2021appReferences

[1]Metin Ata

1]Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan

2]Instituto de Astrofísica de Canarias, s/n, E-38205, La Laguna, Tenerife,
Spain 3]Departamento de Astrofísica, Universidad de La Laguna, E-38206, La
Laguna, Tenerife, Spain 4]INAF - Osservatorio di Astrofisica e Scienza dello
Spazio di Bologna, via Gobetti 93/3, 40129 Bologna, Italy

5]Gemini Observatory, NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, Hawai’i,
96720, USA 6]Department of Physics and Astronomy, University of California,
Davis, One Shields Ave., Davis, CA 95616, USA 7]Institute for Advanced
Research, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8601, Japan
8]Max Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg,
Germany

# Predicted Future Fate of COSMOS Galaxy Protoclusters over 11 Gyrs with
Constrained Simulations

<EMAIL_ADDRESS>Khee-Gan Lee<EMAIL_ADDRESS>Claudio Dalla Vecchia
<EMAIL_ADDRESS>Francisco-Shu Kitaura<EMAIL_ADDRESS>Olga Cucciati
<EMAIL_ADDRESS>Brian C. Lemaux<EMAIL_ADDRESS>Daichi Kashino
<EMAIL_ADDRESS>Thomas Müller<EMAIL_ADDRESS>[  [  [
[  [  [  [  [

###### keywords:

Numerical Simulations, Galaxy Clusters, Cosmology-Theory, High Redshift
Observations

Cosmological simulations are crucial tools in studying the Universe, but they
typically do not directly match real observed structures. Constrained
cosmological simulations, on the other hand, are designed to match the
observed distribution of galaxies. Here, we present constrained simulations
based on spectroscopic surveys at a redshift of $z\sim 2.3$, corresponding to
an epoch of nearly 11 Gyrs ago. This allows us to “fast-forward” the
simulation to our present-day and study the evolution of observed cosmic
structures self-consistently. We confirm that several previously-reported pro-
to clusters will evolve into massive galaxy clusters by our present epoch,
including the ’Hyperion’ structure that we predict will collapse into a giant
filamentary supercluster spanning 100 Mega- parsecs. We also discover
previously unknown protoclusters, with lower final masses than typically
detectable by other methods, that nearly double the number of known
protoclusters within this volume. Constrained simulations, applied to future
high-redshift data sets, represents a unique opportunity for studying early
structure formation and matching galaxy properties between high and low
redshifts.

## Introduction

Understanding the formation of large-scale structures in the Universe,
starting from tiny fluctuations in the matter density, and subsequently
evolving gravitationally into the complex cosmic web seen at the present
epoch, is a key ambition of cosmological science. The formation and evolution
of galaxy clusters throughout different cosmic epochs is a crucial probe of
the current cosmological $\Lambda$CDM concordance model (see Kravtsov2012 for
a review), and is studied with a wide range of observational techniques.

The collapse time of massive galaxy clusters is of the order of the Hubble
time, thus we expect their formation to typically complete at redshifts
$z\lesssim 1$. At higher redshifts $z\gtrsim 1.5$, we usually do not observe
collapsed massive galaxy clusters but instead extended accumulations of
galaxies that do not yet form bound structures Muldrew2015 or only a collapsed
core of the whole structure Miller2018 ; Oteo2018

These diffuse formations in the evolving cosmic web Einasto2011 ;
Suhhonenko2011 , that correspond to Lagrangian volumes of
$V\gtrsim(10\,\,h^{-1}\,{\rm Mpc})^{3}$ Chiang2013 , are called galaxy
protoclusters (see Overzier2016 for a review) and represent the progenitors of
galaxy clusters seen in the Local Universe. As gravitationally evolving
objects, protoclusters are ideal observables to study early structure
formation and to compare to theoretical predictions. Moreover, they are active
sites of star and galaxy formation during the $z\sim 2-3$ Cosmic Noon epoch
Chiang2017 , which makes them excellent laboratories to jointly study the
interplay between baryonic physics and dark matter models.

Observationally, the effort to find and characterize protoclusters is a
lively, ongoing field. In particular, the COSMOS field Capak2007 is an
excellent site for this, as it is covered by deep and coordinated multi-
wavelength observations over a wide field ($>1$ square degree, corresponding
to $>100\,$Mpc in the transverse extent) suited to protocluster studies.

In addition, surveys using Ly$\alpha$ forest tomography — which probes the
large-scale diffuse hydrogen distribution — have targeted parts of this field
Lee2014 ; Lee2018 ; Newman2020 ; Horowitz2021 , offering a complementary way
to study structure formation apart from galaxy surveys. Multiple studies have
identified protoclusters in COSMOS with various claims they are potentially
progenitors of galaxy clusters such as Coma or Virgo, with present-day masses
up to $\sim 10^{15}\,M_{\odot}$ zfire1 ; Yuan2014 ; Franck2016 ; Diener2015 ;
Lee2016 ; Chiang2015 ; Casey2015 ; Wang2016 ; Cucciati2018 ; Darvish2020 ;
Polletta2021 ; Champagne2021 . However, these estimates were done with
different methods that may not be consistent with each other, and typically
did not take into account the large-scale structure environment over
$>10\,h^{-1}\,{\rm Mpc}$, which is known to determine the evolution of the
protoclusters Muldrew2015 ; Chiang2013 ; Overzier2016 . Also, studies show
that density peaks at high redshifts do not necessarily collapse into massive
galaxy clusters at $z=0$. In turn, massive structures at $z=0$ do not always
originate from single high redshift density peaks Cuesta2008 . Up to this
point, there has not been a uniform and self-consistent study dedicated to
these structures in the COSMOS field.

We address this problem with constrained simulations applied towards the rich
legacy of large-scale spectroscopic surveys that have been conducted on the
COSMOS field over nearly a decade, achieving a cosmic volume and number
density unmatched anywhere else on the sky. Standard cosmological simulations
start from randomly created Gaussian matter density fluctuations and produce
structures that are statistically consistent with the observable Universe, but
are not intended to directly match actual observed structures. On the other
hand, constrained cosmological simulations are designed to reproduce specific
large-scale structures in observations. However, hitherto, most applications
have mainly focused on the Local Universe or nearby structures Gottlober2010 ;
Hess2013 ; Wang:2014 ; Jasche:2015 ; Libeskind2020 . By performing
cosmological simulations constrained from high-redshift data ($z>2$), we can
potentially probe epochs when cosmic structure growth was still in the quasi-
linear regime. This allows a more direct mapping between the initial
conditions and the observed galaxy distribution than feasible in the strongly
non-linear regime of later epochs. Moreover, it opens up the exciting
possibility of forward-modeling the evolution of the observed high-redshift
structures to the present time, to predict whether or not they will collapse
into massive cluster halos.

## Constrained Simulations of the COSMOS Field

For our analysis, we use initial density fluctuations at $z=100$ that would
evolve into a matter density field consistent with the 3D distribution of a
large sample of spectroscopically-confirmed galaxies at Cosmic Noon Ata2021 .
This sample was compiled from the zCOSMOS-Deep Lilly2006 , VUDS LeFevre2015
MOSFIRE Kriek2015 , and ZFIRE zfire1 redshift surveys over the redshift range
of $1.4\leq z^{\mathrm{obs}}\leq 3.6$ and lie within the central square degree
of the COSMOS field centered on R.A.$=150.1^{\circ}$, Dec$=2.2^{\circ}$ in the
J2000 ICRS frame. In cosmology, the redshift is often used as a proxy for both
cosmological distance and cosmological time, which in our case is generally
disconnected. To disambiguate, we use $z^{\mathrm{obs}}$ when referring to
observed distance or equivalent comoving line-of-sight position in our
simulations, and $z$ as the time label for the simulation.
Special care was taken to accurately calculate the selection functions of the
individual surveys to ensure an unbiased estimate across the different surveys
Ata2021 . We then applied a multi-survey adaptation Ata2015 of the COSMIC
BIRTH algorithm Kitaura2019 , which is a nested Bayesian inference algorithm
for estimating the initial conditions and underlying matter density field. The
reconstruction grid was chosen to be $256^{3}$ cells with a side length of
$L_{\mathrm{box}}=512\,\,h^{-1}\,{\rm Mpc}$, which is large enough to cover
the comoving line-of-sight extent represented by $2.00\leq
z^{\mathrm{obs}}\leq 2.52$ in COSMOS. This redshift range was specifically
selected to overlap with multiple protoclusters previously reported in the
literature zfire1 ; Yuan2014 ; Diener2015 ; Chiang2015 ; Wang2016 ; Casey2015
; Lee2016 ; Cucciati2018 ; Darvish2020 ; Polletta2021 ).

From the posterior sample of initial conditions, we randomly selected 50
realizations to seed the simulations which were then run using the PKDGRAV3
$N$-body code Potter2017 . We dub the corresponding simulation suite as
“COnstrained Simulations of The COsmos field" (COSTCO).

Figure 1:  Matter density field of the COSMOS volume in the observed redshift
range of $2.00\leq z^{\mathrm{obs}}\leq 2.52$, from one realization out of 50
constrained simulations. Top & Middle: A $z=2.3$ snapshot of our constrained
simulations. The highlighted area on the top panel indicates the observed
region covered by the four galaxy surveys. Black dots show the galaxy
positions. Black crosses show the Literature-Reported Protocluster Candidates
(LRPCs), while the blue diamonds correspond to the center of mass position of
protoclusters seen in COSTCO. The white lines show which LRPC is matched to
each protocluster in a $r=15\,\,h^{-1}\,{\rm Mpc}$ search radius. Bottom: The
density field of the $z=0$ snapshot. The white circles indicate the halos that
were successfully matched with LRPCs while blue circles show unmatched halos
at $z=0$. In both slices, the ordinate axis approximately corresponds to the
Declination dimension in sky coordinates.

The COSTCO multiverse simulations were run from the initial conditions at
$z=100$ until $z=0$, with 6 intermediate time snapshots being output at
$z=2.0-2.5$. Figure 1 shows projected slices of the matter density field from
one of the constrained realizations, centered at approximately
R.A.=$149.99^{\circ}$ in the COSMOS field. We also show as fine dots the
observed galaxy positions in COSMOS, which trace the $z=2.3$ matter density
field as expected.

Evolved to $z=0$, we see that the fuzzy and diffuse matter distributions at
$z=2.3$ have collapsed into much larger structures with higher density
contrast.

The different realizations show variation in the resulting $z=0$ filamentary
structures (see Figure 6 in the Appendix), but massive clusters form
consistently in positions corresponding to large overdensities at $z=2.3$. An
ensemble of simulations like COSTCO, with consistent large-scale structure and
fluctuations on smaller scales is called Local Ensemble Statistics in the
literature and was first studied in Aragon-Calvo2016 . Next, we identified
collapsed halos with the ROCKSTAR halo finder rockstar , that have a minimum
virial mass of $M_{\rm vir}\geq 2\times 10^{14}\,\,h^{-1}\,M_{\odot}$ at
$z=0$, which is approximately equivalent to the mass of the Virgo cluster, the
nearest massive cluster to the Milky Way. We then traced the particles of each
halo back to their Lagrangian positions at the nearest time snapshot to their
observed redshift at $2.00\leq z^{\mathrm{obs}}\leq 2.52$, and computed the
center-of-mass position of the protocluster. Examples of this are shown in
Figure 2, which illustrates the extended ($\gtrsim 20\,\,h^{-1}\,{\rm Mpc}$)
nature of galaxy protoclusters. It is interesting to note that the
protocluster center-of-mass can experience significant displacements ($\sim
10\,\,h^{-1}\,{\rm Mpc}$) as it evolves over 11 Gyr from $z=2.3$ to $z=0$.
Indeed, in the case of the lower-mass cluster, the final halo position is at
an Eulerian coordinate almost outside the envelope of its $z\sim 2.3$
progenitor particles. This illustrates the crucial role that complex
gravitational interactions across the large-scale cosmic web plays in
protocluster evolution, such that oversimplified modeling with e.g. linear
theory can lead to inaccurate results. The variance across different
realizations in the mature $z=0$ masses and Lagrangian center-of-mass
coordinates at $z=2.3$ allow us to calculate uncertainties on the position and
final mass for each identified protocluster.

