# Constraining cosmology with machine learning and galaxy clustering: the
CAMELS-SAM suite

Lucia A. Perez School of Earth and Space Exploration, Arizona State
University, 781 Terrace Mall, Tempe, AZ 85287, USA  Shy Genel Center for
Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY
10010, USA  Columbia Astrophysics Laboratory, Columbia University, 550 West
120th Street, New York, NY 10027, USA  Francisco Villaescusa-Navarro Center
for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY
10010, USA  Department of Astrophysical Sciences, Princeton University, Peyton
Hall, Princeton NJ 08544, USA  Rachel S. Somerville  Center for Computational
Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  Austen
Gabrielpillai Institute for Astrophysics and Computational Sciences, Catholic
University of America, USA  Astrophysics Science Division, NASA GSFC, 8800
Greenbelt Rd, Greenbelt, MD 20771, USA  Center for Research and Exploration in
Space Science and Technology, NASA GSFC, 8800 Greenbelt Rd, Greenbelt, MD
20771, USA  Daniel Anglés-Alcázar Department of Physics, University of
Connecticut, 196 Auditorium Road, Storrs, CT 06269, USA  Center for
Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY
10010, USA  Benjamin D. Wandelt Institut d’Astrophysique de Paris (IAP), UMR
7095, CNRS, Sorbonne Université, France  Center for Computational
Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA  L. Y.
Aaron Yung Astrophysics Science Division, NASA GSFC, 8800 Greenbelt Rd,
Greenbelt, MD 20771, USA

###### Abstract

As the next generation of large galaxy surveys come online, it is becoming
increasingly important to develop and understand the machine learning tools
that analyze big astronomical data. Neural networks are powerful and capable
of probing deep patterns in data, but must be trained carefully on large and
representative data sets. We developed and generated a new ‘hump’ of the
Cosmology and Astrophysics with MachinE Learning Simulations (CAMELS) project:
CAMELS-SAM, encompassing one thousand dark-matter only simulations of (100
$h^{-1}$ cMpc)3 with different cosmological parameters ($\Omega_{m}$ and
$\sigma_{8}$) and run through the Santa Cruz semi-analytic model for galaxy
formation over a broad range of astrophysical parameters. As a proof-of-
concept for the power of this vast suite of simulated galaxies in a large
volume and broad parameter space, we probe the power of simple clustering
summary statistics to marginalize over astrophysics and constrain cosmology
using neural networks. We use the two-point correlation function, count-in-
cells, and the Void Probability Function, and probe non-linear and linear
scales across $0.68<$ R $<27\ h^{-1}$ cMpc. Our cosmological constraints
cluster around 3-8$\%$ error on $\Omega_{\text{M}}$ and $\sigma_{8}$, and we
explore the effect of various galaxy selections, galaxy sampling, and choice
of clustering statistics on these constraints. We additionally explore how
these clustering statistics constrain and inform key stellar and galactic
feedback parameters in the Santa Cruz SAM. CAMELS-SAM has been publicly
released alongside the rest of CAMELS, and offers great potential to many
applications of machine learning in astrophysics: https://camels-
sam.readthedocs.io.

large scale structure, machine learning, cosmology, simulations

††journal: ApJ

##  1 Introduction

Since the earliest galaxy redshift surveys, it has been known that galaxies
are not distributed randomly in space, but trace out vast structures,
including walls, filaments, and voids. Dark matter (DM) makes up the majority
of the mass content of the Universe, and is the dominant driver behind large-
scale structure formation. The distribution of galaxies in space is heavily
influenced by the clustering of dark matter halos, but also carries signatures
of how galaxy properties map to the properties of these dark matter halos
(Peebles 1980; Wechsler & Tinker 2018). Galaxy clustering is a potential key
probe of cosmology, yet accurately describing the baryonic physics that drives
galaxy evolution, and determines this mapping between galaxy and DM halo
properties, is a large ongoing area of research. Astrophysical processes such
as cooling, chemical enrichment, star formation, stellar feedback, black hole
growth and feedback, and galaxy mergers interact in highly non-linear ways.
Active Galactic Nuclei (AGN) expel gas far beyond the center of the host
galaxy, are a crucial element of the feedback cycle in galaxies, and may even
affect the distribution of dark matter itself (e.g. McKee & Ostriker 2007;
Fabian 2012; Kormendy & Ho 2013; Netzer 2015; Borrow et al. 2020). Stellar
feedback in the form of supernovae and radiation from massive stars drives
galactic winds that are key to regulating star formation in galaxies (e.g
Madau & Dickinson 2014; Somerville & Davé 2015; Anglés-Alcázar et al. 2017).
Modern hydrodynamic simulations including these physical processes within the
$\Lambda$CDM cosmological framework have been quite successful at reproducing
many features of the large scale distribution of galaxies. For example,
Springel et al. (2018) measured the matter power spectrum for the dark matter,
gas, and stellar components in the IllustrisTNG simulations, explored the
halo-galaxy connection, and found projected correlation functions that showed
great consistency with diverse observations. However, although a broad
narrative has developed for how these types of feedback tie into the formation
and evolution of galaxies, there is still much uncertainty in our
understanding of the details of how these physical processes operate to shape
galaxy observables (e.g. Steinhardt & Speagle 2014; Somerville & Davé 2015;
Naab & Ostriker 2017; Förster Schreiber & Wuyts 2020). One of the major open
questions in astrophysics is how to disentangle the effects of cosmology and
baryonic physics in order to realize the full potential of galaxies as probes
of both cosmology and astrophysics.

The Cosmology and Astrophysics with MachinE Learning Simulations (CAMELS)
project111https://www.camel-simulations.org/ (Villaescusa-Navarro et al.,
2021a) posits: to constrain cosmology, we should leverage tools that can
marginalize over uncertainties in baryonic physics, and thereby measure the
underlying cosmology. Machine learning has great promise for this goal, as
algorithms can learn relationships between features. However, machine learning
requires large data sets for accurate training and robust results. The CAMELS
project created a large suite of simulations to specifically explore the
potential of machine learning to constrain cosmology. The initial focus of
CAMELS has been developing techniques that can constrain the density of matter
in the universe, $\Omega_{\text{M}}$, and the amplitude of density
fluctuations in the early universe, $\sigma_{8}$, under the broadly supported
$\Lambda$CDM model of cosmology. The project created 4,000+ cosmological
simulations of (25 $h^{-1}$ cMpc)3 spanning thousands of cosmological models.
Half the simulations are dark matter only, and the other half are run with the
IllustrisTNG (Weinberger et al. 2017; Pillepich et al. 2018) and SIMBA (Davé
et al., 2019) hydrodynamic models of galaxy formation, creating 2,000+
simulations also spanning thousands of astrophysical models. All of CAMELS has
been publicly released, as detailed in Villaescusa-Navarro et al.
(2022)222https://camels.readthedocs.io/.

Various studies based on CAMELS explored different methods of constraining
cosmology or astrophysical models with diverse types of machine learning,
computational tools, and astrophysical objects and phenomena. For example,
Villaescusa-Navarro et al. (2021b) obtain constraints on $\Omega_{\text{M}}$
with 3-4% errors using neural networks trained on total matter density maps,
providing robust predictions irrespective of galaxy formation physics
implementation. Similarly, Nicola et al. (2022) used the electron density
power spectrum $P_{ee}(k)$ to obtain cosmological constraints and to probe the
strength of baryonic feedback through the mean baryon fraction
($\bar{f}_{\text{bar}}$). The $z=0\ P_{ee}(k)$ from the IllustrisTNG ‘hump’ of
CAMELS yields constraints with approximately $5-10\%$ error on
$\Omega_{\text{M}}$ but no constraints on $\sigma_{8}$. However, their results
are encouragingly robust across galaxy formation models: the same neural
network is also able to predict $\Omega_{\text{M}}$ and $\bar{f}_{\text{bar}}$
when instead traind on simulations from the SIMBA hump. Nicola et al. (2022)
also confirm that it is possible to obtain constraints on $\Omega_{\text{M}}$
with approximately $4\%$ errors using the CAMELS matter power spectrum
$P_{mm}(k)$s, though no significant constraints on baryon fraction
$\bar{f}_{\text{bar}}$ and $\sigma_{8}$ can be obtained with this method.

Though they represent a valuable resource for many science goals, the original
CAMELS simulations, due to their small volumes and sensitivity to cosmic
variance, are not well-suited for leveraging the most readily used summary
statistics for measuring cosmology from observations: galaxy clustering. For
example, Villaescusa-Navarro et al. (2021b) find that the power spectra
measured from CAMELS 2D maps of the matter density field provided constraints
on $\Omega_{\text{M}}$ with 20$\%$ error–a significant loss of information
compared to using the full maps in their neural networks. The clear path to
improve constraints from galaxy clustering statistics is to increase CAMELS’
volume. However, it is not currently computationally feasible to directly
scale CAMELS and its full hydrodynamic simulations up to larger volumes. In
this work, we present CAMELS-SAM, a third and larger ‘hump’ of the CAMELS
project to address the need for larger volumes of simulated galaxies, and use
it to probe the power of clustering summary statistics towards CAMELS’ goals.

Semi-analytic models are a well-established technique for simulating galaxy
properties in a cosmological context, using simplified but physically
motivated recipes. SAMs are very successful at reproducing a broad range of
galaxy observables, and the predictions of SAMs for many global galaxy
properties have been shown to be in good agreement with the predictions of
hydrodynamic simulations over a broad range of cosmic time (Somerville & Davé,
2015). However, SAMs are more computationally efficient than hydrodynamic
simulations by many orders of magnitude. In this work, we make use of the
well-established Santa Cruz SAM (SC-SAM; Somerville & Primack 1999; Somerville
et al. 2008, 2015, 2021). A detailed, halo by halo comparison of the
predictions of the SC-SAM with the IllustrisTNG hydro simulations has recently
been carried out by Gabrielpillai et al. (2021). Hadzhiyska et al. (2021a)
showed that the SC-SAM produces very similar predictions to IllustrisTNG for
galaxy clustering, including two-point and higher order clustering statistics.

SAMs are set within the backbone of cosmological dark matter merger trees,
which specify how halos grow over time via accretion and mergers. Like most
SAMs, the SC-SAM implements treatments of cooling, partitioning of cold gas
into a molecular, atomic, and ionized phase, star formation, stellar feedback,
chemical enrichment, and black hole growth and feedback. Each of these
processes contain free parameters that represent our incomplete understanding
of the physical processes. Traditionally, these parameters are adjusted to
reproduce a set of observational calibrations for nearby galaxies. In this
work, we run the SC-SAMs within merger trees extracted from the suite of new
DM-only simulations that we have created for the CAMELS-SAM hump. In addition,
we re-run the SAMs for many different values of the parameters controlling
stellar and AGN feedback, in a similar spirit to the CAMELS hydro humps.

There has been much previous work describing galaxy clustering using simple
mappings between galaxy and halo properties, often referred to as the Halo
Occupation Distribution (HOD) framework (Wechsler & Tinker, 2018). A related
approach is the bias formalism, which describes the relationship between
galaxy clustering and dark matter halo clustering. The HOD and bias approach
have the advantage of simplicity and great computational efficiency, and have
been used in many studies that attempt to use galaxy clustering observations
to constrain cosmological parameters such as $\Omega_{\text{M}}$ and
$\sigma_{8}$ (e.g. Zhai et al., 2019b; Sugiyama et al., 2020; Repp & Szapudi,
2020; Uhlemann et al., 2020; Barreira et al., 2021; Kokron et al., 2021; Bayer
et al., 2021; Hahn & Villaescusa-Navarro, 2021; Massara et al., 2021; Mead et
al., 2021). However, most of these approaches focus on larger scales (R
$\gtrsim 7\ h^{-1}$ cMpc or $k_{\max}<1\ h$ cMpc -1) due to the difficulty of
capturing the complexity of clustering on strongly non-linear scales with
these approaches333Some previous works probe cosmology using clustering at
mildly nonlinear scales up to $k_{\max}<5\ h$ Mpc-1 (R $>1.26\ h^{-1}$ cMpc);
e.g. Aricò et al. 2021, and the UNIT simulations of Chuang et al. 2019..
However, smaller, more non-linear scales are expected to be more affected by
the details of feedback and baryonic physics.

An additional advantage of a SAM or hydro based approach is that it has the
potential to provide direct insights into the astrophysical processes, which
HOD/bias models bypass. Additionally, it has been shown that the most basic
and widely used HOD-type models (which assume that the galaxy-halo mapping
depends only on halo mass) do not accurately describe the clustering
predictions of full hydrodynamic simulations (Hadzhiyska et al., 2021b).
Galaxies in hydrodynamic simulations such as IllustrisTNG show a phenomenon
called “assembly bias”, which means that their clustering depends on halo
properties in addition to mass. The SC-SAMs have been shown to closely
reproduce the assembly bias signal seen in IllustrisTNG (Hadzhiyska et al.,
2021a). Other works have also shown indications that HOD models may need more
secondary parameters to accurately recreate the complexity of observed galaxy
clustering (Szewciw et al. 2021; Hahn & Villaescusa-Navarro 2021).

There has been previous work in leveraging machine learning to probe how
galaxy clustering is influenced by cosmology and baryonic physics. Aricò et
al. (2021) created a multidimensional neural network emulator of the
‘baryonification’ of the non-linear matter power spectrum atop the unique
BACCO suite (Angulo et al., 2021) that performs cosmological rescaling of
N-body simulations. The emulator has been tuned to scales within $0.01<k<5\ h$
cMpc-1 and $0<z<1.5$, and yields 1-2$\%$ accuracy when tested against several
dozens of hydrodynamic/N-body simulation pairs. Combined with the emulator of
Contreras et al. (2020) for the dark matter power spectrum from BACCO, it is
expected to give predictions for the non-linear matter power spectrum within
2-4$\%$ accuracy. Additionally, Xu et al. (2021) used machine learning to
predict the HOD, real-space 3D correlation function clustering, and assembly
bias for the SAM of Guo et al. (2011). Finally, Ntampaka et al. (2020)
developed a deep machine learning-based technique for accurately measuring
$\sigma_{8}$ and $\Omega_{\text{M}}$ (to within 3-4 $\%$) from mock galaxy
redshift surveys built atop AbacusCosmos (Garrison et al., 2021) with a HOD.

However, the work presented here with CAMELS-SAM makes four new contributions
to the field. 1) We use a physics-based SAM to model galaxy formation, which
provides predictions of galaxy clustering that agree better and are more
comparable with hydrodynamic simulations than HOD based approaches. 2) We
create an enormous suite of simulated galaxies in large volumes across a wide
range of both cosmological and astrophysical parameter space, specifically
aimed at providing training data for machine learning. 3) We include not only
the two-points correlation function and counts-in-cells (CiC), but also the
less commonly used Void Probability Function (VPF) clustering statistic in our
analysis. 4) We include smaller, more non-linear scales than most works have
attempted – our work probes the two-point correlation function to $R=1.1$ cMpc
(or $R=0.74\ h^{-1}$ cMpc and $k_{\max}<$ 8.5 $h$ cMpc-1), and the VPF and CiC
to $R=1.6$ cMpc ($R=1.1\ h^{-1}$ cMpc and $k_{\max}<$ 5.85 $h$ cMpc-1).

In this work, we present and describe the CAMELS-SAM suite (publicly released
in Villaescusa-Navarro et al. 2022, found at: https://camels-
sam.readthedocs.io). As a proof-of-concept example of CAMELS-SAM’s potential
as a machine learning data set for cosmology, we also probe how well galaxy
clustering can constrain cosmology using neural nets. We present the
constraints that we are able to obtain on $\Omega_{\text{M}}$ and $\sigma_{8}$
using the real-space two-point correlation function (2ptCF), count-in-cells
(CiC), and the Void Probability Function (VPF) at a combined set of redshifts
from $z=0$–1. We investigate how well each of these different clustering
statistics separately can constrain the cosmological parameters, and how the
results are affected by the choice of object selection (halo mass, stellar
mass, or star formation rate) and redshifts included. We also investigate how
well we can constrain the astrophysics parameters that control stellar and AGN
feedback using this approach.

