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Issue
A&A
Volume 689, September 2024
Article Number A171
Number of page(s) 10
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/202348970
Published online 20 September 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Cosmic voids are large under-dense structures that occupy most of the late-time Universe. Unlike other components of the cosmic web (walls, filaments, and galaxy clusters), void interiors are less prone to nonlinear 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 of testing the Lambda cold dark matter (ΛCDM) 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 of constraining the neutrino mass due to their sensitivity to diffuse components (Kreisch et al. 2019; Contarini et al. 2021).

Voids constrain cosmological models through various probes, such as 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. 2022). Moreover, their lensing signals serve as complementary tools for exploring the underlying dark matter distribution, because photons in voids are subject to demagnification effects while traversing them, unlike in overdensities. 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), and that detections of the void lensing signal using stacking methods from large catalogs 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 simulations 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 AISW = ΔTobs/ΔTΛCDM amplitude parameter (the ratio of observed and expected ISW signals) from R ≳ 100 h−1 Mpc voids, or “supervoids”, has often been found to be significantly higher than was expected in the concordance ΛCDM 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 (Aghanim et al. 2020a), moderately significant excess AISW amplitude values (Granett et al. 2008; Cai et al. 2017; Kovács 2018) and good consistency with the ΛCDM 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 the Baryon Oscillation Spectroscopic Survey (BOSS) data to construct void catalogs. The enhanced ISW signals from supervoids are also considered anomalous because two-point correlation analyses do not show significant excess either, compared to ΛCDM predictions (see e.g. Planck Collaboration XXI 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. 2022b). It has been shown, however, that assuming the ΛCDM 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 have also been 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 dataset (Kovács et al. 2022a).

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 cause a demagnification effect, and their stacked signals are expected to show local minima in lensing convergence (see e.g. Raghunathan et al. 2020), estimated from the matter density field, δ(r, θ), via projection as

κ ( θ ) = 3 H 0 2 Ω m 2 c 2 ∫ 0 r max δ ( r , θ ) ( r max − r ) r r max d r $$ \begin{aligned} \kappa (\theta )=\frac{3H_0^2\Omega _m}{2c^2} \int _{0}^{r_{\rm max}} \delta (r,\theta ) \frac{(r_{\rm max}-r)r}{r_{\rm max}}\, \mathrm{d}r \end{aligned} $$(1)

in the Born approximation, with the Hubble constant, H0, and matter density parameter, Ωm, assuming a flat ΛCDM model. The r denotes the co-moving distance to source galaxies in the background of the lenses (with distorted shapes due to lensing), and rmax 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 ΛCDM expectations (Aκ = κdata/κmodel ≈ 1). Measurements using the DES LRG catalogs, however, report a moderately low Aκ ≈ 0.82 ± 0.08 amplitude (Vielzeuf et al. 2021; Kovács et al. 2022c), which is also consistent with the findings of Hang et al. (2021a), who analyzed both voids and superclusters detected in the Dark Energy Spectroscopic Instrument (Dey et al. 2019, DESI) Legacy Survey galaxy dataset, finding an overall Aκ ≈ 0.81 ± 0.06 best-fit amplitude.

In this paper, we perform a novel analysis and identify cosmic voids from the WISE-Pan-STARRS galaxy dataset 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 datasets. 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. Datasets

2.1. WISE-Pan-STARRS galaxies

As tracers of the cosmological large-scale structure (LSS), we used the cross-matched catalog of galaxies from the all-sky Wide-Field Infrared Survey Explorer (WISE, Wright et al. 2010) and 3π 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 broadband 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, and W4) that had effective wavelengths of 3.4, 4.6, 12, and 22 μm, respectively. Regarding the high noise level and the relatively large number of missing error estimates of the W3 and W4 filters, we only considered WISE measurements obtained with the W1 and W2 filters. We then only used W1 information to apply selection cuts among the WISE-PS1 galaxies. The LRG redshifts that we used originate from the work of Beck et al. (2022), who obtained photometric redshifts from a neural network trained on WISE-PS1 data.

