$\displaystyle\leq\int_{0}^{\tau}dt\;\|B^{\mathsf{T}}\varphi(t)\|_{L(q(t))}^{2}\int_{0}^{\tau}dt\;\|B^{\mathsf{T}}\psi(t)\|_{L(q(t))}^{2}$
(232)

for an arbitrary $\varphi$. Hence, by choosing $\varphi(t)=\xi(t)$, the
potential of the duality in Eq. (119) between $q(t)$ and $q(t+dt)$ with $dt$
an infinitesimal time interval, we have

$\displaystyle\mathcal{F}^{\prime}[q]\geq\frac{\pqty{\int_{0}^{\tau}dt\;\xi(t)^{\mathsf{T}}d_{t}q(t)}^{2}}{\int_{0}^{\tau}dt\;\|B^{\mathsf{T}}\xi(t)\|_{L(q(t))}^{2}}.$
(233)

According to the duality (119), if we assume $q(0)=\hat{p}(0)$ and
$q(\tau)=\hat{p}(\tau)$, the numerator reads

$\displaystyle\int_{0}^{\tau}dt\;\xi(t)^{\mathsf{T}}d_{t}q(t)=\int_{0}^{\tau}dt\;\lim_{\Delta
t\to 0}\frac{\mathcal{W}_{1}(q(t),q(t+\Delta t))}{\Delta
t}\geq\mathcal{W}_{1}(\hat{p}(0),\hat{p}(\tau)),$ (234)

where we used the triangle inequality in the last inequality. The denominator
is also bounded as

$\displaystyle\|B^{\mathsf{T}}\xi(t)\|_{L(q(t))}^{2}=\sum_{e}\ell_{e}(q(t))(\xi_{\iota^{\prime}(e)}-\xi_{\iota(e)})^{2}\leq\sum_{e}\ell_{e}(q(t))\leq\frac{1}{2}\sum_{e}\chi_{e}(q(t)),$
(235)

where we used $|\xi_{i}-\xi_{j}|\leq 1$. Recall that
$\chi_{e}(q(t))=k_{e}q_{\iota(e)}(t)+k_{-e}q_{\iota^{\prime}(e)}(t)$ is the
edgewise dynamical activity and the log mean inequality
$\ell_{e}(q(t))\leq(1/2)\chi_{e}(q(t))$. Therefore, we obtain

$\displaystyle\mathcal{F}^{\prime}[q]\geq\frac{\mathcal{W}_{1}(\hat{p}(0),\hat{p}(\tau))^{2}}{\tau\bar{\mathcal{A}}[q]/2}$
(236)

for an arbitrary $q$ with the appropriate initial and final conditions. If we
choose $q=p^{*}$, the minimizer of $\mathcal{F}^{\prime}$ with the initial and
final states $(p(0),p(\tau))$, we have the desired inequality

$\displaystyle\frac{\bar{\mathcal{A}}[p^{*}]}{2}\mathcal{W}_{2}(\hat{p}(0),\hat{p}(\tau))^{2}\geq\mathcal{W}_{1}(\hat{p}(0),\hat{p}(\tau))^{2}.$
(237)

We also find a lower bound for the Onsager excess EP as

$\displaystyle\int_{0}^{\tau}dt\;\sigma^{\mathrm{ex}}\geq\frac{\mathcal{W}_{1}(\hat{p}(0),\hat{p}(\tau))^{2}}{\tau\bar{\mathcal{A}}[\hat{p}]/2}.$
(238)

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# Revealing the Local Cosmic Web by Deep Learning

Sungwook E. Hong Natural Science Research Institute, University of Seoul, 163
Seoulsiripdaero, Dongdaemun-gu, Seoul, 02504, Republic of Korea  Donghui Jeong
Department of Astronomy and Astrophysics, and Institute for Gravitation and
the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
Ho Seong Hwang  Korea Astronomy and Space Science Institute, 776 Daedeokdae-
ro, Yuseong-gu, Daejeon 34055, Republic of Korea  Juhan Kim Center for
Advanced Computation, Korea Institute for Advanced Study, 85 Heogiro,
Dongdaemun-gu, Seoul, 02455, Republic of Korea

###### Abstract

The eighty percent of the matter in the Universe is in the form of dark matter
that comprises the skeleton of the large-scale structure called the Cosmic
Web. As the Cosmic Web dictates the motion of all matters in galaxies and
inter-galactic media through gravity, knowing the distribution of dark matter
is essential for studying the large-scale structure. However, as dominated by
dark matter and warm-hot inter-galactic media, both of which are hard to
trace, the detailed structure of the Cosmic Web is unknown. Here we show that
we can reconstruct the Cosmic Web from the galaxy distribution using the
convolutional-neural-network based deep-learning algorithm. We find the
mapping between the position and velocity of galaxies and the Cosmic Web using
the results of the state-of-the-art cosmological galaxy simulations,
Illustris-TNG. We confirm the mapping by applying it to the EAGLE simulation.
Finally, using the local galaxy sample from Cosmicflows-3, we find the dark-
matter map in the local Universe. We anticipate that the local dark-matter map
will illuminate the studies of nature of dark matter and the formation and
evolution of the Local Group. High-resolution simulations and precise distance
measurements to local galaxies will improve the accuracy of the dark-matter
map.