The layout of this paper is as follows: in §2, we describe the creation of the
N-body simulations, the Santa Cruz SAM, and how we apply it to our
simulations. In §3, we explain how we measure galaxy clustering and our
implementation of neural networks to infer the input cosmological and
astrophysical parameters. In §4, we explore how well our neural networks
constrain the cosmological parameters $\Omega_{\text{M}}$ and $\sigma_{8}$
across various experiments, such as: different galaxy selections, including or
excluding a random down-sampling to fixed number density, and using a single
redshift or combining several. In §5, we explore how these experiments affect
instead the SC-SAM feedback parameters. In §6, we focus particularly on how
each of the clustering statistics we measure perform independently. We discuss
the CAMELS-SAM suite through a ‘meta’-lens in §7, comparing it to CAMELS and
other simulation suites, and discussing the potential in our data release. We
conclude and summarize our results in §8. Our Appendices hold additional
explanatory figures for various parts of the project.

##  2 The new CAMELS-SAM suite

Table 1: Description of all parameters varied in the CAMELS-SAM suite and their significance. The very broad ranges guarantee our neural networks will be unaffected by tight priors near the true expected values of these parameters (Villaescusa-Navarro et al., 2020a). See §2.2 for more detailed descriptions of the parameters, and Somerville et al. (2008) and Somerville et al. (2015) for a full description of their role in the SC-SAM. Param. |  Significance |  IC Range |  Fid. |  Distribution
---|---|---|---|---
$\Omega_{\text{M}}$ |  Fraction of universe’s energy density in dark matter $\&$ baryons |  0.1, 0.5 |  0.3 |  uniform
$\sigma_{8}$ |  Amplitude of the linear power spectrum on the scale of 8 $h^{-1}$ cMpc |  0.6, 1.0 |  0.8 |  uniform
A${}_{\text{SN1}}$ |  Multiplicative pre-factor to $\epsilon_{\text{SN}}=1.7$, amplitude of mass outflow rate due to SNe |  0.25, 4.0 |  1 |  log uniform
A${}_{\text{SN2}}$ |  Additive pre-factor to $\alpha_{rh}=3.0$, power law slope of mass outflow rate due to SNe |  -2.0, +2.0 |  0 |  uniform
A${}_{\text{AGN}}$ |  Multiplicative pre-factor to $\kappa_{\mathrm{radio}}=0.002$, amplitude of mass ejection rate in AGN radio jets |  0.25, 4.0 |  1 |  log uniform

###  2.1 Specifications for the Simulations

The backbone of CAMELS-SAM consists of 1,005 N-body simulations of volume (100
$h^{-1}$ cMpc)3 and N=6403 particles, covering the broad cosmological space of
$\Omega_{\text{M}}$=[0.1, 0.5] and $\sigma_{8}$=[0.6,1.0], and containing 100
snapshots between $20\leq z\leq 0$.

We generated initial conditions with second order Lagrangian perturbation
theory starting at $z=127$, and generated the linear power spectra with CAMB
(Lewis et al., 2000). We specified a periodic box of volume (100 $h^{-1}$
cMpc)3 and with N=$640^{3}$ dark matter particles. The N-body portion of
CAMELS-SAM was run with a setup of Arepo similar to that which ran
Illustris(TNG) (Springel 2010; Vogelsberger et al. 2013; Genel et al. 2014).
The resulting mass resolutions are between roughly $1-5\times 10^{8}\ h^{-1}$
M⊙, and the gravitational softening length was fixed to 4 comoving kpc until
$z=1$, after which it is fixed to a maximum of 2 physical kpc. Apart from
varying $\Omega_{\text{M}}$ and $\sigma_{8}$ for our work, we assume a
standard flat $\Lambda$CDM cosmology with $\Omega_{\rm b}=0.049$, $h=0.6711$,
$n_{s}=0.9624$, $\sum m_{\nu}=0.0$ eV, and $w=-1$. We stored 100 snapshots
between $20\leq z\leq 0$ in the same spacing as IllustrisTNG444See the
complete list within ‘Description of Simulations’ in the documentation:
https://camels-sam.readthedocs.io/en/main/simulations.html..

These 100 snapshots were run through Rockstar555Behroozi et al. (2013a):
https://bitbucket.org/gfcstanford/rockstar and ConsistentTrees666Behroozi et
al. (2013b): https://bitbucket.org/pbehroozi/consistent-trees/src/main/ to
identify dark matter halos and subhalos, and obtain merger trees. Our
implementation of the SC-SAM demands that ‘root’ $z=0$ halos have at least 100
dark matter particles, and also prunes any parts of the tree that are less
than 100 times the mass of the dark matter particles. This guarantees that all
halos and the merger histories the SC-SAM uses are well resolved. Given our
cosmological parameter range, this translates to probing halos of at least
M${}_{\text{halo}}>1-5\times 10^{10}\ h^{-1}$ M⊙. Additionally, the
ConsistentTrees merger trees are post-processed to exclude subhalo trees, as
the SC-SAM models the evolution of sub-halos internally. We refer readers to
§2.2-3.1 of Gabrielpillai et al. (2021) for a contextual explanation of how
Rockstar and ConsistentTrees are used for the SC-SAM, as well as the full
narrative of the SC-SAM astrophysics in the version used to create CAMELS-SAM.

Finally, each N-body simulation was run through multiple iterations of the SC-
SAM, where we varied parameters for stellar and AGN feedback across a
similarly broad hyperspace. Table LABEL:table:ParamDescriptions summarizes all
the parameters we vary in CAMELS-SAM, and we discuss the physical meaning of
these parameters in more detail in the next sub-section. The 1,000+ resulting
SAM catalogs each contain hundreds of thousands to millions of halos and
galaxies at each of the 100 redshift snapshots.

Key CAMELS-SAM simulation data products, such as the halo catalogs, merger
trees, and SC-SAM galaxy catalogs, have been publicly released alongside the
CAMELS suites in Villaescusa-Navarro et al. (2022). We direct readers there
for access details, and to the CAMELS-SAM documentation website for up-to-date
information: https://camels-sam.readthedocs.io.

###  2.2 The Santa Cruz Semi-Analytic Model for CAMELS

Here we briefly summarize the version of the Santa-Cruz semi-analytic model
for galaxy formation used in this work. The core SC-SAM is similar to that
used in Somerville et al. (2015), and the small updates in the version used
here are described in Gabrielpillai et al. (2021). We also direct readers to
the descriptions in Somerville et al. (2008) and Porter et al. (2014), as well
as the recent applications to creating robust mock observations in Somerville
et al. (2021) and Yung et al. (2019a). Our pipeline to go from AREPO N-body
simulations to SC-SAM galaxy catalogs is nearly identical to that described in
Gabrielpillai et al. (2021), who probe how well the SC-SAM run upon the
IllustrisTNG100-1 dark matter only simulation compares to the full
hydrodynamic IllustrisTNG100-1 simulation.

Numerical simulations like IllustrisTNG and SIMBA explicitly solve equations
of gravity, thermodynamics, (magneto-)hydrodynamics, etc. for discrete
particles and fluid elements or cells representing dark matter, gas, stars,
and black holes. However, they must adopt sub-grid recipes for small-scale
processes that are not resolved nor fully understood in galaxy formation (e.g.
stellar feedback). These simulations provide detailed information on the
spatial distribution, kinematics, and composition of the baryonic component as
galaxies evolve which allows for rich and diverse science. (For example, maps
of neutral hydrogen in Hassan et al. 2021 and detailed analyses of the
circumgalactic medium in Moser et al. 2022 from the hydrodynamic CAMELS
suites). However, full hydrodynamic simulations are computationally expensive
to run, often limited to either small volumes or low resolution. This makes it
difficult to thoroughly explore the parameter space of the sub-grid recipes.

SAMs instead work from merger tree histories, either measured directly from
N-body simulations as we have done, or constructed from Extended Press-
Schechter formalism (Somerville & Primack 1999; as in e.g. Yung et al. 2020
recently). Within these halo growth histories, SAMs solve a system of Ordinary
Differential Equations describing the flows of material through different
reservoirs. This approach offers flexibility; for example, the minimum
resolution of a SC-SAM galaxy is not limited to the resolution of stellar mass
particles. As in other SAMs, the SC-SAM contains physically-motivated
prescriptions for how gas is accreted onto halos, how hot halo gas cools and
accretes onto galaxies, how stars form from cooled gas in the ISM, and how
mass and metals are returned to the ISM through stellar winds and SNe. The SC-
SAM partitions the ISM into an atomic, molecular, and ionized phase, and uses
a phenomenological prescription for how stars form out of molecular gas. As
noted previously, several of the key processes in SAMs contain adjustable
parameters, similar to the parameters contained in the sub-grid recipes in
hydrodynamic simulations.

To mimic the stellar feedback parameter variation in CAMELS, we focus on the
SC-SAM parameters $\epsilon_{\text{SN}}$ and $\alpha_{\rm rh}$, which control
the mass outflow rate out of galaxies driven by supernovae and radiation from
massive stars. Like other SAMs and numerical models, which expect winds to
conserve energy and momentum in a galaxy, the SC-SAM assumes that the mass
outflow rate due to stellar feedback scales with the depth of the galaxy’s
potential well. Specifically, the SC-SAM relates the mass outflow rate from
stellar driven winds to the SFR of the galaxy and the circular velocity of the
halo:

$\dot{m}_{\mathrm{out}}=\epsilon_{\mathrm{SN}}\Bigg{(}\frac{V_{0}}{V_{c}}\Bigg{)}^{\alpha_{\mathrm{rh}}}\dot{m}_{*}$
(1)

Here, $\dot{m}_{*}$ is the SFR; $V_{0}$ is a normalization constant set to 200
km s-1; and $V_{c}$ is the maximum circular velocity of galaxy’s disc (assumed
to be the maximum rotational velocity of its host dark matter halo). The
parameters $\epsilon_{\text{SN}}$ and $\alpha_{\rm rh}$ are adjustable. This
is quite similar in spirit to the treatment of kinetic stellar driven winds in
IllustrisTNG and SIMBA.

The SC-SAM, however, employs a somewhat different approach for implementing
AGN feedback compared to the CAMELS TNG and SIMBA model variations. In the
SAM, radiatively inefficient accretion onto black holes is assumed to cause
heating of the hot halo gas via energetic radio jets. The rate of accretion
onto the black hole from the hot halo is given by:

$\dot{m}_{\mathrm{radio}}=\kappa_{\mathrm{radio}}\Bigg{[}\frac{kT}{\Lambda[T,Z_{h}]}\Bigg{]}\Bigg{(}\frac{M_{\mathrm{BH}}}{10^{8}M_{\odot}}\Bigg{)}$
(2)

Here, $kT$ is the temperature of gas within the Bondi accretion radius
($r_{A}\equiv 2GM_{\text{BH}/c_{s}^{2}}$); and $\Lambda[T,Z_{h}]$ is the
temperature- and metallicity-dependent cooling function (Sutherland & Dopita,
1993). This radio mode accretion heats the hot halo gas at a rate that is
proportional to $\dot{m}_{\mathrm{radio}}$, and can partially or completely
offset cooling and accretion into the ISM. Thus, we can control the strength
of the feedback from the jet mode by varying the parameter
$\kappa_{\mathrm{radio}}$. The AGN feedback mainly affects the most massive
galaxies (e.g. Figure 15 in Appendix A).

In the SC-SAM, the default values for the above parameters are:
$\epsilon_{\text{SN}}=1.7$, $\alpha_{rh}=3.0$, and
$\kappa_{\mathrm{radio}}=0.002$. These values, and the other parameter values
that go into the SAM, were selected by tuning the SAM “by hand” to reproduce a
set of key observed relationships (Somerville et al. 2008, 2015, 2021; Yung et
al. 2019a), such as the stellar mass function, cold gas fraction, mass-
metallicity relation for stars, and black hole mass vs. bulge mass relation
(Bernardi et al. 2013; Moustakas et al. 2013; Baldry et al. 2012; Rodríguez-
Puebla et al. 2017; Catinella et al. 2018; Calette et al. 2018; Gallazzi et
al. 2005; Kirby et al. 2011; McConnell & Ma 2013; Kormendy & Ho 2013). We
check that we reproduce these calibrations with our implementation of the SC-
SAM within the CAMELS merger trees in Appendix A Figure 12, finding excellent
agreement with Gabrielpillai et al. (2021). The predictions of the SAM for
these galaxy properties are quite insensitive to the resolution of the input
dark matter merger trees (Gabrielpillai et al. 2021 and Figure 12).

In this work, as in the CAMELS hydrodynamic humps, we vary pre-factors for
each parameter over a fairly broad range. We assign the pre-factor of
$\epsilon_{\text{SN}}$ as the multiplicative A${}_{\text{SN1}}$, and vary it
between [$\frac{1}{4}$,4] with a default of unity. Similarly, the pre-factor
for $\kappa_{\mathrm{radio}}$ is the multiplicative factor A${}_{\text{AGN}}$,
and varies between [$\frac{1}{4}$,4] with a default of unity. Both
A${}_{\text{SN1}}$ and A${}_{\text{AGN}}$ are generated evenly in logarithmic
space to compare the ‘order of magnitude’ scale of effects777When we explore
our neural networks’ constraints for these parameters, A${}_{\text{SN1}}$ and
A${}_{\text{AGN}}$ will therefore appear to bias toward smaller values on the
plots’ linear [1/4,4] $x$-axes.. The parameter $\alpha_{\rm rh}$ appears in
the exponent of a power law, and we therefore define an additive factor
A${}_{\text{SN2}}$. We vary it between [-2,2] and generate it evenly in linear
space with a default value of zero.

Table LABEL:table:ParamDescriptions gives a summary of the parameters and
their ranges. They appear within Equations 1 and 2 in the following manner:

$\begin{split}\dot{m}_{\mathrm{out}}=&\Big{(}\epsilon_{\mathrm{SN}}\times
A_{\text{SN1}}\Big{)}\Bigg{(}\frac{V_{0}}{V_{c}}\Bigg{)}^{(\alpha_{\mathrm{rh}}+A_{\text{SN2}})}\dot{m}_{*}\\\
&\dot{m}_{\mathrm{radio}}=\Big{(}\kappa_{\mathrm{radio}}\times
A_{\text{AGN}}\Big{)}\Bigg{[}\frac{kT}{\Lambda[T,Z_{h}]}\Bigg{]}\end{split}$
(3)

###  2.3 LH, CV, and 1P Simulation Sets

Table 2: Description of the different simulation products across the CAMELS-SAM suite and their significance. Label |  Parameters covered |  Total |  Products
---|---|---|---
LH |  Latin hypercube across {$\Omega_{\text{M}},\sigma_{8}$, A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, A${}_{\text{AGN}}$} = {0.1:0.5, 0.6:1.0, 0.25:4, -2:2, 0.25:4}, unique random seeds. |  1,000 |  ROCKSTAR halo catalogs, ConsistentTrees merger trees, galaxy catalogs
CV |  Fiducial cosmology {$\Omega_{\text{M}},\sigma_{8}$, A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, A${}_{\text{AGN}}$}={0.3, 0.8, 1, 0, 1} and unique random seeds. |  5 |  ROCKSTAR halo catalogs, ConsistentTrees merger trees, galaxy catalogs
1P |  For CV$\\_$0 and CV$\\_$1, the minimum and maximum values of {A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, A${}_{\text{AGN}}$} one at a time. All else held to fiducial values. |  12 |  Additional galaxy catalogs
Ex |  ‘Ex’-treme cosmology and volume simulations of (205 $h^{-1}$ cMpc)3 and (1240)3 at $\Omega_{M}$={0.1, 0.5, 0.5, 0.1, 0.3} and $\sigma_{8}$={0.6, 1.0, 0.6, 1.0, 0.8} |  5 |  Not currently shared; ROCKSTAR catalogs used in Figure 1.
Figure 1: Halo mass functions (HMFs) at $z=0$ (top) and $z=2$ (bottom) for the
CV simulations (black stars; fiducial cosmology and unique random seeds). HMFs
of the (205 $h^{-1}$ cMpc)3, N$=1280^{3}$ ‘extreme’ cosmology volumes, where
Ex{0,1,2,3,4} have $\Omega_{\text{M}}$={0.1, 0.5, 0.5, 0.1, 0.3} and
$\sigma_{8}$={0.6, 1.0, 0.6, 1.0, 0.8}, are shown with {purple, blue, cyan,
orange, red} lines.

