Dark Energy Survey Year 1 results: The relationship between mass and light around cosmic voids
Y. Fang, N. Hamaus, B. Jain, S. Pandey, G. Pollina, C. Sánchez, A. Kovács, C. Chang, J. Carretero, F. J. Castander, A. Choi, M. Crocce, J. DeRose, P. Fosalba, M. Gatti, E. Gaztañaga, D. Gruen, W. G. Hartley, B. Hoyle, N. MacCrann, J. Prat, M. M. Rau, E. S. Rykoff, S. Samuroff, E. Sheldon, M. A. Troxel, P. Vielzeuf, J. Zuntz, J. Annis, S. Avila, E. Bertin, D. Brooks, D. L. Burke, A. Carnero Rosell, M. Carrasco Kind, R. Cawthon, L. N. da Costa, J. De Vicente, S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, S. Everett, A. E. Evrard, B. Flaugher, J. Frieman, J. García-Bellido, D. W. Gerdes, R. A. Gruendl, G. Gutierrez, D. L. Hollowood, D. J. James, M. Jarvis, N. Kuropatkin, O. Lahav, M. A. G. Maia, J. L. Marshall, P. Melchior, F. Menanteau, R. Miquel, A. Palmese, A. A. Plazas, A. K. Romer, A. Roodman, E. Sanchez, S. Serrano, I. Sevilla-Noarbe, M. Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, M. E. C. Swanson, G. Tarle, D. Thomas, V. Vikram, A. R. Walker, J. Weller
DDES-2018-0413FERMILAB-PUB-19-374-AE
MNRAS , 1–16 (2019) Preprint 12 November 2019 Compiled using MNRAS L A TEX style file v3.0
Dark Energy Survey Year 1 results: The relationshipbetween mass and light around cosmic voids
Y. Fang, (cid:63) N. Hamaus, † B. Jain, S. Pandey, G. Pollina, C. S´anchez, A. Kov´acs, , , C. Chang, , J. Carretero, F. J. Castander, , A. Choi, M. Crocce, , J. DeRose, , P. Fosalba, , M. Gatti, E. Gazta˜naga, , D. Gruen, , , W. G. Hartley, , B. Hoyle, , N. MacCrann, , J. Prat, M. M. Rau, E. S. Rykoff, , S. Samuroff, E. Sheldon, M. A. Troxel, P. Vielzeuf, J. Zuntz, J. Annis, S. Avila, E. Bertin, , D. Brooks, D. L. Burke, , A. Carnero Rosell, , M. Car-rasco Kind, , R. Cawthon, L. N. da Costa, , J. De Vicente, S. Desai, H. T. Diehl, J. P. Dietrich, , P. Doel, S. Everett, A. E. Evrard, , B. Flaugher, J. Frieman, , J. Garc´ıa-Bellido, D. W. Gerdes, , R. A. Gruendl, , G. Gutierrez, D. L. Hollowood, D. J. James, M. Jarvis, N. Kuropatkin, O. Lahav, M. A. G. Maia, , J. L. Marshall, P. Melchior, F. Menanteau, , R. Miquel, , A. Palmese, A. A. Plazas, A. K. Romer, A. Roodman, , E. Sanchez, S. Serrano, , I. Sevilla-Noarbe, M. Smith, M. Soares-Santos, F. Sobreira, , E. Suchyta, M. E. C. Swanson, G. Tarle, D. Thomas, V. Vikram, A. R. Walker, andJ. Weller , , (The DES Collaboration) Author affiliations are listed at the end of the paper
Accepted 2019 October 01. Received 2019 September 24; in original form 2019 September 02
ABSTRACT
What are the mass and galaxy profiles of cosmic voids? In this paper we use twomethods to extract voids in the Dark Energy Survey (DES) Year 1 redMaGiC galaxysample to address this question. We use either 2D slices in projection, or the 3D dis-tribution of galaxies based on photometric redshifts to identify voids. For the massprofile, we measure the tangential shear profiles of background galaxies to infer theexcess surface mass density. The signal-to-noise ratio for our lensing measurementranges between 10.7 and 14.0 for the two void samples. We infer their 3D densityprofiles by fitting models based on N-body simulations and find good agreement forvoid radii in the range 15-85 Mpc. Comparison with their galaxy profiles then allowsus to test the relation between mass and light at the 10%-level, the most stringent testto date. We find very similar shapes for the two profiles, consistent with a linear rela-tionship between mass and light both within and outside the void radius. We validateour analysis with the help of simulated mock catalogues and estimate the impact ofphotometric redshift uncertainties on the measurement. Our methodology can be usedfor cosmological applications, including tests of gravity with voids. This is especiallypromising when the lensing profiles are combined with spectroscopic measurements ofvoid dynamics via redshift-space distortions.
Key words: large-scale structure of Universe – cosmology: observations – gravita-tional lensing: weak (cid:63)
Corresponding author: [email protected] † Corresponding author: [email protected]
Cosmic voids are the most underdense regions of the Uni-verse and constitute its dominant volume fraction. Unlikecollapsed structures, which are strongly affected by non-linear gravitational effects and galaxy formation physics, © a r X i v : . [ a s t r o - ph . C O ] N ov DES Collaboration cosmic voids feature less non-linear dynamics (e.g., Hamauset al. 2014a) and are marginally affected by baryons (e.g.,Paillas et al. 2017). This suggests voids to be particu-larly clean probes for constraining cosmological parame-ters, which has already been exploited in the recent lit-erature (e.g. Sutter et al. 2012; Hamaus et al. 2016; Maoet al. 2017). Observational studies on cosmic voids haveseen a rapid increase in recent years, leading to the discov-ery of the uncharted cosmological signals they carry. Theserange from weak lensing (WL) imprints (e.g., Melchior et al.2014; Clampitt & Jain 2015; S´anchez et al. 2017), over theintegrated Sachs-Wolfe (ISW) effect (e.g., Granett et al.2008; Nadathur & Crittenden 2016; Cai et al. 2017; Kov´acset al. 2019), the Sunyaev-Zel’dovich (SZ) effect (Alonsoet al. 2018), to baryon acoustic oscillations (BAO) (Ki-taura et al. 2016), the Alcock-Paczy´nski (AP) effect (e.g.,Sutter et al. 2012, 2014b; Hamaus et al. 2014c, 2016; Maoet al. 2017; Correa et al. 2019) and redshift-space distortions(RSD) (e.g., Paz et al. 2013; Hamaus et al. 2015, 2017; Caiet al. 2016; Achitouv et al. 2017; Hawken et al. 2017). More-over, the intrinsically low-density environments that cosmicvoids provide make them ideal testbeds for theories of mod-ified gravity. It has been shown that Chameleon models pre-dict repulsive and stronger fifth forces inside voids, such thatthe abundance of large voids can be much higher and theircentral density lower than in Λ CDM (Li et al. 2012; Clampittet al. 2013; Zivick et al. 2015; Cai et al. 2015; Falck et al.2015; Achitouv 2016; Falck et al. 2018; Perico et al. 2019).Thus, gravitational lensing by voids opens up the possibilityto probe the distribution of mass inside those low-densityenvironments (Krause et al. 2013; Higuchi et al. 2013) andfurnishes a promising tool to test modified gravity (Barreiraet al. 2015; Baker et al. 2018).However, ‘generic low-density regions in the Universe’is far from a precise definition of cosmic voids. There isno unique prescription of how to determine the boundaryof such regions, especially when considering sparsely dis-tributed tracers of the large-scale structure, such as galax-ies, to identify voids (Sutter et al. 2014a). A considerablenumber of void finding algorithms based on different oper-ative void definitions have been developed and tested overthe last decade. To name a few, Padilla et al. (2005) intro-duced a method to identify spherical volumes with particle-density contrasts below a particular threshold, Lavaux &Wandelt (2010) use Lagrangian orbit reconstruction andRicciardelli et al. (2013) exploit the velocity divergence oftracer fields to obtain a dynamical void definition. Anotherpopular method involves Voronoi tessellations of tracer par-ticles to construct density fields, combined with the water-shed transform to define a void hierarchy (Platen et al. 2007;Neyrinck 2008; Sutter et al. 2015). Furthermore, Delaunaytesselations have been used to identify empty spheres intracer distributions (Zhao et al. 2016). Colberg et al. (2008)compared a total of 13 void finders identifying voids from theMillennium simulation. More recent studies by Cautun et al.(2018) and Paillas et al. (2019) compared various void defi-nitions, focussing on their potential to differentiate betweeneither Chameleon-, or Vainshtein-type modified gravity and Λ CDM via weak lensing. But not only discrete tracer distri-butions have been considered for this purpose, as demon-strated by Davies et al. (2018, 2019) using weak-lensing maps and by Krolewski et al. (2018) using the Lyman- α forest to identify voids.Most of the above void finders have either been ap-plied to simulations, or galaxy survey data with spectro-scopic redshifts (spec-z), where the precise positions of trac-ers are available in 3D. However, spectroscopic surveys like2dF (Colless et al. 2001) or BOSS (Dawson et al. 2013)are expensive in terms of observational time. The result-ing galaxy catalogues typically contain less objects than theones obtained with photometric surveys and may further suf-fer from selection effects, incompleteness and limited depth.Conversely, photometric surveys like HSC (Miyazaki et al.2012), KiDS (de Jong et al. 2013) or DES (Flaugher et al.2015; Dark Energy Survey Collaboration et al. 2016), whichare more efficient, more complete and deeper, can only pro-vide photometric redshifts (photo-z) that are less precise.Therefore, in order to use photo-z galaxies as void tracers,the redshift dispersion along the line of sight (LOS) must bedealt with very carefully.Because of this limitation, void finders for the identi-fication of circular under-densities in 2D projected galaxymaps have been the preferred choice in weak-lensing stud-ies on cosmic voids (Clampitt & Jain 2015; S´anchez et al.2017). For example, S´anchez et al. (2017) employed a tech-nique that splits the sample of tracer galaxies into 2D to-mographic photo-z bins with a width of at least twice thetypical photo-z scatter. These projected maps are then usedto identify voids in 2D as lenses, and to measure the tangen-tial shear of the background galaxies as a function of theirprojected distance to the void centres. A related approachhas used projections of the entire photo-z distribution tostudy troughs in the so obtained 2D density map (Gruenet al. 2016, 2018; Friedrich et al. 2018; Brouwer et al. 2018).Gruen et al. (2016) and Brouwer et al. (2018) also study2D voids tomographically, by splitting the tracer galaxiesinto two redshift bins and defining troughs as a function ofredshift.In this work, we explore the impact of photo-z scatteron watershed-type void finders in 3D, both for the measure-ment of projected two-point correlations between voids andgalaxies, as well as for weak-lensing imprints from voids.Based on hydrodynamical simulations, recent work by Pol-lina et al. (2017) has shown that these two statistics areclosely connected to each other. They find that the tracer-density contrast around voids can be related to the voidmatter-density profile (which is responsible for gravitationallensing) by a single multiplicative constant b slope that coin-cides with the large-scale linear tracer bias for the largestvoids in the measurement; for smaller voids this constantattains higher values, but remains independent of scale. Thesame conclusion has recently been drawn regarding the rela-tive bias between clusters and galaxies around voids in Pol-lina et al. (2019), who partly analyzed the same data thatare used in this work.Understanding the tracer bias around voids is crucialfor many other cosmological tests involving voids, for exam-ple when modeling their abundance (Jennings et al. 2013;Chan et al. 2014; Pisani et al. 2015; Achitouv et al. 2015;Ronconi & Marulli 2017; Ronconi et al. 2019; Contarini et al.2019; Verza et al. 2019), or RSDs (Hamaus et al. 2015, 2016,2017; Cai et al. 2016; Chuang et al. 2017; Achitouv et al.2017; Hawken et al. 2017; Achitouv 2019; Correa et al. 2019). MNRAS000
