# The Gravitational Lensing Imprints of DES Y3 Superstructures on the CMB: A
Matched Filtering Approach

U. Demirbozan,1 S. Nadathur,2 I. Ferrero,3 P. Fosalba,4 A. Kovacs,5,54 R.
Miquel,1,6 C. T. Davies,7 S. Pandey,8 M. Adamow,9 K. Bechtol,10 A. Drlica-
Wagner,11 R. A. Gruendl,12 W. G. Hartley,13 A. Pieres,14 A. J. Ross,15 E. S.
Rykoff,16 E. Sheldon,17 B. Yanny,18 T. M. C. Abbott,19 M. Aguena,20 S.
Allam,21 O. Alves,22 D. Bacon,23 E. Bertin,24,25 S. Bocquet,26 D. Brooks,27 A.
Carnero Rosell,28,20 J. Carretero,29 R. Cawthon,30 L. N. da Costa,20,28 M. E.
S. Pereira,31 J. De Vicente,32 S. Desai,33 P. Doel,27 S. Everett,34 B.
Flaugher,21 D. Friedel,9 J. Frieman,21,11 M. Gatti,35 E. Gaztanaga,4,23 G.
Giannini,29 G. Gutierrez,21 S. R. Hinton,36 D. L. Hollowood,37 D. J. James,38
N. Jeffrey,27 K. Kuehn,39,40 O. Lahav,27 S. Lee,34 J. L. Marshall,41 J. Mena-
Fernández,42 J. J. Mohr,43,26 J. Myles,44,45 R. L. C. Ogando,20,28 A. A.
Plazas Malagón,46,47 A. Roodman,46,47 E. Sanchez,32 I. Sevilla-Noarbe,32 M.
Smith,48 M. Soares-Santos,49,22 E. Suchyta,50 M. E. C. Swanson,9 G. Tarle,22
N. Weaverdyck,51,52 J. Weller,53,43 P. Wiseman,48 ( DES Collaboration) The
authors’ affiliations are shown in Appendix A. E-mail<EMAIL_ADDRESS>mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Low density cosmic voids gravitationally lens the cosmic microwave background
(CMB), leaving a negative imprint on the CMB convergence $\kappa$. This effect
provides insight into the distribution of matter within voids, and can also be
used to study the growth of structure. We measure this lensing imprint by
cross-correlating the _Planck_ CMB lensing convergence map with voids
identified in the Dark Energy Survey Year 3 data set, covering approximately
4,200 deg2 of the sky. We use two distinct void-finding algorithms: a 2D void-
finder which operates on the projected galaxy density field in thin redshift
shells, and a new code, Voxel, which operates on the full 3D map of galaxy
positions. We employ an optimal matched filtering method for cross-
correlation, using the MICE N-body simulation both to establish the template
for the matched filter and to calibrate detection significances. Using the DES
Y3 photometric luminous red galaxy sample, we measure $A_{\kappa}$, the
amplitude of the observed lensing signal relative to the simulation template,
obtaining $A_{\kappa}=1.03\pm 0.22$ ($4.6\sigma$ significance) for Voxel and
$A_{\kappa}=1.02\pm 0.17$ ($5.9\sigma$ significance) for 2D voids, both
consistent with $\Lambda$CDM expectations. We additionally invert the 2D void-
finding process to identify superclusters in the projected density field, for
which we measure $A_{\kappa}=0.87\pm 0.15$ ($5.9\sigma$ significance). The
leading source of noise in our measurements is _Planck_ noise, implying that
future data from the Atacama Cosmology Telescope (ACT), South Pole Telescope
(SPT) and CMB-S4 will increase sensitivity and allow for more precise
measurements.

###### keywords:

cosmology – voids – cosmic microwave background – lensing – dark energy –
large scale structure – superclusters

††pubyear: 2024††pagerange: The Gravitational Lensing Imprints of DES Y3
Superstructures on the CMB: A Matched Filtering Approach

##  1 Introduction

Early large-scale galaxy surveys have revealed that the Universe’s structure
forms a cosmic web, featuring dense filaments and galaxy clusters alongside
under-dense regions known as cosmic voids. (Peebles, 1980). As observational
data has expanded, these vast regions, primarily devoid of matter and dark
matter, have gained heightened attention (e.g., see Hamaus et al., 2016;
Raghunathan et al., 2019; Nadathur et al., 2019a, 2020a; Vielzeuf et al.,
2021; Kovács et al., 2022a; Woodfinden et al., 2023).

In particular, cosmic voids have proven to be useful tools for advancing
cosmological studies. They offer a means to constrain the Neutrino mass sum
(Lesgourgues & Pastor, 2006; Lesgourgues et al., 2013; Villaescusa-Navarro et
al., 2013; Massara et al., 2015; Kreisch et al., 2019; Vielzeuf et al., 2023).
Additionally, their abundance and density profiles aid in distinguishing
Modified Gravity models from General Relativity (GR), with voids’ ability to
bypass screening mechanisms in high-density environments being particularly
noteworthy (Li et al., 2012; Martino & Sheth, 2009; Pisani et al., 2019;
Khoury & Weltman, 2004; Vainshtein, 1972).

As a result, voids offer an environment that is sensitive to physics beyond
the standard model of cosmology. They are sensitive to many effects, such as
the growth rate of cosmic structure and redshift space distortions (Hamaus et
al., 2016, 2017; Nadathur et al., 2020b; Woodfinden et al., 2022, 2023),
Alcock-Paczynski distortions (Lavaux & Wandelt, 2012; Hamaus et al., 2016;
Nadathur et al., 2019a), weak gravitational lensing (Melchior et al., 2014;
Clampitt & Jain, 2015; Sánchez et al., 2016; Fang et al., 2019; Raghunathan et
al., 2019), baryon acoustic oscillations (Kitaura et al., 2016), and the
integrated Sachs-Wolfe effect(Sachs & Wolfe, 1967; Granett et al., 2008;
Nadathur & Crittenden, 2016; Kovács et al., 2019a, 2022a).

Large-scale structure of the Universe can influence the observed cosmic
microwave background (CMB) and imprint secondary anisotropies. Notably, weak
secondary CMB anisotropies, shaped by the evolving low-$z$ structure, provide
key observational tests for dark energy. Notably, Kovács et al. 2019a, 2022a
have shown an excess ISW signal from large voids that deviates from the
predictions of the $\Lambda$CDM model. This has further intensified interest
in CMB lensing anomalies related to voids. Additionally, the weak
gravitational lensing of the CMB by voids represents a significant component
of these secondary anisotropies.

Several recent works (e.g. Schmittfull & Seljak, 2018; Ade et al., 2019;
Tanimura et al., 2020) have studied the cross-correlation between constructed
weak CMB lensing convergence maps ($\kappa$) and large-scale structure. For
example, it was shown that CMB lensing can be used to measure the masses of
dark matter haloes (initial detections include Madhavacheril et al., 2015;
Baxter et al., 2015; Planck Collaboration et al., 2016b) and to explore the
non-linearities in structure formation through its correlation with cosmic
filaments (He et al., 2018). However, unlike filaments and clusters, cosmic
voids, as under-dense regions, cause an anti-lensing effect and imprint a
negative $\kappa$ signal on the constructed CMB lensing map.

The CMB lensing imprints of voids have been measured by various authors (Cai
et al., 2017; Raghunathan et al., 2019; Vielzeuf et al., 2020; Kovács et al.,
2022b; Hang et al., 2021) with detection significances ranging from $\sim
3\sigma$ up to $\sim 9\sigma$. However, Kovács et al. (2022b) found the
amplitude of the lensing signal to be low, in moderate $\sim 2\sigma$ tension
with predictions from simulations, and Hang et al. (2021) also found hints of
lower-than-expected lensing from voids. Given the other tensions in the late-
time measurements of cosmological parameters with the predicted values from
$\Lambda$CDM (Riess et al., 2019; Verde et al., 2019; Heymans et al., 2021;
Abdalla et al., 2022), any discrepancy in the CMB lensing signal from voids
would be of great interest.

