11institutetext: Universitäts-Sternwarte, Fakultät für Physik, Ludwig-
Maximilians-Universität München, Scheinerstr. 1, 81679 München, Germany
22institutetext: Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-
Straße 1, 85741 Garching, Germany 33institutetext: Univ. Lille, CNRS, Centrale
Lille, UMR 9189 CRIStAL, F-59000 Lille, France 44institutetext: Université
Paris-Saclay, CNRS, Institut d’Astrophysique Spatiale, 91405, Orsay, France
55institutetext: Leibniz-Institut für Astrophysik (AIP), An der Sternwarte 16,
14482 Potsdam, Germany 66institutetext: P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Profsojuznaja 84/32 Moscow, Russia, 117997

# Simulating the LOcal Web (SLOW)

I. Anomalies in the local density field

Klaus Dolag 1122 Jenny G. Sorce 334455 Sergey Pilipenko 66 Elena Hernández-
Martínez 11 Milena Valentini 11 Stefan Gottlöber 55 Nabila Aghanim 44 Ildar
Khabibullin 1122

(Received December 1, 2022; submitted)

###### Abstract

Context. Several observations of the local Universe point towards the
existence of very prominent structures. The presence of massive galaxy
clusters and local super clusters on the one hand, but also large local voids
and under-densities on the other hand. However, it is highly non trivial to
connect such different observational selected tracers to the underlying dark
matter (DM) distribution.

Aims. Therefore, it is needed to construct mock catalogues of such observable
tracers using cosmological hydro-dynamical simulations. Such simulations have
to follow galaxy formation physics on the one hand and on the other hand have
to be constrained to reproduce the local Universe. Such constraints should be
based on observables which directly probe the full underlying gravitational
field, like the observed peculiar velocity field, to provide an independent
test on the robustness of those distinctive structures.

Methods. We used a 500 $h^{-1}$Mpc large constrained simulation of the local
Universe to investigate the anomalies in the local density field as found in
observations. Constructing the initial conditions based on peculiar velocities
derived from the CosmicFlows-2 catalogue makes the predictions of the
simulations completely independent from the distribution of the observed
tracer population and following galaxy formation physics directly in the
hydro-dynamical simulations allows to additionally base the comparison
directly on stellar masses of galaxies or X-ray luminosity of clusters. We
also used the 2668 $h^{-1}$Mpc large cosmological box from the Magneticum
simulations to evaluate the frequency of finding such anomalies in random
patches within simulations.

Results. We demonstrate that haloes and galaxies in our constrained simulation
trace the local dark matter density field very differently. Thereby, this
simulation reproduces the observed 50% under-density of galaxy clusters and
groups within the sphere of $\approx$100 Mpc when applying the same mass or
X-ray luminosity limit used in the observed cluster sample (CLASSIX), which is
consistent with a $\approx 1.5\sigma$ feature. At the same time, the
simulation reproduces the observed over-density of massive galaxy clusters
within the same sphere, which on its own also corresponds to a $\approx
1.5\sigma$ feature. Interestingly, we find that only 44 out of 15635 random
realizations (i.e. 0.28%) are matching both anomalies, making the local
Universe to be a $\approx 3\sigma$ environment. We finally compared a mock
galaxy catalogue with the observed distribution of galaxies in the local
Universe, finding also a match to the observed factor of two over-density at
$\sim 16$ Mpc as well as the observed 15% under-density at $\sim$40 Mpc
distance.

Conclusions. Constrained simulations of the local Universe which reproduce the
main features of the local density field open a new window for local field
cosmology, where the imprint of the specific density field and the impact on
the bias through the observational specific tracers can be investigated in
detail.

###### Key Words.:

local Universe – cosmological simulations

##  1 Introduction

The neighborhood in the immediate vicinity of the Milky Way (MW) is known as
the “Local Group”. It is a binary system composed of two average-sized
galaxies (the MW and Andromeda) that occupies a volume that is roughly $\sim$7
Mpc in diameter. At a distance of around 16 Mpc, the Virgo cluster comes into
view as the main defining feature of our neighborhood on these scales. Beyond
Virgo, a number of well-known and well-observed clusters like Centaurus,
Fornax, Hydra, Norma, Perseus, and Coma dominate the local volume, among them
a significant number of very massive clusters.

Therefore, our local Universe, that is centered on us and extends over 150
$h^{-1}$Mpc, is not only a formidable site for detailed observations, but also
appears as a very particular region of the Universe. Indeed, starting from a
local void (Tully & Fisher, 1987), bordered by the local sheet (Tully et al.,
2008), there are also a large number of super-cluster structures identified
within the local Universe, among them Perseus-Pisces, Centaurus, Coma and
Hercules (see recent work by Böhringer & Chon, 2021, and references therein).
Several of these most prominent structures in the local Universe form the so-
called supergalactic plane, which was already recognized by de Vaucouleurs
(1953). See also Lahav et al. (2000); Peebles (2022) for a summary of our
current understanding of these structures as well as Flin (1986); Rubin (1989)
for historical reviews. The impact of these structures is also recognized to
form a global, pancake like structure out to a scale of $\sim$100 Mpc
(Böhringer et al., 2021) as well as large differences in the mean stellar
density between the Northern and Southern hemispheres (Karachentsev &
Telikova, 2018) out to distances of $\sim$60 Mpc. It is often argued, that
such particular structures show unusual over- (Makarov & Karachentsev, 2011)
or under- (Böhringer et al., 2020) density when using luminous matter on
different scales, with different conclusions when evaluating the underlying
dark matter density field from them. As this could play a significant role in
some of the current tensions in cosmology, for example the larger $H_{0}$
value locally inferred compared to CMB measurements (see Freedman, 2021, for a
most recent compilation of the values), it is important to understand how
galaxies and galaxy clusters with different masses and properties trace the
underlying dark matter density field. Also, the selection effect of different
observations has to be understood in detail to answer such questions.

Trying to understand such features in the local Universe motivated various
campaigns producing constrained simulations in the past. However, for a more
detailed study one has to cover large enough volume, covering several hundreds
of Mpc and initial conditions which are not directly based on the distribution
of the traces to allow an independent comparison. In addition, one needs to
include galaxy formation physics into such simulation to properly select
objects by observables like stellar masses of galaxies or X-ray luminosity of
galaxy clusters. Here we present simulations, which for the first time matches
all the three criteria.

The paper is structured as follows: Section 2 gives an extended summary of
previous constrained simulations followed by section 3, which describes the
details of the used simulations, the included galaxy formation physics and
details on the building of the initial conditions, followed by a qualitative
comparison to observations in section 4. In section 5, we then present the
predicted, spatial distribution of matter, halos and galaxies from the
simulations and compare the peculiar features extensively with various
observations in section 6. In addition to the conclusions presented in section
7, we also repeated some of the analysis on previous constrained simulations
in the appendix A to demonstrate the potential inherent in our constrained
simulations which does not depend on assuming a bias between tracer particles
and dark matter but rather predicts this bias quite accurately.

Figure 1: Positions in super-galactic x/y coordinates of all galaxies in the
simulations. To make the simulations comparable, we restricted to galaxies
with stellar masses above $10^{10}M_{\odot}$ within a 50 Mpc thick slice
around the center of the simulation volume. In addition, the positions of
Coma, Virgo and Perseus are marked. The virial masses of the according halos
in the simulations are listed in table 1. From top to bottom: SLOW, SIBELIUS
and CORUSCANT.

##  2 Constrained Simulations

There are two approaches to study the problems mentioned above with numerical
simulations. One could run simulations of very large volumes with very high
resolution and find the objects of interest in these simulations in similar
environment as the observed ones. For example, in a statistical approach, the
scatter in the $H_{0}$ value locally inferred can be studied in a box of (6
$h^{-1}$Gpc)3 (Wojtak et al., 2014) or Local Group candidates can be drawn
from a large sample of isolated halo pairs identified in a set of cosmological
simulations (APOSTLE, Fattahi et al., 2016). An alternative approach is to use
so-called constrained cosmological simulations. The goal of these simulations
is to reproduce as well as possible the positions, velocities, masses and
internal properties of the objects of interest, i.e. to reduce the cosmic
variance in the region of interest as much as possible. Then such simulations
can go well beyond the statistical approach and even trace back evolutionary
paths of the studied objects. Naturally, constrained simulations are based on
observations. There are two different approaches to obtain initial conditions
of cosmological simulations based on the observed matter distribution at
present. The first one goes backwards in time and is based on the Hoffman-
Ribak algorithm (Hoffman & Ribak, 1991). We will use this algorithm throughout
our paper. An alternative approach to study the local Universe with
constrained simulations has been developed during the last decade, namely a
Bayesian forward modelling (e.g.,Kitaura et al. (2012), [ELUCID] Wang et al.
(2016), [SIBELIUS] Sawala et al. (2022); McAlpine et al. (2022), for a recent
comprehensive discussion of this Bayesian modeling see Jasche & Lavaux
(2019)). The quality of the constrained simulations depends on the number and
the quality of the constraints, see for example the discussion of the optimal
sampling of velocity constraints for Wiener Filter reconstructions by Sorce et
al. (2017).

One of the first constrained simulation was based on data derived from
redshift surveys, which estimate the local, total matter density from the
distribution of galaxies using their observed redshift. Here, the main
uncertainties are the assumed, constant relation between galaxy density and
total density as well as the imposed uncertainty in the distances by presence
of peculiar line-of-sight velocity. Initial conditions for a 240 $h^{-1}$ Mpc
box have been constructed, which contains a sphere of 160 $h^{-1}$ Mpc
diameter sampled with higher resolution (Mathis et al., 2002). Simulations
based on these initial conditions cover dark matter only simulations combined
with semi analytic models where used to study the local galaxy population
(Mathis et al., 2002) and pure magneto-hydrodynamical simulations to study the
propagation of cosmic rays in the local Universe
(CORUSCANT111https://www.usm.lmu.de/~dolag/Simulations/#CORUSCANT, Dolag et
al., 2004a) as well as the imprint of the local super-cluster onto the CMB
(Dolag et al., 2005a). Subsequently, these initial conditions have been
evolved with more advanced physics including radiative cooling, star-
formation, stellar evolution and chemical enrichment to study density and
temperature fluctuations in the local and prominent galaxy clusters
(SALACIOUS222https://www.usm.lmu.de/~dolag/Simulations/#SALACIOUS, Kawahara et
al., 2007). Including, additionally, AGN feedback from supermassive black
holes, subsequent simulations based on these initial conditions have been used
to contrast the observed Sunyaev-Zel’dovich effect (SZ) of the Coma and Virgo
clusters as measured by the Planck satellite (Planck Collaboration et al.,
2013b, 2016a).

