Dark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map
C. Chang, A. Pujol, B. Mawdsley, D. Bacon, J. Elvin-Poole, P. Melchior, A. Kovács, B. Jain, B. Leistedt, T. Giannantonio, A. Alarcon, E. Baxter, K. Bechtol, M. R. Becker, A. Benoit-Lévy, G. M. Bernstein, C. Bonnett, M. T. Busha, A. Carnero Rosell, F. J. Castander, R. Cawthon, L. N. da Costa, C. Davis, J. De Vicente, J. DeRose, A. Drlica-Wagner, P. Fosalba, M. Gatti, E. Gaztanaga, D. Gruen, J. Gschwend, W. G. Hartley, B. Hoyle, E. M. Huff, M. Jarvis, N. Jeffrey, T. Kacprzak, H. Lin, N. MacCrann, M. A. G. Maia, R. L. C. Ogando, J. Prat, M. M. Rau, R. P. Rollins, A. Roodman, E. Rozo, E. S. Rykoff, S. Samuroff, C. Sánchez, I. Sevilla-Noarbe, E. Sheldon, M. A. Troxel, T. N. Varga, P. Vielzeuf, V. Vikram, R. H. Wechsler, J. Zuntz, T. M. C. Abbott, F. B. Abdalla, S. Allam, J. Annis, E. Bertin, D. Brooks, E. Buckley-Geer, D. L. Burke, M. Carrasco Kind, J. Carretero, M. Crocce, C. E. Cunha, C. B. D'Andrea, S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, J. Estrada, A. Fausti Neto, E. Fernandez, B. Flaugher, J. Frieman, J. García-Bellido, R. A. Gruendl, G. Gutierrez, K. Honscheid, D. J. James, T. Jeltema, M. W. G. Johnson, M. D. Johnson, S. Kent, D. Kirk, E. Krause, K. Kuehn, S. Kuhlmann, O. Lahav, T. S. Li, M. Lima, M. March, P. Martini, F. Menanteau, R. Miquel, J. J. Mohr, et al. (20 additional authors not shown)
DDES 2016-0212FERMILAB-PUB-17-295-AE
MNRAS , 000–000 (0000) Preprint 21 December 2017 Compiled using MNRAS L A TEX style file v3.0
Dark Energy Survey Year 1 Results:Curved-Sky Weak Lensing Mass Map
C. Chang ∗ , A. Pujol , , , B. Mawdsley , D. Bacon , J. Elvin-Poole , P. Melchior , A. Kov´acs , B. Jain ,B. Leistedt , , T. Giannantonio , , , A. Alarcon , E. Baxter , K. Bechtol , M. R. Becker , , A. Benoit-L´evy , , , G. M. Bernstein , C. Bonnett , M. T. Busha , A. Carnero Rosell , , F. J. Castander ,R. Cawthon , L. N. da Costa , , C. Davis , J. De Vicente , J. DeRose , , A. Drlica-Wagner ,P. Fosalba , M. Gatti , E. Gaztanaga , D. Gruen , , , J. Gschwend , , W. G. Hartley , , B. Hoyle ,E. M. Huff , M. Jarvis , N. Jeffrey , T. Kacprzak , H. Lin , N. MacCrann , , M. A. G. Maia , ,R. L. C. Ogando , , J. Prat , M. M. Rau , R. P. Rollins , A. Roodman , , E. Rozo , E. S. Rykoff , ,S. Samuroff , C. S´anchez , I. Sevilla-Noarbe , E. Sheldon , M. A. Troxel , , T. N. Varga , ,P. Vielzeuf , V. Vikram , R. H. Wechsler , , , J. Zuntz , T. M. C. Abbott , F. B. Abdalla , , S. Allam ,J. Annis , E. Bertin , , D. Brooks , E. Buckley-Geer , D. L. Burke , , M. Carrasco Kind , ,J. Carretero , M. Crocce , C. E. Cunha , C. B. D’Andrea , S. Desai , H. T. Diehl , J. P. Dietrich , ,P. Doel , J. Estrada , A. Fausti Neto , E. Fernandez , B. Flaugher , J. Frieman , , J. Garc´ıa-Bellido ,R. A. Gruendl , , G. Gutierrez , K. Honscheid , , D. J. James , T. Jeltema , M. W. G. Johnson ,M. D. Johnson , S. Kent , , D. Kirk , E. Krause , K. Kuehn , S. Kuhlmann , O. Lahav , T. S. Li ,M. Lima , , M. March , P. Martini , , F. Menanteau , , R. Miquel , , J. J. Mohr , , , E. Neilsen ,R. C. Nichol , D. Petravick , A. A. Plazas , A. K. Romer , M. Sako , E. Sanchez , V. Scarpine ,M. Schubnell , M. Smith , R. C. Smith , M. Soares-Santos , F. Sobreira , , E. Suchyta , G. Tarle ,D. Thomas , D. L. Tucker , A. R. Walker , W. Wester , Y. Zhang (DES Collaboration) The authors’ affiliations are shown in Appendix D. ∗ e-mail address: [email protected]
21 December 2017
ABSTRACT
We construct the largest curved-sky galaxy weak lensing mass map to date from the DES first-year (DES Y1) data. The map, about 10 times larger than previous work, is constructed over acontiguous ≈ ,
500 deg , covering a comoving volume of ≈
10 Gpc . The effects of masking,sampling, and noise are tested using simulations. We generate weak lensing maps from twoDES Y1 shear catalogs, M ETA C ALIBRATION and I M SHAPE , with sources at redshift 0 . < z < . , and in each of four bins in this range. In the highest signal-to-noise map, the ratiobetween the mean signal-to-noise in the E-mode and the B-mode map is ∼ ∼
2) whensmoothed with a Gaussian filter of σ G =
30 (80) arcminutes. The second and third moments ofthe convergence κ in the maps are in agreement with simulations. We also find no significantcorrelation of κ with maps of potential systematic contaminants. Finally, we demonstrate twoapplications of the mass maps: (1) cross-correlation with different foreground tracers of massand (2) exploration of the largest peaks and voids in the maps. Key words: gravitational lensing: weak, cosmology: dark matter, surveys
One way to map the mass distribution of the Universe is by us-ing the technique of weak gravitational lensing. (Kaiser & Squires1993; Massey et al. 2007; Van Waerbeke et al. 2013; Vikram et al. 2015; Chang et al. 2015; Oguri et al. 2017). The motivations forgenerating these mass maps using weak lensing are twofold. First,it is easy to pick out distinct features and understand the qualitativecharacteristics of the mass distribution from maps. Second, as the c (cid:13) a r X i v : . [ a s t r o - ph . C O ] D ec C. Chang et al. maps ideally preserve the full, uncompressed information for thefield, they enable the extraction of non-Gaussian information be-yond the standard two-point statistics used in cosmology (e.g. Ab-bott et al. 2016; Kwan et al. 2017; Hildebrandt et al. 2017). Thesenon-Gaussian statistics are being explored using 3-point statistics(Cooray & Hu 2001; Dodelson & Zhang 2005), peak counts (Di-etrich & Hartlap 2010; Kratochvil et al. 2010; Kacprzak et al.2016), and the full Probability Density Function (PDF) of the map(Clerkin et al. 2015; Patton et al. 2016). As the statistical uncer-tainties in the current and future data sets decrease, we expect thesehigher-order statistics to offer new constraints that are complemen-tary to the more traditional two-point approaches.Physically, a weak lensing mass map, or convergence map,represents the integrated total matter density along the line-of-sight,weighted by a broad lensing kernel that peaks roughly half-waybetween the observer and the source galaxies from which the mea-surement is made. Since lensing does not distinguish between thespecies and dynamical state of the mass, or the “lens”, one can di-rectly probe mass with weak lensing, which is a key difference frommaps constructed from biased tracers of mass such as galaxies. Thetheoretical framework of constructing weak lensing convergencemaps from the weak lensing observable, shear, has been developedsince Kaiser & Squires (1993, hereafter KS) and Schneider (1996).Shear and convergence are second derivatives of the same lensingpotential field, which makes it possible to convert between them upto a constant.Small-field weak lensing mass maps have been commonlyused in galaxy cluster fields to study the detailed structure of thecluster mass distribution and compare with the distribution of bary-onic matter (Clowe et al. 2006; von der Linden et al. 2014; Mel-chior et al. 2015). These maps have relatively high signal-to-noisebecause the cluster lensing signal is ∼
10 times larger than the lens-ing signal from the large-scale structure (Bartelmann & Schneider2001), and the information about the fields was obtained using deepimaging to achieve a high number density of source galaxies forweak lensing measurements. A number of algorithms beyond KSwere developed to specifically tackle the mass reconstruction withclusters and have been successfully implemented on data (Seitzet al. 1998; Marshall et al. 2002; Leonard et al. 2014).Wide-field convergence maps, on the other hand, have onlybeen constructed recently, thanks to the development of dedicatedweak lensing surveys that cover patches of sky on the order of hun-dreds of square degrees or larger. This includes the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS, Erben et al. 2013),the KIlo-Degree Survey (KIDS, de Jong et al. 2015), the HyperSuprimeCam Survey (HSC, Aihara et al. 2017) and the Dark En-ergy Survey (DES, Flaugher 2005). Van Waerbeke et al. (2013) wasthe first to study in detail these wide-field weak lensing mass mapsin four fields (adding up to a total of 154 deg ) of the CFHTLenSdata, including the noise properties, high-order moments, and thecross-correlation with galaxies. In Vikram et al. (2015) and Changet al. (2015), we carried out a similar analysis with early DES Sci-ence Verification (SV) data using a 139 deg contiguous region ofthe sky. Recent work from HSC (Mandelbaum et al. 2017; Oguriet al. 2017) also carried out an analysis of mass map reconstructionusing the HSC data in both 2D and 3D. Although the area of thesemaps are not as large (the total area of the data set is 136.9 deg ,split into six separate fields), the number density of the sourcesis several times larger than in the other data sets (25 galaxies perarcmin ), which allows for reconstruction on much smaller scales.Oguri et al. (2017) looked at cross-correlation of the mass mapswith galaxy distributions and several systematics tests. All three studies described above use the KS method under flat-sky approxi-mation, and show that the mass maps contain significant extractablecosmological information.Continuing from the SV work described above to the first yearof DES data (DES Y1), we present in this paper a weak lensingmass map of ∼ ,
500 deg , more than ten times larger than theSV map. A few advances over the SV studies were made: First,given the large area of the mass map on the sky, it was necessaryto go beyond the flat-sky approximation and employ curved-skyestimators. The implementation of the curved-sky reconstructionborrows from tools developed for CMB polarisation analyses andhas been studied in detail in the context of weak lensing mass map-ping and cosmic shear (Heavens 2003; Castro et al. 2005; Heavenset al. 2006; Kitching et al. 2014; Leistedt et al. 2017; Wallis et al.2017). The first all sky curved weak lensing maps constructed fromsimulations were presented in Fosalba et al. (2008), which was anextension from the work on constructing mock galaxy catalogs inGaztanaga & Bernardeau (1998). Second, we move from a singleredshift bin to multiple redshift bins, a first step towards construct-ing a 3D weak lensing map. These tomographic bins match thoseused in the DES Y1 cosmology analysis, thus making our mapsvery complementary to the series of DES Y1 papers that focus ontwo-point statistics (DES Collaboration et al. 2017; Troxel et al.2017; Prat et al. 2017; MacCrann et al. 2017). Specifically, thispaper presents the spatial configuration and phase information ofthe data that goes into these cosmological analyses. Finally, we ex-plore for the first time the possibility of constructing the lensingpotential and deflection fields. These fields are commonly stud-ied in the CMB lensing community, but seldom constructed andvisualised using measurements of galaxy lensing except in sometheoretical studies (Vallinotto et al. 2007; Dodelson et al. 2008;Chang & Jain 2014). The primary reason that potential and deflec-tion fields are seldom used in galaxy lensing is that the informationof the potential and deflection fields are on scales much larger (orlower (cid:96) modes) than the convergence field. This means that in pre-vious smaller data sets, there is not enough low (cid:96) information inthe data to reconstruct the potential and deflection fields. However,with the wide-field data used in this work, we are just beginningto enter the era where the reconstruction is not dominated by noiseand interesting applications can be explored. For example, with anaccurate deflection field, one can “delens” the galaxy fields andmove the observed galaxy positions back to their unlensed position,which would improve measurements such as galaxy-galaxy lens-ing (Chang & Jain 2014). Similarly, having a good estimate of thelensing potential could in principle provide foreground informationfor delensing the CMB (Marian & Bernstein 2007; Manzotti et al.2017; Yu et al. 2017).This paper is organised as follows. In Sec. 2 we introduce theformalism used for constructing the curved-sky convergence mapfrom shear maps. In Sec. 3, the data and simulations used in thispaper are described. We then outline in Sec. 4 the practical proce-dure of constructing the maps from the DES Y1 shear catalogs. InSec. 5 we present a series of tests using simulated data to quan-titatively understand the performance of the map-making methodas well as how that method interacts with the different sources ofnoise in the data. We then present our final DES Y1 mass mapsin Sec. 6 for different redshift bins and test for residual systematiceffects by cross-correlating the maps with observational quantities.We follow up with two applications of the mass maps in Sec. 7:(1) cross-correlation of the mass maps with different foregroundgalaxy samples, and (2) examination of the largest peaks and voidsin the maps. We conclude in Sec. 8. In Appendix A we investigate MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map the different approaches of masking and their effect on the recon-struction. In Appendix B we demonstrate the possibility of recon-structing the weak lensing potential and deflection maps in additionto the convergence map, which will become more interesting in fu-ture datasets as the sky coverage increases. Finally in Appendix Cwe present convergence maps from the I M SHAPE shear catalog(in addition to the maps from the M
ETA C ALIBRATION shear cata-log presented in the main text) to show the consistency between thecatalogs.
