# Line-of-sight structure of troughs identified in Subaru Hyper Suprime-Cam
Year 3 weak lensing mass maps

Takumi Shimasue1,2,3, Ken Osato4,3,5, Masamune Oguri4,3,5, Rhythm Shimakawa6,
Atsushi J. Nishizawa7,8
1Department of Physics, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
2Research Center for the Early Universe, School of Science, The University of
Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
3Department of Physics, Graduate School of Science, Chiba University, 1-33
Yayoicho, Inage, Chiba 263-8522, Japan
4Center for Frontier Science, Chiba University, 1-33 Yayoicho, Inage, Chiba
263-8522, Japan
5Kavli Institute for the Physics and Mathematics of the Universe, The
University of Tokyo Institutes for Advanced Study,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
6Waseda Institute for Advanced Study (WIAS), Waseda University, 1-21-1 Nishi-
Waseda, Shinjuku, Tokyo 169-0051, Japan
7DX center, Gifu Shotoku Gakuen University, 1-1 Takakuwanishi, Yanaizucho,
Gifu 501-6194, Japan
8Kobayashi Maskawa Institute/Institute for Advanced Research, Nagoya
University, Furocho, Chikusa, Nagoya, Aichi 464-8602, Japan E-mail: shimasue-
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We perform the weak lensing mass mapping analysis to identify troughs, which
are defined as local minima in the mass map. Since weak lensing probes the
projected matter distribution along the line-of-sight, these troughs can be
produced by single or multiple voids projected along the line-of-sight. To
scrutinise the origins of the weak lensing troughs, we systematically
investigate the line-of-sight structure of troughs selected from the latest
Subaru Hyper Suprime-Cam (HSC) Year 3 weak lensing data covering
$433.48\,\mathrm{deg}^{2}$. From a curved sky mass map constructed with the
HSC data, we identify 15 troughs with the signal-to-noise ratio higher than
$5.7$ and address their line-of-sight density structure utilizing redshift
distributions of two galaxy samples, photometric luminous red galaxies
observed by HSC and spectroscopic galaxies detected by Baryon Oscillation
Spectroscopic Survey. While most weak lensing signals due to the troughs are
explained by multiple voids aligned along the line-of-sight, we find that two
of the 15 troughs potentially originate from single voids at redshift $\sim
0.3$. The single void interpretation appears to be consistent with our three-
dimensional mass mapping analysis. We argue that single voids can indeed
reproduce observed weak lensing signals at the troughs if these voids are not
spherical but are highly elongated along the line-of-sight direction.

###### keywords:

gravitational lensing: weak – large-scale structure of Universe – cosmology:
observations

††pubyear: 2023††pagerange: Line-of-sight structure of troughs identified in
Subaru Hyper Suprime-Cam Year 3 weak lensing mass maps–Line-of-sight structure
of troughs identified in Subaru Hyper Suprime-Cam Year 3 weak lensing mass
maps

##  1 Introduction

The shapes and sizes of distant galaxies are distorted due to the
gravitational potential sourced by the intervening matter distribution. This
phenomenon is called weak gravitational lensing. In particular, the coherent
shape distortions induced by the large-scale structure of the Universe are
referred to as cosmic shear (for reviews, see Bartelmann & Schneider, 2001;
Kilbinger, 2015; Mandelbaum, 2018). Since cosmic shear measurements and
analyses do not rely on the uncertain relationship between galaxies and matter
as adopted in galaxy clustering analysis, the matter distribution can be
probed in an unbiased manner. Thus, cosmic shear is recognized as a powerful
and unique approach to studying the nature of dark matter and dark energy in
modern cosmology.

The standard approach to extracting cosmological information from weak lensing
measurements is the correlation analysis of galaxy shapes. The widely used
statistics are two-point statistics, i.e., the correlation function and power
spectrum, which are sensitive to the dark matter density and the fluctuation
amplitude of matter inhomogeneities and contain complete information for the
Gaussian random field (Schneider et al., 2002). While the matter distribution
in the very early Universe is close to Gaussian, the late-time matter density
field becomes highly non-Gaussian due to the non-linear gravitational growth
of the matter distribution. To capture the non-Gaussian information,
statistics beyond two-point correlations are required. Among them, peak
statistics of weak lensing mass maps (Jain & Van Waerbeke, 2000; Hamana et
al., 2004; Yang et al., 2011; Liu & Haiman, 2016; Fluri et al., 2018), i.e.,
the number density of peaks as a function of peak height, is thought to be one
of the most promising statistics since weak lensing peaks are easily
identified, and high peaks have clear correspondence with the density peaks of
the matter distribution (see e.g., Miyazaki et al., 2018b; Oguri et al.,
2021). Indeed, the peak statistics have been employed to constrain
cosmological parameters in weak lensing measurements (Liu et al., 2015;
Kacprzak et al., 2016; Shan et al., 2018; Harnois-Déraps et al., 2021).

Because of their potential association with voids (for a review, see van de
Weygaert & Platen, 2011), the low-density region in the Universe, troughs,
which are defined as local minima in weak lensing mass maps, are also expected
to convey cosmological information (Jain & Van Waerbeke, 2000; Miyazaki et
al., 2002). The first identifications of the void have been reported in
Gregory & Thompson (1978); Jõeveer et al. (1978) followed by detections with
larger spectroscopic surveys (Kirshner et al., 1981; de Lapparent et al.,
1986). As a recent study, Douglass et al. (2023) present the void catalogue
constructed from Sloan Digital Sky Survey (SDSS) Main Sample in Data Release
7, in which more than 1,000 voids with radii larger than
$10\,h^{-1}\,\mathrm{Mpc}$ are identified. Such a large sample of voids
enables a robust measurement of the void statistics, which offers a new avenue
to study modified gravity theories (e.g., Cai et al., 2015; Nadathur, 2016).
The structure and abundance of voids are less sensitive to baryonic physics
due to the scarcity of gas and the fact that the formation and evolution are
purely driven by gravity. The clustering analysis of voids is employed to
constrain the geometry of the Universe thorugh the Alcock–Paczynski test
(Lavaux & Wandelt, 2012; Hamaus et al., 2016; Mao et al., 2017; Nadathur et
al., 2019; Hamaus et al., 2020) and to measure the baryon acoustic oscillation
scale (Kitaura et al., 2016; Liang et al., 2016; Nadathur et al., 2019; Zhao
et al., 2020, 2022). Accordingly, the same feature is expected to hold for
weak lensing trough statistics, and the trough statistics contain information
complementary to peak statistics (Gruen et al., 2016; Barreira et al., 2017;
Coulton et al., 2020; Davies et al., 2021; Osato et al., 2021).

The voids have attracted a lot of attention due to their potential association
with the cold spot of cosmic microwave background (CMB), which is a large low-
temperature region located at $(l,b)\simeq(209\degr,-57\degr)$ in the Galactic
coordinate. The cold spot is first reported by Wilkinson Microwave Anisotropy
Probe (Bennett et al., 2013) and further confirmed by Planck (Planck
Collaboration et al., 2014). One of the plausible explanations for the origin
of the cold spot is a supervoid, which is a large-scale ($\gtrsim
100\,h^{-1}\,\mathrm{Mpc}$) underdense region and possibly consists of
multiple voids (Inoue & Silk, 2006). The decaying gravitational potential of
the supervoid can generate the decrement of the CMB temperature through the
integrated Sachs–Wolfe effect. Indeed, at the position of the cold spot, an
extended void region ($\simeq 200\,h^{-1}\,\mathrm{Mpc}$) called the Eridanus
supervoid is identified at the redshift $z\simeq 0.2$ (Szapudi et al., 2015;
Kovács et al., 2022). However, the shallow underdensity of the Eridanus
supervoid with the density contrast of $\delta\simeq-0.2$ can account for only
$10\text{--}20$ per cent of the decrement signal assuming the $\Lambda$ cold
dark matter cosmological model. A further extensive study of the supervoid
region is required to fully confirm the hypothesis that the origin of the cold
spot is the supervoid.

