# Deep Realistic Extragalactic Model (DREaM) Galaxy Catalogs: Predictions for
a _Roman_ Ultra-Deep Field

Nicole E. Drakos Department of Astronomy and Astrophysics, University of
California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA  Bruno
Villasenor Department of Astronomy and Astrophysics, University of California,
Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA  Brant E. Robertson
Department of Astronomy and Astrophysics, University of California, Santa
Cruz, 1156 High Street, Santa Cruz, CA 95064, USA  Ryan Hausen Department of
Computer Science and Engineering, University of California, Santa Cruz, 1156
High Street, Santa Cruz, CA 95064, USA  Mark E. Dickinson National Optical-
Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ
85719, USA  Henry C. Ferguson Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21218, USA  Steven R. Furlanetto Department of
Physics and Astronomy, University of California, Los Angeles, CA 90024, USA
Jenny E. Greene Department of Astrophysical Sciences, Princeton University,
Princeton, New Jersey 08544, USA  Piero Madau Department of Astronomy and
Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa
Cruz, CA 95064, USA  Alice E. Shapley Department of Physics and Astronomy,
University of California, Los Angeles, 430 Portola Plaza, Los Angeles, CA
90095, USA  Daniel P. Stark  Department of Astronomy, Steward Observatory,
University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA  Risa H.
Wechsler Department of Physics, Stanford University, 382 Via Pueblo Mall,
Stanford, CA 94305, USA  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94305, USA  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA

###### Abstract

In the next decade, deep galaxy surveys from telescopes such as the _James
Webb Space Telescope_ and _Roman Space Telescope_ will provide
transformational data sets that will greatly enhance the understanding of
galaxy formation during the epoch of reionization (EoR). In this work, we
present the Deep Realistic Extragalactic Model (DREaM) for creating synthetic
galaxy catalogs. Our model combines dark matter simulations, subhalo abundance
matching and empirical models, and includes galaxy positions, morphologies,
and spectral energy distributions (SEDs). The resulting synthetic catalog
extends to redshifts $z\sim 12$, and galaxy masses $\log_{10}(M/M_{\odot})=5$
covering an area of $1\,\deg^{2}$ on the sky. We use DREaM to explore the
science returns of a $1\,\deg^{2}$ _Roman_ UDF, and to provide a resource for
optimizing ultra-deep survey designs. We find that a _Roman_ UDF to $\sim
30\,m_{\rm AB}$ will potentially detect more than $10^{6}$ $M_{\rm UV}<-17$
galaxies, with more than $10^{4}$ at redshifts $z>7$, offering an unparalleled
dataset for constraining galaxy properties during the EoR. Our synthetic
catalogs and simulated images are made publicly available to provide the
community with a tool to prepare for upcoming data.

galaxies: high redshift – galaxies: evolution – cosmology: reionization

##  1 Introduction

The basic picture of galaxy formation is well-established. Galaxies first form
within the gravitational potential wells of dark matter halos, and continue to
grow through the accretion of surrounding matter. Galaxies eventually produce
enough radiation to ionize the intergalactic medium (IGM)—an epoch called
reionization (for a review see Robertson et al., 2010; Stark, 2016).
Constraints from a variety of probes, including the cosmic microwave
background (CMB; e.g. Planck Collaboration et al., 2018) and quasar absorption
lines (e.g. Becker et al., 2015), indicate that reionization happens between
$z=6$ and $z=9$. If galaxies dominate the contribution of photoionizing
radiation, the cosmic star-formation rate density (CSFRD) provides a measure
of the photoionizing rate. Observing high-redshift galaxies, to study galaxy
formation and their role in reionization, requires very deep imaging.

Extragalactic ultra-deep surveys such as the Hubble Ultra-Deep Field (HUDF;
e.g. Beckwith et al., 2006) and Hubble Frontier Fields (HFF; e.g. Lotz et al.,
2017) have detected galaxies to magnitudes $m_{\rm AB}\sim 30$ and have begun
to measure galaxy properties out to redshifts of $z\sim 10$. However, there
are still many open questions at these high redshifts including the emergence
of quiescent galaxies, the evolution of the UV luminosity function (UVLF; e.g.
Bouwens et al., 2021), and the exact timeline and mechanism of cosmic
reionization (e.g. Bunker et al., 2004; Finkelstein et al., 2012; Robertson et
al., 2015). Upcoming telescopes, including _James Webb Space Telescope (JWST)_
and the _Nancy Grace Roman Telescope (Roman)_ will produce a large influx of
data in the coming years that will greatly advance our understanding of galaxy
evolution in the epoch of reionization (EoR). Given that _Roman_ is scheduled
to launch in a few years, the purpose of this paper is to examine the science
returns of an ultra-deep survey with _Roman_.

The main advantage of _Roman_ compared to other space telescopes is its wide
field of view (FOV); the _Roman_ Wide-Field Instrument (WFI) FOV is more than
100x larger than _HST_ ’s WFC3 and _JWST_ ’s NIRCam. This large area will
increase the number of detected galaxies, discover bright and rare sources,
reduce cosmic variance, and probe the environment around galaxies and active
galactic nuclei (AGN) at unprecedented redshifts. As outlined in Koekemoer et
al. (2019), a potential Roman Ultra-Deep Field (UDF) survey could cover $\sim
1\,{\rm deg}^{2}$ and image to $m_{\rm AB}\sim 30$ in $\sim 600$ hours of
exposure time per filter. This survey would elucidate the properties of the
dominant ionizing sources at the time of reionization, allow tests for
variations in the high-$z$ faint-end slope of the UVLF with environment, and
likely provide the first galaxy clustering constraints at early times for
faint galaxies.

A prerequisite to understanding in detail what a _Roman_ UDF will be able to
detect is accurate modeling of the expected observations. In particular,
synthetic galaxy catalogs are useful for predicting the science returns of an
upcoming survey, to test analysis tools, and identify potential observational
biases (e.g. Williams et al., 2018; Yung et al., 2019a; Korytov et al., 2019;
Behroozi et al., 2020; Somerville et al., 2021). To make accurate predictions,
the quality and complexity of synthetic observations needs to increase with
expanding theoretical and observational knowledge. This work presents the Deep
Realistic Extragalactic Model (DREaM), a model for generating synthetic
galaxies out to redshifts past the EoR. DREaM accurately reproduces a wide
range of theoretical and observational trends, including stellar mass
functions (SMFs), and the CSFRD. We use DREaM to create synthetic data for a
potential Roman UDF, to provide synthetic catalogs for the community to help
develop analysis and pipeline tools, and quantify the potential scientific
returns of a Roman UDF.

To accurately capture the environment around each galaxy, we begin with dark
matter simulations, and then create an observed lightcone by stitching
together the discrete simulation outputs. The outline of this paper is as
follows: an overview of DREaM is given in Section 2. The underlying dark
matter simulations, and the method for assigning galaxies to dark matter halos
is given in Section 3, and the process of creating the observed lightcone in
Section 4. The galaxy morphologies and spectral energy distributions (SEDs)
are assigned as outlined in Sections 5 and Section 6, respectively. The
resulting star formation history of the universe is discussed in Section 7,
and preliminary predictions for the science returns of a $1\deg^{2}$ UDF with
_Roman_ are given in Section 8. Finally, the implications of this study and
future work are discussed in Section 9.

##  2 Overview of methods

While knowledge of galaxy distributions and properties comes primarily from
observations, dark matter structure is typically studied through numerical
simulations. The galaxy–halo connection describes how visible galaxies relate
to the underlying dark matter structure (for a recent review see Wechsler &
Tinker, 2018). Many different approaches have been used to link halo
properties to galaxies, including hydrodynamical simulations (e.g. Katz &
Gunn, 1991; Katz, 1992; Vogelsberger et al., 2014), semianalytic models (SAMs)
(e.g. White & Frenk, 1991; Kauffmann et al., 1993; Somerville & Primack, 1999;
Guo et al., 2013), halo occupation distribution models (HOD models; Jing et
al., 1998; Berlind & Weinberg, 2002; Wechsler et al., 2002), abundance
matching (AM; e.g. Kravtsov et al., 2004; Conroy et al., 2006; Vale &
Ostriker, 2006), and machine learning (e.g. Jo & Kim, 2019; Moster et al.,
2020; Wechsler et al., 2021). These different techniques range from physically
driven, computationally expensive approaches to empirical models designed to
reproduce known observational trends.

More physically based approaches are theoretically more predictive than
empirical models. However, both hydrodynamical simulations and SAMs typically
struggle to reproduce observed trends to the same accuracy as empirical models
(which match observations by construction).111Though there has been success
with the Santa Cruz SAM (Somerville & Davé, 2015) in making predictions for
upcoming _JWST_ observations Yung et al. (2019a, b, 2020b, 2020a). Therefore,
to successfully reproduce observational trends (including luminosity
functions, star formation histories, and galaxy clustering), synthetic galaxy
catalogs most commonly use empirical models that place galaxies in dark matter
structure using AM related methods (e.g. Moster et al., 2018; DeRose et al.,
2019; Behroozi et al., 2019, 2020; DeRose et al., 2021) or HOD models (e.g.
van den Bosch et al., 2005; Zu & Mandelbaum, 2015).

For the synthetic catalog presented in this work, we begin with dark matter
simulations, and then use subhalo abundance matching (SHAM) to model the
galaxy–halo connection, as outlined in Section 3. The main advantages to this
approach are that by beginning with dark matter simulations, we can accurately
capture the large-scale structure, and provide host dark matter halo
properties for each galaxy. SHAM methods reproduce the proper SMFs by
construction, and are known to reproduce the spatial distribution of galaxies
in the local Universe (e.g. Kravtsov et al., 2004; Reddick et al., 2013;
Lehmann et al., 2017).

To simulate a $1\,{\rm deg}^{2}$ survey that extends past $z\sim 10$, we use a
simulation volume with sides of comoving length $115\,h^{-1}{\rm Mpc}$. Fig. 1
shows the comoving tranverse size of a patch of sky covering $1\deg^{2}$ as a
function of redshift, and the corresponding volume of the survey. To reach the
desired depth of the synthetic realization, we tile 60 boxes in the line-of-
sight direction, as detailed in Section 4.1.

Figure 1: Comoving transverse size, $L$, and volume, $V$, of a survey with a
square $1\,{\rm deg}^{2}$ field of view. We calculate distances assuming a
Planck 2018 cosmology (Planck Collaboration et al., 2018). A redshift of
$z=10$ corresponds to a comoving size of $\sim 115$Mpc$h^{-1}$ in the
transverse direction.

When constructing synthetic observations from simulations, the discrete time
snapshots of the simulations need to be related to the observable sky, in
which the distance of the object corresponds to its observed time. This
relation between simulation data and the observable sky can be achieved by
creating a lightcone. Our lightcone pipeline is described in detail in Section
4, but, in brief, given merger histories for each halo we calculate if, and at
which time, each halo crosses the observers past lightcone. If the position of
the galaxy on the lightcone falls within the survey volume, the galaxy is
included in our synthetic catalog.

After creating the lightcone, we assign galaxy morphological properties and
SEDs in a manner similar to the phenomenological model from Williams et al.
(2018). Details of these procedures are outlined in Sections 5, and Section 6.
We use the publicly available Flexible Stellar Population Synthesis (FSPS;
Conroy et al., 2009; Conroy & Gunn, 2010) code to generate galaxy SEDs and
calculate galaxy fluxes in each of the proposed _Roman_ filters.

A summary of the methods used to generate the synthetic catalogs is shown in
Fig. 2. Section 3.1 describes the underlying dark matter simulation and
creation of the halo catalog. Sections 3.2 and 4 outline the SHAM and
lightcone procedure, respectively. We assign morphologies and SEDs to the
galaxies as described in Sections 5 and 6, respectively). Then, Section 8
outlines the _Roman_ photometry.

Figure 2: Overview of methods used to create the _Roman_ DREaM galaxy catalog,
with the corresponding paper sections labeled. The galaxy catalog is based on
a dark matter simulation. SHAM is used to assign galaxy masses to each dark
matter halo, and then galaxy properties are assigned using empirical
relations. Data products are shown with light blue rectangles, and pipelines
are shown with dark blue ovals. The SED pipeline is shown in more detail in
Fig. 9.

##  3 Generating synthetic galaxies

As described above, we begin our catalog by running dark matter only
simulations, and then use SHAM to assign stellar masses, resulting in halo
catalogs with corresponding galaxy masses for every simulation time output.
This section describes the underlying dark matter simulations, halo catalogs,
and SHAM procedure. In the following section, we use these galaxies to create
a lightcone realization of the catalog. We treat host halos and their subhalos
(i.e. halos that exist inside a larger halo as a self-bound structure)
separately, and use the term “halo” to refer to both host and subhalos.

###  3.1 Dark matter simulation

We use a Planck 2018 cosmology (Planck Collaboration et al., 2018,
$\Omega_{b}=0.04893$, $\Omega_{0}=0.3111$, $\Omega_{\Lambda}=0.6889$,
$H_{0}=67.66\,{\rm km/s/Mpc}$, $\sigma_{8}=0.8102$ and $n_{s}=0.9665$), with a
box size of $115\,h^{-1}{\rm Mpc}$ and $N=2048^{3}$ particles. This
corresponds to a particle mass of $1.5\times 10^{7}M_{\odot}h^{-1}$, The
initial conditions are created using Music (Hahn & Abel, 2011), and the
simulations are run in Gadget-2 (Springel, 2005), with a softening length of
$1.13$ comoving $h^{-1}{\rm kpc}$. The simulation outputs are shown in Fig. 3.
We generate halo catalogs and merger trees using Rockstar (Behroozi et al.,
2013a) and Consistent Trees (Behroozi et al., 2013b). We define halo masses
using the Bryan & Norman (1998) virial definition. Halos are required to have
at least 20 particles, which corresponds to a minimum mass of
$8.50M_{\odot}h^{-1}$.

Figure 3: Cosmological $N$-body simulation with a box size of $115h^{-1}{\rm
Mpc}$. The figure shows the projected mass density at various redshifts (as
labeled), in a $7\,{\rm Mpc/h}$ slice in the radial direction. These
simulations serve as the basic input for finding halos and modeling the galaxy
population.

