Weak Lensing Measurement of Filamentary Structure with the SDSS BOSS and Subaru Hyper Suprime-Cam Data
Hiroto Kondo, Hironao Miyatake, Masato Shirasaki, Naoshi Sugiyama, Atsushi J. Nishizawa
MMNRAS , 1–10 (2019) Preprint 23 May 2019 Compiled using MNRAS L A TEX style file v3.0
Weak Lensing Measurement of Filamentary Structure withthe SDSS BOSS and Subaru Hyper Suprime-Cam Data
Hiroto Kondo, (cid:63) Hironao Miyatake, , , Masato Shirasaki, Naoshi Sugiyama, , Atsushi J. Nishizawa , Division of Particle and Astrophysical Science, Graduate School of Science, Nagoya University, Furo-cho, Nagoya 464-8602, Japan Institute for Advanced Research, Nagoya University, Furo-cho, Nagoya 464-8601, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), UTIAS, The University of Tokyo, Kashiwa,Chiba 277-8583, Japan National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We report the weak lensing measurement of filaments between Sloan Digital Sky Sur-vey (SDSS) III/Baryon Oscillation Spectroscopic Survey (BOSS) CMASS galaxy pairsat z ∼ . , using the Subaru Hyper Suprime-Cam (HSC) first-year galaxy shape cat-alogue. Despite of the small overlap of deg between these surveys we detect thefilament lensing signal at 3.9 σ significance, which is the highest signal-to-noise lensingmeasurement of filaments between galaxy-scale halos at this redshift range. We derivea theoretical prediction and covariance using mock catalogues based on full-sky ray-tracing simulations. We find that the intrinsic scatter of filament properties and thefluctuations in large scale structure along the line-of-sight are the primary componentof the covariance and the intrinsic shape noise from source galaxies no longer limitsour lensing measurement. This fact demonstrates the statistical power of the HSCsurvey due to its deep observations and high number density of source galaxies. Ourresult is consistent with the theoretical prediction and supports the “thick” filamentmodel. As the HSC survey area increases, we will be able to study detailed filamentproperties such as the dark matter distributions and redshift evolution of filaments. Key words: large-scale structure of Universe – cosmology: observations – dark matter– gravitational lensing: weak
One of the key features of what the Λ cold dark matter( Λ CDM) model predicts is that the large scale structureof the Universe, which is formed from small scales such asstars and galaxies to large scales such as galaxy groups andclusters. In this process, the CDM collapses into dark matterhalos and then galaxies are formed within a dark matter haloas a result of gas accretion. After a dark matter halo evolves,it hosts more galaxies and then galaxy groups and clustersare formed. Such massive halos are connected to each othervia filamentary structure, the so-called cosmic web. Suchfilamentary structures have been clearly seen in the three-dimensional galaxy distribution observed by galaxy redshiftsurveys such as the 2dF Galaxy Redshift Survey (2dFGRS;Colless et al. 2001) and Sloan Digital Sky Survey (SDSS;York et al. 2000). Statistical measurement of the filamentary (cid:63)
E-mail: [email protected] structure in the galaxy overdensity between massive haloshas also been performed by stacking pairs of galaxy groupsand clusters (e.g., Zhang et al. 2013)In addition to these measurements, it is of great impor-tance to measure the dark matter filaments, which will be amore direct proof of the Λ CDM cosmology. Such a measure-ment can be made possible by weak gravitational lensingwhich manifests as a coherent distortion in observed galaxyshapes caused by foreground dark matter structures.Several weak lensing measurements of a filament eitherbetween massive galaxy clusters or connected to a massivegalaxy cluster are reported. Dietrich et al. (2012) detected adark matter filament in the super cluster system consistingof Abell 222 and Abell 223. Jauzac et al. (2012) claimedthe detection of a dark matter filament funnelling onto acore of massive galaxy cluster MACSJ0717.5+3745. Higuchiet al. (2015) measured the average surface mass density ofa dark matter filament between galaxy clusters at z = . ,CL0015.9+1609 and RX J0018.3+1618. © a r X i v : . [ a s t r o - ph . C O ] M a y H. Kondo et al.
Several attempts to make a statistical detection of darkmatter filaments by stacking pairs of galaxies have beenmade. Clampitt et al. (2016) reported the detection of darkmatter filaments at 4.5 σ significance by stacking 135,000Luminous Red Galaxy (LRG) pairs from SDSS with thetypical redshift z ∼ . , and claimed that their filamentlensing signal is consistent with a “thick” filament modelpredicted from dark matter N -body simulations rather thana “thin” filament model by a string of halo along the line con-necting the two LRGs. Epps & Hudson (2017) pushed thepair redshift to (cid:104) z (cid:105) ∼ . using the SDSS-III/Baryon Os-cillation Spectroscopic Survey (BOSS; Dawson et al. 2013)LOWZ and CMASS galaxy sample, and claimed the detec-tion at 5 σ significance. He et al. (2018) claimed the filamentlensing detection at 5 σ significance using the filament cat-alogue constructed from density ridges of CMASS galaxies(Chen et al. 2016) and weak lensing of cosmic microwavebackground (CMB) measured from the Planck satellite ex-periment (Planck Collaboration et al. 2014a,b).In this paper, we report the weak lensing measurementof filaments between the SDSS-III/BOSS CMASS galaxypairs using the Subaru Hyper Suprime-Cam (HSC; Miyazakiet al. 2018; Komiyama et al. 2018; Kawanomoto et al. 2018;Furusawa et al. 2018) Subaru Strategic Program (SSP; Ai-hara et al. 2018a) first-year data (Aihara et al. 2018b).Among the ongoing weak lensing surveys such as Dark En-ergy Survey (DES; Dark Energy Survey Collaboration et al.2016) and the Kilo-Degree Survey (KiDS; de Jong et al.2013), the HSC survey provides the deepest images, whichenables us to perform one of the highest signal-to-noise mea-surements of filament weak lensing. We also use the mockCMASS and source galaxy catalogue to estimate theoreticalprediction and covariance.This paper is organised as follows. In Section 2, we de-scribe the details of the data set used for our measurement,including CMASS galaxy pair generation, HSC galaxy shapecatalogue, and mock catalogue generation. In Section 3,we describe measurement methods such as filament lens-ing estimator and covariance. We then present the resultsof our filament lensing measurement in Section 4, and con-clude in Section 5. Throughout the paper, we assume thePlanck 2015 cosmology (Planck Collaboration et al. 2016),i.e., Ω m = . with the flat Λ CDM, unless otherwise stated.
