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Cluster-counterpart Voids: Void Identification from Galaxy Density Field
Junsup Shim
1,2
, Changbom Park
1
, Juhan Kim
3
, and Sungwook E. Hong (홍성욱)
4,5
1
School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
2
Institute of Astronomy and Astrophysics, Academia Sinica, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan
3
Center for Advanced Computation, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea; kjhan@kias.re.kr
4
Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea
5
Astronomy Campus, University of Science & Technology, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
Received 2022 October 6; revised 2023 April 29; accepted 2023 May 15; published 2023 July 18
Abstract
We identify cosmic voids from galaxy density fields under the theory of void–cluster correspondence. We extend
the previous novel void-identification method developed for the matter density field to the galaxy density field for
practical applications. From cosmological N-body simulations, we construct galaxy number- and mass-weighted
density fields to identify cosmic voids that are counterparts of galaxy clusters of a specific mass. The parameters for
the cluster-counterpart void identification such as Gaussian smoothing scale, density threshold, and core volume
fraction are found for galaxy density fields. We achieve about 60%–67% of completeness and reliability for
identifying the voids of corresponding cluster mass above 3 ×10
14
h
−1
M
e
from a galaxy sample with the mean
number density,
n
h4.4 10 Mpc
31 3
¯()=´
-- -
. When the mean density is increased to
n
h10 Mpc
21 3
¯()=-- -
, the
detection rate is enhanced by ∼2%–7% depending on the mass scale of voids. We find that the detectability is
insensitive to the density weighting scheme applied to generate the density field. Our result demonstrates that we
can apply this method to the galaxy redshift survey data to identify cosmic voids corresponding statistically to the
galaxy clusters in a given mass range.
Unified Astronomy Thesaurus concepts: Large-scale structure of the universe (902);Voids (1779)
1. Introduction
Cosmic voids are the largest volume component of the large-
scale structures in the universe (Joeveer & Einasto 1978;
Einasto et al. 1980; de Lapparent et al. 1986; Gott et al. 1986;
Vogeley et al. 1994)but contain a relatively small amount of
mass. Because of their underdense nature, voids have been
utilized as a probe for dark energy (Lee & Park 2009; Lavaux
& Wandelt 2010;Li2011; Pisani et al. 2015; Sutter et al. 2015;
Achitouv 2017), gravitation (Nusser et al. 2005; Li et al. 2012;
Cai et al. 2015; Achitouv 2019), initial conditions (Kim &
Park 1998), as well as a laboratory for studying the
environmental effect on galaxy formation and evolution
(Ceccarelli et al. 2008; Park & Lee 2009; Kreckel et al.
2011; Beygu et al. 2013; Shim et al. 2015; Ceccarelli et al.
2022).
For void identification from simulation and obser-
vation, various void-finding methods have been developed
(Kauffmann & Fairall 1991; El-Ad et al. 1996; Colberg et al.
2005; Patiri et al. 2006b; Hahn et al. 2007; Platen et al. 2007;
Neyrinck 2008; Forero-Romero et al. 2009; Lavaux &
Wandelt 2010; Sutter et al. 2015; Shim et al. 2021). With
reasonable void-finding parameter values, a void-finder may
identify voids that are aspherical and hierarchical. However,
neither the asphericity nor hierarchical structure of voids
are taken into account in the spherical void formation
model (Fillmore & Goldreich 1984; Suto et al. 1984;
Bertschinger 1985). Thus, it is not unnatural to find some
discrepancy between voids in data and the spherical void
model. For example, the shell-crossing density threshold
derived from the statistics of voids in data is inconsistent with
the prediction based on the spherical void approximation (Chan
et al. 2014; Achitouv et al. 2015; Nadathur & Hotchkiss 2015).
This motivates a search for a new definition and identification
method for voids that mitigate the inconsistency between voids
in theory and data.
