Dark Energy Survey Year 1 Results: Galaxy-Galaxy Lensing
J. Prat, C. Sánchez, Y. Fang, D. Gruen, J. Elvin-Poole, N. Kokron, L. F. Secco, B. Jain, R. Miquel, N. MacCrann, M. A. Troxel, A. Alarcon, D. Bacon, G. M. Bernstein, J. Blazek, R. Cawthon, C. Chang, M. Crocce, C. Davis, J. De Vicente, J. P. Dietrich, A. Drlica-Wagner, O. Friedrich, M. Gatti, W. G. Hartley, B. Hoyle, E. M. Huff, M. Jarvis, M. M. Rau, R. P. Rollins, A. J. Ross, E. Rozo, E. S. Rykoff, S. Samuroff, E. Sheldon, T. N. Varga, P. Vielzeuf, J. Zuntz, T. M. C. Abbott, F. B. Abdalla, S. Allam, J. Annis, K. Bechtol, A. Benoit-Lévy, E. Bertin, D. Brooks, E. Buckley-Geer, D. L. Burke, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, F. J. Castander, C. E. Cunha, C. B. D'Andrea, L. N. da Costa, S. Desai, H. T. Diehl, S. Dodelson, T. F. Eifler, E. Fernandez, B. Flaugher, P. Fosalba, J. Frieman, J. García-Bellido, E. Gaztanaga, D. W. Gerdes, T. Giannantonio, D. A. Goldstein, R. A. Gruendl, J. Gschwend, G. Gutierrez, K. Honscheid, D. J. James, T. Jeltema, M. W. G. Johnson, M. D. Johnson, D. Kirk, E. Krause, K. Kuehn, S. Kuhlmann, O. Lahav, T. S. Li, M. Lima, M. A. G. Maia, M. March, J. L. Marshall, P. Martini, P. Melchior, F. Menanteau, J. J. Mohr, R. C. Nichol, B. Nord, A. A. Plazas, A. K. Romer, A. Roodman, M. Sako, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, et al. (15 additional authors not shown)
DDES-2016-0210FERMILAB-PUB-17-277-PPD
Dark Energy Survey Year 1 Results: Galaxy-Galaxy Lensing
J. Prat, ∗ C. Sánchez, † Y. Fang, D. Gruen,
3, 4, 5
J. Elvin-Poole, N. Kokron,
7, 8
L. F. Secco, B. Jain, R. Miquel,
9, 1
N. MacCrann,
10, 11
M. A. Troxel,
10, 11
A. Alarcon, D. Bacon, G. M. Bernstein, J. Blazek,
10, 14
R. Cawthon, C. Chang, M. Crocce, C. Davis, J. De Vicente, J. P. Dietrich,
17, 18
A. Drlica-Wagner, O. Friedrich,
20, 21
M. Gatti, W. G. Hartley,
22, 23
B. Hoyle, E. M. Huff, M. Jarvis, M. M. Rau,
17, 21
R. P. Rollins, A. J. Ross, E. Rozo, E. S. Rykoff,
3, 4
S. Samuroff, E. Sheldon, T. N. Varga,
20, 21
P. Vielzeuf, J. Zuntz, T. M. C. Abbott, F. B. Abdalla,
22, 29
S. Allam, J. Annis, K. Bechtol, A. Benoit-Lévy,
31, 22, 32
E. Bertin,
31, 32
D. Brooks, E. Buckley-Geer, D. L. Burke,
3, 4
A. Carnero Rosell,
8, 33
M. Carrasco Kind,
34, 35
J. Carretero, F. J. Castander, C. E. Cunha, C. B. D’Andrea, L. N. da Costa,
8, 33
S. Desai, H. T. Diehl, S. Dodelson,
19, 15
T. F. Eifler,
37, 24
E. Fernandez, B. Flaugher, P. Fosalba, J. Frieman,
19, 15
J. García-Bellido, E. Gaztanaga, D. W. Gerdes,
39, 40
T. Giannantonio,
41, 42, 21
D. A. Goldstein,
43, 44
R. A. Gruendl,
34, 35
J. Gschwend,
8, 33
G. Gutierrez, K. Honscheid,
10, 11
D. J. James, T. Jeltema, M. W. G. Johnson, M. D. Johnson, D. Kirk, E. Krause, K. Kuehn, S. Kuhlmann, O. Lahav, T. S. Li, M. Lima,
7, 8
M. A. G. Maia,
8, 33
M. March, J. L. Marshall, P. Martini,
10, 50
P. Melchior, F. Menanteau,
34, 35
J. J. Mohr,
18, 17, 20
R. C. Nichol, B. Nord, A. A. Plazas, A. K. Romer, A. Roodman,
3, 4
M. Sako, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, I. Sevilla-Noarbe, M. Smith, R. C. Smith, M. Soares-Santos, F. Sobreira,
54, 8
E. Suchyta, M. E. C. Swanson, G. Tarle, D. Thomas, D. L. Tucker, V. Vikram, A. R. Walker, R. H. Wechsler,
56, 3, 4
B. Yanny, and Y. Zhang (DES Collaboration) Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology,Campus UAB, 08193 Bellaterra (Barcelona) Spain Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA Kavli Institute for Particle Astrophysics & Cosmology,P. O. Box 2450, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Einstein Fellow Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,University of Manchester, Oxford Road, Manchester, M13 9PL, UK Departamento de Física Matemática, Instituto de Física,Universidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil Laboratório Interinstitucional de e-Astronomia - LIneA,Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain Center for Cosmology and Astro-Particle Physics,The Ohio State University, Columbus, OH 43210, USA Department of Physics, The Ohio State University, Columbus, OH 43210, USA Institute of Space Sciences, IEEC-CSIC, Campus UAB,Carrer de Can Magrans, s/n, 08193 Barcelona, Spain Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK Institute of Physics, Laboratory of Astrophysics,École Polytechnique Fédérale de Lausanne (EPFL),Observatoire de Sauverny, 1290 Versoix, Switzerland Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany Universitäts-Sternwarte, Fakultät für Physik, Ludwig-MaximiliansUniversität München, Scheinerstr. 1, 81679 München, Germany Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Dr., Pasadena, CA 91109, USA Department of Physics, University of Arizona, Tucson, AZ 85721, USA Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France Sorbonne Universités, UPMC Univ Paris 06, UMR 7095,Institut d’Astrophysique de Paris, F-75014, Paris, France Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil a r X i v : . [ a s t r o - ph . C O ] S e p 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 Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Department of Astronomy, University of California,Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA 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 Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA Department of Physics and Astronomy, Pevensey Building,University of Sussex, Brighton, BN1 9QH, UK School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil Computer Science and Mathematics Division,Oak Ridge National Laboratory, Oak Ridge, TN 37831 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA (Dated: September 5, 2018)We present galaxy-galaxy lensing measurements from 1321 sq. deg. of the Dark Energy Survey(DES) Year 1 (Y1) data. The lens sample consists of a selection of 660,000 red galaxies with high-precision photometric redshifts, known as redMaGiC, split into five tomographic bins in the redshiftrange . < z < . . We use two different source samples, obtained from the Metacalibration (26 million galaxies) and im3shape (18 million galaxies) shear estimation codes, which are splitinto four photometric redshift bins in the range . < z < . . We perform extensive testing ofpotential systematic effects that can bias the galaxy-galaxy lensing signal, including those fromshear estimation, photometric redshifts, and observational properties. Covariances are obtainedfrom jackknife subsamples of the data and validated with a suite of log-normal simulations. We usethe shear-ratio geometric test to obtain independent constraints on the mean of the source redshiftdistributions, providing validation of those obtained from other photo- z studies with the same data.We find consistency between the galaxy bias estimates obtained from our galaxy-galaxy lensingmeasurements and from galaxy clustering, therefore showing the galaxy-matter cross-correlationcoefficient r to be consistent with one, measured over the scales used for the cosmological analysis.The results in this work present one of the three two-point correlation functions, along with galaxyclustering and cosmic shear, used in the DES cosmological analysis of Y1 data, and hence themethodology and the systematics tests presented here provide a critical input for that study as wellas for future cosmological analyses in DES and other photometric galaxy surveys. I. INTRODUCTION
Weak gravitational lensing refers to the small distor-tions in the images of distant galaxies by interveningmass along the line of sight. Galaxy-galaxy lensingrefers to the cross-correlation between foreground (lens)galaxy positions and the lensing shear of background(source) galaxies at higher redshifts [1–3]. The compo-nent of the shear that is tangential to the perpendicularline connecting the lens and source galaxies is a mea-sure of the projected, excess mass distribution aroundthe lens galaxies. Galaxy-galaxy lensing at small scaleshas been used to characterize the properties of darkmatter halos hosting lens galaxies, while at large scales ∗ Corresponding author: [email protected] † Corresponding author: [email protected] it measures the cross correlation between galaxy andmatter densities. The measurements have many appli-cations, ranging from constraining halo mass profiles[4] to estimating the large-scale bias of a given galaxypopulation to obtaining cosmological constraints [5–10].Recent surveys such as CFHTLenS [11, 12] have pre-sented measurements on galaxy-galaxy lensing [13–15].Similarly, measurements from KiDS [16, 17] have alsostudied the galaxy-mass connection using galaxy-galaxylensing [18–21]. The galaxy-mass connection has alsobeen studied in [22, 23] and by [24] at high redshift.In this paper we present measurements and extensivetests of the tomographic galaxy-galaxy lensing signalfrom Year 1 data of the Dark Energy Survey (DES).DES is an ongoing wide-field multi-band imaging sur-vey that will cover 5000 sq. deg. of the Southern sky overfive years. Our goals are to present the measurementsof galaxy-galaxy lensing with DES, carry out a series ofnull tests of our measurement pipeline and the data, andcarry out related analyses of the lensing and photomet-ric redshift (photo- z ) performance that are critical forthe Y1 cosmological analysis [25]. We use five redshiftbins for the lens galaxies and four bins for the sourcegalaxies. The detailed tests presented here will serve asa foundation for future