The goal of this paper is to measure the filamentary darkmatter distribution between massive galaxies by stackingweak lensing signal around the galaxy pairs. To construct thegalaxy pair catalogue, we use the SDSS-III/BOSS CMASSlarge-scale structure (LSS) galaxy catalogue (Alam et al.2015) with the redshift cut . < z < . Note that we donot impose any selections other than the redshift cut.The spectroscopic redshift of the CMASS sample en-ables us to select galaxy pairs with the robust measurementof the line-of-sight distance. In contrast, if we rely on a pho-tometric sample, the typical statistical uncertainty of a pho-tometric redshift ∆ z p ∼ . , which corresponds to the co-moving distance ∆ χ ∼ h − Mpc at the typical CMASS redshift, introduces more galaxies pairs without filamentsand thus dilutes the filament lensing signal.We basically follow the process described in Clampittet al. (2016) to create a galaxy pair sample. We first ap-ply a spatial cut to select CMASS galaxies within the HSCfirst year full-depth full-color footprints defined in Mandel-baum et al. (2018a), which leaves 14,422 CMASS galaxies.We then apply the line-of-sight distance cut | ∆ χ | < h − Mpc and the physical transverse separation cut h − Mpc < R < h − Mpc . As a result we obtain 70,210 pairs over the HSCfirst-year fields. We define the position and redshift of thepair as the avarage of the CMASS galaxy positions and red-shifts, respectively. The number of pairs is larger than thatof galaxies since a galaxy can be shared among multiplepairs. Figure 1 shows the distribution of transverse sepa-ration between the CMASS pairs. Figure 2 shows the red-shift distribution of the CMASS pairs. In our analysis, weuse the weight provided by the BOSS LSS catalogue, i.e., w l = w seeing w star ( w noz + w cp − ) , where w seeing accounts forthe variation in number density due to local seeing, w star accounts for the contamination from stars which is a func-tion of stellar density, w noz accounts for the redshift failure,and w cp accounts for the fibre collision. Note that we do notuse the FKP weight w FKP , since we do not need to optimisegalaxy clustering measurement when measuring the filamentlensing signal. We define the pair weight as a product of theweights of the CMASS galaxiesWe also create random pairs using the random cata-logue associated with the CMASS LSS catalogue, followingthe same redshift and separation cuts as the galaxy pairs.We divide the random catalogue such that each sample hasthe same number of points as the CMASS galaxies, whichresults in 96 samples of random pairs. The random pairs areused for correcting observational systematics as described inSection 3.2.For null tests, we create a separated galaxy pair cat-alogue with CMASS galaxies. Since galaxies largely sepa-rated along the line-of-sight are not expected to have fil-ament between them, we select galaxies with the line-of-sight distance cut h − Mpc < | ∆ χ | < h − Mpc , whilekeeping the same projected physical transverse separation h − Mpc < R < h − Mpc as our pair sample describedabove.
We use the HSC first-year galaxy shape catalogue as sourcesto measure filament lensing signal. The details of galaxyshape catalogue is described in Mandelbaum et al. (2018a)and Mandelbaum et al. (2018b), and thus we describe a sum-mary of these papers.The HSC first-year lensing catalogue is based on thedata taken during March 2014 through April 2016 withabout 90 nights in total. The total sky coverage is 136.9 deg which consists of six distinct fields named GAMA09H,GAMA15H, HECTOMAP, VVDS, WIDE12H, and XMM,and fully overlaps with the BOSS footprint.The shapes are measured using co-added i -band im-ages with a mean seeing FWHM 0.58 (cid:48)(cid:48) and 5 σ point-source detection limit i lim ∼ . Despite of the conserva-tive magnitude cut i < . to achieve robust shape cali-bration, the weighted number density of source galaxies is MNRAS000
We use the HSC first-year galaxy shape catalogue as sourcesto measure filament lensing signal. The details of galaxyshape catalogue is described in Mandelbaum et al. (2018a)and Mandelbaum et al. (2018b), and thus we describe a sum-mary of these papers.The HSC first-year lensing catalogue is based on thedata taken during March 2014 through April 2016 withabout 90 nights in total. The total sky coverage is 136.9 deg which consists of six distinct fields named GAMA09H,GAMA15H, HECTOMAP, VVDS, WIDE12H, and XMM,and fully overlaps with the BOSS footprint.The shapes are measured using co-added i -band im-ages with a mean seeing FWHM 0.58 (cid:48)(cid:48) and 5 σ point-source detection limit i lim ∼ . Despite of the conserva-tive magnitude cut i < . to achieve robust shape cali-bration, the weighted number density of source galaxies is MNRAS000 , 1–10 (2019) ilament lensing with SDSS BOSS and HSC Pair Separation R pair [Mpc/h] N o m a li s e d N u m b e r Figure 1.