Local Group; dark matter; large-scale structure of universe

##  1 Introduction

Since Fritz Zwicky inferred its existence from the large velocity dispersion
of the Coma cluster (Zwicky, 1933) and Vera Rubin confirmed it with the flat
rotation curve of galaxies (Rubin & Ford, 1970), astronomers have been only
strengthening the necessity of the non-baryonic matter providing excess
gravity. We call that dark matter. The most substantial pieces of evidence
include an excessive mass-to-light ratio in the dwarf galaxies (Aaronson,
1983), the mismatch between the X-ray map (gas distribution) and the weak
gravitational lensing map (mass distribution; Clowe et al., 2006), as well as
the disparity between the heights of even- and odd-acoustic peaks in the
temperature power spectrum of the cosmic microwave background (Larson et al.,
2011). Dark matter is also an indispensable component of the concordance
cosmological model. Accounting for the measured expansion rate of the Universe
(Planck Collaboration et al., 2018) requires the matter component whose energy
density is over five times larger than that of atoms for which the robust
upper limit comes from big-bang nucleosynthesis (Cooke et al., 2014). The
observed large-scale distribution of galaxies (Anderson et al., 2014) and the
map of weak-gravitational lensing potential (Abbott et al., 2018) also require
the dark matter providing the skeleton of the large-scale structure within
which atoms collapse to form stars and galaxies (Davis et al., 1985).

With the essential role that dark matter plays in modern astronomy and
cosmology, in the past few decades, there have been continuous efforts for
searching for the nature of dark-matter particles in the particle accelerators
(Atlas Collaboration, 2019; Vannerom, 2019), cosmic rays (Giesen et al.,
2015), gamma-rays (Ackermann et al., 2015), and high-energy neutrinos (Aartsen
et al., 2018). Beyond the Milky-way halo, there have also been recent studies
focusing on the dark-matter signals from the extra-galactic sources by cross-
correlating the high-energy cosmic rays with the distribution of galaxies
(Fornasa et al., 2016; Fang et al., 2020) and dark matters traced by weak-
gravitational lensing (Tröster et al., 2017; Ammazzalorso et al., 2020). All
searches for the dark matter particles thus far, however, have not concluded
with a firm detection. They have been only narrowing down the possible dark-
matter masses and the interaction strengths among dark matter particles as
well as between dark matter and atoms (Akerib et al., 2017; Arcadi et al.,
2018). For these efforts of searching for the nature of dark matter, the most
basic information currently lacking is the distribution of the dark matters,
or Cosmic Web, in the local large-scale structure beyond the Milky-way halo.
Of course, we have a good reason to believe that dark-matter halos surround
each galaxy in the Universe. It is, however, also well known that the galaxies
are biased, rather than faithful, tracers of the large-scale structure
(Desjacques et al., 2018).

In this article, we shall present a novel method of unveiling the Cosmic Web
in the local Universe. As dark matters are dark, of course, we cannot observe
them directly from the telescope. The only guaranteed way of searching for the
dark matters is the same method for their discovery, through their
gravitational influence on visible objects. On the inter-galactic scales, it
is dark matter that dominates the gravitational interaction and determines the
cosmic velocity flow. We can, therefore, infer the distribution of dark
matters by carefully studying the distribution and motion of galaxies. Taking
the observed distribution of galaxies and their peculiar velocity flow, in
what follows, we shall decipher the dark matter distribution, or Cosmic Web,
within local $\sim 20\mathinner{{\rm Mpc}/h}$.

When reconstructing the local dark-matter distribution directly from observed
galaxy distributions, we face the following challenges. First, the local
galaxy distribution at the low Galactic latitudes is hidden behind the intense
radiation from the Galactic disk and contaminated by the interstellar gas and
dust, which makes it hard to obtain the complete map of the galaxy
distribution. Second, even if we had the complete map of galaxies, they are
biased tracers of the large-scale structure; that is, the distribution of
galaxies does not necessarily reflect the distribution of dark matters.
Previous attempts (Gottloeber et al., 2010; Libeskind et al., 2010; Carrick et
al., 2015) of making the local dark-matter map, therefore, have relied on the
cosmological simulations constrained by the smoothed (with smoothing radius
$\geq 4\mathinner{{\rm Mpc}/h}$) density field at high-Galactic latitudes.
This observational constraint for the fully evolved galaxy distribution is
non-trivial to implement in the simulation because what the simulation needs
is the density distribution at the initial time.

