11institutetext: MTA-CSFK Lendület ”Momentum” Large-Scale Structure (LSS)
Research Group, 1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary
22institutetext: Konkoly Observatory, HUN-REN CSFK, MTA Centre of Excellence,
Budapest, Konkoly Thege Miklós út 15-17. H-1121 Hungary 33institutetext:
Department of Physics of Complex Systems, ELTE Eötvös Loránd University, Pf.
32, H-1518 Budapest, Hungary 44institutetext: Institute of Astronomy and NAO,
Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee Blvd., 1784 Sofia,
Bulgaria 55institutetext: Institute for Astronomy, University of Hawaii, 2680
Woodlawn Drive, Honolulu, HI, 96822, USA

# The CMB lensing imprint of cosmic voids detected in the WISE-Pan-STARRS
luminous red galaxy catalog

G. Camacho-Ciurana,, 112233 P. Lee, 1122 N. Arsenov,, 441122 A. Kovács, 1122
I. Szapudi, MTA Guest Professor 20235511 I. Csabai 33

(Received December 2023)

###### Abstract

Context. The cross-correlation of cosmic voids with the lensing convergence
($\kappa$) map of the Cosmic Microwave Background (CMB) fluctuations offers a
powerful tool to refine our understanding of the dark sector in the consensus
cosmological model.

Aims. Our principal aim is to compare the lensing signature of our galaxy data
set with simulations based on the concordance model and characterize the
results with an $A_{\kappa}$ consistency parameter normalized to unity. In
particular, our measurements contribute to the understanding of the ”lensing-
is-low” tension of the $\Lambda$CDM model.

Methods. In this analysis, we selected luminous red galaxies (LRGs) from the
WISE-Pan-STARSS data set, allowing an extended cross-correlation measurement
using 14,200 deg2 sky area, that offers a more precise measurement compared to
previous studies. We created 2D and 3D void catalogs to cross-correlate their
locations with the Planck CMB lensing map and studied their average imprint
signal using a stacking methodology. Applying the same procedure, we also
generated a mock galaxy catalog from the WebSky simulation to serve as a basis
for comparison.

Results. The 2D void analysis revealed good agreement with the standard
cosmological model with $A_{\kappa}\approx 1.06\pm 0.08$ amplitude, i.e.
$S/N=13.3$, showing a higher signal-to-noise than previous studies using voids
detected in the Dark Energy Survey (DES) data set. The 3D void analysis
exhibited a lower signal-to-noise ratio and demonstrated worse agreement with
our mock catalog than the 2D voids. These deviations might be attributed to
limitations in the mock catalog, such as imperfections in the LRG selection,
as well as a potential asymmetry between the North and South patches of the
WISE-Pan-STARSS data set in terms of data quality.

Conclusions. Overall, we present a significant detection of a CMB lensing
signal associated with cosmic voids, largely consistent with the concordance
model. Future analyses using even larger data sets also hold great promise of
further sharpening these results, given their complementary nature to large-
scale structure analyses.

###### Key Words.:

cosmic microwave background, gravitational lensing

##  1 Introduction

_Cosmic voids_ are large under-dense structures occupying most of the late-
time Universe. Unlike other components of the cosmic web (walls, filaments and
galaxy clusters), void interiors are less prone to non-linear gravitational
effects since they contain significantly less dark matter (see e.g. Pisani et
al., 2019, for a recent review). As a consequence, voids are dominated by dark
energy and therefore these regions are becoming promising laboratories for
extracting cosmological information. In particular, they offer novel ways to
test the $\Lambda$CDM (Lambda-Cold Dark Matter) model, modified gravity
scenarios (see e.g. Clampitt et al., 2013; Cai et al., 2015; Cautun et al.,
2018; Baker et al., 2018; Schuster et al., 2019; Davies et al., 2021), or to
constrain the neutrino mass due their sensitivity to diffuse components
(Kreisch et al., 2019; Contarini et al., 2021).

Voids constrain cosmological models through various probes, e.g. the void size
function, density and velocity profiles, and also their evolution with
redshift (see e.g. Pisani et al., 2015; Verza et al., 2019; Nadathur et al.,
2020; Aubert et al., 2022; Hamaus et al., 2021). Moreover, their lensing
signals serve as complementary tools for exploring the underlying dark matter
distribution, showing a de-magnification effect photons are deflected while
traversing cosmic voids. We note, however, that the detection of cosmic shear
or convergence signal from an individual void is challenging due to the
important uncertainties (Amendola et al., 1999; Krause et al., 2013),
detections of the void lensing signal using _stacking_ methods from large
catalogues of voids have already been reported (Melchior et al., 2014; Sánchez
et al., 2017; Gruen et al., 2016; Clampitt & Jain, 2015; Brouwer et al., 2018;
Fang et al., 2019; Jeffrey et al., 2021), including lensing analyses of given
different void definitions and simulation with different cosmological models
(see e.g. Cautun et al., 2016; Davies et al., 2018, 2021).

To further probe the properties of dark matter and dark energy, an alternative
strategy is to stack the Cosmic Microwave Background (CMB) on the positions of
cosmic voids and thus measure their imprints in temperature or lensing
convergence maps. Along these lines, various studies have probed the
integrated Sachs-Wolfe (ISW) effect (Sachs & Wolfe, 1967) by stacking the
temperature anisotropy maps on void positions. See also Alonso et al. (2018)
and Li et al. (2024) for explorations of void signals in Compton y-maps to
study the thermal Sunyaev-Zeldovich (tSZ) effect.

