# The Three Hundred project: galaxy cluster mergers and their impact on the
stellar component of brightest cluster galaxies

Ana Contreras-Santos,1 Alexander Knebe,1,2,3 Frazer Pearce,4 Roan Haggar,4
Meghan Gray,4 Weiguang Cui,5 Gustavo Yepes,1,2 Marco De Petris,6,7 Federico De
Luca,8 Chris Power,3 Robert Mostoghiu,1 Sebastián E. Nuza,9,10 and Matthias
Hoeft11
1Departamento de Física Teórica, Módulo 15, Facultad de Ciencias, Universidad
Autónoma de Madrid, 28049 Madrid, Spain
2Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de
Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
3International Centre for Radio Astronomy Research, University of Western
Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
4School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD,
United Kingdom
5Institute for Astronomy, University of Edinburgh, Royal Observatory,
Edinburgh EH9 3HJ, United Kingdom
6Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5,
00185 Roma, Italy
7I.N.A.F. - Osservatorio Astronomico di Roma, Via Frascati 33, 00040
Monteporzio Catone, Roma, Italy
8Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca
Scientifica 1, 00133 Roma, Italy
9Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), CC 67, Suc.
28, 1428 Buenos Aires, Argentina
10Facultad de Ciencias Exactas y Naturales (FCEyN), Universidad de Buenos
Aires (UBA), Buenos Aires, Argentina
11Thüringer Landessternwarte, Sternwarte 5, 07778 Tautenburg, Germany
Contact e-mail<EMAIL_ADDRESS>
(Last updated 2015 May 22; in original form 2013 September 5)

###### Abstract

Using the data set of The Three Hundred project, i.e. a suite of 324
hydrodynamical resimulations of cluster-sized haloes, we study galaxy cluster
mergers and their effect on colour and luminosity changes of their brightest
cluster galaxies (BCG). We track the main progenitor of each halo at $z=0$ and
search for merger situations based on its mass accretion history, defining
mergers as very rapid increases in the halo mass. Based upon the evolution of
the dynamical state of the cluster we define a pre- and post-merger phase. We
create a list of all these events and statistically study their mass ratio and
timescales, with the former verifying that all instances are in fact major
mergers. By comparing to a control sample of clusters without mergers, we
study the effect mergers have on the stellar component of the BCG. Analysing
the mass, age and metallicity of the BCG stellar particles, we find that the
stellar content of BCGs grows significantly during mergers and, even though
the main growth mechanism is the accretion of older stars, there is even a
burst in star formation induced by the merger. In our simulations, BCGs in
mergers form in median around 70 per cent more stars than those normally
growing, although this depends on the radius considered for defining the BCG.
Regarding observable properties, we see an increase in SDSS-$u$ luminosity of
20 per cent during mergers, accompanied by a slightly slower increase of the
galaxy $g-r$ colour as compared to the control sample.

###### keywords:

methods: numerical – galaxies: clusters: general – galaxies: haloes –
cosmology: theory – large-scale structure of the universe

††pubyear: 2021††pagerange: The Three Hundred project: galaxy cluster mergers
and their impact on the stellar component of brightest cluster galaxies–The
Three Hundred project: galaxy cluster mergers and their impact on the stellar
component of brightest cluster galaxies

##  1 Introduction

Galaxy clusters are essential in our comprehension of the Universe. They are
the largest gravitationally bound systems and, as such, they can be used to
probe the large scale structure of the Universe, as well as the formation and
evolution of galaxies. On cosmological scales, they are dark matter dominated,
and so their physics are only driven by gravity. On smaller scales, not only
gravity is taken into account but also the interaction of the baryonic
components of clusters plays an important role, leading to several different
phenomena that regulate for instance the properties of the hot gas in the
intracluster medium (ICM). The study of galaxy clusters can therefore yield
results ranging from cosmological parameters to models of the astrophysical
processes that drive galaxy evolution.

Regarding the formation of galaxy clusters, the $\Lambda$ cold dark matter
($\Lambda$CDM) model of the Universe describes a hierarchical model of
structure formation (Blumenthal et al., 1984). Small bound structures are
formed via gravitational collapse, which then grow through mergers with other
haloes or via accretion of smaller systems (White & Rees, 1978; Frenk & White,
2012). This way, large haloes are a result of merger processes throughout
their history and, thus, the understanding of these events becomes crucial in
understanding the formation of clusters and their properties. Besides, due to
the high binding energy and the huge energy releases, cluster mergers are one
of the most energetic events in the Universe, which also makes them a very
relevant field of research.

Several investigations have been conducted in this field both observationally
and theoretically. Multiwavelength observations of galaxy clusters can probe
differently the underlying physics. Observations in radio and far infrared
probe the cold gas (Giard et al., 2008), while the optical (and near infrared)
emission comes from stars, with two main observables being the luminosity and
color (see Bahcall, 1977 or Biviano, 2000 for reviews of optical studies of
galaxy clusters). X-ray observations probe the hot intracluster medium (ICM),
which shines brightly at these wavelengths (see Sarazin, 1988 or Böhringer &
Werner, 2010 for reviews of X-ray observations of clusters). The ICM is also
explored by the Sunyaev-Zeldovich (SZ) signal at millimetre wavelengths,
reaching even larger radii and distant clusters (see review by Mroczkowski et
al., 2019).

With these different observations, cluster surveys can be constructed, see
e.g. for X-ray: ROSAT All-Sky Survey RASS, (Voges et al., 1996), CHEX-MATE,
(The CHEX-MATE Collaboration et al., 2020), eROSITA, (Liu et al., 2021); for
SZ: Planck (Planck Collaboration et al., 2016b), ACT (Hilton et al., 2018),
SPT (Bleem et al., 2020); or Wen et al., 2009 for optical. Then, via their
morphology, clusters undergoing mergers can be identified (Schombert, 1987;
Mann & Ebeling, 2012). Observations of merging clusters allow then a more in-
depth study of these events (Belsole et al., 2005; Okabe & Umetsu, 2008;
Golovich et al., 2019) and the different phases in which they can be observed
(Wilber et al., 2019). They also facilitate the analysis of the relationship
between mergers and different cluster properties, such as kinetic energy and
entropy (Markevitch et al., 1999) or star formation rates for the individual
galaxies in the cluster (Johnston-Hollitt et al., 2008; Deshev et al., 2017).
Other studies investigate the influence of mergers on the presence or absence
of cool cores in clusters and on the different scaling relations like the
X-ray luminosity-temperature relation (Hallman & Markevitch, 2004; O’Hara et
al., 2006; Wang et al., 2016).

Numerical simulations have also been used to study galaxy clusters from a more
theoretical approach (see Borgani & Kravtsov, 2011 for a review), and their
properties can be compared to those from observations (Borgani et al., 2004;
Fabjan et al., 2011). Regarding galaxy cluster mergers, they can be studied in
controlled simulations, where only two clusters are simulated to merge, so
that the initial conditions and outcomes can be thoroughly studied (Poole et
al., 2006, 2007; Valdarnini & Sarazin, 2021). This allows for a very detailed
study, with the possibility to change the initial conditions as desired.
However, it can be of even more interest to study mergers in the frame of
cosmological simulations, where they happen naturally during the evolution of
clusters (e.g. the Millennium Simulation, Springel et al., 2005). Unlike
observations, this kind of simulations allow for tracking the whole history of
a cluster, identifying a merger the moment it takes place. They are also
useful to study the merger rate of dark matter haloes in the universe, and
find its dependence with redshift or mass of the haloes (Fakhouri & Ma, 2008;
Fakhouri et al., 2010). As with observations, data from simulations can be
used to study the effects that mergers have on the already mentioned scaling
relations or presence of cool cores (Ritchie & Thomas, 2002; Kay et al., 2007;
Planelles & Quilis, 2009; ZuHone, 2011), and on different cluster properties
such as their magnetic field (Roettiger et al., 1999; Brzycki & ZuHone, 2019),
halo shape and spin (Vitvitska et al., 2002; Moore et al., 2004; McMillan et
al., 2007; Drakos et al., 2019a), DM density profiles (Kazantzidis et al.,
2006; Drakos et al., 2019b) or the alignment between the DM halo and the
central galaxy (Ragone-Figueroa et al., 2020).

In this paper we focus on the impact that mergers have on the stellar
component of clusters and, particularly, in the stellar component of the
brightest cluster galaxy (BCG). BCGs include the most massive and luminous
galaxies in the Universe. They are large and elliptical galaxies, generally
located right at the centre of their host cluster. Consequently, their
formation and evolution are closely linked to those of the cluster, and hence
different from typical elliptical galaxies (Lin & Mohr, 2004; Brough et al.,
2005). In the hierarchical formation scenario, theoretical models have
predicted BCGs to assemble most of their mass through dry mergers with other
galaxies (Dubinski, 1998; Ruszkowski & Springel, 2009). Regarding the stellar
component of BCGs, recent studies – based on full physics hydrodynamical
simulations – have investigated BCG growth and compared to observational data,
finding good correspondence. They find that the mass growth was moderate since
$z=1$, with a growth factor $\lesssim 2$ in this period (Martizzi et al.,
2016; Ragone-Figueroa et al., 2018). Other studies, based on semi-analytical
models (SAMs), show that the greatest part of star formation in BCGs took
place before $z\sim 2$ (De Lucia et al., 2006; De Lucia & Blaizot, 2007). In
this sense, most BCGs in the observed universe are passive (Fraser-McKelvie et
al., 2014), with star formation occurring mostly at high redshifts. However,
it is also known that some BCGs have a significant amount of star formation at
low redshifts. Although this is in general related to clusters with cool cores
(Donahue et al., 2010; Liu et al., 2012), this is not a closed topic and the
fraction of BCGs with star formation is unclear (Runge & Yan, 2018).

In this work we use ‘The Three Hundred’ data set, that consists of regions of
diameter 30${{h^{-1}{\rm Mpc}}}$ centred on the 324 most massive objects found
within a cosmological dark matter only simulation of side length
1${{h^{-1}{\rm Gpc}}}$. Those regions have been re-simulated with Gadget-X,
i.e. full physics hydrodynamical code for cosmological simulations based upon
a modern SPH (Smoothed-Particle Hydrodynamics) solver (see Cui et al., 2018
for more details about the code and data set). Using these simulations, we
track the central object in each region from $z=0$ up to the highest redshift
where it can be found. By looking at the mass of each object and its
progenitors we find merger events and then study the effect they have on the
involved clusters. To study how the stellar component of the BCGs is affected
by cluster mergers we will first analyse directly the stellar particles in our
simulations. Then, we will also study the luminosity and colour of these
galaxies, to assess how mergers influence observations of BCGs.