The straightforward method to find underdense regions in the Universe is based
on the galaxy number density field, which is the biased tracer of matter,
measured from spectroscopic surveys. However, voids are regions with fewer or
no galaxies by nature, and thus, the identification of voids from galaxy
catalogues entails large statistical uncertainty. Furthermore, the different
void-finding algorithms (El-Ad & Piran, 1997; Hoyle & Vogeley, 2002; Neyrinck,
2008; Sutter et al., 2015; Nadathur et al., 2019) lead to different void
populations, resulting in large systematic noise. The comparison of the
observed void catalogue with simulations also depends on how galaxies are
populated in the simulations, and therefore is subject to the baryon physics
uncertainty.

The void search in weak lensing mass maps has a potential to identify
underdense regions that galaxy surveys cannot detect and settle the debate
about the origin of the cold spot. However, simple analytic estimates indicate
that single voids cannot produce significant weak lensing signals unless their
size is extremely large, $\gtrsim 100\,h^{-1}\,\mathrm{Mpc}$ (Amendola et al.,
1999). Therefore, it is likely that the detection of a single large underdense
region is difficult with weak lensing. This is why previous work of weak
lensing by voids has primarily focused on the stacked weak lensing analysis of
voids identified from galaxy distributions (Higuchi et al., 2013; Krause et
al., 2013; Melchior et al., 2014; Clampitt & Jain, 2015; Sánchez et al., 2017;
Fang et al., 2019; Vielzeuf et al., 2021). Troughs in weak lensing mass maps
tend to be associated with multiple underdense regions along the line-of-sight
(see, e.g., Chang et al., 2018), although possible weak lensing troughs
associated with single large voids have also been identified (see, e.g.,
Jeffrey et al., 2021; Shimakawa et al., 2021). In either case, studies of the
line-of-sight structure of weak lensing troughs have been limited to those in
a few curious cases, and there have not been any systematic studies of the
line-of-sight structures of weak lensing troughs.

In this paper, we employ the latest Subaru Hyper Suprime-Cam (HSC) Year 3 (Y3)
weak lensing shape catalogue (Li et al., 2022) to systematically study line-
of-sight structures of the most significant troughs identified in weak lensing
mass maps. The weak lensing shape catalogue spans $433.48\,\mathrm{deg}^{2}$
with the mean source galaxy number density $22.9\,\mathrm{arcmin}^{-2}$. The
high number density from the deep imaging data enables us to probe the density
field out to higher redshifts. It offers an ideal tool to map the cosmic web
structure with high statistical significance. We study the line-of-sight
structures of the most significant troughs employing two galaxy catalogues;
one is the photometric luminous red galaxy (LRG) catalogue selected with
CAMIRA algorithm (Oguri, 2014; Oguri et al., 2018a, b) and the other is the
LOWZ and CMASS spectroscopic galaxy samples from Baryon Oscillation
Spectroscopic Survey of SDSS Data Release 12 (Reid et al., 2016). Furthermore,
we perform the three-dimensional weak lensing mass mapping (Simon et al.,
2009; Oguri et al., 2018b) to probe the large-scale density field around the
identified troughs.

This paper is organized as follows. In Section 2, we briefly overview the
basics of weak lensing and the mass mapping analysis. In Section 3, we present
the weak lensing shape catalogue from HSC Y3 data and galaxy catalogues to
identify the weak lensing troughs and investigate the line-of-sight structures
at trough positions. In Section 4, we present the results on identifying weak
lensing troughs and line-of-sight galaxy number densities at the trough
positions. We discuss our results in Section 5 and give conclusions in Section
6. Throughout this paper, we adopt a flat $\Lambda$ cold dark matter cosmology
with the matter density $\Omega_{\mathrm{m}}=0.3$, the baryon density
$\Omega_{\mathrm{b}}=0.05$, the Hubble constant
$H_{0}=100h=70\,\mathrm{km}\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, the tilt of
the scalar perturbation $n_{\mathrm{s}}=0.96$, and the present amplitude of
the matter fluctuation at the scale of $8h^{-1}\,\mathrm{Mpc}$
$\sigma_{8}=0.81$.

##  2 Weak lensing mass mapping with Subaru HSC Y3 data

This Section overviews the basics of weak lensing mass mapping analysis and
the HSC Y3 shear catalogue. Throughout the paper, weak lensing mass maps are
constructed in a curved sky without adopting the flat sky approximation, in
contrast to the previous weak lensing mass map analyses using the HSC survey
data (Oguri et al., 2018b, 2021; Miyazaki et al., 2018b) for which the flat
sky approximation has been adopted.

###  2.1 Two-dimensional mass mapping

To formulate weak lensing, we begin by defining the lensing potential for a
single source located at the comoving distance of $\chi_{\mathrm{s}}$:

$\psi(\chi_{\mathrm{s}},\bm{\theta})=\frac{2}{c^{2}}\int_{0}^{\chi_{\mathrm{s}}}\mathrm{d}\chi\frac{f_{K}(\chi_{\mathrm{s}}-\chi)}{f_{K}(\chi_{\mathrm{s}})f_{K}(\chi)}\Phi(\chi,\bm{\theta}),$
(1)

where $\bm{\theta}=(\theta,\phi)$ specifies the position on the sky, $c$ is
the speed of light, $\Phi$ is the gravitational potential, and $f_{K}(\chi)$
is the comoving angular diameter distance with the curvature $K$:

$f_{K}(\chi)=\begin{cases}\sin(\sqrt{K}\chi)/\sqrt{K}&(K>0),\\\ \chi&(K=0),\\\
\sinh(\sqrt{-K}\chi)/\sqrt{-K}&(K<0),\end{cases}$ (2)

although we note that the flat Universe ($K=0$) is assumed throughout the
paper. The gravitational potential can be derived from the Poisson equation:

$\nabla_{x}^{2}\Phi(\chi,\bm{\theta})=\frac{3\Omega_{\mathrm{m}}H_{0}^{2}}{2a}\delta(\chi,\bm{\theta}),$
(3)

where $\nabla_{x}$ is the differential operator in the comoving coordinate,
$a$ is the scale factor, and $\delta(\chi,\bm{\theta})$ is the density
contrast. The convergence $\kappa$ and shear $\gamma$ fields, which correspond
to the isotropic and anisotropic deformations of images, respectively, are
introduced in the curved sky (Heavens, 2003; Castro et al., 2005; Chang et
al., 2018; Jeffrey et al., 2021):

$\kappa=\frac{1}{4}(\eth\bar{\eth}+\bar{\eth}\eth)\psi,\
\gamma=\frac{1}{2}\eth\bar{\eth}\psi,$ (4)

where $\eth$ and $\bar{\eth}$ are the diffrential raising and lowering
operators in spin-$s$ spherical harmonics ${}_{s}Y_{\ell m}$. The convergence
field $\kappa(\bm{\theta},\chi_{\mathrm{s}})$ with the single source at the
comoving distance $\chi_{\mathrm{s}}$ can be computed as

$\kappa(\bm{\theta},\chi_{\mathrm{s}})=\\\
\frac{3H_{0}^{2}\Omega_{\mathrm{m}}}{2c^{2}}\int_{0}^{\chi_{\mathrm{s}}}\\!\\!\frac{\mathrm{d}\chi}{a(\chi)}\frac{f_{K}(\chi_{\mathrm{s}}-\chi)f_{K}(\chi)}{f_{K}(\chi_{\mathrm{s}})}\delta(f_{K}(\chi)\bm{\theta},\chi).$
(5)

Hence, the convergence field is the projected density contrast with the
distance kernel, referred to as the mass map. In real measurements, we use
multiple source galaxies with different redshifts and therefore the observed
convergence field $\kappa(\bm{\theta})$ should be weighted with the redshift
distribution:

$\kappa(\bm{\theta})=\int_{0}^{\chi_{\mathrm{H}}}\\!\\!\mathrm{d}\chi_{\mathrm{s}}\,p(\chi_{\mathrm{s}})\kappa(\bm{\theta},\chi_{\mathrm{s}}),$
(6)

where $\chi_{\mathrm{H}}$ is the comoving distance to the horizon and
$p(\chi_{\mathrm{s}})$ is the redshift distribution of source galaxies
normalised as
$\int_{0}^{\chi_{\mathrm{H}}}\mathrm{d}\chi_{\mathrm{s}}\,p(\chi_{\mathrm{s}})=1$.
The expression can be recast as