This tiny CMB foreground signal has generated considerable interest, as the
$A_{\scaleto{\rm ISW}{4pt}}$ = $\Delta T_{\mathrm{obs}}/\Delta
T_{\scaleto{\rm\Lambda CDM}{4pt}}$ amplitude parameter (the ratio of observed
and expected ISW signals) has often been found significantly higher from
$R\gtrsim 100\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$ voids, or _supervoids_ ,
than expected in the concordance $\Lambda$CDM model (see e.g. Nadathur et al.,
2012; Flender et al., 2013; Ilić et al., 2013; Hernández-Monteagudo & Smith,
2013, for discussions about the expected ISW signal from voids, and on the
role of selection effects). Relying first on Wilkinson Microwave Anisotropy
Probe (Hinshaw et al., 2013, WMAP) and then _Planck_ CMB data (Planck
Collaboration, 2018a), moderately significant excess $A_{\scaleto{\rm
ISW}{4pt}}$ amplitude values (Granett et al., 2008; Cai et al., 2017; Kovács,
2018) and good consistency with the $\Lambda$CDM model predictions (Hotchkiss
et al., 2015; Nadathur & Crittenden, 2016) have both been reported, using the
Sloan Digital Sky Survey (SDSS) luminous red galaxy (LRG) catalogs, and than
Baryon Oscillation Spectroscopic Survey (BOSS) data to construct void
catalogs. The enhanced ISW signals from supervoids are also considered
anomalous because 2-point correlation analyses do _not_ show significant
excess either, compared to $\Lambda$CDM predictions (see e.g. Planck
Collaboration, 2016; Stölzner et al., 2018; Hang et al., 2021b).

We note that the ISW excess problem has also been linked to the CMB _Cold
Spot_ anomaly (Cruz et al., 2005), and significant evidence exists for the
presence of the low-$z$ _Eridanus_ supervoid in its direction (see e.g.
Szapudi et al., 2015; Kovács et al., 2022a). It was shown, however, that
assuming the $\Lambda$CDM model, the size and under-density of the aligned
supervoid is not sufficient to fully explain the observed temperature
depression (Nadathur & Hotchkiss, 2015; Naidoo et al., 2016; Mackenzie et al.,
2017).

Further complicating the picture, similar excess ISW signals were also
reported using the Dark Energy Survey (The Dark Energy Survey Collaboration,
2005, DES) LRG catalogs (Kovács et al., 2017, 2019) and the extended Baryon
Oscillation Spectroscopic Survey (eBOSS) quasar data set (Kovács et al.,
2022).

These results have also generated interest in supplementing the ISW analyses
with the CMB lensing signal of voids, since the convergence maps provide
different but related information about the evolution of the underlying
gravitational potentials. Voids correspond to local minima in the lensing
convergence ($\kappa$) maps, estimated from the matter density field
$\delta(r,\theta)$ via projection as

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

in the Born approximation with the Hubble constant $H_{\mathrm{0}}$ and matter
density parameter $\Omega_{\mathrm{m}}$, where $r$ denotes the co-moving
distance to source galaxies in the background of the lenses (with distorted
shapes due to lensing), and $r_{\rm max}$ determines the maximum distance
considered.

Using BOSS data, Cai et al. (2017) and Raghunathan et al. (2020) both detected
the CMB lensing imprint of voids, finding good consistency with the
simulation-based $\Lambda$CDM expectations ($A_{\kappa}=\kappa_{\scaleto{\rm
data}{4pt}}/\kappa_{\scaleto{\rm model}{4pt}}\approx 1$). Measurements using
the DES LRG catalogs, however, reported a moderately low
$A_{\scaleto{\rm\kappa}{4pt}}\approx 0.82\pm 0.08$ amplitude (Vielzeuf et al.,
2021; Kovács et al., 2022b), that is also consistent with the findings by Hang
et al. (2021a) who analysed both voids and superclusters detected in the Dark
Energy Spectroscopic Instrument (Dey et al., 2019, DESI) Legacy Survey galaxy
data set, finding an overall $A_{\scaleto{\rm\kappa}{4pt}}\approx 0.81\pm
0.06$ best-fit amplitude.

In this paper, we perform a novel analysis and identify cosmic voids from the
WISE-Pan-STARRS galaxy data set to better understand previously reported
tensions concerning the ISW and lensing imprints of these large-scale
structures in the _Planck_ maps. The paper is organized as follows. In Section
2, we introduce our mock and observational data sets. Then, Section 3 contains
a description of our stacking methodology and error analysis. We then present
our main observational results in Section 4, followed by a summary of our main
conclusions in Section 5.

##  2 Data sets

###  2.1 WISE-Pan-STARRS galaxies

As tracers of the cosmological large-scale structure (LSS), we used the
_cross-matched_ catalogue of galaxies from the all-sky Wide-Field Infrared
Survey Explorer (WISE, Wright et al., 2010) and 3$\pi$ Panoramic Survey
Telescope and Rapid Response System (Pan-STARRS1, PS1 for short) sky surveys
presented by Beck et al. (2022). The PS1 Data Release 2 (DR2) catalogs provide
broad-band photometric measurements (Kron and PSF magnitudes) of about three
quarters of the sky using the $g,r,i,z,y$ filters in the optical range (see
Tonry et al., 2012; Chambers et al., 2016; Magnier et al., 2020a, b, c; Waters
et al., 2020, for further details). The WISE survey scanned the full sky in
four infrared photometric bands (W1, W2, W3, W4) having effective wavelengths
of 3.4, 4.6, 12 and 22 $\mu$m, respectively. Regarding the high noise level
and the relatively large number of missing error estimates of the W3, W4
filters, we only considered WISE measurements obtained in the W1 and W2
filters. We then only used W1 information to apply selection cuts among the
WISE-PS1 galaxies.