Table LABEL:table:SimSets describes all products within CAMELS-SAM and their
significance and use. The core “latin hypercube” (LH) set consists of 1,000
simulations, each with different values of $\Omega_{\text{M}}$, $\sigma_{8}$,
A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, and A${}_{\text{AGN}}$. We first
generated 1,000 N-body simulations with Arepo over a latin hypercube of
$\Omega_{\text{M}}$=[0.1,0.5] and $\sigma_{8}$=[0.6,1.0] (yielding individual
dark matter particle masses of approximately $1.3-6\times 10^{8}$ $h^{-1}$
M⊙). The random phases of the initial conditions in the N-body simulations are
allowed to vary. The parameters in the latin hypercube are randomly generated:
linearly across $\Omega_{\text{M}}$=[0.1, 0.5], $\sigma_{8}$=[0.6, 1.0], and
A${}_{\text{SN2}}$=[-2, +2]; and logarithmically across both
A${}_{\text{SN1}}$ and A${}_{\text{AGN}}$=[0.25, 4.0]. These SC-SAM pre-factor
parameters are described in §2.2 and Table LABEL:table:ParamDescriptions.

In addition to the full CAMELS-SAM LH suite of 1,000 simulations, we also
created a small set of 5 ‘cosmic variance’ (CV) simulations, run with
different random seeds for the initial conditions while using the same
fiducial parameters of {$\Omega_{\text{M}},\sigma_{8}$, A${}_{\text{SN1}}$,
A${}_{\text{SN2}}$, A${}_{\text{AGN}}$}={0.3, 0.8, 1, 0, 1}. The CV set allows
us to evaluate how cosmic variance in our (100 $h^{-1}$ Mpc)3 volumes may
affect clustering statistics.

Atop the first two CV N-body simulations, CV$\\_$0 and CV$\\_$1, we also
created twelve min-max ‘one-parameter’ (1P) simulations to serve a similar
purpose as in the hydrodynamic CAMELS suites (Villaescusa-Navarro et al.,
2022). The 1P SC-SAM galaxy catalogs cover the minimum and maximum pre-factor
parameter values, or {A${}_{\text{SN1}}$}={0.25, 4.0},
{A${}_{\text{AGN}}$}={0.25, 4.0}, and {A${}_{\text{SN2}}$}={-2, 2}. These
simulations are useful to investigate each parameter’s effect on our
clustering statistics. For example, in Appendix A (Figures 13, 14, and 15), we
examine how the extreme ends of these parameters affect key galaxy summary
statistics, including the stellar mass function and the ratio of cold gas
fraction to stellar mass in galaxies.

We additionally created five ‘extreme’ simulations with the same resolution
but eight times larger volume. These simulations have box size length L=205
$h^{-1}$ cMpc and N=12803 particles. One was created with the fiducial
cosmology of {$\Omega_{\text{M}}$, $\sigma_{8}$}={0.3, 0.8}, and the other
four were created at the corners of the full cosmological parameter space of
$\Omega_{\text{M}}$=[0.1,0.5] and $\sigma_{8}$=[0.6,1.0]. We ran the SC-SAM on
each of these ‘extreme’ simulations using the fiducial parameters that best
recreate $z=0$ observations. This allowed us to confirm that our selected
galaxy clustering statistics show the expected influence of cosmological
parameter variations.888At the time of writing, the ‘extreme’(-ly large)
simulations are not released with the rest of CAMELS-SAM.

###  2.4 Verifying the Simulations

As a confirmation of our N-body volumes and the ROCKSTAR products, we first
examine the behavior of the halo mass functions (HMFs). Figure 1 shows $z=0$
and $z=2$ HMFs for the five CV simulations. We compare the CV set against the
HMFs of the ‘extreme’ simulations, 4 of which exist at the extreme corners of
our cosmological parameter space and 1 of which is at the same fiducial
cosmology as the CV set. These simulations show well converged halo statistics
for our smaller fiducial volume and illustrate the broad range of conditions
probed by CAMELS-SAM. We also note that the relevant HMFs of the
IllustrisTNG300-1 and -2 simulations are very consistent with the ‘fiducial’
cosmology Ex4 simulation and all the CV simulations (confirmed in Appendix A
Figure 12).

The SC-SAM has been tested using the largest and highest resolution
IllustrisTNG simulations, with box side length of L=205 $h^{-1}$ cMpc and
25003 particles (Gabrielpillai et al., 2021). We confirmed the SC-SAM would
still robustly match key observational statistics for high-mass halos at our
lower resolution, similar to IllustrisTNG300-2 (of L${}_{\text{box}}$=205
$h^{-1}$ Mpc and 12503 particles), with all parameters held to the
IllustrisTNG cosmology and the best-fit SC-SAM parameters from Somerville et
al. (2021).

In Appendix A, we confirm that our suite setup recreates key observed summary
statistics under the fiducial model. We compare the result of our CV
simulations to the SAM outputs of the two highest resolution IllustrisTNG300
volumes, and various near-universe observations for: the stellar mass
function, the stellar mass-halo mass, the cold gas fraction vs. stellar mass
of disk-dominated galaxies, stellar metallicity-stellar mass, and black hole
mass-bulge mass relationships. The consistency with the larger IllustrisTNG300
SAM catalogs and the overall SC-SAM agreement with $z=0$ observations support
the choice of volume and resolution for our CAMELS-SAM suite. In Appendix A,
we also probe these relationships for the ‘1P’ galaxy catalogs, to further
understand how each of the SC-SAM parameters we vary affects astrophysical
summary statistics.

##  3 Methodology

In this section, we describe our methodology for constraining cosmology and
astrophysics using clustering summary statistics and neural networks. In §3.1,
we describe the clustering statistics we test in this work, how they were
measured, and how they are prepared for the neural network pipeline. In §3.2,
we describe our neural network architecture and process.

###  3.1 Galaxy Clustering

(a) VPF (left), CiC (middle), 2ptCF (right) clustering of $z=0$ galaxies with
nonzero stellar mass, down-sampled to 0.005 ($h^{-1}$ cMpc)3.

(b) VPF (left), CiC (middle), 2ptCF (right) clustering of $z=0$ galaxies with
SFR $>$ 1.25 M⊙ yr-1, down-sampled to 0.001 ($h^{-1}$ cMpc)3.

Figure 2: Clustering statistics for CAMELS-SAM simulations 630 through 649
(each in a unique color). We show the Void Probability Function (VPF, left),
count-in-cells at 27.2 cMpc (CiC, center), and real-space 3D two-point
correlation function (2ptCF, right), for two different selections of $z=0$
galaxies (top: stellar mass, bottom: SFR). The shaded black area in the top
figure indicates the clustering across 5 simulations using the ‘fiducial’
cosmological and SAM parameters but generated with different random seeds, to
show the effect of cosmic variance for these volumes.

####  3.1.1 Introduction to Clustering Statistics

The spatial distribution of galaxies traces the structure of underlying dark
matter, and carries signatures of both the cosmology as well as details of how
galaxies interact with their environment and each other. There are many ways
to measure the clustering of galaxies, each with unique strengths, uses,
theoretical foundations, and connections to other physical concepts. In this
proof of concept work, we use the Void Probability Function (VPF), count-in-
cells (CiC), and (real space) two-point correlation function (2ptCF).

The widely used 2ptCF quantifies the probability of finding two galaxies
within a certain distance from each other (compared to a random distribution;
e.g. Peebles 1980; Landy & Szalay 1993). CiC quantifies the number of galaxies
within a randomly placed cell of a given size and theoretically includes all
higher-order $n$-point clustering statistics (Peebles, 1980), but is the most
computationally costly of these three statistics. The VPF is a less commonly
used clustering statistic that simply asks: how likely is a sphere of a given
size to contain zero objects for a given galaxy selection criterion? The VPF
is simple and efficient to calculate, is tied to all higher order correlation
functions as the 0${}^{\text{th}}$ moment of count-in-cells, and encodes
information from higher order clustering that is not captured in the 2ptCF
(Perez et al. 2021; Conroy et al. 2005; White 1979).

These clustering statistics are known to be powerful tools for constraining
cosmology in observations and N-body simulations. CiC has been used
extensively in several recent works constraining cosmology. Uhlemann et al.
(2020) note the inclusion of CiC improved their constraints on
$\Omega_{\text{M}}$ and $\sigma_{8}$ by factors of 5 and 2 respectively
compared to the matter power spectrum alone, and also broke the degeneracy
between massive neutrino mass and $\sigma_{8}$. Salvador et al. (2019) also
probed how well CiC can constrain linear and higher-order galaxy bias, finding
its constraints are consistent with measurements of the bias from
galaxy–galaxy clustering, galaxy–galaxy lensing, cosmic microwave background
lensing, and shear-clustering measurements. Repp & Szapudi (2020) developed a
theoretical description for the CiC distribution, and found it breaks the
degeneracy between $\sigma_{8}$ and the bias parameter, and yields an 11$\%$
error on their $\sigma_{8}$ measurement for the SDSS Main Galaxy Sample. And
lastly, Wen et al. (2020) developed a new technique with the CiC PDF to probe
different cosmological models with dark energy, focusing on dark matter halos
in the DEUS simulations (Alimi et al., 2012) at $0<z<4$ and scales $2-25\
h^{-1}$ cMpc.

Additionally, types of the 2ptCF have been recently found to help constrain
dark energy observables (Zhai et al., 2019a, b). van Daalen et al. (2016) also
used the projected 2ptCF, creating a clustering estimator that improved
constraints on astrophysics within SAMs (specifically, the Munich SAM of Guo
et al. 2013). Our work is among the first to use the VPF999We emphasize here
the VPF is separate from work with cosmic voids, which are large underdense
regions in the cosmic web that require large detailed sky surveys to map and
catalog. The VPF is a simple statistical tool that counts empty
circles/spheres, where cosmic voids can have complex shapes and contain rare
and interesting galaxies (e.g. Habouzit et al., 2020). Cosmic voids are being
used as alternate cosmological probes (Pisani et al., 2015; Hamaus et al.,
2016; Zhang et al., 2020). as the 2ptCF and CiC have been used, and to
systematically compare the constraints that can be obtained from all three
statistics. We also note that Wang et al. (2019) used the VPF and CiC in
conjunction with the projected 2ptCF to help constrain galaxy assembly bias
using halo occupation distributions, and McCullagh et al. (2017) and Walsh &
Tinker (2019) used the VPF to refine generalized HOD fitting.

We note how Wang et al. (2019) used the VPF and CiC in conjunction with the
projected 2ptCF to help constrain galaxy assembly bias using halo occupation
distributions, and how McCullagh et al. (2017) and Walsh & Tinker (2019) used
the VPF to refine generalized HOD fitting.

####  3.1.2 Measuring Clustering in CAMELS-SAM

We measure all clustering statistics between 1 and 40 cMpc (0.6711 to 26.84
$h^{-1}$ cMpc) using the CORRFUNC package (Sinha & Garrison, 2020). We
constrain our analysis to distances larger than 1 cMpc due to known
inaccuracies of the assignment of satellite positions in the SC-SAM relative
to hydro simulations (Hadzhiyska et al., 2021a). This is because the SC-SAM
does not use the subhalo positions within their host halo provided by the
N-body simulation, but treats subhalo merging and tidal destruction using a
semi-analytic recipe (see §2.2.1 in Somerville et al. 2021). Moreover, it is
expected that the presence of baryons will affect the orbits of satellites and
the efficiency of their tidal heating and destruction, in a manner not
currently modeled in the SC-SAM (Jiang et al., 2021). Our analysis still
probes smaller nonlinear scales than most similar studies, reaching the
equivalent of $k_{\text{max}}<5.85h$ cMpc-1 (VPF/CiC) or $8.5\ h$ cMpc-1
(2ptCF), and leverages additional information from non-linear terms of galaxy
bias.

For the 2ptCF, we generate 20 distance scales evenly in logarithmic space
between $10^{0}$ and $10^{1.6}$ cMpc, yielding 19 distance bins centered
between 1.1 cMpc and 36.1 cMpc. With CORRFUNC, we measure the 2ptCF in three-
dimensional real space using the Landy & Szalay (1993) estimator:
$\xi(R)=(DD-2D\mathcal{R}+\mathcal{RR})/\mathcal{RR}$, where $DD$ are the
number of data-data pairs in the bin that encloses the distance scale $R$,
$D\mathcal{R}$ data-random pairs, and $\mathcal{RR}$ random-random pairs. For
the 2ptCF, we use 100 times as many random points as there are galaxies.

For the VPF and CiC, we use CORRFUNC to perform the calculation for 25
distance scales with a maximum of 40 cMpc, yielding 25 linearly spaces radii
between 1.6 and 40 cMpc. We measure the VPF by randomly dropping 100,000 or
500,000 spheres of each tested distance scale (for galaxy densities of 0.001
or 0.005 $h^{3}$ cMpc-3, respectively), and counting those with no galaxies
(see Conroy et al. 2005 for a discussion on the effects of the number of
dropped spheres). For CiC, we also drop 100,000 or 500,000 spheres randomly of
each tested distance scale, and count how many have 0, 1, 2, … $n=500$
galaxies inside them. Measuring to $n=500$ captures nearly the entire CiC
distribution at these densities over almost all scales. The VPF is
additionally calculated and verified101010During this work, we identified a
feature of how CORRFUNC generated the ‘random test spheres’ for the VPF and
CiC that led to inaccurate VPF measurements for small-to-medium sized galaxy
samples. This has now been corrected fully, allowing the powerful CORRFUNC to
be applied to even more samples than it was originally created for. using the
swift k-nearest neighbor method of Banerjee & Abel (2020). Future expansions
to this work will also extend to include higher-order nearest neighbor
statistics, shown to give additional cosmological constraints by Banerjee &
Abel (2020).

Figure 2 displays these clustering measurements for twenty of the LH galaxy
catalogs, under two types of ‘selections’ that vary the SC-SAM galaxy
property, the threshold value, and number density down-sampling (described in
the next section). For the twenty simulations shown (LH$\\_630-649$), the span
of covered parameters is: $\Omega_{\text{M}}=[0.131,0.479]$,
$\sigma_{8}=[0.6342,0.9862]$, A${}_{\text{SN1}}=[0.266,2.45]$,
A${}_{\text{SN2}}=[-1.506,+1.998]$, A${}_{\text{AGN}}=[0.26,3.84]$, and
therefore representative of nearly all of our parameter space. For some
selections, we measured the clustering of the CV catalogs to confirm that the
change in clustering caused by varying the parameters exceeded that caused by
cosmic variance (e.g. Figure 2a).

####  3.1.3 Context: Galaxy Selections within CAMELS-SAM

Before measuring any clustering statistic, we apply a selection across CAMELS-
SAM to yield a coherent sample of galaxies across all cosmologies and feedback
parameters. In this initial proof of concept study, galaxies are selected by:

  * •

halo mass, usually $\log_{10}(\text{M}_{\text{halo}}$/M${}_{\odot})>$ 10-12

  * •

stellar mass, usually $\log_{10}(\text{M}_{\text{stellar}}$/M${}_{\odot})>$
9-10

  * •

instantaneous star formation rate (SFR), M⊙ yr-1

  * •

or specific star formation rate (sSFR = SFR / M${}_{\text{stellar}}$), Gyr-1.