Here we overcome the challenges by taking a novel approach based on the deep
learning (DL) that allows us to incorporate all information in the observed
galaxy data: the spatial distribution and the radial peculiar velocity of
galaxies. That is, we use the convolutional neural network (CNN)-based DL
algorithm to find the mapping between the local dark-matter distribution and
the observed positions and the radial peculiar velocities of local galaxies.

The structure of this paper is as follows. In Section 2, we describe the
simulation and observational data used for DL training and prediction,
respectively. In Section 3, we will briefly describe our DL architecture and
the evaluation of our DL model. In Section 4, we will show the reconstructed
local dark matter map and its statistical robustness. We will summarize our
result in Section 5.

Throughout the paper, we assume a standard $\Lambda$CDM cosmology in
concordance with the Planck 2018 analysis (Planck Collaboration et al., 2018):
$(\Omega_{\rm m}^{0},\Omega_{\Lambda}^{0},h)=(0.31,0.69,0.6777)$. It is
similar to the standard cosmologies adopted in Illustris-TNG and EAGLE
simulations: $(\Omega_{\rm
m}^{0},\Omega_{\Lambda}^{0},h)=(0.3089,0.6911,0.6774)$ and
$(0.307,0.693,0.6777)$, respectively. (Springel et al., 2018; Schaye et al.,
2015)

##  2 Data

###  2.1 Observational Data: Cosmicflows-3

We use the Cosmicflows-3 galaxy catalogue (Tully et al., 2016, CF3 hereafter),
one of the most comprehensive galaxy catalogue that provides distance, radial
peculiar velocity, and luminosity of 17,647 galaxies up to $200~{}{\rm Mpc}$.
To produce a fair galaxy sample over the given region, we make the volume-
limited sub-sample of the CF3 as follows. First, as the number density of the
CF3 galaxies close to the Galactic plane (Galactic latitude $|b|<10^{\circ}$)
is lower than average, we only use the galaxies at $|b|>10^{\circ}$. Also, we
use the $B$-band absolute magnitude ($M_{B}$) compiled from Lyon Extragalactic
Database (LEDA; Paturel et al., 2003) as a proxy of the stellar mass
($M_{\star}$; Wilman & Erwin, 2012) We set the $B$-band magnitude $-15$ as the
selection criterion, which is sufficient for covering the $20\mathinner{{\rm
Mpc}/h}$\- and $40\mathinner{{\rm Mpc}/h}$-cubic volume around the Milky-way
galaxy. We have also tested the cases with $M_{B}<-16$ and $-17$ and find no
noticeable difference to the predictions from the fiducial choice. Note that
we have not used the $K_{S}$-band absolute magnitude, one of the best-known
tracer of the stellar mass (Bell et al., 2003), because that information is
missing for about 30% of the galaxies in our sample (Lavaux & Hudson, 2011;
Huchra et al., 2012).

We calculate the radial peculiar velocity by subtracting the Hubble flow from
the velocity in the Galactic standard of rest ($V_{\rm GSR}$; Kourkchi et al.,
2020). Note that we do not use the velocity in the CMB standard of rest
($V_{\rm CMB}$), to reduce any bias that might be introduced in the
conversion. Instead, when generating training and test samples from simulation
data, we include the peculiar motion of the Milky-way corresponding galaxy in
each simulation. There exists a difference on the Hubble constant between
recent CMB observations ($H_{0}=67.77\,{\rm km/s/Mpc}$; Planck Collaboration
et al., 2018) and the best-fit from the CF3 ($H_{0}=75\,{\rm km/s/Mpc}$; Tully
et al., 2016). In this study, we have tested both values and find that the
effect from the different Hubble constants stays within the uncertainty of the
dark-matter map.

###  2.2 Simulation Data: Illustris-TNG & EAGLE

We use the TNG100-1, a simulation with $V=(75\mathinner{{\rm Mpc}/h})^{3}$ and
$1820^{3}$ dark-matter and gas particles from the Illustris-TNG simulation
suite (Springel et al., 2018), as our high-resolution simulation data (TNG100
hereafter). To mimic the observation from the Milky-way galaxy, we select 988
galaxies with stellar mass

$4\times 10^{10}\mathinner{{\rm M}_{\odot}}<M_{\star}<10^{11}\mathinner{{\rm
M}_{\odot}}$ (1)

(“center galaxies” hereafter), by adopting that the Galactic stellar mass is
about $5.2\times 10^{10}\mathinner{{\rm M}_{\odot}}$ (Licquia & Newman, 2015).
Around each center galaxy, we make a sub-cube with $20\mathinner{{\rm Mpc}/h}$
box-size and calculate the dark-matter density field within the $64^{3}$
uniform grid. We also calculate the relative position of galaxies with
$M_{B}<-15$ (“target galaxies” hereafter), as well as the difference of
peculiar velocity between the target galaxy and centre galaxy.