Our analysis applied further selection cuts to the WISE-PS1 galaxy catalog and
focused on a subset of LRGs. While their exact 3D position information is not
accessible using photo-$z$ data, the expected $\sigma_{z}\approx 0.02(1+z_{\rm
spec})$ precision allows a robust reconstruction of the largest cosmic voids
(see e.g. Sánchez et al., 2017, for further details). We defined a sample of
WISE-PS1 LRGs using the following color and magnitude cuts, inspired to some
extent by the LRG target selection strategy (Zhou et al., 2023) applied by the
Dark Energy Spectroscopic Instrument (DESI, Levi et al., 2013) survey:

  * •

$r-z>(z-17.3)/2$ (_PS1 color cut_)

  * •

$z<20.7$ (_PS1 magnitude cut_)

  * •

$i-W1>0.5$ (_WISE-PS1 color cut_)

  * •

$ext\\_flg\neq 0$ (_WISE extended object cut_)

  * •

$cc\\_flags=^{\prime}0000^{\prime}$ (_WISE cut for flagged data_)

Figure 1:  The Planck CMB lensing map (with a FWHM=$1^{\circ}$ Gaussian
smoothing) is highlighted within the WISE-PS1 sky area, where we performed a
robust measurement of voids $\times$ CMB $\kappa$ cross-correlations. The data
is divided into two larger patches, labelled as North and South, comprising an
unmasked area of approximately $A=14,200$ $\rm deg^{2}$.

Observed magnitudes have been corrected for the galactic dust extinction using
the related extinction coefficients ($\alpha_{g}=3.172$, $\alpha_{r}=2.271$,
$\alpha_{i}=1.682$, $\alpha_{z}=1.322$, $\alpha_{y}=1.087$,
$\alpha_{W1}=0.18$) and the $E(B-V$) dust extinction values of a map that is
based on PS1 observations of Milky Way stars (Schlafly et al., 2014).

We also applied an additional $0.42<z_{\rm phot}<0.7$ photometric redshift cut
to focus on the most dense part of the WISE-PS1 data with the highest galaxy
density, where the resulting sample did not show any visually obvious
contamination or inhomogeneity.

Finally, we created a sky-mask to minimize both Galactic and extra-galactic
sources of possible contamination and biases. Our healpix mask (Gorski et al.,
2005), created at $N_{\rm side}=512$ resolution, was constructed by the
following process:

  * •

we applied the point source mask based on the Planck 2018 measurements (Planck
Collaboration, 2018b)

  * •

dusty pixels with E(B-V) ¿ 0.1 were removed from the analysis (see Schlegel et
al. (1998) for further details)

  * •

we removed pixels with $|b|<25^{\circ}$, i.e. close to the Milky Way

  * •

pixels near the North Galactic cap were removed ($b>60^{\circ}$)

The above LRG selection cuts and sky-masking strategy resulted in a sample of
about $N_{g}=2.46\times 10^{6}$ WISE-PS1 galaxies, distributed in an unmasked
area of approximately $A=14,200$ $\rm deg^{2}$ i.e. covering about $1/3$ of
the sky (see Figure 1 for details).

###  2.2 CMB lensing map from the _Planck_ mission

In our LSS$\times$CMB cross-correlation measurements, we relied on the
reconstructed CMB lensing convergence ($\kappa$) map provided by the Planck
collaboration. This data set is released in the form of $\kappa_{\rm lm}$
spherical harmonic coefficients (see Planck Collaboration, 2018c, for details)
up to $\ell_{\rm max}=2048$. We created a $\kappa$ map at $N_{\rm side}=512$
resolution by converting the $\kappa_{\rm lm}$ values into healpix maps. We
also used a corresponding mask from the publicly available Planck data (part
of our analysis mask described above). In Figure 1, we show the common WISE-
PS1 survey area on top of the Planck $\kappa$ map, indicating our conservative
choices to avoid possibly contaminated regions. Additionally, the Dec ¿
$-30^{\circ}$ PS1 data-taking limit is also marked.

We note that, while higher resolution maps can be constructed from the
available $\kappa_{\rm\ell m}$ coefficients, the chosen $N_{\rm side}=512$ is
a sufficient choice given the degree-size angular scales of cosmic voids in
our analysis. We also applied a FWHM=$1^{\circ}$ Gaussian smoothing to the
convergence map to suppress small-scale noise patterns, following the S/N
optimisation results by Vielzeuf et al. (2021). Another filtering we
implemented was removing the large-scale modes from the $\kappa$ map
($\ell<20$) to remove any remnant wide-angle systematic effect in the map. Our
analyses consistently applied these smoothing and filtering steps to the
observed and simulated $\kappa$ maps.

Finally, we also removed a slight remnant bias from the $\kappa$ map
($\bar{\kappa}\approx-10^{-4}$), which is the result of using the WISE-PS1 sky
mask instead of the _Planck_ mask such that the construction provides a zero-
mean convergence map.

###  2.3 Simulations: mock galaxies and mock CMB $\kappa$ map

To create a simulated data set for calibrating our pipeline, we analysed the
publicly available WebSky
simulation111https://mocks.cita.utoronto.ca/data/websky/ (Stein et al., 2020),
that was constructed assuming a Planck 2018 cosmology (Planck Collaboration,
2018a). Based on the peak-patch method (i.e. not a full N-body run, see Stein
et al., 2019, for further detail), the WebSky simulation encloses a vast total
cosmic volume of $V\simeq 1,900$ $({\rm Gpc})^{3}$ with mass resolution of
about $M_{\rm{min}}\simeq 1.2\times 10^{12}M_{\odot}$ at $0<z<4$. It provides
a full-sky light-cone catalogue that contains about $9\times 10^{8}$ dark
matter halos (each with RA, Dec, redshift, an initial Lagrangian position
[Mpc], final Eulerian position [Mpc], velocity [km/s], and mass
[$M_{200\overline{\rho}_{m}}$]).

Notably, another key objective of the WebSky simulation was to provide a set
of secondary CMB foreground maps correlated with the simulated halo catalogs.
Therefore, to model the WISE-PS1 $\times$ Planck cross-correlations, we also
used the WebSky CMB lensing convergence map in our analyses, that was
constructed using the field particles and a Navarro-Frenk-White (NFW) profile
(NFW, Navarro et al., 1996) for all halos in the peak-patch halo catalogue
that subtend more than a pixel. Stein et al. (2020) added another component to
the convergence, as an uncorrelated Gaussian random field, to account for the
power originating from redshifts larger than the maximum included in the
simulation, zmax=4.5.

