# Identifying the progenitors of present-day early-type galaxies in
observational surveys: correcting ‘progenitor bias’ using the Horizon-AGN
simulation

G. Martin,1 S. Kaviraj,1 J. E. G. Devriendt,2 Y. Dubois,3 C. Pichon,3,4 and C.
Laigle2
1Centre for Astrophysics Research, School of Physics, Astronomy and
Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK
2Dept of Physics, University of Oxford, Keble Road, Oxford OX1 3RH UK
3Institut d’Astrophysique de Paris, Sorbonne Universités, UMPC Univ Paris 06
et CNRS, UMP 7095, 98 bis bd Arago, 75014 Paris, France
4Korea Institute of Advanced Studies (KIAS) 85 Hoegiro, Dongdaemun-gu, Seoul,
02455, Republic of Korea E-mail<EMAIL_ADDRESS>
###### Abstract

As endpoints of the hierarchical mass-assembly process, the stellar
populations of local early-type galaxies encode the assembly history of
galaxies over cosmic time. We use Horizon-AGN, a cosmological hydrodynamical
simulation, to study the merger histories of local early-type galaxies and
track how the morphological mix of their progenitors evolves over time. We
provide a framework for alleviating ‘progenitor bias’ – the bias that occurs
if one uses only early-type galaxies to study the progenitor population.
Early-types attain their final morphology at relatively early epochs – by
$z\sim 1$, around 60 per cent of today’s early-types have had their last
significant merger. At all redshifts, the majority of mergers have one late-
type progenitor, with late-late mergers dominating at $z>1.5$ and early-early
mergers becoming significant only at $z<0.5$. Progenitor bias is severe at all
but the lowest redshifts – e.g. at $z\sim 0.6$, less than 50 per cent of the
stellar mass in today’s early-types is actually in progenitors with early-type
morphology, while, at $z\sim 2$, studying only early-types misses almost all
(80 per cent) of the stellar mass that eventually ends up in local early-type
systems. At high redshift, almost all massive late-type galaxies, regardless
of their local environment or star-formation rate, are progenitors of local
early-type galaxies, as are lower-mass (M⋆ $<$ 1010.5 M⊙) late types as long
as they reside in high density environments. In this new era of large
observational surveys (e.g. LSST, JWST), this study provides a framework for
studying how today’s early-type galaxies have been built up over cosmic time.

###### keywords:

methods: numerical – galaxies: evolution – galaxies: formation – galaxies:
high-redshift – galaxies: statistics

††pubyear: 2017††pagerange: Identifying the progenitors of present-day early-
type galaxies in observational surveys: correcting ‘progenitor bias’ using the
Horizon-AGN simulation–A

##  1 Introduction

In the standard $\Lambda$CDM paradigm, galaxy formation proceeds
hierarchically. Dark matter halos, which arise as a result of primordial
fluctuations in the initial matter density field (Starobinsky, 1982; Guth &
Pi, 1982; Hawking, 1982), merge to form progressively more massive haloes over
cosmic time (e.g. Blumenthal et al., 1984; Kauffmann et al., 1993; Somerville
& Primack, 1999). Cold gas condenses into these halos where it forms
rotationally-supported discs (Franx & van Dokkum, 1996). The rate of star-
formation is determined by the local density of this cold gas (Kennicutt,
1998), with feedback from supernovae (Scannapieco et al., 2008) and active
galactic nuclei (AGN) (e.g. Silk & Rees, 1998; Kaviraj et al., 2017)
regulating the process of stellar mass growth. A consequence of this paradigm
is that the stellar mass of an individual galaxy is assembled through a
combination of in-situ star-formation i.e. by gas turning into stars within a
galaxy’s own halo, and ex-situ star-formation i.e. stars formed in another
halo which have become members of the halo in question as a result of merging
(e.g. Kauffmann et al., 1993). As ‘end-points’ of this hierarchical assembly
process, local ‘early-type’ (i.e. elliptical and S0) galaxies are a
particularly significant class of objects (e.g. Kaviraj et al., 2007). These
galaxies dominate the stellar mass density in today’s Universe (e.g. Bernardi
et al., 2003) and thus encode, in their stellar populations, the signatures of
galaxy mass assembly over cosmic time (e.g. Worthey, 1994; van Dokkum & Franx,
2001). Studying these galaxies offers unique insights into the build up of the
observable Universe (e.g. Barrientos & Lilly, 2003; Longhetti et al., 2005;
McDermid et al., 2015) and significant effort in the literature has,
therefore, been dedicated to understanding these systems.

Galaxies in the early universe form with disc-like (late-type) morphologies
and, through interactions and secular processes, acquire more spheroidal
(early-type) morphologies over time (e.g. Franx & van Dokkum, 1996; van Dokkum
& Franx, 2001; Buitrago et al., 2014; Conselice, 2014; Kaviraj, 2014b).
Consequently, at progressively higher redshift, the progenitors of today’s
early-types become increasingly dominated by late-type systems. Understanding
the formation and evolution of today’s early-types therefore requires us to
consider their entire progenitor population, especially their late-type
progenitors at earlier epochs. If, as is often assumed, early-type galaxies do
not revert back to late-types (e.g. Hau et al., 2008; Fang et al.,
2012)111Dubois et al. (2016) have shown that transformation from early-type to
late-type morphology is prevented by AGN feedback., techniques that identify
late-type galaxies that are progenitors of today’s early-types in
observational surveys are essential. This becomes particularly important at
high redshift, e.g. $z>2$, the redshift regime at which the early-type
population is rapidly assembled (e.g. Conselice, 2014), and which can be
routinely accessed by forthcoming facilities such as JWST (Gardner et al.,
2006), EUCLID (Laureijs et al., 2011), etc.