Recently, a new concept of defining voids has been
presented (Pontzen et al. 2016; Shim et al. 2021; Stopyra
et al. 2021; Desmond et al. 2022). The key idea of void
identification in these studies is that a void of a certain size can
be related with a dark matter halo of a particular mass because a
halo region would have become a void if the initial overdensity
field were inverted. The correspondence between voids and
dark matter halos in the inverted field is more reliable on the
scale of massive galaxy clusters. This argument leads us to
define cosmic voids as a counterpart to galaxy clusters, and we
will call the void of a certain size, identified and related with
the cluster of the corresponding mass, the cluster-counterpart
void (CCV). Other interesting features of CCVs include (1)
almost universal density profiles (Shim et al. 2021)and (2)
their mean density contrast being close to the prediction from
the spherical expansion model (Stopyra et al. 2021; Desmond
et al. 2022). Interestingly, (3)they are still in the quasi-linear
regime, and information on the initial conditions is better
preserved relative to clusters (Kimzx & Park 1998; Stopyra
et al. 2021).
In order to identify CCVs from a given density field, two
different approaches have been suggested. The major differ-
ence between the two approaches is whether the void
identification method requires the reconstruction of the initial
density fields. Without utilizing the initial density field, Shim
et al. (2021)established a simple CCV-identification method
adopting free parameters for smoothing scale, density, and
volume thresholds. On the other hand, reversely evolving a
The Astrophysical Journal, 952:59 (8pp), 2023 July 20 https://doi.org/10.3847/1538-4357/acd852
© 2023. The Author(s). Published by the American Astronomical Society.
Original content from this work may be used under the terms
of the Creative Commons Attribution 4.0 licence. Any further
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1
given density field to the initial state and then forwardly
evolving its sign-inverted density fluctuation field to the present
epoch to identify CCVs was suggested by Stopyra et al. (2021)
and implemented by Desmond et al. (2022). In this study, we
extend and apply the former approach adopted in Shim et al.
(2021)to galaxy density fields.
This paper is organized as follows. We describe our
simulations and the scheme for assigning galaxies to halos in
Section 2. In Section 3, we briefly review the identification of
CCVs from dark matter density fields and describe how we
apply this approach to galaxy density fields in Section 4.We
discuss our results and conclude in Section 5.
2. Simulation and Galaxy Mocks
We use a pair of N-body simulations with inverted initial
overdensity fields to study the CCVs and their corresponding
clusters. The Mirror simulation used in this work is one of the
Multiverse simulations introduced by Park et al. (2019),
Tonegawa et al. (2020). The Reference simulation starts with
the initial density fluctuations that have the same amplitude as in
the Mirror simulation but have the opposite sign. The simulations
adopt the GOTPM (Dubinski et al. 2004)code and WMAP 5 yr
ΛCDM cosmology (Dunkley et al. 2009)with the matter, baryon,
dark energy density parameters set to 0.26, 0.044, and 0.74,
respectively. Each simulation evolves N
p
=2048
3
dark matter
particles, with mass m
p
;9×10
9
h
−1
M
e
, in a periodic cubic box
of side length L
box
=1024 h
−1
Mpc.
We identify dark matter halos with 30 or more particles
using the friend-of-friend algorithm with the linking length
ll0.2
link p
=, where lpis the mean particle separation. We then
populate the identified halos with mock galaxies based on the
most bound member particle (MBP)—galaxy correspondence
model (Hong et al. 2016). In this galaxy assignment scheme, all
MBPs marked in halo merger trees are the proxies for galaxies
(De Lucia et al. 2004; Faltenbacher & Diemand 2006). The
position and velocity of a galaxy are taken from those of an
MBP, whereas galaxy luminosity is determined by the
abundance matching between the mass function of mock
galaxies (see Hong et al. 2016, for the definition of galaxy
mass)and the luminosity function of the SDSS main galaxies
(Choi et al. 2007). The survival time of a satellite halo is given
by the merger timescale modeled with a modified version of the
fitting formula described in Jiang et al. (2008). We set the
power-law index of the host-to-satellite mass ratio to 1.5. This
is to yield a good match for the projected two-point correlation
functions between the mock galaxies in our simulations and
SDSS main galaxies down to scales below 1 h
−1
Mpc (Zehavi
et al. 2011).