$\kappa(\bm{\theta})=\frac{3H_{0}^{2}\Omega_{\mathrm{m}}}{2c^{2}}\int_{0}^{\chi_{\mathrm{H}}}\\!\\!\frac{\mathrm{d}\chi}{a(\chi)}q(\chi)f_{K}(\chi)\delta(f_{K}(\chi)\bm{\theta},\chi),$
(7)

where the lens efficiency $q(\chi)$ is defined as

$q(\chi)=\int_{\chi}^{\chi_{\mathrm{H}}}\\!\\!\mathrm{d}\chi_{\mathrm{s}}\,p(\chi_{\mathrm{s}})\frac{f_{K}(\chi_{\mathrm{s}}-\chi)}{f_{K}(\chi_{\mathrm{s}})}.$
(8)

The shear and convergence fields are not independent but related in the
harmonic space. The harmonic expansions of the lensing potential, convergence
and shear fields are given as

$\displaystyle\psi$ $\displaystyle=\sum_{\ell m}\tilde{\psi}_{\ell m}\,Y_{\ell
m}(\theta,\phi),$ (9) $\displaystyle\kappa$ $\displaystyle=\sum_{\ell
m}\tilde{\kappa}_{\ell m}\,Y_{\ell m}(\theta,\phi),$ (10)
$\displaystyle\gamma$ $\displaystyle=\sum_{\ell m}\tilde{\gamma}_{\ell
m}\,{}_{2}Y_{\ell m}(\theta,\phi),$ (11)

where $\tilde{\psi}$, $\tilde{\kappa}$, and $\tilde{\gamma}$ are the lensing
potential, convergence, and shear in harmonic space, respectively, $Y_{\ell
m}$ is the spherical harmonics, and ${}_{2}Y_{\ell m}$ is the spin-2 spherical
harmonics. By plugging the harmonic expansions into Eq. (4), the convergence
and shear fields and the lensing potential are related linearly:

$\displaystyle\tilde{\kappa}_{\ell m}$
$\displaystyle=-\frac{1}{2}\ell(\ell+1)\tilde{\psi}_{\ell m},$ (12)
$\displaystyle\tilde{\gamma}_{\ell m}$
$\displaystyle=\frac{1}{2}\sqrt{(\ell-1)\ell(\ell+1)(\ell+2)}\tilde{\psi}_{\ell
m}$ (13)
$\displaystyle=-\sqrt{\frac{(\ell-1)(\ell+2)}{\ell(\ell+1)}}\tilde{\kappa}_{\ell
m}.$ (14)

Thus, the convergence field, or equivalently, the mass map, can be obtained
once the shear field is estimated from the shape catalogue. The real and
imaginary parts of convergence and shear field in harmonic space are referred
to as E-mode and B-mode, respectively:

$\displaystyle\tilde{\kappa}_{\ell m}$ $\displaystyle=\tilde{\kappa}_{E,\ell
m}+i\tilde{\kappa}_{B,\ell m},$ (15) $\displaystyle\tilde{\gamma}_{\ell m}$
$\displaystyle=\tilde{\gamma}_{E,\ell m}+i\tilde{\gamma}_{B,\ell m}.$ (16)

Since the scalar potential $\psi$ induces weak lensing effect, the E-mode
convergence $\kappa_{E}$ prevails over the B-mode convegence $\kappa_{B}$.
However, the systematic effect, such as incomplete point spread function
correction results in non-zero B-mode convergence. Thus, the B-mode
convergence should be zero within statistical uncertainty and can be used as a
null test.

In the practical analysis of weak lensing, we estimate the shear field
$\hat{\gamma}_{\alpha}(\bm{\theta})$ as

$\hat{\gamma}_{\alpha}(\bm{\theta})=\frac{\sum_{i}w_{i}(\gamma_{\alpha}(\bm{\theta}_{i})-c_{\alpha,i})W(\bm{\theta};\bm{\theta}_{i})}{\sum_{i}w_{i}(1+m_{i})W(\bm{\theta};\bm{\theta}_{i})},$
(17)

where the sum runs over all source galaxies,
$\gamma_{\alpha}(\bm{\theta}_{i})$ is the local shear field at the position of
the $i$-th galaxy, $w_{i}$ is the lensing weight (Mandelbaum, 2018),
$c_{\alpha,i}$ is the additive bias, $m_{i}$ is the multiplicative bias (see
Section 2.3), $W$ is the Gaussian smoothing kernel. The local shear field is
estimated as

$\displaystyle\gamma_{\alpha}(\bm{\theta}_{i})=\frac{e_{\alpha}(\bm{\theta}_{i})}{2\mathcal{R}},$
(18)

where $e_{\alpha}(\bm{\theta}_{i})$ is the galaxy shape ellipticity and
$\mathcal{R}$ is the shear responsivity given as

$\mathcal{R}=1-\frac{\sum_{i}w_{i}e_{\mathrm{rms},i}^{2}}{\sum_{i}w_{i}},$
(19)

where $e_{\mathrm{rms},i}$ is the intrinsic shape dispersion. The smoothed
shear field is estimated from the shape catalogue (Eq. 17) and pixellated
based on the HEALPix (Górski et al., 2005) pixellization. Spherical harmonic
coefficients of the estimated shear are then calculated with the map2alm_spin
routine of healpy (Zonca et al., 2019). Finally, the convergence field can be
computed from the smoothed shear field in harmonic space through the relation
(Eq. 14). We mask pixels with few galaxies because the shear estimate in such
pixels suffers from a large statistical uncertainty. In practice, we construct
the source galaxy number density-weighted with lensing weights and the pixels
with less than $20$ per cent of the mean of the number density map are masked.

The size of the Gaussian smoothing kernel $W$ should be carefully considered.
In principle, this smoothing kernel size sets the minimum size of troughs or
voids we can find from mass maps. On the one hand, a large kernel size is
desired to find large voids or supervoids. Still, on the other hand, the
effects of residual systematics arising from imperfect shape measurements on
weak lensing mass maps become more critical when the smoothing size is larger
(Oguri et al., 2018b; Li et al., 2022). The large smoothing size also
significantly reduces the effective search area because many troughs touch the
edge of the survey footprint and, therefore, will be discarded. In addition,
an advantage of the HSC survey is its high number density of source galaxies,
which enables us to construct mass maps down to smaller scales accurately. In
this paper, we choose the Gaussian smoothing kernel with the full-width half
maximum (FWHM) scale of $40\,\mathrm{arcmin}$ ($\simeq
11\,h^{-1}\,\mathrm{Mpc}$ at $z=0.2$) as a compromise between the size of
troughs and voids and possible systematic errors in weak lensing mass maps.
Note that the analysis is repeated with the smoothing scale of
$20\,\mathrm{arcmin}$ and $60\,\mathrm{arcmin}$ to confirm that our results
are insensitive to the choice of the smoothing size. In our analysis, we adopt
the Healpix pixelization with $N_{\mathrm{side}}=512$, which is sufficient to
resolve the smoothing kernel.

To estimate the statistical significance of weak lensing mass maps, we
construct the signal-to-noise ratio (S/N) map defined as the convergence map
divided by the standard deviation of the random noise maps (Oguri et al.,
2018b). First, we create random mass maps, which are mass maps computed from
randomly rotated galaxy shapes, and repeat generating random noise maps $100$
times by changing the random seed. The random rotation erases the cosmological
signal and thus, random noise maps reflect the statistical uncertainties of
mass mapping. Then, we measure the standard deviation of the $100$ random mass
maps, which corresponds to the statistical noise at the pixel level. Finally,
we obtain the S/N maps by dividing the convergence map with the noise standard
deviation map. We define troughs as pixels in the S/N map that are lower than
all neighbour pixels. We note that the convergence at troughs is negative in
most cases, and correspondingly, S/N values of troughs are negative.

Figure 1 shows the probability distribution functions (PDFs) of the S/N map of
E-mode and B-mode convergence, where the E-mode convergence map corresponds to
the mass map. For the B-mode convergence map, the PDF is well approximated as
the normal distribution across all ranges. While the width of the best-fitting
normal distribution is slightly larger than unity, this feature is expected
when the map is dominated by the shape noise. Such broadening of the B-mode
PDF is also seen in the analysis of mock weak lensing shape catalogues due to
the boundary effect that partly mixes E- and B-mode signals (Li et al., 2022).
The slight deviation of the B-mode PDF from the Gaussian distribution at the
lower tail ($\mathrm{S/N}\simeq-6$) is due to the residual systematics in weak
lensing shape measurements and the similar feature is also found in Li et al.
(2022). We find that the E-mode PDF near the peak is much broader than the
B-mode PDF, indicating that the weak lensing signal is clearly detected.
Furthermore, the PDF is highly skewed at $\mathrm{S/N}\simeq 10$,
corresponding to weak lensing peaks created by massive structures such as
galaxy clusters.