Cosmic voids are the most underdense regions of the Uni-verse and constitute its dominant volume fraction. Unlikecollapsed structures, which are strongly affected by non-linear gravitational effects and galaxy formation physics, © a r X i v : . [ a s t r o - ph . C O ] N ov DES Collaboration cosmic voids feature less non-linear dynamics (e.g., Hamauset al. 2014a) and are marginally affected by baryons (e.g.,Paillas et al. 2017). This suggests voids to be particu-larly clean probes for constraining cosmological parame-ters, which has already been exploited in the recent lit-erature (e.g. Sutter et al. 2012; Hamaus et al. 2016; Maoet al. 2017). Observational studies on cosmic voids haveseen a rapid increase in recent years, leading to the discov-ery of the uncharted cosmological signals they carry. Theserange from weak lensing (WL) imprints (e.g., Melchior et al.2014; Clampitt & Jain 2015; S´anchez et al. 2017), over theintegrated Sachs-Wolfe (ISW) effect (e.g., Granett et al.2008; Nadathur & Crittenden 2016; Cai et al. 2017; Kov´acset al. 2019), the Sunyaev-Zel’dovich (SZ) effect (Alonsoet al. 2018), to baryon acoustic oscillations (BAO) (Ki-taura et al. 2016), the Alcock-Paczy´nski (AP) effect (e.g.,Sutter et al. 2012, 2014b; Hamaus et al. 2014c, 2016; Maoet al. 2017; Correa et al. 2019) and redshift-space distortions(RSD) (e.g., Paz et al. 2013; Hamaus et al. 2015, 2017; Caiet al. 2016; Achitouv et al. 2017; Hawken et al. 2017). More-over, the intrinsically low-density environments that cosmicvoids provide make them ideal testbeds for theories of mod-ified gravity. It has been shown that Chameleon models pre-dict repulsive and stronger fifth forces inside voids, such thatthe abundance of large voids can be much higher and theircentral density lower than in Λ CDM (Li et al. 2012; Clampittet al. 2013; Zivick et al. 2015; Cai et al. 2015; Falck et al.2015; Achitouv 2016; Falck et al. 2018; Perico et al. 2019).Thus, gravitational lensing by voids opens up the possibilityto probe the distribution of mass inside those low-densityenvironments (Krause et al. 2013; Higuchi et al. 2013) andfurnishes a promising tool to test modified gravity (Barreiraet al. 2015; Baker et al. 2018).However, ‘generic low-density regions in the Universe’is far from a precise definition of cosmic voids. There isno unique prescription of how to determine the boundaryof such regions, especially when considering sparsely dis-tributed tracers of the large-scale structure, such as galax-ies, to identify voids (Sutter et al. 2014a). A considerablenumber of void finding algorithms based on different oper-ative void definitions have been developed and tested overthe last decade. To name a few, Padilla et al. (2005) intro-duced a method to identify spherical volumes with particle-density contrasts below a particular threshold, Lavaux &Wandelt (2010) use Lagrangian orbit reconstruction andRicciardelli et al. (2013) exploit the velocity divergence oftracer fields to obtain a dynamical void definition. Anotherpopular method involves Voronoi tessellations of tracer par-ticles to construct density fields, combined with the water-shed transform to define a void hierarchy (Platen et al. 2007;Neyrinck 2008; Sutter et al. 2015). Furthermore, Delaunaytesselations have been used to identify empty spheres intracer distributions (Zhao et al. 2016). Colberg et al. (2008)compared a total of 13 void finders identifying voids from theMillennium simulation. More recent studies by Cautun et al.(2018) and Paillas et al. (2019) compared various void defi-nitions, focussing on their potential to differentiate betweeneither Chameleon-, or Vainshtein-type modified gravity and Λ CDM via weak lensing. But not only discrete tracer distri-butions have been considered for this purpose, as demon-strated by Davies et al. (2018, 2019) using weak-lensing maps and by Krolewski et al. (2018) using the Lyman- α forest to identify voids.Most of the above void finders have either been ap-plied to simulations, or galaxy survey data with spectro-scopic redshifts (spec-z), where the precise positions of trac-ers are available in 3D. However, spectroscopic surveys like2dF (Colless et al. 2001) or BOSS (Dawson et al. 2013)are expensive in terms of observational time. The result-ing galaxy catalogues typically contain less objects than theones obtained with photometric surveys and may further suf-fer from selection effects, incompleteness and limited depth.Conversely, photometric surveys like HSC (Miyazaki et al.2012), KiDS (de Jong et al. 2013) or DES (Flaugher et al.2015; Dark Energy Survey Collaboration et al. 2016), whichare more efficient, more complete and deeper, can only pro-vide photometric redshifts (photo-z) that are less precise.Therefore, in order to use photo-z galaxies as void tracers,the redshift dispersion along the line of sight (LOS) must bedealt with very carefully.Because of this limitation, void finders for the identi-fication of circular under-densities in 2D projected galaxymaps have been the preferred choice in weak-lensing stud-ies on cosmic voids (Clampitt & Jain 2015; S´anchez et al.2017). For example, S´anchez et al. (2017) employed a tech-nique that splits the sample of tracer galaxies into 2D to-mographic photo-z bins with a width of at least twice thetypical photo-z scatter. These projected maps are then usedto identify voids in 2D as lenses, and to measure the tangen-tial shear of the background galaxies as a function of theirprojected distance to the void centres. A related approachhas used projections of the entire photo-z distribution tostudy troughs in the so obtained 2D density map (Gruenet al. 2016, 2018; Friedrich et al. 2018; Brouwer et al. 2018).Gruen et al. (2016) and Brouwer et al. (2018) also study2D voids tomographically, by splitting the tracer galaxiesinto two redshift bins and defining troughs as a function ofredshift.In this work, we explore the impact of photo-z scatteron watershed-type void finders in 3D, both for the measure-ment of projected two-point correlations between voids andgalaxies, as well as for weak-lensing imprints from voids.Based on hydrodynamical simulations, recent work by Pol-lina et al. (2017) has shown that these two statistics areclosely connected to each other. They find that the tracer-density contrast around voids can be related to the voidmatter-density profile (which is responsible for gravitationallensing) by a single multiplicative constant b slope that coin-cides with the large-scale linear tracer bias for the largestvoids in the measurement; for smaller voids this constantattains higher values, but remains independent of scale. Thesame conclusion has recently been drawn regarding the rela-tive bias between clusters and galaxies around voids in Pol-lina et al. (2019), who partly analyzed the same data thatare used in this work.Understanding the tracer bias around voids is crucialfor many other cosmological tests involving voids, for exam-ple when modeling their abundance (Jennings et al. 2013;Chan et al. 2014; Pisani et al. 2015; Achitouv et al. 2015;Ronconi & Marulli 2017; Ronconi et al. 2019; Contarini et al.2019; Verza et al. 2019), or RSDs (Hamaus et al. 2015, 2016,2017; Cai et al. 2016; Chuang et al. 2017; Achitouv et al.2017; Hawken et al. 2017; Achitouv 2019; Correa et al. 2019). MNRAS000 , 1–16 (2019)
ES Y1 void lensing Thanks to the state-of-the-art DES Year 1 (Y1) shear cata-logue (Zuntz et al. 2018), we have access to the lensing signalby both 2D and 3D voids with unprecedented accuracy. Thisenables us to test the linearity of tracer bias around voidsby comparing their mass- and galaxy-density profiles, andwhether it is affected by the choice of void definition.This paper is organised as follows: in Section 2 we de-scribe the data and mocks used for this work, in Section 3 webriefly introduce the employed void finding algorithms (both2D and 3D). Section 4 outlines our methods for obtaininggalaxy-density and weak-lensing profiles from the availabledata. In Section 5 the detailed measurements are presentedand tests on the impact of photo-z scatter on our results from3D voids are performed. We further discuss the relation be-tween void density profiles from galaxy clustering and weaklensing, and examine the behaviour of galaxy bias aroundvoids. Finally, we summarize our results in Section 6.
The Dark Energy Survey (DES) is a photometric survey thathas recently finished observing 5000 sq. deg. of the southernhemisphere to a depth of r > , imaging about 300 milliongalaxies in 5 broadband filters (grizY) up to redshift z = . .In this work, we use data from a large contiguous region of1321 sq. deg. of DES Y1 observations, reaching a limitingmagnitude of about 23 in the r -band (with a mean of 3exposures out of the planned 10 for the full survey). The tracer galaxies used to identify voids in this work are asubset of the DES Y1 Gold catalogue (Drlica-Wagner et al.2018) selected by redMaGiC (red-sequence Matched-filterGalaxy Catalogue, Rozo et al. 2016), an algorithm used toprovide a sample of Luminous Red Galaxies (LRGs) withexcellent photo-z performance. It obtains a median bias of | z spec − z photo | ≈ . , and a scatter of σ z /( + z ) (cid:39) . . The redMaGiC algorithm selects galaxies above some luminositythreshold based on how well they fit a red-sequence templatethat is calibrated using redMaPPer (Rozo et al. 2015) anda subset of galaxies with spectroscopic redshifts (see Rozoet al. 2016, for a list of external survey data used). The cutoffin the goodness of fit to the template is imposed as a func-tion of redshift and adjusted such that a constant comovingdensity of galaxies is maintained.In Pollina et al. (2019), both redMaGiC galaxies, as wellas redMaPPer clusters have been considered as void tracers.Although clusters ensure a more robust void identification(more specifically, the void-size function identified by clus-ters has been shown to be only mildly affected by photo-z scatter), in this work we are interested in optimizing thelensing signal. For this purpose we have chosen the high den-sity sample (brighter than 0.5 L ∗ and density − h Mpc − )of redMaGiC galaxies as tracers to identify voids. Thesegalaxies are spread from z min (cid:39) . to z max (cid:39) . in red-shift space. We found that voids traced in this manner havedisplayed a significantly stronger lensing signal than voidstraced by redMaPPer clusters. In Section 5.1.1 we argue thatthis is partly due to the lower bias of redMaGiC galaxies,allowing access to deeper voids in the matter-density field, and partly a selection bias in the void sample caused by LOSsmearing in photometric redshifts. For measuring image distortions caused by gravitationallensing we use metacalibration (Huff & Mandelbaum2017; Sheldon & Huff 2017), a recently developed methodto accurately measure weak-lensing shear without using anyprior information about galaxy properties or calibrationfrom simulations. The method involves distorting the imagewith a small known shear, and calculating the response of ashear estimator to the distorted image. It can be applied toany shear estimation pipeline. For the catalogue used in thiswork it has been applied to the ngmix shear pipeline (Shel-don 2014), which uses sums of Gaussians to approximategalaxy profiles in the riz bands to measure the ellipticitiesof galaxies (Zuntz et al. 2018). Multiband ( griz ) photome-try is used to estimate the galaxy redshifts in DES. A mod-ified version of the Bayesian Photometric Redshifts (BPZ)code is applied on measurements of multiband fluxes to ob-tain the fiducial photometric redshifts used in this work (seeHoyle et al. (2018) and Drlica-Wagner et al. (2018) for moredetails). We ignore systematic errors in the source redshiftcalibration, which is justified by the significance of our mea-surements and the small calibration uncertainties. The final metacalibration catalogue consists of 35 million galaxyshape estimates up to photometric redshift z = . We haveonly used source galaxies with mean redshifts higher than0.55 in this study. Aside from the data samples presented above, the redMaGiC algorithm has also been run on a mock catalogue fromthe MICE2 simulation project. The
MICE Grand Chal-lenge (MICE-GC Fosalba et al. 2015b) is an all-sky lightcone N -body simulation evolving dark-matter particles ina ( / h ) comoving volume, assuming a flat concordance Λ CDM cosmology with Ω m = . , Ω Λ = . , Ω b = . , n s = . , σ = . and h = . . The resulting mock cat-alogue includes extensive galaxy and lensing properties for ∼ million galaxies over 5000 sq. deg. up to a redshift z = . (Crocce et al. 2015; Fosalba et al. 2015a; Carreteroet al. 2015). Photometric redshift errors and error distribu-tions are modelled according to the redMaGiC algorithmby fitting every synthetic galaxy to a red-sequence tem-plate (Rozo et al. 2016). The simulated dark matter light-cones are divided into sets of all-sky concentric sphericalshells. Instead of applying a computationally expensive ray-tracing algorithm, the all-sky lensing maps are approximatedby a discrete sum of projected 2D dark matter density mapsmultiplied by the appropriate lensing weights. In this section we introduce the void finding algorithms ap-plied to DES data and mocks. As briefly mentioned above, https://github.com/esheldon/ngmixMNRAS , 1–16 (2019) DES Collaboration we employ one void finder that traces voids in 2D projec-tions of the tracer-density field (2D voids), and a second onethat identifies voids in all three dimensions (3D voids).