Our analysis applied further selection cuts to the WISE-PS1 galaxy catalog and focused on a subset of LRGs. While their exact three-dimensional (3D) position information is not accessible using photo-z data, the expected σz ≈ 0.02(1 + zspec) precision enables 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 ≠ 0 (WISE extended object cut)

  • cc_flags =‘0000’ (WISE cut for flagged data)

Observed magnitudes were corrected for the galactic dust extinction using the related extinction coefficients (αg = 3.172, αr = 2.271, αi = 1.682, αz = 1.322, αy = 1.087, αW1 = 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 <  zphot <  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 extragalactic sources of possible contamination and biases. Our healpix mask (Gorski et al. 2005), created at Nside = 512 resolution, was constructed by the following process:

  • We applied the point source mask based on the Planck 2018 measurements (Aghanim et al. 2020a).

  • 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°; in other words, close to the Milky Way.

  • Pixels near the North Galactic Cap were removed (b >  60°).

The above LRG selection cuts and sky-masking strategy resulted in a sample of about Ng = 2.46 × 106 WISE-PS1 galaxies, distributed in an unmasked area of approximately A = 14 200 deg2; that is, covering about 1/3 of the sky (see Figure 1 for details).

thumbnail Fig. 1.

Planck CMB lensing map (with a FWHM  =  1° Gaussian smoothing) highlighted within the WISE-PS1 sky area, where we performed a robust measurement of voids × CMB κ cross-correlations. The data is divided into two larger patches, labeled as north and south, comprising an unmasked area of approximately A = 14 200 deg2.

2.2. Cosmic microwave background lensing map from the Planck mission

In our LSS×CMB cross-correlation measurements, we relied on the reconstructed CMB lensing convergence (κ) map provided by the Planck Collaboration. This dataset was released in the form of κlm spherical harmonic coefficients (see Aghanim et al. 2020b, for details) up to ℓmax = 2048. We created a κ map at Nside = 512 resolution by converting the κlm 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 Planckκ map, indicating our conservative choices to avoid possibly contaminated regions. Additionally, the Dec > −30° PS1 data-taking limit is also marked.

We note that, while higher-resolution maps can be constructed from the available κℓm coefficients, the chosen Nside = 512 is a sufficient choice given the degree-size angular scales of cosmic voids in our analysis. We also applied a FWHM  =  1° Gaussian smoothing to the convergence map to suppress small-scale noise patterns, following the signal-to-noise (S/N) optimization results of Vielzeuf et al. (2021). Another filtering we implemented was removing the large-scale modes from the κ map (ℓ <  20) to remove any remaining systematic wide-angle effect from the map. Our analyses consistently applied these smoothing and filtering steps to the observed and simulated κ maps. Finally, we also removed a slight remnant bias from the κ map ( κ ¯ ≈ − 10 − 4 $ \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 cosmic microwave background κ map

To create a simulated dataset for calibrating our pipeline, we analyzed the publicly available WebSky simulation1 (Stein et al. 2020), which was constructed assuming a Planck 2018 cosmology (Aghanim et al. 2020a). Based on the peak-patch method, the WebSky simulation encloses a vast total cosmic volume of V ≃ 1900 (Gpc)3 with a mass resolution of about Mmin ≃ 1.2 × 1012M⊙ at 0 <  z <  4. It provides a full-sky light-cone catalog that contains about 9 × 108 dark matter halos (each with RA, Dec, redshift, an initial Lagrangian position [Mpc], final Eulerian position [Mpc], velocity [km/s], and mass [ M 200 ρ ¯ m $ M_{200\overline{\rho}_m} $]). We note that WebSky is not a full N-body simulation, but at the relatively large scales of the voids that we use (scales of a few degrees), the expected differences are at the level of a few percentage points (see Blot et al. 2019, for discussion about WebSky’s performance compared to N-body and other approximate methods).