Naturally, this signal-only WebSky CMB lensing map is insufficient to account
for observational noise in the Planck $\kappa$ reconstructions. In the
following section, we provide details on how we added Planck-like noise to
estimate the measurement errors, which ultimately limit the precision of the
void lensing profiles.

##  3 Methodology

Our main objective is to measure the imprint of cosmic voids on CMB lensing
convergence maps, and below we detail our main tools. We closely follow the
methodology by Vielzeuf et al. (2021) and Kovács et al. (2022b), building on
their findings regarding e.g. optimisation techniques and free parameters in
the analysis.

###  3.1 Void finder methods

To complement standard LSS-CMB cross-correlation methods using clusters or
filaments (see e.g. Baxter et al., 2015; Madhavacheril et al., 2015; He et
al., 2018; Baxter et al., 2018), the under-dense regions of the Universe might
also be identified by void finder algorithms. Identifying cosmic voids is a
complex process, affected by survey properties including tracer quality,
tracer density, and masking effects (see e.g. Sutter et al., 2014; Nadathur &
Hotchkiss, 2015). In particular, void properties depend significantly on the
methodology used to define the voids and thus working with different void
finders make the analysis more robust. In this study, we employed two distinct
void finding methods that are standard methods in the field (see e.g. Fang et
al., 2019, for details).

####  3.1.1 Cosmic voids in 2D maps

Our main analysis pipeline uses a 2D void finder method, developed by Sánchez
et al. (2017) for the analysis photo-$z$ data sets. The algorithm works by
projecting galaxies in two-dimensional redshift slices specified by the user.
We followed the original methodology (see also Vielzeuf et al., 2021, for more
recent applications) and chose a thickness of $100\leavevmode\nobreak\
h^{-1}\mathrm{Mpc}$, which corresponds to about twice the size of typical
photo-$z$ errors. Then, we projected the data into healpix maps with
$\text{N}_{\text{\rm side}}=512$ resolution in 7 redshift slices in the
$0.42<z<0.7$ range covered by the WISE-PS1 LRGs.

The next step is a Gaussian smoothing applied to the projected galaxy count
maps, and voids are defined by _minima_ of the smoothed tracer density field
in the given slice (Sánchez et al., 2017). Annuli of increasing radii are
defined around the prospective void centers, and the algorithm stops when mean
density within the slice ($\delta$ = 0) is reached in a given annulus, where
the void radius ($R_{v}$) is defined. Void catalogs produced by the above
algorithm depend on two main free parameters that we set to the following
values:

  * •

Smoothing scale ($\sigma$): this parameter defines the Gaussian smoothing
scale applied to the density map to define density minima. Here we employed a
$\sigma=10\leavevmode\nobreak\ h^{-1}\mathrm{Mpc}$, also used by e.g. Kovács
et al. (2022a), which enables a reliable identification of larger voids that
carry most of the detectable lensing signal (without smoothing out the density
troughs too much, which would reduce the void lensing signal’s amplitude in
the centre).

  * •

Under-density threshold ($\delta_{\rm min}$): in a smoothed density map, this
parameter defines the minimum threshold to consider a given local density
minimum as a possible void centre. Naturally, significant voids correspond to
lower $\delta$ values, and we again followed Vielzeuf et al. (2021) by setting
$\delta_{\rm min}=-0.2$ to eliminate shallow under-densities.

####  3.1.2 Cosmic voids in 3D

To compare with 2D voids detected from the same galaxy catalogs, we used the
REVOLVER (REal-space VOid Locations from surVEy Reconstruction) void-finding
code222https://github.com/seshnadathur/Revolver Nadathur et al. (2019) to also
construct a 3D voids catalog. The algorithm is based on the ZOBOV technique
(Neyrinck, 2008) to reconstruct the local density of tracers, employing a
Voronoi tessellation and subsequently identifying density minima by comparing
Voronoi cells with their neighboring cells.

Presumably, the analysis of 3D voids is less efficient in detecting lensing
signals than using 2D voids, which guarantee longer photon paths in a coherent
gravitational potential (see e.g. Fang et al., 2019; Davies et al., 2021). Yet
these alternative void definitions and sub-classes may also offer important
checks of the fiducial 2D voids analyses.

Moreover, voids detected using the ZOBOV method also provide a useful proxy
for the sign of the gravitational potential in and around voids, defined as

$\lambda_{v}\equiv\overline{\delta}_{g}\left(\frac{R_{\mathrm{eff}}}{1\;h^{-1}\mathrm{Mpc}}\right)^{1.2}$
(2)

using the average galaxy density contrast,
$\overline{\delta}_{g}=\frac{1}{V}\int_{V}\delta_{g}\,\mathrm{d}^{3}\mathbf{x}$,
and the effective spherical radius,
$R_{\mathrm{eff}}=\left(\frac{3}{4\pi}V\right)^{1/3}$, where the volume $V$ is
determined from the sum of the volumes of Voronoi cells making up the void
(see e.g. Nadathur & Crittenden, 2016).

We note that two consecutive runs of ZOBOV pipeline do not result in identical
void catalogs. While differences are small in total the number of voids and in
the main void parameters, in our analysis we created 10 slightly different
versions of our WISE-PS1 and WebSky 3D void catalogs, and used their median
signals in our CMB lensing cross-correlation measurements for more robust
results.

Figure 2:  Comparison of the redshift distribution of LRGs in the WISE-PS1
data set versus in the WebSky mock. We followed an HOD modelling approach to
create a realistic model of the source density as a function of redshift,
reaching a few-per-cent match throughout the whole range. Figure 3:  Void
properties in the WISE-PS1 data set and in the WebSky mock catalog. The void
radii (left) and $\lambda_{v}$ parameter (right) both show great agreement,
which, after further pruning steps, facilitates a detailed comparison of the
simulated and observed lensing signals from these voids.

