# Galaxy pairs in The Three Hundred simulations: a study on the performance of
observational pair-finding techniques

Ana Contreras-Santos,1 Alexander Knebe,1,2,3 Weiguang Cui,1,4 Roan Haggar,5
Frazer Pearce,5 Meghan Gray,5 Marco De Petris,6,7 and Gustavo Yepes1,2
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
4Institute for Astronomy, University of Edinburgh, Royal Observatory,
Edinburgh EH9 3HJ, United Kingdom
5School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD,
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
Contact e-mail<EMAIL_ADDRESS>
(Last updated 2015 May 22; in original form 2013 September 5)

###### Abstract

Close pairs of galaxies have been broadly studied in the literature as a way
to understand galaxy interactions and mergers. In observations they are
usually defined by setting a maximum separation in the sky and in velocity
along the line of sight, and finding galaxies within these ranges. However,
when observing the sky, projection effects can affect the results, by creating
spurious pairs that are not close in physical distance. In this work we mimic
these observational techniques to find pairs in The Three Hundred simulations
of clusters of galaxies. The galaxies’ 3D coordinates are projected into 2D,
with Hubble flow included for their line-of-sight velocities. The pairs found
are classified into “good” or “bad” depending on whether their 3D separations
are within the 2D spatial limit or not. We find that the fraction of good
pairs can be between 30 and 60 per cent depending on the thresholds used in
observations. Studying the ratios of observable properties between the pair
member galaxies, we find that the likelihood of a pair being “good” can be
increased by around 40, 20 and 30 per cent if the given pair has,
respectively, a mass ratio below 0.2, metallicity ratio above 0.8, or colour
ratio below 0.8. Moreover, shape and stellar-to-halo mass ratios respectively
below 0.4 and 0.2 can increase the likelihood by 50 to 100 per cent. These
results suggest that these properties can be used to increase the chance of
finding good pairs in observations of galaxy clusters and their environment.

###### keywords:

methods: numerical – galaxies: clusters: general – galaxies: general –
galaxies: interactions

††pubyear: 2021††pagerange: Galaxy pairs in The Three Hundred simulations: a
study on the performance of observational pair-finding techniques–Galaxy pairs
in The Three Hundred simulations: a study on the performance of observational
pair-finding techniques

##  1 Introduction

The $\Lambda$ cold dark matter ($\Lambda$CDM) growth paradigm for the Universe
describes a hierarchical model of structure formation, where mergers between
lower mass objects yield more massive systems (White & Rees, 1978; Frenk &
White, 2012). In this context, interactions and mergers between galaxies are
expected and these processes can play a very important role in galaxy
formation and evolution. As such, several efforts have been devoted to
studying the effects of galaxy interactions that can result in property
changes of the involved galaxies.

An essential quantity for studying the effects of mergers on galaxy formation
and evolution is the fraction of galaxies that are undergoing a merger event.
In observations, if high resolution images are available, galaxies undergoing
mergers can be identified via their morphology (Conselice et al., 2003; Lotz
et al., 2004; Lotz et al., 2008; López-Sanjuan et al., 2009). Some studies
even identify the stage of the merger the galaxy is in (Pawlik et al., 2016).
However, this kind of data is not always acquirable, even more so if we want
to study higher redshifts. A common approach taken at this point is to study
galaxies that are close in the sky – close pairs (e.g. Carlberg et al., 1994;
Patton et al., 2000; Kartaltepe et al., 2007; Duncan et al., 2019). Although
they do not directly trace mergers, close galaxies are more likely to merge in
the near future. Assuming a certain merger timescale, the close-pair fraction
can also be used to estimate the merger rate of galaxies (Kitzbichler & White,
2008; Xu et al., 2012; Mundy et al., 2017).

When compared to more isolated galaxies, galaxies with close companions have
been shown to have enhanced star formation (Barton et al., 2000; Li et al.,
2008a; Scudder et al., 2012; Patton et al., 2013; Pan et al., 2018) and
diluted metallicities (Kewley et al., 2010; Rupke et al., 2010; Bustamante et
al., 2020). Some studies also show that close companions can induce the
triggering of active galactic nuclei (AGN) (Silverman et al., 2011; Cotini et
al., 2013; Ellison et al., 2019), although this has been a more debated topic,
with some studies also failing to find a correlation between galaxy
interactions and AGN enhancement (Li et al., 2008b; Cisternas et al., 2011;
Shah et al., 2020). The dependence of these results on the environment the
galaxies are in has also received some attention in the literature, suggesting
that, in general, the tendencies are kept even when entering in high-density
regions (Perez et al., 2006; Alonso et al., 2006, 2012).

To clarify this kind of matters and gain a better understanding of the topic,
a more theoretical approach like the one given by numerical simulations can be
very useful. Compared to observations, simulations have the advantage that
they allow us to access different snapshots, and thus study what happens to
the galaxies in a future or past moment. Previous works have studied the boost
of star formation due to close companions (for example Patton et al. 2020 in
IllustrisTNG or Rodríguez Montero et al. 2019 in the Simba simulation).
Bustamante et al. (2018) show that the metallicity dilution due to galaxy
interactions is also seen in cosmological simulations. Idealized simulations
of galaxy mergers have also found an increased AGN activity (Di Matteo et al.,
2005; Torrey et al., 2012).

When trying to unify all these studies about close pairs, one problem that
arises is regarding the selection criteria. In observations, galaxies are
defined as “close” based on their separation in the sky and their separation
in velocity along the line of sight. However, no unified threshold exists for
these parameters to define a close pair. For the separation in the sky, the
values range from 5 ${{h^{-1}{\rm kpc}}}$ up to even 100 ${{h^{-1}{\rm
kpc}}}$, depending on the particulars of the study. For the velocity
separation, the threshold is usually set to be between 300 and 1000 km/s
difference. Studies like Huško et al. (2022) convert from one definition to
another by assuming a power-law dependence of the close pair fraction on the
maximum separation, but the comparisons remain difficult due to differing
methodology.

In simulations, the 3D separation alone can be used to find close galaxies,
although again the thresholds can differ depending on the study. By using
directly the 3D coordinates, rather than 2D separation, simulations eliminate
the problem of projection effects which can create spurious pairs of galaxies.
Simulations also allow for checking if the two galaxies are in fact
gravitationally bound, although we will not focus on that in this particular
work.

The motivation for this project is to reduce this gap between observations and
simulations, by answering the question of how likely it is that an observed
galaxy pair (or group) in the sky is actually close in 3D. We aim to do this
by using hydrodynamical simulations of galaxy clusters to mock the
observational techniques used to find pairs in 2D. By measuring then the 3D
separation between the galaxies in these pairs, we can assess the performance
of these methods and thus the importance of the projection effects. While the
effects of spurious pairs are expected to be stronger in high-density regions
(Mamon, 1986, 1987), a study by Alonso et al. (2004) found that real pairs
dominate the statistics regardless of the environment, given the thresholds
$r_{\mathrm{sep}}<100$ ${{h^{-1}{\rm kpc}}}$ in distance and
$v_{\mathrm{sep}}<350$ km/s in velocity. We intend to investigate this in more
detail, exploring how it depends on the definition of “close” chosen, and if
there are any properties of the pairs that influence it. We will also consider
the circumstance of a galaxy having more than one close companion, i.e., being
in a group, and similarly explore these situations.

In this work we are going to use The Three Hundred
simulations111https://the300-project.org/, which are 324 re-simulations of the
most massive clusters in a dark matter only cosmological simulation. These
clusters provide a high-density environment in where the interactions between
galaxies are more frequent and can be especially important. On large scales,
clusters are dominated by dark matter and hence gravity. But on smaller
scales, also the interaction of the baryonic components of clusters plays an
important role (see Kravtsov & Borgani, 2012 for a review on galaxy clusters).
This leads to several different phenomena that drive galaxy evolution, also
making clusters very interesting environments to study galaxy interactions
(Gnedin, 2003; Park & Hwang, 2009; Boselli et al., 2014). Using The Three
Hundred simulations allows us to have lots of statistics, with many pairs and
groups to study, belonging to clusters with different properties, and hence
different environments.

