# A declining major merger fraction with redshift in the local Universe from
the largest-yet catalog of major and minor mergers in SDSS

R. Nevin,1 L. Blecha,2 J. Comerford,3 J. Simon,3 B. A. Terrazas,4 R. S.
Barrows,3J. A. Vázquez-Mata5
1Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA
2Department of Physics, University of Florida, Gainesville, FL 32611, USA
3Department of Astrophysical and Planetary Sciences, University of Colorado,
Boulder, CO 80309, USA
4Columbia Astrophysics Laboratory, Columbia University, 550 West 120th Street,
New York, NY 10027, USA
5Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma
de México, Ciudad Universitaria, CDMX, 04510, México
E-mail<EMAIL_ADDRESS>Astronomy and Astrophysics Postdoctoral Fellow

(Accepted Feb 16, 2023. Received YYY; in original form ZZZ)

###### Abstract

It is difficult to accurately identify galaxy mergers and it is an even larger
challenge to classify them by their mass ratio or merger stage. In previous
work we used a suite of simulated mergers to create a classification technique
that uses linear discriminant analysis (LDA) to identify major and minor
mergers. Here, we apply this technique to 1.3 million galaxies from the SDSS
DR16 photometric catalog and present the probability that each galaxy is a
major or minor merger, splitting the classifications by merger stages (early,
late, post-coalescence). We present publicly-available imaging predictor
values and all of the above classifications for one of the largest-yet samples
of galaxies. We measure the major and minor merger fraction
($f_{\mathrm{merg}}$) and build a mass-complete sample of galaxies, which we
bin as a function of stellar mass and redshift. For the major mergers, we find
a positive slope of $f_{\mathrm{merg}}$ with stellar mass and negative slope
of $f_{\mathrm{merg}}$ with redshift between stellar masses of
$10.5<M_{*}(log\ M_{\odot})<11.6$ and redshifts of $0.03<z<0.19$. We are able
to reproduce an artificial positive slope of the major merger fraction with
redshift when we do not bin for mass or craft a complete sample, demonstrating
the importance of mass completeness and mass binning. We determine that the
positive trend of the major merger fraction with stellar mass is consistent
with a hierarchical assembly scenario. The negative trend with redshift
requires that an additional assembly mechanism, such as baryonic feedback,
dominates in the local Universe.

###### keywords:

galaxies: interactions – galaxies: evolution – surveys – catalogues –methods:
statistical – techniques: image processing

††pubyear: 2022††pagerange: A declining major merger fraction with redshift in
the local Universe from the largest-yet catalog of major and minor mergers in
SDSS–B.6

##  1 Introduction

The $\Lambda$CDM model of structure growth predicts that galaxies grow
hierarchically through mergers, but uncertainty still surrounds the impact of
mergers on physical processes in galaxies. For instance, while theory predicts
that mergers contribute to the growth of stellar bulges and elliptical
galaxies Springel 2000; Cox et al. 2008, trigger star formation (Di Matteo et
al. 2008) and active galactic nuclei (AGN, Hopkins et al. 2006), and even
quench star formation (Di Matteo et al. 2005; Hopkins et al. 2008),
observational work often disagrees about the importance of mergers for driving
these evolutionary processes (e.g. for whether mergers trigger AGN and/or star
formation, see Cisternas et al. 2011; Knapen et al. 2015; Ellison et al. 2019;
Pearson et al. 2019). This is a critical tension: the implication is that our
models and/or our current methods for identifying mergers are incorrect.

In order to determine the role of mergers in driving galaxy evolution,
reconcile simulations with observations, and test the $\Lambda$CDM
cosmological model, the galaxy-galaxy merger rate and merger fraction are key
diagnostic tools. The merger rate, which will be the focus of future work
(Simon et al. 2023, in prep), is measured using the merger fraction and the
merger observability timescale (Lotz et al. 2011), both of which vary as a
function of redshift, mass, mass ratio, and critically, the technique used to
identify mergers.

Characterizing the merger fraction as a function of mass, redshift, and mass
ratio is critical for understanding the relative contributions of both major
and minor mergers to the growth of different types of galaxies over cosmic
time. For instance, we can use the mass- and redshift-dependent merger
fraction to constrain the relative contribution of major and minor mergers to
the growth of the most massive galaxies, which are predicted to assemble at
late times by $\Lambda$CDM. It is therefore an important test of $\Lambda$CDM
cosmology. We can also use the merger fraction to test the predictions of
other structure formation channels (see §5.1 for a review).

Many different techniques exist to measure the evolution of the major merger
fraction with redshift, including close-pair (e.g. Patton et al. 1997; Lin et
al. 2004; Kartaltepe et al. 2007; Bundy et al. 2009), clustering (e.g. Bell et
al. 2006; Robaina et al. 2010) and morphological techniques (e.g. Lotz et al.
2008; Conselice et al. 2009). The majority of these studies find that the
major merger fraction peaks at earlier times, in agreement with the above
theoretical measurements. Other work focuses on the evolution of the major
merger fraction with stellar mass (e.g. Xu et al. 2012; Casteels et al. 2014),
finding either an increasing or decreasing merger fraction with stellar mass.
For a thorough review of past results, see §5.2.

Most of the literature has focused either on the mass- or redshift-dependence
of the merger fraction separately. Also, most of the redshift-dependent
studies only cover higher redshifts. In this work we focus on constraining the
mass- and redshift-dependent merger fraction for galaxies in the Sloan Digital
Sky Survey (SDSS). Our focus is on the local Universe, which will allow us to
avoid the uncertainties that plague many of the above studies due to small
sample sizes. We additionally use a carefully calibrated morphologically-based
technique that avoids incompleteness issues due to fiber overlap.

While most past work has focused on the more easily measured major merger
fraction, the minor merger fraction is also an important quantity. Past work
finds that the minor merger fraction is several times higher than the major
merger fraction (e.g. Lotz et al. 2011; López-Sanjuan et al. 2011; Bluck et
al. 2012; Kaviraj 2014b, a; Rodriguez-Gomez et al. 2015), indicating that
minor mergers have a critical role to play in building mass in disk galaxies,
the envelopes of massive ellipticals, and the bulges of lower mass galaxies
without destroying the merger remnant (Hopkins et al. 2010). In this work we
set out to constrain not only the major merger fraction but also the minor
merger fraction and how they both vary as a function of stellar mass and
redshift.

In addition to providing constraints on the importance of galaxy mergers for
galaxy evolution, the galaxy-galaxy merger fraction and rate are crucial for
constraining the predicted supermassive black hole (SMBH) merger rate. The
SMBH merger rate will be measured by upcoming gravitational wave observatories
such as the (evolved) Laser Interferometer Space Antenna (eLISA, LISA), which
is anticipated to detect SMBH mergers out to $z\sim 10$ (Amaro-Seoane et al.
2017; Mueller & Gravitational Observatory Advisory Team 2016; Arun et al.
2022), and indirectly measured by pulsar timing arrays through the
gravitational wave background (e.g. Hobbs et al. 2010; NANOGrav Collaboration
et al. 2015; Arzoumanian et al. 2020), which is dominated by the signal from
binary SMBHs, which form following major galaxy mergers, with masses
$M_{SMBHB}>10^{7}\ M_{\odot}$ out to $z\sim 2$ (e.g. Sesana 2013; Simon &
Burke-Spolaor 2016).

The galaxy-galaxy merger rate is also important for breaking degeneracies in
the gravitational wave signal. For instance, Siwek et al. (2020) find that the
chirp mass of SMBH binaries is degenerate with the merger rate, so separately
constraining the galaxy-galaxy merger rate can complement gravitational wave
background measurements, break these degeneracies, and constrain SMBH
accretion models. A strength of the LDA technique used in this work to
identify mergers is that it is created from detailed temporal simulations of
mergers, hence we have a solid understanding of the merger observability
timescale. In future work (Simon et al. 2023, in prep), we plan to combine the
observability timescales from this work with the merger fractions also
measured in this work to derive the galaxy-galaxy merger rate and make
predictions for the expected gravitational wave background signal from merging
binary SMBHs in the local universe.

In this paper, we address the above challenges using a statistical learning
tool calibrated on well-understood hydrodynamical models of merging galaxies
from Nevin et al. (2019) (henceforth N19). We apply this automated merger
classification technique to the 1.3 million galaxies in the Sloan Digital Sky
Survey (SDSS) DR16 photometric sample (§2). The strength of this approach lies
in the massive statistical sample of mergers identified using a morphological-
based technique that exceeds previous morphological techniques in accuracy and
completeness to classify different types of mergers (§3). The focus of this
paper is twofold: 1) We present publicly-available catalogs of different types
of mergers identified by both stage and mass ratio (major/minor, early, late,
and post-coalescence) and 2) We estimate the galaxy merger fraction as a
function of mass ratio, mass, and redshift (§4). We end by discussing our
results in the context of cosmological models, past empirical studies of the
merger fraction, and future directions (§5). A cosmology with
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and $h=0.7$ is assumed throughout.

##  2 Data

Here we present an overview of the data set. We describe how we create image
cutouts and the properties of the photometric sample in §2.1. We present our
process for measuring imaging predictor values from these image cutouts in
§2.2.

###  2.1 Creating image cutouts of galaxies in SDSS

The Sloan digital sky survey (SDSS, Gunn et al. 2006) is an all-sky
spectroscopic and imaging survey. To construct our sample of galaxies, we use
the $r-$band imaging data from data release 16 (DR16, Ahumada et al. 2020),
which is the fourth data release of SDSS-IV (Blanton et al. 2017). Using
CasJobs, we select all galaxies from the DR16 photometric catalog that have an
$r-$band magnitude less than or equal to 17.77, the completeness limit of
SDSS. We do not restrict the selection to objects that also have a
spectroscopic object ID, maximizing the number of objects in the sample. We
also do not restrict the sample by redshift. The redshift range of the mass
complete sample (described in §3.6) is $0.03<z<0.19$.

The exact SQL search is as follows:

select po.objID, po.ra, po.dec,
(po.petroMag_r - po.extinction_r) as dered_petro_r
into MyDB.five_sigma_detection_saturated_mode1
from PhotoObj as po
where (po.petroMag_r)<=17.77 and po.type = 3
and ((flags_r & 0x10000000) != 0)
and (flags_r & 0x40000) = 0 and mode=1

This query restricts the search to galaxies (po.type=3), eliminates galaxies
that are detected at less than 5$\sigma$ (flags_r 0x10000000!=0) and galaxies
for which no petrosian radius could be determined in the $r-$band (flag_r
0x40000 = 0), and removes duplicates using mode=1. This search returns 1393923
galaxies.

We use the Skycoords utility from astropy to create
$80\aas@@fstack{\prime\prime}0$ by $80\aas@@fstack{\prime\prime}0$ square
cutout $r-$band images for each galaxy from the frame images. After
eliminating a small fraction ($\sim$0.4%) of the cutouts that are blank,
corrupted, or at the edge of the frame, we have a total of 1388533 galaxy
cutout images.

###  2.2 Measuring predictor values from the SDSS cutout images

For each galaxy image, we measure seven imaging predictor values: $Gini$, M20,
Concentration ($C$), Asymmetry ($A$), Clumpiness ($S$), Sersic index ($n$),
and shape asymmetry ($A_{s}$). We use the same procedure as N19 to measure the
imaging predictors, which incorporates SourceExtractor (Bertin & Arnouts
1996), GALFIT (Peng et al. 2002, 2010), and statmorph (Rodriguez-Gomez et al.
2019). We also use statmorph to measure the average S/N value (<S/N>) within
the segmentation maps. After extracting the imaging predictors, the sample
size is 1344677 galaxies; we lose about 3% of the sample due to either GALFIT
or statmorph failing to converge on a good fit.

We next flag galaxies for unreliable predictor values; these galaxies are
included in both the predictor and the classification tables but are excluded
from our analysis of the merger fraction. Excluding the galaxies with one or
more flags, there are 938892 galaxies with clean photometry. We employ three
separate flags:

  1. 1.

The ‘low S/N’ flag is thrown when the average S/N value is below 2.5, which is
the cutoff value quoted in N19 below which the classification is significantly
different.

  2. 2.

The ‘outlier predictor’ flag is thrown when one or more imaging predictors are
outside the range of predictor values from the simulated galaxies. The range
of simulated values is: $0.44<Gini<0.72$, $-2.70<M_{20}<-0.50$, $1.32<C<5.57$,
$-0.24<A<0.76$, $-0.24<S<0.16$, $0.47<n<5.14$, and $0.0<A_{s}<1.21$.

  3. 3.

The ‘segmap’ flag is thrown when the segmentation map does not include the
central pixel or for when the segmentation map extends beyond the edge of a
clipped image. This identifies images for which the predictor values are
actually measuring a brighter foreground galaxy or star.

We present the predictor values for six galaxies in Table 1. We plot the
distributions of predictor values for the full sample in Figure 1 alongside
the six example galaxies from Table 1 identified with capital letters A-F.

| Predictor Valuesb |  | Flagsd
---|---|---|---
SDSS ObjIDa | $Gini$ | $M_{20}$ | $C$ | $A$ | $S$ | $n$ | $A_{s}$ | S/Nc | low S/N | outlier predictor | segmap
1237665179521187863 (A) | 0.54 | -2.15 | 3.62 | -0.04 | -0.01 | 1.49 | 0.13 | 9.98 | 0 | 0 | 0
1237661852010283046 (B) | 0.69 | -0.96 | 3.59 | 0.22 | 0.01 | 1.32 | 0.78 | 12.49 | 0 | 0 | 0
1237648720718463286 (C) | 0.56 | -1.0 | 3.66 | 0.43 | -0.16 | 0.58 | 0.89 | 6.4 | 0 | 0 | 0
1237662306186428502 (D) | 0.56 | -2.16 | 3.59 | 0.14 | 0.02 | 1.38 | 0.57 | 16.35 | 0 | 0 | 0
1237653589018018166 (E) | 0.56 | -2.07 | 3.53 | 0.02 | 0.01 | 1.47 | 0.40 | 14.31 | 0 | 0 | 0
1237654383587492073 (F) | 0.58 | -0.81 | 1.61 | 0.54 | 0.06 | 0.97 | 0.12 | 54.27 | 0 | 0 | 0
Table 1: Six galaxies from the table of predictor values alongside their
identification letters (A-F) that will be used throughout this paper.
aThe SDSS photometric object ID from DR16
bThe pre-standardized predictor values
cAverage S/N for the area of the galaxy enclosed by the segmentation mask
dFlags have a value of 1 when activated Figure 1: Distributions of predictor
values for the full SDSS DR16 sample of galaxies (top, grey distribution), the
simulated galaxies (black contours), and the selected non-flagged sample of
galaxies (color distribution). We show six example galaxies with predictor
values and segmentation maps (bottom) and overplot the locations of these
galaxies on the top panels. All galaxy image panels are
$80\aas@@fstack{\prime\prime}0\times 80\aas@@fstack{\prime\prime}0$.