As mentioned in Sec. 1, the construction of convergence ( κ ) mapsfrom shear ( γ ) maps in data has been done assuming the flat-skyapproximation in most previous work (Van Waerbeke et al. 2013;Vikram et al. 2015; Chang et al. 2015) due to the relatively smallsky coverage involved. In fact, as shown in Wallis et al. (2017),the gain in moving from flat-sky to curved-sky is very marginal inthe case where the data is on the order of 100 deg . In this pa-per, our data set is sufficiently large to warrant a curved-sky treat-ment, which also prepares us for future, even larger, data sets. Thefundamental mathematical operation that we are interested in isto decompose a spin-2 field ( γ ) into a curl-free component and adivergence-free component. The curl-free component correspondsto the convergence signal, which is also referred to as the E-modeconvergence field κ E . The divergence-free component, which werefer to as κ B , is expected to be negligible compared to κ E for grav-itational lensing, but can arise from noise and systematics in theshear estimates. Mathematically, this operation is the same as theclassical Helmholtz decomposition, but generalised onto the spher-ical coordinates. We sketch below the formalism of converting be-tween the κ and γ maps as well as the deflection field η and thepotential field ψ . For detailed derivations, we refer the reader toBartelmann (2010); Castro et al. (2005); Wallis et al. (2017).Consider the 3D Newtonian potential Ψ defined at every givencomoving distance χ and angular position ( θ , φ ) on the sky. Theeffective lensing potential ψ is defined by projecting Ψ along theline-of-sight. That is (Bartelmann & Schneider 2001), ψ ( χ s , θ , φ ) = (cid:90) d χ (cid:48) f K ( χ s − χ (cid:48) ) f K ( χ (cid:48) ) f K ( χ s ) Ψ ( χ (cid:48) , θ , φ ) , (1)where f K depends on the curvature k of the Universe: f K ( χ ) = sin χ , χ , sinh χ for closed ( k = k =
0) and open ( k = − δ ( χ , θ , φ ) via the Poisson equation ∇ χ Ψ ( χ , θ , φ ) = Ω m H a δ ( χ , θ , φ ) , (2)where Ω m is the total matter density today, H is the Hubble con-stant today, and a = / ( + z ) is the scale factor. Note that the gra-dient ∇ χ is taken in the comoving radial direction.Expanding the lensing potential at a given comoving distance χ in spherical harmonics, we have ψ ( χ ) = ∑ (cid:96) m ψ (cid:96) m ( χ ) Y (cid:96) m ( θ , φ ) , ψ (cid:96) m ( χ ) = (cid:90) d Ω ψ ( χ ) Y ∗ (cid:96) m ( θ , φ ) , (3)where Y (cid:96) m are the spin-0 spherical harmonic basis set and ψ (cid:96) m ( χ ) is the coefficient associated with Y (cid:96) m at χ . Below we will omitthe χ reference in our notation for simplicity, but note that theseequations apply to the fields on a given redshift shell. To derive the spherical harmonic representation of shear andconvergence, we have κ = ( ð ð + ð ð ) ψ , (4) γ = γ + i γ = ð ð ψ , (5)where ð and ð are the raising and lowering operators that act onspin-weighted spherical harmonics, s Y (cid:96) m and follow a certain set ofrules (see e.g., Castro et al. 2005, for details). We can now definethe spherical representation of the convergence field and the shearfield to be κ = κ E + i κ B = ∑ (cid:96) m ( ˆ κ E ,(cid:96) m + i ˆ κ B ,(cid:96) m ) Y (cid:96) m , (6)and γ = γ + i γ = ∑ (cid:96) m ˆ γ (cid:96) m Y (cid:96) m . (7)Here Y (cid:96) m are spin-2 spherical harmonics. From Eq. (4) and Eq. (5)it follows thatˆ κ E ,(cid:96) m + i ˆ κ B ,(cid:96) m = − (cid:96) ( (cid:96) + ) ψ (cid:96) m , (8)ˆ γ (cid:96) m = ˆ γ E ,(cid:96) m + i ˆ γ B ,(cid:96) m = [ (cid:96) ( (cid:96) + )( (cid:96) − )( (cid:96) + )] ψ (cid:96) m = − (cid:115) ( (cid:96) + )( (cid:96) − ) (cid:96) ( (cid:96) + ) ( ˆ κ E ,(cid:96) m + i ˆ κ B ,(cid:96) m ) . (9)That is, one can convert between the three fields: κ , γ and ψ bymanipulating their spherical harmonics decompositions. The math-ematical operation described above is entirely analogous to a de-scription of linear polarisation such as that in the CMB polarisationmeasurements. In this analogy, the Q and U Stokes parameters cor-respond to the γ and γ . In the flat-sky limit, we have ð → ∂ and the decomposition into spherical harmonics is replaced by theFourier transform, Σ ψ (cid:96) m Y (cid:96) m → (cid:82) d (cid:96) ( π ) ψ ( (cid:96) ) e i (cid:96) · θ . The above equa-tions then reduce to the usual KS formalism.One can derive the lensing deflection field η in a similar fash-ion. The lensing deflection field is defined as the first derivative ofthe lensing potential η = η + i η = ð ψ , (10)so the deflection field is a spin-1 field and can be expressed as η = η + i η = ∑ (cid:96) m ˆ η (cid:96) m Y (cid:96) m . (11)Carrying through the derivation, we getˆ η (cid:96) m = [ (cid:96) ( (cid:96) + )] ψ (cid:96) m , (12)which is again related to the other lensing quantities via a simplelinear operation in the spin-harmonic space. That is, once γ is mea-sured, the other fields ( κ , η and ψ ) can be constructed using theformalism described above.From Eq. (8) and Eq. (9) we observe from which (cid:96) modes κ , η and ψ receive their dominant contributions: ψ receives most con-tribution from the lowest (cid:96) modes, η receives contribution fromslightly higher (cid:96) modes, and κ receives contribution from evenhigher (cid:96) modes. Therefore, κ is more strongly influenced by the MNRAS000
0) and open ( k = − δ ( χ , θ , φ ) via the Poisson equation ∇ χ Ψ ( χ , θ , φ ) = Ω m H a δ ( χ , θ , φ ) , (2)where Ω m is the total matter density today, H is the Hubble con-stant today, and a = / ( + z ) is the scale factor. Note that the gra-dient ∇ χ is taken in the comoving radial direction.Expanding the lensing potential at a given comoving distance χ in spherical harmonics, we have ψ ( χ ) = ∑ (cid:96) m ψ (cid:96) m ( χ ) Y (cid:96) m ( θ , φ ) , ψ (cid:96) m ( χ ) = (cid:90) d Ω ψ ( χ ) Y ∗ (cid:96) m ( θ , φ ) , (3)where Y (cid:96) m are the spin-0 spherical harmonic basis set and ψ (cid:96) m ( χ ) is the coefficient associated with Y (cid:96) m at χ . Below we will omitthe χ reference in our notation for simplicity, but note that theseequations apply to the fields on a given redshift shell. To derive the spherical harmonic representation of shear andconvergence, we have κ = ( ð ð + ð ð ) ψ , (4) γ = γ + i γ = ð ð ψ , (5)where ð and ð are the raising and lowering operators that act onspin-weighted spherical harmonics, s Y (cid:96) m and follow a certain set ofrules (see e.g., Castro et al. 2005, for details). We can now definethe spherical representation of the convergence field and the shearfield to be κ = κ E + i κ B = ∑ (cid:96) m ( ˆ κ E ,(cid:96) m + i ˆ κ B ,(cid:96) m ) Y (cid:96) m , (6)and γ = γ + i γ = ∑ (cid:96) m ˆ γ (cid:96) m Y (cid:96) m . (7)Here Y (cid:96) m are spin-2 spherical harmonics. From Eq. (4) and Eq. (5)it follows thatˆ κ E ,(cid:96) m + i ˆ κ B ,(cid:96) m = − (cid:96) ( (cid:96) + ) ψ (cid:96) m , (8)ˆ γ (cid:96) m = ˆ γ E ,(cid:96) m + i ˆ γ B ,(cid:96) m = [ (cid:96) ( (cid:96) + )( (cid:96) − )( (cid:96) + )] ψ (cid:96) m = − (cid:115) ( (cid:96) + )( (cid:96) − ) (cid:96) ( (cid:96) + ) ( ˆ κ E ,(cid:96) m + i ˆ κ B ,(cid:96) m ) . (9)That is, one can convert between the three fields: κ , γ and ψ bymanipulating their spherical harmonics decompositions. The math-ematical operation described above is entirely analogous to a de-scription of linear polarisation such as that in the CMB polarisationmeasurements. In this analogy, the Q and U Stokes parameters cor-respond to the γ and γ . In the flat-sky limit, we have ð → ∂ and the decomposition into spherical harmonics is replaced by theFourier transform, Σ ψ (cid:96) m Y (cid:96) m → (cid:82) d (cid:96) ( π ) ψ ( (cid:96) ) e i (cid:96) · θ . The above equa-tions then reduce to the usual KS formalism.One can derive the lensing deflection field η in a similar fash-ion. The lensing deflection field is defined as the first derivative ofthe lensing potential η = η + i η = ð ψ , (10)so the deflection field is a spin-1 field and can be expressed as η = η + i η = ∑ (cid:96) m ˆ η (cid:96) m Y (cid:96) m . (11)Carrying through the derivation, we getˆ η (cid:96) m = [ (cid:96) ( (cid:96) + )] ψ (cid:96) m , (12)which is again related to the other lensing quantities via a simplelinear operation in the spin-harmonic space. That is, once γ is mea-sured, the other fields ( κ , η and ψ ) can be constructed using theformalism described above.From Eq. (8) and Eq. (9) we observe from which (cid:96) modes κ , η and ψ receive their dominant contributions: ψ receives most con-tribution from the lowest (cid:96) modes, η receives contribution fromslightly higher (cid:96) modes, and κ receives contribution from evenhigher (cid:96) modes. Therefore, κ is more strongly influenced by the MNRAS000 , 000–000 (0000)
C. Chang et al. smaller scale effects (e.g. noise) and ψ is affected by large scale ef-fects (e.g. masking). This can also be seen from the fact that the κ ( η ) field is derived from applying a Laplacian (derivative) operatoron the ψ field, which means that the power spectrum of κ ( η ) scaleslike (cid:96) ( (cid:96) ) times the power spectrum of ψ . The main focus of thispaper is to construct the κ map. However, we also explore the con-struction of the η and ψ in Appendix B to show that the quality ofthe reconstruction for these fields is indeed sensitive to the maskon large-scales and less sensitive to shape noise on small scales.We also show that with the 1,500 deg sky coverage of DES Y1,reconstructing the η and ψ maps are just starting to be interesting.In practice, the main observable for weak lensing is the galaxyshape ε , which in the weak lensing regime, is a noisy estimate of γ . When averaged over a large number of galaxies, (cid:104) ε (cid:105) ≈ g = γ − κ ,where g is the reduced shear. As κ (cid:28) ε ≈ γ . The noise in ε is dominated by the intrinsic shape of thegalaxies, or “shape noise”, but also includes measurement noise.That is, ε = γ + ε int + ε m , (13)where ε int is the intrinsic shape of the galaxy and ε m is the error onthe measured shape due to the measurement. One often quantifiesthe combined effect of ε int and ε m using σ ε , the standard deviationof the distribution of ε int + ε m . As we will see in Sec. 4, one needsto average ε over a large number of galaxies to suppress this noise.Note that here we have not considered the effect of intrinsic align-ment (IA, Troxel & Ishak 2015; Blazek et al. 2015), where (cid:104) ε (cid:105) ≈ g no longer holds. DES is an ongoing wide-field galaxy and supernova survey that be-gan in August 2013 and aims to cover a total of 5000 deg in fivefilter bands ( grizY ) to a final median depth of g ∼ r ∼ i ∼ z ∼ Y ∼ σ PSF limiting magnitude, see DarkEnergy Survey Collaboration et al. 2016) at the end of the survey.The survey instrument is the Dark Energy Camera (Flaugher et al.2015) installed on the 4m Blanco telescope at the Cerro TololoInter-American Observatory (CTIO) in Chile. This work is basedon the DES first-year cosmology data set (Y1A1 GOLD) includingphotometrically calibrated object catalogs and associate ancillarycoverage and depth maps (Drlica-Wagner et al. 2017). We focus onthe Southern footprint of the DES Y1 data, which overlaps with theSouth Pole Telescope survey (Carlstrom et al. 2011). This is thelargest contiguous area in the Y1 data set and ideal for construct-ing weak lensing mass maps. We briefly describe below the dataproducts and simulations used in this work. z ) catalog We use the photometric redshifts (photo- z ’s) derived using a codeclosely following the Bayesian Photometric Redshifts (BPZ) algo-rithm developed in Ben´ıtez (2000) and Coe et al. (2006). BPZ is atemplate-fitting code using templates from Coleman et al. (1980);Kinney et al. (1996); Bruzual & Charlot (2003). The catalog gener-ation in Y1 is similar to the SV analysis (Bonnett et al. 2016), butwith several improvements described in Hoyle et al. (2017).BPZ calculates a redshift PDF for each galaxy in that sample.The mean of this PDF, z mean , is used to place source galaxies intoredshift bins, while the n ( z ) for each of the samples is estimated byrandomly drawing a redshift from the PDF of each galaxy. These n ( z ) ’s are validated in Hoyle et al. (2017) using two orthogonalmethodologies: comparison with precise redshifts and clustering-based inference, see Hoyle et al. (2017); Cawthon et al. (2017);Gatti et al. (2017); Davis et al. (2017). Two DES Y1 weak lensing shape catalogs are used in this paper— the M
ETA C ALIBRATION catalog based on Huff & Mandelbaum(2017) and Sheldon & Huff (2017), and the I M SHAPE catalogbased on Zuntz et al. (2013). Both catalogs have been tested thor-oughly in Zuntz et al. (2017, hereafter Z17). Given that the twoalgorithms are fundamentally different and that the pipelines weredeveloped independently, obtaining consistent results from the twocatalogs is a non-trivial test of the catalogs themselves.Briefly, the M
ETA C ALIBRATION algorithm relies on a self-calibration framework using the data itself, instead of a large num-ber of image simulations as is used in many other algorithms (e.g.I M SHAPE , Bruderer et al. 2016; Fenech Conti et al. 2016). The ba-sic idea is to apply a small, known shear on the deconvolved galaxyimages in different directions and re-measure the post-shear recon-volved galaxy shapes. Since the input shear is known, the changein the measured galaxy shapes due to the artificial shearing givesa direct measure of how the shear estimators responds to shear.This quantity is referred to as the response . In addition, selectioneffects can be easily corrected in this framework by measuringthe response for the full sample and for the subsample. The finalsignal-to-noise and size selection for the catalog is S/N >
10 and T / T PSF > . T and T PSF are the sizes of the galaxy and thePSF, respectively). Following Z17, the residual systematic errorsare quoted in terms of m (the multiplicative bias), α (the additivebias associated with the PSF model ellipticity ε PSF ) and β (the ad-ditive bias associated with the errors on the PSF model ellipticity δ ε PSF ). For M
ETA C ALIBRATION , Z17 estimated m = . ± . α ∼
0, and β ∼ −
1. In Troxel et al. (2017), however, it is foundthat the β correction has very little effect on the final measure-ments. We therefore do not correct for β when making the massmaps. We have also checked that setting β = − ∼ , ,
000 galaxies in the full Y1 catalog. The shearmeasurement method in M
ETA C ALIBRATION is based on the NG - MIX method (Sheldon 2014). The full implementation of M
ETA -C ALIBRATION is publicly available and hosted under the
NGMIX code repository .The I M SHAPE algorithm is one of the algorithms also usedin the DES SV analyses (Jarvis et al. 2015). It is a maximum like-lihood fitting code using the Levenberg-Marquardt minimisationthat models the galaxies either as an exponential disk or a de Vau-couleurs profile — fitting is done with both models and the onewith a better likelihood goes into the final catalog. Calibration ofbias in the shear estimate associated with noise (Kacprzak et al.2012; Refregier et al. 2012) is based on the image simulation pack-age G AL S IM , but is significantly more complex and incorporatesmany effects seen in the DES Y1 data as described in Z17 andSamuroff et al. (2017). The final signal-to-noise and size selec-tions are 12 < S / N <
200 and 1 . < R gp / R p <
3, where R gp is Here we refer to the fact that the response is different when one selects asubsample of the galaxies based on signal-to-noise, sizes, redshift etc.. https://github.com/esheldon/ngmix https://github.com/GalSim-developers/GalSim MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map the size of the galaxy and R p is the size of the PSF. The cataloghas an estimated m ∼ . ± . α ∼
0, and β ∼ −
1. Similarto M
ETA C ALIBRATION , we do not correct for β as Troxel et al.(2017) showed that the correction has a negligible effect on themeasurements. The final catalog contains ∼ , ,
000 galaxies.The lower number relative to M
ETA C ALIBRATION is due to thefact that I M SHAPE operates on r -band images while M ETA C ALI - BRATION use all images from the bands r , i and z . The I M SHAPE code is publicly available .Details for both shape catalogs and the tests performed onthese catalogs can be found in Z17. We mainly show results forM ETA C ALIBRATION as it has the higher S/N, but also constructedI M SHAPE maps and performed several systematics tests withthese. Also, as noted above, we only use the SPT wide-field re-gion with Dec < −
35 as it has been the region where most testingwas done for both the shear and the photo- z catalogs. We gener-ate 5 maps for each catalog with different source z mean selections:0 . < z < .
3, 0 . < z < .
43, 0 . < z < .
63, 0 . < z < . . < z < .
3. The first redshift bin combines galaxies in a broadredshift range to allow for a large source number density and there-fore higher signal-to-noise for the mass maps. This is the mapwith which most quantitative studies are done in this paper. Theother four redshift bins match those defined by Troxel et al. (2017),which are well-tested samples that meet the criterion for cosmicshear measurements. These maps are noisier, but allow us to ex-plore the 3D tomographic aspect of the maps. Basic characteristicsof the samples associated with the five maps are listed in Table 1and Table 1 of Troxel et al. (2017).Finally, both shear catalogs were blinded with a multiplicativefactor during the entire analysis and only unblinded after all testswere finalised. See Z17 for the detailed blinding procedure.