Figure 1: Probability distribution functions of E-mode (upper) and B-mode
(lower) S/N maps with errors from the noise map. The orange solid line shows
the normal distribution with the sample mean and standard deviation.

###  2.2 Three-dimensional mass mapping

The weak lensing mass map is the density distribution convolved with the
lensing kernel along the line-of-sight direction, and hence, the line-of-sight
information of the density structure is not accessible due to the projection.
On the other hand, the three-dimensional structure of the density distribution
can be inferred with lensing tomography (Hu, 1999) technique, where the source
galaxy sample is divided into subsamples according to the photometric,
redshifts and mass maps are created for individual subsamples. We can further
push this idea to reconstruct the line-of-sight structures since the resultant
mass maps have different lensing kernels (Hu & Keeton, 2002).

For the reconstruction of three-dimensional mass maps, we follow the
methodology presented in Simon et al. (2009) and Oguri et al. (2018b).
Specifically, we consider convergence fields for source galaxies with
redshifts in the ranges of $[z_{i,\mathrm{min}},z_{i,\mathrm{max}}]$, where
the subscript $i=1,\ldots,N_{\kappa}$ denotes the source redshift bin and
$N_{\kappa}$ is the number of the source redshift bins. In this paper, we
define the source redshift bin as the linear spacing with respect to the
comoving distance:

$\displaystyle\chi(z_{i,\mathrm{min}})$ $\displaystyle=(300+300\times
i)\,h^{-1}\,\mathrm{Mpc},$ (20) $\displaystyle\chi(z_{i,\mathrm{max}})$
$\displaystyle=\chi(z_{i,\mathrm{min}})+300\,h^{-1}\,\mathrm{Mpc},$ (21)

with $N_{\kappa}=14$. The convergence $\kappa_{i}$ at the $i$-th source
redshift bin is expressed as the weighted sum of density contrasts in lens
redshift bins. We consider density contrasts with redshifts in the range of
$[z_{j,\mathrm{min}},z_{j,\mathrm{max}}]$, where the subscript
$j=1,\ldots,N_{\delta}$ denotes the lens redshift bin. In this paper, we
define the lens redshift bin as the linear spacing with respect to the
comoving distance:

$\displaystyle\chi(z_{j,\mathrm{min}})$ $\displaystyle=(100+200\times
j)\,h^{-1}\,\mathrm{Mpc},$ (22) $\displaystyle\chi(z_{j,\mathrm{max}})$
$\displaystyle=\chi(z_{j,\mathrm{min}})+200\,h^{-1}\,\mathrm{Mpc},$ (23)

with $N_{\delta}=10$. Using this expression of the three-dimensional density
field, the convergence $\kappa_{i}$ at the $i$-th source redshift bin is
written as

$\displaystyle\kappa_{i}$
$\displaystyle\approx\sum_{z_{j}<z_{i}}\left[\int_{z_{j,\mathrm{min}}}^{z_{j,\mathrm{max}}}\frac{\bar{\rho}(z)}{H(z)(1+z)\Sigma_{\mathrm{crit},i}(z)}\mathrm{d}z\right]\delta_{j}$
$\displaystyle\equiv\sum_{z_{j}<z_{i}}Q_{ij}\delta_{j},$ (24)

where $H(z)$ is the Hubble parameter. The summation runs over bins that
satisfy $z_{j}<z_{i}$ because only the foreground structure contributes to the
convergence. To put it another way, we set $Q_{ij}=0$ if $z_{j}>z_{i}$. The
critical surface mass density $\Sigma_{\mathrm{crit},i}$ for $i$-th source
redshift bin is approximated as

$\Sigma_{\mathrm{crit},i}^{-1}(z)\approx\frac{4\pi
G}{c^{2}}f_{K}(\chi(z))\frac{f_{K}(\chi(z)-\chi(\bar{z}_{i}))}{f_{K}(\chi(\bar{z}_{i}))},$
(25)

where $\bar{z}=(z_{i,\mathrm{min}}+z_{i,\mathrm{max}})/2$ is the mean
redshift.

In principle, the three-dimensional density distribution $\delta_{j}$ can be
obtained by inverting the linear equation Eq. (24). However, this operation is
numerically unstable in general, and thus, we employ the Wiener filtering to
reduce the noise in Fourier space. To this end, we compute the power spectrum
signal $S_{lm}$ and noise $N_{lm}$ for $l$-th and $m$-th redshift bins:

$\displaystyle S_{lm}$
$\displaystyle=\delta^{\mathrm{K}}_{lm}\frac{1}{(\Delta\chi_{l})^{2}}\int_{z_{l,\mathrm{min}}}^{z_{l,\mathrm{max}}}\frac{c\mathrm{d}z}{H(z)}\frac{1}{\chi^{2}(z)}P_{\mathrm{m}}(k=\ell/\chi,z),$
(26) $\displaystyle N_{lm}$
$\displaystyle=\delta^{\mathrm{K}}_{lm}\frac{\sigma_{e}^{2}}{\bar{n}_{l}},$
(27)

where $\delta^{\mathrm{K}}_{lm}$ is the Kronecker delta,
$\Delta\chi_{l}\approx
c/H(\bar{z}_{l})(z_{l,\mathrm{max}}-z_{l,\mathrm{min}})$ is the width of
comoving distance for the $l$-th redshift bin, $P_{\mathrm{m}}(k,z)$ is the
non-linear matter power spectrum computed with halofit prescription (Smith et
al., 2003) with the updated parameters (Takahashi et al., 2012),
$\sigma_{e}\approx 0.35$ is the root mean square of the ellipticity computed
directly from the weak lensing shape catalogue, and $\bar{n}_{k}$ is the mean
number density of source galaxies in $k$-th redshift bin, which is again
computed directly from the weak lensing shape catalogue. The reconstructed
three-dimensional density field in harmonic space is given by the minimum-
variance estimator:

$\tilde{\bm{\delta}}_{\ell m}=\tilde{W}(\ell)D(\ell)\left[\alpha
S^{-1}+Q^{\mathrm{T}}N^{-1}Q\right]^{-1}Q^{\mathrm{T}}N^{-1}\tilde{\bm{\gamma}}_{\ell
m},$ (28)

where $\tilde{W}(\ell)$ is the Gaussian filter in harmonic space,
$D(\ell)\equiv-\sqrt{\ell(\ell+1)/((\ell-1)(\ell+2))}$ (Eq. 14), and
$\tilde{\bm{\gamma}}_{\ell m}\equiv(\tilde{\gamma}_{1,\ell
m},\ldots,\tilde{\gamma}_{N_{z},\ell m})$ is the shear field in harmonic space
binned with the redshifts. We have introduced the parameter $\alpha$ which
controls the regularization with the signal power spectrum and we adopt
$\alpha=0.03$ following Oguri et al. (2018b). Finally, the three-dimensional
density field $\delta_{i}$ can be obtained by applying the inverse spherical
transformation to $\tilde{\bm{\delta}}_{\ell m}\equiv(\tilde{\delta}_{1,\ell
m},\ldots,\tilde{\delta}_{N_{\kappa},\ell m})$.

###  2.3 Subaru Hyper Suprime-Cam weak lensing shape catalogue

The Hyper Suprime-Cam (HSC) installed on the Subaru Telescope is a wide-field
imaging camera (Miyazaki et al., 2018a), with the field-of-view of
$1.77\,\mathrm{deg}^{2}$. In this work, we use the Year 3 (Y3) weak lensing
shape catalogue based on the $i$-band coadded images from the wide layer of
the HSC Strategic Survey Program (HSC-SSP; Aihara et al., 2018a, b, 2019,
2022) taken from March 2014 to April 2019. The full details of the production
pipeline of the Y3 weak lensing shape catalogue are described in Li et al.
(2022). Here, we overview the basics of the shape catalogue. The galaxy
ellipticities $\bm{e}$ are estimated with the GalSim code (Rowe et al., 2014)
based on the re-Gaussianization method (Hirata & Seljak, 2003):

$\bm{e}=\frac{1-(b/a)^{2}}{1+(b/a)^{2}}(\cos 2\phi,\sin 2\phi),$ (29)

where $b/a$ is the major-to-minor axis ratio of the galaxy isophotes and
$\phi$ is the polar angle measured from the major axis. The re-Guassianization
method is subject to the shear estimation bias and these biases are corrected
by introducing the multiplicative bias $m$ and additive biases $c_{1}$ and
$c_{2}$. These bias terms are calibrated with image simulations as a function
of galaxy properties, e.g., signal-to-noise ratio and redshift.