We construct two galaxy samples with different mean number
densities to test the effect of sample density on the CCV
identification. The mean number densities of the sparse and
dense samples are
n
4.4 10
gal 3
¯=´
-and 1.0 ×10
−2
h
3
Mpc
−3
,
respectively. We chose the sparse sample density as such
assuming that the sparse galaxy sample mimics a volume-limited
sample from the SDSS main galaxy sample with the largest
survey volume (Choi et al. 2010). For the dense sample, its mean
density is set close to the highest achievable density from the
simulations. The dense sample-like data will be available from
upcoming surveys such as Dark Energy Spectroscopic Instru-
ment (DESI Collaboration et al. 2016)and SPHEREx (Doré
et al. 2014)given that their expected number densities of
observed galaxies up to z≈0.2 are higher than our dense sample
density.
3. Identifying CCVs with Dark Matter
We briefly summarize how CCVs are defined using paired
simulations in Section 3.1 and are recovered in dark matter
density fields without using the inverted simulation in
Section 3.2.
3.1. Cluster-counterpart Voids
In Figure 1, we illustrate how CCVs are identified using the
Reference and Mirror simulations. We note that the cluster-
scale dark matter halos in the Mirror simulation become large
voids in the Reference simulation. Therefore, the dark matter
particles belonging to each cluster-scale halo in one simulation
can be used to define the void region in the other one. Thus,
each cluster in the Mirror simulation uniquely defines a CCV in
the Reference simulation.
We generate a catalog containing the position, effective
radius, and mean and central densities of CCVs in the
Reference simulation that correspond to 422,818 halos with
halo mass M
h
10
13
h
−1
M
e
,atz=0 in the Mirror simulation.
The volume of a CCV is defined as the region occupied by the
void member particles in the Reference simulation. A CCV
associated with a more massive dark matter halo tends to have
a larger volume (Shim et al. 2021).
3.2. CCVs from Dark Matter Density Fields
In this subsection, we describe a method developed by Shim
et al. (2021)for finding the CCVs in the Reference simulation
without resorting to the Mirror simulation. First, we find the
most underdense regions in the smoothed density field below a
certain threshold density, and then select those whose volume
is larger than a certain fraction of the typical void volume. The
expected typical or mean void volume is directly related with
the corresponding cluster mass. Namely, a choice of cluster
mass determines the corresponding CCV size, which is used to
select the void core regions. Our method is based on the finding
that a larger CCV tends to develop a larger void core, the
central region with density below the specific threshold value.
There are three parameters used in this method: Gaussian
smoothing scale (R
s
), core density threshold (δ
c
), and core
volume fraction (f
c
)with respect to the CCV volume at that
mass scale. The optimal values of these parameters change
depending on the mean void size or corresponding cluster
mass. The matter density field is smoothed with a Gaussian
filter over R
s
, and the void core regions with overdensity below
δ
c
are found. Among them, those with volume larger than f
c
times the typical volume of the CCVs under interest are
selected. Because void cores only cover the innermost volume
of CCVs, we then grow the cores by repeatedly attaching
neighboring higher-density volume elements. This growing
process stops when the total volume of the recovered voids
reaches that of the CCVs of the interested mass scale in the
Reference simulation.
4. Identifying CCVs from Galaxy Density Fields
We test whether or not the CCV-identification method
originally developed for dark matter density fields could be also
applied to galaxy density fields. The success of the extension of
2
The Astrophysical Journal, 952:59 (8pp), 2023 July 20 Shim et al.
the prescription depends on how accurately the galaxy density
field follows the underlying dark matter density field, especially
in underdense regions. To be more specific, the rank order of
pixel density should be preserved; a region with a higher dark
matter density should have a higher galaxy density value if we
aim to identify the same voids in the galaxy and dark matter
density fields using this method. Any reversal in the rank order
will disturb the correspondence. Thus, we first examine the
relation between galaxy and dark matter density fields in
Section 4.1 and determine the optimal values of free parameters
for galaxy density fields in Section 4.2.
4.1. Dark Matter versus Galaxy Density Fields
Before we develop a void-finding algorithm for a galaxy
sample, we first compare between the number- and mass-
weighted galaxy density fields to study which one has the
tighter relation with the underlying matter distribution. The
weighted density is calculated at the center of each pixel with a
volume h2Mpc
13
(
)
-using the cloud-in-cell assignment
scheme. We then smooth these density fields with Gaussian
smoothing kernel. In Figure 2, we compare between smoothed
dark matter and galaxy density fields. As can be seen, the
overdensities of both galaxy density fields are enhanced
compared to that of the dark matter density field, which
reflects galaxies being the biased tracers of the underlying
matter field (Kaiser 1984; Bardeen et al. 1986; Desjacques et al.