###  3.2 Modelling void properties with mock LRGs

Considering the realistic modelling of the WISE-PS1 LRG sample of galaxies, we
followed a halo occupation distribution (HOD) methodology (see e.g. Tinker et
al., 2012), using the publicly available pyHOD
code333https://github.com/Yucheng-Zhang/pyhod. The HOD approach describes the
average number of central and satellite galaxies residing in halos as a
function of halo mass, $M$. The expected total number of galaxies in a dark
matter halo is the sum of the central and satellite galaxy probabilities,
expressed as

$\langle N_{\mathrm{tot}}(M)\rangle=\langle N_{\mathrm{cen}}(M)\rangle+\langle
N_{\mathrm{sat}}(M)\rangle.$ (3)

The probability that a dark matter halo, typically a massive one
($M_{\rm{min}}\simeq 10^{13}M_{\odot}$ halo mass), contains a central LRG is
given by the smooth step function

$\langle
N_{\mathrm{cen}}(M)\rangle=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\log
M-\log M_{\mathrm{min}}}{\sigma_{\mathrm{\log M}}}\right)\right],$ (4)

where the position of this step is set by $M_{\mathrm{min}}$. A halo with mass
$M\ll M_{\mathrm{min}}$ hosts no central galaxy, which then transitions
according to the value of the $\sigma_{\mathrm{\log M}}$ parameter to $P=1$
probability for $M\gg M_{\mathrm{min}}$ halos (see e.g. Zhou et al., 2020, for
a recent application).

The number of satellite galaxies in each halo is Poisson distributed, with a
mean value given by a power law,

$\langle
N_{\mathrm{sat}}(M)\rangle=\left(\frac{M-M_{\mathrm{0}}}{M_{\mathrm{1}}}\right)^{\alpha},$
(5)

with $\alpha$, $M_{\mathrm{0}}$, and $M_{\mathrm{1}}$ as free parameters of
the satellite occupation model, where satellite galaxies are randomly
positioned in the halo following an NFW profile.

We then tuned the free parameters of the HOD model. Our main objective was to
make the void parameters in the simulated WebSky catalogs entirely consistent
with those of the WISE-PS1 voids.

Figure 4:  The number of voids present in the WISE-PS1 and WebSky mock
catalogs is compared in the North (left) and South (right) regions in the
figure. The difference in the coverage areas of the North and South masks
accounts for the variation in the number of voids observed. Furthermore, there
is a 13% difference in the overall number of voids between the LRG catalogs,
with 6090 voids in the WISE-PS1 catalog and 6992 voids in the WebSky mock
catalog. We use the filter definition $R_{v}>20h^{-1}\text{Mpc}$,
$\delta_{c}$¡-0.3 and $\delta<-0.05$, following Sánchez et al. (2017).

We followed the DESI LRG target selection strategy (Zhou et al., 2023) in our
overall strategy to populate the most massive halos with mock LRGs. In
particular, we found that splitting the halo catalogs to three redshift bins
and setting the following HOD parameters results in the best match between the
WISE-PS1 data and the WebSky simulation:

  * •

$0.4<z<0.55$: $\log M_{\mathrm{min}}=12.87$, $\log M_{\mathrm{0}}=13.67$

  * •

$0.55<z<0.65$: $\log M_{\mathrm{min}}=12.98$, $\log M_{\mathrm{0}}=13.78$

  * •

$0.65<z<0.76$: $\log M_{\mathrm{min}}=13.06$, $\log M_{\mathrm{0}}=13.69$

Further HOD parameter values were identical in the three redshift bins (with
marginal influence on the results), with $\log M_{\mathrm{1}}=13.9$,
$\sigma_{\mathrm{\mathrm{\log M}}}=0.3$, and $\alpha=1.29$.

As a final step, we added Gaussian photo-$z$ errors to the accurately known
simulated galaxy redshifts, using a $\sigma_{z}\approx 0.02(1+z_{\rm spec})$
scatter that is realistic for the WISE-PS1 LRGs (Beck et al., 2022). When
applying the WISE-PS1 sky mask on the full-sky WebSky simulation, we populated
the halo distribution with about $N_{g}=2.46\times 10^{6}$ mock galaxies, that
is in close agreement with the observed WISE-PS1 data set, as demonstrated in
Figure 2.

Our best-fit version of mock catalog of 3D voids is shown in Figure 3. We note
that our choice to choose 3D voids as a basis of mock-data comparison was
motivated by the availability of more details than in 2D catalogs. The
satellite fraction of the resulting WISE-PS1-like mock LRG sample is about
$5\%$ with $f_{\mathrm{sat}}\approx 0.05$. We found that alternative versions
of the mock LRG catalog with higher satellite fractions ($10\%,20\%$) resulted
in slight mismatches in the void radius and under-density distribution, and
this baseline $5\%$ choice guaranteed the best agreement between mock and
data.

We then also compared the properties of 2D voids in the WebSky and WISE-PS1
data sets. In Figure 4, we show a comparison of the observed and simulated 2D
void catalogs, where the average radius for both catalogs is about
$R_{v}\approx 40h^{-1}\text{Mpc}$, with similar minimum and maximum values.

We carried out the void finding process separately in the North and South
patches, in order to minimize any possible large-scale contamination effect
from different survey data quality in the two main survey areas. Yet, the
agreement in 2D voids is less perfect compared to the 3D voids, and we first
detected a $20\%$ mismatch between WISE-PS1 and WebSky in the total number of
2D voids. We then performed a few tests to increase the level of the agreement
(prior to looking at the lensing signals), including the removal of the large-
scale modes ($\ell<20$) from the galaxy density maps. We found that their
absence reduces the disparity in the number of 2D voids to approximately
$13\%$, suggesting that remnant large-scale systematic effects might still be
present in the WISE-PS1 data set. We note that the typical size of a void is
$\sim 1^{\circ}$, thus the lensing measurements are not significantly affected
by this filtering.