work relying on galaxy-galaxylensing measurements, such as Halo Occupation Dis-tribution (HOD) analyses [26, 27]. The galaxy-galaxylensing studies with the DES Science Verification (SV)data serve as precursors to this paper [9, 28–30].The lens galaxy sample used is the red-sequenceMatched-filter Galaxy Catalog (redMaGiC, [31]), whichis a catalog of photometrically selected luminous redgalaxies (LRGs). The redMaGiC algorithm uses theredMaPPer-calibrated model for the color of red-sequence galaxies as a function of magnitude and red-shift [32, 33]. This algorithm constructs a galaxy samplewith far more reliable redshift estimates than is achiev-able for a typical galaxy in DES.For the source galaxy redshifts, we rely on less well-constrained photo- z estimates, calibrated in two inde-pendent ways [34–36]. In this paper, we use the ex-pected behavior of the galaxy-galaxy lensing signal withthe distance to source galaxies (the shear-ratio test)to validate the photo- z estimates and calibration. Thescaling of the galaxy-galaxy lensing signal with sourceredshift for a given lens bin is mostly driven by the ge-ometry of the lens-source configuration, with cosmologydependence being subdominant to potential biases inthe redshift estimation of the galaxies involved. There-fore, such measurements provide useful constraints onthe redshift distribution of source galaxies, which wethen compare to findings by independent studies.The DES Y1 cosmological analysis [25] relies on theassumption that the cross-correlation coefficient be-tween galaxies and matter is unity on the scales used forthis analysis. In this work we provide validation for thisassumption by showing the linear galaxy bias estimatesfrom galaxy-galaxy lensing to be consistent with thoseobtained from galaxy clustering using the same galaxysample [37].The plan of the paper is as follows. In Section II, wepresent the modelling. Section III describes our data,including basic details of DES, descriptions of the lensgalaxy sample, pipelines for source galaxy shape mea-surements, and the photometric redshift estimation oflens and source galaxies. We also describe a set of log-normal simulations used for tests of the measurementmethodology. The details of the measurement and co-variance estimation, together with our galaxy-galaxylensing measurements, are presented in Section IV.Tests of potential systematic effects on the measure-ment are shown in Section V. Section VI presents theuse of tomographic galaxy-galaxy lensing to test thephoto- z ’s of source galaxies. Finally, in Section VII wecompare the galaxy bias estimates from galaxy-galaxylensing to those obtained using the angular clusteringof galaxies [37], and we conclude in Section VIII. II. THEORY
Galaxy-galaxy lensing is the measurement of the tan-gential shear of background (source) galaxies aroundforeground (lens) galaxies (see [38] for a review). Theamplitude of distortion in the shapes of source galax-ies is correlated with the amount of mass that causespassing light rays to bend. Assuming that lens galax-ies trace the mass distribution following a simple linearbiasing model ( δ g = b δ m ), the galaxy-matter powerspectrum relates to the matter power spectrum by asingle multiplicative bias factor. In this case, the tan-gential shear of background galaxies in redshift bin j around foreground galaxy positions in redshift bin i atan angular separation θ can be written as the followingintegral over the matter power spectrum P δδ : γ ijt ( θ ) = b i
32 Ω m (cid:18) H c (cid:19) (cid:90) d(cid:96) π (cid:96) J ( θ(cid:96) ) ×× (cid:90) dz (cid:20) g j ( z ) n il ( z ) a ( z ) χ ( z ) P δδ (cid:18) k = (cid:96)χ ( z ) , χ ( z ) (cid:19)(cid:21) , (1)where we are assuming b i ( z ) = b i within a lens redshiftbin, J is the second order Bessel function, l is the mul-tipole moment, k is the 3D wavenumber, a is the scalefactor, χ is the comoving distance to redshift z , n il ( z ) isthe redshift distribution of foreground (lens) galaxies inbin i and g j ( z ) is the lensing efficiency for backgroundgalaxies in bin j , computed as g j ( z ) = (cid:90) ∞ z dz (cid:48) n js ( z (cid:48) ) χ ( z (cid:48) ) − χ ( z ) χ ( z (cid:48) ) , (2)where n js ( z ) is the corresponding redshift distribution ofbackground (source) galaxies in bin j . The tangentialshear in Eq. (1) depends on the cosmological parame-ters not only through the explicit dependencies but alsothrough the matter power spectrum P δδ . Nonetheless,the dependence on the cosmological parameters is heav-ily degenerate with the galaxy bias of the lens galaxypopulation, b i .It is also useful to express the tangential shear interms of the excess surface mass density ∆Σ . This esti-mator is typically used to study the properties of darkmatter halos (see for instance [23]). However, with thelarge scales used in this analysis, the lensing effect iscaused by general matter overdensities which are tracedby galaxies. In this work, we make use of this estima-tor because the geometrical dependence of the lensingsignal becomes more evident. The estimator reads: γ t = ∆ΣΣ crit , (3)where the lensing strength Σ − is a geometrical factorthat depends on the angular diameter distance to thelens D l , the source D s and the relative distance betweenthem D ls : Σ − ( z l , z s ) = 4 πGc D ls D l D s , (4)with Σ − ( z l , z s ) = 0 for z s < z l , and where z l and z s are the lens and source galaxy redshifts, respectively. N o r m a li z e d c o un t s Lenses redMaGiC . . . . . . . . . . Redshift N o r m a li z e d c o un t s Sources M ETACALIBRATIONIM SHAPE
FIG. 1. (
Top panel ): Redshift distributions of redMaGiClens galaxies divided in tomographic bins ( colors ) and forthe combination of all of them ( black ). The n ( z ) ’s are ob-tained stacking individual Gaussian distributions for eachgalaxy. ( Bottom panel ): The same, but for our two weaklensing source samples,
Metacalibration and im3shape ,using the BPZ photometric redshift code.
Since the redshift distributions of our lens and sourcesamples, n l ( z ) , n s ( z ) respectively, have a non-negligiblewidth and even overlap, we take this into account bydefining an effective Σ − integrating over the corre-sponding redshift distributions. For a given lens bin i and source bin j , this has the following form: Σ − i,j crit , eff = (cid:90) (cid:90) dz l dz s n il ( z l ) n js ( z s ) Σ − ( z l , z s ) . (5)We need to assume a certain cosmology (flat Λ CDMwith Ω m = 0 . ) when calculating the angular diameterdistances in Σ − . The results presented in this analysisdepend only weakly on this choice of cosmology, as wewill further discuss in the relevant sections (see Sec. VI). III. DATA AND SIMULATIONS
The Dark Energy Survey is a photometric survey thatwill cover about one quarter of the southern sky (5000sq. deg.) to a depth of r > , imaging about 300million galaxies in 5 broadband filters ( grizY ) up toredshift z = 1 . [39, 40]. In this work we use data froma large contiguous region of 1321 sq. deg. of DES Year 1observations which overlaps with the South Pole Tele-scope footprint − deg. < δ < − deg. and reaches alimiting magnitude of ≈ in the r -band (with a meanof 3 exposures out of the planned 10 for the full survey).Y1 images were taken between 31 Aug 2013 and 9 Feb2014. A. Lens sample: redMaGiC
The lens galaxy sample used in this work is a subsetof the DES Y1 Gold Catalog [41] selected by redMaGiC[31], which is an algorithm designed to define a sampleof luminous red galaxies (LRGs) with minimal photo- z uncertainties. It selects galaxies above some luminositythreshold based on how well they fit a red sequence tem-plate, calibrated using redMaPPer [32, 33] and a sub-set of galaxies with spectroscopically verified redshifts.The cutoff in the goodness of fit to the red sequenceis imposed as a function of redshift and adjusted suchthat a constant comoving number density of galaxies ismaintained. The redMaGiC photo- z ’s show excellentperformance, with a scatter of σ z / (1 + z ) = 0 . [37].Furthermore, their errors are very well characterizedand approximately Gaussian, enabling the redshift dis-tribution of a sample, n ( z ) , to be obtained by stackingeach galaxy’s Gaussian redshift probability distributionfunction (see [31] for more details).The sample used in this work is a combination ofthree redMaGiC galaxy samples, each of them definedto be complete down to a given luminosity thresh-old L min . We split the lens sample into five equally-spaced tomographic redshift bins between z = 0 . and z = 0 . , with the three lower redshift bins using thelowest luminosity threshold of L min = 0 . L (cid:63) (namedHigh Density sample) and the two highest redshift binsusing higher luminosity thresholds of L min = 1 . L (cid:63) and L min = 1 . L (cid:63) (named High Luminosity and HigherLuminosity samples, respectively). Using the stack-ing procedure mentioned above, redshift distributionsare obtained and shown in Fig. 1. Furthermore, red-MaGiC samples have been produced with two differentphotometric reduction techniques, MAG_AUTO and Multi-object fitting photometry (
MOF ), both described in [41].We follow the analysis of [37] and we use
MAG_AUTO pho-tometry for the three lower redshift bins and
MOF pho-tometry for the rest, as it was found in [37] that thiscombination was optimal in minimizing systematic ef-fects that introduce spurious angular galaxy clustering.