Figure 2:  Two examples of massive $M_{\rm vir}\geq 2\times
10^{14}\,h^{-1}\,M_{\odot}$ halos in our constrained simulations. The $z=0$
center of the virialized halo is shown as a red circle, with the member dark
matter particles shown as red dots. The blue dots show the same dark matter
particle positions at the $z=2.3$ snapshot. Additionally, we mark the $z=2.3$
center of mass position of the dark matter particles with a cyan diamond. The
coordinate axes are the same as in the matter density plot shown in Figure 1.

## Results

To compare with observed protoclusters in the literature (which we term
Literature-Reported Protocluster Candidates or ‘LRPCs’), we convert the
reported coordinates for each LRPC into equivalent Cartesian coordinates in
our simulation grid, where the observed redshift, $z^{\mathrm{obs}}$, is
converted into a line-of-sight comoving distance using a fiducial Planck 2018
cosmology Planck2018 . We then search for the center-of-mass coordinates of
COSTCO protoclusters within a spherical aperture of radius $15\,\,h^{-1}\,{\rm
Mpc}$. The following are our constrained simulations results for the LRPCs
(which are also summarized in Table 1):

ZFOURGE/ZFIRE:

This protocluster was first identified through medium-band imaging zfire1 ,
and subsequently confirmed through spectroscopic follow-up at $z=2.095$
Yuan2014 . We find a matched protocluster in all our realizations and predict
it to evolve into a Coma-like cluster halo of $M_{\rm vir}=(1.2\pm 0.3)\times
10^{15}\,h^{-1}\,M_{\odot}$. The COSTCO simulation ensemble predicts that the
center-of-mass will experience only very small displacements along the line-
of-sight between $z=2.1$ and $z=0,$ but there will be a significant systematic
displacement toward lower declination values at $z=0$. This suggests the
presence of another significant overdensity to the South of the current data
sample, which we will follow-up in a separate paper.

Hyperion:

Various authors had successively discovered protoclusters at $2.423\leq
z^{\mathrm{obs}}\leq 2.507$, across a region spanning approximately
$\Delta\theta=0.5^{\circ}$ on the sky Diener2015 ; Chiang2015 ; Wang2016 ;
Casey2015 ; Lee2016 . It was then argued by Cucciati2018 that these
protoclusters likely form a connected structure, dubbed ‘Hyperion’, with seven
distinct density peaks that will evolve into a super-cluster by $z=0$, with a
total final mass of $4.8\times 10^{15}\,M_{\odot}$. COSTCO successfully
identifies massive cluster halos at the reported position of Hyperion in all
the realizations. Our constrained simulations show that a partial merging of
Hyperion is likely by $z=0$, but we never witnessed a complete merging of all
the constituent peaks to form one cluster only. Instead, COSTCO indicates
that, on average, four virialized clusters will coalesce out of Hyperion to
form a massive filamentary group of clusters with an aggregate mass of $M_{\rm
vir}=(2.5\pm 0.5)\times 10^{15}\,h^{-1}\,M_{\odot}$ within collapsed cluster
halos, spanning over a distance of $d_{\rm Hyp}=(65\pm 10)\,h^{-1}\,{\rm
Mpc}$. The most massive halo is found to have a virial mass of $M_{\rm
vir}(z=0)=(1.25\pm 0.35)\times 10^{15}\,h^{-1}\,M_{\odot}$. The final
collapsed structure is approximately equivalent in spatial extent and mass to
the Coma/A1367 filament in the Local Universe Fontanelli:1984 . Another analog
is the elongated supercluster core of the Sloan Great Wall structure
Einasto2016 , although the limited volume of the current COSMOS data does not
permit us to investigate whether the analogy holds to the full scale of the
Great Wall. Examples of 3D visualizations of Hyperion are shown in Figure 3
for 4 different realizations.

CC2.2:

In 42 out of 50 cases, we find a cluster of $M_{\rm vir}(z=0)=(4.2\pm
1.9)\times 10^{14}\,h^{-1}\,M_{\odot}$ eventually forming out of an
overdensity very close to the CC2.2 protocluster Darvish2020 . Our mean final
halo mass for CC2.2 is less massive than estimated by Darvish2020 , although
the tension is relatively weak at only $1.44\,\sigma$ and could easily be due
to the different methods and data sets.

G237:

We do not find a consistently forming cluster that arises from the reported
position of the recently discovered G237 protocluster, which was initially
detected as a far-IR excess in the Planck satellite data and is reported to
have $M_{\rm vir}(z=0)\sim 3\times 10^{14}\,\,h^{-1}\,M_{\odot}$. A closer
inspection of our galaxy data shows only 2 galaxies in a sphere of 5 arcmin
radius centered at R.A = 150.507∘, Dec = $2.31204^{\circ}$, and $z^{{\rm
obs}}=2.16$, where G237 is observed. We analyzed the same region in our
$z=2.3$ snapshots and searched for $z=0$ halos with $M_{\rm vir}\geq 2\times
10^{14}\,\,h^{-1}\,M_{\odot}$ that had their origin in this area. Throughout
the 50 realizations, this only happens twice within the observed volume.
However, we note that G237 is reported at a position that is near the margins
of our observed region and covered by only the zCOSMOS-Deep survey, i.e. a
region where we do not expect strong constraints. Nevertheless, our
constrained simulations only disfavor a protocluster at the $1.75\,\sigma$
level which is not significant especially considering the low mass reported by
Polletta2021 .

CCPC-z22-006:

At the reported coordinates by Franck2016 , namely $\rm{R.A.}=149.93^{\circ}$,
$\rm{Dec}=2.2^{\circ}$, and an observed redshift of $z^{\rm obs}=2.283$, we
cannot confirm a consistently forming protocluster. In the $z=2.3$ snapshot,
we can only see in 2 out of 50 cases progenitor particles corresponding to a
$z=0$ cluster within the search radius. However, in these cases the particles
are associated with a nearby forming protocluster $25\,\,h^{-1}\,{\rm Mpc}$
away, which we will discuss below. Also, in contrast to G237, CCPC-z22-006
lies within the well observed region of the used galaxy surveys and therefore,
our analysis tends to disfavor a protocluster at the exact position of
CCPC-z22-006.

Figure 3:  3D visualizations of the Hyperion supercluster filament for 4 realizations at $z=0$, showing that the filamentary structure occurs consistently within our ensemble of realizations. The dots indicate the comoving positions of the observed $z^{\mathrm{obs}}\approx 2.4-2.5$ galaxies that were used in the reconstructions.  Name [Ref.] | R.A. [deg] | Dec [deg] | $z^{\rm obs}$ |  Final Mass [$h^{-1}\,M_{\odot}$]
---|---|---|---|---
ZFOURGE/ZFIRE zfire1 ; Yuan2014 | $150.094$ | $2.251$ | $2.095$ | $(1.2\pm 0.3)\times 10^{15}$
G237* Polletta2021 | $150.507$ | $2.312$ | $2.16$ | $--$
CC2.2* Darvish2020 | $150.197$ | $2.003$ | $2.232$ | $(4.2\pm 1.9)\times 10^{14}$
CCPC-z22-006 Franck2016 | $149.930$ | $2.200$ | $2.283$ | $--$
Hyperion Cucciati2018 ; Chiang2015 ; Wang2016 ; Casey2015 ; Lee2016 ; Diener2015 | $150.093$ | $2.404$ | $2.468$ |  }720pt[$(2.5\pm 0.5)\times 10^{15}$]
$149.976$ | $2.112$ | $2.426$
$149.999$ | $2.253$ | $2.444$
$150.255$ | $2.342$ | $2.469$
$150.229$ | $2.338$ | $2.507$
$150.331$ | $2.242$ | $2.492$
$149.958$ | $2.218$ | $2.423$
|  |  |  |
COSTCO J100026.4+020940 | $150.110$ | $2.161$ | $2.298$ | $(4.6\pm 2.2)\times 10^{14}$
COSTCO J095924.0+021435 | $149.871$ | $2.229$ | $2.047$ | $(6.1\pm 2.5)\times 10^{14}$
COSTCO J100031.0+021630 | $150.129$ | $2.275$ | $2.160$ | $(5.3\pm 2.6)\times 10^{14}$
COSTCO J095849.4+020126 | $149.706$ | $2.024$ | $2.391$ | $(6.6\pm 2.3)\times 10^{14}$
COSTCO J095945.1+020528 | $149.938$ | $2.091$ | $2.283$ | $(4.3\pm 2.4)\times 10^{14}$
Table 1:  COSMOS field protoclusters in the redshift range of $2\leq z^{\rm
obs}\leq 2.52$. For protoclusters previously reported in literature, we report
the published coordinates but the final mass is from our analysis. An asterisk
(*) denotes that our study does not include the recent spectroscopic follow-up
observational data used for protoclusters G237 and CC2.2. The dashed line
($--$) denotes that our analysis does not find a $z=0$ halo consistently
forming at the reported position. The lower part of the table summarizes the
protoclusters found in this work with the $z=2.3$ center-of-mass positions and
the estimated $z=0$ final masses.

Apart from the LRPCs, we also find several protoclusters in the COSMOS field
that are, to our knowledge, newly-discovered through COSTCO. These typically
do not show up as strong overdensities at $2.00\leq z^{\mathrm{obs}}\leq
2.52$, but rather as extended milder overdensities that nevertheless contain
sufficient enclosed mass to collapse into clusters by $z=0$. This is a new
class of extended lower-mass protoclusters — with Virgo-like final masses of
$<10^{15}\,M_{\odot}$ — that have previously eluded detections but is now
revealed by our combination of large-scale spectroscopic data and constrained
simulation modelling. Since the observational data has slightly less
constraining power at these milder overdensities, this leads to larger
relative errors in the derived masses. Nevertheless, all the newly-discovered
protoclusters are robust $\sim 5\sigma$ detections that show up in the
majority of the constrained realizations:

COSTCO J100026.4+020940:

In 48 out of 50 cases, we find a cluster progenitor forming at
$\rm{R.A.}=150.110^{\circ}\pm 0.042^{\circ}$, $\rm{Dec}=2.161^{\circ}\pm
0.040^{\circ}$, and $z^{\rm obs}=2.298\pm 0.007$, with a final mass of $M_{\rm
vir}(z=0)=(4.6\pm 2.2)\times 10^{14}\,h^{-1}\,M_{\odot}$. An overdensity of
galaxies was first noted at this location by Lee2016 , but they did not pursue
a more detailed analysis. This halo forms $\sim 28\,h^{-1}\,{\rm Mpc}$ away
from the reported position of CCPC-z22-006. In the $z=2.3$ snapshot, we
noticed that the progenitor particles corresponding to COSTCO J100014.4+020748
and CCPC-z22-006 were partially overlapping in 2 out of these 48 cases.
However, the resulting cluster emerged at the position of COSTCO
J100014.4+020748.

COSTCO J095924.0+021435:

In 42/50 realizations we find a protocluster at $\rm{R.A.}=149.871^{\circ}\pm
0.055^{\circ}$, $\rm{Dec}=2.229^{\circ}\pm 0.069^{\circ}$, at a distance of
$z^{\rm obs}=2.047\pm 0.008$. The final mass is $M_{\rm vir}(z=0)=(6.1\pm
2.5)\times 10^{14}\,h^{-1}\,M_{\odot}$.

COSTCO J100031.0+021630:

In 49/50 cases we see a cluster progenitor forming at
$\rm{R.A.}=150.129^{\circ}\pm 0.051^{\circ}$, $\rm{Dec}=2.275^{\circ}\pm
0.082^{\circ}$, and $z^{\rm obs}=2.160\pm 0.009$. We find a $z=0$ mass of
$M_{\rm vir}=(5.3\pm 2.6)\times 10^{14}\,h^{-1}\,M_{\odot}$.

