# Fast Forward Modelling of Galaxy Spatial and Statistical Distributions

Pascale Berner, 11footnotetext: Corresponding author. Alexandre Refregier
Beatrice Moser  Luca Tortorelli  Luis Fernando Machado Poletti Valle  and
Tomasz Kacprzak

###### Abstract

A forward modelling approach provides simple, fast and realistic simulations
of galaxy surveys, without a complex underlying model. For this purpose,
galaxy clustering needs to be simulated accurately, both for the usage of
clustering as its own probe and to control systematics. We present a forward
model to simulate galaxy surveys, where we extend the Ultra-Fast Image
Generator to include galaxy clustering. We use the distribution functions of
the galaxy properties, derived from a forward model adjusted to observations.
This population model jointly describes the luminosity functions, sizes,
ellipticities, SEDs and apparent magnitudes. To simulate the positions of
galaxies, we then use a two-parameter relation between galaxies and halos with
Subhalo Abundance Matching (SHAM). We simulate the halos and subhalos using
the fast PINOCCHIO code, and a method to extract the surviving subhalos from
the merger history. Our simulations contain a red and a blue galaxy
population, for which we build a SHAM model based on star formation quenching.
For central galaxies, mass quenching is controlled with the parameter Mlimit,
with blue galaxies residing in smaller halos. For satellite galaxies,
environmental quenching is implemented with the parameter tquench, where blue
galaxies occupy only recently merged subhalos. We build and test our model by
comparing to imaging data from the Dark Energy Survey Year 1. To ensure
completeness in our simulations, we consider the brightest galaxies with
$i<20$. We find statistical agreement between our simulations and the data for
two-point correlation functions on medium to large scales. Our model provides
constraints on the two SHAM parameters Mlimit and tquench and offers great
prospects for the quick generation of galaxy mock catalogues, optimized to
agree with observations.

##  1 Introduction

One of the main focuses of modern cosmology is the study of the distribution
of matter in the universe. We know that most of the matter in the universe is
composed of dark matter, which can only be observed indirectly, for example
with gravitational lensing. Visible matter on the other hand is in the form of
gas and galaxies and can be observed in different ranges of the
electromagnetic spectrum.
Galaxy surveys are one of the most important ways to probe the universe.
Galaxy shapes, distortions and magnifications allow for weak lensing analyses
(e.g. Dark Energy Survey (DES, [1]), Kilo-Degree Survey (KiDS, [2])). The
clustering of galaxies is a powerful cosmological probe and also a systematic
effect for lensing analyses.
To understand and evaluate the observed galaxy clustering we rely on
simulations. They are important for example to estimate uncertainties of
observational data, to build analysis pipelines, and for cosmological
parameter estimation. For a simulation to be useful, it must be realistic and
accurate enough and yet cover a large enough volume. This becomes a trade-off
for the finite computational resources.
Simulations that contain only dark matter are simpler, since they do not
require a treatment of the baryons and therefore allow larger numbers of
simulation particles. These N-body simulations describe the motion of the
particles, often through Newtonian gravity, thereby predicting structure
formation (e.g. GADGET-2 [3, 4], GADGET-4 [5], PKDGRAV3 [6] and ABACUS [7]). A
way to extract single objects from the resulting cosmic web is to use dark
matter halos. A halo is a bound overdense region consisting of multiple
simulation particles, describing objects, like the dark matter halo
surrounding our Milky Way. Substructures are then described as subhalos, i.e.,
collapsed halos within another halo. Baryonic matter can be simulated using
hydrodynamical simulations, but these are computationally expensive and
therefore usually applied to smaller simulation volumes. Furthermore, the
modelling of baryons is less well understood since the underlying theory is
more complex than with dark matter only.
An alternative is to simulate the dark matter distribution and relate it to
the distribution of baryons. This is possible for example for neutral hydrogen
mass using a HI to halo mass relation (e.g. [8]) or for galaxies hosted by
halos (e.g. [9]). The latter requires an understanding of the relation between
the dark matter and the galaxy distribution, or the relation between the
galaxy properties and the properties of the dark matter halos they reside in.
Since the field studying galaxy formation and evolution is still evolving and
cannot easily be simplified, the galaxy to halo relation is an approximate
model, which can nonetheless generate realistic galaxy distributions [9].
Commonly used methods for simulations to populate dark matter halos with
galaxies are the Halo Occupation Distribution HOD (e.g. [10, 11, 12, 13, 14,
15, 16, 17, 18, 19]), (Sub-)Halo Abundance Matching (S)HAM (e.g. [20, 21, 22,
23, 24, 25, 26, 27, 28]) and semi-analytic models (e.g. GAEA [29, 30, 31,
32]). While semi-analytic models are powerful for many applications, they
require treatment of the formation and clustering of galaxies and are thus
more complicated. For HOD, one needs to specify the functions Ncen and Nsat,
standing for the average number of central and satellite galaxies respectively
residing in a halo of a given mass at a given redshift. Since Halo Abundance
Matching assigns only one galaxy to each halo or subhalo and derives a
monotonic relation between halo and galaxy properties, SHAM does not require
these two functions. Additionally, SHAM has the advantage of giving positions
and velocities for all the galaxies, since every galaxy is assigned to a
separate halo.
In this work, we simulate the distribution of galaxies with a forward model
consisting of two steps. In the first step, the distribution functions of the
galaxy properties were forward modelled and constrained with an ABC [33, 34],
comparing to data using the image simulations by the Ultra-Fast Image
Generator UFig [35, 36]. For this purpose, the whole galaxy population is
split into two categories: Red galaxies are more quiescent and blue galaxies
have more star formation. In this model, the luminosity functions, sizes,
ellipticities, SEDs and apparent magnitudes in observed filter bands were
jointly constrained.
As a second step, we are left to derive the relation between halo and subhalo
mass and galaxy luminosity, using a SHAM approach for the red and blue
galaxies. The dark matter simulation we use was described in [37]. It allows
us to efficiently simulate halos and subhalos at high resolution in a
lightcone and is based on PINOCCHIO [38, 39, 40]. We describe the halo-subhalo
simulation in more detail in section 2.1. The (sub)halo mass we use for the
SHAM is the one at the observed redshift for halos and the one at accretion
time for subhalos. It has been shown by e.g. [41] that the subhalo mass at
accretion is a good property to be used for matching galaxies to halos. The
galaxy to halo relation could also be derived by using other halo properties
(for example the highest mass in its history, the maximum velocity at
observation, the peak maximum velocity in its history, or a combination of
mass and virial velocity) and other galaxy properties (for example the galaxy
size or the stellar mass) [41]. The dark matter simulation from [37] does not
have information on velocities within the halos, though, and our model to draw
galaxies is based on luminosity functions.
Previous work has been done in this field. The project PTHalos [16] used halo
occupation numbers to simulate the galaxy spatial distribution. The ELUCID
project [42, 43] derives the galaxy-subhalo connection using a neighbourhood
abundance matching method. The UniverseMachine [44] derives the star formation
rate of galaxies from different halo properties, constrained by observed
galaxy properties. ScamPy [45] uses subhalo clustering and abundance matching
(SCAM) to paint galaxies onto halos and subhalos. In [46] they compare HOD
with SHAM as well as SCAM. Another extension of subhalo abundance matching
using hydrodynamical simulations and recipes for star formation rate was
presented by [47]. Several SHAM models were developed by comparing to SDSS and
BOSS/eBOSS [48, 49, 50, 51], e.g. [52, 53]. Recent work using the first data
of DESI [54] (EDR, Early Data Release [55]) includes projects on SHAM [56, 57,
58] and on HOD [59, 60].
Our goal is to develop a forward model that can be run many times due to its
low computational cost. A discussion on forward modelling can be seen in e.g.
[61]. The idea in forward modelling is to generate the same data format as
provided by observations, evaluate both consistently for comparison, and then
loop back to adapt the model parameters. Contrary to a semi-analytical model
or a hydrodynamical simulation, our forward model is simple and in particular
involves only two free SHAM parameters. In order to be able to use Approximate
Bayesian Computation (ABC, e.g. [62, 63, 33, 34]) to optimize the galaxy model
in the future, the galaxy to halo assignment needs to be very fast. Only two
additional free parameters are added to the existing galaxy model of UFig [35,
36]. We also use the approximate dark matter halo catalogues described in [37]
instead of a full N-body simulation with a halo finder to speed up our method,
since the underlying simulation for the dark matter halos and subhalos is by
far the most expensive part of the pipeline.
This paper is structured as follows. In section 2, we describe our procedure
for assigning simulated galaxies to halos and subhalos. We describe the galaxy
survey that we use to compare to our simulated galaxy catalogue in section 3.
In section 4, we show the results of the statistical properties and of the
clustering of our simulated galaxy catalogue. Our conclusions are summarised
in section 5.