The stacked CMB lensing signal from voids is strongly influenced by specific
void parameters. A typical void features a low-density core (with overdensity
$\delta<0$) which may be surrounded by a compensating marginally overdense
region ($\delta>0$). For most voids, the low central matter density gives rise
to a de-lensing effect, characterized by $\kappa<0$ around the line of sight
through the void center. However, depending on the relative amplitudes of the
density fluctuations in the core and the surrounding overdensity and their
physical extents, the convergence profile $\kappa(\theta)$ at angle $\theta$
from the void centre may differ, showing dips and rings of different widths
(Nadathur et al., 2017a; Raghunathan et al., 2020). This in turn can depend on
the properties of the void-finding algorithm used for identification.

The objective of our study is to re-examine the CMB lensing imprint of voids
in the DES Y3 data using a matched filtering method, as outlined in
(Raghunathan et al., 2019), and to employ the Voxel void finder algorithm,
which has not been previously utilized in DES. Moreover, we explore the
variation of the CMB lensing signal with respect to redshift and the type of
void finder employed. In contrast to prior studies by (Raghunathan et al.,
2019; Hang et al., 2021; Kovács et al., 2022b), which each employed a single
type of void finder, our study implements two different void finders to
enhance the depth and range of our findings.

This paper is structured as follows: Section 2 explains our observed and
simulated data sets. Section 3 outlines our methods for identifying voids and
superclusters, as well as our matched filtering technique. Our primary
observational findings are detailed in Section 4, while Section 5 offers a
comprehensive discussion and concludes the study.

##  2 DATA AND SIMULATIONS

###  2.1 Observational Data

####  2.1.1 Dark Energy Survey Year-3 Data

We identify cosmic voids using photometric redshift data from the first three
years (Y3) of the Dark Energy Survey (DES). DES is a six-year sky survey, with
Y3 covering approximately one-eighth of the sky (5000 deg2) to a depth in the
$i_{AB}$ band of less than 24, imaging around 300 million galaxies in five
broadband filters ($grizY$) up to redshift $z=1.4$ (for details see e.g.
Flaugher et al., 2015; Dark Energy Survey Collaboration, 2016). While the DES
will release all data from its six-year observations as DES Y6, the survey
area of DES Y3 already covers the full DES footprint. However, the depth of
DES Y6 is expected to be higher.

We use a luminous red galaxy sample from the DES Y3 dataset. This Red-sequence
MAtched-filter Galaxy Catalogue (redMaGiC) (see Rozo et al., 2016a, for DES
science verification (SV) test results) is a catalog of photometrically
identified luminous red galaxies, with the red-sequence MAtched-filter
Probabilistic Percolation (redMaPPer) cluster finder algorithm (Rykoff et al.,
2014). This algorithm has been used in many DES analyses (Gruen et al., 2016;
Sánchez et al., 2016; Kovács, 2018; Fang et al., 2019; Vielzeuf et al., 2021;
Kovács et al., 2022b).

The redMaGiC galaxies offer the benefit of a low photo-$z$ error,
approximately $\sigma_{z}/(1+z)\approx 0.013$. This error rate is half that of
the MagLim galaxy sample from DES Y3, which stands at $\sigma_{z}/(1+z)\approx
0.027$ (Porredon et al., 2022). For comparison, we consider the study by
(Kovács et al., 2022b) which utilized an earlier version of redMaGiC. Hence,
we adopt the updated redMaGiC v0.5.1 for this research.

The redMaGiC algorithm produces various catalogs, distinguished by the
densities and luminosities of the galaxies. Specifically, the High Density
(HD) catalog maintains a consistent galaxy density, roughly $\bar{n}\approx
10^{-3}h^{3}\,\text{Mpc}^{-3}$ along the redshift range. Conversely, the High
Luminosity (HL) catalog exhibits a galaxy density of $\bar{n}\approx 4\times
10^{-4}h^{3}\,\text{Mpc}^{-3}$, which is considerably lower than its HD
counterpart.

In our analysis, we focus on the redshift range 0.2 < $z$ < 0.8 and make use
of the empirically constructed DES Y3 survey mask, which excludes contaminated
pixels, mainly due to nearby stars. This low photo-$z$ error of redMaGiC
allows us to robustly identify void environments. The details of the
clustering analysis of the DES Y3 redMaGiC sample are documented in (Pandey et
al., 2022). It is also important to mention that this study identified certain
systematic errors, to which our measurements are not sensitive, as we
correlate them with an external LSS tracer, the CMB lensing map.

####  2.1.2 Planck CMB Lensing Map

We utilize the full-sky public Cosmic Microwave Background (CMB) lensing
convergence ($\kappa$) maps from the Planck survey’s 2018 data release (Planck
Collaboration et al., 2020b).111Downloaded from
https://pla.esac.esa.int/#cosmology

More specifically, we employ the COM_Lensing_4096_R3.00 map, which is
reconstructed using a minimum-variance (MV) quadratic estimator (Hu & Okamoto,
2002). This estimator is based on a combination of foreground-cleaned SMICA
(Planck Collaboration et al., 2016a) CMB temperature and polarization maps,
with the mean field subtracted and a conservative mask applied to galaxy
clusters to reduce contamination from thermal Sunyaev-Zel’dovich (tSZ)
contributions. Throughout our analysis, we use $N_{\mathrm{side}}=512$ HEALPix
maps (Górski et al., 2005). We note that this $N_{\mathrm{side}}=512$
resolution is an appropriate choice considering the degree-scale imprints of
voids.

The gravitational lensing of the CMB, occurs due to spatial variations in the
gravitational potential field, $\Phi(r,\theta)$, as CMB photons traverse the
Universe. The convergence can be described by the following equation:

$\kappa(\theta)=\frac{3H_{0}^{2}\Omega_{m}}{2c^{2}}\int_{0}^{r_{\text{max}}}\frac{(r_{\text{max}}-r)r}{r_{\text{max}}}\delta(r,\theta)\,dr$
(1)

where $r$ is the comoving distance and $r_{\rm max}$ is the comoving distance
to the last scattering surface of the CMB. This $\kappa$ is a dimensionless
quantity and measures all the projected matter density up to the CMB surface
and is weighted by a kernel for a given angular direction depending on the
distance. The gravitational potential is also related to the matter density
fluctuation $\delta$ via the Poisson equation:

$\nabla^{2}\Phi=\frac{3H_{0}^{2}\Omega_{m}}{2a}\delta,$ (2)

where $\delta$ is the perturbation in the matter density and $a(t)$ is the
dimensionless scale factor.

###  2.2 Simulation - MICE CMB Lensing Map and redMaGiC Tracers

We utilize the publicly available MICE (Marenostrum Institut de Ciencias de
l’Espai) simulation, which is an N-body light-cone extracted from the MICE
Grand Challenge (MICE-GC). The MICE-GC contains approximately 70 billion dark-
matter particles in a $(3h^{-1}\mathrm{Gpc})^{3}$ comoving volume. This
simulation was created using the Marenostrum supercomputer at the Barcelona
Supercomputing Center (BSC)222www.bsc.es, running the GADGET2 code (Springel,
2005). For details on the creation of the MICE simulation, see Fosalba et al.
(2015b), Crocce et al. (2015), and Fosalba et al. (2015a).

The MICE simulation assumes a flat standard $\Lambda$CDM model with input
fiducial parameters: $\Omega_{m}=0.25$, $\Omega_{\Lambda}=0.75$,
$\Omega_{b}=0.044$, $n_{s}=0.95$, $\sigma_{8}=0.8$, and $h=0.7$, derived from
the Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) best fit results
(Dunkley et al., 2009).

In this study, we utilize the CMB lensing map from the MICE simulation,
generated using the “Onion Universe” methodology as detailed in Fosalba et al.
(2008). The validity of this lensing map was subsequently confirmed through
auto- and cross-correlations with foreground MICE galaxy and dark matter
particles (refer to Fosalba et al. (2015a) for an in-depth description of the
map creation process). Initially, the MICE $\kappa$ map was provided with a
HEALPix pixel resolution of $N_{\rm side}=2048$. However, we downgraded the
map to a lower resolution of $N_{\rm side}=512$. This adjustment significantly
reduces the computational expense without causing a loss of much information,
given that voids are degree-scale objects. We also downgrade the resolution of
the Planck $\kappa$ map to $N_{\rm side}=512$ employed in our analysis.