COSTCO J095849.4+020126:

We find a consistently forming protocluster at $\rm{R.A.}=149.706^{\circ}\pm
0.053^{\circ}$, $\rm{Dec}=2.024^{\circ}\pm 0.081^{\circ}$, and observed
redshift of $z^{\rm obs}=2.391\pm 0.008$ in 46 out of 50 simulations, with a
final mass of $M_{\rm vir}(z=0)=(6.6\pm 2.3)\times
10^{14}\,h^{-1}\,M_{\odot}$.

COSTCO J095945.1+020528:

We find a protocluster slightly to the South of Hyperion, which forms as a
separate cluster in 27 out of 50 cases at $\rm{R.A.}=149.938^{\circ}\pm
0.105^{\circ}$, $\rm{Dec}=2.091^{\circ}\pm 0.081^{\circ}$, and $z^{\rm
obs}=2.495\pm 0.006$. Our simulations predict an average mass of $M_{\rm
vir}(z=0)=(4.3\pm 2.4)\times 10^{14}\,h^{-1}\,M_{\odot}$. Closer inspection
reveals that even though in 40 realizations at $z^{\mathrm{obs}}=2.48$ a clear
overdensity forms in this vicinity, a cluster emerges only in 27 cases whereas
in the other cases the overdensity is tidally disrupted by the gravitational
drag of Hyperion. This cluster can potentially become a substructure of the
Hyperion supercluster.

We show the corresponding confidence map of the identified protoclusters in
the top panel of Figure 4 , numbered in the same order as in Table 1. The
bottom panel of Figure 4 shows the mean density of all 50 COSTCO $z=2.3$
snapshots with the galaxy positions overplotted as black dots.

Figure 4:  Top: Confidence contours of the identified protoclusters (center-
of-mass positions) with the DBSCAN method for all 50 realizations stacked into
the same volume. ZFIRE and Hyperion show clearly the strongest recurrence,
while the contours for COSTCO protoclusters are less distinct. Bottom:
Averaged density field of all 50 COSTCO $z=2.3$ snapshots. The galaxy
positions are superimposed with back dots. The mean density and protocluster
contours agree well with the galaxy distribution at $z=2.3$.

## Summary & Discussion

Based only on analysis of existing large-volume spectroscopic surveys, we have
increased the number of known galaxy protoclusters at $2<z<2.5$ in the COSMOS
field by 1.5-2$\times$, while pushing down to final masses of $M_{\rm
vir}(z=0)\approx 4\times 10^{14}\,h^{-1}\,M_{\odot}$. This represents a lower
mass detection limit than hitherto feasible with survey data sets, from which
protocluster detections were typically in the range of $M_{\rm vir}(z=0)\sim
10^{15}\,h^{-1}\,M_{\odot}$ Overzier2016 . This ability to detect and
simultaneously model more representative lower-mass protoclusters with large-
volume surveys will allow a more uniform characterization of their
demographics. In addition to the new objects, we have confirmed and directly
modeled the fate of several previously-known galaxy protoclusters in the
volume. The Hyperion aggregation Cucciati2018 , in particular, had been the
subject of scientific and public curiosity: its elongated, extended nature
made it difficult to ascertain whether it would collapse into a singular
massive cluster by $z=0$. Our results clearly show that it will not, and
instead points to a fate as multiple massive cluster cores embedded within a
giant filamentary supercluster.

These results illustrate the efficacy of constrained simulations applied to
high-redshift data, although it does require wide and deep spectroscopic
surveys that do not currently exist elsewhere on the sky apart from COSMOS.
However, in the next few years, wide-field massively multiplexed fiber
spectrographs deployed on 8m-class telescopes, such as the Subaru PFS
sugai:2015 and VLT-MOONS moons:2012 , will carry out large-scale high-redshift
surveys across angular footprints $\gtrsim 10\times$ larger than the combined
COSMOS data set that we used in our analysis. These upcoming surveys will
allow constrained realization techniques to be even more effective in
connecting cluster formation across cosmic time as well as the evolution of
their constituent galaxies and gas. Constrained simulations of these upcoming
high-redshift galaxy surveys will also allow us to probe early structure
formation for consistency with the $\Lambda$CDM model with increasing
sensitivity to lower mass galaxy (proto)clusters. Moreover, the identification
of individual protoclusters (as presented in this work) combined with
complementary studies such as X-ray and Ly$\alpha$ observations will allow us
to study gas accretion of star-forming galaxies within these protoclusters and
can provide vital model constraints for high precision hydrodynamical
simulations Overzier2016 . Each identified protocluster represents a unique
environment to study the morphologies and merger rates of member galaxies in
high-density environments at $z\geq 2$, which is, to date, only possible in
theoretical studies or random cosmological simulations.

## References

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## Methods

### Input Galaxy Surveys

In this section, we summarize the galaxy catalogs that have been used in this
study to compute the initial conditions. In total, we employed the data of
four distinct spectroscopic galaxy redshift surveys, namely:

  * •

The new release of the zCOSMOS-Deep Lilly2006 catalog, in which the astrometry
has been updated to match the COSMOS2015 photometric galaxy catalog
Laigle2016. It also incorporates detailed visual inspections of the spectra
that has led to more reliable redshift confidence flags compared to the
original catalog (Lilly et al., 2021 in prep).

  * •

The VIMOS Ultra Deep Survey (VUDS) LeFevre2015 catalog, which was also covered
substantial parts of the COSMOS field. The galaxies were originally selected
for spectroscopy based on photometric redshifts from Ilbert2013.

  * •

The final data release from MOSFIRE Deep Evolution Field (MOSDEF) Survey
Kriek2015, whose target selection is based on Hubble Space Telescope grism
observations Brammer2012.

  * •

The KECK/MOSFIRE Spectroscopic Survey of Galaxies in Rich Environments at
$z\sim 2$ (ZFIRE) zfire2, which is a spectroscopic follow-up of the ZFOURGE
near-IR medium-band imaging program zfourge.

Supplementary Figure 5:  Left: Galaxy positions on the sky for the four
different surveys. The black crosses show the observationally reported
overdensities.The blue diamonds are protoclusters identified in this work,
dubbed as COSTCO. Right: Redshift distribution of the four surveys in the same
colour code as the left plot. We chose a bin width of $\Delta z^{\rm
obs}=0.1$. The grey dashed lines indicate the used redshift range.

Among the different surveys, we consider galaxies that are closer than
$\theta<0.1^{\prime\prime}$ on the sky and additionally have a redshift
separation of less than $\Delta z^{\rm obs}<0.01$ as duplicates and remove the
one with lower redshift confidence. The corresponding footprints and galaxy
number density as a function of redshift are shown in Supplementary Figure 5.
Further, we show the angular positions of the Literature Reported Protocluster
Candidates (LRPCs) and the additional COSTCO protocluster positions (this
work) as black crosses and blue diamonds, respectively. The redshift selection
criteria and the calculations of the survey selection functions are described
in detail in Ata2021.

### Initial Conditions

In this section, we describe the initial conditions that were used to seed the
constrained simulations.

In contrast to standard simulations starting from randomly-generated Gaussian
fluctuations, Crocce2006, Hahn2011, Pilipenko2017, Tatekawa2020, we seek to
find initial conditions that resemble the COSMOS galaxy distribution.

First attempts to infer initial conditions from Gaussian random fields came
into being nearly three decades ago
Dekel1990,Hoffman1991,Gramann1993,Kolatt1997 as starting point for
cosmological simulations. Even though conceptually pioneering works, these
methods suffered from low resolution and over-simplified assumptions, impeding
further scientific applications. The technique has received renewed interest
over the past decade thanks to advances in computing hardware as well as novel
sampling techniques Jasche2013,Wang:2014,Ata2015,Jasche:2015.

In Ata2021 the relevant initial Gaussian density field
$\delta(\boldsymbol{q})$ for this work was reconstructed corresponding to the
COSMOS field using a modified multi-survey version of the COSMIC BIRTH code
Kitaura2019, covering the observed redshift range $1.4\leq z^{\rm obs}\leq
3.6$. COSMIC BIRTH relies on the Hamiltonian Monte-Carlo (HMC) sampling
Neal2012 technique, working on a discretized volume where each voxel of the
reconstructed density field is a parameter of the posterior probability
function. The algorithm estimates the posterior distribution of initial
conditions that, after gravitationally evolving, would resemble the binned
galaxy density field. The binned galaxy density field is therefore connected
to the initial density field via a bias model, and the survey selection
function at each grid cell. This process is fully described in Ata2021. We
computed five separate HMC chains with COSMIC BIRTH, starting at different
initial seeds. Each chain ran at the minimum for $\sim 10,000$ iterations.
After the burn-in phase, when the Markov Chain Monte Carlo (MCMC) reaches the
stationary distribution, we randomly selected 50 individual realizations. We
verified that the iterations were not closer than 500 MCMC steps within the
chain to guarantee independent samples and accurately evaluate the posterior
distribution.

We perform the initial conditions inference in a cubical box with
$L_{\rm{Box}}=512\,\,h^{-1}\,{\rm Mpc}$ comoving side length and a mesh grid
of $N_{\rm C}=256^{3}$, resulting a cell resolution of
$d_{\rm{L}}=2\,h^{-1}\,{\rm Mpc}$. For the current analysis, we follow the
same overall method as in Ata2021 but restrict the reconstruction region to a
redshift range of $2.00\leq z^{\rm obs}\leq 2.52$, in which a large number of
reported protoclusters are located. We transform the coordinate system of the
surveys so that they are aligned with the Cartesian axes of the COSMIC BIRTH
box in the plane-parallel approximation and place the data region in the
center of our reconstructed volume. The initial density fields are then
calculated corresponding to a redshift of $z=100$.

To translate the initial density fields $\delta(\boldsymbol{q})$ into initial
seeds for cosmological simulations, we compute the three-dimensional white
noise field $wn(\boldsymbol{k})$ in Fourier space, defined as:

$wn(\boldsymbol{k})=\frac{\delta(\boldsymbol{k})}{\sqrt{P(\boldsymbol{k})}}\,,$
(1)

where $\delta(\boldsymbol{k})$ is the Fourier-transformed initial density
field and $P(\boldsymbol{k})$ is the theoretical three-dimensional
$\Lambda$CDM power spectrum obtained with the CAMB code Lewis2000. We read the
white noise field into the MUSIC code (MUlti-Scale Initial Conditions)
Hahn2011. MUSIC places a dark matter particle in each cell of the initial
density mesh grid and uses the density contrast to calculate the initial
displacement and velocity field for each particle with second order Lagrangian
perturbation theory.

### Constrained Simulation Methodology

Over the last decade constrained simulations have helped to understand
structure formation and the dynamics of the local Universe
Sawala2016,Sorce2016, the cosmic density from Lyman-$\alpha$ tomographic
observations Horowitz:2019, and also more recently to study massive galaxies
2020MNRAS.493.4607R

In this work, we extend this concept to Cosmic Noon by running $N$-body
simulations based on the COSMOS initial conditions introduced in section for
initial conditions using the massively parallelized PKDGRAV3 $N$-body code
Potter2017, which utilizes the Fast Multipole Method for gravity calculations
and has been tested in a 2 trillion particle cosmological simulation. MUSIC
returns the initial conditions files in the TIPSY format, that can be directly
read in PKDGRAV3. In this study, we are mainly interested in the galaxy
cluster evolution from the observed redshifts of the galaxy surveys, at
$z=2.3$, until $z=0$.

However, to precisely describe the observed structures over the time-evolution
of their lightcones, we also output six PKDGRAV3 snapshots in the interval of
$z=2-2.5$ with a separation of $\Delta z=0.1$.

Supplementary Figure 6:  Pairs of matter density field slice plots showing the
redshift snapshot of $z=2.3$ (which is the mean redshift of the input
galaxies) on the top panel and the $z=0$ snapshot on the bottom. We signpost
the galaxy positions with black dots on top of the $z=2.3$ snapshot. We
transformed the galaxy surveys to a coordinate frame where the line of sight
direction is aligned with the abscissa.