###  3.2 Galaxy masses

Given the halo lightcone catalog constructed in Section 3, we assign a galaxy
stellar mass, $M_{\rm gal}$, to each halo. The SHAM procedure we use can be
expressed mathematically as:

$\int_{x}^{\infty}n(x^{\prime},z){\rm d}x^{\prime}=\int_{M_{\rm
gal}}^{\infty}\phi(M_{\rm gal}^{\prime},z){\rm d}M_{\rm gal}^{\prime}$ (1)

where $\phi(M_{\rm gal},z)$ is the SMF (i.e. the comoving number density of
galaxies per unit galaxy mass, per unit redshift), and $n(x)$ is the number
density of halos as a function of some mass proxy, $x$.

In the simplest formulation of SHAM, galaxy mass is directly mapped to halo
mass. However, subhalo properties at the time of accretion are a better
indicator of galaxy mass, since as subhalos undergo tidal stripping, they lose
a significant amount of dark matter before the galaxy is disrupted (e.g.
Conroy et al., 2006; Vale & Ostriker, 2006). There are many different choices
of mass proxies, and each of these choices produces slightly different spatial
distributions of galaxies. We adopt the peak of the maximum circular velocity
over the entire merger history, $V_{\rm peak}$, as the halo mass proxy,
because SHAM with $V_{\rm peak}$ as the mass proxy is known to reproduce the
small scale clustering of galaxies (Reddick et al., 2013; Hearin et al., 2013;
Lehmann et al., 2017; Campbell et al., 2018).

We use the SMFs from Williams et al. (2018), which are continuously evolving
double-Schechter functions for both star-forming and quiescent galaxies out to
redshifts $z>10$. By construction, these SMFs match observational constraints
at low redshifts ($z\leq 4$), and reproduce known UVLFs when convolved with
the distribution of rest-frame UV magnitudes, $M_{\rm UV}$, i.e.:

$\displaystyle\Phi_{\rm UV}($ $\displaystyle M_{\rm UV},z)$ (2)
$\displaystyle=\int_{0}^{\infty}\phi(M_{\rm gal}^{\prime})\mathcal{N}[M_{\rm
UV},\bar{M}_{\rm UV}(M_{\rm gal}^{\prime},z),\sigma_{\rm UV}]{\rm d}M_{\rm
gal}^{\prime}\,\,\,,$

where $\mathcal{N}[M_{\rm UV},\bar{M}_{\rm UV}(M_{\rm gal},z),\sigma_{\rm
UV}]$ is a normal distribution centered on the average UV magnitude
$\bar{M}_{\rm UV}$ and $\sigma_{\rm UV}$ is the scatter in the $M_{\rm
gal}$–$\bar{M}_{\rm UV}$ relation. Williams et al. (2018) provide a relation
to describe the $M_{\rm gal}$–$\bar{M}_{\rm UV}$ relation and scatter, which
we use to model our galaxies (see Section 6.4).

For redshifts $z\leq 4$, Williams et al. (2018) fit the double-Schechter
function to the data from Tomczak et al. (2014). For quiescent galaxies (QGs),
above $z\sim 4$ the Schechter parameters are extrapolated to higher redshifts.
This extrapolation is in agreement with the few constraints for quiescent
galaxies at $z>3.5$. For star-forming galaxies (SFGs), the high redshift SMFs
are inferred from UVLFs. Specifically, Williams et al. (2018) used data from
Bouwens et al. (2015) for $4\leq z\leq 8$ and from Oesch et al. (2018) at
$z=10$ to fit the SMF in Equation (2). Beyond $z\sim 10$, the SMFs for the
SFGs are extrapolated.

We perform SHAM on the halo catalogs using the SMFs described above.
Specifically, given $N$ halos, we first find the minimum galaxy mass, $M_{\rm
min}$, such that $N_{\rm gal}(>M_{\rm min})=N$, where

$N_{\rm gal}(>M_{\rm min})=V\int_{M_{\rm min}}^{\infty}\phi(M_{\rm
gal}^{\prime}){\rm d}M_{\rm gal}^{\prime}\,\,\,,$ (3)

and $V$ is the comoving volume of the simulation. We then sample $N$ galaxy
masses from the SMF above $M_{\rm min}$. Finally, we rank-order the galaxy
masses and assign galaxy masses to halos (such that the halo with the largest
$V_{\rm peak}$ value gets the largest galaxy mass).

Scatter is commonly introduced in the relation between galaxy and halo
properties by either deconvolving the SMF (e.g. Behroozi et al., 2010) or by
directly adding scatter to the stellar masses, re-ranking, and iteratively
solving for the galaxy mass (Hearin et al., 2013). Since our main goals are
not to reproduce the observed scatter in the stellar-mass–halo-mass (SMHM)
relation, we did not include scatter in the SHAM procedure. As illustrated in
Section 4.3, we are able to produce realistic statistical galaxy properties,
such as galaxy clustering with our approach.

We classify each galaxy as either a star-forming galaxy (SFG) or a quiescent
galaxy (QG) by randomly generating a number and comparing it to the
probability a galaxy of that mass is star-forming, as calculated from the
SMFs. This method does ignore a possible correlation between SFR and mass
accretion histories (e.g. Behroozi et al., 2019), which could potentially
impact galaxy clustering with SFR. In particular, QGs are more likely to be
found in denser environments (e.g. Kauffmann et al., 2004; Kimm et al., 2009).
Finally we remove all galaxies from the catalog with $M_{\rm
gal}<10^{5}M_{\odot}$. As shown in Section 8, the fraction of detectable
galaxies below this mass is zero for redshifts $z>1$.222Though faint high-
redshift galaxies can also be detected through galaxy lensing, we expect this
number to be very low for low-mass galaxies. For example, Kikuchihara et al.
(2020) used gravitational lensing techniques to detect very faint drop out
galaxies in the HFFs to redshifts $z\sim 6$, and detected galaxies with
$M_{\rm gal}>10^{6}M_{\odot}$

##  4 Lightcone pipeline

Thus far we have created halo catalogs with galaxy masses at different time
outputs. In this section we detail how we construct a lightcone from these
discrete snapshots to create a $1\deg^{2}$ survey of galaxies (the “Lightcone
Catalog”). In Section 4.3, we demonstrate that we are able to reproduce
essential statistical properties of galaxy populations with this Lightcone
Catalog.

###  4.1 Lightcone crossing

For this work, we consider a survey with a $1\,\deg^{2}$ field of view. To
reach redshifts $z\gtrsim 10$, we tile 60 boxes in the line-of-sight
direction. This corresponds to a maximum redshift of 13 and a comoving
distance of $6900\,{\rm Mpc}$. As in Bernyk et al. (e.g. 2016), to avoid the
replication of structures viewed by the observer, we randomly translate,
reflect, and permute the axes of every tiled simulation box. We construct the
halo lightcone by finding where each halo first crossed the observers past
lightcone (i.e., the location at which light has had just enough time to reach
the observer). Technical aspects associated with this procedure are outlined
extensively in the literature (e.g. Evrard et al., 2002; Blaizot et al., 2005;
Kitzbichler & White, 2007; Merson et al., 2013; Bernyk et al., 2016; Smith et
al., 2017; Korytov et al., 2019). In this section, we closely follow the
procedure used to generate the CosmoDC2 sky catalog for _LSST_ (Hollowed,
2019; Korytov et al., 2019).

We assume that the observer is at the origin of the coordinate system
$(x,y,z)=(0,0,0)$. For each tiled box, we begin by placing all the host halos
on the lightcone. For every host halo in snapshot $j$, we determine its
position in the subsequent snapshot, $r_{j+1}$. We extrapolate the halo
position from $j$ by assuming constant velocity to find its extrapolated
position in snapshot $j+1$, $r_{j+1,{\rm extrap}}$. If the halo did not have a
descendant located in snapshot $j+1$, we set $r_{j+1}=r_{j+1,{\rm extrap}}$.
Otherwise, we use:

$r_{j+1}=r_{j+1,{\rm extrap}}+{\rm int}\left(\dfrac{r_{j+1,{\rm
extrap}}-r_{j+1,{\rm desc}}}{L}\right)\times L\,\,\,,$ (4)

where $L$ is the length of the simulation box, ${\rm int}$ represents integer,
and $r_{j+1,{\rm desc}}$ is the position of the descendant in snapshot $j+1$.
This approach allows the halo to cross the edge of the simulation box between
snapshots $j$ and $j+1$, and thus $r_{j+1}$ might be outside the domain of the
tiled box. In these cases, we also consider the scenario where the halo
crosses the lightcone on the other side of the box; i.e., we apply periodic
boundary conditions, such that

$\displaystyle r_{j+1}^{p}$ $\displaystyle=r_{j+1}\,{\rm mod}\,L$ (5)
$\displaystyle r_{j}^{p}$ $\displaystyle=r_{j}+(r_{j+1}^{p}-r_{j+1})\,\,\,,$

to allow the halo to cross the observer lightcone between positions
$r_{j}^{p}$ and $r_{j+1}^{p}$.

Given the positions $r_{j}$ and $r_{j+1}$ for each host halo, we calculate the
time the halo would cross the past lightcone, $t_{e}$ (Equations 27-29 in
Korytov et al., 2019). If the halo crosses between the snapshot times $t_{j}$
and $t_{j+1}$, we calculate the position on the lightcone, $r_{e}$ from:

$\displaystyle v_{\rm lin}=(r_{j+1}-r_{j})/(t_{j+1}-t_{j})$ (6) $\displaystyle
r_{e}=r_{j}+v_{\rm lin}(t_{e}-t_{j})\,\,\,.$

We do not include any halos where $r_{e}$ is beyond the domain of the tiled
box.

The final step to generating the lightcone is to assign halo properties (e.g.
mass and substructure) to each object. One approach to assigning halo
properties is to allow the merger of halos to happen at a time randomly
between the two snapshot (e.g. Smith et al., 2017). However, this approach
will result in halos being double counted, and the lightcone needs to be
carefully pruned. Since our simulations have very fine time resolution (500
snapshots between redshifts $z=0$ and $z=20$), we can avoid the technical
difficulties and assumptions introduced by needing to prune the catalog, and
follow the same approach from Korytov et al. (2019). Specifically, every halo
that crosses the lightcone between snapshots $j$ and $j+1$ is assigned
properties from snapshot $j$. When including the substructure, we ensure
subhalos have the same position and velocity offset from the host as in
snapshot $j$.

###  4.2 Survey volume

Given the lightcone catalog consisting of halo properties, galaxy masses, and
positions, we cut out a wedge in the survey volume corresponding to $1\,{\rm
deg}^{2}$. Following Bernyk et al. (2016), we convert the comoving $(x,y,z)$
positions of each galaxy to an angular position as follows:

$\begin{gathered}d=\sqrt{x^{2}+y^{2}+z^{2}}\\\ {\rm
RA}=\dfrac{360}{2\pi}\arctan(y/x)\\\ {\rm
Dec}=\dfrac{360}{2\pi}\arcsin(z/d)\end{gathered}$ (7)

We only consider galaxies with ${\rm RA}<1\,{\rm deg}$ and ${\rm Dec}<1\,{\rm
deg}$, and then center the galaxies on $({\rm RA},{\rm Dec})=(0,0)$.

Fig. 4 shows the resulting galaxy mass density as a function of position. The
lightcone procedure results in a realistic distribution of galaxies that trace
the underlying cosmic web. Since the survey will reach very deep distances,
the survey is tiled in 6 rows, with each row showing 10 tiled boxes. The
survey wedge is complete to a maximum distance of $6588\,\rm{Mpc/h}$ (which
corresponds to a cosmological redshift of $\sim 10.5$); at higher reshifts,
the angle of the survey is larger than the width of the simulation box. There
are discontinuities in the galaxy density where the simulation boxes are
tiled, which is a common feature for lightcones. As addressed in Bernyk et al.
(2016), statistical properties of the galaxy catalog will be accurate on
scales smaller than the box size.

Figure 4: Mass density of galaxies in survey volume, with a field of view of
$1\,{\rm deg}^{2}$. The $x$-axis shows the cosmological redshift (top) and the
comoving distance (bottom). We have plotted 10 tiled boxes in each row, for a
total of 60 boxes to reach the desired redshift range. This figure
demonstrates the depth and distribution of galaxies of a $1\,\deg^{2}$ UDF.

###  4.3 Statistical properties of galaxy lightcone

To verify the lightcone pipeline results in a realistic galaxy population, we
calculate the halo mass functions (HMFs), SMFs, stellar-to-halo mass relations
(SHMRs) and galaxy clustering of the synthetic galaxy catalog.

####  4.3.1 Halo mass functions

To demonstrate the HMF of the halo lightcone catalog is correct, we plot the
recovered HMF of host halos in Fig. 5. The theoretical curve is from the
parameterization from Despali et al. (2016), and calculated using the HMF
routine in Colossus (Diemer, 2018). The theoretical curve and simulation
results agree very well. The catalog was constructed to be complete above
galaxy masses $M_{\rm gal}=10^{5}M_{\odot}$, which corresponds to a halo mass
limit of approximately $M_{\rm halo}\sim 10^{9.5}h^{-1}M_{\odot}$

Figure 5: Expected HMF from the Despali et al. (2016) parameterization (dotted
lines) compared to the host halos in the lightcone catalog (solid lines). The
theoretical curve is the volume-weighted average over the redshift band. The
lightcone catalog agrees with the expected HMF, verifying that the lightcone
procedure reproduces the correct redshift and mass distribution of dark matter
halos.

####  4.3.2 Stellar mass functions and the stellar-to-halo mass relations

Additionally, we show the SMF of the lightcone catalog in Fig. 6 for both SFGs
and QGs. For comparison, we show the volume-weighted average SMFs from
Williams et al. (2018) (dotted lines), averaged over the redshift bin. The
catalog is complete at all redshifts for galaxy masses greater than
$10^{5}\,M_{\odot}$. The galaxies in the lightcone agree with the SMFs from
Williams et al. (2018), which verifies that the lightcone pipeline and
abundance matching procedure reproduce realistic galaxy counts. As discussed
in Section 3.2, producing the desired SMF is important for reproducing
observed luminosity functions.