B. Source samples:
Metacalibration and im3shape
Metacalibration [42, 43] is a recently developedmethod to accurately measure weak lensing shear usingonly the available imaging data, without need for priorinformation about galaxy properties or calibration fromsimulations. The method involves distorting the imagewith a small known shear, and calculating the responseof a shear estimator to that applied shear. This newtechnique can be applied to any shear estimation codeprovided it fulfills certain requirements. For this work,it has been applied to the ngmix shear pipeline [44],which fits a Gaussian model simultaneously in the riz bands to measure the ellipticities of the galaxies. Thedetails of this implementation can be found in [45]. Wewill refer to the ngmix shear catalog calibrated usingthat procedure as
Metacalibration . im3shape is based on the algorithm by [46], modi-fied according to [47] and [45]. It performs a maximumlikelihood fit using a bulge-or-disk galaxy model to esti-mate the ellipticity of a galaxy, i.e. it fits de Vaucouleursbulge and exponential disk components to galaxy im-ages in the r band, with shear biases calibrated fromrealistic simulations [45, 48].Due to conservative cuts on measured galaxy proper-ties, e. g. signal-to-noise ratio and size, that have beenapplied to both Metacalibration and im3shape , thenumber of galaxies comprised in each shear catalog issignificantly reduced compared to that of the full Y1Gold catalog. Still, the number of source galaxies isunprecedented for an analysis of this kind.
Metacali-bration consists of 35 million galaxy shape estimates,of which 26 are used in the cosmological analysis dueto redshift and area cuts, and im3shape is composedof 22 million galaxies, of which 18 are used for cosmol-ogy. The fiducial results in this paper, for instance inSec. VI and Sec. VII, utilize
Metacalibration due tothe higher number of galaxies included the catalog.
C. Photometric redshifts for the source sample
Galaxy redshifts in DES are estimated from griz multiband photometry. The performance and accuracyof these estimates was extensively tested with ScienceVerfication (SV) data, using a variety of photometricredshift algorithms and matched spectroscopy from dif-ferent surveys [49, 50].The fiducial photometric redshifts used in this workare estimated with a modified version of the BayesianPhotometric Redshifts (BPZ) code [34, 51]. BPZ de-fines the mapping between color and redshift by drawingupon physical knowledge of stellar population models,galaxy evolution and empirical spectral energy distri-butions of galaxies at a range of redshifts.Such photo- z ’s are used to split our source samplesinto four tomographic bins by the mean of the estimatedindividual redshift probability density functions ( p ( z ) )between z = 0 . and z = 1 . . For Metacalibration in particular, where potential selection biases need tobe corrected for (cf. section IV A 1), this is done usingphoto- z estimates based on Metacalibration mea-surements of multiband fluxes. For both shear cata-logs, the corresponding redshift distributions come fromstacking random draws from the p ( z ) and are shown inFig. 1. Details of this procedure are described in section3.3 of [34].The photo- z calibration procedure we follow in Y1is no longer based on spectroscopic data, since exist-ing spectroscopic surveys are not sufficiently completeover the magnitude range of the DES Y1 source galax-ies. Instead, we rely on complementary comparisons to1) matched COSMOS high-precision photometric red-shifts and 2) constraints on our redshift distributionsfrom DES galaxy clustering cross-correlations. We re-fer the reader to the four dedicated redshift papers [34–36, 52]. In addition, in this work we will provide furtherindependent validation of their calibration, using weakgravitational lensing (Sec. VI). D. Lognormal simulations
Lognormal models of cosmological fields, such as mat-ter density and cosmic shear, have been shown to ac-curately describe two-point statistics such as galaxy-galaxy lensing on sufficiently large scales. Furthermore,the production of lognormal mock catalogs that repro-duce properties of our sample is significantly less de-manding in terms of computational expenses than N -body simulations such as those detailed in [53]. One ofthe first descriptions of lognormal fields in cosmologicalanalyses was outlined in [54]. The assumption of log-normality for these cosmological fields has shown goodagreement with N -body simulations and real data up tononlinear scales [55–57]. Thus, lognormal mock simula-tions provide a way to assess properties of the galaxy-galaxy lensing covariance matrix that are particularlydependent on the number of simulations produced, dueto their low-cost nature of production.We use the publicly available code FLASK [58], togenerate galaxy position and convergence fields consis-tent with our lens and source samples, and produce 150full-sky shear and density mock catalogs. The maps arepixelated on a HEALPix grid with resolution set byan N side parameter of 4096. At this N side , the typi-cal pixel area is 0.73 arcmin and the maximum mul-tipoles resolved for clustering and shear are (cid:96) = 8192 and (cid:96) = 4096 , respectively. We mask out regions ofthe grid to then produce eight DES Y1 footprints fora given full-sky mock. This produces a total of 1200mock surveys that mimic our sample.To correctly capture the covariance properties of thissample, such as shot noise, we match the number den-sity of the mock tomographic bins to those of the data.We add noise properties to the shear fields accordingto the same procedure detailed in [59]. Galaxy bias isintroduced in the lens samples through the input an-gular auto and cross power spectra between bins, andis also chosen to approximately match the data. Thetracer density fields are subsequently Poisson sampledto yield discrete galaxy positions. IV. MEASUREMENT AND COVARIANCEA. Measurement methodology
Here we describe the details of the tangential shearmeasurement (cid:104) γ t (cid:105) . Similarly, we can measure the cross-component of the shear (cid:104) γ × (cid:105) , which is a useful test ofpossible systematic errors in the measurement as it isnot produced by gravitational lensing. For a given lens-source galaxy pair j we define the tangential ( e t ) andcross ( e × ) components of the ellipticity of the sourcegalaxy as e t,j = − Re (cid:2) e j e − iφ j (cid:3) , e × ,j = − Im (cid:2) e j e − iφ j (cid:3) , (6)where e j = e ,j + i e ,j , with e ,j and e ,j being thetwo components of the ellipticity of the source galaxy ∼ flask/ measured with respect to a Cartesian coordinate sys-tem centered on the lens, and φ j being the positionangle of the source galaxy with respect to the horizon-tal axis of the Cartesian coordinate system. Assumingthe intrinsic ellipticities of individual source galaxies arerandomly aligned, we can obtain the mean weak lensingshear (cid:10) γ t/ × (cid:11) averaging the ellipticity measurements foreach component over many such lens-source pairs. How-ever, note that the assumption of random galaxy orien-tations is broken by intrinsic galaxy alignments (IA),which lead to non-lensing shape correlations (e.g. [60]),which are included in the modelling of the combinedprobes cosmology analysis [25]). Then: (cid:104) γ α ( θ ) (cid:105) = (cid:80) j ω j e α,j (cid:80) j ω j , (7)where θ is the angular separation, α = t or × de-notes the two possible components of the shear and w j = w l w s w e is a weight associated with each lens-source pair, which will depend on the lens ( w l , see V D),on the source weight assigned by the shear catalog ( w s ,see IV A 1 & IV A 2) and on a weight assigned by theestimator ( w e , see App. A). These estimates need to becorrected for shear responsivity (in the case of Meta-calibration shears, IV A 1) or multiplicative and ad-ditive bias (in the case of im3shape , IV A 2). Also notethat in this work w e = 1 because we are using the γ t es-timator, which weights all sources uniformly. Anotheroption would be to choose an optimal weighting schemethat takes into account the redshift estimate of thesource galaxies to maximize the lensing efficiency, as itis the case of the ∆Σ estimator. In the context of a cos-mological analysis combining galaxy-galaxy lensing andcosmic shear, using uniform weighting for the sourceshas the considerable advantage that nuisance parame-ters describing the systematic uncertainty of shear andredshift estimates of the sources are the same for bothprobes. In Appendix A, we find the increase in signal-to-noise ratio due to the optimal weighting scheme tobe small given the photo- z precision of source galaxiesin DES, and hence we use the γ t estimator in this workto minimize the number of nuisance parameters in theDES Y1 cosmological analysis [25].In all measurements in this work, we grouped thegalaxy pairs in 20 log-spaced angular separation binsbetween 2.5 and 250 arcmin. We use TreeCorr [61] tocompute all galaxy-galaxy lensing measurements in thiswork.One advantage of galaxy-shear cross-correlation overshear-shear correlations is that additive shear systemat-ics (with constant γ or γ ) average to zero in the tan-gential coordinate system. However, this cancellationonly occurs when sources are distributed isotropicallyaround the lens and additive shear is spatially constant,two assumptions that are not accurate in practice, espe-cially near the survey edge or in heavily masked regions,where there is a lack of symmetry on the source distri-bution around the lens. To remove additive systematicsrobustly, we also measure the tangential shear around https://github.com/rmjarvis/TreeCorr random points: such points have no net lensing signal(see Sec. V A), yet they sample the survey edge andmasked regions in the same way as the lenses. Our fullestimator of tangential shear can then be written as: (cid:104) γ α ( θ ) (cid:105) = (cid:104) γ α ( θ ) Lens (cid:105) − (cid:104) γ α ( θ ) Random (cid:105) . (8)Besides accounting for additive shear systematics, re-moving the measurement around random points fromthe measurement around the lenses has other benefits,such as leading to a significant decrease of the uncer-tainty on large scales, as was studied in detail in [62].We further discuss the implications the random pointsubtraction has on our measurement and covariance inApp. B. Metacalibration responses