The paper is organized as follows. In Section 2, we present the details of the
simulation and the halo catalogues and merger trees used to track the haloes.
In Section 3 we present the definition used to find mergers and how to define
their duration within our cluster sample. We present the mergers found and
describe them in terms of some of their properties. In Section 4 we analyse
the effects of cluster mergers on their stellar component by comparing a
merger sample and a control sample. Finally, in Section 5, we summarize and
discuss our results.

##  2 The Data

###  2.1 The Three Hundred Clusters

The 324 clusters in The Three Hundred data set were created upon the DM-only
MDPL2 MultiDark Simulation (Klypin et al., 2016), which is a periodic cube of
comoving length 1 ${{h^{-1}{\rm Gpc}}}$ containing $3840^{3}$ DM particles,
each of mass $1.5\cdot 10^{9}$ ${{h^{-1}{\rm{M_{\odot}}}}}$. The Plummer
equivalent softening of this simulation is 6.5 ${{h^{-1}{\rm kpc}}}$. The
cosmological parameters of the MDPL2 simulation are based on the Planck 2015
cosmology (Planck Collaboration et al., 2016a). From this simulation, the 324
clusters with the largest halo virial mass111The halo virial mass is defined
as the mass enclosed inside an overdensity of $\sim$98 times the critical
density of the universe (Bryan & Norman, 1998) at $z=0$ with
$M_{\mathrm{vir}}\gtrapprox 8\cdot 10^{14}{{h^{-1}{\rm{M_{\odot}}}}}$ were
selected. These clusters serve as the centre of spherical regions with radius
15 ${{h^{-1}{\rm Mpc}}}$, where the initial DM particles were split into dark
matter and gas particles (with masses $m_{\mathrm{DM}}=1.27\times 10^{9}$
${{h^{-1}{\rm{M_{\odot}}}}}$ and $m_{\mathrm{gas}}=2.36\times 10^{8}$
${{h^{-1}{\rm{M_{\odot}}}}}$ respectively), according to the cosmological
baryon fraction. These regions were then re-simulated from their initial
conditions including now full hydrodynamics using the SPH code Gadget-X.
Lower-resolution particles were used beyond 15 ${{h^{-1}{\rm Mpc}}}$, to
replicate any large-scale tidal effects on the cluster at a lower
computational cost. The output includes, for each of the 324 clusters, 129
snapshots between $z=0$ and $z=16.98$. At $z=0$, the 324 galaxy clusters have
a mass range from $M_{200}=6.4\cdot 10^{14}{{h^{-1}{\rm{M_{\odot}}}}}$ to
$M_{200}=2.65\cdot 10^{15}{{h^{-1}{\rm{M_{\odot}}}}}$. The size of our sample
is such that it allows for statistically significant subsamples to be
constructed. The Three Hundred data set was presented in an introductory paper
by Cui et al. (2018), and several other papers have been published based on
this data (see e.g. Wang et al., 2018; Mostoghiu et al., 2019; Haggar et al.,
2020; Herbonnet et al., 2021), to which we refer the reader for more details
about this project.

Regarding the code used for the re-simulations, Gadget-X is a modified version
of the non-public Gadget3 code (Murante et al., 2010; Rasia et al., 2015;
Planelles et al., 2017; Biffi et al., 2017), which evolves dark matter with
the Gadget3 Tree-PM gravity solver (an advanced version of the Gadget2 code;
Springel, 2005). It uses an improved SPH scheme that includes artificial
thermal diffusion, time-dependent artificial viscosity, high-order Wendland C4
interpolating kernel and wake-up scheme (see Beck et al., 2016 and Sembolini
et al., 2016 for a presentation of the performance of this SPH algorithm).
Star formation is carried out as in Tornatore et al. (2007), and follows the
star formation algorithm presented in Springel & Hernquist (2003). Black hole
(BH) growth and AGN feedback are implemented following Steinborn et al.
(2015), where super massive black holes (SMBHs) grow via Bondi-Hoyle like gas
accretion (Eddington limited), with the model distinguishing between a cold
and a hot component.

###  2.2 The Halo Catalogues & Merger Trees

All data were analysed with the open-source AHF halo finder (Gill et al.,
2004; Knollmann & Knebe, 2009), which includes both gas and stars in the halo
finding process. Haloes, as well as substructures, are found by locating
overdensities in an adaptively smoothed density field (see e.g. Knebe et al.,
2011 for more details on halo finders). For each halo identified, AHF computes
its $R_{200}$ radius, which is the radius $r$ at which the density
$\rho(r)=M(<r)/(4\pi r^{3}/3)$ drops below $200\rho_{\mathrm{crit}}$, where
$\rho_{\mathrm{crit}}$ is the critical density of the Universe at the
respective redshift. $R_{500}$, as well as the corresponding enclosed masses
$M_{200}$ and $M_{500}$, is defined accordingly. Subhaloes are defined as
haloes which lie within the $R_{200}$ region of a more massive halo, the so-
called host halo.

The luminosity (and magnitude) in any spectral band from the stars within the
haloes is calculated by applying the stellar population synthesis code
stardust (see Devriendt et al., 1999, and references therein for more
details). This code computes the spectral energy distribution (SED) from far-
UV to radio, for an instantaneous starburst of a given mass, age, and
metallicity. The stellar contribution to the total flux is calculated assuming
a Kennicutt initial mass function (Kennicutt, 1998).

Finally, in order to follow the evolution of haloes with redshift, we need to
trace them through the different snapshots. Merger trees are built for this
purpose with MergerTree, a tool that comes with the AHF package. MergerTree
follows each halo identified at $z=0$ backwards in time, using a merit
function to identify the main progenitor, as well as other progenitors, in the
previous snapshot. This tool also allows for skipping snapshots, so that the
halo merger tree does not have to be truncated if no suitable progenitor is
found in the immediately preceding snapshot. The main progenitor of halo A is
the halo B (at a previous redshift) that maximizes the merit function:
$\mathcal{M}=N^{2}_{AB}/(N_{A}N_{B})$, where $N_{A}$ and $N_{B}$ are the
number of particles in haloes A and B respectively, and $N_{AB}$ is the number
of particles that are in both haloes A and B. For more details on the
performance of MergerTree (and also different treebuilders) see Srisawat et
al. (2013).

##  3 Galaxy Cluster Mergers

In contrast to binary merger simulations, where all the focus is on the two
merging clusters (e.g. Poole et al., 2006, 2007; Donnert et al., 2013),
cosmological simulations require a clear method to find mergers and
distinguish them from other events in the evolution of a halo (e.g. Yu et al.,
2015; Nuza et al., 2012, 2017). Simulations also allow to determine with
clarity different merger properties, that can then be compared against other
works or even observations. These properties include the mass ratio between
the two merging clusters, which can significantly influence the outcome of a
merger (see Ricker & Sarazin, 2001; ZuHone, 2011), and the length of the whole
merger event. Detailed studies of the relaxation process after mergers can be
found in Valluri et al., 2007 (N-body simulations of controlled mergers) and
Faltenbacher et al., 2006 (study of one merging event at $z=0.6$ from a high
resolution cosmological N-body simulation).

In this Section we address the question of how to describe and study mergers
in The Three Hundred dataset. We start by defining mergers and how to find
them, and then classify them based on the duration of their effects in the
respective cluster.

###  3.1 Merger Definition: Mass Accretion History

The mass accretion history (MAH) of a cluster is obtained from the merger
trees, which track haloes from $z=0$ up to the highest redshift where their
progenitors are found (always smaller than the maximum redshift in the
snapshots, $z=16.98$). In this work, we are going to focus only on the central
halo at $z=0$ of each of the 324 regions and, initially and for defining
mergers, only on the main branch of its MAH. Similarly to other works (see
Wetzel et al., 2007 and Cohn & White, 2005), we can define a merger as a very
rapid increase in the mass of a halo, as opposed to a slow accretion of many
small objects over a long period of time. This way, in order to find mergers
in our simulated data, we need a clear definition of this ‘rapid’ increase.
The previously mentioned works simply compare the mass of a halo in two
consecutive snapshots, and require a minimum increase in the mass (e.g.
$25-50$ per cent). However, the time elapsed between two snapshots can change
with redshift and might not have any relation to timescales of physical
processes. We therefore see this decision as somewhat arbitrary. For this
reason, to set a time in which we can look for significant mass increases we
are going to use the dynamical time of clusters, $t_{d}$. The most common way
to define this (see e.g. Binney & Tremaine, 2008) is the crossing time, i.e.,
the time it takes for a particle to complete a significant fraction of its
orbit, which can be written as:

$t_{d}\simeq\frac{R_{200}}{V_{\mathrm{circ}}},$ (1)

where $V_{\mathrm{circ}}$ is the circular velocity, obtained as
$\sqrt{(GM_{200})/R_{200}}$. Given that the mass can be written in terms of
the critical density $\rho_{\mathrm{crit}}$ as
$M_{200}=200\rho_{\mathrm{crit}}(z)\frac{4\pi}{3}R_{\mathrm{200}}^{3}$, the
dynamical time can also be written as:

$t_{d}=\sqrt{\frac{3}{4\pi}\frac{1}{200G\rho_{\mathrm{crit}}}}.$ (2)

And since the critical density depends only on the cosmology as
$\rho_{\mathrm{crit}}(z)=3/(8\pi G)\cdot
H_{0}^{2}(\Omega_{M,0}(1+z)^{3}+\Omega_{\Lambda,0})$ we see that the dynamical
time evolves with redshift, but it is the same for all the clusters,
regardless of their mass or radius.

