#  SimBIG: The First Cosmological Constraints from Non-Gaussian and Non-Linear
Galaxy Clustering

ChangHoon Hahn<EMAIL_ADDRESS>Department of Astrophysical
Sciences, Princeton University, Princeton NJ 08544, USA  Pablo Lemos
Department of Physics, Université de Montréal, Montréal, 1375 Avenue Thérèse-
Lavoie-Roux, QC H2V 0B3, Canada  Mila - Quebec Artificial Intelligence
Institute, Montréal, 6666 Rue Saint-Urbain, QC H2S 3H1, Canada  Center for
Computational Mathematics, Flatiron Institute, 162 5th Avenue, New York, NY
10010, USA  Liam Parker  Department of Astrophysical Sciences, Princeton
University, Princeton NJ 08544, USA  Bruno Régaldo-Saint Blancard Center for
Computational Mathematics, Flatiron Institute, 162 5th Avenue, New York, NY
10010, USA  Michael Eickenberg  Center for Computational Mathematics, Flatiron
Institute, 162 5th Avenue, New York, NY 10010, USA  Shirley Ho  Center for
Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY
10010, USA  Jiamin Hou  Department of Astronomy, University of Florida, 211
Bryant Space Science Center, Gainesville, FL 32611, USA  Max-Planck-Institut
für Extraterrestrische Physik, Postfach 1312, Giessenbachstrasse 1, 85748
Garching bei München, Germany  Elena Massara Waterloo Centre for Astrophysics,
University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada
Department of Physics and Astronomy, University of Waterloo, 200 University
Ave W, Waterloo, ON N2L 3G1, Canada  Chirag Modi  Center for Computational
Mathematics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New
York, NY 10010, USA  Azadeh Moradinezhad Dizgah Département de Physique
Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4,
Switzerland  David Spergel  Center for Computational Astrophysics, Flatiron
Institute, 162 5th Avenue, New York, NY 10010, USA  Department of
Astrophysical Sciences, Princeton University, Princeton NJ 08544, USA

###### Abstract

The 3D distribution of galaxies encodes detailed cosmological information on
the expansion and growth history of the Universe. We present the first
cosmological constraints that exploit non-Gaussian cosmological information on
non-linear scales from galaxy clustering, inaccessible with current standard
analyses. We analyze a subset of the BOSS galaxy survey using SimBIG, a new
framework for cosmological inference that leverages high-fidelity simulations
and deep generative models. We use two clustering statistics beyond the
standard power spectrum: the bispectrum and a convolutional neural network
based summary of the galaxy field. We infer constraints on $\Lambda$CDM
parameters, $\Omega_{b}$, $h$, $n_{s}$, $\Omega_{m}$, and $\sigma_{8}$, that
are 1.6, 1.5, 1.7, 1.2, and 2.3$\times$ tighter than power spectrum analyses.
With this increased precision, we derive constraints on the Hubble constant,
$H_{0}$, and $S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$ that are competitive with
other cosmological probes, even with a sample that only spans 10% of the full
BOSS volume. Our $H_{0}$ constraints, imposing the Big Bang Nucleosynthesis
prior on the baryon density, are consistent with the early time constraints
from the cosmic microwave background (CMB). Meanwhile, our $S_{8}$ constraints
are consistent with weak lensing experiments and similarly lie below CMB
constraints. Lastly, we present forecasts to show that future work extending
SimBIG to upcoming spectroscopic galaxy surveys (DESI, PFS, Euclid) will
produce leading $H_{0}$ and $S_{8}$ constraints that bridge the gap between
early and late time measurements and shed light on current cosmic tensions.

cosmological parameters from LSS — Machine learning — cosmological simulations
— galaxy surveys

##  1 Introduction

The standard cosmological model ($\Lambda$CDM) has been remarkably successful
at describing a wide range of cosmological observations, including the CMB
(Page et al., 2003; Bennett et al., 2013; Planck Collab. et al., 2020a; Aiola
et al., 2020; Dutcher et al., 2021), the large-scale structure (LSS;
Bernardeau et al., 2002; Alam et al., 2017), distances to Type Ia supernovae
(SN-Ia; Perlmutter et al., 1999; Riess et al., 1998; Scolnic et al., 2018;
Brout et al., 2022), and the observed abundance of light elements (Schramm &
Turner, 1998; Steigman, 2007; Iocco et al., 2009; Cyburt et al., 2016).
Despite its success, $\Lambda$CDM has recently come under scrutiny from
‘cosmic tensions’, whose statistical significance has continued to persist
with the latest observations (see Abdalla et al., 2022, for a review).

The most statistically significant of the tensions is the “Hubble tension”. It
refers to the disagreement between the late-time measurements of the Hubble
constant, $H_{0}$, with early-time measurements inferred from CMB assuming
$\Lambda$CDM (see Freedman, 2021; Abdalla et al., 2022; Kamionkowski & Riess,
2022, for recent reviews). With the latest observations (Riess et al., 2022;
Planck Collab. et al., 2020a), the disagreement has a statistical significance
of >5$\sigma$. Meanwhile, there is also an “$S_{8}$ tension”, which refers to
the disagreement among measurements of the amplitude of the matter clustering,
parameterized as $S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$. Weak lensing
analyses that probe LSS at $z\sim 0.5$ have systematically found $S_{8}$
values $\sim$2$\sigma$ lower than expected from the best-fit CMB $\Lambda$CDM
cosmology (Troxel et al., 2018; Asgari et al., 2021; Amon et al., 2022; Secco
et al., 2022; Dalal et al., 2023; Sugiyama et al., 2023). Their $S_{8}$ values
are also lower than the recent constraints from CMB lensing, which probes LSS
at higher redshifts, $z=0.5-5$ (Madhavacheril et al., 2023). The $H_{0}$ and
$S_{8}$ tension between late- and early-time constraints, have led a number of
theoretical works to explore alternatives to $\Lambda$CDM (_e.g._ Meerburg,
2014; Chudaykin et al., 2018; Di Valentino et al., 2020; Abellán et al., 2022;
Kamionkowski & Riess, 2022).