In the
Metacalibration shear catalog [42, 43, 45],shears are calibrated using the measured response of theshear estimator to shear, which is usually the ellipticity e = ( e , e . Expanding this estimator in a Taylor seriesabout zero shear e = e | γ =0 + ∂ e ∂ γ (cid:12)(cid:12)(cid:12)(cid:12) γ =0 γ + ... ≡ e | γ =0 + R γ γ + ... , (9)we can define the shear response R γ , which can be mea-sured for each galaxy by artificially shearing the imagesand remeasuring the ellipticity: R γ,i,j = e + i − e − i ∆ γ j , (10)where e + i , e − i are the measurements made on an imagesheared by + γ j and − γ j , respectively, and ∆ γ j = 2 γ j .In the Y1 Metacalibration catalog, γ j = 0 . . Ifthe estimator e is unbiased, the mean response matrix (cid:104) R γ,i,j (cid:105) will be equal to the identity matrix.Then, averaging Eq. (9) over a sample of galaxiesand assuming the intrinsic ellipticities of galaxies arerandomly oriented, we can express the mean shear as: (cid:104) γ (cid:105) ≈ (cid:104) R γ (cid:105) − (cid:104) e (cid:105) (11)It is important to note that any shear statistic will beeffectively weighted by the same responses. Therefore,such weighting needs to be included when averagingover quantities associated with the source sample, forinstance when estimating redshift distributions (cf. [34],their section 3.3). We are including these weights in allthe redshift distributions measured on Metacalibra-tion used in this work.Besides the shear response correction describedabove, in the
Metacalibration framework, whenmaking a selection on the original catalog using a quan-tity that could modify the distribution of ellipticities,for instance a cut in S/N, it is possible to correct forselection effects. In this work, we are taking this into ac-count when cutting on S/N and size (used in Sec. V C totest for systematics effects) and in BPZ photo- z ’s (usedto construct the source redshift tomographic bins). This
10 100 ✓ [arcmin] t ( ✓ ) . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z s < . . < z s < . . < z s < . . < z s < . ✓ [arcmin] . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens
10 100 ✓ [arcmin] . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens
10 100 ✓ [arcmin] t ( ✓ ) . < z l < . redMaGiC HiLum . < z l < . redMaGiC HiLum . < z l < . redMaGiC HiLum . < z l < . redMaGiC HiLum
10 100 ✓ [arcmin] . < z l < . redMaGiC HigherLum . < z l < . redMaGiC HigherLum . < z l < . redMaGiC HigherLum . < z l < . redMaGiC HigherLum M ETACALIBRATION
10 100 ✓ [arcmin] t ( ✓ ) . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens
10 100 ✓ [arcmin] . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens
10 100 ✓ [arcmin] . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens . < z l < . redMaGiC HiDens
10 100 ✓ [arcmin] t ( ✓ ) . < z l < . redMaGiC HiLum . < z l < . redMaGiC HiLum . < z l < . redMaGiC HiLum . < z l < . redMaGiC HiLum
10 100 ✓ [arcmin] . < z l < . redMaGiC HigherLum . < z l < . redMaGiC HigherLum . < z l < . redMaGiC HigherLum . < z l < . redMaGiC HigherLum IM SHAPE
FIG. 2. Tangential shear measurements for
Metacalibration and im3shape together with the best-fit theory lines fromthe DES Y1 multiprobe cosmological analysis [25]. Scales discarded for the cosmological analysis, smaller than h − Mpc in comoving distance, but which are used for the shear-ratio test, are shown as shaded regions. Unfilled points correspondto negative values in the tangential shear measurement, which are mostly present in the lens-source combinations withlow signal-to-noise due to the lenses being at higher redshift than the majority of sources.
HiDens , HiLum and
HigherLum correspond to the three redmagic samples (High Density, High Luminosity and Higher Luminosity) described in Sec. III A. is performed by measuring the mean response of theestimator to the selection, repeating the selections onquantities measured on sheared images. Following onthe example of the mean shear, the mean selection re-sponse matrix (cid:104) R S (cid:105) is (cid:104) R S,i,j (cid:105) = (cid:104) e i (cid:105) S + − (cid:104) e i (cid:105) S − ∆ γ j , (12)where (cid:104) e i (cid:105) S + represents the mean of ellipticities mea-sured on images without applied shearing in component j , but with selection based on parameters from posi-tively sheared images. (cid:104) e i (cid:105) S − is the analogue quantityfor negatively sheared images. In the absence of selec-tion biases, (cid:104) R S (cid:105) would be zero. Otherwise, the fullresponse is given by the sum of the shear and selectionresponse: (cid:104) R (cid:105) = (cid:104) R γ (cid:105) + (cid:104) R S (cid:105) . (13)The application of the response corrections depends onthe shear statistic that is being calibrated; a generic cor-rection for the two point functions, including the tan-gential shear, which is our particular case of interest, isderived in [43]. In this work we make use of two ap-proximations that significantly simplify the calculationof the shear responses. First, in principle we shouldtake the average in Eq. (13) over the sources used ineach bin of θ , but we find no significant variation with θ and use a constant value (see App. C). Therefore, thecorrection to the tangential shear becomes just the av-erage response over the ensemble. Second, we assumethe correction to be independent of the relative orienta-tion of galaxies, so that we do not rotate the responsematrix as we do with the shears in Eq. (6). Overall, oursimplified estimator of the tangential shear for Meta-calibration , which replaces the previous expressionfrom Eq. (7) is: (cid:104) γ t, mcal (cid:105) = 1 (cid:104) R γ (cid:105) + (cid:104) R S (cid:105) (cid:80) j ω l ,j e t,j (cid:80) j ω l ,j , (14)summing over lens-source or random-source pairs j andwhere ω l ,j are the weights associated with the lenses.The measured selection effects due to sample selec-tion and photo- z binning for each tomographic bin are . , . , . and . , which represent 0.99%,2.1%, 1.5% and 2.4% of the total response in each bin. im3shape calibration For the im3shape shear catalog, additive and multi-plicative corrections need to be implemented in the fol-lowing manner, replacing the previous expression fromEq. (7) [45]: (cid:104) γ t, im3shape (cid:105) = (cid:80) j ω l ,j ω s ,j e t,j (cid:80) j ω l ,j ω s ,j (1 + m j ) , (15)summing over lens-source or random-source pairs j ,where m j is the multiplicative correction and the ad-ditive correction c j has to be applied to the Cartesian components of the ellipticity, before the rotation to thetangential component, defined in Eq. (6), has been per-formed. ω l ,j are the weights associated with the lensesand ω s ,j the ones associated with the im3shape catalog.From here on, we will refer to the mean tangentialshear (cid:104) γ t (cid:105) as γ t for simplicity. B. Measurement results
We present the DES Y1 galaxy-galaxy lensingmeasurements in Fig. 2. The total detection sig-nificance using all angular scales for the fiducial
Metacalibration catalog corresponds to
S/N =73 . Signal-to-noise is computed as in [59],
S/N =( γ data t C − γ model t ) / ( (cid:112) γ data t C − γ model t ) , where C and γ model t are the covariance matrix and the best-fit modelsfor galaxy-galaxy lensing measurements in the DES Y1cosmological analysis [25]. A series of companion pa-pers present other two-point functions of galaxies andshear on the same data sample, as well as the associ-ated cosmological parameter constraints from the com-bination of all these two-point function measurements[25, 37, 59]. The shaded regions from this figure corre-spond to scales that are excluded in the multiprobe cos-mological analysis, i.e., scales smaller than h − Mpc in comoving distance for the galaxy-galaxy lensing ob-servable [63]. In the top panel we present the measure-ments for the
Metacalibration shear catalog, andfor im3shape in the bottom panel. Note that the mea-surements from the two shear catalogs cannot be di-rectly compared, since their populations and thus theircorresponding redshift distributions differ. For each ofthe five lens redshift bins, we measure the tangentialshear for four tomographic source bins, which result in20 lens-source redshift bin combinations. The relativestrength of the galaxy-galaxy lensing signal for a givenlens bin depends on the geometry of the lens-source con-figuration. This feature is exploited in the shear-ratiotest, presented in Sec. VI, where we constrain the meanof the source redshift distributions using the small scalesthat are not used in the cosmological analysis (shadedin Fig 2).