We chose the redMaGiC tracers on our mock galaxy catalog from MICE,
maintaining consistency with the methodology utilized in the analysis of the
observed DES Y3 data Rozo et al. (2016b). It is pertinent to note that the
coverage of MICE dark matter halos is confined to an octant of the sky
(5169.25 deg2), which is larger than with the effective footprint of DES Y3
(4147.15 deg2). This MICE-redMaGiC mock galaxy catalog was constructed to
match the number density of the DES Y3 redMaGiC sample. It served as our tool
to trace the distribution of galaxies on a large scale and to identify voids.

It is important to emphasize that the MICE cosmological parameters are
relatively distant from the best-fit Planck cosmological parameters Planck
Collaboration et al. (2020a). For example, the difference in the values of
$\Omega_{m}$ and the Hubble constant $H_{0}$ can affect the amplitude of the
lensing signal. However, we assume that variances in cosmological parameters,
particularly $\Omega_{m}$, negligibly impact our lensing signal measurements.
This postulation aligns with findings from Vielzeuf et al. (2021) and Kovács
et al. (2022b), who, utilizing the WebSky simulation Stein et al. (2020),
showed minimal influence of $\Omega_{m}$ on the CMB lensing signal’s
amplitude.Additionally, our methodology encompasses a comprehensive error
analysis, particularly addressing the MICE template uncertainties, ensuring
the accuracy of our findings despite potential $\Omega_{m}$ discrepancies.

In this context, it is important to mention that, as detailed in Vielzeuf et
al. (2023); Nadathur et al. (2019b), the parameter that seems to have the most
significant impact on the determination of the matter content and the lensing
convergence of voids is $\sigma_{8}$. The value of $\sigma_{8}$ in the MICE
simulation, which is 0.8, is not far off from the best-fit Planck value of
$\sigma_{8}=0.811\pm 0.006$. Additionally, (Nadathur, 2016) and (Nadathur &
Hotchkiss, 2015) have identified the primary factors influencing the size and
number of voids in any galaxy sample as the mean galaxy number density, the
amplitude of galaxy clustering and the linear galaxy bias. Furthermore,
$\sigma_{8}$ also affects void density profiles, especially close to void
center (e.g., see Figure 5 in Nadathur et al., 2019b).

##  3 METHOD

Our approach aligns with the matched filtering technique detailed in (Nadathur
& Crittenden, 2016; Raghunathan et al., 2019). Notably, Nadathur & Crittenden
(2016) evaluated the ISW imprint of voids, emphasizing that this technique
avoids dependence on arbitrary choices of additional tuning parameters (such
as the smoothing scale for Gaussian filtering of the CMB) that could introduce
biases.

Utilizing the Voxel void parameters, $\bar{\delta}_{g}$ and $R_{v}$, we
introduce a dimensionless parameter:

$\lambda_{v}\equiv\bar{\delta}_{g}\left(\frac{R_{v}}{1\
\text{Mpc}/h}\right)^{1.2}\,.$ (3)

This parameter, as empirically demonstrated by Nadathur et al. (2017b),
exhibits a strong correlation with void density profiles and their macroscopic
environments. As such, $\lambda_{v}$ serves as a pertinent proxy for the
gravitational potential associated with voids. Given this relationship, we
expect notable variations in the lensing convergence profiles of voids based
on their respective $\lambda_{v}$ values.

###  3.1 Void and Supercluster Finding

####  3.1.1 Voxel Voids

The main goal of this research is to measure the CMB lensing signal from voids
using two different void definitions: Voxel and 2D. The Voxel method is
designed specifically for datasets with fragmented survey footprints, like DES
Y3. One of the key benefits of Voxel is that it estimates the galaxy density
field by computing number counts on a mesh, normalised by the counts of
unclustered random points whose distribution accounts for the survey window
function. This is the same method as used for estimating densities when
computing power spectra: its use provides a natural way to account for
variations in the survey selection function, and makes the Voxel algorithm
better at handling gaps or fragmented survey masks than algorithms that employ
Voronoi tessellations to estimate the density field. However, after the
density field has been estimated, Voxel identifies voids using a watershed
algorithm similar to that used by other, tesellation-based, algorithms
(Neyrinck, 2008; Sutter et al., 2015; Nadathur & Percival, 2019, e.g.).

The algorithm generates the mesh size, denoted as $N_{\text{mesh}}$, based on
the tracer mean number density. The size of the mesh is set based on the
condition that every cubic grid unit, known as a voxel, should possess a side
length represented by
$a_{\text{vox}}=0.5\times\left(\frac{4\pi\bar{n}}{3}\right)^{-\frac{1}{3}}$.
Here, $\bar{n}$ stands for the estimated average density of galaxies. The
galaxy density field is then subsequently smoothed using a Gaussian filter of
size $n_{t}^{-1/3}$, and local minima are identified across the voxels. Basins
are formed around each local minimum, mirroring the process used in ZOBOV. The
addition of adjacent voxels with increasing overdensity to the basin halts
when the next voxel shows a lower density than its predecessor. Each resulting
basin signifies a Voxel void.

For each identified Voxel void, we compute an average galaxy overdensity
$\bar{\delta}_{g}$ and define an effective spherical radius $R_{v}$, which
equates to the radius of a sphere with a volume equivalent to that of the
void.

We generate the Voxel void catalogs for both DES Y3 and MICE simulation using
the open-source REVOLVER void-finding code Nadathur et al. (2019a)333Available
at https://github.com/seshnadathur/Revolver.

Void sizes in the MICE simulation range from $1.95h^{-1}{\rm Mpc}\leq
R_{v}\leq 61.96\ h^{-1}{\rm Mpc}$, peaking around the median value
$R_{v}=19.37\ h^{-1}{\rm Mpc}$. Meanwhile, for the DES, void sizes range from
$2.04h^{-1}{\rm Mpc}\leq R_{v}\leq 59.41\ h^{-1}{\rm Mpc}$, peaking around the
median value $R_{v}=19.98\ h^{-1}{\rm Mpc}$. Both the MICE and DES have a
median void redshift of $z=0.57$.

By applying the Voxel void finder to the updated redMaGiC galaxies, we
obtained notable insights. Our preference for HD tracers of redMaGiC in
identifying Voxel voids was influenced by their enhanced CMB lensing signal-
to-noise ratio. Our tests confirmed that HD voids exhibit a higher S/N than HL
voids, enhancing the efficacy of the matched filter method over other tracer
densities. This outcome is primarily attributed to the reduced galaxy density
in the HL.

Table 1 outlines our findings: 44,426 voids in the MICE catalog and 33,427
voids in the DES Y3 catalog, both identified using High Density (HD) tracers.
The disparity in void counts between MICE and DES Y3 is due to differences in
effective sky coverage: MICE covers $5169.25\,\text{deg}^{2}$ while DES Y3
covers $4143.17\,\text{deg}^{2}$. Consequently, the total voxel void number
densities for MICE and DES Y3 are $8.59\,\text{voids/deg}^{2}$ and
$8.07\,\text{voids/deg}^{2}$, respectively. Although our primary interest lies
in voids within the $0.2<z<0.8$ redshift range, data constraints restricted
our Voxel void analysis in the MICE catalog to a maximum redshift of $z=0.75$.

Figure 1: Histogram illustrating the distribution of Voxel void number density
per unit volume in both the MICE and DES Y3 HD catalogs.

####  3.1.2 2D Voids

We also employ a 2D void finder, an algorithm initially developed for
photometric redshift surveys. This algorithm identifies voids within
tomographic redshift bins, which are slices of space at different redshift
ranges (Sánchez et al., 2016). The 2D void finder has been employed in several
DES studies Vielzeuf et al. (2021); Kovács et al. (2019a, 2022b).