Table 3: Radii and bins of count-in-cells given to the neural network for our various tests. The spread of $n$ refers to which ’counts’ we consider–e.g. cells with 0 to 50 galaxies only. When doing a single redshift at a time, we broadened our choices to encompass the parts of the distribution with the most divergence across our simulations. To maximize the data we include for a single redshift, we skip every other $n$ value. Therefore, each simulation’s single-redshift count-in-cells sampling yields 510 total data points for the neural network to work upon. For all redshifts combined, we sample the first 50 $n$ values at each distance scale, since the distance scales we selected encompass most of their variety in those regions. The all-redshifts CiC sampling yields 600 total data points. All redshifts, $\mathcal{N}$=0.001 $h^{3}$ cMpc-3 | All redshifts, $\mathcal{N}$=0.005 $h^{3}$ cMpc-3 | Single redshift, any $\mathcal{N}$
---|---|---
Radius, cMpc |  Spread of $n$ |  Radius, cMpc |  Spread of $n$ |  Radius, cMpc |  Spread of $n$
11.2 |  0-50 |  16.0 |  0-50 |  16.0 |  0-50
16.0 |  0-50 |  22.4 |  0-50 |  25.6 |  0-170
20.8 |  0-50 |  28.8 |  0-50 |  30.4 |  50-250
– |  – |  – |  – |  35.2 |  100-350
– |  – |  – |  – |  40.0 |  250-500

These properties are predicted directly by the SC-SAM, and are relatively easy
to compare directly to other simulations. Although they are not directly
observable, there is extensive work in the literature to estimate stellar
masses and SFR from observed samples using various methods (e.g. see reviews
such as Blanton & Moustakas 2009; Madau & Dickinson 2014; Somerville & Davé
2015). For example, the stellar mass of galaxies is often measured with
spectral energy distribution fitting to broad or medium band photometry
(Walcher et al., 2011; Barro et al., 2013; Conroy, 2013; Duncan et al., 2014;
Mobasher et al., 2015), or spectra (e.g. Brinchmann et al. 2004; Brammer et
al. 2011). Star formation rate can be measured with flux in several different
emission lines or bands targeting different sources or tracers of star
formation, though many are sensitive to dust or contamination, and all carry
the uncertainties of assumed timescales (e.g. Calzetti 2013; Ellis 2008).
Specific SFR gives deep insight into the process of galaxy evolution over
cosmic history (e.g. Fig. 11 in Speagle et al. 2014), and is sometimes
measured by proxy with emission lines or with careful analysis of the stellar
mass function (e.g. Davidzon et al. 2018). Finally, the clear bimodal nature
of galaxy colors (and the separation between star-forming vs. quiescent
galaxies) is easily seen in the sSFR vs. stellar mass plane (e.g. Muzzin et
al. 2013).

Ultimately, our goal is to carry out this type of analysis with selections
that closely mimic observed galaxy samples, but simple selections on these
fundamental galaxy properties serve as good initial steps toward this goal.
However, we note these selections that we use are consistent with past studies
and/or ongoing or future planned wide field surveys. For example, several
studies have implemented or found similar halo mass limits as we use: Springel
et al. (2018, log10 M${}_{\text{halo}}>10.9-13\ h^{-1}$ M⊙), Xu et al. (2021,
log10 M${}_{\text{halo}}>10.7\ h^{-1}$ M⊙); Wen et al. (2020, log10
M${}_{\text{halo}}>11.38\ h^{-1}$ M⊙); Gonzalez-Perez et al. (2018, log10
M${}_{\text{halo}}>10.3\ h^{-1}$ M⊙). Our probed SFR and stellar mass cuts are
achievable for future surveys; for example, SFR $>0.1$ M⊙ yr-1 will likely be
observable within $z<1$ for wide Roman and Euclid surveys, assuming surveys to
(dust corrected) limiting magnitudes of 27 in Roman’s WFI F062 filter at 62
$\mu$m and Euclid’s VIS instrument (from Yung et al. (in prep.) Roman light
cones created with the SC-SAM, and forecasts similar to those in Yung et al.
2019a, b). Our M${}_{\text{star}}>10^{9}$ M⊙ selection should similarly be
widely detected with Roman within $z<1$, and to a similar extent with Euclid
(though more easily at closer to $z\sim 1$; stellar masses above $10^{9}$ M⊙
will likely correspond to the brightest Euclid galaxies at $z<0.1$)
Additionally, our higher stellar mass selections complement the limiting
stellar masses derived for 14,000 deg2 across various Legacy Surveys serving
the DESI project: M${}_{\text{star}}>10^{9.5}$ M⊙ at $z\sim 0.1$,
M${}_{\text{star}}>10^{10.5-11}$ M⊙ above $z>0.5$ (Zou et al., 2019). Finally,
these selections are consistent with what the Vera Rubin Observatory will
measure with LSST: M${}_{\text{star}}>10^{9.5}$ M⊙ at $z<0.5$,
M${}_{\text{star}}>10^{10}$ M⊙ at $0.5<z<1.0$ (Riccio et al., 2021, Figure 6).

When finalizing the selection criteria for our chosen galaxy or halo property,
we confirm each criterion will be met by enough CAMELS-SAM simulations to
obtain robust statistics across most of the 1,000 LH simulations. As we later
discuss in §3.2.1, we split up our 1,000 CAMELS-SAM LH simulations into
roughly 70/15/15 percent training, validation, and testing sets, respectively.
However, not all simulations yield a large enough sample to compute the
clustering statistics with all selection criteria, and we choose to prioritize
having at least 700 simulations in the training set, while still keeping
moderately large validation and testing sets.

For all halo or galaxy selections we present in this work, enough simulations
meet the criteria to guarantee training sets of at least 700 samples, and
validation and testing sets of at least 80-100 each. This range is still large
enough to sample the parameter space well, while allowing the flexibility to
use selection criteria that pick out rarer objects, potentially revealing
unique relationships between the parameters and galaxy clustering. The
simulations that do not meet the selection criteria tend to have little
structure formation–very small $\Omega_{\text{M}}$ and $\sigma_{8}$–meaning
they cluster in a specific corner of the parameter space. CAMELS and CAMELS-
SAM intentionally cover such a large parameter space to avoid being affected
by the distribution priors in the central, more realistic region of the
parameter space. Therefore, missing a single corner will still yield robust
constraints near the values of the cosmological parameters that are favored by
observational constraints.

The final aspect of our selection is whether or not we normalize to a specific
number density of objects. Number density has a strong effect on the CiC and
the VPF, and some stellar mass or SFR selections yield samples with very large
numbers of objects that pose a computational challenge. We address this in
this work by randomly “down-sampling” the objects that pass a given selection
to either 0.001 $h^{3}$ cMpc-3 or 0.005 $h^{3}$ cMpc-3, corresponding to 1,000
or 5,000 galaxies in our (100 $h^{-1}$ cMpc)3 volumes. These chosen densities
mimic the galaxy number densities many observations have measured or are
expected to measure111111Assorted examples: SDSS red/blue galaxies (Zehavi et
al., 2005, Table 4), SDSS-IV BOSS for ELGs (Raichoor et al., 2017, Figure 11),
EUCLID within $z<1.5$ (Amendola et al., 2018, Table 3), red sequence galaxies
in the Kilo Degree Survey (Vakili et al., 2020).. They are also large enough
to mitigate Poisson noise while producing samples small enough to calculate
the clustering statistics within a reasonable computational cost. In §4.2, we
examine how our constraints on cosmology and SAM astrophysics change if we do
not randomly down-sample to a fixed number density after applying our mass or
SFR-based selections.

We note that a different approach to fixing the number density of a sample is
more common in the literature: instead choosing the objects above a threshold
in mass or SFR in order to obtain a desired number density (sometimes referred
to as abundance matching). For example, Hadzhiyska et al. (2021b) select the
most massive galaxies such that they reach their desired number density. This
has a very different effect on clustering, as it is selecting _a differently
clustered/biased population of dark matter halos_ , while our random down-
sampling selects halos with the same clustering properties, but just sparsely
samples them to reduce computational load.

Finally, as a contextual example to compare against our selections, the
Springel et al. (2018) clustering measurements of IllustrisTNG300 found
thresholds for these selections at these densities: for a galaxy space density
of 0.001 $h^{3}$ cMpc-3, log10M${}_{\text{stellar}}>10.18\ h^{-1}$ M⊙, SFR
$>1.55$ M⊙ yr-1, sSFR $>1.27\ h$ Gyr-1. For a galaxy space density of 0.003
$h^{3}$ cMpc-3, log10M${}_{\text{stellar}}>10.49\ h^{-1}$ M⊙, SFR $>3.03$ M⊙
yr-1, sSFR $>4.43\ h$ Gyr-1.

###  3.2 Neural Network Implementation

####  3.2.1 Preparing CAMELS-SAM clustering for neural networks

We use a 70/15/15 percent split of the suite for training/validation/testing,
meaning each network was trained on the first 700 of the CAMELS-SAM LH
simulations, validated for performance on the next 150, and tested on the
final 150. (See §3.1.3 for what we do if not all 1,000 simulations are
usable.) The ‘best’ model is whichever has the lowest error value when applied
to the validation set, though we often found that at least 5-10 models
performed quite similarly, meaning these results are not tied to a specific
architecture. In the figures that follow, we show the performance of these
best models on the test set of simulations.

In this fiducial case of ‘all’ clustering, the data given to the neural
networks consists of: for $z=\\{0,0.1,0.5,1.0\\}$, the 2ptCF and the VPF
between roughly 1 and 40 cMpc, and the CiC probability distribution for
several distance scales (between 11-21 cMpc for our lower density samples, and
between 16-30 cMpc for the larger density samples; see Table
LABEL:table:CiCRadii). The clustering measurements at all four redshifts are
strung into a 1D array121212Though explained later in the relevant sections,
when we are not looking at ‘all’ clustering statistics at all four redshifts,
the neural network is instead given in a 1D array: all statistics at a single
redshift (§4.3), or an individual clustering statistic at the four redshifts
(§6).; all statistics are measured at exactly the same distance scales across
all 1000 simulations, so the radius of each measurement is irrelevant for the
neural network. Additionally, we randomly re-sample to our selected number
density for each redshift131313There is no way to fully know how a neural
network learns what it does. However, our choice to re-sample at each
redshift, and the additional noise introduced, may help prevent the networks
from focusing upon the growth factor (Hamilton, 2001) if it e.g. decides to
try taking ratios between clustering statistics. In future work, shuffling the
order of different redshifts’ clustering when training may help further
counteract this.. See §4.3 for an exploration of how our results change if we
use only one redshift at a time, and §6 for the performance when the
clustering statistics are used separately.

Neural networks perform best when trained on normalized data, where the mean
is about zero and the standard deviation is one. We normalize the clustering
1D array so: for each value (corresponding to e.g. the 2ptCF at $R\approx$ 10
cMpc), we take the base-ten logarithm across all 1000 values, subtract the
mean from each, and then divide each by the standard deviation. For galaxy
selections that yield values of zero for a given simulation (e.g. a
particularly unclustered simulation that finds no voids at large scales), the
value is set to $10^{-12}$ before normalizing. For galaxy selections where all
1000 simulations yielded zero (rare, and often at the largest distance scales
for extreme selections), we set the first simulation to a value of $2\times
10^{-12}$ to guarantee the normalization does not fail, and continue. The
cosmological and astrophysical parameters are normalized in almost the same
way—by their mean and standard deviation in linear space—and are returned by
the neural network as 1D output arrays.

####  3.2.2 Loss Functions

In this work, we use the CAMELS-SAM simulations to do parameter inference with
galaxy clustering statistics: how do the input parameter(s) relate to the
statistics, and how do we measure the input parameter(s) given only the
statistics? This marginal posterior we seek to learn is
$p(\vec{\theta}|\vec{C})$, which relates the parameter space $\vec{\theta}$ to
the 1D array of clustering measurements $\vec{C}$. In our work, we often have
$\vec{\theta}$ = {$\Omega_{\text{M}}$, $\sigma_{8}$, $A_{\text{SN1}}$,
$A_{\text{SN2}}$, $A_{\text{AGN}}$} for all 5 input parameters of each
simulation.

The marginal posterior over a single parameter $\theta_{i}$ (out of e.g. all 5
we constrain) can be defined as:

$p(\theta_{i}|\vec{C})=\int_{\theta}p(\theta_{1},\theta_{2},...\theta_{n}|\vec{C})d\theta_{1}...d\theta_{i-1}d\theta_{i+1}...d\theta_{n}$
(4)

The marginal posterior describes the probability that a simulation (and its
array of clustering measurements) were created with a particular combination
of parameters $\vec{\theta}$. Here with CAMELS-SAM and in many applications of
CAMELS, we give the network completely flat priors, or no prior knowledge of
the underlying distribution of parameters (e.g. the measurement of cosmology
from Planck Collaboration et al. 2016). The estimated mean of the marginal
posterior for a given parameter $\theta_{i}$ is:

$\mu_{i}(\vec{C})=\int_{\theta_{i}}p(\theta_{i}|\vec{C})\theta_{i}d\theta_{i}$
(5)

The estimated standard deviation of the marginal posterior for a given
parameter $\theta_{i}$ is:

$\sigma_{i}(\vec{C})=\int_{\theta_{i}}p(\theta_{i}|\vec{C})(\theta_{i}-\mu_{i})^{2}d\theta_{i}$
(6)

The goal of a neural network is to learn the posterior accurately enough that
the mean $\mu_{i}$ and standard deviation $\sigma_{i}$ it predicts are
consistent with estimated posterior values given the input parameters. Our
neural network here assumes a single-peak posterior with one mean and standard
deviation. The actual marginal posterior may not have these properties; for
example, there may be a degenerate combination of our parameters that yields
very similar clustering measurements. In these cases, we can expect a large
standard deviation measurement that will attempt to cover the multiple peaks
in the posterior.

For a neural network to learn a posterior, it requires a loss function to
measure its performance (i.e. calculate the gradients it uses to update the
weights between neurons in order to eventually converge on values closest to
the true ones). We perform both parameter regression with a standard mean-
squared error (MSE) validation criterion, and likelihood-free inference (LFI)
with the method from Jeffrey & Wandelt (2020), updated for CAMELS in
Villaescusa-Navarro et al. (2021c). Our parameter regression is a fast and
straightforward way to measure the mean of the posterior and therefore roughly
approximate the network’s accuracy, while the latter trains for longer to also
measure the posterior’s standard deviation. Finally, our loss functions are
assessed over a given batch of input data, with a batch size of
N${}_{\text{batch}}=64$ as default, meaning that sixty-four random simulations
in the training set are trained at a time per node worker.

The MSE loss function141414A default option in Pytorch (Paszke et al., 2019)
we use simply measures the mean squared error between each element of the
neural network’s predictions and its true value, and is built to approximate
the mean of the marginal posterior (even if non-Gaussian). For a given batch
size, the MSE loss on the model’s predictions is:

$\mathcal{L}_{\text{MSE}}=\sum_{i=1}^{5}\Bigg{(}\frac{1}{\text{N}_{\text{batch}}}\sum_{j\in{\text{N}_{\text{batch}}}}(\theta_{i,j}-\mu_{i,j})^{2}\Bigg{)},$
(7)

where $\mu_{i,j}$ is the network’s prediction for the mean of parameter $i$’s
posterior for simulation $j$; and $\theta_{i,j}$ is the true input value of
parameter $i$ for simulation $j$. We note that this loss sums over all the
cosmological and astrophysical parameters being constrained, meaning it
attempts to measure the marginal posterior means for each parameter at once.