Past observational studies (e.g. Gladders et al., 1998; Stanford et al., 1998)
have attempted to trace the assembly of present-day early-types by focussing
only on the population of early-type galaxies at high-redshift. Since the
stellar populations of present-day early-types are largely in place in the
early Universe (e.g. Trager et al., 2000; Thomas et al., 2005; Kaviraj et al.,
2011; Kaviraj, 2014a), this is a reasonable approximation at low and
intermediate redshift ($z\sim 1$), assuming that galaxies, once they achieve
early-type morphology, cannot revert back to being late-type systems. However,
as noted above, at earlier epochs, an increasing proportion of the mass in
present-day early-types is contained in late-type progenitors. Thus,
considering only early-type galaxies introduces a bias (the so-called
‘progenitor bias’) in any study of their evolution, which becomes
progressively more severe with increasing redshift.

In a similar vein, other observational work (e.g. Bell et al., 2004) has often
used the optical red sequence (Faber et al., 2007; Barro et al., 2013) as a
proxy for the population of progenitors of early-type galaxies. However, since
the stellar populations in late-type galaxies tend to be younger (and
therefore bluer), such a colour cut misses the majority of the late-type
galaxies that are progenitors of local early-types. In addition, since a wide
variety of star-formation histories are observed in early-type galaxies
themselves, particularly at high redshift (Kaviraj et al., 2013; Fitzpatrick &
Graves, 2015; Lofthouse et al., 2017a), a large fraction of blue galaxies that
already have early-type morphology will also be missed (Shankar et al., 2015),
if such a colour cut is employed. While the red sequence traces the
progenitors of early-type galaxies well at the highest end of the luminosity
function (Kaviraj et al., 2009), it becomes less reliable at low masses (where
galaxies are bluer) and fails to identify early and late-type progenitors (of
all stellar masses) that lie blueward of the red sequence. In a general sense,
therefore, a simple colour cut is not a reliable approximation for the
progenitor population of today’s early-type galaxies. And, in a similar vein
to using an early-type selection to identify the progenitor population, the
red-sequence approximation becomes progressively less effective with
increasing redshift.

Progenitor bias is difficult to overcome observationally, since individual
galaxies cannot be observed as they evolve and, therefore, in any given
survey, it is difficult to directly identify objects that will end up in
early-type galaxies at $z\sim 0$. Nevertheless, observational methods have
been applied to reduce or mitigate this issue. For example, a widely-used
technique, proposed by van Dokkum et al. (2010) assumes that cumulative co-
moving number density is conserved as the galaxy mass function evolves with
time (i.e. as a galaxy evolves, the number of galaxies more massive than it
remains constant), so that a given galaxy maintains the same rank. However,
while this method is able to account for the mass evolution of the galaxy
population, the assumptions made may be too simplistic. For example, it
ignores the fact that a galaxy’s rank may change over time, resulting in the
evolution of its co-moving number density. This may occur if, for instance,
galaxies are removed from the population as a result of mergers (Ownsworth et
al., 2014), or as a result of the spread in specific star-formation rate
(sSFR) and its dependence on mass (Leja et al., 2013; Shankar et al., 2015).

Improvements on the method of van Dokkum et al. (2010) which include
prescriptions for the number density evolution of galaxies have been proposed.
For example, Behroozi et al. (2013) uses abundance matching in order to match
observed galaxies to corresponding dark matter halos in $\Lambda$CDM
simulations, allowing the median and dispersion of the cumulative co-moving
number density tracks to be quantified for the progenitors of galaxies of a
given mass. Work by Torrey et al. (2015); Torrey et al. (2017) introduces an
analytic framework, which accounts for the effects of the merger rate
(‘coagulation’) and stochasticity of galaxy growth rates (‘scatter rate’), and
describes the median (Torrey et al., 2015) and intrinsic scatter (Torrey et
al., 2017) of the evolution of the galaxy population in co-moving density
space. Wellons & Torrey (2017) present a probabilistic method based on this
framework, which they show is able to more effectively predict progenitor
properties than the methods of van Dokkum et al. (2010) and Behroozi et al.
(2013).

While this group of methods is widely applicable to any given galaxy property
(e.g. Torrey et al., 2015; Clauwens et al., 2016), they may lose predictive
power in cases where there is no expectation that the rank of the property of
interest will be conserved (e.g. morphology). In such cases, leveraging the
constraining power of additional galaxy properties becomes essential.
Additionally, all of these methods retain a weakness of the van Dokkum et al.
(2010) method, in that they still assume that early-type progenitors follow
the same distribution of properties as the general population at a given
redshift.

An appealing alternative is to employ a simulation that reproduces the
properties of galaxies over cosmic time. Since the identities of the
progenitors of local early-type galaxies are precisely known in the model,
they can be used to calculate the probability that a galaxy is a progenitor,
as a function of its observable properties (e.g. redshift, stellar mass, star-
formation rate (SFR), local environment). Kaviraj et al. (2009) have
previously addressed the problem of progenitor bias using the GalICS semi-
analytical model (Hatton et al., 2003). However, while the semi-analytical
approach has successfully reproduced the phenomenology of many aspects of the
galaxy formation process (e.g. Somerville & Primack, 1999; Cole et al., 2000;
Benson et al., 2003; Bower et al., 2006; Croton et al., 2006), the recent
advent of hydrodynamical simulations in cosmological volumes provides a more
accurate route to addressing the problem. Unlike their semi-analytical
counterparts, hydrodynamical models resolve the gas and baryonic content of
galaxies, typically on kpc scales, (e.g. Devriendt et al., 2010; Dubois et
al., 2014; Vogelsberger et al., 2014; Schaye et al., 2015; Khandai et al.,
2015; Taylor & Kobayashi, 2016; Kaviraj et al., 2017). This enables them to
more accurately model a greater range of physical processes, without the need
for semi-analytical recipes (Schaye et al., 2015), although some processes,
such as AGN feedback, that cannot be resolved, must still be described using
sub-grid models (e.g. Katz, 1992; Booth & Schaye, 2009; Kimm et al., 2015).
Such hydrodynamical simulations typically rely on a smaller number of free
parameters and offer a more realistic treatment of the physical processes
involved in the formation and evolution of galaxies, yielding better agreement
with the observed Universe without the need for tuning.