In Supplementary Figure 6 we show four different constrained simulations in
pairs of $z=2.3$ and $z=0$. We also display the galaxy positions with black
dots on top of the $z=2.3$ snapshot. The density contrast of the $z=2.3$
snapshot closely follows the galaxy distribution in all cases. Outside the
observed region, i.e. $\sim\pm 20\,h^{-1}\,{\rm Mpc}$ from the edges along
line of sight and $\sim\pm 10\,h^{-1}\,{\rm Mpc}$ in transverse direction, the
realizations have been substituted by a random field created from the
$\Lambda$CDM power spectrum. Differences in the small-scale structure of the
realizations, in addition to the variation of random fields outside the
observed area, lead to notably different final density fields at $z=0$.
However, we see consistently forming large-scale structure, such as the strong
overdensity centered at $z^{\rm obs}\sim 2.1$, and an extended filamentary
structure centered at $\langle z^{\rm obs}\rangle\sim 2.45$. Other, less well
constrained regions in the observed volume, show that the $z=0$ outcome can be
very different. For example, within the four $z=2.3$ snapshot shown in
Supplementary Figure 6, we can see different fates of one of the overdensities
at $z^{\mathrm{obs}}\sim 2.05$: in one case, it collapses into a singular
cluster at $z=0$; in two cases becomes part of a wider overdense region; and
in one case diffuses into a shallow density. Also, we can see that in the
region between $2.12\lesssim z^{\mathrm{obs}}\lesssim 2.18$, a few density
peaks consistently form at the $z=2.3$ snapshots. However, their final state
can be a singular high density filament at $z=0$, or, several aligned
filaments that run in parallel to each other. Finally, it is worth mentioning
that even though in all realizations the Hyperion supercluster is forming a
massive filament, the internal structure changes. We can see a different
number of member halos forming within the filament, mostly consisting of 4-5
massive halos (see also Figure 3 for a better visual impression). Future deep
surveys with higher galaxy sampling compared to our dataset, in conjunction
with improved modelling of redshift-space distortions and other effects,
should allow more consistent predictions on the evolution of these structures.

We also tested for a possible systematic influence of the simulations’ grid
resolution on the final results. To do so, we increase the resolution of our
initial conditions from $N_{\rm C}=256^{3}$ to $1024^{3}$ by augmenting small-
scale modes in the Gaussian random field, while preserving the large-scale
ones. This procedure is justified since we are interested in the most massive
halos of these simulations, regardless of the halo substructures. The results
of the high-resolution simulation is consistent with those of the low-
resolution run performed with the same initial conditions, yielding clusters
that agree to within $\sim 2\%$ in position and virial mass. This is clearly a
subdominant effect as compared to the variance among the different
realizations, and we consider the grid resolution of $N_{\rm C}=256^{3}$ to be
converged for our purposes. The resemblance is shown in Supplementary Figure
7.

Supplementary Figure 7:  Comparison of the $256^{3}$ (top) and the $1024^{3}$
(bottom) simulation for the same initial conditions. We find a clear agreement
of the large-scale structure of both simulations and a matching of cluster
masses and positions with a $\sim 2\%$ accuracy.

### Halo Analysis

We identify halos from the PKDGRAV3 simulation snapshots using the ROCKSTAR
phase-space halo finder rockstar. Subsequently, we calculate the merger trees
across the PKDGRAV3 snapshots using the consistent-trees package
consistenttrees. We then select halos at $z=0$ that have a minimum virial mass
of $M_{\rm vir}\geq 2\times 10^{14}\,h^{-1}\,M_{\odot}$ Chiang2017. The
theoretical $\Lambda$CDM halo mass function at $z=0$ predicts $N_{\rm
Halos}\sim 26$ halos of $M_{\rm vir}\geq 2\times 10^{14}\,h^{-1}\,M_{\odot}$
in the considered COSMOS field volume hmf (see also Supplementary Figure 8).

Next, we trace the $z=0$ dark matter particles that ROCKSTAR associates with
each halo back to $z=2.3$, allowing us to self-consistently define a
protocluster as the set of all progenitor dark matter particles of a $z=0$
halo in its Lagrangian volume. The center-of-mass positions for each
protocluster (in the observed volume) are shown with blue diamonds in the
upper panel of Figure 1, while two examples of the virialized halos and their
Lagrangian volumes are shown in Figure 2.

#### Matching the LRPCs

Protocluster searches are a very important research topic for understanding
structure formation at high redshifts
protoclustersearch1,protoclustersearch2,protoclustersearch3,protoclustersearch4,protoclustersearch5,protoclustersearch6.

In this section, we match the protocluster’s center-of-mass position to the
LRPCs for all realizations by performing a nearest-neighbor search. For the
choice of the search radius we must take into account the following
conditions:

  * •

Cosmological displacements:
The most massive halos, that grow in an isolated environment, grow nearly in-
place with a final position at $z=0$ that is nearly identical to the center-
of-mass position of the protocluster. In contrast, less massive halos in dense
environments show significant displacements from $z=2.3$ to $z=0$ that can be
of order $\sim 10\,h^{-1}\,{\rm Mpc}$.

  * •

Incomplete information:
The observed galaxy catalogs contain limited information in two aspects.
First, the phase-space distribution of the galaxies is incomplete, as we only
have spatial information. The unknown peculiar velocities result in a
distortion along line of sight, called redshift space distortions (RSD). Also,
the velocity autocorrelation of galaxies at cosmological distances of
$r=100\,h^{-1}\,{\rm Mpc}$ can still be high Turner2021. Consequently,
structures outside of our observed volume have a significant influence on the
velocities of our halos, called super-survey modes in the literature
Takada2013.

  * •

Sampling variance:
The different initial conditions inferred with the COSMIC BIRTH code are
individual Markov-Chain realizations drawn from a posterior probability
distribution. This chain variance depends on the statistical model, as well as
on the marginalized nuisance parameters Kitaura2019, Ata2021, Ata2015.
Therefore, we naturally expect a certain variance within the Markov-Chain
samples.

To account for these displacements we define a search radius of $r_{\rm
search}=15\,h^{-1}\,{\rm Mpc}$ around the LRPCs and scan for the center-of-
mass positions of the $z=2.3$ protoclusters in their vicinity. Details on the
robustness and the justification for this search radius are given below. If
multiple candidates are found, we choose the most massive among them, or, if
their Lagrangian volume overlaps at $z=2.3$, we sum their masses at $z=0$. If
a matching of protoclusters and LRPCs is successful, we show the distance as
white line in the upper panel of Figure 1.

#### COSTCO Protoclusters

Besides identifying the LRPCs that were previously known, we look for other
recurringly forming halos within the COSTCO suite. For this purpose, we stack
all halos of the COSTCO multiverse into one volume and apply the Density-Based
Spatial Clustering of Applications with Noise (DBSCAN) algorithm, implemented
in the Scikit-learn package. This algorithm identifies clustered points given
the halo density of the COSTCO multiverse and allows to model noise of the
point cloud with an additional parameter. To tune this noise parameter, we
first apply DBSCAN to the LRPCs only. Once we find a best fit setup such that
the LRPCs are robustly identified, we apply the exact same scheme to the
remaining volume and look for additional clusters in the COSTCO multiverse. We
then use the center-of-mass position of these newly identified clusters as
anchor points and scan the 50 realizations at these positions for massive
halos. This has also been studied with the MIP suite using a signal-to-noise
ratio Aragon-Calvo2016. MIP is a set of correlated cosmological simulations
and is based on random initial conditions. The initial conditions of the MIP
realizations are consistent in their large-scale Fourier modes, but were
randomized on the small scales. This represents an interesting analogy to the
constrained COSTCO simulations. We show the corresponding confidence map of
the identified protoclusters on the top panel of Supplementary Figure 4 for
all 50 realizations stacked into the same volume. We write the names of the
LRPCs and COSTCO protoclusters (the same order as in Table 1) on top of the
identified regions. It is clearly noticeable that the ZFIRE and Hyperion
structures show the strongest significance, while the other structures have
more extended and less distinct contour regions, which means that the
uncertainty of these structures is higher. Nevertheless, all identified COSTCO
protoclusters form in the majority of our realizations. This shows that
constrained simulations are a well-suited method for the identification of
less massive protoclusters, that would be less efficient with other methods.

The bottom panel of Figure 1 shows the $z=0$ snapshot and the two kinds of
encircled halos. White circles indicate halos for which a protocluster-LRPC
matching was successful. This means that the white circles represent the
predicted $z=0$ state of the LRPCs. Blue circles show massive halos that could
not be linked to a LRPC partner and are newly discovered by our analysis.

On the bottom panel of Figure 4 we show the mean density of all 50 COSTCO
$z=2.3$ snapshots. While densities outside the observed volume average out, we
see an excellent agreement of the averaged structures and galaxy distribution
within the constrained volume. Interestingly, on average COSTCO V arises from
a stronger overdensity than COSTCO II, but does not evolve to a higher mass
than COSTCO II (Table 1). The main difference of these two structures lies in
their respective large-scale environments: COSTCO II forms in a more isolated
environment and can accrete surrounding matter without disruption, whereas
COSTCO V is within the vicinity of the Hyperion filament/supercluster and
likely experiences some form of large-scale mass stripping by its more massive
neighbor. This stresses the importance of the large-scale environment for
determining the evolution of a protocluster rather than purely relying on
local galaxy number density.

#### Robustness of the Halo Identifications

We asses the robustness of our protocluster identifications. In particular, we
want to know the probability of finding a massive halos at any given fixed
position in a search radius of $r_{\rm Search}=15\,h^{-1}\,{\rm Mpc}$ within
our simulations by chance. The theoretical $\Lambda$CDM halo mass function
predicts about $\mathrm{n}_{\rm H}~{}\sim 8\times
10^{-6}\,h^{3}\,\rm{Mpc}^{-3}$ halos per unit volume with $M_{\rm vir}\geq
2\times 10^{14}\,h^{-1}\,M_{\odot}$. Thus, we estimate the expected number of
these halos in a volume of $V_{\rm H}=4/3\,\pi\,r_{\rm Search}^{3}$ to be
$V_{\rm H}\times\mathrm{n}_{\rm H}\sim 0.1$. Thus, assuming that the halo
distribution in 50 separate volumes is an independent and identically
distributed random variable, we expect 5 halo detections in this search radius
by chance.

To confirm this estimate numerically, we ran five simulations with the same
resolution and same set of cosmological parameters as COSTCO, but with random
initial conditions. We then chose 10 fixed positions inside these random
simulations and searched for $M_{\rm vir}\geq 2\times
10^{14}\,h^{-1}\,M_{\odot}$ halos in a radius of $r_{\rm
Search}=15\,h^{-1}\,{\rm Mpc}$, ensuring that these points were separated by
at least $100\,h^{-1}\,{\rm Mpc}$. On average we found $\sim 0.5$ halos per
position. Extrapolating this number to 50 realizations, we therefore expect to
find $\sim 5$ halos by chance at a random position inside the COSTCO suite.
This agrees with the analytical estimate from the halo mass function.

Relying on Poisson statistics, we consider a COSTCO halo to be a robust
$5\,\sigma$ detection, if its counterparts are found in at least $\geq 27.5$
realizations at the same position given the search radius of $15\,h^{-1}\,{\rm
Mpc}$.

All our claimed COSTCO protoclusters shown in Table 1 fulfill this detection
threshold, except for COSTCO J095945.1+020528 which was found in 27 out of 50
realizations ($4.9\,\sigma$), but we have nevertheless decided to include it
as a detection. Note that apart from this object and the other listed
protoclusters, the next most significant candidate is found in only 8 out of
50 realizations ($1.45\,\sigma$). This further validates our decision to
include COSTCO J095945.1+020528, since it clearly rises above the random
background noise.