###  3.3 Stacking measurements

Studying cosmic voids individually can be challenging as their lensing signals
are weak and therefore noisy in real-world measurements (Krause et al., 2013).
To address this issue, an established technique is to use stacking methods
(see e.g. Granett et al., 2008; Kovács & Granett, 2015; Hang et al., 2021a),
which nonetheless also requires a large number of voids to reduce the
observational noise and detect a clear void lensing signal.

The stacking method uses square-shaped cut-outs from the CMB $\kappa$ map
aligned with the center of each void, followed by a relative re-scaling of
cut-out images by the known angular size of the voids. Any subsequent
statistic is then expressed in the re-scaled $R/R_{v}$ units (see the inset in
Figure 5 for example).

Beyond the stacking image, one may also measure the radial profile with the
number of bins and the corresponding resolution as free parameters. We set
$\text{N}_{\text{bin}}=11$ and up to $(R/R_{v})=3.5$ (i.e. significantly
beyond the void radius to also capture environmental effects), resulting in
$\Delta(R/R_{v})=0.3$ bin width in our analysis.

###  3.4 Error analysis

The most significant source of uncertainty in the CMB lensing signal of the
stacked cosmic voids comes from the instrumental noise in the _Planck_ data.
In addition, the total error has subdominant contributions from overlap
effects of multiple voids along the same line-of-sight, adding an intrinsic
variation to the measurable signal.

Figure 5:  Left. Stacked CMB $\kappa$ radial profile and image of the Websky
mock and WISE-PS1 catalogs using all 2D voids. The $\kappa$ profiles (with 11
radial bins) show great agreement, with an 6% higher than expected signal at
the center, and a clear negative $\kappa$ value inside of the voids, followed
by a clear detection of the void edges in the lensing signal too. We also find
good agreement when looking at the stacked images, where a random noise map
realisation was added to the WebSky mock for a more realistic impression.
Right. Here we show the best-fit $A_{\kappa}$ parameter values for the
fiducial all-voids case (A${}_{\kappa}=1.05\pm 0.08$, i.e. S/N=13.1), and we
also present our separate the North and South results. We also show how
splitting the void catalogs into two halves at R$=34h^{-1}$Mpc also shows
consistent signals. Finally, we also present our main findings when splitting
the voids into three redshift bins using $\triangle z\approx 0.1$ width, with
a weak trend for stronger signals at higher redshifts.

To factor both error sources, we created 2000 _Planck_ -like noise
realizations ($N_{\kappa}^{i}$) and 2000 WebSky-like CMB $\kappa$ map
realizations ($S_{\kappa}^{i}$). Based on the study by Vielzeuf et al. 2021,
we randomly produced the noise maps using the power spectrum released by
Planck Collaboration 2018a with the synfast routine. Then, we extracted the
power spectrum of the WebSky CMB lensing map using the anafast routine, and
generated random signal maps that are _uncorrelated_ with the voids. These
maps are then added to the _Planck_ -like noise maps
($N_{\kappa}^{i}$+$S_{\kappa}^{i}$). Finally, all the 2000 random maps are
stacked on the positions of the voids to characterize the covariance of our
cross-correlation measurements.

During this procedure we note that the maps were pre-processed before the
stacking measurements, using the same methodology as in the case of the
_Planck_ $\kappa$ map. We removed the fluctuations at the largest scales in
the map ($\ell<20$) and applied a Gaussian smoothing with $FWHM=1^{\circ}$.

Following the above steps, the level of consistency between WISE-PS1 and
WebSky becomes quantifiable, which results in constraints for the best-fitting
CMB lensing amplitude parameter $A_{\kappa}=\kappa_{\scaleto{\rm WISE-
PS1}{4pt}}/\kappa_{\scaleto{\rm WebSky}{4pt}}$ and its corresponding
uncertainty ($\sigma_{A_{\kappa}}$). To measure this parameter, we followed
the analysis steps by Vielzeuf et al. 2021, evaluating the statistic

$\chi^{2}=\sum_{ij}(\kappa_{i}^{\rm WISE-PS1}-A_{\kappa}\cdot\kappa_{i}^{\rm
WebSky})C_{ij}^{-1}(\kappa_{j}^{\rm WISE-PS1}-A_{\kappa}\cdot\kappa_{j}^{\rm
WebSky})$ (6)

where $\kappa_{i}$ is the average CMB lensing signal in a bin radius $i$ and
$C$ is the covariance matrix. By minimizing the $\chi^{2}$ values, we are able
to constrain the best-fitting $A_{\kappa}\pm\sigma_{A_{\kappa}}$, which
characterizes the agreement between the WISE-PS1 and WebSky signals.

##  4 Results & Discussion

In this section, we compare the lensing signal from the WISE-PS1 and WebSky
void catalogs, using 2D and 3D methods. In both analyses, we worked separately
in the North and the South patches (see Figure 1), using a simple weighting
approach to generate the $\kappa$ signal of the entire sky area, based on the
number of voids in each patch. With this procedure, we aimed to determine the
level of agreement between the WISE-PS1 data and the _Planck_ -like
$\Lambda$CDM cosmology used in the WebSky simulation (where an
$A_{\kappa}\approx 1$ indicates a good concordance).

###  4.1 2D voids

Following the above methodology for the 2D void finding algorithm, we detected
6090 voids in the WISE-PS1 data, and we found 6992 voids in the WebSky mock
within the same sky area ($14,200$ $\rm deg^{2}$).

In Figure 5, we present the main results for the 2D void catalogs. We show the
stacked images and the reconstructed lensing profiles for all voids detected
in the WISE-PS1 data set and in the WebSky mock catalog, as well as the best-
fit lensing amplitude results for different bins of redshift, radii, and sky
patches.