Figure 1: The predicted early-type fraction in Horizon-AGN, for the
$\nicefrac{{V}}{{\sigma}}$ value (0.55$\pm^{0.03}_{0.02}$) that best
reproduces the observed early-type fractions in the local Universe (Conselice,
2006). The inset shows the value of $\chi^{2}$ between the predicted and
observed early-type fractions for different $\nicefrac{{V}}{{\sigma}}$
thresholds (the red dashed lines show the minimum $\chi^{2}$ and
$\Delta\chi^{2}=1$).

In this study, we use the Horizon-AGN222http://www.horizon-simulation.org
cosmological simulation (Dubois et al., 2014; Kaviraj et al., 2017) to (1)
quantify the evolution of the progenitor population of today’s early-type
galaxies and (2) provide a route for identifying late-type galaxies that are
progenitors of present-day early-types in observational surveys, by estimating
the probability of a given late-type to be the progenitor of a local early-
type system, as a function of measurable observables like redshift, stellar
mass, star-formation rate and local density.

The structure of this paper is as follows. In Section 2, we describe the main
characteristics of the simulation, describe our simulated galaxy sample and
define how observables are measured. In Section 3, we probe the redshift
evolution of the progenitors of present-day early-type galaxies. In Section 4,
we present probabilistic prescriptions to identify late-type galaxies that are
progenitors of local early-types as a function of redshift, stellar mass,
environment and star-formation rate. We summarize our findings in Section 5.

Figure 2: Greyscale map showing the projected gas density in a $\sim$ 20 Mpc
slice through the simulation volume at the final time output ($z=0.06$).
Green, blue and red points correspond to the positions of galaxies
($\mathrm{M_{\star}}>10^{9.5}~{}\mathrm{M}_{\astrosun}$) in low, intermediate
and high density environments respectively (see text in Section 2.3.2 for more
details). Axes are in units of proper Mpc.

##  2 The simulation

We begin by briefly describing the Horizon-AGN simulation, the extraction of
galaxies and merger trees and the prediction of observable quantities.

###  2.1 Horizon-AGN

Horizon-AGN is a cosmological hydrodynamical simulation (Dubois et al., 2014)
that employs the adaptive mesh refinement Eulerian hydrodynamics code, RAMSES
(Teyssier, 2002). The size of the simulation box is 100 $h^{-1}\,\rm coMpc$,
which contains $1024^{3}$ dark matter particles and uses initial conditions
from a _WMAP7_ $\Lambda$CDM cosmology (Komatsu et al., 2011). The simulation
has a dark matter mass resolution of $8\times 10^{7}$ M⊙, a stellar-mass
resolution of $2\times 10^{6}$ M⊙ and a spatial resolution of $\sim$1 kpc. We
direct readers to Kaviraj et al. (2017) for details of the recipes (e.g. star-
formation and stellar and AGN feedback) employed to model the baryonic
evolution of galaxies. Briefly, star-formation follows a standard Schmidt-
Kennicutt law (Kennicutt, 1998), with the model implementing continuous
stellar feedback, that includes momentum, mechanical energy and metals from
stellar winds, Type II SNe and Type Ia SNe. Black-hole (BH) feedback on
ambient gas operates via two separate channels, depending on the gas accretion
rate. For Eddington ratios $>0.01$ (high accretion rates), 1.5 per cent of the
accretion energy is injected as thermal energy (a quasar-like feedback mode),
whilst for Eddington ratios $<0.01$ (low accretion rates), bipolar jets are
employed with a 10 per cent efficiency. The parameters are chosen to produce
agreement with the local cosmic black-hole mass density, and the MBH – M⋆ and
MBH – $\sigma_{\star}$ relations (Dubois et al., 2012). Apart from choosing
the BH-feedback parameters to match the MBH – M⋆ and MBH – $\sigma_{\star}$
relations at $z=0$, Horizon-AGN is not otherwise tuned to reproduce the bulk
properties of galaxies at $z\sim 0$. As described in Kaviraj et al. (2017),
the simulation reproduces key quantities that trace the aggregate stellar mass
growth of galaxies: stellar mass and luminosity functions, rest-frame UV-
optical-near infrared colours, the star-formation main sequence and the cosmic
star-formation history.

###  2.2 Identifying galaxies and building merger trees

In order to track their progenitors, we build merger histories for each early-
type galaxy in the final snapshot of the simulation ($z=0.06$). In the
sections below, we describe the process of galaxy identification, followed by
the process of building merger trees.

####  2.2.1 Identifying the galaxy sample

For each snapshot, we produce a catalogue of galaxies, using the AdaptaHOP
structure finder (Aubert et al., 2004; Tweed et al., 2009), operating directly
on the star particles. Galactic structures are selected using a local
threshold of 178 times the average matter density, where the local density of
individual particles is calculated using the 20 nearest neighbours. Only
galactic structures with more than 100 star particles are considered,
corresponding to a minimum galaxy stellar mass of M${}_{\star}\sim 2\times$108
M☉, which is a consequence of a minimum star particle mass of $m_{\star}\sim
2\times 10^{6}$ M☉. We identify an average of $\sim$150,000 galaxies above the
100 particle threshold in each snapshot. We restrict our study to galaxies
with stellar mass M${}_{\star}>10^{9.5}$ M☉, for reasons outlined in Section
2.2.2.

####  2.2.2 Producing merger histories

Using the catalogue of galaxies identified by AdaptaHOP, we extract merger
histories for each early-type galaxy at the final snapshot. We produce merger
trees using 91 snapshots in the range $z\in[0.06,7.04]$, with an average time-
step of $\sim$130 Myr. Merger trees are produced for each early-type, by
identifying their progenitors at each snapshot, using the method described in
Tweed et al. (2009). Since our threshold for identifying structures is 100
star particles, only mergers where the satellite galaxy has
M${}_{\star}\gtrsim 2\times 10^{8}$ M☉ are considered, regardless of mass
ratio. Given that our sample excludes galaxies less massive than
M${}_{\star}\sim 10^{9.5}$ M☉, and the minimum galaxy mass identified is
M${}_{\star}\sim 2\times 10^{8}$ M☉, our sample is complete for mergers down
to a mass ratio of at least 1:15.