The 3D distribution of galaxies encodes cosmological information that
addresses both tensions. We can compare the angular and physical scales of
features in the galaxy distribution, _e.g._ Baryon Acoustic Oscillations (BAO;
Eisenstein et al., 1998, 2005; Cole et al., 2005), to compute physical
distances and extract the Hubble constant (see Ivanov & Philcox, 2023, for a
recent review). We can also measure the growth of structure from redshift-
space distortions in the galaxy distribution to infer $S_{8}$ parameter. With
these aims, spectroscopic galaxy surveys of the next decade, the Dark Energy
Spectroscopic Instrument (DESI; DESI Collaboration et al., 2016a, b; Abareshi
et al., 2022), Subaru Prime Focus Spectrograph (PFS; Takada et al., 2014;
Tamura et al., 2016), the ESA Euclid satellite mission (Laureijs et al.,
2011), and the Nancy Grace Roman Space Telescope (Roman; Spergel et al., 2015;
Wang et al., 2022a), will probe galaxies over unprecedented cosmic volumes.
They will span >10 Gyrs of cosmic history, $0<z\lesssim 3$, and precisely
constrain $H_{0}$ and $S_{8}$ from the early- to late-times of the Universe.

To extract the cosmological information in galaxy distributions, current
analyses primarily use the power spectrum, which captures all the information
in a Gaussian random field, as the summary statistic of galaxy clustering
(_e.g._ Alam et al., 2017; Beutler et al., 2017; Ivanov et al., 2020; Chen et
al., 2022; Kobayashi et al., 2022). Furthermore, they model the galaxy
clustering using the perturbation theory (PT) of LSS and, as a result, are
limited to large scales where deviations from linear theory are small ($k_{\rm
max}\lesssim 0.2\,h/{\rm Mpc}$). However, recent studies have now established
that there is additional cosmological information beyond these regimes.
Analyses of the bispectrum, the first higher-order statistic, have shown that
there is significant non-Gaussian information (Gil-Marín et al., 2017; Philcox
& Ivanov, 2022; Ivanov et al., 2023). Forecasts in Hahn et al. (2020) and Hahn
& Villaescusa-Navarro (2021) have further shown that cosmological constraints
improve by a factor of $\sim$2 by analyzing the bispectrum down to non-linear
scales ($k_{\rm max}=0.5\,h/{\rm Mpc}$). Forecasts of other clustering
statistics beyond the power spectrum find consistent improvements (Massara et
al., 2020, 2022; Wang et al., 2022b; Hou et al., 2022; Eickenberg et al.,
2022).

Despite its promise, non-Gaussian cosmological information on non-linear
scales cannot be robustly extracted using standard approaches. PT struggles to
accurately model galaxy clustering beyond weakly non-linear scales. It also
cannot model many of the newly proposed summary statistics (_e.g._ Banerjee &
Abel, 2021; Eickenberg et al., 2022; Valogiannis & Dvorkin, 2022; Naidoo et
al., 2022). Observational systematics that significantly impact observed
clustering are also a major challenge (_e.g._ Ross et al., 2012, 2017). Fiber
collisions, for example, prevent galaxy surveys that use fiber-fed
spectrographs (_e.g._ DESI, PFS) from successfully measuring redshifts of more
than one galaxy within some angular scale. Even for the power spectrum, they
cause significant bias on weakly non-linear scales, $k>0.1\,h/{\rm Mpc}$ (Guo
et al., 2012). While proposed corrections may be sufficient for the power
spectrum (Hahn et al., 2017; Pinol et al., 2017; Bianchi et al., 2018; Smith
et al., 2019), no correction has yet been designed for statistics beyond the
power spectrum.

With the SIMulation-Based Inference of Galaxies (SimBIG), we go beyond the
state-of-the-art galaxy clustering analyses. As we present in the first papers
of the series (Hahn et al. 2022 and Hahn et al. 2023b111hereafter H22a and
H23), SimBIG is a forward modeling framework that uses simulation-based
inference222also known as “likelihood-free inference” (LFI) or “implicit
likelihood inference” (ILI) (SBI; see Cranmer et al., 2020, for a review) with
deep generative models from machine learning to perform highly efficient
cosmological inference. SimBIG leverages high-fidelity $N$-body simulations
that accurately model non-linear galaxy clustering and includes observational
effects. With this approach, H22a analyzed the power spectrum multipoles,
$P_{\ell}(k)$, of galaxies333We used 109,636 galaxies in the Southern Galactic
Cap (SGC) of the BOSS CMASS sample within $0.45<z<0.6$ and over $\sim$3,600
${\rm deg}^{2}$, which corresponds to roughly $\sim$10% of the full BOSS
volume. in the Sloan Digital Sky Survey-III Baryon Oscillation Spectroscopic
Survey (BOSS; Eisenstein et al., 2011; Dawson et al., 2013) to demonstrate
that we can robustly analyze the power spectrum down to non-linear scales. In
this work, we go further and present the cosmological constraints that exploit
non-Gaussian information and non-linear scales. We use two summary statistics
of galaxy clustering beyond $P_{\ell}$: the galaxy bispectrum and a field-
level summary based on convolutional neural networks. We explore the
cosmological implications of our constraints and address the $H_{0}$ and
$S_{8}$ tensions.