The paper is organized as follows. In Section 2, we present the details of the
simulation and the halo catalogues used to identify the haloes. In Section 3
we present the method used to find close pairs of galaxies. For the pairs
found this way, in Section 4 we present some statistics regarding the number
of pairs found. In Section 5 we separate our projected pairs into “good” and
“bad” by measuring their 3D distance compared to the 2D one. In the following
section, Section 6, we study if there is any correlation between observational
properties of the pairs and their “goodness”. Finally, in Section 7, we
summarize and discuss our results.

##  2 The Data

###  2.1 The Three Hundred Clusters

The clusters in The Three Hundred data set were created upon the DM-only MDPL2
MultiDark Simulation222The MultiDark simulations – incl. the MDPL2 used here –
are publicly available at https://www.cosmosim.org (Klypin et al., 2016). This
simulation is a periodic cube of comoving length 1 ${{h^{-1}{\rm Gpc}}}$
containing $3840^{3}$ DM particles, each of mass $1.5\times 10^{9}$
${{h^{-1}{\rm{M_{\odot}}}}}$, with cosmological parameters based on the Planck
2015 cosmology (Planck Collaboration et al., 2016). From this simulation, the
324 clusters with the largest halo virial mass333The 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 $z=0$ were
selected. These clusters serve as the centre of spherical regions with radius
15 ${{h^{-1}{\rm Mpc}}}$, where the initial DM particles were traced back to
their initial conditions and then split into dark matter and gas particles
according to the cosmological baryon fraction; leading to a dark matter and
gas particle mass resolution of $m_{\mathrm{DM}}=1.27\times 10^{9}$
${{h^{-1}{\rm{M_{\odot}}}}}$ and $m_{\mathrm{gas}}=2.36\times 10^{8}$
${{h^{-1}{\rm{M_{\odot}}}}}$, respectively. Outside these regions, dark matter
particles were degraded with lower mass resolution particles to reduce the
computational cost but keeping large scale tidal effects. From these initial
conditions, each cluster region was then re-simulated including now full
hydrodynamics with the SPH code Gadget-X, using a Plummer equivalent softening
of 6.5 ${{h^{-1}{\rm kpc}}}$ for both the dark matter and baryonic component.
The output includes, for each of the 324 clusters, 129 snapshots between
$z=16.98$ and $z=0$. At $z=0$, the 324 central galaxy clusters have a mass
range from $M_{200}=6.4\times 10^{14}{{h^{-1}{\rm{M_{\odot}}}}}$ to
$M_{200}=2.65\times 10^{15}{{h^{-1}{\rm{M_{\odot}}}}}$. 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), to which we refer the
reader for more details about this project.

Gadget-X, the code used for the re-simulations, 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

The halo analysis was done using 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). AHF computes the radius $R_{200}$ of
each halo identified, which is the radius $r$ at which the density
$\rho(r)=M(<r)/(4\pi r^{3}/3)$ drops below 200 times the critical density of
the Universe at the given redshift, $\rho_{\mathrm{crit}}$. $R_{500}$ is
defined accordingly, together with the corresponding enclosed masses $M_{200}$
and $M_{500}$. Subhaloes are defined as haloes which lie within the $R_{200}$
region of a more massive halo, the so-called host halo. This way, the mass of
this host halo includes the masses of all the subhaloes contained within it.

AHF also allows other properties to be generated for each halo and subhalo.
The luminosity (and magnitude) in any spectral band is calculated by applying
the stellar population synthesis code stardust (see Devriendt et al., 1999,
and references therein for more details). This code uses the given age and
metallicity of each stellar particle within a given halo and obtains from a
catalogue the full spectrum of a star with those properties. Then, each
spectrum is weighted by the mass of the particle, and the sum of all of them
yields the galaxy SED. A Kennicutt initial mass function (Kennicutt, 1998) is
assumed for these calculations.

To trace the haloes through the different snapshots of the simulations, merger
trees are built using the tree-builder MergerTree, which comes with the AHF
package. However, in this work we will only work with the $z=0$ snapshots.

##  3 Methodology

In this Section we first present the way observers find close pairs of
galaxies in the sky. Then we explain how we apply this same method to The
Three Hundred data set, and how we intend to assess its limitations by using
the simulations’ data.

###  3.1 Finding pairs in observations

In observational studies, the way of finding close pairs is based on two
quantities that relate two different galaxies: their projected separation in
the sky and the separation in velocity along the line of sight. Defining a
maximum threshold for these two quantities, $r_{\mathrm{sep}}$ and
$v_{\mathrm{sep}}$, it can be determined if two galaxies are close. After
applying some kind of selection criterion for the galaxies, e.g. based on
their luminosity, the pairs can be easily found by checking if these
parameters are below the maximum set. In the literature, the values used as
thresholds depend a lot on the specific purpose of the study, so that works
focusing more on mergers themselves use smaller separations (e.g. Robotham et
al. (2014) use $r_{\mathrm{sep}}=20$ kpc/h and $v_{\mathrm{sep}}=500$ km/s),
while works on the effects of interactions and companions can use separations
up to the order of Mpc (e.g. in Patton et al. (2016) they use
$r_{\mathrm{sep}}=2$ Mpc and $v_{\mathrm{sep}}=1000$ km/s).

###  3.2 Application to simulations

We will now apply the same observational method to our simulations, trying to
mimic the procedure as much as possible. Before doing any further calculation,
we will first make a selection of the objects we are going to work with from
the simulations. In this work, we will use the word “galaxy” to refer to the
objects in the hydrodynamical simulations, including both their stellar and
dark matter components. For each of the 324 clusters in The Three Hundred data
set, we first select the region within $5R_{200}$ of the cluster centre, to
also include the cluster outskirts. To all the galaxies within this region, we
apply a stellar mass cut similarly to observational studies, so that we work
only with the galaxies with
$M_{\mathrm{*}}>10^{9.5}{{h^{-1}{\rm{M_{\odot}}}}}$ (see Cui et al., 2018 for
the stellar mass function of all the galaxies in our data set). This way we
end up with a total number of galaxies between $\sim 400$ and 1200 depending
on the cluster, which is still statistically high enough. Note that,
throughout this work we only use the AHF catalogues at $z=0$, i.e. only the
last snapshot of the simulations.

Once we have done the galaxy selection, we can proceed to find the pairs
between them. In order to produce results that are comparable to observations,
we create different 2D projections of the 3D coordinates. We do this by
rotating the coordinates about two orthogonal random axes after placing the
cluster at the centre of the coordinate system, and then projecting the
coordinates into a 2D plane. This way, we obtain 100 random projections for
each cluster, to which we can apply our pair-finding algorithm, based on the
two parameters $r_{\mathrm{sep}}$ and $v_{\mathrm{sep}}$. The separation
between two galaxies is simply computed as their distance in the 2D plane,
while the velocity separation comes from two contributions: the peculiar
velocity of the galaxies along the remaining axis and the difference in
recession velocities due to the Hubble flow. To include both terms in our
calculations, before doing the projections we add to the peculiar velocity
(given by AHF) the contribution of the Hubble flow as $H\cdot\textbf{r}$,
where $H$ is the Hubble constant and r are the coordinates of each object.
Then, after the coordinates are projected, the velocity separation is simply
the difference in velocities along the line-of-sight axis.

In order to study the dependence of the results on the values chosen, and to
have some flexibility in our pair definition, we have decided to use several
different values for the thresholds $r_{\mathrm{sep}}$ and $v_{\mathrm{sep}}$
that determine if two galaxies are close. Based on the literature (see e.g.
Huško et al., 2022 for a summary of different studies) we have used three
different values for each of the parameters, which can thus be combined in 9
different ways. For the separation in the sky we use the values: 20, 50 and
100 kpc/h, while for the velocity separation we use 300, 500 and 1000 km/s.