The other method to constrain initial conditions is based on peculiar
velocities derived from direct distance measures. This has the advantage that
tracing the velocity field is independent of the tracer population and that
the velocity traces the potential caused by the large scale density field of
all matter. However, the data sets of galaxies with measured distance
indicators are much smaller and the intrinsically large uncertainties
associated with these distance indicators have to be dealt with in a very
careful way. Early attempts therefore have tried a hybrid approach, using
velocity data combined with constraints coming from local galaxy clusters
which allowed to span large enough volumes to cover the most important galaxy
clusters like Virgo, Perseus and Coma. This led to the CLUES333www.clues-
project.org (Yepes et al., 2009) dark matter only simulation of a 160 $h^{-1}$
Mpc box centered on the MW position to study the local galaxy population
(Klypin et al., 2003) and a non-radiative, hydrodynamical simulation of the
Virgo cluster within that box to study the properties of the intergalactic
medium (Kravtsov et al., 2002). Based on the same constraints, high resolution
simulations of a 64 $h^{-1}$ Mpc box (Gottlöber et al., 2010) have been
performed and used for example to study the reionisation of the Local Group
(Dixon et al., 2018) while zoomed full hydrodynamical simulations within a few
Mpc large, central region of the same box have been used for many studies of
the Local Group, including simulations following in detail the formation of
isolated dwarfs in the neighborhood of the Local Group (see for example
Benítez-Llambay et al., 2015).

| LU2016 | SLOW/CLONES | CORUSCANT | SIBELIUS
---|---|---|---|---
Cluster | $M_{\mathrm{vir}}[10^{14}M_{\odot}]$
Coma |  | 18. | 7.6 | 15.
Perseus |  | 10. | 13. | 33.
Virgo | $6.5\pm 1$ | 9.7 | 5.5 | 4.3
Table 1: Virial masses of Coma, Perseus and Virgo in the SLOW, CORUSCANT and
SIBELIUS simulations, as well as the distribution from the 200 Virgo-like
clusters from the LU2016 simulations (Sorce et al., 2020). Note that we
converted $M_{200c}$ as given for the SIBELIUS simulation to
$M_{\mathrm{vir}}$ following Ragagnin et al. (2021) and using the same
cosmology as that used in the SIBELIUS simulation. A more detailed comparison
of cluster properties from the SLOW simulation with observations will be
presented in Hernandez-Matinez et al. (in prep).

The growing data of direct distances from the CosmicFlow project (e.g.
Courtois et al., 2012; Tully et al., 2013, 2016, 2022) has driven new
approaches within the CLUES project since then. In a series of papers,
techniques have been developed to deal with the increasing number of
constraints and to handle the biases inherent to velocity data (Doumler et
al., 2013c, a, b; Sorce et al., 2014, 2016b; Sorce, 2015). Based on this
approach, initial conditions for a 64 $h^{-1}$ Mpc box have been constructed.
They have been used within the CosmicDawn project to run a fully coupled
radiation-hydrodynamics simulation of cosmic reionisation and galaxy formation
until redshift $z=6$ (Ocvirk et al., 2020) as well as to run a dark matter
only simulation with $4096^{3}$ particles down to redshift $z=0$ within the
MultiDark project. In this simulation, Milky Way and Andromeda galaxies can be
identified so that their reionisation history can be inferred from the
radiation-hydrodynamics simulation (Sorce et al., 2022). Moreover, initial
conditions based on the same approach have been used to simulate a 100
$h^{-1}$ Mpc box, where a high resolution region of about 10 $h^{-1}$ Mpc is
placed in the central region around the local group, including detailed galaxy
formation physics (HESTIA, Libeskind et al., 2020). First dark matter only
simulations based on a larger 500 $h^{-1}$ Mpc box already allowed to
reproduce the Virgo cluster and its formation history (LU2016, Sorce et al.,
2016a). Consecutive improvement of dark matter simulation of this volume
allowed to reproduce more and more prominent galaxy clusters like Perseus and
Coma (CLONES, Constrained LOcal and Nesting Environment Simulations, Sorce,
2018), to study the formation history of clusters (Olchanski & Sorce, 2018;
Sorce et al., 2020), and to study the large-scale cosmic web in which clusters
like Coma are embedded by comparing observations (Malavasi et al., 2020) and
simulations (Malavasi et al. 2022). Subsequent placing high resolution regions
around such prominent clusters allows direct comparison of galaxy properties
within the Virgo cluster from hydrodynamical simulations with full galaxy
formation physics with their observed counterparts (Sorce et al., 2021). For
the first time, the CLONE (Constrained LOcal and Nesting Environment)
simulation results undoubtedly show that the constrained formation history of
the Virgo cluster significantly differs from averaged clusters of the same
mass (Sorce et al., 2021). Moreover, the simulations indicate phase-space
positions of recently in-falling galaxy groups. It also matches the specific
amplitude and shapes of the appearing velocity waves on the lines-of-sight
towards various, well known galaxy clusters (Sorce et al., 2023).

The SLOW (Simulating the Local Web)
simulation444https://www.usm.lmu.de/~dolag/Simulations/#SLOW is a 500
$h^{-1}$Mpc box, using the realization number 8 of CLONES, as described in
Sect. 3.3 and assumes a Planck like cosmology (Planck Collaboration et al.,
2014), with a Hubble constant $H_{0}=67.77\mathrm{km/s/Mpc}$, a total matter
density of $\Omega_{\mathrm{Matter}}=0.307115$, a cosmological constant of
$\Omega_{\Lambda}=0.692885$ and a baryon fraction corresponding to
$\Omega_{\mathrm{baryon}}=0.0480217$. It follows the evolution of dark and
baryonic matter within a (500 $h^{-1}$Mpc)3 simulation volume centred on the
position of the MW and that stands for our cosmic neighborhood. The initial
conditions for such simulations are obtained with sophisticated algorithms
(see section 3.3) that take into account the position and motion estimates of
thousands of galaxies within our local volume. These local measurements allow
us to constrain the initial conditions that, in turn, lead to the observed
local large scale structure, when motions since early times until today are
followed according to the gravity laws. In addition, the baryonic matter is
treated via hydro-dynamics together with various, state-of-the-art sub-grid
models (see section 3.2). There are variants which follow the evolution of the
magnetic field and cosmic rays, others are following the formation of the
stellar population as well as black holes (BHs) and associated active galactic
nuclei (AGN) physics. Here we are following the prescriptions as used for the
Magneticum simulations (see Hirschmann et al. (2014); Dolag et al. (2016)).
This first paper in a series mainly introduces the general properties, based
on the simulation following galaxy formation processes and therefore a
realistic galaxy population is present in the simulations as well as galaxy
clusters with proper intracluster medium (ICM) properties. Therefore, galaxies
together with the population of galaxy clusters can be directly compared by
the same means to the observed counterparts. This also allows us to study the
effects of the local environment on observational properties and their
cosmological impact.

To give a better impression on the differences in some of the constrained
simulations, we show the predicted distribution of galaxies from the three
simulations SLOW (upper panel), SIBELIUS (middle panel) and SALACIOUS (lower
panel) in figure 1. In addition, the positions of Coma, Perseus and Virgo
within the simulations are shown. In general, the environment of these
clusters are similar in the different simulations (like the large scale
structure leaving Perseus to the lower left, or the horizontal structures
leaving Virgo and Coma. However, the details in these structures are different
as well as there are striking differences between the massive clusters and
their immediate environment. In SLOW/CLONES for example, the immediate
outskirts of Perseus indicate a quite fossil, relax system with a mass which
is very close to that expected from observations, while in CORUSCANT/SALACIOUS
simulation the Perseus structure appears as a twin system with an even more
massive companion, while in the SIBELIUS simulation Perseus appears to be
three times too massive. The obtained virial masses of these clusters in the
different simulations are listed in table 1, where we converted all values to
the same mass definition using the scaling relations given in Ragagnin et al.
(2021).

Figure 2: Shown is a full sky map of the projected dark matter distribution
from the SLOW simulation up to a distance of 350 Mpc. To distinguish the
constrained part of the local Universe simulation, we labeled some of the
cross-identified clusters in the Local Universe. Figure 3: The different
panels show observations in different wavebands. From top to down:
distribution of galaxies in the 2MRS catalog, X-ray surface brightness in the
ROSAT’s 1.5 keV band (R6+R7=0.76-2.04 keV) and Compton-Y map based on Planck
data (produced with the MILCA algorithm Planck Collaboration et al., 2016b)
with the CO mask by Khatri (2016). The inlays in addition show a zoom on the
Perseus and Hercules regions. The labels in the inlays are giving the Name
together with radial velocity (in km/s) as redshift distance indicator. The
yellow labels are indicating clusters which are outside the slice used in the
simulations shown in figure 4. Figure 4: The different panels show the SLOW
simulation in different wavebands. From top to down: distribution of stellar
mass in SLOW, X-ray surface brightness and Compton-$Y$. We used here a much
larger dynamical ranges than in the observations shown on figure 3 for the
color scaling to emphasize the large angular imprint on the very local galaxy
clusters. Note also that we here always show the full simulation volume, out
to a distance of 350 Mpc. The inlays in addition show a zoom on the Perseus
and Hercules regions. Here we used a much more narrow range around Perseus and
A2147, e.g. a 10 respectively 40 Mpc thick region to emphasize the local
structures. The labels in the inlays are giving the virial mass (in
$M_{\odot}$) of the haloes, together with radial velocity (in km/s) as
redshift distance indicator.