###  2.3 Prediction of observables

We produce observables that can be used in conjunction with contemporary and
future observational datasets. These are stellar mass (derived using the total
mass of the star particles in a galaxy), star-formation rate, local number
density and stellar kinematics (which we use as a proxy for morphology). The
following sections describe how we derive each of these measures.

####  2.3.1 Morphology

The morphology of each model galaxy in our analysis is inferred using its
stellar kinematics. Morphology is defined using $\nicefrac{{V}}{{\sigma}}$,
which is the ratio of the mean rotational velocity ($V$) to the mean velocity
dispersion ($\sigma$), both measured using the entire star particle
distribution. Higher values of this ratio correspond to more late-type (disc-
like) morphologies. $\nicefrac{{V}}{{\sigma}}$ is calculated by first rotating
the coordinate system so that the $z$-axis is oriented along the angular
momentum vector of the galaxy. Rotational velocity is defined as the mean
tangential velocity component in cylindrical co-ordinates, $V_{\theta}$, and
velocity dispersion is computed using the standard deviations of the radial,
tangential and vertical star particle velocities, $\sigma_{r},\sigma_{\theta}$
and $\sigma_{z}$, summed in quadrature. $\nicefrac{{V}}{{\sigma}}$ is then
given by

$\nicefrac{{V}}{{\sigma}}=\frac{\sqrt{3}\bar{V}_{\theta}}{\sqrt{\sigma^{2}_{r}+\sigma^{2}_{\theta}+\sigma^{2}_{z}}}.$
(1)

To separate galaxies morphologically into early and late types using
$\nicefrac{{V}}{{\sigma}}$, we consider a range of values for
$\nicefrac{{V}}{{\sigma}}$ and compare how the resulting predicted early-type
fractions compare to their observed counterparts at low redshift. The value of
$\nicefrac{{V}}{{\sigma}}$ which produces the best agreement with the
observational data is then selected as the threshold value that separates
early-types and late-types in the model. Figure 1 shows the predicted early-
type fractions in Horizon-AGN for this best-fitting $\nicefrac{{V}}{{\sigma}}$
value, 0.55$\pm^{0.03}_{0.02}$, compared with early-type fractions derived
from observations (Conselice, 2006). The largest discrepancy between the
observations and the simulation occurs at the low mass end, and is likely a
result of insufficient mass resolution (Dubois et al., 2016). Nevertheless,
over the mass range considered in this study
($\mathrm{M_{\star}}>10^{9.5}~{}\mathrm{M}_{\astrosun}$), the early-type
fractions predicted by Horizon-AGN, for our $\nicefrac{{V}}{{\sigma}}$
threshold of 0.55, are in reasonable agreement with the observations. We note
that the progenitor fractions presented in Section 4 are resistant even to
relatively large changes in our $\nicefrac{{V}}{{\sigma}}$ threshold. Varying
the $\nicefrac{{V}}{{\sigma}}$ threshold by as much as 50 per cent introduces
only a $\pm 0.05$ variation in our calculated progenitor fractions.

We note that the minimum refinement of the AMR grid is increased at $z=$4 ,1.5
and 0.25, in order to keep the minimum physical cell size approximately
constant (Dubois et al., 2014; Peirani et al., 2016). While it is possible
that this refinement may result in the production of sudden instabilities in
previously stable discs that increase galaxy velocity dispersions, the smooth
nature of the $\nicefrac{{\mathrm{V}}}{{\sigma}}$ evolution of the galaxy
population (Dubois et al., 2014), indicates that this is not a significant
effect.

Figure 3: Left: Density plots showing the redshift at which local early-type
galaxies in different environments had their last significant merger (i.e. a
merger with a mass ratio greater than 1:10). Green, blue and red colours
correspond to low, intermediate and high density environments. Over-plotted
are lines showing the mean redshift in each bin and its error. Right:
Histograms (top) and associated cumulative distribution functions (bottom) of
last merger redshifts. The arrows in the bottom panel indicate the redshift
where the cumulative distribution functions reach values of 0.5 (i.e the
median value) for each environment.

####  2.3.2 Local environment

We compute the local environment of each galaxy using an estimate of the 3-D
local number density. For each galaxy, we estimate the surrounding number
density of galaxies above our mass cut ($10^{9.5}$ M☉). This is achieved using
the adaptive kernel density estimation method of Breiman et al. (1977) which
utilises a finite-support Epanechnikov kernel (instead of the more typically-
used infinite-support Gaussian kernel) and using the density estimator itself
as a pilot to steer the local kernel size (Wilkinson & Meijer, 1995; Ferdosi
et al., 2011). Galaxies at our final snapshot ($z=0.06$) are classified into
‘high’, ‘intermediate’ and ‘low’ density environments, by comparing the
density percentile they occupy to the corresponding percentiles that observed
galaxies inhabiting cluster, group and field environments occupy in the low-
redshift universe.

Observations indicate that around 10 per cent of galaxies occupy rich
clusters, while around 50 per cent of galaxies occupy groups and poor clusters
(e.g Bahcall, 1996; Dekel & Ostriker, 1999; Tempel et al., 2012), with the
remaining 40 per cent of galaxies occupying the field. We use these values as
a guide to make density cuts, in order to separate our simulated galaxies into
our three density bins (high, intermediate and low), that are roughly
analogous to cluster, group and field environments. To allocate our simulated
galaxies into these density bins, we first rank the objects by density. Then,
the 10 per cent of galaxies with the highest local density (i.e. the
90${}^{\mathrm{th}}-100^{\mathrm{th}}$ percentile range) are classified as
occupying high density environments, galaxies in the
40${}^{\mathrm{th}}-90^{\mathrm{th}}$ range are classified as occupying
intermediate density and the remaining galaxies are classified as occupying
low-density environments. Figure 2 shows the environment classifications of
galaxies with $\mathrm{M_{\star}}>10^{9.5}~{}\mathrm{M}_{\astrosun}$
superimposed over a map of the gas density. Not unexpectedly, galaxies
classified as being in high density environments lie at the nodes of the
cosmic web, while intermediate density galaxies lie largely along filaments,
with low-density galaxies typically being found in voids. Regions of high gas
density also host a high number density of galaxies which, as we show later in
Section 3, leads to more rapid morphological transformation. More massive
galaxies (those above our detection threshold of $2\times
10^{8}~{}\mathrm{M}_{\astrosun}$) first appear in the simulation around nodes,
owing to a greater abundance of gas from which to accrete, and also undergo a
higher incidence of interactions or mergers. These galaxies are thus likely to
experience earlier and accelerated evolution compared to galaxies in less
dense environments.