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 × Planck cross-correlations, we also used the WebSky CMB lensing convergence map in our analyses, which 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 catalog 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 Planckκ 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 followed the methodology of Vielzeuf et al. (2021) and Kovács et al. (2022c), building on their findings regarding, for example, optimization techniques and free parameters in the analysis.

3.1. Void-finding 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 with void-finding 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 makes 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 two-dimensional maps

Our main analysis pipeline uses a two-dimensional (2D) void-finding method, developed by Sánchez et al. (2017) for the analysis of photo-z datasets. The algorithm works by projecting galaxies in 2D 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 h−1 Mpc, which corresponds to about twice the size of typical photo-z errors. Then, we projected the data into healpix maps with Nside = 512 resolution in 7 redshift slices in the 0.42 <  z <  0.7 range covered by the WISE-PS1 LRGs.

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

  • Smoothing scale (σ): this parameter defines the Gaussian smoothing scale applied to the density map to define density minima. Here we employed σ = 10 h−1Mpc (also used by, for example, Kovács et al. 2022b), which enables the reliable identification of the 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 center).

  • Under-density threshold (δmax): in a smoothed density map, this parameter defines the threshold to consider a given local density minimum as a possible void center. Naturally, significant voids correspond to lower δ values, and we again followed Vielzeuf et al. (2021) by setting δmax = −0.2 to eliminate shallow under-densities.

3.1.2. Cosmic voids in three dimensions

To compare with 2D voids detected from the same galaxy catalogs, we used the REVOLVER (REal-space VOid Locations from surVEy Reconstruction) void-finding code2 Nadathur et al. (2019) to also construct a 3D void 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 subclasses may also offer important checks of the fiducial 2D void analyses.

In the work by Nadathur & Crittenden (2016), a dimensionless parameter, denoted as λv, was constructed empirically. This parameter is closely tied to the sign and magnitude of the gravitational potential, and provides information about void density profiles and the large-scale void environment, defined as

λ v ≡ δ ¯ g ( R eff 1 h − 1 Mpc ) 1.2 $$ \begin{aligned} \lambda _v\equiv \overline{\delta }_g\left(\frac{R_{\mathrm{eff} }}{1\;h^{-1}\mathrm{Mpc} }\right)^{1.2} \end{aligned} $$(2)

using the average galaxy density contrast, δ ¯ g = 1 V ∫ V δ g d 3 x $ \overline\delta_g = \frac{1}{V}\int_{V}\delta_g\,\mathrm{d}^3\mathbf{x} $, and the effective spherical radius, R eff = ( 3 4 π V ) 1 / 3 $ 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 the Voronoi cells that make up the void.

We note that two consecutive runs of the ZOBOV pipeline do not result in identical void catalogs. This feature is due to the random process to place buffer particles around the survey volume, in order to control the void identification at the survey’s edges. While differences are small in terms of the total number of voids and the main void parameters, in our analysis we created ten 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.

3.2. Modeling void properties with mock luminous red galaxies

Considering the realistic modeling 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 code3. 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

⟨ N tot ( M ) ⟩ = ⟨ N cen ( M ) ⟩ + ⟨ N sat ( M ) ⟩ . $$ \begin{aligned} \langle N_\mathrm{tot} (M) \rangle = \langle N_\mathrm{cen} (M) \rangle + \langle N_\mathrm{sat} (M) \rangle . \end{aligned} $$(3)

The probability that a dark matter halo, typically a massive one (Mmin ≃ 1013M⊙ halo mass), contains a central LRG is given by the smooth step function

⟨ N cen ( M ) ⟩ = 1 2 [ 1 + erf ( log M − log M min σ log M ) ] , $$ \begin{aligned} \langle N_\mathrm{cen} (M) \rangle = \frac{1}{2} \left[1 + \mathrm{erf} \left( \frac{\log M - \log M_\mathrm{min} }{\sigma _\mathrm \log M } \right) \right], \end{aligned} $$(4)

where the position of this step is set by Mmin. A halo with mass M ≪ Mmin hosts no central galaxy, which then transitions according to the value of the σlogM parameter to a probability of P = 1 for M ≫ Mmin 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,