##  3 Simulations

###  3.1 Magneticum simulations

The Magneticum555www.magneticum.org Simulations follow for the first time the
evolution of up to 1011 particles in a series of hydro-dynamical simulations
of cosmological volumes ranging in size from (68 Mpc)3 to almost (4 Gpc)3. To
evaluate the significance of features in the local Universe, we use the
largest simulation volume, Box0/mr of the Magneticum simulation set as
reference for the general presence of such features within the $\Lambda$CDM
cosmological model. This simulation covers a box of 2.688$h^{-1}$GPc in size,
resolved with a total of $2\times 4536^{3}$ dark matter and gas resolution
elements, having a mass resolution of $1.3\times 10^{10}h^{-1}M_{\odot}$ and
$2.6\times 10^{9}h^{-1}M_{\odot}$ respectively and featuring full galaxy
formation physics. It is therefore well equipped to reflect galaxy cluster and
group properties. Having $\approx 150$ times the volume of the 500$h^{-1}$Mpc
box, Box0/mr of the Magneticum simulations allow us to compare a large number
of random patches with the constrained part of the SLOW simulation with a very
high statistical significance. The cosmology adopted for these simulations is
slightly different than for SLOW, as it follows the WMAP7 from Komatsu et al.
(2011), with Hubble constant $H_{0}=70.4\mathrm{km/s/Mpc}$, a total matter
density of $\Omega_{\mathrm{Matter}}=0.272$, a cosmological constant of
$\Omega_{\Lambda}=0.728$ and a baryon fraction corresponding to
$\Omega_{\mathrm{baryon}}=0.0459$, as well as an overall normalisation of the
power spectrum of $\sigma_{8}=0.809$ and a slope of the primordial fluctuation
spectra of $n=0.963$. However, these small differences do not play any
significant role for the comparisons presented in this study.

###  3.2 The sub-grid model used

Both, the SLOW as well as the Magneticum simulations are using an updated
formulation of SPH (Beck et al., 2016) with modern, high order Kernels (Dehnen
& Aly, 2012) and include the treatment of the relevant models to describe the
physical processes needed for galaxy formation like cooling, star formation
and winds. This is based on the multi-phase model (Springel & Hernquist,
2003), but extended to follow in detail the stellar population and chemical
enrichment by SN-Ia, SN-II, AGB (Tornatore et al., 2003, 2007), uses metal
depending cooling tables from Wiersma et al. (2009) and a galactic wind
velocity of $\rm 350\ km/s$ for the kinetic feedback. Another important aspect
to follow are the evolution of super massive black holes and the associated
AGN feedback, where we follow (Springel et al., 2005) with various
improvements (Fabjan et al., 2010; Hirschmann et al., 2014) for the treatment
of the black hole sink particles and the different feedback modes. Importantly
for the treatment of the ICM they include isotropic thermal conduction of 1/20
of standard Spitzer value (Dolag et al., 2004b) and a low viscosity scheme to
track turbulence (Dolag et al., 2005b; Beck et al., 2016).

It has been intensively demonstrated, that this sub-grid model leads to galaxy
and ICM properties in galaxy clusters, which largely follow the observational
trends and properties. Thereby, the Magneticum simulations have been compared
to Sunyaev-Zel’dovich (SZ) data from Planck (Planck Collaboration et al.,
2013a) and SPT (McDonald et al., 2014). It has been demonstrated to reproduce
the observable X-ray luminosity-relation (Biffi et al., 2013), the chemical
composition (Dolag et al., 2017; Biffi et al., 2018b) of the ICM and the high
concentration observed in fossil groups (Ragagnin et al., 2019). On larger
scales, the Magneticum simulations demonstrated to reproduce the observed SZ
power spectrum Dolag et al. (2016) as well as the observed thermal history of
the Universe (Young et al., 2021) and the gas properties in between galaxy
clusters (Biffi et al., 2022). On galaxy scales, the simulations lead to an
overall successful reproduction of the basic galaxy properties, like the
stellar mass function at low (Naab & Ostriker, 2017) and high (Lustig et al.,
2023; Remus et al., 2022) redshifts, the environmental impact of galaxy
clusters on galaxy properties (Lotz et al., 2019; Lustig et al., 2023), the
azimuthal distribution of matter around clusters compared with findings in
SDSS (Gouin et al., 2020) and the appearance of post-starburst galaxies (Lotz
et al., 2021) as well as the associated AGN population at various redshifts
(Hirschmann et al., 2014; Steinborn et al., 2016; Biffi et al., 2018a).

###  3.3 SLOW

The large-scale structure of the Universe is effectively described by the
(peculiar) velocity and real space distribution of observable galaxies. A
Wiener Filter (WF) algorithm666Linear minimum variance estimator, in abridged
form WF (Zaroubi et al., 1995, 1999). is needed to reconstruct the true
underlying cosmography from the noisy and incomplete galaxy data reaped from
surveys. The first attempt to construct constrained realizations of Gaussian
random fields subject to linear constraints was made by Hoffman & Ribak
(1991). In the intervening two decades the technique as well as the input
constraints have been refined – peculiar velocity, in fact distance modulus,
surveys such as CosmicFlows-2 (CF2, Tully et al., 2013) are particularly
useful since the cosmic velocity field is directly related to the matter
density field in the linear regime. Wiener Filter reconstructions of CF2 have
already been carried out successfully estimating the density field within
$\sim$100 Mpc (e.g. Laniakea, Tully et al., 2014). Sorce (2018) describes in
details the steps of the method to build the constrained initial conditions
and introduces those used in this paper. The main steps are summarized
hereafter:

  1. 1.

Before deriving the peculiar velocities, grouping (Sorce & Tempel, 2018) of
the distance modulus catalog to remove non-linear virial motions that would
affect the linear reconstruction obtained with the linear method (e.g. Sorce
et al. (2017); Sorce & Tempel (2017)). Typically, when several distance moduli
are available for several galaxies within the same galaxy cluster, they are
replaced by the distance modulus of the cluster.

  2. 2.

Minimizing the biases (Sorce, 2015) inherent to any observational radial
peculiar velocity catalog (e.g. Malmquist biases and lognormal error). An
iterative algorithm permits retrieving the Gaussian radial peculiar velocity
distribution, expected from the theory, from a flat distribution with large
tails. Additionally, uncertainties are derived for these new peculiar
velocities to filter the smoothing effect with the distance (dilution of the
information with the distance) of the subsequent algorithms (Sorce et al.,
2016b; Sorce, 2018).

  3. 3.

Reconstructing the cosmic displacement field with the WF technique applied to
the peculiar velocity constraints.

  4. 4.

Accounting for the cosmic displacement by relocating constraints to the
positions of their progenitors using the Reverse Zel’dovich Approximation and
the reconstructed cosmic displacement field (Doumler et al., 2013c, a, b) and
replacing noisy radial peculiar velocities by their WF 3D reconstructions
(Sorce et al., 2014).

  5. 5.

Producing density fields constrained by the modified observational peculiar
velocities combined with a random realization to restore statistically the
missing structures using the Constrained Realization technique (CR, Hoffman &
Ribak, 1991, 1992).

  6. 6.

Rescaling the density fields to build constrained initial
conditions777GINNUNGAGAP: https://github.com/ginnungagapgroup/ginnungagap,
where increasing the resolution implies adding random small scale features.

Here, as well as for the actual simulations, we assume the standard,
$\Lambda$CDM cosmological model with the Hubble constant
$H_{0}=67.77\mathrm{km/s/Mpc}$, a total matter density of
$\Omega_{\mathrm{Matter}}=0.307115$, a cosmological constant of
$\Omega_{\Lambda}=0.692885$ and a baryon fraction corresponding to
$\Omega_{\mathrm{baryon}}=0.0480217$, as well as an overall normalisation of
the power spectrum of $\sigma_{8}=0.829$ and a slope of the primordial
fluctuation spectra of $n=0.961$ (Planck Collaboration et al., 2014).

###  3.4 SLOW set of simulations

To create the initial conditions, SLOW use the CLONE-500 Mpc/h-5123grid,
realization number 8, to which different small scale features for different
resolutions are added with Ginnungagap. Simulations are then performed with
different levels of additional physics:

  * •

Dark matter only simulations were performed using $768^{3}$, $1576^{3}$,
$3072^{3}$ and $6144^{3}$ particles.

  * •

Magneto-hydrodynamical, non radiative simulations but following also a cosmic
ray component (Böss et al., 2023) using $2\times 3072^{3}$ gas and dark matter
particles.

  * •

Hydrodynamical, full galaxy formation physics simulation using $2\times
768^{3}$, $2\times 1576^{3}$ and $2\times 3072^{3}$ gas and dark matter
particles (last one only down to $z=2$).

Halos are identified using SubFind (Springel et al., 2001; Dolag et al., 2009)
which detects halos based on the standard Friends-of-Friends algorithm (Davis
et al., 1985) and subhalos as self-bound regions around local density peaks
within the main halos. The centre of halos and subhalos are defined as the
position of the particle with the (local) minimum of the gravitational
potential. The virial mass, $M_{\mathrm{vir}}$ is defined through the
spherical-overdensity around a halo as predicted by the generalised spherical
top-hat collapse model (Eke et al., 1996) with an over-density for the chosen
cosmology following Bryan & Norman (1998). For subhaloes, individual
properties are computed based on the particles which are associated to the
individual subhalos. To compute the K-band magnitudes in our simulation which
include galaxy formation physics, we used the K-band stellar mass to light
ratio as obtained from SDSS (Bell et al., 2003)

$\mathrm{log}_{10}(M/L_{K})=-0.42+0.033\mathrm{log}_{10}(Mh^{2}/M_{\odot}),$
(1)

while in the case of dark matter only simulations, we are using the
corresponding Tully-Fisher relation based on the maximum circular velocity
computed within the subhaloes.