The distribution of physical transverse separation be-tween CMASS galaxy pairs used for our analysis. The y-axis isnormalised such that (cid:205) ∆ R pair n ( R pair ) = , where n ( R pair ) is thenormalised number of pairs shown in this figure and ∆ R pair is thebin width with our choice of ∆ R pair = . . Redshift z N o r m a li s e d N u m b e r lens filamentssource galaxies Figure 2.
The redshift distribution of lens filaments and sourcegalaxies. Both distributions are normalised for the maximumnumbers to be unity. The filament redshift is defined to be amean redshift of pair galaxies. The source population is muchdeeper than that of lens filaments, which enables us to performhigh signal-to-noise filament lensing measurement. − . The galaxy shapes ( e , e ) are estimated us-ing the re-Gaussianization PSF correction method (Hirata &Seljak 2003), which was extensively used and characterisedin weak lensing studies with the Sloan Digital Sky Survey(SDSS) data (Mandelbaum et al. 2005; Reyes et al. 2012;Mandelbaum et al. 2013), and then calibrated with the im-age simulations (Mandelbaum et al. 2018b) generated bythe open-source galaxy image simulation package GalSim (Rowe et al. 2015). The calibration factors consist of themultiplicative factor m , which is shared among e and e ,and the additive bias for each ellipticity component ( c , c ) .The image simulations are also used to calibrate other quan- tities which are necessary for lensing signal measurement,i.e., the per-component rms shapes e rms , the measurementnoise of galaxy shapes σ e , and the inverse variance weightfrom both e rms and σ e . Note that we use the shape cataloguewith a star mask called Sirius.Among the photo- z s provided by the HSC first-year cat-alogue (Tanaka et al. 2018), we use the photo- z estimatedby a machine learning method based on self-organising mapcalled MLZ (Carrasco Kind & Brunner 2014) . This is be-cause MLZ is one of the algorithms which yields the smallestsystematic uncertainties in lensing measurement within theredshift range of CMASS galaxies (Miyatake et al. 2019)and is used for creating the mock catalogues which will bedescribed in Section 2.3. The redshift distribution of sourcegalaxies are shown in Fig. 2. The large fraction of sourcegalaxies are at higher redshifts compared to filaments, whichallows for high signal-to-noise filement lensing measurement. Predicting a filament lensing signal analytically is notstraightforward since filaments are not entirely in the linearregime. Analytical estimation of covariance is not straight-forward, either, since our filament lensing estimator, whichwill be described in Section 3.2, uses a source galaxy mul-tiple times and thus strong correlation between spatial binsare expected. In addition, since the HSC first-year field isonly . deg , we expect to have large cosmic variancewhich can contaminate the lensing signal through the in-trinsic scatter of filament properties and fluctuations in largescale structure along the line-of-sight. For these reasons, weuse the CMASS and source mock catalogue based on N -bodydark matter only simulations to estimate the theoretical pre-diction and covariance of filament lensing signal. When es-timating these quantities, we use 108 realisations of mocks,each of which corresponds to the measurement with the realdata performed in this paper.For mock source galaxies, we use the mock cataloguegenerated by Shirasaki et al. (2019). Here we summarise howthe source mock catalogue was created. Those who are in-terested in details should refer to the aforementioned paper.The weak lensing shear in mock catalogue is simulated byfull-sky ray-tracing through multiple lens planes (Takahashiet al. 2017) which are constructed from 14 boxes with a dif-ferent box size. Neighbouring lens planes have the comovingdistance interval of ∆ χ = h − Mpc . Source galaxies aredistributed in the light-cone simulations following the spa-tial distribution extracted from the actual HSC shape cata-logue and the photo- z estimated by MLZ. The ellipticity ofeach galaxy is also extracted from the HSC shape catalogue,but is randomly rotated to vanish coherent shear caused byweak lensing in the actual data. The lensing shear obtainedfrom the ray-tracing simulation is then added to the ran-domised galaxy shapes to mimic the observed galaxy shapes.Note that the weak lensing shear from ray-tracing simula-tions includes all light deflections by the density fluctuationalong the line-of-sight, and thus can be used to derive co-variance including the line-of-sight fluctuations. The mocksource catalogue also includes the lensing shear without in-trinsic shapes of galaxies, which will be used for estimatinga theoretical prediction of our filament lensing signal.We create our CMASS mocks by populating galaxies MNRAS , 1–10 (2019)
H. Kondo et al. in dark matter halos in the N-body simulations used for theray-tracing simulations, following the halo occupation distri-bution (HOD; Zheng et al. 2005) estimated from the CMASSgalaxy-galaxy clustering and abundance. First, we divide thefull CMASS LSS sample into five redshift bins so that thebinning scheme should be the same as the lens plane defini-tion in the ray-tracing simulations. When one of the edges ofthe full CMASS sample, i.e., z min = . or z max = . , fallswithin a redshift bin, we truncate the bin at the edge. There-fore, our final redshift binning is z ∈ [ . , . ] , [ . , . ] , [ . , . ] , [ . , . ] , and [ . , . ] . We then measure thegalaxy abundance and the projected galaxy-galaxy cluster-ing for each redshift bin in the same manner as Miyatakeet al. (2015). While using jackknife covariance for clustering,we assume conservative statistical uncertainties for abun-dance, i.e., of the measured abundance, to absorb thesystematic uncertainties in the CMASS selection functiondue to the depth variations of the survey. To obtain HODparameters for the CMASS sample within a given redshiftbin, we simultaneously fit the projected galaxy-galaxy clus-tering and abundance signals with cosmological parametersfixed. To model the clustering and abundance, we use themodel based on Dark Emulator , which is described in Ap-pendix G in Nishimichi et al. (2018) in detail, without off-centering parameters and incompleteness parameters. Wepopulate central galaxies at the centre of halos and satellitegalaxies using the measured HOD. When populating satel-lites, we assume their radial distribution follows that of darkmatter in each host halo. The detail of generation of mockHOD galaxies is found in Shirasaki et al. (2017). We thengenerate mock pairs in the same manner as described in Sec-tion 2.1, which will be used to estimate theoretical predictionand covariance in Section 4.1 and Section 3.3, respectively.