2018). Interestingly, the nonlinear relation between dark matter
and galaxy density fields is well modeled by the second-order
polynomials of logarithmic densities as discovered in Jee et al.
(2012). Note that this relation reduces to a linear bias model
when |δ|=1.
We find that the mass-weighted galaxy density is more
tightly correlated with the dark matter density than the number-
weighted case. The standard deviation of the logarithmic dark
matter overdensity for a given logarithmic galaxy overdensity
bin is on average 10% smaller in the mass-weighted galaxy
field than that in the number-weighted one. This is consistent
Figure 1. A schematic diagram showing how CCVs are identified under the paradigm of the void–cluster correspondence theory using the Reference and Mirror
simulations. Top panels: initial density fluctuations of the same regions within the Reference (left)and Mirror (right)simulations. The initial density troughs (blue
regions)in the Reference simulation correspond to the initial density peaks (red regions)in the Mirror simulation because the initial overdensity fields of the two
simulations are designed to be the sign-inverted version of the other. Bottom panels: distributions of dark matter particles forming voids (left)and their counterpart
clusters (right)through gravitational evolution from their initial particle distributions representing initial underdensities (top left)and overdensities (top right),
respectively. Note that we illustrate only the density troughs and voids for the Reference simulation, and their corresponding density peaks and clusters for the Mirror
simulation, for simplicity.
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The Astrophysical Journal, 952:59 (8pp), 2023 July 20 Shim et al.
with the previous results showing that mass weighting reduces
the scatter between dark matter and halo density field (Park
et al. 2007,2010; Seljak et al. 2009). However, the standard
deviation for the mass-weighted galaxy density field becomes
comparable to that for the number-weighted case at an
extremely low-density range. For this reason, we consider
both the number- and mass-weighted galaxy density fields even
though the scatter is smaller overall in the mass-weighted case.
The scatter in the relation between the galaxy and dark matter
fields reflects the stochastic nature of bias (Pen 1998; Dekel &
Lahav 1999; Matsubara 1999)and the shot-noise effect on
density measurement. Because of these two effects, the rank
order of pixel values in the density array changes when we
move from the matter density field to galaxy density field.
In Figure 3, we show how much the degree of randomization
in the rank order depends on the density-weighting scheme and
smoothing scale. We compute the volume overlap fraction
(f
overlap
)of the lowest-density regions between galaxy and dark
matter density fields as
fV
V,1
overlap
overlap
low
()=
where V
low
is the volume with density lower than the input
threshold, and V
overlap
is the overlap volume between those
lowest-density regions in the two density fields. And the fraction
of volume occupied by the lowest-density regions is given as
fV
V,2
low
low
sim
()=
where
V
sim is the entire simulation volume. For example,
f
overlap
=0, at f
low
=0.02, means that the most underdense 2%
volumes in galaxy and dark matter density field have no
overlap between them.
We find the decreasing trend of the volume overlap fraction
toward a lower f
low
, which becomes steeper for a smaller
smoothing scale. This implies that the order shuffling between
two density fields is severe at the center of voids, or in the most
underdense regions. It is also shown that the overlap fraction is
higher for a larger smoothing scale implying that the shuffling
becomes less significant when galaxy density fields are
smoothed on relatively larger smoothing scales. However, the
density order at the extreme low-density range is still changed
by ∼10% for the largest smoothing scale.
4.2. Optimal Parameter Values for CCV Identification
We need to fine-tune the free parameters for CCV
identification in the galaxy density field. This is because the
galaxy density field is nonlinearly biased with respect to the
underlying dark matter density field, and hence the fine-tuned
parameters obtained for the dark matter field cannot be applied
to the galaxy density field without adjustment. We thus follow
the approach described in Shim et al. (2021). The optimal
Figure 2. Relation between dark matter and galaxy overdensities. The upper
(lower)panel shows the case for the number- (mass-)weighted galaxy density
field. The Gaussian smoothing scale applied is R
s
=7.3 h
−1
Mpc. Solid lines
are the second-order polynomial fits to the relations (Jee et al. 2012), whereas
dash lines represent the identity relation, δ
DM
=δ
gal
.