##  4 A qualitative comparison with observations

Within the SLOW simulations, more than 40 local clusters have been cross
identified with their observational counterpart, showing good agreement in
global properties (like total mass) as well as profiles of the ICM properties
(like pressure and temperature), see Hernandez Martinez et al. (in prep) for
details. All the results presented in this paper are based on the
hydrodynamical, full galaxy formation simulation with $2\times 1576^{3}$
particles. In Figure 2 we show the distribution of the dark matter within the
SLOW simulation as a full sky map up to a distance of 350 Mpc, where the color
coding is according to the total matter content in each pixel. This shows how
the constrained region of the local Universe (where we labeled a subset of the
cross identified clusters) is embedded in the larger cosmological structures.
In Figure 3 we show a collection of observations. First, we show the
distribution of galaxies in the 2MRS catalog. Note that here the observations
are limited to more local galaxies as the 2MRS gets very sparse at distances
beyond 100 Mpc. On the other hand, X-ray and Compton-$Y$ map contain
significant foreground and other, non-cluster related emission. Here the
selection of the 1.5 keV (R6+R7=0.76-2.04 keV) band for the X-ray surface
brightness map, based on the data of the ROSAT All-Sky Survey (RASS, Snowden
et al., 1997), maximises the visibility of the cluster signal. In the
Compton-$Y$ map, the exclusion of regions with significant CO emission
(according to the mask by Khatri, 2016) from the Planck data (produced with
the MILCA algorithm Planck Collaboration et al., 2016b) is reducing the non-
cluster related foreground. Therefore one has to keep in mind that both
observational tracers contain objects which are outside the distance range of
the simulation counterparts (some prominent ones are marked with yellow
labels), and even some much more distant objects, even outside the simulation
volume, as well as residuals of local, galactic foreground features and are
limited on the faint end by the observational noise. This can be compared with
the according counterparts as derived from the simulation, shown in figure 4.
Here we show the distribution of the stellar component (upper panel), the
X-ray surface brightness (middle panel) and the Compton-$Y$ map (lower panel)
as obtained from the SLOW simulation. Here we can use a much larger dynamical
ranges in the color scaling to emphasize the large angular imprint on the very
local galaxy clusters. Also here we always used the full simulation volume out
to the distance of 350 Mpc for producing the counterpart sky maps from the
simulations. In addition, we show two special regions (Perseus and Hercules)
in more detail in the inlays. Therefore we always center on the cross
identified halo of the prominent cluster (e.g. Perseus and A2147) and used a
much more narrow range around them, e.g. a 10 respectively 40 Mpc, to
emphasize the local structures. The virial mass (in solar masses) are given as
labels for the cross identified halos, while in addition the radial velocity
(in km/s) is given as a distance indicator. A more detailed comparison of the
individual cluster properties across multiple wavebands will be released in a
series of papers. In table 1 we give the virial masses for Coma, Virgo and
Perseus as a reference. As can be seen when comparing the full sky maps from
the observations and the simulations, uncertainties in the constrains which
went into the construction of the initial condition lead to a noticeable shift
in the positions (see also Sorce, 2018). The largest contribution to this
positional discrepancies are the still relatively large uncertainties in the
observed distance modules. This gets even more evident when comparing the
inlays. For example in the Hercules region, A2147 has a clear match and also
A2107 has a corresponding halo. There is also a halo resembling the
A2063/A2052 complex, but this is only a single halo in the simulation and
shows also a significant shift (e.g. several degrees on sky). A similar
situation arises for the Perseus region. Perseus itself has a quite well
matching counterpart, however, a possible counterpart to AWM7 is already
significantly displaced. Nevertheless, this comparison across multiple
wavebands clearly reveals that appearance of clusters can look significantly
different across the different wavebands and demonstrate the need of full,
hydro dynamical simulations for such comparison, as the apparent significance
of structures often largely differs when comparing galaxy and ICM properties.
Still, overall clear similarities in the large scale structure and the
appearance of galaxy clusters are visible across the different wavebands when
comparing the SLOW simulations with the observations, confirming Sorce (2018)
and Sorce et al. (2023) assertions.

Figure 5: Cumulative, relative number density of halos (left) and number
density of galaxies (right) as function of distance from the MW position as
obtained from the SLOW simulation. The different colors are lines obtained for
different lower mass cuts, as indicated in the color bars: virial masses of
the halos (left) and stellar masses of the galaxies (right). The black line in
both panels marks the same when using directly dark matter distribution.

##  5 Mean density for Halos in SLOW

There are various density anomalies reported in the Local Universe, ranging
from the Council of Giants (McCall, 2014), the local void (Tully & Fisher,
1987), close structures, a local deficit of galaxy clusters (Böhringer et al.,
2020) and an overabundance of very massive galaxy clusters like Norma,
Perseus, Coma, Ophiuchus and A2199 or A119. As SLOW is a constrained
simulation, based on peculiar velocity observations, we can investigate how
these structures are present in the predicted density field. Given the large
volume of (500 $h^{-1}$Mpc)3 covered by the simulation, we can investigate the
full range of observed anomalies, given the resolution one can nowadays reach
for such volumes. Having a full galaxy formation physics run allows also
distinguishing between the dark matter density field, halos and galaxies,
including observational properties like stellar mass for galaxies and
properties of the intracluster medium (ICM) within galaxy clusters like their
temperature, X-ray luminosity and more.

Figure 5 shows the cumulative number density within spheres with growing
radius centered on the position of the MW. Here the left panel is using the
distribution of halos with different lower limits in their virial mass as
indicated by the legend, while the right panel uses the galaxies (e.g. sub-
halos) with different lower limits in stellar mass. In both cases the spacing
of the lines corresponds to a change in the mass threshold of
$\Delta\mathrm{log}_{10}(M)=0.05$. The solid black line is obtained directly
from the dark matter particles within the simulation.

Several features are immediately visible. Starting with a quite empty region
within the very close vicinity around the MW position888Here we defined the
position of the MW in this realization, so that the Virgo cluster is exactly
at the correct position., there are several relatively massive galaxies at a
distance of 5 Mpc, which build the equivalent of the so called Council of
Giants, which are observed at $\approx$3.75 Mpc (McCall, 2014). After that,
the local void is clearly visible and filling the space till the Virgo galaxy
cluster at $\approx$16 Mpc and the associated super cluster (the Virgo super
cluster) comes into place. Then, between $\approx$ 30 and 140 Mpc, a clear
under density is present in all tracers, except the very massive galaxies and
massive galaxy clusters, which is in agreement with our findings based on the
halo mass functions as presented in (Sorce et al., 2016b). he high start of
the blue, upper lines reflect the fact that we have a very massive cluster
like Virgo very close. Note that a typical, mean separation of clusters with a
virial mass of $10^{15}M_{\odot}$ is $\approx 180$ Mpc. These lines stay high,
as we approach other very massive clusters (among them Perseus and Coma) well
before the distance reflecting the mean density of such systems. The main
obstacle in relating the density of tracers to the underlying dark matter
distribution depends on the physics of gravitational clustering and the more
complex physics of galaxy formation and is often referred generically as
”bias” (see for example Weinberg et al., 2004, and references therein). It is
interesting to note that the bias between the tracer population and the dark
matter for the galaxy population in the normal mass-range shows the expected,
regular behaviour with only mild dependency on actual mass. In contrast, the
bias using halos shows a strong mass dependency and also large fluctuations in
relative amplitude when comparing halos to the dark matter distribution. It is
equally interesting to note that the factor of 2 over-density associated with
the Virgo supercluster structure as seen in dark matter shows only up in
galaxies and only if galaxies down to stellar masses of at least
$10^{11}\mathrm{M}_{\odot}$ are used and does not show up in the halo number
density distribution at all. This means that in this case, the halo of the
Virgo clusters seems generally quite isolated and therefore the halo itself
traces the general over-density, but not the associated structures in the
environment. At the very high mass end, galaxies and halos align, which just
reflects that the central, very massive galaxies (BCGs) are strictly related
to massive galaxy clusters and their halo, marking the point where galaxies of
a certain mass can no longer be formed by internal processes but mainly grow
by (mostly dry) mergers.

##  6 Comparison to Observations

Figure 6: Cumulative relative density of halos/galaxies with different masses
as a function of distance (computed from the redshift) compared to different
observations. Top panel comparing galaxy clusters to the X-ray sample from
Böhringer et al. (2021), using the same mass (solid) and the same X-ray
luminosity (dashed line) cut. Middle panel, comparing massive galaxy clusters
(M${}_{\rm vir}>10^{15}$ blue line, T${}_{500c}>5$ keV red line) to the five
closest clusters exceeding this mass or temperature from the Tully catalogue
(Tully, 2015) or the BAX database Sadat et al. (2004), respectively. Bottom
panel, comparing our simulated galaxies with the results on the stellar mass
density presented in Karachentsev & Telikova (2018). In the upper two panels,
the gray lines mark the according 1,2 and 3 $\sigma$ lines obtained from the
Magneticum simulation and the black line marks the median (to display the
cosmic variance left).

As discussed in the previous section, the different features in the Local
Universe can be compared more directly to observations to understand better
how unique our Local Universe is. Figure 6 shows the comparison to different
observational tracers as discussed in the following sub-sections in detail.

###  6.1 X-ray cluster sample

The upper panel of figure 6 compares the results from the SLOW simulation to
the findings by Böhringer et al. (2020), who reported a 50% under-density of
X-ray selected galaxy cluster from the CLASSIX catalogue (shown as blue data
points). The luminosity cut of $10^{42}$erg/s in the (0.1-2.4) keV band
translates into a virial mass of $\approx 10^{13.4}\mathrm{M}_{\odot}$ when
using a canonical X-ray luminosity-mass relation (Böhringer et al., 2014). The
blue solid line resembles the result from the SLOW simulation for halos with
this mass threshold, nicely reproducing the data points. However, having a
full hydro-dynamical simulation we can also directly use the predicted X-ray
luminosity of the clusters in the simulations. Here, we started from the
predicted bolometric luminosity within $R_{500c}$ and applied the correction
for the used energy band based on the mass weighted $T_{500c}$ of the cluster.
When applying the X-ray luminosity cut to the simulated clusters, we get the
dashed line, which is very close to the one where we used the virial mass cut
to select the clusters. This shows that the observed signal in the simulation
– and therefore also the signal in the real data – is not driven by the X-ray
selection of galaxy clusters.

Given this signal, it is interesting to investigate how peculiar it is to have
a local environment which features such a 50% under-density within the given
volume. Therefore, we took the very large, general cosmological simulation
from the Magneticum simulations set, namely Box0/mr, which covers a volume of
($2688h^{-1}$Mpc)3. Here we randomly selected more than fifteen thousand
points within the volume, and computed the cumulative over-density profiles
out to 360 Mpc radius. The gray shaded regions mark the one, two and three
sigma regions occupied by these profiles from a random cosmological
simulation. The black solid line is the median of the distribution to indicate
the statistical error left due to the still somewhat limited sample size from
the large simulation. We clearly see that the under-density we live in is not
very uncommon in the cosmological sense, representing a $\approx$1.5 sigma
event, similar to what Sorce et al. (2016b) concluded from comparing the halo
mass function.

###  6.2 Massive galaxy clusters

Although we are living in a large scale under-dense region, there are several
very massive galaxy clusters within that region, exceeding virial masses of
$10^{15}\mathrm{M}_{\odot}$, among them Coma, Perseus and Ophiuchus. Using the
five closest galaxy clusters from the Tully galaxy groups catalogue (Tully,
2015) with masses above $10^{15}M_{\odot}$, the middle part of figure 6 shows
that this corresponds to a very large over-density in the Local Universe (blue
diamonds). We also selected the five closest galaxy clusters from the
BAX999http://bax.irap.omp.eu and references therein. database with X-ray
measured temperatures exceeding 5 keV and showing them as red triangles in the
figure, basically confirming the presence of an overdensity of high mass
systems. Selecting such clusters predicted by the SLOW simulation (blue line
for the mass selection, red line for the temperature selection) again follows
the observational data points extremely well, confirming this significant
overdensity of massive clusters, as it was already clearly visible in figure
5. Note that there is a large overlap between the selected clusters when
switching the selection criteria in both, observations (4/5) and simulations
(3/5). As many of the prominent clusters can be cross identified between the
simulations and observations (like Coma, Virgo, Perseus and many more) and
even show very similar virial masses (for details see Hernandez Martinez et
al. in prep) there is also a significant overlap between the most massive
clusters in simulations and observations which therefore lead to the very
similar behaviour of both sets.