In all of our analysis below, we always cast the local environment in terms of
the density percentile of individual galaxies. This is because observers (and
theorists) inevitably use different metrics for measuring local density.
However, while the absolute values of density depends on the actual metric
being used, the density percentile that a galaxy occupies is likely to be
roughly independent of the actual estimation method (as we demonstrate later
in Section 4).

####  2.3.3 Star-formation rate

Star-formation rates are calculated by computing the change in stellar mass,
$m_{\star}$, of the galaxy in question between two snapshots, dividing by the
time difference between those snapshots, and subtracting the mass of stars
formed _ex-situ_ that has merged with the galaxy between the two snapshots,
$m_{\star,merged}$:

$\mathrm{SFR}=\frac{m_{\star,t=t_{i}}-m_{\star,t=t_{i-1}}-m_{\star,merged}}{\Delta
t},$ (2)

where $t_{i}$ is the time at the current snapshot and $t_{i-1}$ is the time at
the previous snapshot. $\Delta t$ is equal to $t_{i}-t_{i-1}$ and, as
discussed in Section 2.2.2, the average time-step used in the simulation is
$\sim$130 Myr. We note that the SFR in not sensitive to the exact value of
$\Delta t$. For example, if we double or halve the value $\Delta t$, in either
case, our calculated SFRs change by less than 30 per cent.

Figure 4: Top: The number fraction of progenitors that already have early-type
morphology, split by the local environment of the early-type galaxy that the
progenitor ends up in at the present day. Bottom: The mass fraction contained
in progenitors that have early-type morphology, split by the environment of
the early-type galaxy that the progenitor ends up in at the present day.
Poisson error bars are shown (note that most lie within the data points).

##  3 Redshift evolution of the progenitors of local early-type galaxies

We begin our analysis by performing a broad exploration of the redshift
evolution of the progenitors of local early-type galaxies in the simulation.
The left-hand panel of Figure 3 presents density plots showing the last-merger
redshift of each local early-type galaxy. We define this as a galaxy’s last
significant merger, i.e. a merger which has a mass ratio greater than 1:10.
The right-hand panels of Figure 3 show histograms (top-right) and their
associated cumulative density functions (bottom-right) as a function of
redshift for each environment. The arrows in the bottom-right panel indicate
the redshift where the cumulative distribution functions reach values of 0.5
(i.e. the median value) for each environment.

Local early-type galaxies that inhabit denser environments tend to have higher
dynamical ages i.e. their last significant mergers take place at earlier
epochs. While 50 per cent of galaxies in the high density bin have their last
significant merger by $z=1.5$, this is only the case for galaxies in the
lowest density bin at $z=1.1$ (as indicated by the coloured arrows in the
cumulative density function plot (bottom-right panel in Figure 3)). Local
early-types in higher-density environments also tend to have higher final
masses indicating that the bulk of their evolution takes place earlier and is
more rapid, although within a specific environment (high, intermediate or
low), the most massive galaxies finish assembling at later epochs (see also De
Lucia & Blaizot, 2007; Dubois et al., 2016).

In Figure 4, we quantify the extent of progenitor bias at various redshifts,
as a function of galaxy mass and local environment. We show both the fraction
of progenitor galaxies that have already acquired early-type morphology (top
panel) and the mass fraction in the progenitor population that is contained in
progenitors with early-type morphology (bottom panel). We find that, across
all environments, only $\sim 50$ per cent of the progenitors have acquired
early-type morphology by $z\sim 0.6$. This is also true of the mass fraction
contained in these early-type progenitors i.e. at $z\sim 0.6$ only half of the
stellar mass that eventually ends up in early-type galaxies today is contained
in progenitors that have early type morphology. In other words, looking only
at early-type systems to trace the evolution of today’s early-types would miss
half of the progenitor population at $z\sim 0.6$.

Since morphological transformation is more rapid in regions of higher density,
progenitor bias is less severe. Thus, 50 per cent of the progenitors of local
early-types in high-density environments (i.e. today’s clusters) have already
acquired early-type morphology by $z=0.7$ (compared to $z=0.6$ across all
environments). Note, however, that the bias remains reasonably high regardless
of environment. Very similar trends are seen when quantifying progenitor bias
as a function of stellar mass (not shown in Figure 4), with more massive
galaxies (M${}_{\star}>10^{11.5}$ M⊙) following the same trend as the ‘high’
density environment in Figure 4. This is simply because the most massive
galaxies occur overwhelmingly in dense environments.