In this paper, we use the filament lensing signal estimatorproposed by Clampitt et al. (2016). This estimator is com-puted as follows. First, we stretch the pairs such that theline connecting the pairs has the same length. This stretchis done in the same way to both directions of the connectingline and perpendicular to the connecting line. Second, wealign the pairs, set up two-dimensional spatial bins by di-viding the stretched coordinates into grid cells, and computethe stacked lensing shear in each bin. Finally, we combinethe lensing shear computed in multiple bins to cancel outthe contribution from dark matter halos. In the followingsections, we describe how we compute the lensing signal ina given two-dimensional bin and explain how the Clampittet al. (2016) estimator cancels out the contribution from ha-los. We also describe how we estimate the covariance of thefilament lensing signal.
In this section, we describe how to measure the stacked weaklensing signal at a given 2-dimensional bin x . Using the HSCgalaxy shape catalogue, the stacked excess surface mass den- sity (cid:104) ∆Σ (cid:105) is estimated as (cid:104) ∆Σ i (cid:105)( x ) = + ˆ m (cid:205) ls ∈ x (cid:104) w ls (cid:104) Σ − (cid:105) − ls (cid:0) e i , ls / R − c i , ls (cid:1)(cid:105)(cid:205) ls ∈ x w ls , (1)where i denotes the shear component, i.e., i = + is parallelto the line connecting two CMASS galaxies and i = × is theone rotated by 45 degrees against the + component, ls runsover lens-source pairs whose source position falls in the 2-dimensional bin x , w ls is the weight function of a lens-sourcepair based on the CMASS weight and source weight writtenas w ls = w l w l (cid:104) Σ − (cid:105) ls e , ls + σ e , ls , (2)where w l and w l is the CMASS weight of each galaxy ina CMASS pair, (cid:104) Σ − (cid:105) ls is the expected critical surface massdensity for a lens-source pair, which corrects for dilutionof lensing signal due to foreground source contamination,defined as (cid:104) Σ − (cid:105) ls = ∫ ∞ dz Σ − ( z l , z ) P s ( z ) , (3)where P s ( z ) is the normalised probability distribution func-tion of a photo- z of the source galaxy and Σ − ( z l , z s ) is de-fined as Σ − ( z l , z s ) = π Gc D A ( z l ) D A ( z l , z s ) D A ( z s ) , (4)for z s > z l and Σ − ( z l , z s ) = for z s < z l , where D A ( z ) isthe angular diameter distance at redshift z , R is the shearresponsivity which corrects for the non-linear operation inthe summation process of galaxy ellipticity (e.g., Bernstein& Jarvis 2002) R = − (cid:205) ls ∈ x w ls e , ls (cid:205) ls ∈ x w ls , (5) ˆ m is the weighted mean of the multiplicative bias factor ˆ m = (cid:205) ls ∈ x w ls m ls (cid:205) ls ∈ x w ls , (6)and c i , ls is the additive bias for a lens-source pair ls .In Section 4.3, we will show two-dimensional shear map.To compute stacked reduced shear at bin x , we use theslightly modified version of Eq (1) (cid:104) γ i (cid:105)( x ) ∼ (cid:104) g i (cid:105)( x ) = + ˆ m (cid:205) ls ∈ x (cid:104) w (cid:48) ls (cid:0) e i , ls / R − c i , ls (cid:1)(cid:105)(cid:205) ls ∈ x w (cid:48) ls , (7)where w (cid:48) ls denotes the inverse-variance weight optimised forthe shacked shear measurement w (cid:48) ls = w l w l e , ls + σ e , ls . (8) To extract filament lensing signal from weak lensing shearcaused by paired galaxies, we use the estimator proposed byClampitt et al. (2016). This estimator is constructed to can-cel out the contribution from lensing signal caused by darkmatter halos. Figure 3 shows how this estimator works. Letus suppose to pick up a bin“p1”. Since the weak lensing shear
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MNRAS000 , 1–10 (2019) ilament lensing with SDSS BOSS and HSC caused by a spherically symmetric dark matter halo exhibitstangential alignment to the line from the halo centre to thesource galaxy , adding the lensing signal at the point rotatedby 90 degrees from “p1”, which is denoted as “p2”, cancelsout the contribution from the halo on the left-hand side. Inaddition, both “p1” and “p2” are also contaminated by thehalo on the right-hand side, which can be cancelled out byadding “p3” and “p4”, respectively. Note that the combina-tion of “p3” and “p4” also cancels out the contribution fromthe halo on the left-hand side. Assuming the filament hastranslational and line symmetry along the line connectingthe halos, the filament lensing estimator can be written asa function of the distance from the connecting line, which isdenoted as y , ∆Σ fil i ( y a ) ≡ (cid:213) b = [ ∆Σ i ( x b , y a ) + ∆Σ i ( y a , − x b ) + ∆Σ i ( − x b , − y a ) + ∆Σ i ( − y a , x b − ) + ∆Σ i ( x b , − y a ) + ∆Σ i ( y a , x b ) + ∆Σ i ( − x b , y a − ) + ∆Σ i ( − y a , − x b )] . (9)Here, the centre of halo on the left-hand side is taken as theorigin of the stretched coordinates and the distance betweentwo halos is normalised to unity. We divide the connectingline into 10 bins. The each term in the summation on theright-hand side corresponds to Eq. (1) where we omit (cid:104)(cid:105) forsimplicity. For example, when a = and b = , the first, sec-ond, third, and fourth term corresponds to “p1”, “p2”, “p4”,and “p3.”. The summation for the first y bin is denoted asfour arrows in Figure 3, which is denoted as the summationof the first four terms in Eq. (9). Under the assumption ofline symmetry, the shear at positions mirrored against theconnecting line are also in the summation. These shears cor-respond to the last four terms in Eq. (9). Note that we takethe range of a as ≤ a ≤ , since ≤ a ≤ are alreadyincluded in calculating the estimator (see the top arrow inFig. 