Emerge: Empirical predictions of galaxy merger rates since 6

MNRAS 000, 1–23 (2020) Preprint 19 January 2021 Compiled using MNRAS LATEX style file v3.0

 Emerge: Empirical predictions of galaxy merger rates since ∼ 6

 Joseph A. O’Leary,1★ Benjamin P. Moster,1,2 Thorsten Naab,2 Rachel S. Somerville3
 1 Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 München, Germany
 2 Max-Planck Institut für Astrophysik, Karl-Schwarzschild Straße 1, 85748 Garching, Germany
 3 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, US

 Accepted XXX. Received YYY; in original form ZZZ
arXiv:2001.02687v3 [astro-ph.GA] 17 Jan 2021

 ABSTRACT
 We explore the galaxy-galaxy merger rate with the empirical model for galaxy formation,
 emerge. On average, we find that between 2 per cent and 20 per cent of massive galaxies
 (log10 ( ∗ / ) ≥ 10.3) will experience a major merger per Gyr. Our model predicts galaxy
 merger rates that do not scale as a power-law with redshift when selected by descendant stellar
 mass, and exhibit a clear stellar mass and mass-ratio dependence. Specifically, major mergers
 are more frequent at high masses and at low redshift. We show mergers are significant for the
 stellar mass growth of galaxies log10 ( ∗ / ) & 11.0. For the most massive galaxies major
 mergers dominate the accreted mass fraction, contributing as much as 90 per cent of the total
 accreted stellar mass. We reinforce that these phenomena are a direct result of the stellar-to-halo
 mass relation, which results in massive galaxies having a higher likelihood of experiencing
 major mergers than low mass galaxies. Our model produces a galaxy pair fraction consistent
 with recent observations, exhibiting a form best described by a power-law exponential function.
 Translating these pair fractions into merger rates results in an inaccurate prediction compared
 to the model intrinsic values when using published observation timescales. We find the pair
 fraction can be well mapped to the intrinsic merger rate by adopting an observation timescale
 that decreases linearly with redshift as obs = −0.36(1 + ) + 2.39 [Gyr], assuming all observed
 pairs merge by = 0.
 Key words: cosmology: dark matter – galaxies: formation, evolution, stellar content

 1 INTRODUCTION Blaizot 2007; Font et al. 2008; Somerville et al. 2008; Khochfar &
 Silk 2009; Stewart et al. 2009; Hopkins et al. 2010a; Hopkins et al.
 In the hierarchical picture of galaxy formation within the ΛCDM
 2010b; Rodriguez-Gomez et al. 2015).
 framework, mergers play a critical role in the formation and contin-
 ued evolution of galaxies. Consequently the galaxy-galaxy merger Similarly, observed merger rates have not converged so far, with
 rate and its dependence on mass, mass ratio, and redshift are of different rates even derived from the same fields (Mantha et al. 2018;
 fundamental interest. The frequency of galaxy mergers cannot be Duncan et al. 2019). Many of the discrepancies can be attributed
 observed directly and so we must rely on theoretical models for to varying definitions on the merger rate, including whether galaxy
 galaxy formation along with a robust set of observations to ascer- pairs are selected based on their stellar mass or their luminosity
 tain the cosmological galaxy-galaxy merger rate. (Lotz et al. 2011; Man et al. 2016). Furthermore unreliable redshift
 Many theoretical models build upon the foundation laid by dark measurements introduce considerable uncertainty in the selection of
 matter (DM) only -body simulations, with each model applying physically associated pairs. Additionally, merging timescales must
 a different method for populating DM haloes with their constituent be separately derived using theoretical models. However, consider-
 galaxies. The underlying halo-halo merger has largely converged able uncertainty remains in these merging timescales and how they
 among various theoretical models (Fakhouri & Ma 2008; Fakhouri might scale with redshift.
 et al. 2010; Genel et al. 2009, 2010). Despite this agreement in the Theoretical models differ in their approach to linking dark
 foundational structure of galaxy evolution, theoretical models for matter haloes with galaxies. Ab initio methods provide a complete
 galaxy formation have yet to establish a sufficiently accurate value treatment of baryonic physics to building galaxies through directly
 for the galaxy-galaxy merger rate. There remains as a much as an computing the physical processes. These simulations are thus re-
 order of magnitude discrepancy in the predicted values depending liant on accurate treatments of the physics, such as gas cooling,
 on mass, mass-ratio, redshift, and theoretical framework (Bower star formation, and the relevant feedback mechanisms (Somerville
 et al. 2006; Croton et al. 2006; Maller et al. 2006; De Lucia & & Davé 2015; Naab & Ostriker 2017). Due to their sophisticated
 nature, they are time consuming and costly to run, which limits the
 resolution that can be achieved. As it is impossible to resolve the
 ★ E-mail: joleary@usm.lmu.de scales on which the fundamental forces act, most physical processes