We begin by briefly summarizing our methods in Section 2. We present and
discuss our cosmological results in Section 3 and compare them to the
literature. Lastly, we discuss future steps and prospects of extending SimBIG
to upcoming galaxy surveys in Section 3.2.

##  2 Methods

SimBIG uses SBI to infer posteriors of $\Lambda$CDM cosmological parameters
with only a forward model that can generate mock observations, _i.e._ the 3D
galaxy spatial distribution. In this section, we briefly describe the forward
model, the SBI methodology, the considered summaries of the galaxy
distribution, and the validation of our analyses.

###  2.1 Forward Model

The Simbig forward model constructs simulated galaxy catalogs from Quijote
$N$-body simulations (Villaescusa-Navarro et al., 2020) run at different
cosmologies in a Latin-hypercube configuration. Each simulation has a volume
of $1~{}(h^{-1}{\rm Gpc})^{3}$ and is constructed using $1024^{3}$ cold dark
matter particles initialized at $z=127$ and gravitationally evolved until
$z=0.5$. These high-resolution $N$-body simulations accurately model the
clustering of matter down to non-linear scales beyond $k>0.5\,h/{\rm Mpc}$.
From the dark matter particles, halos are identified using the Rockstar halo
finder (Behroozi et al., 2013), which accurately determines the location of
halos and resolve their substructure (Knebe et al., 2011). Afterwards, the
halos are populated using the halo occupation distribution (HOD; _e.g._
Berlind & Weinberg, 2002) framework, which provides a flexible statistical
prescription for determining the number of galaxies as well as their positions
and velocities within halos. SimBIG uses a state-of-the art 9-parameter HOD
model that supplements the standard Zheng et al. (2007) model with assembly,
concentration, and velocity biases for even more flexibility. From the HOD
galaxy catalog, SimBIG adds full BOSS survey realism by applying the survey
geometry and observational systematics. The forward-modeled catalogs have the
same redshift range and the angular footprint of our observed sample. In
summary, the SimBIG forward model aims to generate mock galaxy catalogs that
are statistically indistinguishable from the observations. For further
details, we refer readers to H22a and H23.

###  2.2 Simulation-Based Inference

From the 20,000 forward-modeled galaxy catalogs, we use the SimBIG SBI
framework to infer posterior distributions of cosmological parameters,
$\bm{\theta}$, for given a summary statistic, x, of the observations:
$p(\bm{\theta}\,|\,\textbf{{x}})$. The SBI in SimBIG is based on neural
density estimation (NDE) from deep generative models and uses “normalizing
flow” (Tabak & Vanden-Eijnden, 2010; Tabak & Turner, 2013). It enables
cosmological inference with a limited number of simulated forward models so
that we can extract cosmological information on small, non-linear, scales and
using any statistic.

Among various normalizing flow architectures, SimBIG uses Masked
Autoregressive Flow (MAF; Papamakarios et al., 2017) and Neural Spline Flow
(NSF; Durkan et al., 2019) models444We use the normalizing flow implementation
in $\mathtt{sbi}$ Python package (Greenberg et al., 2019; Tejero-Cantero et
al., 2020).. We train a flow that best approximates the posterior,
$q_{\bm{\phi}}(\bm{\theta}\,|\,\textbf{{x}})\approx
p(\bm{\theta}\,|\,\textbf{{x}})$, by minimizing the KL divergence between
$p(\bm{\theta},\textbf{{x}})=p(\bm{\theta}\,|\,\textbf{{x}})p(\textbf{{x}})$
and $q_{\bm{\phi}}(\bm{\theta}\,|\,\textbf{{x}})p(\textbf{{x}})$. In practice,
we maximize the total log-likelihood $\sum_{i}\log
q_{\bm{\phi}}(\bm{\theta}_{i}\,|\,\textbf{{x}}_{i})$ over the training set. We
determine the training and architecture of our flow through experimentation,
tailored to each summary statistic.

The prior of our posterior estimate is set by the parameter distribution of
our training set, which corresponds to a uniform prior over the cosmological
parameters, $\\{\Omega_{m},\Omega_{b},h,n_{s},\sigma_{8}\\}$, with ranges that
extend well beyond the Planck posterior. For the HOD parameters, we use the
conservative priors centered around previous HOD analyses of the BOSS CMASS
sample.

###  2.3 Summary Statistics Beyond $P_{\ell}$

We use two summary statistics of galaxy clustering beyond $P_{\ell}$: the
bispectrum ($B_{0}$) and a field-level summary using convolutional neural
networks (CNNs). The bispectrum is the first higher-order statistic that
measures the excess probability of finding galaxies in different triangle
configurations over a random distribution. For a near-Gaussian galaxy
distribution, the bispectrum extracts nearly all of its cosmological
information (_e.g._ Fry, 1994; Matarrese et al., 1997; Scoccimarro, 2000). We
also analyze the galaxy distribution at the field-level, using CNNs to perform
massive data compression and extract maximally relevant features. By learning
the relevant features, we aim to extract even more cosmological information
than conventional summary statistics. We provide additional details on these
summary statistics in Appendix A.1. We also refer readers to Hahn et al.
(2023a) and Lemos et al. (2023) for further details.