Figure 6:  Left. Stacked CMB $\kappa$ profiles measured for 3D voids in the
WebSky and the WISE-PS1 data sets. We generally split the all-voids sample
into voids-in-voids ($\lambda_{v}<-3.72$) and voids-in-clouds
($\lambda>12.58$) bins, which are expected to carry most of the observable
void lensing signal among all $\lambda_{v}$ values. Right. We show the best-
fit $A_{\kappa}$ parameters for different cases. The main results are
$A_{\kappa}\approx 0.81\pm 0.12$ for voids-in-voids and A${}_{\kappa}\approx
0.86\pm 0.12$ for voids-in-clouds, both with $S/N\approx 7$.

As expected, we found a clear negative lensing signal in the void interiors,
although the signal of WISE-PS1 voids is about $8\%$ higher in the centre than
the WebSky-based expectations. At $\rm R/R_{v}>1$ the $\kappa$ signal starts
to be positive (where the compensating over-dense regions is located), marking
the edge of the voids. Far from the centre, the $\kappa$ signal of both the
observational and simulated profiles converge to zero. This trend is also
visible in the stacked images, where we added a random noise map realization
to the WebSky mock image, in order to look more similar to the observational
WISE-PS1 image.

To calculate the agreement between WebSky and WISE-PS1 voids catalogs, we
estimated the best-fit value of the $A_{\kappa}$ parameter, that describes the
ratio of the observed and the expected void lensing profiles. For our fiducial
result using all voids, we obtained $A_{\kappa}\approx 1.06\pm 0.08$, i.e. a
$\textup{S/N}\approx 13.3$ detection of a cross-correlation signal between
voids and CMB lensing convergence.

For further insights, we then performed a similar analysis for different
subsets of the void catalogs. As illustrated in Figure 5, the $A_{\kappa}$
values measured in the North and South WISE-PS1 patches are consistent with
each other. We also applied a radius binning by splitting the void catalog
into two identical halves at $R=34h^{-1}$Mpc. The results again show good
consistency between the larger and smaller voids. Finally, we also split the
$0.42<z<0.7$ void catalog into three redshift bins, with $\triangle z\approx
0.1$. In this test, we found a weak trend with stronger void lensing signals
at the higher redshift bins, but overall all results are broadly consistent
with $A_{\kappa}\approx 1$, as demonstrated in Figure 5. For numerical values
of the best-fit lensing amplitudes, see Table 1 below.

###  4.2 3D voids

Considering the 3D void catalog, we make further selection cuts on the
$\lambda_{v}$ parameter to study the specific sub-classes of voids. Typically
surrounded by other voids, the voids-in-voids exhibit $\lambda_{v}<0$ values,
while the voids-in-clouds tend to be embedded in generally over-dense
environments with $\lambda_{v}>0$, i.e. these two classes also correspond to
distinct CMB lensing imprints (see Raghunathan et al., 2020, for further
details). We note that shallow and small voids are expected to produce weaker
lensing signals than the ones with more extreme $\lambda_{v}$ values, and the
associated noise is also greater due to their smaller angular size.

Therefore, we chose $\lambda_{v}<-3.72$ to study the most significant 20% of
voids-in-voids, or under-compensated void class, while we set
$\lambda_{v}>12.58$ to also define a voids-in-clouds (over-compensated)
sample, with 7430 voids in both subsets. The remaining 60% of the 3D voids
with lower absolute $\lambda_{v}$ values are excluded from our analysis.

The summary of these results is depicted in Figure 6. Unlike the good
consistency with the $A_{\kappa}\approx 1$ expectations for 2D voids, we found
that from the 3D void analysis, most $A_{\kappa}$ values are below unity. For
voids-in-voids ($\lambda_{v}<-3.72$ subset), we obtained $A_{\kappa}\approx
0.81\pm 0.12$, i.e. S/N=6.8, that is broadly consistent with our best-fit
amplitude in the voids-in-clouds regime ($\lambda_{v}>12.58$) with
$A_{\kappa}\approx 0.86\pm 0.12$ (S/N=7.2). We also noticed, however, that the
North and the South patches for these two void types show substantial
differences. The South patch for the $\lambda_{v}>12.58$ sub-sample shows
$A_{\kappa}\approx 0.52\pm 0.15$ (i.e. a rather low S/N=3.3), indicating
possible remnant systematic effects in the input data sets.

As in the case of the 2D voids, we again implemented a redshift-binning
approach using the same bin edges and width. For the voids-in-voids sub-class,
we observed a trend with $A_{\kappa}$ values increasing with redshift, with
higher significance than for the 2D results. However, we do not observe this
apparent trend for voids-in-clouds ($\lambda_{v}>12.58$), which suggests that
this feature is again a consequence of remnant systematic effects in the data.