Since we are interested in significant mass increases over a (short enough)
period of time, we are going to use half of this dynamical time, looking for
an increase in the mass of 100 per cent during that period. In terms of the
mass of the cluster, this can also be written as $M_{f}\geq 2M_{i}$, where
$M_{i}$ ($M_{f}$) is the mass of the cluster at time $t_{i}$ ($t_{f}$) with
the difference between those times obeying $t_{f}-t_{i}\leq t_{d}/2$. In The
Three Hundred data set, with its chosen spacing between snapshots, we find
that for all redshifts, two snapshots correspond to $\lesssim t_{d}/2$. Using
the fractional mass change

$\frac{\Delta M}{M}=\frac{M_{f}-M_{i}}{M_{i}},$ (3)

our condition for a merger (i.e. $M_{f}/M_{i}\geq 2$) translates into $\Delta
M/M\geq 1$. We will study the evolution of $\Delta M/M$, using it to identify
all the mergers a cluster undergoes, but also the periods of slower accretion.
We further like to remark that using two snapshots is an upper limit: if the
required mass increase is reached in only one snapshot, then we also identify
that as a merger. Besides, it can also be the case that after identifying all
the mergers in the MAH of a cluster, we end up with consecutive mergers, with
one starting in the snapshot right after the one where the previous merger
ended. We do not consider these to be two different mergers and hence combine
them into a single one.

In summary, our definition of mergers consists of finding a mass increase of
100 per cent happening within half the cluster’s dynamical time, allowing for
possible extension over longer periods in case a ‘new’ merger was found right
afterwards. We search for such instances starting at the highest available
redshift for the MAH of the central halo of each region, and iterating forward
until $z=0$, obtaining a list of situations classified as mergers, with their
respective redshifts and mass increase ratios. Other works, like Planelles &
Quilis, 2009 and Chen et al., 2019, additionally compare the masses of the two
main progenitors, requiring for a merger that they are comparable. We will
return to this mass ratio later, using it instead as a way to characterise
mergers as opposed to adding it to its definition.

It is also worth mentioning that, when studying the mass evolution of haloes,
we have to be aware of their pseudo-evolution (see Diemer et al., 2013). Given
that the definition of $M_{200}$ is based on the critical density
$\rho_{\mathrm{crit}}$, there is an evolution of halo mass only due to this
reference density evolving with time, even if there is no physical accretion.
This is called pseudo-evolution, and in Diemer et al. (2013) it is shown that
it can account for a very significant amount of the mass growth. However, this
effect is specially important for galaxy-sized haloes ($M_{200}\lesssim
10^{12}{{{\rm{M_{\odot}}}}}$), and not for cluster-sized haloes like the ones
we are working with ($M_{200}>10^{14}{{{\rm{M_{\odot}}}}}$). Besides, pseudo-
evolution implies a slower and more continuous mass growth than what we are
defining as mergers, which involve short but significant mass increases. In
this sense, given our required $\Delta M/M\geq 1$ threshold, we can be
confident that the mass increases found are not due to pseudo-evolution but to
physical merger events.

###  3.2 Dynamical State Evolution

Mergers are not only associated with a mass growth but also with a disturbance
of the clusters’ dynamical state. Since mergers are fast increases in the mass
of a cluster, they are expected to significantly disturb clusters. In turn,
after enough time has elapsed and these effects weaken, clusters should be
dynamically relaxed again. For this reason, we are also going to study how the
dynamical state of the clusters in our sample evolves with time, so that we
can thus use its evolution around mergers to define pre- and post-merger
phases.

To quantify the dynamical state of clusters we use the so-called ‘relaxation
parameter’, introduced by Haggar et al. (2020) as:

$\chi_{\mathrm{DS}}=\sqrt{\frac{3}{\left(\frac{\Delta_{r}}{0.04}\right)^{2}+\left(\frac{f_{s}}{0.1}\right)^{2}+\left(\frac{|1-\eta|}{0.15}\right)^{2}}}.$
(4)

This equation is based on the three parameters initially introduced by Neto et
al. (2007) as proxies for relaxation, which were later used by Cui et al.
(2018) to study the relaxation of the clusters in The Three Hundred sample
(another study by De Luca et al., 2021 uses only the first two parameters).
The parameters are:

  * •

centre of mass offset, $\Delta_{r}$: offset of the centre of mass of the
cluster from the density peak of the cluster halo, as a fraction of the
cluster radius $R_{200}$;

  * •

subhalo mass fraction, $f_{s}$: fraction of the cluster mass contained in
subhaloes; and

  * •

virial ratio, $\eta$, defined as $\eta=(2T-E_{s})/|W|$, where $T$ is the total
kinetic energy of the cluster, $E_{s}$ its energy from surface pressure for
both gas and collision-less particles (Cui et al., 2017) and $W$ its total
potential energy.

Figure 1: Top panel, relaxation parameter $\chi_{\mathrm{DS,200}}$ as a
function of lookback time for two clusters selected as examples, 68 and 217.
Middle panel, $\Delta M/M=(M_{f}-M_{i})/M_{i}$, where the values in the x-axis
correspond to the time $t_{i}$. Bottom panel, time evolution of $M_{200}$ in
units of its value at $z=0$. We also provide the corresponding redshifts at
the top of the plots. The horizontal line indicates the threshold used to
identify mergers, i.e. $\Delta M/M=1$. In all panels, the start and end point
of the merger are indicated as red shaded regions that go (from right to left)
from $z_{\mathrm{start}}$ to $z_{\mathrm{end}}$ (vertical red dashed lines).
The yellow regions that overlap with them go from $z_{\mathrm{before}}$ to
$z_{\mathrm{after}}$ (vertical brown dashed lines): we identify this
particular region as ‘merger phase’, i.e the time interval a cluster is
disturbed by a merger.

For a cluster to be most relaxed, $\Delta_{r}$ and $f_{s}$ have to be
minimised, and $\eta\rightarrow 1$, and so ‘dynamically relaxed’ clusters have
$\chi_{\mathrm{DS}}\gtrsim 1$. Moreover, a cluster with increasing
$\chi_{\mathrm{DS}}$ is a relaxing cluster, whereas a sudden decrease in
$\chi_{\mathrm{DS}}$ means that the cluster’s equilibrium was disturbed.

To gain a better understanding of this $\chi_{\mathrm{DS}}$ parameter, we have
studied the influence of each of the individual parameters in the final value.
In Neto et al. (2007), where these parameters were first introduced, the
criterion of minimizing $\Delta_{r}$ is found to be the more restricting one;
while in Power et al. (2012), where both $\Delta_{r}$ and $\eta$ are studied,
$\Delta_{r}$ is found to be the most robust measure of dynamical state in
cosmological N-body simulations, showing a strong correlation with merger
activity. This is also reflected in our Eq. (4), where we have seen by
plotting different $\chi_{\mathrm{DS}}(z)$ curves (not shown here though) that
the parameter that carries the general shape, and therefore the one that
contributes the most is the centre of mass offset $\Delta_{r}$. Regarding
$f_{s}$, even though they follow different approaches, it depends on the
position of haloes and subhaloes like $\Delta_{r}$ does, and so these two
parameters show some correlation. On the other hand, the virial ratio $\eta$
makes use of kinematical information and thus probes the cluster in a
different way. It can be proven to be responsible for the more dramatic
changes in the $\chi_{\mathrm{DS}}(z)$ curve, rather than the general shape
like the other two parameters. Given all this, we are confident that using the
whole $\chi_{\mathrm{DS}}$ as in Eq. (4) gives robustness to our measurement
of the relaxation parameter, while reflecting all the relevant information
about the dynamical state of clusters; we will in turn use it now to quantify
characteristic merger times.

###  3.3 Characteristic merger times

As mentioned before, mergers are events that extend over a certain period of
time (according to our definition at most over half the cluster’s dynamical
time); neither the mass growth they produce nor the related effects they have
on clusters are instantaneous. In this sub-section we now like to define four
characteristic times of a merger:

  * a)

$z_{\rm before}$ marking the time right before the merger takes place and when
the cluster is still in equilibrium,

  * b)

$z_{\rm start}$ indicative of the onset of the merger (as defined via a pre-
defined jump in its MAH),

  * c)

$z_{\rm end}$ marking the end of the actual merger, and

  * d)

$z_{\rm after}$ that characterizes the time when the cluster experiences a new
relaxed phase.

To find $z_{\rm start}$ we need nothing else but the MAH: we define it to be
the redshift of the snapshot at time $t_{i}$ (see Eq. (3) above). For all
other characteristic redshifts we additionally inspect the evolution of the
dynamical state parameter $\chi_{\mathrm{DS}}$ as defined in the previous sub-
section. Below we provide more (technical) details about the actual
calculation of these four characteristic times.

In Fig. 1 we compare the dynamical state evolution of a cluster to its MAH for
two representative regions. The top panel of this figure shows the evolution
of $\chi_{\mathrm{DS}}(t)$222In this paper, $\chi_{\mathrm{DS}}(t)$ always
refers to the parameter estimated inside $R_{200}$. Although the sub-index is
indicated in Fig. 1, we drop it in the text for simplicity., whereas the two
lower panels show the corresponding MAH as quantified by $(\Delta M/M)(t)$
(middle panel) and $M_{200}(t)/M_{\mathrm{200,z=0}}$ (bottom panel). For
$\chi_{\mathrm{DS}}$, the curves shown are the result of taking a moving
average of the values for every three snapshots and then interpolating, in
order to filter the scatter. The curves show that, for every peak in $\Delta
M/M$, there is a corresponding decrease in $\chi_{\mathrm{DS}}$ that then
starts growing again, if there are no other similar events happening. As
expected, there is a clear correlation between dynamical state and MAH, such
that fast mass growths disturb clusters from their relaxed situations, but
only for a short period before they return to a relaxed phase. Even though we
only show two sample cases in Fig. 1, we confirm the same trends for all the
clusters in our sample.

We further highlight in Fig. 1 the aforementioned four characteristic times
(with the corresponding redshift ranges as shaded regions, see figure caption
for more details) whose definition in turn stems from the inspection of these
curves for our clusters:

  * •

$z_{\mathrm{start}}$: this time marks the beginning of the merger, which is
the moment when the cluster starts growing in mass. This can be simply
identified with the initial snapshot where the merger conditions are
satisfied, obtained as described in Section 3.1.

  * •

$z_{\mathrm{end}}$: this is the end point of the merger. To compute it, we
could similarly use the final snapshot identified in the process of finding
mergers, which, depending on the case, could be the snapshot that is strictly
consecutive to the initial one, two snapshots after the initial one, or even
longer if the merger is produced combining other mergers. However, this is
only a measure of when the mass growth rate has fallen below the threshold
set, which does not mean that the merger finished yet. For this reason, we use
instead the dynamical state information and, since the relaxation of clusters
decreases due to merger, we identify $z_{\mathrm{end}}$ with the first minimum
in the $\chi_{\mathrm{DS}}(z)$ curve after $z_{\mathrm{start}}$. This is the
moment the relaxation process restarts and the merger ends and its disturbing
effects end, respectively.