3 for the a = case).We also subtract filament signals estimated with therandom pairs from Eq. (9). Although the random pairs donot exhibit any filament signal, the random signal subtrac-tion corrects for observational systematics, such as imperfectPSF correction and imperfect shape noise cancellation at theedge of survey fields. Estimating the covariance of the filament lensing estimatoris not straightforward, since the filament signal at differentbins is expected to be significantly correlated due to intrinsicscatter of filament properties, large scale structure projectedalong the line-of-sight, and multiple use of source galaxies inthe stacking process. Using the mock catalogue of CMASSgalaxies and source galaxies populated in the ray-tracingsimulations described in Section 2.3, we can estimate thecovariance naturally taking into account these effects. More As described in Appendix of Clampitt et al. (2016), ellipticaldark matter halo would not significantly affect their filament es-timator.
Figure 3.
Schematics of the Clampitt et al. (2016) estimator.Two grey circles denote halos of a pair, and the origin of coor-dinates is at the centre of the halo on the left-hand side. Forthe shear at a given position “p1”, adding the shear at “p2” willcancel out the contribution from the halo on the left-hand side.Likewise, adding the shear at “p3” and “p4” to the shear at “p1”and “p2” cancels out the contribution from the halo on the right-hand side, respectively. Furthermore, adding the shear at “p3” to“p4” cancels out the contribution from the halo on the left-handside. Thus the combination of these four points cancels out allthe contribution form the halos. Assuming the translational sym-metry of the filament along the line connecting the halos, shearsat the positions along the x -axis are summed up, which is shownas the green arrow connecting the halos. Taking the combinationof four points, shear at the positions along all green arrows aresummed up. In addition, assuming the line symmetry of the fil-ament along the connecting line, the positions mirrored againstthe connecting line are summed up, although it is not shown inthis figure for simplicity. The combination of shear at these po-sitions composes the first point of the the Clampitt et al. (2016)estimator. The other points are computed in the same manner asa function of y as shown in this figure. precisely, we estimate the filament lensing covariance as Cov i ( y a , y b ) = N r − (cid:213) r (cid:20) ∆Σ fil , mock , ri ( y a ) − ∆Σ fil , mock i ( y a ) (cid:21) × (cid:20) ∆Σ fil , mock , ri ( y b ) − ∆Σ fil , mock i ( y b ) (cid:21) , (10) MNRAS , 1–10 (2019)
H. Kondo et al. where i denotes the + or × component, r runs the 108 mockrealisations, ∆Σ fil , mock , ri ( y a ) is the filament lensing estima-tor of r -th mock realisation, and ∆Σ fil , mock i ( y ) is the estima-tor averaged over the 108 mock realisations. The correlationmatrix of the estimated covariance which is defined as Cor i ( y a , y b ) = Cov i ( y a , y b ) (cid:112) Cov i ( y a , y a ) Cov i ( y b , y b ) . (11)The correlation matrix for the + component is shown in theleft panel of Fig. 4, which exhibits the strong correlationbetween bins beyond neighbours.In addition to the observed ellipticity, the mock sourcegalaxy catalogue has the pure weak lensing shear from theray-tracing simulations. This allows us to estimate covari-ance without intrinsic shape noise of source galaxies. Theright panel of Fig. 4 shows the ratio of the mock shear covari-ance to the total covariance. The mock shear covariance oc-cupies about 60 percent of the total covariance and is almostconstant over both diagonal and off-diagonal components.This means that our filament lensing measurement is nolonger limited by the intrinsic shape noise of source galaxies.Rather, the shape noise is comparable to the combination ofother sources of covariance which is not relevant to observa-tional conditions, i.e., the fluctuation of large scale structurealong the line of sight and the intrinsic scatter of filamentproperties. This is made possible by the source number den-sity of the HSC survey which is the highest among the on-going weak lensing surveys. Figure 5 shows our filament lensing measurement of theCMASS pairs with the error bars estimated by the 108 re-alisations of mock catalogue. We detect the filament signalat 3.9 σ significance against null hypothesis. As described inSection 3.3, the intrinsic shape noise of source galaxies is al-ready comparable to the combination of other componentsin the covariance. Therefore, increasing the number densityof galaxies would not help to increase signal-to-noise ratio.Rather, increasing survey area will reduce both the shapenoise and other components in the covariance.The theoretical prediction estimated from the weaklensing shear recorded in the mock catalogues is also shownin Fig. 5. The χ value of the data against the theoreti-cal prediction is χ = . with the degree of freedom of 5,which leads to the p -value of . . Hence, our measurementis consistent with the theoretical prediction. Our data pointsare consistently smaller from the theoretical prediction, andreaders may think our χ value is too small. This is due tothe fact that the filament lensing signals between bins arehighly correlated as shown in Section 3.3. To confirm our analysis is not dominated by systematic un-certainties we perform null tests by measureing the × modeof the CMASS pair lensing signal and the + and × modes ofthe separated CMASS pair lensing signal.We expect the × mode of the filament lensing estimator Table 1.