For the robust identification of halos with smaller masses, we would need to
shrink the search radius in order not to be contaminated by random detections.
However, this requires a better knowledge of the $z=2.3$ to $z=0$ displacement
field, which in turn requires a higher number density of tracers that
constrains the initial conditions. In a follow-up paper, we will forecast the
sensitivity of our methods toward lower-mass clusters in future high-redshift
spectroscopic surveys.

#### COSTCO Halo Mass Function

We compare the halo mass functions (HMF) from the COSTCO realizations to the
five random simulations and also to the theoretical HMF that we calculated
with the HMFcalc package. As we can see in Supplementary Figure 8, the COSTCO
as well as the random initial conditions simulations show close agreement with
the theoretical HMF within the uncertainties. Also we can see that the high
mass end at $M_{\rm{Halo}}>10^{15}\,\,h^{-1}\,M_{\odot}$ gets very noisy due
to the low number of halos.

Supplementary Figure 8:  Mean halo mass function of the COSTCO runs (blue
dashed line), five simulations with random initial conditions (orange dashed
line) and the theoretical prediction (red line). Error bars in the same color
as solid lines.

Within the statistical uncertainty of the HMF, we can see that the COSTCO
simulations suite is consistent with the $\Lambda$CDM predictions. However, in
contrast to the random simulations, the COSTCO HMF shows two bumps in the
high-mass tail that slightly exceed the random simulations. We find that these
bumps are caused by the ZFIRE and Hyperion cluster and may hint towards an
overabundance of massive halos in the studied volume. We will the study the
significance of the massive halos in more detail in a subsequent work.

#### Effective Scale of the COSTCO Realizations

The COSTCO simulation ensemble shares the same large-scale structure, while
small scales can notably be different. Therefore, we evaluate the effective
scale on which the different realizations are correlated.

This is done as follows.

  * •

We compute the mean of all 50 COSTCO $z=2.3$ snapshots (see Supplementary
Figure 4 bottom panel). We select a rectangular volume that contains the
observed region.

  * •

We apply Gaussian smoothing with kernel sizes of $r_{\rm S}=3-6\,h^{-1}\,{\rm
Mpc}$ on a simulation with random initial conditions and select the same
volume.

  * •

We compute the power spectra of the COSTCO mean field and the variously
smoothed random simulation in the selected volume.

Supplementary Figure 9:  Assessment of the effective scale calculated from the
COSTCO simulations with the power spectra of the COSTCO mean field (dashed
black line) and the smoothed reference simulation with random initial
conditions (solid colored lines). The COSTCO realizations agree on scales that
correspond to a Gaussian smoothing of $r_{\rm S}\sim 4.5\,h^{-1}\,{\rm Mpc}$.

We show the resulting power spectra in Supplementary Figure 9. The dashed
black line shows the power spectrum for the COSTCO mean field, while the solid
lines show the different smoothing scales of the reference simulation that we
ran with random initial conditions. Comparing the power spectra we can
directly read off, that the COSTCO simulations are effectively correlated on
scales of $r_{\rm{S}}\sim 4.5\,h^{-1}\,{\rm Mpc}$ (compare green and orange
solid line) while fluctuating for smaller scales. The correlated scales are
well suited to analyze the formation and fate of galaxy protoclusters,
focusing on the final masses and positions.

## Software

This research made use of matplotlib Hunter:2007, SciPy Virtanen_2020, Scikit-
learn scikit-learn, Astropy 2018AJ….156..123A, 2013A&A…558A..33A, HMFcalc hmf,
Dark Emulator Nishimichi2019, pynbody pynbody, and ytree ytree.

sn-mathphys sn-bibliography

## Acknowledgements

The authors thank Peter Behroozi and Doug Potter for their support with
Rockstar and PKDGRAV3, respectively. MA thanks Takahiro Nishimichi, Masahiro
Takada, and Valeri Vardanyan for useful discussions. MA was supported by JSPS
Kakenhi Grant JP21K13911. KGL acknowledges support from JSPS Kakenhi Grants
JP18H05868 and JP19K14755. CDV acknowledges support from the Spanish Ministry
of Science and Innovation (MICIU/FEDER) through research grants
PGC2018-094975-C22 and RYC-2015-18078.

This research is based on observations undertaken at the European Southern
Observatory (ESO) Very Large Telescope (VLT) under Large Program 175.A-0839
and has been supported by the Swiss National Science Foundation (SNF). Some of
the material presented in this paper is based upon work supported by the
National Science Foundation under Grant No. 1908422. This work was made
possible by the World Premier International Research Center Initiative (WPI),
MEXT, Japan.

## Author contributions statement

MA calculated the selection functions, initial conditions, simulations, halo
catalogs, and conducted the final analysis. MA and KGL conceptualized the
analysis and wrote the paper. CDV helped performing the simulations. FSK
provided expertise about density field reconstructions. OC, BCL, and DK
provided the (partly) proprietary data and relevant literature. TM provided
the visualizations. All authors reviewed the manuscript.

## Data availability

The data products generated for this study are available at
https://zenodo.org/record/6425013.

## Software availability

Analysis codes used for this study are available at
https://github.com/gmetin/COSTCO.

## Competing Interests

The authors declare that they have no competing financial interests.

## Correspondence

Correspondence and requests for materials can be addressed to the
corresponding author M.A. (email<EMAIL_ADDRESS>


# Fixed Point Theorem: Variants, Affine Context and Some Consequences

Anderson L. A. de Araujo  Universidade Federal de Viçosa, Departamento de
Matemática
Av. Peter Henry Rolfs, s/n, Viçosa, MG, Brazil, CEP 36570-900
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Edir J. F. Leite
Universidade Federal de Viçosa, Departamento de Matemática
Av. Peter Henry Rolfs, s/n, Viçosa, MG, Brazil, CEP 36570-900
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we will present variants Fixed Point Theorem for the affine and
classical contexts, as a consequence of general Brouwer’s Fixed Point Theorem.
For instance, the affine results will allow working on affine balls, which are
defined through the affine $\displaystyle L^{p}$ functional
$\displaystyle\mathcal{E}_{p,\Omega}^{p}$ introduced by Lutwak, Yang and Zhang
in the work Sharp affine $\displaystyle L_{p}$ Sobolev inequalities, J.
Differential Geom. 62 (2002), 17-38 for $\displaystyle p>1$ that is non convex
and does not represent a norm in $\displaystyle\mathbb{R}^{m}$. Moreover, we
address results for discontinuous functional at a point. As an application, we
study critical points of the sequence of affine functionals
$\displaystyle\Phi_{m}$ on a subspace $\displaystyle W_{m}$ of dimension
$\displaystyle m$ given by

$\displaystyle\Phi_{m}(u)=\frac{1}{p}\mathcal{E}_{p,\Omega}^{p}(u)-\frac{1}{\alpha}\|u\|^{\alpha}_{L^{\alpha}(\Omega)}-\int_{\Omega}f(x)udx,$

where $\displaystyle 1<\alpha<p$, $\displaystyle[W_{m}]_{m\in\mathbb{N}}$ is
dense in $\displaystyle W^{1,p}_{0}(\Omega)$ and $\displaystyle f\in
L^{p^{\prime}}(\Omega)$, with
$\displaystyle\frac{1}{p}+\frac{1}{p^{\prime}}=1$.

Keywords: Fixed Point Theorems; Affine Balls; Affine $\displaystyle L^{p}$
Energy; Nontrivial Critical Point
_2020 MSC_ : 47H10; 93B24; 46T20; 35B38

##  1 Introduction

Let $\displaystyle\Omega\subset\mathbb{R}^{n}$, $\displaystyle n\geq 2$ be a
bounded open subset. The affine $\displaystyle L^{p}$ functional (or energy)
for functions $\displaystyle u\in W^{1,p}_{0}(\Omega)$ is given by

$\displaystyle\mathcal{E}_{p,\Omega}(u):=\gamma_{n,p}\left(\int_{\mathbb{S}^{n-1}}\|\nabla_{\xi}u\|_{L^{p}(\Omega)}^{-n}\
d\sigma(\xi)\right)^{-\frac{1}{n}},$

where
$\displaystyle\gamma_{n,p}=\left(2\omega_{n+p-2}\right)^{-1}\left(n\omega_{n}\omega_{p-1}\right)\left(n\omega_{n}\right)^{p/n}$.
In this paper, $\displaystyle\nabla_{\xi}u(x)$ represents the directional
derivative $\displaystyle\nabla u(x)\cdot\xi$ with respect to the direction
$\displaystyle\xi\in\mathbb{S}^{n-1}$ and $\displaystyle\omega_{\kappa}$ is
the volume of the unit Euclidean ball in $\displaystyle\mathbb{R}^{\kappa}$.

Below are some important properties related to this functional on
$\displaystyle W^{1,p}_{0}(\Omega)$, with $\displaystyle p>1$. Namely:

  * (I)

$\displaystyle{\cal E}^{p}_{p,\Omega}$ does not define a norm in
$\displaystyle W^{1,p}_{0}(\Omega)$ (see Proposition 4.1 of [15]);

  * (II)

The affine $\displaystyle L^{p}$ energy $\displaystyle{\cal E}^{p}_{p,\Omega}$
is non convex on $\displaystyle W^{1,p}_{0}(\Omega)$ (see Proposition 4.1 of
[15]);

  * (III)

The affine ball $\displaystyle\overline{B}_{\varrho}(0):=\\{u\in
W^{1,p}_{0}(\Omega);{\cal E}_{p,\Omega}(u)\leq\varrho\\}$ is compactly
immersed into $\displaystyle L^{s}(\Omega)$ for any $\displaystyle 1\leq
s<p^{*}$, where $\displaystyle p^{*}:=\frac{np}{n-p}$ in case $\displaystyle
1<p<n$, and for any $\displaystyle s\geq 1$, in case $\displaystyle p\geq n$,
(see Theorem 6.5.3 of [21]) and is unbounded in $\displaystyle
W^{1,p}_{0}(\Omega)$ (see Theorem 2 of [11]);

  * (IV)

The affine $\displaystyle L^{p}$ functional
$\displaystyle\mathcal{E}_{p,\Omega}^{p}$ is strongly continuous and Fréchet
differentiable in $\displaystyle W^{1,p}_{0}(\Omega)$ (see Theorem 1.1 of
[16]);

  * (V)

The affine $\displaystyle L^{p}$ energy
$\displaystyle\mathcal{E}_{p,\Omega}^{p}$ is $\displaystyle C^{1}$
differentiable in $\displaystyle W^{1,p}_{0}(\Omega)\backslash\\{0\\}$ (see
Theorem 1.1 of [16]).

In 2002, the affine $\displaystyle L^{p}$ energy
$\displaystyle\mathcal{E}_{p,\Omega}$ was introduced in [18] for
$\displaystyle p>1$. In the sequence several related results were developed
(see for example in [5, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 21]). For
$\displaystyle p=1$, we refer to [14] and references therein.

Let $\displaystyle\mathcal{B}=\\{w_{1},w_{2},\dots\\}$ be a Schauder basis of
$\displaystyle W^{1,p}_{0}(\Omega)$ (see [6, 2]). For each $\displaystyle
m\geq 1$, let

$\displaystyle W_{m}:=[w_{1},w_{2},\dots,w_{m}]$

be the $\displaystyle m$-dimensional subspace of $\displaystyle
W^{1,p}_{0}(\Omega)$ generated by $\displaystyle\\{w_{1},w_{2},\dots,w_{m}\\}$
with norm induced from $\displaystyle W^{1,p}_{0}(\Omega)$ and consider
$\displaystyle\zeta=(\zeta_{1},\zeta_{2},\dots,\zeta_{m})\in\mathbb{R}^{m}$.
We defined the functions

$\displaystyle\lfloor\zeta\rfloor_{m}:=\mathcal{E}_{p,\Omega}\left(\sum_{j=1}^{m}\zeta_{j}w_{j}\right)$

and

$\displaystyle\|\zeta\|_{1,p,m}:=\|\sum_{j=1}^{m}\zeta_{j}w_{j}\|_{W^{1,p}_{0}(\Omega)}.$

It is clear that $\displaystyle\|\cdot\|_{1,p,m}$ define a norm in
$\displaystyle\mathbb{R}^{m}$ (see [3] for the details).

