11institutetext: Dipartimento di Fisica e Astronomia, Università di Padova,
Vicolo dell’Osservatorio 3, I-35122, Padova, Italy 22institutetext: INAF -
Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122,
Padova, Italy 33institutetext: Associazione Big Data, via Piero Gobetti 101,
40129 Bologna, Italy 44institutetext: Cavendish Laboratory, University of
Cambridge, 19 J. J. Thomson Ave., Cambridge CB3 0HE, UK 55institutetext: Kavli
Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK 66institutetext: European Southern Observatory, Karl-
Schwarzschild-Strasse 2, 85748, Garching, Germany 77institutetext: Aix
Marseille Univ, CNRS, CNES, LAM, Marseille, France 88institutetext: Infrared
Processing and Analysis Center, California Institute of Technology, Pasadena,
CA 91125, USA 99institutetext: IPAC, California Institute of Technology, 1200
East California Boulevard, Pasadena, CA 91125, USA 1010institutetext:
Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, 1290
Versoix, Switzerland 1111institutetext: Kavli Institute for the Physics and
Mathematics of the Universe, The University of Tokyo Kashiwa, Chiba 277-8583,
Japan 1212institutetext: Department of Astronomy, School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
1313institutetext: The Caltech Optical Observatories, California Institute of
Technology, Pasadena, CA 91125, USA 1414institutetext: INAF - Osservatorio di
Astrofisica e Scienza dello Spazio, via Gobetti 93/3, I-40129 Bologna, Italy
1515institutetext: Centro de Astronomía (CITEVA), Universidad de Antofagasta,
Avenida Angamos 601, Antofagasta, Chile 1616institutetext: University of
Bologna, Department of Physics and Astronomy ”Augusto Righi”, Via Gobetti
93/2, I-40129, Bologna, Italy 1717institutetext: INAF - Osservatorio
Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy
1818institutetext: Cosmic Dawn Center (DAWN), Jagtvej 128, DK22000 Copenhagen
N, Denmark 1919institutetext: Niels Bohr Institute, University of Copenhagen,
Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark 2020institutetext: Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
2121institutetext: Instituto de Física y Astronomía, Universidad de
Valparaíso, Avda. Gran Bretaña 1111, Valparaíso, Chile 2222institutetext:
Department of Physics and Astronomy, University of California, Davis, One
Shields Ave., Davis, CA 95616, USA

# The ALPINE-ALMA [CII] survey: The contribution of major mergers to the
galaxy mass assembly at z$\sim$5

M. Romano E-mail<EMAIL_ADDRESS>P. Cassata 1122 L.
Morselli 33 G. C. Jones 4455 M. Ginolfi 66 A. Zanella 22 M. Béthermin 77 P.
Capak 88
A. Faisst 99 O. Le Fèvre Passed away.77 D. Schaerer 1010 J. D. Silverman
11111212 L. Yan 1313 S. Bardelli 1414 M. Boquien 1515 A. Cimatti 16161717
M. Dessauges-Zavadsky 1010 A. Enia 14141616 S. Fujimoto 18181919 C. Gruppioni
1414 N. P. Hathi 2020 E. Ibar 2121 A. M. Koekemoer 2020 B. C. Lemaux 2222 G.
Rodighiero 1122 D. Vergani 1414 G. Zamorani 1414 E. Zucca 1414

(Accepted July 22, 2021)

###### Abstract

Context. Galaxy mergers are thought to be one of the main mechanisms of the
mass assembly of galaxies in the Universe, but there is still little direct
observational evidence of how frequent they are at $z\gtrsim 4$. Recently,
many works have suggested a possible increase in the fraction of major mergers
in the early Universe, reviving the debate on which processes (e.g., cold
accretion, star formation, mergers) most contribute to the mass build-up of
galaxies through cosmic time.

Aims. To estimate the importance of major mergers in this context, we make use
of the new data collected by the ALMA Large Program to INvestigate [CII] at
Early times (ALPINE) survey, which attempted to observe the [CII] 158 $\mu$m
emission line from a sample of 75 main-sequence star-forming galaxies at
$4.4<z<5.9$.

Methods. We used, for the first time, the morpho-kinematic information
provided by the [CII] emission, along with archival multiwavelength data to
obtain the fraction of major mergers ($f_{\text{MM}}$) at $z\sim 5$. By
combining the results from ALPINE with those at lower redshifts from the
literature, we also studied the evolution of the merger fraction through
cosmic time. We then used different redshift-evolving merger timescales
($T_{\text{MM}}$) to convert this fraction into the merger rate per galaxy
($R_{\text{MM}}$) and in the volume-averaged merger rate
($\Gamma_{\text{MM}}$).

Results. We find a merger fraction of $f_{\text{MM}}\sim 0.44$ (0.34) at
$z\sim 4.5$ (5.5) from ALPINE. By combining our results with those at lower
redshifts, we computed the cosmic evolution of the merger fraction which is
described by a rapid increase from the local Universe to higher redshifts, a
peak at $z\sim 3$, and a slow decrease toward earlier epochs. Depending on the
timescale prescription used, this fraction translates into a merger rate
ranging between $\sim 0.1$ and $\sim 4.0$ Gyr-1 at $z\sim 5$, which in turn
corresponds to an average number of major mergers per galaxy between 1 and 8
in $\sim 12.5$ Gyr (from $z=6$ to the local Universe). When convolved with the
galaxy number density at different epochs, the merger rate density becomes
approximately constant over time at $1<z<4$, including values from $10^{-4}$
to $10^{-3}$ Gyr-1 Mpc-3, depending on the assumed $T_{\text{MM}}$. We finally
compare the specific star formation and star-formation rate density with the
analogous quantities from major mergers, finding a good agreement at $z>4$ if
we assume a merger timescale that quickly decreases with increasing redshift.

Conclusions. Our new constraints on the merger fraction from the ALPINE survey
at $z\sim 5$ reveal the presence of a significant merging activity in the
early Universe. Whether this population of mergers can provide a relevant
contribution to the galaxy mass assembly at these redshifts and through the
cosmic epochs is strongly dependent on the assumption of the merger timescale.
However, our results show that an evolving $T_{\text{MM}}\propto(1+z)^{-2}$
agrees well with state-of-the-art cosmological simulations, suggesting a
considerable role of mergers in the build-up of galaxies at early times.

###### Key Words.:

Galaxies: evolution – Galaxies: formation – Galaxies: high-redshift –
Galaxies: kinematics and dynamics – Cosmology: early Universe

##  1 Introduction

How galaxies grow their stellar mass through cosmic time is still one of the
most puzzling questions of modern cosmology. During the last decades, two
major physical mechanisms have been proposed to contribute or dominate the
build-up of galaxies at different epochs of the Universe: the accretion of
cold gas (both from the internal reservoir or from the outer environment of
galaxies) and the merging of galaxies.

The first process requires the supply of new cold gas to the galaxy from the
filaments of the cosmic web, refreshing and/or enhancing the star formation
(e.g., Kereš et al. 2005; Dekel et al. 2009). Although this could be one of
the main modes of galaxy assembly (e.g., Kereš et al. 2009; Bouché et al.
2010), direct observations of gas flowing through the filaments are tricky
while indirect evidence has increased over the last years (Bouché et al. 2013,
2016; Zabl et al. 2019). However, in situ star formation is observable through
a variety of different proxies and up to the earliest epochs (e.g., Madau &
Dickinson 2014 and references therein).

Galaxy mergers are also thought to play an important role in the stellar mass
growth over cosmic time. They are relatively common through the history of the
Universe and are observable with different methods and at different stages.
The merging of galaxies is a natural prediction of the hierarchical structure
formation model (White & Frenk 1991; Springel et al. 2005; Klypin et al. 2011)
for which dark matter halos grow their mass and affect the build-up of
galaxies in the Universe (Kauffmann et al. 1993; Khochfar & Burkert 2005). Not
only can mergers increase the stellar mass of galaxies (up to a factor of 2
for major mergers, i.e., for galaxies of nearly equal stellar mass; López-
Sanjuan et al. 2012; Oser et al. 2012; Kaviraj et al. 2014), but they can also
trigger starbursts and active galactic nuclei (AGN; Silk & Rees 1998;
Kartaltepe et al. 2012; Chiaberge et al. 2015; see also Shah et al. 2020 who
found that galaxy interactions do not significantly enhance AGN activity up to
$z\sim 3$), in some cases blowing the gas out from the galaxy which arrests,
at least temporarily, the formation of new stars (Silk & Rees 1998; Fabian et
al. 2006; Cattaneo et al. 2009). Therefore, observing mergers at different
epochs is key to shed light on the relative contribution of the distinct
processes that rule the assembly of galaxies in the Universe.

Depending on which phase the merger is passing through, there are two main
ways to estimate the incidence of such events at a given epoch. On-going and
post-mergers could lead to the formation of post-starburst (PSB) galaxies (at
least in groups or clusters; e.g., Wu et al. 2014; Lemaux et al. 2017) and
leave morphological or kinematic imprint on the merging or coalesced system
such as tails of stripped material, irregular shapes, disturbed velocity
fields, and/or deep absorption lines in the spectra of PSBs (e.g., Le Fèvre et
al. 2000; Conselice et al. 2003, 2008; Jogee et al. 2009; Wild et al. 2009;
Casteels et al. 2014). However, observations of these features could be
hampered by the spatial resolution needed to resolve the morphological details
highlighting the presence of a merger (especially at high redshifts), or
confused by diverse galaxy types that can mimic the kinematics or morphology
of a merging system.

On the other hand, galaxy pairs that are going to merge in a certain timescale
can also be identified based on some selection criteria regarding their
spatial and velocity separation (e.g., Le Fèvre et al. 2000; Patton et al.
2000; Lin et al. 2008; de Ravel et al. 2009; López-Sanjuan et al. 2012; Ventou
et al. 2017; Mantha et al. 2018; Duncan et al. 2019). Typically, close pairs
must lie within a defined radius with a projected separation in the sky that
could be as wide as 50 kpc and, in case of spectroscopic surveys, have a
relative velocity not larger than $\Delta v=500$ km s-1 (i.e., $\Delta z\sim
0.0017$) which excludes those pairs that are not gravitationally bound (e.g.,
Patton et al. 2000).

Both these methods can provide an estimate of the merger fraction through
cosmic time but, as they probe different phases and timescales of the merging,
they can lead to discordant results if not interpreted properly. However, even
by adopting the same observational method, a large scatter is present in the
literature for the merger fraction and the corresponding merger rate at all
redshifts. This could be attributed to several reasons: i) the different
criteria used for selecting galaxy pairs; ii) the distinction between major
and minor mergers for which a dual uncertainty has to be taken into account,
both in the pair ratio which can be obtained from the stellar mass (López-
Sanjuan et al. 2013; Tasca et al. 2014; Man et al. 2016; Ventou et al. 2017;
Duncan et al. 2019) or the flux (Bluck et al. 2012; Man et al. 2012, 2016;
López-Sanjuan et al. 2015), and in the major (or minor) merger definition
having a threshold in mass or flux ratio ranging between 2-6 (Xu et al. 2012;
Ventou et al. 2017, 2019); iii) the merger timescale used to convert the pair
fraction into a merger rate, which in turn depends on the method used for
characterizing mergers, on their selection criteria, and on the properties of
the galaxies undergoing a merger (Kitzbichler & White 2008; Jiang et al. 2014;
Snyder et al. 2017).

Several studies tried to constrain the fraction of major mergers and the
merger rate out to $z\lesssim 2$ (e.g., de Ravel et al. 2009; López-Sanjuan et
al. 2011, 2013; Lotz et al. 2011; Xu et al. 2012). During this time, the
merger fraction is found to increase with redshift as $\propto(1+z)^{m}$,
where $1\lesssim m\lesssim 5$, depending on the luminosity, the stellar mass
and the spectral type of galaxies, leading to a large scatter in the
literature. Only a handful of works have extended these measurements to higher
redshifts. For instance, Conselice & Arnold (2009) used both pair counts and
morphological studies to investigate the presence of mergers at $4<z<6$ and
their contribution to the galaxy assembly, finding a possible peak in the
merger fraction at $z\sim 3$ followed by a decline which suggested a deviation
from the power-law trend found at lower redshifts. However, the results by
Conselice & Arnold (2009) do not distinguish between major and minor mergers
and could have been subject to large uncertainties due to photometric
redshifts. A peak in the cosmic evolution of the merger fraction is also found
at $1\lesssim z\lesssim 3$ by more recent observational studies (e.g., Mantha
et al. 2018; Duncan et al. 2019) which, however, do not provide conclusive
statements given the large above-mentioned differences on the selection and
definition of the mergers (see for example the results by Mantha et al. 2018
for both stellar mass- or flux-selected pairs).

In this paper we take advantage of the recent data collected by the ALMA Large
Program to INvestigate [CII] at Early times (ALPINE; Béthermin et al. 2020;
Faisst et al. 2020; Le Fèvre et al. 2020) to further constrain the importance
of major mergers to the galaxy mass assembly at the end of the Reionization
epoch and through cosmic time. This survey allows us to add a key additional
piece of information to the wealth of multiwavelength data already available
for each targeted galaxy thanks to the ALMA observations of the ionized carbon
[CII] at 158 $\mu$m rest-frame and the surrounding continuum. The 3D (i.e.,
RA, Dec, velocity) information enclosed in each ALPINE data cube is of great
importance for identifying morphological and kinematic disturbances
highlighting the presence of possible merging components. In particular, Le
Fèvre et al. (2020) made a first fundamental step in the morpho-kinematic
classification of the ALPINE galaxies, finding a high fraction ($\sim 40\%$)
of mergers (both minor and major) at $z\sim 5$ among the ALPINE sample. An in-
depth analysis of two merging systems found in ALPINE at $z\sim 4.6$ was made
later by Jones et al. (2020) and Ginolfi et al. (2020), showing the strength
of these observations in the identification and characterization of this kind
of sources in the high-redshift Universe.