 © 2020 The Authors

2 J. A. O’Leary et al.
have to be combined into effective models – so-called subgrid mod- This paper is organised as follows; In Section 2 we provide an
els – with a number of free parameters, which are tuned in order to overview of the -body simulation we use, as well as the empiri-
reproduce a number of observational constraints. In this sense, there cal model used to populate the simulated dark matter haloes with
are currently no true ab initio methods in galaxy formation, as all galaxies. We outline our methodologies and fundamental results
simulations include free parameters in some form that need to be fit- in Section 3, and discuss how the merger rate scales with stellar
ted or constrained by observational data, and thus rely on empirical mass, mass ratio, redshift, and star formation rate. Here we also we
evidence. The two most commonly used methods that aim to model illustrate how our model compares with other theoretical predic-
the baryonic physics are hydrodynamical simulations (Dubois et al. tions. In Section 4 we discuss our results in the context of recent
2014; Hirschmann et al. 2014; Vogelsberger et al. 2014a,b; Crain observations. Additionally we provide mock observations using our
et al. 2015; Schaye et al. 2015; Weinberger et al. 2017; Hopkins simulation data to create a more thorough evaluation of our model
et al. 2018; Pillepich et al. 2018), which calculate the gas physics intrinsic results. Finally, in Section 5 we explore the merging his-
along with the gravitational forces at the level of the resolution el- tory of present day galaxies. Here, we determine which galaxies
ements, and semi-analytical models (Kauffmann et al. 1993; Cole are grown through merging, and which type of mergers matter for
et al. 1994; Kauffmann et al. 1999; Somerville & Primack 1999; stellar mass growth.
Cole et al. 2000; Somerville et al. 2001; Baugh 2006; Bower et al.
2006; Somerville et al. 2008; Benson 2012; Henriques et al. 2015;
Somerville et al. 2015), which post-process DM-only simulations 2 DARK MATTER SIMULATIONS AND EMERGE
and populate dark matter haloes with galaxies using analytic pre-
scriptions at the level of individual haloes. Both approaches have Our analysis of galaxy-galaxy merger rates relies on producing
made vast progress in recent years, but still struggle to reproduce a galaxy merger trees encompassing a large dynamic range, occu-
large number of observations simultaneously, as it is very difficult pying an appropriately large cosmic volume. We employ the em-
to explore the parameter space of the subgrid models due to the pirical model emerge to populate dark matter haloes with galaxies
computational cost. based on individual halo growth histories. In this section we discuss
 In this paper, we use an alternative approach known as empir- the fundamental tools used to ultimately produce galaxy merger
ical models of galaxy formation (Moster et al. 2013, 2018, 2020; trees. Throughout this paper we adopt Planck ΛCDM cosmology
Conroy & Wechsler 2009; Behroozi et al. 2013d, 2019). Instead of (Planck Collaboration 2016) where Ω = 0.3070, ΩΛ = 0.6930,
aiming to directly model the baryonic processes, these models use Ω = 0.0485, where 0 = 67.77 km s−1 Mpc−1 , = 0.9677, and
parameterised relations between the properties of observed galaxies 8 = 0.8149.
and those of simulated DM haloes. The parameters of these relations
are then constrained by requiring a number of statistical observa-
 2.1 Obtaining halo merger trees
tions be reproduced. This approach has the advantage of accurately
matching observations by construction, allowing us to analyse the We utilise a cosmological dark matter only -body simulation in a
evolution of galaxy properties with cosmic time, and investigate the periodic box with side lengths of 200 Mpc. The initial conditions for
different growth channels. Furthermore, as these models can very this simulation were generated using Music (Hahn & Abel 2011)
efficiently post-process DM-only simulations it is easy to probe with a power spectrum obtained from CAMB (Lewis et al. 2000).
large volumes to gather statistics across a large dynamic range. The simulation contains 10243 dark matter particles with parti-
 The primary goal of this paper is to determine the cosmolog- cle mass 2.92 × 108 M . The simulation was run from = 63 to 0
ical galaxy-galaxy merger rate, and its dependence on properties using the Tree-PM code Gadget3 (Springel 2005). In total 94 snap-
such as the stellar mass of the main galaxy, the stellar mass ratio shots were created evenly spaced in scale factor (Δ = 0.01). Dark
between both galaxies, and the redshift of the merger. Instead of matter haloes are identified in each simulation snapshot using the
predicting the merger rate with a model that makes assumptions on phase space halo finder, Rockstar (Behroozi et al. 2013a). Halo
the baryonic physics, we derive it empirically, solely based on the merger trees are constructed using ConsistentTrees (Behroozi
evolution of observables such as the stellar mass function and star et al. 2013c), providing detailed evolution of physical halo prop-
formation rates, within a ΛCDM cosmology. To this end we employ erties across time steps. Throughout this paper we use the term
the empirical galaxy formation model emerge1 (Moster et al. 2018, ’main halo’ to designate haloes which do not reside within some
2020). Additionally, we compare our results for the intrinsic merger other larger halo, and ’subhalo’ to refer to haloes contained within
rates, i.e. actual mergers in the model, with those produced via mock another halo.
observations of galaxy pairs in order to provide a better translation
between observables and underlying merger rates. We further in-
vestigate how the stellar mass of galaxies grows over cosmic time, 2.2 Halo-Halo mergers
and whether this growth mainly comes from star-formation within Prior to evaluating the galaxy-galaxy merger rate we take a look
the galaxy, major mergers, or minor mergers. In this context, we at the halo-halo merger rate. Due to our model’s reliance on the
also study how many major and minor mergers a galaxy typically individual growth histories of dark matter haloes, it is important to
has over its lifetime. We perform our analysis in the context of our verify that our simulation is assembling haloes in a manner con-
empirical model, but we compare our results to observational evi- sistent with other theoretical predictions and models (Genel et al.
dence and other theoretical work to estimate the robustness of our 2009, 2010; Fakhouri et al. 2010). We compute the halo-halo merger
conclusions2 . rate directly using the trees constructed with ConsistentTrees. We
 define halo mergers at the time when a halo first becomes a subhalo,
 not when the subhalo becomes distrupted. This is consistent with
1 The code can be obtained at https://github.com/bmoster/emerge the definition of halo mergers adopted by Genel et al. (2009, 2010);
2 Scripts for reproducing our primary results can be obtained at https: Fakhouri et al. (2010). The merger rate is then calculated at each
//github.com/jaoleary redshift as a function of the descendant halo mass 0 , and the mass

 MNRAS 000, 1–23 (2020)

Galaxy merger rates with Emerge 3

 ξ > 0.25 provide a statistical link between galaxy and halo properties without
 M0 = 10 11 M¯
 the need to directly model baryonic physics. In this way, emerge is
 M0 = 10 12 M¯
 able to produce accurate galaxy catalogues exhibiting the range of
 100 M0 = 10 13 M¯
dN/dt [Gyr −1 ]

 physical properties observed in large galaxy surveys. This model ad-
 M0 = 10 14 M¯
 ditionally allows us to self-consistently track galaxies across times
 steps, providing the opportunity to explore and evaluate their indi-
 vidual growth histories.
 10-1
 The primary avenue for galaxy growth in emerge is through
 in-situ star formation. Each galaxy is seeded at the center of a dark
 matter halo with a SFR directly driven by the growth of the dark
 matter halo, .¤ On large scales, baryons are assumed to uniformly
 10-20 1 2 3 4 5 trace the underlying cosmic dark matter distribution such that each
 Redshift halo contains a fixed baryon fraction = Ω /Ω . From this it
 108 ¤ should be directly
 M0 = 10 11 M¯ follows that the growth rate of each halo ,
 107 FMBK10
 proportional to the rate of baryonic growth within the halo, and the
 M0 = 10 12 M¯ M0 = 10 12 M¯
 106 M0 = 10 13 M¯ SFR in the central galaxy is given by:
 G09
 105 M0 = 10 12 M¯ M0 = 10 14 M¯
dN/dξ/dz

 104 d ∗ ( , ) d bary d 
 = ( , ) = bary ( , ) . (1)
 103 d d d 
 102 Here, ¤ bary ( , ) is the baryonic growth rate which describes how
 101 much baryonic material is becoming available, and ( , ) is the in-
 100 stantaneous conversion efficiency, which determines how efficiently
 z = 0.10 this material can be converted into stars.
 10-110-5 10-4 10-3 10-2 10-1 100 The instantaneous baryon conversion efficiency is impacted
 ξ = Mi /M1 by a variety of physical processes, gas cooling, AGN feedback, su-
 pernova feedback, etc. (Somerville & Davé 2015; Naab & Ostriker
 2017) emerge seeks to establish the minimally viable parametrisa-
Figure 1. Halo-halo merger rates. The coloured lines indicate halo-halo tion necessary to replicate observations. In the most basic picture,
merger rates from our simulation for the noted descendant halo masses. For
 the instantaneous efficiency is governed only by redshift and halo
ease of comparison we adopt the mass ratio definition of Fakhouri et al.
(2010), where = / 1 = 1/ (see Section 3). Top panel: Major ( >
 mass. However, the model remains flexible as additional param-
0.25) Halo-halo merger rates per Gyr, scaling with redshift. The bumps for eters can be added on an "as-needed" basis. In particular, it was
higher mass haloes is due to low number statistics. Bottom panel: Halo-halo determined that a double power-law parametrisation is sufficient to
merger rates scaling with mass ratio at = 0.1. The black dash-dash and model the instantaneous baryon conversion efficiency as a function
dash-dot lines shows the best fit merger rate for a halo with 0 = 1012 , of halo mass at any redshift (Behroozi et al. 2013b; Moster et al.
from Genel et al. (2009) and Fakhouri et al. (2010) respectively. 2018),
 "  # −1
 − 
  
