# Galaxy Evolution in all Five CANDELS Fields and IllustrisTNG: Morphological,
Structural, and the Major Merger Evolution to $z\sim 3$

A. Whitney  University of Nottingham, School of Physics & Astronomy,
Nottingham, NG7 2RD, UK  L. Ferreira  University of Nottingham, School of
Physics & Astronomy, Nottingham, NG7 2RD, UK  C. J. Conselice  University of
Nottingham, School of Physics & Astronomy, Nottingham, NG7 2RD, UK  Jodrell
Bank Centre for Astrophysics, University of Manchester, Oxford Road,
Manchester UK  K. Duncan  SUPA, Institute for Astronomy, Royal Observatory,
Blackford Hill, Edinburgh, EH9 3HJ, UK  Leiden Observatory, Leiden University,
PO Box 9513, NL-2300 RA Leiden, The Netherlands Amy Whitney
<EMAIL_ADDRESS>
(Received XXX; Revised YYY; Accepted ZZZ)

###### Abstract

A fundamental feature of galaxies is their structure, yet we are just now
understanding the evolution of structural properties in quantitative ways. As
such, we explore the quantitative non-parametric structural evolution of
16,778 galaxies up to $z\sim 3$ in all five CANDELS fields, the largest
collection of high resolution images of distant galaxies to date. Our goal is
to investigate how the structure, as opposed to size, surface brightness, or
mass, changes with time. In particular, we investigate how the concentration
and asymmetry of light evolve in the rest-frame optical. To interpret our
galaxy structure measurements, we also run and analyse 300 simulation
realisations from IllustrisTNG to determine the timescale of mergers for the
CAS system. We measure that from $z=0-3$, the median asymmetry merger
timescale is $0.56^{+0.23}_{-0.18}$Gyr, and find it does not vary with
redshift. Using this data, we find that galaxies become progressively
asymmetric at a given mass at higher redshifts and we derive merger rates
which scale as $\sim(1+z)^{1.87\pm 0.04}$Gyr-1, which agrees well with recent
machine learning and galaxy pair approaches, removing previous
inconsistencies. We also show that far-infrared selected galaxies that are
invisible to HST have a negligible effect on our measurements. We also find
that galaxies are more concentrated at higher redshifts. We interpret this as
a sign of how their formation occurs from a smaller initial galaxy that later
grows into a larger one through mergers, consistent with the size growth of
galaxies from ‘inside-out’, suggesting that the centres are the oldest parts
of most galaxies.

##  1 Introduction

Throughout the history of galaxy studies, the most common way to derive galaxy
evolution is through examining some property as a function of time. This
famously includes the evolution of star formation, stellar mass, metallicity,
and other properties. One of the most fundamental properties that we are still
exploring is the morphological or structural evolution of galaxies, which is
an integrated result of the many different galaxy properties and formation
processes (e.g., Conselice et al. 2008; Mortlock et al. 2013; Conselice 2014;
Huertas-Company et al. 2016).

There are many ways in which to trace the structural evolution of galaxies.
The most simplistic and direct way is investigating the size evolution (e.g.,
Trujillo et al. 2007; Buitrago et al. 2008; Allen et al. 2017; Whitney et al.
2019), and the surface brightness evolution (e.g., Whitney et al. 2020), as
well as simply the evolution of apparent morphology classified into Hubble
types/peculiars (e.g., Conselice et al. 2005). Another way to examine the
evolution of galaxy structure is to examine the bulge and disk components of
galaxies and how these evolve together (Bruce et al. 2014; Margalef-Bentabol
et al. 2018). What has not been carried out in any detail is the quantitative
evolution of galaxies as measured with non-parametric parameters. These
parameters, including concentration and asymmetry (Kent 1985; Conselice 2003),
reveal the processes of galaxy assembly through the systematic change of
galaxy light over time.

Throughout a galaxy’s lifetime, it will undergo several processes that will
alter its structure and its morphology. Within a cosmological context of
$\Lambda$-CDM, this includes the formation of bulges and then disks. In the
simplest paradigm galaxies collapse into small systems that grow through star
formation and mergers with other galaxies. At some point gas accretion will
also occur and this is a primary method by which spiral arms and disks are
formed. This process includes the accumulation of gas forming into stars that
will expand galaxies in their outer parts (Whitney et al. 2019). This also
includes mergers that will lead to structural peculiarities, and eventually,
for some, into more concentrated systems. Furthermore, within clusters of
galaxies processes such as ram pressure stripping (Gunn & Gott 1972),
starvation (Larson et al. 1980), and harassment (Moore et al. 1996) are all
tied to galaxy star formation history and can strongly influence the physical
and morphological properties of a galaxy.

A galaxy’s morphology is traditionally defined as the point at which it lies
on the ‘tuning fork’ diagram, first described by Edwin Hubble (Hubble 1926),
whereby galaxies are defined as either spirals or ellipticals/S0s.
Historically, these morphological classifications were done visually (e.g., de
Vaucouleurs 1959; Sandage 1975; van den Bergh 1976; Lintott et al. 2008,
2011), whereby an individual examines galaxy images and assigns labels to
those images based in their visual appearance. However, this method gives rise
to errors and biases and the sheer sample size of current and upcoming surveys
mean this is quickly becoming inefficient even for large citizen-science
projects such as Galaxy Zoo (Cheng et al. 2020). In order to remove some of
these problems it is important to use a less subjective and more quantitative
way of classifying galaxies. One such method is a non-parametric system that
seeks to measure the concentration, asymmetry, clumpiness, Gini and $M_{20}$
(CAS parameters) of galaxies by using measured light distribution. This system
is described in papers such as Conselice et al. (2000b, 2002); Lotz et al.
(2004). Using this method, galaxies can be placed in parameter space and from
this, it can be seen that all major classes of galaxies in various phases of
evolution are easily differentiated (Conselice 2003). Furthermore, classical
classifications of galaxies are unable to be used at higher redshifts, whereby
most galaxies are not elliptical or spirals (e.g., Conselice et al. 2005;
Mortlock et al. 2013).

An important stage of galaxy evolution that can be measured with these
parameters is when a galaxy undergoes a merger. Mergers can be identified
using methods such as pair fractions (Man et al. 2016; Mundy et al. 2017;
Duncan et al. 2019; Ventou et al. 2019), deep learning models (Ferreira et al.
2020), and by using the CAS parameter space (Conselice 2003; Lotz et al. 2004;
Conselice et al. 2008; López-Sanjuan et al. 2009). Major mergers lie in a
specific areas of these parameter spaces and as such, they are a useful tool
in determining whether a galaxy is a merger or not. The merger rate can then
be calculated from this to determine the role of mergers in forming galaxies.

In this paper we investigate the general evolution of galaxy structure through
cosmic time. We start with visual estimates of morphology and structure and
then we examine the quantitative structural evolution of these systems. We use
IllustrisTNG simulations to help us interpret these structures from which we
derive the rates of galaxy formation processes such as those driven by mergers
and how light concentration in a galaxy changes with time.

Throughout this paper we use AB magnitudes and assume a $\Lambda$-CDM
cosmology with H0 = 70 kms-1Mpc-1, $\Omega_{m}$ = 0.3, and $\Omega_{\Lambda}$
= 0.7.

##  2 Data & Methods

###  2.1 Data

We use a sample of 16,778 galaxies at redshifts in the range 0.5 $<z<$ 3 with
stellar masses from $10^{9.5}M_{\odot}<M_{*}<10^{12.2}M_{\odot}$ and only
select galaxies with S/N $>$ 10\. This signal-to-noise cut removes only 340
galaxies from the initial sample. The mass and redshift distribution can be
seen in Figure 1 where the yellow regions indicate a higher density of
galaxies and purple indicates a lower density of galaxies. We select two
samples from the 16,778 galaxies; a low mass sample with
$10^{9.5}M_{\odot}<M_{*}<10^{10.5}M_{\odot}$ and a high mass sample with
$M_{*}>10^{10.5}M_{\odot}$. We later also consider how our results change if
we used a constant co-moving number density selected sample. The mass limits
for our mass-selected sample are shown as horizontal dashed lines on Figure 1.
Masses and redshifts are determined using the method described in Duncan et
al. (2019). A brief description of the process used to obtain the measurements
is given in §2.4.

Figure 1: Mass and redshift distribution of the sample of galaxies. Yellow
indicates a higher density of galaxies and purple indicates the lowest
density. White areas indicates there are no galaxies are present in this
region of parameter space. The dashed horizontal lines at
log${}_{10}(M_{*}/M_{\odot})$ = 9.5 and log${}_{10}(M_{*}/M_{\odot})$ = 10.5
indicate the boundaries between the two mass bins used in this study.

This study makes use of data from the Cosmic Assembly Near-infrared Deep
Extragalactic Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011).
This survey covers 800 arcmin2 over five fields; GOODS-North, GOODS-South,
COSMOS, UDS, and EGS. Our sample consists of galaxies from each of the five
fields. CANDELS makes use of the Advanced Camera for Surveys (ACS) and the
Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST). For this work
we use imaging data from the F814W, F125W, and F160W filters. This filters
will be referred to as $I_{814}$, $J_{125}$, and $H_{160}$ from this point
onward.

We measure all parameters in the optical rest-frame at approximately
$\lambda_{rest}\sim 4000$Å. We do this by locating the observed filter where
$\lambda_{rest}\sim 4000$Å falls as determined by its redshift. We then use
that filter in our further analysis to have a consistent rest-frame optical
view of our galaxies. The idea here is to measure everything at a constant
optical rest-frame wavelength, to avoid structural biases produced by changes
in rest-frame wavelength. The rest-frame wavelength and associated filter are
given for each redshift bin in Table 1.

Table 1: The bands corresponding to the rest-frame optical at each redshift.

$z$ |  $O_{i,j}^{raw}$ |  $\lambda_{rest}$
---|---|---
0.75 |  $I_{814}$ |  4650Å
1.25 |  $I_{814}$ |  3620Å
1.75 |  $J_{125}$ |  4550Å
2.25 |  $J_{125}$ |  3850Å
2.75 |  $H_{160}$ |  4270Å
|  |

  * •

Note. Column 1 gives the midpoint of the redshift bin. Each redshift bin spans
a redshift range of $\Delta z=0.5$. Column 2 gives the band corresponding to
the optical rest-frame and column 3 gives the rest frame wavelength probed.

We use visual morphological classifications from Kartaltepe et al. (2015) who
base the majority of their classifications on $H$-band images, however $J$\-
and $V$-band images are also used for some features such as clumpy light. For
a summary of how these classifications are done and in which morphological
classes they are placed, see Kartaltepe et al. (2015).