Previous works estimated the merger fraction by exploiting morphological
information or direct pair counts obtained through rest-frame optical
observations and photometric or spectroscopic redshifts, as derived from
optical ionized gas tracers (e.g., Conselice & Arnold 2009; Xu et al. 2012;
Ventou et al. 2017). Here, for the first time, we make use of the rest-frame
FIR [CII] line to compute the major merger fraction in the early Universe. As
this line is less affected by dust extinction than optical tracers,
morphological and kinematic analysis of this emission can reveal the presence
of mergers with dusty components that are partially or completely obscured in
the optical bands. Following these previous studies, we thus refine the
fraction of mergers at the redshifts explored by the ALPINE survey and analyze
in detail their [CII] emission combined with optical data for estimating their
incidence at $z>4$ and, along with other values in the literature, through
cosmic time.

This paper is structured as follows: in Section 2 we describe the data used to
estimate the merger fraction, and the previous works on the morpho-kinematic
classification of the ALPINE galaxies on which we based our analysis. The
criteria used for selecting mergers from the ALPINE sample, the methods
employed for their characterization and the estimated parameters are stated in
Section 3. The results on the merger fraction and merger rate, their evolution
with the cosmic time and their contribution to the galaxy mass assembly are
reported in Section 4. We discuss our results and summarize them in Section 5
and Section 6, respectively.

Throughout this work, we adopt a $\mathrm{\Lambda-CDM}$ cosmology with
$H_{0}=70$ km s-1 Mpc-1, $\mathrm{\Omega_{m}}=0.3$ and
$\mathrm{\Omega_{\Lambda}}=0.7$. Stellar masses are normalized to a Chabrier
(2003) initial mass function (IMF).

##  2 Data

The ALPINE survey was designed to detect the [CII] line at 158 $\mu$m rest-
frame and the surrounding FIR continuum emission from a sample of 118 star-
forming galaxies at $4.4<z<5.9$, avoiding the redshift range $4.6<z<5.1$ due
to a low-transmission atmospheric window. The campaign took $\sim 70$ hours of
observation with ALMA in Band 7 (275-373 GHz) during cycles 5 and 6. The
ALPINE targets are drawn from the Cosmic Evolution Survey (COSMOS; Scoville et
al. 2007a, b) and Extended Chandra Deep Field South (E-CDFS; Giavalisco et al.
2004; Cardamone et al. 2010) fields and have secure spectroscopic redshifts
from the VIMOS UltraDeep Survey (VUDS; Le Fèvre et al. 2015; Tasca et al.
2017) and DEIMOS 10K Spectroscopic Survey (Hasinger et al. 2018). The rest-
frame UV selection ($\mathrm{L_{UV}>0.6\leavevmode\nobreak\ L^{*}}$) ensures
that these sources lie on the so-called main-sequence of star-forming galaxies
(e.g., Noeske et al. 2007; Rodighiero et al. 2011; Tasca et al. 2015), being
thus representative of the underlying star-forming, UV-detected galaxy
population at $z\sim 5$ (e.g., Speagle et al. 2014). Due to the location of
the ALPINE targets in these well-studied fields, archival multiwavelength data
are available from the UV to the near-IR (e.g., Koekemoer et al. 2007; Sanders
et al. 2007; McCracken et al. 2012; Guo et al. 2013; Nayyeri et al. 2017),
reaching the X-ray and radio bands (e.g., Hasinger et al. 2007; Smolčić et al.
2017), allowing us to recover physical quantities such as stellar masses
($\mathrm{9\lesssim log(M_{*}/M_{\odot})\lesssim 11}$) and star-formation
rates (SFRs; $\mathrm{1\lesssim log(SFR/M_{\odot}yr^{-1})\lesssim 3}$) through
spectral energy distribution (SED)-fitting (Faisst et al. 2020).

The collected ALMA data were reduced and calibrated using the standard Common
Astronomy Software Applications (CASA; McMullin et al. 2007) pipeline and each
data cube was continuum-subtracted to obtain line-only cubes with channel
width of $\sim 25\leavevmode\nobreak\ \mathrm{km\leavevmode\nobreak\ s^{-1}}$
and beam size $\sim 1^{\prime\prime}$ (with a pixel scale of $\sim
0.15^{\prime\prime}$; Béthermin et al. 2020). A line search algorithm was then
applied to each continuum-subtracted cube resulting in 75 [CII] detections
(S/N$>$3.5) out of 118 ALPINE targets (of which 23 sources were also detected
in continuum).

For a more in-depth description of the overall ALPINE survey, the observation
and data processing, and the multiwavelength analysis see Le Fèvre et al.
(2020), Béthermin et al. (2020) and Faisst et al. (2020), respectively.

###  2.1 Previous morpho-kinematic classifications

Le Fèvre et al. (2020) performed a preliminary morphological and kinematic
classification of the 75 ALPINE targets showing [CII] emission. In particular,
the channel maps around the emission line, the intensity (moment-0) and
velocity (moment-1) maps, the integrated [CII] spectra, the position-velocity
diagrams (PVDs) along the major and minor axis of velocity maps, and the
multiwavelength data at the position of the source were visually inspected by
an internal team of the ALPINE collaboration that placed these galaxies in the
following classes:

  1. 1.

Rotators. The galaxies in this class are characterized by a single source in
the ancillary data and show a clear velocity gradient in the moment-1 maps
which then implies a tilted (straight) PVD along the major (minor) axis. They
can also exhibit a double-horned profile in the integrated spectra.

  2. 2.

Mergers. In case of interacting systems the presence of two or more components
in the optical images within the ALMA field of view and/or in the moment-0
maps and PVDs is expected. A complex behavior in the channel maps and in the
[CII] spectra could be visible.

  3. 3.

Extended Dispersion Dominated. These sources extend in [CII] emission over
multiple ALMA beams and are typically characterized by straight PVDs along the
major and minor axis and by a Gaussian line profile.

  4. 4.

Compact Dispersion Dominated. As opposed to the class 3 galaxies, these
objects are unresolved in the moment-0 maps.

  5. 5.

Weak. These targets are too weak to be visually classified into one of the
above classes.

Le Fèvre et al. (2020) found that $\sim 40\%$ of the ALPINE galaxies are
undergoing a merging phase and that $\sim 13\%$ of the targets are possibly
forming an ordered disk at $z\sim 5$. Extended and compact sources make up
20$\%$ and 11$\%$ of the sample, respectively, with the remaining galaxies
(16$\%$) being too complex or weak to be classified. These results possibly
suggest a large contribution of the mergers to the galaxy mass assembly at
$z\sim 5$. However, such a qualitative classification could be prone to
different kinds of uncertainties arising from the subjectivity of the method.

On the other hand, Jones et al. (2021) expanded this initial classification of
the ALPINE targets by using the tilted ring fitting code
$\mathrm{{}^{3D}Barolo}$ (3D-Based Analysis of Rotating Objects from Line
Observations; Di Teodoro & Fraternali 2015). Their selection criteria led to a
successful fit of $39\%$ of the ALPINE parent sample (i.e., 29 out of 75
sources) and to a robust classification of 14 sources. This new quantitative
analysis confirmed the morpho-kinematic diversity of the ALPINE galaxies found
by Le Fèvre et al. (2020) but resulted in a somewhat different statistics with
$43\%$ of rotators, $36\%$ of mergers, and $21\%$ of dispersion-dominated
galaxies.

As we aim to compute the contribution of mergers to the galaxy mass assembly
in the early Universe, a robust estimate of the fraction of mergers at $z\sim
5$ is required. We thus perform a further and in-depth morpho-kinematic
investigation of the ALPINE [CII]-detected galaxies based on the same
classification criteria used in Le Fèvre et al. (2020) and on the results by
Jones et al. (2021).

##  3 Methods

To examine the merger fraction at $z\sim 5$, we take advantage of the
continuum and [CII] emission maps of the targets from the ALPINE data release
1 (DR1; Béthermin et al. 2020)111DR1 data are available at
https://cesam.lam.fr/a2c2s/data_release.php.. Briefly, for each source the
CASA task UVCONTSUB was used to obtain line-only cubes. The continuum was
first estimated in the uv-plane by masking the spectral channels containing
line emission222For a detailed description of the procedure used to estimate
the range of spectral channels containing line emission for all the
[CII]-detected galaxies, see Section 6.1 in Béthermin et al. (2020). and
fitting a model to the visibilities, and it was then subtracted to the data
cubes. With the CASA IMMOMENTS task the spectral channels that included the
emission line were summed up to produce the velocity-integrated intensity
(moment-0) maps, that is $M_{0}(x,y)=\Delta v\sum_{i}I_{i}(x,y)$, where
$I_{i}$ is the intensity of the $i$-th pixel at the position $(x,y)$, $\Delta
v$ is the velocity width between two consecutive channels, and the sum is over
all the channels containing the spectral line (Béthermin et al. 2020). By
masking all the pixels below $3\sigma$ (where $\sigma$ is the rms estimated in
the moment-0 map), we also compute the intensity-weighted velocity (moment-1)
and velocity dispersion (moment-2) maps, defined as
$M_{1}=\sum_{i}I_{i}v_{i}/M_{0}$ and
$M_{2}=\sqrt{\sum_{i}I_{i}(v_{i}-M_{1})^{2}/M_{0}}$, respectively. It is worth
noting that in the analysis by Le Fèvre et al. (2020) the morphology and
kinematic of the [CII] line were inspected within the $2\sigma$ emission
contours of the moment maps. In this work, we decide to select only the pixels
above the $3\sigma$ level in order to be less affected by possible spurious
emissions that could alter the effective morphology and velocity field of the
targets (however, we find that this choice does not significantly change our
final results).

With the CASA IMFIT task we fit one 2D Gaussian component to each source in
the moment-0 map, retrieving best-fit parameters such as the peak intensity
and integrated flux density, and morphological information like the
coordinates of the [CII] emission peak, the full width at half maximum (FWHM)
of the major and minor axis of the Gaussian and its position angle (PA). If
the [CII] morphology shows the presence of different spatially-resolved
components in the vicinity of the main ALPINE target (as in case of mergers),
we fit an individual 2D Gaussian to each of them in order to retrieve the
above morphological and kinematic information for all the observed sub-
structures. We then make use of the morphological central position provided by
the 2D Gaussian best-fit on each ALPINE target333In case of multiple
spatially-resolved [CII] emissions, we use the position provided by the
Gaussian fit on the global system rather than the positions obtained by
fitting individual 2D Gaussians to each of the observed components. to produce
the PVDs along the major and minor axis with the CASA IMPV task, with an
averaging width of five pixels and a pseudo-slit length of $3^{\prime\prime}$
(if not otherwise stated). Conversely to the analysis made by Le Fèvre et al.
(2020) in which the major axis was aligned with the morphological PA, we
decide to produce the two PVDs with the major axis oriented along the
direction of major velocity gradient in the moment-1 map, and with the minor
axis perpendicular to the major one. In this way, we increase the chances of
identifying possible mergers or rotating galaxies in the ALPINE sample444Two
merging systems with different velocity fields or a rotating galaxy with a
clear velocity gradient would be ideally represented with separated components
in the PVDs and with an “S-shape” in the PVD along the major axis,
respectively..

Lastly, we extract the spatially-integrated 1D [CII] spectrum of each target
within the $3\sigma$ contours at the position of the source from the moment-0
map using the CASA task SPECFLUX.

Figure 1: Left: Morpho-kinematic analysis of the ALPINE target
$\mathrm{DC\\_773957}$ at $z_{\mathrm{[CII]}}=5.68$. First row: HST/ACS F814W
(left) and UltraVISTA DR4 $K_{s}$-band (right) images centered on the UV rest-
frame position of the target. Each cutout is $6^{\prime\prime}\times
6^{\prime\prime}$ wide. The cyan contours show the [CII] ALMA emission
starting from $3\sigma$ above the noise level. Yellow contours in the optical
maps represent 3, 5 and 7$\sigma$ emission. In the lower left corner, the ALMA
beam (white) and HST or UltraVISTA resolutions (green) are displayed. Second
row: moment-1 (left) and moment-2 (right) maps color-coded for the velocity
and velocity dispersion in $\mathrm{km\leavevmode\nobreak\ s^{-1}}$. The
velocity map reports the direction of the major (solid) and minor (dashed)
axis (centered on the coordinates returned by the best-fit 2D Gaussian model
on the moment-0 map) along which the PVDs are computed. Third row: PVDs along
the major (left) and minor (right) axis color-coded for the flux intensity in
each pixel. Dashed contours include the $2\sigma$ emission in the maps while
3, 5 and 7$\sigma$ emission is represented by solid lines. Fourth row: [CII]
spectrum (black histogram) extracted within the 3$\sigma$ contours of the
intensity map. The gray shaded band marks the 1$\sigma$ level of the spectrum
while the dashed vertical line shows the zero velocity offset computed with
respect to the redshift of the [CII] line. Purple, blue, and green lines are
three individual possible components of the merging system, resulting in the
global profile in red. The shaded areas under the curves represent the
channels used to compute the [CII] intensity maps of the corresponding
individual components. A single Gaussian fit is also visible with a dashed-
orange line. Right: Morpho-kinematic analysis of the ALPINE target
$\mathrm{vc\\_5100822662}$ at $z_{\mathrm{[CII]}}=4.52$. Same panels as in the
left figure. In the bottom panel we report the [CII] spectra of the major
(blue) and minor (green) merger components extracted at the positions of the
two resolved emissions, as further described in the text.

###  3.1 Mergers characterization

Figure 1 shows two examples of the data analyzed for each [CII]-detected
source. Taking as a reference the left set of panels, we display at the top
the [CII] morphology superimposed on the HST/ACS F814W (left) and UltraVISTA
(UVista) $K_{s}$ (right) cutouts centered on the UV rest-frame coordinates of
the ALPINE target. We also show the $>3\sigma$ emission contours from the
optical images. In the second row, the moment-1 and moment-2 maps are reported
color-coded for the velocity and velocity dispersion, respectively. In the
velocity field map the major and minor axis used to produce the underlying
PVDs (color-coded for the flux intensity in Jy/beam) are shown. The axis are
centered on the coordinates retrieved from the 2D Gaussian fit on the
intensity map and are represented with the direction along which the PVDs are
computed. The PA of the major axis is measured from north through east. Both
the moment-1 and moment-2 maps are smoothed by using a bilinear interpolation.
Finally, the lower panel reports the [CII] spectrum (in black) extracted from
the continuum-subtracted data cube and modeled with a single and a multiple-
component Gaussian function (in this latter case, we show also the possible
single components shaping the global spectral profile).