##  3 Methods

With predictor values in hand for 1.344 million galaxies, we are ready to
classify the galaxies using the LDA imaging classification technique (Nevin et
al. 2019). We review the classification technique and discuss some relevant
changes in §3.1. We describe how we further split the classification by merger
stage in §3.2. We apply the different classifications to the measured
predictor values in §3.3 and describe how we account for all possible merger
priors in §3.4, which is critical for the direct comparison of
$p_{\mathrm{merg}}$ values across the different classifications as well as the
calculation of the merger fraction. We present the MergerMonger suite in §3.5.
Finally, we describe how we create a mass-complete sample in §3.6.

###  3.1 Review of the LDA merger identification technique

The merger classification technique is built on a Linear Discriminant Analysis
(LDA) framework that is trained to separate mock images of simulated
nonmerging from merging galaxies using their imaging predictors. The full
details of the technique are presented in Nevin et al. (2019) and Nevin et al.
(2021) (henceforth, N19 and N21). N19 presents the imaging side of the
approach, and N21 presents the kinematic side of the approach and some
relevant changes to the N19 method. Here we will briefly review the results of
these earlier papers.

The classification was trained using a suite of five SUNRISE/GADGET-3
simulations of merging galaxies. The galaxies in this suite are best described
as initially disk-dominated intermediate mass galaxies ($3.9-4.7\times
10^{10}\ \mathrm{M}_{\odot}$). They span a range of stellar mass ratios
($\mu_{*}={0.1,0.2,0.333,0.333,0.5}$), have gas fractions of 0.1 and 0.3, and
have initial bulge-to-total-mass ratios of 0 and 0.2. While the simulated
training set is limited in morphological parameter space, this does not
significantly affect our main results (see §5.6).

Each simulation spans 3-10 Gyr and contains a total of 100-200 snapshots in
time, with a spacing of $\sim$10 Gyr. For each snapshot in time, we sample the
merger at seven isotropically spaced viewpoints. We show example snapshots
from the $\mu_{*}=0.5$ major merger and the $\mu_{*}=0.1$ minor merger in
Figure 2, where $\mu_{*}$ is the stellar mass ratio of the two merging
galaxies.

In order to build the classification, we also required a set of simulated
nonmerging galaxies which consist of isolated galaxies that were matched in
gas fraction and stellar mass to each simulated merger as well as merging
snapshots before first pericentric passage and 0.5 Gyr after final coalescence
(pre- and post-merger snapshots).

Figure 2: Snapshots from the $\mu=0.5$ major merger (top row) and $\mu=0.1$
minor merger (bottom row) with merging snapshots in pink and orange,
respectively, and non-merging snapshots in blue. The non-merging snapshots
include the pre-merger snapshots (before first pericentric passage), the post-
merger snapshots ($>0.5$Gyr after coalescence), and the matched isolated
galaxies (right column), which are matched to the initial conditions of each
merger simulation in mass and gas fraction.

We created mock images from the simulated galaxies that match the
specifications of SDSS $r-$band images and measured the seven imaging
predictors from the mock images. We trained seven separate LDA classifiers to
identify mergers (one each for the five simulations and one each for a
combined major and combined minor merger simulation).

Relevant details of the LDA classification include:

  * •

The LDA relies on a prior to correct for the larger fraction of merging
relative to nonmerging galaxies in the simulations. In N19, we use fiducial
merger fraction priors of $f_{\mathrm{merg}}=0.1$ and 0.3 for the major and
minor merger classifications, respectively. We explore how changing the merger
fraction prior affects our measured posterior merger fraction in §4.7.

  * •

We include interaction terms to explore correlations between predictors.

  * •

We use $k$-fold cross-validation to obtain 1$\sigma$ errors on the predictor
coefficients and to measure the performance statistics of the classifications.

  * •

In order to select which coefficients are necessary for the classification, we
use a forward step-wise selection technique, which orders and includes only
the relevant terms and interaction terms.

For complete details, including the full mathematical formulation for the LDA,
see N19 and N21.

There are two key differences between the imaging LDA presented in N19 and the
classification we use in this work that result in slightly different merger
classifications and performance metrics. First, updates to the scikit-learn
software (we are now using version 0.24.2, Pedregosa et al. 2011) including
bug fixes and enhancements to the modeling logic result in classifications
with different coefficients, terms, and slightly different performance
metrics. Second, the training sets are slightly different from those used in
N19; in N21 and here, we use the predictor values from all of the simulated
snapshots that have measured values of imaging and kinematic predictors.

After rerunning the analysis from N19 with all of the above updates, the major
merger classification is:

$\begin{split}\mathrm{LD1}_{\mathrm{major}}=&{\color[rgb]{0,0,1}+13.9\
A_{s}}{\color[rgb]{1,0,0}-8.0\ C*A_{s}-5.4\ A*A_{s}}{\color[rgb]{0,0,1}+5.1\
A}\\\ &{\color[rgb]{0,0,1}+4.8\ C}{\color[rgb]{1,0,0}-2.9\
Gini*A_{s}}{\color[rgb]{0,0,1}+0.6\ M_{20}*A}\\\ &{\color[rgb]{0,0,1}+0.4\
M_{20}*n+0.4\ Gini}-0.6\end{split}$ (1)

Terms with positive/negative contributions to the LD1 value are blue/red.

The minor merger classification is:

$\begin{split}\mathrm{LD1}_{\mathrm{minor}}=&{\color[rgb]{1,0,0}-10.4\
C*A_{s}}{\color[rgb]{0,0,1}+8.8\ C*A}{\color[rgb]{1,0,0}-7.8\ Gini*S-7.8\
A}\\\ &{\color[rgb]{0,0,1}+6.6\ A_{s}+6.5\
Gini*M_{20}}{\color[rgb]{1,0,0}-6.0\ M_{20}*S}\\\ &{\color[rgb]{1,0,0}-5.7\
M_{20}*A_{s}}{\color[rgb]{0,0,1}+4.9\ S}{\color[rgb]{1,0,0}-4.4\
M_{20}}{\color[rgb]{0,0,1}+3.7\ Gini*C}\\\ &{\color[rgb]{1,0,0}-2.9\ S*n-1.0\
n*A_{s}-0.2\ A*S}-0.7\end{split}$ (2)

We present the four leading coefficients for each LDA run alongside their
uncertainties in Table 2.

We quantify the observability timescales and performance metrics for the LDA
classifications using the cross-validation set of simulated mergers. We
measure the observability timescale by applying each classification to the
corresponding simulation and determining the length of time where the average
LD1 value for consecutive snapshots is greater than zero. The observability
timescale of the major/minor merger classifications is 2.31/5.36 Gyr. It is
important to emphasize that the observability timescale is a performance
metric that is measured by applying the derived LDA classifications applied to
the simulated images. This is why the observability timescales from the early
and late stage classifications do not sum to the observability timescale of
the pre-coalescence classification.

Accuracy ($A$) is the fraction of true positive (TP) and true negative (TN)
classifications relative to all classifications:

$A=\frac{TP+TN}{TP+TN+FP+FN}$

where FP are false positive and FN are false negative classifications.

Precision ($P$) quantifies the fraction of true positive classifications to
all positive classifications:

$P=\frac{TP}{TP+FP}$

Recall is also known as the completeness and quantifies the ability of the
classifier to retrieve mergers:

$R=\frac{TP}{TP+FN}$

The F1 score is the harmonic mean of precision and recall:

$F1=\frac{2TP}{2TP+FN+FP}$

The major merger combined simulation has an accuracy of 0.86, a precision of
0.96, and a recall of 0.83. The minor merger combined simulation has an
accuracy of 0.77, a precision of 0.93, and a recall of 0.63. We present these
performance metrics and the observability timescales for all classifications
in Table 3.

Classification | Term 1 | Term 2 | Term 3 | Term 4
---|---|---|---|---
All Major Mergers | 13.9 $\pm$1.0 $A_{s}$ | -8.0 $\pm$ 0.7 $A_{s}*C$ | -5.4 $\pm$ 0.4 $A_{s}*A$ | 5.1 $\pm$ 0.4 $A$
Major, pre-coalescence | 10.0 $\pm$ 0.6 $A_{s}$ | 7.5 $\pm$ 0.2 $A$ | -6.3 $\pm$ 0.2 $A_{s}*A$ | -6.1 $\pm$ 0.5 $A_{s}*C$
Major, early stage | 9.1 $\pm$ 0.4 $A_{s}$ | -5.8 $\pm$ 0.4 $A_{s}*C$ | 5.3 $\pm$ 0.6 $C$ | 4.9 $\pm$ 0.5 $A$
Major, late stage | -8.9 $\pm$ 0.8 $A_{s}*A$ | 7.9 $\pm$ 0.4 $A_{s}$ | 7.2 $\pm$ 0.7 $Gini*A$ | 1.2 $\pm$ 0.2 $A*S$
Major, post-coalescence (0.5) | -10.8 $\pm$ 0.9 $A_{s}*M_{20}$ | 10.1 $\pm$ 1.1 $C*Gini$ | -10.0 $\pm$ 1.1 $A_{s}*C$ | 5.0 $\pm$ 0.9 $Gini*M_{20}$
Major, post-coalescence (1.0) | -14.3 $\pm$ 0.9 $C*n$ | 11.7 $\pm$ 1.4 $C$ | 5.9 $\pm$ 0.9 $Gini*n$ | -1.3 $\pm$ 0.2 $A_{s}*M_{20}$
All Minor Mergers | -10.4 $\pm$ 1.9 $A_{S}*C$ | 8.8 $\pm$ 0.7 $A*C$ | -7.8 $\pm$ 3.3 $Gini*S$ | -7.8 $\pm$ 0.6 $A$
Minor, pre-coalescence | -31.3 $\pm$ 7.7 $Gini*S$ | -28.6 $\pm$ 6.0 $Gini*n$ | 27.4 $\pm$ 5.7 $n$ | 21.0 $\pm$ 2.8 $C$
Minor, early stage | 20.8 $\pm$ 3.6 $C$ | -20.5 $\pm$ 5.4 $Gini*C$ | -18.0 $\pm$ 2.2 $n*M_{20}$ | -16.7 $\pm$ 2.2 $n*C$
Minor, late stage | 10.1 $\pm$ 1.4 $A_{s}*C$ | -5.3 $\pm$ 1.0 $A_{s}*Gini$ | 1.9 $\pm$ 0.1 $A_{s}*A$ | –
Minor, post-coalescence (0.5) | 2.3 $\pm$ 0.2 $A_{s}$ | – | – | –
Minor, post-coalescence (1.0) | 2.0 $\pm$ 0.1 $Gini$ | -1.1 $\pm$ 0.1 $A*S$ | 0.6 $\pm$ 0.1 $n$ | –
Table 2: The four leading coefficients and terms and for each classification.
The LD1 value for each classification is constructed by multiplying the
standardized predictor value by each coefficient and summing all terms. We
distinguish between the post-coalescence classifications with a 0.5 Gyr and
1.0 Gyr cutoffs after coalescence.

###  3.2 Classifying by Merger Stage

Classification | Accuracy | Precision | Recall | F1 | $t_{\mathrm{obs}}$
---|---|---|---|---|---
All Major Mergers | 0.86 | 0.96 | 0.83 | 0.89 | 2.31
Major, pre-coalescence | 0.87 | 0.96 | 0.83 | 0.89 | 2.16
Major, early stage | 0.86 | 0.95 | 0.78 | 0.86 | 1.72
Major, late stage | 0.94 | 0.97 | 0.84 | 0.90 | 0.83
Major, post-coalescence (0.5) | 0.84 | 0.89 | 0.65 | 0.75 | 0.40
Major, post-coalescence (1.0) | 0.90 | 0.94 | 0.85 | 0.89 | 1.26
All Minor Mergers | 0.77 | 0.93 | 0.63 | 0.75 | 5.36
Minor, pre-coalescence | 0.80 | 0.89 | 0.71 | 0.79 | 5.75
Minor, early stage | 0.83 | 0.89 | 0.73 | 0.80 | 3.11
Minor, late stage | 0.93 | 0.79 | 0.79 | 0.79 | 5.85
Minor, post-coalescence (0.5) | 0.85 | 0.53 | 0.60 | 0.56 | 0.19
Minor, post-coalescence (1.0) | 0.85 | 0.84 | 0.71 | 0.77 | 0.96
Table 3: Accuracy, precision, recall, F1 score, and observability timescale
for each classification measured from the cross-validation sample of simulated
mergers.

In N19, the classification is applied to the entire duration of the merger
(from early to post-coalescence stages). In this work, we further split the
classification into multiple different stages (pre-coalescence, further
subdivided into early and late, and post-coalescence). Splitting the
classification by merger stage will enable other work to address if and how
galaxy mergers drive time-dependent evolutionary processes.

Our definitions of merger stage are based on previous theoretical and
observational work that define merger stages using both morphological and
evolutionary (i.e. star formation) properties. Moreno et al. (2015) establish
a sequence of merger stages for the pre-coalescence stages of the merger based
on triggered star formation: a) Incoming, b) First pericentric passage, c)
Apocenter, and d) Second approach. Other theoretical work to identify mergers
in cosmological simulations such as IllustrisTNG is limited in temporal
sampling and tends to distinguish more coarsely between pre-coalescence and
post-coalescence mergers, where the time since merger varies based on the
study (Hani et al. 2020; Bickley et al. 2021).