The 2D algorithm inspects possible minima in the galaxy density field, which
has been projected and smoothed for each redshift slice. The void radius
$R_{v}$ is defined when the density inside a thin annulus around the void
center reaches the mean density of the redshift slice, increasing the annuli
by $1\;h^{-1}$Mpc. While $R_{v}$ is typically measured in degrees and then
converted to $h^{-1}{\rm Mpc}$ for consistency, we opt to use degrees in our
matched filtering analysis.

Key parameters for the 2D void finder include the smoothing scale for the
galaxy density maps, the redshift slice thickness, and the central minimum
pixel density. We adopt a smoothing parameter of $\sigma=10$ $h^{-1}{\rm
Mpc}$, a central pixel density that is at least $10\%$ of the most underdense
pixels in the redshift slice, and a slice thickness of $s\approx 100h^{-1}{\rm
Mpc}$. This results in 12610 and 10904 voids for MICE and DES Y3,
respectively.

To remove potentially spurious objects due to variations in photometric
redshift, a measure of redshift derived from the photometric observations of
an object, we apply a cut of $R_{v}>20h^{-1}{\rm Mpc}$, following the
precedent set in Vielzeuf et al. (2021) and Kovács et al. (2022b). After this
cut, the number of 2D voids reduces to 6295 for MICE2 and 5198 for DES Y3.

Normalizing by the effective area, the 2D void densities are approximately
$1.22\,\text{voids/deg}^{2}$ for MICE2 and $1.25\,\text{voids/deg}^{2}$ for
DES Y3 within the total redshift range. Just as with the Voxel voids, the 2D
void densities of MICE and DES Y3 align closely. This underscores that the
discrepancies in absolute void count largely arise from the differences in
effective areas: 5169 $\text{deg}^{2}$ for MICE2 and 4147 $\text{deg}^{2}$ for
DESY3. Figure 2 shows the 2D void number density per unit volume as a function
of redshift bins. Specifically, we find that MICE (voids/$\text{deg}^{2}$) =
1.22, while DES Y3 (voids/$\text{deg}^{2}$) = 1.25, which indicate a close
alignment between the two. For a detailed view, refer to Table 2.

Figure 2: Histogram illustrating the distribution of 2D void number density
per unit volume as a function of tomographic redshift bins for the MICE and
DES Y3.

For this analysis, we utilize the High Luminosity (HL) tracers from our
updated redMaGiC v0.5.1 sample, this tracer density type choice is consistent
with Kovács et al. (2022b) and has also been used with 2D voids in other DES
studies (Kovács et al., 2022a; Kovacs et al., 2021). The study conducted by
Kovács et al. (2022b) is particularly relevant to our analysis as it also
examined DES Y3 2D voids defined on HL tracers and used the MICE simulation,
similar to our approach. The key difference lies in their filtering and
stacking method, which differs from ours. Therefore, it becomes intriguing to
investigate the extent of the difference these distinct methods can cause,
given the similarity in both the dataset and the simulation used. This choice
was made with the aim of examining the effects of an alternative method, the
matched filter, on our measurements.

####  3.1.3 2D Superclusters

In order to supplement our analysis of 2D voids and provide further validation
of our results, we have expanded our study to include 2D superclusters. To
identify these superclusters, we applied the same 2D void finder to the
tomographic galaxy density maps used in the previous void analysis, but with a
twist: we inverted the density maps by multiplying them by $-1$. This
operation effectively flips the galaxy density map, facilitating the detection
of voids in the newly inverted galaxy density landscape. As such, the detected
’voids’ in this inverted landscape correspond to superclusters in the original
data. It is important to mention that our main reason for using this
supercluster definition is to be consistent with (Kovács et al., 2022b). The
superclusters found in this way represent overdense structures on a large
scale within the distribution of galaxies, though they may not be bound by
gravity. Similar to how voids are associated with extensive troughs in the
density field, superclusters are linked to prominent peaks in this
field.(Nadathur & Hotchkiss, 2014)

Using this method, we compiled a catalog of 2D superclusters, finding 5424
superclusters in the MICE simulation and 4432 superclusters in the DES Y3
dataset after applying a size cut of $R_{v}>20h^{-1}{\rm Mpc}$. We then apply
the same redshift binning as in 2D voids and we obtain the 2D supercluster
numbers in Table 2

###  3.2 Void Lensing in MICE Simulation

To begin, we inspect the void lensing imprints within the MICE simulation to
calibrate the void lensing profiles $\kappa(\theta)$. This involves stacking
10∘ x 10∘ patches, derived from the full-sky MICE $\kappa$ map and centered on
each void. These stacks reveal a negative $\kappa$ imprint (divergent lensing)
at the center of the void, signifying the void’s central underdensity. In
contrast, a less conspicuous positive imprint encircles this, denoting matter
overdensities at the outer periphery of the void. The measured radial profiles
from these stacks can be seen in Figure 3.

Unlike some studies (Kovács et al., 2022b; Cai et al., 2017; Hotchkiss et al.,
2015; Vielzeuf et al., 2021; Kovács et al., 2019a; Vielzeuf et al., 2023) that
adopt a re-scaling method, our measure of void lensing signals is in units of
degrees $(\theta)$. This choice stems from our use of the matched filtering
approach, akin to (Raghunathan et al., 2019), which necessitates that these
simulation template void lensing profiles are defined as $\kappa(\theta)$.
Importantly, by not re-scaling, we maintain uniformity in the noise power from
the CMB lensing map across all void measurements. This is critical because the
noise in the CMB lensing map is scale-dependent; thus, re-scaling might
inadvertently mix different levels of noise power across various scales.

Additionally, (Nadathur et al., 2017b) has shown a stronger correlation
between the void lensing signal and $\lambda_{v}$, as defined in Equation 3—a
function of void radius $R_{v}$ and void overdensity $\bar{\delta}_{g}$—than
with the angular size of voids.

We organize the MICE void sample into three distinct redshift categories:
$\mathbf{LOWZ}$, $\mathbf{MIDZ}$, and $\mathbf{HIGHZ}$. This segmentation by
redshift is strategic; it enables us to investigate the nuanced redshift-
dependent impacts of CMB lensing imprints on voids.

Table 1: This table presents the number of Voxel voids alongside the mean void size within each redshift bin for both MICE and DESY3 datasets. For the detailed Voxel void size distribution across all 9 void bins see Figure 8 in the Appendix. | MICE Void Count | DES Y3 Void Count | MICE Mean Void Size (Mpc/h) | DES Y3 Mean Void Size (Mpc/h)
---|---|---|---|---
LOWZ (0.2 - 0.43) | 9298 | 6821 | 20.01 | 20.57
MIDZ (0.43 - 0.59) | 14679 | 10861 | 20.22 | 20.54
HIGHZ (0.59 - 0.75) | 20449 | 15027 | 19.58 | 20.02

Further granularity is achieved by subdividing each redshift bin into three
separate $\lambda_{v}$ bins. Importantly, each $\lambda_{v}$ bin is populated
with an approximately equal quantity of voids, a methodological decision that
aligns with the approach delineated in (Raghunathan et al., 2019). By
proceeding in this way, we derive a total of nine bins for our Voxel void
sample.

As anticipated, the void lensing imprint strongly relies on the value of
$\lambda_{v}$. More explicitly, voids with negative $\lambda_{v}$ values, such
as those in bins 1 and 2, correlate with slightly larger, lower-density voids,
exhibiting $\kappa\leq 0$. On the other hand, voids with higher $\lambda_{v}$
values (as in bin 3) equate to "voids within voids" Sheth & van de Weygaert
(2004), predominantly smaller voids enveloped within large-scale
underdensities, which present a positive $\kappa$ ring around the void
boundary.

We also take into account the fact that the MICE footprint is significantly
larger than the DES Y3 footprint and as a result, contains more voids. To
assess the impact of this difference, we apply the DES Y3 mask to the MICE
octant and identify the voids within this overlapping area. Subsequently, we
carry out our stacking analysis using these voids. As expected, and in line
with previous studies, we do not observe any substantial differences in the
stacked profiles, despite the varying footprints. Consequently, we opt to use
all the voids in the MICE simulation without implementing any footprint cut.
This approach is consistent with the methodologies adopted by (Vielzeuf et
al., 2020; Kovács et al., 2022b), providing further validation to our study.