The described panels report the morpho-kinematic analysis of one of the
sources classified as merger both by Le Fèvre et al. (2020) and by the
analysis undertaken in this work555We note that Jones et al. (2021) classified
this target as ‘uncertain’, following their classification criteria., that is
$\mathrm{DEIMOS\\_COSMOS\\_773957}$ ($\mathrm{DC\\_773957}$) at
$z_{\mathrm{[CII]}}=5.68$. There are many lines of evidence for considering
this galaxy in a merging phase. First, the [CII] emission is elongated toward
the East at $>3\sigma$, resulting in a clearly disturbed morphology.
Furthermore, two peaks of emission at $3\sigma$ are resolved in the UVista
$K_{s}$ band, which could indicate a close pair of objects or the presence of
a dust screen. The velocity field is also quite disturbed, with the PVD along
the major axis presenting two $3\sigma$ components (solid contours) separated
in velocity by $\sim 140\leavevmode\nobreak\ \mathrm{km\leavevmode\nobreak\
s^{-1}}$ and with a slight spatial offset. These two components are also
visible in the [CII] spectrum with green and blue lines. A third minor
component at $\sim-200\leavevmode\nobreak\ \mathrm{km\leavevmode\nobreak\
s^{-1}}$ could be instead due to a faint satellite nearby the ALPINE target
and it is fairly visible in the PVDs within the $2\sigma$ contours. It is
worth noting that, even though with a single Gaussian function we find an
integrated flux of the line that is comparable with that obtained from the
multicomponent fit, the above-mentioned three components are needed in order
to reproduce the observed [CII] spectrum.

Moreover, to obtain a more robust sample of mergers we compute, for each of
the analyzed merger candidate, the rms of the observed spectra and use it to
compute the signal-to-noise ratio (S/N) of the possible merging components. In
particular, we obtain the integrated [CII] fluxes of the two principal
Gaussian components of each spectrum and divide them by the product between
the rms and the corresponding FWHM. We thus consider as mergers only those
systems for which the minor (in terms of the [CII] integrated flux) component
has S/N$\geq 3$. The S/N for the major and minor components of each system are
reported in Table 1.

Another example of merging galaxies is provided on the right of Figure 1 by
the morpho-kinematic analysis of $\mathrm{vuds\\_cosmos\\_5100822662}$
($\mathrm{vc\\_5100822662}$) at $z_{\mathrm{[CII]}}=4.52$. This source was
classified as a merger both by Le Fèvre et al. (2020) and Jones et al. (2021).
Indeed, two close and spatially resolved components are clearly visible in the
moment-0 and optical maps. From the PVD along the major axis666In this case,
we use a pseudo-slight of 4” of length in order to cover all the $3\sigma$
emission from the velocity map., the two merging systems are at the same
velocity but spatially separated by $\sim 1.5^{\prime\prime}$ (i.e., $\sim 10$
kpc at $z\sim 4.5$). In such a case, it is not possible to compute the
contribution supplied by each individual source to the global shape of the
line directly from the [CII] spectrum. However, we can provide an estimate of
the integrated flux from the spectrum of each single component by modeling the
total [CII] emission in the moment-0 map with two 2D Gaussian functions. In
this way, we can extract the spectrum of the resolved component from the
continuum-subtracted data cube at the position and with the shape provided by
the best-fit parameters from the model (i.e., centroid, major and minor FWHM).
For this specific ALPINE target, the spectra of the two merging components are
shown in the bottom row of the figure as the blue and green histograms. As can
be seen, their sum is quite consistent with the shape of the original [CII]
spectrum.

A complete description of each individual merging system found from this work
in the ALPINE survey is reported in Appendix A.

###  3.2 Comparison with previous classifications

With the methodology introduced in the previous section, we proceed for a new
morpho-kinematic classification of the ALPINE targets. As the main aim of this
work is to find the contribution of mergers to the galaxy mass assembly at
$z\sim 5$ from the ALPINE survey, we particularly focus on this class of
sources, in order to obtain the final sample of mergers needed to estimate the
major merger fraction in the early Universe.

As done by Le Fèvre et al. (2020), we visually inspect the ancillary data, the
intensity maps, the velocity and velocity dispersion fields presented in
Section 3.1 to search for the presence of multiple components or disturbed
morphology near the position of the targets. The channel maps, the spectra and
the PVDs are checked together searching for consistent emission features. By
taking into account the results of the initial qualitative classification by
Le Fèvre et al. (2020) and of the more recent quantitative analysis of a
subsample of the ALPINE targets by Jones et al. (2021), we proceed with a more
in-depth characterization of the [CII]-detected galaxies aimed at obtaining a
robust merger fraction at $z\sim 5$. Adopting the same criteria described in
Section 2.1 to differentiate the targets and considering the S/N of the minor
merger component as described in Section 3.1, we find a slightly lower
fraction ($\sim 31\%$, 23 out of 75 [CII]-detected sources) of mergers777Some
of the mergers found by Le Fèvre et al. (2020) were also analyzed in terms of
the tilted ring model fitting by Jones et al. (2021) who found that the
kinematic of those sources could be even compatible with rotating disks or
dispersion dominated galaxies, or that the sensitivity and resolution of the
data are too low for a conclusive classification (see Appendix A). if compared
to the $40\%$ found by Le Fèvre et al. (2020), with 12, 20 and $7\%$ of the
sample made by rotating, extended and compact dispersion dominated sources,
respectively. To be more conservative in the classification of the galaxies
(especially for obtaining a more robust merger statistics), we define the
remaining $30\%$ of the sample as ‘uncertain’, a new category that includes
the weak galaxies (as described in Le Fèvre et al. 2020) and also objects
that, by visual inspection, present features that are intermediate to those of
various classes. This category is similar to the ‘uncertain’ (UNC) class
introduced in Jones et al. (2021) that, based on the results of the
$\mathrm{{}^{3D}Barolo}$ fits, contains sources they are unable to classify
because of the low S/N and/or spectral resolution, or contrasting evidence in
their classification criteria. Although the uncertain category is populated by
a significantly larger fraction of sources with respect to the weak class
($\sim 16\%$) in Le Fèvre et al. (2020), we recover the same qualitative
morpho-kinematic distribution of the previous analysis, confirming the high
fraction of rotators and mergers at these early epochs.

Table 1: Physical parameters of the mergers.

|  |  |  |  |  |  | Minor | Major
---|---|---|---|---|---|---|---|---
Source | $z_{1}$ | $z_{2}$ | $\Delta v$ | $r_{p}$ | $\mu_{\mathrm{[CII]}}$ | $\mu_{K}$ | FWHM | $r_{\mathrm{e}}$ | S/N | FWHM | $r_{\mathrm{e}}$ | S/N
|  |  | [km s-1] | [kpc] |  |  | [km s-1] | [kpc] |  | [km s-1] | [kpc] |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
CG$\\_$38 | 5.5731 | 5.5698 | 152.3 | 2.7 | 1.8 | - | 96.9 | 1.4 | 3.8 | 165.9 | 2.0 | 3.9
DC$\\_$308643 | 4.5238 | 4.5221 | 92.3 | 3.0 | 1.2 | - | 85.4 | 1.0 | 4.5 | 85.6 | 1.4 | 5.3
DC$\\_$372292 | 5.1345 | 5.1374 | 144.0 | 1.8 | 2.7 | 2.9 | 82.8 | 2.2 | 4.1 | 157.9 | 3.7 | 5.7
DC$\\_$378903 | 5.4311 | 5.4293 | 94.5 | 0.9 | 1.9 | - | 70.0 | 4.4(∗) | 3.6 | 111.2 | 1.3 | 4.2
DC$\\_$417567 | 5.6676 | 5.6700 | 106.7 | 5.3 | 2.1 | - | 84.8 | 1.0 | 3.1 | 163.9 | 4.0 | 3.4
DC$\\_$422677 | 4.4361 | 4.4378 | 92.8 | 0.0 | 1.4 | - | 95.7 | 1.0 | 4.7 | 93.1 | 0.5 | 6.7
DC$\\_$434239 | 4.4914 | 4.4876 | 206.1 | 5.5 | 6.9 | - | 103.5 | 3.9 | 4.5 | 441.8 | 3.4 | 7.2
DC$\\_$493583 | 4.5122 | 4.5141 | 103.4 | 1.0 | 4.7 | - | 64.8 | 3.3 | 3.6 | 139.5 | 1.0 | 8.0
DC$\\_$519281 | 5.5731 | 5.5765 | 158.0 | 0.9 | 4.5 | - | 95.7 | 0.9 | 3.1 | 163.8 | 2.3 | 8.3
DC$\\_$536534 | 5.6834 | 5.6886 | 234.6 | 4.4 | 2.7 | - | 113.9 | 1.9 | 3.6 | 228.5 | 2.6 | 4.9
DC$\\_$665509 | 4.5244 | 4.5261 | 95.9 | 1.0 | 2.0 | - | 70.5 | 4.5(∗) | 3.0 | 107.0 | 2.4 | 4.0
DC$\\_$680104 | 4.5288 | 4.5308 | 106.4 | 0.0 | 1.0 | - | 89.5 | 0.5 | 3.7 | 111.6 | 8.7(∗) | 4.7
DC$\\_$773957 | 5.6802 | 5.6770 | 141.2 | 3.0 | 1.6 | 5.6 | 118.0 | 1.3 | 5.9 | 179.5 | 1.7 | 6.2
DC$\\_$814483 | 4.5823 | 4.5796 | 145.5 | 7.9 | 2.4 | - | 128.1 | 1.1 | 3.3 | 192.7 | 2.8 | 4.2
DC$\\_$818760 | 4.5626 | 4.5609 | 92.3 | 9.9 | 1.3 | 2.6 | 127.1 | 2.4 | 10.4 | 271.0 | 2.9 | 18.3
DC$\\_$834764 | 4.5076 | 4.5055 | 119.0 | 1.0 | 1.7 | - | 96.2 | 1.4(∗) | 5.1 | 143.8 | 3.1 | 5.8
DC$\\_$842313 | 4.5547 | 4.5406 | 751.5 | 11.6 | 32.9 | 1.5 | 229.9 | 4.3(∗) | 5.1 | 1089.7 | 1.7 | 5.9
DC$\\_$859732 | 4.5353 | 4.5315 | 205.2 | 4.9 | 3.9 | - | 55.4 | 1.7(∗) | 3.0 | 176.4 | 3.3(∗) | 3.7
DC$\\_$873321 | 5.1545 | 5.1544 | 4.5 | 6.5 | 1.2 | 3.1 | 171.2 | 1.3 | 4.9 | 219.3 | 2.8 | 7.6
vc$\\_$5100541407 | 4.5628 | 4.5628 | 1.9 | 13.8 | 1.6 | 1.4 | 203.8 | 4.4(∗) | 3.5 | 166.5 | 1.2 | 6.9
vc$\\_$5100822662 | 4.5210 | 4.5205 | 22.3 | 10.9 | 1.6 | 1.7 | 189.6 | 4.0 | 5.1 | 192.7 | 2.0 | 8.0
vc$\\_$5101209780 | 4.5724 | 4.5684 | 217.3 | 10.8 | 4.1 | 2.5 | 126.1 | 2.3 | 4.4 | 303.9 | 6.6 | 7.4
vc$\\_$5180966608 | 4.5294 | 4.5293 | 8.9 | 7.2 | 3.0 | 3.7 | 243.6 | 1.8 | 3.4 | 231.6 | 4.3 | 10.9

  * •

Notes. The right side of the table shows the measurements of FWHM (in km s-1),
[CII] emission size (in kpc), and S/N for both the minor and major components,
intended as such in terms of their integrated [CII] fluxes. The asterisk near
the size of some of the sources indicates a bad $r_{e}$ estimate from the
best-fit parameters of the 2D Gaussian fit on the moment-0 maps of the
individual components.

Figure 2: Top panel: Velocity separation between the components of each ALPINE
merger as a function of their projected distance. Each target is represented
by its own color while the symbols are different for $z\sim 4.5$ (circles) and
$z\sim 5.5$ (hexagons) galaxies. The gray shaded region shows the area of the
plot populated by galaxies which are more likely to be real mergers, with a
projected distance larger than $>2.1$ kpc and $\mathrm{\Delta
v<500\leavevmode\nobreak\ km\leavevmode\nobreak\ s^{-1}}$. The error bar on
the top-left corner represents the typical error on each value. Bottom panel:
FWHM as a function of $r_{\mathrm{e}}$ for the minor (left) and major (right)
components of each merger. The colors and symbols are the same as in the top
panel. Hatched markers identify the sources for which the best-fit size
estimate is not reliable, after a visual inspection of the residuals. The
dashed black lines mark the average [CII] size of the ALPINE targets (as
estimated in this work), represented with small light squares in the
background.

###  3.3 Physical parameters estimate

For each source classified as merger we measure several physical quantities
that help us to estimate the merger fraction at $z\sim 5$. From the spectra,
we compute the [CII] flux ratio between the major and minor components of the
merger888In the case the merging system is composed by more than two sources,
we only consider the two major components in terms of their [CII] fluxes.,
that is $\mu_{\mathrm{[CII]}}=F_{\mathrm{1,[CII]}}/F_{\mathrm{2,[CII]}}$,
finding a good agreement with the corresponding ratio computed from the PVDs.
To obtain the integrated fluxes of the individual galaxies, we use a Gaussian
decomposition of the global [CII] spectrum, as described in Section 3.1. When
allowed by the resolution, we also take the UVista $K_{s}$-band flux ratio of
the two merging sources, namely $\mu_{K}\equiv F_{1,K}/F_{2,K}$, where
$F_{1,K}$ and $F_{2,K}$ are the integrated fluxes obtained by fitting a 2D
Gaussian function to the major and minor components of the merger in the
considered band, respectively. Then, from the spectral features, we also
obtain the separation in velocity ($\Delta v$) between the intensity peaks of
the merger components, and the FWHM both for the individual objects and single
(overall spectrum) source.