Finally, to be able to assess if a galaxy has more than one close companion,
we also allow for groups to be created connecting the galaxies. This way, if
there are more than two galaxies grouped together, we will define that as a
group rather than a pair. Regarding this, our algorithm links galaxies (or
pairs) with existing pairs by establishing friends-of-friends connections.
This means that if there is a pair of close galaxies A and B, and galaxy C is
found to be close to only one of them, the three of them will still be
classified as a group. Then, if a galaxy D is close to any of the three
galaxies, it will also be included in the group, and so on.

Figure 1: Example of one random projection of the $R_{200}$ region of cluster
45 in The Three Hundred data set. The blue dots are isolated galaxies, the red
pentagons are galaxies in pairs, and the green diamonds represent those in
groups (with three or more members). For comparison, the plot is divided in
half, with the definition of close used for each half indicated in the
corresponding corner. The two circles delimitate the $R_{200}$ and $0.5\cdot
R_{200}$ regions. The axes are centred so that the cluster centre is at $X=0$,
$Y=0$ Mpc/h.

Fig. 1 shows an example of the pairs and groups found for one random 2D
projection for cluster 45 in The Three Hundred sample, for two different
definitions of close. In the left for $r_{\mathrm{sep}}=20$ kpc/h and in the
right for $r_{\mathrm{sep}}=100$ kpc/h, while $v_{\mathrm{sep}}$ was set at
$500$ km/s for both of them. The blue dots show the isolated galaxies, the red
pentagons are the galaxies in pairs and the green diamonds are those in groups
(with three or more members). Comparing both halves in Fig. 1 we can see the
effects of increasing the maximum separation, finding not only more pairs but
also a significant amount of groups.

Once the pairs and groups are found for all the clusters and projections, they
can be analysed in terms of their statistics and properties, making use of all
the information available in the simulations. For that, we will stack together
these results (distribution and properties of pairs and groups) for the 100
projections for each of the 324 clusters. This way, in our results we can
distinguish between the clusters but not between projections for the same
cluster.

####  3.2.1 Defining 2D pairs as ‘good’

Using the previous methodology we can order all the projected galaxies into
different pairs and groups. Then, taking advantage of the power of
simulations, the 3D coordinates of the galaxies can be used to determine if
the projected pairs are also close in real space. For simplicity, for this
part of the study we will focus only on galaxy pairs. We will see in the
following section (Section 4) that we are not losing much information by doing
this, since pairs dominate the statistics.

In order to address the question of the “goodness” of a pair, we use the
physical (3D) separation between the involved galaxies. Our idea is to check
if the 3D separation between the galaxies is within the range allowed by the
$r_{\mathrm{sep}}$ threshold in projected distance. To quantify this in a
single parameter we compute the ratio $r_{\mathrm{3D}}/r_{\mathrm{sep}}$,
where $r_{\mathrm{3D}}$ is the physical separation between the galaxies in the
simulation and $r_{\mathrm{sep}}$ is fixed to 20, 50 or 100 kpc/h, depending
on the value chosen for the pair-finding method. Using this ratio, we can
simply set a threshold at $r_{\mathrm{3D}}/r_{\mathrm{sep}}=1$, so that pairs
with this ratio below (or equal to) 1 can be tagged as “good” pairs, with a 3D
separation within the allowed range; and pairs with
$r_{\mathrm{3D}}/r_{\mathrm{sep}}>1$ are classified as “bad” pairs. Note that
we call them “good” and “bad” pairs, since our criterion is just based on the
projected and physical separations between the galaxies in the pair (rather
than e.g. “true” and “false”), and thus it only allows us to assess the degree
to which the 3D distance is within the maximum separation allowed. Other
criteria, like the one used in Haggar et al. (2021), can be used to determine
if the two galaxies are gravitationally bound, but for this work we prefer to
focus on this simple comparison, which already gives a significant amount of
knowledge about the pairs. We will take a different approach and investigate a
separation between “true” and “false” pairs in a follow-up work (Contreras-
Santos et al., in prep.).

In the following sections we will first show some statistics of the way the
galaxies are connected using the methodology described, regarding the number
of pairs and groups found and the galaxies that belong to them. Then, for the
pairs, we will separate them into “good” and “bad” according to the criterion
we just described. We will study the fraction of pairs that belong to each
class and analyse their properties based on this division.

##  4 Statistics of 2D grouping

Using the methodology described in the previous section, we can link the
galaxies to their close projected companions in each of the 324 clusters. In
this section we will analyse the statistics of the pairs and groups found, by
computing the fraction of galaxies that are in a pair or group, the number of
members that each association of galaxies has (i.e. if they are pairs or
groups) and finally by seeing if there is any dependence of these results on
the properties of the specific cluster.

Figure 2: Left, fraction of galaxies that are in a group or pair, i.e. that
have at least one close companion, as a function of maximum separation chosen,
$r_{\mathrm{sep}}$. The dots show the median values of the distribution for
the 324 clusters in The Three Hundred data set, with the shaded regions
indicating the 16-84 percentiles. For each cluster, the value is computed as
the mean percentage for 100 random projections. The colours indicate the
velocity threshold $v_{\mathrm{sep}}$ used, in magenta 300 km/s, in green 500
km/s and in cyan 1000 km/s. Previous studies have also been included, as
indicated in the legend. Right, same as the left panel but for the mean number
of members per galaxy association (see explanation in the text).

###  4.1 Fraction of galaxies in pairs and groups

For each of the 324 clusters in The Three Hundred data set, we connect the
galaxies as described in Section 3 and count the fraction of them that remain
in a pair or group (in 2D), i.e., the fraction of galaxies that have at least
one close companion. We do this for the 100 random 2D projections for each
cluster. This way we can obtain the mean fraction for each cluster, together
with the standard deviation. By putting all the mean values together, we can
compute the median value of this distribution, together with its 16-84
percentiles. This is shown in the left panel of Fig. 2 (dots for the median
values and shaded regions for the percentiles). To compare the results, we
show this median value as a function of the maximum projected separation used,
$r_{\mathrm{sep}}$, and for the three different values of the velocity
separation used, $v_{\mathrm{sep}}$, as indicated in the legend.

In the plot we can see that the results show a strong dependence on the chosen
value of the separation. For the lowest threshold $r_{\mathrm{sep}}=20$ kpc/h,
the fraction is below 8 per cent, with very small dependence on the velocity
threshold. For the higher values of $r_{\mathrm{sep}}$ the velocity threshold
starts to make a difference. The fraction of galaxies with a close companion
is between $\sim 12$ and 21 per cent for 50 kpc/h separation, while for 100
kpc/h the range becomes even greater, going from $\sim 27$ for
$v_{\mathrm{sep}}=300$ km/s (in magenta) to more than 40 per cent for
$v_{\mathrm{sep}}=1000$ km/s (in cyan).

We also include some previous observational results in the left panel of Fig.
2. The green diamond is extracted from López-Sanjuan et al. (2012), where data
from the COSMOS survey are analysed. They use the thresholds
$r_{\mathrm{sep}}=30$ kpc/h and $v_{\mathrm{sep}}=500$ km/s to measure the
close pair fraction for galaxies with $M_{\mathrm{*}}\geq
10^{11}{{{\rm{M_{\odot}}}}}$. In spite of this different selection criterion,
we can see that, although a bit smaller, their value is compatible with the
expected result for our data. We also show, as stars, the results from Lin et
al. (2008), where they study the pair fraction of galaxies from the DEEP2
Redshift Survey, applying a cut by luminosity. In this case, they employ three
different maximum separations, $r_{\mathrm{sep}}=30$, 50 and 100 kpc/h, with
$v_{\mathrm{sep}}=500$ km/s. We see that their results are in general smaller
than ours, even compared to our $v_{\mathrm{sep}}=300$ km/s results. However,
we have to note here that Lin et al. (2008) results come from a wide survey
rather than a cluster environment like ours, where the galaxy density is
higher. In this same work by Lin et al. (2008), and also in later works (Lin
et al., 2010; de Ravel et al., 2011), the pair fraction is found to be
dependent on the galaxy number density, so that our results are in fact
expected to be higher, as seen in Fig. 2.