Figure 6: SMFs for the lightcone catalog. SFGs and QGs are shown in the top
and bottom panels, respectively. The Williams et al. (2018) SMFs that were
used in the abundance matching procedure are shown with dotted lines,
calculated as a volume-weighted average over the redshift bin. The lightcone
pipeline and abundance matching procedure reproduce the desired SMFs.

####  4.3.3 Galaxy clustering

Galaxy clustering is commonly described using the two-point correlation
function (2PCF), $\xi(r)$, defined as

${\rm d}P=n_{g}(r)[1+\xi(r)]{\rm d}V\,\,\,,$ (8)

where ${\rm d}P$ is the excess probability above Poisson noise, $n_{g}(r)$ is
the mean density of galaxies at separation $r$, and ${\rm d}V$ is the
differential volume. In practice, the projected 2PCF, $w_{p}$, is often used.
The 2PCF depends on the projected separation, $r_{p}$, and the line-of-sight
separation, $\pi$:

$w_{p}=\int_{-\infty}^{\infty}\xi(r_{p},\pi){\rm d}\pi\approx
2\times\int_{0}^{\pi_{\rm max}}\xi(r_{p},\pi){\rm d}\pi\,\,\,,$ (9)

where the second equality assumes isotropy. The upper limit, $\pi_{\rm max}$,
needs to be chosen large enough to nullify the effect of peculiar velocities,
but not so large that it creates artificial edge effects from the survey
boundary. The optimal value for $\pi_{\rm max}$ depends on the underlying
survey volume, but is typically in the range 40–80 Mpc (Zehavi et al., 2005;
van den Bosch et al., 2013).

We calculate $\xi$ based on the commonly used Landy-Szalay estimator (Landy &
Szalay, 1993),

$\xi\approx\dfrac{DD-2DR+RR}{RR}\,\,\,,$ (10)

where $DD$, $DR$ and $RR$ are normalized number of counts from data–data,
data–random and random–random pairs. The random catalog contains $10^{3}$
points per ${\rm arcmin}^{2}$, with distances assigned such that the random
catalog has the same redshift distribution as the synthetic catalog, i.e.,

$\displaystyle N$ $\displaystyle(z_{\rm min}<z<z_{\rm max})$ (11)
$\displaystyle=\theta\sin(\theta)\int_{\chi(z_{\rm min})}^{\chi(z_{\rm
max})}\chi^{2}\int_{M_{1}}^{M_{2}}\phi(M,z(\chi)){\rm d}M{\rm d}\chi$

where $\theta$ is the survey angle, and $\phi$ is the Williams et al. (2018)
SMF. Values for RA and Dec coordinates for each random galaxy were selected so
that galaxies are distributed isotropically:

$\displaystyle{\rm RA}$ $\displaystyle=r_{1}\theta$ (12) $\displaystyle{\rm
Dec}$
$\displaystyle=\arcsin\left[2r_{2}\sin\left(\dfrac{\theta}{2}\right)\right]\,\,\,.$

where $r_{1}$, $r_{2}$ are random numbers selected uniformly between -0.5 and
0.5.

To measure the projected 2PCF we use the package Corrfunc (Sinha & Garrison,
2020), with $\pi_{\rm max}=60\,h^{-1}\,{\rm Mpc}$. We estimateerrors in the
measured 2PCF by bootstrapping the galaxy catalog data with 200 subsamples. We
present this measured 2PCF in Fig. 7, along with SDSS data for redshift
$z\approx 0.1$ galaxies from Yang et al. (2012). To compare our synthetic
catalog 2PCF to the data, we use galaxies in the redshift range $0.1<z<0.2$
within a $10\deg\times 10\deg$ survey. The low redshift clustering of the
synthetic galaxies agree with observations within the $1-\sigma$ error bars,
indicating that the lightcone halo clustering is realistic.

Figure 7: 2PCF in a $100\deg^{2}$ lightcone catalog for all galaxies
$0.1<z<0.2$ (lines). We estimate uncertainties in the synthetic catalog 2PCF
by bootstrapping the data. In comparison, SDSS data from Yang et al. (2012),
for galaxies at redshifts $z\approx 0.1$ are shown with black points. The
agreement between the simulation and data shows that the lightcone catalog
matches the observed galaxy clustering at low redshifts.

##  5 Galaxy morphologies

Galaxy morphologies offer key insights into galaxy evolution, and are
important for understanding observational systematics. We model all galaxies
as Sérsic (1968) profiles with an index, $n_{s}$. Additionally, all galaxies
are assigned a projected size $R_{\rm eff}$, a projected axis ratio $q=b/a$,
and a position angle (PA). We follow the morphological prescriptions from
Williams et al. (2018)—which are based on the empirical distributions measured
in _HST_ images—with the exception of galaxy sizes.

###  5.1 Size

Galaxies sizes are known to decrease with increasing redshift at a fixed
stellar mass (e.g. van der Wel et al., 2014; Shibuya et al., 2015; Curtis-Lake
et al., 2016), typically evolving as

$R_{e}\propto(1+z)^{-\alpha}\,\,\,,$ (13)

where $R_{e}$ is the half-light radius. Further, SFGs and QGs evolve
differently, with star-forming galaxies having lower values of $\alpha$ (e.g.
van der Wel et al., 2014; Ma et al., 2018).333However, Miller et al. (2019)
recently showed that differences between size trends for SFGs and QGs may
disappear if you use $r_{80}$—the radius enclosing 80 % of stellar
light—rather than $R_{e}$. In addition to these observed trends, theoretical
models commonly show a correlation between the size of a galaxy and that of
its host dark matter halo (e.g. Mo et al., 1998; Kravtsov, 2013; Jiang et al.,
2019). Given that our synthetic catalog begins with the underlying dark matter
structure, we assign galaxy sizes based on the dark matter halo sizes, and
then test that the size–redshift–mass relation is consistent with current
data.

Specifically, we characterize the size of each galaxy as the half-light radius
in the semi-major axis $R_{\rm eff}$.444$R_{\rm eff}$ is related to the
commonly used circularized half-mass radius, $R_{\rm eff,circ}$, by $R_{\rm
eff,circ}=\sqrt{b/a}R_{\rm eff}$, where $b/a$ is the minor-to-major axis
ratio. To assign $R_{\rm eff}$ to each galaxy, we use the relation between
halo and galaxy size (e.g. Kravtsov, 2013; Zanisi et al., 2020):

$R_{\rm eff}=A\,R_{\rm vir},$ (14)

where $R_{\rm vir}$ is the radius of the halo in physical units. We assign
$R_{\rm eff}$ values from this mean relation, with lognormal scatter,
$\sigma_{R}$. We use the coefficients $A$ and scatter $\sigma_{R}$ from Zanisi
et al. (2020), as summarized in Table 1. Zanisi et al. (2020) used size
distributions of central galaxies from the Sloan Digital Sky Survey (SDSS) DR7
sample (Abazajian et al., 2009; Meert et al., 2016). While these coefficients
capture the dependence of $R_{\rm eff}$ on mass and on whether a galaxy is
star-forming or quiescent, they are calibrated for $z=0$ galaxies.

Table 1: Coefficients, $A$, for the $R_{\rm eff}$–$R_{\rm vir}$ relation (Equation 14) and the scatter $\sigma_{R}$; values are from Zanisi et al. (2020). $\log_{10}(M_{\rm gal}/M_{\odot})$ |  $A$ (SFGs) |  $A$ (QGs) | $\sigma_{R}$
---|---|---|---
$\leq 9.5$ | 0.018 | 0.006 | 0.2
$(9.5,10]$ | 0.019 | 0.007 | 0.2
$(10,10.5]$ | 0.019 | 0.010 | 0.15
$(10.5,11]$ | 0.019 | 0.011 | 0.15
$(11,11.5]$ | 0.019 | 0.015 | 0.15
$>11.5$ | 0.024 | 0.016 | 0.1

To test whether this relation extends reasonably to higher redshifts, we
compare the median $R_{\rm eff}$–$z$ relation from our catalog to data from
van der Wel et al. (2014) and Shibuya et al. (2015), as shown in Fig. 8. The
datasets shown in Fig. 8 consist of galaxies from the 3D-_HST_ +CANDELS
catalog. Redshifts and masses are from Skelton et al. (2014), and galaxies are
classified as either star-forming or quiescent using UVJ diagram criteria.

Figure 8: Median half-light semi-major radius, $R_{\rm eff}$ as a function of
redshift from the synthetic galaxy catalog for SFGs (top) and QGs (bottom).
Different columns show different mass bins. The shaded regions are the
$1-\sigma$ standard deviation in each bin for the synthetic galaxies. The
points show median data $R_{\rm eff}$ from van der Wel et al. (2014); Shibuya
et al. (2015), with the 16th and 84th percentiles of the data point
distribution. Our synthetic galaxies match observed trends of galaxy sizes
with mass, redshift, and galaxy classification.

Fig. 8 demonstrates that our simple assumption that Equation (14) holds for
all redshifts agrees remarkably well with the data. We capture the decrease in
$R_{\rm eff}$ with both decreasing mass and increasing redshift. We also
capture the effect that QGs are smaller than SFGs on average. Our ability to
capture redshift trends, despite only using known relations between $R_{\rm
eff}$ and $R_{\rm vir}$ at $z=0$, likely reflects the fact that virial radii
evolve as $R_{\rm vir}\propto 1/(1+z)$, and therefore higher redshift galaxies
have smaller virial radii. Our finding that the redshift evolution in the
mass-size relation can be captured through the redshift evolution $R_{\rm
vir}$ has also been found by Mowla et al. (2019), who show that the ratio
$r_{80}/R_{\rm vir}$ is constant for redshifts $z<3$.

###  5.2 Axis ratios, Sérsic indices and Position Angles

The distribution of galaxy axis ratios and Sérsic indices should differ
between SFGs and QGs (e.g. Franx et al., 2008; Bell et al., 2012; Mortlock et
al., 2013) and with redshift (e.g. van der Wel et al., 2011; Guo et al.,
2015). To assign shape (defined as the projected axis ratio $q=b/a$, where $a$
is the semi-major half-light size and $b$ is the semi-minor half-light size)
and Sérsic indices, $n_{s}$, we use the method directly from Williams et al.
(2018). Specifically, using data compiled from van der Wel et al. (2012) and
Skelton et al. (2014), Williams et al. (2018) finds the distribution of $q$
and $n_{s}$ as a function fo redshift. Due to limited data, morphologies for
QGs with $z>4$ are drawn from the $3\leq z\leq 4$ distribution, and
morphologies for SFGs with $z>6$ are drawn from the $5\leq z\leq 6$
distributions. We use these resulting distributions to draw $q$ and $n_{s}$
values to assign to each galaxy.

We assign PAs uniformly between $0$ and $2\pi$. Synthetic galaxy catalogs
often assume that galaxies are oriented isotropically (e.g. Williams et al.,
2018), and this assumption will not affect the galaxy clustering, number
counts, or cosmic star formation history of our synthetic catalog.

##  6 Galaxy SEDs

In this section we outline the methods we use to generate galaxy SEDs. The SED
pipeline aims to reproduce observed star formation histories and UVLFs, as
these quantities dictate the number of ionizing photons produced by galaxies
(see Section 7). The SEDs also have realistic SFG and QG properties, including
their observed colors, ages, and metallicity (see Appendix B).

###  6.1 Overview

One of the main observables included in the synthetic catalog is the
photometry in the _Roman_ filters, which requires accurate modeling of the
galaxy SEDs. We model the SEDs for each galaxy using the software FSPS (Conroy
et al., 2009; Conroy & Gunn, 2010). This section serves as an overview of the
model used to generate the galaxy SEDs.

We use a Chabrier (2003) initial mass function (IMF), and include the IGM
absorption model from Madau (1995). The star-formation history model is
described in Section 6.2.1. Additionally, for SFGs, we include the FSPS
nebular emission model (Byler et al., 2017) which is controlled by the gas
ionization parameter and the gas metallicity. As in Williams et al. (2018), we
approximate the stellar and ISM metallicities as a single metallicity value,
$Z_{\rm met}$.

The dust modeling includes dust emission and dust absorption. Dust emission
follows Draine & Li (2007), which is a silicate-graphite-PAH grain model. The
general FSPS dust absorption model follows the commonly used Charlot & Fall
(2000) model. We use the Calzetti et al. (2000) attenuation curve, where dust
attenuation is applied to all starlight equally, and therefore depends on one
parameter, the total attenuation optical depth $\bar{\tau}_{v}$. We also
include the asymptotic giant branch (AGB) circumstellar dust models from
Villaume et al. (2015). Including circumstellar dust can make a significant
contribution to the IR emission from galaxies with little diffuse gas, and
will need to be included in stellar modeling to accurately interpret data from
upcoming IR facilities, including _JWST_.

Overall, this leaves seven free parameters to describe the galaxy SEDs:

  * •

$M_{\rm gal}$: the stellar mass of the galaxy.

  * •

$z$: the redshift of the galaxy (or alternatively, $t_{\rm age}$, the age of
the universe at that redshift).

  * •

$Z_{\rm met}$: the galaxy metallicity. The gas-phase and stellar metallicities
are assumed to be equal.

  * •

$t_{\rm start}$: the age of the universe at the start of star formation.

  * •

$\tau$: the $e$-folding time for star-formation.

  * •

$\bar{\tau}_{v}$: the dust attenuation parameter, defined as the opacity at
$5500\AA$.

  * •

$U_{S}$: the gas ionization parameter.

In addition to these seven parameters, other relevant quantities included in
the synthetic catalog include the star-formation rate (SFR), the UV magnitude,
$M_{\rm UV}$, the slope of the UV continuum, $\beta$, and the rest-frame
colors of the galaxies. Section 6.2 describes the calculation of each of these
quantities.

Each galaxy begins with a redshift $z$ (from the lightcone pipeline), a mass,
$M_{\rm gal}$ (from abundance matching), and is labeled as either a SFG or a
QG, as sampled from the SMF (see Section 3.2). Sections 6.4 and 6.5 below
describe SED pipeline for assigning the remaining FSPS parameters ($Z_{\rm
met}$, $t_{\rm start}$, $\tau$, $\bar{\tau}_{v}$ and $U_{S}$) for SFGs and
QGs, respectively. The pipeline is summarized in Fig. 9.