We only consider the area where the data in all five HSC broadbands
($g,r,i,z,y$) reach the full-depth and the error of the point spread function
is small. The total area of the catalogues is $433.48\,\mathrm{deg}^{2}$,
which are divided into six patches: XMM, VVDS, GAMA09H, GAMA15H, WIDE12H, and
HECTOMAP. Figure 2 shows the S/N map derived from the HSC Y3 shape catalogue.

Figure 2: The S/N maps of E-mode convergence fields for six HSC survey
patches: XMM, VVDS, GAMA09H, GAMA15H, WIDE12H, HECTOMAP.

The photometric redshifts of source galaxies in Y3 shape catalogues are
estimated with various approaches (for HSC Y1 weak lensing analysis, see
Tanaka et al., 2018). We adopt dNNz method (Nishizawa, et al., in prep.),
which utilizes neural networks with inputs of cmodel magnitudes, size, and
point spread function matched aperture magnitudes in all five HSC bands. For
each galaxy, the probability distribution function of the redshift in the
range of $0<z<7$ with $100$ bins is computed. The performance is evaluated
with the test sample, which consists of $10$ per cent of the whole sample and
is not used in the training process; the bias is $<10^{-4}$, the scatter is
$3$ per cent, and the outlier fraction is $<10$ per cent.

##  3 Photometric and spectroscopic galaxy catalogues

To study the line-of-sight density structures at the positions of troughs in
detail, we utilize the three-dimensional galaxy distributions, which are the
biased tracers of the large-scale structures, adopting two different galaxy
catalogues. First, we employ the photometric LRG catalogue constructed from
the HSC photometric data. Since only the photometric redshifts are available
for the LRG catalogue, the uncertainty of the line-of-sight distance of
galaxies is relatively large. On the other hand, an advantage of the HSC
photometric galaxy catalogue is that it uniformly covers the entire weak
lensing mass map regions with the high galaxy number density out to the high
redshift of $z\simeq 1.2$, enabling us to address the line-of-sight density
structures with less statistical noise. Next, the spectroscopic galaxy samples
from SDSS are used. This spectroscopic catalogue is complementary to the
photometric LRG catalogue because the redshift is robustly determined with
spectroscopy. Hence, the line-of-sight structure is not smeared by the
redshift uncertainty. On the other hand, the spectroscopic catalogue contains
fewer galaxies with a narrower range of redshifts ($0.1<z<0.8$). Leveraging
these two catalogues, we investigate the three-dimensional structures of high-
significance troughs in weak lensing mass maps.

###  3.1 Photometric galaxy catalogue

First, we employ the photometric LRG catalogue in HSC survey regions
identified using the stellar population synthesis model fitting of galaxy
colours used for Cluster-finding Algorithm based on Multi-band Identification
of Red-sequence gAlaxies (CAMIRA; Oguri, 2014; Oguri et al., 2018a, b). With
the help of the deep multi-band imaging by HSC, the photometric LRG catalogue
is constructed out to the redshift of $z\sim 1$ with redshift accuracy of
$\Delta z/(1+z)\sim 0.03$ (Oguri et al., 2018b; Ishikawa et al., 2021). We use
the photometric LRG catalogue based on the latest internal HSC-SSP data that
contains 5,479,879 LRGs in the redshift range of $0.05<z<1.25$. The redshift
distribution of the photometric LRGs is shown in Figure 3. Although the
redshift accuracy of the photometric catalogue is not so high as compared with
the spectroscopic catalogue, its high number density is helpful for the robust
investigation of the density structure in the line-of-sight direction at the
positions of weak lensing troughs.

Figure 3: The redshift distribution of the number density of the HSC
photometric LRGs.

###  3.2 Spectroscopic galaxy catalogue

Next, we employ the spectroscopic galaxy catalogue that provides highly
accurate galaxy redshifts. To this end, we use the galaxy catalogue of SDSS-
III Baryon Oscillation Spectroscopic Survey (BOSS) DR12 data (Reid et al.,
2016). In particular, CMASS and LOWZ galaxy samples are employed to address
the line-of-sight density structures. These two catalogues are constructed
based on the different colour and magnitude cuts (Eisenstein et al., 2001;
Cannon et al., 2006). These selection criteria are designed so that the
redshift ranges of CMASS and LOWZ are $0.4<z<0.8$ and $z\lesssim 0.4$,
respectively. The effective areas of CMASS and LOWZ are
$9376\,\mathrm{deg}^{2}$ and $8337\,\mathrm{deg}^{2}$, respectively. We note
that the HSC and BOSS survey regions are almost overlapped. The total number
of galaxies is 777,202 for CMASS and 361,762 for LOWZ. Figure 4 shows the
redshift distributions of CMASS and LOWZ galaxy samples. Although the number
densities of CMASS and LOWZ samples are substantially lower than the HSC
photometric LRG sample, the precise redshift determination of CMASS and LOWZ
samples complements the analysis with the HSC photometric LRG sample.

Figure 4: The redshift distributions of the number densities of LOWZ (green
dashed), CMASS (orange dotted), and the sum of LOWZ and CMASS samples (blue
solid) in SDSS DR12.

##  4 Results

In this Section, we present troughs identified in mass maps constructed from
the HSC Y3 shape catalogue. We then measure the radial profiles of the
convergence for representative troughs to see how the underdense regions
extend in the sky. Next, we measure the redshift distributions of photometric
LRGs and CMASS/LOWZ galaxies at the positions of selected troughs to
investigate whether the trough originates from single or multiple underdense
regions along the line-of-sight. Finally, we present the three-dimensional
mass maps around the troughs and discuss their consistency with the galaxy
distributions.

###  4.1 Properties of identified troughs

First, we present the S/N and positions of troughs identified in our analysis
in descending order of significance in Table 1. The possible origin of each
trough, i.e., a single void or alignment of multiple voids along the line-of-
sight, can be discriminated by the redshift distribution of the photometric
LRGs and the CMASS/LOWZ galaxies at the position of the trough, which will be
shown in Section 4.3. Some troughs are located close to the edge of the survey
region. The troughs in the edge regions may be spurious because the mass map
is noisier due to the mixing of E- and B-mode near the edge. We classify
troughs which are located less than $60\,\mathrm{arcmin}$ from the nearest
masked region as “Edge origin”.

The redshift distribution describes the number density at the trough position
divided by the mean number density as a function of redshift, to which we
refer to the line-of-sight number density contrast. Specifically, it is
defined as $n_{\mathrm{trough}}/n_{\mathrm{mean}}$, where
$n_{\mathrm{trough}}(z_{i})$ is the number of galaxies within the $z_{i}$ bin
and the top-hat aperture radius of $10\,\mathrm{arcmin}$
($50\,\mathrm{arcmin}$) for HSC photometric LRGs (CMASS/LOWZ galaxies111The
aperture radius of $50\,\mathrm{arcmin}$ for CMASS/LOWZ galaxies is much
larger than the fibre collision scale $\sim 1\,\mathrm{arcmin}$.) from the
centre of the trough divided by the aperture area. The aperture radii for HSC
and SDSS are determined so that the similar number of galaxies are included
within the aperture, which yields the similar Poisson noise.

The mean number density $n_{\mathrm{mean}}(z_{i})$ for HSC photometric LRGs is
calculated as the number of galaxies in each redshift bin divided by the total
survey area. For CMASS/LOWZ galaxies, the random catalogues are provided in
SDSS DR12. We employ the random catalogues to compute the mean number density
to incorporate the survey mask effect and completeness of the CMASS/LOWZ
galaxies. The redshift bins are equally spaced in the range of $0.1<z<1.2$
with $10$ bins. The error is estimated as Poisson noise.