###  2.2 Morfometryka

We measure the non-parametric concentration, asymmetry, and clumpiness
parameters using the Morfometryka code (Ferrari et al. 2015). These
measurements are made within the Petrosian region (e.g., Conselice et al.
2000b). This is defined as the area with the same axis ratio and position
angle as the galaxy and with major axis equal to
$N_{R_{\textup{Petr}}}R_{\textup{Petr}}$, where $N_{R_{\textup{Petr}}}=1.5$
and $R_{\textup{Petr}}$ is the Petrosian radius. Below we give a description
of each index we use, although for more details see Conselice (2003).

####  2.2.1 Asymmetry

The asymmetry, $A$, is calculated in the same way as in Conselice (2003)
whereby a galaxy image rotated by 180∘ is subtracted from the original source
image, and the absolute value of the total light in this self-subtracted image
is divided by the total light in the original image. Asymmetry through various
tests has been shown to be one of the most robust non-parametric structural
parameters to measure and use in analyses. The formula for calculating the
asymmetry index ($A$) we use is outlined in detail in Conselice (2003) and is
given by:

$A={\rm min}\left(\frac{\Sigma|I_{0}-I_{180}|}{\Sigma|I_{0}|}\right)-A_{bkg}$
(1)

where $I_{0}$ represents the original galaxy image, and $I_{180}$ is this
image after rotating it by 180 deg from its centre. The asymmetry value is
calculated through an iterative approach to find the centre of the rotation
which is altered to find the one that gives the minimum asymmetry value. This
minimum asymmetry which is used as the final asymmetry value (e.g., Conselice
et al. 2000b) with a search radius typically a pixel or half pixel. $A_{bkg}$
is the background asymmetry of the image. This background term differs from
the original application of the CAS parameter measurements whereby the
background asymmetry is determined from a single region. Morfometryka
initially did not include such a background correction. In this case we use
the same method for determining $A_{bkg}$ as in Tohill et al. (2020) whereby a
$10\times 10$ grid is overlayed on the image in an area outside of the galaxy
segmentation map, the asymmetry of the individual cells are measured and then
the median of these values is taken for $A_{bkg}$. This removes the bias
present in the original method of measuring the asymmetry of a single
background region and ensures the measurement is more robust.

####  2.2.2 Concentration

The concentration of a galaxy is used to distinguish between the different
types of galaxies; for example, early type galaxies and their immediate
progenitors tend to have a higher concentration than late type galaxies (e.g.,
Bershady et al. 2000; Conselice 2003). Concentration, $C$, is defined as the
ratio between two circular radii containing certain fractions of the total
flux of the galaxy (Kent 1985). We use $R_{20}$, the radius containing 20% of
the total light, and $R_{80}$, the radius containing 80% of the total light.
Therefore, the concentration is given as

$C=5\times\textup{log}_{10}\left(\frac{R_{80}}{R_{20}}\right).$ (2)

This calculation of the concentration is sensitive to seeing effects that are
more prominent in the central regions and is therefore sensitive to the value
of the inner radii $R_{20}$ (Ferrari et al. 2015). We however, investigate
other forms of concentration indices, and fully explore the idea of light
concentration and what it actually implies for our galaxies. We also consider
the $R_{50}$ and $R_{90}$ radii to avoid problems with the inner parts of
galaxies, as well as examine the Petrosian radii measures of light
concentration.

####  2.2.3 Clumpiness

The clumpiness, $S$, is a measure of the small scale structure within a
galaxy. A higher clumpiness indicates that there are clumps of material within
a galaxy, for example spiral galaxies contain many star forming regions and
therefore contain many clumps of material. Elliptical galaxies on the other
hand are generally smooth and therefore have a clumpiness that is close to
zero. $S$ is calculated in the same way as in Conselice (2003) whereby the
image is first smoothed by a filter and then subtracted from the original
image. The flux contained within this residual is then divided by the flux
contained with the original image. In this case, the filter used to smooth the
original image is a Hamming window of size $R_{\textup{Petr}}$/4. Formally,
the clumpiness can be described by the following

$S=\sum_{i,j}\frac{|I(i,j)-I_{S}(i,j)|}{|I(i,j)|}-S_{bkg}$ (3)

where $I(i,j)$ is the original image, $I_{S}(i,j)$ is the smoothed image, and
$S_{bkg}$ is the average smoothness of the background.

###  2.3 Correcting CAS Values for Redshift Effects

Galaxy structure will change for a given galaxy when that same galaxy is
viewed at higher redshifts. To decouple evolution from this effect, we need to
account for this. In order to correct for redshift effects, we artificially
redshift a sample of galaxies at 0.5 $<z<$ 1 in redshift intervals of 0.5 to a
maximum redshift of $z=2.75$. We take the galaxies in this initial bin from
both GOODS fields and consider this our fiducial sample. We simulate these
galaxies to higher redshifts and in the other CANDELS fields. We do this using
the method described in Tohill et al. (2020). This method considers a number
of effects. Firstly, it considers the rebinning factor $b$, which is the
decrease in the apparent size of the galaxy when viewed at a higher redshift.
This is done by following the method outlined in Conselice (2003) and de
Albernaz Ferreira & Ferrari (2018). Luminosity evolution is also considered
and this is implemented in the form found in Whitney et al. (2020) whereby the
intrinsic surface brightness goes as

$\mu_{\textup{int}}\propto(1+z)^{-0.18}$ (4)

for a mass-selected sample of galaxies. After applying the rebinning factor,
cosmological dimming of the form $(1+z)^{-4}$, as found by Tolman (1930), is
applied. Finally, the image is convolved with the PSF corresponding to the
rest-frame filter and inserted into an actual CANDELS background. For the
first redshifting interval ($z=0.75$ to $z=1.25$) we do not convolve with the
PSF as the filter corresponding to the optical rest-frame is the same for both
redshifts. We then use Morfometryka to measure the CAS parameters at each
redshift and compare these new values to the parameters measured at the
original redshift. The differences are then applied to the real galaxies at
the corresponding redshift. This change is dependent on the field due to the
depth reached by each of the fields. The GOODS fields reach a greater depth
than the COSMOS, EGS, and UDS fields and as such, we apply a different
correction to each field. We also ensure that we are comparing the same
galaxies at each redshift interval. To do this we only consider at all
redshifts those galaxies that can be measured in the highest redshift as this
ensures that we are comparing exactly the same galaxies at all redshifts to
create a fair comparison and relevant correction factors.

The left panel of Figure 2 shows the average asymmetry corrections applied to
all CANDELS fields. The right panel of Figure 2 shows the average
concentration corrections applied to the five CANDELS fields. Table
LABEL:tab:corrections gives the values of these corrections for each redshift
bin within each field.

Table 2: The asymmetry and concentration corrections applied to each field at each redshift interval. | Asymmetry | Concentration
---|---|---
$z$ | GOODS-N | GOODS-S | COSMOS | EGS | UDS | GOODS-N | GOODS-S | COSMOS | EGS | UDS
0.75 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
1.25 | -0.010 | -0.010 | -0.009 | -0.004 | -0.007 | -0.27 | -0.29 | -0.30 | -0.40 | -0.20
1.75 | 0.008 | -0.004 | -0.008 | -0.007 | -0.023 | -0.39 | -0.43 | -0.32 | -0.46 | -0.43
2.25 | -0.032 | -0.016 | -0.036 | -0.027 | -0.060 | -0.83 | -0.87 | -0.75 | -0.85 | -0.84
2.75 | -0.084 | -0.079 | -0.104 | -0.100 | -0.118 | -1.08 | -1.10 | -1.08 | -1.09 | -1.14

|
---|---

(a)  (b)

Figure 2: Left: mean asymmetry correction applied to all five CANDELS fields.
Right: mean concentration correction for all CANDELS fields. The error bars
are equal to one standard deviation from the mean. In both panels, the GOODS-
North correction is shown by a blue circle, GOODS-South by an orange diamond,
COSMOS by a green triangle, EGS by a red square, and UDS by a purple inverted
triangle.

###  2.4 Photometric Redshifts and Stellar Masses

The photometric redshifts of the galaxies within our sample are calculated
using the method described in Duncan et al. (2019). The photometric redshift
software eazy (Brammer et al. 2008) is used to determine the template-fitting
estimates and three separate template sets are used and fit to all available
photometric bands. The templates used include zero-point offsets to the input
fluxes and additional wavelength-dependent errors. A Gaussian process code
(GPz; Almosallam et al. (2016)) is then used to calculate further empirical
estimates using a subset of the available photometric bands. Individual
redshift posteriors are calibrated and the four estimates are combined in a
statistical framework via a hierachical Bayesian combination to produce a
final redshift estimate. For a more in-depth description of the process, see
section 2.4 of Duncan et al. (2019).

The galaxy stellar masses we use are measured by using a modified version of
the spectral energy distribution (SED) code described in Duncan et al. (2014).
Instead of finding the best-fit mass for a fixed input redshift, the stellar
mass is estimated at all redshifts in the photo-$z$ fitting range. Also
included in these estimates is a so-called ‘template error function’ (the
method for this is described in Brammer et al. (2008)) to account for
uncertainties introduced by the limited template set and any wavelength
effects.

This mass-fitting technique uses Bruzual & Charlot (2003) templates and
includes a wide range of stellar population parameters and assumes a Chabrier
(2003) initial mass function. The assumed star formation histories follow
exponential $\tau$-models for both positive and negative values of $\tau$.
Characteristic timescales of $\left|\tau\right|$ = 0.25, 0.5, 1, 2.5, 5, and
10 are used, along with a short burst ($\tau$ = 0.05) and continuous star
formation models ($\tau\gg$ 1/$H_{0}$).

We compare the mass measurements we make to the average of those determined by
the several teams within the CANDELS collaboration (Santini et al. 2015). This
is done in order to ensure that the stellar mass estimates do not suffer from
systematic biases. There is some scatter between the two mass estimates,
however our mass estimates are not affected by any significant biases compared
to others. For further details on the method and models used, see §2.5 of
Duncan et al. (2019) for an extensive discussion of the masses we use here.

###  2.5 IllustrisTNG Simulations

To interpret our results, we investigate how non-parametric morphologies
change with redshift in a controlled manner with simulated galaxies. We do
this as we want to disentangle real evolution effects from redshift effects.
Additionally, by measuring these structural properties in simulated data we
can estimate the observability timescales of galaxy mergers since it is
possible to follow the time evolution of particular galaxies in this way
(e.g., Lotz et al. 2008). With this goal in mind, we use data from the
IllustrisTNG simulations (Pillepich et al. 2018; Springel et al. 2018; Nelson
et al. 2018; Naiman et al. 2018; Marinacci et al. 2018; Nelson et al. 2019,
2019; Pillepich et al. 2019), which is a suite of cosmological, gravo-
magnetohydrodynamical simulation runs, ranging within a diverse set of
particle resolutions for three comoving simulation boxes of size,
$50,100,300\rm\ Mpc\ h^{-1}$, named TNG50, TNG100 and TNG300, respectively.