Figure 3. Volume overlap fraction (f
overlap
)of underdense regions between
dark matter and galaxy density fields as a function of the volume fraction (f
low
)
of the low-density regions. The galaxy density fields are constructed using the
dense galaxy sample. Solid (dashed)lines represent number- (mass-)weighted
galaxy density fields. Different colors indicate various Gaussian smoothing
scales.
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The Astrophysical Journal, 952:59 (8pp), 2023 July 20 Shim et al.
values of the free parameters are determined so that complete-
ness and reliability are maximized. We calculate the complete-
ness and reliability of void finding given by
NN,3
sv
()º
and
NN,4
sc
()º
respectively. Here, N
v
,N
c
, and N
s
represent the number of the
CCVs of a target mass scale, all the identified void cores, and
successfully reproduced CCVs, respectively. From here on, the
mass scale of a CCV refers to the corresponding cluster mass.
The completeness represents the successfully recovered
fraction among the CCVs with the corresponding cluster mass
above Mmi
n
. On the other hand, the reliability is the fraction of
recovered voids with the corresponding MM0.8 min
(to allow
for a buffer in mass)among all the identified voids by this
method. Here, it is counted as a successful detection when the
nearest CCV from the identified void core satisfies the mass
criterion or when the most massive CCV within 1.5r
c
from the
void core satisfies the mass criterion. Here, r
c
is the effective
radius of the void core and is measured from the void core
volume V
c
using the following relation
rV3
4.5
cc
1
3
⎛
⎝⎞
⎠()
p
=
We repeat calculating the detection completeness and
reliability in three-dimensional parameter space of R
s
,δ
c
, and
f
c
, and provide their optimal values in Table 1. Note that we
limit the smoothing scales to be the reciprocals of integers
multiplied by the effective radius of the CCVs of the target
mass scale. The optimal length scales for smoothing galaxy
density fields are R
s
=R
v
/3 in most cases, where R
v
is the
effective radius of a CCV of a particular mass scale. This
implies that the optimized smoothing scale of a given galaxy
density field is determined by the target mass scale, or
equivalently the target size, of voids. Interestingly, the relation
between the smoothing scale and void size is consistent with
the case for the dark matter density fields (Shim et al. 2021).
In Figure 4, we show the detection completeness and
reliability for different minimum target mass scales of CCVs. It
is shown that the detection completeness, and equivalently the
reliability, does not depend much on the density-weighting
Table 1
Optimal Values of the Free Parameters for the CCV Identification from Galaxy Density Fields Constructed Using the Dense and Sparse Galaxy Samples
ngal
¯
M
min R
v
R
s
δ
c
δ
upper
f
c
((h
−1
Mpc )
−3
)(10
14
h
−1
M
e
)(h
−1
Mpc)(R
v
)
number-weighted density field 1.0 ×10
−2
3 14.5 1/4−0.981 −0.785 0.02
5 17.2 1/3−0.915 −0.738 0.02
7 19.3 1/3−0.923 −0.725 0.01
10 21.7 1/3−0.890 −0.727 0.05
4.4 ×10
−3
3 14.5 1/4−0.995 −0.853 0.01
5 17.2 1/4−0.981 −0.858 0.05
7 19.3 1/4−0.967 −0.861 0.10
10 21.7 1/2−0.835 −0.619 0.01
mass-weighted density field 1.0 ×10
−2
3 14.5 1/4−0.987 −0.835 0.02
5 17.2 1/3−0.928 −0.797 0.03
7 19.3 1/3−0.942 −0.779 0.01
10 21.7 1/3−0.921 −0.771 0.04
4.4 ×10
−3
3 14.5 1/4−0.996 −0.894 0.01
5 17.2 1/3−0.957 −0.829 0.02
7 19.3 1/3−0.950 −0.824 0.03
10 21.7 1/3−0.937 −0.816 0.05
Note. The upper (lower)half of the table corresponds to the number- (mass-)weighted galaxy density fields. Listed in columns are the mean galaxy number density
n
gal
¯, minimum target mass scale of a CCV
M
min, effective radius of the CCV R
v
, smoothing scale R
s
, core density threshold δ
c
, upper density threshold δ
upper
when
core growing stops, and minimum core fraction f
c
.