Again we investigate the statistical significance of such excess of massive
systems. Here, the gray shaded regions again mark the one, two and three sigma
regions as obtained from the large cosmological simulation Box0/mr from the
Magneticum set. The comparison indicates that such excess in massive system is
approximately a 1.5 sigma coincidence. The black line marks the median from
the 15635 samples we used and gives an indication on the remaining statistical
uncertainty, which in this case is somewhat larger due to the general low
number density of such massive galaxy clusters.

One might speculate, that the general under-density of clusters might be
related to the over-density of massive systems, however, a closer
investigation based on Box0/mr indicates the opposite. In fact, when requiring
a very conservative limit to have at 90 Mpc a mean density below 0.65 for
clusters with viral masses of $\approx 10^{13.4}M_{\odot}$ and having an over-
density of massive systems of 2.3 at 160 Mpc (corresponding to the
observational value of the the closest system), we find only 44 out of the
15635 samples to match. This corresponds to a more than $3\sigma$ case.
Besides, it clearly demonstrates that these two peculiarities of the local
Universe are not related.

###  6.3 Galaxies

Figure 7: Comparing the flattening of the galaxy clusters distribution to the
X-ray sample from Böhringer et al. (2021) (see text for description), to the
simulation, using the same mass (black) and the same X-ray luminosity (blue
solid line) cut. The dashed and the dashed dotted red lines are applying the
same measurement in the other, super galactic coordinates for comparison. The
dashed and dotted dashed blue lines are slightly different cuts in X-ray
luminosity.

Finally we can also compare the density fields as traced with galaxies, as
this might be closer to the actual, underlying distribution of dark matter, as
shown in the previous section. Here, in the lower panel of figure 6 we compare
the stellar mass in all simulated galaxies in SLOW (solid blue line) with
observed stellar density as reported in (Karachentsev & Telikova, 2018). Here
the red band correspond to the overall mean stellar density reported in
Karachentsev & Telikova (2018), normalized to the global stellar density of
$\Omega_{*}=0.0027$ (Fukugita & Peebles, 2004) or to the value at the largest
distance. The orange and the dark orange bands correspond to the observed
stellar density for the northern and southern (Karachentsev & Telikova, 2018),
normalized to these two general values respectively. In addition, the blue
dashed line marks the relative, mean number density of galaxies with
$M_{*}>10^{10}M_{\odot}$, while the gray line is the relative dark matter
density. Comparing to the expectations from the SLOW simulation in both cases
matched the large over-density feature of factor $\approx 2$ of the local
structures associated with the Virgo cluster at distances around $\approx 16$
Mpc quite well. It also shows a very similar shape further out with two minima
and two maxima, although the first under-density in the simulation appears at
a distance of $\approx 40$ Mpc, while in the observations the minimum in the
relative density is at $\approx 55$ Mpc. As seen in comparison with the dark
matter density, these features are clearly directly related to the underlying
dark matter density field. Closer than the distance to Virgo, the simulations
show a very prominent, large under-density of the local void (already visible
in figure 5, however, the stellar density as reported in Karachentsev &
Telikova (2018) features a larger, local density. Interestingly to note, the
SLOW simulation predict a global stellar density of $\Omega_{*}=0.0031$, which
is close to the value of $\Omega_{*}=0.0027$ as reported in Fukugita & Peebles
(2004). In general, the good match of the gray line for the dark matter
density in the bottom panel with the observed as well as the simulated stellar
density at distances beyond $\approx 20$ Mpc indicates that the observed
stellar densities obtained from galaxies quite robustly trace the underlying
dark matter field, which is not the case when using galaxy clusters as
tracers, as shown in the previous sub-sections.

###  6.4 Anisotropies

Figure 8: Comparison of our simulated galaxies with the results on the stellar
mass density presented in Karachentsev & Telikova (2018) divided in the
Northern Galactic hemisphere (blue) and Southern Galactic hemisphere (red). As
comparison, the thinner, dark red dashed and dotted dashed line show the
result when splitting along super galactic SGX and SGZ coordinates.

As mentioned in the introduction, the local structures also lead to observed
non isotropic distribution of tracers of the large scale structures. Here we
want to compare the results of the SLOW simulations with two of these reported
features. In figure 7 we compare the distribution of galaxy clusters in the
simulations to the pancake like structure out to a scale of $\approx$ 100 Mpc
as reported in Böhringer et al. (2021). We follow the characterization of the
flattened superstructure as in the observations. Starting from a cylinder with
100 Mpc radius, centered at the MW position and oriented along the
supergalactig SGY direction, we compute the over-density within slices of
thickness $\pm w_{1/2}$ Mpc and compare this with the mean density of the same
objects within a sphere of 100 Mpc radius, centred on the MW. To mimic the
observational selection of clusters we selected clusters either by their X-ray
luminosity (e.g. $10^{42}$erg/s as in the observations) or by the according
virial mass (e.g. $M_{\mathrm{vir}}=10^{13.4}M_{\odot}$) as shown by the blue
and black solid line respectively. This qualitatively agrees well with the
observational data points (blue symbols with error bars) which are also shown.
In addition, the blue dashed and dashed-dotted lines show the dependence on
the luminosity threshold used, indicating that choosing a slightly larger
luminosity would increase the agreement with the observational data further.
To further strengthen this result, the red dashed and dotted-dashed lines are
showing the absence (or even reverse signal) obtained when doing the split
along the supergalactic SGZ and SGX coordinate, respectively. We can conclude
here, that the pancake like structure is a solid prediction by the SLOW
simulation and well aligns qualitatively with the observational finding
without fine tuning selection parameters.

In figure 8, we repeat the comparison of the stellar density within the Local
Universe with the finding of Karachentsev & Telikova (2018), where a large
differences between the Northern and Southern Galactic hemisphere was
reported. Splitting our galaxies from the SLOW simulation in the same way (red
and blue solid line) the SLOW simulation shows some features very similar to
the observations (red and blue bands). Clearly, there is an over respectively
under density present in the northern respectively southern hemisphere at
large distances. This qualitatively agrees with observations, although here
the quantitative agreement is not as good as for the pancake like structure
traced by galaxy clusters. Also here we add in addition a split along the
super-galactic x and z coordinate (dashed and dashed-dotted lines) to show the
dependence of the signal onto the directional split. As seen before, the lower
stellar density within the SLOW simulation out to a distance of $\approx 15$
Mpc is present and independent on the direction. Interestingly the split along
the super-galactic x coordinate follows the observations even better and may
indicate that the positional uncertainty of some prominent structure in the
simulated Local Universe might influence the actual selection and might be
worth further studies.

##  7 Conclusions

We presented the first results from a long standing initiative, to perform
hydrodynamical, cosmological simulations of the Local Universe (a) with high
enough resolution and detailed enough galaxy formation physics to reproduce
reliable galaxies as well as intracluster medium properties; (b) which extends
to large distances to properly cover the transition of the local structures
towards the cosmological average properties; (c) to be constructed not on
total density estimates based on galaxy densities in redshift surveys. The
latter makes the comparisons with the observed galaxy distribution and galaxy
clusters properties independent from observations, that where already used to
construct the constraints for the initial condition and also allow to better
compare their evolution. Initial conditions of our SLOW simulation are based
on peculiar velocities (more precisely, a complex interplay and combination of
observed redshift and distance modulus from the CosmicFlows-2 catalogue),
applying various improvements in their creation over the last years (Sorce &
Tempel, 2018; Sorce, 2020) leading to a simulation which captures a volume of
(500 $h^{-1}$Mpc)3 in which various clusters of the Local Universe can be
cross identified (see Sorce (2018) and Hernandez Martinez et al, in prep, for
details).

The predicted density field of the simulation shows various distinct features
and indicates that the Local Universe transits into the cosmic mean at
distance of $\approx$ 200 Mpc (with significant variances depending on the
tracers used). Within this region, several ranges with relative under- and
over-density are present, which can be compared to observational indications.
Especially we find that

  1. 1.

Up to the distance of Virgo cluster (i.e. $\approx 16$ Mpc), the SLOW
simulation predicts the mean dark matter density in the Local Universe to be
at $\approx$ 0.5 of the mean value. With the Virgo super cluster this transits
into a factor of two over-density region until $\approx$ 30 Mpc, after which
the Local Universe seems to be $\approx$ 20% under-dense out to $\approx$ 200
Mpc.

  2. 2.

While this is traced by normal galaxies within the simulation with an expected
bias of order of 30%, using halos or galaxy clusters can result in quite
different, sometimes even opposite conclusions, depending on the halo mass
cut.

  3. 3.

Applying the same mass (or alternatively X-ray luminosity) cut like the
CLASSIX galaxy cluster sample, the simulation very closely reproduces the
observed 50% under-density of galaxy clusters in the Local Universe.

  4. 4.

Using clusters with virial masses above $10^{15}M_{\odot}$, simulation and
observations consistently show a significant over-density of such objects
within the same volume out to $\approx 200$ Mpc within the Local Universe.

  5. 5.

Comparing with hydro-dynamical simulations of very large cosmological volumes,
we find that both these features individually are not such uncommon and
correspond to $\approx 18$% of cases (e.g. fall into the $\approx$1.5 sigma
region). However, they appear to be unrelated to each other and the
combination is only found in $\approx 0.28$% of random selections from our (4
Gpc)3 reference simulation Box0/mr of the Magneticum simulation set, and
therefore would correspond to a three sigma case).

  6. 6.

The SLOW simulation also shows the pancake-like distribution of galaxy
clusters within 100 Mpc of the Local Universe Böhringer et al. (2021). Thereby
the radial distribution of the number counts of galaxy clusters follows quite
closely the observations when splitting in super-galactic north/south
direction.

  7. 7.

The SLOW simulation predicts a global stellar density of $\Omega_{*}=0.0031$,
which is close to the value of $\Omega_{*}=0.0027$ as reported in Fukugita &
Peebles (2004). At distances larger than $\approx 15$ Mpc the stellar density
obtained from the SLOW simulation follows the one reported in Karachentsev &
Telikova (2018) remarkably well and confirms both, the large, relative over-
density induced by Virgo as well as the $\approx 20$% under-density beyond
distances of 100 Mpc.