Observational work most often defines merger stage based on projected
separation. Ellison et al. (2013) distinguish between pre- and post-
coalescence mergers in a sample of 10,800 spectroscopic close pairs in SDSS,
where pre-coalescence mergers have projected separations less than 80 kpc. Pan
et al. (2019) define a merger sequence based on morphological disturbance and
separation; 1) well-separated pairs without disturbance, 2) close pairs with
strong interaction signs, 3) well-separated pairs with weak distortion
(apocenter), and 4) strong distortion (final coalescence) and single galaxies
with morphological remnants from merging (post-mergers).

We divide our classification into pre- and post-coalescence stages to match
the methodology of cosmological merger identification schemes. The early and
late stages roughly correspond to the stages from Moreno et al. (2015) and Pan
et al. (2019) of first pericentric passage and apocenter (early) and final
approach (late). We also implement a sliding timescale for the definition of
the post-coalescence stage; we use the time cutoff of 0.5 Gyr after
coalescence and then additionally implement a time cutoff of 1 Gyr. The 1 Gyr
cutoff is motivated by the work of Bickley et al. (2021), who find that the
morphology of IllustrisTNG galaxies is disturbed for up to 2.5 Gyr following a
merger.

To reconstruct the separate classifications, we eliminate all merger snapshots
that are not from the stage in question. For example, for the major merger
combined early stage classification, we eliminate all of the merger snapshots
belonging to the late and post-coalescence stages, but we retain the pre- and
post-merger snapshots as examples of nonmergers. In this way, we are training
the classification to recognize traits of a specific stage while discouraging
it from learning a strict cutoff between stages.

It is important to mention that since the merger stage classifications are all
trained separately, there may be overlap between stages, i.e. certain galaxies
will have high probabilities of belonging to multiple merger stages. We
discuss how to directly compare $p_{\mathrm{merg}}$ values from different
classifications in §4.3 and quantify this overlap in §4.7.

We present the accuracy, precision, recall and F1 score for the new
classifications in Table 3 and the four leading coefficients for each new
classification in Table 2.

###  3.3 Classifying SDSS image cutouts

The next step is to measure the LD1 values for each SDSS galaxy and to assign
each galaxy a probability of merging for each merger classification. To
calculate LD1 for each galaxy, we standardize the measured predictor values
using the mean and standard deviation for each classification. We then
determine the value of LD1 for each galaxy by summing the coefficients and
standardized predictor values for each classification. We present a schematic
of this process in Figure 3, which demonstrates how this process works for one
example image for the major merger combined classification.

Figure 3: Schematic showing the classification steps for an example galaxy
(left). Our first step is to measure the imaging predictor values (top
middle). We then standardize these values and plug them into each LD1 formula.
We show this (top right) for the major merger classification. The LD1 value
for this galaxy is 4.781, which places it to the right of the decision value
in the histogram of LD1 values (right). Our final step is to assign each
galaxy a probability value (bottom left).

We assign a probability of merging to each galaxy. From N19, the probability
of a galaxy belonging to the merging class is:

$p_{\mathrm{merg}}=\frac{e^{\hat{\delta}_{\mathrm{merg}}}}{e^{\hat{\delta}_{\mathrm{merg}}}+e^{\hat{\delta}_{\mathrm{nonmerg}}}}$
(3)

where $\hat{\delta}_{\mathrm{merg}}$/$\hat{\delta}_{\mathrm{nonmerg}}$ is the
score of a galaxy for the merging/nonmerging class.

Linear discriminant axis 1, or LD1, can be written in terms of
$\hat{\delta}_{\mathrm{merg}}$ and $\hat{\delta}_{\mathrm{nonmerg}}$:

$\mathrm{LD1}=\hat{\delta}_{\mathrm{merg}}-\hat{\delta}_{\mathrm{nonmerg}}$
(4)

where the decision boundary is at LD1 = 0 and if
$\hat{\delta}_{\mathrm{merg}}>\hat{\delta}_{\mathrm{nonmerg}}$, then the
galaxy will be classified as merging.

Using equation 4, equation 3 can be re-written in terms of LD1:

$p_{\mathrm{merg}}=\frac{1}{1+e^{-\mathrm{LD1}}}$ (5)

For the 1344677 galaxies in SDSS DR16, we calculate the value of LD1 and the
merger probability for the major and minor merger classifications and for all
of the stage-specific classifications (early/late/pre-coalescence/post-
coalescence). We present these results in §4.1.

###  3.4 Marginalizing the calculation of the merger fraction over all merger
priors

Critical to this paper is a discussion of the merger fraction priors ($\pi$)
that are incorporated into the calculation of the $p_{\mathrm{merg}}$ values.
In N19, we adopt a fiducial merger fraction prior of $\pi=0.1$ for the major
merger classifications and $\pi=0.3$ for the minor merger classifications,
meaning that we expect 10% and 30% of galaxies in the local Universe to be
experiencing major and minor mergers, respectively. These priors are based on
observations and simulations (e.g. Rodriguez-Gomez et al. 2015; Lotz et al.
2011; Conselice et al. 2009; López-Sanjuan et al. 2009; Shi et al. 2009;
Bertone & Conselice 2009).

The fiducial priors are used to measure the LD1 and the $p_{\mathrm{merg}}$
values in the previous section. The choice of this input prior affects the
distribution of LD1 and $p_{\mathrm{merg}}$ values for the full sample and
therefore also affects the individual values. It is therefore particularly
important to consider which $\pi$ value is used when comparing
$p_{\mathrm{merg}}$ values between classifications and when calculating the
merger fraction $f_{\mathrm{merg}}$, which is the focus of this paper.

To approach the comparison of $p_{\mathrm{merg}}$ values and the calculation
of $f_{\mathrm{merg}}$ in the cleanest and most agnostic (to input prior) way
possible, we perform a Bayesian marginalization where we re-calculate the
$p_{\mathrm{merg}}$ values for all possible input priors in a range
$0.05<\pi<0.5$ (we fully justify this range of priors in §4.7). The
implication is that we redo the previous $p_{\mathrm{merg}}$ calculation for
46 different input priors, returning 46 different $p_{\mathrm{merg}}$ values
for each galaxy in SDSS. From these, we calculate the 16th, 50th (median), and
84th percentile of the posterior distribution for each galaxy, which we
present in §4.1. We present the results for the overall merger fraction
calculation based on these measurements in §4.7.

###  3.5 The MergerMonger Suite

We prepare a suite of tools
(MergerMonger)111https://github.com/beckynevin/MergerMonger that applies the
LDA method to classify major and minor merging galaxies from optical images.
MergerMonger includes four main utilities:

  1. 1.

GalaxySmelter: A tool for measuring imaging predictors from simulated or
observed galaxy images.

  2. 2.

Classify: A tool that creates the LDA classification using the predictor
values from the simulated training set.

  3. 3.

MergerMonger: A tool that applies the LDA classification to observed galaxies,
measuring merger probabilities.

  4. 4.

Utilities that help with the interpretation of the predictor and probability
values for each galaxy.

In this work we apply the MergerMonger suite to SDSS $r-$band imaging.
However, the classification is designed with broader use in mind. The
classification can be re-created using new sets of simulated images (i.e.
simulated images created to match the specifications of LSST or DESI imaging)
or new imaging filters. For example, to apply the classification to LSST
images, one could design their own set of mock LSST mergers and extract the
training data using GalaxySmelter. Then, they could use Classify to train
their own LDA classification and finally classify LSST galaxies using
MergerMonger.

###  3.6 Galaxy stellar masses

To measure the stellar masses for the SDSS galaxies, we use the empirical
relation from Bell et al. (2003) that relates the SDSS $u$, $g$, $r$, $i$, and
$z$ band luminosities and colors to the stellar mass-to-light (M/L) ratio
using the k-correction: $log_{10}(M/L)=a_{\lambda}+(b_{\lambda}\times color)$,
where the color in units of AB magnitudes and the luminosity is in solar
units. We use the values for SDSS $g-r$ color because Du et al. (2019) find
that the $g-r$ color provides an almost unbiased $M/L$ value for many
different galaxy types and regions. We use the $a_{r}=-0.840$ and
$b_{r}=1.654$ from Zibetti et al. (2009), which incorporates an TP-AGB star
correction and revised SFHs for bursty galaxies, improving upon the
prescription from Bell et al. (2003).

To conduct this calculation, we rely on photometric-based redshifts, which are
available for the full SDSS sample (1035607 available photometric redshifts
versus 437094 spectroscopic redshifts). In Appendix A we further explore the
differences between using photometric and spectroscopic redshifts to determine
the stellar mass. Although there are biases inherent to using the photometric-
based redshifts (especially at low redshift), we find that our results remain
unchanged when we measure the merger fraction as a function of redshift
(§4.8).

Our method for measuring stellar mass shows good agreement with the SED-based
approach of Mendel et al. (2014), which uses a stellar population synthesis
approach to measure the stellar mass using SDSS SEDs and Sérsic models of the
bulge and disk components. We present this comparison in Figure 4, where the
mean stellar masses agree above a stellar mass of $\sim 10^{9}$.

Figure 4: Comparing the stellar masses derived from the Mendel et al. (2014)
method (x-axis) to those derived from using the empirical color method from
Zibetti et al. (2009).

Next, we determine the mass completeness limit as a function of redshift using
the technique from Darvish et al. (2015). For each redshift bin222We use the
redshift bins presented in §4.8., we compute the lowest stellar mass
($M_{\mathrm{lim}}$) that could be detected for each galaxy given the
magnitude limit of SDSS ($r=17.77$):
$log(M_{\mathrm{lim}})=log(M)+0.4\times(r-17.77)$, where $r$ is the apparent
(rest-frame) $r-$band magnitude of each galaxy and $M$ is the stellar mass.
The mass completeness limit at each redshift bin is the mass at which 95% of
the limiting masses are below the mass completeness limit, meaning that only
5% of galaxies would be missed in the lowest mass end of the mass function.

Our final step is to eliminate all galaxies below the mass completeness limit
at each redshift bin. We show this process in Figure 5. This reduces our
sample by roughly a factor of three from 958840 photometrically clean galaxies
with measured masses to 362216 galaxies in a mass-complete sample. The factor
of $\sim$3 reduction in sample size induced by the mass completeness
correction is similar to the sample reduction in Cebrián & Trujillo 2014,
which applies a similar mass completeness correction to the NYU-VAGC catalog
of SDSS DR7 galaxies.

Figure 5: Mass completeness as a function of redshift for redshift bins with
spacing $\Delta z=0.02$. For each redshift bin, we determine the 95%
completeness limit (pink line) and eliminate all galaxies below this point.
For the distribution of masses at each redshift bin, see Appendix A.

##  4 Results

We present the classification results in §4.1, and provide a guide for
interpreting the predictors that influence the classification in §4.2. We also
provide a guide for deciding between merger stages and types in §4.3 and a
guide for dealing with cases where by-eye classification and the LDA
classification are in conflict in §4.4.

We then analyze the properties of the merger sample in §4.5 and compare our
results to previous SDSS merger selections in §4.6.

We constrain the observed merger fraction using all of the different merger
classifications in §4.7 and explore how the major merger fraction varies as a
function of galaxy mass and redshift in §4.8. We explore if S/N or galaxy
morphology (bulge-to-total mass ratio and color) are confounding the redshift-
dependent major merger fraction in §4.9 and 4.10, respectively. We explore how
the minor merger fraction varies as a function of stellar mass and redshift in
§4.11. We discuss the influence of contamination of the major and minor merger
fraction calculations by mergers of the opposite type (minor and major,
respectively) in §4.12. We run numerous sanity checks in §4.13 (more details
can be found in Appendix B) to confirm the main result of how the major merger
fraction trends with mass and redshift. Finally, we end with a discussion of
the importance of mass binning to our result in §4.14, where we find a
different result in the absense of mass binning.

|  |  |  |  |  |  |  |  |  | low S/N
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  | / outlier predictor /
ID | LD1 | $p_{merg}$ | CDF | Leading term 1 | Leading coef 1 | Leading term 2 | Leading coef 2 | Leading term 3 | Leading coef 3 | segmap
1237665179521187863 (A) | -4.137 | 0.016 | 0.510 | $A_{s}$ | -11.3 | $A$ | -3.8 | $C$ | -0.5 | 0/0/0
1237661852010283046 (B) | 4.781 | 0.992 | 0.919 | $A_{s}$ | 31.8 | $A$ | 5.2 | $Gini$ | 0.9 | 0/0/0
1237648720718463286 (C) | 2.081 | 0.889 | 0.839 | $A_{s}$ | 39.4 | $A$ | 12.4 | $n*M_{20}$ | 0.5 | 0/0/0
1237662306186428502 (D) | 4.235 | 0.986 | 0.907 | $A_{s}$ | 17.9 | $A$ | 2.2 | $n*M_{20}$ | 0.2 | 0/0/0
1237653589018018166 (E) | 1.784 | 0.856 | 0.830 | $A_{s}$ | 6.9 | $A_{s}*A$ | 2.2 | $A*M_{20}$ | 0.2 | 0/0/0
1237654383587492073 (F) | 0.678 | 0.663 | 0.792 | $A$ | 16.0 | $A_{s}*C$ | 9.1 | $A_{s}*Gini$ | 2.4 | 0/0/0
Table 4: Classification results for the six galaxies presented in Figure 1.
Here we provide the LD1 value and corresponding $p_{\mathrm{merg}}$ value for
the major merger classification. We also list the leading (most influential)
term in each classification and the contribution from this term, which is the
product of the standardized predictor value and the LD1 coefficient for that
term. We bold the classifications where a galaxy is classified as a merger
($p_{\mathrm{merg}}>0.5$). The online-available tables provide these values
for all six merger classifications (major, major pre-coalescence, major post-
coalescence, minor, minor pre-coalescence, and minor post-coalescence).

###  4.1 LDA classification results

Here we present three data products:

  1. 1.