With the LFI loss function, our goal is to instead get the neural network
output to converge on both the mean and standard deviation of the marginal
posterior. With slight modifications to the loss function presented in Jeffrey
& Wandelt (2020), one can define the loss function so that the neural network
outputs converge over both $\mu_{i}$ and $\sigma_{i}$ (see Villaescusa-Navarro
et al. 2021c §3.1.2):

$\begin{split}\mathcal{L}_{\text{LFI}}=&\sum_{i=1}^{5}\log\Bigg{(}\sum_{j\in{\text{N}_{\text{batch}}}}(\theta_{i,j}-\mu_{i,j})^{2}\Bigg{)}\\\
&+\sum_{i=1}^{5}\log\Bigg{(}\sum_{j\in{\text{N}_{\text{batch}}}}\Big{(}(\theta_{i,j}-\mu_{i,j})^{2}-\sigma_{i,j}^{2}\Big{)}^{2}\Bigg{)}\end{split}$
(8)

As described in Villaescusa-Navarro et al. (2021c), the effect of this loss
function is that the neural network ignores noisy parameters until it has
learned the well-determined parameters first. This is important in
circumstances where a feature is very sensitive to specific parameters and not
at all for others (e.g. the CAMELS dark matter density field, which very
mildly detects TNG/SIMBA astrophysical parameter variations). This LFI loss
function removes the dependence on the overall scale in the scatter of a
parameter with the included logarithm, therefore inverse-variance weighting
the combination of gradients from the different terms compared to the MSE loss
in Eq. 7. The clustering of galaxies is dominated by that of dark matter
halos, meaning the influence of the cosmological parameters will likely be
much stronger than any of the SC-SAM astrophysical parameters. Therefore, we
prioritize the use of the LFI loss method throughout this work.

We use the MSE loss only on the single-redshift exploration in §4.3 due to its
faster training time, and LFI everywhere else for more detailed constraints.
We generally find the MSE and LFI results are broadly consistent, though the
LFI are often slightly more precise, likely due to the introduction of the
logarithm discussed above.

We do, however, leverage the root mean square error (rMSE) to gauge the
accuracy of our neural network predictions. In Tables LABEL:table:oneatatime-
LABEL:table:ClusteringStats_CosmoConstraints reporting our results, we list
two types of errors, ‘rMSE’ and $\bar{\sigma}$. $\bar{\sigma}_{i}$ is the mean
across the test set’s $\sigma_{i}$ values after going through the best-
performing neural network under the LFI loss, and reflects the actual
1$\sigma$ error on the constraint. ‘rMSE’ refers to the rMSE calculated on the
mean $\mu_{i}$ values from the relevant loss function, and roughly measures
the accuracy of the predictions. Following the common definitions, the rMSE
and $\bar{\sigma}_{i}$ for the predicted parameter $\theta_{i}$ across the
test set of length N${}_{\text{test}}$ are:

$\text{rMSE}_{i}=\sqrt{\frac{1}{\text{N}_{\text{test}}}\sum_{j\in{\text{N}_{\text{test}}}}(\theta_{i,j}-\mu_{i,j})^{2}}$
(9)

$\bar{\sigma}_{i}=\frac{1}{\text{N}_{\text{test}}}\sum_{j\in{\text{N}_{\text{test}}}}\sigma_{i,j}$
(10)

####  3.2.3 Tests for accuracy

We generally find that the rMSE calculated on the LFI-loss predicted means is
often very close to the LFI-loss predicted error, though that often rMSE
$<\bar{\sigma}$. To confirm the general behavior of our LFI-loss networks is
accurate and is not under-predicting the error, we implement a few simple
quantitative tests.

First, we include a calculation of each individual test simulation’s
‘$Z$-value’ to exclude the rare outlier(s), according to the behavior of the
entire test set’s LFI predicted means. We begin by calculating the absolute
value difference between actual and predicted parameter $i$ values across the
whole test set of length $t$:

$\vec{\phi}_{i}=\\{\phi_{i,1},\phi_{i,2},...,\phi_{i,t}\\},\quad\phi_{i,j}=|\theta_{i,j}-\mu_{i,j}|$
(11)

Then for each simulation $j$ in the test set, the ‘$Z$ value’ for a parameter
$i$ is:

$Z_{i,j}=\frac{\Big{(}\phi_{i,j}-\textrm{mean}(\vec{\phi})\Big{)}}{(\text{std}(\vec{\phi}))}$
(12)

We remove simulations in the test set with $Z>6$ before calculating the rMSE
on the LFI predicted means that are listed in future tables; visually, these
are the extreme outliers clear to the eye whose predictions are often
unrealistic or beyond the bounds of the parameters (e.g. see
A${}_{\text{AGN}}$ for Figure 19b). Though very rare to find an outlier of
$Z>6$, none of the selections we have implemented had more than a single
outlier in the test set, and we have found it is nearly always the same LH
simulation. We remind readers that the best-performing neural network
minimizes the loss across all simulations in the given set; therefore, the
network as a whole still performs well across the parameter space even with
the outlier. This simple ‘$Z$-value’ test helps generate rMSE assessments for
the networks that are reflective of their visual performance in our Figures.

Next, we examine whether our neural networks are under- or over-estimating the
$\bar{\sigma}$ errors, as we find the rMSE is often slightly larger. As did
Jeffrey et al. (2022) for their moment networks, we define for a given
parameter $\theta_{i}$ and simulation $j$ $\chi_{j}(\theta_{i})$ as:

$\chi_{j}(\theta_{i})=\frac{\theta_{i,j\ \text{true}}\
-\mu_{i,j}}{\sigma_{i,j}}$ (13)

The distribution of $\chi(\theta_{i})$ can help qualify the accuracy of the
neural network. Parameters that find accurate constraints–or, that follow the
1:1 slope on our figures, regardless of how closely–have $\chi(\theta_{i})$
roughly consistent to a Gaussian centered at zero with a variance of 1. For
parameters that cannot be constrained by the network, e.g. the SC-SAM
parameters in halo-mass selections, the distributions of $\chi(\theta_{i})$
are either flat distributions within $-2<\chi(\theta_{i})<2$, or whose peak is
$\chi(\theta_{i})\approx\pm 1.5$.

In Appendix B’s Figure 16, we plot the distributions of $\chi(\theta_{i})$ for
the two types of mass-selected clustering in Figures 4 and 5 across all 5
parameters. We find that our good constraints, even if their $\bar{\sigma}$
trends slightly smaller than rMSE, have $\chi(\theta_{i})$ consistent with
neural networks that are accurate and not over- or under- predicted. In Tables
where relevant, we list both the rMSE and $\bar{\sigma}$ calculated for
various parameters; and we show only the rMSE errors in summary Figures 3 \-
11 for clarity.

####  3.2.4 Architecture

We used the Optuna package (Akiba et al., 2019) to quickly train and validate
1D:1D fully-connected neural networks, all while identifying the best-
performing hyperparameters (e.g. number of hidden layers, neurons per layer,
learning rate, etc.). When using Optuna to create neural networks, we limit it
to:

  * •

take in a 1D array of normalized clustering values for a given galaxy
selection, and predict the 5 parameters of $\Omega_{\text{M}}$, $\sigma_{8}$,
A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, A${}_{\text{AGN}}$ (or a single one,
in the specific experiment of §5.3)

  * •

have no more than 5 layers total, each with no more than 1,000 neurons

  * •

assess 250-1,000 trials (i.e. sample neural networks within the hyperparameter
space), each with 500 training epochs per trial

  * •

use with Leaky ReLu activation in each layer

  * •

and use the Adam optimizer with $\beta$ parameters equal to 0.5, 0.999.

We often find 3-4 layers of 500-700 neurons do very well, testing more than
250 trials was often unnecessary, and most trials converge within 300 epochs
and after approximately 2-3 GPU days. The MSE loss training and hyper-
parameter selection often took less than 24 GPU hours, while the LFI loss
averaged 3-3.5 GPU days.

##  4 Constraining Cosmology with Clustering using CAMELS-SAM

Figure 3: A summary of the $1\sigma$ rMSE errors we find on our five
parameters when using ‘all’ clustering (the 2ptCF, VPF, and CiC at
$z=\\{0.0,0.1,0.5,1.0\\}$) across our various galaxy selections. The color of
the points indicates the property we selected by: halo mass (black), stellar
mass (magenta), SFR (cyan), sSFR (blue). The shape of the marker indicates
what density we down-sampled to after the selection: 0.001 $h^{3}$ cMpc-3
(downward pointing triangle), 0.005 $h^{3}$ cMpc-3 (upward pointing triangle),
or no density down-sampling (star). The plus sign markers in deeper colors
indicate the rMSE constraints found by having a neural network focus only on
the single parameter (see Table LABEL:table:oneatatime); these symbols have
been slightly shifted left for clarity. The grey dash-dotted lines indicate
the mean of the parameter’s prior space; errors around this value indicate
poor to no constraints found by the neural networks. The density plotted on
the $x$-axis is the $z=0$ value averaged across the 5 CV simulations after
applying each selection, but before randomly down-sampling. See Tables
LABEL:table:GlxSelections_Constraints, LABEL:table:GlxSelections_noDS, and
LABEL:table:oneatatime for complete details.

In this section, we explore the constraints on the cosmological parameters
$\Omega_{\text{M}}$ and $\sigma_{8}$ that we obtain using clustering
statistics from the CAMELS-SAM simulation suite. In §4.1, we explore how the
constraints respond to various galaxy selections, especially between halo and
stellar mass. In §4.2, we test how our cosmological constraints respond to
including density down-sampling or not. Finally, in §4.3, we examine the
constraints at each of the individual redshifts we probe between $0<z<1$.

Throughout this section, our Figures and Tables will often contain the results
for constraints on the SC-SAM astrophysical parameters; those will be reported
and discussed independently in the following §5. For the best comparison, we
compare only the results of ‘all’ clustering, which includes the VPF, 2ptCF,
and CiC at $z={0,0.1,0.5,1.0}$. The 2ptCF and VPF are measured between roughly
$1<R<40$ cMpc, the CiC at select distance scales within that range (see Table
LABEL:table:CiCRadii). See §6 for statistic-by-statistic comparisons, and §4.3
for redshift-by-redshift comparisons.

###  4.1 Cosmological Constraints: Comparing Galaxy Selections

The SC-SAM properties we select upon are halo mass, stellar mass, star
formation rate, and specific star formation rate. In this section, we
specifically focus on cosmological constraints. In §5, we shift to discussing
the constraints on the SC-SAM astrophysical parameters. Throughout both
sections, we will first describe and report the results of our experiments,
and then shift to discussing their meaning and significance.

In Table LABEL:table:GlxSelections_Constraints, we list the various selections
applied: the SC-SAM galaxy property, the cut-off value applied, and the down-
sampling density. We then list for each of the 5 parameters: the rMSE error
across the entire test set, and the mean $1\bar{\sigma}$ error estimation from
the LFI loss method across the test set. Good fits are low-value and show a
tight 1:1 slope; poor fits are often either flat and centered around the mean
of the distribution that the neural network was given. For cosmology, poor
fits correspond to an rMSE or $\bar{\sigma}$ of 0.1.

Figure 3 summarizes these results visually, and also includes constraints that
will be discussed in §4.2, where we do not randomly down-sample to a specific
number density before measuring the galaxies’ clustering.

####  4.1.1 Dark Matter Halo vs. Stellar Mass Selections

In Figure 4, we show the best constraints we find on $\Omega_{\text{M}}$ and
$\sigma_{8}$, from ‘all’ clustering of halos with halo mass greater than
$2\times 10^{11}$ M⊙ (or log10 M${}_{\text{halo}}>11.3$ M⊙), randomly down-
sampled to a density of 0.005 $h^{3}$ cMpc-3. The best-performing neural
network produces $\Omega_{\text{M}}$ predictions accurate to rMSE = 0.014, or
approximately 5$\%$ about the fiducial value $\Omega_{\text{M}}=0.3$. Our LFI
method also measures an average 1$\sigma$ standard deviation error of
$\bar{\sigma}=0.014$ on $\Omega_{\text{M}}$. For $\sigma_{8}$, the rMSE error
of the mean values across the test set is 0.032 (4$\%$ for $\sigma_{8}=0.8$),
while $\bar{\sigma}$ is 0.024 (3$\%$).

We also probe a much higher halo mass selection of $1.2\times 10^{12}$ M⊙ with
a lower density down-sampling of 0.001 $h^{3}$ cMpc-3, and find comparable
though slightly worse constraints (likely due to Poisson noise introduced from
the lower density). We take the tightest constraints from our dark matter
selections as the ‘best’ throughout this work.

Figure 4: Our best cosmological constraints with CAMELS-SAM, using ‘all’
N-body only halo clustering and the LFI loss function. Here we selected halos
with mass greater than $2\times 10^{11}$ M⊙, down-sampled to a density of
0.005 $h^{3}$ cMpc-3 at $z=\\{0.0,0.1,0.5,1.0\\}$.

Of the SC-SAM galaxy property selections we test, stellar mass can be expected
to give the tightest cosmological constraints. The stellar and halo masses of
galaxies are the most correlated of the selections we probe, meaning the tight
constraints from halo mass clustering will likely also be seen somewhat in
stellar mass.

In Figure 5, we show the constraints on $\Omega_{\text{M}}$ and $\sigma_{8}$
from ‘all’ clustering at $z=\\{0.0,0.1,0.5,1.0\\}$ of galaxies with stellar
mass greater than $1\times 10^{9}$ M⊙, randomly down-sampled to 0.005 $h^{3}$
cMpc-3. Other stellar mass selections we also tested include stellar mass
greater than $1\times 10^{10}$ M⊙ to a density of 0.001 $h^{3}$ cMpc-3; and
stellar mass greater than $2\times 10^{10}$ M⊙ to a density of 0.001 $h^{3}$
cMpc-3. We found the lower-threshold and higher down-sampling density
selection yielded the tightest constraints on cosmology.

The best-performing neural network, using the clustering of galaxies with
stellar mass greater than $1\times 10^{9}$ M⊙, produces $\Omega_{\text{M}}$
predictions accurate to rMSE = 0.02, or approximately 7$\%$ about the fiducial
value $\Omega_{\text{M}}=0.3$. The LFI loss measures an average 1$\sigma$
standard deviation error of $\bar{\sigma}=0.014$ on $\Omega_{\text{M}}$, the
same value as it found with dark-matter only clustering. For $\sigma_{8}$, the
rMSE error of the mean values across the test set is 0.038 (approximately
5$\%$ for $\sigma_{8}=0.8$), while $\bar{\sigma}=0.021$ (3$\%$).

Figure 5: Our best cosmological constraints using a stellar mass selected
sample and the LFI loss function. Here, we selected SAM galaxies with stellar
mass greater than $10^{9}$ M⊙, down-sampled to a density of 0.005 $h^{3}$
cMpc-3 at $z=\\{0.0,0.1,0.5,1.0\\}$.  Figure 6: Comparing how well a neural
network trained on the clustering of a dark matter halo mass selected sample
can predict cosmological parameters for a stellar mass-selected sample. This
network was trained with ‘all’ clustering at $z=\\{0.0,0.1,0.5,1.0\\}$ for
dark matter halos with dark matter halo mass greater than $2\times 10^{11}$
M⊙, down-sampled to a density of 0.005 $h^{3}$ cMpc-3. Blue circles show the
results of giving the network the test set for the same type of clustering
with which it was trained and validated. Red stars show the results from
instead evaluating the best-performing model on the test set of simulations
using the clustering of SAM galaxies of stellar mass greater than $1\times
10^{9}$ M⊙, down-sampled to the same density of 0.005 $h^{3}$ cMpc-3.

This section has so far focused on the best cosmological constraints we find
from clustering of samples with halo mass and stellar mass selection criteria.
However, exactly how much information about cosmology is contained within the
clustering of SAM galaxies? We can probe the relationship between halo and
galaxy clustering in our sample with a simple experiment: if we trained a
neural network to measure cosmology with only the clustering of dark matter
halos, how well can it predict cosmology when tested using galaxy clustering
instead?