We classify troughs into two different classes according to their origin:
“single-void candidate” or “multiple-voids”. We define the single-void origin
as the trough region where both LRG and CMASS/LOWZ catalogues exhibit the
single underdense regions deeper than $4\sigma$ with respect to the mean in
the line-of-sight redshift distribution and multiple-void origin as those
having multiple underdense regions, respectively. Here, we regard the single-
void origin troughs as candidates because the line-of-sight density structure
probed by galaxy samples employed in this analysis has large uncertainties. To
identify the void origins, we mainly use the photometric HSC catalogue because
the CMASS/LOWZ catalogue does not fully cover the HSC Y3 weak lensing survey
footprint. We use CMASS/LOWZ galaxies for cross-checking in the region where
they are available. Note that the LOWZ and CMASS survey regions are not fully
overlapped, and as a result, the number densities of spectroscopic galaxies
can be extremely small for some of the troughs.

Excluding troughs near the edge, there are four troughs with
$|\mathrm{S/N}|>7$, all of which turn out to be the multiple-voids origin. By
lowering the $|\mathrm{S/N}|$ threshold, we find 15 troughs with
$|\mathrm{S/N}|>5.7$, two of which are single-void origin candidates.

Table 1: Troughs with $|\mathrm{S}/\mathrm{N}|>5.7$ in the mass maps constructed from the HSC Y3 weak lensing shape catalogue. The label is numbered in descending order with respect to $|\mathrm{S}/\mathrm{N}|$. Note that $\mathrm{S}/\mathrm{N}$ is negative in general because the convergence at a trough is negative. The possible source of each trough is judged by visual inspections of the redshift distribution of photometric LRGs and CMASS/LOWZ galaxies at the position of each trough. Label | $|\mathrm{S}/\mathrm{N}|$ | RA | Dec | Source | Redshifts of source voids
---|---|---|---|---|---
T1 | $8.77$ | $12^{\mathrm{h}}29^{\mathrm{m}}53^{\mathrm{s}}$ | $-00\degr 35\arcmin 49\arcsec$ | Multiple voids | 0.6, 1.0
T2 | $7.83$ | $10^{\mathrm{h}}04^{\mathrm{m}}41^{\mathrm{s}}$ | $+01\degr 25\arcmin 03\arcsec$ | Multiple voids | 0.2, 0.5
T3 | $7.21$ | $10^{\mathrm{h}}57^{\mathrm{m}}04^{\mathrm{s}}$ | $-00\degr 08\arcmin 57\arcsec$ | Multiple voids | 0.6, 0.7
T4 | $7.15$ | $11^{\mathrm{h}}36^{\mathrm{m}}06^{\mathrm{s}}$ | $-00\degr 31\arcmin 20\arcsec$ | Multiple voids | 0.7, 1.0
T5 | $7.01$ | $22^{\mathrm{h}}53^{\mathrm{m}}33^{\mathrm{s}}$ | $+02\degr 44\arcmin 12\arcsec$ | Edge | —
T6 | $6.92$ | $14^{\mathrm{h}}54^{\mathrm{m}}07^{\mathrm{s}}$ | $+44\degr 06\arcmin 08\arcsec$ | Multiple voids | 0.1, 0.3
T7 | $6.79$ | $22^{\mathrm{h}}43^{\mathrm{m}}22^{\mathrm{s}}$ | $+02\degr 54\arcmin 39\arcsec$ | Single void (candidate) | 0.2
T8 | $6.67$ | $02^{\mathrm{h}}08^{\mathrm{m}}19^{\mathrm{s}}$ | $-02\degr 14\arcmin 19\arcsec$ | Edge | —
T9 | $6.67$ | $02^{\mathrm{h}}18^{\mathrm{m}}52^{\mathrm{s}}$ | $-04\degr 55\arcmin 48\arcsec$ | Multiple voids | 0.3, 0.6
T10 | $6.42$ | $12^{\mathrm{h}}00^{\mathrm{m}}00^{\mathrm{s}}$ | $+04\degr 15\arcmin 23\arcsec$ | Edge | —
T11 | $6.37$ | $11^{\mathrm{h}}42^{\mathrm{m}}46^{\mathrm{s}}$ | $+02\degr 14\arcmin 19\arcsec$ | Multiple voids | 0.2, 0.5
T12 | $6.36$ | $10^{\mathrm{h}}38^{\mathrm{m}}05^{\mathrm{s}}$ | $+00\degr 00\arcmin 00\arcsec$ | Multiple voids | 0.6, 1.0
T13 | $6.24$ | $09^{\mathrm{h}}06^{\mathrm{m}}41^{\mathrm{s}}$ | $-01\degr 02\arcmin 40\arcsec$ | Edge | —
T14 | $6.24$ | $14^{\mathrm{h}}16^{\mathrm{m}}45^{\mathrm{s}}$ | $-01\degr 29\arcmin 32\arcsec$ | Edge | —
T15 | $6.05$ | $22^{\mathrm{h}}54^{\mathrm{m}}37^{\mathrm{s}}$ | $+00\degr 22\arcmin 23\arcsec$ | Multiple voids | 0.4, 0.6
T16 | $6.05$ | $12^{\mathrm{h}}00^{\mathrm{m}}21^{\mathrm{s}}$ | $+04\degr 28\arcmin 51\arcsec$ | Edge | —
T17 | $6.01$ | $22^{\mathrm{h}}03^{\mathrm{m}}38^{\mathrm{s}}$ | $+01\degr 38\arcmin 29\arcsec$ | Edge | —
T18 | $5.96$ | $09^{\mathrm{h}}10^{\mathrm{m}}54^{\mathrm{s}}$ | $+01\degr 02\arcmin 40\arcsec$ | Multiple voids | 0.1, 0.6, 0.7, 0.8
T19 | $5.92$ | $12^{\mathrm{h}}17^{\mathrm{m}}14^{\mathrm{s}}$ | $+02\degr 32\arcmin 15\arcsec$ | Multiple voids | 0.2, 0.3
T20 | $5.89$ | $11^{\mathrm{h}}16^{\mathrm{m}}45^{\mathrm{s}}$ | $+00\degr 53\arcmin 43\arcsec$ | Multiple voids | 0.2, 0.6
T21 | $5.84$ | $02^{\mathrm{h}}18^{\mathrm{m}}52^{\mathrm{s}}$ | $-04\degr 19\arcmin 52\arcsec$ | Multiple voids | 0.3, 0.6
T22 | $5.73$ | $11^{\mathrm{h}}03^{\mathrm{m}}03^{\mathrm{s}}$ | $+01\degr 16\arcmin 06\arcsec$ | Edge | —
T23 | $5.73$ | $22^{\mathrm{h}}19^{\mathrm{m}}27^{\mathrm{s}}$ | $+04\degr 06\arcmin 24\arcsec$ | Single void (candidate) | 0.3

In what follows, we present a line-of-sight property of the most prominent two
single-void origin candidates, T7 and T23. For comparison, we also show the
line-of-sight property of T2 and T9 as representative examples of multiple-
voids origin troughs. Figures 5 and 6 show the S/N maps around these troughs.
It is seen that these troughs are located far ($>1\,\mathrm{deg}$) from the
survey edge, and thus, they are not likely to be artefacts.

Figure 5: The S/N maps around the troughs T7 (circle symbol) and T23 (cross
symbol), which are single-void origin candidates.

Figure 6: Similar to Figure. 5, but for T2 (upper) and T9 (lower).

###  4.2 Radial profile of the convergence around troughs

We measure radial profiles of the convergence centred at the troughs. The
radial profile is derived by computing the mean of convergence in the annuli,
which are equally spaced in the range of $0<\theta[\mathrm{arcmin}]<110$ with
$19$ bins. Figures 7 and 8 show the measured radial profiles. The decrement of
the convergence extends to $\simeq 1\,\mathrm{deg}$, which is larger than the
smoothing scale of $40\,\mathrm{arcmin}$ in FWHM, implying that the troughs
originate from spatially extended underdense regions.