Finally, we note that the early-type fraction appears to decrease (somewhat
counter-intuitively) from $z=5$ and begins to increase again around $z=3$.
This is not an artefact of our $2\times 10^{8}$ M⊙ detection threshold, which
might cause the most rapidly evolving (and more massive) galaxies to be
detected first, potentially biasing our result. If we limit our study to
narrow mass bins or follow only the evolution of galaxies that are detected by
the structure finder at $z\sim 5$, we observe the same non-monotonic
evolution. This is partially the result of generally more clumpy star
formation and more disturbed morphologies at high redshift (e.g. Ceverino et
al., 2010), and is consistent with observational work (e.g. Kassin et al.,
2012; Kassin et al., 2014), which has shown that star-forming galaxies
steadily settle into flat, rotationally supported discs, through the process
of ‘disc settling’ at these epochs. This kinematic settling is driven by the
fact that many processes which are able to ‘puff up’ (i.e. increase the
dispersional motion) of gas in the disc without disturbing stellar orbits
significantly, become less frequent and/or less intense with time - examples
of this include (minor) mergers and gas accretion episodes (e.g. Covington et
al., 2010; Bournaud et al., 2011; Lofthouse et al., 2017b; Martin et al.,
2017), strong stellar feedback as a result of the high star-formation rates at
high redshift (e.g. Silk & Norman, 2009) and the high gas fractions at early
epochs that lead to increased disc instability. In essence, the more gentle
evolution that galaxies undergo at later times is thought to allow the gas in
the disc to settle into a more ordered state (Kassin et al., 2012). Star
formation then proceeds primarily in a planar disc, gradually reducing the
mean $\nicefrac{{V}}{{\sigma}}$ as more stars form. Indeed, for Horizon-AGN
galaxies that still host a significant disc at $z=0.06$, we find that old
stars (those formed before $z=3$) are more likely to be found in orbits
outside of the plane of the disc, symptomatic of the fact that, at these early
epochs, gas fractions and merger rates were typically higher on average. Disc
settling has also been observed in other simulations (e.g. Kassin et al.,
2014; Ceverino et al., 2017). In the case of Horizon-AGN, this effect may be
compounded slightly by changes to the maximum refinement of the gas grid,
which, as mentioned in Section 2.3.1 is increased towards lower redshifts in
order to keep the minimum physical cell size approximately constant. This can
have the effect of artificially thickening discs of some galaxies, although we
still observe non-monotonic evolution regardless of refinement level.

We conclude this section by exploring the morphologies of the progenitors of
early-type galaxies that are involved in mergers. We focus only on binary
mergers because, although non-binary mergers do occur, they are rare (around
two orders of magnitude less frequent than binary mergers). In Figure 5, we
show the fraction of mergers at a given redshift that involve two late-type
galaxies (‘late-late’), one late-type and one early-type galaxy (‘mixed’) and
two early-type galaxies (‘early-early’). Mergers between two late-type
galaxies dominate in the early Universe i.e. around the epoch of peak cosmic
star-formation and beyond ($z>1.5$). The fraction of mergers involving two
early-type galaxies climbs rapidly in the low redshift Universe ($z<0.5$).
However, at all redshifts, the majority of mergers involve at least one late-
type galaxy.

Figure 5: The morphological composition of binary mergers within the
progenitor population, that have mass ratios greater than 1:10, as a function
of redshift. ‘Late-late’ indicates mergers between two late-type galaxies,
‘mixed’ indicates mergers between one early-type and one late-type galaxy and
‘early-early’ indicates a merger between two early-type galaxies. Poisson
errors are shown.

##  4 Identifying late-type galaxies that are progenitors of present day
early-types

We proceed by constructing probabilistic prescriptions for identifying late-
type progenitors of local early-type galaxies in observational surveys, as a
function of quantities that are measurable in today’s datasets: redshift,
stellar mass, local environment and star-formation rate. As noted in the
introduction, the overall aim is to provide a means for correcting progenitor
bias in observational studies, by allowing for the inclusion of late-type
progenitors of today’s early-type systems. We do this by calculating the
fraction of late-type galaxies that are progenitors of local early-types, as a
function of the measurable quantities mentioned above. This fraction can then
be thought of as a probability that a galaxy with the given properties is the
progenitor of an early-type galaxy at present day. Observers who wish to
include late-type progenitors of early-type galaxies can then use these
probabilities to ‘weight’ objects in observational surveys, thus enabling them
to reduce progenitor bias by including, in a probabilistic sense, the late-
type members of the progenitor population. These probabilistic prescriptions
are likely to be particularly useful in the new era of deep-wide surveys (e.g.
DES (Abbott et al., 2005), EUCLID (Laureijs et al., 2011), LSST (Abell et al.,
2009), JWST (Gardner et al., 2006) etc.) which will routinely offer large
datasets that probe the early Universe, where progenitor bias becomes most
severe, and simplifying assumptions, such as using only early-type galaxies or
the red sequence to trace the progenitor population, break down. In what
follows, we first explore 1-D progenitor probabilities as a function of
stellar mass (split by local environment) and then 2-D probabilities as a
function of mass and environment and mass and star-formation rate.

Figure 6: The evolution with redshift of the fraction of late-type galaxies
that are progenitors of early-type galaxies at $z\sim 0$. Each panel shows, in
black, the fraction of late-type galaxies that are progenitors of local early-
types for a given redshift, as a function of the stellar mass of the
progenitor. The sample of late-type galaxies is split further into high (red),
intermediate (blue) and low (green) density environments. Error bars show
Poisson errors. The sample becomes smaller (leading to a corresponding
increase in the size of the Poisson error bars) towards higher redshifts,
because there are fewer galaxies with stellar masses above
M${}_{\star}>10^{9.5}$ in the simulation. For clarity, errors bars and points
are are not shown where the errors are larger than 0.5.

Since we are interested in probing progenitor probabilities as a function of
local environment, we calculate, at each redshift of interest, the 3-D local
number density using the method described in Section 2.3.2. As in Section 3,
we consider galaxies in the 90${}^{\mathrm{th}}-100^{\mathrm{th}}$ percentile
range to be inhabiting high density environments, those in the
40${}^{\mathrm{th}}-90^{\mathrm{th}}$ percentiles to be inhabiting
intermediate-density environments and those in the
0${}^{\mathrm{th}}-40^{\mathrm{th}}$ percentile range to be inhabiting low
density environments. As noted before, the density percentile in which a
galaxy lies (which is driven by its rank in density) is likely to be
reasonably resistant to the exact method used for the density estimation.