###  2.4 Posterior Validation

For each summary statistic, we validate the trained $q_{\phi}$ in two steps.
First, we perform a “NDE accuracy test” on a held-out subset of the training
data, either using simulation-based calibration (Talts et al., 2020) or a
“distance to random point” test (Lemos et al., 2023a). With this test, we
check whether $q_{\phi}$ accurately estimates the posterior throughout the
entire prior range of the cosmological parameters. Second, we assess the
robustness of $q_{\phi}$ by applying it to the SimBIG “mock challenge”
simulations (H23). These simulations include three different sets of test
simulations evaluated at some fiducial cosmology: 500 constructed using the
same forward model as the training data, 500 constructed using a different
halo finder and HOD model, and 1,000 constructed using a different $N$-body
simulation and halo finder. We apply $q_{\phi}$ to the simulations and compare
the resulting posteriors to their true parameter values. Since two of the test
sets are constructed using different forward models than the training
simulations, if the $q_{\phi}$ posteriors derived from them are statistically
consistent with the true values, we conclude that the $q_{\phi}$ is
sufficiently robust.

Figure 1:  Left: Posteriors of cosmological parameters inferred from $B_{0}$
(blue) and CNN (orange) using SimBIG. All posteriors include a $\omega_{b}$
prior from BBN studies. The contours mark the 68 and 95 percentiles. For
comparison, we include the posterior from the SimBIG $P_{\ell}$ analysis
(gray). Right: We focus on the posteriors of $\Omega_{m}$ and $\sigma_{8}$,
the parameters that can be most significantly constrained by galaxy
clustering. We include the posterior from the Ivanov et al. (2020) PT-based
$P_{\ell}(k<0.25\,h/{\rm Mpc})$ analysis for reference (black dashed). The
SimBIG $B_{0}$ and CNN constraints are significantly tighter yet consistent
with $P_{\ell}$ constraints.

##  3 Results

We present the posteriors of the $\Lambda$CDM cosmological parameters for the
$B_{0}$ (blue) and CNN (orange) SimBIG analyses in Figure 1. In the right
panels, we focus on the constraints on the growth of structure parameters:
$\Omega_{m}$ and $\sigma_{8}$. For comparison, we include the posteriors from
the H22a SimBIG (gray) and Ivanov et al. (2020) (black dashed) $P_{\ell}$
analyses. The Ivanov et al. (2020) constraints are inferred from roughly the
same galaxy sample and is based on PT. In each of the contours, we mark the 68
and 95 percentiles of the posteriors. For all posteriors, we impose a Gaussian
prior on $\omega_{b}=\Omega_{b}h^{2}=0.02268\pm 0.00038$ from BBN studies
(Aver et al., 2015; Cooke et al., 2018; Schöneberg et al., 2019), typically
included in clustering analyses (_e.g._ Ivanov et al., 2020; Philcox & Ivanov,
2022). We list the marginalized parameter constraints in Table 1.

Focusing first on $B_{0}(k_{123}<0.5\,h/{\rm Mpc})$, we find that our
$\Lambda$CDM constraints are significantly tighter yet consistent with
$P_{\ell}$ constraints. Compared to the SimBIG $P_{\ell}(k<0.5\,h/{\rm Mpc})$
constraints, the SimBIG $B_{0}$ constraints are 2.78, 3.17, 1.91, 1.77, and
1.53$\times$ tighter for $\Omega_{b}$, $h$, $n_{s}$, $\Omega_{m}$, and
$\sigma_{8}$. Compare to the Ivanov et al. (2020) $P_{\ell}(k<0.25\,h/{\rm
Mpc})$ constraints, the $B_{0}$ constraints are 1.59, 1.51, 1.68, 1.17, and
2.04$\times$ tighter for $\Omega_{b}$, $h$, $n_{s}$, $\Omega_{m}$, and
$\sigma_{8}$, respectively. The improvement on $\sigma_{8}$ is larger because
Ivanov et al. (2020) does not include the non-linear scales (0.25 < $k$ < 0.5
$h/{\rm Mpc}$) included in the SimBIG $P_{\ell}$ analyses.

Similarly, our constraints from the SimBIG CNN analysis are significantly
tighter yet consistent with the $P_{\ell}$ constraints. For $\Omega_{b}$, $h$,
$n_{s}$, $\Omega_{m}$, and $\sigma_{8}$, the CNN constraints are 2.51,
2.51,1.46, 1.42, and 1.75$\times$ tighter than the SimBIG $P_{\ell}$
constraints. In comparison to the PT $P_{\ell}$ analysis, the $\Omega_{b}$,
$h$, $n_{s}$, and $\sigma_{8}$ constraints are 1.44, 1.20, 1.28, and
2.33$\times$ tighter while the $\Omega_{m}$ constraint is slightly
(1.07$\times$) broader.