For each galaxy in the 1,344,577 DR16 sample, we provide all of the predictor
values and the flag values. This table was previously described in §2 and
presented in Table 1.

  2. 2.

For each merger classification, we provide the fiducial LD1,
$p_{\mathrm{merg}}$, and CDF (described below) values for each galaxy in the
1,344,577 SDSS DR16 galaxy sample accompanied by explanatory information such
as the most important (leading) terms in the classification and the
coefficients associated with these leading terms. Our intent is that these
tables can be used to ascertain why a galaxy is classified as a merging or
non-merging galaxy according to the different fiducial classifications. We
describe how this explanatory analysis might work in §4.2. Table 4 presents
the major merger classification results for the six galaxies from Figure 1.

  3. 3.

We also provide a table (Table 5) that presents the 16th, 50th, and 84th
percentile of the posterior $p_{\mathrm{merg}}$ distribution (and accompanying
CDF value) for all photometrically clean galaxies (958,840) from the
marginalization analysis described in §3.4. This single table includes these
results for all of the merger classifications. Using this table, the user can
directly compare $p_{\mathrm{merg},50}$ values across different
classifications.

In Figure 6, we present histograms of the fiducial LD1 values and the
corresponding $p_{\mathrm{merg}}$ values for the training set of simulated
galaxies and the SDSS galaxies classified by the major and minor merger
classifications. Since the LDA technique is designed to find the hyperplane of
maximal separation between two populations, the distribution of probability
values in the bottom panels of Figure 6 peak very near to 0 and 1. This makes
direct interpretation of these probability values very difficult. We therefore
provide a complementary cumulative distribution function (CDF) analysis (which
is part of data products 2 and 3) to compare individual $p_{\mathrm{merg}}$
values to the the $p_{\mathrm{merg}}$ values of all SDSS galaxies for a given
classification. For instance, if we examine the major merger classifications
in Table 4, galaxy A has a $p_{\mathrm{merg}}$ value of 0.016, which
corresponds to an CDF value of 0.510, meaning that 51% of galaxies in SDSS
have a lower $p_{\mathrm{merg}}$ value. In Table 6, we list the
$p_{\mathrm{merg}}$ values that correspond to the 5%, 10%, 90%, and 95% values
of the CDF for the fiducial merger classifications.

|  |  $p_{\mathrm{merg,16}}$/$p_{\mathrm{merg,50}}$/$p_{\mathrm{merg,84}}$ (CDF)
---|---|---
ID | Type | All | Pre-coalescence | Post-coalescence (1.0)
1237648720718463286 | Major | 0.84/0.99/1.0 (0.96) | 0.67/0.88/0.99 (0.85) | 0.0/1.0/1.0 (0.99)
| Minor | 0.0/0.12/1.0 (0.51) | 0.0/0.04/0.84 (0.44) | 0.46/1.0/1.0 (0.98)
1237653589018018166 | Major | 0.79/0.88/0.92 (0.89) | 0.81/0.89/0.94 (0.85) | 0.88/0.97/0.99 (0.83)
| Minor | 0.88/0.95/0.98 (0.88) | 0.88/0.96/0.99 (0.87) | 0.74/0.93/1.0 (0.74)
1237654383587492073 | Major | 0.52/1.0/1.0 (0.98) | 0.96/1.0/1.0 (0.97) | 0.0/0.0/0.0 (0.0)
| Minor | 0.0/0.0/0.91 (0.18) | 0.0/0.0/1.0 (0.17) | 0.1/0.89/1.0 (0.7)
1237661852010283046 | Major | 0.93/0.98/0.99 (0.94) | 0.99/1.0/1.0 (0.94) | 0.02/1.0/1.0 (0.92)
| Minor | 0.04/0.89/1.0 (0.85) | 0.33/0.98/1.0 (0.89) | 0.19/1.0/1.0 (1.0)
1237662306186428502 | Major | 0.99/1.0/1.0 (0.98) | 0.99/1.0/1.0 (0.95) | 0.98/1.0/1.0 (0.93)
| Minor | 0.78/0.98/1.0 (0.91) | 0.71/0.99/1.0 (0.91) | 0.63/0.98/1.0 (0.84)
1237665179521187863 | Major | 0.03/0.09/0.17 (0.56) | 0.01/0.02/0.07 (0.51) | 0.29/0.63/0.76 (0.68)
| Minor | 0.13/0.36/0.56 (0.67) | 0.18/0.37/0.57 (0.68) | 0.17/0.46/0.62 (0.55)
Table 5: Marginalized $p_{\mathrm{merg}}$ values and accompanying CDF values
for the six galaxies from Figure 1. We list the $p_{\mathrm{merg}}$
corresponding to the 16th, 50th (median), and 84th percentiles of the
marginalized posterior $p_{\mathrm{merg}}$ distributions for each galaxy using
each classification. We also list the CDF value that corresponds to the 50th
percentile in parenthesis. For each galaxy, we list only the combined
minor/major merger classifications and the pre- and post-coalescence (1.0)
results. In the online-available table, we also include the early, late, and
post-coalescence (0.5) results. In the online-available table, each of the
16/50/84 percentile values is its own column.

Figure 6: Distribution of LD1 values for the simulated suite (top panel) and the SDSS sample (middle panel) and the corresponding distribution of $p_{\mathrm{merg}}$ values for the SDSS sample (bottom panel) for the major (left) and minor (right) merger classifications. In all cases, the y-axis is the number of galaxies. In the bottom panels, we zoom in on the distributions and the inset numbers give the number of galaxies in the largest bin.  | CDF threshold value
---|---
Classification | 0.01 | 0.05 | 0.1 | 0.25 | 0.5 | 0.75 | 0.9 | 0.95 | 0.99
Major merger | 3.2e-8 | 1.6e-7 | 3.2e-7 | 7.4e-7 | 0.01 | 0.39 | 0.9891260 | 0.999999353 | 0.9999998720
Minor merger | 5.6e-8 | 2.8e-7 | 5.7e-7 | 0.02 | 0.24 | 0.79 | 0.996 | 0.99999950 | 0.99999989
Table 6: Probability of merging ($p_{\mathrm{merg}}$) that correspond to
different thresholds of the CDF for all of the merger classifications. This
table is provided to enable user interpretation of individual
$p_{\mathrm{merg}}$ values, which evolve exponentially and their
interpretation can be assisted with careful consideration of the CDF values.
For instance, if a galaxy has a $p_{\mathrm{merg}}$ value of 0.01 for the
major merger classification, this corresponds to a CDF value of 0.5, meaning
that about half of our SDSS sample is more likely to be a non-merger.

Finally, we provide visual examples of a randomly selected sample of merging
galaxies (Figure 7) and non-merging galaxies (Figure 8) according to the
fiducial major merger LDA classification.

Figure 7: Merging galaxies ($p_{\mathrm{merg}}>0.5$) according to the major
merger LDA technique. The inset panels list the LD1 value and its accompanying
$p_{\mathrm{merg}}$ value and CDF value. All panels are
$80\aas@@fstack{\prime\prime}0\times 80\aas@@fstack{\prime\prime}0$. Figure 8:
Non-merging galaxies ($p_{\mathrm{merg}}<0.5$) according to the major merger
LDA technique. The inset panels are described in Figure 7.

###  4.2 A guide for interpreting classification results

The LDA classification method was designed with the interpretability of
individual results as one of its central goals. In this section, we discuss
how to use the additive linear terms that compose LD1 to understand why a
galaxy is classified as merging or non-merging. To assist users with this
interpretation, we provide Table 4, which lists the $p_{\mathrm{merg}}$ and
CDF values for the major merger classification for individual galaxies
alongside the most influential predictors and coefficients.

We include an utility within MergerMonger that calculates CDF values for
$p_{\mathrm{merg}}$ values and vice versa. This is useful if the user wants to
create a ‘superclean’ merger sample that has minimal non-merger contamination.
They could either do this by defining an CDF threshold or by deciding on a
$p_{\mathrm{merg}}$ threshold (i.e. $p_{\mathrm{merg}}>0.95$) and using Table
6 or the MergerMonger utility to determine the corresponding
$p_{\mathrm{merg}}$ or CDF value. It is then possible to re-run the LDA
classifications using MergerMonger and a different $p_{\mathrm{merg}}$ value
as the threshold to identify mergers.

We also provide a diagnostic tool within MergerMonger (find_galaxy.py) that
accepts single or multiple galaxy SDSS Object ID(s) as input. This utility
then presents the predictor values, the most influential predictors in the
classification, and the classification results in a diagnostic diagram that
includes the individual galaxy image and segmentation map. We show an example
of two diagnostic diagrams in Figure 9 for the major (top) and minor (bottom)
merger classifications for galaxy F from Figure 1.

This galaxy is classified as a merger by both major and minor merger fiducial
classifications, with high LD1 and corresponding $p_{\mathrm{merg}}$ values in
the upper left informational panel. The lower panel on the left hand image
lists the three leading terms and their corresponding contribution to the
value of LD1; here, shape asymmetry and asymmetry are important predictors for
both classifications. The inset informational panel for the right hand
segmentation maps lists all of the pre-standardized predictor values.

These diagnostic diagrams can help the user interpret why the classifications
have determined that this galaxy is likely to be a merger. Looking first at
the major merger panels, shape asymmetry followed by the $A_{s}*A$ cross term
are the most influential terms. In the right panel, the asymmetry for this
galaxy is low while the shape asymmetry is high. This is due to the low
surface brightness of the shell feature. Since the coefficient of the $A_{s}$
term is positive in Equation 1, this boosts the LD1 score. The coefficient of
the $A_{s}*A$ term is negative in Equation 1. This coefficient will be
multiplied by the standardized $A_{s}$ and $A$ values, which will be positive
and negative respectively (recall, the $A$ value is relatively low). The net
result will be a positive contribution to LD1, meaning that this galaxy is
even more likely to be detected as a merger. In this case, the $A_{s}*A$ term
allows the LDA to better distinguish between asymmetric bright features such
as spiral arms and low surface brightness asymmetric features that are more
likely to be caused by a merger.

For the minor merger classification, the $M_{20}*A_{s}$ cross term is
influential; this term has a negative coefficient in Equation 2, so for this
term to have a large positive influence, either the standardized value of
$M_{20}$ or $A_{s}$ must be very negative (meaning relatively low for SDSS
galaxies). Here, this is because $M_{20}$ is quite negative, meaning that the
light is concentrated. By eye, this galaxy looks like a post-coalescence
merger with a shell from the merger event; the minor merger technique is
relying both upon the high concentration (also measured by $M_{20}$) and the
shell feature to identify it as a merger. This galaxy and others like it
demonstrate that the LDA classification succeeds in the case of concentrated
early-type galaxies.

Figure 9: Diagnostic classification diagrams for the major (top) and minor
(bottom) merger classification results for galaxy E. The three most
influential terms and their contribution to LD1 are given in the bottom left
panels. We describe how to interpret these leading terms in the text.

###  4.3 A guide for distinguishing between merger types and stages

Here we discuss the overlap between different merger stages and types and how
to directly compare $p_{\mathrm{merg}}$ values across different
classifications. Directly comparing $p_{\mathrm{merg}}$ values between the
fiducial runs is not encouraged, especially between minor and major
classifications. These different classifications were prepared assuming
different priors, meaning that the distribution of $p_{\mathrm{merg}}$ values
will be affected by this choice. We also do not recommend directly comparing
the $p_{\mathrm{merg}}$ values from Table 4 between different stages of the
same merger type (i.e. early versus late stage major mergers) because these
tables assume the same fiducial merger prior. As we will show in §B.6, this is
not a safe assumption.

Best practice is therefore to use the marginalized $p_{\mathrm{merg}}$ values
from Table 5 to decide which stage or which merger type is most likely for a
given galaxy. This table includes the $p_{\mathrm{merg}}$ values that
corresponds to the 16th, 50th , and 84th percentile of the posterior
distribution of $p_{\mathrm{merg}}$ for each galaxy for the major, minor, and
pre- and post-coalescence (1.0 Gyr) stages. The online-available table also
includes the early, late and post-coalescence (0.5) stage results.

Here we walk the user through the process of distinguishing between merger
types and stages using Table 5 and the (compare_classifications.py) utility
within MergerMonger, which plots an image of a galaxy and compares the
$p_{\mathrm{merg}}$ values between different classifications.

Using galaxy E from Figure 1 and Table 5, we show a diagnostic diagram in
Figure 10 created using compare_classifications.py as an informative example
for how to decide between merger type and stage for an individual galaxy. The
compare_classifications.py utility decides the most likely classifications in
a hierarchical manner; first, it determines if the galaxy is more likely to be
a major or minor merger by directly comparing the $p_{\mathrm{merg,50}}$
values from each classification. The utility then decides whether the galaxy
is more likely to be a pre-coalescence merger or a post-coalescence (1.0 Gyr)
merger. It does this for both the major and minor classifications. All of
these rankings occur regardless of if the $p_{\mathrm{merg,50}}$ values are
greater than 0.5.

For galaxy E using the major and minor merger $p_{\mathrm{merg,50}}$ values,
we are able to conclude that it is more likely a minor merger. We can then
further distinguish between the minor merger stages, finding that it is more
likely a pre-coalescence minor merger.

In general, we recommend following the hierarchical framework of
compare_classifications.py; first decide between the all-inclusive major and
minor merger classifications and then decide between the sub-stages of each.
We also recommend using the post-coalescence (1.0) classification as opposed
to the post-coalescence (0.5) classification, which has lower performance
statistics due to its short observability timescale. If the use case is to
identify all early-stage major and minor mergers, then we recommend creating a
new process using the code framework of compare_classifications.py that
requires that $p_{\mathrm{merg,50}}$ from the early stage classifications is
greater than the $p_{\mathrm{merg,50}}$ values corresponding to the late and
post-coalescence (1.0) stage classifications. In this case, we recommend
comparing the stages of the major/minor merger classification directly to one
another (i.e. major merger early is compared to major merger late and post-
coalescence).