##  2 Method

In this section, we describe our approach to simulate galaxies including their
spatial distribution. Our approach includes both the simulation of dark matter
halos and subhalos and the subsequent assignment of galaxies to them.

###  2.1 Halo and subhalo catalog

For the spatial distribution of galaxies, we use the simulated distribution of
dark matter halos and subhalos by applying Subhalo Abundance Matching. The
simulation of faint galaxies with accurate clustering requires populating both
halos and subhalos with galaxies. This helps reproduce the fact that galaxies
are also associated with subhalos, especially in high density regions such as
galaxy clusters.
More distant galaxies are observed at an earlier stage of the universe. We
thus need lightcones of halos and subhalos in order to populate them with
galaxies. Since one of our main goals is to create fast simulations, we cannot
depend on full N-body simulations with halo finders. They are computationally
too expensive, especially since we need simulations with a high enough
resolution for halos associated to single galaxies. We therefore use the halo-
subhalo clustering simulations presented in [37].
These simulations are based on PINOCCHIO [38, 39, 40], a dark matter
simulation that efficiently produces halos in simulation boxes and lightcones.
PINOCCHIO is much faster than an N-body simulation, as it uses Lagrangian
Perturbation Theory and builds groups of particles while running. It can thus
be run at high resolution, while still using a large enough simulation volume.
However, it does not directly simulate the subhalo catalogues.
In a previous work in this series [37], the subhalo progenitors for each halo
in a snapshot or lightcone were extracted from the merger history. The
survival of subhalos is modelled using a fit for the subhalo merger time.
Radial density profiles are used for the spatial distribution of the surviving
subhalos within their hosts. A comparison with simulations using the N-body
simulation GADGET-2 [3, 4] and the halo finder Rockstar [64, 44] showed that
the resulting subhalo velocity function and the two-point correlation function
of subhalos are realistic. The halo-subhalo simulations by [37] were shown to
be of the order $\sim 700$ faster than a full N-body simulation.
We use the lightcones calculated directly within PINOCCHIO, combined with the
subhalo code by [37]. With PINOCCHIO, we can create high resolution halo
simulations in large cosmological volumes. For this work, we simulate
lightcones with a half opening angle of 60 $\deg$, resulting in one quarter of
the sky. We simulate until redshift $z=0.75$, which is sufficient given our
selection of relatively bright galaxies in this work. To reach the necessary
resolution of halos corresponding to single galaxies and even dwarf galaxies,
we use a box size of 500 h-1Mpc and 20483 simulation particles, and include
halos and subhalos with at least 10 particles. This results in a minimum
(sub-)halo mass of $1.1\cdot 10^{10}$ h-1M⊙. Since PINOCCHIO is not a fully
numerical simulation, it is possible for us to use halos with very few
particles, as described in [37]. For the simulations, we use the same
cosmological parameters as in [37], namely $\Omega_{\Lambda}=0.73$,
$\Omega_{m}=0.27$, $h=0.7$, $\Omega_{b}=0.045$, $\sigma_{8}=0.811$, $n=0.961$
and $w=-1$, which is close to the WMAP Cosmology [65]. To estimate the
statistical uncertainty, we run multiple simulations with the same cosmology,
the same resolution, the same lightcone settings but different initial
conditions.