###  4.2 Number of members per galaxy association

In addition to the percentage of galaxies that have at least one close
companion, i.e. are in a pair or group, it can also be of interest to know the
size of these associations of galaxies. Are the galaxies mainly in pairs or in
bigger groups? Similarly to the previous subsection, we have computed the mean
number of galaxies per association (we use the word “association” here to
refer to pairs and groups at the same time), again for each of the 324
clusters as the mean value for the 100 projections. In the right panel of Fig.
2 we show the median and 16-84 percentiles of the distribution of this value
for the 324 clusters. The format is the same as in the left panel, with the
X-axis indicating the maximum separation used and the colours indicating the
velocity threshold. Looking first at the minimum value of $r_{\mathrm{sep}}$,
we see that the mean number of galaxies is very slightly above 2 (regardless
of the velocity), meaning that almost all non-isolated galaxies are in pairs,
with very few groups. In fact, we find that groups represent around 7 per cent
of all the galaxy associations for this $r_{\mathrm{sep}}$ value. For
$r_{\mathrm{sep}}=50$ kpc/h the increase is less significant than in the left
panel of Fig. 2, with the number ranging between $2.2$ and $2.4$. This means
that for this separation the galaxy associations are still very predominantly
pairs, the velocity threshold still not being too determining. The percentage
of groups is now between 15 and 20 per cent of all the associations found.
Finally, for $r_{\mathrm{sep}}=100$ kpc/h, the median of the mean number of
galaxies is $2.5$ for $v_{\mathrm{sep}}=300$ km/s (in magenta), showing that
galaxy pairs are still predominant but more groups are found now. We can see
that the value increases notably with the maximum velocity allowed, the mean
number of galaxies even reaching $3.2$ for $v_{\mathrm{sep}}=1000$ km/s,
although the scatter in this value is also becoming bigger. In this case, the
percentage of groups ranges between $\sim$ 25 and 35 per cent for the
different velocity thresholds, showing that pairs dominate the statistics even
for the least restrictive thresholds.

In this case we limit the plot to our own results, since this quantity has not
been that thoroughly studied in other works and thus the comparison is more
difficult. However we believe the results in the right panel of Fig. 2 are
important by themselves, to see how the values of $r_{\mathrm{sep}}$ and
$v_{\mathrm{sep}}$ affect the way the galaxies are ordered in groups and
pairs, and to gain a better understanding of the results shown throughout this
work.

###  4.3 Dependence of the 2D statistics on cluster properties

In this subsection we study the correlation of the previous statistics, the
fraction of galaxies in a pair or group, $f_{\mathrm{gal}}$ and the mean
number of members per galaxy association, $n_{\mathrm{gal}}$, with two
different cluster properties: its mass and dynamical state, which can be
derived in observational studies, too. This way, we can investigate if these
two properties of clusters have an effect on the distribution of projected
galaxies in pairs and groups, an information that can be useful for future
studies and to gain a better understanding of close pairs of galaxies in
cluster environments.

####  4.3.1 Mass

In the top panel in Fig. 3 we show the correlation between the fraction of
galaxies that are found to be in a pair or group and the mass of the cluster.
We use the mass $M_{200}$ at $z=0$, given directly by AHF. For simplicity, we
only show the results using $r_{\mathrm{sep}}=100$ kpc/h and
$v_{\mathrm{sep}}=500$ km/s to link the galaxies, but we will comment on the
results for different values. In the plot we can see that there is no
correlation at all, which could be expected since to obtain this fraction we
are already normalising by the total number of galaxies. For different
definitions of proximity, i.e. different values of $r_{\mathrm{sep}}$ and
$v_{\mathrm{sep}}$ the results are the same, with essentially no dependence of
the results on the mass. Although in the plot we only show the results for the
fraction of galaxies with at least one close companion, the conclusion still
holds for the mean number of galaxies per association, with no correlation
with the mass found.

####  4.3.2 Dynamical state

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

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

This equation is based on the three parameters initially introduced by Neto et
al. (2007) as proxies for relaxation. Later studies by Cui et al. (2018) or De
Luca et al. (2021) also used these parameters to study the relaxation of the
clusters in The Three Hundred sample. The parameters are the centre of mass
offset, $\Delta_{r}$, which is the offset between the positions of the centre
of mass of the cluster and the density peak, normalised to the halo radius;
the subhalo mass fraction, $f_{s}$, which is the fraction of cluster mass
contained in subhaloes; and the virial ratio $\eta$ (see Cui et al., 2017, for
an updated calculation for hydrodynamic simulations). 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 $\chi_{\mathrm{DS}}\gtrsim 1$.
The relaxation parameter $\chi_{\mathrm{DS}}$ thus provides a continuous way
to classify the dynamical state of the clusters.

In the lower panel of Fig. 3 we show a scatter plot of the mean number of
galaxies per association against the relaxation parameter at $z=0$. As with
the mass, we only show the results for $r_{\mathrm{sep}}=100$ kpc/h and
$v_{\mathrm{sep}}=500$ km/s. In this case we see that there is a clear
negative correlation, meaning that in more disturbed clusters the associations
of galaxies are more likely to be bigger. Although not shown here, we see a
very similar correlation with the fraction of galaxies in a pair or group,
$f_{\mathrm{gal}}$, meaning that galaxies are less likely to be isolated in
these disturbed clusters. These results are in line with what could be
expected, since relaxed clusters have less substructure it is logical for them
to have fewer pairs and groups. The value of the Spearman correlation
coefficient is $r_{S}\lesssim-0.5$ in both cases, and remains the same when
changing $v_{\mathrm{sep}}$ for the same separation threshold. However, the
strength of the correlation decreases if we use lower values of
$r_{\mathrm{sep}}$, even disappearing for the most ‘restrictive’ definition
with $r_{\mathrm{sep}}=20$ kpc/h and $v_{\mathrm{sep}}=300$ km/s. So in
general we can say there is a negative correlation with the dynamical state of
the cluster, so that more relaxed clusters have less galaxies connected to
other galaxies and the groups are smaller, but this does not hold for all the
possible definitions of galaxy pair.

Figure 3: Top, fraction of galaxies in a group, computed for
$r_{\mathrm{sep}}=100$ kpc/h and $v_{\mathrm{sep}}=500$ km/s, as a function of
mass of the cluster, in logarithmic scale. Bottom, mean number of galaxies per
group as a function of the relaxation parameter $\chi_{\mathrm{DS}}$, also in
logarithmic scale. In both panels, the Spearman correlation coefficient,
$r_{S}$, is indicated in the bottom left corner.

##  5 Fraction of ‘good’ 2D pairs

Now that we have studied in a quantitative way the projected pairs and groups
of close galaxies found following the methodology in Section 3, we can focus
the study on the pairs themselves, concentrating on their particular
properties. As a reminder, the objective of the work presented here is the
quantification of the likelihood for a pair seen in 2D to be actually close in
3D. As it has also been said before, we will now work only with pairs, i.e.
only with the two-member groups, since in Section 4 we showed that they are
very predominant in most of the cases, and they are also much easier to
analyse.

In order to assess the “goodness” of a pair, we use the method described in
Section 3.2.1, based on $r_{\mathrm{3D}}/r_{\mathrm{sep}}$, the ratio of the
3D distance to the maximum 2D separation allowed. We set the threshold for
classifying the pairs at 1, so that “good” pairs are those with
$r_{\mathrm{3D}}\leq r_{\mathrm{sep}}$, with $r_{\mathrm{sep}}$ being fixed
for the given definition. Here we will show some of the results obtained for
the given pairs.