C. Covariance matrix validation
Galaxy-galaxy lensing measurements are generallycorrelated across angular bins. The correct estimationof the covariance matrix is crucial not only in the us-age of these measurements for cosmological studies butalso in the assessment of potential systematic effectsthat may contaminate the signal. While a validatedhalo-model covariance is used for the DES Y1 multi-probe cosmological analysis [63], in this work we usejackknife (JK) covariance matrices given the require-ments of some systematics tests performed here, suchas splits in area, size or S/N. A set of 1200 lognormalsimulations, described in Section III D, is used to val-idate the jackknife approach in the estimation of thegalaxy-galaxy lensing covariances. We estimate the JK
10 100 ✓ [arcmin] ✓ [ a rc m i n ] Data JK
10 100 ✓ [arcmin]FLASK JK
10 100 ✓ [arcmin]FLASK True . . . . . . . . . C i j / p C ii ⇤ C jj C ij / ( C ij )0 . . . . . . ✓ [ a rc m i n ] FLASK JK - Data JKFLASK JK - FLASK True N (0 ,
10 100 ✓ [arcmin]FLASK JK - Data JK
10 100 ✓ [arcmin]FLASK JK - FLASK True C i j / ( C i j ) FIG. 3. Correlation matrices obtained from the jackknife method on the data (top-left panel), from the mean of jackknifecovariances using 100 FLASK realizations (top-middle panel) and from the 1200 lognormal simulations FLASK (top-rightpanel), for an example redshift bin ( . < z l < . and . < z s < . ). In the bottom-middle and bottom-rightpanels, we show the differences between the covariance matrices shown in the upper panels normalized by the uncertaintyon the difference, for the same example redshift bin. On the bottom-left panel, we display the normalized histograms ofthese differences ( × for each covariance, corresponding to 20 angular bins) for all the × lens-source redshift bincombinations, compared to a Gaussian distribution centered at zero with a width of one. covariance using the following expression: C JK ij ( γ i , γ j ) = N JK − N JK N JK (cid:88) k =1 (cid:0) γ ki − γ i (cid:1) (cid:0) γ kj − γ j (cid:1) , (16)where the complete sample is split into a total of N JK regions, γ i represents either γ t ( θ i ) or γ × ( θ i ) , γ ki denotesthe measurement from the k th realization and the i th angular bin, and γ i is the mean of N JK resamplings.Jackknife regions are obtained using the kmeans al-gorithm run on a homogeneous random point cata-log with the same survey geometry and, then, all fore-ground catalogs (lenses and random points) are splitin N JK = 100 subsamples. Specifically, kmeans is aclustering algorithm that subdivides n objects into N groups (see Appendix B in [64] for further details).In the upper panels of Fig. 3 we present the differentcovariance estimates considered in this work, namelythe jackknife covariance in the data ( Data JK ), themean of 100 jackknife covariances measured on the log-normal simulations (
FLASK JK ) and the true covariancefrom 1200 lognormal simulations (
FLASK True ), for agiven lens-source redshift bin combination ( . < z l < . and . < z s < . ). On the lower panels of this https://github.com/esheldon/kmeans_radec figure, we show the differences between them normal-ized by the corresponding uncertainty. The lower leftpanel shows the distribution of these differences andits agreement with a normal distribution with µ = 0 and σ = 1 , as expected from a pure noise contribu-tion, using all possible lens-source bin combinations,and the lower middle and right panels show the samequantity element-by-element for the redshift bin com-bination used in the upper panels. The uncertainty onthe data jackknife covariance comes from the standarddeviation of the jackknife covariances measured on 100lognormal simulations. The uncertainties on the twoother covariance estimates are significantly smaller; inthe mean of 100 jackknife covariances it is √ N timessmaller, where N = 100 in our case. On the otherhand, the uncertainty on each element of the true co-variance from 1200 lognormal simulations is calculatedusing (∆ C ij ) = ( C ii C jj + C ij C ij ) / ( N − , where N = 1200 in our case. The lower left panel showsan overall good agreement between the covariance es-timates, even though the larger tail of the orange his-togram with respect to a normal distribution indicatesa potential slight overestimation of the covariance ob-tained with the jackknife method.In Fig. 4 we compare the diagonal elements of thecovariance for the 20 lens-source redshift bin combina-tions, obtaining good agreement for all cases and scales.As in Fig. 3, the uncertainty on the data jackknife co-0 − − . < z s < . σ ( γ t ) . < z l < . Data JKFLASK JKFLASK True . < z l < .
45 0 . < z l < .
60 0 . < z l < .
75 0 . < z l < . − − . < z s < . σ ( γ t ) − − . < z s < . σ ( γ t )
10 100 θ [arcmin] − − . < z s < . σ ( γ t )
10 100 θ [arcmin]
10 100 θ [arcmin]
10 100 θ [arcmin]
10 100 θ [arcmin] FIG. 4. Comparison of the diagonal elements of the covariance obtained from the jackknife method on the data (
Data JK ),from the mean of jackknife covariances using 100 FLASK realizations (
FLASK JK ) and from the 1200 lognormal simulationsFLASK (
FLASK True ), for all the lens-source combinations. variance comes from the standard deviation of the jack-knife covariances measured on 100 lognormal simula-tions. The uncertainties on the two other error esti-mates are also shown on the plot, but are of the sameorder or smaller than the width of the lines.Overall, we have validated the implementation of thejackknife method on the data by comparing this co-variance to the application of the same method on 100lognormal simulations and to the true covariance ob-tained from 1200 lognormal simulations, and findinggood agreement among them, both for the diagonal andoff-diagonal elements.
V. DATA SYSTEMATICS TESTS
In order to fully exploit the power of weak gravita-tional lensing, we need to measure the shapes of millionsof tiny, faint galaxies to exceptional accuracy, and pos-sible biases may arise from observational, hardware andsoftware systematic effects. Fortunately, weak lensingprovides us with observables that are very sensitive tocosmology and the physical properties of the objectsinvolved but also with others for which we expect no cosmological signal. By measuring such observables,we can characterize and correct for systematic effectsin the data. In this section, we perform a series of teststhat should produce a null signal when applied to truegravitational shear, but whose non-zero measurement,if significant, would be an indication of systematic er-rors leaking into the galaxy-galaxy lensing observable.
A. Cross-component
The mean cross-component of the shear γ × , which isrotated 45 degrees with respect to the tangential shearand is defined in Eq. (6), should be compatible with zeroif the shear is only produced by gravitational lensing,since the tangential shear captures all the galaxy-galaxylensing signal. Note that the cross-component wouldalso be null in the presence of a systematic error that isinvariant under parity.In the top panel of Fig. 5 we show the resultingcross-shear measured around redMaGiC lenses (includ-ing random point subtraction) for one lens-source red-shift bin combination and for both shear catalogs. Inthe bottom panel we display the null χ histogram1
10 100 θ [arcmin] − − − − γ × × θ × − . < z l < . . < z s < . . < z l < . . < z s < . M ETACALIBRATIONIM SHAPE χ . . . . . . M ETACALIBRATIONIM SHAPE χ pdf (ndf = 20 ) FIG. 5. (
Top panel ): Cross-component of the galaxy-galaxylensing signal with random points subtraction for one lens-source redshift bin combination. (
Bottom panel ): The null χ histogram from all × lens-source redshift bins com-binations computed with the jackknife covariance correctedwith the Hartlap factor [65], compared to the χ distribu-tion with 20 degrees of freedom corresponding to 20 angularbins. We find the cross-component to be consistent withzero. coming from all × lens-source γ × measurements,computed using the jackknife covariance for the cross-component, described and validated in Sec. IV C. Tocompute the null χ , i.e. χ null = γ T × C − γ × , we needan estimate of the inverse of the covariance matrix,but since jackknife covariance matrices contain a non-negligible level of noise, we need to correct for the factthat the inverse of an unbiased but noisy estimate of thecovariance matrix is not an unbiased estimator of theinverse of the covariance matrix [65]. Thus, we applythe Hartlap correcting factor ( N JK − p − / ( N JK − to the inverse covariance, where N JK is the number ofjackknife regions and p the number of angular bins. Ourresults indicate the cross-component is consistent withzero.
10 100 θ [arcmin] − . − . − . . . . . γ t , P S F r e s i du a l s × − . < z < . redMaGiC Null χ /ndf . / FIG. 6. PSF residuals for
PSFEx model, using a single non-tomographic lens bin, including random-point subtraction.It is consistent with a null measurement and much smallerthan the signal.
B. Impact of PSF residuals
The estimation of source galaxy shapes involves mod-eling them convolved with the PSF pattern, which de-pends on the atmosphere and the telescope optics andwhich we characterize using stars in our sample. Next,we test the impact of residuals in the PSF modeling onthe galaxy-galaxy lensing estimator, and we comparethe size of this error to the actual cosmological signal.Explicitly, the PSF residuals are the differences be-tween the measured shape of the stars and the
PSFEx model [45, 66] at those same locations. In Fig. 6 weshow the measured mean of the tangential componentof the PSF residuals around redMaGiC galaxies, includ-ing the subtraction of the same quantity around randompoints, in the same manner as for the tangential shearsignal. We find it is consistent with zero, and also muchsmaller than the signal (cf. Figure 2).