  * •

$z_{\mathrm{before}}$: even before we reach $\Delta M/M=1$ the two merging
objects start to influence each other. This is reflected in the decrease of
$\chi_{\mathrm{DS}}$ prior to $z_{\rm start}$. Therefore,
$z_{\mathrm{before}}$ can be identified with the first peak in the
$\chi_{\mathrm{DS}}(z)$ curve before the minimum which was identified as
$z_{\mathrm{end}}$.

  * •

$z_{\mathrm{after}}$: the last characteristic time is associated with the end
of the whole merger phase, i.e. the time when the cluster returned to a
relaxed position. As we have seen this is not the case for $z_{\rm end}$ which
marked the start of the relaxation after the merger. We therefore define
$z_{\mathrm{after}}$ to coincide with the first peak in
$\chi_{\mathrm{DS}}(z)$ after the minimum $z_{\mathrm{end}}$. This is an
indicator of when we can say that the cluster has ‘recovered’ from the merger.

In summary, we have defined four characteristic times during a merger event,
that can be computed for every merger using the MAH and dynamical state
information of the cluster. Note that the start of a merger $z_{\rm start}$ is
computed using only the MAH of the clusters, it does not include the mass
ratio of the two most massive progenitors: this ratio enters later into our
analysis (see Section 3.4.2, where we consider the masses of the main
progenitors that belong to the merging clusters). The other three times are
computed using dynamical state information $\chi_{\mathrm{DS}}$. Finally,
since we are working with discrete snapshots in the simulations, their
distributions will not be strictly continuous.

We further like to remark that our mass growth condition used in the
definition of a merger is certainly a free parameter and we also investigated
how our results are affected when lowering (or even increasing) it. Using
different threshold values simply means shifting the dot-dashed line shown in
the middle panel of Fig. 1 up or down. Here one can clearly see that our
choice rejects multiple minor mergers readily seen in the MAH as well as in
the dynamical state parameter. Our methodology and merger time definitions
will clearly hold for any such value. But for this particular study presented
here we are primarily interested in how major mergers affect the internal
properties of clusters and hence our choice of using mass growth by 100 per
cent.

###  3.4 Merger Sample

Applying our MAH-based criterion to the full range of available cluster
regions, we here specify the characteristics of the identified mergers a bit
more. This includes quantifying the mass ratio of the two most massive
progenitors, investigating the merger timescales, and defining a control
sample that did not undergo a merger. We will also introduce a lower mass
limit in order to only focus on mergers that are reasonably resolved in our
simulations.

####  3.4.1 Pre-selection of mergers

We have described in detail our methodology to find and characterise mergers
in our clusters sample. Before further analysing them we have to take into
account the numerical resolution. First, we apply a cut in the identified
value of $\Delta M/M$ for our mergers, and consider only instances where
$\Delta M/M\leq 3$. This is to ensure that the mass growth seen is physical
and not due to a misidentification by the halo finder (see Behroozi et al.,
2015). Then, to make sure that all of our mergers are well resolved, we
introduce a lower limit, $N_{\mathrm{cut}}$, in the number of particles in the
halo at $z_{\mathrm{start}}$, i.e. we discard all mergers where
$N_{\mathrm{part}}(z_{\mathrm{start}})<N_{\mathrm{cut}}$. We will further
investigate the effect of lowering $\Delta M/M$ from its usual value 1.

In Table 1 we show the total number of mergers found for three different
values of $N_{\mathrm{cut}}$; $1000$, $5000$ and $10000$, together with the
values for no limitation, “No $N_{\mathrm{cut}}$”333Note that the number of
particles per object is limited to 20 as per mode of operation of AHF.. We can
see how the number of found mergers decreases when increasing the cut in
number of particles, but even for the most conservative choice of $N_{\rm
cut}=10000$ (and the requirement to find a mass increase of a factor of 2) we
still find more than 170 mergers across the whole sample. In previous
convergence studies (Trenti et al., 2010) it is shown that, for individual
halo masses, $N_{\mathrm{part}}\gtrsim 5000$ is required to have an
uncertainty smaller than 10$\%$ in the obtained value. Although some
properties like the virial radius $R_{\mathrm{vir}}$ are very stable for low
numbers of particles (see Trenti et al., 2010), other properties like halo
shape (Allgood et al., 2006) or halo spin (Benson, 2017), may require even
more particles to have such a low uncertainty. As the prime objective of this
work is to quantify the influence of mergers on internal properties of
clusters, we prefer to apply the most conservative criterion and hence choose
$N_{\mathrm{part}}\gtrsim 10000$ for the remainder of this work: comparing the
columns for $N_{\mathrm{cut}}=5000$ and $10000$, we see that the differences
between them are relatively small, and using $N_{\mathrm{cut}}=10000$ we are
not losing that many mergers in our statistics.

We have additionally included in Table 1 possible variations of the other free
parameter we have in our merger definition, the mass increase limit. Again, as
we are primarily interested in major merger events, our usual choice is
$\Delta M/M=1$, but to gain insight into how our statistics might increase
when lowering this fractional mass increase we also provide the number of
mergers for $\Delta M/M=0.75$ and $\Delta M/M=0.5$. We see that in the three
cases the trends are the same, meaning that a lower mass growth criterion is
only reflected in the total number of mergers, and not in their distribution
regarding the number of particles in the halo.

Table 1: Total number of mergers depending on the different parameters chosen. Rows are different mass increase ratios, $\Delta M/M$, and columns show the threshold in the number of particles in the halo. Each threshold in $N_{\mathrm{part}}$ is also associated with a cut in redshift, where this number of particles is reached. The median redshift for each cut, computed including all the regions, is also indicated here. |  No $\bm{\mathrm{N_{cut}}}$ |  $\bm{\mathrm{N}_{\mathrm{cut}}=}$ 1000 | 5000 | 10000
---|---|---|---|---
median($z$): | 16.98 | 5.29 | 3.93 | 3.41
$\Delta M/M=1$ | 575 | 339 | 225 | 178
$\Delta M/M=0.75$ | 875 | 533 | 368 | 308
$\Delta M/M=0.5$ | 1440 | 885 | 666 | 560

In summary, with our selections, $\Delta M/M=1$ and $N_{\mathrm{cut}}=10000$
(which corresponds to approximately $M_{200}=8\cdot
10^{12}\,{{{\rm{M_{\odot}}}}}$), we have a sample of 178 mergers, where we
have also excluded situations where $\Delta M/M>3$ to guarantee that the
growth is physical. This way, even though we have been conservative in our
choices to ensure the quality of the mergers, we end up with a sample that is
still statistically significant.

####  3.4.2 Mass ratio between merging haloes

Using our MAH-based criterion to find mergers, we cannot yet say anything
about the ‘type’ of mergers, e.g. is it a ‘major’ or rather a ‘minor’ merger.
Therefore, and following other works (e.g. Planelles & Quilis, 2009; Chen et
al., 2019), a quantity that is very interesting to characterise mergers is the
mass ratio between the merging objects, which allows for the classification
into minor ($0.1\leq M_{2}/M_{1}<0.33$) and major mergers ($M_{2}/M_{1}\geq
0.33$), where $M_{2}$ and $M_{1}$ are simply the masses of the two main
progenitors, chosen so that this ratio is always $\leq 1$. To obtain this
ratio we use the trees created with MergerTree to find a list of all the
progenitors of the central halo at the initial snapshot of the merger. To find
the progenitor that is merging with the main progenitor, we select the one
that maximizes the function $M_{\mathrm{prog,i}}/M_{\mathrm{prog,1}}$, where
$M_{\mathrm{prog,1}}$ is the mass of the main progenitor and
$M_{\mathrm{prog,i}}$ the mass of each of the other progenitors.

Fig. 2 shows the distribution of this value for all the mergers in our sample,
for the three different $\Delta M/M$ thresholds. For $\Delta M/M=1$, it can be
seen that almost all the mergers are major mergers ($M_{2}/M_{1}>0.33$), with
a peak in the distribution at around $M_{2}/M_{1}=0.7$. This is reasonable,
since the main branch halo is increasing its mass by 100 per cent, and thus we
expect it to be merging with a very massive halo (and the remaining mass
coming from minor mergers and accretion). Looking at the other values of
$\Delta M/M$, we see that the distributions are shifted towards lower
$M_{2}/M_{1}$ ratios, meaning that the objects merging with the main branch
are now smaller – as expected. However, the differences between the three
distributions are quite small, and even for the $\Delta M/M=0.5$, where we
have 560 mergers (see Table 1), we can see that most of them are classified as
major mergers. In spite of this, for further calculations we are still going
to use the $\Delta M/M=1$ sample, since this is where we get the highest
number of major mergers, which is the scenario we are most interested in for
this particular study.

We like to restate that our ‘merger’ definition only uses the MAH of the
clusters (see Section 3.1), and hence the situations we find and classify as
‘mergers’ are in fact just very fast and significant mass growths. In this
subsection, by comparing the mass of the main and the (second) most massive
progenitors of the main halo, we have verified that these mass growths can in
fact be associated with (major) mergers, as demonstrated in Fig. 2. For the
cases where the ratio $M_{2}/M_{1}$ is lower, we have to take into
consideration that there are other (less frequent) situations that can lead to
a significant mass growth with a low $M_{2}/M_{1}$ ratio associated. These
include multiple-object mergers or simple accretion of material, which we are
not considering here.

Figure 2: Distributions of mass ratio between the two merging haloes,
$M_{2}/M_{1}$, computed as described in Section 3.4.2. The plot compares the
distributions for the three different mass increase ratios indicated, all with
the limit $N_{\mathrm{cut}}=10\,000$ in the number of particles in the halo.
The vertical lines designate the limits for the definitions of minor and major
mergers (0.10 and 0.33).

####  3.4.3 Merger timescales

Another interesting quantity to calculate is the timescale of the merger.
Using our characteristic merger times defined above in Section 3.3, we have
two options, i.e.

  * •

$z_{\rm end}-z_{\rm start}$: the length of the actual merger event (referred
to as ‘time of merger’), and

  * •

$z_{\rm after}-z_{\rm before}$: the ‘merger phase’, i.e. the time it takes to
go from the unperturbed pre-merger state to the new post-merger state.