Table 1: Results from the error analysis for the 2D and the 3D voids with different binning, comparing the Aκ, signal-to-noise and number of voids. 2D voids | $A_{\kappa}\pm\sigma_{\textup{A}_{\kappa}}$ | S/N | $\textup{N}_{\textup{voids}}$
---|---|---|---
All | $1.06\pm 0.08$ | 13.3 | 6090
North Patch | $1.02\pm 0.10$ | 10.2 | 3812
South Patch | $1.13\pm 0.14$ | 8.1 | 2278
$R_{v}<34h^{-1}\textup{Mpc}$ | $1.10\pm 0.12$ | 9.2 | 2936
$R_{v}>34h^{-1}\textup{Mpc}$ | $1.00\pm 0.10$ | 10.0 | 3154
0.42 ¡ z ¡ 0.51 | $0.82\pm 0.14$ | 5.9 | 1998
0.51 ¡ z ¡ 0.61 | $1.10\pm 0.12$ | 9.2 | 1800
0.61 ¡ z ¡ 0.70 | $1.05\pm 0.11$ | 9.5 | 2292
3D voids | $A_{\kappa}\pm\sigma_{\textup{A}_{\kappa}}$ | S/N | $\textup{N}_{\textup{voids}}$
All low ${}_{\lambda_{v}<-3.72}$ | $0.81\pm 0.12$ | 6.8 | 7430
All high ${}_{\lambda_{v}>12.58}$ | $0.86\pm 0.12$ | 7.2 | 7430
North Patch ${}_{\lambda_{v}<-3.72}$ | $0.63\pm 0.15$ | 4.2 | 4749
South Patch ${}_{\lambda_{v}<-3.72}$ | $1.05\pm 0.23$ | 4.6 | 2681
North Patch ${}_{\lambda_{v}>12.58}$ | $0.93\pm 0.16$ | 5.8 | 4572
South Patch ${}_{\lambda_{v}>12.58}$ | $0.52\pm 0.16$ | 3.3 | 2858
0.42 ¡ z${}_{\lambda_{v}<-3.72}$ ¡ 0.51 | $0.44\pm 0.16$ | 2.8 | 2279
0.51 ¡ z${}_{\lambda_{v}<-3.72}$ ¡ 0.61 | $0.73\pm 0.17$ | 4.3 | 2724
0.61 ¡ z${}_{\lambda_{v}<-3.72}$ ¡ 0.70 | $1.20\pm 0.21$ | 5.7 | 2427
0.42 ¡ z${}_{\lambda_{v}>12.58}$ ¡ 0.51 | $0.74\pm 0.17$ | 4.3 | 2512
0.51 ¡ z${}_{\lambda_{v}>12.58}$ ¡ 0.61 | $0.77\pm 0.15$ | 5.1 | 2537
0.61 ¡ z${}_{\lambda_{v}>12.58}$ ¡ 0.70 | $0.82\pm 0.17$ | 4.8 | 2381

##  5 Conclusions

In this paper, we extended the series of CMB lensing $\times$ voids cross-
correlation measurements. Our motivation was to learn more about the dark
matter content of cosmic voids by reducing the statistical uncertainties, and
to possibly learn about the validity of the puzzling ”lensing-is-low” anomaly
(see e.g. Heymans et al., 2021).

We analyzed a WISE-PS1 LRGs sample with 14,200 deg2 sky area, allowing a more
precise cross-correlation measurement than achieved in most previous studies.
We cross-correlated both, 2D and 3D void catalogs with the Planck lensing map,
and thus studied their average $\kappa$ imprint using a stacking methodology,
relying on a mock galaxy catalog as reference, built from the WebSky dark
matter simulation (Stein et al., 2019).

Our fiducial analysis revealed good agreement with the standard $\Lambda$CDM,
finding an $A_{\kappa}\approx 1.06\pm 0.08$ lensing amplitude for 2D voids,
i.e. $S/N\approx 13.3$ (see Figure 5). We did not find significant evidence
for deviations from this main results in the North vs. South patches of the
WISE-PS1 catalog, splitting into large vs small voids, or when binning the
voids into 3 redshift slices (see Table 1). We report that our detection is of
higher $S/N$ than previous studies using voids detected in the DES Year-3 data
set (see e.g. Vielzeuf et al., 2021; Kovács et al., 2022b), and it is
comparable to DESI Legacy Survey results (Hang et al., 2021a).

The 3D void analysis exhibited lower $S/N$ and it agrees less precisely with
the WebSky mock than our findings based on 2D voids. We found
$A_{\kappa}\approx 0.81\pm 0.12$ amplitude for the voids-in-voids subset
($\lambda_{v}<-3.72$) and $A_{\kappa}\approx 0.86\pm 0.12$ for voids-in-clouds
($\lambda_{v}>12.58$), i.e. about $1\sigma$ lower than the expected
$A_{\kappa}\approx 1$ (see Figure 6). However, we detected even lower
amplitudes when we analyzed the North and South patches separately, and
especially when we analyzed different redshift bins in the case of the voids-
in-voids sample (see Table 1). These findings lead us to the conclusion that,
despite the lack of similar trends in the fiducial 2D void analysis, our WISE-
PS1 LRG catalog might feature remnant systematic effects that affect the
outcomes.

The observed mild deviations from $A_{\kappa}\approx 1$ might be attributed to
limitations in our simulation approach, or to imperfections in the WISE-PS1
LRG selection. We thus conclude that the explanation of the observed
moderately significant $2-3\sigma$ tensions in some of our binned void lensing
signals may well be observational effects, rather than cosmological in their
nature.

Overall, we presented a significant detection of a CMB lensing signal
associated with about 7000 cosmic voids at $0.42<z<0.7$, largely consistent
with the concordance model. Future analyses using even larger data sets (e.g.
_Euclid_ , Vera C. Rubin Observatory, Roman, SPHEREx, J-PAS, etc) will provide
further insights into cosmological tensions, and to the nature of voids in
particular.

## Data availability

The WISE-PS1 galaxy catalogues and masks are publicly available as part of the
PS1 DR2 release (using freely available WISE data for the cross-matching), as
well as the WebSky simulation, and also the _Planck_ 2018 CMB lensing map.
Void catalogs and CMB cross-correlation measurement code will be made
available upon reasonable request to the corresponding author.

## Acknowledgments

The Large-Scale Structure (LSS) research group at Konkoly Observatory has been
supported by a _Lendület_ excellence grant by the Hungarian Academy of
Sciences (MTA). This project has received funding from the European Union’s
Horizon Europe research and innovation programme under the Marie Skłodowska-
Curie grant agreement number 101130774. Funding for this project was also
available in part through the Hungarian Ministry of Innovation and Technology
NRDI Office grant OTKA NN129148. The authors thank Conor McPartland and Róbert
Beck for their help with the general WISE-PS1 galaxy catalogs that we used to
select LRGs for our analysis.

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