ratio = / 1 of the progenitor haloes (for > 1), where 1 is ( , ) = 2 + , (2)
the most massive progenitor to 0 . 1 1
 Figure 1 shows the mean halo-halo merger rate per descen-
dant halo. When taking the halo-halo merger rate per halo, we find where the normalisation , the characteristic mass 1 , and the low
rates that adopt the same nearly mass-independent scaling shown and high-mass slopes and are the free parameters used for the
in previous works (Genel et al. 2009, 2010; Fakhouri et al. 2010), fitting. Furthermore, the model parameters are linearly dependent
The top panel shows that the merger rate per Gyr exhibits a strong on the scale factor:
power-law scaling with redshift. The bottom panel shows that, just
 
as in previous works, we find a scaling ∝ −2 for a fixed redshift log10 1 ( ) = 0 + , (3)
 +1
interval. Our results indicate that our underlying -body simulation 
is in agreement with other works. = 0 + , (4)
 +1
 At this point we are free to implement our galaxy formation 
model. In this process we will see how this simple universal halo ( ) = 0 + , (5)
 +1
merger rate becomes transformed through the complex connection ( ) = 0 . (6)
between galaxies and their haloes.
 These parameters are allowed to vary freely within their boundary
 conditions in order to produce a fit in agreement with observation.
2.3 Connecting galaxies to haloes
 Observables are chosen such that model parameters can be iso-
In the hierarchical view of galaxy formation, each galaxy starts its lated and independently constrained, thus avoiding degeneracy. In
life at the center of an isolated halo. As the dark matter haloes grow particular, the characteristic mass ( 0 and )is constrained by
and cannibalise one another, so too will their occupant galaxies. stellar mass functions (SMFs). The efficiency normalisation param-
Empirical models populate simulated DM haloes with galaxies, and eters ( 0 and ) can be constrained by the cosmic star formation
evolve each galaxy according to physically motivated parametrisa- rate density (CSFRD). The efficiency slopes ( 0 , and 0 ) are
tions, directly constrained by real observables. Thus, these models constrained by specific star formation rates (sSFRs).

MNRAS 000, 1–23 (2020)

4 J. A. O’Leary et al.
2.4 Galaxy growth through mergers orphan halo mass is updated at each time step, declining at the same
 average rate since peak mass:
Aside from in-situ star formation, galaxy mergers are the other pri-
mary mechanism contributing to galaxy growth in emerge. In the
context of emerge, we specify galaxies of three types; central, satel- peak −Δ /( loss − peak )
  
lite and orphan. Central galaxies exist in the center of main haloes. = −1 (10)
 loss
While, satellite galaxies sit at the center of sub-haloes, orbiting
within some larger main halo. Orphan galaxies were formed in the Here is the index of the current snapshot and − 1 is the previ-
same way as satellite galaxies, however, their subhalo has since been ous snapshot index, and Δ is the amount of time elapsed between
stripped below the resolution of the halo finder. As orphans are no these snapshots in Gyr. The halo is assumed to lose mass at the
longer traceable in the simulation, they require special numerical same average rate, with a slope defined by halo peak mass peak
treatments to address their continued evolution. at peak and subhalo disruption mass loss at loss . In the initial
 When a galaxy first becomes an orphan, a dynamical friction emerge release, orphans inherited their last resolved halo mass.
clock is set. We use its last known orbital parameters to compute the This static halo mass made orphans impervious to stripping which
dynamical friction time. Specifically, we use the dynamical friction leads to a more resolution dependent model. One consequence of
formulation specified by Boylan-Kolchin et al. (2008) to control that implementation is that a galaxy would need to be stripped prior
orphan orbital decay: to entering the orphan phase, necessitating a large value for in
 order to reproduce the observed clustering values.
 ( 0 / 1 ) 2 2 The stripping fraction is a free parameter in this model, and is
  
 = 0.0216 ( ) −1 exp(1.9 ) 1 , (7) largely driven by galaxy clustering observations. As with their halo
 ln(1 + 0 / 1 ) vir
 mass, orphans also have their positions updated at each timestep
where ( ) is the Hubble parameter, vir is the virial radius of the until has elapsed. Orphans are placed randomly on a sphere
main halo ( 0 ), 1 is the radial position of the subhalo ( 1 ) with √︁
 with radius = 0 dec where dec = 1 − Δ / , and Δ is the
with respect to the center of the main halo, and is a measure for time elapsed since the subhalo was last resolved (Binney & Tremaine
the orbital circularity of the subhalo. When the dynamical friction 1987).
time has elapsed the orphan galaxy will be merged with the central The interplay between halo mass, stripping, and clustering
system where a portion of the satellite stellar mass will be added to make a treatment for orphan halo mass critical in fitting clustering
the descendant galaxy as down to 10 kpc, which is important for determining merger rates
 derived through projected galaxy pairs, Section 4.1. Figure 2 shows
 desc = main + orphan (1 − esc ), (8)
 the projected galaxy correlation function in several mass bins under
where desc is the mass of the descendant galaxy, main is the this new model variation.
mass of the main progenitor galaxy, orphan is the mass of the When implementing these model improvements we did not
progenitor orphan galaxy, and esc is the fraction of mass that will alter the fitting procedure. Parameter uncertainties are determined
be distributed to the ICM during the merger. The escape fraction through the same affine invariant MCMC described by Goodman
is a free parameter in the model and is largely constrained by the & Weare (2010) and employed in Moster et al. (2018). Lastly, we
low redshift behaviour on the massive end of SMFs along with the utilise the same observed data sets listed in Moster et al. (2018),
sSFR of massive galaxies. as well as the additional data sets noted in Moster et al. (2020).
 If however, an orphan is on its way to merge with a satellite The addition of this halo mass loss formulation along with new
galaxy and that target itself becomes an orphan before has observational data naturally resulted in a set of best fit parameters
elapsed, then the orphan galaxy will have its dynamical friction different from previously published results (see Table 1). While
clock reset according to the mass of the new central system. In the most notable parameter change was , other parameters have
this special case where must be reset, we rely on a recently moved beyond the 1 uncertainty ranges quoted in Moster et al.
implemented approximation for orphan-halo mass loss. (2018). This movement does not reflect tension in the observed
 datasets, but is more likely the result of the complex topology of the
 high-dimensional parameter space. The posterior surfaces contains
 local minima separated by many sigma from one another with each
2.5 Orphan-halo mass loss exhibiting a reasonable fit to the observations.
A prescription for orphan-halo mass loss is important for a few
reasons. The first is the dependence of on the mass of both
systems involved, as shown in eq. 7. The other reason is as a means of 2.6 Satellite quenching
defining the gravitational potential of the system, which is important
 Galaxy quenching is one other mechanism that affects the growth
for galaxy stripping. When a halo has lost enough mass the galaxy
 of galaxies. If a dark matter halo begins to become accreted by
at the center can also become subject to tidal forces and experiences
 a larger halo, its own growth rate will decline. At some point the
stripping. This model implements a simple halo mass threshold
 halo will reach its peak mass peak after which the halo will not
below which the galaxy can no longer remain bound and will be
 grow, consequently reducing the ‘inflow’ of gas. After some time
distributed to the ICM.
 the galaxy at the center of such a halo will deplete the remaining
 < peak (9) cold gas supply through star formation and become quenched.
 To address star formation in these galaxies emerge invokes a
Here, peak is the halo peak mass, and is the stripping fraction. ‘delayed-then-rapid’ model for quenching (Wetzel et al. 2013). In
 To ensure that all galaxies, including orphans, are subject to this model, after a halo has reached peak mass the central galaxy will
stripping, we apply a simplified formula as a stand-in for the physical continue to form stars at a constant rate equal to the star formation
tidal stripping process experienced by sub-haloes. In this approach rate at peak . After a time the cold gas supply is assumed depleted