To quantify the length of these periods, we convert the difference in redshift
to $\Delta t$ which will further be normalized by the dynamical time $t_{d}$
at the larger redshift 444To give the reader an approximate idea of the actual
values of these timescales, the dynamical time, $t_{d}$, of clusters (which
depends only on redshift, see Section 3.1) ranges from $\sim$1.4 Gyr at $z=0$
to $\sim$0.5 Gyr at $z=2$.. Fig. 3 shows the distribution of $\Delta t/t_{d}$
for the two different time spans. The plot is based upon our usual sample
(i.e. $\Delta M/M=1$ and $N_{\rm cut}=10000$), but only includes the results
for all the mergers for which the four characteristic times could be computed:
mergers that happened very recently may have not had enough time to reach
either the end of the merger or to relax again. This happens for 14 mergers
leaving us hence with 164 for that plot (our ‘reduced’ merger sample). In the
figure it can be seen that the distribution of the time of the merger peaks at
$\Delta t\sim 0.7t_{d}$, which could be expected given our merger definition,
that uses $0.5t_{d}$ to search for mass jumps. For the merger phase, the
distribution peaks at substantially higher times ($\sim 3t_{d}$) and is also
much wider, with values up to $8t_{d}$. This means that, while mergers are
rather rapid, returning to equilibrium takes clusters quite some time.

We have further correlated the time of merger with the merger phase (though
not explicitly shown here) where we find that the Spearman correlation
coefficient between these two quantities is $r_{s}\simeq 0.4$, which shows
that, in general, clusters that undergo a long lasting merger, also take more
time to relax afterwards, as expected.

Figure 3: Distributions of $\Delta t=t_{\mathrm{snap,i}}-t_{\mathrm{snap,f}}$
in units of the dynamical time, $t_{d}$, at $z_{\mathrm{start}}$. The two
different timescales go from $z_{\mathrm{start}}$ to $z_{\mathrm{end}}$ (solid
black) and from $z_{\mathrm{before}}$ to $z_{\mathrm{after}}$ (dash-dotted
blue).

###  3.5 Control Sample

Now that we have defined and characterised mergers, we have a very well-
defined merger sample. But before proceeding to study the effect mergers have
on different cluster properties, we require a ‘control sample’. Simply
comparing cluster properties before and after the merger is not sufficient for
drawing any conclusions about the effect of mergers on clusters. We need to
compare our results against a sample of clusters that did not undergo
(comparable) merger. For this reason, we need to define a control sample to be
compared against.

To create the control sample, we first obtain the values of
$z_{\mathrm{start}}$ and $M_{\mathrm{start}}=M(z_{\mathrm{start}})$ for each
merger in the merger sample. Then we check each of the other 323 clusters in
The Three Hundred sample, and select all those that, at
$z_{*}=z_{\mathrm{start}}$, have a mass within 1$\%$ of $M_{\mathrm{start}}$,
i.e., those which satisfy
$|M(z_{*})-M_{\mathrm{start}}|<0.01M_{\mathrm{start}}$. This way we obtain a
list of objects that has a comparable mass distribution at
$z_{\mathrm{start}}$ as the merger sample, and also the same distribution for
$z_{\mathrm{start}}$.

We further need to make sure that – within the same time span as characterized
by the ‘merger time’ – those objects do not undergo a merger. In order to
achieve this, we first need to set the corresponding characteristic times.
Since the values of $z_{\mathrm{start}}$ are already defined, we need to
define the other three characteristic redshifts. Focusing first on
$z_{\mathrm{end}}$, our aim is to mimic the distribution of the length of the
time of the merger itself. For that, we obtain the distribution
$z_{\mathrm{end}}-z_{\mathrm{start}}$ for the merger sample and resample from
there. This is, for each item in the control sample we pick a random value
from this distribution for the difference between these two times. We then
compute the value of the control $z_{\mathrm{end}}$ based on
$z_{\mathrm{start}}$. We can just repeat this process for
$z_{\mathrm{before}}$. For $z_{\mathrm{after}}$, since we want to also mimic
the distribution of the length of the whole merger phase, we resample from the
distribution of $z_{\mathrm{after}}-z_{\mathrm{before}}$, and derive the value
of the control $z_{\mathrm{after}}$. As a result, we obtain a control sample
consisting of a list of clusters with four characteristic times corresponding
to the values found for the merger sample.

We finally need to ensure that our objects selected this way do not undergo a
merger. For this purpose, we check the value of $\Delta M/M$ for every
instance in the control sample. We obtain this value by computing the value
for each snapshot from $z_{\mathrm{before}}$ to $z_{\mathrm{after}}$ and then
selecting the maximum. The mergers in our merger sample were identified
following the criterion $\Delta M/M\geq 1$ and so, to set an even more
restrictive criterion, we verify that every instance in our control sample
satisfies $\Delta M/M<0.8$. Besides, we also check that they satisfy the
resolution condition $N_{\mathrm{part}}(z_{\mathrm{start}})>N_{\mathrm{cut}}$
(see Section 3.4.1). We discard every instance where any of these two
conditions is not satisfied, obtaining our final control sample.

It is important to note that this method has a random component due to the
random resampling of the redshifts, and thus a different control sample can be
obtained in each execution. For consistency, we are going to work with a fixed
random seed, so that our sample is always the same for all the calculations
(we have checked different samples and got essentially the same results for
all). With this certain seed, we have created the control sample described,
and obtained a list of 250 items, which mimic the mergers in our merger
sample, but where no significant mass growth is happening.

As a confirmation of this, the top panel of Fig. 4 shows the distribution of
the value of $\Delta M/M$ for the mergers in the merger sample, together with
the same values obtained for the control sample. As we have imposed, the
values in the control sample range from 0 to 0.8, peaking around the centre of
this range, whereas the merger sample peaks at 1 (which is a lower limit due
to our choice), but shows a wider and decreasing distribution until the cut
made at 3. The bottom panel of Fig. 4 shows the distribution of the mass of
the cluster at $z_{\mathrm{start}}$, again comparing the merger sample with
our control sample. In this case we can see that the distributions are
similar, the range of masses being the same for both. This is further
confirmed by conducting a two-sample Kolmogórov-Smirnov (K-S) test, which
yields a p-value of 0.40 that implies that the two distributions could have
been drawn from the same parent distribution. This way we ensure that there
will be no mass bias in future comparisons between the merger and the control
sample. Note that in these plots we are working again with the ‘reduced’
samples, i.e. excluding the situations where the merger is not over yet (and
thus $z_{\mathrm{end}}$ or $z_{\mathrm{after}}$ are not found). This leaves us
with 164 mergers in the merger sample and 236 items in the control sample.

Figure 4: Top, distributions of the value of $\Delta M/M$ for the merger
sample (solid black) and the control sample (dash-dotted blue) described in
Section 3.5. Bottom, distributions of the mass of the cluster at
$z_{\mathrm{start}}$ for the same two samples.

##  4 Effect of mergers on BCGs

We will now focus on the brightest cluster galaxies (BCGs) – and particularly
on their stellar component – studying how mergers are affecting them. Based on
the previously described merger and control samples, we investigate properties
like colour and luminosity and assess the impact of (major) galaxy cluster
mergers on them. This is relevant for understanding the evolution of these
giant galaxies in the context of the hierarchichal formation scenario.

###  4.1 Stellar mass in BCGs

A simple way to analyse the stellar component is by measuring the mass in
stars in the central galaxy. Since the edge of a BCG is not clearly defined,
we determine it by selecting all the stars inside a certain radius, using the
halo centre as the origin. We call this parameter the radius of the central
aperture, $R_{ca}$, and we employ three different values for it: 30, 50 and 70
kpc, so that we can study the dependence of the results on the region
considered. This way of defining BCGs, based on a fixed physical (not
comoving) radius, has often been adopted in simulations. For instance,
McCarthy et al. (2010) used a radius of 30${{h^{-1}{\rm kpc}}}$, whereas in
Ragone-Figueroa et al. (2018) three different radii were used too (30 kpc, 50
kpc and $0.1R_{500}$). Kravtsov et al. (2018) advocate the use of stellar
masses defined this way for comparison with observations, and they provide
values for nine BCGs using several different radii. Besides, Stott et al.
(2010) showed that, using a 50 kpc aperture radius, BCG luminosities can be
recovered with less than 5 per cent difference from those obtained in some
observational analyses. Hence, using $R_{ca}$ allows us to easily count and
identify in the simulation the stellar particles that are in this central
region at every stage of the merger. We like to remark that we are only going
to use this definition at the characteristic merger times previously
described. This way, although because of the mergers the mass can be ill-
defined by this approach in-between these times, the halo centre is always
stable at these times, and thus the BCG is well identified.

As a way to check the obtained values and compare with the halo mass (shown in
Fig. 4), we have plotted in Fig. 5 the distribution of the stellar mass of the
BCGs at $z_{\mathrm{start}}$, only focusing on $R_{ca}=50$ kpc for simplicity.
In this figure, similarly to the lower panel in Fig. 4, we compare the
distributions for the merger and the control sample. We confirm again that our
control sample is not biased towards more (or less) massive BCGs.

Figure 5: Distribution of the stellar mass of the BCG at $z_{\mathrm{start}}$
for $R_{ca}=50$ kpc. Values for the merger sample (solid black) and the
control sample (dash-dotted blue). Figure 6: Top, median ‘new stellar mass’ in
the BCG of the clusters, defined by the physical aperture radius $R_{ca}$.
This mass is defined as the sum of the mass of all the stellar particles
identified at $z_{\mathrm{after}}$ that were not identified at
$z_{\mathrm{before}}$. In magenta the median for all the mergers in the merger
sample, and in green for the control sample. Shaded regions indicate the 25
and 75 percentiles. Bottom, median stellar mass that was formed during the
merger. Computed as the sum of the new stellar particles whose age is smaller
than the length of the merger phase.

To study the effect mergers have on BCGs, we analyse the stellar components
before and after the merger (see merger times as defined in Section 3.4.3),
comparing them to the control sample where no particularly fast mass growth is
taking place. In Fig. 6 we quantify the mass in new and newly formed stars as
a function of the applied $R_{ca}$ for both the merger and control sample. The
upper panel shows the values for the mass of new stars (i.e. stars that did
not belong to the BCG prior to the merger) that are identified in the central
region (BCG) of each cluster after a merger. To obtain this quantity we
identify all the stellar particles whose ID is identified at
$z_{\mathrm{after}}$ in the corresponding central region, but not at
$z_{\mathrm{before}}$, and then we sum all their masses. Note that the number
of ‘new stellar particles’ is not necessarily the difference between the
number of stars before and after the merger. We explicitly count new stars
that previously were not inside $R_{ca}$, not accounting for possible stellar
mass loss (which we find to be negligible, see discussion below). We compute
this mass for all the clusters in each sample and then plot the median values
(green and magenta dots) and the 25 and 75 percentiles (shaded regions). We
find that, as expected, the stellar mass of BCGs grows more in mergers than in
the control sample, especially when considering the outer regions up to
$R_{ca}=70$ kpc.