In Sec. 7.1 we use a flux-limited galaxy sample as a tracer of theforeground mass of the mass maps. This sample is constructed tobe a simple, clean flux-limited sample from the DES Y1 catalog(Drlica-Wagner et al. 2017), which is easier to compare with sim-ulations as it puts less pressure on having other galaxy properties(colour, galaxy type) in the catalogs being matched to the data.The catalog is built by applying the following selections to theDES Y1 catalog: 17 . < i < . − < g − r < − < r − i < . − < i − z < z ’s; flags gold = to remove any blended, satu-rated, incomplete or problematic galaxies; flags badregion ≤ to remove problematic regions with e.g. high stellar contamination; modest class = to select objects as galaxies. The full catalogcontains ∼ z cuts to construct two samples, 0 . < z < . . < z < .
6, to-gether with a cut in Dec < −
35 to select the SPT region. The twosamples are then pixelated into H
EALPIX maps of nside = Two types of simulations are used in this work to investigate theperformance of the convergence map reconstruction and the effectsof noise and masking. First, we generate fully sampled, Gaussian https://bitbucket.org/joezuntz/im3shape/ maps with a given power spectrum using the synfast routine inH EALPIX (G´orski et al. 2005). We use the software package C OS - MOSIS (Zuntz et al. 2015), which wraps around the CAMB soft-ware (Lewis & Bridle 2002), to generate the input power spec-trum with the cosmological parameters: Ω m = . Ω b = . σ = . h = . n s = .
97, and w = −
1, although the particulardetails of the power spectrum are not very important for the testswe perform with these Gaussian simulations.Second, we use the “Buzzard v1.3” mock galaxy catalogsbased on N-body simulations as described in DeRose et al. (inprep). Briefly, three flat Λ CDM dark-matter-only N-body simula-tions were used, with 1050 , 2600 and 4000 M pc h − boxesand 1400 , 2048 and 2048 particles, respectively. These boxeswere run with LG ADGET -2 (Springel 2005) with 2LPT IC initialconditions from (Crocce et al. 2006) and CAMB. The cosmol-ogy assumed was Ω m = . Ω b = . σ = . h = . n s = .
96, and w = − × grizY magnitudes and shapes are assigned to each galaxy using the al-gorithm Adding Density Determined Galaxies to Lightcone Sim-ulations (ADDGALS, Wechsler et al. in prep., DeRose et al. inprep.). Galaxies are assigned to dark matter particles and given r-band absolute magnitudes based on the distribution p ( δ | M r ) mea-sured from a high resolution simulation populated with galaxiesusing subhalo abundance matching (SHAM) (Conroy et al. 2006;Reddick et al. 2013), where δ is a large scale density proxy. Eachgalaxy is assigned an SED from SDSS DR6 (Cooper 2006) by find-ing neighbors in the space of M r − Σ , where Σ is the projecteddistance to the fifth nearest neighbor in redshift slices of width δ z = .
02. These SEDs are k-corrected and integrated over the ap-propriate bandpasses to generate grizY magnitudes.Finally, the weak lensing parameters ( κ and γ ) in the simu-lations are based on the ray-tracing algorithm Curved-sky grAvita-tional Lensing for Cosmological Light conE simulatioNS (CAL-CLENS; Becker 2013) which builds on previous work by Gaz-tanaga & Bernardeau (1998) and Fosalba et al. (2008) to make allsky weak lensing maps from projected density fields in simulations.The ray-tracing resolution is accurate to (cid:39) . z pipeline (BPZ)on the photometry, adding shape noise , imposing redshift, size,signal-to-noise cuts to match the shear catalog described in Sec. 3.2(here the cuts are tailored to the M ETA C ALIBRATION catalog) andthe flux-limited galaxy catalog described in Appendix 3.3. We note,however, that due to the setup of the simulation box, the footprintof the simulations is 26% smaller than the data, with the area ofRA > ◦ removed. For the purpose of testing in this work, thisdoes not impose a significant problem. We also note that the galaxynumber density is 20% lower than our data set. To account forthat, we scale the shape noise by a factor of (cid:112) n gal , Buzzard / n gal , DES ,where n gal , Buzzard and n gal , DES are the number density of sourcegalaxies in the simulations and data respectively. The Buzzard catalogs include shape noise that are modeled from externalSubaru data sets, which are not fully representative of our data. In order tohave a better matching between simulation and data, we instead randomlydraw the galaxy shapes from the M
ETA C ALIBRATION catalog and add thesimulated shear to the galaxy shape.MNRAS000
ETA C ALIBRATION catalog and add thesimulated shear to the galaxy shape.MNRAS000 , 000–000 (0000)
C. Chang et al.
We describe here the steps taken to construct the convergence mapfor the two shear catalogs. The only difference between the two cat-alogs is that different calibration schemes are applied to the shearestimates prior to making the maps.All the maps are constructed using H
EALPIX pixelisation,which is a natural choice for map making on the sphere and in-cludes the necessary tools to manipulate the data on a sphere. Thisincludes the decomposition of the spin fields into spin harmonics,which is essential for the transformation between shear γ , conver-gence κ , the lensing potential ψ and the deflection angle η , as weoutlined in Sec. 2. We use a H EALPIX map of nside = ∼ ,
500 deg , which appears larger than the naive footprint ofour data in the SPT region ( ∼ ,
300 deg , Troxel et al. 2017). Thisis because we are using a coarser pixel resolution than what is usedto estimate the footprint (nside = ε and ε from the galaxy shape catalogs. To do this, we follow the proce-dure outlined in Section 7 of Z17 for calculating the mean shearper pixel. Note that both the response R for the M ETA C ALIBRA - TION catalog and the multiplicative noise-bias calibration (NBC)factor m for the I M SHAPE catalog are noisy within each pixel ofour maps. We therefore use the mean R and m values for each sam-ple instead of calculating them in each pixel when constructing themaps. That is, for M ETA C ALIBRATION , we have ε ν i = ∑ n i j ε ν i j n i ¯ R ν , ν = , , (14)where n i is the number of source galaxies in pixel i and ε ν i j is theshape estimate of each individual galaxy j in that pixel. ¯ R ν is themean response of the full sample. The ¯ R ν values vary from ∼ . ∼ . R ν comesfrom the correction of the selection effects. For the I M SHAPE , wehave ε ν i = n i ( + ¯ m ) ∑ n i j = ( ε ν i j − c ν i j ) w i j ∑ n i j = w i j , ν = , , (15)where c ν i j and w i j are the additive NBC factor and weight for galaxy j in pixel i , and ¯ m is the average multiplicative NBC factor for eachsample. Typical m values range from -0.08 to -0.18 going from lowto high redshift. We then subtract the mean shear for each samplefrom the maps as suggested by Z17 Section 7.1.Next, we perform the spin transformation which converts theellipticity maps (which combine to form a spin-2 field ε + i ε )into a curl-free E-mode convergence map κ E and a divergence-freeB-mode convergence map κ B . The H EALPIX functions map2alm performs this decomposition in spherical harmonic space and re-
Table 1.
Characteristics of the source galaxy samples and the maps. Thenumber preceding the semicolon is for the M
ETA C ALIBRATION catalogwhile the number after the semicolon is for the I M SHAPE catalog. ¯ z is themean redshift estimate from BPZ for each sample, while σ ε is the mean of σ ε and σ ε , the standard deviation of the weighted galaxy shapes reportedfrom the catalogs (see last column in Table 1 of Troxel et al. 2017). The areaof the map is ∼ for both M ETA C ALIBRATION and I M SHAPE ,where the exact size changes slightly from different photo-z bins and shearcatalogs. The H
EALPIX maps have a resolution of nside = z σ ε . < z < . . < z < .
43 0.38; 0.36 0.26; 0.260 . < z < .
63 0.51; 0.52 0.30; 0.280 . < z < . . < z < . turns the E- and B-mode coefficients, which are equivalent to theˆ γ E ,(cid:96) m and ˆ γ B ,(cid:96) m in Eq. (9). We calculate ˆ κ E ,(cid:96) m and ˆ κ B ,(cid:96) m , then usethe H EALPIX function alm2map to convert these coefficients backto the real space κ E and κ B maps. Similarly, ψ and η maps can beconstructed using Eq. (9) and Eq. (12).For all the convergence map visualisation in this paper, wefurther smooth the maps with a Gaussian kernel. The noise asso-ciated with each pixel after smoothing can be calculated through(Van Waerbeke et al. 2013) σ κ = σ ε + σ ε πσ G n gal , (16)where σ ε and σ ε are the standard deviation of the two compo-nents for the measured galaxy shapes, σ G is the width of the Gaus-sian filter used to smooth the maps, and n gal is the number densityof the source galaxies.Finally, for all measurements in this work, we estimate theerror bars and the covariance matrix using a standard Jackknifeapproach. We divide the footprint into N JK Jackknife regions us-ing a kmeans clustering code and divide the mask into N JK ap-proximately equal-area regions. Throughout this paper, we use N JK = TREECORR is used. In this section we present a series of simulation tests to validateour procedure for map generation and quantify the uncertaintiesassociated with the various source of systematics and noise. Westart with an idealised setup of a Gaussian, fully-sampled, full-skymap in Sec. 5.1 to quantify the errors associated with the shear-to-convergence conversion alone, then we impose a DES Y1-likemask to investigate the degradation introduced by the mask. Next inSec. 5.2, we repeat the tests in Sec. 5.1 with a mock galaxy catalogbased on an N-body simulation. We test the effect of shot noise(finite sampling) and shape noise.For both Sec. 5.1 and Sec. 5.2, we quantify the quality of the https://github.com/esheldon/kmeans_radec https://github.com/rmjarvis/TreeCorr/wiki MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Gaussian κ sm (a) ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Gaussian full sky κ E - (a)(b) − . − . − . . . . . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Gaussian cut sky κ E (c) ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ (c) - (a)(d) − . − . − . . . . . Figure 1.
Tests from maps of simulated Gaussian fields. All maps are smoothed with a Gaussian filter of σ G =
30 arcminutes, mean-subtracted and projectedonto a conic projection. Panel (a) shows the original Gaussian κ sm map; panel (b) is the difference map between the full-sky reconstructed κ E map and panel(a); panel (c) shows the cut-sky reconstructed κ E map, and panel (d) shows the difference map between panel (c) and panel (a). The series of maps shows thatthe reconstruction on the edges is degraded when introducing the mask. reconstruction using the following statistics: F = (cid:115) (cid:104) κ E (cid:105)(cid:104) κ (cid:105) ; F = (cid:104) κ E κ sm (cid:105)(cid:104) κ (cid:105) , (17)where κ E is the reconstructed map, κ sm is the true convergence mapdegraded to the same resolution as κ E (see Sec. 5.1 for details), (cid:104) XY (cid:105) is the zero-lag cross-correlation between two maps X and Y ,or (cid:104) XY (cid:105) = N N ∑ i = X i Y i . (18)The index i runs over all pixels in the map where the pixels are notmasked. F is the square-root of the ratio of the second moments ofthe map. F on the other hand, tests in addition that the phases (inaddition to the amplitudes) of the map are reconstructed correctly,or in other words, that the patterns in the maps are correctly recon-structed. F and F are designed to have the same units as κ E / κ sm .We require that for our final reconstruction (including all noise andsystematics effects) of both F and F be consistent with 1 withinthe 2 σ measurement errors. In Appendix B, we perform a subset ofthe tests above on reconstructing the lensing potential and deflec-tion field described in Sec. 2.In Sec. 5.3, we take the maps in Sec. 5.2 one step further andexamine the PSF of the maps and the second and third momentsas a function of smoothing. We require the reconstructed map tohave second and third moments consistent with expectation fromsimulations within 2 σ of the measurement errors, which then as-sures that the reconstruction preserves the distribution of structureson different scales.We note that the requirements on the reconstruction perfor-mance depends on the specific application. Passing the require-ments on F , F and the moments means that the mean variance,phase, and distribution of power on different scales (on the scales we tested) in the maps are robust. Extending to further applicationswould require additional tests. We consider a set of full-sky, noiseless, Gaussian lensing maps ( γ and κ ) generated using the H EALPIX routine synfast . These mapsare constructed using an input lensing power spectrum for a flat Λ CDM model with cosmological parameters: Ω m = . h = . Ω b = . σ = . w = −
1. The source redshift distribution n s ( z ) is approximately matched to the redshift estimate of BPZ forredshift bin 0 . < z < . = (cid:96) max = × nside. Note that the (cid:96) max cut is necessary for furtherH EALPIX manipulations, since the modes close to the pixel scalecan introduce undesired noise. This means that these maps do notcontain information on scales beyond (cid:96) max . The synfast routineoutputs three maps that are consistent with the input power spec-trum: a spin-0 map and two maps for the two components of thespin-2 field. We can then identify the spin-0 map as the conver-gence map κ sm and the spin-2 maps as the shear maps γ sm . Sinceall the lensing maps are effectively smoothed, we use the ‘sm’ sub-script to distinguish these maps (which do not contain informationon scales beyond (cid:96) max ) from the true underlying field with infiniteresolution. We denote κ E and κ B to be the E- and B-mode conver-gence generated from the smoothed shear maps γ sm .For visualisation purpose, all maps presented in this paper arefirst smoothed with a Gaussian filter of σ G =
30 arcminutes, then
MNRAS000
MNRAS000 , 000–000 (0000)
C. Chang et al. . . . . . . . . . Excluded edges (arcmin) . . . . . . . . F o r F F Gaussian F Buzzard F Buzzardwith shape noise F Gaussian F Buzzard F Buzzardwith shape noise
Figure 2. F = (cid:113) (cid:104) κ E (cid:105) / (cid:104) κ (cid:105) (circle markers) and F = (cid:104) κ E κ sm (cid:105) / (cid:104) κ (cid:105) (triangle markers) for the reconstructed convergence map from the Gaussian simu-lations and the Buzzard mock galaxy catalogs. F measures how well the variance of the map is reconstructed, while F measures in addition how well thephase information is reconstructed. F = F = F and F changes when we exclude pixels within a certaindistance from the edge of the mask. The larger the exclusion, the less effected the reconstruction is from the edge effects. mean-subtracted , and finally projected onto a plane with Albersequal-area conic using the code S KY M APPER (for quantitativeanalyses later we use the raw map themselves). The smoothingscale is chosen so that the highest peaks in the E-mode S/N mapshave S/N values greater than ∼ • Panel (a): noiseless κ sm map directly from synfast , cutout inthe Y1 footprint. • Panel (b): subtracting panel (a) from a full-sky, fully sampled,noiseless κ E reconstruction. This shows that in this ideal situation,the reconstructed κ E is able to recover κ sm very well with negligi-ble residuals, validating our basic implementation of the shear-to-convergence transformation. • Panel (c): κ E reconstruction when applying the Y1 mask tothe shear maps. This illustrates overall good reconstruction of thespatial pattern of the maps compared to panel (a). As we have setthe mask to zero, the amplitude of the κ E map is slightly lower thanpanel (a) at this relatively large smoothing scale. • Panel (d): subtracting panel (a) from panel (c). We can seeedge effects resulting from the Y1 mask, as the pixels on the edgehave less information to infer the convergence than the pixels inthe centre of the field. In addition, the residuals are small but anti-correlated with the real structure, since the overall amplitude ofpanel (c) is lower than panel (a).In Fig. 2 we show in black and green how F and F (Eq. (17))change when we exclude regions up to 30 arcminutes away fromthe mask edge. For F , we find a value ∼ .