We employ the 2D void finding algorithm describedin S´anchez et al. (2017), which is similar to that utilizedby Clampitt & Jain (2015). This void finder identifies under-densities in 2D galaxy-density fields, which are constructedby projecting galaxies in redshift slices. We use relativelythick redshift shells of width
100 Mpc / h to minimize the ef-fect of photo-z scatter. This choice has proven to be optimalin previous studies, because it amounts to at least twice thetypical photo-z scatter in DES. The algorithm implementsthe following steps (see S´anchez et al. 2017, for more details):(i) It projects tracer galaxies in a redshift slice of giventhickness into a HEALpix map (G´orski & Hivon 2011). Thesetting is kept the same as in S´anchez et al. (2017): N side = , which corresponds to an angular resolution of 0.1 deg.(ii) For each slice, it divides the map by its mean tracerdensity and subtracts unity to obtain a density-contrastmap. The latter is then smoothed with a Gaussian filterwith comoving smoothing scale σ s =
10 Mpc / h .(iii) The most underdense pixel in the smoothed map ofeach slice is identified as the first void centre. Then a circle ofradius R v is grown around the void centre until the densityinside it reaches the mean density.(iv) All pixels within this circle are now removed fromthe list of potential void centres. Steps (iii) and (iv) arerepeated until all pixels below some density threshold haveeither been identified as a void centre, or removed.(v) Finally, the resulting void catalogue is pruned by join-ing voids in neighboring redshift slices that are angularlyclose. More specifically, two voids in neighbouring slices willbe grouped together, if the angular separation between theircenters is smaller than half the mean angular radii of thetwo voids. Meanwhile, voids extending beyond the surveyedge will be cut out from the final catalogue. We discardthose that contain a significantly lower number density ofmasked random points than average, which indicates an in-tersection with survey boundaries (Clampitt & Jain 2015;S´anchez et al. 2017). In order to identify voids in 3D, we use the publicly availableVoid IDentification and Examination toolkit ( vide , Sutteret al. 2015), which is a wrapper for an enhanced version ofZOnes Bordering On Voidness ( zobov , Neyrinck 2008). vide provides functionality for the identification of voids from realobservations, while zobov was originally intended for void-finding in simulations with periodic boundary conditions.The algorithm can be summarized by the following steps:(i) A Voronoi tessellation is applied to the entire tracerdistribution in 3D. This procedure assigns a unique Voronoicell around each tracer particle, delineating the region closerto it than to any other particle. The density of any locationin each cell is calculated as the inverse of its cell volume.(ii) Density minima in the Voronoi density field are found. R v [Mpc /h ] nu m b e r d e n s i t y ( M p c / h ) −
2D voids3D voids
Figure 1.
Distribution of comoving effective void radii in theDES Y1 void catalogues. 2D voids are identified using projectedredshift slices of thickness
100 Mpc / h and 3D voids are found withthe watershed algorithm vide . The vertical lines indicate the binedges we use to divide our void catalogues into sub-samples. A density minimum is located at the tracer particle with aVoronoi cell larger than all its adjacent cells.(iii) Starting from a density minimum, the algorithmjoins together adjacent cells with increasing density untilno higher-density cell can be found. The resulting basins aredenoted as zones , local depressions in the density field.(iv) A watershed transform (Platen et al. 2007) is per-formed to join zones into larger voids, and to define a hier-archy of voids and sub-voids. To prevent voids from growinginto very overdense structures, we set a density thresholdabove which the merging of two zones is stopped (Neyrinck2008): the ridge between any two zones has to be lower than20% of the average tracer density.(v) Each void is assigned an effective radius R v of a sphereof the same total void volume. Void centres are defined asvolume-weighted barycentres of all Voronoi cells that makeup each void. Applying the void finding algorithms to the DES Y1 red-MaGiC sample of galaxies, we find a total of 443 2D voidsand 4754 3D voids between z = . and z = . . We discardvoids outside this range to avoid the redshift boundariesof the redMaGiC sample. Figure 1 shows the effective voidradius distributions for both void catalogues. Note that thetwo void samples are not expected to yield similar size distri-butions, due to their different definition criteria. We divideeach catalogue into 3 sub-samples based on the effective ra-dius. For 2D voids we define three bins: R v = −
40 Mpc / h , R v = −
60 Mpc / h , and R v = −
120 Mpc / h , each bin of in-creasing R v has 267, 100, and 76 voids. For 3D voids we alsodefine three bins: R v = −
20 Mpc / h , R v = −
30 Mpc / h ,and R v = −
60 Mpc / h , each bin of increasing R v has 2214,1873, and 667 voids (see table 1 for a summary). The binedges have been chosen so as to obtain reasonable statisticsfor the available range of effective void radii in each bin. MNRAS000
60 Mpc / h , each bin of increasing R v has 2214,1873, and 667 voids (see table 1 for a summary). The binedges have been chosen so as to obtain reasonable statisticsfor the available range of effective void radii in each bin. MNRAS000 , 1–16 (2019)
ES Y1 void lensing Table 1.
Summary of DES Y1 void sample properties.bin 1 bin 2 bin 3 all bins2Dvoids R v [ Mpc / h ] R v [ Mpc / h ] With the void catalogues at hand, we are ready to measurethe tangential shear, as well as the galaxy density contrastaround voids in DES. A measurement of the lensing sig-nal allows us to validate the ability of the employed voidfinders to identify underdense regions in the matter distri-bution of the Universe. It furthermore provides us with thenecessary information to constrain the radial mass-densityprofiles of voids. In this section, we present our methodol-ogy for obtaining the lensing measurement, an estimate ofits covariance, and the measurement of the clustering signalof galaxies around voids.
The tangential shear γ + of background galaxies (sources) in-duced by voids (lenses) is a direct probe of the excess surfacemass density ∆Σ around voids, defined as ∆Σ ( r p / R v ) ≡ Σ ( < r p / R v ) − Σ ( r p / R v ) = Σ crit γ + ( r p / R v ) , (1)where Σ ( < r p ) = r p ∫ r p r (cid:48) p Σ ( r (cid:48) p ) d r (cid:48) p (2)is the average surface mass density enclosed inside a circleof projected radius r p from the void centre. Distances areexpressed in units of effective void radius R v and the criticalsurface mass density is given by Σ crit = c π G D A ( z s ) D A ( z l ) D A ( z l , z s ) , (3)with comoving angular diameter distance D A and the lensand source redshifts z l and z s , respectively. Note that Σ − ( z l , z s ) = for z s < z l . All distances and densities aregiven in comoving coordinates assuming a flat ΛCDM cos-mology with Ω m = . (for the mocks we use the in-put cosmology with Ω m = . ). We apply inverse-varianceweights (Sheldon et al. 2004; Mandelbaum et al. 2013) andfollow the approach of McClintock et al. (2019) to estimateour lensing observable via ∆Σ ( + , ×) ( r p / R v ) = (cid:205) ls Σ − ( z l , (cid:104) z s (cid:105)) γ ( + , ×) , ls ( r p / R v ) (cid:205) ls Σ − ( z l , (cid:104) z s (cid:105)) (cid:0) R γ, s + (cid:104) R sel (cid:105) (cid:1) (4)where ( + , ×) denotes the two possible components of theshear: tangential and cross. The sum runs over all lens-source pairs ls in the radial bin r p / R v , and we require themean of the source photo-z distribution per galaxy to obey (cid:104) z s (cid:105) > z l + . . Note that for the DES Y1 data, we are using the metacalibration shear catalogue (Huff & Mandelbaum2017; Sheldon & Huff 2017), so we need to apply responsecorrections, namely the shear response R γ and selection re-sponse R sel to the shear statistics as described in McClintocket al. (2019). In essence we stack the excess surface mass den-sities of all voids within the redshift range of . ≤ z l ≤ . to obtain an average ∆Σ profile at an effective lens redshift of (cid:104) z l (cid:105) = . . This is a reasonable approximation, given thatthe density profile of voids in simulations does not evolvemuch within the considered redshift range (Hamaus et al.2014a). To estimate the covariance of our lensing measurement, weperform a void-by-void jackknife resampling technique asdescribed in S´anchez et al. (2017). We therefore repeat ourmeasurement N v times (the number of voids in our sample),each time omitting one void in turn to obtain N v jackkniferealizations. The covariance of the measurement is thereforegiven by C ( ∆Σ i , ∆Σ j ) = N v − N v × N v (cid:213) k = (cid:16) ∆Σ ki − (cid:104) ∆Σ i (cid:105) (cid:17) (cid:16) ∆Σ kj − (cid:10) ∆Σ j (cid:11)(cid:17) , (5)where ∆Σ ki denotes the excess surface mass density from the k -th jackknife realization in the i -th radial bin, with a mean (cid:104) ∆Σ i (cid:105) = N v N v (cid:213) k = ∆Σ ki . (6)The signal-to-noise ratio (SNR) for our lensing measurementcan be calculated as (Becker et al. 2016) S / N = (cid:205) i , j ∆Σ data i C − ij ∆Σ model j (cid:113)(cid:205) i , j ∆Σ model i C − ij ∆Σ model j , (7)where i , j are indices for the N bin radial bins of the measuredexcess surface mass density ∆Σ data with model expectation ∆Σ model (see section 5.1.2 below), and C − is an estimate ofits inverse covariance matrix including the Hartlap correc-tion factor (Hartlap et al. 2007). Apart from their ability to act as gravitational lenses dueto their low matter content as compared to the mean back-ground density, voids are also underdense in terms of galax-ies. In fact, this property is used for their definition in thefirst place. It is therefore interesting to extract the averageradial galaxy distribution around voids, and to compare it tothe lensing signal. The stacked galaxy-density profile aroundvoids is equivalent to the void-galaxy cross-correlation func-tion in 3D (e.g., Hamaus et al. 2015), ξ vg ( r ) = n vg ( r ) (cid:10) n g (cid:11) − , (8)where n vg ( r ) is the density profile of galaxies around voids atdistance r (in 3D), and (cid:104) n g (cid:105) the mean density of tracers at agiven redshift. Gravitational lensing, however, provides theprojected surface mass density along the LOS, as defined MNRAS , 1–16 (2019)
DES Collaboration in equation (1). For a more direct comparison it is there-fore instructive to project all galaxies along the LOS and tomeasure the 2D void-galaxy correlation function instead, ξ vg ( r p ) = Σ g ( r p ) (cid:10) Σ g (cid:11) − , (9)where Σ g ( r p ) is the projected surface density of galaxiesaround void centres at projected distance r p , and (cid:10) Σ g (cid:11) isthe mean projected surface density of galaxies in the red-shift slice.In order to estimate the 2D void-galaxy cross-correlation function from the data we have to take into ac-count the survey geometry. This can be achieved with thehelp of a random galaxy catalogue with the same mask andselection function as the original galaxy sample, albeit ahigher density of unclustered objects. With that the Davis &Peebles estimator (Davis & Peebles 1983) provides the pro-jected excess-probability of finding a void-galaxy pair, i.e.the 2D void-galaxy cross-correlation function, via ξ vg ( r p ) = N r N g Σ g ( r p ) Σ r ( r p ) − , (10)where N g and N r are the total numbers of galaxies and ran-doms, respectively, and Σ r ( r p ) is the projected 2D surface-density of randoms around the same voids. We have alsotested the Landy & Szalay estimator (Landy & Szalay 1993)and found negligible differences to using equation (10). In this section we present measurements of lensing and clus-tering around 2D and 3D voids in DES Y1 data. With thehelp of the MICE2 mocks we first investigate the impact ofphoto-z scatter on the observables.