In brief, we first generate “parent catalogs” for both SFGs and QGs, spanning
a realistic range of parameters. These parent catalogs serve as a lookup table
for realistic galaxy SEDs. Details on how the parent catalogs are calculated
are given in Section 6.3. We propose parameters for each galaxy, and find the
10 nearest neighbors in the parent catalog for each set of galaxy parameters
using k-d trees555k-d trees are $k$-dimensional data structures arranged as a
binary tree. This structure allows for rapid searches to find the nearest
neighbors of a given point in the $k$-dimensional space.. For SFGs, the
proposed parameters are $M_{\rm gal}$, $z$, $M_{\rm UV}$ and $\beta$ while for
QGs, the proposed parameters are $M_{\rm gal}$ and $z$. The distance metrics
used to find the nearest neighbours are detailed in Sections 6.4 and 6.5.

We assign the five free FSPS parameters ($Z_{\rm met}$, $t_{\rm start}$,
$\tau$, $\bar{\tau}_{v}$ and $U_{S}$) to each galaxy by taking the weighted
average of the nearest neighbors. Given these five parameters, along with the
mass and redshift of the galaxy, we use FSPS to generate an SED. We then
calculate the UV properties, SFRs and rest-frame colors directly from the
spectrum. These calculated properties will not be identical to the proposed
parameters, however we still retain the imposed scaling relations in the
parent catalogs (see Appendix B).

Figure 9: Overview of the SED pipeline. We begin with a galaxy catalog
consisting of redshift and masses for each galaxy, and parent catalogs for the
SFGs and QGs. We propose parameters for every galaxy in the catalog and find
the nearest neighbors in the parent catalogs. We then assign FSPS parameters
based on a weighted average of the nearest neighbors.

###  6.2 Calculated quantities

For the SED pipeline outlined above, we need to accurately measure the SFR,
rest-frame colors, and UV properties of the galaxy SEDs. This section details
the methods we use to measure these quantities. Since FSPS normalizes the
stellar modeling, such that one stellar mass is created over the formation
history, we scale all the spectra. Specifically, we scale all fluxes and the
SFR by $M_{\rm gal}/f$, where $f$ is the surviving mass fraction in the
stellar population, not including stellar remnants.

####  6.2.1 Star formation history

Galaxy SFRs dictate the cosmic star formation rate density (CSFRD) of the
universe, which in turn constrains the amount of ionizing photons produced by
galaxies in the EoR. Given the importance of SFRs to galaxy evolution and the
sources of ionizing photons during the EoR, we need to ensure our synthetic
galaxies have SFRs consistent with observations.

The SFR depends on the age, $e$-folding time and mass of the galaxy. We model
the star formation history, $\psi(t)$, using a “delayed tau model”, in which:

$\psi(t)\propto t\exp{-t/\tau}\,\,\,,$ (15)

where $t$ is the time since the start of star formation, $t_{\rm start}$. As
discussed in Williams et al. (2018), this parameterization achieves the
expectation from simulations that high-redshift galaxies have rising star
formation histories (Finlator et al., 2011), and accurately reproduces the
colors and mass-to-light ratios of galaxies in smoothed particle hydrodynamics
(SPH) simulations (e.g. Simha et al., 2014).

To determine the SFR for each galaxy, we use the FSPS calculated SFR, and
scale it by $M_{\rm gal}/f$, as described above.

####  6.2.2 Rest-frame colors

UVJ diagram galaxy classifications allow for the separation of QGs and dusty
SFGs (e.g. Williams et al., 2009; Whitaker et al., 2013; Papovich et al.,
2018). To examine the distribution of the synthetic galaxies in UVJ space, we
use calculated U-V and V-J colors. Specifically, we use the Johnson U and V
filters, and the WFC3 F125W J filter included in FSPS.

####  6.2.3 UV properties

_Roman_ will constrain the number of ionizing photons that are produced by
galaxies in the EOR by precisely measuring the UVLF at high redshift. The UVLF
depends on the SMF and the UV magnitude, $M_{\rm UV}$ (see Equation 2).

Following Robertson et al. (2013) we define the UV magnitude, $M_{\rm UV}$, as
the average magnitude at rest-frame wavelength in a flat filter, in the range
1450-1550Å. We measure $M_{\rm UV}$ directly from the rest-frame galaxy SEDs
by calculating the average flux density:

$\langle
f_{\lambda}\rangle=\dfrac{\int_{\lambda_{1}}^{\lambda_{2}}f_{\lambda}d\lambda}{\lambda_{2}-\lambda_{1}}\,\,\,,$
(16)

with $\lambda_{1}=1450$Åand $\lambda_{2}=1550$Å.

In addition to the UV magnitude, the slope of the UV continuum, $\beta$, helps
determine the role of galaxies in ionizing the universe. The slope $\beta$ is
defined as $f_{\lambda}\propto\lambda^{\beta}$ (e.g. Meurer et al., 1999). The
presence of very blue UV continuum slopes in Roman imaging may provide a
signpost of very high escape fractions in early galaxies. We use the method
from Dunlop et al. (2012) to determine $\beta$. Specifically, we take the SEDs
shifted to redshift $z=7$, and calculate

$\beta=4.43(J125-H160)-2$ (17)

using the FSPS included filters F125W and F160W for the Wide Field Camera 3
(WFC3).

###  6.3 Parent Catalogs

We create the parent catalogs by sampling from known scaling relations. By
only populating regions of parameter space corresponding to the desired galaxy
population, we can fully sample the relevant space, while also limiting the
size of the parent catalog. The distributions we use to construct the parent
catalogs are summarized in Tables 2 and 3 for SFGs and QGs, respectively, and
outlined in detail in this section.

Table 2: Parameters for SEDs of SFGs. The synthetic galaxies have a fixed parameters $M_{\rm gal}$ and $z$ and “proposed” parameters $M_{\rm UV}$ and $\beta$. We identify the 10 closest neighbours in the parent catalog (defined by the distance metric), and assign the final catalog properties $Z_{\rm met}$, $U_{S}$, $\bar{\tau}_{V}$, $t_{\rm start}$ and $\tau$ from the nearest neighbors. Parameter | Proposed Parameter | Parent Catalog | Distance Metric | Assigned Parameter
---|---|---|---|---
$M_{\rm gal}$ | fixed | uniform in $\log_{10}(M_{\rm gal}/M_{\odot})\in[5,12]$ | $\log_{10}(M_{\rm gal}/10M_{\odot})$ | fixed
$z$ | fixed | uniform in $a=1/(1+z)\in[0.07,1]$ | $a=10/(1+z)$ | fixed
$Z_{\rm met}$ | … | FMR | … | nearest neighbors
$U_{S}$ | … |  $U_{S}$–$Z_{\rm met}$ relation | … | nearest neighbors
$\bar{\tau}_{V}$ | … | $\mathcal{N}(\mu=0,\sigma=0.5)\in[0,4]$ | … | nearest neighbors
$t_{\rm start}$ | … | uniform in $[1\,{\rm Myr},t_{\rm age}]$ | … | nearest neighbors
$\tau$ | … | uniform in $[0.1,100]$ Gyr | … | nearest neighbors
$\psi$ | … | calculated; cut $\log_{\rm 10}(\psi/M_{\odot}{\rm yr}^{-1})\in[-5,4]$ | … | calculated
$M_{\rm UV}$ |  $M_{\rm UV}$–$M_{\rm gal}$ | calculated; cut in $[-25,-5]$ |  $M_{\rm UV}$ \+ 20 | calculated
$\beta$ |  $\beta$–$M_{\rm UV}$ | calculated; cut in $[-3,-1]$ | $\beta$ | calculated
$U$-$V$ | … | calculated | … | calculated
$V$-$J$ | … | calculated | … | calculated
Table 3: Parameters for SEDs of QGs. The synthetic galaxies have a fixed parameters $M_{\rm gal}$ and $z$. We identify the 10 closest neighbours in the parent catalog (defined by the distance metric), and assign the final catalog properties $Z_{\rm met}$, $\bar{\tau}_{V}$, $t_{\rm start}$ and $\tau$ from the nearest neighbors. Parameter | Proposed Parameter | Parent Catalog | Distance Metric | Assigned Parameter
---|---|---|---|---
$M_{\rm gal}$ | fixed | uniform in $\log_{10}(M_{\rm gal}/M_{\odot})\in[5,12]$ | $\log_{10}(M_{\rm gal}/10M_{\odot})$ | fixed
$z$ | fixed | uniform in $[0,13]$ | $a=10/(1+z)$ | fixed
$Z_{\rm met}$ | … | FMR | … | nearest neighbors
$\bar{\tau}_{V}$ | … | $\mathcal{N}(\mu=0,\sigma=0.5)\in[0,4]$ | … | nearest neighbors
$t_{\rm start}$ | … | uniform in $[1\,{\rm Myr},t_{\rm age}]$ | … | nearest neighbors
$\tau$ | … | uniform in $[0.01,10]$ Gyr | … | nearest neighbors
$\psi$ | … | calculated; cut $\log_{10}(\psi/M_{\odot}{\rm yr}^{-1})\in[-4,1]$ | … | calculated
$U$-$V$ | … | calculated; cut in UVJ diagram | … | calculated
$V$-$J$ | … | calculated; cut in UVJ diagram | … | calculated

####  6.3.1 Star-forming galaxy parent catalog

For the SFG parent catalog, we sample the seven FSPS parameters for $10^{6}$
galaxies. The mass, redshift, and SFH parameters $t_{\rm start}$ and $\tau$
are sampled from uniform distributions, as summarized in Table 2. We sample
the dust parameter, $\bar{\tau}_{V}$ uniformly from a normal distribution
centered on zero with a standard deviation of 0.5, truncated between 0 and 4.

Since the metallicity of galaxies depends on the metal production from stars,
SFR is related to metallicity. More massive galaxies have higher metallicities
on average (the mass-metallicity relation; e.g Lequeux et al., 1979; Tremonti
et al., 2004), and the scatter in this relation depends strongly on SFR. The
relation between $M_{\rm gal}$–$Z_{\rm met}$–$\psi$ is known as the
fundamental metallicity relation (FMR; e.g. Mannucci et al., 2010; Hunt et
al., 2016), and does not display any significant redshift evolution up to at
least redshift $z>3$ (e.g. Cresci et al., 2019, and references therein). Given
these observed trends, we first assign realistic SFRs for galaxies, before
assigning metallicities from the FMR.

To assign realistic SFRs, we use the parameterization from Schreiber et al.
(2017), constructed to match observations in the redshift range $0<z<7$. For
SFGs,

$\begin{gathered}\log_{10}[\psi/(M_{\odot}{\rm yr}^{-1})]=\\\ \log_{10}(M_{\rm
gal}/M_{\odot})-9.5+1.5\log_{10}(1+z)\\\ 0.3[\max(0,\log_{10}(M_{\rm
gal}/M_{\odot})-9.36-2.5\log_{10}(1+z))]^{2}\end{gathered}$ (18)

with a log-normal scatter of 0.3 dex.

We then propose $Z_{\rm met}$ by drawing from the FMR presented in Williams et
al. (2018) (based off of data from Hunt et al., 2016)

$\begin{gathered}\log_{10}(Z_{\rm met}/Z_{\odot})+8.7\approx 12+\log_{10}({\rm
O}/{\rm H})\\\ =-0.14\log+{10}(\psi/M_{\odot}{\rm yr}^{-1})\\\ \hskip
85.35826pt+0.37\log_{10}(M_{\rm gal}/M_{\odot})+4.82\end{gathered}$ (19)

with scatter from a Student’s $t$-distribution with 3 degrees of freedom, a
standard deviation of $0.3$ in $\log_{10}(Z_{\rm met}/Z_{\odot})$, and
truncated between $-2.2<Z_{\rm met}<0.24$. A Student’s $t$-distribution is
more heavily tailed than a Gaussian, and agrees better with observational
data. As discussed in Williams et al. (2018), imposing a maximum metallicity,
$\log_{10}(Z_{\rm met}/Z_{\odot})=0.24$, in the synthetic galaxies reproduces
the observed turnover in the mass–metallicity relation, despite the fact that
Equation 19 is linear in mass. We use the same FMR to assign metallicities to
both SFGs and QGs, as we find that existing measurements of quiescent galaxies
are consistent with the FMR used to create this synthetic catalog (e.g Peng et
al., 2015; Leethochawalit et al., 2018).

Gas-phase metallicity is also an indirect tracer of gas flows in galaxies.
Metallicity and stellar mass both correlate with star formation history, but
gas flows will introduce scatter in the metallicity and stellar mass relation.
Thus, the gas ionization parameter, $U_{S}$ (which represents the ratio of the
number density of ionizing photons to the number density of hydrogen atoms)
correlates with metallicity. To assign the gas ionization parameter, we use
the low-redshift relation from Carton et al. (2017):

$\log_{10}U_{S}=-0.8\log_{10}(Z_{\rm met}/Z_{\odot})-3.58$ (20)

with a scatter of 0.3 dex sampled from a Student’s $t$-distribution with three
degrees of freedom, truncated between $-4<\log_{10}{U_{S}}<-1$. As in Williams
et al. (2018), we assume the relation holds at all redshifts.

Given the seven FSPS parameters, we then calculate $M_{\rm UV}$, $\beta$,
$\psi$ as outlined in Section 6.2. We discard galaxies that have unrealistic
$\psi$, $M_{\rm UV}$ or $\beta$ values, as summarized in Table 2. The
resulting SFG parent catalog has $\sim 9\times 10^{5}$ galaxies.

####  6.3.2 Quiescent galaxy parent catalog

We generate the parent catalog for the QGs in a similar way to the parent
catalog for the SFGs. For $10^{7}$ galaxies, we assign galaxy masss,
redshifts, and SFH parameters $t_{\rm start}$ and $\tau$ from uniform
distributions, as summarized in Table 3. We sample the dust parameter,
$\bar{\tau}_{V}$ uniformly from a normal distribution centered on zero with a
standard deviation of 0.5, truncated between 0 and 4.