C. Size and S/N splits
Potential biases in shape measurements are likely tobe more important for galaxies which are either smallor detected at low signal-to-noise (S/N). Even thoughthe shape measurement codes utilized in this work arecalibrated in a way such that these effects are taken intoaccount, it is important to test for any residual biasesin that calibration. In order to perform such a test,we split the source galaxy samples in halves of eitherlow or high size or S/N, and examine the differencesbetween the galaxy-galaxy lensing measurements usingthe different halves of the source galaxy samples. Forthis test, we use the lower redshift lens bin to mini-mize the overlap in redshift with the source samples.2 . . . . . . . . . Redshift . . . . . . . n ( z ) S/N split . . . . . . . . . . Redshift
Size split redMaGiC (0 . < z l < . Low M
ETACALIBRATION
High M
ETACALIBRATION
Low IM SHAPE
High IM SHAPE . . . . . . . γ h i g h t / γ l o w t Σ − , highcrit , eff / Σ − , lowcrit , eff M ETACALIBRATION Σ − , highcrit , eff / Σ − , lowcrit , eff IM SHAPE M ETACALIBRATIONIM SHAPE
FIG. 7. S/N ( left ) and size ( right ) splits tests for
Metacalibration and im3shape , using scales employed in the cosmologyanalysis ( > h − Mpc ). (
Top panels ): Redshift distributions of the lens and source samples used for this test. (
Bottompanels ): Comparison between the ratio of Σ − , eff using the above redshift distributions ( boxes ) to the ratio between theamplitudes coming from the fit of the tangential shear measurement for each half to the smooth template from the lognormalsimulations ( points ). The sources are all combined into a single bin, to maxi-mize the sensitivity to potential differences between thehalves.In order to estimate the size of galaxies, for
Meta-calibration we use round measure of size (
T_r ), andfor im3shape we use the R gpp /R p size parameter, bothdefined in [47]. We estimate the S/N of galaxies us-ing the round measure of S/N for Metacalibration ,( s2n_r ), and the snr quantity for im3shape , both de-fined in [47]. Splitting the source galaxy samples inhalves of low and high galaxy S/N or size, we measurethe corresponding galaxy-galaxy lensing signals, and wecheck their consistency.Since these quantities can correlate with redshift, dif-ferences can arise in the redshift distributions betweenthe halves of S/N and size splits, as seen in the up-per panels of Fig. 7. When comparing the tangentialshear signals of each half of the split, we therefore needto account for the differences in the lensing efficiencygiven by the two redshift distributions. We do this inthe following way. From Eq. (3), the ratio between thetangential shear measurements for each half of the splitin the absence of systematics effects is γ l,s high t γ l,s low t = Σ − l,s high crit , eff Σ − l,s low crit , eff , (17)since γ l,s high t and γ l,s low t share the same lens sample andthus the same ∆Σ . Σ − , eff , defined in Eq. (5), is a double integral over the lens and source redshift distri-butions and the geometrical factor Σ − , which dependson the distance to the lenses, the sources and the rel-ative distance between them. Then, to check the con-sistency between the tangential shear measurements foreach half of the source split we will compare the ratiobetween them to the ratio between the corresponding Σ − , eff ’s.Then, the validity of this test to flag potential bi-ases in shape measurements related to S/N and size islinked to an accurate characterization of the redshiftdistributions. The ensemble redshift distributions areestimated by stacking the redshift probability densityfunctions of individual galaxies in each split, as givenby the BPZ photo- z code. As described in [34] and aseries of companion papers [35, 36, 52] we do not relyon these estimated redshift distributions to be accu-rate, but rather calibrate their expectation values usingtwo independent methods: a matched sample with high-precision photometric redshifts from COSMOS, and theclustering of lensing sources with redMaGiC galaxies ofwell-constrained redshift. These offsets to the BPZ es-timate of the ensemble mean redshift, however, couldwell be different for the two halves of each of the splits.To estimate these calibration differences between thesubsamples, we repeat the COSMOS calibration of theredshift distributions (see [34] for details), splitting thematched COSMOS samples by Metacalibration sizeand signal-to-noise ratio at the same thresholds as inour data. We find that the shifts required to match3the mean redshifts of the subsamples with the meanredshifts of the matched COSMOS galaxies are differentby up to | ∆(∆ z ) | = 0 . for the overall source sample.In the upper panels of Fig. 7, the mean values ofthe redshift distributions have been corrected using theresults found in the analysis described above, and thesecorrected n ( z ) ’s are the ones that have been used in thecalculation of Σ − , eff in Eq. (17). The ratio of Σ − , eff ’sis shown in the lower panels of Fig. 7 and its uncertaintycomes from the propagation of the error in the meanof the source redshift distributions for each half of thesplit, i.e. √ times the non-tomographic uncertainty asestimated in [34] using COSMOS.Regarding the left-hand side of Eq. (17), to avoid in-ducing biases from taking the ratio between two noisyquantities, we fit an amplitude for each half of the splitto a smooth tangential shear measurement that we ob-tain from the mean of tangential shear measurementson 100 independent log-normal simulations. Then, wetake the ratio between the amplitudes fitted for eachhalf of the split. We repeat this procedure for eachdata jackknife resampling, obtaining a ratio for each ofthose, whose mean and standard deviation are shownin the lower panels of Fig. 7 ( points ), compared to theratio of Σ − , eff ’s ( boxes ).Given the uncertainties in both the measurementsand the photometric redshift distributions presentedin Fig. 7, we find no significant evidence of a differ-ence in the galaxy-galaxy lensing signal when splittingthe Metacalibration or im3shape source samples bysize or S/N. Specifically, we find a 1.6 σ (0.24 σ ) differ-ence for the Metacalibration ( im3shape ) S/N splitand a 0.90 σ (1.3 σ ) difference for the Metacalibra-tion ( im3shape ) size split. D. Impact of observing conditions
Time-dependent observing conditions are intrinsic tophotometric surveys, and they may impact the derivedgalaxy catalogs, for instance, introducing galaxy den-sity variations across the survey footprint. In this sec-tion we test for potential biases in the galaxy-galaxylensing measurements due to these differences in ob-serving conditions and their effect in the survey galaxydensity. We use projected
HEALPix [67] sky maps (withresolution N side = 4096 ) in the r band for the followingquantities: • AIRMASS : Mean airmass, computed as the opti-cal path length for light from a celestial objectthrough Earth’s atmosphere (in the secant ap-proximation), relative to that at the zenith forthe altitude of CTIO. • FWHM : Mean seeing, i.e., full width at half maxi-mum of the flux profile. • MAGLIMIT : Mean magnitude for which galaxies aredetected at
S/N = 10 . • SKYBRITE : Mean sky brightness.More information on these maps can be found in [41]and [37]. . . . . . γ h i g h t / γ l o w t AIRMASSAIRMASS FWHMFWHM . . . . . γ h i g h t / γ l o w t MAGLIMITMAGLIMIT Σ − , highcrit , eff / Σ − , lowcrit , eff M ETACAL Σ − , highcrit , eff / Σ − , lowcrit , eff IM SHAPE M ETACALIBRATIONIM SHAPE
SKYBRITESKYBRITE
FIG. 8. Results for the tests involving area splits in halves ofdifferent observational systematics maps in the r band, withangular scales used in the cosmology analysis ( h − Mpc ).We compare the ratio of Σ − , eff ( boxes ) using the redshiftdistributions for each split to the ratio between the ampli-tudes coming from the fit of the tangential shear measure-ment for each half to the smooth template derived from thelognormal simulations ( points ), following the same proce-dure as for the S/N and size splits, described in Sec. V Cand shown in Fig. 7. In order to test for potential systematic effects, wesplit each map into halves of high and low values of agiven quantity, and measure the galaxy-galaxy lensingsignal in each half. We are using the same configurationas in the S/N and size splits, i.e. the lower redshift lensbin and a single non-tomographic source bin between . < z s < . . In this case, we are splitting both thelens and the source samples, since the split is performedin area.To check the consistency between the measurementsin each half we follow the same approach as for theS/N and size splits, described in detail in the previoussection, where we take into account the differences inthe redshift distributions of the sources. We find thecorrelation between observing conditions and redshiftto be very mild for the source sample, as can be seenin Fig. 8, where the ratios of Σ − , eff ’s are all compat-ible with unity. For the lens sample this correlation iseven smaller, consistent with the lens sample contain-ing brighter and lower-redshift galaxies. The differenceson the mean redshift between the lens redshift distri-butions of the two halves are of the order of . orsmaller for all maps, which is negligible for this test, al-though we have not performed independent calibrationof redshift biases for these split samples.The results for these area splits are shown in Fig. 8for Metacalibration and im3shape . In most cases,the ratio between the measurements on each half ofthe splits lie within 1 σ of the corresponding ratio of Σ − , eff ’s, and at slightly more than 1 σ in the remainingcases. Thus, we do not encounter any significant biaseson the galaxy-galaxy lensing signal due to differences in4observing conditions.The effect of the same variable observing conditionsin the galaxy clustering measurements using the sameDES redMaGiC sample is studied in detail in [37]. Inthat analysis, maps which significantly correlate withgalaxy density are first identified, and then a set ofweights is computed and applied to the galaxy sam-ple so that such dependency is removed, following amethod similar to that presented in [68, 69]. The re-sulting set of weights from that analysis has been alsoused in this work, for consistency in the combinationof two-point correlation functions for the DES Y1 cos-mological analysis. Nonetheless, the impact of such aweighting scheme in the galaxy-galaxy lensing observ-ables is found to be insignificant, consistent with thetests presented above in this section and with previousstudies (see [9]). VI. SHEAR-RATIO TEST