For the low-resolution dark-matter map with $V=(40\mathinner{{\rm
Mpc}/h})^{3}$, we use the TNG300-1 from the Illustris-TNG simulations, whose
volume and number of particles are $V=(205\mathinner{{\rm Mpc}/h})^{3}$ and
$2500^{3}$, respectively (TNG300 hereafter). Note that the amplitude of the
luminosity function of TNG300 is lower than the observation and TNG100, mainly
due to the lower spatial resolution of the simulation (Pillepich et al.,
2018). Therefore, we additionally apply the “resolution correction” to find
the center and target galaxies by using the number density obtained from
TNG100 rather than directly using the face values of $M_{\star}$ or $M_{B}$.
We also use the TNG300-1-Dark, a dark-matter-only counterpart of the TNG300,
to test how baryonic physics affects our result. We select the center and
target galaxies by finding the mass cut of dark matter halos with the same
number density. The result from the TNG300-1-Dark is similar to or slightly
worse than TNG300.

Also, we use the RefL0100N1504, a reference simulation with
$V=(67.77\mathinner{{\rm Mpc}/h})^{3}$ and $1504^{3}$ dark-matter and gas
particles from the EAGLE simulation suite (Schaye et al., 2015; Crain et al.,
2015, EAGLE hereafter), to check the generality of our result. For the center
galaxies, we use the same selection criterion to TNG100 and find 478 center
galaxies. For the target galaxies, however, we do not directly use $M_{B}$. It
is because the luminosity function of EAGLE is reliable only for bright
galaxies ($M_{B}\lesssim-18$), since the EAGLE simulations calculate the
luminosity only to massive galaxies ($M_{\star}\geq 10^{8.5}{\rm M_{\odot}}$;
Camps et al., 2018). Instead, similar to TNG300, we use the galaxy number
density obtained from TNG100 to find the stellar mass cut of target galaxies.

##  3 Methods

###  3.1 Deep Learning Architecture

Figure 1: The convolutional neural network (CNN) architecture used for TNG300.
The size (except the number of filter) of each layer for TNG100 is half of
TNG300.

We use the convolutional neural network (CNN) similar to the U-Net
(Ronneberger et al., 2015) or V-Net (Milletari et al., 2016) to predict the
dark-matter density field from the galaxy position and radial peculiar
velocity (see Figure 1). Our CNN architecture consists of the following two
stages. The encoding stage (Input–Conv#) extracts features by increasing the
number of filters and decreasing the layer size. The decoding stage
(UpConv#–Output), on the other hand, constructs a new 3-dimensional map by
increasing the layer size and decreasing the number of filters. The decoding
stage uses the layer with the same size from the encoding stage by
concatenation to increase the output resolution.

In practice, the encoding stage consists of multiple Conv# layers, where #
denotes the layer size. Conv# consists of (1) 2-pixel reflection padding; (2)
3-dimensional convolution with 5-pixel filters and 2-pixel strides; (3)
rectified linear unit (ReLU) activation (Hahnloser et al., 2000; Glorot et
al., 2011); and (4) batch normalization (Ioffe & Szegedy, 2015). On the other
hand, UpConv# layers in the decoding stage contain (1) up-sampling; (2)
concatenation with a layer from the encoding stage; (3) batch normalization;
(4) 1-pixel reflection padding; (5) 3-dimensional convolution with 3-pixel
filters; and (6) ReLU activation. For the output layer, however, we use
hyperbolic tangent so that the values stay between $-1$ and $+1$.

We apply two different types of layer size for TNG100 and TNG300. In the case
of TNG100, we perform the encoding stage to the $64^{3}$-grid input layer to
produce 2,048 channels of the $2^{3}$-grid layer (Conv2) and apply it to the
decoding stage to produce $64^{3}$-grid output layer. In the case of TNG300,
on the other hand, we start with the $128^{3}$-grid input layer to 2,048
channels of the $4^{3}$-grid layer, then to the $128^{3}$-grid output layer.
We have also tested other CNN architectures with smaller values of the maximum
number of channel, and find that the CNN architecture shown in Figure 1
performs the best.

###  3.2 Training

We divide the training and validation samples from TNG100 so that all sub-
cubes from the validation sample do not overlap with those from the training
sample. As a result, we only use 525 sub-cubes — 432 for training and 93 for
validation. For each sub-cube, we make two $64^{3}$ uniform grids as a two-
channel input layer: the number of target galaxies and the averaged radial
peculiar velocity in units of ${\rm km/s}$. For the input layer, we apply the
same Galactic-latitude mask to the CF3 data (masking out $|b|<10^{\circ}$).
For the output layer, we normalize the logarithm of dark-matter density to be

$y=\frac{1}{4.5}\log_{10}(\rho/\rho_{0})$ (2)

so that all values in the output layer would be between $-1$ and $+1$.