###  2.2 Galaxy assignment

####  2.2.1 UCat

The simulated galaxies for the abundance matching and the final simulation are
taken from the framework of the Ultra-Fast Image Generator (UFig, [35, 66]).
UFig simulates images for astronomical applications by getting a catalogue and
rendering images that include observational and instrumental effects.
Observational properties include spectral coefficients, while instrumental
effects include the point spread function, noise and pixelization. Originally
developed for the Monte-Carlo Control Loops Pipeline (MCCL, [67, 36]), UFig
was designed for forward modelling and therefore to be fast.
The underlying models are simple, which beside code optimization is the reason
for UFig’s speed. The catalogue generator for UFig is called UCat and the
galaxy population model was developed by [35, 33]. The total galaxy population
is split into a redder and a bluer population, with bluer galaxies having
higher star formation rates. Whether a galaxy is red or blue in this
simulation is an intrinsic property and not decided according to a colour cut,
as is custom for galaxy surveys. UCat estimates the number of red and blue
galaxies per volume and applies that to a given redshift range and field of
view (i.e. lightcone aperture or image size). The code then samples red and
blue galaxies separately from luminosity functions, which are expressed in
absolute magnitudes M. The luminosity functions $\mathrm{\Phi(M,z)}$ as a
function of absolute magnitude and redshift were optimized together with the
rest of the model parameters of UFig in [33] and [34] using Approximate
Bayesian Computation (ABC, [63]). A more detailed description of the
luminosity functions can be found in [34]. We use the posterior from the ABC
of [33] for this work.
UCat then calculates apparent magnitudes in different filter bands from the
absolute magnitudes and redshifts by using spectral coefficients, filter
throughput and extinction values. The five spectral coefficients for each
galaxy give a combination of the kcorrect spectral templates [68]. The
procedure is shown schematically in figure 1. Note that additionally, the
spectra simulator USpec [69, 61] can calculate realistic spectra from the
spectral coefficients by adding a sky model, atmospheric transmission and
readout noise.
Before this work, the simulated galaxy positions were randomly selected. In
this work, we introduce a method to simulate galaxy catalogues with realistic
clustering properties.

Figure 1: Diagram showing the workings of UCat, UFig and USpec. UCat draws the
galaxies on catalogue level without observational influence, UFig adds survey-
specific properties to simulate realistic images and USpec adds a sky model
and noise sources to simulate spectra.

####  2.2.2 Abundance matching

Abundance Matching derives a monotonic relation between (sub)halo and galaxy
properties by matching the numbers of halos and galaxies per volume, assuming
that each halo in the catalogue hosts one galaxy. In our case, the matching is
done based on the assumption that more luminous galaxies are hosted by more
massive halos. The number density of (sub)halos and galaxies, as well as the
luminosity function of galaxies, depends on redshift. We therefore match the
abundance in small bins of redshift. To ensure that also the simulated volume
is the same, we match the field of view.
The clustering of galaxies does not only depend on the intrinsic luminosity,
but also on the galaxy type. A simple way to split the whole population is to
distinguish between red and blue galaxies. Red galaxies are passive, with
little star formation, while blue galaxies have more star formation and are
typically younger. The spatial distribution of halos depends on (sub)halo mass
and on whether that halo is a halo or a subhalo. A galaxy assigned to a halo
is called a central galaxy (especially if there are subhalos in this halo),
while the galaxies assigned to subhalos are satellites. There has been plenty
of research into the relation between red and blue and central and satellite
galaxies (see e.g. [70, 71, 72, 73, 9, 41, 74]).
Red galaxies are typically older than blue galaxies, having already lost most
of their active star formation due to being stripped of their cold gas. Galaxy
clusters also have a higher abundance of red galaxies compared to the whole
Universe. Furthermore, the central galaxy in a group or cluster, therefore the
galaxy associated to a massive halo, is more often red than blue. Overall,
blue galaxies on average reside in less dense regions than red galaxies (see
e.g. [75, 76, 77]).
We therefore assign red galaxies to older subhalos and to halos above a
certain halo mass Mlimit. The distinction between red and blue satellites is
done using the satellite quenching time tquench, where a satellite is blue
until it has been part of its host for longer than tquench. This procedure is
shown in figure 2. A step function without any spread is a simplification, but
our model is designed to be as simple as possible. The mass limit Mlimit and
the quenching time of satellites tquench are free parameters of our model and
can be determined to ensure the galaxy statistics are conserved. Subhalos that
have been part of their host for longer than their expected survival time
according to the survival model from [37] are not included in our simulation,
and therefore not assigned a galaxy.

(a) Red vs. blue for halos

(b) Red vs. blue for subhalos

Figure 2: Diagrams depicting the SHAM model based on star formation quenching.
Halos host blue central galaxies below a certain mass limit (left figure,
2(a)). Subhalos host blue satellite galaxies until a certain quenching time
(right figure, 2(b)).

Each halo and subhalo in the lightcone is flagged to either host a red or a
blue galaxy. The galaxies drawn from UCat also have a well-defined flag to be
either red or blue. The relation between halo mass and galaxy luminosity is
therefore done separately for red and blue galaxies.
We bin the halos in redshift and in logarithmic mass, to get a two-dimensional
histogram. The number of bins, both in redshift and in
$\log(\mathrm{M}_{\mathrm{halo}})$, has to be adjusted. Too few bins give a
relation that is too approximated, which is especially problematic when too
few mass bins are used. Too many bins lead to gaps and instability in the halo
histograms. We found that a good bin width in redshift is $\Delta z=0.01$. In
mass, we’re using 60 bins with logarithmic spacing between the minimal and
maximal halo mass. After binning the halos and subhalos in mass and redshift,
we bin the galaxies in redshift, with the same binning edges.
For each redshift bin, we then start at the highest mass bin and match the
numbers with those of the most luminous galaxies and save the minimum and
maximum absolute magnitude for this mass bin. Removing the matched galaxies,
we proceed with the next lower mass bin and do the same, continuing until the
lowest mass bin that contains halos.
When creating the mass to luminosity relation, it is important to have a large
enough halo-subhalo catalogue with a high enough resolution. Otherwise, when
applying the mass to luminosity relation to a catalogue for a different
simulation, there may be halos or subhalos for which the corresponding
absolute magnitude is not specified.