As a sanity check, we measure the radial profiles from weak lensing mass maps
with smoothing scales of $20$ and $60\,\mathrm{arcmin}$. Their convergence
profiles are similar for different smoothing scales, although the
$|\mathrm{S/N}|$ at the centres are different for different smoothing scales
because the noise maps depend on the smoothing scale. The radii of voids for
T7 and T23 are not sensitive to the smoothing scale, and the locations of the
troughs are also similar with small shifts by one pixel, corresponding to
$6.9\,\mathrm{arcmin}$. Thus, we conclude that the identified troughs are of
physical origin.

Figure 7: Radial profiles of the convergence centred at T7 (upper) and T23
(lower).

Figure 8: Similar to Figure 7, but for T2 (upper) and T9 (lower).

###  4.3 Line-of-sight structure probed by galaxy distributions

To study the line-of-sight density structure of each trough, we utilize the
galaxy distribution, which is a biased tracer of the underlying dark matter
distribution. Specifically, we employ HSC photometric LRGs and CMASS/LOWZ
catalogues, which are described in Section 3, and visually inspect line-of-
sight distributions of HSC photometric LRGs and CMASS/LOWZ galaxies around the
troughs.

Figures 9 and 10 show the line-of-sight number density contrasts of HSC
photometric LRGs and CMASS/LOWZ galaxies. A strong depression appears both for
LRGs and CMASS/LOWZ galaxies at $z\simeq 0.2$ for T7 and $z\simeq 0.3$ for
T23, which makes these troughs strong candidates of single-void origin
troughs. In addition, a decrement of the number density for T23 appears at
$z\simeq 0.6$ in CMASS/LOWZ. However, the decrement is significant within the
$2\sigma$ level, and the decrement of the number density is not observed for
HSC galaxies. Therefore, the decrement may be caused by the statistical
fluctuation.

On the other hand, both T2 and T9 have multiple decrements at different
redshifts in their number density contrasts. Specifically, there are
decrements at $z\simeq 0.2$ and $z\simeq 0.5$ for T2 and at $z\simeq 0.3$ and
$z\simeq 0.6$ for T9. These features indicate that trough signals of T2 and T9
are induced by multiple voids aligned along the line-of-sight.

Figure 9: The line-of-sight number density contrasts of HSC photometric LRGs
(upper panels) and CMASS/LOWZ galaxies (lower panels) as a function of
redshift for trough T7 (left panels) and T23 (right panels). The error bar
corresponds to the Poisson noise from the number of galaxies in each bin. The
green horizontal line shows the cosmic mean.

Figure 10: Similar to Figure 9, but for T2 (left panels) and T9 (right
panels).

###  4.4 Three-dimensional mass map around troughs

An independent and complementary approach to probing the line-of-sight density
structure is the three-dimensional mass mapping formulated in Section 2.2.
This method does not require external data sets such as HSC photometric LRG or
CMASS/LOWZ catalogues. Also, it is not affected by the uncertainty of the
connection between galaxy and dark matter distributions. However,
reconstructed three-dimensional mass maps tend to be noisy, and our
reconstruction method described in Section 2.2 suffers from smearing along the
line-of-sight direction as well as the redshift bias in reconstructed mass
maps (Simon et al., 2009; Oguri et al., 2018b). Nevertheless, we study three-
dimensional mass maps around single-void origin candidates for further cross-
checking.

Figure 11 shows the reconstructed three-dimensional mass density fields around
the region of single-void origin candidates, T7 and T23. According to the
analysis of line-of-sight number density contrasts, voids that produce trough
signals at T7 and T23 are located at $z\simeq 0.2$ and $z\simeq 0.3$,
respectively. For the trough T23, there is a clear correspondence in the
three-dimensional mass map; the mass density field of $z=0.25\text{--}0.33$
exhibits low-density structures around the trough. On the other hand, the
source void of T7 is ambiguous from the three-dimensional mass maps. The
density field at $z=0.17\text{--}0.25$ around the trough is not underdense but
rather, low-density regions are found at $z=0.10\text{--}0.17$ and
$z=0.25\text{--}0.33,z=0.33\text{--}0.41$.

There is an angular offset between the trough position and the low-density
region found in the three-dimensional mass maps. The low-density regions
appear at $z=0.10\text{--}0.25$ at the northern side from the centre of T7 but
also at the southern side at $z=0.25\text{--}0.41$. A similar feature appears
for T23; the low-density regions are found at $z=0.03\text{--}0.10$ at the
northern side and at $z=0.25\text{--}0.41$ at the southern side. To
investigate the global density structure around the troughs, we measure the
redshift distribution of galaxies at the positions shifted from the centres of
T7 and T23 to the north and south by $0.5\,\mathrm{deg}$ in declination in
Figure 12. Although three-dimensional mass mapping suffers from large
statistical uncertainty, the low-density region at the south of T23 leads to
the decrement of photometric LRGs and spectroscopic galaxies at redshifts
$z\sim 0.2\text{--}0.3$. From the three-dimensional mass mapping, the size of
the low-density region is more than $10\,h^{-1}\,\mathrm{Mpc}$, which should
be an interesting target for future spectroscopic surveys.

Figure 11: The three-dimensional mass density fluctuation maps covering both
of the single-void origin candidates, T7 (circles) and T23 (crosses), with six
redshift bins in the range of $z=0.03\text{--}0.49$.

Figure 12: Similar to Figure 9, but the redshift distributions shifted from T7
(left) and T23 (right) to the north (red) and south (blue) by
$0.5\,\mathrm{deg}$ are shown.

##  5 Discussions

Troughs are expected to originate from multiple voids aligned along the line-
of-sight because single voids with modest sizes cannot produce significant
weak lensing signals (Amendola et al., 1999). Our analysis confirms this
expectation by finding that most of the troughs in our sample are multiple-
void origin.

However, our sample of troughs also includes two troughs (T7 and T23)
classified as single-void origin candidates. We estimate the weak lensing
signals produced by single voids and compare them with observed signals to
discuss the possible origin. For a spherical top-hat void with the radius
$r_{\mathrm{v}}$ and a constant density contrast $\delta_{\mathrm{v}}$, the
convergence at the void centre $\kappa_{\mathrm{v}}$ is estimated simply as

$\kappa_{\mathrm{v}}=\frac{2\delta_{\mathrm{v}}\bar{\rho}(z_{\mathrm{v}})r_{\mathrm{v}}}{\Sigma_{\mathrm{crit}}},$
(30)

where $\Sigma_{\mathrm{crit}}$ is the critical surface mass density given by
Eq. (25). $\bar{\rho}(z)=(1+z)^{3}\Omega_{\mathrm{m}}\rho_{\mathrm{crit}}$ is
the mean mass density at the redshift $z$, $\rho_{\mathrm{crit}}$ is the
critical density in the present Universe, and $z_{\mathrm{v}}$ is the redshift
of the void. We take into account the redshift distribution of the source
galaxies in the HSC Y3 weak lensing shape catalogue.

Under the assumption of a spherical void, the void radius $r_{\mathrm{v}}$ can
be estimated based on the perpendicular size of the trough in the sky. The
radial profile of the convergence around the troughs shown in Figure 7
indicates that the transverse angular size of T7 and T23 is estimated as
$1\,\mathrm{deg}$. Together with the redshift of the void candidates,
$z_{\mathrm{v}}\sim 0.3$, the approximate size of the voids can be estimated
as $\sim 11\,h^{-1}\,\mathrm{Mpc}$. Since simulations predict that the average
central density contrast of voids with this size is quite small,
$\delta_{\mathrm{v}}\sim-1$ (Hamaus et al., 2014), for brevity, we simply
assume $\delta_{\mathrm{v}}=-1$.

Assuming the parameter values mentioned above, from Eq. (30), we find that the
convergence signal at the void centre is computed as
$\kappa_{\mathrm{v}}\sim-1\times 10^{-3}$. In contrast, Figure 7 indicates
that the observed convergence values at the centres of single-void origin
candidates are $\kappa_{\mathrm{v}}\sim-8\times 10^{-3}$. To put it another
way, the expected S/N from the candidate single void for troughs T7 and T23 is
found to be small, with $|\mathrm{S/N}|<1$, in contrast to the observed S/N of
$|\mathrm{S/N}|\sim 6$. This simple analytic estimate confirms the previous
argument (Amendola et al., 1999) that single voids cannot produce significant
signals in weak lensing mass maps.