Here we use data from both TNG50-1 and TNG300-1. For estimating the merging
timescales we focus in the largest simulation box TNG300-1, as it provides us
with a more mass complete sample at higher redshifts. However, when doing
direct comparisons between the morphology measured in CANDELS and in the
simulations we use data from TNG50-1 as its mass and spatial resolution
produce more realistic morphologies. This enable us to generate images of
simulated galaxies that are embedded in a cosmological context.

To measure the observability timescales of pair galaxy mergers discussed in
Section §3.4.3, we select galaxies using the TNG300-1 merger trees by
searching and locating galaxies in the simulation that have had only one major
merger event within redshifts $z=2$ to $z\sim 0$. This is to avoid
contamination in the structure from past merger events. To balance any
potential issues with the mass resolution of TNG300-1, we limit our analysis
only to massive galaxies with $M_{*}>10^{10}M_{\odot}$, which are in general
represented by thousands of stellar particles. For these galaxies, we select
all the snapshots and subhalos that are $\pm\ 2\rm\ Gyr$ of the snapshot from
where the merger event takes place. We then narrow down our selection to 300
distinct galaxies, each with 35 different snapshots, resulting in a total of
$\sim 10,000$ distinct objects. This ensures that we extensively probe not
only around the merging event, but also the stages where the galaxies does not
show signs of merging.

For a comparison between the non-parametric morphology and merger
classifications of CANDELS galaxies and simulated galaxies we use the small
box, high mass resolution TNG50-1 simulation, which is capable of producing
output images in higher resolutions due to the high mass resolution and
smaller gravitational softening length. Here we do two different selections on
TNG50-1 galaxies, which are described in §3.5.

The IllustrisTNG data contains the information from the stellar, gas, and dark
matter particles for each source. However, to create mock broadband images
from this information we post-process each stellar particle with a population
synthesis process, as each particle represents a large region that can be
described by a rich stellar population based on its age, mass, and
metallicity. Instead of using the approach outlined in Ferreira et al. (2020),
we follow the recipes from Trayford et al. (2017) and Vogelsberger et al.
(2020) to post-process the simulation data with the Monte Carlo dusty
radiative transfer code SKIRT (Camps & Baes 2015, 2020). We also include the
resampling of the star-forming regions outlined in Camps et al. (2016) and
Trayford et al. (2017), as this is particularly important to avoid problems
with the coarse representation of star forming regions. For each source
simulated with SKIRT we produce observations in four different orientations in
the 3D volume containing the galaxy cutout. Three of those are aligned with
the simulation box axis: xy, xz and yz. We also include a fourth orientation
covering an octant of the 3D volume. This post-processing step is independent
of the simulation run used, the only difference being the number of stellar
and gas particles available and the output size of the datacube.

The output from SKIRT is a datacube with a spectral energy distribution (SED)
for each pixel, which is then convolved with the filter response function for
the HST CANDELS filters used in this work, namely $I_{814}$, $J_{125}$, and
$H_{160}$. The resulting broadband images are stored in two ways: without any
observation effects and with HST-matched properties, including PSF, noise
level, and sky background. The final images are then processed with
Morfometryka to measure their CAS values. We later use these simulation
outputs as a method for understanding our observational results.

##  3 Results

In the following subsections we discuss the results of our study in terms of
the morphological and structural evolution of galaxies to $z=3$. We first
describe the visual morphological classification evolution of our sample of
galaxies, while later we discuss the quantitative evolution using the CAS
parameters. Finally we use these results to derive the merger fraction and
merger rate evolution for our sample and the resulting number of mergers and
mass accretion from mergers over this cosmic time.

###  3.1 Visual Classifications

Using the visual classifications from the CANDELS survey from Kartaltepe et
al. (2015), we examine the fraction of each galaxy type within our sample. We
consider those galaxies that are classified as the following: spheroid, disk,
peculiar, and other. A galaxy is considered to have a classification if the
fraction of classifiers within Kartaltepe et al. (2015) that deem it to be
that particular type of object is greater than 0.6, and the fraction of all
other classifications is less than 0.6. For peculiar galaxies, we consider any
galaxy where the fraction of classifiers that consider that galaxy to be
irregular in shape is greater than 0.6; no other conditions are required for
this group. The other group consists of galaxies for which no consensus could
be reached as to its morphological type, galaxies that were deemed
unclassifiable, or galaxies that had no classification given. Figure 3 shows
the evolution of the fraction of galaxies in each of these groups within our
sample across the redshift range $0.5<z<2.5$. The spheroid galaxies are shown
as red circles, disk galaxies as blue crosses, and peculiar galaxies as green
triangles. The unclassifiable objects are shown as grey inverted triangles. We
also fit power laws of the form $\alpha(1+z)^{\beta}$ to the spheroid (solid
line), disk (dashed line), and peculiar (dotted line) categories. We find that
the fractions evolve as:

$f_{\textup{sp}}=(0.73\pm 0.09)(1+z)^{-0.95\pm 0.16}$ (5)

for the spheroid galaxies,

$f_{\textup{di}}=(0.48\pm 0.05)(1+z)^{-0.60\pm 0.13}$ (6)

for the disk galaxies, and

$f_{\textup{pe}}=(0.11\pm 0.01)(1+z)^{1.04\pm 0.10}$ (7)

for the peculiar galaxies. The fraction of both disk and spheroid galaxies
increase with cosmic time, whereas the fraction of peculiar galaxies decreases
as redshift decreases. Note that these fits are only valid down to $z\sim
0.5$. It is clear that extrapolating these to $z\sim 0$ would over-predict the
number for each type. The fraction of ‘other’ galaxies evolves from $\sim$20%
at $z\sim 2.25$ and decreases to $\sim$6% at $z\sim 0.75$ due to the fact that
galaxy features will be more distinguishable at lower redshifts due to
increased resolution and fewer effects such as surface brightness dimming.

Figure 3: Evolution of the fractions of galaxies in our sample within a given
visually morphological classification. Spheroid galaxies are shown as red red
circles, disk galaxies as blue crosses, peculiar galaxies as green diamonds,
and unclassifiable objects as grey inverted triangles. Also shown are the
power law fits to the spheroid (solid line), disk (dashed line), and peculiar
categories (dotted line). For the error bars on these fitted parameters see
the text.

###  3.2 Asymmetry and Concentration

We next explore the data by comparing the measured asymmetry and concentration
values with the visual classifications from Kartaltepe et al. (2015) as
described in the previous section. In this section we examine those galaxies
that are identified as disk, spheroid, or peculiar galaxies. We first examine
the uncorrected CAS parameters here, and later investigate a way to correct
these parameters for redshift effects. Essentially, the uncorrected values are
comparable to each other at a given redshift (internally consistent), whereas
the corrected values are comparable across all redshifts (externally and
internally consistent).

In Figure 4, we show the evolution of the uncorrected concentration versus
asymmetry for the galaxies within the sample that have one of our three main
classifications. Red circles indicate a spheroid galaxy, blue crosses a disk
galaxy, and green triangles a peculiar galaxy. Galaxies that fit none of these
categories are shown as small grey circles. Also shown are the cuts between
galaxy types taken from Bershady et al. (2000); the solid black line is the
boundary between intermediate- and late-type galaxies whereby the galaxies
that lay above this line are considered to be late-type galaxies, the dot
dashed line is the boundary between early- and intermediate-type galaxies
whereby galaxies below this line are considered to be early-type galaxies, and
the dashed line is the boundary between mergers and non-mergers whereby
galaxies that lie above this line are mergers.

Figure 4: Evolution of the concentration-asymmetry plane with redshift,
colour-coded by visual classification. Red circles are galaxies classified as
spheroids, blue crosses as disks, and green triangles as peculiar galaxies.
Any galaxies that do not fit within these three classifications are shown as
small grey circles. The lines are boundaries between galaxy types derived from
Bershady et al. (2000). Galaxies that are above the dashed line are considered
to be mergers. The dot-dashed line is the boundary between early- and
intermediate-type galaxies and the solid line is the boundary between late-
and intermediate-type galaxies. The redshift indicated in the top right corner
of each panel is the midpoint of the redshift bin whereby the upper and lower
limits are 0.25 either side of this point.

At all redshifts, galaxies classified as spheroids tend to have a higher
concentration and lower asymmetry, than the two other classifications. Whereas
peculiar galaxies have a higher asymmetry but lower concentration. This is
consistent with the results found by Conselice (2003) for nearby galaxies. The
range of concentration values increases with time but the overall distribution
of these values for the spheroid and disk galaxies remains roughly constant
with redshift. The asymmetry of the peculiar galaxies is centred on the $A>$
0.35 merger condition, however, the mean concentration changes from $\langle
C\rangle$ = 3.52 at $2.5<z<3.0$ to $\langle C\rangle$ = 2.49 at $0.5<z<1.0$.
The mean values for spheroid galaxies are $\langle C\rangle$ = 2.67 and
$\langle A\rangle$ = 0.16 at the highest redshift. These values change to
$\langle C\rangle$ = 3.10 and $\langle A\rangle$ = 0.12 at the lowest
redshift. Galaxies classified as being disks have average concentration and
asymmetry values of $\langle C\rangle$ = 2.52 and $\langle A\rangle$ = 0.20 at
$2.5<z<3.0$ and these values do not change significantly within the lowest
redshift bin of $0.5<z<1.0$ where $\langle C\rangle$ = 2.50 and $\langle
A\rangle$ = 0.23. Thus the disks change their value the least and the
spheroids the most. These results also show that galaxy classification is
consistent and can be carried out to $z\sim 3$.

####  3.2.1 Disk/Bulge Dominated

Within the catalog of Kartaltepe et al. (2015) every image has flags based on
a number of interesting structural features present. Two of these flags
indicated whether the galaxy appears to be bulge or disk dominated. As with
the visual morphological classifications, we assume a galaxy is bulge or disk
dominated if the fraction of classifications that designate it to be such is
greater than 0.6, and the other classification is less than 0.6. We show the
evolution with redshift of the concentration-asymmetry plane for those
galaxies classified as being bulge or disk dominated in Figure 5. Disk
dominated galaxies are shown as blue points and bulge dominated galaxies are
shown as red points. As with Figure 4, we also plot the boundaries between
galaxy types. At higher redshifts there are fewer galaxies with either of the
classifications in consideration here, likely due to greater noise and surface
brightness dimming causing there to be less consensus among those classifying
the images. This highlights the issues in classifying galaxy images and the
need for a less subjective method of classifying galaxies.