Figure 3: Schematic description of the clustered galaxy simulations. The halo
catalogue is created separately, while the interpolation and the drawing of
galaxies is within UCat. The output in the end can be used for UFig the same
way as in the version without clustering.

In section 4.3.3, we show the derived relation between halo mass and absolute
magnitude for red and blue galaxies. For each halo, we draw the absolute
magnitude of its hosted galaxy using the derived two-dimensional interpolation
in mass and redshift. The number of galaxies simulated this way is thus the
same as the number of halos in the lightcone.
The halo redshift and the assigned absolute magnitude are then used to
calculate the apparent magnitude with spectral coefficients, extinction values
and filter throughputs within UCat. The filters bands have to be set according
to the galaxy survey one wishes to simulate, and the filter throughputs are
integrated once in UFig using the kcorrect templates and K corrections. Figure
3 shows a schematic of the procedure of creating the halo catalogue, deriving
the interpolation, populating each halo with a galaxy and getting the apparent
magnitudes.
This halo and subhalo catalogue is complete in mass, therefore the new galaxy
catalogue is complete in absolute magnitude. To simulate images with UFig or
to simulate a galaxy survey in general, the catalogue can be cut by setting an
upper limit for the apparent magnitude. It is thereby important to have a dark
matter simulation with a high enough resolution, otherwise some of the
faintest galaxies will be missing, especially at low redshifts.

###  2.3 Magnitude shift calibration

Figure 4: Calibration of the shift in apparent magnitude $i$ with Source
Extractor, using simulations. The solid orange line is the mean offset as a
function of magnitude; the dotted lines are the 2$\sigma$ intervals. The
calibration is done by simulating images using UFig and Source Extractor and
then comparing the resulting magnitudes to the ones before creating images. On
the left, the shift between real and Source Extractor magnitude is shown, on
the right the opposite. We show dashed lines at $\pm$0.5 for reference.

Our comparison to data is done at catalogue level. The data we compare to is
collected by a telescope and in images, as described in section 3. The survey
collaboration has run Source Extractor [78] on their imaging data to produce
catalogues. Source Extractor retrieves the sources from images and measures
their fluxes and therefore their apparent magnitudes. Depending on the exact
settings used in Source Extractor, the values for the apparent magnitudes are
offset differently. Even small offsets in apparent magnitude that are not
calibrated correctly will lead to shifted clustering results, especially when
magnitude cuts or binning are involved.
The apparent magnitudes that we calculate using UFig for our catalogues do not
have such an offset. UFig allows us to create realistic images. We therefore
produced a small set of images, then ran Source Extractor with the same
settings as the survey, and matched the retrieved objects to the ones in the
original catalogue before the images. This allows us to calibrate the effect
of Source Extractor on the apparent magnitudes in different filter bands.
For the i-band, the results of this calibration are shown in figure 4. We have
performed the same calibration also for the other magnitude bands. On the
left, the shift from simulated magnitudes to observed magnitudes due to Source
Extractor is shown. On the right, the opposite is shown, calibrating the shift
to get from observed magnitude to true magnitude. The latter is useful for us,
as we can either perform the correction by applying the first shift to the
simulated catalog or by applying the second shift to the data. Each blue dot
corresponds to one galaxy, after removing stars. We calculate the mean and
standard deviation of the offset in narrow bins of apparent magnitude since
both shift and scatter are strongly magnitude dependent. The orange lines show
the mean and 2$\sigma$ intervals of the offset. The black lines show -0.5, 0.0
and 0.5 shifts for clarity. For this calibration, we used objects reaching
observed magnitudes of $i=22$, but our later analysis is restricted to
brighter objects.
We apply the shift whenever comparing data to simulations. For figures 7, 11
and 12, we have shifted and added a scatter to the i-band apparent magnitude
according to the calibration shown in figure 4. For figure 8, the shift of
each magnitude band is applied, but not the scatter, since the scatter would
have to be correctly correlated for the different bands for an analysis in
colour space. For simplicity, and since applying the scatter has no noticeable
effect on the measured clustering, we applied the shift to the data instead of
the simulations for the chi-squared analysis presented in figure 13.

##  3 Data

To constrain the free model parameters Mlimit and tquench described in section
2.2.2, we compare our simulations to observational data. Unlike imaging
surveys, spectroscopic surveys are usually not complete in a certain magnitude
range. The Bright Galaxy Survey (BGS) of the Dark Energy Spectroscopic
Instrument (DESI) [79] will provide a magnitude complete spectroscopic data
set, but the survey is still running. Such incompleteness is complicated to
forward model, as the same kind of objects have to be missing in the
simulations as in the data, otherwise systematic errors are created.
Furthermore, our model includes all galaxy types, not just a specific
selection according to colour cuts. Since we do not want to build a model for
just certain types of galaxies, we avoid any colour cuts, and therefore work
with imaging data.
We compare our simulations to galaxies with $i<20$ from DES Y1, the public
data release of the first year of observations of the Dark Energy Survey [80,
81, 1, 82, 83, 84, 85]. Both the data in catalogue form and the spatial,
pixelized mask we use are publicly available from the DES Data
Management.222https://des.ncsa.illinois.edu/releases/y1a1/gold
For accurate forward modelling, we simulate the same cut-out of the sky as is
covered by the data we use. The mask made publicly available by DES is in
HEALPix [86] format with NSIDE=4096. For simplicity, since a smaller lightcone
is easier to simulate, we reduce ourselves to the larger connected part of DES
Y1 south of DEC < $-30\deg$. This results in an area of 1685 $\deg^{2}$, after
removing the pixels that are flagged with badmask.
Our simulated galaxy catalogue is by construction free of stars and of any
other objects besides galaxies. We therefore use the star-galaxy
classification provided by DES and select only the objects classified as
galaxies. Furthermore, we remove all the flagged objects, to ensure a clean
data set. The corresponding columns are called MODEST_CLASS and FLAGS_GOLD in
the data release. We note that UFig does simulate stars when rendering images,
but we are only using the galaxy catalogues here.

Figure 5: Spatial mask used in this work, with the galaxies from DES Y1 with
$i<20$. This figure uses NSIDE=256, while the mask used throughout this work
has NSIDE=4096.