We assume QG metallicities, $Z_{\rm met}$, follow the same FMR as SFGs, as
given in Equation 19, which is consistent with existing measurements of
quiescent galaxies (e.g Peng et al., 2015; Leethochawalit et al., 2018). We
propose SFRs from Schreiber et al. (2017) as before. For quiescent galaxies,

$\log_{10}[\psi/(M_{\odot}{\rm
yr}^{-1})]=0.5\log_{10}(M/M_{\odot})+\log_{10}(1+z)-6.1\,\,\,,$ (21)

with a log-normal scatter of 0.45 dex.

As before, given the seven FSPS parameters, we then calculate $\psi$ and the
UVJ colors, as outlined in Section 6.2. We remove any galaxies in the QG
parent catalog that do not fall in the proper space in the UVJ diagram, to
avoid including any galaxies with unrealistic colors. Specifically, we use the
criteria that a galaxy can be classified as SF if it satisfies any of the
following conditions

$\displaystyle U-V$ $\displaystyle<1.3$ (22) $\displaystyle V-J$
$\displaystyle>1.6$ $\displaystyle U-V$ $\displaystyle<0.88(V-J)+0.49$

(Williams et al., 2009). After cuts in SFR and UVJ, the resulting parent
catalog has $5\times 10^{4}$ galaxies. Though SFR not set explicitly for QGs,
these cuts ensure that QGs have a low SFR, differentiating them from the SFG
population (see Appendix B).

###  6.4 Star-forming galaxies

This section outlines our procedure for assigning realistic SEDs for SFGs. To
accurately capture the observed CSFRD and UVLFs, galaxies must have realistic
SFRs and UV magnitudes. We also aim to produce realistic UV continuum slopes,
$\beta$, the fundamental mass relation (FMR), and the relationship between
metallicity and gas ionization. Overall, we propose five parameters to define
the SFG SEDs, $\psi$, $M_{\rm UV}$, $\beta$, $Z_{\rm met}$, and $U_{S}$.
Details of how these proposed parameters are generated, and how the FSPS
parameters are determined for the SFGs are given in this section, and
summarized in Table 2.

To reproduce observed UVLFs, the synthetic galaxies need to follow the same
$\bar{M}_{\rm UV}$–$M_{\rm gal}$ relation that Williams et al. (2018) used to
generate the high-redshift SMFs used in this work:

$\displaystyle\bar{M}_{\rm UV}$ $\displaystyle=-1.66(M_{\rm gal}-b_{M_{\rm
UV}})-20$ (23) $\displaystyle b_{M_{\rm UV}}$
$\displaystyle=\begin{cases}0.12z^{2}-0.98z+11.4175&{\rm for}\,z\leq 3.75\\\
-0.12z+9.88&{\rm for}\,3.75<z<8\\\ 8.92&{\rm for}\,z\geq 8\\\ \end{cases}$

Additionally, Williams et al. (2018) found that scatter observed in 3D-_HST_
data (Skelton et al., 2014) is constant in both stellar mass and redshift,
with an average value of $\sigma_{\rm UV}=0.7$. Therefore, we propose $M_{\rm
UV}$ values from Equation (23) with a Gaussian scatter of $\sigma_{\rm
UV}=0.7$.

We also use the relations from Williams et al. (2018) to propose UV continuum
slopes, $\beta$,

$\displaystyle\beta$ $\displaystyle=m_{\beta}(M_{\rm UV}+19.5)+b_{\beta}$ (24)
$\displaystyle b_{\beta}$ $\displaystyle=\begin{cases}-0.09z-1.49&{\rm
for}\,z\leq 8\\\ -0.77&{\rm for}\,z>8\\\ \end{cases}$ $\displaystyle
m_{\beta}$ $\displaystyle=\begin{cases}-0.007z-0.09&{\rm for}\,z\leq 8\\\
-0.146&{\rm for}\,z>8\,\,\,.\end{cases}$

Given the proposed galaxy properties, we next determine which FSPS parameters
to assign. To generate SEDs that match closely to the proposed parameters, we
find the 10 nearest neighbors the SFG parent catalog. Table 2 outlines the
distance metric we use to find the nearest neighbors. We average the five free
FSPS parameters ($Z_{\rm met}$, $t_{\rm start}$, $\tau$, $\bar{\tau}_{v}$ and
$U_{S}$) of these neighbors, with a weight based on their distance to the
proposed parameters. In cases where the assigned $t_{\rm start}$ value is
greater than $t_{\rm age}$, we set $t_{\rm start}=t_{\rm age}$. This method
allows us to “fit” the target (proposed) parameters, even with the very large
number of galaxies, something that would not be computationally realistic to
achieve iteratively.

Given the large degeneracies between different galaxy properties (e.g. age,
metallicity, dust, star formation history) we find that our solution depends
sensitively on how realistically we populate the parent catalogs, and also how
we define the distance metric. In particular, we find that to ensure the final
parameters were close to the proposed parameters, it is especially important
to find neighbors that are close in redshift.

###  6.5 Quiescent galaxies

In addition to SFGs, we include a QG population. We turn off the nebular
emission model for QGs, as low redshift galaxies are known to have very low
gas content. Though galaxy modeling often assumes QGs do not contain any gas
or dust (e.g. Williams et al., 2018), recent results from Gobat et al. (2018)
suggest that higher redshift galaxies should have significant dust content. We
therefore include dust for the QGs. We assign QG properties in a similar we
assign SFG properties. Given the QG parent catalog described in Section 6.3,
we determine FSPS parameters using the 10 nearest neighbors in mass and
redshift. We summarize the details of this procedure in Table 3.

###  6.6 Example SEDs

We present the resulting SEDs for two example galaxies at redshift $z\sim 8$
in Fig. 10. The _Roman_ WFI filters include seven photometric filters: R062,
Z087, Y106, J129, H158, F184, F213666The F213 was recently added; the science
justifications for this filter are outlined in Stauffer et al. (2018). The
higher background in F213 will result in brighter flux limits compared to
bluer Roman bands., a wide-field filter W146, a grism, and a prism. We plot
the flux calculated in the seven photometric filters R062, Z087, Y106, J129,
H158, F184, and F213. Both galaxies show a clear Lyman break in Y106.

Figure 10: Example SEDs for a SFG (top) and a QG (bottom) at redshift
$z\approx 8$. The Roman photometric filters (R062, Z087, Y106, J129, H158,
F184, and F213) are over-plotted. The observed magnitudes in each band are
shown as colored points. The SFG would be selected in this UDF as a LBG using
_Roman_ photometry.

The SFG (top panel) should be selected as a Lyman break galaxy (LBG) in
_Roman_ using the common Lyman-break method. This example also displays strong
Ly-$\alpha$ emission. Ly-$\alpha$ emitters (LAEs) are a common way to study
study the EoR, and _Roman_ will detect many such galaxies. Quantifying the
science returns of LAEs in a _Roman_ UDF will be the subject of future work.

Due to the SMFs we used to construct our catalog, we model a small population
of high-redshift QGs, such as the one shown in the bottom panel of Fig. 10. It
is unclear whether redshift $z=8$ galaxies exist, or what the mechanism of
their formation would be. Due to the large area a deep survey with _Roman_
will cover, a _Roman_ UDF could help identify rare high-redshift quiescent
galaxies if they exist. This will be discussed further in Section 9.2.

##  7 Cosmic star formation history

Since one of the main goals of a _Roman_ UDF is to study the ionizing photon
contribution from galaxies, we ensure we reproduce observed UVLFs (Section
7.1) and the CSFRD (Section 7.2). Additional galaxy properties are discussed
in Appendix B to show the range of scientific questions our synthetic galaxy
catalog can address.

###  7.1 UV properties

Placing strong constraints on the faint end of the UVLF during the EoR is a
main goal of a _Roman_ UDF. To reproduce observed UVLFs, the synthetic
galaxies must follow the correct distribution of $M_{\rm UV}$, as discussed in
Section 6.2.3. On average, the UV magnitude, $M_{\rm UV}$, decreases (becomes
brighter) with galaxy mass, $M_{\rm gal}$ (e.g. Stark et al., 2009). For
galaxies with $\log_{10}(M_{\rm gal}/M_{\odot})>10$ the relationship flattens
(e.g. Spitler et al., 2014). Apart from this flattening at high masses, the
observed $M_{\rm UV}$–$M_{\rm gal}$ relation has a constant slope, with the
normalization evolving with redshift (e.g. Duncan et al., 2014; Stefanon et
al., 2017). Similarly, the UV continuum slope, $\beta$, correlates with
$M_{\rm UV}$. Though the exact relation between $\beta$ and $M_{\rm UV}$ is
still being studied, $\beta$ appears bluer (decreases) with increasing
(fainter) $M_{\rm UV}$ and increasing redshift (e.g. Bouwens et al., 2014).

We calculate the UV properties, $M_{\rm UV}$ and $\beta$, of the synthetic
SFGs as described in Section 6.2.3, and presented in Fig. 11. We reproduce
trends for both $M_{\rm UV}$ and $\beta$ imposed from Williams et al. (2018).
The $M_{\rm UV}$-mass relation enables us to reproduce observed UVLFs, as
discussed below. We also closely reproduce the $\beta$–$M_{\rm UV}$ relation
for $\beta>-2.5$. Below $\beta<-2.5$, our galaxy catalog agrees with studies
that indicate that the $\beta$–$M_{\rm UV}$ relationship flattens for $M_{\rm
UV}\gtrsim-19$ galaxies (e.g. Bouwens et al., 2014).

Figure 11: Average $M_{\rm UV}$–mass relation (top) and $\beta$–$M_{\rm UV}$
relation (bottom) for SFGs in the synthetic catalog (solid lines) and the
standard deviation in the bin (shaded region). We compare the synthetic
catalog to the relations from Williams et al. (2018), calculated as the
volume-weighted average over the redshift bin (dotted lines). The synthetic
galaxy catalogs are constructed to match the underlying $M_{\rm UV}$–mass
relations, to reproduce observed UVLFs. The synthetic galaxy catalog matches
the underlying relations closely, except for a flattening in the
$\beta$–$M_{\rm UV}$ relation for faint galaxies. This flattening is
consistent with current constraints, as described in Section 7.1.

We show the UVLFs of our synthetic SFGs in Fig. 12, compared to the UVLFs that
are expected from convolving the SMF with a normal distribution, centred on
$M_{\rm UV}$ (Equation 2). The galaxy catalog agrees very well with the
imposed relation, and is consistent with current observations, as shown in
Section 8.3. Our catalog has slightly less galaxies at faint magnitudes
($M_{\rm UV}>-17$) than expected due to the fact that we do not perfectly
recreate the $M_{\rm UV}$–$M_{\rm gal}$ relationship. These magnitudes are
fainter than the detection limits of a $30\,m_{\rm AB}$ survey, and our
synthetic catalog UVLF is consistent with current observations, as discussed
in Section 8.3.

Figure 12: UV luminosity function of the synthetic galaxy catalog. The dotted
lines show the expected UVLF from convolving the Williams et al. (2018) SMF
with the $M_{\rm UV}$–$M_{\rm gal}$ relationship (Equation 2), and then
averaged over the redshift bin. Overall, the data matches the underlying
relations very well, and agrees with current observations (see Fig. 18).

###  7.2 Cosmic star formation rate density

As discussed in Section 6.2.1, the CSFRD describes the ionizing photon
contribution of galaxies as a function of cosmic time (for a review, see e.g.
Madau & Dickinson, 2014). We display the evolution of the CSFRD in the top
panel of Fig. 13. We calculate the CSFRD by taking the sum of galaxy SFRs,
$\psi$ (averaged over the last 100 Myr), in each bin divided by the comoving
volume of the bin. As in Williams et al. (2018), to compare to Madau &
Dickinson (2014), we imposed a luminosity limit of $0.03\,L_{*}$.777The
equivalent limit in $M_{\rm UV}$ is $M_{\rm UV}\sim-14.5$ at $z<1$ (Cucciati
et al., 2012), $M_{\rm UV}\sim-15.5$ at $1<z<2$ (Cucciati et al., 2012),
$M_{\rm UV}\sim-16.89$ at $2<z<2.7$ (Reddy & Steidel, 2009), and $M_{\rm
UV}\sim-17$ at $z>2.7$ (Bouwens et al., 2015, 2016; Finkelstein et al., 2015;
Oesch et al., 2018).

Figure 13: Evolution of star formation in the universe. Top: evolution of the
CSFRD of synthetic galaxies with a luminosity limit of $0.03\,L_{*}$ (dark
blue). Error bars are smaller than the width of the line. The dashed black
line is the CSFR density of the universe compiled by Madau & Dickinson (2014),
and colored points are measurements from the literature (Cucciati et al.,
2012; Bouwens et al., 2015; Oesch et al., 2018). The data are converted from a
Salpeter (1955) to a Chabrier (2003) IMF by dividing the CSFRD by a factor of
1.7. Bottom: the redshift evolution of the median sSFR of
$8.8<\log_{10}(M_{\rm gal}/M_{\odot})<10$ synthetic galaxies (solid line)
compared to measurements from the literature (colored points; Tasca et al.,
2015). Overall, the cosmic star formation history of the synthetic galaxy
catalogs is in close agreement with observational data.

Similar to Williams et al. (2018), we find that the CSFRD agrees roughly with
the model presented in Madau & Dickinson (2014), but more closely to data
(e.g. Cucciati et al., 2012; Oesch et al., 2018). Most notably, our synthetic
galaxies agree with high-redshift ($z>8$) measurements of the CSFRD. As in
Williams et al. (2018), our synthetic galaxies have a slight deficit at
redshifts $1<z<3$, which reflect the underlying SMFs. As discussed in Williams
et al. (2018), the SMFs we used do not include the dusty SFGs currently missed
in the UV-selected samples.

We show the evolution of the specific star formation rate (sSFR) in the bottom
panel of Fig. 13, for galaxies with $8.8<\log_{10}(M_{\rm gal}/M_{\odot})<10$.
We calculate the median sSFR by dividing the SFR averaged over the last 100
Myr by the galaxy mass. For comparison, we plot data from Tasca et al. (2015).
The synthetic galaxies and observations agree very well. The sSFR of galaxies
at redshifts $z>4$ is a matter of active research, but we find an increase in
sSFR, consistent with current studies (Stark, 2016). Given that our galaxy
catalog is consistent with observations for both galaxy clustering and cosmic
star formation histories, we now turn our attention to making preliminary
predictions for the science returns of a _Roman_ UDF.