Figure 6 shows the results of this experiment. We first train a neural network
with ‘all’ clustering for SAM galaxies with host dark matter halo mass greater
than $2\times 10^{11}$ M⊙, down-sampled to a density of 0.005 $h^{3}$ cMpc-3.
As expected, the network finds tight constraints if it is tested with the same
type of dark matter only clustering. Notably, the LFI loss gives very small
1$\sigma$ error measurements that are consistent with the predictions on the
means, for dark matter only clustering: $\Omega_{\text{M}}$ rMSE = 0.014
(4.7$\%$ about the fiducial $\Omega_{\text{M}}=0.3$) and $\sigma_{8}$ rMSE =
0.032 (4$\%$ about the fiducial $\sigma_{8}=0.8$).

We then have the best-performing model try to measure the cosmology when given
only the clustering of SAM galaxies of stellar mass greater than $1\times
10^{9}$ M⊙, down-sampled to the same density of 0.005 $h^{3}$ cMpc-3. We note
that this stellar mass selection was chosen because it performed the best of
those we tried, and not through any connection between the halo mass above and
the stellar mass of galaxies within them (making this a more challenging
problem for the neural network). We carry out exactly the same calculations,
only changing the objects in the test set. The constraints are less precise,
but still fairly accurate: $\Omega_{\text{M}}$ rMSE = 0.062 (21$\%$) and
$\sigma_{8}$ rMSE = 0.059 (7.4$\%$). The LFI errors remain very small,
consistent with what the network achieved on its training set. We discuss the
significance and lessons of this experiment later in §4.1.4.

(a) Constraints from ‘all’ clustering at $z=\\{0.0,0.1,0.5,1.0\\}$ for
galaxies with stellar mass greater than $2\times 10^{10}$ M⊙ and down-sampled
to a density of 0.001 $h^{3}$ cMpc-3.

(b) Constraints from ‘all’ clustering at $z=\\{0.0,0.1,0.5,1.0\\}$ for
galaxies with star formation rate greater than 1.25 M⊙ yr-1 and down-sampled
to a density of 0.001 $h^{3}$ cMpc-3.

(c) Constraints from ‘all’ clustering at $z=\\{0.0,0.1,0.5,1.0\\}$ for
galaxies with specific star formation rate greater than 0.2 Gyr-1 and down-
sampled to a density of 0.005 $h^{3}$ cMpc-3.

Figure 7: Comparing how selections on basic galaxy properties generated by the
SC-SAM affect the predictive power of a neural network trained on galaxy
clustering: (a) stellar mass, (b) star formation rate, and (c) specific star
formation rate. Detailed quantitative comparisons can be found in Table
LABEL:table:GlxSelections_Constraints. Outliers (red stars, often out of
range) are simulations in the test set whose ‘$Z$-value’ (Eq. 12) are greater
than 6; see §3.2.3 for more details.

####  4.1.2 SFR and sSFR Selections

Next, we explore the constraints obtained when galaxies are selected via their
star formation rate (SFR) and specific star formation rate (sSFR, or SFR
divided by stellar mass). In this Section, we focus on cosmological
constraints when using these selections; see §5.1 for the constraints on the
SC-SAM parameters controlling baryonic processes of stellar and AGN feedback.
Our results are summarized in the bottom panels of Figure 3, and some examples
of these constraints are shown in Figure 7.

Much like with stellar mass, our experiments here test both the selection
threshold, and the number density to which we randomly down-sample. Under the
set of choices we have adopted so far, with ‘all’ clustering statistics at
$z=\\{0.0,0.1,0.5,1.0\\}$ and random down-sampling to a fixed number density,
we probe SC-SAM galaxies with SFR $>0.2$ M⊙ yr-1 randomly sampled to
$\mathcal{N}=0.005\ h^{3}$ cMpc-3, and also SFR $>1.25$ M⊙ yr-1 randomly
sampled to $\mathcal{N}=0.001\ h^{3}$ cMpc-3. For sSFR, we probe SC-SAM
galaxies with sSFR $>0.1$ Gyr-1 and sSFR $>0.2$ Gyr-1, both randomly sampled
to $\mathcal{N}=0.005\ h^{3}$ cMpc-3.

For SFR and sSFR, errors on the cosmological parameters span 9-15% for
$\Omega_{\text{M}}$, and 5-8% for $\sigma_{8}$ (though we note the high
threshold, low density SFR selection finds essentially no constraints). The
tightest constraints on cosmology consistently come from stellar mass.

Table 4: Constraints for cosmological and astrophysical parameters across different SC-SAM galaxy selections with ‘all’ galaxy clustering statistics at $z=\\{0.0,0.1,0.5,1.0\\}$, when down sampling to a single number density (in $h^{3}$ cMpc -3).  Galaxy Selections | $\Omega_{\text{M}}$ | $\sigma_{8}$ | A${}_{\text{SN1}}$ | A${}_{\text{SN2}}$ | A${}_{\text{AGN}}$
---|---|---|---|---|---
Property |  Value |  Density |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$
M${}_{\text{halo}}$, log10M⊙ |  11.3 |  0.005 |  0.014 |  0.014 |  0.032 |  0.024 |  N/A |  N/A |  N/A |  N/A |  N/A |  N/A
… |  12.08 |  0.001 |  0.019 |  0.016 |  0.064 |  0.040 |  N/A |  N/A |  N/A |  N/A |  N/A |  N/A
M${}_{\text{stellar}}$, log10M⊙ |  9.0 |  0.005 |  0.02 |  0.014 |  0.038 |  0.021 |  0.908 |  0.777 |  0.812 |  0.522 |  0.995 |  0.962
… |  10.0 |  0.001 |  0.032 |  0.03 |  0.054 |  0.049 |  0.793 |  0.717 |  0.965 |  0.969 |  1.058 |  1.003
… |  10.3 |  0.001 |  0.037 |  0.026 |  0.064 |  0.053 |  0.753 |  0.621 |  1.013 |  0.881 |  0.986 |  0.923
SFR, M⊙ yr-1 |  0.2 |  0.005 |  0.03 |  0.033 |  0.057 |  0.051 |  0.78 |  0.562 |  0.699 |  0.621 |  1.111 |  1.112
… |  1.25 |  0.001 |  0.047 |  0.03 |  0.102 |  0.103 |  0.821 |  0.721 |  1.008 |  0.932 |  1.029 |  0.963
sSFR, log10yr |  -10.0 |  0.005 |  0.028 |  0.027 |  0.055 |  0.043 |  0.878 |  0.956 |  0.867 |  0.826 |  1.051 |  1.073
… |  -9.7 |  0.005 |  0.027 |  0.025 |  0.043 |  0.037 |  1.029 |  0.936 |  0.803 |  0.570 |  1.077 |  1.024

####  4.1.3 Discussion: Selections for Cosmological Constraints

The good accuracy and precision found on $\Omega_{\text{M}}$ (5-7%) and
$\sigma_{8}$ (3-5%) with SC-SAM stellar mass-selected clustering are quite
encouraging. First, we reach accuracy comparable to the dark matter only
clustering (5% on $\Omega_{\text{M}}$, 3-4% on $\sigma_{8}$), even with the
introduction of an astrophysical model. Secondly, it is heartening to see that
though this neural network was trained on simulations with a broad range of
feedback from the SAM baryonic prescriptions, and was asked to constrain all 5
parameters at once, it is still able to obtain robust cosmological constraints
and focus on their strong influence on clustering.

Of the basic galaxy properties we select by, stellar mass tends to give
slightly better constraints across the board. This is expected, as dark matter
halo mass clustering would be expected to give the best constraints, and
stellar mass tracks halo mass the most closely of the selections we probe. SFR
provides good constraints on $\Omega_{\text{M}}$, but interestingly much worse
constraints on $\sigma_{8}$, even obtaining no constraint at the strongest
selection of SFR $>1.25$ M⊙ yr-1 with $\mathcal{N}=0.001\ h^{3}$ cMpc-3.
Specific SFR shows good constraints on both cosmological parameters,
benefiting from the simple combination of stellar mass and star formation.

We also find that more extreme selections on these parameters have mixed
effects, occasionally occluded by the necessity of a lower down-sampling
density and the introduction of more Poisson noise. For example, increasing
the stellar mass threshold at $\mathcal{N}=0.001\ h^{3}$ cMpc-3 from $1\times
10^{10}$ M⊙ to $2\times 10^{10}$ M⊙ has inconclusive effects on
$\Omega_{\text{M}}$ (increasing rMSE but decreasing $\bar{\sigma}$). However,
increasing the sSFR threshold at $\mathcal{N}=0.005\ h^{3}$ cMpc-3 from 0.1
Gyr-1 to 0.2 Gyr-1 improves both cosmological constraints.

Finally, there is a possible trend to obtain better constraints for
simulations with parameter values —notably $\Omega_{\text{M}}$ and
A${}_{\text{SN2}}$— that yield higher densities across the suite (i.e. better
constraints moving rightwards on the $x$-axis of Figure 3). Why would
$\Omega_{\text{M}}$ be better constrained with the clustering of galaxies
under a selection that yields higher densities, especially for stellar mass? A
lower stellar mass threshold combined with our density down-sampling means
that the objects whose clustering we measure will tend to be lower mass, and
therefore may better probe the general range of structure formation within the
simulations. Additionally, poorer constraints at higher stellar mass may come
from a combination of more sensitivity to Poisson noise (at the lower
density), as well as a possible degeneracy with $\sigma_{8}$ that may mask the
effect of $\Omega_{\text{M}}$.

####  4.1.4 Discussion: Information in Galaxy vs. Halo Clustering

Finally, the §4.1.1 exercise of training on halo mass clustering and then
testing on stellar mass clustering is helpful in two ways. First, it humbly
reminds us that the precision of machine learning results is only as good as
the training samples. The LFI error predictions underestimate the true error
on the stellar mass clustering constraints, as the network simply assumes the
data it is given works the same way that the dark matter only clustering did.
Second, this exercise emphasizes the strength of using clustering: having
networks trained only on dark matter halos (which we theoretically understand
well and can simulate cheaply) can yield accurate (if imprecise) constraints
when given galaxy clustering. The neural network is retaining some
information, even when being tested on the ‘wrong’ thing, precisely because
galaxies generally follow the clustering of dark matter.

However, this is not to understate the significance of how much our
cosmological constraints improve when training directly with SAM galaxy
clustering (as in Figure 5). The clustering of dark matter halos dominates the
signal of galaxy clustering, but it does not completely describe it. Without
the additional information of the SC-SAM galaxy formation model, and without
allowing the neural network to marginalize over the effects and uncertainties
they introduce, there would be no improving the dark-matter-only constraints
and no path forward in this field. Galaxies and halos have complicated
relationships that are not well understood, yet even with the many variations
of the SC-SAM models that we include and the broad parameter space that our
suite covers, there is still more information to be learned than exists in a
simplistic galaxy-halo model. Information about cosmology is lost when the
clustering of galaxies is assumed to be related to dark matter in an overly
simplistic way, and much of it may be gained back when including a robust
galaxy formation model.

This exercise therefore reinforces the importance of one of CAMELS’ central
tenets as a project: to teach neural networks to marginalize over the
uncertain galaxy formation physics. Neural networks are able to learn the
complicated ways that galaxy physics affect clustering statistics, and yield
constraints within 3-10$\%$ on cosmology from galaxy clustering. Finally, this
exercise also strongly motivates implementing additional methods of linking
galaxy properties to dark matter (e.g. various HODs, more SAMs, sub-halo
abundance matching), to more fully explore the marginalizing power of the
neural networks.

###  4.2 Cosmological Constraints: Effect of Number Density Down-Sampling

Next, we examine how the choice to down-sample to a given density affects our
cosmological constraints. CiC and VPF are sensitive to number density and
encode its effect very strongly, so we chose to correct for its influence to
pull out the effect our parameters have on the large-scale structure more
clearly. Additionally, randomly down-sampling after a selection means the same
underlying clustering is maintained, and reduces the computational load of
clustering measurements for our 1000+ simulations. However, how do constraints
change if we allow number density to vary, and therefore allow the neural
networks to use that information for predictions?

To test constraints without our density down-sampling, we applied high
threshold cuts for select SC-SAM properties and measured the clustering on all
resulting galaxies. To reduce the computational load while still maintaining
robust number statistics for the sparsest volumes, we chose high value cuts
for our simulations that yielded at least several hundred to a few thousand
galaxies in more than 95$\%$ of the simulations. Table
LABEL:table:GlxSelections_noDS details our selections and the resulting
constraints, Figure 8 highlights some of the best-performing of these.

The only repeated selection from §4.1 is M${}_{\text{stellar}}>2\times
10^{10}$ M⊙. We also include M${}_{\text{stellar}}>7\times 10^{9}$ M⊙ as a
small test of threshold sensitivity, as well as SFR$>1$ M⊙ yr-1. These
selections yield galaxy densities mostly between $10^{-4}-10^{-2}\ h^{3}$
cMpc-3, which corresponds to as much as fifty to one hundred thousand galaxies
in a volume across the entire breadth of our parameter space. For more
grounded context, the CV simulations with the fiducial parameters yield ten to
twenty thousand galaxies under these selections. These selections therefore
likely have minimal Poisson error in the bulk of their clustering
measurements.

(a) Constraints from ‘all’ clustering at $z=\\{0.0,0.1,0.5,1.0\\}$ of SC-SAM
galaxies with stellar mass $>2\times 10^{10}h^{-1}$ M⊙, with no density down-
sampling.

(b) Constraints from ‘all’ clustering at $z=\\{0.0,0.1,0.5,1.0\\}$ of SC-SAM
galaxies with star formation rate $>1$ M⊙ yr-1, with no density down-sampling.

Figure 8: Exploring how not down-sampling to a fixed density affects constraints on cosmology and the SC-SAM feedback parameters, for ‘all’ clustering of galaxies selected by (a) stellar mass (magenta) or (b) star formation rate (cyan). Detailed quantitative comparisons can be found in Table LABEL:table:GlxSelections_noDS. Outliers (red stars, often out of range) are simulations in the test set whose ‘$Z$-value’ are greater than 6 (§3.2.3). Table 5: Constraints for cosmological and astrophysical parameters from different SAM galaxy selections with ‘all’ galaxy clustering statistics at $z=\\{0.0,0.1,0.5,1.0\\}$, when not correcting to a fixed number density. The density ranges give a rough idea of the number of galaxies passing each selection across the LH suite, and are in units of $h^{3}$ cMpc-3.  Galaxy Selections | $\Omega_{\text{M}}$ | $\sigma_{8}$ | A${}_{\text{SN1}}$ | A${}_{\text{SN2}}$ | A${}_{\text{AGN}}$
---|---|---|---|---|---
Property |  Value |  Density range |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$
SFR, M⊙ yr-1 |  1.0 |  1.9e-4 to 1.9e-2 |  0.033 |  0.03 |  0.076 |  0.065 |  0.648 |  0.676 |  0.874 |  0.886 |  0.889 |  0.758
M${}_{\text{stellar}}$, log10M⊙ |  9.845 |  2.4e-4 to 2.2e-2 |  0.025 |  0.021 |  0.036 |  0.030 |  0.648 |  0.549 |  0.747 |  0.612 |  1.048 |  0.857
… |  10.3 |  1e-4 to 1e-2 |  0.028 |  0.029 |  0.036 |  0.043 |  0.452 |  0.478 |  0.757 |  0.717 |  1.04 |  0.978

The results from the best-performing neural networks for two of these
selections are shown in Figure 8. The cosmological constraints found for these
three selections are detailed in Table LABEL:table:GlxSelections_noDS and are
included in the summary Figure 3 for direct comparison with earlier
selections. We discuss the effect of not down-sampling on the SC-SAM
astrophysical parameters in §5.2.