Overall, the CNN and $B_{0}$ constraints are in excellent agreement. There is
a slight offset in $n_{s}$; however it is not statistically significant. The
CNN constraints are slightly broader overall than $B_{0}$ except for
$\sigma_{8}$, even though the CNN aims to maximally extract the cosmological
information in the galaxy field. This is in part due to the fact that, in
practice, we limit the CNN’s constraining power of the CNN to ensure
robustness and generalizability (Appendix A.1).

| $\Omega_{m}$ | $\sigma_{8}$ | $\Omega_{b}$ | $h$ | $n_{s}$ | $S_{8}$ | $H_{0}$
---|---|---|---|---|---|---|---
$P_{\ell}(k<0.5)$ | $0.287^{+0.059}_{-0.038}$ | $0.808^{+0.068}_{-0.066}$ | $0.044^{+0.009}_{-0.006}$ | $0.713^{+0.056}_{-0.059}$ | $0.935^{+0.070}_{-0.073}$ | $0.797^{+0.107}_{-0.093}$ | $71.255^{+5.615}_{-5.921}$
$B_{0}(k_{123}<0.5)$ | $0.297^{+0.020}_{-0.035}$ | $0.763^{+0.054}_{-0.033}$ | $0.050^{+0.003}_{-0.002}$ | $0.676^{+0.018}_{-0.018}$ | $0.952^{+0.035}_{-0.040}$ | $0.757^{+0.048}_{-0.053}$ | $67.556^{+1.801}_{-1.843}$
CNN | $0.295^{+0.036}_{-0.033}$ | $0.753^{+0.040}_{-0.036}$ | $0.046^{+0.003}_{-0.003}$ | $0.702^{+0.024}_{-0.022}$ | $1.039^{+0.053}_{-0.044}$ | $0.746^{+0.055}_{-0.051}$ | $70.199^{+2.387}_{-2.208}$
Table 1: Posteriors of $\Lambda$CDM cosmological parameters inferred from the
power spectrum multipoles, the bispectrum monopole, and the CNN using SimBIG.
We present the median and 68 percentile uncertainties of the parameters. All
posteriors include a $\omega_{b}$ prior from BBN.

Overall, the SimBIG analyses beyond $P_{\ell}$ produce significantly tighter
cosmological constraints than $P_{\ell}$ analyses. Taking the best constraints
from the $B_{0}$ and CNN analyses, we improve $\Omega_{b}$, $h$, $n_{s}$,
$\Omega_{m}$, and $\sigma_{8}$ by 1.59, 1.51, 1.68, 1.17, and 2.33$\times$
over the PT $P_{\ell}$ analysis. This firmly shows that there is substantial
non-Gaussian cosmological information in galaxy clustering. Furthermore, our
posteriors are in excellent agreement with previous galaxy clustering analyses
that use an entirely different approach. This consistency serves as additional
corroboration that SimBIG provides a robust framework for analyzing clustering
beyond $P_{\ell}$.

Figure 2:  $S_{8}$ and $H_{0}$ posteriors for the SimBIG $B_{0}$ (blue) and
CNN (orange) analyses. We include the posterior from the Ivanov et al. (2020)
$P_{\ell}$ analyses of CMASS-SGC for comparison (black dashed). All posteriors
include a $\omega_{b}$ prior from BBN studies. With the $B_{0}$ and CNN, we
infer consistent posteriors on $S_{8}$ and $H_{0}$ that are $\sim$1.9 and
1.5$\times$ tighter than the PT $P_{\ell}$ analyses. Our tighter constraints
enable us to inform the $S_{8}$ and $H_{0}$ “tensions” found between
cosmological probes in the early versus late Universe.  Figure 3:  Left:
Comparison of the SimBIG $S_{8}$ constraints (black) to existing bounds in the
literature. We include constraints from galaxy clustering (red), CMB (blue),
weak lensing (purple), and multi-probe (brown) analyses. Despite only using
10% of the full BOSS volume, we derive $S_{8}$ constraints comparable to full
BOSS $P_{\ell}$ analysis. Our $S_{8}$ constraints are in good agreement with
weak lensing experiments and, thus, lie below the CMB constraints. Right:
Comparison of the SimBIG $H_{0}$ constraints (black) to the literature. We
include constraints from galaxy clustering (red), CMB (blue), SN (purple), and
other probes (brown). Our $H_{0}$ constraint from SimBIG $B_{0}$ is in
excellent agreement with Planck. We infer a higher $H_{0}$ with the CNN that
is consistent with both Planck and SN-Ia constraints.

###  3.1 Cosmic Tensions

Our tighter constraints on cosmological parameters enable us to also inform
the $S_{8}$ and $H_{0}$ cosmic tensions. In Figure 2, we present the $S_{8}$
and $H_{0}$ posteriors from the SimBIG $B_{0}$ (blue) and CNN (orange)
analyses. For comparison, we also plot the posterior for the Ivanov et al.
(2020) $P_{\ell}$ PT analysis (black dashed). We again include a $\omega_{b}$
prior from BBN studies. The $S_{8}$ and $H_{0}$ constraints from $B_{0}$ and
CNN are consistent with each other (Table 1). Compared to Ivanov et al.
(2020), $B_{0}$ improves $S_{8}$ and $H_{0}$ constraints by 1.92 and
1.51$\times$ and the CNN improves the $S_{8}$ and $H_{0}$ by 1.83 and
1.20$\times$.