In Fig. 4 we show the distribution of the ratio
$r_{\mathrm{3D}}/r_{\mathrm{sep}}$ for all the pairs found for the given
definition $r_{\mathrm{sep}}=50$ kpc/h and $v_{\mathrm{sep}}=500$ km/s. We
stacked all the pairs together, regardless of the cluster. We can see that the
distribution shows a peak at $\sim 1$, but extends to quite higher values of
the ratio, corresponding to pairs that are much farther away in 3D distance
than allowed by the $r_{\mathrm{sep}}$ criterion. For simplicity, in this plot
we only show the results for one definition, but the conclusions hold for all
of them. In the plot we have indicated this threshold at
$r_{\mathrm{3D}}=r_{\mathrm{sep}}$ with a red dashed line, that separates the
distribution into “good” (to the left of this value) and “bad” (to the right)
pairs. This way, we can easily measure the fraction of good pairs.

In Fig. 5 we show this fraction for all the different values of
$r_{\mathrm{sep}}$ and $v_{\mathrm{sep}}$, computed using
$r_{\mathrm{3D}}/r_{\mathrm{sep}}\leq 1$ to identify good pairs. We computed
this fraction separately for each cluster, so that the dots show the median
value of the distribution for the 324 clusters, while the shaded regions are
the 16-84 percentiles. As in Fig. 2, the results are shown as a function of
$r_{\mathrm{sep}}$ in the $x$-axis, while the different colours of the lines
indicate the maximum velocity separation allowed.

We can see in the plot that the results are dependent both on the velocity and
on the distance separation, with the good fraction decreasing notably for
increasing values of $v_{\mathrm{sep}}$ and $r_{\mathrm{sep}}$. The difference
is more significant for $r_{\mathrm{sep}}=20$ kpc/h, where the percentage of
good pairs goes from 49 to 64 per cent for decreasing values of
$v_{\mathrm{sep}}$. When increasing the 2D separation threshold to
$r_{\mathrm{sep}}=50$ kpc/h, the fraction is notably reduced, ranging between
33 per cent for the highest velocity separation and 47 for the lowest one.
Moving to $r_{\mathrm{sep}}=100$ kpc/h, the variation of the values is
smaller, and they also become less dependent on the velocity threshold. For
1000 km/s, the value remains almost constant at 33 per cent, while for
$v_{\mathrm{sep}}=500$ km/s and 300 km/s the fraction decreases to 35 and 39
per cent respectively. We also note that the scatter increases for lower
values of $r_{\mathrm{sep}}$ due to the decreasing statistics.

Looking at the values of the fraction in general, we have to highlight that
they can be significantly low. For the most restrictive criterion, with
$r_{\mathrm{sep}}=20$ kpc/h and $v_{\mathrm{sep}}=300$ km/s, the fraction of
good pairs is 64 per cent, but increasing the separation threshold reduces the
fraction below 50 per cent, meaning that still more than half of the close
pairs observed in projections have a 3D separation in real space that is above
the limit set. We will devote the next section to the study of ways to improve
this result, this is, to improve the likelihood that an observed pair will be
classified as good, using observable properties of the pairs.

Figure 4: Distribution of the $r_{\mathrm{3D}}/r_{\mathrm{sep}}$ ratio for all
the pairs found with $r_{\mathrm{sep}}=50$ kpc/h and $v_{\mathrm{sep}}=500$
km/s, stacking all the 324 clusters together. The red dashed line indicates
the threshold used to separate good and pairs,
$r_{\mathrm{3D}}/r_{\mathrm{sep}}=1$. Figure 5: Same as Fig. 2 but for the
fraction of 2D pairs that are “good” in 3D, as a function of maximum
separation chosen, $r_{\mathrm{sep}}$. The dots show the median values of the
distribution for the 324 clusters in The Three Hundred data set, with the
shaded regions indicating the 16-84 percentiles. The colours indicate the
velocity threshold $v_{\mathrm{sep}}$ used, in magenta 300 km/s, in green 500
km/s and in cyan 1000 km/s

##  6 Separating the ‘good’ from the ‘bad’

In the previous section we found that, following the methodology described in
Section 3, the fraction of close pairs that are “good”, according to our
definition, is in general below 40 per cent. We like to remind here that our
definition is only based on the 3D separation between the galaxies in the pair
being below the 2D separation threshold set. In this section we will study
some observable properties of the given pairs. By separating them into “good”
and “bad” pairs, we intend to find significant differences that can be used to
distinguish them. Our aim is to provide observers with an additional test to
pick the “good” pairs from their original selection, thus improving the
quality of the identified close pairs of galaxies in the sky.

Figure 6: Comparison of the properties of galaxies in pairs (solid lines) to
all the galaxies in general (dashed lines). From top to bottom: stellar mass,
stellar metallicity, $g-r$ colour, $c_{\mathrm{*}}$ parameter and stellar-to-
halo mass ratio.

###  6.1 Properties of galaxies in pairs

First of all, we explore some overall properties of the galaxies we are
working with. This way, we gain some general understanding of our data set,
and check that the values are within reasonable ranges. We focus on properties
that are accessible to observers as well. By comparing their distribution for
the galaxies in pairs as opposed to that for the whole distribution of
galaxies, we can also see if there is any particular difference between them,
which could be useful for further studies.

We present in Fig. 6 the distributions of five different properties of the
galaxies in pairs (solid line), together with the distributions for all the
galaxies in the simulations (dashed line) that have been used throughout this
work (see Section 3 for the initial galaxy selection criteria). The first
three panels in Fig. 6 show, from top to bottom, the stellar mass of the
galaxies (given by AHF), their stellar metallicity (also given in the AHF
catalogues), and the $g-r$ colour (derived from the SDSS magnitudes of the
galaxies that are computed using the stardust code, see Section 2). Although
there are some very slight differences between the distributions, in general
we see in this plot that the galaxies in pairs show no particular distribution
for these properties. As before, we only show the results for one definition
of proximity, with $r_{\mathrm{sep}}=50$ kpc/h and $v_{\mathrm{sep}}=500$
km/s, because the results are very similar for all of them.

The fourth panel in Fig. 6 shows the value of $c_{\rm*}$ defined as the ratio
of the minor to major axis of the moment of inertia tensor as calculated only
for the star particles. This parameter is an indicator of the shape of the
galaxy, with a value of $c_{\mathrm{*}}$ equal to 1 indicating perfect
sphericity. The bottom panel of Fig. 6 shows the stellar-to-halo mass ratio of
the galaxies in our study, which can give relevant information about the dark
matter haloes of galaxies in pairs, and is computed as $M_{\rm*}/M_{200}$
using the (sub-)halo masses $M_{\rm*}$ and $M_{200}$ as given by AHF. We can
see in these plots that, although galaxies in pairs seem more likely to be
less spherical and with more relevance of the stellar component than the
general population, the differences between the two distributions remain
minor.

Figure 7: Left column, distribution of the ratio of the given property (from
top to bottom: stellar mass, metallicity, $g-r$ colour, $c_{*}$ and stellar-
to-halo mass ratio) between the two galaxies in the pair, computed for all the
pairs found and separating into “good” and “bad” pairs, depending on their
$r_{\mathrm{3D}}/r_{\mathrm{sep}}$ quotient. For the colour, the inset shows a
zoom-in to better show the differences between the distributions. Right
column, for the same five properties, fraction of good pairs within each bin
(gray bars). The lines indicate the fraction of good pairs for the ratios
being above (black) and below (gray) the given $x$-value, while the dashed
horizontal line is the total good pair fraction.