Without the normalization to a fixed number density across all simulations and
models, SFR- and stellar mass- selected clustering yields slightly improved
cosmological constraints. For the stellar mass selections, $\Omega_{\text{M}}$
constraints stay the same or improve slightly, while $\sigma_{8}$ constraints
notably improve by a few percent (dropping from approximately 6.3$\%$ to
4-5.5$\%$). For SFR selection, $\Omega_{\text{M}}$ constraints are about as
good as when down-sampling is applied, when using the LFI loss. However,
removing the down-sampling greatly improves the $\sigma_{8}$ constraints from
the strong SFR selection. The clustering of all galaxies with SFR$>1$ M⊙ yr-1
yields $\bar{\sigma}=0.065$ (8.1$\%$ error), much improved over the
essentially unconstrained $\bar{\sigma}=0.103$ when taking a similarly strong
SFR cut and then randomly down-sampling. We cannot confirm how much of the
improvement comes from reduced Poisson noise, and how much comes from the
additional information contained in the predicted number density values.

The results of this experiment, at least with regard to cosmology, can be
interpreted in a mixed way: constraints are often improved by including number
density, especially for $\sigma_{8}$ when SFR selected samples are used,
though not a significant amount of information about $\Omega_{\text{M}}$ is
lost when randomly down-sampling to a fixed number density. The computational
effort required to measure so many galaxies’ clustering is not negligible,
meaning that perhaps for an initial assessment of cosmology, randomly down-
sampling could be a practical choice. We revisit this assessment when
analyzing the constraints on the SC-SAM astrophysical parameters in 5.2.

###  4.3 Cosmological Constraints: Effect of Redshift Choice

Next, we briefly examine the effect of combining multiple redshifts when
training our neural networks151515We note that e.g. Nicola et al. (2022) and
Uhlemann et al. (2020) have also included a few redshifts at a time in their
analyses.. With real galaxy observations, it would be difficult to make
identical galaxy selections at multiple redshifts, especially those as
observationally different as $z=0$ vs. $z=1$. Therefore, we create single-
redshift samples at $z=\\{0.0,0.1,0.5,1.0\\}$ to examine the constraints at
single redshifts for a few types of selections.

At each redshift, we down-sample to our selected number density, meaning each
redshift measures the clustering of a different randomly selected sample of
objects. We also include more radii in the CiC distribution, as using one
redshift at a time allows for more spatial scales to be included (see Table
LABEL:table:CiCRadii for full details; in essence, we sample more distance
scales for more $n$ count-in-cells measurements). We use the same VPF and
2ptCF as in previous sections.

Table 6: Constraints from clustering statistics at a single redshift; using the same VPF and 2ptCF as with all four redshifts, but more of the CiC distribution (see §3.2.1, §4.3, and Table LABEL:table:CiCRadii). Masses are log10M⊙, and densities for each selection are indicated with superscript symbols: ∗ for 0.001, † for 0.005 $h^{3}$ cMpc -3. ‘All’ here indicates the four combined redshifts.  Galaxy Selections | $\Omega_{\text{M}}$ | $\sigma_{8}$ | A${}_{\text{SN1}}$ | A${}_{\text{SN2}}$
---|---|---|---|---
Property |  Value |  $z$ |  rMSE |  rMSE |  rMSE |  rMSE
M${}_{\text{stellar}}$ |  10.3∗ |  0.0 |  0.057 |  0.072 |  0.716 |  0.987
… |  10.3∗ |  0.1 |  0.047 |  0.075 |  0.681 |  1.025
… |  10.3∗ |  0.5 |  0.048 |  0.097 |  0.613 |  1.069
… |  10.3∗ |  1.0 |  0.046 |  0.098 |  0.655 |  1.078
… |  10.3∗ |  All |  0.037 |  0.064 |  0.753 |  1.013
M${}_{\text{halo}}$ |  11.3† |  0.0 |  0.027 |  0.05 |  N/A |  N/A
… |  11.3† |  0.1 |  0.026 |  0.056 |  N/A |  N/A
… |  11.3† |  0.5 |  0.023 |  0.057 |  N/A |  N/A
… |  11.3† |  1.0 |  0.021 |  0.079 |  N/A |  N/A
… |  11.3† |  All |  0.020 |  0.032 |  N/A |  N/A
Figure 9: Similar to Figure 3, but instead summarizing the constraints of
Table LABEL:table:SingleRedshift_rMSE, §4.3, and Appendix C, illustrating the
effects of using clustering constraints at a single redshift vs. four
redshifts combined (All).

Table LABEL:table:SingleRedshift_rMSE and summary Figure 9 present the
calculated rMSE errors across the test set for all selections probed. We
discuss the SC-SAM parameter constraints when data from a single redshift at a
time is used in §5.2.1. We place the detailed neural network prediction
figures for this experiment in Appendix C, as they are quite similar to
figures already shown. Figures 17 and 18 focus on each redshift’s (expanded)
‘all’ clustering for dark matter halos with halo mass greater than $2\times
10^{11}$ M⊙ and down-sampled to 0.005 $h^{3}$ cMpc-3; and Appendix C Figure 19
focuses on each redshift’s (expanded) ‘all’ clustering for galaxies with
stellar mass greater than $2\times 10^{10}$ M⊙ and down-sampled to 0.001
$h^{3}$ cMpc-3.

As expected, combining the information from multiple redshifts improves the
constraints. Networks trained on clustering from a single redshift find
slightly worse constraints, with fractional errors increasing by at least
5$\%$ on $\Omega_{\text{M}}$ and 1-5$\%$ on $\sigma_{8}$. We note that one or
two extreme outlier test simulations are more common in the neural networks
trained on single redshifts (see the visible red star outliers in several of
the SC-SAM parameters in Figures 19c and 19b). Constraints on
$\Omega_{\text{M}}$ and $\sigma_{8}$ worsen by at least a few percent within
the two types of mass selection we probe for individual redshifts. The
individual redshifts have comparable rMSE errors across all parameters. There
may be a degradation in $\Omega_{\text{M}}$ constraints with decreasing
redshift, but the pattern is not conclusive.

We remind readers that, for the single redshift experiment, the VPF and 2ptCF
were kept identical, but the CiC distributions were expanded to use the
maximum array length the neural network can handle (§3.2.1 and Table
LABEL:table:CiCRadii). Interestingly, we note here that including clustering
at more redshifts improves the constraints much more than including more of
the CiC information at a single redshift. As Table
LABEL:table:SingleRedshift_rMSE and summary Figure 9 show, the constraints
from ‘all’ clustering at four redshifts always improve for $\sigma_{8}$ and
nearly always improve for $\Omega_{\text{M}}$, often dropping several
percentage points. However, this may be due to the neural network
independently learning the growth factor (a factor that determines the growth
of density perturbations as a function of cosmology, especially
$\Omega_{\text{M}}$). We do not find strong evidence for a particular redshift
outperforming another, though perhaps $\Omega_{\text{M}}$ constraints worsen
slightly with decreasing redshift.

Though we have not run experiments for each iteration and combination of
clustering selections, these initial results lend credence to prioritizing
getting samples of the same type of galaxies at different redshifts rather
than measuring more detailed clustering statistics for a single sample, or
measuring clustering across a broader range of scales. For example, a galaxy
sample for which this is likely feasible in the near future is H$\alpha$ and
[OIII] emission line galaxies confirmed with photometric redshifts, either
from large scale structure surveys like DES (Abbott et al., 2018), or
narrowband surveys like HiZELS (Khostovan et al., 2018) and LAGER (Khostovan
et al., 2020).

Finally, we note that each CAMELS-SAM simulation has halo and galaxy
information at 100 snapshots between $20<z<0$, and that we have only probed
$z<1$ in this initial work. We also re-emphasize the caveat that we did not
use exactly the same CiC for the single redshift training as we did with all
four redshifts (see Table LABEL:table:CiCRadii): we include more distributions
at a slightly larger range, in an attempt to leverage as much information as
we could give the simple 1D:1D network at a single redshift.

##  5 Constraining SC-SAM parameters for stellar and AGN feedback

Next, we explore in detail the constraints our neural networks obtain on the
SC-SAM parameters controlling baryonic processes related to stellar and AGN
feedback. All selections we tested were described in detail in the previous
section, and most of the results we examine were quantified in Tables
LABEL:table:GlxSelections_Constraints, LABEL:table:GlxSelections_noDS, and
LABEL:table:SingleRedshift_rMSE, and Figures 3, and 9.

###  5.1 Astrophysical Constraints: Comparing Galaxy Selections

First, we compare how different galaxy properties perform under the same type
of clustering–‘all’ statistics at $z=\\{0.0,0.1,0.5,1.0\\}$, and randomly
down-sampled to either $\mathcal{N}=0.001$ or $0.005\ h^{3}$ cMpc-3. See
upcoming §5.2 for a discussion about how constraints on the SC-SAM parameters
change with no down-selection and at individual redshifts, and §6.2 for how
individual clustering statistics perform for these parameters.

As introduced in §4.1, we select SC-SAM galaxies based on stellar mass, star
formation rate, and specific star formation rate. Here, we examine the
constraints that our experiments find on the SC-SAM astrophysical parameters
A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, and A${}_{\text{AGN}}$. We remind
readers that all neural networks have been asked to constrain all five
parameters at the same time (with the exception of the focused experiment in
the upcoming §5.3), and that our LFI loss function (described in §3.2.2) has
been used specifically for its strength in pulling out the influence of weaker
parameters. The results for constraints on the SC-SAM astrophysical parameters
are summarized in Figure 3 and Table LABEL:table:GlxSelections_Constraints.
Select neural network constraint examples can be seen in Figure 7. We note
that poor fits on the SC-SAM parameters correspond to rMSE and $\bar{\sigma}$
errors near 1.

Of the core selections with ‘all’ clustering with density down-sampling, we
find all selections find good constraints in different circumstances. Stellar
mass and sSFR do well at constraining the parameter A${}_{\text{SN2}}$
($0.5<\bar{\sigma}<0.8$), and find moderate ($0.6<\bar{\sigma}<0.8$, stellar
mass) or very poor ($\bar{\sigma}=1$, sSFR) constraints on A${}_{\text{SN1}}$.
Both are outperformed by SFR selections, especially at high density down-
samplings. All selections in this category find poor to no constraints on
A${}_{\text{AGN}}$. As we will explore later, removing the density down-
sampling vastly improves constraints on the SC-SAM parameters.

The high-density SFR selection does well at constraining both SC-SAM
A${}_{\text{SN}}$ parameters–not surprisingly, since those parameters control
the normalization and slope of the mass outflow rate driven by stellar
feedback, regulating star formation in galaxies. Specific SFR, as the simple
combination of stellar mass and star formation, unfortunately does not show
“the best of both worlds” and does not improve on the results from stellar
mass or SFR selections separately (and, as explored in §4.1.2, does not stand
out for cosmological constraints).

The poor to non-existent constraints on A${}_{\text{AGN}}$ may perhaps be due
to the fact that it only effects the properties of galaxies that are much more
massive than our selection limit (see Figure 15 in Appendix A). In §5.3, we
attempt to improve these astrophysical constraints by having the neural
networks learn one parameter at a time.

Finally, we note some trends on the constraints obtained with different galaxy
selections. As seen in Figure 3, the parameter A${}_{\text{SN2}}$ (like
$\Omega_{\text{M}}$) is better constrained with lower stellar mass and lower
SFR selections (higher number density), and the dependence on mass or SFR is
fairly strong. A${}_{\text{SN1}}$ appears to show the inverse behavior, with
better constraints from higher mass selections that yield fewer galaxies (but
less of a clear trend with SFR selection). Marginally better constraints on
A${}_{\text{AGN}}$ may also be obtained with higher mass (lower density)
selections, likely because this parameter has the greatest effect on the
highest mass (and therefore most star-forming) galaxies (e.g. Appendix A
Figure 15).

###  5.2 Astrophysical Constraints: Effect of Number Density and Redshift

Next, we report the effects of using clustering at one redshift, and not down-
sampling to a single number density, on the constraints on the astrophysical
parameters A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, and A${}_{\text{AGN}}$.

####  5.2.1 One Redshift vs. Multiple

Table LABEL:table:SingleRedshift_rMSE and Appendix C Figure 19 show that there
is no strong trend on the quality of the constraint on the astrophysics
parameters with redshift, much like what was found with the cosmological
parameters in §4.3.

A${}_{\text{SN2}}$ constraints are non-informative (i.e. they are close to the
mean of the prior) at the individual redshifts, and only somewhat constrained
with all four redshifts combined. The constraints on A${}_{\text{AGN}}$ remain
very poor at individual redshifts. A${}_{\text{SN1}}$ is still decently
constrained across redshifts. There is an interesting phenomenon of ‘all’ four
redshifts combined finding slightly worse rMSE constraints on
A${}_{\text{SN1}}$ than the individual redshifts, though we note the LFI loss
prediction finds $\bar{\sigma}=0.621$, slightly better than three of the four
constraints. For the SC-SAM supernova parameters, there therefore may be no
loss of constraining power when focusing on a single redshift. Further
analysis with other selections can confirm this is not due to changes in
Poisson noise due to the lower number density down-sampling.

####  5.2.2 Effect of Density Down-Sampling

Throughout our process of narrowing down what galaxy selections we would
undertake, we found that varying the SC-SAM parameters for stellar feedback
created large variations in the number of galaxies that pass a given stellar
mass or star formation rate cut. This is not surprising, especially with e.g.
the strong influence the A${}_{\text{SN}}$ parameters have on the stellar mass
functions, as seen in Appendix A. However, do they also affect the clustering
of galaxies?

We revisit summary Figure 3, Figure 8, and Table
LABEL:table:GlxSelections_noDS to assess the constraints found on the SC-SAM
parameters A${}_{\text{SN1}}$, A${}_{\text{SN2}}$, and A${}_{\text{AGN}}$.
Without down-selecting to fixed number density, we find significantly improved
constraints on the SC-SAM parameters for both SFR and stellar mass selected
samples: often decreasing $\bar{\sigma}$ by a tenth or more. Both types of
selections get imprecise but not inaccurate constraints on A${}_{\text{SN2}}$.
Stellar mass selected samples produce better constraints on
A${}_{\text{SN1}}$, while stellar mass and SFR selected samples show the first
signs of finding any constraint on the elusive A${}_{\text{AGN}}$ parameter.

Appendix A uses the 1P catalogs to explore how the SC-SAM parameters affect
key galaxy relationships at either very low or high values. We find that the
SN parameters show strong influence on relationships involving stellar mass,
especially the stellar mass function (SMF) and stellar mass-halo mass
function. The AGN parameter has the strongest influence at high halo masses.
Therefore, it is not surprising that including the additional information on
galaxy number density would improve the SC-SAM parameter constraints.

The VPF and CiC are quite sensitive to number density (the VPF especially,
essentially constraining it alone). Including the number density as additional
information for a neural network to leverage gives it a lot of knowledge about
the SMF, and therefore allows it to more easily learn the effect of the SC-SAM
parameters. This is perhaps why A${}_{\text{AGN}}$ is finally being
constrained at all: the VPF and CiC sense whatever small influence it may have
on galaxies’ stellar mass and sSFR, and the neural network has more
information to learn its relationships.

###  5.3 Astrophysical Constraints: Focused Neural Networks

We have explored how various galaxy selections and choices around clustering
statistics affect constraints on cosmological parameters and SC-SAM
astrophysics parameters when all are constrained at once. However, how much
improvement can we find on our constraints for the SC-SAM parameters around
stellar and AGN feedback by focusing on them one at a time?

####  5.3.1 Experiment: Focused NNs

We first confirm that this method works by isolating the cosmological
parameters; our constraints do not improve when a neural network focuses on
$\Omega_{\text{M}}$ or $\sigma_{8}$ alone given dark-matter only or stellar
mass-selected galaxy clustering. This likely indicates that the full
5-parameter predictions are accurately pulling out their influence. See Table
LABEL:table:oneatatime for detailed comparisons.