Figure 5: Evolution of the concentration-asymmetry plane with redshift for
galaxies that have been given either a bulge dominated flag (red circles) or a
disk dominated flag (blue crosses). Galaxies that have neither flag are not
plotted. As with Figure 4, the lines are boundaries between galaxy types
derived from Bershady et al. (2000). The redshift indicated in the top right
corner of each panel is the midpoint of the redshift bin whereby the upper and
lower limits are 0.25 either side of this point.

Galaxies classified visually as disk-dominated predominantly lie in the late
region of the concentration-asymmetry plane and the average values in the
$0.5<z<1.0$ bin are $\langle A\rangle$ = 0.22 and $\langle C\rangle$ = 2.68.
These values remain roughly constant across redshift and become $\langle
A\rangle$ = 0.20 and $\langle C\rangle$ = 2.69 at $2.5<z<3.0$. At high
redshifts, the bulge-dominated galaxies appear to lie in the late region but
as redshift decreases, there are more galaxies in the intermediate and early
regions of the plane. The average values of asymmetry and concentration of
these bulge-dominated galaxies are $\langle A\rangle$ = 0.13 and $\langle
C\rangle$ = 3.14 at $0.5<z<1.0$. The asymmetry value remains roughly constant
to higher redshifts and becomes $\langle A\rangle$ = 0.14 at $2.5<z<3.0$ but
the concentration decreases with redshift and becomes $\langle C\rangle$ =
2.80 at this redshift.

####  3.2.2 Clumpiness/Patchiness Matrix

Along with visual classifications of galaxies, classifiers involved in the
work of Kartaltepe et al. (2015) also assigned flags based on how clumpy or
patchy the distribution of the light within the galaxies is. Clumps are
defined to be concentrated knots of light and patches are defined to be more
diffuse structures of light. There are 9 flags associated with this method of
identifying features, with each flag denoting a different combination of the
clumpiness (C) and patchiness (P). For example, a galaxy that exhibits no
clumpiness and no patchiness will be labelled as 0C0P. A galaxy that appears
to be extremely clumpy and patchy will be labelled as 2C2P. The levels of
clumpiness and patchiness range between 0 and 2. As such we are able to
organise the flags into a 3$\times$3 grid with each square of the grid
denoting the level of clumpiness or patchiness. Note, this clumpiness is not
the same as the clumpiness, $S$, described in §2.2.3 and is a purely visual
descriptor. A score of 0 indicates no clumpiness or patchiness and a score of
2 indicates a large amount of clumpiness or patchiness. In Figure 6, we show
this matrix of values and in each square of the grid, give the average
concentration (top left, green), asymmetry (top right, red), and clumpiness
(bottom left, blue). These average values are across all redshifts so any
evolution in these parameters is not considered here. The lighter shades of
colour denote a lower value of the CAS parameters while a darker shade denotes
a higher value. In terms of concentration, the less clumpy or patchy a galaxy
is, the higher its concentration on average. The opposite is true for the
asymmetry and clumpiness; the more clumpy/patchy an image is, the higher the
measured asymmetry and clumpiness.

|
---|---
|

(a)  (b)

(c)  (d)

Figure 6: This figure shows the clumpiness (C) and patchiness (P) distribution
visually vs. the CAS parameters. The values on the $x$\- and $y$-axes are the
visual estimates of clumpy galaxies, with high numbers denoting more visually
looking clumpy systems. The clumpiness values on the $y$-axes are different to
the clumpiness calculated by Morfometryka and are a purely visual score. Top
left, green: mean concentration ($C$) values for each region of the
clumpiness/patchiness matrix defined by Kartaltepe et al. (2015). Top right,
red: mean asymmetry ($A$) values for each region of the clumpiness/patchiness
matrix. Bottom left, blue: mean clumpiness ($S$) values for each region of the
clumpiness/patchiness matrix. In each panel, a darker colour indicates a
larger value of the relevant CAS parameter. Each three by three grid
represents the clumpiness/patchiness matrix whereby a higher value on each
axis represents a higher clumpiness or patchiness. The visual
clumpiness/patchiness criteria appear to correlate well with the measured $A$
and $S$ values whilst concentrated galaxies are not visually clumpy or patchy.

###  3.3 CAS Parameter Evolution

We now investigate with our data the evolution of the CAS parameters. We
ultimately use image simulations to determine how these parameters change with
the effects of resolution and noise removed. Ultimately the evolution of these
parameters will lead to a physical understanding of the driving forces behind
galaxy formation over the epoch $0.5<z<3$.

####  3.3.1 Concentration Evolution

The concentration of a galaxy tells us important information about how the
light is distributed within a galaxy, relative to its centre. We show the
evolution of the corrected concentration index defined in §2.2.2 in Figure 7
for two different mass ranges across the redshift range $0.5<z<3$. The
concentration values have been corrected using the method described in §2.3.

The first includes galaxies in the mass range $10^{9.5}M_{\odot}\leq
M_{*}<10^{10.5}M_{\odot}$ and the evolution of the mean concentration for this
mass bin is shown as red crosses (see Figure 7). The second mass bin includes
galaxies that have a mass $M_{*}\geq 10^{10.5}M_{\odot}$. This mass bin is
plotted as the blue triangles. For both, the error bars represent one standard
deviation from the mean. The lower mass galaxies have a lower concentration
than the higher mass galaxies, but both samples on average exhibit a decrease
in concentration with time. This is opposite to what we find when we examine
the uncorrected concentration index evolution. This therefore deserves some
attention to try to understand the origin of this, and whether it is in fact a
real effect.

Figure 7: Evolution of the mean corrected concentration for both mass bins.
The lower mass bin is shown as a red cross and the higher mass bin is shown by
the blue triangles. The error bars represent one standard deviation from the
mean. On average, the higher mass bin has a greater concentration than the
lower mass bin at all redshifts, however both bins show a decrease in
concentration with cosmic time.

####  3.3.2 Asymmetry Evolution

The asymmetry of a galaxy is a useful indicator as to whether a galaxy is
undergoing any interactions or mergers with other galaxies (Conselice et al.
2000a, b). We explore its use to define mergers in the following section, but
first we examine the evolution of the corrected asymmetry for our sample of
galaxies in the redshift range $0.5<z<3$. Figure 8 shows this evolution for
two different mass ranges. As in Figure 7, the lower mass bin is shown as red
crosses, the higher mass bin is shown as blue triangles, and the error bars
represent one standard deviation from the mean. On average, the asymmetry
decreases with cosmic time for both mass bins, with the higher mass bin
exhibiting a steeper decrease.

Figure 8: Evolution of the mean corrected asymmetry with redshift. As with
Figure 7, the lower mass bin is shown as a red cross and the higher mass bin
is shown by the blue triangles. The error bars represent one standard
deviation from the mean. Both mass bins show a decrease in asymmetry with
decreasing redshift.

####  3.3.3 Concentration-Asymmetry Plane

In Figure 9, we show the evolution of the concentration-asymmetry plane for
the corrected values with redshift. As with Figures 4 and 5, we show the
boundaries between the galaxy types. We also plot the three visual
classifications for each galaxy, along with the position of any galaxy that
does not fit these three categories within this plane. Spheroid galaxies are
shown as red circles, disks as blue crosses, and peculiar galaxies as green
triangles. Any other galaxies are shown as small grey circles. The spread in
distribution of both concentration and asymmetry increases at lower redshifts,
leading to the evolution of the average of both these parameters as seen in
Figures 7 and 8.

Figure 9: Evolution of the corrected concentration-asymmetry plane with
redshift. As with Figure 4, the lines indicates the boundaries between galaxy
types. Galaxies that lie above the dashed line are considered to be mergers.
The dot-dashed line is the boundary between early- and intermediate-type
galaxies and the solid line is the boundary between late- and intermediate-
type galaxies. Spheroid galaxies are indicated by red circles, disks by blue
crosses, and peculiar galaxies by green triangles. Galaxies that do not fall
into any of the three categories are shown as small grey circles. The redshift
indicated in the top right corner of each panel is the midpoint of the
redshift bin whereby the upper and lower limits are 0.25 either side of this
point.

There are few galaxies that lie within the early region of the concentration-
asymmetry plane and those that are are typically spheroid galaxies. This
suggests that the galaxies within our sample are more asymmetric than is
typical for spheroids. Spheroid galaxies that do not lie in this early region
are mostly within the intermediate region. Within the lowest redshift bin
($0.5<z<1.0$), spheroids have a mean asymmetry of $\langle A\rangle$ = 0.14
and a mean concentration of $\langle C\rangle$ = 3.03. At the highest redshift
($2.5<z<3.0$), these values change to $\langle A\rangle$ = 0.25 and $\langle
C\rangle$ = 3.77 with a steady increase in both $A$ and $C$ at intermediate
redshifts. Disk galaxies are primarily located within the late region, as to
be expected. The mean values are $\langle A\rangle$ = 0.21 and $\langle
C\rangle$ = 2.52 at $0.5<z<1.0$ and change to $\langle A\rangle$ = 0.28 and
$\langle C\rangle$ = 3.62 at $2.5<z<3.0$. The change in $C$ is a steady
increase however asymmetry remains at an approximately constant value until
the highest redshift bin. Peculiar galaxies lie around the merger limit of
$A>$ 0.35 with 29.20 $\pm$ 0.01% of galaxies classified as being peculiar
lying within this merger region. If these corrections are right, we can see
that there is a diversity already at $2.5<z<3.0$ in the galaxy population and
that galaxies of all types are more asymmetric and more concentrated at higher
redshifts.

###  3.4 Merger Fractions and Rates

This section of the paper describes our results of the merger history of
galaxies within the five CANDELS fields using a CAS approach. The outline of
this section is as follows: first we give a background description of the
merger history of galaxies, include definitions of the merger process. We then
describe in some detail how to measure the timescale for mergers and how these
can then be used for measuring the merger rate of galaxies. Finally, from the
merger rates we are able to say how many mergers these galaxies undergo on
average and how much mass is added to galaxies through this process.

####  3.4.1 Merger Fractions

In this subsection we investigate the observational quantity that we can
obtain directly from the data. This is the merger fraction, which in this
paper we use the CAS parameters described in §2.2 to define. We later compare
the merger fractions and the merger rates we derive using merger timescales to
other measurements of the merger history using machine learning methods
(Ferreira et al. 2020) and galaxies in pairs (Duncan et al. 2019). First we
define what we mean by a merger fraction - that is, the fraction of galaxies
within some galaxy sample which is undergoing a major merger.