In previous sections we have seen that the variationof the galaxy-galaxy lensing signal with source redshiftdepends solely on the angular diameter distances rela-tive to foreground and background galaxy populations.Such dependency was initially proposed as a probe fordark energy evolution in [70]. The shear-ratio is, how-ever, a weak function of cosmological parameters, andmore sensitive to errors in the assignment of source orlens redshifts [17]. Since redshift assignment is a crucialbut difficult aspect of robust cosmological estimate fora photometric survey like DES, the shear-ratio test isa valuable cross-check on redshift assignment. In thecontext of the DES Y1 cosmological analysis, the usageof high-quality photometric redshifts for lens galaxiesallows us to put constraints on the mean redshift ofsource galaxy distributions.In this section we present a general method to con-strain potential shifts on redshift distributions using thecombination of ratios of galaxy-galaxy lensing measure-ments. First, we present the details of the implementa-tion, and we test it on lognormal simulations. Then, weuse the galaxy-galaxy lensing measurements shown inFig. 2, restricted to angular scales which are not usedin the DES Y1 cosmological analysis, to place indepen-dent constraints on the mean of the source redshift dis-tributions shown in the lower panel of Fig. 1. Finally,we compare our findings with those obtained from aphotometric redshift analysis in the COSMOS field andfrom galaxy angular cross-correlations.The ratio of two galaxy-galaxy lensing measurementsaround the same lens bin, hence having equivalent ∆Σ ,can be derived from Eq. (3) and is given by: γ l,s i t γ l,s j t = Σ − l,s i crit , eff Σ − l,s j crit , eff , (18)where Σ − , eff is the double integral over lens and sourceredshift distributions defined in Eq. (4). Therefore, fortwo given γ t measurements sharing the same lens popu-lation but using two different source bins, we can predicttheir ratio from theory by using the estimated redshiftdistributions involved. In addition, we can allow for a shift in each of those redshift distributions and use the γ t measurements to place constraints on them.In this section we generalize this approach by includ-ing all possible combinations of ratios of galaxy-galaxylensing measurements sharing a given lens bin, and al-lowing for independent shifts in their redshift distribu-tions. With the purpose of providing constraints onthe shifts of redshift distributions which are indepen-dent of the measurement involved in the fiducial DESY1 cosmological analysis, we restrict the galaxy-galaxylensing measurements used for this shear-ratio test toscales smaller than the ones used by the cosmologicalanalysis but which have still been tested against sys-tematic effects in this work.In order to estimate the ratio of galaxy-galaxy lensingmeasurements, which can be noisy and thus bias theirratio, we fit each measurement involved in the ratio,both around the same lens bin, to a power law fit of thehighest signal-to-noise γ t measurement for the same lensbin. That fixes the shape of the galaxy-galaxy lensingsignal around that lens galaxy sample. Then, fits tothe amplitude of this power law are used to obtain theshear ratio. A. Testing the method on simulations
With the purpose of testing our method to estimateratios of galaxy-galaxy lensing measurements and ourability to recover the expected values from theory, weuse the lognormal simulations described in Sec. III D,where we know the true lens and source redshift distri-butions. For that case, we should be able to find goodagreement between measurements and theory, withoutthe necessity of allowing for any shifts in the redshiftdistributions.Figure 9 shows all the possible ratios of two γ t mea-surements in the FLASK simulations sharing the samelens bin using the lens-source binning configuration usedthroughout this paper (as depicted in Fig. 1), with theerror bars coming from the variance of the 1200 simu-lations. It also shows the expected values for the ratiosgiven from theory, using the true corresponding redshiftdistributions with no shifts applied. The agreement be-tween measurements and theory is excellent, demon-strating that the method described in this section isable to recover the true values of γ t measurements fromtheory when the redshift distributions are known. B. Application to data
Now we turn to data, and utilize this shear-ratiomethod to constrain possible biases in the mean of red-shift distributions. The lens and source redshift binsconsidered and their fiducial estimated redshift distri-butions are depicted in Figure 1. The high-precisionphotometric redshifts of the redMaGiC sample ensurethe lens redshift distributions are well known, with po-tential shifts found to be very small and consistent withzero in [52], and hence we keep them fixed. On the con-trary, source galaxies are generally fainter and have amuch larger uncertainty in their redshift distributions.5 ( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , )( , , ) Lens-Source bin combinations ( l, s i , s j )0 . . . . . . . γ l , s i t / γ l , s j t Theory (No shifts)Flask
FIG. 9. Comparison between the mean ratio of tangential shear measurements using 1200 independent log-normal simulationsand the ones calculated from theory, for all lens-source bin ratio combinations sharing the same lens bin. The errorbarscorrespond to the standard deviation of the measurement on individual simulations, thus being representative of the errorsthat we will obtain from the data. . . . . . . . . . < z s < . B oo s t f a c t o r s . < z l < .
30 0 . < z l < .
45 0 . < z l < .
60 0 . < z l < .
75 0 . < z l < . . . . . . . . . . < z s < . B oo s t f a c t o r s . . . . . . . . . < z s < . B oo s t f a c t o r s
10 100 θ [arcmin] . . . . . . . . . < z s < . B oo s t f a c t o r s
10 100 θ [arcmin]
10 100 θ [arcmin]
10 100 θ [arcmin]
10 100 θ [arcmin] FIG. 10. Boost factor correction accounting for clustering between lenses and sources in each lens-source bin used in thisanalysis. Non-shaded scales correspond to scales used in the DES Y1 cosmological analysis, while shaded regions are usedfor the shear-ratio geometrical test in this Section. Boost factors are unity or percent-level for the former, but can besignificantly larger for the latter in some cases, and hence they are applied in our analysis of the shear-ratio test. ∆ z i in eachof the measured source redshift distributions n i obs ( z ) ,such that n i pred ( z ) = n i obs ( z − ∆ z i ) , (19)to be constrained from the combination of ratios ofgalaxy-galaxy lensing measurements through their im-pact in the Σ − , eff factors in Eq. (18).When turning to the data case, we also have to con-sider effects which are not included in the simulations.In particular, next we take into account the effects ofpotential boost factors and multiplicative shear biasesin the measurements.
1. Boost factors
The calculation of the mean galaxy-galaxy lensingsignal in Eq. (1) correctly accounts for the fact thatsome source galaxies are in front of lenses due to over-lapping lens and source redshift distributions, but onlyunder the assumption that the galaxies in those distri-butions are homogeneously distributed across the sky.As galaxies are not homogeneously distributed but theyare clustered in space, a number of sources larger thanthe n obs ( z ) suggests may be physically associated withlenses. These sources are not lensed, causing a dilutionof the lensing signal which can be significant at smallscales. In order to estimate the importance of this ef-fect, we compute the excess of sources around lensescompared to random points [22]: B ( θ ) = N r N l (cid:80) l,s w l,s (cid:80) r,s w r,s (20)where l, s ( r, s ) denotes sources around lenses (randompoints), w l,s ( w r,s ) is the weight for the lens-source(random-source) pair, and the sums are performed overan angular bin θ . Figure 10 shows this calculation forevery lens-source bin in this analysis. The shaded re-gions in the plot mark the scales used for the shear-ratiotest (unused by the cosmological analysis). The impor-tance of boost factors at small scales can be as largeas 10%, while on the large scales used for cosmologyit does not depart from unity above the percent level.The data measurements used for the shear-ratio test inthis section have been corrected for this effect.
2. Multiplicative shear biases
Multiplicative shear biases are expected to be presentin the galaxy-galaxy lensing signal and need to be takeninto account. This potential effect is included as anindependent parameter m i for each source redshift bin,parametrized such that the shear ratios in Eq. (18) looklike the following: γ l,s i t γ l,s j t = (1 + m j ) Σ − l,s i crit , eff (1 + m i ) Σ − l,s j crit , eff . (21) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) Lens-Source bin combinations ( l, s i , s j )0 . . . . . . . γ l , s i t / γ l , s j t Theory (No shifts)Best-fit ± σ Data
FIG. 11. Comparison between the ratio of tangential shearmeasurements on
Metacalibration ( blue points ) to theones calculated from theory, both without applying any shiftto the original source n ( z ) (cid:48) s ( dashed orange line ) and ap-plying the best-fit shifts with a σ uncertainty band ( grayband ).
3. Results
In practice, the sensitivity of the shear-ratio geomet-rical test to shifts in the mean of redshift distributionsdecreases significantly the higher the distribution is inredshift, due to the relative differences in distance withrespect to the lenses and the observer being smaller forthat case. For that reason, the sensitivity to shifts inthe highest source redshift bin defined in this work isvery small, and as there are strong correlations withthe other shifts, we left out the fourth source bin. Wealso leave out the two highest lens redshift bins as thegalaxy-galaxy lensing S/N for these cases is very smalland they add little information to this test.In order to find the best-fit shifts for all combina-tions of fixed-lens γ t ratios using these redshift bins, weset a Monte-Carlo Markov Chain (MCMC) to let theshifts vary, with a broad flat prior of [-0.5,0.5] for eachshift ∆ z i . We follow the recommendations in [45] andinclude a Gaussian prior of µ = 0 . and σ = 0 . on the multiplicative shear biases m i for each sourcebin i . As the covariance is estimated from JK resam-pling, the corresponding Hartlap factor is applied to thecovariance. Some recent studies have discussed and pre-sented further corrections to that procedure [71]. Giventhat in our case the Hartlap factor is (cid:39) . , such correc-tions would result in a small change to the parametercontours, and have not been considered in this analysis.However, a more detailed treatment of noisy covariancesmay need to be considered in forthcoming, more sensi-tive, DES analyses.Figure 11 shows the equivalent of Fig. 9 for the datacase, including the theory prediction with no shifts andwith best-fit shifts from the MCMC run, and the shear-7 Shear-ratioCOSMOSWZ − . − . . . ∆ z − .
04 0 .
00 0 .
04 0 . ∆ z . . . . ∆ z − . − .
03 0 .
00 0 . ∆ z .
00 0 .
15 0 .