⟨ N sat ( M ) ⟩ = ( M − M 0 M 1 ) α , $$ \begin{aligned} \langle N_\mathrm{sat} (M) \rangle = \left( \frac{M-M_\mathrm{0} }{M_\mathrm{1} } \right)^{\alpha }, \end{aligned} $$(5)

with α, M0, and M1 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.

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: logMmin = 12.87, logM0 = 13.67

  • 0.55 <  z <  0.65: logMmin = 12.98, logM0 = 13.78

  • 0.65 <  z <  0.76: logMmin = 13.06, logM0 = 13.69.

Further HOD parameter values were identical in the three redshift bins (with a marginal influence on the results), with logM1 = 13.9, σlogM = 0.3, and α = 1.29.

As a final step, we added Gaussian photo-z errors to the accurately known simulated galaxy redshifts, using a σz ≈ 0.02(1 + zspec) 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 Ng = 2.46 × 106 mock galaxies, which is in close agreement with the observed WISE-PS1 dataset, as is demonstrated in Figure 2.

thumbnail Fig. 2.

Comparison of the redshift distribution of LRGs in the WISE-PS1 dataset versus the WebSky mock. We followed an HOD modeling approach to create a realistic model of the source density as a function of redshift, reaching a match within a few percent throughout the whole range.

Our best-fit version of a mock catalog of 3D voids is shown in Figure 3. We note that our choice to use 3D voids as a basis of mock data comparison was motivated by the greater availability of details than in 2D catalogs. The satellite fraction of the resulting WISE-PS1-like mock LRG sample is about 5%, with fsat ≈ 0.05. We find that alternative versions of the mock LRG catalog with higher satellite fractions (10%,20%) result in slight mismatches in the void radius and under-density distribution, and this baseline 5% choice guarantees the best agreement between the mock data and the actual data.

thumbnail Fig. 3.

Void properties in the WISE-PS1 dataset and in the WebSky mock catalog. The void radii (left) and λv 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.

We then also compared the properties of 2D voids in the WebSky and WISE-PS1 datasets. In Figure 4, we show a comparison of the observed and simulated 2D void catalogs, where the average radius for both catalogs is about Rv ≈ 40 h−1 Mpc, with similar minimum and maximum values.

thumbnail Fig. 4.

Comparison of the number of voids present in the WISE-PS1 and WebSky mock catalogs in the north (left) and south (right) regions. The difference in the coverage areas of the northern and southern 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 used the filter definition Rv >  20 h−1 Mpc, δc <  −0.3 and δ <  −0.05, following Sánchez et al. (2017).

We carried out the void-finding process separately in the northern and southern patches in order to minimize any possible large-scale contamination effect from different survey data quality in the two main survey areas. However, the agreement in the 2D voids is less perfect than in 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 (ℓ <  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 large-scale, leftover systematic effects might still be present in the WISE-PS1 dataset. We note that the typical size of a void is ∼1°; 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 κ map aligned with the center of each void, followed by a relative rescaling of cut-out images by the known angular size of the voids. Any subsequent statistic is then expressed in the rescaled R/Rv units (see the inset in Figure 5 for example).

thumbnail Fig. 5.

Left. Stacked CMB κ radial profile and image of the Websky mock and WISE-PS1 catalogs using all 2D voids. The κ profiles (with 11 radial bins) show great agreement, with a 7%-higher-than-expected signal at the center, and a clear negative κ 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 realization was added to the WebSky mock for a more realistic impression. Right. Here we show the best-fit Aκ parameter values for the fiducial all-voids case (Aκ = 1.05 ± 0.08, i.e., S/N  =  13.1), and we also present our separate northern and southern results. We also show how splitting the void catalogs into two halves at R = 34 h−1 Mpc shows consistent signals. Finally, we also present our main findings when splitting the voids into three redshift bins using a △z ≈ 0.1 width.