Next, from our various experiments, we determine which galaxy and clustering
selections may give the most useful constraints on the SC-SAM parameters. For
the A${}_{\text{SN}}$ parameters, we select ‘all’ clustering of SAM galaxies
with stellar mass greater than $2\times 10^{10}$ M⊙ across 4 redshifts (no
down-sampling). For A${}_{\text{AGN}}$, we select the ‘all’ clustering of all
SAM galaxies with star formation rate greater than 1 M⊙ yr-1 across 4
redshifts (also no down-sampling).

(a) Neural network predicts only A${}_{\text{SN1}}$. (b) Neural network
predicts only A${}_{\text{SN2}}$. (c) Neural network predicts only
A${}_{\text{AGN}}$.

Figure 10: Results from neural networks that attempt to predict one SAM
parameter at a time. These networks were given ‘all’ clustering statistics at
$z=\\{0.0,0.1,0.5,1.0\\}$ for all SAM galaxies with stellar mass greater than
$2\times 10^{10}$ M⊙ (a,b; magenta) or star formation rate greater than 1 M⊙
yr-1 (c; cyan), with no down-sampling to fixed number density. Predictions for
each parameter when all 5 parameters are constrained at once are given in
black with $\bar{\sigma}$ errors for comparison. The constraints do not
demonstrably improve with a focused neural network—this improves our
confidence that the neural networks fitting all 5 parameters at once are
extracting close to the full available information.

Our best constraints for the SC-SAM parameters when training networks to focus
on them alone are detailed in Figure 10 and Table LABEL:table:oneatatime. They
are also included in summary Figure 3 for easier comparison against the neural
networks that constrained all parameters at once (slightly darker plus signs).
The rMSE constraints A${}_{\text{SN1}}$ and A${}_{\text{SN2}}$ are only
slightly improved by individualized training. Similarly, the rMSE constraints
on A${}_{\text{AGN}}$ are slightly improved with the focused approach, which
we partially attribute to this parameter being the most subtle of the SC-SAM
parameters with weak influence (see Appendix A’s Figure 15). The LFI-predicted
errors $\bar{\sigma}$ when constraining all five parameters at once give very
similar or slightly better constraints than the focused parameter rMSEs.

Table 7: Constraints from learning the parameters one at a time. The neural networks are given ‘all’ clustering across $0<z<1$ for the following selections on galaxy properties. We list the rMSE of the LFI posterior means when predicting the parameters alone of alongside the other 4. Best-Performing Galaxy Selections | Alone | All 5
---|---|---
Param. |  Prop. |  Value |  $\mathcal{N}$ |  rMSE |  rMSE |  $\bar{\sigma}$
$\Omega_{\text{M}}$ |  M${}_{\text{halo}}$ |  2e11 M⊙ |  0.005 |  0.014 |  0.014 |  0.014
… |  M${}_{\text{star}}$ |  2e10 M⊙ |  N/A |  0.031 |  0.028 |  0.029
$\sigma_{8}$ |  M${}_{\text{halo}}$ |  2e11 M⊙ |  0.005 |  0.031 |  0.032 |  0.024
… |  M${}_{\text{star}}$ |  2e10 M⊙ |  N/A |  0.039 |  0.036 |  0.043
A${}_{\text{SN1}}$ |  M${}_{\text{star}}$ |  2e10 M⊙ |  N/A |  0.478 |  0.452 |  0.478
A${}_{\text{SN2}}$ |  M${}_{\text{star}}$ |  2e10 M⊙ |  N/A |  0.745 |  0.757 |  0.717
A${}_{\text{AGN}}$ |  SFR |  1 M⊙ yr-1 |  N/A |  0.766 |  0.889 |  0.758

####  5.3.2 Discussion: Focused NNs

The results of the focused neural networks did not necessarily yield the
desired outcome—stronger constraints on the SC-SAM astrophysical
parameters—but the exercise communicates the strength of our neural network
implementation. The ‘Focused NN’ results we summarize in Figure 3 are all
either nearly identical to or slightly worse than the $\bar{\sigma}$ that our
LFI loss predicts when predicting all five parameters at the same time. In
theory, focusing the entire breadth of a neural network to predict only one
parameter will lead to the tightest constraints the architecture is able to
find. This result therefore indicates that the LFI loss is behaving exactly as
advertised—it can, indeed, learn noisy and less sensitive parameters even in
the presence of strongly influential parameters.

As we describe in §3.2.2, the LFI loss function removes from the scatter of a
parameter the dependence of the overall space, which in practice in CAMELS and
CAMELS-SAM means it is able to learn parameters with relatively subtle
effects, such as A${}_{\text{AGN}}$, while also learning $\Omega_{\text{M}}$
and $\sigma_{8}$. Finally, we also ran focused neural networks for the
cosmological parameters, and found their constraints showed very little
difference from those obtained when constraining all five parameters,
additionally confirming the LFI loss method has optimized the constraints on
the strongest parameters.

##  6 Comparing Constraints from Different Statistics: 2ptCF vs. VPF vs. CiC

Throughout this work, we have obtained constraints using the combined results
of the two-point correlation function (2ptCF), count-in-cells (CiC), and the
Void Probability Function (VPF). This leads to natural questions: how much is
each statistic contributing? Is one better than the others for specific
constraints? What statistics are worth investing computational time into?

In this section, we compare the constraining power of each of the clustering
statistics that we have used. We train neural networks keeping all choices but
the clustering statistics the same. We test several galaxy selections, making
sure to keep the density, the radii tested, and redshifts selected
($z={0.0,0.1,0.5,1.0}$) the same for each clustering statistic. We compare
each clustering statistic’s constraints against the combination of ‘all’
clustering to determine which statistic may be dominating a given constraint
for a given selection.

Summary Figure 11 shows the 1$\sigma$ error predictions for
$\Omega_{\text{M}}$ and $\sigma_{8}$ (top half), and A${}_{\text{SN1}}$ and
A${}_{\text{SN2}}$ (bottom half). Tables
LABEL:table:ClusteringStats_CosmoConstraints and
LABEL:table:ClusteringStats_AsnConstraints contain all test result.

###  6.1 Clustering Statistic Experiment Setup

We test the constraints from individual clustering statistics across several
halo and galaxy selections: two thresholds of halo mass, three thresholds of
stellar mass, and one of SFR, all randomly down-sampled to either
$\mathcal{N}$=0.001 $h^{3}$ cMpc-3 or 0.005 $h^{3}$ cMpc-3. The exact details
of how we measure and prepare the 2ptCF, CiC, and VPF for our neural networks
are explained in §3.1.2-3.2.1. In this experiment, when giving a neural
network the independent clustering statistics, we pass exactly the same values
that go into ‘all’ clustering, allowing for fair comparison.

As representative examples, we plot the constraints from different clustering
statistics under two mass selections: the clustering of dark matter halos with
halo mass greater than $2\times 10^{11}$ M⊙, randomly sampled to a density of
0.005 $h^{3}$ cMpc-3; and the clustering of SAM galaxies with stellar mass
greater than $1\times 10^{9}$ M⊙, randomly sampled to a density of 0.005
$h^{3}$ cMpc-3. Figures 20, 21, and 22 showing these constraints are shown in
Appendix D, as the exact constraints closely resemble Figures already shown.

###  6.2 Results: Constraints by Statistic

Table 8: We compare the cosmology ($\Omega_{\text{M}}$ and $\sigma_{8}$) constraints from the best-performing neural networks across clustering statistics in Figures 20, 21, and 22. For $z=\\{0.0,0.1,0.5,1.0\\}$, we use either the two-point correlation function, count-in-cells, the Void Probability Function, or ‘all’ combined. The densities for each selection are indicated with superscript symbols: ∗ means a density of 0.001 $h^{3}$ cMpc -3, † means 0.005 $h^{3}$ cMpc -3. We note that for these parameters, a rMSE on the LFI-loss means or the mean standard deviation $\bar{\sigma}$ around 0.1 indicate imprecise and inaccurate constraints, with error bars that span half the parameter space and predictions that are flat and around the mean of the prior.  Clustering Statistic: | 2pt Correlation Function | Count-in-Cells | Void Probability Function | All Statistics
---|---|---|---|---
Galaxy Selections | $\Omega_{\text{M}}$ | $\sigma_{8}$ | $\Omega_{\text{M}}$ | $\sigma_{8}$ | $\Omega_{\text{M}}$ | $\sigma_{8}$ | $\Omega_{\text{M}}$ | $\sigma_{8}$
Property |  Value |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$
M${}_{\text{halo}}$, $\log_{10}$M⊙ |  11.3† |  0.016 |  0.016 |  0.035 |  0.035 |  0.015 |  0.016 |  0.042 |  0.031 |  0.013 |  0.019 |  0.041 |  0.039 |  0.014 |  0.014 |  0.032 |  0.024
… |  12.08∗ |  0.026 |  0.022 |  0.069 |  0.071 |  0.023 |  0.024 |  0.089 |  0.08 |  0.018 |  0.016 |  0.07 |  0.062 |  0.019 |  0.016 |  0.064 |  0.04
M${}_{\text{stellar}}$, $\log_{10}$M⊙ |  9.0† |  0.034 |  0.03 |  0.053 |  0.045 |  0.035 |  0.032 |  0.044 |  0.041 |  0.027 |  0.023 |  0.055 |  0.033 |  0.02 |  0.014 |  0.038 |  0.021
… |  10.0∗ |  0.04 |  0.04 |  0.067 |  0.06 |  0.038 |  0.042 |  0.08 |  0.068 |  0.041 |  0.038 |  0.078 |  0.066 |  0.032 |  0.03 |  0.054 |  0.049
… |  10.3∗ |  0.049 |  0.047 |  0.066 |  0.061 |  0.05 |  0.042 |  0.087 |  0.075 |  0.047 |  0.042 |  0.071 |  0.07 |  0.037 |  0.026 |  0.064 |  0.053
SFR, M⊙ yr-1 |  1.25∗ |  0.054 |  0.044 |  0.102 |  0.109 |  0.05 |  0.036 |  0.102 |  0.104 |  0.047 |  0.041 |  0.105 |  0.113 |  0.047 |  0.03 |  0.102 |  0.103
Table 9: We compare the SC-SAM supernova parameter constraints from the best-performing neural networks across clustering statistics in Figure 22. For $z=\\{0.0,0.1,0.5,1.0\\}$, we use either the two-point correlation function, count-in-cells, the Void Probability Function, or ‘all’ combined. The densities for each selection are indicated with superscript symbols: ∗ means a density of 0.001 $h^{3}$ cMpc -3, † means 0.005 $h^{3}$ cMpc -3. We note that for these parameters, rMSE errors for the LFI means around 1.0 indicate imprecise and inaccurate constraints, with error bars that span half the parameter space and predictions that are flat and around the mean of prior. Parameters with rMSE less than 0.8 tend to show a rough 1:1 relationship but with considerable 1$\sigma$ errors. We dive further into constraining the A${}_{\text{AGN}}$ parameter in §5.3. Clustering Statistic: | 2pt Correlation Function | Count-in-Cells | Void Probability Function | All Statistics
---|---|---|---|---
Galaxy Selections |  A${}_{\text{SN1}}$ |  A${}_{\text{SN2}}$ |  A${}_{\text{SN1}}$ |  A${}_{\text{SN2}}$ |  A${}_{\text{SN1}}$ |  A${}_{\text{SN2}}$ |  A${}_{\text{SN1}}$ |  A${}_{\text{SN2}}$
Property |  Value |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$ |  rMSE |  $\bar{\sigma}$
M${}_{\text{stellar}}$, $\log_{10}$M⊙ |  9.0† |  0.94 |  0.853 |  0.815 |  0.725 |  0.948 |  0.866 |  0.873 |  0.768 |  0.906 |  0.491 |  0.749 |  0.476 |  0.908 |  0.777 |  0.812 |  0.522
… |  10.0∗ |  0.923 |  0.878 |  1.029 |  1.079 |  0.924 |  0.893 |  1.052 |  1.083 |  0.791 |  0.683 |  0.988 |  1.034 |  0.793 |  0.717 |  0.965 |  0.969
… |  10.3∗ |  0.916 |  0.919 |  0.916 |  0.919 |  1.09 |  1.02 |  0.932 |  0.924 |  1.104 |  1.022 |  1.02 |  0.931 |  0.753 |  0.621 |  1.013 |  0.881
SFR, M⊙ yr-1 |  1.25∗ |  0.981 |  0.928 |  0.991 |  0.973 |  0.885 |  0.775 |  1.082 |  1.024 |  0.795 |  0.777 |  0.995 |  1.009 |  0.821 |  0.721 |  1.008 |  0.932
Figure 11: Similar to Figures 3 and 9, instead showing the constraints
obtained when different clustering statistics are used separately (see §6,
Tables LABEL:table:ClusteringStats_CosmoConstraints and
LABEL:table:ClusteringStats_AsnConstraints, and Appendix D). Dotted lines
connect the same galaxy selection scenarios for ease of comparison. The dash-
dotted lines indicate the mean of the prior, where constraints are not
informative.

We detail how the 2ptCF, CiC, and VPF constrain $\Omega_{\text{M}}$ and
$\sigma_{8}$ in Table LABEL:table:ClusteringStats_CosmoConstraints, and the
upper half of summary Figure 11. Table
LABEL:table:ClusteringStats_AsnConstraints focuses instead on the SC-SAM
parameters controlling stellar feedback in Equation 1. The lower half of
Figure 11 summarizes the constraints in context, and Figure 22 shows some
neural network results in detail. We note that the AGN feedback parameter
A${}_{\text{AGN}}$ from Equation 2 is generally difficult to constrain, so
that each clustering statistic alone (or even all combined) cannot constrain
it beyond the mean of the prior under the selections we probe.

Across the many selections we attempt, patterns emerge in how cosmological
constraints are affected by the choice of clustering statistic. First, as
expected, the combination of multiple clustering statistics nearly always
improves the constraints found, especially for the galaxy selections. The
clustering statistics often find similar constraints on $\sigma_{8}$. The
2ptCF tends to do best at $\sigma_{8}$ for lower density down-sampling
selections, which is not unexpected: the 2ptCF, via its connection to the
power spectrum, is a close measurement of the density fluctuations of the
universe and does not inherently vary with number density. Interestingly, the
VPF often yields better constraints on $\Omega_{\text{M}}$ than the 2ptCF and
CiC; our thoughts on why are discussed in 6.3.

Unlike the cosmological parameters, there is notable improvement in
constraining A${}_{\text{SN1}}$ and A${}_{\text{SN2}}$ when all clustering
statistics are combined. There is some evidence across the galaxy selections
that the VPF may drive the bulk of the constraints on A${}_{\text{SN1}}$ and
A${}_{\text{SN2}}$. CiC and 2ptCF mostly perform comparably to each other, and
the VPF tends to also perform similarly for A${}_{\text{SN2}}$ across the bulk
of our selections.

Examination of Figure 22 helps give important context to some of the
constraints we find. That specific neural network predicted low-value
A${}_{\text{SN1}}$ slightly more accurately than the high A${}_{\text{SN1}}$
values, while maintaining a still generally flat dependence across the whole
parameter space; this leads to deceptively lower errors given the performance.
For a visual example of this phenomenon, compare this parameter’s constraints
in Figure 22c against the others. Table
LABEL:table:ClusteringStats_AsnConstraints indicates that the VPF yields the
best constraints on the A${}_{\text{SN1}}$ and A${}_{\text{SN2}}$ feedback
parameters (and therefore likely dominates the ‘all’ clustering neural network
results), but this is not an evident pattern in the Figures themselves and
could be a statistical anomaly.

###  6.3 Discussion: Comparing Clustering Statistics

As discussed, the clustering statistics often find similar constraints on
$\sigma_{8}$, with some evidence that the 2ptCF tends to do best at
$\sigma_{8}$ for lower density down-sampling selections. As the Fourier