###  6.2 Properties of galaxy pairs

In Fig. 6 we showed the properties of all the galaxies in pairs together with
those for all the galaxies in our data set. Now, we are going to work instead
with the properties of the galaxy pairs themselves. This is, we are going to
compute, for each property, the ratio between the values of this property for
both galaxies in the pair: $P_{2}/P_{1}$, where $P_{2}$ and $P_{1}$ are simply
the specific properties of the two galaxies in the pair, chosen so that this
ratio is always $\leq 1$. For instance, by comparing the stellar masses of the
galaxies that constitute a pair, we compute their stellar mass ratio,
$M_{\mathrm{*2}}/M_{\mathrm{*1}}$, which gives us information about how
similar they are in mass. We compute this ratio for each pair for the five
properties presented in the previous subsection.

The distributions of this ratio for all the close pairs found are shown in the
left column of Fig. 7. The properties are the same as in Fig. 6, from top to
bottom: stellar mass, metallicity, colour, shape and stellar-to-halo mass
ratio; although in this case we are showing the ratio between the property for
the two galaxies that constitute a pair. Note again that this ratio is
computed so that the value is always $\leq 1$. The values are computed for all
the close pairs found for one definition ($r_{\mathrm{sep}}=50$ kpc/$h$ and
$v_{\mathrm{sep}}=500$ km/s as before), for the 324 clusters and 100
projections, stacking them all together as previously.

In order to analyse how the probability of a pair being “good” is affected by
either of these properties, in the left column of Fig. 7 we have separated the
galaxies according to our definition of good presented in Section 3, with the
results shown in Section 5. The blue lines are the normalised distributions of
the ratios for the good pairs, while the orange ones are for the bad pairs.
The green dashed line is for all the pairs together. Each row in this figure
represents the results for each of the properties in Fig. 6.

Starting with the first row in Fig. 7, in the left panel we see that there is
a clear difference between the distributions for good and bad pairs, with the
former showing a more prominent peak at low values of the mass ratio. This
means that good pairs tend to have a lower value than bad ones, i.e., the
masses of the galaxies in good pairs are more different between each other.
This is quantified in the plot in the right, which shows the fraction of good
pairs for each mass ratio bin, this is, the likelihood of a pair being good
given a certain value of its stellar mass ratio. The gray bars show the
likelihood for the ratio being within each bin, while the black line is for a
mass ratio above the given $x$-value, and the gray line for a ratio below it.
The horizontal dashed line shows the overall fraction of good pairs, as shown
in Fig. 5, so that whenever the bars or lines are above this line, this means
the probability of finding a good pair is increased in that range. In the case
of the stellar mass ratio, we see that for the first bin,
$M_{\mathrm{*2}}/M_{\mathrm{*1}}\leq 0.2$, the likelihood increases by 46 per
cent (from 41 to 60 per cent), while for ratios above this the probability
decreases to 10 per cent for $0.6<M_{\mathrm{*2}}/M_{\mathrm{*1}}\leq 0.8$.

For the second row in Fig. 7, for the metallicity ratio
$Z_{\mathrm{*2}}/Z_{\mathrm{*1}}$, we see the opposite difference than for the
mass. Pairs that we defined as good are more likely to have a stellar
metallicity ratio that is close to 1, this is, these galaxies tend to have
more similar metallicities. Looking at the plot to the right to quantify this
we see that, for $Z_{\mathrm{*2}}/Z_{\mathrm{*1}}>0.8$ the likelihood is
increased by $\sim 23$ per cent (from 41 to 50 per cent).

For the colour, we refer to the third row in Fig. 7. The inset in the left
plot shows a zoom-in for the interesting region, for ratios below 0.8. We see
that the distribution here seems to be dominated by good pairs, meaning that
pairs with a colour ratio below this value are more likely to be good than
bad. This is confirmed in the plot beside, where we see that a colour ratio
below 0.8 increases the good fraction from 41 to 54 per cent (an increase of
around 30 per cent, see gray line). This likelihood increases even more when
going to lower colour ratios, reaching more than 60 per cent for a ratio below
0.2, although we see in the upper panel that very few pairs are found within
this range.

Regarding the two final properties, the fourth row in Fig. 7 shows the same
distributions for $c_{\mathrm{*}}$. Here we can see that, while bad pairs
dominate at ratios around 1, for lower values the fraction of good pairs is
increased, reaching more than 70 per cent for
$c_{\mathrm{*2}}/c_{\mathrm{*1}}$ < 0.4 (see right panel). This means that
good pairs tend to have galaxies with different shapes, i.e. one spherical and
the other one more disky, presumably due to the interaction between them. In
the final row of Fig. 7 we study the situation for the ratio of the stellar-
to-halo mass ratio between the two galaxies in the pair. We see an even
clearer situation here, where good pairs prominently dominate the region of
lower values in the $x$-axis, with the good pair fraction growing to almost 70
per cent for the first $x$-bin (ratios below 0.2). This can also be
interpreted as an effect of the interactions between the galaxies in the pair,
where one of the DM haloes is stripped by the other one, creating this
difference in the stellar-to-halo mass ratio of the two involved galaxies.

We summarise our main results, as discussed in the text and extracted from
Fig. 7, in Table 1. Since Fig. 7 shows the results only for
$r_{\mathrm{sep}}=50$ kpc/h and $v_{\mathrm{sep}}=500$ km/s, we include in
this table the results for all the different definitions of proximity used
throughout the paper, for the three different values for the parameters
$r_{\mathrm{sep}}$ and $v_{\mathrm{sep}}$. The first column in this table
simply shows the fraction of good pairs, although, unlike in Fig. 5, it is
computed by stacking all the pairs for all the clusters together, rather than
computing $f_{\mathrm{good}}$ for each cluster and then getting the median
value for the 324 clusters. The following columns indicate by which amount is
this fraction increased when only the pairs within certain ranges of the
studied properties are considered. As an example, for the first row, the
fraction of good pairs increases by 21.4 per cent when limiting the pairs to
those with $M_{\mathrm{*2}}/M_{\mathrm{*1}}\leq 0.2$. This means that it goes
from $f_{\mathrm{good}}=0.642$ (see first column) to
$f_{\mathrm{good}}\times(1+0.214)=0.780$. In this table we see that the
overall situation is similar for the different values of $r_{\mathrm{sep}}$
and $v_{\mathrm{sep}}$, the main difference being for $r_{\mathrm{sep}}=20$
kpc/h, where the colour and $c_{*}$ ratios (fourth and fifth columns) do not
separate good and bad pairs for the given range, especially for the two lower
$v_{\mathrm{sep}}$ value, where the negative sign means the probability
decreases. We like to note here that, although selecting a lower threshold in
the colour ratio would mean a greater increase in the likelihood (see Fig. 7),
it can also be seen in this figure that the number of pairs is very low for
these bins, and so we prefer to use 0.8 as the threshold for the table. For
the shape parameter, $c_{*}$, we can see in the table that the results are
very dependent on the maximum 2D separation allowed and, while in the given
range the likelihood slightly decreases for $r_{\mathrm{sep}}=20$ kpc/h, it
can be increased by more than 100 per cent for $r_{\mathrm{sep}}=100$ kpc/h.

We conclude from this subsection that, using these five observable properties
and computing the ratio for an observed pair, its value can be used as an
indicator of the probability of that pair being good according to our
definition. To further increase the likelihoods, the criteria for the
different properties can be combined. For instance, for $r_{\mathrm{sep}}=50$
kpc/h and $v_{\mathrm{sep}}=500$ km/s, if a pair has both mass ratio below 0.2
and metallicity ratio above 0.8, the likelihood of it being good increases
from the initial 41 to more than 75 per cent (higher than for each of them
individually). For simplicity, we do not include these numbers in the table,
but we note that every combination of them results in an ever increased
percentage of good pairs.