##  8 Predictions for a Roman ultra-deep field survey

In this section, we demonstrate the capabilities of a $1\,\deg^{2}$ _Roman_
UDF to study the earliest galaxies at the EoR and beyond. We begin by
discussing our method for determining whether a galaxy is selected (Section
8.1), and the expected galaxy number counts (Section 8.2), before presenting
our projected constraints (Section 8.3). Throughout this section we compare
our predictions for a _Roman_ UDF to the best current constraints in the
literature. However, by the time _Roman_ launches _JWST_ will have provided an
extraordinary amount of data, greatly enhancing our understanding of the
universe. We discuss synergies between a _Roman_ UDF and _JWST_ in Section
9.4.

###  8.1 Selection criteria

Understanding of galaxy formation increased immensely when techniques to
select high redshift ($z>5$) galaxies were developed (for review see, e.g.
Dunlop, 2013). The most commonly used method is the Lyman-break technique,
which selects LBGs based on a step in the blue UV continuum emission at
$\lambda_{\rm rest}=1216$ Å. To determine whether a UDF with _Roman_ would
select each galaxy in the synthetic catalog, we use a magnitude limit based on
the Lyman-break, in the same manner as Williams et al. (2018). Specifically,
for each filter, we assign a non-overlapping wavelength range (the dividing
point was taken to be halfway between the centres of two adjacent bands). Each
galaxy is assigned a “dropout” filter, according to the wavelength that
corresponds to its Lyman-break, $\lambda=1216(1+z)$Å. If a galaxy is brighter
than the magnitude limit in the band immediately redder to its dropout filter,
we consider it to be detectable. We use the magnitude limit $m_{\rm AB}=30$
for a $5\sigma$ detection.

We show the fraction of detectable galaxies in Fig. 14. The mass at which
$\sim 50$% of galaxies ranges from $\sim 10^{6}$ at $z\approx 1$ to $10^{7}$
at $z=10$. We find that nearly 100% of galaxies are detectable above masses of
$M_{\rm gal}=10^{8}M_{\odot}$, which indicates that there should be very tight
constraints on galaxy properties (such as galaxy clustering and UVLFs) above
this mass. Given this, we can measure the clustering of faint galaxies
($M_{\rm UV}\lesssim-18$) at the EoR. With completeness corrections, we can
measure quantities such as the UVLFs for galaxies brighter than $M_{\rm
UV}\approx-17$ out to redshifts $z\approx 10$. Section 8.3 will explore these
potential constraints in more detail.

Figure 14: Fraction of detectable galaxies ($5\sigma$) in the Roman synthetic
catalog. Nearly all galaxies with masses $M_{\rm gal}>10^{8}M_{\odot}$
(corresponding to $M_{\rm UV}<-18$) are detectable out to redshifts $z\approx
10$.

In the following sections, we assume that any detectable galaxy is selected.
This approach assumes that there is no scatter in the photometric redshifts,
and no galaxies are obscured by forefront galaxies. Future work will examine
_Roman_ UDF constraints in more detail, taking into account these effects. The
catalogs and images presented in this work will be instrumental to explore
these sources of systematic errors.

###  8.2 Number counts

Here we provide number counts of the detectable galaxies ($5\sigma$), to
understand the science returns of a _Roman_ UDF and measure the completeness
of the survey. Fig. 15 shows both the cumulative number counts (top), and the
differential number counts (bottom). We find that a $1\deg^{2}$ _Roman_ UDF
could detect more than $10^{4}$ galaxies at redshifts $z>7$. For comparison,
_HST_ has detected on the order of $10^{3}$ galaxies at redshifts $z\sim
4$–$10$ (Koekemoer et al., 2013; Lotz et al., 2017), and Cycle 1 _JWST_
programs are expected to detect $\sim 5000$ galaxies at redshifts $z>6$
(Williams et al., 2018).

Figure 15: The cumulative (top) and differential (bottom) number counts of the
detectable galaxies ($5\sigma$) in our _Roman_ UDF synthetic catalog. A
_Roman_ UDF will contain over a million galaxies, with tens of thousands in
the EoR.

Though a UDF catalog will consist largely of SFGs, there will also be a number
of QGs, as demonstrated in Fig. 16. These predictions indicate that a
$1\,\deg^{2}$ will contain a handful of detectable QGs above redshift $z=7$ if
they exist. In combination with e.g. _JWST_ spectroscopy, this could identify
$z>7$ QGs. Given that we currently have only detected QGs out to redshifts of
$z\sim 4$–$5$ (Merlin et al., 2019; Valentino et al., 2020), this will greatly
enhance our understanding of the QG population. Determining the number and
mass distribution of high-redshift QGs is fundamental to understanding galaxy
evolution, and for testing different models for the emergence of QGs (see,
e.g. Cortese et al., 2021, for a recent review on quenching).

Figure 16: Number counts of detectable galaxies ($5\sigma$) for SFGs (top) and
QGs (bottom) in our _Roman_ UDF synthetic catalog. A _Roman_ UDF will detect
QGs out to very high redshifts, possibly even detecting a few at the EoR.

Due to the nature of these kinds of predictions, the underlying galaxy
properties in the catalog are extrapolated beyond our current knowledge (in
particular, the number of low-mass objects is dependent on extrapolations of
the SMFs). Therefore, the number counts presented in this section represent a
reasonable estimate of what a UDF might see, but there is much to be learned
in the future. Regardless, the large number of galaxies that _Roman_ will
detect will provide the best census to date of faint high-redshift galaxies,
including those thought responsible for reionizing the universe.

###  8.3 The science returns of a Roman UDF

Some of the quantities that a _Roman_ UDF will aim to constrain include the
SMF, the UV luminosity function, galaxy clustering and the galaxy–halo
connection at high redshifts. In this section we provide preliminary projected
constraints for a $1\deg^{2}$ _Roman_ UDF, based on the detectable galaxies
identified in Section 8.1. We have not included uncertainties from cosmic
variance or from errors on photometric redshifts in our predictions, as that
is beyond the scope this paper. However, we note that our synthetic catalog
provides the basis for rigorous studies of systematic errors in a _Roman_ UDF,
and will be the focus of future work.

####  8.3.1 Stellar mass functions

The evolution of the SMF of galaxies over cosmic time provides information on
how galaxy populations have evolved, the star-formation history of the
universe, and the galaxy–halo connection. Fig. 17 shows the SMF for the
detectable galaxies in the synthetic catalog, with the underlying SMFs shown
as dotted lines for both SFGs (top) and QGs (bottom). We perform a
completeness correction by dividing the number of detectable galaxies by the
fraction of detectable galaxies at the corresponding mass and redshift. As
shown in Fig. 6, our synthetic catalog matches the SMF nearly exactly by
construction.

Figure 17: SMF of the detectable galaxies ($5\sigma$) in our synthetic galaxy
catalog for SFGs (top) and QGs (bottom), with completeness corrections. The
synthetic galaxies agree with the Williams et al. (2018) SMFs (dotted lines)
by construction. A _Roman_ UDF can potentially constrain the SMF beyond
redshift $z=10$ ($M_{\rm gal}>10^{6.5}\,M_{\odot}$) for SFGs, and to redshift
$z\approx 7$ ($M_{\rm gal}>10^{9.5}\,M_{\odot}$) for QGs.

For our catalog, the SMF for the detectable SFGs is measurable to masses of
$M_{\rm gal}>10^{6.5}\,M_{\odot}$ for all redshifts. For the detectable QGs,
the SMF is measurable to redshifts $z\sim 7$ (masses $M_{\rm
gal}>10^{9.5}\,M_{\odot}$). In comparison, observations have only recently
placed _any_ constraints on the SFG SMF at redshift $z=8$–$10$. Further, these
constraints only exist for galaxies with $M_{\rm gal}<10^{8}\,M_{\odot}$, and
have very large uncertainties (about 100%; Stefanon et al., 2021).
Measurements of QG SMFs only extend to redshifts $z\approx 4$ (e.g. Girelli et
al., 2019).

The exact constraints on a SMF from _Roman_ will depend on stellar mass
measurements and the accuracy of completeness corrections. This work provides
a first calculation of the completeness of a _Roman_ UDF (Section 8.1), and
highly realistic simulated data to enable future rigorous calculations. In
practice, stellar mass estimates of high redshift galaxies will require
photometry redshifts beyond the reddest _Roman_ filters. However, _JWST_ will
provide tight constraints on the $M_{\rm UV}$–$M_{\rm gal}$ relation which can
be used to estimate stellar masses of galaxies in a _Roman_ UDF.

####  8.3.2 UV luminosity functions

To clearly answer whether there were enough galaxies during the EoR to
reionize the universe, we need strong constraints on the faint end of the UVLF
during the EoR. Bouwens et al. (2021) provide the most recent, comprehensive
constraints on the UVLF, and are able to constrain the faint end ($M_{\rm
UV}<-17$) out to redshift $z\sim 10$. At redshift $z\sim 10$, only 8 sources
are used to constrain the UVLF, so the uncertainties are very large
($\sim$50–100%). In comparison, a _Roman_ UDF could detect $\sim 10^{3}$
sources at $z\sim 10$, and thus greatly reduce the uncertainties in the UVLF.

We show the UVLF of the detectable synthetic galaxies in Fig. 18, compared to
the data from Bouwens et al. (2021). As in the previous section, we perform a
completeness correction by dividing the number of selected galaxies by the
fraction of selected galaxies at the corresponding mass and redshift. We
predict a _Roman_ UDF will be able to constrain the UVLF beyond redshift
$z\approx 10$, with galaxies $M_{\rm UV}<-17$. Due to the vast number of
galaxies in a _Roman_ UDF, it could constrain the faint-end UVLF to within
$\sim 1$ percent to redshifts $z\gtrsim 10$, far tighter than existing limits.

Figure 18: The UVLF of the detectable galaxies in our synthetic galaxy catalog
(blue lines) and the 1-$\sigma$ Poisson noise (shaded region). We show the
recent Bouwens et al. (2021) data for comparison (black points). We also
indicate the number of sources, $N$, in each panel used to calculate the UVLF.
A _Roman_ UDF will provide remarkably tight constraints on the faint end of
the UVLF at high redshifts.

We note that the bright end of the synthetic galaxy catalog UVLF is higher
than the Bouwens et al. (2021) data at low redshifts. As shown in Fig. 12, our
galaxy catalog agrees with the underlying UVLF model we used at the bright
end, and therefore this difference can be attributed to the SMFs used in this
work. This excess of low-redshift ($z<4$), bright galaxies is consistent with
the possibility of inefficient mass quenching, low dust obscuration, or hidden
AGN activity, as suggested by Harikane et al. (2021).

####  8.3.3 Galaxy clustering

_Roman_ ’s wide, contiguous field of view will allow measurements of the 2PCF
of faint galaxies at the time of reionization and thus their underlying dark
matter halo masses. To date, halo masses have only been strongly constrained
to redshifts $z\sim 6$, and are limited to bright ($M_{\rm UV}\sim-20$)
galaxies at redshifts $z=4$–$6$ (e.g. Harikane et al., 2018). JWST will
potentially provide the first measurement of the clustering of high-redshift
galaxies (Endsley et al., 2020), but due to its smaller field of view, it will
likely suffer from cosmic variance, and will not have the same capability as
_Roman_ to study clustering in different environments.

We measure the 2PCF of the detectable galaxies in the $1\deg^{2}$ survey, as
outlined in Section 4.3. We included all galaxies brighter than $M_{\rm
UV}<-18$. Fig. 19 shows the measured 2PCF. The 2PCF for these faint galaxies
is measured to very high accuracy (within 1%) out to redshifts $z\approx 7$,
with constraints out to $z\approx 10$ (to within 10%).

Figure 19: The 2PCF of the $M_{\rm UV}<-18$ detectable galaxies in our
synthetic galaxy catalog (blue lines), with the uncertainty calculated from
bootstrapping the data. (shaded region). The 2PCF of the detectable galaxies
is measured to within 1% for galaxies $z\lesssim 7$, and to within 10% at
$z\approx 10$.

####  8.3.4 Galaxy–halo connection

In addition to constraining the number and spatial distribution of faint,
high-redshift galaxies, _Roman_ will also provide constraints on the
galaxy–halo connection. Galaxy clustering can be used to infer dark matter
halo mass, as the underlying dark matter will dictate the gravitational field
(Mo & White, 1996). Together with galaxy mass measurements, galaxy clustering
gives a direct measure of the SHMR (which summarizes the connection between
galaxy masses and their host dark matter halos).

As discussed above, halo masses have only been measured directly for redshifts
$z\lesssim 6$ (e.g. Harikane et al., 2018), while at high redshifts, SHMR
constraints have been achieved with abundance matching between observed
stellar masses and dark matter only simulations (e.g. Stefanon et al., 2021).
There is a potential disagreement between these two techniques—the SHMR
measured from abundance matching is typically 3–4 times higher than that
measured from clustering (Stefanon et al., 2021). The difference between SHMRs
derived from clustering and AM indicates that direct halo mass measurements
are needed to understand the galaxy–halo connection at high redshifts. A
_Roman_ UDF will provide extraordinary data, which may elucidate the origin of
this discrepancy.

We show the SHMR for our synthetic galaxy catalog in Fig. 20, and recent
constraints from Harikane et al. (2018) and Stefanon et al. (2021). The
Stefanon et al. (2021) data were scaled down by a factor of 1.7 to convert
from a Salpeter (1955) IMF to a Chabrier (2003) IMF. We use all the $5\sigma$
Lyman-break selected galaxies in the catalog. Our synthetic catalog clustering
measurements agree well with the Stefanon et al. (2021) data (within $\sim
1\sigma$), and are slightly higher than the Harikane et al. (2018) data.

Figure 20: SHMR of the detectable galaxies in our synthetic galaxy catalog
(blue solid line), and the 1$\sigma$ spread (shaded blue region). We compare
to data from Harikane et al. (2018) (orange points; error bars show error in
the mean) and Stefanon et al. (2021) (red points; error bars denote the
scatter). The Stefanon et al. (2021) data has been scaled by a factor of
$1/1.7$ to convert to a Chabrier (2003) IMF. Stefanon et al. (2021) derived
halo masses from abundance matching, while Harikane et al. (2018) derived halo
masses from galaxy clustering data.