Here we also considered the stellar mass that is ‘lost’ during the merger, in
the form of stellar particles that leave the central region. However, we find
this quantity to be nearly independent of the radius $R_{ca}$ considered. The
median value of this ‘lost mass’ is $\sim 0.8\cdot
10^{11}{{{\rm{M_{\odot}}}}}$ for the merger sample and $\sim 0.6\cdot
10^{11}{{{\rm{M_{\odot}}}}}$ for the control sample. Comparing these values to
those in the upper plot of Fig. 6 we can see that the mass of new stars is
much greater, confirming that BCGs are growing significantly during this time.

The lower panel of Fig. 6 shows the median stellar mass formed during the
merger in each BCG. This was computed by comparing the age of all the new
stellar particles with the length of the merger phase in Gyr. The plot has the
same features as the upper panel plot, i.e. we see more star formation in the
merger sample, specially for $R_{ca}=50$ and $70$ kpc, where the difference is
around 70 per cent with the control sample. Comparing the quantities in both
plots, we note newly formed stars only contribute in $\sim 20$ per cent to the
total stellar mass growth. This leads to the conclusion that the main reason
for BCGs to grow during a merger is the accretion of existing stars, although
we also appreciate that actual star formation takes place during the merger.

Figure 7: 2D histogram of the (mass-weighted) age and metallicity of all the
BCG stellar particles for the indicated sample and time. For the first two
columns on the left: first column for the time before the merger, second
column after. Top row for the control sample and bottom row merger sample. The
two panels on the third column show the ratios between the two times (after
over before) for the control (top) and merger sample (bottom). The last plot,
on the bottom right corner, shows the ratio between the two plots on the third
column, i.e., the ratio after/before for the merger sample over the same ratio
for the control sample.

Finding both that stars are being lost and formed during the merger, it raises
the question if the two partaking BCGs actually merge to form a new BCG. To
shed light into this, we have compared the two BCG masses before the merger
and the resulting BCG mass after the merger. We found that the sum of the pre-
merger BCG masses in general only accounts for about 60 per cent of the new
BCG mass. This is in line with our previous finding that the new BCG contains
in fact new stars. We further confirm that in most of the cases one of the two
merging BCGs contributes nearly all of its stars with the other supplying only
a small fraction, whereas in some cases no merger between the BCGs occurs at
all. In the end we deduce that the situation is not as simple as ‘two BCGs
merge to form a new one’. There are stars accreted from the surroundings, new
stars are being formed, and this happens primarily in one of the partaking
BCGs.

An important concern emerging at this point relates the exact particulars of
the BH treatment, which has been shown to have relevant consequences not only
on BH properties themselves, but also on other components of the cosmological
simulation. For instance, Bahé et al. (2021) study the influence of BH
repositioning on baryonic components of galaxies. A different work by Ragone-
Figueroa et al. (2018) found that an improved ‘BH pinning’ can result in
smaller BCG masses, eventually leading to better agreement with observations.
Bassini et al. (2020) emphasise that BHs being moved from the centre is a
relevant problem in cosmological simulations, in particular for galaxy
clusters, where the absence of AGN feedback – due to exactly such ‘BH
displacement’ – can produce non-physically high star formation rates. As
merger events can be drivers for such displacements, we have therefore
carefully investigated any possible correlation between them and the star
formation observed during the merger. We find that during the merger the
overall BH mass inside the considered BCG aperture increases, but not
necessarily right at the centre. There is a median increase in the BH distance
to the centre of $2.0,4.5$ and $7.8$ kpc for the $30,50$ and $70$ kpc
apertures, respectively 555For more detail, the first and third quartiles of
these distributions are, respectively: -1.8 and 7.1, -2.1 and 12.3, -3.3 and
15.7 kpc. The negative values here indicate that the BH is becoming closer to
the centre.. Such an increase is not found for the control sample though. We
further cannot confirm any correlation of this increase with the observed rise
in newly formed stars during the merger. But we also need to remark that the
overall BH mass increase mentioned here not only refers to a mass growth of an
individual BH, it could also mean that additional BHs have been captured by
the BCG. They either directly come from the merger component(s) or constitute
BHs lost by satellite galaxies during their passages near the BCG. In any
case, there are hardly ever more than 2-3 BHs inside a BCG – irrespective of
the applied aperture – and their velocity distribution follows the same as
that of the stars making up the BCG. Considering the numerical challenges in
holding the BHs at the centre of their galaxies during a merging process, it
is not surprising to find, in some cases, that several BHs were not merged
into a single one.

###  4.2 Stellar age and metallicity

As we eventually aim at studying the effect of mergers on luminosities and
colours, we will first investigate here those values that are being used by
stardust to calculate these properties, i.e. stellar mass, age, and
metallicity. By conducting this study we make sure we understand the
properties of our BCG samples as given by the simulation before using them to
derive more observable properties in post-processing. We focus again on
stellar particles as found within a fixed aperture around the halo centre.

Our results are summarized in Fig. 7 where we show in the first $2\times 2$
plots the mass-weighted distributions of metallicity and age for the control
(top row) and merger (bottom row) sample before (first column) and after
(second column) merger. We chose to show 2D histograms as the pair age-
metallicity is relevant for the selection of the spectra in stardust (and the
stellar mass is also used as a weight in that code). For simplicity, we only
show here the values for the BCGs obtained using $R_{ca}=50$ kpc, but similar
results are obtained for the other two apertures. By comparing the first and
second columns, the aging of the stars can be seen in both samples. This
corresponds to the time elapsed during the merger phase, from ‘before’ to
‘after’ the merger: looking at Fig. 3 in Section 3.3, we can see that the peak
of this distribution is at $3\times t_{d}$. For $t_{d}\sim 1$ Gyr, the aging
is around 3 Gyr, in agreement with what is shown in the first two columns in
Fig. 7.

The temporal evolution of the 2D histograms has been quantified in the third
column, which shows the ratios between the distributions after and before the
merger for the control (top) and the merger (bottom) samples. The blue
regions, with ‘Ratio < 1’, indicate a decrease of stars for these pairs of age
and metallicity, while we find an increase in the red regions. Note that the
ratios are between the normalised histograms, rather than the absolute number
of star particles. This way, a ratio greater than 1 means that the percentage
of stars in a sample with that particular age-metallicity is increasing. If
the histograms were not normalised, the ratio would be > 1 in most of the
regions, since, in general, the number of stars in the BCG is increasing
during the merger, as we saw in the previous subsection.

Looking in detail at the third column in Fig. 7, the main finding is indeed
the aging of the stars. For both samples, the percentage of older stars is
higher after the merger, specially above 8 Gyr. Focusing now on the younger
stars, with ages below 2 Gyr, we see that the situation is very similar for
both samples, with the percentage of stars being higher before the merger than
after (blue region in the histograms). There is only a slight difference in
the region with ratio < 0.5 for ages below 2 Gyr, with this region being
smaller for the merger sample (bottom row). This can be interpreted as a hint
of what we saw in the previous subsection, that BCGs in the merger sample are
actually forming more new stars, and thus the difference for these young ages
with the control sample. To further observe this effect, the plot in the
bottom right corner of Fig. 7 shows the ratio between the two plots of the
third column. That is, it shows the ratio between the change in the stellar
distribution for the stellar particles in the merger sample over the same
change for the control sample. We observe that for all metallicities, the
increase in stars younger than 2 Gyr is greater for the merger sample, which
also validates our previous findings.

To summarize this subsection, we have studied the mass-weighted age-
metallicity distributions of all the BCG stellar particles, comparing the
merger and the control samples before and after the merger. We have only found
some small differences between them that nevertheless can have an impact on
observable properties like colour and luminosity, something we will analyse in
the following sub-subsection.

###  4.3 Colour and luminosity

Figure 8: Colour-magnitude diagram for the BCG of the 324 clusters in The
Three Hundred sample at $z=0$. The results for the three different values for
$R_{ca}$ are shown: 30 (dark blue), 50 (light blue) and 70 kpc (white). The
inset shows the distribution of BCG magnitudes for the observational SDSS
sample from Pipino et al. (2011). Figure 9: Comparison of colour-magnitude
relations for the merger (black dots) and the control sample (blue diamonds).
The values are for the BCG at $z_{\mathrm{before}}$ on the left and at
$z_{\mathrm{after}}$ on the right. The distributions of the $g-r$ colour and
the SDSS$-r$ magnitude are also shown in the plot for each sample (black solid
lines for merger and dash-dotted blue for control) and time, together with
their median values.

In the two previous subsections we have seen that the stellar component of
BCGs is growing significantly more in the merger sample than in the control
sample. Moreover, we have found that mergers produce a burst in star
formation, which we have confirmed by looking at the distribution of all the
stellar particles in age and metallicity. Although we have seen that these
bursts are, in general, weak (the mass formed is a small fraction of the final
stellar mass, leaving a small imprint on the age-metallicity distributions),
previous studies (Pérez-González et al., 2003) have shown that, even for
bursts as weak as 1 per cent, the young population has a major influence on
the optical colours of galaxies.

For this reason, in this subsection we are going to analyse the impact mergers
have on observable properties of the BCG. As new stars are formed during the
merger, we might find an effect on luminosity and – in particular – colour,
since young stars are hot and blue, as opposed to the red colours of the older
stars. In order to check this, we can use the stellar code stardust again
(Devriendt et al., 1999, see also Section 2), but this time only applying it
to the stars found in the BCG. stardust uses the given age and metallicity of
each stellar particle and obtains from a catalogue the full spectrum of a star
with those properties. Then, each spectrum is weighted by the mass of each
particle, and the sum of all of them yields the galaxy SED. This way, we can
compute the luminosities and magnitudes for the BCG stars, considering all the
identified stellar particles within each $R_{ca}$ region at each snapshot,
with their properties we already analysed. In particular, we have used this
code to obtain the luminosities in the SDSS bands, and from the magnitudes we
compute the widely used $g-r$ colour index.