In our sample, a merger is defined within the CAS system as a galaxy that
satisfies the following criteria:

$(A>0.35)\ \&\ (A>S)$ (8)

where both $A$ and $S$ are the asymmetry and clumpiness values corrected for
redshift effects. This method predominantly identifies only major mergers
where the ratio of the stellar masses of the progenitors is at least 1:4
(Conselice 2003, 2006; Lotz et al. 2008). $H$-band images of examples of
galaxies we consider to be a merger, based on this system, are shown in Figure
10. The left two columns show lower redshift ($1<z<2$) galaxies, and the right
two columns show higher redshift ($2<z<3$) galaxies. The redshift and
asymmetry of each example are indicated on each postage stamp.

Figure 10: Examples of galaxies within our sample that are classified as a
merger using the criteria given in Equation 8. Lower redshift ($1<z<2$)
galaxies are shown in the left two columns and higher redshift ($2<z<3$)
galaxies are shown in the right two columns. The redshift and asymmetry are
given for each example.

We calculate the merger fraction using this equation for each redshift bin,
the evolution of which is shown in the left panel of Figure 12. The results
for the low mass bin ($10^{9.5}M_{\odot}\leq M_{*}<10^{10.5}M_{\odot}$) are
shown as red crosses and the results for the higher mass bin ($M_{*}\geq
10^{9.5}M_{\odot}$) are shown as blue triangles. We fit a power law to the
merger fraction evolution given by the form:

$f_{m}(z)=f_{0}(1+z)^{\beta}$ (9)

by applying a least squares fit to our data and setting $f_{0}$ to a fixed
value of the merger fraction at $z=0$. We take this $z\sim 0$ value to be
0.0193 $\pm$ 0.0028 from Casteels et al. (2014). This point is shown as a grey
inverted triangle. We also include both mass bins in this fit. We find that
the merger fraction, $f_{m}$ evolves as

$f_{m}=0.0193\pm 0.004(1+z)^{1.87\pm 0.04}.$ (10)

This fit is shown in the left panel of Figure 12 as a solid black line and one
standard deviation from this fit is indicated by the grey shaded area. Also
shown are the results of Duncan et al. (2019) (yellow circles) who identify
mergers using pair statistics and fit a power law of the form
$(1+z)^{1.775^{+0.205}_{-0.196}}$. We also compare to Ferreira et al. (2020)
(green diamonds) who identify mergers using a convolutional neural network and
find that the merger fraction evolves as $(1+z)^{2.82\pm 0.46}$. Our merger
fractions lie between these two examples. The merger fractions and their
errors are given in Columns 2 and 4 of Table LABEL:tab:merge_frac for both the
lower and higher mass bins respectively.

####  3.4.2 IR Luminous Galaxies

One of the things that we need to consider in our approach is that some
galaxies are invisible to optical/NIR light and are only detected in the far-
IR, sub-mm or radio (e.g., Wang et al. 2019). These are sometimes called
$H$-band drop-outs and are of interest as they may in fact be galaxies that
are massive, within our limits, but are not considered part of the merger
fraction as they would not be added into our $H$-band selected sample.

A simple argument can be made to show that this concern is insignificant for
our merger fractions. The study of Wang et al. (2019) found 39 galaxies across
the same fields we study, which are not detected in the $H$-band, and thus
would likely not be within our sample. Wang et al. (2019) claim that these
galaxies are typically massive ones at $z>2$, with a median stellar mass of
$10^{10.6}M_{\odot}$. Let us examine the results of assuming that all of these
galaxies are mergers.

For our CANDELS sample, we have 6273 massive galaxies at $z>2$, whilst the
merger fraction we measure is $f_{m}=0.183\pm 0.005$ for these galaxies. This
gives in total 1135 mergers within our sample. Then if we consider all 39 Wang
et al. (2019) galaxies to be mergers at $z>2$ and within the mass range we are
looking at, then the new merger fraction with the Wang et al. (2019) drop-outs
included would be (1135+39)/(6273+39) = 0.186. This gives a very small merger
difference of $\sim 0.003$. The Poisson error on our measurements is also
0.005, at the same level. If we consider that these galaxies have the same
merger fraction as the bulk of our systems, at $f_{m}=0.183$, then the merger
fraction would be: $f_{m}=0.183$, a difference of $<0.001$ \- almost unchanged
and vastly lower than our counting errors.

Regardless, if we consider these FIR galaxies as mergers or that have a merger
fraction higher than our measured one, it would only increase our values
extremely slightly and well within our current measurement errors.

####  3.4.3 Merger Timescales from IllustrisTNG

In this section we investigate the merger timescales for a sample of galaxies
in the IllustrisTNG 300-1 simulation. The process we use to generate mock
images is outlined in §2.5 and here we use only HST-matched mocks. We measure
the observability timescales with a process similar to the one developed in
Lotz et al. (2008) and Nevin et al. (2019). The main difference here is that
the galaxies used in this evaluation are generated as a result of a
cosmological simulation instead of an isolated galaxy-galaxy merger
simulation. Our sample is limited to massive $M_{*}>10^{10}M_{\odot}$ major-
mergers. We briefly outline our steps to measure the timescales below.

First, we follow each galaxy in the simulation for a variety of snapshots
spanning $\pm\ 2\ \rm Gyr$ around each merger event (limited to $0<z<2$). The
snapshot where the merger event happens is defined as the the central
snapshot, $S_{c}$. This information is extracted from the merger trees in the
simulation and is originally defined by the SubLink friends-of-friends
algorithm (Rodriguez-Gomez et al. 2015). The snapshots after $S_{c}$ are
considered to be after the merger event, while the snapshot previous to
$S_{c}$ are classified as before the merger event. This is the only
distinction done here for particular merger stages, which is fundamentally
different to the stages assigned in Lotz et al. (2008).

Secondly, we select only the broadband images that correspond to the rest-
frame optical at those redshifts. For these images, we measure the asymmetry
using Morfometryka. Thus, for each source simulated in §2.5 we have $A$ for
each snapshot in four different orientations. By averaging $A$ by the viewing
angle, we find where the asymmetries of each individual galaxy falls below the
$A>0.35$ threshold for both sides of the merger event.

Finally, by comparing the time difference between the redshift where the
asymmetry threshold is no longer valid and the redshift of the central
snapshot, we estimate an observability timescale for the asymmetry, both for
the post-merger stage and the before merger stage, and, if combined, for the
total merger event. By doing this for all our sample, we have a statistical
estimation of the merger timescales for the asymmetry in IllustrisTNG, as
shown in Figure 11. We find a $\tau=0.56$, lower than the values reported in
Lotz et al. (2008), but higher than the ones found in Nevin et al. (2019) for
major-mergers. We find an asymmetric distribution in the timescales that is
broad but not deep; there is an asymmetric tail at higher timescales. The mean
timescale of this distribution is $\sim$20% greater than the median at 0.67
Gyr so the tail does not significantly impact the timescale used. We consider
the tail in our error in the timescale by finding the differences between the
median of the full sample of simulations and the median of the sample both
above and below 0.56 Gyr. From this, we yield a timescale of
$0.56^{+0.23}_{-0.18}$ Gyr. We find that this timescale does not vary with
redshift unlike in the case of pair statistics (Snyder et al. 2017).

Figure 11: Distribution of the timescales for the total merger events within
the IllustrisTNG sample. The dashed line indicates the median timescale used
to determine the merger rates.

####  3.4.4 Merger Rates

Merger fractions are a solely observational quantity and can tell us little
about the evolution of galaxy formation and evolution. The merger rate (time
between mergers per galaxy) on the other hand is a fundamental parameter in
galaxy evolution. We are able to convert from a merger fraction to a the
merger rate ($\mathcal{R}$) at each redshift using

$\mathcal{R}=\frac{f_{m}}{\tau}$ (11)

where $f_{m}$ is the merger fraction and $\tau$ is the merger timescale.

We calculate the merger rate evolution for our sample of galaxies using a
timescale of $\tau=0.56^{+0.23}_{-0.18}$ Gyr found using mergers within the
IllustrisTNG simulations as described above. The merger rate evolution is
shown in the right panel of Figure 12. As with the merger fraction, we fit a
power law to both the low-mass (red crosses) and high-mass (blue triangles)
data of the form

$\mathcal{R}(z)=\mathcal{R}_{0}(1+z)^{\beta}\ \rm Gyr^{-1}$ (12)

where $\mathcal{R}_{0}$ is the local merger rate calculated using the local
merger fraction, $f_{0}$, from Casteels et al. (2014) and the merger timescale
we find using the IllustrisTNG simulations. This local merger rate is shown as
a grey inverted triangle. We find that the merger rate evolves as:

$\mathcal{R}(z)=0.03\pm 0.02(1+z)^{1.87\pm 0.04}\ \rm Gyr^{-1}.$ (13)

This fit is shown by a solid black line and one standard deviation from this
fit is indicated by the grey shaded area in the right panel of Figure 12. As
with the merger fractions, we also show the results of Duncan et al. (2019)
and Ferreira et al. (2020). Due to the different methods used to identify
mergers, different timescales must be used; Ferreira et al. (2020) use
$\tau=0.6$ Gyr and Duncan et al. (2019) use a timescale that varies with
redshift such that $\tau$ = 0.17 Gyr at $2.5<z<3.0$ and $\tau$ = 0.77 Gyr at
$0.5<z<1.0$ in order to calculate their merger rates.