  8. 8.

At distances between $\approx 15$ and $100$ Mpc, the predicted stellar density
in the slow SLOW simulation shows also qualitatively similar difference
between the northern and southern Galactic hemisphere than reported in
Karachentsev & Telikova (2018).

The presented SLOW simulation of the Local Universe reproduces some of the
main features of the local density field and therefore opens a new window for
local field cosmology. It allows us to better verify and interpret
observations of the local structures and their tracers. In future this will
allow us to evaluate in detail the imprint of the specific density field of
the Local Universe on the local estimations of cosmological parameters like H0
as well detailed studies of the imprint of the formation history on actual
properties of galaxy clusters.

###### Acknowledgements.

We want to thank T. Hoffmann for the help when extracting observational data
points from the according publication. This work was supported by the grant
agreements ANR-21-CE31-0019 / 490702358 from the French Agence Nationale de la
Recherche / DFG for the LOCALIZATION project. NA acknowledges support from the
European Union’s Horizon 2020 research and innovation program grant agreement
ERC-2015-AdG 695561. KD and MV acknowledge support by the Excellence Cluster
ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC-2094 –
390783311 and funding for the COMPLEX project from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program grant agreement ERC-2019-AdG 882679. MV also acknowledges support from
the Alexander von Humboldt Stiftung and the Carl Friedrich von Siemens
Stiftung. The calculations for the hydro-dynamical simulations were carried
out at the Leibniz Supercomputer Center (LRZ) under the project pr83li
(Magneticum) and pn68na (SLOW). We are especially grateful for the support by
M. Petkova through the Computational Center for Particle and Astrophysics
(C2PAP).

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##  Appendix A Comparison to other simulations

Here we present some comparison with other, previous or more recent
constrained simulations, like CORUSCANT (Dolag et al. 2004a) and its SALACIOUS
variant (Kawahara et al. 2007) which includes galaxy formation physics, the
early CLUES (Klypin et al. 2003) simulation (label as B160_WM3) as well as the
SIBELIUS (McAlpine et al. 2022) simulation.

Figure 9: Cumulative, relative dark matter density as function of distance
from the MW position as obtained from the SLOW (black), CORUSCANT (blue) and
early CLUES (red) simulation.

###  A.1 Dark matter distribution

Similar to figure 5, figure 9 shows the dark matter density in the Local
Universe but comparing different simulation: SLOW, CORUSCANT, and B160_WM3.
Unfortunately, for SIBELIUS the dark matter data are not publicly available.
All simulations show the over density related to the Virgo complex, but only
SLOW shows the large scale under density towards a distance of 100 Mpc, mainly
because the other two simulations cover a too small volume. However, B160_WM3
shows a significant over density within a spherical region of 100 Mpc radius.
Interestingly, both, SLOW and B160_WM3 show a clear local under density in
dark matter within 10 Mpc, while CORUSCANT shows a larger over density in this
region.

###  A.2 Galaxy cluster distribution

Figure 10: Equivalent to top and middle panel of figure 6 but for the
different simulations, showing the cumulative, relative cluster density as
function of distance from the MW position as obtained from the SLOW (solid),
CORUSCANT (dashed), SIBELIUS (long dashed) and early CLUES (dashed dotted)
simulation. The blue lines are for very massive halos, the red line are for
clusters and correspond to the observations from the CLASSIX Böhringer et al.
(2020) sample.

Similar to figure 6, in figure 10 we show the density of galaxy clusters in
the Local Universe as obtained from the different simulations, divided into
massive (blue) and all clusters (red), compared to the observational data
points. While in SLOW and B160_WM3 reproducing the high number of local, very
massive system, CORUSCANT and SIBELIUS are falling somewhat short in having
the unusual large number of such massive systems. Note however that this is
not really significant, given the low number of halos in this case. However,
when comparing to the CLASSIX sample, only SLOW is able to match the reported
significant under density out to a distance of 100 Mpc.

###  A.3 Galaxy density profiles

Figure 11: Same as right panel of figure 5 but for SIBELIUS (left panel) and
SALACIOUS (right panel). Shown is the mean density of galaxies within the
Local Universe for different stellar masses (as indicated in the color bar).
For SALACIOUS, additionally the dark matter density is also shown.

Finally we repeat the right panel of figure 5 for SIBELIUS (upper panel) and
SALACIOUS (lower panel) in figure 11. Shown is (as before) the mean density of
galaxies as function of distance for different stellar masses, as indicated in
the color bar. Note, that from this we obtain a $\Omega_{*}=0.0014$ for
SIBELIUS and $\Omega_{*}=0.0021$ for CORUSCANT, compared to
$\Omega_{*}=0.0031$ for SLOW and the observed value of $\Omega_{*}=0.0027$ as
reported in Fukugita & Peebles (2004). Compared to the SLOW simulations, both
simulations are displaying significant less of variation in the mean density,
and especially the SIBELIUS does not show one sided deviation from the mean
density over very large scales. As already discussed in figure 10, SIBELIUS
does not show the large scale under density out to a distance of 100 Mpc, as
reported in the CLASSIX sample of galaxy clusters. Here it gets clear, that
contrary to SLOW, no galaxy or galaxy cluster selection in SIBELIUS does
feature this observed anomaly in the Local Universe. Once more this
demonstrates that the assumption of a constant bias within reconstructions
based on galaxy densities limits the predictive power of the resulting
constrained simulations. This can in principle be overcome with
reconstructions based on the observed velocity field, as demonstrated through
the SLOW simulation, which seems to match the observational findings of
various different tracers of the large scale structure very well. However, as
shown, they come with their own obstacles which are difficult to overcome in a
satisfactory manner, as demonstrated by the overall effort which has been
spent to bring forward the constrained simulations to this point.


††thanks: P.N. Choudhury is supported by National Post-Doctoral Fellowship
(PDF/2019/000275), from SERB, Government of India. K.C. Sivakumar acknowledges
funds received from MATRICS (MTR/2018/001132) of SERB, Government of India.

# Nesbitt and Shapiro Cyclic Sum Inequalities for Positive Definite Matrices

Projesh Nath Choudhury  Department of Mathematics
Indian Institute of Science
Bengaluru 560012
India<EMAIL_ADDRESS><EMAIL_ADDRESS>K.C. Sivakumar
Department of Mathematics
Indian Institute of Technology Madras
Chennai 600036
India<EMAIL_ADDRESS>
(Date: October 18, 2021)

###### Abstract.

The aim of this note is to show that certain number theoretic inequalities due
to Nesbitt and Shapiro have noncommutative counterparts involving positive
definite matrices.

###### Key words and phrases:

Positive definite matrices, trace inequality, eigenvalue inequality, Nesbitt’s
inequality, Shapiro’s inequality

###### 1991 Mathematics Subject Classification:

Primary 15A45, 47A63; Secondary 15B57, 26D15

##  1\. Introduction

Let $\mathbb{C}^{n\times n}$ denote the set of all $n\times n$ matrices over
the complex numbers. A Hermitian matrix $A\in\mathbb{C}^{n\times n}$ is called
a positive definite, if $x^{*}Ax>0$ for all $0\neq x\in\mathbb{C}^{n}$. An
interesting and rather classical topic of study in matrix analysis concerns
inequalities for positive definite matrices. Roughly speaking, matrix
inequalities are noncommutative versions of the corresponding scalar
inequalities. Most of the work in this direction considers the comparison of
eigenvalues or singular values or the traces, of various combinations of two
positive definite matrices [3, 4, 9, 18]. The main objective of this short
note is to present matrix versions of certain inequalities involving more than
two positive real numbers.

We begin by stating the inequalities that we will be interested in obtaining
matrix analogues of, in this work. For three positive real numbers $a,b$ and
$c$, Nesbitt’s cyclic sum inequality [12], is the following:

$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq\frac{3}{2}.$ (1.1)

For positive real numbers $a_{1},\ldots,a_{p}~{}(p\geq 3)$, Shapiro [15] posed
the problem of proving the inequality

$\frac{a_{1}}{a_{2}+a_{3}}+\frac{a_{2}}{a_{3}+a_{4}}+\cdots+\frac{a_{p}}{a_{1}+a_{2}}\geq\frac{p}{2}.$
(1.2)

Note that if $p=3$, then Shapiro’s inequality is the same as Nesbitt’s
inequality. Let us denote

$S_{p}:=S(a_{1},\ldots,a_{p})=\frac{a_{1}}{a_{2}+a_{3}}+\frac{a_{2}}{a_{3}+a_{4}}+\cdots+\frac{a_{p}}{a_{1}+a_{2}}.$

Then, inequality (1.2) is the same as:

$S_{p}\geq\frac{p}{2}.$

Inequality (1.2) has a rich history. In 1958, Mordell [11] proved Shapiro’s
inequality for $p=3,4,5,6$. The inequality was subsequently proved for $p=8$
[7], $p=10$ [13], and $p=11,12$ [8]. It follows from a remarkable result by
Diananda [6] that $S_{p}\geq p/2$ for all $p\leq 12$. On the other hand,
Zulauf [19] showed that the inequality does not hold for $p=14$, and for
$p=25$, Daykin [5] and Malcolm [10] gave a counterexample. The difference in
the behavior between even and odd $p$ was explained by Searcy and Troesch
[14]. Finally, Shapiro’s inequality was completely settled by Troesch [16,
17]. To summarize, Shapiro’s inequality holds for positive numbers
$a_{1},a_{2},\ldots,a_{p}$ precisely for the following values of $p:$

$3,4,5,\ldots,12,\quad 13,15,17,\ldots,23.$

Let us turn our attention to the present work. A matrix counterpart of
Nesbitt’s inequality is obtained in Theorem 2.5. In Theorem 2.7, we obtain a
matrix version of the Shapiro’s inequality for four variables. In Theorem 2.8,
we give a necessary condition on the validity of matrix version of Shapiro’s
inequality. In the last part, two matrix analogues of the Cauchy-Schwarz
inequality are proved. After a brief discussion on the notation and two
preliminary results, we prove our results.

Notation: The trace of a matrix $A\in\mathbb{C}^{n\times n}$, denoted by
$Tr(A)$ is the sum of its diagonal entries/eigenvalues. If $A$ and $B$ are
Hermitian matrices, and $A-B$ is positive semidefinite, then this will be
denoted by $A\succeq B.$ Let $\lambda(A)$ denote an arbitrary eigenvalue of
$A\in\mathbb{C}^{n\times n}$. The next two results are only stated. Their
proofs follow easily.