The area on the sky considered needs to be large enough to provide stable
clustering information on small, medium, and large scales. Although an even
larger area would increase the stability, the data provided by DES Y1 is
enough to build our model. Furthermore, a wider survey would require a bigger
lightcone, leading to more difficulties when simulating and processing it.
Figure 5 shows the spatial distribution of the galaxies from DES Y1 used in
our comparison.

As described in section 2.2.2, we assign one galaxy to each halo and subhalo.
At low redshifts, completeness in apparent magnitude requires a higher mass
resolution for faint galaxies. Faint galaxies at low redshifts are small dwarf
galaxies residing in very small halos and subhalos. Achieving the same
completeness in the simulations as in the data is important for our forward
model. Considering only the brightest galaxies of the dataset allows us to
ensure that we do not miss galaxies in our simulations due to lack of
resolution. We therefore apply a simple cut in apparent magnitude in the
i-band, at $i=20$. At low redshifts, this data set still includes very small
and faint galaxies.

##  4 Results

In this section, we present our results derived from simulations and
comparisons to the data described in the previous section. Our focus lies on
the spatial distribution of galaxies and on the dependence of our simulations
on the two free SHAM parameters.

###  4.1 Galaxy distribution functions

The galaxy population model in UCat and UFig is tuned via ABC such that the
distribution of photometric properties of the galaxy sample agrees with
observations (see [34]). In this work, the number of galaxies is given by the
number of halos and subhalos. The abundance matching technique ensures the
relation between halo mass and absolute magnitude, but does not maintain e.g.,
the number ratio between red and blue galaxies. We therefore perform some
consistency checks of UCat before and after including clustering by using
halos. We also compare the distribution of simulations and data.

(a) Redshift distribution

(b) Absolute magnitude distribution

Figure 6: Histogram distributions of the simulated galaxies, comparing UCat
only (light colours) and UCat with PINOCCHIO (dark colours). The redshift
distribution is shown on the left with the vertical lines indicating the mean
redshifts, the distribution of absolute magnitudes on the right.

Figure 6(a) shows the number of galaxies as a function of redshift, where the
galaxies have an apparent magnitude below $i=20$. The orange line shows the
distribution from UCat only, the black line shows it after including
populating halos with galaxies.
By construction, the luminosity function of both red and blue galaxies is
preserved. What may be changed is the minimum luminosity (maximum absolute
magnitude) of the simulated galaxies. This can also be different for red and
blue galaxies, if the number of halos set to host blue or red galaxies
respectively is incorrect. Figure 6(b) shows the distribution of absolute
magnitudes for sampled red and blue galaxies when applying the cut at $i=20$,
before and after using the abundance matching method. The consistency check
presented in figure 6 shows good agreement.
For forward modelling, the galaxy distributions of our simulations must agree
with the data that we wish to simulate. The easiest way to check this is by
looking at the histogram of apparent magnitudes. We do not have direct access
to absolute magnitudes and estimating them for the data is difficult since we
do not have accurate redshifts. Figure 7 thus shows the histograms for the
apparent magnitudes in the r and in the i band. The i-band apparent magnitude
is sharply cut at $i=20$, by construction. We see that the simulation agrees
well to the data. The luminosity function was fine-tuned in previous works
involving UFig and UCat, and we use the posterior from the ABC presented in
[33].

Figure 7: Apparent magnitude histograms in the r and i band, for SHAM
simulations (in orange) and the data (in blue). Figure 8: Colour-colour
scatter plot, comparing SHAM simulations (on the left) and the data (on the
right). For the simulations, we additionally colour code the blue and the red
galaxies, according to the two UCat galaxy populations. The lines indicate the
68% and 95% contours.

Figure 8 displays the distribution of galaxies in colour space, for our
simulation on the left and for the data on the right. Only a small number of
galaxies is shown, for easier inspection. For the simulation, the catalogue is
split into red and blue galaxies, according to the intrinsic types from UCat.
The distribution in colour space agrees well with the data, though the data is
more spread out and scattered due to uncertainties and scatter from Source
Extractor. A possible way to improve this in the future could be to add
scatter when applying the calibrated magnitude shift due to Source Extractor,
but the scatter would have to be correctly correlated between the different
magnitude bands.

###  4.2 Galaxy clustering

####  4.2.1 Spatial distribution

We assign the simulated galaxies differently to (sub-)halos depending on
whether they are from the red or from the blue population within UCat.
Therefore, the two populations do not have the same clustering. This can be
seen in figure 9 where blue galaxies are shown on the left and red galaxies on
the right. In both cases, we show the same cutout of the sky in right
ascension and declination spanning 9 $\deg^{2}$, and the same redshift range
$0.40<z<0.41$. The simulated red galaxies are visibly more clustered, while
the blue galaxies are more spread out to under-dense regions. This comes from
the fact that we place red galaxies in highly clustered massive halos and into
the majority of subhalos.

Figure 9: Visualization of the clustering of blue (left) and red galaxies
(right), within a sky area of 9 $\deg^{2}$ in a narrow redshift range
$0.40<z<0.41$. Galaxies are represented by circles, with both size and colour
representing their absolute magnitudes. Figure 10: Polar plot displaying the
spatial galaxy distribution as a function of redshift and angle, for both red
and blue galaxies. Redshift is shown as the radial coordinate, and right
ascension as the angular coordinate. For visual clarity, we only display
galaxies within the declination range -55.2 deg < DEC < -55 deg.

Figure 10 shows a visual representation of the resulting simulated galaxy
clustering in the lightcone. A cut-out through the lightcone along the
redshift direction is shown, with redshift or comoving distance plotted as the
radial coordinate. Only a small cut-out along DEC is shown for better
visibility, -55.2 deg < DEC < -55. The structure on large scales including
filaments, underdense and overdense regions is visible. Red and blue galaxies
are shown in different colours.