Summary table of null tests. Note that we use the signalafter the random signal correction.null test χ /dof p -value × mode 7.27/5 0.20separated pair + mode 2.34/5 0.80separated pair × mode 6.05/5 0.30 is consistent with zero due to the translational symmetry ofthe filament along the line connecting the pair. The × -modesignal is shown in Fig. 6 and the χ value and p -value aresummarised in Table 1, which confirms that the the × modeis consistent with null.We also expect both the + mode and the × mode ofthe separated pair signal are consistent with null, since theseparated pair is defined such that their line-of-sight distanceis h − Mpc < | ∆ χ | < h − Mpc as described in Section 2.1and we do not expect massive filaments exist between thehalos. These signals are shown in Fig. 6 and the χ valuesand p -values are summarised in Table 1. We also confirmthat separated pair signals are consistent with null. The left panel of Fig. 7 shows two-dimensional stacked shearmap of our CMASS pair sample. The vertical shear betweenCMASS halos are the sum of the tangential shear fromCMASS halos and filament shear. Just from this figure itis not clear if there is a filament between halos. This demon-strates that it is hard to distinguish the filaments componentfrom dark matter halos in two-dimensional shear map. How-ever, the Clampitt et al. (2016) estimator nicely eliminatesthe contributions from halos, and thus isolates the signalfrom the filaments, as described in Section 4.1. In Fig. 7, weobserve lensing shear surrounding the pair at large scales.This is part of the galaxy-galaxy-shear correlation function,which was measured by Simon et al. (2013).From this shear map, we compute the convergence map,which is shown in the right panel of Fig. 7. In doing so, weuse the prescription by Kaiser & Squires (1993) with theconvolution kernel adopted in Miyazaki et al. (2015), i.e., κ ( x ) = ∫ d y γ t ( y ; x )) Q G (| y |) , (12)where γ t is the tangential component of the shear at theposition y with respect to x and Q (| x |) is the convolutionkernel Q G ( x ) = π x (cid:34) − (cid:32) + x x G (cid:33) exp (cid:32) − x x G (cid:33)(cid:35) . (13)We choose the width of kernel as x G = R pair / . In thisconvergence map, we can clearly see the diffuse filament be-tween circularly symmetric galaxy halos. This independentlysupports the “thick” filament predicted by the N -body sim-ulations rather than the “thin” filament modelled by a lineof small halos. MNRAS000
Summary table of null tests. Note that we use the signalafter the random signal correction.null test χ /dof p -value × mode 7.27/5 0.20separated pair + mode 2.34/5 0.80separated pair × mode 6.05/5 0.30 is consistent with zero due to the translational symmetry ofthe filament along the line connecting the pair. The × -modesignal is shown in Fig. 6 and the χ value and p -value aresummarised in Table 1, which confirms that the the × modeis consistent with null.We also expect both the + mode and the × mode ofthe separated pair signal are consistent with null, since theseparated pair is defined such that their line-of-sight distanceis h − Mpc < | ∆ χ | < h − Mpc as described in Section 2.1and we do not expect massive filaments exist between thehalos. These signals are shown in Fig. 6 and the χ valuesand p -values are summarised in Table 1. We also confirmthat separated pair signals are consistent with null. The left panel of Fig. 7 shows two-dimensional stacked shearmap of our CMASS pair sample. The vertical shear betweenCMASS halos are the sum of the tangential shear fromCMASS halos and filament shear. Just from this figure itis not clear if there is a filament between halos. This demon-strates that it is hard to distinguish the filaments componentfrom dark matter halos in two-dimensional shear map. How-ever, the Clampitt et al. (2016) estimator nicely eliminatesthe contributions from halos, and thus isolates the signalfrom the filaments, as described in Section 4.1. In Fig. 7, weobserve lensing shear surrounding the pair at large scales.This is part of the galaxy-galaxy-shear correlation function,which was measured by Simon et al. (2013).From this shear map, we compute the convergence map,which is shown in the right panel of Fig. 7. In doing so, weuse the prescription by Kaiser & Squires (1993) with theconvolution kernel adopted in Miyazaki et al. (2015), i.e., κ ( x ) = ∫ d y γ t ( y ; x )) Q G (| y |) , (12)where γ t is the tangential component of the shear at theposition y with respect to x and Q (| x |) is the convolutionkernel Q G ( x ) = π x (cid:34) − (cid:32) + x x G (cid:33) exp (cid:32) − x x G (cid:33)(cid:35) . (13)We choose the width of kernel as x G = R pair / . In thisconvergence map, we can clearly see the diffuse filament be-tween circularly symmetric galaxy halos. This independentlysupports the “thick” filament predicted by the N -body sim-ulations rather than the “thin” filament modelled by a lineof small halos. MNRAS000 , 1–10 (2019) ilament lensing with SDSS BOSS and HSC y / R pair y / R p a i r C o r ij = C o v ij C o v ii C o v jj y / R pair y / R p a i r C o v s h e a r / C o v t o t a l Figure 4. left panel:
Correlation matrix of our filament lensing measurement. The covariance is estimated from from 108 mock realisations.The correlation matrix is higher than 0.77, which means our filament lensing measurement is highly correlated due to the multiple use ofsource galaxies, the intrinsic scatter of filament properties, and the fluctuations in the projected large scale structure. right panel:
Ratiobetween covariance estimated from the pure lensing shear in the mock catalogue to the total covariance including intrinsic galaxy shapenoise. The covariance from the intrinsic scatter of filament properties and fluctuations in the projected large scale structure occupies60% of the covariance, which means our measurement is no longer limited solely by the shape noise.