The black points in figure 2 represent the excess surface massdensity profiles inferred via equation (4) using the tangen-tial component of shear from a weak-lensing measurementaround a subsample of our 3D voids from the MICE2 mocks.To determine the impact of photo-z scatter on the observ-ables, we validate our pipeline on the MICE2 mocks by ex-changing photometric with spectroscopic redshift estimates,which are known in the simulated galaxy catalogue. Hence,we repeat our entire measurement including the void iden-tification step with vide . For the 2D voids the impact ofphoto-z scatter has already been investigated in S´anchezet al. (2017), and we have adopted a projection width ofsufficient size to minimize its impact. Figure 2 shows a com-parison of excess surface density profiles inferred via weaklensing by vide voids identified using either photometric, orspectroscopic redshifts. Evidently, the two profiles are quitedifferent and the signal obtained from photometric voids isstronger.A possible origin for this difference is due to the ‘smear-ing’ of galaxies along the LOS in photometric space. Thiscauses under-densities that are elongated along the LOSto be more likely identified as voids, whereas structures − r p /R v − . − . − . − . . . . . . ∆ Σ [ M (cid:12) h / p c ] R v = 30 −
60 [Mpc /h ] spectroscopicphotometric Figure 2.
Comparison of excess surface mass density profilesinferred via weak lensing by 3D voids found in spec-z (red) andphoto-z (black) redMaGiC mocks in MICE2. r /R v r / R v - . - . - . . R v = 30 −
60 [Mpc /h ] -0.3-0.2-0.2-0.1-0.1+0.0+0.1+0.1+0.2+0.2+0.3 Figure 3.
Stack of the true positions (spec-z’s) of MICE2 red-MaGiC galaxies around the centres of 3D voids that have beenidentified using photo-z’s of the same mock galaxies. The colourcoding reflects the excess density of galaxies, n vg / (cid:10) n g (cid:11) − , as afunction of the void-centric distances along ( r (cid:107) ) and perpendicu-lar ( r ⊥ ) to the LOS. As discussed in section 5.1.1, the stack givesa misleading impression of void elongation due to photo-z scatter. oriented perpendicular to the LOS may get smoothed outmore easily (Granett et al. 2015; Kov´acs et al. 2017). Lightpassing along an elongated void gets deflected more, hencethe stronger lensing signal. By means of the MICE2 mocks,which provide both photo-z and spec-z information, we maydirectly test this conjecture. In particular, we stack the red-MaGiC galaxy positions based on their spectroscopic red-shifts around the centres of 3D voids that have been identi-fied in the corresponding photo-z galaxy distribution. Thisstack is performed in two directions, along and perpendicu-lar to the LOS, to isolate the smearing effect. The result ispresented in figure 3, featuring a very significant LOS elon-gation with an axis ratio of about 4.This does not imply that every individual void exhibits MNRAS000
Stack of the true positions (spec-z’s) of MICE2 red-MaGiC galaxies around the centres of 3D voids that have beenidentified using photo-z’s of the same mock galaxies. The colourcoding reflects the excess density of galaxies, n vg / (cid:10) n g (cid:11) − , as afunction of the void-centric distances along ( r (cid:107) ) and perpendicu-lar ( r ⊥ ) to the LOS. As discussed in section 5.1.1, the stack givesa misleading impression of void elongation due to photo-z scatter. oriented perpendicular to the LOS may get smoothed outmore easily (Granett et al. 2015; Kov´acs et al. 2017). Lightpassing along an elongated void gets deflected more, hencethe stronger lensing signal. By means of the MICE2 mocks,which provide both photo-z and spec-z information, we maydirectly test this conjecture. In particular, we stack the red-MaGiC galaxy positions based on their spectroscopic red-shifts around the centres of 3D voids that have been identi-fied in the corresponding photo-z galaxy distribution. Thisstack is performed in two directions, along and perpendicu-lar to the LOS, to isolate the smearing effect. The result ispresented in figure 3, featuring a very significant LOS elon-gation with an axis ratio of about 4.This does not imply that every individual void exhibits MNRAS000 , 1–16 (2019)
ES Y1 void lensing λ max /λ min -3 -2 -1 P ( λ m a x / λ m i n ) R v = 30 −
60 [Mpc /h ] spectroscopicphotometric | cos ϑ | P ( | c o s ϑ | ) spectroscopicphotometric Figure 4.
Normalized probability distributions for the elonga-tion (top, defined as the ratio between the largest and the small-est eigenvalue of the inertia tensor) and the orientation (bottom,defined as the cosine of the angle ϑ between the LOS and theprincipal inertia tensor eigenvector) of 3D voids found in spectro-scopic (red) and photometric (black) redMaGiC mocks in MICE2.Vertical lines indicate the mean of each distribution (solid red forspectroscopic, dashed black for photometric mocks). such an extreme stretch. Rather, photo-z smearing breaksisotropy in the distribution of detected voids, which are morelikely to be aligned with the LOS. Stacking such a distribu-tion of aligned voids with varying shapes smears out theirboundaries along the LOS and results in a very elongatedaverage profile shape. We have verified that the distributionof void elongations is only marginally affected by photo-zscatter, so the 3D nature of our vide void samples is pre-served. This is demonstrated in the top panel of figure 4,where we plot the normalized distribution of void elonga-tions defined via the ratio λ max / λ min , the largest and thesmallest eigenvalue of each void’s inertia tensor (see Sutteret al. 2014a, for more details on its definition). As appar-ent from the close agreement of the two distributions, theelongation of individual voids is only marginally changed bythe influence of photo-z scatter. In contrast, the statisticallyuniform distribution of void orientations is affected, as canbe appreciated from the bottom panel of figure 4. Here wecalculate the angles between each void centre’s LOS direc-tion and its inertia tensor eigenvector corresponding to thelargest eigenvalue λ max . Obviously, photo-z selected voids ex- hibit a non-uniform orientation distribution that peaks to-wards angles aligned with the LOS. This explains the smear-ing effect shown in figure 3. However, the slightly overdenseridges located at r ⊥ / R v (cid:39) in that figure imply that theeffective and the projected void radii agree well, supportingthe conclusion that our individual 3D voids are not severelyelongated by photo-z scatter. Thus, naively applying a 3Dvoid finder on photometric data can bias the identified voidsample towards a population of voids elongated in the red-shift direction, which in turn yields a boosted lensing signal.The goal of this work is to compare the lensing and cluster-ing properties around voids within a given sample, and wehave no reason to expect that the selection bias on void ori-entation impacts the relation between these two statistics.In principle we could also use the results on mock cataloguesto recalibrate the measured profiles, but we do not attemptthat here.In figure 5 we present the stacked lensing profiles forour entire samples of both 2D and 3D voids found in theDES Y1 data. The significantly negative tangential shearcomponent clearly indicates these voids to be underdense intheir interior matter content compared to the average. Thetangential shear SNR is 10.7 and 14.0 for 2D and 3D voids,respectively. In contrast, the cross component of the shear isvery close to zero, consistent with expectation. This servesas a nice sanity check that systematics in the measurementare under control. We also note that the lensing signal from2D voids features a slightly higher (more negative) ampli-tude than the one from 3D voids, but also larger scatter andbigger error bars. The lensing imprint from 3D vide voidsin DES is remarkably smooth and precise, it constitutes themost significant void-lensing measurement in the literatureto date, thanks to the large number of 3D void lenses andbackground source galaxies available in DES. Figure 6 showsthe corresponding covariance matrices for ∆Σ ( r p ) calculatedvia equation (5) and normalized by their diagonals.We further divide our void catalogues into three binsin void radius to investigate the dependence of the lensingsignal on void size. The corresponding lensing profiles areshown in figure 7 for 2D, and figure 8 for 3D voids. Table 1summarizes the results from all void samples. While it ishard to discern a definite trend from 2D voids, 3D voidsexhibit more negative excess surface mass densities towardslarger R v . Moreover, the positive ∆Σ at distances beyondthe void radius is most distinct for smaller 3D voids, butdisappears for the largest ones. This is a known feature of3D voids that has been predicted by theory (Sheth & van deWeygaert 2004) and observed in simulations (Hamaus et al.2014a,b) before: smaller voids tend to be compensated byoverdense ridges, while larger voids are not. In order to establish a quantitative comparison to existingresults in the literature, we consider the void density profilefunction of Hamaus et al. (2014a, HSW), ρ v ( r )(cid:104) ρ (cid:105) − = δ c − ( r / r s ) α + ( r / R v ) β , (11)which has been shown to accurately describe the densityfluctuations around voids in both simulations and observa-tions (e.g., Hamaus et al. 2014a, 2016; Sutter et al. 2014a; MNRAS , 1–16 (2019)
DES Collaboration − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 2 . R v = 20 −
120 [Mpc /h ] fit: δ c =-1.00, r s /R v =1.26, α =0.98, β =19.53data tangentialdata cross − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 0 . R v = 10 −
60 [Mpc /h ] fit: δ c =-1.00, r s /R v =0.81, α =2.16, β =7.83data tangentialdata cross Figure 5.
Excess surface mass density profiles inferred via weak-lensing tangential shear by stacking all 2D (left) and 3D (right) voidsidentified in DES Y1 data (black points). The cross components of shear are depicted as blue crosses. Error-bars represent σ confidenceintervals obtained via jackknife resampling of the void catalogues. Red dashed lines show the fits of equation (11) to the data, withbest-fit parameters and corresponding reduced chi-square values shown in each panel. . . . . . . r p /R v . . . . . . r p / R v R v = 20 −
120 [Mpc /h ] − . − . − . − . . . . . . r i , j ≡ C i j / p C ii , C jj . . . . . . r p /R v . . . . . . r p / R v R v = 10 −
60 [Mpc /h ] − . − . − . − . . . . . . r i , j ≡ C i j / p C ii , C jj Figure 6.
Covariance matrices of ∆Σ ( r p ) for 2D (left) and 3D void samples (right), normalized by their diagonal. − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 0 . R v = 20 −
40 [Mpc /h ] fit: δ c =-1.00, r s /R v =1.19, α =2.76, β =14.67data − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 1 . R v = 40 −
60 [Mpc /h ] fit: δ c =-0.74, r s /R v =1.13, α =1.20, β =4.73data − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 2 . R v = 60 −
120 [Mpc /h ] fit: δ c =-0.17, r s /R v =1.10, α =16.13, β =18.97data Figure 7.
Lensing profiles for 2D voids in DES data, similar to the left panel of figure 5, but here the voids are divided into threedifferent radius bins. The red dashed lines show the fits of equation (11) to the data, with best-fit parameters shown in each panel legend.MNRAS000
Lensing profiles for 2D voids in DES data, similar to the left panel of figure 5, but here the voids are divided into threedifferent radius bins. The red dashed lines show the fits of equation (11) to the data, with best-fit parameters shown in each panel legend.MNRAS000 , 1–16 (2019)
ES Y1 void lensing − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 0 . R v = 10 −
20 [Mpc /h ] fit: δ c =-0.56, r s /R v =0.75, α =3.48, β =8.28data − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 0 . R v = 20 −
30 [Mpc /h ] fit: δ c =-0.87, r s /R v =0.79, α =2.06, β =8.90data − r p /R v − − − − ∆ Σ [ M (cid:12) h / p c ] χ = 2 . R v = 30 −
60 [Mpc /h ] fit: δ c =-1.00, r s /R v =0.89, α =1.43, β =6.38data Figure 8.
Lensing profiles for 3D voids in DES data, similar to the right panel of figure 5, but here the voids are divided into threedifferent radius bins. The red dashed lines show the fits of equation (11) to the data, with best-fit parameters shown in each panel legend.