By using the above notation, we can identify the spaces
$\displaystyle(W_{m},\|\cdot\|_{W^{1,p}_{0}(\Omega)})$ and
$\displaystyle(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$ by the isometric linear
transformation

$u=\sum_{j=1}^{m}\zeta_{j}w_{j}\in
W_{m}\mapsto\zeta=(\zeta_{1},\ldots,\zeta_{m})\in\mathbb{R}^{m}.$ (1)

Below are some fundamental inequalities for the development of the paper: We
start with the affine $\displaystyle L^{q}$ Poincaré-Sobolev inequality on
$\displaystyle W^{1,p}_{0}(\Omega)$, $\displaystyle q\in[1,p^{*}]$ (see
inequality (4) in [15]): There is an optimal constant
$\displaystyle\mu_{p,q}^{\cal A}=\mu_{p,q}^{\cal A}(\Omega)>0$ such that

$\mu_{p,q}^{\cal A}\|u\|_{L^{q}(\Omega)}\leq{\cal E}_{p,\Omega}(u).$ (2)

Now, by Theorem 9 of [11], we have

$\displaystyle C_{n,p}(\Omega)\|u\|_{L^{p}(\Omega)}^{\frac{n-1}{n}}\|\nabla
u\|_{L^{p}(\Omega)}^{\frac{1}{n}}\leq\mathcal{E}_{p,\Omega}(u)\leq\|\nabla
u\|_{L^{p}(\Omega)}$

for all $\displaystyle u\in W^{1,p}_{0}(\Omega)$, where $\displaystyle
C_{n,p}(\Omega)$ is a positive constant. For a proof of second inequality see
for example to page 33 in [18]. Thus, there exists a constant $\displaystyle
C=C(n,p,m,\Omega)>0$ such that

$C\|\nabla u\|_{L^{p}(\Omega)}\leq\mathcal{E}_{p,\Omega}(u)\leq\|\nabla
u\|_{L^{p}(\Omega)}$ (3)

for all $\displaystyle u\in W_{m}$. It is clear that the norms
$\displaystyle\|\nabla\cdot\|_{L^{p}(\Omega)}$,
$\displaystyle\|\cdot\|_{L^{p}(\Omega)}$ and
$\displaystyle\|\nabla_{\xi}\cdot\|_{L^{p}(\Omega)}$ are equivalents in
$\displaystyle W_{m}$ for every $\displaystyle\xi\in\mathbb{S}^{n-1}$.
Moreover, by Lemma 1 of [11], there is a positive constant $\displaystyle
D_{1}>0$ such that

$\displaystyle
D_{1}\|u\|_{L^{p}(\Omega)}\leq\|\nabla_{\xi}u\|_{L^{p}(\Omega)}\leq\|\nabla
u\|_{L^{p}(\Omega)}$

for every $\displaystyle\xi\in\mathbb{S}^{n-1}$ and for all $\displaystyle
u\in W^{1,p}_{0}(\Omega)$. Thus, there exists $\displaystyle D_{2}>0$
independent of $\displaystyle\xi$, such that

$D_{2}\|\nabla
u\|_{L^{p}(\Omega)}\leq\|\nabla_{\xi}u\|_{L^{p}(\Omega)}\leq\|\nabla
u\|_{L^{p}(\Omega)}$ (4)

for every $\displaystyle\xi\in\mathbb{S}^{n-1}$ and for all $\displaystyle
u\in W_{m}$. From inequalities (3) and (4), we get

$C\|\nabla_{\xi}u\|_{L^{p}(\Omega)}\leq\mathcal{E}_{p,\Omega}(u)\leq
D_{3}\|\nabla_{\xi}u\|_{L^{p}(\Omega)}$ (5)

for every $\displaystyle\xi\in\mathbb{S}^{n-1}$ and for all $\displaystyle
u\in W_{m}$, where $\displaystyle C,D_{3}>0$ are independents of
$\displaystyle\xi$.

To prove our main result, we will need to understand some results of the
affine theory in finite dimension. In this sense, we concentrate attention on
the following central ingredients. Namely:

  * (i)

$\displaystyle\lfloor\cdot\rfloor_{m}$ does not define a norm in
$\displaystyle\mathbb{R}^{m}$;

  * (ii)

The affine ball
$\displaystyle\overline{B}^{m}_{\varrho}(0):=\\{z\in\mathbb{R}^{m};\lfloor
z\rfloor_{m}\leq\varrho\\}$ is a non convex subset of
$\displaystyle\mathbb{R}^{m}$;

  * (iii)

The affine ball
$\displaystyle\overline{B}^{m}_{\varrho}(\zeta_{0}):=\\{z\in\mathbb{R}^{m};\lfloor
z-\zeta_{0}\rfloor_{m}\leq\varrho\\}$ is a closed and bounded subset of
$\displaystyle\mathbb{R}^{m}$;

  * (iv)

$\displaystyle\overline{B}^{m}_{\varrho}(\zeta_{0})$ and
$\displaystyle\overline{B}^{p,m}_{1}(0):=\\{z\in\mathbb{R}^{m};\|z\|_{1,p,m}\leq
1\\}$ are homeomorphic in $\displaystyle(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$
(see Lemma 2.2 in Section 2).

Note that (i) and (ii) follow of isometric linear transformation (1) and
Proposition 4.1 of [15]. Note also that $\displaystyle\lfloor\cdot\rfloor_{m}$
does not satisfy the triangular inequality in $\displaystyle\mathbb{R}^{m}$.
For (iii) applying the weak lower semi-continuity of the functional
$\displaystyle u\in W^{1,p}_{0}(\Omega)\rightarrow{\cal E}_{p,\Omega}^{p}(u)$
(see Theorem 2.1 of [15]), we obtain
$\displaystyle\overline{B}^{m}_{\varrho}(\zeta_{0})$ is a closed subset of
$\displaystyle\mathbb{R}^{m}$. Now, the boundedness follows from the
inequality (3).

Our main objective in this work is to obtain a topological tool to be able to
attack problems in the affine and classical contexts. Our first result is an
affine Fixed Point Theorem. Namely:

######  Theorem 1.1.

Let $\displaystyle
F:(\mathbb{R}^{m},\|\cdot\|_{1,p,m})\rightarrow(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$
be a continuous function such that $\displaystyle\left\langle
F(\zeta),\zeta\right\rangle\geq 0$ for every
$\displaystyle\zeta\in\mathbb{R}^{m}$ with
$\displaystyle\lfloor\zeta\rfloor_{m}=\varrho$ for some
$\displaystyle\varrho>0$, and
$\displaystyle\langle\cdot,\cdot\rangle^{1/2}=|\cdot|_{2}$, where
$\displaystyle|x|_{2}$ denote the usual euclidean norm in
$\displaystyle\mathbb{R}^{m}$. Then, there exists $\displaystyle z_{0}$ in the
closed affine ball
$\displaystyle\overline{B}^{m}_{\varrho}(0)=\\{z\in\mathbb{R}^{m};\lfloor
z\rfloor_{m}\leq\varrho\\}$ such that $\displaystyle F(z_{0})=0$.

This result is well known when the function involved is a norm and the ball is
a convex subset of $\displaystyle\mathbb{R}^{m}$ (see [4] and [13]). As a
counterpart of the classical context (see for instance in [1, 4, 3, 20] and
references therein), Theorem 1.1 associated with the Galerkin method will be
very useful for solving affine problems. We refer to [11, 15, 16] for papers
focused on affine problems via variational methods.

Our second result is a affine Fixed Point Theorem for discontinuous functional
at a point, that is, a variant of the Theorem 1.1. Precisely:

######  Theorem 1.2.

Let $\displaystyle
F:(\mathbb{R}^{m}\setminus\\{y_{0}\\},\|\cdot\|_{1,p,m})\rightarrow(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$
be a continuous function such that $\displaystyle\left\langle
F(\zeta),\zeta-\zeta_{0}\right\rangle\geq 0$ for every
$\displaystyle\zeta\in\mathbb{R}^{m}\setminus\\{y_{0}\\}$ with
$\displaystyle\lfloor\zeta-\zeta_{0}\rfloor_{m}=\varrho$ for some
$\displaystyle 0<\varrho<\lfloor\zeta_{0}-y_{0}\rfloor_{m}$ and
$\displaystyle\langle\cdot,\cdot\rangle^{1/2}=|\cdot|_{2}$. Then, there exists
$\displaystyle z_{0}$ in the closed affine ball
$\displaystyle\overline{B}^{m}_{\varrho}(\zeta_{0})=\\{z\in\mathbb{R}^{m};\lfloor
z-\zeta_{0}\rfloor_{m}\leq\varrho\\}$ such that $\displaystyle F(z_{0})=0$.

Now, our third result is a new variant of classical Fixed Point Theorems, that
is, when the function involved represents a norm. Namely:

######  Theorem 1.3.

Let $\displaystyle\|\cdot\|_{m}$ be a general norm in
$\displaystyle\mathbb{R}^{m}$ and $\displaystyle
F:(\mathbb{R}^{m}\setminus\\{y_{0}\\},\|\cdot\|_{1,p,m})\rightarrow(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$
be a continuous function such that $\displaystyle\left\langle
F(\zeta),\zeta-\zeta_{0}\right\rangle\geq 0$ for every
$\displaystyle\zeta\in\mathbb{R}^{m}\setminus\\{y_{0}\\}$ with
$\displaystyle\|\zeta-\zeta_{0}\|_{m}=\varrho$ for some $\displaystyle
0<\varrho<\|\zeta_{0}-y_{0}\|_{m}$ and
$\displaystyle\langle\cdot,\cdot\rangle^{1/2}=|\cdot|_{2}$. Then, there exists
$\displaystyle z_{0}$ in the closed ball
$\displaystyle\overline{B}_{\varrho}(\zeta_{0}):=\\{z\in\mathbb{R}^{m};\|z-\zeta_{0}\|_{m}\leq\varrho\\}$
such that $\displaystyle F(z_{0})=0$.

Note that, if $\displaystyle
F:(\mathbb{R}^{m},\|\cdot\|_{1,p,m})\rightarrow(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$
is a continuous function, then Theorems 1.2 and 1.3 hold for all
$\displaystyle\varrho>0$, that is, we can remove the upper bound of
$\displaystyle\varrho$ required on the hypotheses of these results.