The first detection of the filament between massive galaxypairs was made by Clampitt et al. (2016) using SDSS LRGpairs at a typical redshift z ∼ . and SDSS source galax-ies, where they used the exactly same estimator as ours.We significantly extend the redshift range of galaxy pairsto z ∼ . . The significance level of their detection intheir measurement was 4.5 σ , which is slightly higher thanours despite of their low number density of source galax-ies ( ∼ . arcmin − ). This is because their survey area( ∼ deg ) is much larger than ours ( ∼ deg ), andthus their measurement is not limited by the large scalestructure noise. This in turn means that we will soon be ableto measure the lensing signal at a higher signal-to-noise ra-tio as the HSC survey area increases. The amplitude of theirfilament signal is as twice as large as ours. This is likely dueto the difference in halo mass between the LRG and CMASSsample, i.e., the CMASS sample is less massive than LRG,which results in less massive filaments. This might be alsobecause the structure growth is larger at z = . than thoseat z = . , so the filaments may have more masses even ifthe mass of the halos are the same. We may check this effectby using mock simulations, but this is beyond the scope ofthis paper.Epps & Hudson (2017) reported the filament measure-ment of galaxy pairs of the combined BOSS LOWZ andCMASS sample with the mean redshift of (cid:104) z (cid:105) ∼ . , us-ing the CFHTLenS source galaxies (Heymans et al. 2012).Using the two-dimensional convergence map where the con-tribution from halos was removed by using their “control” sample, they claimed the detection at 5 σ significance. Theyexplored modelling the filament signal by the galaxy-galaxy-convergence three point correlation function, and found rea-sonable agreement with their measurement. In contrast, ourpair sample purely consists of the CMASS sample whichpushed the mean redshift to (cid:104) z (cid:105) ∼ . . We also derive themore robust theoretical prediction using the mock CMASSpairs and source galaxies based on ray-tracing simulations.He et al. (2018) reported the first detection of filamentlensing signal imprinted in CMB by cross-correlating the fil-ament map derived from the CMASS galaxies (Chen et al.2016) with the Planck CMB convergence map (Planck Col-laboration et al. 2014b) in Fourier space. Note that their fila-ment sample is intrinsically different from our sample. Theyobserved filaments where CMASS galaxies reside whereaswe observe filaments between CMASS pairs, and thus theirfilament sample is expected to be more massive than ours.The main focus of de Graaff et al. (2019) was probingmissing baryons through thermal Sunyaev-Zel’dovich effect(tSZ; Sunyaev & Zeldovich 1969, 1972) from filaments be-tween CMASS pairs, but they also measured lensing signalusing the Planck convergence map to investigate the con-nection between gas and dark matter filaments. Their two-dimensional convergence measurement yielded 1.9 σ , whichis smaller than our measurement despite of their use of theentire CMASS sample without any spatial cut. This factdemonstrates the statistical power of the HSC survey en-abled by the high number density or source galaxies. In ad-dition, HSC source galaxies has the lensing kernel whichmatches to the CMASS sample better than CMB lensing. MNRAS , 1–10 (2019)
H. Kondo et al. y / R pair [ M h / p c ] datatheory Figure 5.
Result of our filament lensing measurement and the-oretical prediction. The light-blue solid line denotes our filamentlensing measurement of CMASS galaxy pairs. The covariance isestimated from 108 realisations of mock measurements. The fila-ment lensing is detected at 3.9 σ significance. The orange dashedline is the theoretical prediction obtained by averaging 108 mockmeasurements. When estimating the theoretical prediction, weuse pure weak lensing shear in mocks, i.e., shear without shapenoise, to efficiently reduce the statistical uncertainties. The orangeband denotes the statistical uncertainties of the averaged mockmeasurements. The χ value and p -value between the measure-ment and theoretical prediction is 8.1 with the degree of freedomof 5 and 0.15, respectively. Although the data points are con-sistently smaller than theoretical prediction, the small χ valuestill suggests the data is consistent with theory. This is becausethe the filament estimators between the spatial bins are highlycorrelated. We have measured weak gravitational lensing caused by fila-mentary structures between the CMASS galaxy pairs usingthe Subaru HSC first-year galaxy shape catalogue. Desiteof the small overlap between BOSS and HSC ( ∼ deg ),the high number density of the HSC shape catalogue ( ∼ arcmin − ) enabled us to measure the filament lensingsignal at 3.9 σ significance, which is the highest signal-to-noise measurement of the filament between high-redshift( z ∼ . ) galaxy-scale halos to date. We have used theClampitt et al. (2016) estimator to cancel out the contribu-tion from the dark matter halos and extract the signal fromthe filaments. We have found that the covariance is highlycorrelated and the contribution from the combination of theintrinsic scatter of filament properties and the fluctuationsin large scale structure is already comparable to the contri-bution from the intrinsic shape noise from source galaxies.This demonstrates the statistical power of the HSC survey.We expect the signal-to-noise ratio will rapidly increase asthe survey field grows. At the end of the HSC survey, thesignal-to-noise ratio of our measurement will be improved bythe factor of three, which will allows us to do more detailedstudies such as the dark matter distribution and redshiftevolution of filaments. We have observed negative value ofthe Clampitt et al. (2016) estimator, which supports the“thick” filament model and consistent with what Clampitt y / R pair [ M h / p c ] × modeseparated pair + modeseparated pair × mode Figure 6.