To check this, we compare two different density estimation methods. These are
the adaptive kernel density estimator used in Section 2.3.2 and the
$k^{\mathrm{th}}$ nearest neighbours density estimator, that is commonly used
in many observational studies (e.g. Dressler, 1980; Baldry et al., 2006;
Ferdosi et al., 2011; Shattow et al., 2013). Note that for consistency with
the adaptive kernel method, we choose a definition for the $k^{\mathrm{th}}$
nearest neighbour algorithm whereby each galaxy is considered to be its own
neighbour. Specifically, we choose, $k=6$, which is almost equivalent333i.e.
both methods calculate the density within the same radius, but the number of
objects differs by 1, meaning the $k=5$ (not-own-neighbour) number density
estimate simply differs by a factor of $5/6$ from the $k=6$ (own-neighbour)
estimate. to the commonly used case where $k=5$ and each galaxy is not
considered to be its own neighbour (e.g. Baldry et al., 2006). We test the two
methods on a 7 Mpc (proper) slice through the simulation snapshot at $z=0.5$,
which corresponds to a difference in velocity due to the Hubble flow of
$\Delta$V $=500$ km s-1, and implies a requisite precision in redshift of
$\Delta z=0.002$. Such precision will be achievable at intermediate and high
redshift using spectroscopic and grism redshifts from future instruments, such
as MOONS (Cirasuolo et al., 2011), PFS (Takada et al., 2014), 4MOST (de Jong
et al., 2012) and JWST (Gardner et al., 2006). We explore estimates of both
the 2-D surface density and the 3-D density. We find that the rank of each
galaxy indeed remains approximately constant, regardless of either the exact
estimator used, or whether we consider the 2-D or 3-D densities. Typically,
the rank of a galaxy does not change by more than 10 per cent and, therefore,
changing the density estimator leaves our conclusions unchanged.

Figure 6 shows the fraction of late-type galaxies at a given redshift that are
the progenitors of a local early-type, as a function of stellar mass and split
by local environment. We show the 2-D progenitor probability as a function of
both stellar mass and local density in Figure 7, with the colour bar
indicating the progenitor probabilities. At all redshifts, there is a positive
trend of progenitor probability with stellar mass i.e. more massive late-type
galaxies are more likely to be progenitors of local early-type remnants.

At high redshifts, almost all massive galaxies, regardless of their local
environment, are progenitors of present-day early-types. While the progenitor
probabilities increase with redshift, for the most massive galaxies the
progenitor probability remains close to $\sim 1$ until $z\sim 0.5$. The
principal reason for an increase in the progenitor fraction with redshift is
simply the fact that late-type galaxies have more time to merge with other
galaxies and undergo morphological transformation before the present day. The
rate of morphological transformations is regulated by the merger rate per
galaxy, which rises with redshift (e.g. Welker et al., 2015; Rodriguez-Gomez
et al., 2015; Kaviraj et al., 2015) and thus controls the rate of change in
the progenitor fraction as a function of redshift. Lower-mass (M⋆ $<$ 1010.5
M⊙) galaxies can also exhibit high progenitor probabilities at high redshift,
but _only_ if they occupy regions of high density (e.g. the 80-100th density
percentile, see Figure 7). Note that the progenitor probabilities decline for
all galaxies towards low redshifts, because these systems will not have had
time to undergo enough merging to achieve early-type morphology.

Figure 7: Density plots showing the redshift evolution of the fraction of
late-type galaxies that are progenitors of early-types at the present day, as
a function of the stellar mass and density percentile of the late-type
galaxies in question. The late-type progenitor fraction is represented by the
colour bar. We do not plot bins containing 3 or fewer galaxies. Figure 8:
Left: Star-formation main sequence (i.e. star-formation rate plotted against
stellar mass) for late-type galaxies, as a function of redshift, colour-coded
by the fraction of late-type galaxies that are progenitors of early-types at
the present day. We do not plot bins containing 3 or fewer galaxies.

We proceed, in Figure 8, by exploring the progenitor probabilities as a
function of stellar mass and star-formation rate (the parameter space that is
commonly referred to as the ‘star-formation main sequence’). The colour bar
indicates the progenitor probabilities. Mirroring the trends found earlier,
massive late-type galaxies are more likely to be progenitors of early-types.
At high redshift, these massive late-type progenitors can be some of the most
highly star-forming systems in the Universe. However, this is simply a
consequence of star-formation activity being, on average, more vigorous in the
early Universe. It is worth noting, however, that at all redshifts, progenitor
fractions increase at fixed stellar mass as the star-formation rate decreases.
In other words, at fixed stellar mass, late-type galaxies with lower specific
star-formation rates are more likely to be progenitors. For example, at $z\sim
2$ (top-right hand panel of Figure 8), a late-type galaxy with a stellar mass
of 1010.7 $\mathrm{M}_{\astrosun}$, which resides at the upper end of the
star-formation main sequence, has a progenitor probability of around 70 per
cent. A galaxy with a similar mass which sits at the bottom of the star-
formation main sequence has a progenitor probability of close to 100 per cent.

We release the progenitor probabilities calculated here, as a function of
different observables, as described in the Appendix A. For a series of
redshifts, we provide tables of progenitor probabilities as a function of the
three principal observables studied here: stellar mass, star-formation rate
and local environment. Since not all observational datasets may offer access
to all three quantities, we also provide separate tables for projections of
this 3-D parameter space i.e. progenitor probabilities as a function of
stellar mass only, stellar mass and local environment and stellar mass and
star-formation rate. As noted above, the properties of individual galaxies in
current and future observational surveys which provide these observables can
be compared to these tables in order to estimate the probability that they are
progenitors of an early-type galaxy in the local Universe. The structure of
the files containing these tables, and scripts to read them, are provided in
the Appendix A.

##  5 Summary

As end-points of the hierarchical mass assembly process, early-type galaxies
host more than half of the stellar mass density in the local Universe, their
stellar populations encoding the assembly history of galaxies over cosmic
time. Studying these galaxies in the local Universe and probing their
progenitors at earlier epochs offers a unique perspective on the evolution of
the observable Universe. However, since morphological transformations
progressively convert late-type (disc-like) galaxies into early-type systems,
the progenitors of today’s early-type galaxies become increasingly dominated
by late-types at high redshift. Understanding the evolution of early-types
over cosmic time therefore requires a reliable method for identifying these
late-type progenitors of local early-types. Here, we have used the Horizon-AGN
cosmological hydrodynamical simulation, which produces good agreement with the
observed properties of galaxies in the redshift range $0<z<5$, to study how
the progenitors of local early-type galaxies evolve over cosmic time.