For each positive integer $\displaystyle m$, we define the affine functional
$\displaystyle\Phi_{m}:W_{m}\rightarrow\mathbb{R}$ given by

$\Phi_{m}(u)=\frac{1}{p}\mathcal{E}_{p,\Omega}^{p}(u)-\frac{1}{\alpha}\|u\|^{\alpha}_{L^{\alpha}(\Omega)}-\int_{\Omega}f(x)udx,$
(6)

whose Gâteaux derivative $\displaystyle\Phi_{m}^{\prime}:W_{m}\rightarrow
W_{m}^{\prime}$ is such that (see Theorem 10 of [11])

$\displaystyle\displaystyle\langle\Phi_{m}^{\prime}(u),\varphi\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\displaystyle\int_{\Omega}H_{u}^{p-1}\left(\nabla
u\right)\nabla H_{u}\left(\nabla u\right)\cdot\nabla\varphi
dx-\int_{\Omega}|u|^{\alpha-2}u\varphi dx-\int_{\Omega}f(x)\varphi dx$

for each $\displaystyle\varphi\in W_{m}$, where $\displaystyle 1<\alpha<p$,

$\displaystyle H_{u}^{p}(\varsigma)=\gamma_{n,p}^{-\frac{n}{p}}\;{\cal
E}_{p,\Omega}^{n+p}(u)\int_{\mathbb{S}^{n-1}}\left(\int_{\Omega}|\nabla_{\xi}u(x)|^{p}\,dx\right)^{-\frac{n+p}{p}}|\langle\xi,\varsigma\rangle|^{p}\,d\sigma(\xi)\
\ {\rm for}\ \varsigma\in\mathbb{R}^{n}\,$

and $\displaystyle f\in L^{p^{\prime}}(\Omega)$, with $\displaystyle f\neq 0$
in $\displaystyle\Omega$ and
$\displaystyle\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Note that

$\displaystyle\int_{\Omega}H_{u}^{p-1}\left(\nabla u\right)\nabla
H_{u}\left(\nabla u\right)\cdot\nabla\varphi dx=$

$\displaystyle\gamma_{n,p}^{-n}\;\mathcal{E}_{p,\Omega}^{n+p}(u)\int_{\mathbb{S}^{n-1}}\left(\int_{\Omega}|\nabla_{\xi}u(x)|^{p}\,dx\right)^{-\frac{n+p}{p}}\int_{\mathbb{R}^{n}}\\{\nabla_{\xi}u\\}^{p-1}\langle\nabla\varphi,\xi\rangle
dxd\xi,$

where $\displaystyle\\{x\\}^{p}:=|x|^{p}sg(x)$ and so

$\displaystyle\displaystyle\int_{\Omega}H_{u}^{p-1}\left(\nabla u\right)\nabla
H_{u}\left(\nabla u\right)\cdot\nabla udx=\mathcal{E}_{p,\Omega}^{p}(u)$

for all $\displaystyle u\in W_{0}^{1,p}(\Omega)$ (see proof of Theorem 10 of
[11]). Notice also that $\displaystyle[W_{m}]_{m\in\mathbb{N}}$ is dense in
$\displaystyle W^{1,p}_{0}(\Omega)$.

As a application of the Theorem 1.1, we obtain a nontrivial critical point
$\displaystyle u_{m}$ of $\displaystyle\Phi_{m}$ for each $\displaystyle m\geq
1$. Moreover, the sequence $\displaystyle\mathcal{E}_{p,\Omega}^{p}(u_{m})$ is
bounded and so $\displaystyle(u_{m})$ admits a strongly convergence
subsequence in $\displaystyle L^{s}(\Omega)$ once $\displaystyle s<p^{*}$.

##  2 Proof of Theorems 1.1, 1.2 and 1.3

Recently, Leite and Montenegro [16] showed that
$\displaystyle\mathcal{E}_{p,\Omega}^{p}$ is strongly continuous in
$\displaystyle W^{1,p}_{0}(\Omega)$. Evidently, the affine $\displaystyle
L^{p}$ functional $\displaystyle\mathcal{E}_{p,\Omega}^{p}$ is continuous in
$\displaystyle W_{m}$, for each $\displaystyle m\geq 1$. To make this work
more complete, we will make a proof below following the same ideas, but a
little more direct, since we are in a finite dimensional space.

######  Lemma 2.1.

Let $\displaystyle m\in\mathbb{N}$. The affine $\displaystyle L^{p}$
functional $\displaystyle\mathcal{E}_{p,\Omega}$ is continuous in
$\displaystyle W_{m}$.

###### Proof.

Let $\displaystyle u_{k}\rightarrow u$ in $\displaystyle W_{m}$. For
$\displaystyle u=0$, the inequality (3) implies that
$\displaystyle\mathcal{E}_{p,\Omega}(u_{k})\rightarrow
0=\mathcal{E}_{p,\Omega}(0)$.

Suppose now that $\displaystyle u\neq 0$. Note that, by inequality (3), we get
$\displaystyle\mathcal{E}_{p,\Omega}(u)>0$. Then, from inequality (4), we have
$\displaystyle\|\nabla_{\xi}u_{k}\|_{L^{p}(\Omega)}\rightarrow\|\nabla_{\xi}u\|_{L^{p}(\Omega)}$
uniform on $\displaystyle\xi\in\mathbb{S}^{n-1}$,
$\displaystyle\|\nabla_{\xi}u\|_{L^{p}(\Omega)}>c_{1}$ for every
$\displaystyle\xi\in\mathbb{S}^{n-1}$ and
$\displaystyle\|\nabla_{\xi}u_{k}\|_{L^{p}(\Omega)}^{-n}\leq c_{2}$, where
$\displaystyle c_{1},c_{2}>0$ are independents of $\displaystyle\xi$. Thus,
$\displaystyle\|\nabla_{\xi}u_{k}\|_{L^{p}(\Omega)}^{-n}\rightarrow\|\nabla_{\xi}u\|_{L^{p}(\Omega)}^{-n}$
uniform on $\displaystyle\xi\in\mathbb{S}^{n-1}$ and so, by the dominated
convergence theorem, we conclude
$\displaystyle\mathcal{E}_{p,\Omega}(u_{k})\rightarrow\mathcal{E}_{p,\Omega}(u)$.
This finish the proof. ∎

The following lemma will be important to apply the general Brouwer’s Fixed
Point Theorem. This theorem states that if $\displaystyle E$ is a bounded and
closed subset in $\displaystyle\mathbb{R}^{m}$ and homeomorphic to the closed
unit ball, then any continuous mapping $\displaystyle S:E\rightarrow E$ admits
a fixed point (see [22]).

######  Lemma 2.2.

The affine ball
$\displaystyle\overline{B}^{m}_{\varrho}(\zeta_{0})=\\{z\in\mathbb{R}^{m};\lfloor
z-\zeta_{0}\rfloor_{m}\leq\varrho\\}$ and
$\displaystyle\overline{B}^{p,m}_{1}(0)=\\{z\in\mathbb{R}^{m};\|z\|_{1,p,m}\leq
1\\}$ are homeomorphic in $\displaystyle(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$.

###### Proof.

Let us consider the applications $\displaystyle
T:\overline{B}^{m}_{\varrho}(\zeta_{0})\to\overline{B}^{p,m}_{1}(0)$ defined
by

$\displaystyle
T(x):=\left\\{\begin{array}[]{lcl}\frac{x-\zeta_{0}}{\varrho}\frac{\lfloor
x-\zeta_{0}\rfloor_{m}}{\|x-\zeta_{0}\|_{1,p,m}}&\mbox{ if
}&x\neq\zeta_{0},\\\ 0&\mbox{ if }&x=\zeta_{0}\end{array}\right.$

and $\displaystyle
G:\overline{B}^{p,m}_{1}(0)\to\overline{B}^{m}_{\varrho}(\zeta_{0})$ defined
by

$\displaystyle G(x):=\left\\{\begin{array}[]{lcl}\varrho
x\frac{\|x\|_{1,p,m}}{\lfloor x\rfloor_{m}}+\zeta_{0}&\mbox{ if }&x\neq 0,\\\
\zeta_{0}&\mbox{ if }&x=0.\end{array}\right.$

By standard computations we obtain that $\displaystyle G=T^{-1}$, that is,
$\displaystyle T\circ G=G\circ T=Id$. By the continuity of the functional
$\displaystyle u\in W_{m}\mapsto\mathcal{E}_{p,\Omega}(u)$ follows the
continuity of the function $\displaystyle T$ in $\displaystyle x\neq\zeta_{0}$
and $\displaystyle G$ in $\displaystyle x\neq 0$. Now, $\displaystyle T$ is
continuous in $\displaystyle x=\zeta_{0}$ since applying inequality (3), we
have

$\displaystyle\|T(x)-T(\zeta_{0})\|_{1,p,m}=\left\|\frac{x-\zeta_{0}}{\varrho}\frac{\lfloor
x-\zeta_{0}\rfloor_{m}}{\|x-\zeta_{0}\|_{1,p,m}}\right\|_{1,p,m}=\frac{1}{\varrho}\lfloor
x-\zeta_{0}\rfloor_{m}\leq\frac{1}{\varrho}\|x-\zeta_{0}\|_{1,p,m}.$

To proof the continuity of $\displaystyle G$ in $\displaystyle x=0$, let us
consider a sequence $\displaystyle(x_{k})$ in
$\displaystyle\overline{B}^{p,m}_{1}(0)$, with $\displaystyle x_{k}\to 0$.
Lets suppose that the exists a subsequence $\displaystyle(x_{k_{j}})$ of
$\displaystyle(x_{k})$ such that

$\displaystyle\lim_{j\to\infty}G(x_{k_{j}})=\theta\neq\zeta_{0}.$

Hence, by continuity of $\displaystyle T$, we obtain

$\displaystyle\lim_{j\to\infty}x_{k_{j}}=\lim_{j\to\infty}T\circ
G(x_{k_{j}})=T(\theta)\neq 0$

which contradicts $\displaystyle x_{k}\to 0$. In conclusion

$\displaystyle\lim_{k\to\infty}G(x_{k})=\zeta_{0}=G(0).$

Therefore, $\displaystyle T$ and $\displaystyle G$ are continuous, hence
$\displaystyle\overline{B}^{m}_{\varrho}(\zeta_{0})$ and
$\displaystyle\overline{B}^{p,m}_{1}(0)$ are homeomorphic in
$\displaystyle(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$. ∎

Proof of Theorem 1.1: Firstly, there exists $\displaystyle c(m)>0$ such that

$\lfloor
x\rfloor_{m}=\mathcal{E}_{p,\Omega}\left(\sum_{j=1}^{m}x_{j}w_{j}\right)\leq\left\|\sum_{j=1}^{m}x_{j}w_{j}\right\|_{1,p,m}\leq
c(m)|x|_{2},\,\,\ \forall x\in\mathbb{R}^{m}.$ (7)

We assume $\displaystyle F(x)\neq 0$ for all $\displaystyle
x\in\overline{B}^{m}_{\varrho}(0)$. From (7), we have

$\displaystyle\overline{B}^{p,m}_{\varrho}(0):=\\{z\in\mathbb{R}^{m};\|z\|_{1,p,m}\leq\varrho\\}\subset\overline{B}^{m}_{\varrho}(0).$

We define

$\displaystyle
g:(\overline{B}^{m}_{\varrho}(0),\|\cdot\|_{1,p,m})\rightarrow(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$

by

$\displaystyle g(x)=-\frac{\varrho}{\lfloor F(x)\rfloor_{m}}F(x)$

which maps $\displaystyle\overline{B}^{m}_{\varrho}(0)$ into itself and is
continuous. Therefore, by general Brouwer’s Fixed Point Theorem, we have
$\displaystyle g$ admits a fixed point $\displaystyle x_{0}$. Since
$\displaystyle x_{0}=g(x_{0})$, we obtain $\displaystyle\lfloor
x_{0}\rfloor_{m}=\lfloor g(x_{0})\rfloor_{m}=\varrho>0$. But then by (7) and
assumptions, we get

$\displaystyle 0<\varrho^{2}=\lfloor x_{0}\rfloor^{2}_{m}\leq
c(m)^{2}|x_{0}|^{2}_{2}=c(m)^{2}\left\langle
x_{0},x_{0}\right\rangle=c(m)^{2}\left\langle x_{0},g(x_{0})\right\rangle$
$\displaystyle=-c(m)^{2}\frac{\varrho}{\lfloor
F(x_{0})\rfloor_{m}}\left\langle x_{0},F(x_{0})\right\rangle\leq 0,$

which is a contradiction.