For data augmentation, we allow swapping the $(x,y,z)$-axes of each sub-cube
so that the number of samples becomes three times larger. We also make more
data augmentation by flipping the axis direction, which in return increases
the number of samples eight times. Note that, unlike U-Net or V-Net, we have
not split a single cube into multiple smaller cubes for data augmentation, to
prevent the change of Galactic-latitude mask and the radial peculiar velocity.
In the end, we obtain the total 10,368 and 2,232 samples, respectively, in
training and validation sets.

We build our CNN architecture by using the Keras (Chollet et al., 2015) with
the Tensorflow backend (Abadi et al., 2015). We use the Adam optimizer (Kingma
& Ba, 2014) with learning rate $10^{-3}$ and adopt the mean squared error as
the loss function. We use the NVIDIA Tesla V100-PCIE 16GB GPU card for
training, and we set mini-batch size as 6 to be able to run within the given
GPU memory. For each training, we run 200 epochs by using 1,728 mini-batches
per epoch, and it takes about 73 hours for a single run.

We perform a similar training for the TNG300 outcome, except the following
differences. First, we have 10,629 training sub-cubes and 1,256 validation
sub-cubes, with each sub-cube having $128^{3}$-grids. Unlike TNG100, we do not
apply further data augmentation, mainly due to the expensive computational
cost from large CNN architecture size. Second, since the dynamic range of
dark-matter density of TNG300 is wider than TNG100, we use

$y=\frac{1}{5}\log_{10}(\rho/\rho_{0})$ (3)

for the output layer instead. Third, for the predictions of logarithmic dark-
matter density, we use the cyclic learning rate between $3\times 10^{-8}$ and
$4\times 10^{-5}$, which we find by additional learning rate tests (Smith,
2015). Finally, due to the large CNN architecture size, we use four NVIDIA
Tesla V100-SXM2 32GB GPU cards with mini-batch size as 8. For each training,
we run 400 epochs by using only 157 mini-batches per epoch, and it takes about
90 hours for a single run.

For TNG300, we perform two more alternative trainings. One alternative
training uses the logarithm of the total stellar mass-weighted galaxy number
density rather than the simple galaxy number density as an input layer, and
its result is similar to our fiducial choice. The other one, one the other
hand, does not use the radial peculiar velocity, and its result does not show
any small-scale structure.

In both cases, the validation loss function reaches its global minimum, and it
does not increase more than 1.1 times its global minimum until the last epoch.
From each run, we select three models from three different epochs for the
following performance test: at the minimum validation loss, at the minimum
training loss, and the last epoch.

###  3.3 Performance Test

Figure 2: 3-way projections of a single TNG300 validation sample with
$5\mathinner{{\rm Mpc}/h}$-thickness. From left to right: galaxy number,
radial peculiar velocity, true dark-matter density, and reconstructed dark-
matter density.

Figure 3: Result of the performance tests for the deep learning result using
the three-dimensional dark-matter density field of simulations. Top panel:
statistical comparison between the ground truth and the predicted dark-matter
density from the entire TNG300 validation sample. From left to right: joint
probability distribution (colors) with $1,2,3$-$\sigma$ certainty level
contours (lines), median (lines) and 1-$\sigma$ deviation (shades) of
histograms, and median (lines) and 1-$\sigma$ deviation (shades) of the two-
point correlation functions. Bottom panel: similar to the top panel, but by
applying the TNG100 training to the entire EAGLE test sample.

For both TNG100 and TNG300 training outcomes, we test the resulting models by
applying them to the validation samples, so that we can compare the resulting
dark-matter density cube with the ground truth. Specifically, we use the
following four methods for the performance test—visual comparison, joint
probability distribution, histogram, and two-point correlation function (2pCF)
(see Figures 2 & 3). We find that 2pCF is the most efficient method to test
the performance of each model. For both TNG100 and TNG300, the models at the
minimum training loss provide the closest distribution of the 2pCF predictions
to their truth, and we adopt them as our optimal models.

After choosing the optimal models, we perform the convergence test between
models with different simulation resolutions and setups. First, we compare the
local dark-matter density field predictions from TNG100 and TNG300 within the
radius $r=10\mathinner{{\rm Mpc}/h}$. We find that they show similar
distribution up to $r\sim 4\mathinner{{\rm Mpc}/h}$, while the dark-matter map
from TNG100 shows finer small-scale structures than TNG300 (see Figure 3).
Also, we apply the CNN model from TNG100 to the test sample of EAGLE and find
that its performance test result is similar to the TNG100 validation sample
(see Figure 3). Note that we do not apply the CNN model from TNG300 to EAGLE
because the volume of EAGLE is not sufficiently larger than the volume of
TNG300 sub-cubes.