 MNRAS 000, 1–23 (2020)

Galaxy merger rates with Emerge 5

 Table 1. The best fit model parameters used for this work.
 104 Emerge - z = 0.08
 Observed mean
 103
wp / Mpc

 Parameter Best-fit Upper 1 lower 1 

 102 0 11.34829 +0.03925 -0.04153
 0.654238 +0.08005 -0.07242
 101
 
 0 0.009010 +0.00657 -0.00451
 100 9.34 log 10 (m ∗ /M¯ ) < 9.84 
 0
 0.596666
 3.094621
 +0.02880
 +0.15251
 -0.02366
 -0.14964
 104 -2.019841 +0.22206 -0.20921
 0 1.107304 +0.05880 -0.05280
 103
wp / Mpc

 0.562183 +0.02840 -0.03160
 102 0.004015 +0.00209 -0.00141
 4.461039 +0.42511 -0.40187
 101
 0
 0.346817 +0.04501 -0.04265
 100 9.84 log 10 (m ∗ /M¯ ) < 10.34
 104 3 THE GALAXY-GALAXY MERGER RATE
 103
wp / Mpc

 In this section we discuss the galaxy-galaxy merger rates intrinsic
 to and derived from emerge. First we present the intrinsic merger
 102 rate, that is the rate at which galaxies are merged using the processes
 101 outlined in Section 2.4. The intrinsic merger rate provides insight
 into the actual buildup of stellar material using the internal mechan-
 100 10.34 log 10 (m ∗ /M¯ ) < 10.84 ics of the empirical model. We then present a merger rate derived
 104 using mock observations applied to mock galaxy catalogues. This
 provides a bridge to more completely address any discrepancies
 103
wp / Mpc

 between theoretical models and observations.
 102 First we should address some terminology common to both
 approaches. Each galaxy merger can be classified in terms of stellar
 101 mass and stellar mass ratio:
 100 10.84 log 10 (m ∗ /M¯ ) < 11.34 0 : The stellar mass of the descendant galaxy at the snapshot
 following the merger.
 104 1 : The stellar mass of the main progenitor galaxy defined at the
 103
wp / Mpc

 snapshot just prior to the merger.
 2 : The stellar mass of the co-progenitor galaxy at the snapshot
 102 just prior to the merger.
 101 : The stellar mass ratio taken with respect to the progenitor
 galaxies, ≡ 1 / 2 . In the most general case the main progenitor
 100 11.34 log 10 (m ∗ /M¯ ) < 11.84 1 is also the most massive progenitor. Due to scatter in the stellar-
 0.01 0.1 1.0 10.0 to-halo mass relation (SHMR) there are some scenarios under which
 rp / Mpc 2 > 1 . In these cases we invert this relation such that ≥ 1. These
 special cases represent fewer than 5 per cent of all mergers with a
 descendant mass larger than 109 .
Figure 2. Clustering in the updated model down to 10 kpc. Shaded regions
illustrate the 1 uncertainty range determined though jackknife sampling.
Red crosses and associated error bars represent the mean observed 3.1 Intrinsic merger rate
value for each bin. The observational points represent average quantities Having constructed galaxy merger trees, computing the merger rate
constructed from Li et al. (2006); Guo et al. (2011) and Yang et al. (2012).
 is straight forward. In the trees we identify galaxy mergers as any
 pair of galaxies sharing an identical descendant galaxy. In each case
and the star formation rate will be set to 0. The quenching timescale we assume every merger is binary and occurs instantaneously at
can be parameterised as: .3 We provide two measures for the merger rate: merger rate
   − s per comoving volume and merger rate per galaxy. The process is
 ∗ similar in each case, with the difference arising in how the rate
 = dyn 0 . (11)
 1010 is normalised. The first step is to construct bins for time, stellar
Here dyn is the halo’s dynamical time and 0 is normalization, mass and mass ratio. For each time bin we count the number of
which together specifying a minimum quenching time of dyn 0 .
The normalisation determines the quenching timescale for galaxies 3 This binary assumption holds true for > 98 per cent of all mergers with a
with ∗ ≥ 1010 while the slope s describes the quenching descendant mass > 109 . Despite this small number of cases we nonethe-
timescale for low mass galaxies. These parameters are largely con- less adjust the descendant mass of each merger to ensure the descendant
strained by the observed fraction of quenched galaxies at several mass only reflects mass contributed by the most massive progenitor and the
redshifts. merging satellite.

MNRAS 000, 1–23 (2020)