Beyond the stacking image, one can also measure the radial profile with the number of bins and the corresponding resolution as free parameters. We set Nbin = 11 and up to (R/Rv)=3.5 (i.e., significantly beyond the void radius to also capture environmental effects), resulting in a Δ(R/Rv)=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, a secondary, subdominant error arises from random fluctuations in the κ map, attributed to overlap effects of multiple cosmic structures along the same line of sight. Based on the study by Vielzeuf et al. 2021, we randomly produced Planck-like instrumental noise maps ( N κ i $ N_{\kappa}^i $) with the synfast routine, using the power spectrum released by Aghanim et al. 2020a. Then, we extracted the power spectrum of the WebSky CMB lensing map using the anafast routine, and generated random convergence maps ( S κ i $ S_{\kappa}^i $) to characterize the overlap effects. These power-spectrum-based random maps are by construction Gaussian, which offers a good approximation at the large scales we are interested in. We then added these two different kinds of noise maps ( N κ i + S κ i $ N_{\kappa}^i + S_{\kappa}^i $) to account for both relevant error sources in our measurements.

We created 2000 Planck-like noise realizations ( N κ i $ N_{\kappa}^i $) and 2000 WebSky-like CMB κ map realizations ( S κ i $ S_{\kappa}^i $). Finally, all the 2000 random maps ( N κ i + S κ i $ N_{\kappa}^i + S_{\kappa}^i $) were 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 preprocessed before the stacking measurements, using the same methodology as in the case of the Planckκ map. We removed the fluctuations at the largest scales in the map (ℓ <  20) and applied a Gaussian smoothing with FWHM = 1°.

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 Aκ = κWISE − PS1/κWebSky and its corresponding uncertainty (σAκ). To measure this parameter, we followed the analysis steps by Vielzeuf et al. 2021, evaluating the statistic

χ 2 = ∑ ij ( κ i WISE − PS 1 − A κ · κ i WebSky ) C ij − 1 ( κ j WISE − PS 1 − A κ · κ j WebSky ) , $$ \begin{aligned} \chi ^2 = \sum _{ij} \left(\kappa _i^\mathrm{WISE-PS1}-A_{\kappa }\cdot \kappa _i^\mathrm{WebSky}\right)C_{ij}^{-1}\left(\kappa _j^\mathrm{WISE-PS1}-A_{\kappa }\cdot \kappa _j^\mathrm{WebSky}\right) ,\end{aligned} $$(6)

where κi is the average CMB lensing signal in a bin radius, i, and C is the covariance matrix. By minimizing the χ2 values, we are able to constrain the best-fitting Aκ ± σAκ, which characterizes the agreement between the WISE-PS1 and WebSky signals.

4. Results and 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 northern and the southern patches (see Figure 1), using a simple weighting approach to generate the κ 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 ΛCDM cosmology used in the WebSky simulation (where Aκ ≈ 1 indicates a good concordance).

4.1. Two-dimensional 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 deg2).

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 dataset and in the WebSky mock catalog, as well as the best-fit lensing amplitude results for different bins of redshift, radii, and sky patches.

As was expected, we found a clear negative lensing signal in the void interiors, although the signal of WISE-PS1 voids is about 7% higher in the center than the WebSky-based expectations. At R/Rv >  1, the κ signal starts to be positive (where the compensating over-dense regions are located), marking the edge of the voids. Far from the center, the κ signal of both the observational and simulated profiles converges 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 make it look more similar to the observational WISE-PS1 image.