We have studied the merger histories of local early-types and the morphologies
of galaxies that are involved in these mergers and traced how the
morphological mix of galaxies in the progenitor population changes over cosmic
time. We have then used the simulation to study the fraction of late-type
galaxies that are progenitors of present-day early-types, as a function of
redshift, stellar mass, local environment and star-formation rate: observables
that can be routinely measured in current and future datasets. As noted
earlier, these fractions can be treated as probabilities that can then be used
to include late-type progenitors of local early-types in observational surveys
by ‘weighting’ these late-type systems by these probabilities. The benefit of
this approach is to alleviate progenitor bias i.e. the bias that occurs if one
considers only early-type galaxies (or proxies like the red sequence) to study
the progenitor population of today’s early-types. Our key conclusions are as
follows:

  * •

The merger history of early-type galaxies indicates that these systems finish
assembling their stellar mass at relatively early epochs. By $z\sim 1$, around
60 per cent of today’s massive early-types, averaged over all environments,
have had their last significant merger (i.e. a merger with mass ratio greater
than 1:10). For early-type galaxies that inhabit high density environments at
the present day (e.g. clusters) this value is 70 per cent, while it is
$\sim$50 per cent in early-types that inhabit low-density environments (e.g.
the field). On average, morphological transformation is $\sim$50 per cent
faster in high-density environments compared to low-density regions.

  * •

Progenitor bias is significant at all but the lowest redshifts. Until $z\sim
0.6$ less than half of the progenitors of today’s early-types actually have
early-type morphology. Similarly, less than half of the stellar mass that ends
up in an early-type today is actually hosted by a progenitor that has early-
type morphology at this redshift. Around the epoch of peak cosmic star-
formation, which is also the epoch at which morphological transformation
occurs most rapidly, studying only early-type galaxies misses almost all (at
least 80 per cent) of the stellar mass that eventually ends up in early-types
at the present day.

  * •

The morphological mix of progenitor galaxies that are involved in mergers
evolves over time. At all redshifts, the majority of mergers have at least one
late-type progenitor. Mergers between two late-type galaxies dominate at early
times i.e. around the epoch of peak cosmic star-formation and beyond ($z>1.5$)
and the fraction of mergers involving two early-type galaxies climbs rapidly
at low redshift ($z<0.5$).

  * •

At all redshifts, late-type galaxies with larger stellar masses are more
likely to be progenitors of local early-type remnants. At high redshifts,
almost all massive (M⋆ $<$ 1011 M⊙) late-type galaxies, regardless of their
local environment, are progenitors of present-day early-type galaxies. While
the progenitor probabilities increase with redshift, for these massive
galaxies, the progenitor probability remains close to $\sim 1$ until $z\sim
0.5$. Lower-mass (M⋆ $<$ 1010.5 M⊙) galaxies also exhibit high progenitor
probabilities at high redshift, as long as they occupy regions of high density
(e.g. the 80-100th percentiles in density).

  * •

At high-redshift, massive late-type galaxies that are progenitors of present-
day early-types can be some of the most highly star-forming systems in the
Universe, simply because star-formation activity is, on average, more vigorous
in the early Universe. However, at fixed stellar mass, progenitor fractions
increase as the star-formation rate decreases i.e. late-type galaxies with
lower specific star-formation rates are more likely to be progenitors of
early-type galaxies.

In the impending era of large observational surveys (e.g. LSST, EUCLID, JWST),
this paper provides a framework for studies of how the stellar mass hosted by
the local early-type galaxy population is built up over cosmic time.

## Acknowledgements

We thank the anonymous referee for constructive comments that improved this
paper. GM acknowledges support from the Science and Technology Facilities
Council [ST/N504105/1]. SK acknowledges a Senior Research Fellowship from
Worcester College Oxford. JD acknowledges funding support from Adrian
Beecroft, the Oxford Martin School and the STFC. CL is supported by a Beecroft
Fellowship. This research has used the DiRAC facility, jointly funded by the
STFC and the Large Facilities Capital Fund of BIS, and has been partially
supported by grant Spin(e) ANR-13-BS05-0005 of the French ANR. This work was
granted access to the HPC resources of CINES under the allocations 2013047012,
2014047012 and 2015047012 made by GENCI. We thank Stephane Rouberol for
running and maintaining the Horizon cluster hosted by the Institut
d’Astrophysique de Paris. Elias Brinks and Chiaki Kobayashi are thanked for
useful comments. This work is part of the Horizon-UK project.

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##  Appendix A Tables of progenitor probabilities

The joint progenitor probability distribution is tabulated as a function of
redshift, stellar mass, local environment and star-formation rate and stored
in an $a\times b\times c\times d$ binary file. The python routine
tabulate_progenitor_probability.py outputs progenitor probabilities from the
joint probability distribution and can be run as follows:

python tabulate_progenitor_probability.py -z <redshift> -m <mass> -p
<percentile> -s <SFR>,

where mass is in units of log10(M⋆/M☉) and SFR is in units of M☉ yr-1. At
least 1 of the 4 keywords must be supplied, if fewer than 4 keywords are
supplied the routine outputs the progenitor probability marginalised over the
missing dimension(s).

e.g. calling ‘python tabulate_progenitor_probability.py -z 2.0 -m 10.0 -s 3’
returns the joint progenitor probability for $z=2$, $M_{\star}=10^{10}$ M⊙ and
$SFR=3$ M☉ yr-1 with environment marginalised out.

We also provide progenitor probabilities in ASCII format. For each redshift
$z\in$[0,5] progenitor probabilities are tabulated in 4 files:

  * •

m_rho_SFR.txt contains 4 columns listing stellar mass [log10(M⋆/M☉)],
percentile, star-formation rate [M☉ yr-1] and the joint progenitor
probability.

  * •

m_rho.txt contains 3 columns listing stellar mass, percentile and the
progenitor probability marginalised over SFR.

  * •

m_SFR.txt contains 3 columns listing stellar mass, SFR and the progenitor
probability marginalised over percentile.

  * •

m.txt contains 2 columns listing stellar mass and the progenitor probability
marginalised over percentile and SFR.

The routine and files above are available from
http://www.star.herts.ac.uk/~gmartin/bias.