Despite using a timescale that differs from other merger identification
methods, our results are largely consistent with these previous results. Our
merger rates and errors for the low and high mass bins are given in Columns 3
and 5 of Table LABEL:tab:merge_frac respectively. We also compare our results
to those found by Rodriguez-Gomez et al. (2015) for the Illustris simulation.
This fit is shown by a dashed line here. The fit, given in Table 1 of
Rodriguez-Gomez et al. (2015), is a complex function of redshift, the
descendent galaxy stellar mass $M_{*}$, and the progenitor stellar mass ratio
$\mu_{*}$. We show the results for a stellar mass of $1\times
10^{10}M_{\odot}$ and a stellar mass ratio of 1/4 (consistent with major
mergers, the type of mergers we are able to probe with our asymmetry
selection). The Illustris fit is steeper than the results we find in this
paper and can be approximated as a power law of the form $\sim(1+z)^{2.7}$.
These simulations also predict a smaller merger rate, particularly at the
lowest redshifts, with the local merger rate predicted to be $\sim$3 times
smaller than the value we use here.

|
---|---

(a)  (b)

Figure 12: Left: evolution of the merger fraction. Right: evolution of the merger rate. The lower mass bin from this work is shown as a red cross and the higher mass bin from this work is shown as a blue triangle. The fit to both these points and a known local value from Casteels et al. (2014) (grey inverted triangle) is shown as a solid black line with the grey shaded area indicating one standard deviation from the fit. We find a fit of the form $f_{m}=0.0193\pm 0.004(1+z)^{1.87\pm 0.04}$ for the merger fractions. For the merger rates, we find a fit of the form $\mathcal{R}(z)=0.03\pm 0.02(1+z)^{1.87\pm 0.04}$ Gyr-1. We compare our results to Duncan et al. (2019) (yellow circles) and Ferreira et al. (2020) (green diamonds). We also compare the merger rate to that found by Rodriguez-Gomez et al. (2015) for the Illustris simulation. This fit is given as a dashed line. Table 3: Merger fractions and rates for each redshift bin and each mass bin. | $10^{9.5}M_{\odot}\leq M_{*}<10^{10.5}M_{\odot}$ | $10^{10.5}M_{\odot}\leq M_{*}$
---|---|---
$z$ | $f_{m}$ |  $\mathcal{R}$ (Gyr-1) | $f_{m}$ |  $\mathcal{R}$ (Gyr-1)
0.75 | 0.091 $\pm$ 0.006 | $0.163^{+0.068}_{-0.053}$ | 0.074 $\pm$ 0.011 | $0.131^{+0.057}_{-0.046}$
1.25 | 0.102 $\pm$ 0.006 | $0.181^{+0.075}_{-0.059}$ | 0.108 $\pm$ 0.013 | $0.194^{+0.083}_{-0.066}$
1.75 | 0.119 $\pm$ 0.006 | $0.213^{+0.088}_{-0.069}$ | 0.131 $\pm$ 0.012 | $0.233^{+0.098}_{-0.078}$
2.25 | 0.140 $\pm$ 0.007 | $0.251^{+0.104}_{-0.082}$ | 0.158 $\pm$ 0.018 | $0.283^{+0.120}_{-0.096}$
2.75 | 0.217 $\pm$ 0.009 | $0.388^{+0.160}_{-0.126}$ | 0.263 $\pm$ 0.028 | $0.469^{+0.199}_{-0.158}$

####  3.4.5 Number of Mergers Since $z\sim 3$

From the merger rate we are able to calculate the number of mergers a galaxy
at $0<z<3$ undergoes by integrating the inverse of the characteristic time
between mergers, $\Gamma(z)$. First, we must convert the merger fraction,
$f_{m}$, to the galaxy merger fraction, $f_{gm}$, which gives the number of
galaxies merging as opposed to the number of mergers (Bluck et al. 2009). The
galaxy merger fraction is given by

$f_{gm}=\frac{2\times f_{m}}{1+f_{m}}.$ (14)

We can then calculate $\Gamma$, which is essentially the time in between
mergers, using this galaxy merger fraction:

$\Gamma=\frac{\tau}{f_{gm}}$ (15)

where $\tau$ is the merger timescale. We are able to fit a power law to
$\Gamma$ of the form $\Gamma(z)=\Gamma_{0}(1+z)^{-m}$ Gyr and find that
$\Gamma_{0}=14.4\pm 0.7$ Gyr and $m=1.96\pm 0.13$. We calculate the number of
mergers a given galaxy will undergo between redshifts $z_{1}$ and $z_{2}$,
$N_{\textup{m}}$, using

$N_{\mathrm{m}}=\int^{t_{2}}_{t_{1}}\frac{1}{\Gamma(z)}dt=\int^{z_{2}}_{z_{1}}\frac{1}{\Gamma(z)}\frac{t_{H}}{1+z}\frac{dz}{E(z)}$
(16)

where $t_{H}$ is the Hubble time, and
$E(z)=[\Omega_{m}(1+z)^{3}+\Omega_{\textup{k}}(1+z)^{2}+\Omega_{\Lambda}]^{\frac{1}{2}}=\frac{H(z)}{H_{0}}$.
We assume $\Omega_{m}=0.3$, $\Omega_{\textup{k}}=0$, and
$\Omega_{\Lambda}=0.7$.

The resulting number of mergers undergone between redshifts $z\sim 3$ and
$z\sim 0$ using the time between mergers of $\Gamma(z)=14.4\pm
0.7(1+z)^{-1.96\pm 0.13}$ Gyr is shown in Figure 13 as a solid red line. We
estimate that a galaxy with mass $M_{*}>10^{9.5}M_{\odot}$ will undergo
$2.90_{-0.41}^{+0.50}$ major mergers on average between $z\sim 3$ and $z\sim
0$, with $1.87_{-0.31}^{+0.39}$ mergers occurring before $z\sim 1$. This
result lies between some previous results; Bluck et al. (2009) who find that
massive ($M_{*}>10^{11}M_{\odot}$) galaxies undergo $1.7\pm 0.5$ major mergers
with a timescale of $\tau=0.4$ Gyr from $z\sim 3$ to $z\sim 0$. Conselice
(2006) find that galaxies with $M_{*}>10^{10}M_{\odot}$ undergo
$4.4^{+1.6}_{-0.9}$ major mergers at $z>1$ but undergo less than a single
merger at $z<1$. For comparison, we also plot the evolution of the number of
mergers for a constant merger rate where we set $\Gamma(z)$ to be that at
$z=3$ to show how many mergers would occur without the observed decrease in
merger rate. This is shown as a blue dashed line. The grey shaded areas on
both lines indicate the error on the number of mergers.

We are then able to determine how much mass is added to a galaxy due to these
mergers. Major mergers as identified by the CAS parameters are defined as
galaxy mergers where the mass ratio between the progenitors is $\mu=1/4$ or
less and so the amount of mass added to a galaxy during a merger must be
similar to the original galaxy’s mass. We take the ratio between progenitor
masses to be an average of 1:1.5, as in Conselice (2006), and we approximate
the amount of stellar mass gained by a galaxy due to mergers by $\delta
M_{\textup{merger}}\sim 1.65^{N_{m}}\times M_{0}$, where $N_{m}$ is the number
of mergers and $M_{0}$ is the mass of the initial galaxy (Conselice 2006).
Therefore, a galaxy of mass $M_{*}=10^{9.5}M_{\odot}$ at $z=3$ will gain a of
mass $M_{*}=1.3\times 10^{10}M_{\odot}$ by $z=0$, giving a relative mass
increase from $z=3$ to $z=0$ of $\Delta M_{\textup{merger}}/M=3.2$. A galaxy
of initial mass $M_{*}=10^{10.5}M_{\odot}$ will, on average, accumulate
$M_{*}=1.4\times 10^{11}M_{\odot}$ by $z=0$, giving a relative mass increase
of $\Delta M_{\textup{merger}}/M=4.1$. The mass accretion rate, in terms of
the amount of mass accreted from mergers per unit time, is $\dot{M}\sim
10^{9}M_{\odot}$ Gyr-1 at $M_{*}=10^{9.5}M_{\odot}$, while for
$M_{*}=10^{10.5}M_{\odot}$ the mass accretion rate is a factor of ten times
high, on average, giving $\dot{M}\sim 10^{10}M_{\odot}$ Gyr-1.

These are simply the average mass accretion rates per Gyr. Naturally, this
process is not smooth and mass will be added in discrete jumps during the 2.9
mergers at $z<3$. However, this mass accretion rate due to mergers per galaxy
is an average value for all galaxies that can be applied to a mass selected
population.

Figure 13: The total number of mergers since $z\sim 3$, as a function of
redshift. The solid red line shows this quantity for $\Gamma=14.4\pm
0.7(1+z)^{-1.96\pm 0.13}$ Gyr. We find that galaxies with
$M_{*}>10^{9.5}M_{\odot}$ undergo $\sim$ 3 mergers on average since $z\sim 3$.
For comparison, also shown is the evolution of the number of mergers for a
constant merger rate of $\Gamma=1.05$ Gyr, which is equal to the merger rate
at $z=3$. This is indicated by a blue dashed line. The grey shaded regions
indicate the error range on the number of mergers.

###  3.5 Comparison to Simulations

We compare our results to simulations within IllustrisTNG. Instead of using
TNG300-1 as we do to determine the merger timescale, we use the higher
resolution TNG50-1. This is done to ensure that morphologies are not affected
by mass or spatial resolution, especially in the central regions. For
TNG300-1, the radius containing $20\%$ of the light, $R_{20}$, is close to the
softening length of the simulation for some galaxies, and as a result could
impact the concentration measurement. From TNG50-1 we randomly select galaxies
in the same mass range as our mass-selected sample, generate the mocks
following the description in §2.5 and measure the CAS parameters using
Morfometryka. In Figure 14, we show a comparison of the concentration
evolution for our results from CANDELS and the IllustrisTNG simulations. The
CANDELS results are corrected for redshift effects, as described in §2.3. The
IllustrisTNG simulations images are produced to have at least 3 times the
resolution of the CANDELS images and have not been matched with HST realism as
the concentration should be universal for all resolutions. The resulting
images are 640 $\times$ 640 pixels and the spectra have been shifted to
coincide with the same filters as in Table 1. As with Figure 7, the CANDELS
low mass ($10^{9.5}M_{\odot}<M_{*}<10^{10.5}M_{\odot}$) galaxies are shown as
blue triangles and the CANDELS high mass ($10^{10.5.5}M_{\odot}<M_{*}$) are
shown as red crosses. The low mass IllustrisTNG simulations are shown as a
dashed orange line and the high mass simulations are shown as a solid green
line. We find that the concentration of the simulations show very little
evolution with redshift for both mass bins. This suggests that the galaxies
are growing in size with the inner and outer regions growing at a similar rate
whereas the trend in the CANDELS data suggests the galaxies are growing from
the inside out.

Figure 14: Comparison of the evolution of the concentration for our results
from CANDELS (corrected for redshift effects) and the galaxies within the
IllustrisTNG simulations. As with Figure 7, the CANDELS low mass
($10^{9.5}M_{\odot}<M_{*}<10^{10.5}M_{\odot}$) galaxies are shown as blue
triangles and the CANDELS high mass ($10^{10.5.5}M_{\odot}<M_{*}$) are shown
as red crosses. The low mass IllustrisTNG simulations are shown as an dashed
orange line and the high mass simulations are shown as a solid green line. The
shaded regions indicate one standard deviation from the mean. The IllustrisTNG
data is based upon high resolution images.