Proof of Theorem 1.2: Suppose, $\displaystyle F(x)\neq 0$ for all
$\displaystyle x\in\overline{B}^{m}_{\varrho}(\zeta_{0})$. Notice that by (7)
we have

$\displaystyle\overline{B}^{p,m}_{\varrho}(\zeta_{0}):=\\{z\in\mathbb{R}^{m};\|z-\zeta_{0}\|_{1,p,m}\leq\varrho\\}\subset\overline{B}^{m}_{\varrho}(\zeta_{0}).$

Define

$\displaystyle
g_{0}:(\overline{B}^{m}_{\varrho}(\zeta_{0}),\|\cdot\|_{1,p,m})\rightarrow(\mathbb{R}^{m},\|\cdot\|_{1,p,m})$

by

$\displaystyle g_{0}(x)=-\frac{\varrho}{\lfloor
F(x)\rfloor_{m}}F(x)+\zeta_{0}$

which maps $\displaystyle\overline{B}^{m}_{\varrho}(\zeta_{0})$ into itself
and is continuous. Hence it has a fixed point $\displaystyle x_{0}$, by
general Brouwer’s Fixed Point Theorem. Since $\displaystyle
x_{0}=g_{0}(x_{0})$, we have $\displaystyle\lfloor
x_{0}-\zeta_{0}\rfloor_{m}=\lfloor g(x_{0})-\zeta_{0}\rfloor_{m}=\varrho>0$.
But then by (7)

$\displaystyle\displaystyle 0<\varrho^{2}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lfloor x_{0}-\zeta_{0}\rfloor^{2}_{m}\leq
c(m)^{2}|x_{0}-\zeta_{0}|^{2}_{2}=c(m)^{2}\left\langle
x_{0}-\zeta_{0},x_{0}-\zeta_{0}\right\rangle$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle c(m)^{2}\left\langle
x_{0}-\zeta_{0},g_{0}(x_{0})-\zeta_{0}\right\rangle=-c(m)^{2}\frac{\varrho}{\lfloor
F(x_{0})\rfloor_{m}}\left\langle x_{0}-\zeta_{0},F(x_{0})\right\rangle\leq 0,$

by assumptions, which is a contradiction.

Proof of Theorem 1.3: The proof of this result follows analogous to that of
Theorem 1.2, by using the equivalence between two norms in
$\displaystyle\mathbb{R}^{m}$ and replacing
$\displaystyle\lfloor\cdot\rfloor_{m}$ by $\displaystyle\|\cdot\|_{m}$.

##  3 Application: existence of nontrivial critical point of the functional
$\displaystyle\Phi_{m}$

Before proving our application, we will show the following continuity result.
Namely:

######  Proposition 3.1.

Let $\displaystyle A:\mathbb{R}^{m}\to\mathbb{R}^{m}$ be a function such that

$\displaystyle A(\zeta)=(A_{1}(\zeta),A_{2}(\zeta),\dots,A_{m}(\zeta)),$

where
$\displaystyle\zeta=(\zeta_{1},\zeta_{2},...,\zeta_{m})\in\mathbb{R}^{m}$,

$\displaystyle\begin{array}[]{lll}A_{j}(\zeta)=\int_{\Omega}H_{u}^{p-1}(\nabla
u)\nabla H_{u}(\nabla u)\cdot\nabla w_{j}\;dx\end{array}$

$\displaystyle j=1,2,\dots,m$, and $\displaystyle
u=\sum_{i=1}^{m}\zeta_{i}w_{i}\in W_{m}$. Then, the function $\displaystyle A$
is continuous.

###### Proof.

For this, let $\displaystyle\zeta^{k}\rightarrow\zeta^{0}$ in
$\displaystyle\mathbb{R}^{m}$ and $\displaystyle
u_{k}=\sum_{i=1}^{m}\zeta^{k}_{i}w_{i}\in W_{m}$, $\displaystyle
k=0,1,2,\dots$. We fixed $\displaystyle j\in\\{1,2,\dots,m\\}$. Note that

$\displaystyle
A_{j}(\zeta^{k})=\gamma_{n,p}^{-n}\;\mathcal{E}_{p,\Omega}^{n+p}(u_{k})\int_{\mathbb{S}^{n-1}}\|\nabla_{\xi}u_{k}\|_{L^{p}(\Omega)}^{-n-p}\int_{\Omega}\\{\nabla_{\xi}u_{k}\\}^{p-1}\langle\nabla
w_{j},\xi\rangle dxd\xi.$

If $\displaystyle\zeta^{0}=0$, then by inequality (4), we have
$\displaystyle\|\nabla_{\xi}u_{k}\|_{L^{p}(\Omega)}\rightarrow 0$ uniform on
$\displaystyle\xi\in\mathbb{S}^{n-1}$. Note that, using inequality (5), there
exists $\displaystyle D_{0}>0$ independent of $\displaystyle\xi$ such that

$\displaystyle|A_{j}(\zeta^{k})|\leq\gamma_{n,p}^{-n}\;D_{0}\int_{\mathbb{S}^{n-1}}\int_{\Omega}|\langle\nabla
u_{k},\xi\rangle|^{p-1}|\langle\nabla w_{j},\xi\rangle|dxd\xi.$

Now, by Hölder and Cauchy-Schwarz inequalities and dominated convergence
theorem, we have

$\displaystyle|A_{j}(\zeta^{k})|\leq\gamma_{n,p}^{-n}\;D_{0}|\Omega|^{\frac{1}{p}}\|\nabla
w_{j}\|_{L^{p}(\Omega)}\int_{\mathbb{S}^{n-1}}\|\nabla_{\xi}u_{k}\|^{p-1}_{L^{p}(\Omega)}d\xi\rightarrow
0=A_{j}(\zeta^{0}).$

If $\displaystyle\zeta^{0}\neq 0$, then by Lemma 2.1, we have
$\displaystyle\mathcal{E}_{p,\Omega}^{n+p}(u_{k})\rightarrow\mathcal{E}_{p,\Omega}^{n+p}(u)$.
Finally, using the same arguments used in the proof of Lemma 2.1 and in the
proof of the previous case, we have $\displaystyle A_{j}(\zeta^{k})\rightarrow
A_{j}(\zeta^{0})$. This concludes the proof. ∎

######  Theorem 3.2.

Suppose that $\displaystyle f\in L^{p^{\prime}}(\Omega)$ with $\displaystyle
f\neq 0$ in $\displaystyle\Omega$, $\displaystyle 1<\alpha<p$. Let
$\displaystyle\Phi_{m}$ be the functional defined in (6). Then there exists a
nontrivial solution $\displaystyle u\in W_{m}$ satisfying
$\displaystyle\Phi^{\prime}_{m}(u)=0$.

###### Proof.

Define the function $\displaystyle F:\mathbb{R}^{m}\to\mathbb{R}^{m}$ such
that

$\displaystyle F(\zeta)=(F_{1}(\zeta),F_{2}(\zeta),\dots,F_{m}(\zeta)),$

where
$\displaystyle\zeta=(\zeta_{1},\zeta_{2},...,\zeta_{m})\in\mathbb{R}^{m}$,

$\displaystyle\begin{array}[]{lll}F_{j}(\zeta)&=&\displaystyle\int_{\Omega}H_{u}^{p-1}\left(\nabla
u\right)\nabla H_{u}\left(\nabla u\right)\cdot\nabla w_{j}dx\\\
&&-\int_{\Omega}|u|^{\alpha-2}uw_{j}dx-\int_{\Omega}f(x)w_{j}dx,\end{array}$

$\displaystyle j=1,2,\dots,m$, and $\displaystyle
u=\sum_{i=1}^{m}\zeta_{i}w_{i}\in W_{m}$.

By Proposition 3.1, we have $\displaystyle F$ is a continuous function.
Therefore, by Hölder’s inequality, we get

$\displaystyle\begin{array}[]{lll}\left\langle
F(\zeta),\zeta\right\rangle&=&\displaystyle\mathcal{E}_{p,\Omega}^{p}(u)-\|u\|^{\alpha}_{L^{\alpha}(\Omega)}-\int_{\Omega}f(x)udx\\\
&\geq&\mathcal{E}_{p,\Omega}^{p}(u)-\|u\|^{\alpha}_{L^{\alpha}(\Omega)}-\|f\|_{L^{p^{\prime}}(\Omega)}\|u\|_{L^{p}(\Omega)},\end{array}$

where $\displaystyle\frac{1}{p^{\prime}}+\frac{1}{p}=1$. By affine Poincaré-
Sobolev inequality (see inequality (2)), we obtain

$\displaystyle\begin{array}[]{lll}\left\langle
F(\zeta),\zeta\right\rangle&\geq&\mathcal{E}_{p,\Omega}^{p}(u)-\left(\mu_{p,\alpha}^{\cal
A}\right)^{-\alpha}\mathcal{E}_{p,\Omega}^{\alpha}(u)-\left(\mu_{p,p}^{\cal
A}\right)^{-1}\|f\|_{L^{p^{\prime}}(\Omega)}\mathcal{E}_{p,\Omega}(u).\end{array}$

Now, with the notations of Theorem 1.1, let
$\displaystyle\lfloor\zeta\rfloor_{m}=\varrho$ for some
$\displaystyle\varrho>0$ to be taken below. Thus, we have

$\displaystyle\begin{array}[]{rcl}\displaystyle\left\langle
F(\zeta),\zeta\right\rangle&\geq&\displaystyle\varrho^{p}-\left(\mu_{p,\alpha}^{\cal
A}\right)^{-\alpha}\varrho^{\alpha}-\left(\mu_{p,p}^{\cal
A}\right)^{-1}\|f\|_{L^{p^{\prime}}(\Omega)}\varrho.\end{array}$

If $\displaystyle\varrho$ is such that

$\displaystyle\varrho>\max\left\\{\left[2\left(\mu_{p,p}^{\cal
A}\right)^{-1}\|f\|_{L^{p^{\prime}}(\Omega)}\right]^{\frac{1}{p-1}},\left[2\left(\mu_{p,\alpha}^{\cal
A}\right)^{-\alpha}\right]^{\frac{1}{p-\alpha}}\right\\}$

then

$\displaystyle\left\langle F(\zeta),\zeta\right\rangle>0.$

We take

$\displaystyle\varrho:=\max\left\\{\left[2\left(\mu_{p,p}^{\cal
A}\right)^{-1}\|f\|_{L^{p^{\prime}}(\Omega)}\right]^{\frac{1}{p-1}},\left[2\left(\mu_{p,\alpha}^{\cal
A}\right)^{-\alpha}\right]^{\frac{1}{p-\alpha}}\right\\}+1.$

Thus, Theorem 1.1 ensure the existence of $\displaystyle
z_{0}\in\mathbb{R}^{m}$ with $\displaystyle\lfloor y\rfloor_{m}\leq\varrho$
and such that $\displaystyle F(z_{0})=0$. In other words, there exists
$\displaystyle u_{m}\in W_{m}$ verifying

$\displaystyle\mathcal{E}_{p,\Omega}(u_{m})\leq\varrho,$

and such that

$\displaystyle\displaystyle\int_{\Omega}H_{u_{m}}^{p-1}\left(\nabla
u_{m}\right)\nabla H_{u_{m}}\left(\nabla u_{m}\right)\cdot\nabla
wdx=\int_{\Omega}|u_{m}|^{\alpha-2}u_{m}wdx+\int_{\Omega}f(x)wdx$

for all $\displaystyle w\in W_{m}$. Then $\displaystyle u_{m}$ is nontrivial.
In particular, considering $\displaystyle w=u_{m}$ we obtain

$\displaystyle\displaystyle\mathcal{E}_{p,\Omega}^{p}(u_{m})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\|u_{m}\|^{\alpha}_{L^{\alpha}(\Omega)}+\int_{\Omega}f(x)u_{m}dx.$

######  Remark 1.

It is important to mention that $\displaystyle\varrho$, does not depend on
$\displaystyle m$.

Thus, $\displaystyle\mathcal{E}_{p,\Omega}^{p}(u_{m})$ is bounded and so, by
Theorem 6.5.3 of [21], there exists $\displaystyle u\in W^{1,p}_{0}(\Omega)$
such that $\displaystyle u_{m}\rightarrow u$ strongly in $\displaystyle
L^{s}(\Omega)$ once $\displaystyle s<p^{*}$, up to a subsequence.

∎

### Acknowledgements

The first author was partially supported by FAPEMIG/ APQ-02375-21,
APQ-04528-22, FAPEMIG/RED-00133-21 and CNPq Process 307575/2019-5 and
101896/2022-0.
The second author was partially supported by CNPq/Brazil (PQ 316526/2021-5)
and Fapemig/Brazil (Universal-APQ-00709-18).

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