97 when no pixels areexcluded and this improves up to about 0.99 when areas 15 arcmin-utes around the edges are excluded. The fact that F < F behaves very similar to F , which confirms that the reconstruction is good to ∼
1% in theseideal scenarios with only small effects coming from the dilutiondue to the edges. We note that the above analysis was evaluated Since lensing reconstruction is only valid up to a constant offset, we sub-tract the mean to avoid this constant additive bias. https://github.com/pmelchior/skymapper for the map at 0 ◦ < RA < ◦ in order to compare to the Buzzardsimulations.Alternative approaches to dealing with the mask and edge ef-fects include filling in the empty pixels via a smooth interpolationfrom neighboring pixels and more sophisticated inpainting tech-niques (Pires et al. 2009). We investigate the former in Appendix Aand find that it does not improve the performance of the map recon-struction significantly given the noise level and mask geometry ofour data, while the latter is beyond the scope of this paper. Next, we turn to using mock galaxy catalogs generated from N-body simulations. The main differences between these and theGaussian simulations are that (1) they only sparsely sample thelensing fields at a given thin redshift slice, effectively introduc-ing shot noise, (2) they are derived from a ray-traced lensing fieldwhich contains non-Gaussian information, and (3) as discussed inthe previous section, the maps naturally contain a small amountof information on scales beyond (cid:96) max = × nside that we cannotreconstruct when we enforce a (cid:96) max smoothing during the recon-struction. We would like to understand how these factors affect thereconstruction of the convergence maps. In this section, we mainlyuse the Buzzard mock galaxy catalogs described in Sec. 3.4 for test-ing, but we have also tested on an independent set of simulations(the Marenostrum Institut de Ciencias de l’Espai Simulations, orthe MICE simulations, Fosalba et al. 2015b,a; Crocce et al. 2015)and found consistent results.We carry out a series of tests using the convergence map gen-erated for redshift bins that are matched to that used for the data(see Sec. 6). That is, we bin the galaxies using the mean redshiftreported by the photo- z code and check that the resulting n ( z ) re-ported by BPZ is close to that of our data. Next, we make threemaps using directly the quantities provided by the simulation: • κ pix : convergence • γ pix : shear • ε pix : galaxy shapes.These maps are constructed with the same resolution (nside = MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Buzzard κ sm (a) ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Buzzard κ γE - (a)(b) − . − . − . . . . . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Buzzard κ (cid:15)E (c) ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ (c) - (a)(d) − . − . − . . . . . Figure 3.
This figure is similar to Fig. 1 but using the Buzzard mock galaxy simulations. Panel (a) shows the original Buzzard κ sm map; panel (b) shows thedifference between the reconstructed κ γ E map (without shape noise) and panel (a); panel (c) shows the reconstructed κ ε E map (with shape noise), and panel (d)shows the difference map between panel (c) and panel (a). • κ sm : to ensure that all maps we compare later have the sameresolution, we smooth the κ pix map by removing all (cid:96) modes be-yond (cid:96) max = × nside; • κ γ E , κ γ B : E- and B-mode convergence constructed using shear γ pix ; • κ ε E , κ ε B : E- and B-mode convergence constructed using galaxyshapes ε pix .In Fig. 3 we compare visually several of these reconstructed maps: • Panel (a): κ sm map from the Buzzard simulation. Comparingwith panel (a) of Fig. 1, one can see that the convergence mapfrom the galaxy catalog has similar amplitudes and characteristicspatial patterns as the Gaussian map. The Buzzard maps appearslightly more clustered, which comes from the non-Gaussian na-ture of these maps compared to the pure Gaussian simulations. • Panel (b): subtracting panel (a) from the reconstructed κ γ E mapfrom Buzzard, which includes shot noise from the finite samplingfrom the galaxies and the Y1 mask but no shape noise. Similarto panel (d) of Fig. 1, there is an anti-correlation of the low-levelresiduals with the true structures. • Panel (c): reconstructed κ ε E map from the Buzzard simulation,which includes shot noise from the finite sampling from the galax-ies, the Y1 mask and shape noise. We find the amplitude of the mapto be higher than the κ sm map in panel (a) and that there are spuri-ous structures that arise from noise which do not correspond to realstructures in the κ sm map. However, the resemblance of the κ ε E mapto the κ sm map is still very obvious, especially the large-scale pat-terns in the maps. This suggests that despite of noise, the majorityof the structures in the κ ε E map are associated with real structureson this smoothing scale. • Panel (d): subtracting panel (a) from panel (c). We see moreclearly the shape noise-induced small-scale noise peaks as well asa large scale pattern that is very similar to that in panel (b). The edge effect, in comparison, becomes less visible in the presence ofshape noise.In Fig. 2 we again show the F (red) and F (blue) statisticsas a function of the area excluded around the mask. We find that F behaves very similar to the Gaussian version shown in green,while F appears systematically higher than the Gaussian simula-tions. This indicates that the reconstruction with the mock galaxycatalogs introduces un-correlated noise in κ E , causing the overallvariance in the map to be larger, while the phase remains the same.This additional noise comes from the finite sampling of the shearfield inside each pixel — the mean shear over all galaxies insideeach pixel is different from the true mean shear in that pixel. Thisnoise can be suppressed by smoothing the maps at a scale slightlylarger than the pixel scale, as can be seen in Fig. 4. Both F and F increase by a few percent when excluding the edges. When in-troducing shape noise, the error bars on F increase, but the ampli-tude stays roughly unchanged, suggesting that on average, shapenoise does not change the phase information. The raw F withshape noise is will be dominated by shape noise in the denomina-tor, therefore we show instead the “de-noised” version F definedas (cid:113) ( (cid:104) κ E (cid:105) − (cid:104) κ E , ran (cid:105) ) / (cid:104) κ (cid:105) , where κ E , ran is a convergence mapconstructed by randomizing the ellipticities. In the remaining ofthe paper, “ F with shape noise” refers to this de-noised quantity.Overall, we find that at the number density and pixel resolu-tion of this particular map (0 . < z < . F . After including shape noise, both F and F are consistent with1 even without exclusion of the edge pixels. We also note that if weperform the same tests on a different redshift bin where the num-ber density of galaxies is lower, the performance of the reconstruc-tion using both the Gaussian and the Buzzard simulations becomes MNRAS000
This figure is similar to Fig. 1 but using the Buzzard mock galaxy simulations. Panel (a) shows the original Buzzard κ sm map; panel (b) shows thedifference between the reconstructed κ γ E map (without shape noise) and panel (a); panel (c) shows the reconstructed κ ε E map (with shape noise), and panel (d)shows the difference map between panel (c) and panel (a). • κ sm : to ensure that all maps we compare later have the sameresolution, we smooth the κ pix map by removing all (cid:96) modes be-yond (cid:96) max = × nside; • κ γ E , κ γ B : E- and B-mode convergence constructed using shear γ pix ; • κ ε E , κ ε B : E- and B-mode convergence constructed using galaxyshapes ε pix .In Fig. 3 we compare visually several of these reconstructed maps: • Panel (a): κ sm map from the Buzzard simulation. Comparingwith panel (a) of Fig. 1, one can see that the convergence mapfrom the galaxy catalog has similar amplitudes and characteristicspatial patterns as the Gaussian map. The Buzzard maps appearslightly more clustered, which comes from the non-Gaussian na-ture of these maps compared to the pure Gaussian simulations. • Panel (b): subtracting panel (a) from the reconstructed κ γ E mapfrom Buzzard, which includes shot noise from the finite samplingfrom the galaxies and the Y1 mask but no shape noise. Similarto panel (d) of Fig. 1, there is an anti-correlation of the low-levelresiduals with the true structures. • Panel (c): reconstructed κ ε E map from the Buzzard simulation,which includes shot noise from the finite sampling from the galax-ies, the Y1 mask and shape noise. We find the amplitude of the mapto be higher than the κ sm map in panel (a) and that there are spuri-ous structures that arise from noise which do not correspond to realstructures in the κ sm map. However, the resemblance of the κ ε E mapto the κ sm map is still very obvious, especially the large-scale pat-terns in the maps. This suggests that despite of noise, the majorityof the structures in the κ ε E map are associated with real structureson this smoothing scale. • Panel (d): subtracting panel (a) from panel (c). We see moreclearly the shape noise-induced small-scale noise peaks as well asa large scale pattern that is very similar to that in panel (b). The edge effect, in comparison, becomes less visible in the presence ofshape noise.In Fig. 2 we again show the F (red) and F (blue) statisticsas a function of the area excluded around the mask. We find that F behaves very similar to the Gaussian version shown in green,while F appears systematically higher than the Gaussian simula-tions. This indicates that the reconstruction with the mock galaxycatalogs introduces un-correlated noise in κ E , causing the overallvariance in the map to be larger, while the phase remains the same.This additional noise comes from the finite sampling of the shearfield inside each pixel — the mean shear over all galaxies insideeach pixel is different from the true mean shear in that pixel. Thisnoise can be suppressed by smoothing the maps at a scale slightlylarger than the pixel scale, as can be seen in Fig. 4. Both F and F increase by a few percent when excluding the edges. When in-troducing shape noise, the error bars on F increase, but the ampli-tude stays roughly unchanged, suggesting that on average, shapenoise does not change the phase information. The raw F withshape noise is will be dominated by shape noise in the denomina-tor, therefore we show instead the “de-noised” version F definedas (cid:113) ( (cid:104) κ E (cid:105) − (cid:104) κ E , ran (cid:105) ) / (cid:104) κ (cid:105) , where κ E , ran is a convergence mapconstructed by randomizing the ellipticities. In the remaining ofthe paper, “ F with shape noise” refers to this de-noised quantity.Overall, we find that at the number density and pixel resolu-tion of this particular map (0 . < z < . F . After including shape noise, both F and F are consistent with1 even without exclusion of the edge pixels. We also note that if weperform the same tests on a different redshift bin where the num-ber density of galaxies is lower, the performance of the reconstruc-tion using both the Gaussian and the Buzzard simulations becomes MNRAS000 , 000–000 (0000) C. Chang et al. − − h κ i κ pix κ sm κ γE κ (cid:15)E − − κ (cid:15)E κ E data0 20 40 60 80Smoothing (arcmin)10 − − − − − h κ i κ pix κ sm κ γE κ (cid:15)E − − − − − κ (cid:15)E κ E data Figure 4.
The upper left panel shows the second moments of the maps as a function of smoothing scale for different κ maps in one Buzzard simulation, fromthe most idealised noiseless case (grey), to two intermediate stages (black and green), and to the κ ε E map that includes observational noise that match to the data(blue). The shaded blue band in the upper right panel shows the mean and standard deviation of the κ ε E measurement for 12 independent Buzzard realisations.The measurement from the data is shown in red. The lower panels show the same as the upper panels, except for the third moments. The grey band in the lowerleft panel marks the scales that we remove for third moments analyses due to noise on small scales. All maps are generated for the redshift bin 0 . < z < . worse with the same pixel resolution. That is, the three factors —resolution of the map, effect of the edges, and number density ofthe source galaxies — are tightly coupled. If the chosen pixel res-olution is sub-optimal for the data set, the reconstruction could besignificantly biased. For example, if the pixel size is much smallerthan the typical separation of source galaxies, there will be a largenumber of empty pixels, which would result in a lower amplitudein the reconstructed maps. For our sample of the DES Y1 shear cat-alog, we perform quantitative studies only on the highest S/N mapat 0 . < z < . F and F statistics for this map in different resolutions and findthat increasing or decreasing the resolution by a factor of 2 in thenoiseless Buzzard simulation changes F and F by at most 3%. One final powerful test of the reconstruction is to look at the mo-ments and the PDF of the maps. In this section, we examine the sec-ond and third moments of the various maps used in Sec. 5.2 as weprogressively smooth the maps on increasingly larger scales. Sincethese moments of the convergence maps as a function of smooth-ing scale are sensitive to cosmology (Bernardeau et al. 1997; Jain& Seljak 1997; Jain & Van Waerbeke 2000), it is important toverify how well the reconstructed maps preserve these character-istics. A similar test was performed in Van Waerbeke et al. (2013), where they checked up to the 5th moment of the maps. We onlyconsider the second and the third moments as the galaxy num-ber density in our maps is lower compared to that used in VanWaerbeke et al. (2013), and the higher moments are more sen-sitive to the noise in the maps. We begin with the set of Buz-zard maps described in the previous section: κ pix , κ sm , κ γ E and κ ε E . For each map, we smooth with a Gaussian filter with σ G =[ . , ., . , . , . , . , . , . , . , . ] arcminutes, wherethe first case is equivalent to the unsmoothed map examined pre-viously. To correct for the effect of smoothing on the edge pixels,we smooth the mask with the same filter and dividing the map bythe smoothed mask. We then calculate the second moment (cid:104) κ (cid:105) and third moments (cid:104) κ (cid:105) of these maps for the different smoothingscales. For κ ε E , we follow the de-noising prescription described inVan Waerbeke et al. (2013). That is (cid:104) ( κ ε E , denoise ) (cid:105) = (cid:104) ( κ ε E ) (cid:105) − (cid:104) ( κ ε E , ran ) (cid:105) , (19)where κ ε E , ran is obtained from shuffling the positions of the galaxieswhile keeping their ellipticities fixed. κ ε E , ran is a measure of the con-tribution from shape noise to the second moments and thus needsto be subtracted from the raw measured second moments.The second and third moments of the various κ maps as afunction of smoothing scale are shown in the left panels of Fig. 4.The error bars are estimated via the standard Jackknife approach.We find that κ pix and κ sm disagree slightly with no smoothing, butonce a small amount of smoothing is applied, which removes the MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map − . − .
02 0 .
00 0 .
02 0 . κ − − κ pix κ sm κ γE κ (cid:15)E κ (cid:15)E, ran κ (cid:15)B − . − .
02 0 .
00 0 .
02 0 . κ − − κ (cid:15)E κ E data Figure 5.
Pixel histograms for various maps in simulation and data when smoothed with a Gaussian filter of σ G = . κ ε E in 12 independent Buzzard realisations, while the red lineshows the pixel histogram for the data κ E map (See Section 6.1 for discussion). All maps are generated for the redshift bin 0 . < z < . very small scale information in the κ pix map, they agree vey well. κ sm and κ γ E are also consistent within the error bars, suggestingthat the reconstruction does not distort the information about howthe structures of different scales are distributed in the maps. Fi-nally, κ γ E and κ ε E agree with each other within 1 σ for the secondmoments on all scales and for the third moments on scales > κ ε E are larger due to shape noise. Wenote that the third moment measurements on small scales are notrecovered due to the noise on small scales (for a shear signal of1%, a smoothing scale of 5 arcminutes would result in an effectiveS/N of ∼ >
40 arcminutes, noise can cause the third moments to benegative. We repeat the measurement for 12 independent realisa-tions of the Buzzard simulations. The mean and standard deviationof the 12 measurements for κ ε E are shown in the right panels ofFig. 4. This provides a measure of the contribution from cosmicvariance. We find that, within the uncertainties from the measure-ment and cosmic variance, we can indeed recover the second andthird moments as a function of smoothing scales with our recon-struction method for scale larger than 5 arcminutes in the map cor-responding to 0 . < z < .