####  4.3.1 Colour-Magnitude Diagrams

Before analysing changes in luminosity and/or colour due to mergers, we first
present our absolute values. In Fig. 8 we display a $g-r$ colour-magnitude (at
SDSS-$r$ band) diagram for all the BCGs in The Three Hundred sample at $z=0$,
computed for the three values adopted for $R_{ca}$. We see that the values for
70 kpc (in white) are in general brighter than those for $R_{ca}=50$ (light
blue) and 30 kpc (dark blue), which is in agreement with the BCGs being more
massive within the larger aperture region. A comparison to observations of the
values for the whole Three Hundred sample (not only BCGs but all galaxies) can
be seen in Cui et al. (2018). For Gadget-X, the code we are using in this
work, good agreement is found in general, with the colours being a bit lower
than observations ($\sim 0.1-0.2$). However, since in this work we are mainly
interested in relative changes, these small differences are not relevant for
us.

Comparing the absolute magnitudes in Fig. 8 to observational studies of BCGs
like Pipino et al. (2011) we see that our values are significantly larger.
This deviation can be due to different reasons. First, due to the nature of
our sample, which is a mass complete sample of the largest clusters in a full
1${{h^{-1}{\rm Gpc}}}$ side box, at $z=0$. The cluster sample in Pipino et al.
(2011) is 85 per cent complete, and spans over redshifts 0.1 to 0.3, so that
our magnitudes can be expected to be higher. Even when taking this into
account, the differences with observations are still significant. However,
this issue is not only seen in The Three Hundred simulations but also in other
simulations. For instance, in Bottrell et al. (2017) they find that galaxies
in the Illustris simulation (Vogelsberger et al., 2014) are roughly twice as
large and 0.7 mag brighter on average than galaxies in the SDSS. In
Vogelsberger et al. (2014) it is also shown that the number of galaxies at the
bright end of the galaxy luminosity function is higher than in observations.
Regarding BCGs, both IllustrisTNG (Springel et al., 2018; Pillepich et al.,
2018) and C-Eagle (Barnes et al., 2017; Bahé et al., 2017) simulations possess
significantly more massive BCGs than observed (Bahé et al., 2017; Henden et
al., 2020). This effect can cause the magnitudes to be higher too, as seen in
Fig. 8. In contrast to this, Ragone-Figueroa et al. (2018) obtain BCG masses
from cosmological simulations that are in good agreement with observations,
ascribing these results to a better control on the SMBH centering in the host
galaxy.

####  4.3.2 Merger-induced Colour-Magnitude Changes

Given the colour-magnitude values, we are interested in how they are affected
by mergers. For a qualitative analysis of this, we can first compare the
colour-magnitude diagrams of the merger and the control sample. Fig. 9 shows
this comparison (black dots symbolise the merger sample and blue diamonds the
control one) at the two relevant times before and after the merger, only for
$R_{ca}=50$ kpc. Above each plot and to its right, the distribution of each
variable is shown (solid line for merger sample and dash-dotted for control),
together with their median values. We can see that the situation before the
merger is quite different to the situation after it. While before the merger
(left panel) both samples are indistinguishable, in the right panel one sample
is moved with respect to the other, which is confirmed by the distinct
distributions in the smaller panels. The BCGs in the merger sample become, in
general, brighter and, although both samples increase their colour (i.e. they
become redder), we can see that the increase is significantly smaller for the
merger sample. Although we restricted the presentation of the results in Fig.
9 to the $R_{ca}=50$ kpc BCGs for clarity, the conclusions are similar for the
other aperture values, and we will discuss them in more detail below.

The results in Fig. 9 are in agreement with our previous findings in Section
4.1, BCGs grow in stellar mass with a certain fraction of it being newly
formed (and hence blue) stars. This entails that they become more luminous and
do not redden as quickly as their counterparts from the control sample. This
can be even better quantified by directly studying the change in these
properties.

In general, given a certain cluster property, a simple way of quantifying its
change between two points $z_{1}$ and $z_{2}$ is using the ‘fractional
difference’, computed as:

$\mathrm{FD}=-\frac{P(z_{1})-P(z_{2})}{P(z_{1})},$ (5)

where $P(z_{i})$ simply means the value of said property labelled as $P$ at
redshift $z_{i}$. As described before, for our merger study we have two
interesting intervals to work with, and so the pair $(z_{1},z_{2})$ can be
$(z_{\mathrm{start}},z_{\mathrm{end}})$ and
$(z_{\mathrm{before}},z_{\mathrm{after}})$, respectively. In the first case,
since the cluster is not relaxed yet at $z_{\mathrm{end}}$, we study the
immediate effects of mergers and how clusters are affected until the moment
the merger is completed. The second case is more relevant for long-term
effects, since it allows us to study the consequences of mergers that last
even when the cluster has dynamically recovered from the merger.

Figure 10: Distributions of fractional difference for SDSS$-u$ luminosity
obtained using Eq. (5) with $z_{1}=z_{\mathrm{before}}$ and
$z_{2}=z_{\mathrm{after}}$ (left) and $z_{1}=z_{\mathrm{start}}$ and
$z_{2}=z_{\mathrm{end}}$ (left). Results for the BCGs in the merger sample
(solid black) and the control sample (dash-dotted blue), using $R_{ca}=50$
kpc. Figure 11: Distributions of fractional difference for colour index $g-r$
obtained using Eq. (5) with $z_{1}=z_{\mathrm{before}}$ and
$z_{2}=z_{\mathrm{after}}$ (left) and $z_{1}=z_{\mathrm{start}}$ and
$z_{2}=z_{\mathrm{end}}$ (left). Results for the BCGs in the merger sample
(solid black) and the control sample (dash-dotted blue), using $R_{ca}=50$ kpc

In Fig. 10 we show the distribution of the fractional difference obtained
using Eq. (5) for the luminosity of the BCGs in the SDSS-$u$ band, which we
chose in order to focus on the young hot stars. In each plot we compare the
distributions obtained for the merger sample (solid black line) and control
sample (dash-dotted blue line). Again, in the plot we only show the results
for $R_{ca}=50$ kpc for clarity, but we will discuss the other values below.
We first focus on the left panel, which studies the whole merger phase between
$z_{\rm before}$ and $z_{\rm after}$. We see that, for the control sample the
distribution peaks at negative values, meaning that for most BCGs the SDSS-$u$
luminosity is decreasing during the merger. On the other hand, looking at the
merger sample, it can be seen that the distribution does not show such a
prominent peak, and it is moved to higher values of the fractional difference,
showing that decreases in luminosity are less frequent in this sample. This
effect is less remarkable when checking the right panel, which compares only
changes taking place between the start and end of the merger itself. Here both
distributions peak at around 0, meaning no significant change in luminosity.
While we quantify this even more below (also including the other apertures),
here we can already draw some conclusions. As seen for the control sample,
BCGs tend to decrease their luminosity in the SDSS-$u$ band in the time
interval studied. This can be attributed to the stars getting older. On the
contrary, newly formed stars are bright at these wavelengths, and hence the
decrease in luminosity is smaller for the merger sample, where a certain
amount of new stars is being formed (see Fig. 6), adding to the SDSS-$u$
luminosity.

Fig. 11 shows the same results as Fig. 10 but for the colour index $g-r$. For
this index we have to keep in mind that an increase in its value means that
the actual colour is becoming redder and vice versa. In this case Fig. 11
shows that, when looking at the whole merger phase (left panel) the difference
is not as significant as with luminosity. However, we still see that the
merger sample is slightly shifted to the left, meaning that BCGs in the
control sample are becoming redder than those in the merger sample. In the
right panel, that focuses only on the interval $[z_{\rm start},z_{\rm end}]$,
the difference between the two distributions is even smaller.

Figure 12: Median values (dots) and 25 and 75 percentiles (shaded regions) of the distributions of the fractional difference computed for two different properties and time periods. All plots compare the results of the merger (magenta) and the control (green) samples. The plots in the upper row are for SDSS-$u$ luminosity, and those in the lower row are for $g-r$ colour index. For both rows, the plots on the left are for the whole merger phase ($z_{\mathrm{before}}$ to $z_{\mathrm{after}}$), while those on the right compare the time of the merger ($z_{\mathrm{start}}$ to $z_{\mathrm{end}}$). Table 2: Resulting $p$-values of the K-S test conducted between the distributions of the indicated fractional differences for the merger and the control sample. | Lum SDSS-$\bm{u}$ |  Colour $\bm{g-r}$
---|---|---
$\bm{R_{ca}}$ (kpc) | Before-After | Start-End | Before-After | Start-End
30 | 0.302 | 0.133 | 0.345 | 0.800
50 | $<10^{-3}$ | 0.015 | 0.295 | 0.256
70 | $<10^{-5}$ | $<10^{-6}$ | 0.098 | 0.005

For a quantitative assessment of the similarity between the distributions, we
have computed the two-sample Kolmogórov-Smirnov (K-S) test between each
merger-control sample pair. We have done this not only for the distributions
shown for the BCGs defined using $R_{ca}=50$ kpc, but also for the same
distributions for 30 and 70 kpc. For $R_{ca}=30$ kpc, we see that for both
luminosity and colour the p-values are high, meaning that the two samples in
each pair could have been drawn from the same distribution. For the luminosity
within $R_{ca}=50$ kpc, the low p-value for the whole merger phase confirms
that mergers do make a difference in BCGs’ luminosity, an effect that is
intensified for 70 kpc. This situation does not remain for the colour, where
the p-values are still high for 50 kpc, and hence we cannot say that the two
distributions are actually distinct. The changes become more significant for
70 kpc, where the p-values decrease. This can also be explained by star
formation, since we have seen that more stars are formed in this region than
in the others.