Figure 3: This figure presents the derived profiles from stacked images, each
centered on identified Voxel voids within the MICE simulation. The data is
segregated into three different redshift bins, and within each of these, there
are three different $\lambda_{v}$ bins. The representation thus illustrates
the behaviors and properties of these Voxel voids across three redshift bins
and different values of $\lambda_{v}$. The shaded areas in the figure
represent $1\sigma$ uncertainty intervals, estimated through standard errors
obtained from "void-by-void Jackknife resampling." The dashed lines illustrate
the aggregate result from all voids within the specified redshift category.

###  3.3 Filtering the CMB lensing map

When handling the CMB lensing map, we focus on lensing modes with $L\leq
2048$. This is because the lensing signals from cosmic voids are typically
found on degree scales, and higher $L$ modes mostly consist of noise rather
than useful signal. While we did consider the exclusion of the largest scale
modes with $L<8$ as was done in Planck Collaboration et al. (2020b), we found
that this had a negligible impact. As a result, we decided to exclude only the
monopole and the dipole.

The key challenges in detecting the lensing signals are the lensing
reconstruction noise present in the Planck $\kappa$ map and the additional
$\kappa$ values contributed by unrelated structures in our line of sight.
These noise sources are about ten times larger than the lensing signal from a
single void, rendering it essentially invisible. Hence, our strategy involves
stacking these signals together and dividing the voids based on their
$\lambda_{v}$ values. We then apply an optimized ’matched filter’ to the
$\kappa$ map before stacking. This filter, derived from templates in the MICE
simulations, combines with the original map in a way that highlights the parts
of the map that match the filter, making the lensing signals stand out more.

The matched filter transforms the original lensing map, represented as
$\kappa$, into a ’filtered’ version, or $\kappa^{\mathrm{MF}}$. This
transformation involves convolving the filter with the original map,
effectively amplifying the signals that match the filter. We created these
filtered maps for each of nine different void ’bins’.

One of the key benefits of this matched filter technique is its neutrality -
it is applied to the whole map uniformly so, it doesn’t unfairly favor certain
parts of the map over others. It also reduces the variability, or ’noise’, at
the locations of voids to the lowest possible level.

By comparing these measurements across all voids stacked together, we can see
the benefits of our approach to split the data into bins based on
$\lambda_{v}$ values. We also find that the lensing signals are most easily
detected in the two extreme $\lambda_{v}$ bins rather than the middle bin.

To construct the matched filters, we initiate by defining the convergence
field at a location $\bm{\theta}$, originating from the position
$\bm{\theta}_{0}$ which corresponds to the $\kappa$ value at the center of a
void as illustrated in Figure 3. This is expressed as:

$\kappa(\theta)=\kappa_{\text{template}}(|\theta-\theta_{0}|;\lambda_{v})+p(\theta)$
(4)

In this context, $p$ is symbolic of the noise component in all the $\kappa$
maps, excluding the void lensing signal itself. Meanwhile, the
$\kappa_{\text{template}}$ symbol represents the corresponding void lensing
template profile, which is derived from the stacked images from the MICE.

It is possible to separate this template profile as:

$\displaystyle\kappa_{\text{template}}(\theta;\lambda_{v})$
$\displaystyle=\kappa_{0}(\lambda_{v})k(\theta;\lambda_{v})$ (5)
$\displaystyle=\kappa_{0}(\lambda_{v})\sum_{L=0}^{\infty}k_{L0}(\lambda_{v})Y_{L}^{0}(\cos\theta),$
(6)

In the given equations,
$\kappa_{0}(\lambda_{v})\equiv\kappa_{\text{template}}(0;\lambda_{v})$ is what
we refer to as the ’amplitude term’. It informs us about the maximum intensity
of the template profile at the void center. Furthermore, $k(\theta)$, which we
call a ’shape function’, standardizes the shape of the template profile based
on the coefficients of its spherical harmonic, $k_{L0}$. To derive the
$k_{L0}$ coefficients, an interpolated univariate spline was employed to
construct a HEALPix template map from the $k(\theta)$ measurements. This map’s
pixel values, calculated from spherical coordinates, were then normalized
relative to the value of $\kappa_{0}$.

By taking these definitions into account, and making the assumption that the
noise component is uniform, shows no directional bias, and averages out to
zero, we are in a position to calculate the spherical harmonic coefficients
for the ideal matched filter.

$\displaystyle\Psi_{\text{L0}}^{MF}(\lambda_{v})=\frac{\alpha
k_{L0}{(\lambda_{v})}}{C_{L}^{N_{\text{tot}}}}$ (7)

In the above eqaution $\alpha$ defined as:

$\displaystyle\alpha^{-1}\equiv\sum_{L=0}^{\infty}\frac{(k_{L0}{(\lambda_{v})})^{2}}{C_{L}^{N_{\text{tot}}}}$
(8)

$C_{L}^{N_{\text{tot}}}=C_{L}^{\kappa\kappa}+N_{L}^{\kappa\kappa}$ (9)

where ${C_{L}^{N_{\text{tot}}}}$ is the total power spectrum, and
${C_{L}^{\kappa\kappa}}$ and $N_{L}^{\kappa\kappa}$ are the lensing and noise
power spectra, respectively, for the Planck lensing map.

In our analysis, we have opted to calculate ${(k_{L0}{(\lambda_{v})})^{2}}$ up
to $L=700$, by convention. We made this choice because the spherical harmonic
coefficients for higher values of $L$ tend to approach zero, as demonstrated
in Figure 4.

The ideal matched filters for each void bin, constructed using previously
acquired template profiles, are displayed in Figure 4. We have also
transferred the Jackknife errors from the template profiles into the matched
filters.

An important takeaway from Figure 3 is that the templates for the
$\lambda_{v}$ bin 1 in every redshift category don’t undergo a sign change,
which suggests that the corresponding matched filters for these bins also stay
the same. For all the remaining bins, a conspicuous crossover point is
observed, resulting in filter profiles that are either partly or entirely
compensated. In this context, a "compensated" profile refers to a matched
filter profile where the areas under the curve with positive and negative
values of kappa cancel each other out, leading to a net zero integral over the
profile. This compensation phenomenon actually reflects a balance between the
regions of under-density and over-density within the voids and superclusters,
as captured by the lensing signal.

Figure 4: This figure shows the optimally determined spherical harmonic
coefficients, known as the kernel of the matched filters $\psi_{LO}^{MF}$,
derived from the MICE template profiles shown in Figure 3. The shaded regions
represent uncertainty estimates calculated from 1000 synthetic datasets using
Jackknife standard errors, as detailed in Equation 15. In our analysis, we
confine the application of spherical harmonics to $L=700$, a convention based
on the observation that the power of these coefficients approaches zero beyond
this threshold.

For each bin defined by the matched filter, based on $\lambda_{v}$, the
lensing map which has been filtered, denoted by $\kappa_{\text{MF}}$, can be
viewed as a convolution of the filter with the original map. Formally,

$\kappa^{\text{MF}}(\eta)=\int
d\Omega\kappa(\theta)\Psi^{\text{MF}}(|\theta-\eta|),$ (10)

This can also be transcribed into the spherical harmonic space (Schäfer et
al., 2006) as follows,

$\kappa_{\text{LM}}^{MF}=\sqrt{\frac{4\pi}{2L+1}}\kappa_{LM}\Psi_{\text{L0}}^{MF}.$
(11)

The matched filter’s design is very unique that it ensures the expected value
of the filtered field at the void locations is,

$\langle\kappa^{\text{MF}}(0;\lambda_{v})\rangle=\kappa_{0}(\lambda_{v}),$
(12)

suggesting that the filter is unbiased. This is the $\kappa_{0}$ measured at
the void center. This matched filter approach also ensures the variance of the
filtered field at this location is minimized, given by,

$\sigma_{\text{MF}}^{2}(0;\lambda_{v})=\sum_{L=0}^{\infty}C_{L}^{N_{\text{tot}}}|\Psi_{\text{L0}}^{MF}|^{2},$
(13)

Furthermore, the maximum detection level of the optimal matched filter, for a
single isolated void, can be quantified as (McEwen et al., 2008),

$\Gamma_{single}(\lambda_{v})\equiv\frac{\langle\kappa_{\text{MF}}(0;\lambda_{v})\rangle}{\sigma_{\text{MF}}(0;\lambda_{v})}=\alpha^{-1/2}\kappa_{0}(\lambda_{v}).$
(14)

In summary, matched filtering is applied to optimize the signal-to-noise
ratio. This technique emphasizes the expected lensing signal in specific
regions of the lensing map, following a predetermined template, while reducing
non-relevant areas. The lensing signal, as represented by the void center
pixel in the filtered map, enables a statistically optimized quantification of
the lensing effect, which is suitable for stacking.