In this work we do not attempt to predict how well a _Roman_ UDF will be able
to constrain the SHMR. As discussed in Section 8.3.1, stellar mass
measurements for high-redshift galaxies likely require photometry at
wavelengths longer than what _Roman_ will provide, but may be possible to
estimate with scaling relations. Since our synthetic galaxy catalogs contain
information regarding host dark matter halo properties, this work can form the
basis of future analyses of the galaxy–halo connection in UDFs.

###  8.4 Synthetic images

In addition to the galaxy catalog, we present synthetic images of a _Roman_
UDF. These images are intended for developing analysis tools, and studying
systematics (see Section 9.5). Full resolution versions of the RGB images
presented in this section in our data release (see Appendix A for details).

We create the synthetic images using GalSim (Rowe et al., 2015). GalSim
contains a module specifically for _Roman_ observations (Troxel et al., 2021),
which includes five of the _Roman_ filters: Z087, Y106, J129, H158 and F184.
We model each galaxy as a Sérsic profile, with an index $n_{s}$, axis ratio
$q$ and PA, as described in Section 5. We truncate the distribution of Sérsic
indices between 0.3 and 6.2, to avoid numerical inaccuracies at more extreme
values. The scale radius of the Sérsic profile, $r_{0}$, is directly related
to the assigned half-light radius, $R_{\rm eff}$:

$r_{0}=\dfrac{R_{\rm eff}}{b^{n_{s}}\sqrt{q}}\,\,\,,$ (25)

where $b\approx 2n_{s}-1/3$ (Moriondo et al., 1998). Fig. 21 shows a composite
of the full synthetic galaxy catalog using Z087 (blue), Y106 (green), and H158
(red) filters. We note that we do not include simulated stars in any of the
released images.

Figure 21: Noise-free composite simulated image of the full $1\deg^{2}$ galaxy
catalog using the Z087 (blue), Y106 (green), and H158 (red) filters. The image
was created with the _Roman_ module in GalSim. A native resolution version is
available online at https://www.nicoledrakos.com/dream.

To simulate a _Roman_ $30~{}m_{\rm AB}$ survey, we convolve the image with the
_Roman_ point spread function (PSF), and add noise to the image. We calculate
the image noise assuming proposed exposure times from Koekemoer et al. (2019),
to reach a $5\sigma$ limit of $m_{\rm AB}\approx 30$ (see Table 4). We add
noise to the image in the same manner as Troxel et al. (2021). First, we
generate the sky background accounting for the stray light and thermal
emission from the telescope which is added to the image. Given this, we add
errors associated with poisson noise, reciprocity failure, dark current, the
calibration of the Roman detectors, inter-pixel capacitance, and instrument
read noise.

Table 4: Approximate exposure times for the simulated filters, as calculated in Koekemoer et al. (2019) to reach a $5\sigma$ limit of $m_{\rm AB}\approx 30$. Filter | Exposure Time (hrs)
---|---
Z087 | 60
Y106 | 70
J129 | 90
H158 | 40
F184 | 60

We present a RGB visualization of the full _Roman_ catalog, with the PSF and
noise included in Fig 22, with one _Roman_ footprint overlaid on top. Each
_Roman_ pointing will have 18 detectors. This visualization shows the rich
amount of structure that will be contained in a _Roman_ UDF. As described in
Koekemoer et al. (2019), a $1\deg^{2}$ UDF would consist of three _Roman_
pointings. Though a UDF based on three tiled WFI pointings would not be
perfectly square, and the exposure time would not be perfectly uniform, these
synthetic images demonstrate the richness of a $1\deg^{2}$ UDF and will be
incredibly useful in designing and preparing for wide, deep surveys.

Figure 22: Composite simulated image of a region of the galaxy catalog using
the Z087 (blue), Y106 (green), and H158 (red) filters convolved with the
_Roman_ PSF. Noise is calculated assuming a depth of $\sim 30\,m_{\rm AB}$ in
each filter. The image was created with the _Roman_ module in GalSim. The
_Roman_ footprint is shown on top, as well insets showing a zoomed in region
of the image. A native resolution version of this $1\deg^{2}$ visualization is
available online at https://www.nicoledrakos.com/dream.

We include FITS maps of the galaxy catalog for all 5 filters currently
included in GalSim in our data release. The top row of Fig. 23 shows the flux
in one WFI detector (each detector has an area of $\sim 7.3\,\arcmin\times
7.3\,\arcmin$). The bottom row shows an example $z=9.4$ galaxy (with the
background subtracted in each panel). This galaxy would be selected as a Y106
dropout. The dropout galaxy has a size of $R_{\rm eff}=0.38$ kpc, and
therefore is not resolved with the WFI (which has an angular resolution of
$0.11\arcsec$ per pixel).

Figure 23: Flux in one WFI detector (top row), and an example of a Y106
dropout galaxy (bottom row). Each column corresponds to a different
photometric filter, as labeled. The WFI detector has 4kx4k pixels, with an
angular resolution of $0.11\arcsec$ per pixel. The example dropout galaxy is
at redshift $z=9.4$, has a galaxy mass of $M_{\rm gal}=10^{7.6}\,M_{\odot}$.

The synthetic images we release are at the resolution of the WFI detector,
$0.11\arcsec$ per pixel. An actual _Roman_ UDF would likely be dithered, which
would sample the sky on a sub-sampled pixel scale, which would improve the PSF
sampling and angular resolution. Since this process would create correlations
between adjacent pixels, we did not include this in our data release. Future
work will build upon DREaM to examine dithering strategies of a _Roman_ UDF.

##  9 Discussion

This work presents synthetic galaxy catalogs for a $1\deg^{2}$ UDF with
_Roman_ , created using DREaM, a model for deep, realistic realizations of
galaxy populations. Our model successfully reproduces a number of well-known
trends, including the size–mass relation, the fundamental metallicity
relation, and UV luminosity functions. Additionally, we reproduce observed
low-redshift galaxy clustering and SMFs. We have made the galaxy catalog, and
synthetic images public, to provide the community tools to prepare for a
potential UDF with _Roman_.

A UDF survey with _Roman_ will address a number of science topics, including
the EoR, the emergence of quiescent galaxies, and the high-redshift
galaxy–halo connection. _Roman_ ’s large field of view will capture enormous
sample sizes, over contiguous fields, imaging multiple ionization bubbles.
This large area will decrease cosmic variance, and probe detailed environments
around high-redshift galaxies.

###  9.1 The epoch of reionization

The EoR is the final frontier for galaxy surveys. Given the difficulty in
measuring galaxies at high redshifts, this period in the universe’s history is
remarkably unconstrained. High-redshift low-mass galaxies were likely the
major source of the ionizing photons in the EoR (e.g. Finkelstein et al.,
2019), and observations indicate that reionization was a “patchy” process
(e.g. Furlanetto & Oh, 2005; Villasenor et al., 2021). To fully understand the
EoR, we need a complete census of galaxies and their ionizing photon
contribution.

To determine whether there are enough high-redshift, faint galaxies to cause
reionization, we must accurately model the CSFRD, which depends on the UVLF.
Though there are some constraints on the UVLF to $z\sim 10$ (e.g. Bouwens et
al., 2021), there is still much improvement to be made on constraining both
the bright steep end of the UVLF and the faint end (e.g. Bowler et al., 2014,
2015) at high redshifts. _JWST_ will make improvements on this front. For
example, Kauffmann et al. (2020) predicts that the $100\,{\rm arcmin}^{2}$
Cosmic Evolution Early Release Science Survey (CEERS), will constrain the
faint-end of the UVLF to a precision of $0.25$ at $z\geq 8$, but that a survey
would need to be at least $300\,{\rm arcmin}^{2}$ to constrain bright end up
to $z=8$. We predict a _Roman_ UDF will capture enough high-redshift galaxies
to constrain the UVLF to within 1% at redshifts beyond $z\sim 10$. This will
either confirm that there are enough faint galaxies to account for the
ionizing photons needed to cause reionization, or indicate that another
source, such as AGN (e.g. Madau & Haardt, 2015) must contribute.

In addition to the abundance of galaxies, the total ionizing photon budget
depends on the Lyman-continuum (LyC) production efficiency, $\xi_{\rm ion}$,
and the LyC escape fraction, $f_{\rm esc}$. To account for reionization,
galaxies need higher escape fractions at high redshifts than have been
observed at low redshifts (Davies et al., 2021). Since the neutral IGM absorbs
LyC photons, $f_{\rm esc}$ is very difficult to constrain at the EoR. However,
$f_{\rm esc}$ could possibly be measured during the EoR using indirect methods
(e.g. Leethochawalit et al., 2016; Zackrisson et al., 2017; Chisholm et al.,
2018, 2020). A _Roman_ UDF will detect tens of thousands of galaxies during
the epoch of reionization, including rare, bright sources that are ideal for
spectroscopic follow-up to measure $f_{\rm esc}$. Further, since _Roman_ can
map out the density around each galaxy, it will allow for the measurement of
the escape fraction in different environments.

###  9.2 The emergence of quiescent galaxies

There exists a clear bimodality in star formation rates, indicating two
distinct populations (star-forming and quiescent). The decline in the CSFRD
since redshift $z\approx 2$ was likely caused by the quenching of galaxies
(e.g. Renzini, 2016). However, the mechanisms that transforms galaxies from
star-forming to quiescent are not fully understood. Possible mechanisms that
have been proposed include feedback from stars and AGN, or the removal of gas
through tidal or ram pressure stripping. Quenching is likely caused by a
combination of these mechanisms, and the dominant processes may be redshift
dependant (Kalita et al., 2021).

Advances in the understanding of the origin of quiescent galaxies is being
greatly improved by surveys such as the spectroscopic survey Gemini
Observations of Galaxies in Rich Early ENvironments (GOGREEN; Balogh et al.,
2021), which targets quiescent galaxies in clusters around redshift $z\sim 1$.
_Roman_ has the potential to identify quiescent galaxy populations out to much
higher redshifts. In particular, the synthetic catalog presented in this work
predicts that an UDF with _Roman_ will contain $\sim 10^{5}$ detectable
quiescent galaxies, including QGs beyond redshift $z=7$. Though the exact
number of quiescent galaxies relies heavily on the underlying assumed SMF,
_Roman_ will likely detect the highest redshift QG to date if they exist,
allowing for the study of quenching mechanisms at high redshifts.

###  9.3 The galaxy–halo connection

A _Roman_ UDF will likely provide the first strong clustering measurements of
the faint, high-redshift galaxies responsible for reionization, which will
enable a measurement the underlying dark matter mass. In our preliminary
predictions, we have shown that the coverage from a UDF will constrain the
2PCF for faint galaxies out to redshifts $z=10$ (at the 10% level). These
measurements will likely provide the first direct measurement of halo masses
at $z=10$, placing constraints the galaxy–halo connection at high redshift.
Since the galaxy–halo connection depends on both galaxy formation physics and
the underlying cosmological model, this will provide important tests for
current models of galaxy formation.

###  9.4 Synergies with JWST

_Roman_ ’s extensive field of view will allow for the contiguous, deep imaging
of galaxies, reducing cosmic variance, and probe the environment around
individual galaxies. The large number of galaxies detected by a _Roman_ UDF
will reduce poisson noise, and increase the number of rare objects that will
be detected. Since a single _Roman_ pointing is wide enough to capture several
reionization bubbles at the height of reionization, a _Roman_ UDF will allow
for the study of differences between galaxies in ionized and neutral regions.
For instance, there may be possible variations in the faint-end slope of the
UVLF with environment, which can be measured with a deep, wide, galaxy survey.

Before the launch of _Roman_ , _JWST_ will begin to address questions
regarding the EoR, galaxy formation and the galaxy–halo connection. In
particular, the JWST Advanced Deep Extragalactic Survey (JADES) is a $236{\rm
arcmin}^{2}$ planned imaging and spectroscopy survey888Other JWST programs,
such as Public Release IMaging for Extragalactic Research (PRIMER), and the
COSMOS-Webb survey will cover larger areas than JADES, and probe the EoR, but
neither will go as deep or as wide as a $1\deg^{2}$_Roman_ UDF. Though JADES
will provide revolutionary data, it will detect less than $\sim 10\times$ the
number of objects at all redshifts compared to a _Roman_ UDF. In addition,
cosmic variance may dominate over Poisson noise for future high-$z$ surveys
(Trapp & Furlanetto, 2020), and JADES will cover a much smaller area.

Though _Roman_ will cover a huge area on the sky, it will be difficult to
obtain photometry at wavelengths greater than 2 microns. At high redshifts,
accurate measurements of stellar masses and QG identification require
supplementary imaging from another observing facility. _JWST_ will probe very
far into the infrared, which will potentially allow for Balmer-break selection
of galaxies, improving redshift measurements and stellar mass estimates
(though JWST is not sensitive to light blueward of 1 micron). Therefore, JWST
may provide valuable spectroscopic followup to rare detections from a _Roman_
UDF. For example, _Roman_ will detect many galaxies, on the bright end of the
UVLF. Bright galaxies (e.g. starburst galaxies, with $M_{\rm UV}<-22$) are
ideal for spectroscopic follow-up with JWST, and can be used to observe
nebular lines and estimate $f_{\rm esc}$.

###  9.5 Applications of synthetic images

In addition to the simulated galaxy catalog, we provide synthetic images of a
_Roman_ UDF. These images can be used to determine the impact of source
blending, line confusion, potential problems with SED fitting (e.g. Borlaff et
al., 2019; Massara et al., 2020; Kauffmann et al., 2020). Having very deep
synthetic images will be useful for studying WFI systematics, processing
issues (e.g. low surface brightness issues), and secondary analysis (e.g.
photo-z studies). Quantifying these issues will also be beneficial for many
other _Roman_ surveys, such as the High Latitude Survey (HLS). For instance,
in Section 8.1, we assumed that all galaxies that are detectable as will be
selected. However, since a UDF will be so richly populated with structure, a
fraction of the high redshift galaxies will be obscured by forefront galaxies.
We intend to use the synthetic images and catalogs in this work to quantify
this effect.