Summary plot of null tests. The light-blue solid linedenotes the × -mode lensing signal of the CMASS pairs. The or-ange dashed line and green dotted line denotes the + -mode and x -mode lensing signal of CMASS pairs whose separation alongthe line-of-sight is h − Mpc < | ∆ χ | < h − Mpc , respectively.The covariance is estimated from the measurements on 108 real-isations of mock simulations which are performed exactly in thesame manner as on the data. All the null tests are passed as sum-marised in Table 1. Note that the orange dashed line and greendotted line are slightly shifted along the x -axis for illustrativepurposes. et al. (2016) found in the LRG pair sample. We have con-firmed this fact by drawing the two-dimensional convergencemap.de Graaff et al. (2019) performed a pioneering work tocombine multi-wavelength data to study the connection be-tween light, gas, and dark matter within filaments. In comingyears, as various surveys for imaging, spectroscopic, X-ray,and SZ will collect more data, more detailed studies to inves-tigate such astrophysical properties of filaments will becomepossible.Using the ongoing and upcoming spectroscopic surveyssuch as SDSS-IV/eBOSS (Dawson et al. 2016), Dark EnergySpectroscopic Instrument (DESI; DESI Collaboration et al.2016), and Subaru Prime Focus Pectrograph (PFS; Takadaet al. 2014), we can extend the redshift range of a filamentsample. In addition, the upcoming imaging surveys such asLarge Synoptic Survey Telescope (LSST; LSST Science Col-laboration et al. 2009), Euclid (Laureijs et al. 2011), Wide-field Infrared Survey Telescope (WFIRST; Spergel et al.2015) will enable us to pursue the weak lensing measurementof these filaments with even higher signal-to-noise ratio. ACKNOWLEDGEMENTS
The Hyper Suprime-Cam (HSC) collaboration includes theastronomical communities of Japan and Taiwan, and Prince-ton University. The HSC instrumentation and software weredeveloped by the National Astronomical Observatory ofJapan (NAOJ), the Kavli Institute for the Physics andMathematics of the Universe (Kavli IPMU), the Univer-sity of Tokyo, the High Energy Accelerator Research Or-
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MNRAS000 , 1–10 (2019) ilament lensing with SDSS BOSS and HSC shear = 0.0011.5 1.0 0.5 0.0 0.5 1.0 1.5 x / R pair y / R p a i r l o g ( + e ) Figure 7. left panel:
Two-dimensional shear map of the stacked CMASS galaxy pairs. The shear between halos is composed of thetangential shear from both halos and the shear from filament which is perpendicular to the line connecting halos. Although the latteris unclear in this map, the filament signal is observed in the Clampitt et al. (2016) estimator as shown in Fig. 5. The tangential shearsurrounding the pair is from the large scale structure associated with the pair, which can be expressed as the galaxy-galaxy-shearcorrelation function. right panel:
Two-dimensional convergence map of the stacked CMASS galaxy pairs computed from the shear map.The colour scale shows log ( κ + − ) , where we add the offset to κ to avoid a negative argument in the logarithmic function. The convergencemap supports the “thick” filament model derived from N -body simulations. ganization (KEK), the Academia Sinica Institute for As-tronomy and Astrophysics in Taiwan (ASIAA), and Prince-ton University. Funding was contributed by the FIRST pro-gram from Japanese Cabinet Office, the Ministry of Educa-tion, Culture, Sports, Science and Technology (MEXT), theJapan Society for the Promotion of Science (JSPS), JapanScience and Technology Agency (JST), the Toray ScienceFoundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Prince-ton University.The Pan-STARRS1 Surveys (PS1) have been made pos-sible through contributions of the Institute for Astronomy,the University of Hawaii, the Pan-STARRS Project Office,the Max-Planck Society and its participating institutes, theMax Planck Institute for Astronomy, Heidelberg and theMax Planck Institute for Extraterrestrial Physics, Garch-ing, The Johns Hopkins University, Durham University,the University of Edinburgh, Queen’s University Belfast,the Harvard-Smithsonian Center for Astrophysics, the LasCumbres Observatory Global Telescope Network Incorpo-rated, the National Central University of Taiwan, the SpaceTelescope Science Institute, the National Aeronautics andSpace Administration under Grant No. NNX08AR22G is-sued through the Planetary Science Division of the NASAScience Mission Directorate, the National Science Founda-tion under Grant No. AST-1238877, the University of Mary-land, and Eotvos Lorand University (ELTE).This paper makes use of software developed for theLarge Synoptic Survey Telescope. We thank the LSSTProject for making their code available as free software athttp://dm.lsst.org.Based in part on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which isoperated by the Subaru Telescope and Astronomy Data Cen-ter at National Astronomical Observatory of Japan.HM acknowledges the support from JSPS KAK-ENHI Grant Number JP18H04350, JP18K13561, andJP19H05100. MS acknowledges the support in part fromby JSPS KAKENHI Grant Number JP18H04358. Numer-ical computations were in part carried out on Cray XC50at Center for Computational Astrophysics, National Astro-nomical Observatory of Japan. AJN acknowledges the sup-port in part by MEXT KAKENHI Grant Number 15H05890. REFERENCES
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