As observed in Figure 4, it is important to note that the power of the filter
primarily lies in the $L$ modes less than 500. This makes the final central
pixel of voids susceptible to smaller-scale noise as we anticipate that high
$L$ modes are dominated by noise. This does not compromise its effectiveness
but rather highlights its specificity. Given that cosmic voids are large-scale
structures, it is reasonable that these $L$ modes capture the most significant
lensing information. Therefore, the efficiency of the matched filter is not
predicated on capturing all possible information, but on maximizing the
detectability of the specific lensing signal, assuming that the MICE template
accurately characterizes this signal.

###  3.4 Error Estimation

We introduce a novel technique to estimate errors in the MICE void lensing
templates and the associated matched filters. This approach contrasts with the
method used by Kovács et al. (2022b) (hereafter referred to as K22), which
assumes that errors in MICE are negligible. Our approach utilizes Jackknife
resampling, setting $N_{jk}=N_{\text{void}}$ for each void bin in the MICE
simulation. This process results in a Jackknife sample of size
$N_{\text{void}}$. The standard errors of the MICE templates, as used in
Figure 3, are given by:

$\sigma_{\text{JK}}=\sqrt{\frac{N_{\text{void}}-1}{N_{\text{void}}}\sum_{i=1}^{N_{\text{void}}}(\theta_{i}-\overline{\theta})^{2}}$
(15)

where $\theta_{i}$ are the individual Jackknife samples, $\overline{\theta}$
is the mean of the Jackknife estimates, and $N_{\text{void}}$ is the number of
voids in that bin.

We then estimate the covariance matrix from these "void by void" Jackknife
samples by using: (Mohammad & Percival, 2022):

$C_{ij}^{(jk)}=\frac{N_{\text{void}}-1}{N_{\text{void}}}\sum_{k=1}^{N_{\text{void}}}(\theta_{k_{i}}-\bar{\theta_{i}})(\theta_{k_{j}}-\bar{\theta_{j}})$
(16)

where $k$ represents the individual Jackknife samples and $\theta_{k_{i}}$ and
$\theta_{j_{i}}$ represents the corresponding $\theta[\degree]$ measurement of
20 bins of $\theta\,[\degree]$ measurement as shown in Figure 3.

After this, a synthetic data vector is created as follows:

$\xi_{\text{mock}}=LZ+\xi_{\text{th}},$ (17)

where $LL^{T}=C_{\text{ij}}$, L is obtained by a Cholesky decomposition of the
covariance matrix $C_{\text{ij}}$, and $Z$ is a vector of independent standard
normal random variables ( $\mu=0$ and $\sigma=1$), and $\xi_{\text{th}}$ is
the mean signal template for the corresponding bin. In this way, we conserve
the structure of the covariance and add random Gaussian noise by using $Z$,
which makes our template error analysis more robust unlike previous CMB x LSS
studies (Kovács et al., 2019b; Kovacs et al., 2021; Vielzeuf et al., 2020;
Camacho-Ciurana et al., 2023)

We then obtained $N_{\text{void}}$ synthetic data vectors using Equation 17
for each void bin and calculated the mean profile of these. This process was
repeated $N=1000$ times to obtain 1000 mean profiles.

We then apply our matched filter methodology to obtain the spherical harmonics
coefficients based on these 1000 mean profiles, as shown in Fig 4.

Subsequently, we perform a convolution of these matched filters with 1000
randomly generated MICE $\kappa$ maps, employing the synfast function from the
Healpy library. This operation introduces an additional layer of randomness,
derived from both the $\kappa$ map and the coefficients of the matched
filters, enabling us to observe the impact of errors in the MICE templates.
Otherwise, we could have just used the mean template profile to convolve with
1000 randomly generated MICE $\kappa$ maps, but we wanted to observe the
effect of the uncertainty in the template profiles.

We further tested our error analysis by applying Jackknife resampling to
different sub-volumes and using various groupings with different $N_{jk}$
values, such as 50, 64, and 100, instead of treating each void as an
individual Jackknife sub-sample. Our findings indicated that when implementing
Equation 17, the standard deviation of the resulting templates remained
consistent across these different groupings. This consistency reinforced our
decision to employ the "void by void" Jackknife approach for our final error
analysis. This error analysis was repeated for 2D superclusters as well.

Intriguingly, we find that the final stacked $\kappa$ value emanating from the
DES Y3 void centers remains unaffected by the randomness in the spherical
harmonics coefficients of the matched filters. This can likely be attributed
to the fact that the influence of this Jackknife randomness is minimal at
smaller scales, such as the center pixels of voids. We, therefore, conclude
that our measurements are predominantly dominated by the noise in the Planck
$\kappa$ map.

We first extract $\kappa_{0}$ values from the randomly generated and matched
filtered Planck-like convergence maps at the location of the central pixels of
each bin of DES Y3 voids. We then average out these values to calculate
$\kappa_{0}$. This process, involving 1000 random instances of $\kappa_{0}$,
allows us to construct a covariance matrix for our measurements.

Next, we determine the $\kappa_{0}$ values at the central pixels of DES Y3
void positions for each bin on the corresponding matched filtered Planck map.
These measured $\kappa_{0}$ values are subsequently compared with the
corresponding MICE $\kappa_{0}$ values. Figure 5 illustrates the measured
$\kappa_{0}$ values for each void bin, encompassing both Voxel and 2D voids.
The standard errors incorporated into the plot correspond to the diagonal
entries of our established covariance matrices.

To measure our CMB lensing detection significance, we use the $\chi^{2}$
minimization technique as in (Kovács et al., 2022b; Vielzeuf et al., 2020). We
make use of the following equation;

$\chi^{2}=\sum_{l,m}\left(\kappa^{\text{DES}}_{l}-A_{\kappa}\cdot\kappa^{\text{MICE}}_{l}\right)C^{-1}_{lm}\left(\kappa^{\text{DES}}_{m}-A_{\kappa}\cdot\kappa^{\text{MICE}}_{m}\right)$
(18)

where $\kappa_{l}$ denotes the mean CMB lensing signal within $\theta$ bin
$l$, and $C$ represents the related covariance matrix. We checked to identify
the best-fitting amplitude, represented as $A_{\kappa}\pm\sigma_{A_{\kappa}}$,
by constraining the shape of the stacked convergence profile from DES Y3 x
Planck to align with the shape calibrated from the MICE simulation. Moreover,
in our matched filtering methodology, we apply the Percival correction factor
when inverting covariance matrices as described in (Percival et al., 2021).
Conversely, for the covariance matrix derived from the template fitting
methodology of K22, we employ the Anderson-Hartlap factor
$h=(n_{\text{randoms}}-1)/(n_{\text{randoms}}-n_{\text{data points}}-2)$
before inverting the covariance matrix (Hartlap et al., 2007). This approach
is adopted to maintain consistency with the methods utilized by K22. We then
obtain the final results using the template fitting methodology of K22 as
depicted in Figure 7.