####  4.2.2 Correlation functions

To compare the simulations with the data and to check whether they are in
statistical agreement, we look at the two-point correlation function. We
measure the projected correlation function $w(\theta)$ as a function of
angular separation using Corrfunc [87, 88]. Given that we do not have radial
information for the data, since DES in an imaging survey and therefore does
not measure accurate redshifts, we cannot compare the 3D two-point correlation
function. We therefore only consider the positions on the projected sphere.
We first evaluate the scales of trust in real space, in order not to over-
interpret our simulations or the data on certain scales. We have defined
certain physical scales corresponding to different effects in the dark matter
simulations. This is related to the work presented in [37], where the
simulations of halos and subhalos used here are described.
The following scales are in decreasing order. At 20 h-1Mpc, we have medium
scales where we trust our halo/subhalo simulations and they are well in
agreement with the reference simulations based on GADGET-2 with Rockstar.
Below about 6 h-1Mpc, PINOCCHIO starts to underestimate the clustering of
halos due to its approximations. This scale is relatively inaccurate as it
depends somewhat on the mass of the considered halos. The one-halo regime
(according to the halo model of the power spectrum) starts at about 2 h-1Mpc.
While our simulations from [37] added the subhalos to the output halo
catalogues from PINOCCHIO, the density profile used to distribute the subhalos
within halos could not be calibrated precisely. This is mainly because the
subhalos from the numerical reference simulations had significant resolution
effects. The work in [37] saw a bump compared to the reference simulation,
meaning a higher correlation function, at around 0.7 h-1Mpc. Scales below 0.3
h-1Mpc are too small to be simulated precisely with the given resolutions.

(a) n(z) for different magnitude bins

(b) Spatial scales of interest for $17<i<18$

Figure 11: Redshift distribution for different apparent magnitude bins on the
left, calculated from the simulations. The dotted vertical lines are the mean
redshifts for each magnitude bin. On the right, we show the two-point
correlation function for data (solid line) vs. simulation (dashed line) for
the magnitude bin $17<i<18$, to indicate certain scales of interest (dotted
vertical lines).

To relate these physical scales to our results, we transform them from
comoving distances to angular distances projected on the sky. As this relation
depends on redshift, figure 11(a) shows the redshift distributions of the
simulated galaxies for different magnitude bins. Given that we place the
galaxies drawn from UCat into halos and subhalos, each resulting galaxy has a
precise redshift in the simulated lightcone. The dotted vertical lines in
figure 11(a) indicate the mean redshifts for each magnitude bin.
For the magnitude bin $17<i<18$, we show the resulting scales in figure 11(b),
along with the two-point correlation functions from the data and from one
simulation as a reference. Given the discussion above, we can only use scales
within the two-halo regime for a stable comparison between data and
simulations. For the magnitude bin shown in 11(b), this corresponds to angular
scales with $\theta>0.15$ degrees. Given that brighter magnitude bins have a
lower mean redshift than fainter magnitude bins, the usable scales correspond
to smaller angles for fainter galaxies. This effect would only be increased if
the mass dependence of the physical scales were included, since brighter
galaxies sit in larger halos. For the following comparison, we only show the
scales within the two-halo regime for each magnitude bin.

Figure 12: Galaxy two-point correlation function, as a function of separation
angle $\theta$, projected along the line of sight. Solid lines are the mean
values from 10 simulations, dashed values correspond to the data. Different
colours refer to different bins in apparent magnitude i. The lower panels show
the ratio of simulations to data, with the 1$\sigma$ and 2$\sigma$ intervals
due to statistical and systematic uncertainties.

In figure 12, we show the comparison of galaxy clustering between data and
simulations. The data is shown as dashed lines, while the simulations are
shown as solid lines. Different colours correspond to different magnitude bins
in the i band. For the simulations, the mean from 10 PINOCCHIO realizations
with the same SHAM model is shown. For this figure, the fiducial SHAM
parameters were chosen as M${}_{\mathrm{limit}}=8\cdot 10^{12}$ h-1M⊙ and
t${}_{\mathrm{quench}}=$ 2.0 Gyr, since tquench is expected be about 2 Gyr for
redshifts $0<z<1$ and the mass limit between $10^{12}$ and $10^{13}$ h-1M⊙
from literature [9].
In the lower panels, the ratio between simulations and data in the
corresponding bands is shown. The coloured areas given are the 1$\sigma$ and
2$\sigma$ intervals from our estimation of uncertainties. We combined the
statistical and systematic uncertainties together. Statistical variability
comes from running different PINOCCHIO runs. Systematic uncertainty comes here
from the galaxy population model, where we run the same simulation with a
fixed set of SHAM parameters and different posterior points from [33]. The
parameters include shape and normalization of the luminosity function, which
directly affect the mass to luminosity relation derived with our SHAM method.
We find that our simulations are well in agreement with the data, mostly
within 1$\sigma$. The least agreement is achieved for the highest magnitude
bin, meaning the faintest galaxies, for which the simulation slightly under-
predicts the clustering, especially for larger separation angles.

###  4.3 Galaxy to halo relation

Our main goal is to build the SHAM model and show that the resulting
clustering agrees with observational data. Since the relation between galaxy
luminosities and halo masses is derived when the halos and subhalos are
matched to red and blue galaxies, it is not assumed or an input. In this
section we present the effect of the SHAM parameters, as well as the resulting
galaxy-halo relation when using the reference SHAM parameters.