We measure the projected distance ($r_{p}$) between the centers of the merger
components as $r_{p}=\theta\times d_{A}(z_{m})$, where $\theta$ is the angular
separation in the sky (in arcsec) between the two galaxies and $d_{A}(z_{m})$
is the angular diameter distance (in kpc arcsec-1) computed at the mean
redshift $z_{m}$ of the two sources. In case the two components are not
spatially resolved in [CII], we consider the distance between the two best-fit
centroids returned by the 2D Gaussian fit on their moment-0 maps. Similarly to
what has been done in Ginolfi et al. (2020), the latter maps are obtained by
collapsing the channels including the emission of the individual components.
Such a case is shown, for instance, in Figure 1 for the target
$\mathrm{DC\\_773957}$. The shaded blue and green areas under the curves in
the spectrum mark the collapsed channels used for building the [CII] intensity
maps of the two components. As it can be seen, the velocity channels assigned
to each component are not overlapping with each other. If instead the two
objects are spatially resolved but at the same velocity (as for
$\mathrm{vc\\_5100822662}$), we are not able to create the [CII] moment-0 maps
of the individual components (i.e., the two sources emit at the same frequency
and we cannot disentangle their contribution in [CII]) and we measure the
distance directly from the [CII] intensity map or from the PVDs.

Finally, we compute the size of the single components by fitting a simple 2D
Gaussian model to the [CII] intensity maps obtained as described above.
Following Fujimoto et al. (2020), we refer to the merging component size as
the circularized effective radius $r_{\mathrm{e}}$ defined as
$r_{\mathrm{e}}\equiv\sqrt{a\leavevmode\nobreak\ b}$, where $a$ and $b$ are
the best-fit semi-major and semi-minor axis, respectively. When the source is
resolved, we use the best-fit beam-deconvolved $a$ and $b$ parameters,
otherwise we take the beam-convolved sizes provided by the fit and correct
them for the ALMA beam (the average beam minor axis of ALPINE is $\sim
0.8^{\prime\prime}$). It is worth noting that Fujimoto et al. (2020) measured
the [CII] effective radii by fitting the line visibilities with the CASA task
UVMULTIFIT and assuming an exponential-disk profile. However, by applying our
more simplistic size measurements to the ALPINE galaxies analyzed by Fujimoto
et al. (2020) and considering the mergers as single components, we found a
good agreement with their results. A larger scatter is found only for some of
the objects classified as mergers whose size measurements are flagged as
unreliable by Fujimoto et al. (2020), likely because of the complicated [CII]
morphology.

From now on, we refer to $r_{\mathrm{e}}$ when talking about the size of a
source. All the information presented in this section is reported in Table 1.

###  3.4 Description of the merger sample

We show in Figure 2 (upper panel) the difference in velocity between the
merger components as a function of their projected distance $r_{p}$ for all
the possible merging systems in ALPINE. In the upper-left corner, we report
the uncertainties assumed for these data. The error on $\Delta v$ is obtained
by the sum in quadrature of the average uncertainty on each spectral element
(i.e., 25 km s-1). To estimate the typical error on the projected distance, we
consider two random centroids corresponding to the initial positions of the
minor and major components of the merger. Then, we obtain the distribution of
the errors on the centroids as provided by the 2D Gaussian best-fits on the
intensity maps of the merging components, and extract from that
$\mathrm{N}=1000$ values used to perturb the initial positions. At each
iteration, we re-compute the projected distance between the perturbed
positions of the two components. We thus estimate the average uncertainty on
$r_{p}$ as the standard deviation of the distribution of the N projected
separations (corresponding to $\sim 0.3\leavevmode\nobreak\ $kpc).

To derive a robust major merger fraction ($f_{\mathrm{MM}}$), we decide to
consider as mergers only those systems in which the components are separated
by no more than $\mathrm{\Delta v\leq 500\leavevmode\nobreak\
km\leavevmode\nobreak\ s^{-1}}$, assuming that this is the limit for a system
to be gravitationally bound (e.g., Patton et al. 2000; Lin et al. 2008; Ventou
et al. 2017). As shown, all the mergers in our sample satisfy this constraint
except one source. As analyzed in detail in Appendix A, this system is
composed by the ALPINE target $\mathrm{DC\\_842313}$ and a peculiar neighbor
source with a large FWHM and [CII] flux, and could even be a merger of three
galaxies with a $\Delta v$ between the target and the further third component
smaller than 500 km s-1. We thus account for this object in the computation of
the merger fraction.

Also, we consider as reliable mergers only those systems with a projected
distance larger than the typical [CII] sizes of individual galaxies at these
redshifts. Indeed, closer components could just represent clumps of star
formation within the same galaxy, faking the presence of a merger by affecting
the morphology and kinematics of the [CII] line. From the size measurements of
the ALPINE galaxies we find an average [CII] size of $\sim 2.1$ kpc, that is
in good agreement with the median value found by Fujimoto et al. (2020) and
with the typical distance between galaxies and satellites or mergers (¿2 kpc)
at $z>4$ (e.g., Carniani et al. 2018; Whitney et al. 2019; Zanella et al.
2021). Therefore, we use this value as a threshold to classify the secure
merging systems.

The two selection criteria on the distance and velocity separations define the
gray region in Figure 2 which thus includes the sources that are classified as
robust mergers. However, we cannot exclude a priori the systems with
$r_{p}<2.1$ kpc as they could be structures in an advanced stage of merging
that we are not able to resolve with ALMA because of their close proximity and
observational limitations (i.e., large synthesized beam, limited sensitivity).
It is worth noting that, in some cases, spatially close clumps are observed
with HST within the typical ALPINE beam size. However, the lack of redshift
information for these sources prevent us from considering them as associate
galaxies experiencing a merging. This could have an impact on our results,
possibly reducing the estimated major merger fraction (and, consequently, the
merger rate) at $z\sim 5$. For these reasons, we also show in Figure 2 (bottom
panels) the individual sizes of the minor and major components of each of the
possible mergers as a function of their FWHM. The colors and marker symbols
are the same as in the top panel, while the hatched markers highlight those
objects with a bad $r_{\mathrm{e}}$ estimate from the fit, after a visual
inspection of the residuals. We also show for comparison the data of the
individual ALPINE galaxies (excluding the mergers) as light squares. As it can
be seen, the sizes of the individual components lie, within the uncertainties,
among those of the ALPINE targets. This suggests that all the individual
components we are analyzing could be merging galaxies and that, with the
current data, we cannot exclude them from the computation of the merger
fraction at $z>4$.

Finally, we provide an estimate of the stellar mass range covered by our
merger components. We only have information on the stellar mass of the merging
system as a whole through SED-fitting measurements (Faisst et al. 2020)
because the majority of the merger candidates are not resolved in most of the
optical and near-IR bands. We make the assumption that $M_{*,1}+M_{*,2}=M_{*}$
and define the mass ratio as $\mu=M_{*,1}/M_{*,2}$, where $M_{*,1}$ and
$M_{*,2}$ are the stellar masses of the primary and secondary galaxy (i.e.,
$M_{*,1}>M_{*,2}$) and $M_{*}$ is the total mass of the system from the SED-
fitting. Then, we set $\mu=4$ as a threshold to separate major ($1\leq\mu\leq
4$) from minor ($\mu>4$) mergers (e.g., Lotz et al. 2011; Mantha et al. 2018;
Duncan et al. 2019) and use this value to solve the above equations for
$M_{*,1}$ and $M_{*,2}$. With the assumptions above, we obtain a stellar mass
distribution of the major components ranging between
$\mathrm{log}(M_{*,1}/M_{\odot})\sim 9.0$ and 10.7, with a mean value
$\mathrm{log}(M_{*,1}/M_{\odot})\sim 10$ (reducing by $\sim 0.2$ dex if
assuming $\mu=1$). We want to stress that this is just an exercise that is
intended to provide an estimate of the stellar mass range of the merger
components we are analyzing and that helps us in the comparison of our
estimates with those of previous works. Indeed, we are not considering that
some of the merger components are resolved in the $K_{s}$ band and have their
own mass ratio (see Section 4.1). Therefore, following the above
considerations, we can conclude that the close pairs of our sample are
characterized by $r_{p}<20$ kpc, $\Delta v<500$ km s-1, and average stellar
masses $\mathrm{log}(M_{*}/M_{\odot})\sim 10$.

###  3.5 Accounting for completeness

When deriving the merger fraction, we have to consider the possibility of
missing a certain number of mergers because of the faintness of the minor
component. Indeed, mergers with a principal component near the threshold of
the observable [CII] flux may imply the presence of a secondary component that
could be lost by instrumental limitations. To account for this, we must
correct for the incompleteness of our sample.

Previous works typically assumed corrections based on the limiting flux or
stellar mass of the corresponding survey (e.g., Patton et al. 2000; López-
Sanjuan et al. 2011; Mundy et al. 2017; Duncan et al. 2019). ALPINE is not a
flux-limited survey as each individual ALMA pointing reaches different depths
(e.g., Béthermin et al. 2020). For this reason, we adopt the classical
completeness corrections found in the literature but considering for each
object its limiting flux. In particular, we apply a weight to each minor
component of our sample defined as

$w_{\mathrm{comp}}(L,z)=\frac{\int_{L_{2}}^{L_{1}}\Phi(L,z)\,dL}{\int_{L_{\mathrm{lim}}}^{L_{1}}\Phi(L,z)\,dL},$
(1)

where $\Phi(L,z)$ is the [CII] luminosity function derived from the UV-
selected central ALPINE targets (Yan et al. 2020), $L_{\mathrm{lim}}$ is the
luminosity corresponding to the limiting flux of each ALPINE pointing, $L_{1}$
is associated to the flux of the primary component $F_{\mathrm{1,[CII]}}$, and
$L_{2}$ corresponds to $F_{\mathrm{2,[CII]}}=F_{\mathrm{1,[CII]}}/4$. We
compute the [CII] luminosities by following Solomon et al. (1992) with the
relation

$L_{\mathrm{[CII]}}=1.04\times 10^{-3}\leavevmode\nobreak\
F_{\mathrm{[CII]}}\leavevmode\nobreak\
D^{2}_{\mathrm{L}}(z)\leavevmode\nobreak\ \nu_{obs}(z)\leavevmode\nobreak\
[L_{\odot}],$ (2)

where $F_{\mathrm{[CII]}}$ is the integrated flux (in Jy km s-1),
$D_{\mathrm{L}}(z)$ is the luminosity distance (in Mpc) at the redshift of the
galaxy, and $\nu_{\mathrm{obs}}(z)$ is the observed frequency of the [CII]
line (in GHz). We also assume that $w_{\mathrm{comp}}=1$ when
$L_{2}>L_{\mathrm{lim}}$. Such a correction corresponds to the ratio between
the number density of galaxies with [CII] flux higher than
$F_{\mathrm{2,[CII]}}$, and the number density of galaxies with a flux higher
than the flux limit of each ALMA pointing.

##  4 Results

###  4.1 The fraction of major mergers in ALPINE

Figure 3: Distributions of the $K_{s}$-band (turquoise histogram) and [CII]
(hatched histogram) flux ratios between the components of each merging system.
Inset: comparison between the [CII] and $K_{s}$-band ratios for the sources
having this information in common, both at $z\sim 4.5$ and $z\sim 5.5$
(represented as yellow and gray circles, respectively). The green area marks
the region of the figure with $\mu_{K}<4$. The dashed black line reports the
1:1 relation between the two quantities.

The aim of this work is to derive the fraction of galaxies undergoing a major
merger at $z\sim 5$ and to estimate their contribution to the galaxy mass
assembly through cosmic time. Although the cosmic evolution of minor mergers
is still poorly constrained at these redshifts, several studies suggest that
they are more frequent in the nearby Universe showing a decreasing fraction
for $z\gtrsim 3$ (López-Sanjuan et al. 2011; Lotz et al. 2011; Ventou et al.
2019). We thus have to take into account a possible contamination from them in
our sample.

First, as we do not have the stellar mass information for the majority of our
merger components, we define as major mergers those candidates for which the
$K_{s}$-band flux ratio is smaller than 4, that is $1<\mu_{K}<4$. The $K_{s}$
band is indeed a good tracer of the stellar mass of galaxies up to $z\sim 4$
(e.g., Laigle et al. 2016). At higher redshifts, the rest-frame $K_{s}$ band
samples the emission below the Balmer break ($K_{s}$-band corresponds to rest-
frame $\sim 3600\leavevmode\nobreak\ \AA$ at $z\sim 5$, taking as a reference
the mean wavelength of the filter) which is no more directly linked to the
stellar mass of the galaxy. In these cases, mid-IR bands could be better
tracers of the galaxy stellar masses than the $K_{s}$ filter, but their worst
resolution do not allow us to spatially resolve the close galaxies for which
we have the $K_{s}$ fluxes. For comparison, López-Sanjuan et al. (2013) used
the $i$-band ratios to identify major mergers. At the redshifts explored in
their work ($0.9<z<1.8$), the observed $i$ band corresponds to $\sim
3000-3500\leavevmode\nobreak\ \AA$ rest-frame (which is quite similar to the
wavelength range covered by the $K_{s}$ band at the highest redshift of our
sample). They compared these ratios with those obtained from the $K_{s}$ band
(sampling the emission between 8000 and 10000 $\AA$ rest-frame, thus tracing
better the stellar mass content of the galaxies), finding no significant
changes in their major merger classification, and then supporting the $i$-band
results. Therefore, following these arguments, we consider $\mu_{K}$
reasonably comparable to the stellar mass ratio of our sources, treating
cautiously these estimates for those galaxies at the highest redshifts of our
sample.

The $K_{s}$-band ratio is available for 9 out of 23 mergers, while for the
remaining sources we only have the [CII] flux ratio, from which we cannot draw
conclusions about the nature of the merger. These two ratios are shown in
Figure 3 where it is evident that the majority of the sources have both
$\mu_{K}$ and $\mu_{\mathrm{[CII]}}$ smaller than 4. Furthermore, in the inset
we compare the [CII] and $K_{s}$-band flux ratios for the objects having these
two measures in common999Both in the main figure and in the inset we do not
report the measure of the [CII] flux ratio for the merger
$\mathrm{DC\\_842313}$ (i.e., $\mu_{\mathrm{[CII]}}\sim 33$). We also do not
consider this source in the statistics described in the text because, as
elaborated in Appendix A, this is a peculiar source.. There is a good
agreement between the two ratios for some of the objects in the figure, but
others show a large scatter from the 1:1 relation, suggesting that complex
physical processes could take place in these systems. However, we find that 7
out of 8 sources (i.e., $\sim 88\%$ of this subsample) have $\mu_{K}<4$ while
only one system shows a larger $K_{s}$-band flux ratio. Although a larger
statistical sample would be needed, we can interpret this as an indication of
the minor merger contamination of our sample. Indeed, if we consider that only
the sources outside the green region contribute to the minor merger
contamination (thus lying at $\mu_{K}>4$), we can assume that the 12% of all
the objects having only the $\mu_{\mathrm{[CII]}}$ information is affected by
minor mergers (or, equivalently, that only the 88% of that sample contains
major mergers). We note that, when excluding from this statistics the sources
at $z\sim 5.5$ (whose $K_{s}$-band ratio is less reliable as a tracer of their
stellar mass), all the objects in the inset of Figure 3 have $\mu_{K}<4$,
raising the fraction of major mergers in this subsample to 100%. This is taken
into account later, when estimating the uncertainty on the merger fraction.