Next, we compare our $S_{8}$ and $H_{0}$ posteriors to constraints in the
literature from galaxy clustering and other cosmological probes (Figure 3).
The constraints from the literature are selected from Abdalla et al. (2022).
In the left panel, we compare the SimBIG $S_{8}$ constraints (black) to
constraints from galaxy clustering (red), CMB (blue), weak lensing (purple),
and multi-probe (brown) analyses. From galaxy clustering, we include the
Ivanov et al. (2020) and Kobayashi et al. (2022) $P_{\ell}$ analyses of the
CMASS-SGC sample. These analyses use nearly the same observational sample so
their constraints are the most consistently comparable to the SimBIG
constraints555The galaxy sample in our SimBIG analyses has a $\sim$30% smaller
footprint than the CMASS-SGC.. We also include constraints from PT analyses of
the full BOSS volume: Philcox & Ivanov (2022) and Chen et al. (2022). Philcox
& Ivanov (2022) analyze both $P_{\ell}$ and $B_{0}$ and includes a BBN prior;
Chen et al. (2022) analyze the pre- and post-reconstruction $P_{\ell}$ with
fixed $\Omega_{b}$ and $n_{s}$. Despite only analyzing 10% of the total BOSS
volume, our analyses produce $S_{8}$ constraints with comparable precision
level to full volume analyses.

For CMB, we include constraints from: Planck TT,TE,EE+lowE with and without
CMB lensing (Planck Collab. et al., 2020b); the Aiola et al. (2020) combined
high-$\ell$ and low-$\ell$ measurements from the Atacama Cosmology Telescope
(ACT) and the Wilkinson Microwave Anisotropy Probe (WMAP); and the ACT CMB
lensing constraint combined with BAO (Madhavacheril et al., 2023). For weak
lensing (WL), we include: the cosmic shear analysis of Kilo Degree Survey
(KiDS; Asgari et al., 2021); the Asgari et al. (2020) combined analysis of
earlier KiDS data, VIKING data, and the Dark Energy Survey (DES) first year
data; the DES analyses of the first (Troxel et al., 2018) and third year data
(Amon et al., 2022; Secco et al., 2022); and the Hyper Suprime Cam (HSC)
cosmic shear analysis (Dalal et al., 2023) and 3$\times$2pt analysis (Sugiyama
et al., 2023) of the third year data. Lastly, we include multi-probe analyses
that combine galaxy clustering and the cross-correlation with the Planck CMB
lensing reconstruction (Krolewski et al., 2021; White et al., 2022).

Next, in the right panel of Figure 3, we compare the SimBIG $H_{0}$ posteriors
to constraints from a wide variety of cosmological probes: galaxy clustering
(red), CMB (blue), SN-Ia (purple), and others (brown). From galaxy clustering,
we include the Ivanov et al. (2020) CMASS-SGC analysis. Again, this is the
analysis most consistent with our analyses. We also include the Philcox &
Ivanov (2021), Chen et al. (2022), and Alam et al. (2017) analyses of the full
BOSS volume. From CMB, we include constraints from WMAP (Hinshaw et al.,
2013), Planck (Planck Collab. et al., 2020b), ACT (Aiola et al., 2020), and
the South Pole Telescope (SPT; Dutcher et al., 2021). We also include $H_{0}$
constraints from distance ladder methods: SN-Ia calibrated using Cepheids from
the SH0ES collaboration (Riess et al., 2022) and SN-Ia calibrated using the
Tip of the Red Giant Branch (TRGB; Freedman, 2021). Lastly, we include
constraints from time-delay strong lensing (Denzel et al., 2021), the Tully-
Fisher relation (Kourkchi et al., 2020), and gravitational wave standard
sirens (Palmese et al., 2020).

The SimBIG $P_{\ell}$ constraints are not precise enough to meaningfully
inform the $S_{8}$ and $H_{0}$ tensions. However, the tighter constraints from
the SimBIG $B_{0}$ and CNN analyses provide interesting insights into the
tension. For $S_{8}$, the SimBIG $B_{0}$ and CNN constraints are in good
agreement with the weak lensing constraints and, thus, lower than the CMB
constraints. Despite the differences, we emphasize that the SimBIG constraints
are statistically consistent with both CMB and weak lensing. For $H_{0}$, the
tighter constraints from SimBIG $B_{0}$ and CNN are in good agreement with
constraints from CMB and LSS. The $B_{0}$ constraint is in excellent agreement
with Planck and, thus, is in tension with the SH0ES measurement. The CNN
constraint is notably higher than the $B_{0}$ constraint and consequently
reduces the tension with SH0ES. Nevertheless, it is statistically consistent
with $B_{0}$ and Planck.

###  3.2 Future Steps and Upcoming Surveys

Despite only using 10% of the BOSS volume, we derive $S_{8}$ and $H_{0}$
constraints that are competitive with other cosmological probes and galaxy
clustering analyses of the full BOSS volume. In short, our forward-modeling
approach improves $S_{8}$ and $H_{0}$ constraints by $\sim$2.0 and 1.5$\times$
over the standard PT $P_{\ell}$ analysis. The improvement on $S_{8}$ is
equivalent to applying the standard PT $P_{\ell}$ analysis to a galaxy sample
four times the cosmic volume.

Yet, the improvement does not include all of the cosmological information that
can be extracted using clustering statistics beyond $P_{\ell}$. We require
SimBIG analyses to derive unbiased cosmological constraints for a suite of
test simulations constructed with different forward models (Section 2.4). To
meet this requirement for the CNN analysis, we use dropout, regularization,
and SWA to reduce overfitting (Appendix A.1). These procedures discard a
substantial amount of cosmological information.