Table 1: For each $r_{\mathrm{sep}}$ and $v_{\mathrm{sep}}$ combination, fraction of good pairs (“All pairs” column) and amount by which this fraction is increased when considering only the pairs within the given range for each property (see second column in Fig. 7 for the values for 50 kpc/h and 500 km/s). Note that the values in the first column, $f_{\mathrm{good}}$, are not exactly the same as in Fig. 5, since for this table they are computed as the fraction of good pairs for all the projections for all the clusters stacked together; while in Fig. 5 the value for each cluster was computed and then the median values (and 16-84 percentiles) for the 324 clusters are shown. |  | $\bm{f_{\mathrm{good}}}$ |  Increase in $\bm{f_{\mathrm{good}}}$
---|---|---|---
$\bm{r_{\mathrm{sep}}}$(kpc/h) |  $\bm{v_{\mathrm{sep}}}$(km/s) | All pairs |  $M_{\mathrm{*}}$ ratio ${\leq 0.2}$ |  $Z_{\mathrm{*}}$ ratio ${\geq 0.8}$ |  $g-r$ ratio ${\leq 0.8}$ |  $c_{*}$ ratio ${\leq 0.4}$ | SMHM ratio ${\leq 0.2}$
20 | 300 | 0.642 | 0.214 | 0.076 | -0.086 | -0.094 | 0.279
| 500 | 0.576 | 0.281 | 0.105 | -0.035 | -0.019 | 0.374
| 1000 | 0.484 | 0.400 | 0.153 | 0.047 | 0.121 | 0.548
50 | 300 | 0.481 | 0.366 | 0.188 | 0.220 | 0.332 | 0.622
| 500 | 0.410 | 0.458 | 0.230 | 0.305 | 0.520 | 0.791
| 1000 | 0.336 | 0.572 | 0.285 | 0.404 | 0.787 | 1.023
100 | 300 | 0.392 | 0.437 | 0.244 | 0.366 | 0.883 | 0.817
| 500 | 0.352 | 0.492 | 0.276 | 0.379 | 1.068 | 0.967
| 1000 | 0.318 | 0.538 | 0.317 | 0.346 | 1.248 | 1.126

###  6.3 Dependence of the separation on distance to cluster centre

Apart from the properties previously discussed, there is another property of
the pairs that is easy to check observationally, and that can give interesting
information about the pair. This is the projected distance to the cluster
centre. We measure this distance for all the pairs, and represent it in units
of the cluster radius $R_{200}$ in Fig. 8. In this figure we basically repeat
the plot in Fig. 7 but only for this distance to centre.

In the upper panel of Fig. 8 we see that there is a very clear difference
between good and bad pairs. While the good ones are almost uniformly
distributed along the $5R_{200}$ zone, the bad ones are preferentially located
inside the cluster. This is something that could be expected, the higher
density of galaxies towards the centre makes it harder to identify real pairs.
However, it is still an interesting result, that already warns that galaxy
pairs close to the cluster centre are much more likely to be farther away in
physical distance. In fact, looking at the lower panel in this figure, we can
see that the likelihood of finding a good pair inside $R_{\mathrm{200}}$
decreases to 25 per cent, while it grows to 57 per cent for distances above
this radius.

Figure 8: Same as Fig. 7 but for the 2D distance from the pair to the cluster
centre, in units of $R_{200}$. Top, distribution for all the pairs separated
into “good” and “bad” pairs, depending on the quotient $r_{\mathrm{sep}}$/3D
distance ratio. Bottom, fraction of good pairs within each bin (gray bars),
and above (black line) and below (gray line) the given $x$-value. The dashed
horizontal line is the total good pair fraction.

Now, given this difference in the number of good pairs depending on the region
we are in, we ask ourselves whether the results obtained in the previous
subsection are just an effect of this result regarding the distance to the
cluster centre. This is, we question if the differences in mass ratio, colour
and metallicity between good and bad pairs are just a consequence of these
pairs being located in different regions of the cluster environment. To answer
this question, in Fig. 9 we present the same plot as the first row in Fig. 7
but separating for the region inside the cluster, $R\leq R_{200}$ (left), and
outside the cluster, $R>R_{200}$ (right).

We see that, although the results weaken, the differences still hold outside
the cluster, with good pairs having in general lower values of the stellar
mass ratio. In the bottom left panel we see that, in the
$M_{\mathrm{*2}}/M_{\mathrm{*1}}\leq 0.2$ bin, the probability increases from
25 to 46 per cent, while in the right one the increase is from 57 to 71 per
cent in this same bin. Although not shown here, we repeated these plots for
the other four different properties analysed throughout the paper, finding
that the trends seen always hold in both regions inside at outside the
cluster. For the metallicity and the colour ratios, a similar situation to the
mass is found, with the results weakening a bit outside the cluster, while for
the shape and the stellar-to-halo mass ratio, the results remain the same
regardless of the region.

Figure 9: Same as first row in Fig. 7 but separating into the region inside
the cluster (left) and outside the cluster (right).

We note that, when talking about the cluster centre, we refer to the maximum
density peak of the main halo, identified by our halo finder, rather than
taking a more observational approach like using the position of the brightest
cluster galaxy (BCG). Nevertheless, several works have already shown that the
offset between these two positions is small, especially in relaxed clusters,
so the same results are expected if using the BCG position or the centroid of
X-ray emission (see Cui et al., 2016 for an in-depth study on how the centre
position is affected by the method chosen to identify it; or De Luca et al.,
2021, where the offset is computed for The Three Hundred clusters).

###  6.4 Dependence of the separation on cluster properties

In the previous subsection we tried to analyse the probability of correctly
identifying a galaxy pair in a 2D observation as a function of the properties
of the pairs themselves. Now, we also want to study if there is a correlation
between this probability of success and the properties of the whole cluster.

In order to do this, we tried to find a correlation between the fraction of
pairs that were correctly identified for each cluster, $f_{\mathrm{good}}$, as
shown in Fig. 5, and any cluster property. We show this in Fig. 10, for the
total halo mass of the cluster, $M_{200}$, and for the cluster’s dynamical
state, quantified by the relaxation parameter $\chi_{\mathrm{DS}}$, as
explained in Section 4.3. In both cases we could not find any significant
correlation, meaning that the probability of an observed pair being good does
not depend on properties of the cluster as a whole but rather of those of the
pair itself. In Section 4.3 we found a significant correlation between the
fraction of galaxies in a pair or group and the dynamical state of the
cluster, with disturbed clusters having more pairs and groups. Now we see that
the fraction of good pairs is not dependent on the dynamical state of the
cluster, so for unrelaxed clusters we find more good pairs but also more bad
pairs, meaning that projection effects affect relaxed and unrelaxed clusters
equally.

Figure 10: Top, fraction of good pairs of galaxies, computed for
$r_{\mathrm{sep}}=50$ kpc/h and $v_{\mathrm{sep}}=500$ km/s, with the
selection criterion $r_{\mathrm{good}}=0.8$ for good pairs, as a function of
mass of the cluster, $M_{200}$, in logarithmic scale. Bottom, same good
fraction as top panel, but as a function of the relaxation parameter
$\chi_{\mathrm{DS}}$. In both panels, the Spearman correlation coefficient,
$r_{S}$, is indicated in the bottom left corner.

##  7 Conclusions

In this work we studied close pairs of galaxies, that are used in observations
as an indicator of galaxy mergers and interactions. We worked with a set of
cosmological simulations of galaxy clusters and their surroundings, allowing
us to additionally investigate any correlation with environment. By taking
advantage of the power of simulations, we evaluated the performance of the
observational techniques used to identify these pairs, and proposed ways,
using observable properties of the galaxies, to improve this performance.