Barreira et al. 2015; Pollina et al. 2017, 2019; Falck et al.2018; Perico et al. 2019). Equation (11) has 4 free parame-ters: a central void under-density δ c , a scale radius r s (typi-cally expressed in units of R v ), and two slopes α and β . Thisfunction does not account for on average anisotropic voidprofiles, which are preferentially obtained by void findersoperating on photometric redshifts (see above). We never-theless use it as a template to describe an effective, spheri-cally symmetric density profile with the same excess surfacemass density when projected along the LOS.For each of our void samples, we perform a 4-parameterfit of equation (11) to the observed excess surface mass den-sities via a Monte Carlo Markov Chain (MCMC). For thiswe need to convert the 3D density ρ ( r ) to a surface massdensity Σ ( r p ) via (Pisani et al. 2014) Σ ( r p ) = ∫ ρ (cid:18)(cid:113) [ r z − D A ( z l )] + r p (cid:19) d r z , (12)where the void lenses are located at redshift z l and we in-tegrate up to a distance of R v away from the void centrealong the LOS coordinate r z . The best-fit HSW-profiles areshown as dashed lines in figures 5, 7 and 8. The agreementwith the data is striking in most cases, except for the largestvoid radius bins. However, this is the most noisy regime ofour data with the fewest voids, featuring a double-dip inthe excess surface mass density profile that cannot be re-produced with equation (11). A possible origin could be thepresence of prominent sub-structures that do not averageout in a void stack with limited statistics. The reduced chi-square values are shown in each panel of figures 5, 7 and 8,calculated as χ = N − (cid:213) i , j (cid:16) ∆Σ data i − ∆Σ model i (cid:17) C − ij (cid:16) ∆Σ data j − ∆Σ model j (cid:17) , (13)where the number of degrees of freedom is N dof = N bin − .An example contour plot of the MCMC posterior prob-ability density function (PDF) for 3D voids of radii −
30 Mpc / h is shown in figure 9. The values of the HSW-profile parameters at the maximum of the PDF are in excel-lent agreement with N -body simulation results (cf. figure 2of Hamaus et al. 2014a) and provide an accurate inferenceof the distribution of dark matter inside our observed voidsamples. However, it should be kept in mind that the pa- δ c = − . +0 . − . . . . . r s / R v r s /R v = 0 . +0 . − . α α = 2 . +0 . − . − . − . − . δ c β .
72 0 .
78 0 .
84 0 . r s /R v α ββ = 8 . +1 . − . Figure 9.
Posterior PDF for the parameters of equation (11),obtained via MCMC fit to the excess surface mass density of 3Dvoids of size ≤ R v <
30 Mpc / h in DES Y1 data. rameters of equation (11) describe a spherically symmet-ric density profile, whereas our voids tend to be orientedalong the LOS. Therefore, our fits should be understood asconstraints on the spherically symmetric equivalent of theanisotropic void density profile, which causes the same lens-ing imprint. This implies that the central under-density ofour voids is less negative than the best-fit values we obtainfor δ c , as evident from figure 3. This also explains why thelower boundary of δ c = − is encountered in some cases.Figure 10 presents the corresponding 3D void densityprofile of equation (11) evaluated for all the posterior pa-rameter values sampled in our MCMC from figure 9, so re-gions of higher density correspond to a higher probability.This measurement can in principle be used to compare pre-dictions from competing models of dark matter and grav-ity (e.g., Barreira et al. 2015; Yang et al. 2015; Baker et al.2018). We note, however, that the effect of anisotropic void MNRAS , 1–16 (2019) DES Collaboration
Figure 10.
3D void density profile from equation (11) evaluatedat each parameter set sampled in the MCMC from figure 9. selection due to the impact of photo-z scatter will need to bemodelled in order to fully interpret the inferred 3D densityprofile.
With the inferred matter distribution around voids from ourcatalogues at hand, we may now directly compare this withthe corresponding distribution of galaxies around the samevoids. Because the lensing data provide us with projectedexcess surface mass densities ∆Σ ( r p ) , we measure the corre-sponding quantity for the clustering of galaxies, namely theexcess surface galaxy density ∆Σ g ( r p ) ≡ Σ g ( < r p ) − Σ g ( r p ) .With the use of equation (9) we can write Σ g ( r p ) (cid:14)(cid:10) Σ g (cid:11) = ξ Dvg ( r p ) + , and thus ∆Σ g ( r p ) (cid:10) Σ g (cid:11) = ξ Dvg ( < r p ) − ξ Dvg ( r p ) ≡ ∆ ξ Dvg ( r p ) . (14)Now, following Pollina et al. (2017), we may relate the 3Dvoid-galaxy and void-matter cross-correlation functions viaa single bias parameter b slope , ξ Dvg ( r ) = b slope ξ Dvm ( r ) . (15)Because b slope is a scale-independent constant, the same re-lation holds for the projected correlation functions ξ D andthus also for ∆ ξ D . Therefore, we have ∆Σ g ( r p ) (cid:10) Σ g (cid:11) = ∆ ξ Dvg ( r p ) = b slope ∆ ξ Dvm ( r p ) = b slope ∆Σ ( r p )(cid:104) Σ (cid:105) . (16)Note that the validity of this equation is compromised inthe case there is a significant redshift evolution in both b slope and the void density profile. However, there is no evidencefor redshift dependence in the bias of the redMaGiC sampleinferred via galaxy-galaxy lensing in DES (Prat et al. 2018).Also the void density profile evolves very little in the con-sidered redshift range in simulations (Hamaus et al. 2014a),so we may safely neglect redshift-evolution effects here.In practice, we measure the quantity ξ Dvg ( r p ) via equa-tion (10) and the quantity ∆Σ ( r p ) via equation (4). Becauseequation (4) involves redshift-dependent inverse-variance weights, but equation (10) does not, the ratio of the quan-tities ξ Dvg ( r p ) and ∆Σ ( r p ) can be biased. This bias wouldbe absorbed by b slope in equation (16), resulting in a wrongvalue. In order to account for this difference, we repeated themeasurement of ξ Dvg applying the same weights as for theestimator in equation (4). We find consistent results withand without weights, with differences far below our mea-surement accuracy. For this reason, we omit any weightingscheme for the estimator in equation (10).Comparing the measurements of ∆ ξ Dvg ( r p ) and ∆Σ ( r p ) allows us to test the linearity of equation (15) via equa-tion (16). In particular, the ratio ∆ ξ Dvg / ∆Σ should be inde-pendent of the projected radius r p , with a constant value c slope ≡ b slope (cid:104) Σ (cid:105) . (17)Taking the ratio of measured quantities that are subject tonoise is sub-optimal and can lead to noise bias. To avoid this,we use an MCMC approach to robustly infer a constant c slope relating ∆ ξ Dvg ( r p ) and ∆Σ ( r p ) . We first test this method on 3D voids identified in theMICE2 mocks. In figure 11, both galaxy-density profiles ∆ ξ Dvg ( r p ) and lensing profiles ∆Σ ( r p ) , multiplied by the best-fit c slope parameter, are shown for the following void-radiusbins: R v ∈ [ , ] ; [ , ] Mpc / h . We omit showing smallvoids whose effective radius is close to the mean galaxyseparation of the redMaGiC sample ( ∼
10 Mpc / h ). Forthose voids the excess void-galaxy correlation function ∆ ξ Dvg may switch sign inside the void radius r p < R v and turnpositive. This is a sampling artefact caused by voids thatare defined by only a few galaxies: their volume-weightedbarycentre tends to coincide with the central Voronoi-cell ofa galaxy, which causes a central overdensity in the estimateof ∆ ξ Dvg . However, this artifact disappears for voids largerthan ∼
30 Mpc / h , where the correspondence between lens-ing and clustering becomes remarkably accurate. In fact, theradial profiles of ∆Σ ( r p ) and ∆ ξ Dvg ( r p ) are consistent withintheir measurement errors everywhere, suggesting the linearrelation from equation (16) between the two holds. In figure 12 we present the same plots as before, but obtainedfrom DES Y1 data. Although the statistical accuracy is lowerdue to the smaller sky area, the agreement between the ex-cess surface density profiles of matter and galaxies aroundvoids is striking. We do observe a few outliers at small pro-jected distances in ∆Σ ( r p ) , but the overall agreement is verygood within the errors. We repeat the same analysis for our2D voids in radius bins of [ , ] ; [ , ] Mpc / h , the re-sults are shown in figure 13. In this case the agreement be-tween mass and light is somewhat degraded compared to the3D voids. However, the sparsity of 2D voids results in a muchnoisier signal for both lensing and clustering measurements,which at least partly may explain the larger discrepancy.With the inferred parameter c slope = b slope /(cid:104) Σ (cid:105) we canalso estimate the value of the galaxy bias around voids, b slope . For this, we need to calculate the mean comoving MNRAS000
30 Mpc / h , where the correspondence between lens-ing and clustering becomes remarkably accurate. In fact, theradial profiles of ∆Σ ( r p ) and ∆ ξ Dvg ( r p ) are consistent withintheir measurement errors everywhere, suggesting the linearrelation from equation (16) between the two holds. In figure 12 we present the same plots as before, but obtainedfrom DES Y1 data. Although the statistical accuracy is lowerdue to the smaller sky area, the agreement between the ex-cess surface density profiles of matter and galaxies aroundvoids is striking. We do observe a few outliers at small pro-jected distances in ∆Σ ( r p ) , but the overall agreement is verygood within the errors. We repeat the same analysis for our2D voids in radius bins of [ , ] ; [ , ] Mpc / h , the re-sults are shown in figure 13. In this case the agreement be-tween mass and light is somewhat degraded compared to the3D voids. However, the sparsity of 2D voids results in a muchnoisier signal for both lensing and clustering measurements,which at least partly may explain the larger discrepancy.With the inferred parameter c slope = b slope /(cid:104) Σ (cid:105) we canalso estimate the value of the galaxy bias around voids, b slope . For this, we need to calculate the mean comoving MNRAS000 , 1–16 (2019)
ES Y1 void lensing -1 r p /R v c s l o p e ∆ Σ R v = [20 ,
30] Mpc /h ∆ ξ Dvg c slope ∆Σ -1 r p /R v c s l o p e ∆ Σ R v = [30 ,
60] Mpc /h ∆ ξ Dvg c slope ∆Σ Figure 11.
Comparison of ∆Σ ( r p ) profiles from weak lensing (black dots with error bars) and projected galaxy-density profiles ∆ ξ Dvg ( r p ) (green area) around 3D voids of different size in MICE2 redMaGiC mocks. ∆Σ ( r p ) has been rescaled by an overall amplitude c slope toyield a best match with ∆ ξ Dvg ( r p ) . The first data point of ∆ ξ Dvg has been fixed to a value of zero and is not used in the fit. -1 r p /R v c s l o p e ∆ Σ R v = [20 ,
30] Mpc /h ∆ ξ Dvg c slope ∆Σ -1 r p /R v c s l o p e ∆ Σ R v = [30 ,
60] Mpc /h ∆ ξ Dvg c slope ∆Σ Figure 12.
Same as figure 11 for 3D voids in DES Y1 data. -1 r p /R v c s l o p e ∆ Σ R v = [40 ,
60] Mpc /h ∆ ξ Dvg c slope ∆Σ -1 r p /R v c s l o p e ∆ Σ R v = [60 , /h ∆ ξ Dvg c slope ∆Σ Figure 13.
Same as figure 11 for 2D voids in DES Y1 data.MNRAS , 1–16 (2019) DES Collaboration
10 20 30 40 50 60 R v [Mpc /h ] G a l a xy b i a s ( b s l o p e ) Prat et al. (2018)datamock
Figure 14.