##  4 RESULTS

The results of our study affirm that both Voxel and 2D voids are in strong
agreement with the observed measurements, as seen in Figure 5. The measured
$A_{k}$ values return $A_{Voxel}=1.03\pm 0.22$ and $A_{2D}=1.02\pm 0.17$.
These findings carry considerable statistical significance, presenting a
$4.61\sigma$ detection of CMB lensing in the case of Voxel voids, and an even
more substantial $5.92\sigma$ detection for 2D Voids. We attribute the
marginally higher detection associated with 2D voids to their intrinsic
elongation along the line of sight due to their projected nature in redshift
shells. We then apply the same redshift binning to compare the results of 2D
superclusters, showing that the amplitude $A_{2D}=0.87\pm 0.15$ corresponds to
a $5.94\sigma$ detection of CMB lensing as seen in Figure 6.

Figure 5: Comparison of Voxel and 2D Voids: This figure illustrates the
correlation between the CMB lensing convergence ($\kappa$) derived from the
Dark Energy Survey Year 3 (DESY3) and Planck data, and the simulated void
lensing $\kappa$ from the MICE simulation at the stacked center pixels of
voids. Each redshift bin (LOWZ, MIDZ, and HIGHZ) is represented by different
markers. Within each redshift category, the bin values of $\lambda_{v}$
increase from the smallest to the largest. The plots include a reference line
at $A_{k}=1.0$ and the best-fit line. Our analysis reveals a significant
correlation between DESY3 and MICE data, with best-fit amplitudes of
$A_{k}=1.03$ for Voxel voids and $A_{k}=1.02$ for 2D voids, corresponding to
$4.61\sigma$ and $5.92\sigma$ detection levels for CMB lensing, respectively.
These findings show a strong agreement with $\Lambda$CDM expectations. Figure
6: Comparative Analysis of 2D superclusters and 2D Voids: This figure presents
stacked $\kappa_{0}$ values for each redshift bin of both 2D superclusters and
2D voids. Our joint fit utilizing different redshift categories yields a
statistical significance of $5.94\sigma$ and an amplitude of $A_{k}=0.87\pm
0.15$ for 2D superclusters, while for 2D voids we find a statistical
significance of $5.92\sigma$ and an amplitude of $A_{k}=1.02\pm 0.17$. The
amplitude $A_{k}$ observed for 2D superclusters is slightly lower than that
for 2D voids, but still falls within $1\sigma$ of $A_{k}=1.00$. These results
are in line with (Hang et al., 2021), which also found that 2D superclusters
exhibit a slightly lower $A_{k}$ value than their void counterparts. However,
it’s important to note that our measured $A_{k}$ values imply a good agreement
with $A_{k}=1.00$, while (Hang et al., 2021) demonstrate a marginally lower
lensing amplitude for voids and superclusters.

###  4.1 Comparison with K22- Voids

In order to understand the differences that the employed method and type of
tracers can make, we compare our results with K22 (Kovács et al., 2022b). K22
uses the same dataset and simulation as in our study. The difference lies in
our updated redMaGiC algorithm to select the galaxies both in DES Y3 and the
MICE simulation.

In addition, we use the matched filtering method, and we do not re-scale when
stacking the CMB cutouts around voids. However, to investigate if the employed
method makes a significant difference in the results, we also use the same
method as in the K22 study.

We, therefore, re-scale the stacked CMB cutouts around void centers and filter
the CMB lensing maps with a Gaussian filter with a size Full Width at Half
Maximum (FWHM) = 1∘.

In pursuit of covariance determination, we generate 1000 random Planck maps in
a manner akin to our matched filtering approach. This procedure involves the
repetition of Gaussian filtering and the re-scaled stacking of DES Y3 voids
and superclusters, as carried out in K22. To estimate the lensing amplitude,
we employ equation 18, consistent with our matched filtering methodology. As
seen in Figure 7, the results show that $A_{\kappa}=0.97\pm 0.28$ with a
$3.53\sigma$ detection of CMB lensing for 2D Clusters, and $A_{\kappa}=0.93\pm
0.30$ with a $3.09\sigma$ CMB lensing detection for 2D voids, in line with the
K22 approach.Despite the lower detection significance compared to our matched
filtering approach, the measured amplitudes are impressively close to
$A_{k}=1.00$, thus affirming the robustness of our findings. However, it is
noteworthy that our result slightly deviates from K22, where authors reported
a lower $A_{k}$ amplitude and a minor tension with MICE expectations, albeit
with a higher detection significance. We associate this discrepancy with our
updated galaxy tracer selection of redMaGiC. Furthermore, it is pertinent to
mention that we identify twice as many voids using the same void finder on the
same dataset, suggesting that the divergence likely originates from our usage
of the updated tracer selection.

Figure 7: Stacked profile of 2D voids and 2D superclusters with the template fitting methodology from K22. The 2D superstructures with radius $R_{v}<20h^{-1}{\rm Mpc}$ are not included in the analysis in accordance with K22.There is no further binning applied. The stacked images are obtained by rescaling the image up to $5R_{v}$ centered on void positions. We then measured the radial kappa profile from the final stacked images. The shaded regions show the $1\sigma$ error bars from the covariance matrix as explained in Section3.4 Table 2: Comparison of 2D Voids and 2D Superclusters in MICE and DES Y3. Total numbers without any cut are as follows: MICE 2D Voids: 12610, MICE 2D Superclusters: 13167, DES Y3 2D Voids: 10904, DES Y3 2D Superclusters: 10592. Category | Redshift | MICE (>20Mpc/h) | DES Y3 (>20Mpc/h)
---|---|---|---
2D Voids | Low | 1201 | 921
| Mid | 1840 | 1512
| High | 3254 | 2715
| Total (>20Mpc/h) | 6295 | 5148
2D Superclusters | Low | 1043 | 809
| Mid | 1556 | 1338
| High | 2885 | 2285
| Total (>20Mpc/h) | 5484 | 4432

##  5 Discussion and Conclusion

In this study, we conducted an in-depth examination of superstructures and
their interrelation with the CMB lensing, zeroing in on a pivotal instrument
for the detection of the CMB lensing effect: the matched filter technique.

For the first time in the literature, we applied the Voxel void-finding
algorithm to photo-z galaxies and gauged their CMB lensing footprints.

The CMB lensing footprints of cosmic voids, sourced from both the DES Y3
dataset and the MICE simulation for Voxel and 2D voids, demonstrated a robust
agreement. This revelation provides compelling insights into the intricacies
of delineating superstructures. It highlights the significance of not only the
methodologies employed but also the noise properties associated with CMB
lensing map and the choice of galaxies utilized in their definition.

Our results indicate a slight divergence from the weaker signal reported for
DES Y3 voids in the study by (Kovács et al., 2022b), but align well with the
findings for DES Y1 voids by (Vielzeuf et al., 2021). The disparity with the
previous DES Y3 results is primarily attributed to the enhancements made in
the redMaGiC galaxy selection algorithm employed in our investigation. It
became evident that the selection of tracer galaxies used to define
superstructures wields a considerable influence on the CMB lensing outcomes of
these, independent of the methodology employed in identifying the
superstructures.

Our research emphasized significant observations of lensing stemming from
Voxel voids, 2D voids, and 2D clusters, each uniformly demonstrating an
amplitude consistent with $A_{k}=1.00$. Such uniformity across a variety of
void types and structures not only reinforces the robustness of our findings
but also aligns seamlessly with the predictions of the $\Lambda$CDM
cosmological model, agreeing well with the results of previous studies
(Raghunathan et al., 2019; Vielzeuf et al., 2021; Cai et al., 2017; Hang et
al., 2021; Camacho-Ciurana et al., 2023).

Furthermore, we identified a notable correspondence between 2D and 3D (Voxel)
voids in our study, where the 2D voids exhibited marginally lower
uncertainties and enhanced levels of CMB lensing detection. This observation
indicates the superior precision of 2D void analysis in capturing lensing
signals. Such precision is presumably owed to the elongated structure of 2D
voids along the line of sight, which are defined on projected tomographic
redshift slices. The 2D algorithm finds structures elongated along the line of
sight because they result in a big impact on the projected density field but
structures aligned perpendicular do not. This structural characteristic
possibly contributes to the increased sensitivity in detecting lensing
phenomena which is the integral effect along the line of sight, further
validating the efficacy of employing 2D void analysis in studies of this
nature.