6 J. A. O’Leary et al.
mergers which satisfy the mass and mass ratio requirements, and The merger rate per comoving volume (Γ) exhibits a nearly
whose merging time ( ) resides within that bin. When computing mass independent shape in the number of mergers occurring. For
the merger rate per comoving volume we then divide the number each mass and mass ratio interval we find a sharp increase in the
of mergers ( merge ) in each bin by the bin widths d and by the number of mergers at low redshift, with a well defined peak 1 .
volume of our box, so the merger rate is given by: . 2. Beyond the peak we see a rapid decay in the total number
 merge h i of mergers towards high redshift. When we instead take the merger
Γ= cMpc−3 Gyr−1 . (12) rate in the context of an evolving galaxy population (ℜmerge ) we find
 d d 
For the merger rate per galaxy we instead divide the number of quite a different trend. In general we find the merger rate increases
mergers in each bin by the bin width, and the number of galaxies with redshift at all masses and mass ratios. However, our results
( gal ) at the central redshift of the bin. At these redshifts gal is do not show a simple power-law scaling with redshift. For major
determined by linearly interpolating counts between the nearest two mergers, at low and intermediate mass, we find an excess over a
snapshots. The merger rate per galaxy is then described by: power-law for & 3. This break from a power-law redshift scaling
 is primarily evident when selecting mergers based on descendant
 merge h i
ℜ= Gyr−1 (13) galaxy stellar mass.
 gal d Additionally, We can observe a discrepancy in the merger rate
 While operating on the same data the two measures for merger when selecting merger rates based on progenitor mass vs. descen-
rate produce qualitative differences due to the scaling of the merger dant mass. In the lowest mass bins the difference in these two
rate per galaxy with the number density of galaxies. For this reason quantities only becomes manifest at higher redshifts ( & 2), where
the merger rate per galaxy is often the preferred measure as it is more selecting based on progenitor mass produces a noticeably lower
robust against cosmic variance. We explore both rate measures to merger rate. However, for the most massive galaxies the difference
provide a more complete comparison with other works. is more dramatic. We find that in this mass range both measures
 produce functionally similar results, with a nearly constant scaling
 offset, where the descendant mass selected major merger rate is a
3.1.1 Scaling with mass and redshift factor of 1.5 − 2 larger than that produced with progenitor selection.
Figure 3 illustrates the merger rate scaling with redshift for three To explain this, we should recall that in our approach the average
different mass bins. For each mass bin we select mergers based number of galaxies remains fixed for any time and mass interval,
on descendant mass 0 (solid lines), or main progenitor mass 1 that is gal of eq. 13 remains the same regardless of whether we
(dashed lines). Selection based on descendant mass probes galaxy compute the rate based on descendant mass or progenitor mass.
mergers after their mass contributions have been added into the Additionally, for a lower mass threshold, there are in general more
descendant galaxy. This is a common approached taken by other galaxies that fit within our mass range at timestep compared to
theoretical works exploring galaxy merger rates. This method pre- − 1. At the highest masses and redshifts this effect is amplified
sumes we have knowledge on how the mass of two galaxies is added due to the low numbers of galaxies present. This is similar to the
together in a merger, something not easily accessible to observa- argument by Genel et al. (2009) to explain such differences in the
tions. Selecting mergers based on progenitor properties provides a context of halo-halo merger rates.
more observer friendly measure of the merger rate as it can be more Lastly, we can look across the mass panels to see how the
directly compared with observationally derived rates (see Section frequency of major mergers changes with redshift. For low-mass
4.1). galaxies it is clear that minor and mini mergers dominate. When
 In each mass band we show the merger rate for three different moving to intermediate-mass ranges, galaxies like the Milky Way,
mass ratio intervals. The first, and arguably the most important, we find that for the first half of cosmic time minor and mini merg-
are called major mergers (blue lines), although the precise mass ers occur at greater frequency than major mergers. Near ≈ 1 this
ratio that defines a major merger is not well defined and varies changes, as major mergers become more frequent than minor merg-
across literature. The key characteristic of major mergers are their ers and occur at nearly the same frequency as mini mergers by = 0.
transformative properties. Such mergers are very disruptive to both Finally, for the most massive galaxies, at . 3 major mergers are
systems, suspected of prompting drastic changes in stellar popu- the most frequent, with mini mergers being the next most common,
lations and descendant morphologies. In this paper we take major and minor mergers making up the smallest fraction of mergers. We
mergers to be 1 ≤ ≤ 4. The next mass ratio interval are similarly explore these strong mass ratio dependencies of the merger rate in
labelled minor mergers (green lines). Minor mergers while not as closer detail in the next section.
individually disruptive as their larger counterparts, still contribute
to the evolutionary process of large galaxies. There is evidence to
 3.1.2 Scaling with mass ratio
suggest that such mergers, if occurring at high enough frequency,
can produce some of the same morphological changes generated In this section we address the galaxy-galaxy merger rate scaling with
through major mergers (Naab et al. 2009; Oser et al. 2010; Hilz mass ratio. We have already seen in the previous section that the
et al. 2012, 2013; Karademir et al. 2019), lead to the thickening of merger rate density and rate per galaxy appear to possess a strong
disc galaxies (Abadi et al. 2003; Kazantzidis et al. 2008; Purcell scaling with mass for a fixed mass ratio interval. Most notably, major
et al. 2009; Moster et al. 2010, 2013), and even drive the rotation mergers dominate at high-mass, and mini/minor mergers dominate
speed of massive early type galaxies (Bois et al. 2011; Moody et al. at low masses. Our goal here is to investigate this inflection with a
2014; Penoyre et al. 2017; Schulze et al. 2018) . The final category finer binning in order to explain this phenomenon.
are so called mini mergers (orange lines). As their name entails, this Figure 4 explores the relationship between descendant stellar
category represents the smallest merging events. While not terribly mass and merger mass ratio. To provide a more granular view of the
transformative, understanding their frequency is helpful for con- mass ratio distribution, without influence from the explicit definition
structing a complete mass budget and internal radial distribution for of major(minor) merger, we express these results as the merger rate
the accreted material of a galaxy. per galaxy per log10 mass ratio interval merge /dlog10 ( )/d . In

 MNRAS 000, 1–23 (2020)

Galaxy merger rates with Emerge 7

 9 log 10 (m/M¯ ) < 10 10 log 10 (m/M¯ ) < 11 11 log 10 (m/M¯ ) < 12
 10-3 1 µ 4
 4 < µ 10
 10 < µ 100
Γ(z)[cMpc −3 Gyr −1 ]

 10-4

 10-5

 descendant mass m0
 10-6 progenitor mass m1

 100
(z)[Gyr −1 ]

 10-1

 10-2

 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 6
 Redshift Redshift Redshift

Figure 3. The galaxy-galaxy merger rates as a function of redshift and mass ratio, ≡ 1 / 2 . The top row of panels show the merger rate density Γ( ), i.e.
the total number of mergers per comoving volume, and the lower panels show the merger rate per galaxy ℜ( ). Solid lines indicate galaxy mergers selected
based on the descendant mass of the merging system, 0 . In this scenario the progenitor galaxies ( 1 , 2 ) are permitted to have masses outside the noted
mass bands. The dashed lines show mergers selected based on the main progenitor mass ( 1 ) of the merging systems. Here only the main progenitor must
reside in the specified mass band. The shaded regions indicate Poisson error in the number of mergers.

the top panel we take mergers of a fixed mass ratio at a fixed minor scaling differences with increasing redshift. Below ≈ 10
redshift. We then bin those galaxies according to the stellar mass we find a greater dependence on mass. In this regime the most
of their descendant system. This way we are able to see the relative massive galaxies maintain a nearly constant slope for all , even
importance of some merger as a function of mass. We selected four experiencing a boost in the major merger rate.. Conversely, for low-
target mass ratios with a uniform log space bin of 0.45 dex. In mass and intermediate-mass systems we see a dropping merger rate,
general each mass ratio shows a similar qualitative trend. At lower illustrating suppressed major mergers. Narrowing in on galaxies
masses the slope stays relatively flat, approaching high masses each with log10 ( 0 / ) = 10.5, we see an interesting evolution. At
curve show an inflection point when more massive galaxies begin to ≈ 2 these galaxies scale with mass ratio similar to the lowest mass
experience more mergers. Looking closely we see that the inflection galaxies, with a suppressed rate for . 10. As we transition to lower
occurs at lower stellar mass for lower redshifts. redshift we can see this relationship change, with major mergers
 becoming more prevalent at lower redshift, and the once decreasing
 Similarly, The bottom panel of Figure 4 shows how merger trend bending up to meet the clean scaling seen for massive galaxies.
mass ratios are distributed for a fixed descendant mass and redshift. Both of these trends can be explained in the context of an evolving
This plot is analogous to Figure 1 for the halo-halo merger rate. In stellar-to-halo mass relation (SHMR).
this comparison we can immediately note some glaring differences
compared to the halo-halo merger rate, most notably the galaxy- First, why are major mergers suppressed for low-mass galaxies?
galaxy merger rate does not show the same mass independent merger The first driver for this effect can be seen directly in the SHMR.
rate with respect to mass ratio. In the case of galaxy-galaxy mergers Figure 5 shows the SHMR present in our model for galaxies at = 0.
we break from a simple power law to a more complex relation with a Once again recalling the simple relation assumed by the halo-halo
clear redshift, mass ratio and strong mass dependence. For mergers merger rate (Figure 1), we can trace how a major merger in halo mass
with & 10 all masses exhibit a similar merger rate, with some would translate to mergers in galaxy stellar mass. In this thought