####  4.3.1 Constraints on the SHAM parameters

We have calculated the covariance matrix for the simulations based on 10
simulation runs with the same cosmological and SHAM parameters but different
underlying PINOCCHIO runs. For this and the following chi-squared analysis, we
have calculated the projected two-point correlation functions $w(\theta)$
using Corrfunc. We split up the data set into 5 magnitude bins: $15<i\leq 16$,
$16<i\leq 17$, $17<i\leq 18$, $18<i\leq 19$ and $19<i\leq 20$. For better
stability, we have reduced the number of bins in angular separation $\theta$.
Since the scales of trust depend on apparent magnitude, as discussed in
section 4.2.2, this gave us 4 to 6 bins in $\theta$ between about $0.1\deg$
and $30\deg$, with the number of bins depending on the magnitude bin.
Using the same bins as for the simulations, both in magnitude and angular
separation, we calculated the correlation functions for the observational
data. We then computed the chi-squared values using $\chi^{2}=D^{T}COV^{-1}D$,
where $D$ is the data vector defined as
$D=w_{\mathrm{data}}-w_{\mathrm{sim}}$. Both for the data vector and for the
covariance matrix $COV$ the $w(\theta)$ vectors for the separate magnitude
bins are stacked together. Since the covariance matrix is ill-conditioned, we
used Tikhonov Regularization [89, 90] for the inversion. Figure 13 shows the
grid points of the SHAM parameters we used, as well as the contours of the
chi-squared surface. The chi-squared values are not normalized here.
We investigated values between 0 and 4 Gyr for tquench, as we expect it to be
around 1-2 Gyr for redshifts $0<z<1$ (e.g. [9]). Values of tquench below 0 Gyr
are impossible, and values above 4 Gyr would make almost all satellite
galaxies in clusters red, which is not what observations show.
For Mlimit, we considered the range $10^{12}$ to $4\cdot 10^{13}$ h-1M⊙.
Literature places the transition between red and blue centrals in the range
$10^{12}$ to $10^{13}$ h-1M⊙ [9]. Values of Mlimit above $2\cdot 10^{13}$
h-1M⊙ lead to too few red central galaxies, inconsistent with observations. On
the other hand, values below $10^{12}$ h-1M⊙ would mean that most medium sized
galaxies should be red, again disagreeing with observations.
As discussed in more details in section 4.3.2, increasing either the quenching
time tquench for satellite galaxies or the mass limit Mlimit for central
galaxies has a similar effect on galaxy clustering. This is the reason for the
degeneracy between the two parameters, as can be seen in figure 13. The best
fit parameters lie along the degeneracy between (Mlimit, tquench) = ($2\cdot
10^{12}$, 4.0), and ($4\cdot 10^{13}$, 0.0). The units are h-1M⊙ for the mass
and Gyr for the time. The constraining power is relatively weak since the
parameters do not have a strong effect on clustering. Therefore, the shape of
the best fit contour is irregular, and its size is large. The exact shape also
depends on the number of layers used for plotting, but the degeneracy remains
the same.

Figure 13: The resulting chi-squared surface for the two SHAM model parameters
Mlimit on the x-axis and tquench on the y-axis. The black dots show the grid
points at which we run a simulation. There is a degeneracy from the top left
to the bottom right.

####  4.3.2 Impact of the SHAM parameters on clustering

The faintest galaxies sit in the smallest halos, by definition of abundance
matching, which have the lowest correlation function. This holds mainly for
halos and not for subhalos, since subhalos are much more clustered. On small
scales, subhalos and therefore satellite galaxies are more clustered due to
the 1-halo term. On large scales, subhalos are more clustered compared to
halos of the same mass since they sit in host halos which are often 100 times
as massive as themselves, and more massive halos are more clustered.
To understand the effect of the two SHAM parameters, it is important to note
that the galaxy sample we look at is cut in magnitude. The whole simulation is
limited in the i-band by $i<20$, and each magnitude bin corresponds to two
cuts according to its bin limits. Therefore, although each halo and subhalo is
populated with a galaxy, many of them are not included. Galaxies are excluded
if they are too faint for the magnitude cut. For a given redshift, galaxies
are effectively cut in absolute magnitude or luminosity. Overall, distant
galaxies appear much fainter and are therefore excluded unless they are very
bright.
Increasing the mass limit Mlimit leads to blue galaxies occupying larger
central halos. Since the luminosity function of blue galaxies is fixed, this
leads to small halos being occupied by fainter galaxies, thus leading to them
being excluded by the selection. Since small halos lower the correlation
function, this increases the overall clustering.
A similar effect happens when the quenching time tquench is increased. This
leads to more blue galaxies occupying subhalos and becoming satellites.
Therefore, the small halos are filled with faint galaxies, making them be
removed by the magnitude cuts. In either case, to increase clustering one
needs to ensure that small halos are not included, therefore increasing Mlimit
and/or tquench.

(a) Perpendicular to the degeneracy

(b) Along the degeneracy

Figure 14: Both figures show the clustering of red vs. blue galaxies for
apparent magnitudes within $18<i<19$. On the left, the dependence of galaxy
clustering across the degeneracy shows an increase in clustering on all scales
with increasing Mlimit and tquench. On the right, the dependence along the
degeneracy is shown, where the effect is mostly on small scales.

This can be seen in figure 14(a), where the correlation function for the
magnitude bin $18<i<19$ is shown. Only results from our simulations are shown
here, and we distinguish between red and blue galaxies according to the two
UCat populations. The values in the 2D parameter plane for this figure were
chosen perpendicular to the degeneracy. The colours from dark to light shades
are from small to high values of Mlimit and tquench. The clustering varies on
all scales for both red and blue galaxies, with the effect being strongest in
the 1-halo regime.
The relation in figure 14(b) is analogous to figure 14(a), but with the
parameter points being chosen along the degeneracy. Since increasing Mlimit
and decreasing tquench or vice versa cancels out the effect of the parameters
on clustering, the 2-halo term in figure 14(b) is almost unchanged. On small
scales, in the 1-halo regime, the overall correlation function would also be
mostly unchanged. Since we split the sample into red and blue galaxies, we can
see an effect. A low value of tquench leads to few or no blue galaxies being
satellites, therefore strongly reducing the 1-halo term. Many red satellites
surviving the cuts on the other hand increases the clustering of red galaxies
on small scales.

####  4.3.3 Mass to luminosity relation

A natural way to show the relation between halos and galaxies for a SHAM model
is to look at the relation between the two properties that are matched. In our
case, the two properties are the halo/subhalo mass and the galaxy absolute
magnitude or luminosity. We show the mass to luminosity relation for red and
blue galaxies separately for our reference SHAM parameters in subfigure 15(a).
The relation depends on redshift, and we show it for redshift z=0.3, which is
close to the mean redshift for our simulations. The non-smooth feature in the
relation comes from the fact that the mass limit Mlimit leads to halos only
contributing above the limit for red galaxies and only below the limit for
blue galaxies. Subhalos are split according to tquench, but this is not a cut
in mass and therefore not directly visible in the mass to luminosity relation.
In subfigure 15(b), we show the same relation, but with the mass-to-light
ratio on the y-axis. When comparing to other works (e.g. [91, 92]), we find
that our resulting relation between galaxies and halos agrees with other
literature.