We define the major merger fractions in the two ALPINE redshift bins (at mean
redshift $z\sim 4.5$ and 5.5, respectively) as

$f_{\text{MM}}=0.88\leavevmode\nobreak\
\frac{\sum_{j=1}^{\mathrm{N_{p}}}w_{\mathrm{comp}}^{j}}{\mathrm{N_{g}}},$ (3)

where $\mathrm{N_{p}}$ represents the number of all the mergers in our sample,
$\mathrm{N_{g}}$ is the number of ALPINE galaxies in the considered redshift
bin, and $w_{\mathrm{comp}}^{j}$ is the weight associated to each merger to
correct for incompleteness (see Section 3.5). The factor 0.88 accounts for the
statistical uncertainty on the real merger nature (the fact that the 12% of
the $\mu_{\mathrm{[CII]}}$-only subsample could be affected by the presence of
minor mergers). Note also that, as detailed in Section 3.4, we are not
excluding any source with $r_{p}<2$ kpc because the individual sizes of these
merger components are comparable with those of the ALPINE galaxies.

Table 2: Number of sources at $z\sim 4.5$ and $z\sim 5.5$ from the ALPINE survey used for the computation of the major merger fraction. | $z\sim 4.5$ | $z\sim 5.5$
---|---|---
$\mathrm{N_{g}}$ | 46 | 29
$\mathrm{N_{p}}$ | 15 | 8
$\mathrm{N_{p,corr}}$ | 23 | 11
$f_{\text{MM}}$ | $0.44^{+0.11}_{-0.16}$ | $0.34^{+0.10}_{-0.13}$

To attribute an uncertainty to $f_{\text{MM}}$ we consider two extreme cases.
At first, as an upper limit to the major merger fraction, we count all the
mergers in each redshift bin and divide them by $\mathrm{N_{g}}$,
independently of their components projected distance and flux ratio. This
uncertainty includes also the case in which our sample is not contaminated by
minor mergers (i.e., all the analyzed mergers are major). As a lower limit
instead, we only take the mergers whose components have $r_{p}>2$ kpc,
assuming that all the individual components with a smaller projected
separation are not major mergers (see discussion above). Then, given the small
statistics on both the total number of ALPINE galaxies and the subsample of
mergers, we compute the corresponding Poissonian errors and add them in
quadrature to the above-mentioned uncertainties.

With these assumptions and from Equation 3 we obtain
$f_{\text{MM}}^{z_{4}}=0.44^{+0.11}_{-0.16}$ and
$f_{\text{MM}}^{z_{5}}=0.34^{+0.10}_{-0.13}$, at $z\sim 4.5$ and $z\sim 5.5$,
respectively. These results are shown in Figure 4 and reported in Table 2
along with the numbers used for their computation at both redshifts. We note
that, by using Equation 3, we recover a merger fraction $\sim 30\%$ higher
than obtained if not accounting for completeness corrections. It is also
interesting that, assuming that 12% of the mergers are minor would imply a
minor merger fraction of at least $\sim 5\%$, in agreement with the estimates
by Ventou et al. (2019) at $z>3$ (i.e., $0.08^{+0.07}_{-0.05}$). It should be
noted that this fraction could be even higher. Indeed, the relatively low
spatial resolution of the ALPINE survey does not allow us to constrain the
minor merger fraction at $z\sim 5$ (Le Fèvre et al. 2020), as we are not able
to detect faint satellites that are instead expected to be in the neighborhood
of $z>4$ galaxies by simulations (e.g., Pallottini et al. 2017; Kohandel et
al. 2019).

Finally, it is worth mentioning that $\sim 10\%$ of the ALPINE targets are
part of a massive proto-cluster of galaxies (PCI J1001+0220) at $z\sim 4.57$
located in the COSMOS field (Lemaux et al. 2018). In principle, high-density
environments like those in high-$z$ proto-clusters, or groups and low-mass
clusters at lower redshift, could enhance the merging activity, resulting in a
merger rate that could be several times larger than what expected in lower-
density regimes (e.g., Lin et al. 2010; Lotz et al. 2013; Tomczak et al. 2017;
Pelliccia et al. 2019). To account for this possible caveat, we compare the
ALPINE members of the proto-cluster with the mergers in Table 1, finding that
only two of them are part of the observed over-density. However, by removing
these two sources from our analysis, we do not find any significant change in
the merger fraction at $z\sim 4.5$ and, consequently, in the estimate of the
merger rate.

Figure 4: Cosmic evolution of the major merger fraction $f_{\text{MM}}$ from
the local to the early Universe. Colored squares show the data collected from
the literature at different cosmic times through pair counts, both with
spectroscopic (de Ravel et al. 2009; López-Sanjuan et al. 2013; Tasca et al.
2014; Ventou et al. 2017) and photometric (Xu et al. 2012; Duncan et al. 2019)
redshifts, and/or morphological studies (Conselice & Arnold 2009). Blue stars
are the $f_{\text{MM}}$ estimates found in this work. The solid black line and
the shaded region are the best-fit to the data with a combined power-law and
exponential function and the associated 1$\sigma$ error, respectively.
Finally, the solid brown line illustrates the parameterized redshift evolution
(up to $z=4$) of the major merger fraction from the EAGLE simulation for
galaxies with log$(M_{*}/M_{\odot})>9.5$. The dashed line is just an extension
of that curve to higher redshifts. Table 3: Description of the selection
criteria and sample properties in previous pair counts studies and in this
work.

Literature | Redshift range | Major/Minor separation | $\Delta v$ | $r_{p}$ | Primary Galaxy Mass
---|---|---|---|---|---
|  |  | [km s-1] | [kpc] |
(1) | (2) | (3) | (4) | (5) | (6)
de Ravel et al. (2009) | $0.5<z<0.9$ | $F_{1}/F_{2}\leq 4.0$ | $\leq 500$ | $<20$ | $\mathrm{log}(M_{*}/M_{\odot})\geq 9.5$
Xu et al. (2012) | $0\leq z\leq 0.6$ | $M_{1}/M_{2}\leq 2.5$ | $\leq 500^{(*)}$ | $5-20$ | $9.8<\mathrm{log}(M_{*}/M_{\odot})\leq 10.2$
López-Sanjuan et al. (2013) | $0.9<z<1.8$ | $F_{1}/F_{2}\leq 4.0$ | $\leq 500$ | $<20$ | $9<\mathrm{log}(M_{*}/M_{\odot})<11$
Tasca et al. (2014) | $1.8\leq z\leq 4.0$ | $M_{1}/M_{2}\leq 4.0$ | $\leq 500$ | $<25$ | $9<\mathrm{log}(M_{*}/M_{\odot})<11$
Ventou et al. (2017) | $0.2\leq z\leq 4.0$ | $M_{1}/M_{2}\leq 6.0$ | $\leq 500$ | $<25$ | $\mathrm{log}(M_{*}/M_{\odot})\geq 9.5$
Duncan et al. (2019) | $0.5\leq z<3.5$ | $M_{1}/M_{2}\leq 4.0$ | $\leq 500^{(*)}$ | $5-30$ | $9.7<\mathrm{log}(M_{*}/M_{\odot})<10.3$
This work | $4.4\leq z<5.9$ | $M_{1}/M_{2}\leq 4.0$ | $\leq 500$ | $<20$ | $9\lesssim\mathrm{log}(M_{*}/M_{\odot})\lesssim 11$

  * •

Notes. The asterisks in column 4 indicate the works in the literature that
made use of photometric redshifts to define the selection criterion on the
velocity separation between pairs.

###  4.2 Cosmic evolution of the major merger fraction

During the last years, a large number of studies focused on the estimate of
the merger fraction at different redshifts and on its evolution through cosmic
time (e.g., Le Fèvre et al. 2000; Conselice et al. 2003; Conselice & Arnold
2009; López-Sanjuan et al. 2013; Mundy et al. 2017; Ventou et al. 2017; Mantha
et al. 2018; Duncan et al. 2019). However, the choice of various selection
criteria on the spatial and velocity separation for the pair counts, the
different mass ranges probed by each author and the diverse threshold in $\mu$
to distinguish between major and minor mergers make a direct comparison
between the many results found in the literature quite problematic. Therefore,
we now compare the major merger fraction found from this work at $4<z<6$ with
those computed from samples of galaxies with similar stellar masses and merger
selection criteria.

At $z<1$, Xu et al. (2012) evaluated the major merger fraction from a sample
of close pairs drawn from the COSMOS survey (Ilbert et al. 2009). From that
work we select only the results in the stellar mass bin
$9.8<\mathrm{log}(M_{*}/M_{\odot})\leq 10.2$, to be consistent with the
average stellar mass of our sample. At this epoch we also account for the
merger fraction computed by de Ravel et al. (2009) who analyzed a sample of
spectroscopically confirmed galaxy pairs from the VIMOS VLT Deep Survey (VVDS;
Le Fèvre et al. 2005) with $\mathrm{log}(M_{*}/M_{\odot})>9.5$. At $1<z<4$,
López-Sanjuan et al. (2013) and Tasca et al. (2014) exploited the VUDS, VVDS
and the Mass Assembly Survey with SINFONI in VVDS (MASSIVE; Contini et al.
2012) surveys to estimate $f_{\text{MM}}$ from merging systems with
spectroscopic measurements and average stellar masses
$\mathrm{log}(M_{*}/M_{\odot})\sim 10$. Ventou et al. (2017) observed the
Hubble Ultra Deep Field (Beckwith et al. 2006) and the Hubble Deep Field South
(Williams et al. 2000) with the Multi Unit Spectroscopic Explorer (MUSE) to
constrain the galaxy major merger fraction up to $z\sim 6$. However, we take
only their data points with $\mathrm{log}(M_{*}/M_{\odot})\geq 9.5$ which
extend up to $z\sim 3$. Finally, we get the pair fraction computed by Duncan
et al. (2019) in a mass-selected ($9.7<\mathrm{log}(M_{*}/M_{\odot})<10.3$)
sample of galaxies drawn from the CANDELS survey (Grogin et al. 2011;
Koekemoer et al. 2011) using a probabilistic approach based on their
photometric redshifts. All these works assume a projected separation
$r_{p}<30$ kpc between the merger components and a difference in velocity
$\mathrm{\Delta v\leq 500\leavevmode\nobreak\ km\leavevmode\nobreak\ s^{-1}}$.
Most of them adopt $\mu<4$ to identify the major merger population, with $\mu$
computed either as the ratio between the stellar mass of the primary and
secondary galaxy in the system or as the difference in magnitude between the
two components. The only exceptions are from Xu et al. (2012) and Ventou et
al. (2017) that make use of a mass ratio of 2.5 and 6, respectively (however,
as stated by Ventou et al. 2017, adopting a mass ratio limit of 4 implies the
loss of only a few sources in their sample, not significantly altering the
final statistics). Table 3 summarizes the main features of the samples from
which the observational data at $z<4$ introduced above are taken.

As shown in Figure 4, we combine these data with our measurements at $z\sim 5$
to provide the cosmic evolution of the major merger fraction. Traditionally,
this is parameterized with a power-law fitting formula of the form

$f_{\text{MM}}(z)=f_{0}\leavevmode\nobreak\ (1+z)^{m},$ (4)

where $f_{0}$ is the merger fraction at $z=0$ and $m$ is the index that rules
the redshift evolution. Our data suggest a slight decline of the merger
fraction at $z>4$, with a possible peak at lower redshift, as also previously
found by other studies (e.g., Conselice & Arnold 2009; Ventou et al. 2017;
Mantha et al. 2018; Duncan et al. 2019). Therefore, we make use of a combined
power-law and exponential function to fit our observational data, such as

$f_{\text{MM}}(z)=\alpha\leavevmode\nobreak\ (1+z)^{m}\leavevmode\nobreak\
\mathrm{exp}(\beta(1+z)),$ (5)

where $\beta$ controls the exponential side of the curve and
$f_{0}=\alpha\leavevmode\nobreak\ \mathrm{exp}(\beta)$. By using a non linear
least square algorithm, we fit our data both with Equation 4 and Equation 5.
Taking advantage of the Bayesian information criterion (BIC), we find a
significant statistical evidence ($\Delta\mathrm{BIC}>10$) in favor of the
combined power-law and exponential form for describing the evolution of the
merger fraction through cosmic time. These results are in contrast with those
by Duncan et al. (2019) who, also relying on the BIC, found that there was no
strong statistical evidence in favor of one of the two above-described
parameterizations. This difference could be attributed to the fact that, for
stellar masses $\mathrm{log}(M_{*}/M_{\odot})\sim 10$, they measured the major
merger fraction up to $z\sim 3$, while in this work our new ALPINE data allow
us to constrain this quantity up to earlier epochs (i.e., $z\sim 5$).