MNRAS 000, 1–23 (2020)

8 J. A. O’Leary et al.

 z ≈ 0.1 z ≈ 1.0 z ≈ 2.0
 100
dN merge /dlog 10 (µ)/dt [Gyr −1 ]

 10-1

 µ≈4
 10-2 µ ≈ 10
 µ ≈ 100
 µ ≈ 1000
 108 109 1010 1011 1012108 109 1010 1011 1012108 109 1010 1011
 Descendent Mass, m0 [M¯ ]

 z ≈ 0.1 z ≈ 1.0 z ≈ 2.0
 100
dN merge /dlog 10 (µ)/dt [Gyr −1 ]

 10-1
 log 10 (m0 /M ¯ ) = 9.0
 10-2 log 10 (m0 /M ¯ ) = 9.5
 log 10 (m0 /M ¯ ) = 10.0
 log 10 (m0 /M ¯ ) = 10.5
 10-3 µ=4 G09 Halo merger rate
 log 10 (m0 /M ¯ ) = 11.0
 FMBK10 Halo merger rate
 101 102 103 104 101 102 103 104 101 102 103 104
 Mass Ratio, µ

Figure 4. The galaxy-galaxy merger rate as a function of descendant stellar mass and progenitor mass ratio. Top panel: The merger rate per galaxy as a function
of descendant stellar mass, 0 . The coloured lines specify the merger rate for a specific mass ratio . Shaded regions indicate Poisson error in the number of
mergers. Bottom panel: The merger rate per galaxy as a function of progenitor mass ratio. Each line includes mergers with a descendant mass noted by the
colour. The x-axis shows the distribution of merger mass ratios experienced within each mass band. The vertical dashed black line denotes the threshold for
major mergers at = 4. Shaded regions surrounding solid lines indicate Poisson error in the number of mergers. The red and blue shaded regions show the
best fit halo-halo merger rate for haloes with 11.25 ≤ log10 ( / ) < 13, from Genel et al. (2009) and Fakhouri et al. (2010) respectively.

experiment we can make the presumption that any halo-halo merger log10 ( 0 / ) ≈ 10.5 have suppressed major mergers at high
will eventually result in a galaxy-galaxy merger. Starting along the redshift but not at low redshift? Here we move beyond the static
low-mass slope of the SHMR, we note that if we select the average low redshift SHMR, and instead focus on how the SHMR evolves.
galaxy masses for a fixed halo mass, a major merger in halo mass These intermediate-mass galaxies reside close to the turnover on
would translate to a mini-merger ( ≈ 30) in galaxy stellar mass. the SHMR. From = 2 to = 0 we see the average halo mass for
Conversely, we can observe the opposite scenario on the high-mass such galaxies increase by ∼ 0.08 dex. That is these galaxies tend
slope, beyond the turnover. Due to the shallow slope in the SHMR to live in larger haloes at lower redshift. Additionally, the turnover
for high masses we can see that a major halo-halo merger has a in the SHMR has a mild shift to lower halo mass by ∼ 0.17 dex.
much larger likelihood of also leading to a major merger in galaxy These combined effects mean these galaxies tend to sit higher along
stellar mass. In short, if the slope of the SHMR is unity, we would the turnover where the slope approaches unity. Thus, these galaxies
expect a major halo-halo merger to directly lead to a major galaxy- begin to experience more major mergers with decreasing redshift.
galaxy merger. Subsequently, where the slope is greater than unity This can be clearly seen in the much flatter major merger rate scaling
we expect a suppression of major mergers, and where less than unity with redshift shown in Figure 3. The suppression and enhancement
we expect an enhanced rate of major mergers. The second driving of major galaxy mergers as a consequence of the SHMR has been
factor is dynamical friction. We see that small satellites orbiting explored in the context of other semi-empirical models (Stewart
a massive central galaxies have much longer dynamical frictions et al. 2009; Hopkins et al. 2010a). However, it is important to stress
times (eq. 7) and would simply not have had enough time to merge. the necessity of a model that can accurately reproduce the observed
This effect prevents minor halo mergers from being transformed data (e.g. SMFs, cosmic and specific SFRs). Models that show sig-
into major mergers where the stellar mass ratio of the two systems nificant deviations from these fundamental observations, or cannot
would otherwise be sufficient. self-consistently track their redshift evolution may arrive at differing
 Now, why do we see that intermediate-mass galaxies with conclusions regarding galaxy merger rates or galaxy mass assembly.

 MNRAS 000, 1–23 (2020)