We also provide the 16th and 84th percentile values if the user wants to
develop a more conservative sample, i.e. requiring that
$p_{\mathrm{merg,major,16}}>p_{\mathrm{merg,minor,84}}$ would be a more
conservative way to compare the classifications. However, there is significant
overlap between different classification samples when using the full range
(16th and 84th percentiles), so we recommend using the 50th percentile
(median) values for simplicity. For instance, in Figure 10, if we were to use
the more conservative technique, all of the classifications and stage-specific
classifications would overlap.

Figure 10: Diagnostic diagram for determining the most likely merger type
(major or minor) and the most likely merger stages for galaxy E from Figure 1.
This diagram is produced by the compare_classifications.py utility. The top
left panel shows the galaxy, segmentation map (yellow), and imaging predictor
values. The top right panel runs through a diagnosis of merger type, beginning
with diagnosing whether $p_{\mathrm{merg,maj,50}}$ or
$p_{\mathrm{merg,min,50}}$ are greater than 0.5. If so, then the galaxy is
identified as a merger. The next step is to identify which is more likely (a
major or minor merger), which is determined using the $p_{\mathrm{merg,50}}$
values from each classification. We provide the $p_{\mathrm{merg,50}}$ values
for all classifications along with the $p_{\mathrm{merg,16}}$ and
$p_{\mathrm{merg,84}}$ values in the following format: $p_{\mathrm{merg,50}}$
($p_{\mathrm{merg,16}}$,$p_{\mathrm{merg,84}}$). Finally, this diagnostic
diagram decides which stage is more likely for the major followed by the minor
merger classifications. Here, the post-coalescence stage is more likely for
the major merger and the pre-coalescence stage is more likely for the minor
merger classification. In the bottom panel, the y-axis is used to order the
classification results, where different colors correspond to the median,
$p_{\mathrm{merg,50}}$ values for each classification and the error bars give
the the range between the 16th and 84th percentile of the $p_{\mathrm{merg}}$
value for each classification.

Note that there is overlap between stages and/or merger types, i.e. there will
be many galaxies that have $p_{\mathrm{merg,50}}$ values that are greater than
0.5 for multiple different classifications. We discuss this overlap in more
detail in §4.7, where we measure the merger fraction.

###  4.4 Interpreting cases where the LDA classification disagrees with by-eye
classification

We acknowledge that as with any merger identification approach that relies on
imaging predictors, the individual classifications may disagree with by-eye
decisions. We therefore recommend that if the user is working with a
relatively small sample they also examine the classifications by eye to
identify potential misclassifications.

A failure mode of the LDA major merger combined classification, for instance,
is classifying equal mass major mergers that happen to be in a symmetric
configuration as non-merging. This happens when the merging galaxies also have
a low overall concentration. This is relatively rare and can be understood by
running the interpretive MergerMonger utilities which reveals which predictors
are responsible for the non-merger classification. We also recommend running
galaxies with surprising classifications through the compare_classification.py
utility in order to examine the results of the different merger stage
classifications. Obvious early or late stage equal mass major mergers might be
classified as having a low probability of being a major merger by the overall
classification (due to the unlikely combination of imaging predictor values)
often have a high probability of being a major merger in the pre-coalescence
stage.

The combined LDA classifications (major and minor) are trained from an
ensemble of images, meaning that they are optimized for high accuracy for all
stages of the merger. The implication is that the combined classifications are
best for determining bulk sample properties such as the overall merger
fraction, while the individual stage classifications may be better suited for
understanding smaller samples of mergers or for cases where determining the
merger stage is important.

###  4.5 Properties of the LDA mergers

The challenge we face in validating our sample of merging galaxies is that
there is no gold standard to rely upon for which galaxies are truly mergers.
We therefore take the approach of checking for large systematic issues by
investigating the global properties of the merger samples. We carry out this
analysis in two parts: first, here we compare the properties of the (mass-
complete) parent sample to those of the merger samples. Second, in §4.6, we
will compare the properties of the merger samples to those of other merger
selection techniques.

In Figure 11 we compare the probability density functions (pdfs) for the major
(pink) and minor (yellow) merger samples to that of the parent SDSS sample
(white) using average S/N, $r-$band magnitude, color ($g-r$), stellar mass,
and redshift. The pdfs are normalized so that all bins from a given
distribution sum to a value of one.

The mergers have properties that span the full range of properties of the
parent distribution. This is a major success when we consider that our
training sample of galaxies was limited in these spaces. For instance, the
training set of galaxies spanned $3.9-4.7\times 10^{10}M_{\odot}$ in stellar
mass and all galaxies in the training set had their surface brightnesses and
apparent sizes adjusted to a redshift of $z=0.03$. The fact that the LDA
techniques identify mergers over a large range in surface brightness, stellar
mass, and redshift indicates that the LDA method is successfully able to
adjust to a wider span of galaxy properties.

Furthermore, we run two-sample Kolmogorov-Smirnov (KS) tests to compare the
cumulative distribution functions (constructed from the pdfs) for each
property and find that the distributions are statistically indistinguishable.
Specifically, we are unable to reject the KS null hypothesis (that the
distributions are identical) when we compare the parent distribution to the
major and minor merger selection and when we compare the major and minor
merger distributions. The implication is that while the major and minor merger
classifications are using different imaging properties to identify mergers,
they are not significantly biased in any of these properties.

This is a massive success of the method; previous studies have uncovered
significant biases, especially related to S/N. For instance, Bickley et al.
(2021) train a Convolutional Neural Network (CNN) to identify post-merger
galaxies in Illustris TNG100. When they test the performance on galaxies in
the Canada-France Imaging Survey, they find a deficit of very faint galaxies
in the post-merger sample. Their merger technique is slightly biased towards
identify more massive, brighter, and higher redshift galaxies (due to the
volume-limited nature of the survey, more massive galaxies are more likely to
appear at higher redshift).

Figure 11: Probability density functions (pdfs) of the properties of the
parent sample of SDSS gal axies (white) compared to properties of the
$p_{\mathrm{merg}}>0.5$ major merger sample (pink) and the minor merger sample
(yellow). All histograms are normalized so that all bins sum to a value of
one. Left to right: the distributions for average S/N ratio, $r-$band
magnitude, color ($g-r$), log stellar mass, and redshift. Using the two-sample
KS test, we confirm that the all distributions are statistically
indistinguishable.

Despite the KS test revealing that the distributions are statistically
indistinguishable, we do notice some slight by-eye differences. The major and
minor classifications have slight excesses at low (brighter) $r-$band
magnitudes compared to the parent distribution. To quantify this, we measure
the offset in the median value of each major/minor distribution compared to
the parent distribution and find values of $\Delta r=0.11/0.12$, where the
major and minor merger distributions are slightly brighter than the parent
distribution. The distributions also differ at low redshift, where the major
and minor merger distributions tend towards lower redshift values ($\Delta
z=0.002/0.005$). In terms of mass, the major mergers tend to have higher
masses ($\Delta logM_{*}(M_{\odot})=0.06$).

The brighter major mergers constitute two populations; one is more massive and
at higher redshift, while one is less massive and at lower redshift. Both of
these populations have slightly lower S/N ratios than the parent sample. These
properties could reflect a slight bias for the merger classifications to
identify galaxies with lower S/N ratios as mergers, which is the opposite bias
as that identified in work such as Bickley et al. (2021). We investigate this
potential bias in more depth in §4.9, where we show that the merger fraction
does increase with decreasing S/N when we control for mass and redshift.
However, we also show in this section that this trend does not change our
finding of a decreasing merger fraction with increasing redshift.

###  4.6 Properties of LDA mergers compared to previous merger samples in SDSS

In order to better understand the biases of our technique, we compare the
mergers selected using the LDA major merger classification with those from two
large SDSS merger samples: GalaxyZoo and the Ackermann et al. (2018) technique
(from here on, A18).

First, we compare our SDSS merger catalog to the GalaxyZoo selection of
mergers in SDSS imaging, which is a large publicly-available catalogs of
mergers (Lintott et al. 2008, 2011). We cross-match the GalaxyZoo catalog from
DR8 (893,163 galaxies) with our clean DR16 sample and find 570,455 matches.
The GalaxyZoo catalog provides $p$, or probability values, for four
morphological categories (mergers, ellipticals, combined spirals, and ‘don’t
know’), corresponding to the percentage of users that selected each
morphological category. We identify the morphological category with the
highest $p$ value for each galaxy. We then identify the number of galaxies in
each category that have a fiducial major merger probability greater than 0.5
from our classification. We use the major merger classifications from our
technique for comparison because the GalaxyZoo classifications are based on
visual inspection, which is more likely to identify the more obvious major
mergers.

The results are as follows: for the GalaxyZoo merger category, 6626/10433
(64%) are LDA major mergers, for the combined spirals, 25467/176213 (14%) are
LDA major mergers, for the ellipticals, 54431/378993 (14%), and for the
ambiguous category, 1413/4816 (29%). We also build a ‘clean’ sample of
GalaxyZoo mergers, where the fraction of users that classify galaxies as
mergers is greater than 95%. Of these, 30/34 (88%) are classified as mergers
by our classification.

These results are reassuring in two ways: first, the LDA classification
returns $\sim$2/3 of mergers identified in GalaxyZoo, and second, the fraction
of spirals and ellipticals that are identified as mergers by the LDA method
are not significantly different. This tells us that the LDA method is not
significantly biased as a function of galaxy morphology.

We visually inspect mergers according to GalaxyZoo that we classify as
nonmergers and find that many of them can be described as double nuclei
galaxies without noticeable tidal tails. Some of these galaxies may be
nonmergers that are superimposed along the line of sight and some of them may
be very early stage mergers (approaching for a first encounter) or gas poor
mergers. For these galaxies, the most important major merger predictors, such
as shape asymmetry, have low values, resulting in a non-merger identification
from the LDA technique. We discuss this particular failure mode of the LDA
classification in §4.4.

We next compare the properties of mergers from our classification technique to
those identified in the SDSS sample using the A18 technique, which uses
transfer learning to retrain a convolutional neural network (CNN) on the Darg
et al. (2010) sample of merging galaxies (from GalaxyZoo). A18 use the 3003
merger objects from Darg et al. (2010) as merger examples ($0.005<z<0.1$) and
10,000 GalaxyZoo galaxies with $f_{m}<0.2$ as examples of nonmergers, where
$f_{m}$ is the fraction of users who identify a galaxy as a merger. We cross-
match the results from the A18 catalog, which is mass complete down to
$10^{10}M_{\odot}$, with those of our mass-complete LDA classifier (we
calculate completeness as a function of redshift, Figure 5), and find an
overlap of 98,645 galaxies. From these, we use the same method as A18 to
identify galaxies with an average $p_{m}$ value above 0.95 as merging, where
$p_{m}$ is the output of the CNN classifier.

Figure 12: Galaxies classified by the LDA classification as major mergers that
are classified by the A18 major merger classification as nonmerging. The
yellow line marks the edge of the segmentation mask and the inset panels
provide the LD1, $p_{\mathrm{merg}}$ and CDF values for each galaxy. The LDA
technique identifies a large fraction of SDSS galaxies as mergers that the A18
technique does not; this is the case even when the A18 threshold is adjusted
to a lower value.

We first compare the overlap of the merger samples. When we measure the
performance statistics of our merger sample relative to the A18
classifications (assuming the A18 classifications to be correct), we find an
accuracy of 0.85, a precision of 0.11, a recall of 0.78, and an F1 score of
0.20. The precision is low because there are a large number of galaxies that
we identify as mergers that A18 does not.

We present a few examples of galaxies that we classify as major mergers that
A18 does not in Figure 12. Using visual inspection, one of the galaxies in
this example (top right) looks like a faint major merger, three appear to be
minor mergers (top left333This merger and others like it could be chance
projections along the line of sight. We discuss this caveat of the method in
more detail in §5.8., top middle, and bottom left), and two appear to be post-
merger remnants (bottom middle and bottom right).

Figure 12 demonstrates something fundamental about the differences between
techniques that are trained using visually-identified samples and the LDA
technique presented here; techniques trained using mergers identified by eye
are biased towards identifying major mergers in the early or late stages. The
LDA technique on the other hand will identify a greater variety of merger
stages (including the post-coalescence stage, see the result of longer merger
observability timescale from N19).

We next use the color-mass diagram (Figure 13) to compare the properties of
galaxies selected as mergers by the LDA technique ($p_{\mathrm{merg}}>0.5$) to
those of the galaxies selected as mergers by the A18 technique
($p_{\mathrm{merg}}>0.95$). The cross-matched sample is incomplete at low
galaxy stellar mass ($M_{*}<10^{10}M_{\odot}$) due to the A18 sample, meaning
that the parent sample is almost entirely composed of red sequence galaxies.
However, over the extent of the cross-matched sample, it is clear that the
mergers identified using the LDA method span the same regions of color-mass
space as those identified using the A18 method, further verifying that the LDA
technique does not introduce significant morphological biases relative to the
A18 method.

Figure 13: Color-mass diagram. We cross-match the merger catalog from A18 with
the LDA catalog; the parent distribution is shown in black contours. We
compare mergers selected using the LDA technique (green) to those selected
using the A18 technique (orange). The mergers identified using the LDA
technique span the same color and mass ranges as those identified using the
transfer learning technique, indicating that the LDA technique does not
introduce significant morphological biases in its merger identification.

We next bin the color-mass diagram by both stellar mass and color to compare
the colors and stellar masses, respectively, of our sample of mergers to the
A18 mergers. Using the KS test to compare the merger distributions, we find
mergers identified using the LDA technique have similar stellar masses (for a
fixed color) and are slightly bluer (for a fixed stellar mass) relative to
mergers identified using the A18 method. Ackermann et al. (2018) compare their
sample of mergers to those of their training set (Darg et al. 2010) and find
that their sample tends towards redder colors relative to the GalaxyZoo-
identified mergers. We also find that the A18 sample is redder relative to our
galaxies.

###  4.7 Merger fraction

We measure the merger fraction ($f_{\mathrm{merg}}$), which is the fraction of
galaxies that have a $p_{\mathrm{merg}}$ value greater than 0.5. We do this
for both the major and minor merger classifications, focusing mostly on the
major merger fraction in our analysis. For the remainder of the paper,
$f_{\mathrm{merg}}$ or ‘merger fraction’ refers to the major merger fraction.
We will specify if we are referring to the minor merger fraction.