The simulations used in this work are provided by The Three Hundred project,
and consist of a set of 324 numerically modelled spherical regions centred on
the most massive clusters found in the DM-only MDPL2 MultiDark Simulation. For
each of these simulations, we limit our study to the region within $5R_{200}$
of the cluster halo centre, and select all the galaxies with
$M_{\mathrm{*}}>10^{9.5}{{h^{-1}{\rm{M_{\odot}}}}}$. To mimic observational
studies, we randomly rotate the 3D coordinates and then project them into a 2D
plane, so that we can get numerous projections of the same cluster. We applied
the same techniques used by observers to find close pairs of galaxies. This
way, we identified close pairs as those that lie within a separation in the
sky smaller than $r_{\mathrm{sep}}$ and a separation in velocity along the
line of sight below $v_{\mathrm{sep}}$ (where we included the Hubble flow).
The values used for $r_{\mathrm{sep}}$ (20, 50 and 100 kpc/h) and for
$v_{\mathrm{sep}}$ (300, 500 and 1000 km/s) were based on previous studies in
the literature, and combined for a broad research of close pairs and the
dependence on these parameters. We also allowed galaxies to have more than one
close companion, and hence being in groups.

After grouping the galaxies for the different clusters and projections, in
Fig. 2 we analysed the statistics of the results. We showed first the fraction
of galaxies having at least one close companion, that depends a lot on the
$r_{\mathrm{sep}}$ threshold used. We saw that our results are overall in line
with previous observational results. We also showed in Fig. 2 that the great
majority of the connected galaxies are in pairs rather than groups, with the
latter becoming more relevant for $r_{\mathrm{sep}}=100$ kpc/h. For this
reason, the following parts of our work were only focused on galaxy pairs,
that also allow for a simpler analysis.

The next step in this study was to use the physical coordinates available in
the simulations to obtain the 3D separations between the galaxies in pairs. We
computed the ratio $r_{\mathrm{3D}}/r_{\mathrm{sep}}$ for all the pairs found,
where $r_{\mathrm{sep}}$ is the fixed threshold set for the projected
separation in the sky for each definition of proximity. This way, we could
divide the pairs into “good” pairs: those with
$r_{\mathrm{3D}}/r_{\mathrm{sep}}\leq 1$, i.e., with a 3D separation within
the allowed $r_{\mathrm{sep}}$ range; and “bad” pairs, which do not satisfy
this condition. Using this threshold, in Fig. 5 we depicted the fraction of
good pairs for all the different definitions. This fraction is shown to be
between 50 and 65 per cent for the smallest $r_{\mathrm{sep}}$ value used,
while it drops significantly as this parameter increases. For maximum
separations of 50 and 100 kpc/h, the fraction of good pairs ranges between 30
and less than 50 per cent depending on the velocity separation limit.

For a better understanding of this separation between good and bad pairs, we
devoted Section 6 to studying observable properties of these pairs and the
galaxies within them, trying to find differences that can be used to
distinguish them. We worked with the stellar mass, stellar metallicity and
colour. We also studied the shape, quantified by $c_{*}$, and stellar-to-halo
mass ratio. In Fig. 7 we analysed the ratio of each property between the two
galaxies in every pair, distinguishing the values for good and bad pairs.
Regarding the stellar mass ratio, we found that good pairs tend to have a
lower value, i.e. the galaxies are more different in mass in these pairs. The
situation is reversed for metallicity, where a ratio close to 1 indicates that
the pair is more likely to be a good one. Concerning the colours, Fig. 7 shows
that for ratios below $\sim 0.8$ the pairs are also more likely to be good.
These results are quantified, and summarised for each
$r_{\mathrm{sep}}-v_{\mathrm{sep}}$ combination, in Table 1. We saw in this
table that, generally speaking, the fraction of good pairs is increased by
30-50 per cent if the given galaxy pair has
$M_{\mathrm{*2}}/M_{\mathrm{*1}}\leq 0.2$,
$Z_{\mathrm{*2}}/Z_{\mathrm{*1}}\geq 0.8$ or a $g-r$ colour ratio below 0.8,
although these results weaken if the 2D separation threshold to find the pairs
is set at 20 kpc/h. For the shape and stellar-to-halo mass ratio, we found in
both cases that good pairs tend to have lower values, again meaning that,
regarding these properties, the situation for the two galaxies involved in the
pair is significantly different. This can be interpreted as a consequence of
the interaction between the galaxies. Table 1 shows that, specially for 50 and
100 kpc/h separations, selecting lower values within these properties can
increase the fraction of good pairs by 50-100 per cent.

Apart from these properties, we also studied the projected distance from the
pair to the cluster centre. We found that separating the good from the bad
pairs also gives a clear difference in the distributions, with the bad pairs
more clearly found near the cluster centre, within $R_{200}$ (see Fig. 8).
However, we found that the results for the previous properties, although
weakened in some cases (for the stellar mass, metallicity and colour ratios),
still hold if we separate for the regions inside and outside $R_{200}$ (see
Fig. 9 for the mass), so they are not just a consequence of bad pairs being
more predominant in the inner region. In this regard, Ellison et al. (2010)
investigated galaxy pairs as a function of the environment, finding that,
although galaxy interactions seem to be ubiquitous in the Universe, their
observational manifestations show a dependence on the environment. Here we
like to note again that our study has been done for galaxy pairs in cluster
environments, reaching up to $5R_{200}$ of the main cluster centre (additional
massive objects can also be found here), so that our results are only valid
for these regions and they may not hold for randomly selected pairs. Future
works repeating this study on a different environment could provide with more
information regarding how the results are affected by this.

Previous works have already used simulations to similarly analyse
observational results. For instance, McConnachie et al. (2008) used mock
galaxy catalogues based on the Millennium simulation to investigate the
spatial properties of groups of galaxies, finding that only $\sim 30$ per cent
of the groups were truly compact in three dimensions. In a later work by the
same authors, McConnachie et al. (2009) showed that selecting groups by
surface brightness reduces this effect significantly, increasing the fraction
of physically compact groups. Regarding the properties of the involved
galaxies, several works have studied how they are affected by close companions
and interactions. Scudder et al. (2012) studied galaxy pairs in the Sloan
Digital Sky Survey, tracing the changes induced in star formation rates and
metallicities. Alonso et al. (2006) also studied star formation and colour of
the galaxies in observed pairs, and how they are affected by interactions. In
this work we take a different approach, based on observational properties of
the pairs rather than the galaxies, i.e. using the ratio $P_{2}/P_{1}$, where
$P_{2}$ and $P_{1}$ are the values of the given property for the two galaxies.
We propose the use of these ratios to increase the likelihood of a pair being
close in physical distance too, and thus reduce these effects that deteriorate
close-pair catalogues.

In conclusion, our results show that, computing the two galaxies’ ratio for
the projected pairs, the stellar mass, metallicity, $g-r$ colour, shape and
stellar-to-halo mass ratio can be used as indicators of the probability of the
pairs being good according to our definition. This is simply based on the 2D
distance threshold used, $r_{\mathrm{sep}}$, and the 3D separation between the
galaxies, $r_{\mathrm{3D}}$. We note again that this definition of “good” is
only a measure of 3D proximity, rather than physically testing if the galaxies
are bound. Previous works like Haggar et al. (2021) or Choque-Challapa et al.
(2019) associate galaxies with a host galaxy based on their relative velocity
and the host gravitational potential. In future works following this study we
aim at using a more theoretical approach to distinguish between these physical
pairs and galaxies that are just passing by. Besides, although in this paper
we mainly focused on pairs of galaxies, a more in-depth study could be done
regarding groups of galaxies and observational techniques to find them. A
similar work has also been performed recently by Taverna et al. (2022),
analysing the completeness and reliability of compact group catalogues using
semi-analytical models of galaxy formation.

## Acknowledgements

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, and GY are supported by the MICIU/FEDER
through grant number PGC2018-094975-C21 as well as by the Ministerio de
Ciencia e Innovación under research grant PID2021-122603NB-C21. AK further
thanks Urge Overkill for sister havanna. WC is supported by the STFC AGP Grant
ST/V000594/1 and by the Atracción de Talento Contract no. 2020-T1/TIC-19882
granted by the Comunidad de Madrid in Spain. He further acknowledges the
science research grants from the China Manned Space Project with NO. CMS-
CSST-2021-A01 and CMS-CSST-2021-B01. RH acknowledges support from STFC through
a studentship.

## 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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