Table 4: Best-fit parameters and 1$\sigma$ uncertainties obtained by fitting the merger fractions and merger rates with a power-law plus exponential functional form of the type $\alpha(1+z)^{m}\leavevmode\nobreak\ \mathrm{exp}(\beta(1+z))$. $T_{\text{MM}}$ | $\alpha$ | $m$ | $\beta$
---|---|---|---
Merger fraction $f_{\text{MM}}$
- | $0.024\pm 0.003$ | $4.083\pm 0.501$ | $-0.797\pm 0.189$
Merger rate $R_{\text{MM}}$
Kitzbichler & White (2008) | $0.020\pm 0.003$ | $4.282\pm 0.488$ | $-1.036\pm 0.182$
Jiang et al. (2014) | $0.015\pm 0.002$ | $4.350\pm 0.492$ | $-0.740\pm 0.184$
Snyder et al. (2017) | $0.010\pm 0.001$ | $6.083\pm 0.501$ | $-0.797\pm 0.189$

For these reasons, we show in Figure 4 the best-fitting function to our data
obtained through Equation 5, and use the results from this parameterization
for the rest of the work. In particular, the parameters of this fit are
$\alpha=0.024\pm 0.003$, $m=4.083\pm 0.501$ and $\beta=-0.797\pm 0.189$ which
are in good agreement with those found by Duncan et al. (2019) when fitting
the CANDELS data points for galaxies with stellar mass
$9.7<\mathrm{log}(M_{*}/M_{\odot})<10.3$. For comparison, we also show the
results obtained by Conselice & Arnold (2009) exploiting morphological
analysis and pair counts for a sample of Lyman-break drop-out galaxies at
$4<z<6$ with masses $\mathrm{log}(M_{*}/M_{\odot})>9-10$. Their data points
are lower than our results but still comparable to them within the
uncertainties. Anyway, as they do not differentiate between major and minor
mergers, we do not include them in the computation of the cosmic evolution of
the merger fraction.

We show in Figure 4 the best-fit parameterization of the major merger fraction
found by Qu et al. (2017) for galaxies with $\mathrm{log}(M_{*}/M_{\odot})\geq
9.5$ from the Evolution and Assembly of Galaxies and their Environments
(EAGLE) hydrodynamical simulation (Schaye et al. 2015). They adopted a stellar
mass ratio $\lesssim 4$ to identify major mergers and the same
parameterization as in Equation 5, obtaining a cosmic evolution of
$f_{\text{MM}}$ that is similar to ours up to $z\sim 4$ (the higher redshift
reached by the simulation). Extending their prediction to earlier epochs, a
slightly milder redshift evolution with respect to our results is found.
However, the two trends are still in good agreement with each other.

Finally, it is worth noting that we are recovering the cosmic evolution of the
merger fraction by comparing galaxies with a constant stellar mass range at
all redshifts. We are aware that, in this way, we are not necessarily tracing
the same galaxy population through cosmic time, as instead achieved with
constant cumulative comoving number density selections (Papovich et al. 2011;
Conselice et al. 2013; Ownsworth et al. 2014; Mundy et al. 2015; Torrey et al.
2015)101010Although this is no more assured in case of major mergers or strong
changes in star formation (e.g., Leja et al. 2013; Ownsworth et al. 2014)..
Nevertheless, as the majority of the observational and theoretical works in
the literature make use of the constant stellar mass selection to estimate the
fraction and rate of major mergers through cosmic time, such a choice allows
us to make a fair comparison to these works and provides a quite easily
quantifiable observational benchmark for future studies.

Figure 5: Left panel: Redshift evolution of the major merger rate. The solid
lines with the shaded regions represent the best-fitting functions to the data
assuming three different merger timescales and their associated 1$\sigma$
uncertainties, respectively. The cosmic evolution of $R_{\text{MM}}$ computed
from the halo-halo merger rate by Dekel et al. (2013), from the empirical
model by Hopkins et al. (2010), and from the Illustris (Rodriguez-Gomez et al.
2015) and EMERGE (O’Leary et al. 2021) simulations are also shown with dashed
dark blue, turquoise, cyan, yellow and orange lines. The dotted turquoise line
shows the extrapolation of the Hopkins et al. (2010) merger rate to higher
redshifts. Right panel: Cumulative number of major mergers per galaxy over
cosmic time. Solid lines represent the cumulative distributions obtained by
integrating Equation 8 adopting the merger timescale prescriptions of
Kitzbichler & White (2008), Jiang et al. (2014) and Snyder et al. (2017). The
shaded regions are the associated uncertainties computed from the best-fit
errors on the merger rate cosmic evolution.

###  4.3 Galaxy major merger rate

The major merger fraction can be translated into the merger rate
$R_{\text{MM}}$ (i.e., the number of mergers per galaxy and Gyr) as

$R_{\text{MM}}(z)=\frac{C_{\text{merg}}\leavevmode\nobreak\
f_{\text{MM}}(z)}{T_{\text{MM}}(z)},$ (6)

where $f_{\text{MM}}(z)$ is the pair fraction at redshift $z$ estimated in
Section 4.2, and $C_{\text{merg}}$ is the fraction of close pairs that will
eventually merge into a single system in a typical timescale
$T_{\text{MM}}(z)$. This last term represents the major source of uncertainty
in the merger rate computation and is usually obtained through simulations
based on the dynamical friction timescales affecting the merging components
(e.g., Kitzbichler & White 2008; Lotz et al. 2010; Jiang et al. 2014; Snyder
et al. 2017).

Kitzbichler & White (2008) applied a semi-analytic model to the Millennium
simulation (Springel et al. 2005) outputs finding an average merging time that
depends linearly on the redshift and projected distance of the pair and weakly
on the stellar mass of the main galaxy, namely $T_{\text{MM}}\propto
r_{p}\leavevmode\nobreak\ M_{*}^{-0.3}\leavevmode\nobreak\ (1+z/8)$. This
relation is valid at $z\leq 1$, while at higher redshift it becomes

$T_{\text{MM}}^{-1/2}=T_{0}^{-1/2}+f_{1}z+f_{2}(\mathrm{log}M_{*}-10),$ (7)

where $T_{0}$ is the merging time at $z=0$, $M_{*}$ is the stellar mass of the
primary galaxy, while $f_{1}$ and $f_{2}$ are two coefficients of the
parameterization. A mass dependence similar to the $z\leq 1$ Kitzbichler &
White (2008) timescale ($T_{\text{MM}}\sim M_{*}^{-0.2}$) was also found by
Lotz et al. (2010) for mergers in the stellar mass range
$9.7<\mathrm{log}(M_{*}/M_{\odot})<10.7$ but in a smaller sample. More recent
works explored a different redshift evolution of the merger timescale. For
instance, Jiang et al. (2014) took into account the mass loss due to dynamical
friction of the virial masses of galaxies experiencing a merger and, by using
a high-resolution $N-$body cosmological simulation, found
$T_{\text{MM}}\propto H(z)^{-1/3}$, where $H(z)$ is the Hubble parameter at
redshift $z$. An even stronger redshift evolution was found by Snyder et al.
(2017) by comparing mass-selected close pairs with the intrinsic galaxy merger
rate from the Illustris simulations (Genel et al. 2014; Vogelsberger et al.
2014). They found $T_{\text{MM}}=2.4\leavevmode\nobreak\ (1+z)^{-2}$ but for
primary galaxies with stellar mass range
$10.5<\mathrm{log}(M_{*}/M_{\odot})\leq 11$. It is better to specify that, in
the case of Snyder et al. (2017), $T_{\text{MM}}$ does not correspond to the
merger timescale of Equation 6, rather it is the merger observability
timescale which already includes the effects of the $C_{\text{merg}}$ term.
Nevertheless, we refer to that as $T_{\text{MM}}$ for simplicity.

For the rest of this work, we compare the results obtained with the merger
timescales provided by Kitzbichler & White (2008), Jiang et al. (2014) and
Snyder et al. (2017), in order to highlight the large differences in the final
outcome due to the choice of this parameter. About the probability for a pair
to merge over a certain timescale (i.e., $C_{\text{merg}}$), it is usually
fixed to 0.6 (e.g., Lotz et al. 2011; Mantha et al. 2018). However, as in most
cases this quantity is already included in the merger timescale prescriptions
provided in different works (e.g., Duncan et al. 2019), we set
$C_{\text{merg}}=1$ throughout.

With these assumptions, we present in Figure 5 (left panel) the cosmic
evolution of the merger rate, as obtained by combining our new $z>4$ ALPINE
data with those at lower redshifts from the literature. For each data point we
compute $R_{\text{MM}}$ from Equation 6, adopting the three different merger
timescales defined above111111When adopting the merger timescale by Jiang et
al. (2014), we just consider the $T_{\text{MM}}$ redshift evolution found in
their work (i.e., $T_{\text{MM}}\sim H(z)^{-1/3}$) and normalize it to
$T_{\text{MM}}(z=0)$ from Kitzbichler & White (2008). This is also similar to
setting a constant merger timescale of $\sim 1$ Gyr. and fit the resulting
data with the same functional form used for the merger fraction. We show the
best-fitting functions to these data (and their $1\sigma$ uncertainties) in
Figure 5 and report the corresponding parameters in Table 4.

As evident, all the three functions show a steep increase of the merger rate
from the local Universe to $z\sim 1$. However, at higher redshifts they lead
to large differences about the incidence of these sources at early epochs.
When considering the Kitzbichler & White (2008) timescale, we find a redshift
evolution which is similar to what obtained in Section 4.2 for the merger
fraction, with a peak at $z\sim 2.5$ and a quite fast decrease at higher
redshifts. Both the merger rate evolutions obtained with $T_{\text{MM}}(z)$ by
Jiang et al. (2014) and Snyder et al. (2017) show instead a milder redshift
attenuation at $z\gtrsim 4$. In particular, the increasingly smaller merger
timescale found by Snyder et al. (2017) involves a very large merger rate in
the early Universe compared to the other functions, possibly implying a
significant role of major mergers in the galaxy mass assembly at $z\sim 5$.

We then compare our results with the cosmic evolution of several expected
merger rates from simulations. At $z<3$, Hopkins et al. (2010) found that
$R_{\text{MM}}\propto(1+z)^{1.8}$ for galaxies with
$\mathrm{log}(M_{*}/M_{\odot})>9$ and $\mu<3$. This redshift evolution is
comparable to our best-fitting function with $T_{\text{MM}}(z)\propto
H(z)^{-1/3}$ at $1\lesssim z\lesssim 3$. However, if we extrapolate their
results up to $z\sim 6$, we find that their predicted major merger rates could
be larger (by $\sim 0.5$ dex) with respect to our observations. A stronger and
increasing redshift dependence was found by Rodriguez-Gomez et al. (2015) for
the merger rate from the Illustris simulations. By exploiting their parametric
fitting formula with a stellar mass121212It is worth noting that, as also
stated by Snyder et al. (2017) and O’Leary et al. (2021), the mass ratio
defined by Rodriguez-Gomez et al. (2015) is evaluated at the time when the
secondary galaxy has achieved its maximum stellar mass. This prevent us to
make a direct comparison among the results of this and the other simulations.
of $\mathrm{log}(M_{*}/M_{\odot})=10$ and an average mass ratio $\mu=2$ over
cosmic time (see Section 4.5), the simulation tends to over-predict our
observed merger rates at $z\sim 0$, while it underestimates them at
intermediate redshifts. We also show the merger rate predictions found by
O’Leary et al. (2021) using the results from the Empirical ModEl for the
foRmation of GalaxiEs (EMERGE; Moster et al. 2018). In particular, they
modeled the merger fraction evolution as in Equation 5 and then convolved it
with a typical merger timescale to obtain $R_{\text{MM}}$ as a function of
redshift. We consider their best-fit parameters describing $f_{\text{MM}}$ for
galaxies with $\mathrm{log}(M_{*}/M_{\odot})\geq 9.7$ and projected distance
$r_{p}<30$ kpc. When convolving this $f_{\text{MM}}(z)$ with a merger
timescale $T_{\text{MM}}\propto H(z)^{-1/3}$ and with a power-law scaling as
$T_{\text{MM}}\propto(1+z)^{-2}$ we obtain a lower and milder $R_{\text{MM}}$
cosmic evolution with respect to our major merger rate results obtained with a
Kitzbichler & White (2008) timescale, and a comparable trend with the Jiang et
al. (2014)-based evolution, respectively. Lastly, we show the predicted halo-
halo merger rate by Dekel et al. (2013) which over-predicts the observed
merger rates at low redshifts but is in good agreement with the Snyder et al.
(2017)-based $R_{\text{MM}}$ at $z\sim 6$.

We compute the average number of mergers a galaxy undergoes between $0<z<6$ by
integrating the merger rate over time as

$N_{\text{MM}}=\int_{z_{1}}^{z_{2}}{\frac{R_{\text{MM}}(z)}{(1+z)H_{0}E(z)}dz},$
(8)

where $E(z)=\sqrt{\mathrm{\Omega_{m}}(1+z)^{3}+\mathrm{\Omega_{\Lambda}}}$,
assuming a flat Universe (i.e., $\mathrm{\Omega_{k}}=0$). We show in Figure 5
(right panel) the cumulative number of major mergers as a function of redshift
adopting three different timescales, with the 1$\sigma$ uncertainties derived
from the errors on the major merger rate cosmic evolution. The average number
of mergers computed with the Kitzbichler & White (2008) and Jiang et al.
(2014) prescriptions is low, ranging between 0.5 and 1.4 down to $z=0$. This
is consistent, for instance, with the results by Mundy et al. (2017) who found
that galaxies with $\mathrm{log}(M_{*}/M_{\odot})>9.5$ at $z\sim 3.5$
(selected at constant number densities, thus representing the progenitors of
$\mathrm{log}(M_{*}/M_{\odot})>11$ galaxies at $z\sim 0$) undergo $\sim 0.5$
major mergers between $0<z<3.5$. However, if we assume an evolving merger
timescale as $2.4\leavevmode\nobreak\ (1+z)^{-2}$ (Snyder et al. 2017), the
average number of mergers since $z=6$ increases significantly, reaching
$N_{\text{MM}}\sim 8$ at the present day. Interestingly, this is similar to
the result obtained by Conselice & Arnold (2009) that found $N_{\text{MM}}\sim
7$ by integrating Equation 8 from $z=6$ to 0, and assuming a constant merger
timescale $T_{\text{MM}}=0.35$ (which is equal to the average timescale by
Snyder et al. 2017 at $z<6$; see their Figure 15). When examining our results
at higher redshifts, we find that the number of mergers diminishes quickly. In
particular, at $z>4$, $N_{\text{MM}}\lesssim 1$ if we consider the Kitzbichler
& White (2008) and Jiang et al. (2014) merger timescale evolution, suggesting
that, in these scenarios, not every galaxy will undergo a merger during this
epoch.