Furthermore, to estimate the uncertainties of the dark-matter map, we perform
a stress test to our CNN models by incorporating the uncertainties of distance
measurement in the CF3. We first use the $1\sigma$ uncertainty of the distance
modulus of each galaxy and generate 1,000 sets of random distance moduli that
follow the normal distribution. Then, we re-calculate the radial peculiar
velocity by subtracting the Hubble flow corresponding to the random distances
from the $V_{\rm GSR}$. Since the distance measurement error exists only along
the radial direction, we have generated the two-dimensional column density map
of the dark matter that is less affected by the error than the three-
dimensional dark-matter density field (see Figure 6). Also, we find that the
dark-matter column density map driven from TNG300 shows significantly less
deviation than that of TNG100, which suffers from some spurious structure
consistently appearing near the Galactic plane.

##  4 Results

###  4.1 Three-dimensional view of the local Cosmic Web

|
---|---
|
Figure 4: Three-dimensional density maps of the local dark matter with
$40\,{\rm Mpc}/h$-boxsize and $4\,{\rm Mpc}/h$-thickness. ‘X’-mark at the
center: Milky-way galaxy. Dots: galaxies with $M_{B}<-15$. Texts: galaxy
groups, clusters, and local structures. Arrows: estimated directions of motion
derived from the gradient of the reconstructed gravitational potential.

Figure 4 shows a sliced view of the reconstructed Cosmic Web integrated over
$4\,{\rm Mpc}/h$ thickness. Each panel shows the Cosmic Web on the plane of
the Supergalactic Cartesian coordinates (SGX, SGY, and SGZ), extended to the
full cube with the side length $40\,{\rm Mpc}/h$. Figure 4 clearly shows known
local objects that we designated by their common name. The figure also
recovers known local large-scale structures. For example, we find a $10\,{\rm
Mpc}/h$-spread along $+{\rm SGY}$-direction in the SGZ-SGY (upper left panel)
and SGY-SGX (lower-right panel) planes. This structure is known as Local Sheet
that connects the Local Group and Virgo cluster and containing M81, NGC5194,
Canes II, and Coma I groups (Tully et al., 2008; Courtois et al., 2013). We
also find that, around the Local Group, the Local Sheet is connected to the
Fornax Wall (Fairall et al., 1994), which is a $20\,{\rm Mpc}/h$-sized spread
along ($-$SGY,$-$SGZ)-direction, containing Fornax cluster, Eridanus cluster,
and Dorado group as members (upper-left panel). At the opposite direction to
the Fornax Wall on the SGZ-SGY plane, the Local Void (Tully & Fisher, 1987) is
also apparent (also shown on the SGZ-SGX plane), which might extend beyond the
boundary of our local universe sample. In Figure 4, we also present the
velocity flow lines derived from the gradient of the reconstructed
gravitational potential with arrows and black lines. The velocity flow shows
the motion of material from the Local Void to nearby filamentary structures
and clusters such as the Local Sheet, Fornax Wall, and Virgo cluster. It
should be noted that we cannot reproduce the velocity flow from the Virgo
cluster to the Great Attractor ($+{\rm SGX}$-direction), because of the
limited extension of the volume that we analyze here. However, we would like
to emphasize that the recovered dark-matter map provides us detailed density
and velocity fields around these known local large-scale structures.

The recovered Cosmic Web also shows a hint of new structures that require
further investigation. For example, the direction of the Local Sheet is
similar to the direction of the so-called vast polar structure (VPOS), which
consists of satellite galaxies, globular clusters and stellar streams around
the Milky-way galaxy (Pawlowski et al., 2012). As shown in Figure 4, the Local
Sheet, being the strongest filamentary structure around the Local Group, is a
source of velocity flow; that might cause a connection between the two. Also,
there are a couple of small filaments visible in our maps, which could be good
targets for systematic examination with deep imaging surveys.

###  4.2 Sky map of the Local Cosmic Web

|
---|---
|
|
|
|
Figure 5: Two-dimensional full-sky map of the local dark-matter column density
with $4\mathinner{{\rm Mpc}/h}$ width. Left panels: predictions from TNG300
training, from the nearest to the farthest radial bin. Right panels:
comparison predictions from TNG100 training (‘TNG100’), training with dark
matter halos from the dark matter-only simulation (‘DMhalo’), and training
without using the radial peculiar velocity (‘noVpec’). Small dots: positions
and peculiar velocity (color) of known local galaxies. Large dots: galaxy
groups and clusters with their names.