Moreover, we demonstrated that Voxel voids can be effectively utilized for
analyses involving the cross-correlation of lensing effects on the CMB. This
study is the first to measure the CMB lensing imprints of Voxel voids. Our
results provide evidence that these voids exist in regions of the Universe
characterized by a true deficit in matter density as shown by their CMB
lensing imprints. Encouragingly, our results align closely with those derived
from 2D voids, a void finder methodology employed extensively in previous
studies. Furthermore, as demonstrated by (Radinović et al., 2023), Voxel voids
serve as a viable choice for void-galaxy correlation analyses. Unlike 2D
voids, whose tomographically projected nature precludes this type of analysis,
Voxel voids do not suffer from this limitation. This key advantage makes Voxel
voids particularly suited to forthcoming large-scale surveys such as the Dark
Energy Spectroscopic Instrument (DESI) (DESI Collaboration et al., 2016a, b)
and the Euclid mission (Laureijs et al., 2011).

In our comparison of the matched filter and Gaussian filtering methods, we
observed consistent results for $A_{k}=1.00$ across 2D voids and clusters, as
detailed in (Kovács et al., 2022b; Vielzeuf et al., 2021). The matched filter
method reduced uncertainties and enhanced detection significance, illustrating
its superior efficacy. Our study demonstrated that the matched filter method
can provide more precise $A_{k}$ estimates, suggesting its potential for
future research. In particular, it can be interesting to utilize the matched
filtering method on the same dataset used in (Kovács et al., 2022a) to examine
the detected ISW signals. The authors of the cited study reported an excess
ISW signal without using matched filtering, which presents an intriguing
opportunity for future analysis and verification of their claim.

This work provides the pivotal groundwork for upcoming studies from the Vera
Rubin Observatory and the Euclid Survey, which aim to further investigate the
Integrated Sachs-Wolfe (ISW) effect and CMB lensing due to cosmic
superstructures. Our research not only contributes to the evolving
understanding of lensing phenomena in the Universe but also lays a strong
foundation for future studies. By applying the insights discussed in our
analysis, future large-scale structure surveys can refine their strategies for
exploring the universe’s large-scale structure, potentially leading to
enhanced accuracy and precision in cosmological constraints derived from these
measurements.

##  6 Appendix

We show additional plots that can be useful for our analysis.

Figure 8: This plot illustrates the Voxel void size distributions for both
MICE and DESY3 using a normalized histogram, segmented across 9 redshift bins.
Notably, the number of voids inreases with increasing redshift bin.
Additionally, voids within the same $\lambda_{v}$ bins, yet across differing
redshift bins, exhibit a consistent size distribution pattern. This
consistency emphasizes the functional relationship between $\lambda_{v}$ and
$R_{v}$, as shown in Equation 3. Figure 9: This figure shows the correlation
matrix for 9 Voxel bins. The bins range from BIN 1, representing LOWZ
$\lambda_{v}$ Bin 1, to BIN 9, representing HIGHZ $\lambda_{v}$ Bin 3.

## Acknowledgements

Author contributions: We would like to acknowledge everyone who made this work
possible. UD and SN made significant contributions to project development,
including paper writing and figures. They also were involved in data analysis
and methods validation. IF was instrumental in the construction and validation
of the MICE redMaGiC catalog, ensuring the accuracy and reliability of this
critical dataset. PF took charge of constructing and validating the MICE CMB
lensing map. CTD and SP conducted an internal review of the paper within the
DES collaboration, offering suggestions that significantly improved its
quality. AK was pivotal in idea generation and providing useful suggestions
that helped shape the project’s direction. SN and RM provided advice, guiding
the project through their expertise in the field. MA, KB, ADW, RAG, WGH, APi,
ESR, ESh and BY were responsible for the construction and validation of the
DES redMagic catalog, a cornerstone of our analysis. The remaining authors
have made contributions to this paper that include, but are not limited to,
the construction of DECam and other aspects of data collection; data
processing and calibration; development of broadly used methods, codes, and
simulations; operation of the pipelines and validation tests; and advancement
of the science analysis.

Funding for the DES Projects has been provided by the U.S. Department of
Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology Facilities Council of the
United Kingdom, the Higher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of Illinois at
Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas AM University, Financiadora de Estudos e Projetos, Fundação
Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho
Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da
Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de
Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-
Maximilians Universität München and the associated Excellence Cluster
Universe, the University of Michigan, NSF’s NOIRLab, the University of
Nottingham, The Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, Texas AM University, and the OzDES
Membership Consortium. Based in part on observations at Cerro Tololo
InterAmerican Observatory at NSF’s NOIRLab (NOIRLab Prop. ID 2012B-0001; PI:
J. Frieman), which is managed by the Association of Universities for Research
in Astronomy (AURA) under a cooperative agreement with the National Science
Foundation.

The DES data management system is supported by the National Science Foundation
under Grant Numbers AST-1138766 and AST-1536171. The DES participants from
Spanish institutions are partially supported by MICINN under grants
ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-2016-0597,
and MDM-2015-0509, some of which include ERDF funds from the European Union.
IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
Research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Program
(FP7/2007-2013) including ERC grant agreements 240672, 291329,and 306478. We
acknowledge support from the Brazilian Instituto Nacional de Ciência e
Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2).

The data production, processing and analysis tools for this paper have been
developed, implemented and operated in collaboration with the Port
d’Informació Científica (PIC) data center. PIC is maintained through a
collaboration agreement between the Institut de Física d’Altes Energies (IFAE)
and the Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT). UD specifically thanks Carles Acosta and Christian Neissner for
their invaluable assistance with the PIC services, and warmly acknowledges the
enlightening discussions with Qianjun Hang that have contributed to this
research.

UD acknowledges the support by PREBIST project and has received funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No.754558. SN acknowledges support from
an STFC Ernest Rutherford Fellowship, grant reference ST/T005009/2. AK
received funding from the European Union’s Horizon Europe research and
innovation programme under the Marie Skłodowska-Curie grant agreement number
101130774, from the Hungarian Ministry of Innovation and Technology NRDI
Office grant OTKA NN147550, and from a _Lendület_ excellence grant by the
Hungarian Academy of Sciences (MTA).

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##  Appendix A Author Affiliations

1 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
2 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, UK
3 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
4 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
5 MTA-CSFK Lendület "Momentum" Large-Scale Structure (LSS) Research Group,
1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary
6 Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
7 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81679 Munich, Germany
8 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
9 Center for Astrophysical Surveys, National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA
10 Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison,
1150 University Avenue Madison, WI 53706-1390
11 Department of Astronomy and Astrophysics, University of Chicago, Chicago,
IL 60637, USA
12 Center for Astrophysical Surveys, National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA
13 Department of Astronomy, University of Geneva, ch. d’Écogia 16, CH-1290
Versoix, Switzerland
14 Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
15 Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA
16 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA
17 Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA
18 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,
USA
19 Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile
20 Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
21 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,
USA
22 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
23 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, UK
24 CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
25 Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut
d’Astrophysique de Paris, F-75014, Paris, France
26 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81679 Munich, Germany
27 Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
28 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain
29 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
30 Physics Department, William Jewell College, Liberty, MO, 64068
31 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029
Hamburg, Germany
32 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
33 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
34 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Dr., Pasadena, CA 91109, USA
35 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
36 School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia
37 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA
38 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
39 Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113,
Australia
40 Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA
41 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, and Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA
42 LPSC Grenoble - 53, Avenue des Martyrs 38026 Grenoble, France
43 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse,
85748 Garching, Germany
44 Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA
45 Kavli Institute for Cosmological Physics, University of Chicago, Chicago,
IL 60637, USA
46 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA
47 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
48 School of Physics and Astronomy, University of Southampton, Southampton,
SO17 1BJ, UK
49 Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil
50 Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831, USA
51 Department of Astronomy, University of California, Berkeley, 501 Campbell
Hall, Berkeley, CA 94720, USA
52 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA
94720, USA
53 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians
Universität München, Scheinerstr. 1, 81679 München, Germany
54 Konkoly Observatory, HUN-REN CSFK, MTA Centre of Excellence, Budapest,
Konkoly Thege Miklós út 15-17. H-1121 Hungary