###  4.4 Merger rate density

Figure 6: Merger rate density as a function of redshift. The squares represent
the volume-averaged merger rates computed from the literature data at $z<4$
adopting the Kitzbichler & White (2008) prescription for the merger timescale.
The blue stars are the ALPINE data at $z\sim 5$ from this work. The error bars
are obtained by combining in quadrature the uncertainties on the merger
fractions and on the number densities. The darker hatched area reports the
results including errors by Mundy et al. (2017) at $z<3.5$. The lighter area
only shows their extrapolation to higher redshifts. The dashed yellow and
brown lines display the evolution of $\Gamma_{\text{MM}}$ found with the
EMERGE simulation (O’Leary et al. 2021) for galaxies with
$\mathrm{log}(M_{*}/M_{\odot})>9$ and $\mathrm{log}(M_{*}/M_{\odot})>10$,
respectively.

As discussed in Section 4.3, the merger rate traces the number of merger
events per galaxy at a given mass and redshift. A more informative quantity is
the volume-averaged merger rate which is defined as

$\Gamma_{\text{MM}}(z)=\frac{f_{\text{MM}}(z)\leavevmode\nobreak\
n(z)}{T_{\text{MM}}(z)},$ (9)

where $f_{\text{MM}}$ and $T_{\text{MM}}$ are the previously defined merger
fraction and merger timescale, and $n(z)$ is the number density of galaxies in
Mpc-3. The latter is computed by integrating the galaxy stellar mass function
(GSMF) in a certain redshift bin and between a minimum and maximum stellar
mass limit in the range $M_{*}^{\text{min}}<M_{*}<M_{*}^{\text{max}}$. In
particular, we exploit the GSMFs best-fit parameters from Mortlock et al.
(2015), Santini et al. (2012) and Davidzon et al. (2017) at $z<3$, $3<z<4$ and
$z>4$, respectively, after converting into a Chabrier (2003) IMF. However, it
is worth noting that, as stated in Mundy et al. (2017), making use of
different GSMF parameterizations does not significantly change the final
results. We thus integrate these functions from
$\mathrm{log}(M_{*}/M_{\odot})=9$ to $\mathrm{log}(M_{*}/M_{\odot})=11$, in
order to include the full possible range of galaxy masses for the considered
data at different redshifts. We estimate the errors on $n(z)$ by perturbing
the corresponding GSMF with the associated uncertainties on its best-fit
parameters at each redshift. We repeat this procedure 104 times, recomputing
the integrated number densities at each step. From the perturbed distributions
we then obtain the 16th and 84th percentiles as the 1$\sigma$ errors on
$n(z)$.

Figure 6 shows the merger rate density computed with Equation 9 for each of
the data introduced in Section 4.2 and for the two $z>4$ ALPINE bins. For the
sake of simplicity, in this case we report only the data points obtained by
adopting the Kitzbichler & White (2008) prescription for the merger timescale.
When using the $T_{\text{MM}}(z)$ redshift evolution from Jiang et al. (2014)
and Snyder et al. (2017), we obtain similar trends but shifted to higher
merger rate densities. The error bar on each point is computed by propagating
the merger fraction and number density uncertainties on Equation 9. At
$1\lesssim z\lesssim 5$, Duncan et al. (2019) found that the volume-averaged
merger rate is quite constant. On the other hand, from our data we find a
slight decrease both at $z<1$ and at $z>4$. At these early epochs, this
difference could be caused by the poor constraints on the GSMFs adopted by
Duncan et al. (2019) which result in large uncertainties on their data, making
it impossible to draw robust conclusions at $z\sim 5$. Nevertheless, our
derived merger rate densities are in agreement, within the uncertainties, with
other results derived in the literature. For example, Mundy et al. (2017)
studied the evolution of the merger rate density up to $z\sim 3.5$ for a large
sample of $\mathrm{log}(M_{*}/M_{\odot})>10$ galaxies. They found that, for
close pairs with projected separations $5<r_{p}<20\leavevmode\nobreak\ $kpc,
the $\Gamma_{\text{MM}}(z)$ evolution is better described by a power-law of
the form $\Gamma_{0}(1+z)^{\gamma}$, with
$\Gamma_{0}=1.64^{+0.58}_{-0.41}\times 10^{-4}$ and
$\gamma={0.48}^{+1.00}_{-1.15}$. We report their results, along with the
uncertainties, in Figure 6 and extrapolate them to the redshifts explored by
ALPINE. As evident, our data points are comparable with the results by Mundy
et al. (2017). If we also fit our merger rate densities (derived assuming the
Kitzbichler & White (2008) merger timescale) with a power-law function we
obtain $\Gamma_{0}=(2.40\pm 0.63)\times 10^{-4}$ and $\gamma=-0.16\pm 0.23$,
that are consistent with the above outcomes. The power-law fit on the data
computed with the timescales by Jiang et al. (2014) is in agreement with the
Mundy et al. (2017) findings, as well. If we instead consider the results
obtained with the Snyder et al. (2017) timescale, we find a rapid increase of
the merger rate density with redshift, which departs from the upper envelope
of Mundy et al. (2017) already at $z\gtrsim 1$.

Finally, we also show the results from the EMERGE simulation (O’Leary et al.
2021) for galaxies in two different stellar mass bins. The two curves in the
figure represent the intrinsic merger rate from the simulation for major
mergers (i.e., $\mu<4$) selected basing on their main progenitor mass, thus
describing a population of galaxies that will undergo a merger after a certain
time. Galaxies with $\mathrm{log}(M_{*}/M_{\odot})>9$ are characterized by an
increase of the merger rate density up to $z\sim 1$, an almost constant
$\Gamma_{\text{MM}}$ until $z\sim 4$ and a slow decrease to the highest
redshifts. At larger stellar masses, a slighter increase of
$\Gamma_{\text{MM}}$ is present at low redshifts, while a steeper decrease in
the early Universe is found. Both these trends are comparable to the one
obtained in this work after exploiting data at different redshifts and
including the new estimates at $z>4$ from the ALPINE survey. The fact that
they are systematically smaller than our results could be ascribed to several
factors, such as the lower merger fraction evolution found in their simulation
or the measure of the intrinsic merger rate which is computed when the merging
process is already happened (thus possibly enhancing the merger timescales
with respect to our assumptions).

###  4.5 The major merger specific mass accretion rate

Figure 7: Left: Comparison between the redshift evolution of the specific mass
accretion rate (as derived from $R_{\text{MM}}$) and the specific star
formation rate. The solid lines with the shaded regions are the best-fitting
functions (and corresponding uncertainties) to the same $R_{\text{MM}}$ data
of the left panel of Figure 5 at different merger timescales divided by the
average mass ratio $\overline{\mu}$ through cosmic time. The dot-dashed lines
are several specific star formation rates from the literature (Dekel et al.
2009; Speagle et al. 2014; Tasca et al. 2015). The sSFR by Dekel et al. (2009)
is normalized to sSFR$(z=0)=0.1$, as in Tasca et al. (2015). The dark cyan
diamonds are the values by Khusanova et al. (2021) from the ALPINE survey.
Right: Stellar mass accretion rate density ($\rho_{\text{MM}}$) as a function
of redshift (top panel). Solid and dashed lines represent the best-fits to the
data assuming double power-law, and combined power-law and exponential
functions, respectively. The shaded regions highlight the uncertainties
resulting from the choice of the fitting form. The colors correspond to
alternative merger timescales and are the same as in the left panel. The
dotted curve reports the mass accretion rate density, $\rho_{\text{ill}}$,
obtained from the Illustris simulation, as described in the text. The dot-
dashed red line shows the SFRD ($\rho_{\mathrm{SFRD}}$) by Madau & Dickinson
(2014). The red diamonds are the total SFRD values obtained from the ALPINE
survey (Khusanova et al. 2021). The bottom panel displays the ratio between
$\rho_{\text{MM}}$ and $\rho_{\mathrm{SFRD}}$ as a function of cosmic time.
The dashed horizontal line marks a ratio equal to 1.

By taking advantage of the merger rates and merger fraction deduced in the
previous sections, we now provide an estimate of the average stellar mass
gained by a galaxy during a merger event over cosmic time. To this aim, we
compute the major merger specific mass accretion rate as
$R_{\text{MM}}\leavevmode\nobreak\ \overline{\mu}^{-1}$, where
$\overline{\mu}$ is the average mass ratio. We find that, for each data sample
used to compute the merger rate at different redshifts,
$\overline{\mu}^{-1}\sim 0.5$, including the average mass ratio from our
sample of major mergers for which we know $\mu_{K}$. Therefore, we just take
the best-fitting functions to the cosmic evolution of the merger rate and
multiply them by a factor $\sim 0.5$ to obtain the specific mass accretion
rate in Gyr-1. Our results are reported in Figure 7 along with the specific
star formation rate (sSFR) evolution obtained by different authors (Dekel et
al. 2009; Speagle et al. 2014; Tasca et al. 2015; Khusanova et al. 2021). As
shown, star formation seems to be the dominant mode of mass growth at all
epochs. However, if we assume a merger timescale
$T_{\text{MM}}\propto(1+z)^{-2}$ (Snyder et al. 2017), the specific mass
accretion rate reaches the sSFR at $z>3$, being comparable to it within the
uncertainties. Therefore, we cannot exclude that major mergers may have
significantly contributed to the galaxy mass assembly in the early Universe.
In particular, if the galaxy growth is dominated by cold gas accretion, the
sSFR should evolve with redshift as $(1+z)^{2.25}$ (e.g., Dekel et al. 2009).
Some authors find instead a flattening of the trend or even a possible
decrease with redshift as from the ALPINE data (Tasca et al. 2015; Faisst et
al. 2016; Khusanova et al. 2021), leaving space to other mechanisms that could
regulate the assembly of galaxies at high redshifts, such as major mergers
(e.g., Tasca et al. 2015; Faisst et al. 2016).

###  4.6 The major merger mass accretion rate density

Similarly to what done in Section 4.5 for the specific mass accretion rate, we
exploit here the major merger rate density to obtain an estimate of the mass
accreted through major mergers per unit time and volume for galaxies in a
given stellar mass range, $\rho_{\text{MM}}(z)$. This quantity can be compared
to the other mechanisms of mass accretion, such as the mass gained through the
process of star formation, and is then of fundamental importance to understand
the role of mergers in the Universe.

Following Duncan et al. (2019), we assume that the increase in stellar mass
for each merger and at each redshift is given by
$\overline{M}_{*}\leavevmode\nobreak\ \overline{\mu}^{-1}$, where
$\overline{\mu}$ is the average mass ratio defined in Section 4.5 and
$\overline{M}_{*}$ is the average stellar mass calculated through the GSMF.
The latter is computed as

$\overline{M}_{*}(z)=\frac{\int_{M_{*}^{\text{min}}}^{M_{*}^{\text{max}}}{\Phi(z,M_{*})\leavevmode\nobreak\
M_{*}\leavevmode\nobreak\
dM_{*}}}{\int_{M_{*}^{\text{min}}}^{M_{*}^{\text{max}}}{\Phi(z,M_{*})\leavevmode\nobreak\
dM_{*}}},$ (10)

where $\Phi(z,M_{*})$ represents the shape of the GSMF in a certain redshift
bin. We estimate the uncertainties on $\overline{M}_{*}(z)$, as previously
done for the number density, by recomputing it N times after perturbing the
corresponding GSMF with its associated errors. At each redshift, we obtain a
typical uncertainty of $\sim 0.5\leavevmode\nobreak\ $dex. In this way, the
stellar mass accretion rate density can be estimated as

$\rho_{\text{MM}}(z)=\overline{\mu}^{-1}\leavevmode\nobreak\
\overline{M}_{*}\leavevmode\nobreak\ \Gamma_{\text{MM}}(z).$ (11)

We report the cosmic evolution of $\rho_{\text{MM}}$ in Figure 7 (right panel,
on top). The dashed lines show the best-fitting curves to the data assuming a
combined power-law and exponential function, as done for the merger fraction
and merger rate, for different merger timescales. As shown, at $z>1$ the
curves start to decrease quite rapidly to lower mass accretion rate densities.

Following Mundy et al. (2017), we then provide the major merger accretion rate
density computed with the results from the Illustris simulation,
$\rho_{\text{ill}}(z)$. We convolve the specific mass accretion rate by
Rodriguez-Gomez et al. (2016) with the number density of galaxies by Torrey et
al. (2015) as

$\rho_{\text{ill}}(z)=\int_{M_{*}^{\text{min}}}^{M_{*}^{\text{max}}}\int_{\mu_{\text{min}}}^{\mu_{\text{max}}}{\Phi_{\text{ill}}(z,M_{*})\leavevmode\nobreak\
{\dot{m}}_{\text{accr}}(z,M_{*},\mu)\leavevmode\nobreak\
d\mu\leavevmode\nobreak\ dM_{*}},$ (12)

where $\Phi_{\text{ill}}$ is the shape of the GSMF at redshift $z$ and stellar
mass $M_{*}$, and $\dot{m}_{\text{accr}}$ represents the average amount of
mass accreted by a single galaxy when the merger occurs. As done for
$\rho_{\text{MM}}$, we integrate the GSMF in the range
$9<\mathrm{log}(M_{*}/M_{\odot})<11$ and the mass accretion rate for mass
ratios between $1<\mu<4$, accounting for major mergers. The result of the
integration is reported in Figure 7 as a function of redshift. As can be seen,
the computed $\rho_{\text{ill}}$ is comparable with our findings, lying
between the best-fit curves obtained by adopting the Kitzbichler & White
(2008) and Snyder et al. (2017) prescriptions for the merger timescale,
respectively. Moreover, the shape of the curve is quite similar to those
deduced by our analysis, with a fast decrease at high redshifts.

To compare the relative contribution of mergers and star formation to the mass
assembly of galaxies through the cosmic time, we also show the star-formation
rate density (SFRD) from Madau & Dickinson (2014), after converting for a
Chabrier IMF131313To convert SFRs from Salpeter (1955) to Chabrier IMF we
multiply by 0.63 (e.g., Madau & Dickinson 2014).. Moreover, we find that our
data are also well fit by a double power-law of the form assumed by Madau &
Dickinson (2014) for the SFRD, although we find no significant differences
between the two functional forms when comparing them with a BIC statistics.
The results of these fits are reported in the figure (solid lines), showing a
flatter trend to the early Universe with respect to the combined power-law and
exponential curves. In the bottom region of the figure we show the ratio
between the major merger and star formation contributions to the stellar mass
accretion in galaxies at different epochs. Looking at both curves, if we
assume the merger timescales by Kitzbichler & White (2008) and Jiang et al.
(2014), major mergers seem to contribute less than $10\%$ to the galaxy mass
assembly compared to the star formation mechanism at all redshifts. For
comparison we also report, in the top panel, the total (UV+IR) SFRD obtained
with the ALPINE data (Khusanova et al. 2021). These measurements indicate a
possible $z>4$ evolution of the SFRD that is shallower than previously
thought, further decreasing the importance of major mergers to the mass
assembly of galaxies at these epochs. On the other hand, adopting the Snyder
et al. (2017) timescale prescription, a larger contribution of mergers in the
early Universe is in place which becomes comparable to that provided by the
star formation at $z\gtrsim 6$.