3. The data point for the third momenton the largest scale is (-2.9 ± × − , which is not shown onthe log plot, but is consistent within 2 σ with the simulation valueof (0.97 ± × − .It is also instructive to look at the PDF of the different mapsfor one smoothing scale in Fig. 4. The left panel of Fig. 5 showsthe distribution of κ pix (grey shaded), κ sm (black) and κ γ E (green)when smoothed by a Gaussian filter of 5.1 arcminutes. We find thatthe three histograms agree very well, and the non-Gaussian natureof the PDF is apparent. These distributions closely resemble thelog-normal distribution and is consistent with the results shown inClerkin et al. (2015). The distribution of κ ε E (blue), κ ε B (orange) and κ ε E , ran (purple) are also shown. Due to the added shape noise, thesethree fields appear much more Gaussian and the shape of the PDFis much broader. The fact that the distribution of κ ε E , ran is consistentwith κ ε B suggests that shape noise is the main contributor of the B- mode map on these smoothing scales, rather than B-mode leakagedue to imperfect reconstruction. We also check by looking at theB-mode signal in the noiseless reconstruction scenario, and find itto be negligible compared to the B-mode from shape noise. Theshape of the of κ ε E PDF is qualitatively different from κ ε E , ran and κ ε B — the κ ε E map contains more extreme high and low values, whichcorrespond to real peaks and voids in the mass distribution. The κ ε E PDF is also slightly skewed towards positive values, which is theimprint of the skewed true κ distribution seem in κ pix . Now we present the main goal of the paper. In Fig. 6 we showthe signal-to-noise (S/N) maps associated with the E-mode and B-mode convergence generated from the M
ETA C ALIBRATION cata-log for galaxies in the redshift range 0 . < z < . σ G =
30 arcminutes. The S/N in these maps apply both to thepositive (peaks) and negative (voids) values — extreme positiveand negative values are significant, while values close to zero aremore likely to be consistent with noise. In Fig. 7, maps for the fourtomographic bins are shown. The I M SHAPE convergence mapsin all the redshift bins are shown in Appendix C for comparison,together with maps generated using the Science Verification data(Vikram et al. 2015; Chang et al. 2015).We first look at the E-mode maps. Fig. 6 includes the full red-shift range (0 . < z < .
3) and thus has much higher signal-to-noisecompared to the tomographic maps in Fig. 7, as expected from thehigher number density of source galaxies. The visual impression ofthe map is very similar to the maps generated from the mock galaxycatalogs shown in Fig. 3, where there is an imprint of large-scalestructure stretched over tens of degrees. The area close to RA ∼ ◦ suffers from a more complicated mask structure as well as shal-lower depth, which results in a lower S/N in the map in that region.In Fig. 7, we find that the redshift bin 0 . < z < . MNRAS000
3) and thus has much higher signal-to-noisecompared to the tomographic maps in Fig. 7, as expected from thehigher number density of source galaxies. The visual impression ofthe map is very similar to the maps generated from the mock galaxycatalogs shown in Fig. 3, where there is an imprint of large-scalestructure stretched over tens of degrees. The area close to RA ∼ ◦ suffers from a more complicated mask structure as well as shal-lower depth, which results in a lower S/N in the map in that region.In Fig. 7, we find that the redshift bin 0 . < z < . MNRAS000 , 000–000 (0000) C. Chang et al. ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N Figure 6.
Pixel signal-to-noise (S/N) κ E / σ ( κ E ) maps (top) and κ B / σ ( κ B ) maps (bottom) constructed from the M ETA C ALIBRATION catalog for galaxies inthe redshift range of 0 . < z < .
3, smoothed by a Gaussian filter of σ G =
30 arcminutes. σ ( κ E ) and σ ( κ B ) are estimated by Eq. (16). the lower noise coming from the higher number density of sourcegalaxies. Structures that show up in a given map are likely to alsoshow up in the neighbouring redshift bins, since the mass that iscontributing to the lensing in one map is likely to also lens galaxiesin neighbouring redshift bins. This is apparent in e.g. the structuresat (RA, Dec)=(35 ◦ , -48 ◦ ) and (58 ◦ , -55 ◦ ). Next, we compare theE-mode maps with their B-mode counterpart in Fig. 6 and Fig. 7.In general, the B-mode maps have lower overall amplitudes. Themean absolute S/N of the E-mode map is ∼ σ G =
80 arcminutes, this ratio increases to ∼
2. There are no sig-nificant correlations between the E- and the B-mode maps in Fig. 6and Fig. 7: we find that the Pearson correlation coefficients are allconsistent with zero, as expected for maps where systematic effectsare not dominant. Comparing the four tomographic B-mode maps The Pearson correlation coefficient two maps X and Y is defined as (cid:104) ( X − ¯ X )( Y − ¯ Y ) (cid:105) / ( σ X σ Y ) , where ¯ X and ¯ Y are the mean pixel values forthe two maps, the (cid:104)(cid:105) averages over all pixels in the map, and σ indicates thestandard deviation of the pixel values in each map. in Fig. 7, there is no obvious correlation between the structures inone map with maps of neighboring redshift bins. We find that thePearson correlation coefficient between the second and third (thirdand fourth) redshift bins for the B-mode maps is 8 (5.5) times lowerthan that for the E-mode maps. The E and B-mode maps for thelowest redshift bin 0 . < z < .
43 have similar levels of S/N, whichis expected since the lensing signal at low redshift is weak and thenoise level is high.We now examine the second and third moments of the κ E maps similar to the tests in Sec. 5.2. For direct comparison withsimulations, the measurements are done using the map with the fullredshift range 0 . < z < . ◦ < RA < ◦ .Our results are shown in the right panels of Fig. 4, where the meanand standard deviation of the 12 noisy simulation results are alsooverlaid.We note that we do not expect perfect agreement between thesimulation and data for several reasons: first, the detailed shapenoise incorporated in the simulations is only an approximation tothe M ETA C ALIBRATION shape noise. In particular, there is no cor-relation of the shape noise with other galaxy properties in our sim-
MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N Figure 7.
Same as Fig. 6 but for the four tomographic maps. The κ E / σ ( κ E ) maps are shown on the left and the κ B / σ ( κ B ) maps are shown on the right. ulations. This, however, should be a second-order effect, since wedo not expect the galaxy properties to correlate with the true con-vergence. Second, the number density and n ( z ) in the simulationsonly approximately match the data as we discussed in Sec. 3.4.This is also a second-order effect since lensing is mainly sensitiveto the mean redshift of the lensing kernel. The detailed shape ofthe n ( z ) will not significantly alter the convergence maps. Finally,the simulations assume a certain cosmology that may not be thetrue one. As (cid:104) κ (cid:105) ∝ σ Ω . m and (cid:104) κ (cid:105) / (cid:104) κ (cid:105) ∝ σ − . (Bernardeauet al. 1997; Jain & Van Waerbeke 2000), these measurements aredirectly sensitive to the cosmological parameters. Given the currentconstraints in σ and Ω m from Planck Collaboration et al. (2016), changing the cosmological parameters by 2 σ does not affect thecomparison carried out here.From Fig. 4, we find very good agreement between the mea-surements from data and simulations in the overall amplitude andtrend of the second and third moments as a function of smooth-ing scale. The fact that our measurements are in agreement withthe simulations suggests that they are also in agreement with thecosmology assumed in the simulations (see Sec. 3.4), though theerror bars are fairly large compared to e.g. Troxel et al. (2017);DES Collaboration et al. (2017). The histograms of the κ E and κ B maps smoothed with a 5.1 arcminute Gaussian filter are shown inthe right panel of Fig. 5, together with the simulation counterpartsgenerated from the 12 Buzzard simulations. Again, we find good MNRAS000
Same as Fig. 6 but for the four tomographic maps. The κ E / σ ( κ E ) maps are shown on the left and the κ B / σ ( κ B ) maps are shown on the right. ulations. This, however, should be a second-order effect, since wedo not expect the galaxy properties to correlate with the true con-vergence. Second, the number density and n ( z ) in the simulationsonly approximately match the data as we discussed in Sec. 3.4.This is also a second-order effect since lensing is mainly sensitiveto the mean redshift of the lensing kernel. The detailed shape ofthe n ( z ) will not significantly alter the convergence maps. Finally,the simulations assume a certain cosmology that may not be thetrue one. As (cid:104) κ (cid:105) ∝ σ Ω . m and (cid:104) κ (cid:105) / (cid:104) κ (cid:105) ∝ σ − . (Bernardeauet al. 1997; Jain & Van Waerbeke 2000), these measurements aredirectly sensitive to the cosmological parameters. Given the currentconstraints in σ and Ω m from Planck Collaboration et al. (2016), changing the cosmological parameters by 2 σ does not affect thecomparison carried out here.From Fig. 4, we find very good agreement between the mea-surements from data and simulations in the overall amplitude andtrend of the second and third moments as a function of smooth-ing scale. The fact that our measurements are in agreement withthe simulations suggests that they are also in agreement with thecosmology assumed in the simulations (see Sec. 3.4), though theerror bars are fairly large compared to e.g. Troxel et al. (2017);DES Collaboration et al. (2017). The histograms of the κ E and κ B maps smoothed with a 5.1 arcminute Gaussian filter are shown inthe right panel of Fig. 5, together with the simulation counterpartsgenerated from the 12 Buzzard simulations. Again, we find good MNRAS000 , 000–000 (0000) C. Chang et al. ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ κ E ; 0 . < z < . RM clusters ( λ > − − − S / N Figure 8.
Top panel shows the κ E map at 0 . < z < .
9, overlaid with
RED M APPER (RM) clusters at λ >
30 and 0 . < z < . λ , or the cluster mass. agreement in the shape and width of the κ E PDF between the sim-ulation and the data. The slightly narrower width of the simulationPDF at the extreme κ E values is likely due to the lack of spatialvariation of shape noise, which is not properly incorporated in thesimulations.Finally, as an additional visual inspection, we overlay a sam-ple of RED M APPER galaxy clusters (Rykoff et al. 2016) onto theE-mode map at 0 . < z < . . < z < . λ >
30 (corresponding to roughly a mass greater than 2 × M (cid:12) ;Melchior et al. 2017). Each circle indicates a cluster, with the sizeof the circle proportion to the richness (mass) of the cluster. Visu-ally we can see the correlation between the cluster positions andthe region of the map with high κ values. It is noticeable that thehigh κ regions in the map are often associated with an ensembleof smaller clusters rather than one large cluster, while there is aclear lack of clusters inside most of the “void” regions in the map.There are exceptions, though, where very high S/N peaks do notline up with the cluster distributions. For example, the peaks at(RA, Dec)=(55.9 ◦ , -53.8 ◦ ) and (34.3 ◦ , -47.5 ◦ ) do not correspond toany clusters at the centre of the peak, and the void area around (RA,Dec)=(60.3 ◦ ,-43.3 ◦ ) overlaps with several clusters. This could bein part due to the shape noise moving the locations of peaks andvoids, as we have seen in Fig. 3. Nevertheless, further investigationof these structures would be interesting in identifying e.g. massivestructures with relatively low luminosity. Overall, in Fig. 8 we findthat there are ∼
30% of clusters in pixels above S/N >
1, and ∼ < -1; ∼
13% of clusters in pixels above S/N >
2, andnone in pixels S/N < -2. We have explored, in Sec. 5, the systematic effects associated withthe reconstruction algorithm, masking, shot noise, and shape noiseusing simulations. We also examined the zeroth order systematiceffects in the data by looking at the B mode convergence maps inSec. 6.1. In this section we concentrate on examining other poten- tial sources of systematic effects that could contaminate our maps.Specifically, we look at whether there exists any spurious correla-tion between our maps and quantities that are not expected to cor-relate with the convergence maps. This technique is similar to thatused in Elvin-Poole et al. (2017).We first identify a number of potential systematics that couldcontaminate the κ E maps. The potential systematics presented hereare listed below: • κ B : B-mode convergence map • ε , ε : the mean galaxy ellipticity • ε , ε PSF : the mean PSF ellipticity • κ E , PSF , κ B , PSF : κ E and κ B maps generated from ε and ε • R PSF : the mean PSF size used for galaxy shape measurement • R PSF , r : the mean r -band PSF FWHM size • depth r : the mean r -band magnitude limit • airmass r : the mean r -band airmass.Note that we have checked the PSF size, depth and airmassquantities for other filter bands but only present here the r -bandquantities. For potential systematics s , we construct map M s .For quantities where we expect the mean to be close to zero( κ B , ε , ε , ε , ε , κ E , PSF , κ B , PSF ), M s is constructed using themean-subtracted values; whereas for the rest of the quantities wherethe mean is non-zero, we use the fractional contrast of the map M s = δ s = s − ¯ s ¯ s .We first look for correlation at the pixel level between the fourtomographic κ E maps with each of the above potential systemat-ics s . That is, we are interested in whether the high κ E values areassociated with a certain systematic quantity being high or low. Todo this, we bin the pixels in the systematics templates into 10 binsdepending on the value of the pixels, and measure the average con-vergence in the pixels assigned to each of the 10 bins. The errorbars are evaluated using a Jackknife approach. We then perform a We use the quantity mean psf fwhm in the I M SHAPE catalog and psfrec T in the M
ETA C ALIBRATION catalog. These are 10 σ detection limits for galaxies.MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map | b | / σ b < z < Linear fit slope: κ E MetaCalibrationIm3shape0123 | b | / σ b < z < | b | / σ b < z < κ B ε ε ε P S F ε P S F κ E , P S F κ B , P S F R P S F d e p t h r a i r m a ss r R P S F , r | b | / σ b < z < | b | /σ b BuzzardData (MetaCalibration)
Figure 9.
The upper four panels show the absolute value of the best-fitslope b divided by the uncertainty of b for the linear fit of κ E vs. varioussystematics templates. | b | / σ b measures the significance of a trend betweenthe convergence and the systematics templates. The four panels correspondto the four tomographic bins which we construct the κ E maps, and the twoset of points correspond to the two shear catalogs. The list of systematicstemplates are labeled for the last redshift bin. The bottom panel shows ahistogram of | b | / σ ( b ) measured from the 12 Buzzard simulations (thin bluelines) compared to M ETA C ALIBRATION data points (thick black line). linear fit with intercept a and slope b to the measurements. In orderto see whether there is a significant correlation between the valueof the convergence and the value of the systematics template, weplot | b | / σ b in Fig. 9. There is one data point that has a | b | / σ b valuelarger than 3 ( κ E , PSF for M
ETA C ALIBRATION in highest redshiftbin), which we show in the histogram in the bottom panel. To un-derstand whether these | b | / σ b values are a cause of concern, weperform the same analysis for the 12 Buzzard simulation maps by χ / N d o f < z < h κ E M s i MetaCalibrationIm3shape0123 χ / N d o f < z < χ / N d o f < z < κ B ε ε ε P S F ε P S F κ B , P S F κ E , P S F R P S F d e p t h r a i r m a ss r R P S F , r χ / N d o f < z < . . . . . . χ /N dof . . . . . . . BuzzardData (MetaCalibration)
Figure 10.
Similar to Fig. 9, the upper four panels show the reduced χ forthe cross-correlation of the κ E maps with various systematics templates tobe consistent with zero. The cross-correlation is measured for a range ofscales, with a total of 8 data points, thus the number of degrees-of-freedom( N dof ) for the χ is 8. The bottom panel shows a histogram of the reduced χ measured from the 12 Buzzard simulations (thin blue lines) comparedto M ETA C ALIBRATION data points (thick black line). cross-correlating them with the systematics templates. Since thesesimulations cannot be possibly correlated with the data, this mea-surement provides a quantitative way to interpret the results. Thedistribution of all | b | / σ b values are shown in the bottom panel ofFig. 9, together with the M ETA C ALIBRATION results (as the sim-ulations are matched to the characteristics of the M
ETA C ALIBRA - TION catalog). We find that 97% (88%) of the points in the simu-lations are below 3 σ (2 σ ), which is in reasonable agreement withthat from the data (98% of the points below 3 σ and 91% of the MNRAS000
ETA C ALIBRA - TION catalog). We find that 97% (88%) of the points in the simu-lations are below 3 σ (2 σ ), which is in reasonable agreement withthat from the data (98% of the points below 3 σ and 91% of the MNRAS000 , 000–000 (0000) C. Chang et al. points below 2 σ ). The overall distribution of | b | / σ b values in thesimulations also agrees well with the data.Next, we compute the two-point angular cross correlation be-tween the convergence maps and the systematics templates. Thismeasurement tests the potential contamination of cross-correlatingthe κ maps with other maps, such as that investigated in Sec. 7.1.We measure (cid:104) κ M s (cid:105) ( θ i ) = N i N i ∑ j ( κ M s ) j , (20)where M s is the systematics template of interest and the sum is overall pairs of pixels j in the maps separated by angular distance withinthe bin θ i . The correlation function is evaluated in 8 logarithmicallyseparated angular bins θ i between 10 and 200 arcminutes. The co-variance matrix is derived from the Jackknife approach. We thencalculate the reduced χ of each correlation for it to be consistentwith null signal, ie. (cid:104) κ M s (cid:105) ( θ i ) =
0, at all θ i , where the χ is definedthrough χ = DCov − D D T (21)where D = (cid:104) κ M s (cid:105) ( θ i ) is the angular correlation function and Cov D is the covariance matrix between the 8 angular bins.The results of the two-point cross-correlation are shown inFig. 10. We also perform similar measurements using the 12 Buz-zard simulations and show the total distribution of the reduced χ in the bottom panel. We find that reduced χ for all combinationsof maps, shear catalogs, and redshift bins, all fall below 3, indicat-ing no significant contamination in the maps directly introducedby these potential systematics quantities on the two-point level.Comparing with simulations also shows that the overall distribu-tion of these reduced χ values are consistent with no correlationbetween the κ E maps and the systematics templates. We find that100% (92%) of the points in the simulations are below 2 σ (1 σ ),which is in reasonable agreement with that from the data (98% ofthe points below 2 σ and 80% of the points below 1 σ ). In this section we present two applications of the convergence mapconstructed from this work. In Sec. 7.1 we cross-correlate the con-vergence maps with foreground mass tracers to demonstrate thatour maps do indeed contain significant signal and is consistent withexpectation. In Sec. 7.2 we take a closer look at some of the highsignal-to-noise structure in the maps and discuss the physical in-terpretation for the largest peaks and voids respectively. We defersome of the more involved applications (e.g. cross-correlation ofthe convergence maps to CMB lensing maps, peak statistics) to fu-ture work.