To better quantify all of these findings, we have computed the median values
of each of the distributions in Figs. 10 and 11, as well as the first and
third quartiles. We have done this for the three different regions defining
BCGs, and we show all these values in Fig. 12. The results for the merger
sample (in magenta) are compared with those for the control sample (in green).
The dots depict the median value of each distribution, while the shaded
regions are the 25 and 75 percentiles. This figure allows for an easy
comparison of the effects of mergers on luminosity and colour of the involved
BCGs. The upper row of Fig. 12 shows the results for the SDSS-$u$ luminosity.
For the whole merger phase (left panel), we see that, regardless of the region
considered, the BCGs in the control sample decrease their luminosity in
$\sim$20 per cent in median during this time. On the other hand, for BCGs
undergoing mergers, the luminosity increases up to $\sim$20 per cent for the
outermost region of 70 kpc, while the difference is significantly smaller for
$R_{ca}=30$ kpc. Similar trends hold when looking at the shorter time interval
between $z_{\mathrm{start}}$ and $z_{\mathrm{end}}$ (right panel), although
the differences between merger and control sample are now smaller. We have
further checked the correlation between the change in luminosity and the
stellar mass growth (not explicitly shown here though) and found a Spearman
correlation coefficient of $\sim$0.40 for the three values of $R_{ca}$.
Moreover, when considering only the growth due to stars recently formed (see
Fig. 6) the correlation grows to $\sim$0.45 while it decreases to $\sim$0.3
when considering only the accreted stars. This suggests that star formation
produces the increase (or better formulated, reduced decrease) in luminosity
in the merger sample, so that for the control sample, where fewer stars are
formed, the luminosity decreases, driven by the aging of stars. The lower row
of Fig. 12 shows the same results as the upper one, but for the colour index
$g-r$. We see again that, for the innermost region, the distributions for the
merger and the control sample are essentially the same (as inferred from Table
2). For $R_{ca}=50$ and 70 kpc we can see that the medians are higher for the
control sample. This is in agreement with our previous findings, as seen in
Fig. 11, that BCGs in mergers become less red than those growing ‘normally’.
However, it is important to note that, although there is a difference in the
medians, the differences are very small for the two time intervals considered.

To sum up, we find that the luminosity of BCGs in clusters that have recently
experienced a major merger is expected to be higher than the luminosity of
those that did not. This is due to the fact that mergers produce an increase
in star formation in the BCG, with new stars being hot and bright. Although a
small difference can already be seen for the innermost region of the BCG, the
effects are significant and stronger in the outer regions of these galaxies.
BCGs in mergers also accrete a significant number of stars ($\sim$4-6 times
more stars than those formed during the merger), and although they contribute
to the luminosity increase too, this contribution is less relevant. Regarding
the colour, the accretion of older stars combined with the aging of all the
stars makes BCGs become redder in both merging and non-merging clusters. This
colour increase is a bit slower in the merger sample (due to newly formed
stars), although the difference might not be significant enough to be
observable for $R_{ca}=30$ and 50 kpc. The effect is stronger for the
$R_{ca}=70$ kpc region, which is also where star formation is the highest.

##  5 Conclusions

In this work we studied the highly energetic events of galaxy cluster mergers
as they happen in cosmological simulations. These events are very important in
the hierarchical model of structure formation, where they play an important
role in the growth of structure. Regarding the multiple effects they can have
on both the cluster itself and its individual galaxies, we focused on the
brightest cluster galaxies, which include the most massive and luminous
galaxies in the universe. We studied how cluster mergers affect their stellar
component, and how this reflects on their observable properties colour and
luminosity.

We used the sample of 324 numerically modelled galaxy cluster haloes provided
by The Three Hundred project, all with $M_{200}\geq 6.4\cdot
10^{14}{{h^{-1}{\rm{M_{\odot}}}}}$. We tracked the evolution of the central
haloes found at $z=0$ back to the highest redshift where they could be found.
Using their mass accretion history, we identified possible merger situations
as mass growths of at least 100 per cent happening in less than half a
dynamical time of the cluster. The relaxation parameter $\chi_{\mathrm{DS}}$
was used to quantify the dynamical state of clusters. By studying its
evolution around the time of the merger as identified in the MAH, we
distinguished different characteristic merger times: these are given by the
times when the clusters are relaxed (i.e. reach a peak in their
$\chi_{\mathrm{DS}}$ curve) before and after the merger, and the time when
they are most disturbed (i.e. reach a minimum) after a merger (see Fig. 1 for
a depiction). As we were interested in the effect of mergers on internal
cluster properties, we discarded, for resolution reasons, objects with
$N<10000$ at the largest redshift found (which in median means a cut at $z\sim
3.4$). We also removed objects for which the mass growth was too high, $\Delta
M/M>3$; such instances are reminiscent of mis-identifications in the merger
tree (Srisawat et al., 2013). We ended up with a list of 178 objects. Ab
initio, they simply define a sample of ‘fast mass growing’ clusters, but – as
verified in Fig. 2 which shows the ratio between the two most massive merging
components – they can be associated with actual major mergers. Regarding their
timescales, while mergers themselves take less than one dynamical time $t_{d}$
(as imposed by our definition), the whole merger phase lasts in median
$3\times t_{d}$ (although the values can be as large as $7-8\times t_{d}$).
Having a well-defined merger sample, we also created a control sample
imitating its features, but with no significant mass growth of the haloes.

Defining the BCG alternatively as the region within 30, 50 and 70 kpc of the
halo centre, we compared the stellar content of BCGs before and after the
merger. By doing this for both the merger and the control sample, we found
that BCGs in mergers accrete around twice as much stellar mass as those just
‘normally’ growing. Comparing the stellar mass formed during the merger we
found, using radii 50 and 70 kpc, that BCGs in mergers also form around 70 per
cent more stars than those in the control sample. This number drops to 30 per
cent when considering only the innermost region of 30 kpc.

It could be argued that BCGs are not strictly always located at the halo
centre, a situation that is in general related to galaxy clusters mergers.
However, given the way our applied halo finder AHF works, we can be confident
that this is not affecting our results. AHF locates the halo centre as a peak
in the density field, and hence $R_{ca}$ will always be placed at the highest
density peak, which is precisely the BCG. Besides, previous studies (Martel et
al., 2014; De Propris et al., 2021) have shown that, although BCGs might not
be in the centre when their host cluster is unrelaxed, they always go back to
the centre after some relaxation time (see also De Luca et al., 2021 for a
similar study using The Three Hundred dataset). And since for this analysis we
have only used the characteristic times $z_{\mathrm{before}}$ and
$z_{\mathrm{after}}$, which are the times when the cluster is most relaxed, we
can be confident with our method. For this same reason, we are not worried
about possible ‘halo swaps’ due to halo finder problems during a merger (see
Behroozi et al., 2015), given that we focus our study on the times before and
after the merger.

We analysed the mass-weighted age-metallicity distributions for all the
stellar particles in the BCGs, as these are the properties that enter into the
calculation of galaxies’ luminosities and magnitudes via stardust. We did this
for the merger and control samples before and after the merger (Fig. 7). We
found that, although the differences were small, there was a slight increase
of young stars after the merger in the merger sample with respect to the
control sample, which we attributed to the newly formed stars previously
found.

To assess the effect of mergers on readily accessible observational
properties, we studied the luminosity in the SDSS-$u$ band and $g-r$ colour of
the involved BCGs (computed with stardust), again for the three different
central aperture radii. For the BCGs within 70 kpc we found that, along the
whole merger phase, the luminosity of BCGs in clusters that underwent a merger
increased by $\sim$20 per cent in median. This was opposed to a decrease of
the same amount in the control sample. For the 50 kpc region we also found an
increase in luminosity but only of 5 per cent in median for the merger sample,
opposed to a 20 per cent decrease in the control sample, while the difference
is even smaller using a 30 kpc radius. We attribute this change in luminosity
to the new stars that are being formed during the merger, which are more
numerous in the outer regions. It has been shown (Pérez-González et al., 2003)
that even a few very young stars can make a difference in galaxies’ observable
properties. There is also a less relevant contribution from the accreted
stars, whose combined mass is also higher in the outer regions (see Fig. 6).

Regarding the colours, we computed the colour index $g-r$ and compared its
values at the four characteristic times. We found that BCGs in both the merger
and control samples become redder with time in general, which is attributed to
the aging of their stellar component. However, we found that this trend is
slightly slowed down by cluster mergers. Again, the more significant effects
occur within 70 kpc of the halo centre, where the colour growth is 14 per cent
slower in the merger sample. Similar differences were found for the smaller
regions within 50 and 30 kpc, although in this case they are not statistically
significant, and thus might not be observed. These results are in agreement
with our previous results regarding stellar mass changes, confirming that
there is a burst in star formation in the BCG due to the cluster merging,
which is stronger in the outer regions of the BCG (up to 70 kpc), and which is
not the main mechanism for the galaxy to grow.

Previous studies have shown that galaxy cluster mergers can produce a burst in
star formation in the individual galaxies, which in turn can change their
observed colours (Roettiger et al., 1996; Miller, 2005; Johnston-Hollitt et
al., 2008). These studies do not focus on BCGs, but on the whole galaxy
population of the cluster. In respect to BCGs, some works, like Pipino et al.
(2009) and Cerulo et al. (2019), have investigated blue BCGs, relating them to
recent star formation. However, the relation between star formation in BCGs
and major cluster mergers has not been that thoroughly studied in the
literature. Even so, we believe it is relevant for understanding both the
evolution of BCGs and the importance of major cluster mergers in the
cosmological context. Nevertheless, in future studies we could also include
these individual galaxies for a more complete analysis.

## Acknowledgements

We thank the anonymous referee for raising interesting questions whose answers
helped to improve the paper.

This work has been made possible by The Three Hundred
(https://the300-project.org) collaboration. The simulations used in this paper
have been performed in the MareNostrum Supercomputer at the Barcelona
Supercomputing Center, thanks to CPU time granted by the Red Española de
Supercomputación. As part of The Three Hundred project, this work has received
financial support from the European Union’s Horizon 2020 Research and
Innovation programme under the Marie Sklodowskaw-Curie grant agreement number
734374, the LACEGAL project.

AC, AK, RM and GY are supported by the MICIU/FEDER through grant number
PGC2018-094975-C21. AK further thanks Prefab Sprout for Steve McQueen. RH
acknowledges support from STFC through a studentship. WC is supported by the
European Research Council under grant number 670193 and by the STFC AGP Grant
ST/V000594/1. He further acknowledges the science research grants from the
China Manned Space Project with NO. CMS-CSST-2021-A01 and CMS-
CSST-2021-B01.MDP acknowledges support from Sapienza University in Rome thanks
to Progetti di Ricerca Medi 2019 and 2020, RM11916B7540DD8D and
RM120172B32D5BE2. SEN is member of the Carrera del Investigador Científico of
CONICET. He acknowledges support by the Agencia Nacional de Promoción
Científica y Tecnológica under the grant PICT-201-0667.

## Data Availability

The results shown in this work use data from The Three Hundred galaxy clusters
sample. These data are available on request following the guidelines of The
Three Hundred collaboration, at https://www.the300-project.org. The data
specifically shown in this paper will be shared upon request to the authors.

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