######  Theorem 1.1.

[1, Exercise 12.14] Let $A$ and $B$ be positive semidefinite matrices. Then

$0\leq Tr(AB)\leq Tr(A)Tr(B).$

######  Theorem 1.2.

[18, Exercise 18, p. 213] Let $X,Y\in\mathbb{C}^{m\times n}$ and let
$A\in\mathbb{C}^{m\times m}$ be a positive definite matrix. Then

$|Tr(X^{*}Y)|^{2}\leq Tr(X^{*}AX)Tr(Y^{*}A^{-1}Y).$

##  2\. Main Results

It is well known that, if $A$ and $B$ are positive definite matrices, then
$Tr\\{(A-B)(B^{-1}-A^{-1})\\}\geq 0$ [1, Exercise 12.28 (c)] and [18, Exercise
21, p. 213]. In our first result, we strengthen this inequality by showing
that each eigenvalue of $(A-B)(B^{-1}-A^{-1})$ is greater than or equal to
$0$.

######  Remark 2.1.

The following facts will be used in our discussion without further reference.
If $A$ and $B$ are positive definite, then the eigenvalues of the product
$X=AB^{-1}$ are positive, and

$\lambda(X+X^{-1})\geq 2.$

Also,

$Tr(X)Tr(X^{-1})\geq n^{2},$

for any positive definite matrix $X.$

######  Proposition 2.2.

Let $A,B\in\mathbb{C}^{n\times n}$ be positive definite matrices. Then

$\lambda\\{(A-B)(B^{-1}-A^{-1})\\}\geq 0.$

###### Proof.

In view of Remark 2.1, one has

$\displaystyle\lambda\\{(A-B)(B^{-1}-A^{-1})\\}$ $\displaystyle=$
$\displaystyle\lambda(AB^{-1}+BA^{-1}-2I)$ $\displaystyle=$
$\displaystyle\lambda(X+X^{-1})-2\geq 0.$

∎

To prove our main results, we need a matrix version of the following
inequality: Let $a_{1},\ldots,a_{p}$ be positive real numbers. Then, one has:

$(a_{1}+\cdots+a_{p})(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{p}})\geq p^{2}.$
(2.1)

######  Theorem 2.3.

Let $A_{1},\ldots,A_{p}\in\mathbb{C}^{n\times n}$ be positive definite
matrices. Then

$A^{-1}_{1}+\cdots+A^{-1}_{p}\succeq p^{2}(A_{1}+\cdots+A_{p})^{-1}$.

###### Proof.

Let $A_{1},\ldots,A_{p}\in\mathbb{C}^{n\times n}$ be positive definite
matrices. For $i=1,\ldots,p$, define the matrices

$M_{i}=\begin{pmatrix}A^{-1}_{i}&I\\\ I&A_{i}\end{pmatrix}.$

By [2, Theorem 1], $M_{i}\succeq 0$ for $i=1,\ldots p$. Thus

$M=\sum\limits_{i=1}^{p}M_{i}=\begin{pmatrix}\sum\limits_{i=1}^{p}A^{-1}_{i}&pI\\\
pI&\sum\limits_{i=1}^{p}A_{i}\end{pmatrix}\succeq 0.$

Since Schur complement preserves positive semidefiniteness, by considering the
Schur complement of $M$ with respect to the fourth block we have,

$\sum\limits_{i=1}^{p}A^{-1}_{i}\succeq
p^{2}\left(\sum\limits_{i=1}^{p}A_{i}\right)^{-1}.$

∎

Next, we prove a matrix counterpart for inequality (2.1).

######  Corollary 2.4.

Let $A_{1},\ldots,A_{p}\in\mathbb{C}^{n\times n}$ be positive definite
matrices. Then

$\lambda\\{(A_{1}+\cdots+A_{p})(A^{-1}_{1}+\cdots+A^{-1}_{p})\\}\geq p^{2}$.

###### Proof.

By Theorem 2.3,

$A^{-1}_{1}+\cdots+A^{-1}_{p}\succeq p^{2}(A_{1}+\cdots+A_{p})^{-1}$.

Since $A_{1}+\cdots+A_{p}$ is positive definite, its square root exists. Thus

$(A_{1}+\cdots+A_{p})^{1/2}(A^{-1}_{1}+\cdots+A^{-1}_{p})(A_{1}+\cdots+A_{p})^{1/2}\succeq
p^{2}I.$

Hence

$\displaystyle\lambda\\{(A_{1}+\cdots+A_{p})(A^{-1}_{1}+\cdots+A^{-1}_{p})\\}$
$\displaystyle=$
$\displaystyle\lambda\\{(A_{1}+\cdots+A_{p})^{1/2}(A^{-1}_{1}+\cdots+A^{-1}_{p})(A_{1}+\cdots+A_{p})^{1/2}\\}$
$\displaystyle\geq$ $\displaystyle p^{2}.$

∎

These results allow us to show a noncommutative version of Nesbitt’s
inequality (1.1).

######  Theorem 2.5.

Let $A,B,C\in\mathbb{C}^{n\times n}$ be positive definite matrices. Then
$\lambda(M)\geq\frac{3}{2}$, where

$M=A(B+C)^{-1}+B(C+A)^{-1}+C(A+B)^{-1}$.

###### Proof.

One may rewrite $M$ as

$M=(A+B+C)[(B+C)^{-1}+(C+A)^{-1}+(A+B)^{-1}]-3I$.

Let $X=B+C$, $Y=C+A$ and $Z=A+B$. Then

$M=\frac{1}{2}(X+Y+Z)(X^{-1}+Y^{-1}+Z^{-1})-3I$.

By Corollary 2.4,

$\lambda(M)\geq\frac{1}{2}3^{2}-3=\frac{3}{2}$.

∎

The next result generalizes the matrix version of Nesbitt’s inequality to $k$
variables. We leave the details of the proof to the interested reader.

######  Theorem 2.6.

Let $A_{1},\ldots,A_{k}\in\mathbb{C}^{n\times n}$ be positive definite
matrices and let $S=\sum\limits_{i=1}^{k}A_{i}$. Then
$\lambda(M)\geq\frac{k}{k-1}$, where

$M=\sum\limits_{i=1}^{k}A_{i}(S-A_{i})^{-1}.$

As was mentioned earlier, Shapiro’s inequality for $p=4$ says

$\displaystyle\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\geq
2,~{}\forall~{}a,b,c,d\in(0,\infty).$ (2.2)

It turns out that a verbatim analogue of (2.2) for the eigenvalues, does not
hold. For example, let $A=\begin{pmatrix}5&6\\\
6&7.5\end{pmatrix},B=\begin{pmatrix}2&1\\\
1&2\end{pmatrix},C=\begin{pmatrix}6&4\\\ 4&3\end{pmatrix}$ and
$D=\begin{pmatrix}3&2\\\ 2&5\end{pmatrix}$. Then $A,B,C,D$ are positive
definite matrices, and the eigenvalues of

$M:=A(B+C)^{-1}+B(C+D)^{-1}+C(D+A)^{-1}+D(A+B)^{-1}$

are $2.6393\pm 0.1871i$. However, in the next result, we prove a
generalization of Shapiro’s inequality, in terms of the trace function. For
positive definite matrices $A_{1},\ldots,A_{p}\in\mathbb{C}^{n\times
n}~{}(p\geq 3)$, define

$F_{p}:=F(A_{1},\ldots,A_{p}):=Tr\left[\sum\limits_{i=1}^{p}A_{i}(A_{i+1}+A_{i+2})^{-1}\right],$

where $A_{p+1}=A_{1}$ and $A_{p+2}=A_{2}$. In our discussion, we shall find it
convenient to not specify the domain of $F$. This allows us to use the same
$F$ simultaneously, even when we are considering different numbers of positive
definite matrices as arguments (see the proof of Proposition 2.8, for
instance). We are interested in studying the following matrix version of
Shapiro’s inequality:

$\displaystyle F_{p}\geq\frac{p}{2}n.$ (2.3)

Note that the case $p=3$ follows from Theorem 2.5. In the next result, we show
that the above inequality holds for $p=4$.

######  Theorem 2.7.

$F_{4}\geq 2n$.

###### Proof.

Since it is less cumbersome to work with matrices without subscripts, we
denote $A,B,C,D$ to be positive definite matrices in $\mathbb{C}^{n\times n}$
(instead of considering $A_{1},A_{2},A_{3}$ and $A_{4}$). Set

$M:=A(B+C)^{-1}+B(C+D)^{-1}+C(D+A)^{-1}+D(A+B)^{-1}.$

We must show that $Tr(M)\geq 2n$. Set

$N:=B(B+C)^{-1}+C(C+D)^{-1}+D(D+A)^{-1}+A(A+B)^{-1}$

and

$P:=C(B+C)^{-1}+D(C+D)^{-1}+A(D+A)^{-1}+B(A+B)^{-1}$.