One of the motivations for generating a convergence map insteadof using the weak lensing shear directly is that in many cases ascalar field is easier to manipulate and cross-correlate with otherdata sets compared to a spin-2 field. Here we demonstrate someof the usages by cross-correlating the convergence maps in Fig. 7with other tracers of mass. Specifically, we look at a flux-limitedgalaxy sample (described in Sec. 3.3) and the
RED M AGIC
Lumi-nous Red Galaxies (LRG) sample (Rykoff et al. 2016). The am-plitudes of these cross-correlations will be a direct measure of thegalaxy bias for the different samples (see e.g. Pujol et al. 2016; Chang et al. 2016). Note that the cross-correlation can naturally ex-tend to include maps of other wavelengths such as X-ray, Gammaray (Shirasaki et al. 2014), H I neutral hydrogen (Kirk et al. 2015),the CMB, CMB lensing (Liu & Hill 2015; Hand et al. 2015; Kirket al. 2016) and others.In this analysis, we opt for calculating the real-space 2-pointcorrelation function similar to that used in Sec. 6.2, (cid:104) κδ X (cid:105) ( θ i ) = N i N i ∑ j ( κδ X ) j , (22)where X denotes the specific sample of interest (flux-limited galaxysample or RED M AGIC galaxies in different redhshift ranges). δ = n − ¯ n ¯ n is the density contrast of the sample, where n is the number ofcounts per pixel and ¯ n is the mean number count over the full map.The average is calculated for all pairs of points j whose angularseparation θ fall in the angular bin θ i . The cross-correlation is cal-culated for scales 2.5 to 250 arcminutes. In later analyses wherewe compare the cross-correlation between the convergence mapand the two foreground samples, we exclude scales larger than 100arcminutes and smaller than 15 arcminutes. The small-scale cut-off corresponds to about 3 times the scale corresponding to (cid:96) max ,while the large-scale cutoff corresponds to the size of the Jackkniferegion.We begin with testing whether the cross-correlation betweenthe κ E and κ B map with a foreground flux-limited galaxy sample isconsistent with expectation from the simulations. We use the sameset of mock galaxy catalogs used in Sec. 5.2, with the addition ofa simulated foreground sample that matches with the flux-limitedsample. We perform the cross-correlation for various redshift com-binations of the κ map and the galaxy map, as well as the two shearcatalogs. We find very good agreement between the two shear cat-alogs and between the simulation and data.In Fig. 11, we show two examples of the measurements: cross-correlation of the M ETA C ALIBRATION κ maps at 0 . < z < . . < z < . δ g maps at0 . < z < . . < z < .
6, respectively. We show the datameasurements together with the mean and standard deviation ofthe 12 measurements from the Buzzard simulations. Both the E-mode and the B-mode cross-correlation show excellent agreementbetween the data and the simulations. As the amplitude of the cross-correlation is sensitive to the cosmological model, galaxy bias, andthe photo- z , the agreement between simulations and data suggeststhat there is no outstanding differences between the simulations andthe data that could be potentially a sign of systematic effects.Next, we measure the cross-correlation of the same κ mapswith foreground RED M AGIC samples. We construct the samplesso that the mean and spread of the n ( z ) distribution is similar tothat of the flux-limited sample. This corresponds to a redshift se-lection of 0 . < z < .
45 (0 . < z < .
6) for the flux-limitedsample at 0 . < z < . . < z < . RED M AGIC sampleand the flux-limited sample would scale directly as the ratio of thegalaxy bias for the two samples. The cross-correlation between the κ maps and the RED M AGIC sample is also shown in Fig. 11. Theerror bars for the
RED M AGIC sample are larger due to the lowernumber density, but overall the shape of the cross-correlation asa function of scale is very similar, with an overall multiplicativefactor that is nearly constant over scales. The value of the multi-plicative factor (within the 15–100 arcminute range) is ∼ .
38 forthe lower redshift bin 0 . < z < . ∼ .
27 for the higher red-shift bin 0 . < z < .
6. Crocce et al. (2016) measured the galaxybias for a flux-limited galaxy sample in the DES SV data using an-
MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map θ (arcmin) . . . . . . . κ : 0 . < z < . δ g : 0 . < z < . h κ E δ g i ; Buzzard h κ B δ g i ; Buzzard h κ E δ g ih κ B δ g ih κ E δ RM i θ (arcmin) . . . . . . . κ : 0 . < z < . δ g : 0 . < z < . h κ E δ g i ; Buzzard h κ B δ g i ; Buzzard h κ E δ g ih κ B δ g ih κ E δ RM i Figure 11.
Cross-correlation of the κ maps with foreground galaxy samples. The blue (red) data points show the cross-correlation between the κ E ( κ B ) mapwith the foreground flux-limited sample for two redshift bins. The shaded band show the mean and standard deviation of the 12 Buzzard simulations, while thedata points show the DES Y1 data. The green data points show (cid:104) κ E δ RM (cid:105) , the cross-correlation of the same κ E maps with the foreground RED M AGIC samplewhich have similar redshift distributions as the two flux-limited samples but higher galaxy bias (therefore higher amplitudes). The grey shaded region is notused for the calculation of galaxy bias. –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 ∆ RA [deg] –1.5–1.0–0.50.00.51.01.5 ∆ D ec [ d e g ] z N < z < ∆ RA [deg] –1.5–1.0–0.50.00.51.01.5 ∆ D ec [ d e g ] z N < z < Figure 12.
This figure shows the cluster galaxy distribution around the biggest overdensity in Fig. 8 at (RA, Dec) = (309 ◦ , − ◦ ). We show in the left panelsall members of RED M APPER -detected galaxy clusters with 0 < z < λ >
5. The top panel is the spatial distribution, projected onto the tangent plane, withshading of each circle indicating the lensing weights. The red dots indicate SPT-detected galaxy clusters from Bleem et al. (2015) in the same redshift range.The bottom panel is the redshift distribution of the cluster member galaxies, before (light) and after (dark) applying the lensing weights. The right panelsshow the same as the left panels, but for a narrower redshift range of z = . ± . gular clustering measurements; their results give a bias of ∼ . ∼ .
29) if we interpolate onto the redshift and magnitude rangeof our sample at 0 . < z < . . < z < . RED M AGIC sample to be ∼ ∼ . ∼ MNRAS000
29) if we interpolate onto the redshift and magnitude rangeof our sample at 0 . < z < . . < z < . RED M AGIC sample to be ∼ ∼ . ∼ MNRAS000 , 000–000 (0000) C. Chang et al.
Another strength of map-level products is that one can visualiseand detect pronounced local over- or underdense regions that wouldotherwise be averaged over in global summary statistics. The abun-dance of the massive peaks is a sensitive cosmological probe, asthey occupy the highest end of the halo mass function (Bahcall& Fan 1998; Haiman et al. 2001; Holder et al. 2001). Some ofthe extreme structures can also help to constrain a certain class ofmodified gravity theories (Knox et al. 2006; Jain & Khoury 2010).On the other hand, the abundance of large voids has been used asa powerful test of Λ CDM cosmology (Plionis & Basilakos 2002;Higuchi & Inoue 2017). In this section we seek to briefly charac-terise the physical nature of peaks and voids that are associated withthe largest over- or underdense regions in the convergence maps.To construct a catalog of peaks and voids, we begin with the4 tomographic S/N maps presented in Fig. 7, which are smoothedwith a σ G =
30 arcminutes Gaussian filter. We place a threshold onthe pixels value at 2.5 σ (S/N > . S / N < − . RED M APPER clusters with richness λ > ◦ , -56 ◦ ). While some correlated struc-ture appears present in the 2D distribution, the redshift distribu-tions of the cluster galaxies in this region appears to be very broad,even after taking the lensing kernel into account. This suggests thatthe large peak cannot be accounted for by one large structure lo-calised in redshift space. In the right panel of Fig. 12 we isolatea particular peak at z ≈ . RED M APPER clusters with z = . ± . z ≈ .
4, falls in this redshiftrange.Performing an analogous analysis at different redshifts oron other peaks yields similar results, namely that overdensitiessmoothed on such a large scale generally do not correspond to mas-sive structures in physical contact. Instead, the broad redshift ker-nel is prone to accumulating multiple layers of mildly overdensestructures along the line of sight. This outcome demonstrates thedifficulty of detecting clusters in weak-lensing mass maps or shearcatalogs, especially when the number density of source galaxies islow and one cannot go to a smaller smoothing scale. This generallyneeds the construction of optimal matched filters in configurationand redshift space (Maturi et al. 2005; Simon et al. 2009; Vander-Plas et al. 2011), which is outside of the scope of this work.For voids the situation is more promising. We use
RED -M AGIC galaxies with relatively good photo- z ’s (same as that usedin Sec. 7.1) as tracers of the foreground matter density and study
600 900 1200 1500 1800 2100
Comoving distance [Mpc/h] − . − . − . − . − . − . . . δ R v =220 Mpc/h R v =100 Mpc/h S o u r ce ga l a x i e s . < z < . Redshift
Figure 13.
The data points show the
RED M AGIC galaxy density contrast δ along the foreground line-of-sight of the largest void identified in themass map in the highest redshift bin. The profile fits very well with a modelconsisting two supervoids with a size of 220 and 100 h − Mpc, as shownwith the cyan line. their radial distribution. We project the data into 2D slices of 50 h − Mpc along the line-of-sight. We then measure the density con-trast of the
RED M AGIC galaxies in these 2D slices where the largevoids in the maps (significant negative convergence values) aremeasured, compared to the mean
RED M AGIC density at that red-shift. The density contrast measurements at different redshifts arethen used to reconstruct the radial density profile of voids. Asan example, we look at the largest void detected in the furthest0 . < z < . ◦ , -43 ◦ ), and count galaxieswithin 2.0 degrees of the void centre, which approximately cor-responds to the full angular size of the void in the map. We showthe resulting line-of-sight density profile measurements of the RED -M AGIC galaxies in Fig. 13. We find two extended underdensitiesthat are consistent with supervoids of radii R v =100 h − Mpc and R v =220 h − Mpc assuming simple Gaussian void profiles (Finelliet al. 2016; Kov´acs & Garc´ıa-Bellido 2016; S´anchez et al. 2017).These supervoids are quite shallow even in their centres but theirsize is comparable to the largest known supervoids. Most probably,these supervoids have substructure at smaller scales but that infor-mation is not accessible even using high quality photo- z data like RED M AGIC .We repeated the above analysis for less significant and lessextended voids, finding that voids identified in the mass maps thatextend beyond ∼ of size can typically be associated withat least one R v (cid:38) h − Mpc supervoid in the
RED M AGIC cata-logue. These are of greatest interest in cosmology and their inte-grated Sachs-Wolfe imprint was also studied using DES Y1 data(see e.g. Kov´acs et al. 2017).
Weak lensing allows us to probe the total mass distribution in theUniverse. One of the most intuitive ways to visualise and compre-hend this information is through weak lensing convergence maps,or mass maps. These maps contain the Gaussian and non-Gaussianinformation for the matter field, which could then either be ex-tracted via various statistical tools, or analyzed locally for regionsof special interest.
MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map In this paper, we construct weak lensing mass maps for thefirst year of Dark Energy Survey data (DES Y1) using two inde-pendent shear catalogs, M
ETA C ALIBRATION and I M SHAPE , inthe redshift range 0 . < z < . ∼ ,
500 deg , corresponding to a total volume of ≈
10 Gpc .With the unprecedented large sky coverage, a spherical reconstruc-tion approach was used based on decomposing the shear field intospin-2 spherical harmonics, followed by an E/B mode separation.The curl-free E-mode and the divergence-free B-mode form the E-and B-mode lensing convergence maps, κ E and κ B . The lensingpotential ψ and deflection angles η can also be reconstructed usingthese decomposed spin harmonics.We test the mass map reconstruction with simulations, startingwith an idealised setup and gradually degrading the simulations tomatch the data. By doing so, we can isolate the effect of individ-ual sources of systematics and noise. We use the F and F statis-tics (Eq. (17)) to quantify the performance of the reconstruction interms of the amplitude and the phase information: for perfect recon-struction F = F =
1. Based on these statistics, we find that (1) wecan reconstruct very well the convergence field in a fully-sampled,full-sky Gaussian simulation for scales larger than the pixel scale,as expected; (2) the DES Y1 mask biases the reconstructed mapsat the few percent level, but the bias mainly comes from the pix-els around the edges; (3) finite sampling from galaxies at the DESY1 density does not degrade the reconstruction significantly for ourmaps at a resolution of 3.44 arcminutes; (4) adding shape noise in-creases the variance of the map and perturbs the phase information,but at the DES Y1 noise level, the signal-to-noise is still signifi-cant and the resulting F and F are consistent with 1; (5) we canreconstruct within measurement uncertainty the second moment ofthe maps on all scales and third moment of the maps for scales > ψ and deflection angles η maps. We explore briefly in Ap-pendix B this application, finding a ∼
70% ( ∼ ψ ( η ). The reconstruction of ψ and η is relatively poor compared to that of the κ maps because informa-tion in η is dominated by scales larger than κ , and the informationin ψ is dominated by even larger scales. This suggests that from κ to η to ψ , the importance of the mask increases while the impor-tance of shot noise and shape noise decreases.After rigorous testing with simulations, we generate weaklensing mass maps from the DES Y1 data with a spatial reso-lution of ∼ .
44 arcminutes. We construct one map that coversthe entire redshift range of 0 . < z < .
3, which carries the high-est S/N, and also four tomographic bins at the redshift intervals0 . < z < .
43, 0 . < z < .
63, 0 . < z < .