To calculate the agreement between WebSky and WISE-PS1 void catalogs, we estimated the best-fit value of the Aκ parameter, which describes the ratio of the observed and expected void lensing profiles. For our fiducial result using all voids, we obtained Aκ ≈ 1.06 ± 0.08; that is, a S/N ≈ 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 is illustrated in Figure 5, the Aκ values measured in the northern and southern 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 = 34 h−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 △z ≈ 0.1. In this test, we found a weak trend with stronger void lensing signals at the higher redshift bins, but overall all of the results are broadly consistent with Aκ ≈ 1, as is demonstrated in Figure 5 (for numerical values of the best-fit lensing amplitudes, see Table 1 below).

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.

4.2. Three-dimensional voids

Considering the 3D void catalog, we made further selection cuts on the λv parameter to study the specific subclasses of voids. Typically surrounded by other voids, the voids-in-voids exhibit λv <  0 values, while the voids-in-clouds tend to be embedded in generally over-dense environments with λv >  0; that is, 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 λv values, and the associated noise is also greater due to their smaller angular size.

Therefore, we chose λv <  −3.72 to study the most significant 20% of voids-in-voids, or an under-compensated void class, while we set λv >  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 λv values were excluded from our analysis.

The summary of these results is depicted in Figure 6. Unlike the good consistency with the Aκ ≈ 1 expectations for 2D voids, we found that from the 3D void analysis, most Aκ values are below unity. For voids-in-voids (the λv <  −3.72 subset), we obtained Aκ ≈ 0.81 ± 0.12; in other words, S/N  =  6.8, which is broadly consistent with our best-fit amplitude in the voids-in-clouds regime (λv >  12.58) with Aκ ≈ 0.86 ± 0.12 (S/N  =  7.2). We also notice, however, that the northern and southern patches for these two void types show substantial differences. The southern patch for the λv >  12.58 sub-sample shows Aκ ≈ 0.52 ± 0.15 (i.e., a rather low S/N  =  3.3), indicating possible remnant systematic effects in the input datasets.

thumbnail Fig. 6.

Left. Stacked CMB κ profiles measured for 3D voids in the WebSky and the WISE-PS1 datasets. We generally split the all-voids sample into voids-in-voids (λv <  −3.72) and voids-in-clouds (λ >  12.58) bins, which are expected to carry most of the observable void lensing signal among all λv values. Right. We show the best-fit Aκ parameters for different cases. The main results are Aκ ≈ 0.81 ± 0.12 for voids-in-voids and Aκ ≈ 0.86 ± 0.12 for voids-in-clouds, both with S/N ≈ 7. Splitting the voids into three redshift bins using a △z ≈ 0.1 width for both voids-in-voids and voids-in-clouds presents a weak trend for stronger signals at higher redshifts.

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 subclass, we observe a trend of Aκ values increasing with redshift, with a higher significance than for the 2D results. However, we do not observe this apparent trend for voids-in-clouds (λv >  12.58), which suggests that this feature is again a consequence of remaining systematic effects in the data.

5. Conclusions

In this paper, we extended the series of CMB lensing × 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 a 14 200 deg2 sky area, which enabled us to achieve a more precise cross-correlation measurement than that in most previous studies. We cross-correlated both 2D and 3D void catalogs with the Planck lensing map, and thus studied their average κ imprint using a stacking methodology, relying on a mock galaxy catalog as a reference, built from the WebSky dark matter simulation (Stein et al. 2019).

Our fiducial analysis reveals good agreement with the standard ΛCDM, finding an Aκ ≈ 1.06 ± 0.08 lensing amplitude for 2D voids; that is, S/N ≈ 13.3 (see Figure 5). We do not find significant evidence for deviations from the main results in the northern and southern patches of the WISE-PS1 catalog, when splitting into large versus small voids, or when binning the voids into three redshift slices (see Table 1). We report that our detection is of a higher S/N than previous studies using voids detected in the DES Year 3 dataset (see e.g. Vielzeuf et al. 2021; Kovács et al. 2022c), and that it is comparable to DESI Legacy Survey results (Hang et al. 2021a).