Galaxy merger rates with Emerge 9
 where is the age of the universe at redshift , and Ψ is the specific
 star formation rate.
 In Figure 7 we illustrate how the major merger rate scales
 when selecting mergers based on the star formation properties of
 their progenitor and descendant systems. We perform this in two
 mass bands for galaxies with log10 ( )/ ≥ 10. For each mass
 bin we compute a global merger rate, only considering the star
 formation of the progenitor galaxies. Additionally, we perform the
 same analysis only considering mergers with a quenched descen-
 dant galaxy. We designate four different scenarios based on the star
 forming properties of the progenitor galaxies:
 Active-active: Both progenitors are active.
 Passive-passive: Both progenitors are passive.
 Passive-active: The main progenitor is passive and the secondary
 progenitor is active.
 Active-passive: The main progenitor is active and the secondary
 progenitor is passive.
 In Figure 7 we compare the total merger rates when consid-
 ering all galaxies (upper panels) versus only considering mergers
 with a quenched descendant galaxy (lower panels). For the redshift
Figure 5. The = 0 stellar-to-halo mass relation produced by emerge. The range shown we find very little difference in the total merger rates.
solid line illustrates the best fit over the mean values of the data. The shaded This suggests that most major mergers are occurring in dense envi-
regions show how a major merger in halo mass ( = 4) translates to ronments around an already quenched central galaxy. For the most
galaxy-stellar mass ratio ( ) on both the low and high-mass end of the
 massive galaxies, by ≈ 1 most mergers are occurring between two
stellar to halo mass relation. Along the best fit curve we can see that a major
 passive galaxies (red lines), or between a passive central galaxy and
halo-halo merger on the low-mass end will result in a very minor merger in
stellar mass. On the high-mass end a major halo-halo merger will result in a an active satellite (yellow lines). Beyond ≈ 1 most mergers are
major merger in stellar mass. occurring between active galaxies (blue lines). When considering
 only mergers with a quenched descendant (bottom panel) we find
 a nearly constant merger rate if the major galaxy in the merger is
 Finally, in Figure 6 we provide the cumulative galaxy-galaxy already quenched. There are several different effects that lead to this
merger rates with respect to mass ratio. The information contained result. The first is the prevalence of gas-rich (active) galaxies at high
in the cumulative merger rates is identical to that of Figure 4. Absent redshift making the likelihood of active-active mergers greater. The
a generalised fitting function for our results, the cumulative rates second being that if a central galaxy is quenched, it is likely that its
provide a quicker reference for determining the number of mergers descendant galaxy will also be quenched. Due to the lack of passive
occurring at some descendant mass for a given mass ratio interval. galaxies at high these results are more uncertain than the results
 shown in Figure 3. For the lower mass bin, mergers involving any
 quenched component cannot be constrained to better than a factor
3.1.3 Active vs. Passive galaxies ∼ 2 − 3 for & 1.5. For the higher mass bin we see a similar degree
 of uncertainty for & 2.25
Finally we address SFR dependencies of the cosmic galaxy-galaxy
merger rate. Once again looking back to the baryon conversion effi-
ciency (eq. 2) of galaxies we can see a characteristic halo mass (see
 3.2 Comparison to other theoretical predictions
Table 1 and eq. 3) at which a galaxy is most efficient at converting gas
into stars. One conclusion from this relation is that larger galaxies Although most theoretical predictions are based on the same ΛCDM
are inefficient at creating new stars. Thus it is important to under- framework, the methods used to link DM haloes and galaxy proper-
stand the merger rate for these specific galaxies to learn how galaxy ties has direct consequences on the predicted merger rates. Figure 8
mergers drive their galaxies’ continued mass growth (Khochfar & displays a side-by-side comparison of galaxy merger rates in three
Silk 2009). Specifically we would like to know if these galaxies are different mass bands produced by a diverse set of models. While
grown through the merging of other large quenched galaxies, or con- our results might initially be surprising, we can see that within
structed more slowly through the accretion of smaller star forming the context of other theoretical predictions we are firmly within a
satellites. Further, understanding these mergers may help explain previously established range of merger rates.
how mergers initiate star formation, or power AGN (Hirschmann Theoretical methods for determining the merger rate can be
et al. 2010, 2014, 2017; Weinberger et al. 2017; Choi et al. 2018; roughly broken down into a few categories:
Steinborn et al. 2018).
 (i) Halo-halo: Assume an average halo-halo merger rate with
 We begin by defining our galaxies in terms of their star forma-
 a dynamical friction delay set at vir . Haloes are populated with
tion properties. Broadly this means designating a galaxy as passive
 galaxies according to a SHMR (e.g. Hopkins et al. 2010a).
or active, where passive galaxies are quenched and active galaxies
 (ii) Subhalo disruption: Subhalo disruption rates are convolved
are actively star forming. We adopt the quenching criteria of Franx
 with some SHMR with no delay applied (e.g. Stewart et al. 2009;
et al. (2008) to make this distinction, where a galaxy is considered
 Rodríguez-Puebla et al. 2017).
quenched if:
 (iii) Halo merger trees without substructure: Halo merger trees
Ψ < 0.3 −1
 , (14) (EPS or -body) are populated with galaxies according to a model.

MNRAS 000, 1–23 (2020)

10 J. A. O’Leary et al.

 z ≈ 0.1 z ≈ 1.0 z ≈ 2.0
 100
dN merge ( µ)/ dt [Gyr −1 ]

 10-1

 10-2 µ=4
 µ = 10
 µ = 100
 µ = 1000
 10-3
 108 109 1010 1011 1012 108 109 1010 1011 1012 108 109 1010 1011
 Descendent Mass, m0 [M¯ ]

 100 z ≈ 0.1 z ≈ 1.0 z ≈ 2.0
dN merge ( µ)/dt [Gyr −1 ]

 10-1

 10-2 log 10 (m0 /M ¯ ) = 9.0
 log 10 (m0 /M ¯ ) = 9.5
 log 10 (m0 /M ¯ ) = 10.0
 10-3 µ=4 log 10 (m0 /M ¯ ) = 10.5
 log 10 (m0 /M ¯ ) = 11.0
 101 102 103 104 101 102 103 104 101 102 103 104
 Mass Ratio, µ

Figure 6. The cumulative galaxy-galaxy merger rate as a function of descendant stellar mass and progenitor mass ratio. The data shown here is the cumulative
histogram of the data displayed in Fig. 4. The cumulative merger rate can be used to determine the merger rate for a desired mass and mass ratio. Top panel: The
cumulative merger rate per galaxy as a function of descendant stellar mass, 0 . The coloured lines each specify a different mass ratio threshold, and include all
mergers below that threshold. Bottom panel: The merger rate per mass ratio interval. Each line includes mergers with a descendant mass noted by the colour.
The x-axis therefore shows the distribution of merger mass ratios experienced within each mass band. The black line denotes the threshold for major mergers
 = 4. As an example (square black point) we can see that at = 0.1 for log10 ( 0 / ) = 11.0 approximately 4 per cent of galaxies will experience a merger
in the next Gyr.

A dynamical friction delay is applied to satellites at vir (Somerville cess in each modelling strategy, but do not classify all model options
et al. 2008; Khochfar & Silk 2009; Font et al. 2008). that could impact the merger rate. In particular the calibrated or
 (iv) Halo merger trees with substructure: Sub-haloes are tracked predicted SMF, as well as the chosen treatments for orphan/satellite
within an -body simulation, where galaxies are populated accord- stripping could play a factor in the resulting merger rates within each
ing to a model. A dynamical friction delay is applied to satellite model. Any model implementations that could impact the SHMR
galaxies when their -body subhalo is disrupted (e.g. this work, or cause premature/prolonged merging would also impact the rates,
Bower et al. 2006; Croton et al. 2006; De Lucia & Blaizot 2007). regardless of the guiding dark matter component of the model. A
 (v) Hydro: The baryonic components of galaxies in hydrody- more complete overview of the models shown in Figure 8, as well
namical simulations are tracked to determine when they coalesce as the systematic uncertainties in determining merger rates can be
(e.g. Rodriguez-Gomez et al. 2015). found in Hopkins et al. (2010b) and Rodriguez-Gomez et al. (2015).
 In the next section we will explore how our own model assumptions
 At low and intermediate masses our model predicts a merger
 and treatments for orphans/satellites can impact the merger rate.
rate as much as an order of magnitude lower than some other predic-
tions. Additionally, due to the more shallow scaling with redshift,
our results deviate from other models more strongly at higher red-
 3.2.1 Impact of model assumptions
shifts, particularly at intermediate masses. We do, however, find our
results to be in good agreement with those of Bower et al. (2006), Although the purpose of this work is not to address the systemics
who employ a semi-analytic model on top of -body merger trees. that impact the predicted merger rates across the range of available
At the highest masses models tend to agree more closely in terms models, we can address known sources of uncertainty, and deter-
of magnitude and redshift scaling of the merger rate. mine what role our model assumptions play in the resulting merger
 The categories noted above broadly describe the merging pro- rates. In this section we will vary core components of emerge and

 MNRAS 000, 1–23 (2020)

Galaxy merger rates with Emerge 11

 10 log 10 (m/M¯ ) < 11 11 log 10 (m/M¯ ) < 12
 100

 10-1
(z)[Gyr −1 ]