Figure 6: Same as Figure 5, but showing statistical maps. Left panels: mean of
the logarithm of dark-matter column-density estimated from 1000 random
realizations incorporating the uncertainties in distance estimate to the local
galaxies. Middle panels: standard deviation from 1,000 random realizations
(${\rm Nside}=4$, $8$, $8$, $16$, $16$ from top to bottom). Right panels:
systematic bias from different simulation input for the deep-learning (TNG300
galaxies vs. dark-matter halos).

The left panels of Figure 5 (labeled as TNG300) show the recovered local dark-
matter map (gray map) on the sky (with the Healpix (Zonca et al., 2019; Górski
et al., 2005) parameter of ${\rm Nside}=128$, which roughly corresponds to the
angular resolution of $27$ arc minutes) along with the location and radial
velocities (color-coded dots) of galaxies that we use for the reconstruction.
The figure also shows the locations of some well-known galaxy groups and
clusters. The map in Figure 5 uses the radial distance and radial peculiar
velocities as reported in the Cosmicflows-3 catalogue (Tully et al., 2016). We
have mitigated the $10\sim 30\%$ uncertainties of distance measurement in the
catalogue by adopting the radial binning of $\Delta r=4\mathinner{{\rm
Mpc}/h}$. We further analyze the statistical uncertainties of the recovered
dark-matter map by generating 1,000 realizations incorporating the
uncertainties of the distance measurement. From the high angular resolution
map (${\rm Nside}=128$), we find that the angular covariance function
indicates a strong pixel-to-pixel correlation below angular scale
$\theta_{0}=20.7^{\circ}$, $9.71^{\circ}$, $6.53^{\circ}$, $5.04^{\circ}$, and
$4.24^{\circ}$, respectively, from the nearest ($r<4\mathinner{{\rm Mpc}/h}$)
to the farthest ($16\mathinner{{\rm Mpc}/h}<r<20\mathinner{{\rm Mpc}/h}$)
radial bins. For the statistical analysis, we degrade the angular resolution
of each map to $\theta_{0}$ and assume that each pixel in the degraded map is
statistically independent. Figure 6 shows the mean and the $1$-$\sigma$
($68\%$ C.L.) uncertainty maps of the recovered local dark-matter
distribution, adopting the distance uncertainties reported in the
Cosmicflows-3 galaxy catalogue. We find that the standard deviation per pixel
stays in the range of $\sigma(\log_{10}\Sigma/\Sigma_{0})\simeq 0.1\sim 0.4$,
regardless of the density contrast, and the signal-to-noise ratio scales
almost linearly as the density contrast, reaching up to signal-to-noise ratio
10 for the density peaks. On average, the signal-to-noise ratios for dark-
matter distribution per pixel at higher Galactic latitudes ($|b|>10^{\circ}$)
are 4.25, 3.76, 3.94, 4.19, and 4.52, respectively, from the nearest to the
farthest radial bin.

Note that, when estimating the uncertainty map in Figure 6, we have not
included the systematic uncertainties in DL mapping itself into the error
budget. For example, the galaxy simulations with different resolutions or
different sub-grid prescriptions can lead to different DL mapping. First, we
check the systematic effect of the resolution by comparing the local dark-
matter map estimated from TNG100 and TNG300. The top-right panel of Figure 5
shows the $r<4\mathinner{{\rm Mpc}/h}$ bin dark-matter map driven from the
TNG100 simulation. The high-resolution result (TNG100) systematically
underestimate the density contrast by $2.3$-$\sigma$ on average. To estimate
the systematic effect from different sub-grid prescriptions, we have repeated
the deep-learning procedure by using the dark-matter halo samples (matching
number density) from the dark-matter only TNG simulation. The right panels of
Figure 6 show the difference between the two dark-matter maps in units of
standard deviation at each pixel. Even with this extreme comparison (full
hydrodynamic simulation against pure $N$-body simulation), we find that
systematic effects lead to $1.7$, $1.4$, $1.2$, $1.1$, $1.0$-$\sigma$
deviations per pixel from the top (nearest) to the bottom (furthest) maps. We
further test the systematic effect due to different Hubble parameters
($H_{0}=75\,{\rm km/s/Mpc}$) and find only $0.15$-$\sigma$ deviation.
Different $B$-band magnitude cuts ($M_{B}<-16$, and $M_{B}<-17$) lead to
$\simeq 1$-$\sigma$ deviation. Most importantly, none of the systematic maps
shows a significant correlation with the derived cosmic web structure,
ensuring the robustness of the derived dark-matter distribution, or the Cosmic
Web.