We also compare the measured merger fractions within the simulations to the
CANDELS data. For this, we randomly select $5\%$ of all available galaxies in
TNG50-1 in this mass range. As this naturally selects more galaxies in the low
mass bin than in the high mass bin, we further select more galaxies for the
high mass bin until both bins have the same number of galaxies. The
simulations in this case are calibrated on HST resolution. This comparison is
shown in Figure 15. As for Figure 12, the CANDELS data is shown as red crosses
(low mass bin) and blue triangles (high mass bin). The IllustrisTNG merger
fraction evolution determined using the same asymmetry condition as used for
the CANDELS data for the low and high mass bins are shown as a solid orange
line and a dashed green line respectively. We also determine the ‘true’ merger
fraction within the simulation by using the merger labels produced based on
the merger trees. We ensure we only select major mergers by selecting mergers
that have a mass ratio greater than 1/4. We also select only those galaxies
that are within 0.65 Gyr of their merging event, a time between the median and
mean of our CAS merger time-scale. These ‘true’ merger fractions are shown as
a purple dotted line and a dot-dashed yellow line for the low and high mass
bins respectively. The shaded regions show the error on the merger fractions
for the IllustrisTNG simulations. We find that both the ‘true’ and asymmetry
defined mergers are consistent with the observations. When selecting galaxies
within a shorter timescale than the one chosen here, the number of galaxies in
each redshift bin are too few and thus the ‘true’ merger fractions decrease
and do not agree very well with the asymmetry selected mergers from
IllustrisTNG or the CANDELS data. This in part happens due to the time
resolution between snapshots in IllustrisTNG, which is roughly $\Delta t\sim
0.16$ Gyr.

Figure 15: Comparison of the evolution of the merger fraction for our results
from CANDELS and the galaxies within the IllustrisTNG simulations. The
IllustrisTNG data is based upon HST matched mocks. As with Figure 7, the
CANDELS low mass ($10^{9.5}M_{\odot}<M_{*}<10^{10.5}M_{\odot}$) galaxies are
shown as blue triangles and the CANDELS high mass galaxies
($10^{10.5.5}M_{\odot}<M_{*}$) are shown as red crosses. The merger fraction
evolution for the simulations determined using the asymmetry condition is
shown as a solid orange line and a dashed green line for the low and high mass
bins respectively. Also shown are the ‘true’ merger fractions - the merger
fraction determined based on the labels from the merger trees within the
simulations and we select galaxies that are within 0.65 Gyr of a merging event
- between the median and mean of the CAS time-scale distribution. These merger
fractions are shown as a purple dotted line and a dot-dashed yellow line for
the low and high mass bins respectively. Our results are consistent with both
the asymmetry defined mergers and the ‘true’ mergers from the simulations.

We explore how many of galaxies within the Illustris simulation have $A>0.35$
and also are considered to be a ‘true’ major merger (with a mass ratio greater
than 0.25). In this case we consider galaxies that are within 0.45 Gyr of a
merger event, or 0.9 Gyr in total, to be ‘true’ mergers in order to account
for the time difference between snapshots within the simulation. We find that
across all redshifts, an average of $\sim 37\%$ of the ‘true’ major mergers
lie above $A>0.35$. This is slightly lower than the $\sim 50\%$ found by
Conselice (2006).

##  4 Discussion

In this paper we have presented a number of observed features of galaxy
evolution, as viewed through the structural changes in galaxies over the past
11 Gyr. We carry this out through an analysis of all five CANDELS fields,
examining the change in light concentration and asymmetry for these systems.
As opposed to other studies of galaxy evolution at these epochs we take a
largely structurally evolutionary view of galaxies, rather than one based on
stellar populations. In this viewing of galaxies we are interested in the
build up of stars in galaxies and how that affects the distribution of light
within galaxies, rather than examining the type and distribution of stars
themselves.

###  4.1 Concentration

We find that galaxies are more concentrated at higher redshifts when probing
galaxy structure in the optical rest-frame. This is an interesting result and
is somewhat different from expectations, so it is important to examine the
reasons for this. We test this result by examining the evolution of the
concentration calculated using various measures of size. First, we correct the
radii $R_{20}$ and $R_{80}$ (where $R_{20}$ is the radius containing 20% of
the total light within a galaxy and $R_{80}$ is the radius containing 80% of
the light) using the same method as described in §2.3 and use the mean of
these corrected values to calculate the concentration using equation 2.

Examining the individual radii for our sample, we find that the evolution of
the corrected $R_{80}$ values remain roughly constant with redshift whereas
$R_{20}$, on average, increases with redshift. Therefore, we see a lower
concentration at lower redshifts. We show the evolution of the concentration
calculated using the corrected values of $R_{20}$ and $R_{80}$ in Figure 16.
As in Figure 7, the low mass bin is shown as red crosses and the high mass bin
is shown as blue triangles. Also shown are power law fits to the
concentrations, with the dotted line being for the low mass bin and the dashed
line being for the high mass bin. For comparison, we show the evolution of the
measured value of $C$ for all masses as grey circles. This decrease in
concentration is consistent with the corrected concentrations shown in Figure
7.

Figure 16: Evolution of the concentration as calculated from the corrected
$R_{20}$ and $R_{80}$ growth radii. The low mass bin is shown as red crosses
and the high mass bin is shown as blue triangles. The grey circles represent
the evolution of the measured concentration across all masses.

We further explore this by examining the Petrosian radius,
$R_{\textup{Petr}}(\eta)$. The Petrosian radius is defined to be the radius at
which the surface brightness at a given radius is a specific fraction ($\eta$)
of the surface brightness within that radius (e.g., Bershady et al. 2000;
Conselice 2003; Whitney et al. 2019). $\eta$ is given by:

$\eta(r)=\frac{I(r)}{\left\langle I(r)\right\rangle}$ (17)

where $I(r)$ is the surface brightness at radius $r$ and $\left\langle
I(r)\right\rangle$ is the mean surface brightness within that radius. Using
this definition, $\eta=0$ at large radii. We consider
$R_{\textup{Petr}}(\eta=0.2)$ (the outer edge of a galaxy) and
$R_{\textup{Petr}}(\eta=0.8)$ (the inner region of a galaxy). Whitney et al.
(2019) show that $R_{\textup{Petr}}(\eta=0.2)$ grows more rapidly than
$R_{\textup{Petr}}(\eta=0.8)$ for a mass-selected sample (where the masses lie
in the range $10^{9}M_{\odot}<M_{*}<10^{10.5}M_{\odot}$) across a redshift
range of $1<z<7$ in the GOODS-North and GOODS-South CANDELS fields. So, one
would expect the concentration at lower redshifts to be larger. In the same
fashion as with the growth radii, we calculate the concentration of the
galaxies in Whitney et al. (2019) with mass
$\textup{log}_{10}(M_{*}/M_{\odot})>9.5$ using the Petrosian radii such that

$C_{\textup{P}}=5\times\textup{log}_{10}\left(\frac{R_{\textup{Petr}}(\eta=0.2)}{R_{\textup{Petr}}(\eta=0.8)}\right).$
(18)

The resulting concentration evolution is shown in Figure 17. We fit a power
law to the entire redshift range from Whitney et al. (2019) (dotted line) and
also the redshift range we consider in this paper (solid line). We find that
$C_{\textup{P}}$ goes as $(1+z)^{-0.09\pm 0.06}$ for the redshift range of
$1<z<6$ and goes as $(1+z)^{0.09\pm 0.09}$ for the smaller redshift range of
$1<z<3$, which is within $2\sigma$ of being considered a flat relation. The
vertical dashed line indicates the redshift limit of this work. Whilst on
average over the redshift range of $1<z<6$, the concentration decreases with
redshift, when we consider the lowest redshift bins, the concentration appears
to increase with redshift, which is consistent with the results we find and
show in Figure 7. This is due to the fact that the inner radius appears to
grow more rapidly compared to the outer radius within this small redshift
range. We find the same result when considering galaxies with
$\textup{log}_{10}(M_{*}/M_{\odot})>10.5$.

Figure 17: Evolution of the concentration calculated using the Petrosian radii
at $\eta=0.2$ and $\eta=0.8$ as per Equation 18 for galaxies of mass
$\textup{log}_{10}(M_{*}/M_{\odot})>9.5$ from Whitney et al. (2019). We fit a
power law to the whole redshift range, shown as a dotted line, and also to the
redshift range we consider here (solid line). We find that $C_{\textup{P}}$
goes as $(1+z)^{-0.09\pm 0.06}$ for the redshift range of $1<z<6$ and as
$(1+z)^{0.09\pm 0.09}$ for the smaller redshift range of $1<z<3$. The vertical
dashed line indicates the redshift limit of this work. On average, the
concentration decreases at higher redshift however when considering the lowest
redshifts, the concentration increases with redshift.

Alternatively, it might be the case that as we go to lower redshifts, we are
sampling more and more galaxies using a constant mass limit (Mundy et al.
2015). The result of this is that intrinsically lower halo mass galaxies will
be entering our sample and these may have lower concentrations. We test this
by considering how the concentration changes for a number density selection
sample. Doing this at number densities of $3\times 10^{-5}$ Mpc-3 we find that
the concentration decreases by a factor of $1.4\pm 0.3$ from the highest
redshift bin of $2.5<z<3$ to the lowest bin of $0.5<z<1$. This is comparable
to the change seen in the mass-selected samples where the concentration
decreases by a factor of $1.4\pm 0.3$ for the lower mass bin and a factor of
$1.2\pm 0.2$ for the higher mass bin over the same redshift range. A
comparison of the evolution for the mass-selected and number density-selected
samples are shown in Appendix A. This suggests that the evolution we see in
the concentration is real, and not due to newer galaxies being introduced to
the lower redshift bins.

Radius measurements of the inner regions can be inaccurate due to factors such
as the size of the PSF. This could create inaccuracies in the concentration
measurement. To ensure the concentration evolution we see is not an artefact
of such issues, we also use $R_{50}$ and $R_{90}$ values in place of $R_{20}$
and $R_{80}$ in Equation 2. We find that the evolution of $C_{59}$ exhibits
similar behaviour to $C_{28}$, whereby the concentration increases by a factor
of $\sim$1.5 over the redshift range $0.5<z<3$. We can therefore conclude that
the evolution in concentration we see is real and not as a result of problems
with the inner radius measurement.

We interpret a decreasing concentration with lower redshift as an indication
that a small initial galaxy will eventually grow in size as a result of galaxy
mergers, as well as galaxy accretion. This growth is occurring in such a way
that the inner regions of a galaxy grow more rapidly in comparison to the
outer regions at $z<3$.