The 3D void analysis exhibits a lower S/N and agrees less precisely with the WebSky mock than our findings based on 2D voids. We find an Aκ ≈ 0.81 ± 0.12 amplitude for the voids-in-voids subset (λv <  −3.72) and Aκ ≈ 0.86 ± 0.12 for voids-in-clouds (λv >  12.58); that is, about 1σ lower than the expected Aκ ≈ 1 (see Figure 6). However, we detected even lower amplitudes when we analyzed the northern and southern 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 Aκ ≈ 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 − 3σ tensions in some of our binned void lensing signals may well be observational effects, rather than cosmological in nature.

Overall, we present 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 datasets (e.g. Euclid, Vera C. Rubin Observatory, Roman, SPHEREx, J-PAS, etc.) will provide further insights into cosmological tensions, and into the nature of voids in particular.

Data availability

The WISE-PS1 galaxy catalogs (https://archive.stsci.edu/hlsp/wise-ps1-strm) 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 (https://mocks.cita.utoronto.ca/data/websky/), and also the Planck 2018 CMB lensing map (https://www.cosmos.esa.int/web/planck/pla). Void catalogs and the CMB cross-correlation measurement code will be made available upon reasonable request to the corresponding author, and they will be shared publicly after a follow-up analysis of other cosmological observables will be finished by our team.

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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All Tables

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.

In the text

All Figures

thumbnail Fig. 1.

Planck CMB lensing map (with a FWHM  =  1° Gaussian smoothing) highlighted within the WISE-PS1 sky area, where we performed a robust measurement of voids × CMB κ cross-correlations. The data is divided into two larger patches, labeled as north and south, comprising an unmasked area of approximately A = 14 200 deg2.
In the text
thumbnail Fig. 2.

Comparison of the redshift distribution of LRGs in the WISE-PS1 dataset versus the WebSky mock. We followed an HOD modeling approach to create a realistic model of the source density as a function of redshift, reaching a match within a few percent throughout the whole range.
In the text
thumbnail Fig. 3.

Void properties in the WISE-PS1 dataset and in the WebSky mock catalog. The void radii (left) and λv 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.
In the text
thumbnail Fig. 4.

Comparison of the number of voids present in the WISE-PS1 and WebSky mock catalogs in the north (left) and south (right) regions. The difference in the coverage areas of the northern and southern 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 used the filter definition Rv >  20 h−1 Mpc, δc <  −0.3 and δ <  −0.05, following Sánchez et al. (2017).
In the text
thumbnail Fig. 5.

Left. Stacked CMB κ radial profile and image of the Websky mock and WISE-PS1 catalogs using all 2D voids. The κ profiles (with 11 radial bins) show great agreement, with a 7%-higher-than-expected signal at the center, and a clear negative κ 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 realization was added to the WebSky mock for a more realistic impression. Right. Here we show the best-fit Aκ parameter values for the fiducial all-voids case (Aκ = 1.05 ± 0.08, i.e., S/N  =  13.1), and we also present our separate northern and southern results. We also show how splitting the void catalogs into two halves at R = 34 h−1 Mpc shows consistent signals. Finally, we also present our main findings when splitting the voids into three redshift bins using a △z ≈ 0.1 width.
In the text
thumbnail Fig. 6.

Left. Stacked CMB κ profiles measured for 3D voids in the WebSky and the WISE-PS1 datasets. We generally split the all-voids sample into voids-in-voids (λv <  −3.72) and voids-in-clouds (λ >  12.58) bins, which are expected to carry most of the observable void lensing signal among all λv values. Right. We show the best-fit Aκ parameters for different cases. The main results are Aκ ≈ 0.81 ± 0.12 for voids-in-voids and Aκ ≈ 0.86 ± 0.12 for voids-in-clouds, both with S/N ≈ 7. Splitting the voids into three redshift bins using a △z ≈ 0.1 width for both voids-in-voids and voids-in-clouds presents a weak trend for stronger signals at higher redshifts.
In the text

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