Galaxy bias parameter values inferred via the relationof galaxy-clustering and lensing measurements around 3D voidsin DES Y1 data (blue points), as well as in MICE2 mocks (blacksquares). The vertical dashed lines represent the boundaries of thevoid-radius bins used, and the horizontal shaded area depicts thelarge-scale galaxy-galaxy lensing constraint by Prat et al. (2018). surface density of the Universe (cid:104) Σ (cid:105) in the relevant projectedredshift range, (cid:104) Σ (cid:105) = ∫ D A ( z max ) D A ( z min ) (cid:104) ρ ( r z )(cid:105) d r z = ∫ z max z min (cid:104) ρ ( z )(cid:105) cH ( z ) d z == H c π G ∫ z max z min Ω m (cid:112) Ω m ( + z ) + − Ω m d z , (18)where we integrate over the entire LOS extension of the lenssample (voids in redMaGiC galaxies) from redshift z min = . to z max = . . The resulting bias parameters b slope from thedifferent radius bins for our 3D void samples in DES Y1 dataand MICE2 mocks are shown in figure 14, along with the re-sult from the galaxy-galaxy lensing analysis by Prat et al.(2018). The inferred b slope around voids is slightly higherin comparison to the large-scale estimates from Prat et al.(2018), but still consistent at the σ -level. Earlier analy-ses have already found that tracer bias can be enhanced invoid environments, especially for smaller voids (Pollina et al.2017, 2019). Moreover, in simulations the halo bias has beenshown to be density dependent, with increasing values atlow densities (see figure 1 in Neyrinck et al. 2014). Upcom-ing data from DES will allow us to more accurately probe theenvironmental dependence of tracer bias around voids. Wehave also repeated the same analysis for our 2D voids. Theresults are consistent with the 3D case, albeit with largerscatter, which is why we do not explicitly show them here. We have measured the lensing shear and galaxy-density pro-files around voids in the Year 1 data of the Dark Energy Sur-vey, and validated our methodology using mock catalogues.The voids were identified using two different void-finding al-gorithms adapted to the photometric redshift accuracy ofDES redMaGiC galaxies: one algorithm operated on pro- jected 2D slices while the other used the estimated 3D po-sitions of galaxies. We summarize our results as follows:(i) We have presented weak-lensing measurements byvoids in the galaxy distribution, revealing their underdensecores and compensation walls at the highest SNR achievedto date, up to a value of . . We further divide both of ourvoid samples into three bins in void radius and thus measuretheir lensing profile as a function of void size.(ii) We have investigated the impact of photo-z scatter onour measurements from 3D voids with the help of MICE2mocks, which provide both photometric as well as spectro-scopic redshift estimates. We find that 3D voids identifiedin a photometric redshift catalogue feature enhanced lensingimprints, which can be explained by a selection bias in thewatershed algorithm we employ, acting in favour of voidswith elongations oriented along the LOS.(iii) The inferred excess surface mass density profilearound our 3D voids is very consistent with the equiv-alent density profile of on average spherically symmetricvoids found in N -body simulations, and is well describedby the universal density profile of equation (11). The pre-sented methodology paves a way to infer various character-istics of voids in the full matter distribution, such as theircentral density. We also confirm smaller voids to be sur-rounded by overcompensated ridges, which disappear grad-ually for larger voids, as anticipated in simulation stud-ies (e.g., Hamaus et al. 2014a; Sutter et al. 2014a; Leclercqet al. 2015).(iv) In order to study the relationship between mass andlight around voids, we have compared galaxy-density profileswith lensing profiles. We find a linear relationship betweenthe mass distribution and the galaxy distribution aroundvoids with effective radii above ∼
30 Mpc / h , as describedby equation (16). For smaller voids deviations arise close tothe void centre due to sparse sampling effects. This is con-sistent with voids identified from hydrodynamical simula-tions, where the void-centric density profiles of galaxies anddark matter were shown to exhibit a linear relation (Pollinaet al. 2017). A similar linearity has also been found betweengalaxy- and cluster-density profiles around voids in DES Y1data (Pollina et al. 2019).(v) A quantitative comparison of mass and light aroundour voids enabled us to constrain the bias of the tracer galax-ies used, namely the redMaGiC sample. We find slightlyhigher values compared to large-scale results from thegalaxy-galaxy lensing analysis of Prat et al. (2018), albeitwith larger uncertainties. An enhanced tracer bias aroundvoids has already been found in Pollina et al. (2017) andmay be related to the environmental dependence of tracerbias. However, a thorough investigation of this effect requireshigher statistical accuracy.The statistical accuracy of the presented results is ex-pected to grow with the improved sky coverage and depthin subsequent DES data releases. Data from planned galaxysurveys of the near future, such as LSST (LSST ScienceCollaboration et al. 2009), Euclid (Laureijs et al. 2011), andWFIRST (Spergel et al. 2013) will further improve the sit-uation. There are several applications of our method. Forexample, the existence of fifth forces in theories of modifiedgravity can affect both the mass profile and, for given massprofile, the lensing signal (Cai et al. 2015; Cautun et al. 2018; MNRAS000
30 Mpc / h , as describedby equation (16). For smaller voids deviations arise close tothe void centre due to sparse sampling effects. This is con-sistent with voids identified from hydrodynamical simula-tions, where the void-centric density profiles of galaxies anddark matter were shown to exhibit a linear relation (Pollinaet al. 2017). A similar linearity has also been found betweengalaxy- and cluster-density profiles around voids in DES Y1data (Pollina et al. 2019).(v) A quantitative comparison of mass and light aroundour voids enabled us to constrain the bias of the tracer galax-ies used, namely the redMaGiC sample. We find slightlyhigher values compared to large-scale results from thegalaxy-galaxy lensing analysis of Prat et al. (2018), albeitwith larger uncertainties. An enhanced tracer bias aroundvoids has already been found in Pollina et al. (2017) andmay be related to the environmental dependence of tracerbias. However, a thorough investigation of this effect requireshigher statistical accuracy.The statistical accuracy of the presented results is ex-pected to grow with the improved sky coverage and depthin subsequent DES data releases. Data from planned galaxysurveys of the near future, such as LSST (LSST ScienceCollaboration et al. 2009), Euclid (Laureijs et al. 2011), andWFIRST (Spergel et al. 2013) will further improve the sit-uation. There are several applications of our method. Forexample, the existence of fifth forces in theories of modifiedgravity can affect both the mass profile and, for given massprofile, the lensing signal (Cai et al. 2015; Cautun et al. 2018; MNRAS000 , 1–16 (2019)
ES Y1 void lensing Barreira et al. 2015; Baker et al. 2018). The inference ofcentral void densities, as well as the linearity between massand light around void centres can therefore provide a consis-tency test of GR. Another example concerns the nature ofdark matter and the impact of massive neutrinos on voids.Warm or hot dark-matter particles (massive neutrinos) havea different distribution in voids than cold dark matter, whichmakes their relative abundance inside voids higher than else-where in the cosmos (Yang et al. 2015; Massara et al. 2015;Banerjee & Dalal 2016; Kreisch et al. 2019; Schuster et al.2019). Similar arguments apply for tests of potential cou-plings between dark matter and dark energy (Pollina et al.2016). While these tests require much higher precision mea-surements, the methodology developed in our study maystimulate further theoretical explorations for signatures ofnew physics in voids.The apparent linear relationship between mass and lightin our data suggests the physics of void environments to beremarkably simple. Similar conclusions have already beendrawn concerning the dynamics in voids, probed via redshift-space distortions (Hamaus et al. 2015, 2016, 2017; Cai et al.2016; Achitouv et al. 2017; Hawken et al. 2017). The combi-nation of dynamical measurements from spectroscopic red-shifts and the lensing mass profiles presented here is apromising probe of cosmology and gravity. It motivates fur-ther methodology for identifying and characterizing voids inspectroscopic and high-quality photometric surveys (Pisaniet al. 2019).
ACKNOWLEDGEMENTS
This paper has gone through internal review by the DEScollaboration. We are grateful to Arka Banerjee, Elena Mas-sara, Alice Pisani and Ravi Sheth for helpful discussions.NH and GP acknowledge support from the DFG cluster ofexcellence “Origins” and the Trans-Regional CollaborativeResearch Center TRR 33 “The Dark Universe” of the DFG.YF and BJ are supported in part by the U.S. Departmentof Energy grant DE-SC0007901.This work has made use of CosmoHub, see Carreteroet al. (2017). CosmoHub has been developed by the Portd’Informaci´o Cient´ıfica (PIC), maintained through a collab-oration of the Institut de F´ısica d’Altes Energies (IFAE)and the Centro de Investigaciones Energ´eticas, Medioambi-entales y Tecnol´ogicas (CIEMAT), and was partially fundedby the “Plan Estatal de Investigaci´on Cient´ıfica y T´ecnica yde Innovaci´on” program of the Spanish government.Funding for the DES Projects has been provided bythe U.S. Department of Energy, the U.S. National Sci-ence Foundation, the Ministry of Science and Education ofSpain, the Science and Technology Facilities Council of theUnited Kingdom, the Higher Education Funding Council forEngland, the National Center for Supercomputing Applica-tions at the University of Illinois at Urbana-Champaign, theKavli Institute of Cosmological Physics at the Universityof Chicago, the Center for Cosmology and Astro-ParticlePhysics at the Ohio State University, the Mitchell Institutefor Fundamental Physics and Astronomy at Texas A&MUniversity, Financiadora de Estudos e Projetos, Funda¸c˜aoCarlos Chagas Filho de Amparo `a Pesquisa do Estado do Riode Janeiro, Conselho Nacional de Desenvolvimento Cient´ı- fico e Tecnol´ogico and the Minist´erio da Ciˆencia, Tecnologiae Inova¸c˜ao, the Deutsche Forschungsgemeinschaft and theCollaborating Institutions in the Dark Energy Survey.The Collaborating Institutions are Argonne NationalLaboratory, the University of California at Santa Cruz,the University of Cambridge, Centro de InvestigacionesEnerg´eticas, Medioambientales y Tecnol´ogicas-Madrid, theUniversity of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Ei-dgen¨ossische Technische Hochschule (ETH) Z¨urich, FermiNational Accelerator Laboratory, the University of Illi-nois at Urbana-Champaign, the Institut de Ci`encies del’Espai (IEEC/CSIC), the Institut de F´ısica d’Altes Ener-gies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit¨at M¨unchen and the associated Ex-cellence Cluster Universe, the University of Michigan, theNational Optical Astronomy Observatory, the University ofNottingham, The Ohio State University, the University ofPennsylvania, the University of Portsmouth, SLAC NationalAccelerator Laboratory, Stanford University, the Universityof Sussex, Texas A&M University, and the OzDES Member-ship Consortium.Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Obser-vatory, which is operated by the Association of Universi-ties for Research in Astronomy (AURA) under a cooperativeagreement with the National Science Foundation.The DES data management system is supported bythe National Science Foundation under Grant NumbersAST-1138766 and AST-1536171. The DES participants fromSpanish institutions are partially supported by MINECOunder grants AYA2015-71825, ESP2015-66861, FPA2015-68048, SEV-2016-0588, SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the Euro-pean Union. IFAE is partially funded by the CERCA pro-gram of the Generalitat de Catalunya. Research leading tothese results has received funding from the European Re-search Council under the European Union’s Seventh Frame-work Program (FP7/2007-2013) including ERC grant agree-ments 240672, 291329, and 306478. We acknowledge supportfrom the Brazilian Instituto Nacional de Ciˆencia e Tecnolo-gia (INCT) e-Universe (CNPq grant 465376/2014-2).This manuscript has been authored by Fermi ResearchAlliance, LLC under Contract No. DE-AC02-07CH11359with the U.S. Department of Energy, Office of Science, Of-fice of High Energy Physics. The United States Governmentretains and the publisher, by accepting the article for pub-lication, acknowledges that the United States Governmentretains a non-exclusive, paid-up, irrevocable, world-wide li-cense to publish or reproduce the published form of thismanuscript, or allow others to do so, for United States Gov-ernment purposes.