###  9.6 Grism predictions

In this paper we have only included photometric predictions, but the
realistically modeled galaxy SEDs can also be used to generate grism
predictions. In particular, the MOSFIRE Deep Evolution Field (MOSDEF) survey
(Kriek et al., 2015) measured the detailed rest-frame optical emission-line
SEDs in galaxies $z=2$–$3$. More recent measurements with Keck/MOSFIRE have
presented spectroscopic measurements out to $z\sim 8$ (Topping et al., 2021).
We can combine these observations with our synthetic catalog, to guide the
_Roman_ grism data reduction pipeline at high redshifts.

The synthetic catalog can also be used to study Lyman-$\alpha$ emitters
(LAEs). LAEs produce a large amount of ionizing photons, and both UVLFs and
clustering of LAEs are important to characterize EoR. The Ly$\alpha$ UVLF
decreases towards early stage of EoR, since neutral hydrogen absorbs
Lyman-$\alpha$ photons (e.g. Ouchi et al., 2018). The clustering of LAEs
emitters constrains the ionized fraction and topology of reionization.

##  10 Summary and Conclusions

This work presents DREaM, a Deep Realistic Extragalactic Model for creating
synthetic galaxy catalogs. We use this model to understand the potential power
of a $1\deg^{2}$ UDF with _Roman_ , and provide publicly available realistic
synthetic galaxy catalogs and images. The synthetic catalogs and images will
aid the community in designing and interpreting a _Roman_ UDF. A summary of
our main predictions are below.

A $1\deg^{2}$ _Roman_ UDF will:

  1. 1.

contain more than $10^{6}$ detectable galaxies, with more than $10^{4}$ during
the EoR ($z>7$).

  2. 2.

contain $\sim 10^{5}$ detectable quiescent galaxies, including a few at
redshifts beyond $z\sim 7$, likely detecting the farthest redshift quiescent
galaxy to date.

  3. 3.

help constrain SMFs for SFGs to redshifts beyond $z\sim 10$, and for QGs past
redshift $z\sim 7$.

  4. 4.

provide tight constraints (within 1% percent) on the faint-end ($M_{\rm
UV}<-17$) of the UV luminosity function, out to redshifts $z\sim 10$.

  5. 5.

provide high redshift ($z>7$) constraints on the clustering of the faint
galaxies thought to be responsible for reionization.

  6. 6.

look for variations in the UVLF in different environments.

Overall, _Roman_ ’s wide field of view offers a unique ability to create wide,
deep surveys. A _Roman_ UDF would enable a tremendous amount of science by
detecting the largest census of high-redshift galaxies to date, and
differentiating between galaxy populations in low and high density regions
during the EoR.

This work was supported by NASA contract NNG16PJ25C. The authors acknowledge
use of the lux supercomputer at UC Santa Cruz, funded by NSF MRI grant AST
1828315, and the NASA supercomputers. In addition, the authors thank Christina
Williams, Yifei Luo, David O. Jones and Kevin Hainline for useful discussion.
_Software_ : numpy (Harris et al., 2020), matplotlib (Hunter, 2007), scipy
(Virtanen et al., 2020), astropy (Astropy Collaboration et al., 2013, 2018),
python-fsps (Foreman-Mackey et al., 2014)

##  Appendix A Data Release

We have made a number of our data products public, as summarized in Table 5.
We release a main galaxy catalog, a catalog containing intrinsic galaxy
properties, and a halo catalog. The galaxy properties that are included in the
catalogs are summarized in Table 6. These data products, and an interactive
online visualization of the synthetic images are available at
https://www.nicoledrakos.com/dream.

Table 5: Data Products. Available online at https://www.nicoledrakos.com/dream. Product |  Filename |  Description
---|---|---
Main Catalog |  DREaM_main.fits |  Contains galaxy masses, positions, morphologies, rest-frame properties, and photometry in _Roman_ and _JWST_ filters. See Table 6 for more information.
Intrinsic Properties |  DREaM_intrinsic.fits |  Contains the FSPS parameters used to generate galaxy SEDs. See Table 6 for more information.
Halo Properties |  DREaM_halos.fits |  Contains the host halo properties, such as the mass, shape, size, spin and peculiar velocity of the host halos. See Table 6 for more information.
Table 6: Catalog content. The galaxy information is contained in either the main catalog (MC), internal properties catalog (IPC) or halo properties catalog (HPC). Variables |  Description |  MC |  IPC |  HPC
---|---|---|---|---
ID |  Galaxy ID. The same across all the catalogs |  ✓ |  ✓ |  ✓
RA, Dec |  Right Ascension and Declination [degrees, $-0.5$ to $0.5$] |  ✓ |  ✓ |  ✓
redshift |  Galaxy redshift |  ✓ |  ✓ |  ✓
M_halo |  Halo mass [$M_{\odot}/h$] |  ✓ |  ✓ |  ✓
M_gal |  Galaxy mass [$M_{\odot}$] |  ✓ |  ✓ |  ✓
logpsi |  Star-formation rate [$\log_{10}(\psi/(M_{\odot}/{\rm yr}))$] |  ✓ |  ✓ |  ✓
M_UV |  Rest-frame UV magnitude |  ✓ |  ✗ |  ✗
R_eff |  Half-light radius in the semi-major axis [kpc/h, physical] |  ✓ |  ✗ |  ✗
n_s |  Sérsic index |  ✓ |  ✗ |  ✗
q |  Projected axis ratio: semi-minor to semi-major half-light size |  ✓ |  ✗ |  ✗
PA |  Position angle [radians, $0$ to $2\pi$] |  ✓ |  ✗ |  ✗
beta |  Rest-frame UV continuum slope |  ✓ |  ✗ |  ✗
U, V, J |  Rest-frame magnitude in U, V and J bands |  ✓ |  ✗ |  ✗
R062, Z087, Y106, J129, H158, F184, F213 |  Apparent AB magnitude in _Roman_ filters |  ✓ |  ✗ |  ✗
f070, f090, f115, f150, f200, f277, f356, f444 |  Apparent AB magnitude in _JWST_ filters |  ✓ |  ✗ |  ✗
SF |  Star-forming (True) or Quiescent (False) |  ✗ |  ✓ |  ✗
t_start |  Age of universe when galaxy started forming [Gyr] |  ✗ |  ✓ |  ✗
tau |  $e$-folding time for star formation [Gyr] |  ✗ |  ✓ |  ✗
logZ |  Metallicity parameter [$\log_{10}(Z/Z_{\odot})$] |  ✗ |  ✓ |  ✗
dust |  Dust attenuation parameter |  ✗ |  ✓ |  ✗
logUS |  Gas ionization parameter [$\log_{10}(U_{s})$] |  ✗ |  ✓ |  ✗
V_max |  Halo maximum circular velocity [km/s, physical] |  ✗ |  ✗ |  ✓
R_s |  Halo scale radius [kpc/$h$, comoving] |  ✗ |  ✗ |  ✓
R_vir |  Halo virial radius [kpc/$h$, comoving] |  ✗ |  ✗ |  ✓
M_peak |  Halo peak mass over accretion history [$M_{\odot}/h$] |  ✗ |  ✗ |  ✓
V_peak |  Halo peak $V_{\rm max}$ over accretion history [km/s, physical] |  ✗ |  ✓ |  ✗
b_a |  Halo axis ratio, $b/a$ |  ✗ |  ✗ |  ✓
c_a |  Halo axis ratio, $c/a$ |  ✗ |  ✗ |  ✓
haloID |  ID of host halo |  ✗ |  ✗ |  ✓
hostID |  ID of least massive host halo (-1 if distinct halo). |  ✗ |  ✗ |  ✓
spin_B |  Bullock et al. (2001) halo spin parameter |  ✗ |  ✗ |  ✓
spin_P |  Peebles (1971) halo spin parameter |  ✗ |  ✗ |  ✓
x,y,z |  Halo positions [Mpc/$h$, comoving] |  ✗ |  ✗ |  ✓
vx,vy,vz |  Halo peculiar velocities [km/s, physical] |  ✗ |  ✗ |  ✓

##  Appendix B Synthetic galaxy properties

Our main goal was to study the ability of a _Roman_ UDF to constrain the
photoionizing contribution of high redshift galaxies, and the environments
around these galaxies. Therefore, we carefully constructed the DREaM galaxy
catalogs to have realistic clustering properties and UV properties. In
addition to these properties, our synthetic galaxies also capture a number of
other observational trends.

In this section we examine some of the properties of the full galaxy catalog.
We begin by looking at the how the UVJ colors (Section B.1), ages (Section
B.3) and SFRs (Section B.2) differ between star-forming and quiescent
galaxies. Additionally we verify that we reproduce the well-known fundamental
metallicity relation (Section B.4).

###  B.1 UVJ colors

As discussed in Section 6.2.2, UVJ diagrams differentiate between star-forming
and quiescent galaxy populations. We display the U-V and V-J colors for our
two galaxy populations in Fig. 24, along with the UVJ selection box from
Williams et al. (2009). The two populations occupy two distinct regions in
this parameter space, demonstrating that the synthetic galaxy catalog does
capture the bimodal population. Observational constraints for QGs do not
currently exist for galaxies past redshifts $z\approx 4$, the UVJ distribution
of the synthetic galaxies at low redshifts do agree with observations (e.g.
Schreiber et al., 2015).

Figure 24: UVJ diagram of synthetic SFGs (blue) and QGs (orange). We used the
selection box from Williams et al. (2009) (black lines). Our synthetic
galaxies demonstrate a clear bimodal distribution, with SFGs falling in the
top left corner of the UVJ diagram.

###  B.2 Star formation rate

We do not explicitly assign SFRs to the galaxies in the mock catalog, but we
reproduce realistic trends in SFRs. By assigning realistic $M_{\rm UV}$ values
to the SFGs, and constraining QGs to the appropriate place in the UVJ color
diagrams, we accurately model a bimodal population, with SFGs having higher
SFRs than QGs. Additionally we reproduce the observed cosmic star formation
rate density (CSFRD) as demonstrated in Section 7.2.

Fig. 25 shows the SFR–Mass relation of the synthetic catalog compared to the
relations from Schreiber et al. (2017). We closely match the Schreiber et al.
(2017) SFG SFR–mass relations. The SFRs for QGs is lower than the Schreiber et
al. (2017) relation. However, the SFRs–mass relations from Schreiber et al.
(2015) assumed that all IR emission from QGs originated from residual star-
formation. Alternative explanation are that the IR emission originates from
active galactic nuclei (AGN) torus emission, dust heating, or incorrect
classification of SFGs. The QGs SFRs still agree with what current
observations can predict.

Figure 25: The average SFR versus galaxy mass for synthetic SFGs (top) and QGs
(bottom). The SFRs increase with mass, and the SFGs have higher SFRs than QGs,
as expected. For comparison, we plot the relation from Schreiber et al. (2017)
(dotted lines). The galaxy catalog agrees closely with the Schreiber et al.
(2017) parameterization for the SFGs. The QGs have lower SFRs than the
Schreiber et al. (2017) parameterization, but are still consistent with
observations (see discussion in text).

###  B.3 Age

Galaxy ages are a measure of star-formation histories (SFHs). SFHs of
individual galaxies, give a direct measurement of the evolution of different
galaxy populations. Trends in galaxy ages have been well-established; for
instance, QGs are older than SFGs, and high-redshift galaxies are older than
low-redshift galaxies (e.g. Webb et al., 2020). However, galaxy ages for
individual galaxies are difficult to measure due to degeneracies between other
parameters (metallicity, dust), sensitivity to priors in fitting (e.g. Leja et
al., 2019), and the similarity between SEDs in galaxies older than $\sim 5$
Gyr (e.g. Gallazzi et al., 2005). Our synthetic galaxy catalogs provide age
estimates, and realistic SEDs to examine this further.

We calculate the mass-weighted age of the synthetic galaxies as

$a_{\rm gal}=\dfrac{\int_{0}^{t_{\rm age}}(t_{\rm age}-t)\psi
dt}{\int_{0}^{t_{\rm age}}\psi dt}\,\,\,.$ (B1)

For the delayed-tau model,

$a_{\rm gal}=t_{\rm tot}-2\tau+\dfrac{t_{\rm tot}^{2}}{\tau e^{t_{\rm
tot}/\tau}-(\tau+t_{\rm tot})}\,\,\,,$ (B2)

where $t_{\rm tot}=t_{\rm age}-t_{\rm start}$ is the total time of star
formation.

Fig. 26 shows the average galaxy ages as a function of redshift for both SFGs
(blue) and QGs (red). QGs are older than SFGs at all redshifts and age
decreases with increasing redshift, as expected. At low redshifts, QGs are
$\approx 3$ Gyr older than SFGs.

Figure 26: Average mass-weighted age of synthetic galaxies for SFGs (blue
points) and QGs (red points), as defined in Equation (B1). QGS are older than
SFGs, and ages decrease with increasing redshift. Galaxy ages are less than
the age of the universe (dashed black line).

###  B.4 Metallicity

Galaxy metallicity can greatly affect quantities such as galaxy color, and
therefore must be accurately modeled. We assigned metallicities from the FMR
(Equation 19, to the parent catalogs used in the SED pipeline. Fig. 27 shows
the FMR for the synthetic galaxies. We show for comparison the FMR (dotted
lines), where we have used the assumption $12+\log_{10}({\rm
O/H})\approx\log_{10}(Z_{\rm met}/Z_{\rm sol})$.

Figure 27: Mass–metallicity relation of the synthetic galaxies (solid lines).
The shaded regions are one standard deviation. Dotted lines show the FMR from
Williams et al. (2018), which were used to generate the parent catalog SEDs.
The synthetic galaxies have increased metallicity with mass and SFR, in
agreement with the underlying FMR.

Our synthetic galaxies follow the expected FMR, where metallicity increases
with mass, with a turn-over at high masses, as seen in observations (see
discussion in Section 6.3). The low-mass galaxies have lower metallicities
than the high-mass galaxies as expected, but do not vary much with SFR.
However, at these low masses, very few galaxies will have SFRs $\psi>-1$.

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