Then $N+P=4I$ and so $Tr(N+P)=4n$. Further, by Theorem 2.3, one has

$(B+C)^{-1}+(D+A)^{-1}\succeq 2^{2}(A+B+C+D)^{-1}$

as well as

$(C+D)^{-1}+(A+B)^{-1}\succeq 2^{2}(A+B+C+D)^{-1}.$

By the monotonicity of the trace function, we then have

$Tr\left[(A+C)((B+C)^{-1}+(D+A)^{-1})\right]\geq
4Tr\left[(A+C)((A+B+C+D)^{-1})\right]$

and

$Tr\left[(B+D)((C+D)^{-1}+(A+B)^{-1})\right]\geq
4Tr\left[(B+D)((A+B+C+D)^{-1})\right].$

Note that

$M+P=(A+C)((B+C)^{-1}+(D+A)^{-1})+(B+D)((C+D)^{-1}+(A+B)^{-1}).$

Thus,

$Tr(M+P)\geq 4Tr\left[(A+B+C+D)((A+B+C+D)^{-1})\right]=4n.$

Next, we estimate a lower bound for $Tr(M+N)$. For the sake of notational
convenience, define $X_{U,V}:=U+V$ (so that $X_{U,V}=X_{V,U}$), given matrices
$U$ and $V$. Then,

$\displaystyle Tr(M+N)$ $\displaystyle=$ $\displaystyle
Tr\left[X_{A,B}X_{B,C}^{-1}+X_{B,C}X_{C,D}^{-1}\right]$
$\displaystyle+Tr\left[X_{C,D}X_{A,D}^{-1}+X_{A,D}X_{A,B}^{-1}\right]$
$\displaystyle\geq$ $\displaystyle
2\sqrt{Tr[X_{A,B}X_{B,C}^{-1}]Tr[X_{B,C}X_{C,D}^{-1}]}$
$\displaystyle+2\sqrt{Tr[X_{C,D}X_{A,D}^{-1}]Tr[X_{A,D}X_{A,B}^{-1}]}$
$\displaystyle=$ $\displaystyle
2\sqrt{Tr[X_{A,B}^{1/2}X_{B,C}^{-1}X_{A,B}^{1/2}]Tr[X_{C,D}^{-1/2}X_{B,C}X_{C,D}^{-1/2}]}$
$\displaystyle+2\sqrt{Tr[X_{C,D}^{1/2}X_{A,D}^{-1}X_{C,D}^{1/2}]Tr[X_{A,B}^{-1/2}X_{A,D}X_{A,B}^{-1/2}]}$
$\displaystyle\geq$ $\displaystyle
2\left[Tr[X_{A,B}^{1/2}X_{C,D}^{-1/2}]+Tr[X_{C,D}^{1/2}X_{A,B}^{-1/2}]\right]$
$\displaystyle\geq$ $\displaystyle
4\left[Tr[X_{A,B}^{1/2}X_{C,D}^{-1/2}]Tr[X_{C,D}^{1/2}X_{A,B}^{-1/2}]\right]^{1/2}$
$\displaystyle=$ $\displaystyle
4\left[Tr[X_{C,D}^{-1/4}X_{A,B}^{1/2}X_{C,D}^{-1/4}]Tr[X_{C,D}^{1/4}X_{A,B}^{-1/2}X_{C,D}^{1/4}]\right]^{1/2}$
$\displaystyle\geq$ $\displaystyle 4n.$

The justifications in the above calculation are: the second and the fifth
inequalities follow by the AM-GM inequality, (the third and the sixth
equalities make use of the fact that $Tr(UV)=Tr(VU)$), the fourth inequality
makes use of Theorem 1.2 (upon taking the square roots), and the last
inequality follows from Remark 2.1. Combining these, one obtains,

$\displaystyle Tr(M)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Tr(M+N)+Tr(M+P)-Tr(N+P))\geq 2n.$

∎

The next result gives additional information on the validity of inequality
(2.3).

######  Proposition 2.8.

If inequality (2.3) does not hold for $p=k$, then it does not hold for
$p=k+2$.

###### Proof.

Supoose that inequality (2.3) is false for $p=k$. Then there exist positive
definite matrices $A_{1},\ldots,A_{k}\in\mathbb{C}^{n\times n}$ such that
$F(A_{1},\ldots,A_{k})<\frac{kn}{2}$. It is easy to observe that

$F(A_{1},\ldots,A_{k},A_{1},A_{2})=F(A_{1},\ldots,A_{k})+n.$

Thus $F(A_{1},\ldots,A_{k},A_{1},A_{2})<\frac{(k+2)n}{2}$. ∎

As mentioned in the introduction, Shapiro’s inequality holds for all $p\leq
13$ and, for odd $p$ in $13\leq p\leq 23$. By Proposition 2.8, if the
inequality (2.3) holds for $p=12$ and $p=23$, then it holds for $5\leq p\leq
12$ (note that the case $p=3$ is Theorem 2.5 and $p=4$ is Theorem 2.7) and,
for all odd $p$ in the interval $13\leq p\leq 23$. This leads us to the
following conjecture.

######  Conjecture 2.9.

Let $p=12$ or $p=23$. Then $F_{p}\geq\frac{p}{2}n$.

Recall that Shapiro’s inequality does not hold for even $p$ in the interval
$14\leq p\leq 22$ and for any $p>23$. It is natural to ask if there exist
positive definite matrices $A_{1},\ldots,A_{p}$ such that inequality (2.3)
holds for some $p>23$ or, for some even $p$ in the interval $14\leq p\leq 22$?
The following result provides an affirmative answer, and says something more.

######  Theorem 2.10.

Let $p\geq 3$ and let $A_{1},\ldots,A_{p}\in\mathbb{C}^{n\times n}$ be
positive definite matrices. Then

$F(A_{1},\ldots,A_{p})+F(A_{p},A_{p-1},\ldots,A_{1})\geq pn$.

###### Proof.

Define

$A_{p+1}:=A_{1},~{}X_{i}:=A_{i}+A_{i+1},~{}X_{p+1}:=X_{1}$ and
$X_{p+2}=X_{2}.$

Then,

$\displaystyle F(A_{1},\ldots,A_{p})+F(A_{p},\ldots,A_{1})$ $\displaystyle=$
$\displaystyle
Tr\left[\sum\limits_{i=1}^{p}(X_{i}-X_{i+1}+X_{i+2})X^{-1}_{i+1}\right],$
$\displaystyle=$ $\displaystyle
Tr\left[\sum\limits_{i=1}^{p}(X_{i}+X_{i+2})X^{-1}_{i+1}-pI\right]$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{p}Tr\left(X_{i}X^{-1}_{i+1}+X_{i+1}X^{-1}_{i}\right)-pn$
$\displaystyle\geq$ $\displaystyle 2pn-pn$ $\displaystyle=$ $\displaystyle
pn,$

where the fourth inequality follows from Remark 2.1. ∎

For $p=4$, we can strengthen the conclusion of Theorem 2.10. We state this
next. The proof is similar to Theorem 2.5 and we leave the details to the
interested reader.

######  Corollary 2.11.

Let $A_{1},A_{2},A_{3}$ and $A_{4}$ be positive definite matrices. Then

$\lambda\left(\sum\limits_{i=1}^{4}A_{i}(A_{i+1}+A_{i+2})^{-1}+\sum\limits_{i=1}^{4}A_{4-(i-1)}(A_{4-i}+A_{4-(i+1)})^{-1}\right)\geq
4,$

where $A_{5}=A_{1},A_{6}=A_{2}$ and $A_{0}=A_{4},A_{-1}=A_{3}$.

In the last part of this article, we consider two consequences of the Cauchy-
Schwarz inequality for matrices. We provide a proof for the first result,
while we simply state the second result, as it follows along similar lines.
Let us first recall the matrix version of Cauchy-Schwarz inequality

######  Theorem 2.12.

Let $A,B\in\mathbb{C}^{m\times n}$. Then

$|Tr(AB^{*})|^{2}\leq Tr(AA^{*})Tr(BB^{*}).$

######  Theorem 2.13.

Define $M$ by

$M=A(2A+B)^{-1}+B(2B+C)^{-1}+C(2C+A)^{-1}$,

where $A,B,C\in\mathbb{C}^{n\times n}$ are positive definite matrices. Then
$Tr(M)\leq(3n-1)/2.$

###### Proof.

Define the matrix

$N=B(2A+B)^{-1}+C(2B+C)^{-1}+A(2C+A)^{-1}$.

First, observe that $2M+N=3I$ and so, $Tr(2M+N)=3n$. Next, we show that
$Tr(N)\geq 1$. In order to make it easy for computations, we define the block
matrices $W,Z\in\mathbb{C}^{n\times 3n}$, each having three column blocks as
follows:

$W=(BW_{1},CW_{2},AW_{3})$ and $Z=(Z_{1},Z_{2},Z_{3}),$

where

$W_{1}:=(2B^{1/2}AB^{1/2}+B^{2})^{-1/2}=Z_{1}^{-1},$
$W_{2}:=(2C^{1/2}BC^{1/2}+C^{2})^{-1/2}=Z_{2}^{-1}$

and

$W_{3}:=(2A^{1/2}CA^{1/2}+A^{2})^{-1/2}=Z_{3}^{-1}.$

Then,

$WZ^{*}=A+B+C,ZZ^{*}=A^{2}+B^{2}+C^{2}+2(A^{\frac{1}{2}}CA^{\frac{1}{2}}+B^{\frac{1}{2}}AB^{\frac{1}{2}}+C^{\frac{1}{2}}BC^{\frac{1}{2}})$

and

$WW^{*}=BW_{1}^{2}B+CW_{2}^{2}C+AW_{3}^{2}A.$

Thus,

$Tr(WZ^{*})=Tr(A+B+C),$ $Tr(ZZ^{*})=Tr(A^{2}+B^{2}+C^{2}+2(AB+BC+CA))$

and

$Tr(WW^{*})=Tr(B^{2}W_{1}^{2}+C^{2}W_{2}^{2}+A^{2}W_{3}^{2}).$

Note that, $Tr(N)=Tr(WW^{*}).$ By Theorem 2.12,

$|Tr(WZ^{*})|^{2}\leq Tr(WW^{*})Tr(ZZ^{*}).$

Employing Theorem 1.1, one observes that the quotient

$\frac{|Tr(WZ^{*})|^{2}}{Tr(ZZ^{*})}\geq 1.$

Thus, $Tr(N)\geq 1$, so that $Tr(M)\leq(3n-1)/2$. ∎

The corollary concluding this note, is an easy consequence.

######  Corollary 2.14.

Let $A_{1},\ldots,A_{p}\in\mathbb{C}^{n\times n}$ be positive definite
matrices. Then

$Tr(A_{1}^{2}A_{2}^{-1}+\cdots+A_{p}^{2}A_{1}^{-1})\geq
Tr(A_{1}+\cdots+A_{p}).$

###### Proof.

Since $A_{1},\ldots,A_{p}$ are positive definite, their square roots and
inverses exist. Define block matrices $W,Z\in\mathbb{C}^{n\times pn}$ by

$W=(A_{1}A_{2}^{-1/2},A_{2}A_{3}^{-1/2},\ldots,A_{p}A_{1}^{-1/2})$

and

$Z=(A_{2}^{1/2},A_{3}^{1/2}\ldots,A_{p}^{1/2},A_{1}^{1/2}).$

Then, $WZ^{*}=A_{1}+\ldots+A_{p}$, $WW^{*}=A_{1}A_{2}^{-1}A_{1}+\ldots
A_{p}A_{1}^{-1}A_{p}$ and $ZZ^{*}=A_{1}+\ldots+A_{p}$. By Theorem 2.12,

$Tr(A_{1}^{2}A_{2}^{-1}+\cdots+A_{p}^{2}A_{1}^{-1})\geq
Tr(A_{1}+\cdots+A_{p}).$

∎

### Acknowledgment

We thank Apoorva Khare for a detailed reading of an earlier draft and for
providing valuable comments and feedback. We also thank anonymous referees for
their helpful comments.

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