REFERENCES
Achitouv I., 2016, Phys. Rev. D, 94, 103524Achitouv I., 2019, arXiv e-prints, p. arXiv:1903.05645Achitouv I., Neyrinck M., Paranjape A., 2015, MNRAS, 451, 3964Achitouv I., Blake C., Carter P., Koda J., Beutler F., 2017, Phys.Rev. D, 95, 083502Alonso D., Hill J. C., Hloˇzek R., Spergel D. N., 2018, Phys.Rev. D, 97, 063514MNRAS , 1–16 (2019) DES Collaboration
Baker T., Clampitt J., Jain B., Trodden M., 2018, Phys. Rev. D,98, 023511Banerjee A., Dalal N., 2016, J. Cosmology Astropart. Phys., 11,015Barreira A., Cautun M., Li B., Baugh C. M., Pascoli S., 2015,J. Cosmology Astropart. Phys., 8, 028Becker M. R., et al., 2016, Phys. Rev. D, 94, 022002Brouwer M. M., et al., 2018, MNRAS, 481, 5189Cai Y.-C., Padilla N., Li B., 2015, MNRAS, 451, 1036Cai Y.-C., Taylor A., Peacock J. A., Padilla N., 2016, MNRAS,462, 2465Cai Y.-C., Neyrinck M., Mao Q., Peacock J. A., Szapudi I.,Berlind A. A., 2017, MNRAS, 466, 3364Carretero J., Castander F. J., Gazta˜naga E., Crocce M., FosalbaP., 2015, MNRAS, 447, 646Carretero J., et al., 2017, in Proceedings of the European PhysicalSociety Conference on High Energy Physics. 5-12 July. p. 488Cautun M., Paillas E., Cai Y.-C., Bose S., Armijo J., Li B., PadillaN., 2018, MNRAS, 476, 3195Chan K. C., Hamaus N., Desjacques V., 2014, Phys. Rev. D, 90,103521Chuang C.-H., Kitaura F.-S., Liang Y., Font-Ribera A., Zhao C.,McDonald P., Tao C., 2017, Phys. Rev. D, 95, 063528Clampitt J., Jain B., 2015, MNRAS, 454, 3357Clampitt J., Cai Y.-C., Li B., 2013, MNRAS, 431, 749Colberg J. M., et al., 2008, MNRAS, 387, 933Colless M., et al., 2001, MNRAS, 328, 1039Contarini S., Ronconi T., Marulli F., Moscardini L., VeropalumboA., Baldi M., 2019, MNRAS, 488, 3526Correa C. M., Paz D. J., Padilla N. D., Ruiz A. N., Angulo R. E.,S´anchez A. G., 2019, MNRAS, 485, 5761Crocce M., Castander F. J., Gazta˜naga E., Fosalba P., CarreteroJ., 2015, MNRAS, 453, 1513Dark Energy Survey Collaboration et al., 2016, MNRAS, 460,1270Davies C. T., Cautun M., Li B., 2018, MNRAS, 480, L101Davies C. T., Cautun M., Li B., 2019, arXiv e-prints, p.arXiv:1907.06657Davis M., Peebles P. J. E., 1983, ApJ, 267, 465Dawson K. S., et al., 2013, AJ, 145, 10Drlica-Wagner A., et al., 2018, ApJS, 235, 33Falck B., Koyama K., Zhao G.-B., 2015, J. Cosmology Astropart.Phys., 7, 049Falck B., Koyama K., Zhao G.-B., Cautun M., 2018, MNRAS,475, 3262Flaugher B., et al., 2015, AJ, 150, 150Fosalba P., Gazta˜naga E., Castander F. J., Crocce M., 2015a,MNRAS, 447, 1319Fosalba P., Crocce M., Gazta˜naga E., Castander F. J., 2015b,MNRAS, 448, 2987Friedrich O., et al., 2018, Phys. Rev. D, 98, 023508G´orski K. M., Hivon E., 2011, HEALPix: Hierarchical EqualArea isoLatitude Pixelization of a sphere, Astrophysics SourceCode Library (ascl:1107.018)Granett B. R., Neyrinck M. C., Szapudi I., 2008, ApJ, 683, L99Granett B. R., Kov´acs A., Hawken A. J., 2015, MNRAS, 454,2804Gruen D., et al., 2016, MNRAS, 455, 3367Gruen D., et al., 2018, Phys. Rev. D, 98, 023507Hamaus N., Sutter P. M., Lavaux G., Wandelt B. D., 2014a,J. Cosmology Astropart. Phys., 12, 013Hamaus N., Wandelt B. D., Sutter P. M., Lavaux G., WarrenM. S., 2014b, Physical Review Letters, 112, 041304Hamaus N., Sutter P. M., Wandelt B. D., 2014c, Physical ReviewLetters, 112, 251302Hamaus N., Sutter P. M., Lavaux G., Wandelt B. D., 2015, J. Cos-mology Astropart. Phys., 11, 036 Hamaus N., Pisani A., Sutter P. M., Lavaux G., Escoffier S.,Wandelt B. D., Weller J., 2016, Physical Review Letters, 117,091302Hamaus N., Cousinou M.-C., Pisani A., Aubert M., Escoffier S.,Weller J., 2017, J. Cosmology Astropart. Phys., 7, 014Hartlap J., Simon P., Schneider P., 2007, A&A, 464, 399Hawken A. J., et al., 2017, A&A, 607, A54Higuchi Y., Oguri M., Hamana T., 2013, MNRAS, 432, 1021Hoyle B., et al., 2018, MNRAS, 478, 592Huff E., Mandelbaum R., 2017, arXiv e-prints, p.arXiv:1702.02600Jennings E., Li Y., Hu W., 2013, MNRAS, 434, 2167Kitaura F.-S., et al., 2016, Physical Review Letters, 116, 171301Kov´acs A., et al., 2017, MNRAS, 465, 4166Kov´acs A., et al., 2019, MNRAS, 484, 5267Krause E., Chang T.-C., Dor´e O., Umetsu K., 2013, ApJ, 762,L20Kreisch C. D., Pisani A., Carbone C., Liu J., Hawken A. J., Mas-sara E., Spergel D. N., Wandelt B. D., 2019, MNRAS, p. 1877Krolewski A., et al., 2018, ApJ, 861, 60LSST Science Collaboration et al., 2009, arXiv e-prints, p.arXiv:0912.0201Landy S. D., Szalay A. S., 1993, ApJ, 412, 64Laureijs R., et al., 2011, arXiv e-prints, p. arXiv:1110.3193Lavaux G., Wandelt B. D., 2010, MNRAS, 403, 1392Leclercq F., Jasche J., Sutter P. M., Hamaus N., Wandelt B.,2015, J. Cosmology Astropart. Phys., 3, 047Li B., Zhao G.-B., Koyama K., 2012, MNRAS, 421, 3481Mandelbaum R., Slosar A., Baldauf T., Seljak U., Hirata C. M.,Nakajima R., Reyes R., Smith R. E., 2013, MNRAS, 432, 1544Mao Q., Berlind A. A., Scherrer R. J., Neyrinck M. C., Scocci-marro R., Tinker J. L., McBride C. K., Schneider D. P., 2017,ApJ, 835, 160Massara E., Villaescusa-Navarro F., Viel M., Sutter P. M., 2015,J. Cosmology Astropart. Phys., 11, 018McClintock T., et al., 2019, MNRAS, 482, 1352Melchior P., Sutter P. M., Sheldon E. S., Krause E., WandeltB. D., 2014, MNRAS, 440, 2922Miyazaki S., et al., 2012, in Proc. SPIE. p. 84460Z,doi:10.1117/12.926844Nadathur S., Crittenden R., 2016, ApJ, 830, L19Neyrinck M. C., 2008, MNRAS, 386, 2101Neyrinck M. C., Arag´on-Calvo M. A., Jeong D., Wang X., 2014,MNRAS, 441, 646Padilla N. D., Ceccarelli L., Lambas D. G., 2005, MNRAS, 363,977Paillas E., Lagos C. D. P., Padilla N., Tissera P., Helly J., SchallerM., 2017, MNRAS, 470, 4434Paillas E., Cautun M., Li B., Cai Y.-C., Padilla N., Armijo J.,Bose S., 2019, MNRAS, 484, 1149Paz D., Lares M., Ceccarelli L., Padilla N., Lambas D. G., 2013,MNRAS, 436, 3480Perico E. L. D., Voivodic R., Lima M., Mota D. F., 2019, arXive-prints, p. arXiv:1905.12450Pisani A., Lavaux G., Sutter P. M., Wandelt B. D., 2014, MNRAS,443, 3238Pisani A., Sutter P. M., Hamaus N., Alizadeh E., Biswas R., Wan-delt B. D., Hirata C. M., 2015, Phys. Rev. D, 92, 083531Pisani A., et al., 2019, in BAAS. p. 40 ( arXiv:1903.05161 )Platen E., van de Weygaert R., Jones B. J. T., 2007, MNRAS,380, 551Pollina G., Baldi M., Marulli F., Moscardini L., 2016, MNRAS,455, 3075Pollina G., Hamaus N., Dolag K., Weller J., Baldi M., MoscardiniL., 2017, MNRAS, 469, 787Pollina G., et al., 2019, MNRAS, 487, 2836Prat J., et al., 2018, Phys. Rev. D, 98, 042005Ricciardelli E., Quilis V., Planelles S., 2013, MNRAS, 434, 1192MNRAS000
Baker T., Clampitt J., Jain B., Trodden M., 2018, Phys. Rev. D,98, 023511Banerjee A., Dalal N., 2016, J. Cosmology Astropart. Phys., 11,015Barreira A., Cautun M., Li B., Baugh C. M., Pascoli S., 2015,J. Cosmology Astropart. Phys., 8, 028Becker M. R., et al., 2016, Phys. Rev. D, 94, 022002Brouwer M. M., et al., 2018, MNRAS, 481, 5189Cai Y.-C., Padilla N., Li B., 2015, MNRAS, 451, 1036Cai Y.-C., Taylor A., Peacock J. A., Padilla N., 2016, MNRAS,462, 2465Cai Y.-C., Neyrinck M., Mao Q., Peacock J. A., Szapudi I.,Berlind A. A., 2017, MNRAS, 466, 3364Carretero J., Castander F. J., Gazta˜naga E., Crocce M., FosalbaP., 2015, MNRAS, 447, 646Carretero J., et al., 2017, in Proceedings of the European PhysicalSociety Conference on High Energy Physics. 5-12 July. p. 488Cautun M., Paillas E., Cai Y.-C., Bose S., Armijo J., Li B., PadillaN., 2018, MNRAS, 476, 3195Chan K. C., Hamaus N., Desjacques V., 2014, Phys. Rev. D, 90,103521Chuang C.-H., Kitaura F.-S., Liang Y., Font-Ribera A., Zhao C.,McDonald P., Tao C., 2017, Phys. Rev. D, 95, 063528Clampitt J., Jain B., 2015, MNRAS, 454, 3357Clampitt J., Cai Y.-C., Li B., 2013, MNRAS, 431, 749Colberg J. M., et al., 2008, MNRAS, 387, 933Colless M., et al., 2001, MNRAS, 328, 1039Contarini S., Ronconi T., Marulli F., Moscardini L., VeropalumboA., Baldi M., 2019, MNRAS, 488, 3526Correa C. M., Paz D. J., Padilla N. D., Ruiz A. N., Angulo R. E.,S´anchez A. G., 2019, MNRAS, 485, 5761Crocce M., Castander F. J., Gazta˜naga E., Fosalba P., CarreteroJ., 2015, MNRAS, 453, 1513Dark Energy Survey Collaboration et al., 2016, MNRAS, 460,1270Davies C. T., Cautun M., Li B., 2018, MNRAS, 480, L101Davies C. T., Cautun M., Li B., 2019, arXiv e-prints, p.arXiv:1907.06657Davis M., Peebles P. J. E., 1983, ApJ, 267, 465Dawson K. S., et al., 2013, AJ, 145, 10Drlica-Wagner A., et al., 2018, ApJS, 235, 33Falck B., Koyama K., Zhao G.-B., 2015, J. Cosmology Astropart.Phys., 7, 049Falck B., Koyama K., Zhao G.-B., Cautun M., 2018, MNRAS,475, 3262Flaugher B., et al., 2015, AJ, 150, 150Fosalba P., Gazta˜naga E., Castander F. J., Crocce M., 2015a,MNRAS, 447, 1319Fosalba P., Crocce M., Gazta˜naga E., Castander F. J., 2015b,MNRAS, 448, 2987Friedrich O., et al., 2018, Phys. Rev. D, 98, 023508G´orski K. M., Hivon E., 2011, HEALPix: Hierarchical EqualArea isoLatitude Pixelization of a sphere, Astrophysics SourceCode Library (ascl:1107.018)Granett B. R., Neyrinck M. C., Szapudi I., 2008, ApJ, 683, L99Granett B. R., Kov´acs A., Hawken A. J., 2015, MNRAS, 454,2804Gruen D., et al., 2016, MNRAS, 455, 3367Gruen D., et al., 2018, Phys. Rev. D, 98, 023507Hamaus N., Sutter P. M., Lavaux G., Wandelt B. D., 2014a,J. Cosmology Astropart. Phys., 12, 013Hamaus N., Wandelt B. D., Sutter P. M., Lavaux G., WarrenM. S., 2014b, Physical Review Letters, 112, 041304Hamaus N., Sutter P. M., Wandelt B. D., 2014c, Physical ReviewLetters, 112, 251302Hamaus N., Sutter P. M., Lavaux G., Wandelt B. D., 2015, J. Cos-mology Astropart. Phys., 11, 036 Hamaus N., Pisani A., Sutter P. M., Lavaux G., Escoffier S.,Wandelt B. D., Weller J., 2016, Physical Review Letters, 117,091302Hamaus N., Cousinou M.