008, The Astrophysical Journal, 688, 709

##  Appendix A 3D cumulative galaxy density profiles

Figure 10: Redshift evolution of 3D cumulative galaxy density profiles for the
four simulation flavours and for the four bin in mass in Table 1.

We present in Figure 10 the 3D cumulative galaxy density distribution for the
four simulation flavors considered – lr_dmonly, lr_hydro, hr_dmonly and
hr_hydro(from left to right). From top to bottom we present results for the
four bins in mass presented in Table 1. We plot the stacked density profiles
for each slice in redshift. Notice that for the hr_hydro simulations we only
dispose of five regions, which explains why profiles are missing at high
redshift for the highest mass bins.