Figure 4. Completeness and reliability of identifying CCVs from the mass-
weighted (dashed)and number-weighted (solid)density fields constructed
using the dense (red)and sparse (blue)galaxy sample. The shaded regions are
the detection rate uncertainties calculated as the standard deviations of the
completeness and reliability measured from bootstrap resampled galaxy density
fields. The detection rates and uncertainties are calculated when the optimal
parameter values in Table 1are adopted. Note that the completeness and
reliability are identical for the optimal parameter values.
5
The Astrophysical Journal, 952:59 (8pp), 2023 July 20 Shim et al.
scheme. On the other hand, the mean galaxy number density
significantly affects the detection quality. This is just because
of the unavoidable nature of the cosmic voids whose
identification is sensitive to the tracer number density. For
the sparse sample, the detection rates increase from
0.60
== to 0.66 with the void mass scale before
MhM710
min 14 1
´-
, and then they drop to 0.62 at
MhM10
min 15 1
=-
. When using the dense sample, the
detection rates are enhanced approximately by 2%–7%. This
is because the dense sample includes low-mass galaxies that are
more likely to be in voids than in other environments (Alonso
et al. 2015). Thus,increasing sample density decreases the
shot-noise effect, and hence the density-order shuffling in
underdense regions. The detection rate enhancement is most
noticeable at the largest mass scale, MhM10
min 15 1
=-
.
Because the halo abundance in a large void is lower than that
in a small void (Patiri et al. 2006a), increasing the sample
density by adding lower-mass galaxies has more significant
effect on decreasing the shot-noise for larger voids. Conse-
quently, the density-order shuffling diminishes, leading to the
rapid increase in the detection rate at the largest mass scale. In
line with this interpretation, we find that the uncertainty of the
detection rate, which increases with the void mass scale, is the
largest for the largest scale voids. We compute the uncertainty
as the standard deviation of the detection rate measured from
1000 realizations of bootstrap resampled galaxy density fields.
More importantly,increasing sample density reduces the
uncertainties of the detection rates approximately by 26% and
7% at the largest void mass scale, whereas they only decreased
by 7.6% and 3.7% on average on smaller mass scales for the
number- and mass-weighted cases, respectively. Thus, the
benefit of increasing galaxy sample density is most significant
in the largest CCV detection.
Finally, we show in Figure 5the spatial correspondence
between the recovered voids (black contours)using this method
and the CCVs (colored regions)defined under the void–cluster
Figure 5. Recovered voids (black contours)from the number-weighted galaxy density fields and CCVs (nonblack)of various mass scales in a 2 h
−1
Mpc thick slice.
The optimal parameter values listed in Table 1are adopted. All panels show the identical field. Recovered voids corresponding to galaxy cluster mass scale above
1×10
15
,7×10
14
,5×10
14
, and 3 ×10
14
h
−1
M
e
are shown in clockwise direction from the upper left panel to bottom left.
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The Astrophysical Journal, 952:59 (8pp), 2023 July 20 Shim et al.
correspondence theory. The voids are recovered from the
number-weighted galaxy density fields constructed using the
dense galaxy sample. It is shown that the recovered void
regions well overlap with the CCVs of the target mass scales. A
recovered void typically corresponds to a single CCV unless
the corresponding CCV has neighboring CCVs. This nearly
one-to-one correspondence between a recovered void and a
CCV becomes relatively weaker for a smaller minimum target
void mass. Thus, the trend of a recovered void encompassing
multiple CCVs is most noticeable for M
void
>3×10
14
h
−1
M
e
.
However, in principle, by comparing the recovered void
regions for two different target mass scales, one can decompose
such void complexes into several smaller voids that respec-
tively correspond to single CCVs.
5. Summary and Discussion
We identify cosmic voids from galaxy density fields under
the void–cluster correspondence theory. The correspondence
between voids and clusters of the same mass scale is
established using paired cosmological simulations with sign-
opposite initial overdensity fields. CCVs in one simulation are
defined as the regions occupied by the same dark matter
particles that form cluster-scale halos in its inverted simulation.