More specifically, a given output merger fraction $f_{\mathrm{merg}}$, is
computed from an individual LDA classification that is calibrated using an
input prior $\pi$ and then applied to all of the galaxies in SDSS. Our
fiducial values of $\pi$ for the major/minor merger classifications are
0.1/0.3, respectively. Therefore, the measured (output) merger fraction for
the fiducial major merger classification is:

$f_{\mathrm{merg,\pi=fiducial}}=\frac{N_{p_{\mathrm{merg}>0.5}}}{N_{\mathrm{all}}}$

where $p_{\mathrm{merg}}$ is the merger probability for each SDSS galaxy
calculated using the major merger classification created using the fiducial
prior of $\pi$ = 0.1, $N_{p_{\mathrm{merg}>0.5}}$ is the number of SDSS
galaxies with probability values greater than the threshold of 0.5, and
$N_{\mathrm{all}}$ is the number of SDSS galaxies in the sample. We perform
this calculation for the 363,644 galaxy subset that are photometrically clean
and mass-complete.

As we discuss in §3.4, adjusting the input prior affects the LDA
classification and distribution of LD1 and $p_{\mathrm{merg}}$ values. We
demonstrate this in Figure 14, where adjusting the prior ($\pi$, x-axis)
affects our measurement of the posterior (merger fraction,
$f_{\mathrm{merg}}$). In order to calculate the overall posterior probability,
we employ the Bayesian approach described in §3.4, marginalizing over the
prior probability. The marginalized output merger fraction is the median of
the individual merger fractions from each of the 46 priors shown in Figure 14.
The error on $f_{\mathrm{merg}}$ is calculated from the standard deviation of
the $f_{\mathrm{merg}}$ values for each input prior.

Figure 14 demonstrates a flattening of the relationship between the input
prior and the output posterior between the range $0.05<\pi<0.25$ for both the
major and minor merger fraction. On the upper end, we rerun this calculation
for major merger priors $\pi>0.5$ and find a similar flattening in the
relationship between the prior and posterior. Furthermore, we find that the
median major merger fraction is unchanged when we widen the prior to
$0.05<\pi<0.85$. This further justifies the 0.5 cutoff of the uniform prior
that we introduced in §3.4 and assures us that we have used the appropriate
prior range to recover the true merger fraction.

For each merger classification, we calculate the fiducial values of
$f_{\mathrm{merg}}$ (which do not have associated errors) and the marginalized
value of $f_{\mathrm{merg}}$ (uses the full posterior distribution of
$p_{\mathrm{merg}}$) for both the clean and the clean and mass-complete
samples of SDSS galaxies. We present these results in Table 7 for the major
and minor combined classification and the pre- and post-coalescence (1.0 Gyr)
classifications for each.

Finally, it is important to note that some galaxies will be counted multiple
times in this approach to calculating merger fraction. For instance, many
galaxies that are classified as major mergers are also classified as minor
mergers. The opposite is slightly less common and may be due to an increased
minor merger fraction. Quantifying the overall major merger fraction
(requiring that $p_{\mathrm{merg,50,maj}}>0.5$), we find a major merger
fraction of 0.21. When we remove all galaxies that are more likely to be minor
mergers ($p_{\mathrm{merg,50,min}}>p_{\mathrm{merg,50,min}}$), we find a clean
major merger fraction of 0.12. We repeat this calculation for the minor merger
fraction and clean minor merger fraction and find values of 0.28 and 0.24,
respectively. We find that these contamination fractions remains the same when
we adjust the $p_{\mathrm{merg}}$ threshold value we use to define mergers.

We also investigate this overlap as it pertains to the calculation of the
merger fraction trends with stellar mass and redshift in more depth in §4.12.
We ultimately find that the contamination of minor mergers in the major merger
sample does not affect our results about the merger fraction trends.

We also find that many galaxies are classified as multiple different stages of
mergers. For instance, users should be aware that if they select for major
mergers in the early stage ($p_{\mathrm{merg,maj,early,50}}>0.5$), many of
these galaxies will also be included when they select for major mergers in the
late stages ($p_{\mathrm{merg,maj,late,50}}>0.5$).

Quantitatively, we find that 0.18 of galaxies are major mergers in the early
stage and that this fraction drops to 0.03 after eliminating galaxies that are
more likely to be late and post-coalescence stage major mergers. Similarly,
0.19 of galaxies are major mergers in the late stage; when we consider the
clean late stage major merger fraction, this fraction drops to 0.13. The post-
coalescence stage has a major merger fraction of 0.35, which drops to 32% when
only considering clean post-coalescence mergers. The implication is that a
significant fraction of early stage mergers are likely to be identified as
mergers in other stages. This result also holds for the merger stages of the
minor mergers. The unclean/clean early stage minor merger fraction is
0.28/0.14. This figure is 0.24/0.16 for the late stage and 0.44/0.32 for the
post-coalescence stage.

Figure 14: Measured merger fraction as a function of the input prior for the major (pink) and minor (yellow) merger classifications for the mass complete sample. We marginalize over the posterior probability (y-axis) to account for the effects of multiple possible input priors (prior probability, x-axis). The horizontal lines and shaded regions show the median and standard deviation of the merger fraction when marginalized over all input priors. The slope of this relationship is flat between a prior range of $0.05<\pi<0.25$ and that the slope flattens out beyond $\pi>0.5$. This justifies the chosen prior range of $0.05<\pi<0.5$ and assures us that this range most likely covers the true merger fraction.  Priors | Major | Major | Major | Minor | Minor | Minor
---|---|---|---|---|---|---
| All | Pre-coalescence | Post-coalescence (1.0) | All | Pre-coalescence | Post-coalescence (1.0)
Fiduciala, all clean SDSS | 0.18 | 0.18 | 0.30 | 0.37 | 0.27 | 0.39
Flat $[0.05,0.5]$, all clean SDSS | 0.22$\pm$0.04 | $0.22\pm 0.03$ | $0.36\pm 0.05$ | 0.31$\pm$0.08 | $0.29\pm 0.09$ | $0.46\pm 0.04$
Fiducial, mass complete | 0.20 | 0.20 | 0.47 | 0.41 | 0.29 | 0.53
Flat $[0.05,0.5]$, mass complete | 0.28$\pm$0.07 | $0.24\pm 0.03$ | $0.53\pm 0.06$ | 0.35$\pm$0.09 | $0.32\pm 0.10$ | $0.60\pm 0.04$
Table 7: Merger fraction for the full sample of SDSS galaxies using different
classification thresholds.
aThe fiducial model is when $p_{\mathrm{merg}}>0.5$ and the priors are $\pi$ =
0.1 and 0.3 for the major and minor mergers, respectively.

###  4.8 Dependence of the major merger fraction on stellar mass and redshift

In this section, we explore how the measured major merger fraction changes
with galaxy stellar mass and redshift. In §4.13 and §B, we further explore if
these dependencies reflect biases of the classification or of the galaxy mass
selection.

First, in Figure 15, we separate the mass-complete sample into 15 evenly-sized
bins in stellar mass (meaning there are the same number of galaxies in each
one-dimensional bin) and bins of $\Delta z=0.02$ in redshift. After binning
the distribution, we eliminate bins where the median values of redshift and
mass for the galaxies in that bin are significantly different from the bin
centers, which we define as $>1\sigma$ above or below the bin center, where
$\sigma$ is the standard deviation of the values for the galaxies in the bin.
This eliminates bins where incompleteness in redshift and/or mass could bias
our results. We show the final binning scheme with the number of galaxies in
each complete bin in Figure 15. All bins (red) have at least 1000 galaxies.
This conservative approach restricts the final sample to 310,012 galaxies.

Figure 15: Redshift and mass distribution of all galaxies in the mass-complete
sample. For the analysis in this section, we select mass and redshift bins
that have $>1000$ galaxies and where the mass and redshift distributions are
complete (the medians are aligned with the bin center). We outline the final
selected bins used for the analysis and annotate the number of galaxies in
each bin.

We determine the median and standard deviation of the $f_{\mathrm{merg}}$
value in each bin by marginalizing across all priors. Next, for each redshift
bin, we fit a line to the data points at each stellar mass by running a Markov
Chain Monte Carlo (MCMC) analysis; we add the standard deviation (error bar)
multiplied by a value drawn from a random normal distribution to each
$f_{\mathrm{merg}}$ value and use statsmodels to fit a linear regression. We
show the key results for the major merger classification in Figure 16 and 17,
where we find a positive slope of $f_{\mathrm{merg}}$ with stellar mass and a
negative slope with redshift, respectively.

Figure 16: Linear fits to the binned $f_{\mathrm{merg}}$ values for the major
merger classification as a function of stellar mass for bins at fixed redshift
(bin spacing is $\Delta z=0.02$). We show the average line fit in color and
the MCMC iterative fits in grey. We conclude that $f_{\mathrm{merg}}$ has a
positive relationship with increasing mass for the majority of the redshift
bins. All panels have the same y range. Figure 17: Same as Figure 16 but
analyzing the slope of the major merger fraction with redshift for bins of
fixed stellar mass. Here the slope is significantly negative with redshift for
all mass bins.

The slope of the major merger fraction with mass (Figure 16) is mostly
positive; for 6/8 bins this is a significantly positive slope to 1$\sigma$,
where $\sigma$ is the variation in the slope value found via the MCMC
iterative analysis. In 2/8 cases, the slope is significantly positive to
2$\sigma$, and in 3/8 cases, it is significantly to 3$\sigma$. The slope of
the major merger fraction with redshift (Figure 17) is significantly negative
in 13/13 bins to 1$\sigma$ confidence.

Considering only the bins that have statistically significant slopes, we find
that the value of the slope ranges between $0.31<\alpha<0.53$ with mass (for
the $z$ bins) and between $-3.35<\alpha<-1.08$ with redshift (for the mass
bins). Generally, the trend is more steeply positive towards higher redshift
and more steeply negative towards low and intermediate masses.

###  4.9 Is S/N confounding the redshift-dependent major merger fraction?

A statistical confound is a variable that distorts the apparent causal
relationship between the independent and dependent variables because it is
independently associated with both. To investigate if S/N is a confound that
is causing the apparent negative slope in the major merger fraction with
redshift, stratify, or bin, by S/N. We first restrict the S/N range to
$0<\mathrm{S/N}<50$ because galaxies with S/N $>50$ have a sparse distribution
in the 3D parameter space. This restricts the sample from 363,644 to 305,321
galaxies.

We present our results in Figure 18, where redshift is the target independent
variable and S/N and mass bins are the y and x-axis of the figure,
respectively. We demonstrate that for almost all 2D bins (in mass and S/N),
the slope of $f_{\mathrm{merg}}$ is significantly negative with increasing
redshift. In many cases, the slope is slightly more negative than the 2D
binning analysis with mass and redshift. We can conclude that a projection of
the S/N-dependence of $f_{\mathrm{merg}}$ does not explain the negative slope
with redshift; when the sample is stratified by S/N, the slope of the major
merger fraction is even more negative with redshift.

We also run this analysis with S/N as the independent variable of interest and
find that when we stratify by mass and redshift that the major merger fraction
has a mostly negative trend with S/N, meaning that we find higher merger
fractions for lower S/N galaxies. This trend is not well fit by a linear
relationship; the slope is either flat or negative but very close to flat.
This result is distinct from many studies that find a positive trend of merger
fraction with S/N, where they are biased to detect brighter galaxies due the
merger identification technique’s reliance on faint tidal features (e.g.
Bickley et al. 2021).

Figure 18: The slope of the major merger fraction with redshift (inset subplot
x-axis) for almost all S/N (figure y-axis) and mass (figure x-axis) bins is
significantly negative. This indicates that the negative redshift trend for
$f_{\mathrm{merg}}$ cannot be attributed to increasing S/N with increasing
redshift.

###  4.10 Are morphology (bulge-to-total mass ratio) or color confounding the
redshift-dependent merger fraction?

We investigate if the negative slope of the major merger fraction with
redshift could be attributed to a sensitivity to galaxy type. For instance,
some studies find a different evolution of the merger fraction with redshift
for early-type galaxies (ETGs) and late-type galaxies (e.g. Lin et al. 2008;
López-Sanjuan et al. 2012). In some cases, the ETGs have a negative slope with
increasing redshift (Lin et al. 2008; Groenewald et al. 2017).

To conduct this analysis, we repeat the analysis of the previous section, this
time treating bulge-to-total mass ratio (B/T) and color ($g-r$) as the suspect
confounding variables. This 3D binning analysis is identical to the S/N
investigation we describe in §4.9; here we replace S/N with B/T and $g-r$
color and re-calculate the major merger fraction. By stratifying by these
nuisance parameters, we remove their influence from the other parameters of
interest (stellar mass and redshift). We find that the slope of the major
merger fraction with mass and redshift does not significantly change as a
function of galaxy color or B/T mass ratio. This is strong evidence that
neither color nor B/T are responsible for the mass and redshift trends. The
exception is our reddest bin, where the slope of the major merger fraction
with redshift is flat or positive.

It is important to make the distinction that while B/T and color are not a
confounding variables that are responsible for the negative redshift
dependence, they can still have independent influence on the merger fraction.
For instance, when we stratify by mass, redshift, and B/T, we find that the
major merger fraction is mostly flat as a function of B/T but increases with
B/T for some bins, peaking around a B/T mass ratio of 0.7. When we stratify by
mass, redshift, and color, the major merger fraction is positive with $g-r$,
meaning the major merger fraction increases for redder galaxies at high masses
and redshifts. At low masses and redshifts, the slope is instead negative or
flat. This is consistent with a picture where the major merger fraction
increases with B/T and $g-r$ mostly for higher mass galaxies.

###  4.11 Dependence of the minor merger fraction on stellar mass and redshift

Here we repeat the analysis, instead using the minor merger classification to
identify merging galaxies. We show the results for the binned analysis in
Figures 19 and 20 for the slope of the merger fraction with mass and with
redshift, respectively. We find that the slope of the merger fraction is
mostly flat with stellar mass except for two redshift bins where it is
negative. The slope of the merger fraction with redshift is flat for all mass
bins. In other words, the minor merger fraction shows little dependence on
mass or redshift. We discuss the implications of this in §5.5.