Solely for reference, in Figure 4 we present the $\Omega_{m}$ and $\sigma_{8}$
posterior for the CNN (red hatched) if we relax our robustness requirement.
This is not a robust posterior; however, it illustrates that the CNN is
capable of capturing significantly more cosmological information than the
posteriors presented in this work. The non-robust CNN has >2$\times$ the
precision of the robust CNN. In Figure 4, we also include posteriors derived
from another summary statistic: the wavelet scattering transform (WST). The
WST is set of descriptive statistics particularly suited for characterizing
non-Gaussian fields. It is motivated by the structure of CNNs while being
completely parameter-free. We exclude the WST analysis in our main comparison
due to concerns about its robustness; however, the WST is capable of deriving
constraints comparable to the $B_{0}$ analysis. For further details on the WST
analysis, see Regaldo-Saint Blancard et al. (2023).

The full constraining power of neither the CNN nor the WST can currently be
robustly exploited. This is because these statistics identify cosmological
imprints that are specific to our forward model. One way to robustly exploit
these statistics is to use a more flexible and generalizable forward model
that can describe the WST and CNN across the different test simulations.
Alternatively, we can also construct clustering statistics more robust to
model misspecification (_e.g._ Huang et al., 2023). Such an approach would
learn the imprints of cosmological parameters while ignoring any model-
specific features. In future work we will explore both approaches.

Figure 4:  The $H_{0}$ and $S_{8}$ posterior for the CNN (red hatched) with
relaxed robustness requirements. We also include the posterior from the SimBIG
WST analysis (green hatched). The posteriors are not robust; however, they
illustrate the potential of the WST and CNN at capturing even more
cosmological information than the posteriors presented above. The WST is
capable of deriving constraints comparable to $B_{0}$ while the non-robust CNN
can double the precision of the robust CNN constraints. In future work, we aim
to extract some of this additional constraining power using SimBIG with a more
flexible and generalizable forward model and with summary statistics more
robust to model misspecifications.  Figure 5:  Expected 1$\sigma$ precision
level of $H_{0}$ ($\sigma_{H_{0}}$; top) and $S_{8}$ ($\sigma_{S_{8}}$;
bottom) constraints from applying SimBIG to upcoming galaxy surveys, DESI,
PFS, Euclid (black). For DESI, we present SimBIG forecasts for the different
galaxy samples individually. The width of $\sigma_{H_{0}}$ and
$\sigma_{S_{8}}$ marks the redshift range of each sample. We include
constraints from SN-Ia (red), CMB and CMB lensing (blue), weak lensing
(green), and previous BOSS/eBOSS $P_{\ell}$ analyses (gray), for comparison.
Future SimBIGanalyses of DESI, PFS, and Euclid will have the precision and
redshift range to provide key input into the $H_{0}$ and $S_{8}$ tensions and
potentially reveal new physics beyond the standard $\Lambda$CDM model.

In future work, we will also extend SimBIG to the next generation
spectroscopic galaxy surveys. DESI will probe 14,000 deg2 across $0<z<2.1$
with >40 million galaxies (Chaussidon et al., 2023; Hahn et al., 2022;
Raichoor et al., 2023; Zhou et al., 2023). DESI has recently completed two out
of its five years of observations. Meanwhile, the PFS Cosmology Survey will
probe 1,200 deg2 across $0.6<z<2.4$ with 4 million emission-line galaxies and
is set to begin observations next year. Lastly, the Euclid spectroscopic
survey (Euclid Collab. et al., 2022) will measure the redshift of about 30
million H$\alpha$ emitters across 15,000 ${\rm deg}^{2}$ over $0.9<z<1.8$. It
was recently launched in July 2023.

These surveys will probe enormous cosmic volumes in epochs without precise
constraints on $H_{0}$ or $S_{8}$ from other cosmological probes. We highlight
this in Figure 5, where we present the forecasted 1$\sigma$ precision levels
of $H_{0}$ (top) and $S_{8}$ (bottom) constraints from applying SimBIG to
DESI, PFS, and Euclid (black). These forecasts are derived by scaling the
SimBIG constraints by $(V_{\rm survey}/V_{\rm CMASS-SGC})^{0.5}$. For $H_{0}$,
upcoming surveys will help bridge the gap between the late-time measurements
from SN-Ia (SH0ES; red) and early-time measurements from the CMB (Planck;
blue). $H_{0}$ constraints from other probes, _e.g._ time delay lensing, have
significantly larger uncertainties and lie well outside the range of the
figure. Similarly, for $S_{8}$, DESI, PFS, and Euclid will bridge the gap
between $S_{8}$ constraints from weak lensing surveys that probe low redshifts
and from CMB lensing and CMB at high redshifts. The shaded regions represent
the lensing efficiency kernel for weak lensing experiments (0.1 < $z$ < 0.7;
Amon et al., 2022) and the mass-map weights for CMB lensing (0.5 < $z$ < 5.0;
Madhavacheril et al., 2023).

SimBIG can produce leading constraints on both $S_{8}$ and $H_{0}$ by more
fully extracting the cosmological information of upcoming galaxy surveys. We
note that the SimBIG forecasts are conservative estimates that do not account
for the fact that the DESI, PFS, and Euclid galaxy samples will have a higher
number density than CMASS. Given their higher number density, we expect even
larger gains from analyzing clustering beyond $P_{\ell}$. With their precision
levels and the extensive redshift range of DESI, PFS, and Euclid, future
SimBIG analyses will provide critical input into the cosmic tensions and
potentially reveal new physics beyond the standard $\Lambda$CDM model.