One of the possible solutions to mitigate the tension is to consider a non-
spherical void elongated along the line-of-sight direction. While the
spherical symmetry is assumed in the calculation above, simulations predict
that the three-dimensional shape of voids is quite non-spherical (e.g., Park &
Lee, 2007; Platen et al., 2008). For instance, the central convergence value
of a prolate void with axis ratios of $f:1:1$ ($f>1$), when observed along the
major axis, is modified as

$\kappa_{\mathrm{v}}=\frac{2f\delta_{\mathrm{v}}\bar{\rho}(z_{\mathrm{v}})r_{\mathrm{v}}}{\Sigma_{\mathrm{crit}}},$
(31)

where $r_{\mathrm{v}}$ is void size along the minor axis, corresponding to the
size of the trough observed on the sky. Considering a highly elongated void
with $f\sim 3\,\text{--}\,4$ or an even larger value, the weak lensing signal
can be significantly enhanced in proportion to $f$. Although such highly
prolate voids are rare, the analysis with $N$-body simulations (Park & Lee,
2007) suggests that there exist a few highly elongated voids with $f\sim
3\,\text{--}\,4$ in the simulation volume of
$(500\,h^{-1}\,\mathrm{Mpc})^{3}$, which is roughly 10 times smaller than the
survey volume of the presented weak lensing analysis. Hence, detecting such
prolate voids in HSC weak lensing analysis is statistically consistent with
the findings of Park & Lee (2007).

Together with the so-called Eddington bias (Eddington, 1913), that can also
enhance the observed S/N. However, it is difficult to predict the Eddington
bias for voids since the probability distribution function of the trough
significance is less known at this moment. Detailed analysis of ray-tracing
simulations is needed to quantitatively address the Eddington bias for
troughs, which is beyond the scope of this paper. On the other hand, for
shear-selected clusters identified from high S/N peaks in mass maps, the
Eddington bias of the cluster masses is shown to be up to around 55%, which
were derived based on detailed simulations (Chen et al., 2020). A similar
level of the Eddington bias may be expected for troughs.

We conclude that weak lensing signals of troughs T7 and T23 can be explained
by single voids elongated along the line-of-sight. This hypothesis can be
tested with dense spectroscopic follow-up observations in the candidate trough
regions with, e.g., Prime Focus Spectrograph (Takada et al., 2014). We present
the troughs with high $|\mathrm{S/N}|>7$ and medium $|\mathrm{S/N}|>5.7$
significance levels. Obviously, a low threshold will yield more troughs. For
example, we have identified 300 troughs with $|\mathrm{S/N}|>2$. However, the
3D density structures and the redshift of the voids are determined with a
visual inspection of HSC and SDSS galaxy number densities, which hampers
thorough analysis of less significant voids. We leave the systematic analysis
of the full sample of troughs for future work.

The selection effect can easily explain the alignment of elongated voids along
the line-of-sight. Since the weak lensing signal is significantly enhanced
when observed along the major axis, the major axes of single voids found from
mass maps should be preferentially aligned along the line-of-sight direction.
This is indeed the case for peaks selected in mass maps. Hamana et al. (2012)
confirm the presence of an orientation bias for weak lensing selected clusters
through a meticulous analysis of ray-tracing simulations. They also find that
this bias is stronger for higher S/N peaks. Since our trough sample represents
the most significant troughs with the highest $|\mathrm{S/N}|$, the
orientation bias is expected to be quite strong.

##  6 Conclusions

Voids are large-scale underdense regions in the Universe and one of the
building blocks of the cosmic web structure. Since there are few or no
galaxies in voids, the void statistics is expected to be less affected by
baryonic physics and hence offers a powerful approach to constrain the
geometry of the Universe through Alcock–Paczynski test (Lavaux & Wandelt,
2012; Hamaus et al., 2016; Mao et al., 2017; Nadathur et al., 2019; Hamaus et
al., 2020) and to measure the BAO scale (Kitaura et al., 2016; Liang et al.,
2016; Nadathur et al., 2019; Zhao et al., 2020, 2022). However, the scarcity
of galaxies in voids makes it challenging to identify voids using
spectroscopic galaxy surveys. Weak gravitational lensing has the potential to
overcome this fundamental problem because it probes the density field in an
unbiased manner.

On the other hand, searching for voids with weak lensing has been considered
problematic because single voids typically cannot produce significant weak
lensing signals. In this paper, we have performed the mass mapping analysis
with the latest Subaru HSC Y3 shape catalogue (Li et al., 2022) with the
survey area of $433.48\,\mathrm{deg}^{2}$ and the mean source galaxy number
density of $22.9\,\mathrm{arcmin}^{-2}$ to identify troughs, which are defined
as local minima in weak lensing mass maps and are created by aligned
underdense regions, i.e. voids. Excluding troughs near the edge of the survey
footprint, we have identified 4 high-significance troughs with
$|\mathrm{S}/\mathrm{N}|>7$ and 11 medium-significance troughs with
$|\mathrm{S}/\mathrm{N}|>5.7$. To study the line-of-sight structure of these
troughs, we have utilized redshift distributions of two galaxy samples, HSC
photometric LRGs (Oguri, 2014; Oguri et al., 2018a, b) and spectroscopic
CMASS/LOWZ galaxies from SDSS DR12 (Reid et al., 2016). We have investigated
the line-of-sight density structures at the trough positions by cross-checking
both galaxy samples. While multiple underdense regions are seen in the
redshift distributions of the galaxies for most of the troughs, we find that
there are two troughs for which only single underdense regions are identified
in their redshift distributions of the galaxies albeit relatively high
threshold $|\mathrm{S/N}|>5.7$. Weak lensing signals of these two troughs are
potentially produced by single voids. We have also carried out the three-
dimensional mass mapping (Simon et al., 2009; Oguri et al., 2018b) around
these two troughs to investigate the global density structures. From this
analysis, we have identified large-scale ($\gtrsim 10\,h^{-1}\,\mathrm{Mpc}$)
underdense regions around one of the troughs.

The convergence radial profiles of these two troughs indicate that their
perpendicular physical sizes are $\sim 11\,h^{-1}\,\mathrm{Mpc}$. Assuming a
zero-density spherical void, this translates into the predicted convergence at
the void centre of $\sim-1\times 10^{-3}$, which is much larger than the
observed convergence at the centres of the troughs, $\sim-8\times 10^{-3}$. We
have argued that these two troughs can still originate from single voids if we
consider single voids highly elongated along the line-of-sight direction.
Simulations predict that voids are not spherical but rather prolate, and their
weak lensing signals can be significantly boosted when their major axes are
aligned with the line-of-sight direction. In addition, the Eddington bias
(Eddington, 1913) can also help mitigate the difference between the predicted
and observed convergence values. Our hypothesis can be tested with dense
spectroscopic follow-up observations in the trough regions to identify voids
with great accuracy, which we leave for future work.

## Acknowledgements

We thank an anonymous referee for useful comments. This work was supported in
part by MEXT/JSPS KAKENHI Grant Number JP22K14036 (K.O.), JP20H05856 (M.O.),
JP23H00108 (A.N.), and JP22K21349 (M.O. and A.N.).

Some of the results in this paper have been derived using the healpy and
HEALPix packages.

The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from the Japanese Cabinet Office,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University.

This paper is based on data collected at the Subaru Telescope and retrieved
from the HSC data archive system, which is operated by Subaru Telescope and
Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out
with the cooperation of Center for Computational Astrophysics (CfCA) at NAOJ.
We are honored and grateful for the opportunity of observing the Universe from
Maunakea, which has the cultural, historical and natural significance in
Hawaii.

This paper makes use of software developed for Vera C. Rubin Observatory. We
thank the Rubin Observatory for making their code available as free software
at http://pipelines.lsst.io/.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been
made possible through contributions by the Institute for Astronomy, the
University of Hawaii, the Pan-STARRS Project Office, the Max Planck Society
and its participating institutes, the Max Planck Institute for Astronomy,
Heidelberg, and the Max Planck Institute for Extraterrestrial Physics,
Garching, The Johns Hopkins University, Durham University, the University of
Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for
Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation grant No.
AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the
Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

## Data Availability

The catalogue and data underlying this article are available on the public
data release site of Hyper Suprime-Cam Subaru Strategic Program
(https://hsc.mtk.nao.ac.jp/ssp/data-release). The HSC Y3 shape catalogue will
be available at the same site later.

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