9, and 0 . < z < . ETA C ALIBRATION , 0 . < z < . ∼ ∼
2) when smoothed with a Gaussian filter of σ G =
30 (80)arcminutes. We examine the PDF of the maps, together with thesecond and third moments of the PDF as a function of smoothingscale and find them to be consistent with realistic simulations thatincorporate similar noise and mask properties as the data. We fur-ther test for systematic effects by cross-correlating the maps withvarious environment and PSF quantities at the one-point and two- point level. We find no significant systematic contamination of themaps beyond what is expected from statistical fluctuations.Finally, we demonstrate two applications of these mass maps.First, we cross-correlate the mass maps with two sets of fore-ground mass tracers constructed to have similar redshift distri-butions: a flux-limited galaxy sample and an LRG sample. Thecross-correlation is done in two redshift bins and shows very goodagreement with simulations. The ratio of the amplitudes of thecross-correlation, which reflects the ratio of the galaxy bias for thetwo samples, are consistent with previous measurements of simi-lar samples in earlier DES data. Second, we examine the extremepeaks and voids identified in the maps. We find that most high S/Npeaks in the maps correspond to an accumulated mass distribu-tion along the line of sight, even though rare filamentary structurescould be found occasionally. For the high-S/N voids, however, mostof them correspond to real void structures with R v (cid:38) h − Mpcin the foreground.The DES Y1 mass maps are the largest weak lensing massmaps to date constructed from galaxy surveys, about ten timeslarger than the previous maps from CFHTLenS (Van Waerbekeet al. 2013) and DES SV (Vikram et al. 2015; Chang et al. 2015).Even though the Y1 depth is shallower (and therefore noisier) thanthe previous maps, these very large maps provide a new perspec-tive on weak lensing map making and the various topics one canexplore with them. Moving onto the larger dataset from DES andother surveys, we expect many of the explorations in this paper tobe carried out and advanced to serve as complementary probes ofcosmology alongside more traditional two-point statistics.
ACKNOWLEDGEMENT
We thank Sandrine Pires for useful discussions. CC was supportedin part by the Kavli Institute for Cosmological Physics at the Uni-versity of Chicago through grant NSF PHY-1125897 and an en-dowment from Kavli Foundation and its founder Fred Kavli. APacknowledges support from beca FI and 2009-SGR-1398 fromGeneralitat de Catalunya, project AYA2012-39620 and AYA2015-71825 from MICINN, and from a European Research CouncilStarting Grant (LENA-678282). Support for DG was providedby NASA through Einstein Postdoctoral Fellowship grant numberPF5-160138 awarded by the Chandra X-ray Center, which is op-erated by the Smithsonian Astrophysical Observatory for NASAunder contract NAS8-03060. BL was supported by NASA throughEinsteinPostdoctoral Fellowship Award Number PF6-170154. BJand MJ are partially supported by the US Department of Energygrant DE-SC0007901 and funds from the University of Pennsylva-nia. ES is supported by DOE grant DE-AC02-98CH10886.Funding for the DES Projects has been provided by theU.S. Department of Energy, the U.S. National Science Founda-tion, the Ministry of Science and Education of Spain, the Sci-ence and Technology Facilities Council of the United Kingdom, theHigher Education Funding Council for England, the National Cen-ter for Supercomputing Applications at the University of Illinois atUrbana-Champaign, the Kavli Institute of Cosmological Physics atthe University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institutefor Fundamental Physics and Astronomy at Texas A&M Univer-sity, Financiadora de Estudos e Projetos, Fundac¸˜ao Carlos ChagasFilho de Amparo `a Pesquisa do Estado do Rio de Janeiro, Con-selho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico andthe Minist´erio da Ciˆencia, Tecnologia e Inovac¸˜ao, the Deutsche
MNRAS000
MNRAS000 , 000–000 (0000) C. Chang et al.
Forschungsgemeinschaft and the Collaborating Institutions in theDark Energy Survey.The Collaborating Institutions are Argonne National Labora-tory, the University of California at Santa Cruz, the University ofCambridge, Centro de Investigaciones Energ´eticas, Medioambien-tales y Tecnol´ogicas-Madrid, the University of Chicago, Univer-sity College London, the DES-Brazil Consortium, the Universityof Edinburgh, the Eidgen¨ossische Technische Hochschule (ETH)Z¨urich, Fermi National Accelerator Laboratory, the University ofIllinois at Urbana-Champaign, the Institut de Ci`encies de l’Espai(IEEC/CSIC), the Institut de F´ısica d’Altes Energies, LawrenceBerkeley National Laboratory, the Ludwig-Maximilians Univer-sit¨at M¨unchen and the associated Excellence Cluster Universe, theUniversity of Michigan, the National Optical Astronomy Observa-tory, the University of Nottingham, The Ohio State University, theUniversity of Pennsylvania, the University of Portsmouth, SLACNational Accelerator Laboratory, Stanford University, the Univer-sity of Sussex, Texas A&M University, and the OzDES Member-ship Consortium.The DES data management system is supported by the Na-tional Science Foundation under Grant Numbers AST-1138766and AST-1536171. The DES participants from Spanish institu-tions are partially supported by MINECO under grants AYA2015-71825, ESP2015-88861, FPA2015-68048, SEV-2012-0234, SEV-2016-0597, and MDM-2015-0509, some of which include ERDFfunds from the European Union. IFAE is partially funded by theCERCA program of the Generalitat de Catalunya. Research leadingto these results has received funding from the European ResearchCouncil under the European Union’s Seventh Framework Pro-gram (FP7/2007-2013) including ERC grant agreements 240672,291329, and 306478. We acknowledge support from the AustralianResearch Council Centre of Excellence for All-sky Astrophysics(CAASTRO), through project number CE110001020.This manuscript has been authored by Fermi Research Al-liance, LLC under Contract No. DE-AC02-07CH11359 with theU.S. Department of Energy, Office of Science, Office of High En-ergy Physics. The United States Government retains and the pub-lisher, by accepting the article for publication, acknowledges thatthe United States Government retains a non-exclusive, paid-up, ir-revocable, world-wide license to publish or reproduce the publishedform of this manuscript, or allow others to do so, for United StatesGovernment purposes.This paper has gone through internal review by the DES col-laboration.
REFERENCES
Abbott T., et al., 2016, Phys.Rev.D, 94, 022001Aihara H., et al., 2017, preprint, ( arXiv:1702.08449 )Bahcall N. A., Fan X., 1998, ApJ, 504, 1Bartelmann M., 2010, Classical and Quantum Gravity, 27, 233001Bartelmann M., Schneider P., 2001, Physics Reports, 340, 291Becker M. R., 2013, MNRAS, 435, 115Ben´ıtez N., 2000, ApJ, 536, 571Bernardeau F., van Waerbeke L., Mellier Y., 1997, A&A, 322, 1Blazek J., Vlah Z., Seljak U., 2015, JCAP, 8, 015Bleem L. E., et al., 2015, ApJS, 216, 27Bonnett C., et al., 2016, Phys.Rev.D, 94, 042005Bruderer C., Chang C., Refregier A., Amara A., Berg´e J., Gamper L., 2016,ApJ, 817, 25Bruzual G., Charlot S., 2003, MNRAS, 344, 1000Carlstrom J. E., et al., 2011, PASP, 123, 568 Castro P. G., Heavens A. F., Kitching T. D., 2005, Phys.Rev.D, 72, 023516Cawthon R., et al., 2017, in prep.Chang C., Jain B., 2014, MNRAS, 443, 102Chang C., et al., 2015, Physical Review Letters, 115, 051301Chang C., et al., 2016, MNRAS, 459, 3203Clerkin L., Kirk D., Lahav O., Abdalla F. B., Gazta˜naga E., 2015, MNRAS,448, 1389Clowe D., Bradaˇc M., Gonzalez A. H., Markevitch M., Randall S. W., JonesC., Zaritsky D., 2006, ApJ, 648, L109Coe D., Ben´ıtez N., S´anchez S. F., Jee M., Bouwens R., Ford H., 2006, AJ,132, 926Coleman G. D., Wu C.-C., Weedman D. W., 1980, ApJS, 43, 393Comaniciu D., Meer P., 2002, IEEE Transactions on Pattern Analysis andMachine Intelligence, 24, 603Conroy C., Wechsler R. H., Kravtsov A. V., 2006, ApJ, 647, 201Cooper M. C., 2006, in American Astronomical Society Meeting Abstracts.p. 1159Cooray A., Hu W., 2001, ApJ, 548, 7Crocce M., Pueblas S., Scoccimarro R., 2006, MNRAS, 373, 369Crocce M., Castander F. J., Gazta˜naga E., Fosalba P., Carretero J., 2015,MNRAS, 453, 1513Crocce M., et al., 2016, MNRAS, 455, 4301DES Collaboration et al., 2017, to be submitted to Phys. Rev. DDark Energy Survey Collaboration et al., 2016, MNRAS, 460, 1270Davis C., et al., 2017, in prep.Dietrich J. P., Hartlap J., 2010, MNRAS, 402, 1049Dodelson S., Zhang P., 2005, Phys.Rev.D, 72, 083001Dodelson S., Schmidt F., Vallinotto A., 2008, Phys.Rev.D, 78, 043508Drlica-Wagner A., et al., 2017, submitted to Astrophys. J. Suppl. Ser.Elvin-Poole J., et al., 2017, to be submitted to Phys. Rev. DErben T., et al., 2013, MNRAS, 433, 2545Fenech Conti I., Herbonnet R., Hoekstra H., Merten J., Miller L., Viola M.,2016, preprint, ( arXiv:1606.05337 )Finelli F., Garc´ıa-Bellido J., Kov´acs A., Paci F., Szapudi I., 2016, MNRAS,455, 1246Flaugher B., 2005, International Journal of Modern Physics A, 20, 3121Flaugher B., et al., 2015, AJ, 150, 150Fosalba P., Gazta˜naga E., Castander F. J., Manera M., 2008, MNRAS, 391,435Fosalba P., Gazta˜naga E., Castander F. J., Crocce M., 2015a, MNRAS, 447,1319Fosalba P., Crocce M., Gazta˜naga E., Castander F. J., 2015b, MNRAS, 448,2987Gatti M., et al., 2017, in prep.Gaztanaga E., Bernardeau F., 1998, A&A, 331, 829G´orski K. M., Hivon E., Banday A. J., Wandelt B. D., Hansen F. K., Rei-necke M., Bartelmann M., 2005, ApJ, 622, 759Haiman Z., Mohr J. J., Holder G. P., 2001, ApJ, 553, 545Hand N., et al., 2015, Phys.Rev.D, 91, 062001Heavens A., 2003, MNRAS, 343, 1327Heavens A. F., Kitching T. D., Taylor A. N., 2006, MNRAS, 373, 105Higuchi Y., Inoue K. T., 2017, preprint, ( arXiv:1707.07535 )Hildebrandt H., et al., 2017, MNRAS, 465, 1454Holder G., Haiman Z., Mohr J. J., 2001, ApJ, 560, L111Hoyle B., et al., 2017, to be submitted to Mon. Not. R. Astron. Soc.Huff E., Mandelbaum R., 2017, preprint, ( arXiv:1702.02600 )Jain B., Khoury J., 2010, Annals of Physics, 325, 1479Jain B., Seljak U., 1997, ApJ, 484, 560Jain B., Van Waerbeke L., 2000, ApJ, 530, L1Jarvis M., et al., 2015, preprint, ( arXiv:1507.05603 )Kacprzak T., Zuntz J., Rowe B., Bridle S., Refregier A., Amara A., VoigtL., Hirsch M., 2012, MNRAS, 427, 2711Kacprzak T., et al., 2016, MNRAS, 463, 3653Kaiser N., Squires G., 1993, ApJ, 404, 441Kinney A. L., Calzetti D., Bohlin R. C., McQuade K., Storchi-BergmannT., Schmitt H. R., 1996, ApJ, 467, 38Kirk D., Abdalla F. B., Benoit-L´evy A., Bull P., Joachimi B., 2015, Ad-MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map vancing Astrophysics with the Square Kilometre Array (AASKA14),p. 20Kirk D., et al., 2016, MNRAS, 459, 21Kitching T. D., et al., 2014, MNRAS, 442, 1326Knox L., Song Y.-S., Tyson J. A., 2006, Phys.Rev.D, 74, 023512Kov´acs A., Garc´ıa-Bellido J., 2016, MNRAS, 462, 1882Kov´acs A., et al., 2017, MNRAS, 465, 4166Kratochvil J. M., Haiman Z., May M., 2010, Phys.Rev.D, 81, 043519Kwan J., et al., 2017, MNRAS, 464, 4045Leistedt B., McEwen J. D., B¨uttner M., Peiris H. V., 2017, MNRAS, 466,3728Leonard A., Lanusse F., Starck J.-L., 2014, MNRAS, 440, 1281Lewis A., Bridle S., 2002, Phys.Rev.D, 66, 103511Liu J., Hill J. C., 2015, Phys.Rev.D, 92, 063517MacCrann N., et al., 2017, in prep.Mandelbaum R., et al., 2017, preprint, ( arXiv:1705.06745 )Manzotti A., et al., 2017, preprint, ( arXiv:1701.04396 )Marian L., Bernstein G. M., 2007, Phys.Rev.D, 76, 123009Marshall P. J., Hobson M. P., Gull S. F., Bridle S. L., 2002, MNRAS, 335,1037Massey R., et al., 2007, ApJS, 172, 239Maturi M., Meneghetti M., Bartelmann M., Dolag K., Moscardini L., 2005,A&A, 442, 851Melchior P., et al., 2015, MNRAS, 449, 2219Melchior P., et al., 2017, MNRAS, 469, 4899Oguri M., et al., 2017, preprint, ( arXiv:1705.06792 )Patton K., Blazek J., Honscheid K., Huff E., Melchior P., Ross A. J.,Suchyta E., 2016, preprint, ( arXiv:1611.01486 )Pires S., Starck J.-L., Amara A., Teyssier R., R´efr´egier A., Fadili J., 2009,MNRAS, 395, 1265Planck Collaboration et al., 2016, A&A, 594, A13Plionis M., Basilakos S., 2002, MNRAS, 330, 399Prat J., et al., 2017, to be submitted to Phys. Rev. DPujol A., et al., 2016, MNRAS, 462, 35Reddick R. M., Wechsler R. H., Tinker J. L., Behroozi P. S., 2013, ApJ,771, 30Refregier A., Kacprzak T., Amara A., Bridle S., Rowe B., 2012, MNRAS,425, 1951Rykoff E. S., et al., 2016, ApJS, 224, 1S´anchez C., et al., 2017, MNRAS, 465, 746Schneider P., 1996, MNRAS, 283, 837Seitz S., Schneider P., Bartelmann M., 1998, A&A, 337, 325Sheldon E. S., 2014, MNRAS, 444, L25Sheldon E. S., Huff E. M., 2017, preprint, ( arXiv:1702.02601 )Shirasaki M., Horiuchi S., Yoshida N., 2014, Phys.Rev.D, 90, 063502Simon P., Taylor A. N., Hartlap J., 2009, MNRAS, 399, 48Springel V., 2005, MNRAS, 364, 1105Troxel M. A., Ishak M., 2015, Physics Reports, 558, 1Troxel M. A., et al., 2017, to be submitted to Phys. Rev. DVallinotto A., Dodelson S., Schimd C., Uzan J.-P., 2007, Phys.Rev.D, 75,103509Van Waerbeke L., et al., 2013, MNRAS, 433, 3373VanderPlas J. T., Connolly A. J., Jain B., Jarvis M., 2011, ApJ, 727, 118Vikram V., et al., 2015, Phys.Rev.D, 92, 022006Wallis C. G. R., McEwen J. D., Kitching T. D., Leistedt B., Plouviez A.,2017, preprint, ( arXiv:1703.09233 )Yu B., Hill J. C., Sherwin B. D., 2017, preprint, ( arXiv:1705.02332 )Zuntz J., Kacprzak T., Voigt L., Hirsch M., Rowe B., Bridle S., 2013, MN-RAS, 434, 1604Zuntz J., et al., 2015, Astronomy and Computing, 12, 45Zuntz J., et al., 2017, submitted to Mon. Not. R. Astron. Soc.de Jong J. T. A., et al., 2015, A&A, 582, A62von der Linden A., et al., 2014, MNRAS, 439, 2 Excluded edges (arcmin) . . . . . . . . F FiducialGaussian interpolationMean interpolationRandom interpolation
Figure A1. F = (cid:104) κ E κ sm (cid:105) / (cid:104) κ (cid:105) for different interpolation schemes forempty pixels inside the footprint. APPENDIX A: INTERPOLATING EMPTY PIXELS
In this appendix we test the impact of the empty pixels insidethe contiguous footprint and different approaches to interpolateover them. We use the same noiseless Buzzard simulations used inSec. 5.2 and test with the redshift bin of 0 . < z < .