(a) Mass to luminosity relation

(b) Relation between mass and mass-to-light ratio

Figure 15: Relation between (sub)halo mass and luminosity, derived using a
two-dimensional interpolation in the mass-redshift plane. Red and blue
galaxies have separate relations. On the left, the relation between mass and
absolute magnitude or luminosity is shown. On the right, the mass-to-light
ratio is shown in relation to the (sub)halo mass. The relations are shown at
redshift z=0.3. The shaded areas correspond to the 1$\sigma$ and 2$\sigma$
uncertainties.

##  5 Conclusion

We presented our galaxy forward model based on Subhalo Abundance Matching
(SHAM). The halo lightcones are generated with PINOCCHIO. The subhalo
extraction from the merger history presented in [37] was used to add the
surviving subhalos to the halo catalogue. This allows us to create halo-
subhalo lightcone catalogues at high resolution with large cosmological
volumes.
The galaxies used to populate the halos and subhalos are drawn from luminosity
functions from UCat, the catalogue component of the Ultra-Fast Image Generator
(UFig). UCat contains two separate galaxy populations, for red and blue
galaxies. The blue galaxies have more star formation, while the red galaxies
are more quiescent.
These two galaxy populations are included in our SHAM model based on star
formation quenching. With time, the formation of stars is quenched due to
several processes (e.g., reduced inflow of gas or tidal stripping), making
blue galaxies become red. For central galaxies, our SHAM model includes mass
quenching, with the parameter Mlimit. For satellite galaxies, the star
formation is mostly reduced due to environmental quenching. Our corresponding
parameter tquench separates into recently and long-since merged subhalos,
hosting blue and red satellite galaxies, respectively.
The properties matched in the abundance matching are the halo or subhalo mass
and the galaxy absolute magnitude, corresponding to the luminosity. The most
luminous galaxies reside in the most massive halos, filling up the halos and
subhalos from massive to light. Our model performs this matching in narrow
redshift bins, leading to an accurate, redshift dependent relation between
mass and luminosity.
We showed the comparison of our simulations with data from the public Dark
Energy Survey Year 1 (DES Y1). To ensure completeness of our simulations at
very low redshifts, we have restricted the comparison to a magnitude range of
$i<20$, leaving only bright galaxies. In section 4.1, we showed that the
histograms of apparent magnitudes and galaxy distribution in colour space are
in agreement with the data.
The results of our comparison for the clustering were shown in section 4.2. We
can only trust the comparison between data and our simulations in the the
2-halo regime, due to the approximate nature of PINOCCHIO and the subhalo
model introduced by [37]. On reliable angular scales, the correlation
functions of the simulations and the data are in statistical agreement.
By comparing to the data, we constrained the two SHAM parameters purely based
on clustering. We found that good values within the best fit contour are
M${}_{\mathrm{limit}}=$ $8\cdot 10^{12}$ h-1M⊙ and t${}_{\mathrm{quench}}=$
2.0 Gyr. There is a degeneracy between the two parameters, since increasing
either parameter increases the clustering of the galaxies. We could not reduce
this degeneracy by distinguishing between red and blue galaxies, since the
effect of changing the parameters along the degeneracy is only measurable in
the 1-halo regime, where we cannot compare data and simulations reliably.
Compared to models matching galaxies to halos based on a full N-body
simulation or even compared to a hydrodynamical simulation, our simulations
take little computational effort. Running the PINOCCHIO lightcone (once) took
19 minutes on 1032 CPU cores of the Euler Cluster from ETH
Zurich333https://scicomp.ethz.ch/wiki/Euler, with 20483 simulation particles.
Creating the subhalo catalogue based on the merger history takes about 10
hours, on one CPU core. The SHAM simulation itself is much faster and can be
run multiple times for one halo/subhalo ligthcone. Sampling from the
luminosity functions before the SHAM takes about 4 minutes, the SHAM model
itself until the final galaxy catalogue about 28 minutes, both on one CPU. The
SHAM model requires about 30 GB of RAM. The reported values are for the
simulations of DES Y1 for the redshift range z<0.75.
Future work may include an extension to fainter galaxies and higher redshifts,
and applications to current wide surveys. Further projects in progress include
the development of a bias model based on the presented SHAM model, for the
galaxies in UCat and UFig. Additionally, the effect of clustering on weak
lensing systematics can be investigated using our framework. Matching the
initial conditions and therefore the cosmic seed of our halo-subhalo lightcone
to a full N-body particle simulation may open doors for further applications,
including probe combinations.

## Acknowledgments

We thank Jörg Herbel at ETH Zurich for his helpful comments on UFig. We
further thank Adam Amara at ETH Zurich and University of Portsmouth for his
input on running simulations in previous work. We would like to thank Pascal
Hitz and Silvan Fischbacher, both at ETH Zurich, for valuable discussions on
simulations and numerical issues. We thank Martina Fagioli at ETH Zurich for
helpful discussions on galaxy populations. This project was supported in part
by SNSF grant 200021_192243.

This project used public archival data from the Dark Energy Survey (DES).
Funding for the DES Projects has been provided by the U.S. Department of
Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology FacilitiesCouncil of the United
Kingdom, the Higher Education Funding Council for England, the National Center
for Supercomputing Applications at the University of Illinois at Urbana-
Champaign, the Kavli Institute of Cosmological Physics at the University of
Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State
University, the Mitchell Institute for Fundamental Physics and Astronomy at
Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos
Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho
Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da
Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the
Collaborating Institutions in the Dark Energy Survey.
The Collaborating Institutions are Argonne National Laboratory, the University
of California at Santa Cruz, the University of Cambridge, Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the
University of Chicago, University College London, the DES-Brazil Consortium,
the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH)
Zürich, Fermi National Accelerator Laboratory, the University of Illinois at
Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the
Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence
Cluster Universe, the University of Michigan, the National Optical Astronomy
Observatory, the University of Nottingham, The Ohio State University, the
OzDES Membership Consortium, the University of Pennsylvania, the University of
Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the
University of Sussex, and Texas A&M University.
Based in part on observations at Cerro Tololo Inter-American Observatory,
National Optical Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation.

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