The most striking feature that we have recovered in this study is the
filamentary Cosmic Web that is apparent in both Figures 5 and 6. We find that
the radial peculiar velocity information is vital to reconstructing the cosmic
web, without which the same DL algorithm can not reproduce the Cosmic Web
structure at all. For example, the right panels in Figure 5, indicated by
‘noVpec’, show the deep-learning result only using galaxy distributions. Note
the absence of the filamentary structure in those maps. We note that the
‘noVpec’ maps simply resemble the smoothed version of the galaxy distribution.
The deep-learning algorithm with stellar-mass weighted galaxy distribution,
without peculiar velocity information, leads to the similarly poor quality
map. Another interesting feature in the map is the dark-matter distribution at
lower Galactic latitudes ($|b|<10^{\circ}$) where we do not have any input
galaxy data. To our surprise, we find that the signal-to-noise ratios per
pixel for this region are $4.18$, $4.73$, $5.31$, $5.80$, and $6.21$,
respectively, from the nearest to the farthest radial bin. We, however,
anticipate that the theoretical uncertainties for the DL mapping would be most
substantial for this region. For example, from the aforementioned studies on
systematic uncertainties, we find that on average lower Galactic latitudes
($|b|<10^{\circ}$) map suffers about $0.5$-$\sigma$ more systematical shifts
than higher Galactic latitudes ($|b|>10^{\circ}$) map. This is indicated in
the top two panels of Figure 5, as well as the systematic shifts shown in the
right panels of Figure 6.

##  5 Discussion

In this paper, we present a novel CNN-based Deep Learning method of
reconstructing the local dark-matter distribution map and discover the local
Cosmic-Web structure traced by the positions and radial peculiar velocities of
Cosmicflow-3 galaxies. We find that including the radial peculiar velocity
field is the key to recover the dark matter distribution in the Cosmic Web.
Incorporating the observational uncertainties in the galaxy distance
measurements, the average detection significance of the dark-matter map
exceeds $4.1$-$\sigma$ for each Healpix pixel at higher Galactic latitudes
($|b|>10^{\circ}$). The quoted statistical significance, however, does not
include the uncertainties in the galaxy-to-dark-matter mapping itself. Here,
we have tested that the DL results stay robustly for three different
simulations: TNG100, TNG300, and EAGLE but future studies must quantify the
theoretical uncertainties by applying the same method to the large-scale
structure simulations with different baryonic prescriptions. The comparison of
the DL results between TNG300 and $N$-body simulations, however, indicates
that the filamentary Cosmic-Web structure may not suffer from the systematic
effects.

The main statistical uncertainty in the galaxy data comes from the uncertainty
in the distance measurement. As the observed shift in the galaxy spectra
constrains the sum of the distance (Hubble flow) and the radial peculiar
velocity, the uncertainty affects both the galaxy distribution and the radial
peculiar velocity field. Therefore, to obtain a dark-matter map with higher
significance, it is necessary to explore the ways to reduce the uncertainties
of the current distance estimators such as the Tip of the Red Giant Branch,
the Type Ia supernova, and the Fundamental Plane through continuous cross
calibration (Tully et al., 2016), and to increase the number of galaxies with
measured distances through systematic surveys (e.g., 6dFGS (Springob et al.,
2014), James Webb Space Telescope (Gardner et al., 2006)).

We anticipate that the reconstructed three-dimensional dark-matter map and
peculiar velocity field will open an entirely new chapter of cosmological
study. For example, the dark-matter map can make it possible to run the
cosmological galaxy simulations with the precise initial condition of the
Local Group for studying the past and future of our cosmic neighborhood. It
will also allow the in-depth study of the nature of dark matter by cross-
correlating the reconstructed dark matter map with the full-sky diffuse
emission maps constructed from the radio-to-gamma-ray electromagnetic spectra
as well as the full-sky map of gravitational wave binaries. The latter can
test the models where black holes in binaries have formed out of dark matter
(Shandera et al., 2018).

Authors acknowledge Christophe Pichon, Changbom Park, Sungryong Hong, Inkyu
Park, Dongsu Bak, Graziano Rossi, Yung-Kyun Noh for discussion. The full-sky
maps in this paper have been derived using the healpy and HEALPix package. All
figures in this paper have been produced with helps of NumPy/SciPy, pandas,
astropy, and Matplotlib packages. The list of nearby galaxy groups and
clusters are derived from www.atlasoftheuniverse.com. Authors acknowledge the
Korea Institute for Advanced Study for providing computing resources (KIAS
Center for Advanced Computation Linux Cluster System). Computational data were
transferred through a high-speed network provided by the Korea Research
Environment Open NETwork (KREONET). SEH was supported by Basic Science
Research Program through the National Research Foundation of Korea funded by
the Ministry of Education (2018R1A6A1A06024977). DJ was supported at
Pennsylvania State University by NASA ATP program (80NSSC18K1103). JK was
supported by a KIAS Individual Grant (KG039603) via the Center for Advanced
Computation at Korea Institute for Advanced Study.

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