3. In this map,the fraction of empty pixels inside the footprint occupies ∼ . F and F statistics defined in Sec. 5: • Fiducial: set the empty pixels to 0. • Gaussian interpolation: interpolate the values of these emptypixels from a Gaussian kernel with a σ corresponding to 3 timesthe pixel size. • Mean interpolation: we assign the empty pixels the meanvalue of their neighbour pixels. • Random interpolation: we assign the empty pixels the value ofa random neighbour pixel.In Fig. A1 we show the F statistics as a function of the scaleexcluded from the edges for all this cases, similar to Fig. 2. The F statistics looks qualitatively similar to F . We see that at ourresolution, the different approaches all give very similar results.We therefore adopt the fiducial approach for simplicity in our mainanalysis. APPENDIX B: RECONSTRUCTING THE LENSINGPOTENTIAL AND DEFLECTION MAP
As discussed in Sec. 2, in addition to the convergence maps κ , wecan also construct the lensing potential ψ and deflection η mapswith similar formalism. In this appendix we show the implementa-tion of the reconstruction for ψ and η . We perform in Appendix B1similar tests on the reconstruction with simulations as in Sec. 5.1.Then we apply the method to DES Y1 data in Appendix B2. Al-though the quality of the reconstruction for ψ and η is not com-parable to that of the κ maps, they point to an area that we canstart to explore as the sky coverage of future weak lensing data setsbecomes increasingly large. MNRAS000
As discussed in Sec. 2, in addition to the convergence maps κ , wecan also construct the lensing potential ψ and deflection η mapswith similar formalism. In this appendix we show the implementa-tion of the reconstruction for ψ and η . We perform in Appendix B1similar tests on the reconstruction with simulations as in Sec. 5.1.Then we apply the method to DES Y1 data in Appendix B2. Al-though the quality of the reconstruction for ψ and η is not com-parable to that of the κ maps, they point to an area that we canstart to explore as the sky coverage of future weak lensing data setsbecomes increasingly large. MNRAS000 , 000–000 (0000) C. Chang et al. ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ True η (1 arcmin) − − ψ × ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Reconstruct without shape noise η (1 arcmin) − − ψ × ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ Reconstruct with shape noise η (1 arcmin) − − ψ × Figure B1.
The truth (top), noiseless reconstruction (middle) and noisy re-construction (bottom) of ψ and η field using the Gaussian simulations anda Y1-like mask. The colours indicate the value of the ψ fields while thearrows indicate the observed deflection angle caused by lensing. The ar-rows are not to-scale — they are enlarged for visualisation purpose. Theamplitude of the reconstructed field is lower than the true field, thereforethe colour bars in the bottom panel span over a range 4 times smaller thanthe top panel; while the arrows in the bottom panel are enlarged 2 timesmore than the top panel, as indicated by the one-arcminute bar on the upperright. B1 Simulation tests
Similar to our test in reconstructing the convergence maps in Sec. 5,we investigate the performance of reconstructing the lensing po-tential field ψ and the deflection field η . The techniques used formapping these quantities are similar to those used for κ and utilisethe H EALPIX routines. The definition of η has been introduced inEq. (10) and related to ψ in Eq. (11), but as η is a spin-1 field it re-quires use of the H EALPIX routine alm2map spin function to pro-duce the final maps. In Fig. B1 we show an example of a ψ and η map generated via synfast . The top panel displays the true fields;the middle panel shows a reconstructed field with the Y1 mask im-posed and RA > ◦ region excluded; the bottom panel shows thereconstruction with the mask and realistic Y1 shape noise. We findthat both the ψ and η fields exhibit significant degradation due tothe mask, as shown in the difference between the upper two panels,even though some level of resemblance remains. The addition ofshape noise has a much less significant effect, as can be seen fromthe bottom panel, which is very similar to the middle panel; this is κ η η φ − . . . . . . . F Gaussian F Gaussian F Gaussian, with shape noise
Figure B2. F (square markers) and F (triangle markers) statistics for thereconstruction of the κ (black), η = ( η , η ) (green and red) and ψ (blue)fields measured excluding pixels within ∼
10 arcminutes from the edge ofthe mask. All measurements are done with the Gaussian simulations and theY1-like mask described in Sec. 5.1. The round markers are the same as thetriangle markers except for the addition of shape noise. expected as shape noise mainly degrades small-scale information,and is less important for the reconstruction of the ψ and η maps.To quantify the degradation caused by the mask, we calculatethe F and F (replacing κ by ψ and η in Eq. (17)) when excluding10 arcminutes from the edges as shown in Fig. B2. We generate 500Gaussian realisations of the sky with the same underlying powerspectrum to account for the effect of cosmic variance, which is animportant factor in the reconstruction of ψ and η since the informa-tion is dominated by large scales. We show the mean and standarddeviation from these 500 simulations in Fig. B2. As expected, wefind that the mask has a stronger effect upon these two maps thanfor κ , as they use a higher proportion of information from the lower (cid:96) modes, which are more poorly constrained. This can be seen asa progressive degradation, from κ to η , to the most adversely af-fected ψ , but significant information is still reconstructed from themaps. The main effect of the mask on ψ can be seen from the lowvalue of F , due to the large unobserved sky regions suppressing thepower inside the masked region by ∼ η also suf-fers from this but to a lesser extent; η is suppressed by ∼
60% and η by ∼ η and η are reconstructing the deflection anglein different directions on the sky — the mask is a non-isotropic andthere is more information in the RA direction, which contributesmainly to η .To measure the reliability of the reconstruction of the phaseinformation we use F . Comparing F to F gives a measurementof the phase reconstruction. These results are also shown in Fig. B2.We find that for both ψ and η , the mean F is at a similar level as F , but the standard deviation of F is much larger than that of F ,which suggests that the quality of the phase reconstruction variesdramatically depending on the specific realisation of the sky. Fur-thermore, we find that the influence of shape noise on F is muchless compared to the influence from the mask, as also suggested byFig. B1.Taken in combination, F and F suggest that considerableinformation can be inferred about η and ψ , although with muchlarger uncertainties than for κ . We do not perform further quanti-tative analyses on these maps, but note that for data sets on areaslarger than DES Y1, the reconstruction of these other lensingfields becomes interesting. In these scenarios, algorithms that MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map specifically deal with the mask will be particularly useful. Forexample, instead of converting the γ field to ψ and η directly, onecan imagine forward-modelling the observed γ field from someunderlying ψ field. We defer the study of a forward-fitting massmapping method to future work. B2 Deflection and potential maps for DES Y1
We now apply the reconstruction method above to DES Y1 data.In Fig. B3 we show these maps constructed using M
ETA C ALIBRA - TION shear measurements with the full redshift range 0 . < z < . κ , η and ψ . On large scales, we find the low convergence (underdensed) re-gions are mostly located on the upper half of the footprint in Fig. 6:those correspond to a high potential value in Fig. B3, and the de-flection angle points from low to high potential. The characteristicscale of ψ is larger than η , which is larger than κ . The amplitude ofthe ψ and η map also agrees well with that expected from the sim-ulation tests in the previous section — the potential field is ∼ − and the deflection angle has a value on the order slightly below anarcminute. This map has implications for the mass distribution be-yond the footprint. For example, the fact that the deflection anglepoints away from the boundary at the lower boundary of the map atRA ∼ ◦ could indicate a large-scale overdensity just outside thefootprint, which will be tested when more data is available. APPENDIX C: CATALOG CONSISTENCY
In this appendix we compare maps from I M SHAPE catalog withthe results presented in the main text from M
ETA C ALIBRATION .We also compare with the map from DES Science Verification data(SV, Vikram et al. 2015; Chang et al. 2015) which partly overlapswith the Y1 footprint.In Fig. C1 and Fig. C2 we show the convergence maps gener-ated using the I M SHAPE shear catalogs. Comparing to Fig. 6 andFig. 7 ,we can see that the broad structures in the κ E maps are sim-ilar, especially for the high S/N maps. The contrast between the E-and B-mode is less strong compared to M ETA C ALIBRATION due tothe overall lower S/N in the I M SHAPE catalog. Nevertheless, theagreement between the two independent catalogs provides a checkof the shear measurement pipeline.In Fig. C3 we show the Y1 map and the SV map in the SVfootprint; both maps use galaxies with a mean redshift 0 . < z < .
3, and smoothed with σ G =
20 arcminutes. The SV map wasconstructed using another independent catalog ngmix and a differ-ent photo-z code, S
KYNET . The SV map is also half a magnitudedeeper than the Y1 map. The visual correspondence between thestructures in the two maps is very good given the differences in theinput data. This again serves as a consistency check between thedifferent catalogs.
APPENDIX D: AUTHOR AFFILIATIONS Kavli Institute for Cosmological Physics, University of Chicago, Chicago,IL 60637, USA DEDIP/DAP, IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France Universit´e Paris Diderot, AIM, Sorbonne Paris Cit´e, CEA, CNRS, F-91191Gif-sur-Yvette, France Institut de Ci`encies de l’Espai, IEEC-CSIC, Campus UAB, Facultat deCi`encies, Torre C5 par-2, 08193 Bellaterra, Barcelona, Spain Institute of Cosmology & Gravitation, University of Portsmouth,Portsmouth, PO1 3FX, UK Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,University of Manchester, Oxford Road, Manchester, M13 9PL, UK Department of Astrophysical Sciences, Princeton University, Peyton Hall,Princeton, NJ 08544, USA Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Institute of Sci-ence and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104, USA New York University, CCPP, New York, NY 10003, USA Einstein Fellow Kavli Institute for Cosmology, University of Cambridge, MadingleyRoad, Cambridge CB3 0HA, UK Institute of Astronomy, University of Cambridge, Madingley Road, Cam-bridge CB3 0HA, UK Faculty of Physics, Ludwig-Maximilians-Universit¨at, Scheinerstr. 1,81679 Munich, Germany LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA Department of Physics, Stanford University, 382 Via Pueblo Mall, Stan-ford, CA 94305, USA Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,Stanford University, Stanford, CA 94305, USA CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris,France Department of Physics & Astronomy, University College London, GowerStreet, London, WC1E 6BT, UK Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, Institutd’Astrophysique de Paris, F-75014, Paris, France Laborat´orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos´eCristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ -20921-400, Brazil Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas(CIEMAT), Madrid, Spain Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland Jet Propulsion Laboratory, California Institute of Technology, 4800 OakGrove Dr., Pasadena, CA 91109, USA Department of Physics, The Ohio State University, Columbus, OH 43210,USA Center for Cosmology and Astro-Particle Physics, The Ohio State Uni-versity, Columbus, OH 43210, USA University of Arizona, Department of Physics, 1118 E. Fourth St., Tuc-son, AZ 85721, USA Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse,85748 Garching, Germany Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL60439, USA Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ,UK Cerro Tololo Inter-American Observatory, National Optical AstronomyMNRAS000
APPENDIX D: AUTHOR AFFILIATIONS Kavli Institute for Cosmological Physics, University of Chicago, Chicago,IL 60637, USA DEDIP/DAP, IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France Universit´e Paris Diderot, AIM, Sorbonne Paris Cit´e, CEA, CNRS, F-91191Gif-sur-Yvette, France Institut de Ci`encies de l’Espai, IEEC-CSIC, Campus UAB, Facultat deCi`encies, Torre C5 par-2, 08193 Bellaterra, Barcelona, Spain Institute of Cosmology & Gravitation, University of Portsmouth,Portsmouth, PO1 3FX, UK Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,University of Manchester, Oxford Road, Manchester, M13 9PL, UK Department of Astrophysical Sciences, Princeton University, Peyton Hall,Princeton, NJ 08544, USA Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Institute of Sci-ence and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain Department of Physics and Astronomy, University of Pennsylvania,Philadelphia, PA 19104, USA New York University, CCPP, New York, NY 10003, USA Einstein Fellow Kavli Institute for Cosmology, University of Cambridge, MadingleyRoad, Cambridge CB3 0HA, UK Institute of Astronomy, University of Cambridge, Madingley Road, Cam-bridge CB3 0HA, UK Faculty of Physics, Ludwig-Maximilians-Universit¨at, Scheinerstr. 1,81679 Munich, Germany LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA Department of Physics, Stanford University, 382 Via Pueblo Mall, Stan-ford, CA 94305, USA Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,Stanford University, Stanford, CA 94305, USA CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris,France Department of Physics & Astronomy, University College London, GowerStreet, London, WC1E 6BT, UK Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, Institutd’Astrophysique de Paris, F-75014, Paris, France Laborat´orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos´eCristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ -20921-400, Brazil Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas(CIEMAT), Madrid, Spain Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland Jet Propulsion Laboratory, California Institute of Technology, 4800 OakGrove Dr., Pasadena, CA 91109, USA Department of Physics, The Ohio State University, Columbus, OH 43210,USA Center for Cosmology and Astro-Particle Physics, The Ohio State Uni-versity, Columbus, OH 43210, USA University of Arizona, Department of Physics, 1118 E. Fourth St., Tuc-son, AZ 85721, USA Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse,85748 Garching, Germany Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL60439, USA Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ,UK Cerro Tololo Inter-American Observatory, National Optical AstronomyMNRAS000 , 000–000 (0000) C. Chang et al. ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ . < z < . η (1 arcmin) − . − . − . − . − . . . . . . ψ × Figure B3.
The lensing potential (colour map) and deflection field (arrows) reconstructed for the DES Y1 M
ETA C ALIBRATION data in the redshift range0 . < z < .
3. The arrows are not to-scale, but can be compared to the one-arcminute arrow in the upper right corner.Observatory, Casilla 603, La Serena, Chile Department of Physics and Electronics, Rhodes University, PO Box 94,Grahamstown, 6140, South Africa Department of Astronomy, University of Illinois, 1002 W. Green Street,Urbana, IL 61801, USA National Center for Supercomputing Applications, 1205 West Clark St.,Urbana, IL 61801, USA Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma deMadrid, 28049 Madrid, Spain Astronomy Department, University of Washington, Box 351580, Seattle,WA 98195, USA Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA Australian Astronomical Observatory, North Ryde, NSW 2113, Australia Departamento de F´ısica Matem´atica, Instituto de F´ısica, Universidade deS˜ao Paulo, CP 66318, S˜ao Paulo, SP, 05314-970, Brazil Instituci´o Catalana de Recerca i Estudis Avanc¸ats, E-08010 Barcelona,Spain Department of Physics and Astronomy, Pevensey Building, University ofSussex, Brighton, BN1 9QH, UK Department of Physics, University of Michigan, Ann Arbor, MI 48109,USA School of Physics and Astronomy, University of Southampton,Southampton, SO17 1BJ, UK Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas,13083-859, Campinas, SP, Brazil Computer Science and Mathematics Division, Oak Ridge National Labo-ratory, Oak Ridge, TN 37831 MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N Figure C1.
Same as Fig. 6, but constructed using the I M SHAPE shear catalog.MNRAS000
Same as Fig. 6, but constructed using the I M SHAPE shear catalog.MNRAS000 , 000–000 (0000) C. Chang et al. ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +340 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +350 ◦ − ◦ − ◦ − ◦ κ E ; 0 . < z < . ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ +60 ◦ +70 ◦ +350 ◦ − ◦ − ◦ − ◦ κ B ; 0 . < z < . − − − S / N Figure C2.
Same as Fig. 7, but constructed using the I M SHAPE shear catalog. MNRAS , 000–000 (0000) ark Energy Survey Year 1 Results: Curved-Sky Weak Lensing Mass Map +70 ◦ +80 ◦ − ◦ − ◦ − ◦ SV − − − S / N +70 ◦ +80 ◦ − ◦ − ◦ − ◦ Y1 − − − S / N Figure C3.
Convergence map constructed using the SV
NGMIX catalog (left) and the Y1 M
ETA C ALIBRATION catalog (right).MNRAS000