Figure 19: Same as Figure 16 but for the minor merger fraction. Figure 20:
Same as Figure 17 but for the minor merger fraction.

###  4.12 Accounting for contamination in the major/minor merger samples by
minor/major mergers

In §4.8 and 4.11, we empirically measure the merger fraction as a function of
stellar mass and redshift for the major and minor merger classifications,
respectively. These results include overlap between classifications, since we
consider all galaxies with median $p_{\mathrm{merg}}$ values greater than 0.5
as mergers. Here we investigate if these results change when we calculate the
merger fraction for the sample of major and minor mergers without overlap
between classification.

To calculate the clean major and minor merger fraction, we require that
$p_{\mathrm{merg,med}}>0.5$ and
$p_{\mathrm{merg,med,maj}}>p_{\mathrm{merg,med,min}}$ for the major mergers
and $p_{\mathrm{merg,med}}>0.5$ and
$p_{\mathrm{merg,med,min}}>p_{\mathrm{merg,med,maj}}$ for the minor mergers.
The second requirement significantly reduces the sample size of major mergers
from 86,843 galaxies to 53,573 galaxies and reduces the sample size of minor
mergers from 103,907 to 86,837 galaxies. The major merger sample therefore has
a greater contamination contribution from minor mergers, which is to be
expected given the larger overall merger fraction for minor mergers.

When we re-calculate the mass- and redshift-dependent merger fraction for the
clean samples, we find similar results. The clean major merger fraction has a
positive slope with mass and a negative slope with redshift for all bins. Most
slopes are slightly flatter than the not clean case; however, this difference
is not statistically significant (to $1\sigma$ errors). This slight flattening
could be due to a contamination from the minor mergers, where the trend with
mass and redshift is flatter. The clean minor merger fraction slopes are
consistent to $1\sigma$ with the not clean minor merger fraction slopes.

In conclusion, while double counting in the major and minor merger samples has
a significant effect on the overall number of mergers, double counting does
not affect our conclusions about the slope of the major and minor merger
fraction with redshift and mass. The implication is that the slope of the
merger fraction is robust to these levels of contamination (38% and 16% of the
major and minor merger samples, respectively, are contaminated).

###  4.13 Sanity checks

As we will address in the discussion section, the result of increasing merger
fraction with stellar mass has precedent in the literature. However, the
result of decreasing merger fraction with redshift is unprecedented. Given
this surprising result, we explore in Appendix B whether the result of
decreasing merger fraction with increasing redshift is physical (real) or
whether we can attribute it to sample systematics (i.e. mass incompleteness at
higher redshift or errors in the mass calculation or determination of the
photometric redshift).

After running our merger sample through multiple sanity checks in Appendix B,
we can conclude that the trend of the major merger fraction increasing with
stellar mass (for constant redshift) and decreasing with $z$ for constant
stellar mass is robust to changes in how we measure redshift and stellar mass.
It is also robust to changes in how we bin the data for this analysis and how
we compute the mass completeness. These steps were all taken to rule out the
leading culprits of systematic bias in the sample that could lead to our
surprising result of the negative evolution of the major merger fraction with
redshift.

Finally, we compare our major merger sample to a different merger sample
(A18). We find a mostly flat result with redshift for the A18 merger sample.
Since we use the same cross-matched sample to rerun the LDA classification and
still find a negative trend with redshift for the cross-matched sample, we are
able to conclude that this result is not due to peculiarities of the galaxy
sample but instead can be attributed to differences due to the merger
selection itself.

###  4.14 In the absence of mass binning the major merger fraction has an
artificial positive trend with redshift

We have taken one final step towards understanding the negative trend of the
major merger fraction with redshift in the context of other work. Here we run
our analysis without mass binning, as other work has done in the past in the
absence of enough data to bin and still retrieve a statistically significant
result.

We re-run the analysis without mass bins to determine the confounding role of
the positive mass trend in the redshift slope when we do not control for mass.
We additionally experiment with eliminating the completeness correction (of
Figure 15) and with using spectroscopic redshifts.

We present our results in Figure 21. We find a significant positive slope of
the major merger fraction with redshift in all cases where we do not bin for
mass. This includes the sample that is mass complete with photometric
redshifts (top), the sample that is mass incomplete with photometric redshifts
(middle), and the sample that is mass complete with spectroscopic redshifts
(bottom). All of plots in this figure use color-derived masses, but we find
similar results with SPS-derived stellar masses. Figure 21 is therefore an
important reminder that what looks like a positive slope with redshift is
actually the projection of a positive slope in mass onto redshift.

This figure additionally highlights that while the overall trend is positive,
there are different features in each plot produced by the slightly different
sample selections. For instance, the peak at low redshift can most likely be
attributed to the bias produced by photometric redshift that artificially
increases the stellar masses of low mass galaxies. Additionally, the peak at
higher redshift in the middle plot (mass incomplete sample) can most likely be
attributed to the mass incompleteness of the sample.

The conclusions from this section can be directly connected to our overall
conclusions from this work. While we cannot completely rule out that our
negative trend with redshift is not the result of some other systematic bias
or a combination of biases (i.e., confounding factors like mass incompleteness
and redshift bias), we can at least clearly show the most simple and likely
scenario: that mass binning versus no mass binning produce dramatically
different trends of the evolution of the major merger fraction with redshift.
This demonstrates the importance of running this type of analysis on large
samples of galaxies and with a merger classification technique such as the LDA
that demonstrates broad reliability across a range of galaxy types. Both of
these elements of this paper were essential to be able to bin the sample in
both redshift and mass and do a careful completeness correction.

Figure 21: Slope of the major merger fraction with redshift when we do not bin
for mass. We show the results for the mass complete sample with photometric
redshifts (top), for the mass incomplete sample with photometric redshifts
(middle), and for the mass complete sample with spectroscopic redshifts
(bottom). The black data points and accompanying error bars give the average
and standard deviation of the merger fraction for each redshift bin, the red
lines show the linear fits under the MCMC realizations (as outlined in §4.8),
and the red line is the average line fit with slope and error given at the
bottom of the plot. The average slope is positive for all samples, yet
experiences a significant downturn at redshifts $z<0.1$ for the two top plots,
which measure redshift using the photometric redshifts. These plots
demonstrates the importance of binning for mass; if this is not considered,
the positive trend of the major merger fraction with mass is projected onto
the redshift axis resulting in an artificial positive trend of the major
merger fraction with redshift.

##  5 Discussion

Our mass-complete binning analysis of a large sample of galaxies using a
carefully calibrated set of classification techniques allows us to make clear
conclusions about the evolution of the merger fraction locally
($0.03<z<0.19$). The additional novelty of this study is that the large sample
size allows us to do this over a finely spaced grid of redshift and mass bins.

Here we discuss our measurements of the mass and redshift-dependent merger
fraction in the broader context of previous work. We focus all discussion on
predictions and measurements of the merger fraction and reserve all discussion
of the merger rate for future work (Simon et al. 2023, in prep). We begin with
a brief review of predictions for the local merger fraction from cosmological
models (§5.1) and a review of recent empirical estimates of the merger
fraction (§5.2). We then discuss the implications of our findings of the mass
and redshift dependence of the major merger fraction for galaxy evolution in
the local Universe (§5.3 and §5.4, respectively). We also discuss the
implications of the distinct results we find for the minor merger fraction
evolution (§5.5). We summarize our precautions throughout this paper to
prevent morphological biases in the results in §5.6. We end with a discussion
of the relative strengths of the methodology presented here (§5.7) as well as
the caveats and future work motivated by this study (§5.8).

###  5.1 Predictions of the redshift and mass-dependence of the merger
fraction from cosmological models

The $\Lambda$CDM model of structural assembly (e.g. White & Rees 1978)
predicts hierarchical, or bottom-up assembly, meaning that mergers assemble
smaller halos first followed by more massive halos at later times (e.g.
Blumenthal et al. 1984). This predicts a merger rate that evolves with the
density of galaxies and space in the Universe. The measured fraction of
merging galaxies should therefore increase with redshift back to $z\sim$ 2 -
3. Additionally, the merger fraction should have a steep dependence on mass in
the local Universe, since the most massive galaxies are predicted to assemble
at later times.

An alternate assembly scenario is cosmic downsizing, where the largest
galaxies form early and then stall (e.g. Cowie et al. 1996; Juneau et al.
2005; Treu et al. 2005; Cowie & Barger 2008). Mergers have been invoked as a
mechanism to drive this compact star formation followed by rapid quenching.
While Bridge et al. (2010) invoke downsizing as a mechanism to drive a
negative mass dependence in the merger fraction between redshifts $0.2<z<1.2$,
Estrada-Carpenter et al. (2020) show that the phenomenon of downsizing has a
minimum redshift that ranges between $1.5<z<8$. The implication is that this
assembly scenario does not apply to the local Universe.

Baryonic evolutionary processes (i.e. feedback) play an important role in
galaxy assembly. Simulation work finds that when baryonic feedback is combined
with the bottom-up formation model (hierarchical assembly), this can manifest
as top-down assembly, i.e. downsizing (Stringer et al. 2009 and references
therein). Baryonic feedback suppresses the growth of stellar mass in galaxies
above and below $\sim 10^{11}M_{\odot}$. This results in a higher number of
intermediate mass galaxies. If this effect is strong, it could result in more
major mergers between equal-mass galaxies locally.

In reality, the picture is probably far more complicated than any one of the
above formation scenarios. Different processes likely dominate for different
mass scales and at various epochs over the age of the Universe. Directly
observing the galaxy-galaxy merger fraction as a function of redshift and mass
and separating major from minor mergers is therefore critical for constraining
the relative contributions of these competing processes.

###  5.2 Reviewing past empirical measurements of the mass- and redshift-
dependent merger fraction

Characterizing the mass-dependence of the major merger fraction can help us
understand how elliptical galaxies and the bulges of galaxies are being built
up over different mass ranges. Accurately measuring the mass-dependent merger
fraction locally is especially important for anchoring the redshift-dependent
merger fraction and directly testing the hierarchical assembly prediction that
the most massive galaxies are assembled locally.

As with the redshift-dependent merger fraction, previous work in this area
relies on either close pair methods (e.g. Xu et al. 2004; Patton & Atfield
2008; Domingue et al. 2009; Xu et al. 2012) or morphological studies (e.g.
Bridge et al. 2010; Casteels et al. 2014) to measure the mass-dependent merger
fraction. Most studies find a constant or slightly increasing merger fraction
with mass (e.g. Xu et al. 2004; Patton & Atfield 2008; Domingue et al. 2009;
Xu et al. 2012; Robotham et al. 2014). Of particular note is the work of
Robotham et al. (2014), which focuses on galaxies in the GAMA survey
($0.05<z<0.2$). They find that the merger fraction increases with mass by a
factor of $\sim$3 between stellar masses of $10^{9}<M_{*}(log\
M_{\odot})<10^{11}$.

Other work finds a decreasing fraction with mass (e.g. the morphological
studies of Bridge et al. 2010 and Casteels et al. 2014). Bridge et al. (2010)
($0.2<z<1.2$) claim that the decreasing fraction with mass is due to cosmic
downsizing, while Casteels et al. (2014) ($0.001<z<0.2$) argue that the
decrease is due to an increasing observability timescale for lower mass
galaxies.

A number of studies have measured the major merger fraction and how it trends
with redshift. It is important to note that most of the past work that has
examined the redshift-dependence of the merger fraction has done so for
redshift intervals that do not overlap with our study. The consensus among
these studies is mostly for a higher merger fraction at higher redshift (Lin
et al. 2008; Conselice et al. 2009; López-Sanjuan et al. 2012; Robotham et al.
2014; Mundy et al. 2017; Mantha et al. 2018; Snyder et al. 2019; Kim et al.
2021), although some studies find a relatively flat merger fraction (Bundy et
al. 2009; Jogee et al. 2009; Keenan et al. 2014).

Only the GAMA-focused studies of Robotham et al. (2014) and Keenan et al.
(2014) have more than one redshift bin below $z=0.2$. Additionally, due to
small sample sizes, the above work is often unable to bin finely in both
stellar mass and redshift, and therefore unable to explore both
simultaneously. Our study is unique in that we have a large enough sample size
to create bins in both stellar mass and redshift and this is the first study
to do so for fine redshift bins locally ($0.03<z<0.19$).

###  5.3 Implications of a positive mass dependence of the major merger
fraction

We find a positive relationship between the major merger fraction and stellar
mass between a range $10.5<M_{*}(log\ M_{\odot})<11.5$. This is consistent
with a hierarchical assembly picture, where more massive halos (hence
galaxies) are assembling locally. In this case, since we observe this trend
for all redshift bins, we can conclude that this trend holds for the last
$\sim$2 Gyrs of galaxy evolution. As mentioned in §5.1, if baryonic processes
such as feedback are coupling with hierarchical assembly, this could manifest
as top-down assembly (cosmic downsizing). While we cannot rule this scenario
out entirely, we can conclude that if this is happening, it is not strong
enough to invert or flatten our observed positive trend for the major merger
fraction.

###  5.4 Implications of a negative redshift dependence for the major merger
fraction

Our key result is a decreasing major merger fraction with redshift over the
range $0.03<z<0.19$ (Figure 17). The implication is that major mergers become
more important in the nearby Universe. This result cannot be explained by
either hierarchical assembly or cosmic downsizing. Hierarchical assembly
predicts an increase of the merger fraction out to high redshifts, while
cosmic downsizing likely does not operate in the local Universe and does not
make explicit predictions for the merger fraction. We find it most likely that
baryonic feedback is dominating locally, overriding the positive slope
predicted by hierarchical assembly.

Here we focus mostly on the implications of our finding in the context of past
studies and how these results merit a revision of the current techniques used
to measure the evolution of the major merger fraction.