 10-2

 all
 10-3 active-active
 passive-passive
 10-4 passive-active
 active-passive

 100 quenched descendant quenched descendant

 10-1
(z)[Gyr −1 ]

 10-2

 10-3

 10-4
 0 1 2 0 1 2 3
 Redshift Redshift

Figure 7. The major merger rate ( ≤ 4) as a function redshift for various active/passive progenitor configurations. We show the four possible scenarios for
active/passive combinations in progenitor systems. Active-active would specify a merger where both progenitor galaxies are actively star forming, passive-active
specifies a configuration where the main progenitor is passive (quenched) and the secondary progenitor is active (star forming), etc. Top Panel: The galaxy
merger rate dependency on star forming properties of the progenitor galaxies. The black line shows the total merger rate for all star forming configurations of
the descendant system. Bottom Panel: The galaxy merger rates only considering mergers with a quenched descendant galaxy. At high redshift most mergers
occur between active progenitors. There is a transition near ≈ 0.5 where mergers start to become dominated by passive progenitors.

analyse the resulting merger rates to determine if any of our model (iv) All satellites as orphans. Orphan galaxies are initiated when
assumptions are suppressing merger rates compared to other em- one halo enters the virial radius of another for the first time.
pirical approaches (i.e. Stewart et al. 2009; Hopkins et al. 2010a). (v) No satellites. Galaxies are merged at the same time as their
Throughout this section we take our standard model implementation host haloes.
with the parameters of Table 1 as our reference. We only explore (vi) Increased resolution. Standard Emerge options applied to a
the impact these model assumptions have on the major merger rate simulation with a much higher particle/halo resolution.
with 1 ≤ ≤ 3. We chose this definition of major merger for ease (vii) Average halo-halo merger rates with rank ordered abun-
of comparison with Hopkins et al. (2010a). Figure 9 illustrates the dance matching, using the SHMR derived directly from our refer-
results of this study alongside Hopkins et al. (2010a). As we vary ence model.
each model element we do not refit for each model permutation. (viii) Average halo-halo merger rates and abundance matching
Consequently, these model variations do not reproduce all observa- with scatter allowed in the SHMR.
tions as accurately as the reference case. Our model permutations
cover the following: In section 2.5 we described our methodology for stripping
 satellite galaxies, and our newly implemented approximation for
 halo mass-loss in orphan galaxies, which presents the possibility
 (i) Impact from our satellite stripping implementation. that our implementation may strip orphan satellites too strongly,
 (ii) Swapping the dynamical friction formula from Boylan- suppressing the merger rate. We can provide a simple check for this
Kolchin et al. (2008) to Binney & Tremaine (1987) . scenario by setting the stripping parameter = 0. This ensures
 (iii) No orphan galaxies. Satellites galaxies are merged onto their that all satellites with a short enough dynamical friction will merge
host when their sub-haloes are lost in the simulation. and contribute to the computed merger rate. In Figure 9 this case

MNRAS 000, 1–23 (2020)

12 J. A. O’Leary et al.

 9 log 10 (m/M¯ ) < 10 10 log 10 (m/M¯ ) < 11 11 log 10 (m/M¯ ) < 12
 100 Emerge 1 µ 3

 10-1
(z)[Gyr −1 ]

 Semi-Analytic:
 10-2 Croton+ (2006)
 De Lucia+ (2007)
 Somerville+ (2008)
 10-3 Empirical: Khochfar+ (2009)
 Hopkins+ (2010) Hydro Simulations: Bower+ (2006)
 Stewart+ (2009) Rodriguez-Gomez+ (2015) Font+ (2008)
 10-40.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
 Redshift Redshift Redshift

Figure 8. A comparison of merger rate per galaxy ( ≤ 3) for other theoretical models, adapted from Hopkins et al. (2010b). Semi-analytic: Croton et al.
(2006); De Lucia & Blaizot (2007); Somerville et al. (2008); Khochfar & Silk (2009); Bower et al. (2006); Font et al. (2008). Empirical: Hopkins et al. (2010a);
Stewart et al. (2009). Hydrodynamical simulations: Rodriguez-Gomez et al. (2015). In general our model produces a rate lower than most other models, and is
in agreement with the results of Bower et al. (2006).

 9 log 10 (m/M¯ ) < 10 10 log 10 (m/M¯ ) < 11 11 log 10 (m/M¯ ) < 12
 100 1 µ 3

 10-1
(z)[Gyr −1 ]

 Reference
 10-2 fs = 0
 BT87 tdf
 No delay from subhalo destruction
 10-3 FMBK10 + AM Delay applied at Rvir
 FMBK10 + AM + scatter No delay from halo-halo merger
 Hopkins+ (2010) TNG100-Dark
 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 6
 Redshift Redshift Redshift

Figure 9. A comparison of major merger rates (1 ≤ ≤ 3) under various model assumptions. Dashed lines indicate merger rates derived using average
halo-halo merger rates convolved with a SMF. Solid lines illustrate results derived directly from emerge galaxy merger trees. The solid black line represents the
reference results using our standard model shown throughout this paper. Coloured solid lines show our results under different implementations for dynamical
friction, satellite evolution, and simulation resolution. Methods using average halo-halo merger rates overpredict galaxy merger rates compared to full tree
based methods in all cases. This over-prediction becomes more extreme at lower stellar masses.

is illustrated by the blue line labelled ‘ = 0’. We find a ∼ 33 per alescence. While this formulation should provide a more physical
cent boost in the major merger rate at intermediate masses, but at description for the satellite decay process, we can nonetheless ex-
low and high masses we see no discernible change in the merger plore merger rate with the more classical dynamical friction formula
rate. Though not displayed in Figure 9 we also verified that our provided by Binney & Tremaine (1987). We find that with respect
merger rates are nearly unchanged within the ± 1 parameter to major mergers the choice of dynamical friction makes little dif-
range noted in Table 1. Also, setting = 0.1 did not reduce the ference for massive galaxies. At intermediate and low mass we find
merger by more than a factor ∼ 2 compared to the reference, though lower merger rates compared to our reference case. This effect is
we note that both the = 0 and = 0.1 cases do not reproduce most pronounced at high redshift and intermediate masses where
the local clustering data well. We can conclude from these tests that we see a ∼ 56 per cent difference from the reference.
our major merger rates are not strongly impacted by our current
 The need for orphans in modelling has been, by this point,
implementation for halo mass loss or stripping.
 thoroughly discussed (e.g. Campbell et al. 2018; Behroozi et al.
 The models shown in Figure 8 adopt a wide range of dynam- 2019). Where models continue to differ is in the precise treatment
ical friction formulations. The use of Boylan-Kolchin et al. (2008) of orphan galaxies, with a core difference being when orphans are
dynamical friction has by now become the standard for control- placed into the simulation. It has been argued that models relying
ling satellite decay, and builds upon earlier work by tuning the on -body trees that track substructure experience lower merger
merging timescales using a suite of idealised -body mergers that rates due to overly effective tidal stripping of subhaloes with the
track haloes and their baryonic components from infall to final co- absence of baryons (Hopkins et al. 2010b). Furthermore, it has

 MNRAS 000, 1–23 (2020)