##  4 Summary

We present a comparison of SimBIG cosmological constraints from analyzing
galaxy clustering beyond the power spectrum: the bispectrum and a field-level
summary statistic based on CNNs. SimBIG provides a forward-modeling framework
that uses simulation-based inference with normalizing flows to perform highly
efficient cosmological inference using high-fidelity $N$-body simulations
(H22a, H23). With SimBIG, we extract non-Gaussian and non-linear cosmological
information, currently inaccessible with standard PT analyses.

The SimBIG $B_{0}$ and CNN analyses tightly constrain all $\Lambda$CDM
parameters and, compared to a standard PT analysis of $P_{\ell}$ (Ivanov et
al., 2020), improve $\Omega_{b}$, $h$, $n_{s}$, $\Omega_{m}$, and $\sigma_{8}$
constraints by 1.6, 1.5, 1.7, 1.2, and 2.3$\times$. With these tighter
constraints, the SimBIG analyses can inform recent “tensions” between the
early and late time measurements of $S_{8}$ and $H_{0}$. Even with a subset
set of the SDSS-III BOSS CMASS-SGC sample that corresponds to only <10% of the
full volume, we derive $S_{8}$ constraints competitive with PT $P_{\ell}$
analyses of the full BOSS volume. Our $S_{8}$ constraint is statistically
consistent with both CMB and weak lensing experiments. We also infer
competitive $H_{0}$ constraints that are consistent with early universe
constraints from CMB and other LSS analyses and in tension with late-time
measurements.

In future work, we will focus on extending SimBIG in two main avenues. First,
we will increase the flexibility of our forward model in order to improve its
robustness and generalizability. With a more robust forward model, we will use
more constraining statistics that can improve cosmological constraints even
further (Figure 4). We will also explore constructing new summary statistics
that are simultaneously more sensitive to cosmological imprints and more
robust to model misspecifications.

In addition, we will extend SimBIG to upcoming spectroscopic galaxy surveys:
DESI, PFS, and Euclid. These surveys will probe unprecedented volumes with >70
million galaxies. SimBIG can produce leading constraints on both $S_{8}$ and
$H_{0}$ across $0<z<2.4$ by more fully extracting the cosmological information
from these surveys (Figure 5). The SimBIG analyses of these surveys will
bridge the gap between the early- and late-time $H_{0}$ and $S_{8}$
constraints to provide key input into the tensions and potentially reveal
deviations from the standard $\Lambda$CDM cosmology model.

## Acknowledgements

It’s a pleasure to thank Peter Melchior, Uros̆ Seljak, Benjamin D. Wandelt,
and the members of the Simons Collaboration on Learning the
Universe666https://www.learning-the-universe.org/ for valuable discussions. We
also thank Mikhail M. Ivanov and Yosuke Kobayashi for providing us with the
posteriors used for comparison. This work was supported by the AI Accelerator
program of the Schmidt Futures Foundation. JH has received funding from the
European Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie grant agreement No 101025187. AMD acknowledges funding from
Tomalla Foundation for Research in Gravity. The work reported on in this paper
was substantially performed using the Princeton Research Computing resources
at Princeton University, which is a consortium of groups led by the Princeton
Institute for Computational Science and Engineering (PICSciE) and Office of
Information Technology’s Research Computing.

##  Appendix A Supplementary Information

###  A.1 Summary Statistics

The bispectrum: $B(k_{1},k_{2},k_{3})$, is the three-point correlation
function in Fourier space and measures the excess probability of different
triangle configurations $(k_{1},k_{2},k_{3})$ over a random distribution. In
this work, we use the monopole of the bispectrum: $B_{0}(k_{1},k_{2},k_{3})$
To measure $B_{0}$, we use the redshift-space bispectrum estimator by
Scoccimarro (2015). The estimator accounts for the survey geometry using a
random catalog that has the same radial and angular selection functions as the
observed catalog but with >4 million objects. For each galaxy, we include the
Feldman et al. (1994) weights. For the observed galaxy sample, we also include
angular systematic weights to account for stellar density and seeing
conditions as well as redshift failure weights. We do not include weights for
fiber collisions, since this effect is included in the SimBIG forward model.
We measure $B_{0}$ in triangle configurations defined by $k_{1},k_{2},k_{3}$
bins of width $\Delta k=0.0105\,h/{\rm Mpc}$. In practice, we use the reduced
bispectrum, $Q_{0}(k_{1},k_{2},k_{3})$, which normalizes the bispectrum to
reduce the dynamic range of $B_{0}$. For $k_{\rm max}=0.5\,h/{\rm Mpc}$,
$Q_{0}$ has 10,052 total triangle configurations.

CNN: Convolutional Neural Networks (CNNs) can be thought of as flexible,
hierarchical, non-linear functions that can be optimized to extract relevant
features from input images or, in this case, 3D galaxy fields. In this work,
we use a 3D CNN to perform a step of massive data compression from the galaxy
sample directly to the cosmological parameters. First, the galaxy
distributions are meshed to $64\times 64\times 128$ voxels using cloud-in-cell
mass assignment. The voxels have dimensions $\sim[11,11,11]~{}h/\textrm{Mpc}$,
which imposes a scale cut of $k<k_{\rm max}=0.28~{}h/\text{Mpc}$. We then
train the CNN to accurately compress the galaxy field by minimizing the mean-
squared-error between the network-predicted cosmological parameters and the
true parameter values of SimBIG training set using stochastic gradient descent
(SGD). In addition, we also perform weight marginalization on the network
using Stochastic Weight Averaging (SWA; Maddox et al., 2019; Wilson &
Izmailov, 2020), to improve the generalizability of our CNN (Lemos et al.,
2023b).

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