Most other close-pair studies find a positive trend of major merger fraction
with redshift, yet it is important to note that the majority of these studies
do not cover the same redshift range as this work ($z<0.2$) and that none of
these studies control for both mass and redshift simultaneously. As we have
shown, the major merger fraction varies as a function of both mass and
redshift and the mass dependence can project onto the redshift axis, resulting
in an artificial positive relationship with redshift.

Our recommendation is for the community to: 1) Revisit past analyses of the
redshift-dependence of the major merger fraction using a careful mass and
redshift binning analysis, and 2) Design future studies that cohesively span
the local Universe and the higher redshift Universe. Currently, it is unclear
if our findings represent a local inversion in the higher redshift ($z>0.2$)
trend of a positive evolution of the major merger fraction with redshift or if
higher redshift studies will be inverted when mass binning and mass
completeness are accounted for.

Additionally, many of the past close-pair studies that find a positive slope
with redshift for the major merger fraction are sampling from a severely
restricted volume. For a more in-depth analysis of the volume probed by
various merger fraction studies, see the discussion of the role of cosmic
variance in the calculation of the merger fraction in López-Sanjuan et al.
(2014). Furthermore, Patton & Atfield (2008) find that the cosmic variance
from the SDSS survey is negligible. While cosmic variance is one important
consideration, surveys that are limited in volume due to survey size, depth,
or additional mass selections, will suffer from the inability to achieve
statistically meaningful results from their decreased number statistics
because they are unable to finely bin in mass and/or redshift.

###  5.5 Implications of distinct trends for the minor merger fraction

It is noteworthy that the minor merger fraction shows remarkably different
mass- and redshift-evolution relative to the major merger fraction. The minor
merger fraction has a flat dependence on both of these properties. While the
slope is flatter, the error bars on the minor merger fraction tend to be
larger than those on the major merger fraction for each bin. This could
reflect the decreased accuracy of the minor merger classification.

We explore a few explanations for the flat trends: 1) the minor mergers are
subject to different structural assembly processes in the local universe
(relative to major mergers), 2) the error bars on the minor merger fraction
are obfuscating trends that are positive with stellar mass and negative with
redshift, or 3) there are some systematic biases at play in the minor merger
classification.

First, we consider option 1. If this result is not due to a bias but is a
physical finding, this demonstrates that minor mergers are about equally
important for the assembly of all galaxy masses locally as well as all
redshifts within our range. This could further motivate the above explanation
explanation that a baryonic process such as feedback is driving the negative
redshift trend for the major merger fraction. A process like feedback that
increases the fraction of intermediate mass galaxies (hence, increasing the
major merger fraction locally) could also lead to a relatively smaller
fraction of minor mergers.

We next consider options 2 and 3. We observe underlying structure in the mass-
dependent minor merger fraction (a peak at intermediate masses). In our
analysis of the properties of the different merger classifications (§4.8), we
find that minor mergers have a tendency towards intermediate masses. This
could reflect a bias against identifying low mass and high mass galaxies,
which could result in a flatter trend for the minor merger fraction since we
should expect to miss both low and high mass galaxies. This makes sense given
that the minor merger classification relies upon shape asymmetry to identify
faint tidal tails or faint companions. In the case of a bright primary galaxy,
this task becomes much more difficult for the classification. This bias is
also related to a lower accuracy for the minor merger classification
(explanation 2). Our hypothesis is that while this slight bias could exist, we
find it unlikely that this slight bias alone is driving the flat evolution. In
future work, it would be worth exploring the biases related to the minor
merger classification; here we choose instead to focus on the biases related
to the major merger classification.

Regardless of whether it is a physical trend or related to classification
biases, the flatness of both of these relations for the minor merger sample
further motivates the importance of separating minor mergers from major
mergers; if there is significant minor merger contamination in a major merger
sample, this would act to flatten out both the mass- and redshift-dependence,
resulting in a flat relationship for both. While we find that this does not
have a significant influence on our results (§4.12), we recommend that future
studies of the redshift dependence of the major merger fraction take this
result into consideration.

###  5.6 Do the limitations of the simulated training set affect the
robustness of these results?

The LDA training set consists of intermediate mass, initially disk-dominated
simulated galaxies with initial stellar masses $3.9\times
10^{10}<M_{*}(M_{\odot})<4.7\times 10^{10}$ and initial B/T mass ratios
$0.0<B$/$T<0.2$. 444The stellar masses and B/T mass ratios evolve throughout
the time duration of the simulations. For instance, the simulated mergers
increase in mass and the major mergers remnants are bulge-dominated (N19).
Since this training set is limited, we have taken measures to minimize
potential biases and find no significant impact on our main result of the
mass- and redshift-dependence of the major merger fraction.

To minimize potential morphological biases, the SDSS galaxies used in our
merger fraction analysis are restricted to regions familiar to the LDA
classifier using the ‘outlier predictor’ flag (Figure 1). While the classifier
may be morphologically biased for galaxy morphologies outside of the training
set, this does not concern the results presented here, which are limited to
morphologies that are familiar to the LDA classifier.

To assess if the galaxies we classify are morphologically biased despite the
above precaution, we explore the properties of the SDSS merger sample in §4.5
and find no distinction in S/N, $r-$band magnitude, $g-r$ color, stellar mass,
and redshift between the merger and parent samples. This reflects a major
success or our method; that the merger classifications are not biased by
galaxy property. In §4.6 when we compare the fraction of ellipticals and
mergers in GalaxyZoo classified as LDA major mergers, we find the same
fraction (14%), which is further evidence that the technique does not retain a
morphological bias.

In §4.10, we confirm that the major merger fraction trends with redshift and
stellar mass persist when we control for galaxy morphology ($g-r$ color or B/T
ratio). This means that our results hold for all galaxy morphologies in the
photometrically clean sample.

Fully investigating the mass and morphological biases of the classification,
especially for galaxies with the ‘outlier predictor’ flag are beyond the scope
of this work. Future work could investigate the performance of the classifier
across different galaxy morphologies.

###  5.7 Strengths of this approach

For many past studies that focus on measuring the major merger fraction, small
number statistics are a concern. Cosmic (or sample) variance due to small
fields (i.e. see the discussion of Xu et al. 2012) can result in large error
bars, leading to a conclusion of flat redshift or mass evolution of the merger
fraction. Of additional concern, many of the close pair studies (which
constitute the bulk of this literature) suffer from spectroscopic
incompleteness at small angular separation, while morphological methods suffer
from surface brightness limitations, and as a result are biased towards
identifying high mass gas-rich major mergers only. Many morphological methods
also suffer from small sample sizes and with a variety of systematics related
to different methodologies.

In this work, we begin from a merger identification technique that is based on
a set of well-understood simulations of mergers. This technique has four
distinct advantages over past merger identification techniques.

  1. 1.

We are able to calibrate our methodology, which will become critically
important in future work (Simon et al. 2023, in prep), where we plan to
constrain the merger rate. In order to determine the merger rate, the merger
observability timescale is important, which we are able to measure from the
set of simulated mergers.

  2. 2.

Since the technique does not rely on spectroscopic detections, we apply the
method to the full SDSS photometric dataset and return the largest-yet sample
of merging galaxies. With this large sample, we are able to control for both
mass and redshift when we measure the merger fraction as a function of both of
these quantities, which we have shown is essential.

  3. 3.

Our technique spans a variety of merger stages, including pre- and post-
coalescence stages. It therefore overlaps in stages with both close-pair and
morphological studies, which will be crucial for comparing different types of
studies when we measure the merger rate.

  4. 4.

Our technique shows significant gains in accuracy and completeness relative to
past work, allowing us to build a more complete (and larger) sample of merging
galaxies.

###  5.8 Caveats and future work

There are three types of double counting of mergers that can occur under this
classification technique: 1) The overlap between major and minor mergers, 2)
The overlap between merger stages, and 3) Sometimes in the early stage of the
merger, the technique identifies both galaxies as mergers, which is double
counting compared to a close-pair technique.

We have already discussed the overlap between merger stages and types in
previous sections (§4.3 and 4.12, respectively). In §4.3 we find that the
early and late stages have significant overlap in classifications but the
post-coalescence stage tends to have less overlap; we discuss the implications
of this in terms of classification interpretation. In §4.12 we conclude that
the slope of the merger fraction with mass and redshift is unchanged when we
account for the double counting of major and minor mergers.

While splitting mergers by stage was not a primary focus of the merger
fraction analysis in this work, in future work (Simon et al. 2023, in prep),
we plan to compare our galaxy sample with the close-pair sample from Simon et
al. 2022, in prep, in order to constrain the absolute merger fraction. This
will be especially important for the early stage mergers, which are most
similar to close pair studies.

In addition, in this work we conduct a brief analysis of the overlap of merger
type classifications, in other words, the contamination of the major merger
fraction by minor mergers. Our focus is primarily on if this affects our
findings related to the merger fraction slope with stellar mass and redshift.
In future work we plan to characterize the overlapping merger populations.

In future work it will also be necessary to further address the third type of
double counting. For early stage mergers, we find that the LDA method
sometimes (but not always) identifies both galaxies in a pair as merging
galaxies. This represents a double count relative to close-pair studies where
the duo of merging galaxies would be considered to be one ‘pair’. On the other
hand, the LDA method also identifies mergers in the late and post-coalescence
stages, which boosts our derived merger fraction relative to that of close-
pair methods. Both of these considerations mean that directly comparing our
method to close-pair studies is difficult. For this reason, in this work, we
have not attempted to directly compare the absolute number of mergers and have
instead compared the slope of $f_{\mathrm{merg}}$ with stellar mass and
redshift. In Simon et al. 2023, in prep, we plan to compare our galaxy sample
with the close-pair sample from Simon et al. 2022, in prep, in order to
constrain the absolute merger fraction. In this future work (Simon et al.
2023, in prep), we will also be able to determine the calibration factor,
$C_{\mathrm{merg}}$, to convert between the close pair fraction and the
fraction of close pairs that will ultimately merge.

It is also important to mention a fundamental difference between
morphologically-reliant merger identification techniques like the LDA
technique and spectroscopic-based techniques like the close-pair method.
Galaxies like those shown in the top left panel of Figure 12 that are
identified as mergers by the LDA technique may in fact be chance projections
of unrelated galaxies along the line of sight. Fully characterizing the
expected frequency of these chance projections is beyond the scope of this
work, although we plan to discuss this in more depth in Simon et al. 2022, in
prep, when we compare our merger sample to that of the close pair method.

##  6 Conclusions

In this work we apply the merger classification method from Nevin et al.
(2019) to the 1.3 million galaxies in the Sloan Digital Sky Survey DR16
photometric catalog. We additionally expand the merger classifications from
N19 to include the different stages of the merger in addition to major versus
minor classifications. This results in twelve different merger
classifications: major and minor and then we further split by stage: early,
late, pre-coalescence (includes early and late), and two different post-
coalescence classifications (one extends to 0.5 Gyr post-merger, one extends
to 1.0 Gyr).

We apply all of these classifications to image cutouts from SDSS, calculate
the $p_{\mathrm{merg}}$ values and repeat this process for a range of
different input priors, marginalizing over these priors to retrieve the
posterior distribution of $p_{\mathrm{merg}}$ values for all galaxies for all
classifications. We provide these classifications to the reader in the form of
online-available tables in addition to an interpretable classification repo
known as MergerMonger. In the text we provide examples for how to interpret
the results and distinguish between different merger types.

We next analyze the properties of the merger samples and compare these
properties to other merger samples in the literature. We conclude that the
properties of the different types of mergers span the full range of properties
of the parent SDSS distribution (in S/N, $r-$band magnitude, color, stellar
mass, and redshift), which is a major success of the method. We also find that
the LDA technique retrieves the majority of the GalaxyZoo and Ackermann et al.
(2018) mergers and further identifies a large sample of galaxies as mergers
that were missed by these techniques, demonstrating its success in finding
less-obvious mergers than visually identified samples.

The main goal of this paper is to retrieve the merger fraction
($f_{\mathrm{merg}}$) as a function of galaxy properties, which we do by
measuring stellar masses, carefully building a mass-complete sample (our final
sample is 310,012 galaxies), and binning by both stellar mass and redshift.
For the major merger sample we find a significantly positive trend
(1-3$\sigma$ confidence) between $f_{\mathrm{merg}}$ and stellar mass and a
significantly negative (to 1$\sigma$ confidence) trend with redshift. We show
these key results in Figures 16 and 17, respectively. This trend is robust
between stellar masses of $10.5<M_{*}(log\ M_{\odot})<11.6$ and redshifts of
$0.03<z<0.19$. We show that when we do not correct for completeness or bin for
mass, the strong dependence of the major merger fraction on mass results in a
positive redshift slope, underscoring the importance of a careful binning
analysis with a large sample size to recover this result.

Examining these results in the context of past theoretical and observational
work, we find that the positive trend of the major merger rate with stellar
mass agrees with past results and is consistent with a hierarchical assembly
scenario for the Universe. On the other hand, this is the first time a study
has focused on the measuring the merger fraction locally ($z<0.2$) for finely
spaced mass and redshift bins, which underscores the uniqueness of the finding
of a negative trend for the major merger fraction with redshift.

In future work (Simon et al. 2023, in prep) we plan to use these results in
combination with the SDSS-derived close pair fraction from Simon et al. 2022,
in prep to calculate a merger rate. From this, we can constrain the
gravitational wave background from SMBH mergers.

## Acknowledgements

To Dr. Aaron Stemo and Dr. Plamen G Krastev for some phenomenal supercomputing
support.

This research is supported by NSF AST-1714503 and NSF AST-1847938.

JS is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship
under award AST-2202388.

The computations in this paper were run on the FASRC Cannon cluster supported
by the FAS Division of Science Research Computing Group at Harvard University.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.

SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

## Data Availability

All data products detailed in §4.1 are available on Zenodo555DOI:
10.5281/zenodo.7438610. For the MergerMonger code, see the Github repo  . This
includes all of the analysis utilities used to generate the results of this
paper.

## Software

astropy (Astropy Collaboration et al., 2013), pandas (pandas development team,
2020), seaborn (Waskom, 2021), scikit-learn (Pedregosa et al., 2011), and gnu
parallel (Tange, 2018)

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