###  4.2 Merger Consistency

Mergers can be identified in multiple ways such as pair statistics (e.g.,
Bundy et al. 2009; Man et al. 2016; Mundy et al. 2017; Duncan et al. 2019),
CAS (e.g., Conselice et al. 2009; Bluck et al. 2012), Gini-$M_{20}$ (e.g.,
Lotz et al. 2008, 2010), and Deep Learning (e.g., Ferreira et al. 2020) with
each of these methods yielding differing merger counts, and therefore
fractions due to the different criteria. Using a power law fit, we find that
the merger fraction goes as $(1+z)^{1.87\pm 0.04}$ when fitting our data and
setting the $z\sim 0$ merger fraction to a fixed value. This is shallower than
that found by Conselice et al. (2009) who derive merger fractions using the
CAS parameters for a sample of galaxies within the Extended Groth Strip and
Cosmic Evolution Survey at $0.2<z<1.2$; they find a relation of $(1+z)^{2.3\pm
0.4}$ using their data and by using a fixed value of the merger fraction at
$z\sim 0$, the relation changes to $(1+z)^{3.8\pm 0.2}$. Ferreira et al.
(2020) use a convolution neural network trained on simulations from
IllustrisTNG and applied to data from all five CANDELS fields and find an
evolution of $(1+z)^{2.82\pm 0.46}$, again slightly steeper than the results
we find. Duncan et al. (2019) on the other hand find a shallower relation for
their pair fractions of $(1+z)^{0.844^{+0.216}_{-0.235}}$ for galaxies with
log${}_{10}(M_{*}/M_{\odot})>10.3$. In general, the morphologically defined
mergers give a higher merger fraction than those derived through pairs
(Conselice et al. 2009). However, this is not unexpected, as different methods
of finding mergers will give different results for the fractions.

Consistency is however expected when converting to merger rates and comparing
these values. Along with different methods giving different merger fractions,
different selection criteria within the same sample also give varying results;
Man et al. (2016) examine the pair fraction evolution up to $z\sim 3$ using
the UltraVISTA/COSMOS $K_{S}$-band selected catalog along with 3DHST+CANDELS
$H$-band imaging and explore the effect of using the flux ratio versus the
mass ratio in the selection. When selecting via the stellar mass ratio, the
merger fraction appears to increase with cosmic time in contradiction to the
results we and others find. A selection via the $H$-band flux ratio however,
the trend is similar to ours in that the merger fraction decreases.

As with merger fractions, merging timescales vary quite drastically depending
on the selection method. For example, Lotz et al. (2008) use GADGET N-Body
hydrodynamical simulations and the radiative transfer code SUNRISE to create a
sample of galaxy mergers. They use these to determine merging timescales for
multiple methods of identifying mergers. Using the same asymmetry condition we
use here, they determine a timescale of 0.94 $\pm$ 0.13 Gyr, whilst using a
Gini and $M_{20}$ condition, they find the timescale to be 0.26 $\pm$ 0.10
Gyr. Conselice et al. (2008) who examine the merger rate up to $z\sim 3$ in
the Hubble Ultra-Deep Field use a merger time-scale of 0.34 Gyr, determined
using the asymmetry condition. These methods all assume a constant timescale
with redshift, however using a close pair method of identifying mergers,
Snyder et al. (2017) find that the timescale varies with redshift as
$\tau\propto(1+z)^{-2}$. These timescales all differ from the constant
timescale of $\tau=0.56^{+0.23}_{-0.18}$ Gyr we find using the asymmetry
criteria.

Despite these differences in timescales and merger fractions for various
methods of identifying mergers, merger rates are largely consistent with each
other, as shown in Figure 12. The merger rates determined using the asymmetry
condition to identify mergers as we have done here are, within the errors,
similar to merger rates determined using other methods such as probabilistic
pair-counts (Duncan et al. 2019) and finding mergers using Deep Learning
(Ferreira et al. 2020). This is despite using different timescales for a
merging event, showing that while different methods identify different numbers
of mergers, our CAS method gives a consistent merger history for galaxies. The
merger rates presented here have greater errors associated with them than
previous work due to the consideration of the error in the merger timescale;
the IllustrisTNG simulations used here to determine the merger timescale have
an asymmetric distribution in the timescale and so we therefore use an
asymmetric error. Ferreira et al. (2020) use a timescale based on their
selection and as such, do not have an associated error. Duncan et al. (2019)
calculate the errors on their merger fractions using the bootstrap technique.
The timescale taken from Snyder et al. (2017) does not have an associated
statistical error so when determining the fit (by convolving the merger
fraction fits with the redshift dependent timescale), the merger rates only
consider the error in merger fraction. Therefore, we have found that using
three independent methods that we are able to measure a consistent merger rate
at $z<3$, and therefore have a firm idea of what the history of galaxy major
mergers are within this redshift range.

##  5 Conclusions

Galaxy structure is one of the most fundamental ways in which galaxies and
their formation and evolution can be understood, yet there have been few
systematic studies of this property at higher redshifts. In this paper we
collate the high resolution HST imaging of galaxies from all five fields of
the CANDELS survey to determine the time evolution of galaxy structure to
derive the processes for galaxy evolution. We combine these 16,778 galaxy
structures, both visually based and quantitatively based, with stellar mass
measurements and photometric redshifts to derive the evolution of galaxy
structure. To aid our investigation of these morphologies we also utilise
Illustris TNG300 simulations to calibrate timescales for morphological
features and Illustris TNG50 for direct morphological comparisons with
observations.

Our conclusions, based on a detailed non-parametric analysis of the structures
of galaxies in all five CANDELS fields are as follows:

  1. 1.

We find that galaxies are distributed in a concentration-asymmetry plane up to
$z\sim 3$ in a similar way as they do at lower redshifts. At these redshifts
there is a continuum of parameters, suggesting unlike the colour-magnitude
plane galaxies on this relation evolve continuously over time, rather than
change dramatically.

  2. 2.

Based on dynamical simulations with the IllustrisTNG simulation, we show that
the evolution of galaxies within the $C-A$ plane over this time period is
largely driven by galaxy formation processes, including the accretion of gas
as well as galaxy mergers.

  3. 3.

We find that galaxies are both more asymmetric and more concentrated at higher
redshifts. This demonstrates that galaxies are assembled in a way where
material is coming in through galaxy formation processes that are relatively
changing the inner portions of galaxy light quicker than the outer radii at
$z<3$.

  4. 4.

Using merger timescales from Illustris TNG300 we determine how the
morphological timescales for mergers evolves with time. We find that for the
CAS system of finding mergers, the mean merger timescale is
$0.56^{+0.23}_{-0.18}$ Gyr. We also find, that unlike in the galaxy pair
situation, this merger timescale does not evolve with time.

  5. 5.

Using these timescales we are able to determine the merger fraction evolution
and the merger rate evolution up to $z\sim 3$. We find a good agreement
between our merger rate calculations in comparison with previous work using
pairs of galaxies and finding mergers based on Deep Learning. This is
therefore a consistent merger history for galaxies as measured in three
different ways. We now have a firm and consistent measure of the galaxy major
merger history at $z<3$.

Our results are the best measurements of the merger history using HST imaging
that will likely be possible unless a new large area survey in the near
infrared is performed. When Euclid and JWST launch we will be able to expand
these results in many ways. We will be able to measure the merger history in a
similar way up to $z\sim 6$, and at lower redshifts we will be able to probe
down to lower masses to find the evolution of a fuller sampling of the galaxy
population. Euclid will allow us to examine the merger history over 15,000
deg2, allowing detailed merger histories to be measured. The techniques and
methods here can be used with both data sets, and others, to fully explore and
measure this merger history, a principle part of the entire galaxy formation
process.

##  6 Acknowledgements

This work is based on observations taken by the CANDELS Multi-Cycle Treasury
Program with the NASA/ESA HST, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
We thank the rest of the CANDELS team for their heroic work making their
products and data available. We acknowledge funding from the Science and
Technology Facilities Council (STFC) to support this work. LF acknowledges
funding from the Coordenaşão de Aperfeiçoamento de Pessoal de Nível Superior -
Brazil (CAPES).

##  Appendix A Number Density-Selected Sample

In order to test whether the mass limits of our sample are creating an
artificial trend, we instead select galaxies using a constant number density
of $3\times 10^{-5}$ Mpc-3. This yields galaxies in the mass range
$10^{10.7}M_{\odot}\leq M_{*}<10^{11.5}M_{\odot}$. This method avoids problems
with having a fixed stellar mass bin that galaxies can enter at lower
redshift, compared with those that are at higher redshifts (e.g., Mundy et al.
2015; Ownsworth et al. 2016). This has an effect such that by $z=0$ only $\sim
5$ % of galaxies with a high mass selection would have been within the same
mass selection at $z>2.5$. However, by selecting through a constant co-moving
number density we are able to remove this bias and always examine what are
statistically the same number of massive galaxies at each redshift. Because
there are so many ‘new’ galaxies in a mass-selected sample, it is possible
that the relations we derive are biased by these new systems.

We determine the concentration evolution for this number density selected
sample, and compare it to the evolution seen for the mass selected-sample. We
find that the evolution for the number density-selected sample is very similar
to that of the mass-selected sample, as shown in Figure 18. Green circles
indicate the concentration for the number density-selected sample, red crosses
indicate the lower mass bin, and blue triangles the higher mass bin. Therefore
our finding of a reduction in concentration using the mass selected sample is
not due to new galaxies entering the criteria.

Figure 18: A comparison of the evolution of concentration for a number
density-selected sample (green circles) and a mass-selected sample. The number
density is kept constant at $3\times 10^{-5}$ Mpc-3. As for Figure 7, red
crosses indicate the lower mass bin and blue triangles indicate the higher
mass bin. We find no significant difference in the evolution of the $C$
parameter using these different selection methods.

As a test for another key result, we examine how the merger fraction evolution
would occur by using a selection based on this number density selection of
$3\times 10^{-5}$ Mpc-3. This is shown in Figure 19 where we find that the
merger fraction is slightly lower for all but the final lowest redshift bin.
However, the fitted merger fraction evolution and thus the corresponding
merger rate is statistically very similar and does not give significantly
different merger histories for galaxies. Thus, our result using a constant
mass selection is a viable way in which to trace the evolution of the massive
galaxy population, and our results are robust to galaxy selection methods.

Figure 19: A comparison of the evolution of the merger rate for a number
density-selected ($n=3\times 10^{-5}$ Mpc-3) sample of galaxies (green
circles), low mass galaxies (red crosses), and higher mass galaxies (blue
triangles). The fit to the number density-selected sample is shown by the
solid line and the fit to the mass-selected sample is shown by the dashed
line. The known local value of the merger fraction from Casteels et al. (2014)
is from a mass-selected sample.

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