-C., Pisani A., Aubert M., Escoffier S.,Weller J., 2017, J. Cosmology Astropart. Phys., 7, 014Hartlap J., Simon P., Schneider P., 2007, A&A, 464, 399Hawken A. J., et al., 2017, A&A, 607, A54Higuchi Y., Oguri M., Hamana T., 2013, MNRAS, 432, 1021Hoyle B., et al., 2018, MNRAS, 478, 592Huff E., Mandelbaum R., 2017, arXiv e-prints, p.arXiv:1702.02600Jennings E., Li Y., Hu W., 2013, MNRAS, 434, 2167Kitaura F.-S., et al., 2016, Physical Review Letters, 116, 171301Kov´acs A., et al., 2017, MNRAS, 465, 4166Kov´acs A., et al., 2019, MNRAS, 484, 5267Krause E., Chang T.-C., Dor´e O., Umetsu K., 2013, ApJ, 762,L20Kreisch C. D., Pisani A., Carbone C., Liu J., Hawken A. J., Mas-sara E., Spergel D. N., Wandelt B. D., 2019, MNRAS, p. 1877Krolewski A., et al., 2018, ApJ, 861, 60LSST Science Collaboration et al., 2009, arXiv e-prints, p.arXiv:0912.0201Landy S. D., Szalay A. S., 1993, ApJ, 412, 64Laureijs R., et al., 2011, arXiv e-prints, p. arXiv:1110.3193Lavaux G., Wandelt B. D., 2010, MNRAS, 403, 1392Leclercq F., Jasche J., Sutter P. M., Hamaus N., Wandelt B.,2015, J. Cosmology Astropart. Phys., 3, 047Li B., Zhao G.-B., Koyama K., 2012, MNRAS, 421, 3481Mandelbaum R., Slosar A., Baldauf T., Seljak U., Hirata C. M.,Nakajima R., Reyes R., Smith R. E., 2013, MNRAS, 432, 1544Mao Q., Berlind A. A., Scherrer R. J., Neyrinck M. C., Scocci-marro R., Tinker J. L., McBride C. K., Schneider D. P., 2017,ApJ, 835, 160Massara E., Villaescusa-Navarro F., Viel M., Sutter P. M., 2015,J. Cosmology Astropart. Phys., 11, 018McClintock T., et al., 2019, MNRAS, 482, 1352Melchior P., Sutter P. M., Sheldon E. S., Krause E., WandeltB. D., 2014, MNRAS, 440, 2922Miyazaki S., et al., 2012, in Proc. SPIE. p. 84460Z,doi:10.1117/12.926844Nadathur S., Crittenden R., 2016, ApJ, 830, L19Neyrinck M. C., 2008, MNRAS, 386, 2101Neyrinck M. C., Arag´on-Calvo M. A., Jeong D., Wang X., 2014,MNRAS, 441, 646Padilla N. D., Ceccarelli L., Lambas D. G., 2005, MNRAS, 363,977Paillas E., Lagos C. D. P., Padilla N., Tissera P., Helly J., SchallerM., 2017, MNRAS, 470, 4434Paillas E., Cautun M., Li B., Cai Y.-C., Padilla N., Armijo J.,Bose S., 2019, MNRAS, 484, 1149Paz D., Lares M., Ceccarelli L., Padilla N., Lambas D. G., 2013,MNRAS, 436, 3480Perico E. L. D., Voivodic R., Lima M., Mota D. F., 2019, arXive-prints, p. arXiv:1905.12450Pisani A., Lavaux G., Sutter P. M., Wandelt B. D., 2014, MNRAS,443, 3238Pisani A., Sutter P. M., Hamaus N., Alizadeh E., Biswas R., Wan-delt B. D., Hirata C. M., 2015, Phys. Rev. D, 92, 083531Pisani A., et al., 2019, in BAAS. p. 40 ( arXiv:1903.05161 )Platen E., van de Weygaert R., Jones B. J. T., 2007, MNRAS,380, 551Pollina G., Baldi M., Marulli F., Moscardini L., 2016, MNRAS,455, 3075Pollina G., Hamaus N., Dolag K., Weller J., Baldi M., MoscardiniL., 2017, MNRAS, 469, 787Pollina G., et al., 2019, MNRAS, 487, 2836Prat J., et al., 2018, Phys. Rev. D, 98, 042005Ricciardelli E., Quilis V., Planelles S., 2013, MNRAS, 434, 1192MNRAS000 , 1–16 (2019)
ES Y1 void lensing Ronconi T., Marulli F., 2017, A&A, 607, A24Ronconi T., Contarini S., Marulli F., Baldi M., Moscardini L.,2019, MNRAS, p. 2038Rozo E., Rykoff E. S., Becker M., Reddick R. M., Wechsler R. H.,2015, MNRAS, 453, 38Rozo E., et al., 2016, MNRAS, 461, 1431S´anchez C., et al., 2017, MNRAS, 465, 746Schuster N., Hamaus N., Pisani A., Carbone C., KreischC. D., Pollina G., Weller J., 2019, arXiv e-prints, p.arXiv:1905.00436Sheldon E. S., 2014, MNRAS, 444, L25Sheldon E. S., Huff E. M., 2017, ApJ, 841, 24Sheldon E. S., et al., 2004, AJ, 127, 2544Sheth R. K., van de Weygaert R., 2004, MNRAS, 350, 517Spergel D., et al., 2013, arXiv e-prints, p. arXiv:1305.5422Sutter P. M., Lavaux G., Wandelt B. D., Weinberg D. H., 2012,ApJ, 761, 187Sutter P. M., Lavaux G., Hamaus N., Wandelt B. D., WeinbergD. H., Warren M. S., 2014a, MNRAS, 442, 462Sutter P. M., Pisani A., Wandelt B. D., Weinberg D. H., 2014b,MNRAS, 443, 2983Sutter P. M., et al., 2015, Astronomy and Computing, 9, 1Verza G., Pisani A., Carbone C., Hamaus N., Guzzo L., 2019,arXiv e-prints, p. arXiv:1906.00409Yang L. F., Neyrinck M. C., Arag´on-Calvo M. A., Falck B., SilkJ., 2015, MNRAS, 451, 3606Zhao C., Tao C., Liang Y., Kitaura F.-S., Chuang C.-H., 2016,MNRAS, 459, 2670Zivick P., Sutter P. M., Wandelt B. D., Li B., Lam T. Y., 2015,MNRAS, 451, 4215Zuntz J., et al., 2018, MNRAS, 481, 1149de Jong J. T. A., et al., 2013, The Messenger, 154, 44
AFFILIATIONS Department of Physics and Astronomy, University of Penn-sylvania, Philadelphia, PA 19104, USA Universit¨ats-Sternwarte, Fakult¨at f¨ur Physik, Ludwig-Maximilians Universit¨at M¨unchen, Scheinerstr. 1, 81679M¨unchen, Germany Institut de F´ısica d’Altes Energies (IFAE), The BarcelonaInstitute of Science and Technology, Campus UAB, 08193Bellaterra (Barcelona) Spain Instituto de Astrof´ısica de Canarias (IAC), Calle V´ıaL´actea, E-38200, La Laguna, Tenerife, Spain Departamento de Astrof´ısica, Universidad de La Laguna(ULL), E-38206, La Laguna, Tenerife, Spain Department of Astronomy and Astrophysics, University ofChicago, Chicago, IL 60637, USA Kavli Institute for Cosmological Physics, University ofChicago, Chicago, IL 60637, USA Institut d’Estudis Espacials de Catalunya (IEEC), 08034Barcelona, Spain Institute of Space Sciences (ICE, CSIC), Campus UAB,Carrer de Can Magrans, s/n, 08193 Barcelona, Spain Center for Cosmology and Astro-Particle Physics, TheOhio State University, Columbus, OH 43210, USA Department of Physics, Stanford University, 382 ViaPueblo Mall, Stanford, CA 94305, USA Kavli Institute for Particle Astrophysics & Cosmology, P.O. Box 2450, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, Menlo Park, CA94025, USA Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland Max Planck Institute for Extraterrestrial Physics,Giessenbachstrasse, 85748 Garching, Germany Department of Physics, The Ohio State University,Columbus, OH 43210, USA Department of Physics, Carnegie Mellon University, Pitts-burgh, Pennsylvania 15312, USA Brookhaven National Laboratory, Bldg 510, Upton, NY11973, USA Department of Physics, Duke University Durham, NC27708, USA Institute for Astronomy, University of Edinburgh, Edin-burgh EH9 3HJ, UK Fermi National Accelerator Laboratory, P. O. Box 500,Batavia, IL 60510, USA Instituto de Fisica Teorica UAM/CSIC, Universidad Au-tonoma de Madrid, 28049 Madrid, Spain CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095,Institut d’Astrophysique de Paris, F-75014, Paris, France Centro de Investigaciones Energ´eticas, Medioambientalesy Tecnol´ogicas (CIEMAT), Madrid, Spain Laborat´orio Interinstitucional de e-Astronomia - LIneA,Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400,Brazil Department of Astronomy, University of Illinois atUrbana-Champaign, 1002 W. Green Street, Urbana, IL61801, USA National Center for Supercomputing Applications, 1205West Clark St., Urbana, IL 61801, USA Physics Department, 2320 Chamberlin Hall, University ofWisconsin-Madison, 1150 University Avenue Madison, WI53706-1390 Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio deJaneiro, RJ - 20921-400, Brazil Department of Physics, IIT Hyderabad, Kandi, Telangana502285, India Excellence Cluster Origins, Boltzmannstr. 2, 85748Garching, Germany Faculty of Physics, Ludwig-Maximilians-Universit¨at,Scheinerstr. 1, 81679 Munich, Germany Santa Cruz Institute for Particle Physics, Santa Cruz, CA95064, USA Department of Astronomy, University of Michigan, AnnArbor, MI 48109, USA Department of Physics, University of Michigan, Ann Ar-bor, MI 48109, USA Center for Astrophysics | Harvard & Smithsonian, 60 Gar-den Street, Cambridge, MA 02138, USA George P. and Cynthia Woods Mitchell Institute forFundamental Physics and Astronomy, and Department ofPhysics and Astronomy, Texas A&M University, CollegeStation, TX 77843, USA Department of Astrophysical Sciences, Princeton Univer-sity, Peyton Hall, Princeton, NJ 08544, USA Instituci´o Catalana de Recerca i Estudis Avan¸cats, E-08010 Barcelona, Spain Department of Physics and Astronomy, Pevensey Build-ing, University of Sussex, Brighton, BN1 9QH, UK
MNRAS , 1–16 (2019) DES Collaboration School of Physics and Astronomy, University ofSouthampton, Southampton, SO17 1BJ, UK Brandeis University, Physics Department, 415 SouthStreet, Waltham MA 02453 Instituto de F´ısica Gleb Wataghin, Universidade Estad-ual de Campinas, 13083-859, Campinas, SP, Brazil Computer Science and Mathematics Division, Oak RidgeNational Laboratory, Oak Ridge, TN 37831 Institute of Cosmology and Gravitation, University ofPortsmouth, Portsmouth, PO1 3FX, UK Argonne National Laboratory, 9700 South Cass Avenue,Lemont, IL 60439, USA Cerro Tololo Inter-American Observatory, National Op-tical Astronomy Observatory, Casilla 603, La Serena, Chile
This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000