##  5 Discussion

###  5.1 The importance of the merger timescale

As evidenced by the results presented in Section 4, one of the major sources
of uncertainty in the investigation of the contribution of major mergers to
the mass assembly of galaxies through cosmic time is the typical time needed
for a pair of objects to coalesce with each other. Indeed, this parameter is
critical for converting the pair fraction into a merger rate, affecting all
the derived quantities as well. Because of this, it could represent the main
reason of disagreement between models and observations noticed in many works.

Previous studies often assumed $T_{\text{MM}}$ as a constant over time.
However, the emerging picture of a possibly decreasing (as found in this work)
or nearly flat merger fraction through high redshifts (Conselice & Arnold
2009; Ventou et al. 2017; Mantha et al. 2018), compared to the increase of the
merger rate found in most simulations over the same epoch (e.g., Hopkins et
al. 2010; Rodriguez-Gomez et al. 2015), suggests that a redshift-dependent
merger timescale is more suitable to reconcile models and observations. Here
we show the results derived from three different merger timescales
(Kitzbichler & White 2008; Jiang et al. 2014; Snyder et al. 2017), as
introduced in Section 4.3, and compare them with state-of-the-art simulations.
Kitzbichler & White (2008) found a merger timescale which slightly depends on
the primary galaxy mass and that increases with redshift, leading to lower
values of the merger rate when moving to early epochs. On the other hand,
Jiang et al. (2014) and Snyder et al. (2017) found a slow and fast decrease of
$T_{\text{MM}}$ over time, respectively, implying a larger contribution from
mergers to the galaxy mass assembly at $z>4$.

Looking at Figure 5 (left panel), the merger rates obtained from the
Kitzbichler & White (2008) timescale prescription are not easily reproducible
by models. In fact, although we find values of $R_{\text{MM}}$ comparable to
those of the EMERGE simulation (O’Leary et al. 2021) at $z\gtrsim 5$, the
latter are computed with a Jiang et al. (2014)-like evolving timescale which
can provide $T_{\text{MM}}$ values up to 5 times lower than the Kitzbichler &
White (2008) ones at these epochs. A good agreement with the
$T_{\text{MM}}\propto H(z)^{-1/3}$ merger rates is found by convolving the
EMERGE pair fraction evolution with the decreasing Snyder et al. (2017)
timescale (again using smaller merging times in the simulation with respect to
the observations) and with the outputs of the Hopkins et al. (2010)
simulations, at least at $z<3$ (indeed, they predicted larger merger rates
with respect to our findings at earlier epochs). However, in this case, we
note that the merger rates from Hopkins et al. (2010) are properly computed up
to $z\sim 3$ and that the high-redshift extrapolation could be different from
the real outputs of their simulation. Moreover, the observed shift between our
findings and the EMERGE results due to the different choice of the merger
timescale could be ascribed to the merger fraction evolution which, from their
simulation, is smaller than ours. Finally, the fast-decreasing merger
timescale by Snyder et al. (2017) lead to merger rates which are more than an
order of magnitude larger than those from the Kitzbichler & White (2008) and
Jiang et al. (2014) prescriptions. Adopting this formalism, we find that our
merger rates are larger than those predicted by the Illustris simulations
(Rodriguez-Gomez et al. 2015) (with an average mass ratio $\sim 2$) and
comparable with the merger rates of $M_{\text{halo}}\sim
10^{12}\leavevmode\nobreak\ M_{\odot}$ by Neistein & Dekel (2008); Dekel et
al. (2013) at $z\sim 6$.

Duncan et al. (2019) found a good agreement between their observations and
simulated merger rates from Rodriguez-Gomez et al. (2015) for galaxies with
$\mathrm{log}(M_{*}/M_{\odot})>10.3$, while they also obtained higher values
than those predicted for lower mass galaxies. However, the results of that
simulation depend significantly on the adopted mass ratio. Indeed, assuming
$\mu>2$, the Illustris simulation is able to reproduce our observations, at
least at $z\gtrsim 4$. Moreover, Mantha et al. (2018) also found merger rates
comparable to those predicted by Rodriguez-Gomez et al. (2015) up to $z\sim 3$
by dividing their observed pair fraction by the Snyder et al. (2017) merger
timescale.

These results suggest that to reconcile observations and simulations, a
redshift-evolving timescale should be employed when converting the pair
fraction into a merger rate, with a $T_{\text{MM}}$ decreasing toward early
epochs likely being the most suitable choice. However, many other factors can
affect the $R_{\text{MM}}$ estimates, like the probability of merging
$C_{\text{merg}}$ or the dependency of the merger timescale on several
properties like the stellar mass of the galaxies, the on-going phase of the
merger or the selection criteria of the close pairs. For these reasons,
further investigation is needed in order to obtain more robust conclusions on
the incidence of merger through cosmic time.

###  5.2 The contribution of major mergers to the galaxy mass-assembly

The relative contribution of the different mechanisms driving the galaxy mass
assembly through cosmic time has yet to be ascertained. The most favored
scenario predicts cold accretion and in situ star formation as the principal
sources of stellar mass increase in galaxies, with major mergers only playing
a minor role in the galaxy build-up (e.g., Dekel et al. 2009; Conselice et al.
2013; Kaviraj et al. 2013; Sánchez Almeida et al. 2014). However, during the
last years several works started to suggest a prominent contribution of
mergers within this context, opening again the debate on the importance of
these events in the framework of galaxy evolution (e.g., Tasca et al. 2014;
Mantha et al. 2018; Duncan et al. 2019).

Depending on the merger timescale adopted, with our new ALPINE data we can
find large merger rates up to high redshifts which could imply a substantial
contribution of these events to the galaxy mass accretion at early epochs. We
find that the stellar mass accretion rate density due to major mergers has a
similar redshift evolution as the cosmic SFRD (Madau & Dickinson 2014) and
that, assuming the Snyder et al. (2017) merger timescale, they become
comparable with each other at redshifts approaching the Reionization epoch.
However, the lack of further constraints on the most suitable choice for the
merger timescale prevent us from providing firm conclusions. Indeed, looking
at Figure 7 (right panel), the importance of major mergers in the assembly of
galaxies could be very different, especially at high redshifts, depending on
all the above-mentioned caveats and also on the adopted fitting function. It
is worth noting that the comparison between the SFRD and the mass accretion
rate density in Figure 7 is not intended to provide the contribution of major
mergers to the global star formation through cosmic time, rather it represents
the relative importance of the two processes to the galaxy mass assembly at a
given epoch.

Despite this, we can certainly affirm that major mergers are frequent
processes in the early Universe and must be accounted for when considering the
mechanisms of galaxy assembly across cosmic time.

##  6 Summary and conclusions

In this work, we investigate the role of major mergers in the stellar mass
assembly of galaxies at $z\sim 5$. To this aim, we take advantage of the
recent [CII] observations of a significant statistical sample of normal star-
forming galaxies in the early Universe, as carried out by the ALMA large
program ALPINE. The large amount of data (both spectroscopic and photometric)
available for these sources combined with their new morphological and
kinematic characterization made possible by the 3D [CII] information, allow us
to put one of the first constraints on the fraction and rate of mergers
shortly after the end of the Epoch of Reionization. Indeed, only a handful of
works have been undertaken so far about the study of the mergers contribution
to the build-up of galaxies at $z>4$, among which some are affected by large
uncertainties due, for instance, to the lack of spectroscopic redshifts which
properly identify close pairs. In the following, we summarize our major
results:

  * $\bullet$

We identify 23 mergers, corresponding to $\sim 31\%$ of the 75 ALPINE
[CII]-detected galaxies, with an average stellar mass of the most massive
galaxies of $\mathrm{log}(M_{*}/M_{\odot})\sim 10$. We find major merger
fractions $f_{\text{MM}}=0.44^{+0.11}_{-0.16}$ and
$f_{\text{MM}}=0.34^{+0.10}_{-0.13}$ at $z\sim 4.5$ and $z\sim 5.5$,
respectively, after taking into account the presence of minor mergers and
correcting for completeness. These results are in good agreement with
morphological studies by Conselice & Arnold (2009) at the same redshifts and,
when combined with previous works down to the local Universe, suggest a cosmic
merger fraction evolution with a rapid increase from $z=0$ to $z\sim 2$, a
peak at $z\sim 2-3$, and a possible slow decline for $z>3$. This trend is well
described by a combined power-law and exponential function and is quite
comparable with the outputs of the EAGLE hydrodynamical simulations from which
Qu et al. (2017) claim that major mergers contribute less than $10\%$ to the
stellar mass gain in galaxies less massive than log$(M_{*}/M_{\odot})=10.5$,
attributing a leading role to the in situ star formation.

  * $\bullet$

We convert the merger fraction into the merger rate per galaxy by adopting
different redshift scaling for the typical time of merging ($T_{\text{MM}}$).
The results we obtain are strongly dependent on the choice of $T_{\text{MM}}$.
For instance, Kitzbichler & White (2008) found a merger timescale evolution
which diverges more and more from that of Snyder et al. (2017) moving to high
redshift. When using the first timescale prescription, we obtain
$R_{\text{MM}}\sim 0.09$ and $R_{\text{MM}}\sim 0.07$ Gyr-1 at $z\sim 4.5$ and
$z\sim 5.5$, respectively. However, these numbers are more than an order of
magnitude larger if $T_{\text{MM}}\propto(1+z)^{-2}$ is used. Accordingly,
this uncertainty propagates to all our subsequent outcomes that involve the
choice of a merger timescale.

  * $\bullet$

By integrating the merger rate over time we obtain the average number of
mergers a galaxy undertakes during its cosmic history from the early to the
local Universe which goes from less than 1 to $\sim 8$ mergers at $z=0$,
depending on the adopted merger timescale prescription. Our results are quite
in agreement with the outputs of several cosmological simulations, especially
when adopting the redshift-dependent Snyder et al. (2017) merger timescale
with which we can match the large merger rates predicted at high redshifts by
the Illustris simulations (Rodriguez-Gomez et al. 2015) and the halo-halo
merger rate by Dekel et al. (2013).

  * $\bullet$

Another important quantity we estimate is the volume-averaged merger rate
$\Gamma_{\text{MM}}$, that is the merger rate $R_{\text{MM}}$ multiplied by
the number density of galaxies at a given redshift and for a specific stellar
mass range. We find a nearly constant $\Gamma_{\text{MM}}$ at intermediate
redshifts in agreement with previous works in the literature (Mundy et al.
2017; Duncan et al. 2019), and a possible decrease at both $z<1$ and $z>4$.

  * $\bullet$

We take advantage of the two observationally-estimated merger rates to provide
a constraint on the contribution of major mergers to the galaxy mass assembly
through the cosmic epochs. From the merger rate per galaxy, we obtain the
average stellar mass accreted per major merger per unit of mass and compare it
with the sSFR computed from different authors. The latter seems to dominate
over the specific mass accretion rate at all times. However, when considering
a rapidly evolving merger timescale like $T_{\text{MM}}\propto(1+z)^{-2}$, the
two quantities become comparable within the uncertainties at $z>4$. We then
estimate the mass accretion rate density from $\Gamma_{\text{MM}}$ and compare
this quantity to the well-known SFRD cosmic evolution (Madau & Dickinson
2014). We note that the contribution of major mergers to the global star
formation rate ranges between being approximately equal to the SFRD to less
than 1% of it, depending on the exact choice of the merging timescale, the
parametric form used to fit the data, and the redshift.

To conclude, major mergers could have played a significant role in the galaxy
mass assembly through cosmic time, especially when merger timescales from
recent literature, in which $T_{\text{MM}}$ decreases rapidly with increasing
redshift, are taken into account. However, future investigation is needed in
order to finally establish the importance of such events in the complex
picture of the galaxy evolution. In particular, a larger statistical sample of
spectroscopically confirmed galaxies, observed with deeper resolution, will
allow us to confirm the large fraction of mergers at early times and their
incidence at different redshifts, unveiling the relative contribution of each
process to the build-up of galaxies through the Universe.

###### Acknowledgements.

We warmly thank the referee for her or his careful reading of the paper and
constructive suggestions that contributed to improve the quality of this work.
This paper is based on data obtained with the ALMA Observatory, under Large
Program 2017.1.00428.L. ALMA is a partnership of ESO (representing its member
states), NSF(USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA
(Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of
Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. A.C.
acknowledges the support from grant PRINMIUR 2017 - 20173ML3WW$\\_$001\. The
Cosmic Dawn Center is funded by the Danish National Research Foundation under
grant No. 140. G.C.J. acknowledges ERC Advanced Grant 695671 “QUENCH” and
support by the Science and Technology Facilities Council (STFC). This work was
supported by the Programme National Cosmology et Galaxies (PNCG) of CNRS/INSU
with INP and IN2P3, co-funded by CEA and CNES. E.I. acknowledges partial
support from FONDECYT through grant N∘ 1171710\. This paper is dedicated to
the memory of Olivier Le Fèvre, PI of the ALPINE survey.

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##  Appendix A Individual description of mergers

We present here the characterization of the mergers analyzed in this work, as
illustrated in Section 3.1 and Figure 1. For each source, we show the moment
maps, the PVDs, and the integrated [CII] spectrum. Due to the peculiarity of
these objects, we briefly comment on them in the following, also highlighting
possible differences and similarities with the previous morpho-kinematic
classifications.

CG$\\_$38: Le Fèvre et al. (2020) classified this galaxy as a merger. Although
the low S/N and small spatial extent, we include this source in our analysis
as it shows multiple peaks in the PVDs that are associated to different
components in the [CII] spectrum.