Extending the CCV-identification method (Shim et al. 2021)
developed for matter density fields, we find the optimal values
of the free parameters for the void identification in galaxy
density fields: Gaussian smoothing scale, density threshold,
and core volume fraction. The optimal length scale for
smoothing galaxy density fields is determined by the target
mass scale of voids, which is related to the target void radius as
R
s
=R
v
/3. The optimal parameter values for the number- and
mass-weighted density fields constructed using the sparse and
dense galaxy samples are listed in Table 1. Using the dense
galaxy sample, we can achieve about 64%–69% of CCV
detection completeness and reliability from the number-
weighted galaxy density fields. The detection rates decrease
by ∼2%–7% when using the sparse galaxy sample due to the
increasing effects of shot-noise on density.
Our result demonstrates that we can identify CCVs from
galaxy density fields in real space. In actual galaxy surveys, on
the other hand, one needs to consider the redshift-space
distortion (RSD)when constructing density fields from the
observed galaxy distribution in redshift space. Directly
identifying CCVs in redshift space without considering the
RSD effect will be less reliable than in real space. This is
because the RSD further introduces the density-rank order
randomization in addition to those induced by the shot-noise
and stochasticity of bias. Thus, it is better to adopt
reconstruction techniques and identify CCVs in real space
than in redshift space. The fingers of God (Jackson 1972)due
to the internal motion of galaxies within clusters can be
effectively removed by forcing the line-of-sight elongation of
clusters equal to their size perpendicular to the line of sight
(Tegmark et al. 2004; Park et al. 2012; Tully 2015; Hwang
et al. 2016). On larger scales, the Kaiser effect (Kaiser 1987)
due to coherent flow induced by large-scale overdensities and/
or underdensities can be corrected using Zel’dovich approx-
imation (Zel’Dovich 1970)or second-order Lagrangian
perturbation theory (Scoccimarro 1998). Using these
approaches, one can place galaxies back to their real space
positions by subtracting the displacement due to their peculiar
motions (Wang et al. 2009; Kitaura et al. 2012; Bos et al.
2019).
Identifying CCVs from observational data could help us
perform a more precise cosmological and astrophysical analysis
using voids. Because CCVs have a universal mass density
distribution (Shim et al. 2021)and common central and average
densities (Pontzen et al. 2016; Stopyra et al. 2021; Desmond
et al. 2022), they represent a relatively more homogeneous void
population than that of arbitrary underdense regions extracted
from density fields. Consequently, when using CCVs, it is
possible to decrease the uncertainties arising due to the scatter
in void properties. This will be the subject of a future study.
One of the possible applications of CCVs in the cosmological
context is to measure the linear growth rate using CCVs. In
addition to the homogeneous properties of the CCVs, their
internal dynamics remain near linear (Stopyra et al. 2021), and
the linear bias relation holds at around CCVs (Pollina et al.
2017). Because the linearity in and around the CCV
environment allows a simple linear modeling for void–galaxy
cross-correlation, an accurate measurement of the linear growth
rate using voids (Achitouv 2019; Hamaus et al. 2022;
Woodfinden et al. 2022)can be made using CCVs. We leave
such cosmological analysis using CCVs from galaxy surveys to
future work.
Acknowledgments
We thank an anonymous referee for helpful comments that
helped improve the original manuscript. J.S. was supported by
Academia Sinica Institute of Astronomy and Astrophysics and
KIAS individual grant PG071202 at Korea Institute for
Advanced Study. C.B.P. was supported by KIAS individual
grant PG016903 at Korea Institute for Advanced Study. J.K.
was supported by a KIAS individual grant (KG039603)via the
Center for Advanced Computation at Korea Institute for
Advanced Study. S.E.H. was supported by the project 우주
거대구조를이용한암흑우주연구(“Understanding
the Dark Universe Using Large Scale Structure of the
Universe”), funded by the Ministry of Science. The computing
resources were kindly provided by the Center for Advanced
Computation at Korea Institute for Advanced Study.
ORCID iDs
Junsup Shim https://orcid.org/0000-0001-7352-6175
Changbom Park https://orcid.org/0000-0001-9521-6397
Juhan Kim https://orcid.org/0000-0002-4391-2275
Sungwook E. Hong (홍성욱)https://orcid.org/0000-0003-
4923-8485
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