30 0 . ∆ z FIG. 12. Comparison of the constraints obtained on the source redshift distribution shifts using different methods: Shear-ratio test, photo- z studies in the COSMOS field (COSMOS, [34]) and cross-correlation redshifts (WZ, [35, 36]).TABLE I. Priors and posteriors on the mean of source redshift distributions ( ∆ z ) and multiplicative shear biases ( m ) forthe first three source bins defined in this work (Fig. 1), using the shear-ratio test. Priors are uniform in ∆ z and Gaussianon m , and posteriors are given as the mean value with 68% constraints. ∆ z Prior ∆ z Posterior m Prior m PosteriorSource bin 1 Uniform( − . ,0.5) . +0 . − . Gaussian(0.012,0.021) . +0 . − . Source bin 2 Uniform( − . ,0.5) − . +0 . − . Gaussian(0.012,0.021) − . +0 . − . Source bin 3 Uniform( − . ,0.5) . +0 . − . Gaussian(0.012,0.021) . +0 . − . ratio case in Fig. 12 shows the ∆ z constraints from theMCMC, marginalizing over multiplicative shear biases m , where very clear correlations can be observed be-tween the different shifts. In addition, Table I presentsthe derived constraints on ∆ z and m for the differentsource bins considered. Even though Σ crit depends oncosmology through Ω m , the results are insensitive tothat parameter to the extent that no significant changeson the shifts are observed when marginalizing over itwith a broad flat prior of . < Ω m < . . Also, theboost factor correction from Eq. (20) has no significanteffect on the derived ∆ z constraints.In the past, several studies have proposed shear self-calibration techniques, either from galaxy-galaxy lens-ing only [72], or using combinations of observables(e. g. [73, 74]). Interestingly, the shear-ratio test canalso be used as a way to calibrate potential multiplica-tive shear biases ( m ) present in the data. Figure 13 dis-plays the m priors and posteriors for the three source redshift bins considered, where the posteriors show areduction of up to 20% in the width of the priors (seealso Table I) for the second and third bins, thereforeshowing potential as a method to internally constrainshear biases in the data.
4. Caveats and future work
The redshift evolution of the ∆Σ profile of the lenssample within a redshift bin could potentially affectthe shear-ratio test and would not be noticeable inthe FLASK simulations. This would especially influ-ence the ratios between lens and source bins that areclose in redshift. However, the usage of relatively thinlens tomographic bins, of 0.15 in redshift, and the littlegalaxy bias evolution of the redMaGiC sample for thefirst three lens bins, as shown in Fig. 14 below and [28],suggest that this effect is small compared to our current8 Bin 1PriorPosterior
Bin 2 − . − . − .
02 0 .
00 0 .
02 0 .
04 0 .
06 0 . Multiplicative shear bias m Bin 3
FIG. 13. Prior and posterior distributions for multiplicativeshear biases ( m ) for the first three source bins defined in thiswork (Fig. 1). The shear-ratio test appears to be informativeon the multiplicative shear biases for the second and thirdsource bins, reducing the prior width by as much as 20%,even though posteriors are all consistent with the priors atbetter than 1- σ level. error bars. On the other hand, mischaracterization ofthe tails in the fiducial (unshifted) redshift distributionsof the source galaxies, especially for those close to thelenses, could also affect the results of the shifts obtainedwith the shear-ratio test. Studying the impact of sucheffects in the shear-ratio geometrical test using N -bodysimulations is beyond the scope of this paper and it isleft for future work.In addition, intrinsic alignment (IA) between physi-cally associated lens-source galaxy pairs can potentiallyaffect the shear ratio measurement (see, e. g., [22, 75]).While IA on larger scales is modeled when measuringcosmology or the galaxy bias, we have not included thiseffect on the small scales used here. The boost factormeasurements in Fig. 10 yield an estimate of the frac-tion of physically associated pairs in all our measure-ments. As seen in [75], for typical lensing sources theimpact of IA contamination on the observed lensing sig-nal is smaller than that of the boosts themselves. Sincethe boost corrections here are small and have a minimaleffect on the derived source photo- z shifts, we expectthe impact of IA to be highly subdominant. However,it will be beneficial in future work to include the impactof IA when performing shear ratio tests.
5. Comparison of ∆ z constraints and conclusions In Fig. 12 we also compare the shear-ratio constraintswith those obtained independently from photo- z stud-ies in the COSMOS field [34] and from galaxy cross-correlations [35, 36], and we find consistency amongthe three independent studies, with χ /dof = 5.57/6for the combination of the three cases. As expected,the constraining power of the shear-ratio test for theshifts on the source distributions decreases rapidly thehigher the redshift of the distributions is, so that the 1-D marginalized constraints on the first tomographicbin are competitive with those from the other probes,and for the third tomographic bin they add very littleinformation. However, on the 2-D space, the shear-ratiocontours show great potential in breaking degeneracieswith other probes. Therefore, the use of this methodwith forthcoming data sets can have a major impact indetermining possible photometric redshift biases, espe-cially from source distributions at low redshift.The importance of an accurate photometric redshiftcalibration in DES was already noticed in the analy-sis of Science Verification data, where it proved to beone of the dominant systematic effects [76]. For this rea-son, showing the consistency of constraints derived fromgalaxy-galaxy lensing only to those from more tradi-tional photo- z methods and from galaxy angular cross-correlations represents an important demonstration ofthe robustness of the companion DES Y1 cosmologicalanalysis. VII. REDMAGIC GALAXY BIAS
Galaxy-galaxy lensing is sensitive to cosmological pa-rameters and the galaxy bias of the corresponding lensgalaxy population, as expressed in Eq. (1). Similarly,the galaxy clustering of the same lens population alsodepends on both cosmology and the galaxy bias, butwith a different power of the latter [37]. Therefore,the combination of galaxy clustering and galaxy-galaxylensing breaks the degeneracy between the galaxy biasand cosmological parameters. This combination is oneof the more promising avenues to understand the under-lying physical mechanism behind dark energy, and hasbeen used together with cosmic shear measurements toproduce cosmological results from DES Y1 [25].Alternatively, fixing all cosmological parameters, themeasurements of galaxy clustering and galaxy-galaxylensing can provide independent measurements of thegalaxy bias of a given lens population. The DES Y1cosmology analysis relies on the assumption that the lin-ear bias from galaxy clustering and from galaxy-galaxylensing is the same, which is known to break down onthe small-scale regime [77]. To verify this assumptionover the scales used in the DES Y1 cosmology analysis,we measure the galaxy bias from each probe separately.In Fig. 14 we show the bias constraints from galaxyclustering (or galaxy autocorrelations, b A ) and galaxy-galaxy lensing (or galaxy-shear cross-correlations, b × )on the five lens redMaGiC tomographic bins definedin this work, fixing all cosmological parameters to thebest-fit obtained in the DES Y1 cosmological analy-sis [25]. We use comoving angular separations largerthan 8 h − Mpc for galaxy clustering, and larger than12 h − Mpc for galaxy-galaxy lensing, which correspondto the scales used in the DES Y1 cosmological anal-ysis. In order to obtain these results, the clusteringmeasurements from [37] and the galaxy-galaxy lensingmeasurements from this work have been analyzed withthe same pipeline used in [25], including the covariancebetween the two probes and marginalizing over all nui-sance parameters like photometric redshift, shear cali-bration and intrinsic alignments uncertainties. We find9the obtained constraints on the galaxy bias from galaxy-galaxy lensing to be in good agreement with those ob-tained from galaxy clustering. . . . . . . . . Redshift . . . . . . . G a l a xy b i a s L min = 0 . L ∗ L ∗ . L ∗ b A b × FIG. 14. Comparison of the galaxy bias results obtainedfrom galaxy clustering measurements ( b A , [37]) and fromthe galaxy-galaxy lensing measurements in this work ( b × ),by fixing all cosmological parameters to the 3x2 cosmologybest-fit from [25]. The vertical dotted lines separate thethree redMaGiC samples, which have different luminositythresholds L min , defined in Sec. III A. . . . . . . . . Redshift . . . . . r DES Y1 AllDES Y1 w ( θ ) + γ t DES Y1 Shear
FIG. 15. Cross-correlation coefficient r between galaxies anddark matter obtained by comparing the galaxy bias fromgalaxy clustering ( b A ) and from galaxy-galaxy lensing only( b × ), fixing all cosmological parameters to three differentcosmologies from DES Y1 cosmological results [25]: (i) 3x2best-fit (All), (ii) ω ( θ ) + γ t best-fit, and (iii) cosmic shearbest-fit. The results in Fig. 14 can also be interpreted by al-lowing a non-unity cross-correlation parameter betweenthe galaxy and matter distributions. This parameteris usually expressed in terms of the matter and galaxypower spectra, P δδ and P gg respectively, and the galaxy-matter power spectrum P gδ , as r ( k, χ ( z )) = P gδ ( k, χ ( z )) (cid:112) P δδ ( k, χ ( z )) P gg ( k, χ ( z )) , (22) where we have explicitly included its possible scale andredshift dependence. In the context of this model, thegalaxy power spectrum remains unchanged with respectto r = 1 , P gg = b P δδ , but the galaxy-matter powerspectrum changes from P gδ = bP δδ to P gδ = b rP δδ .That introduces an r factor in the galaxy-galaxy lensingexpression in Eq. (1), and hence the two estimates ofthe galaxy bias in Fig. 14 can be transformed to: b = b A ; r = b × /b A , (23)and this allows us to place constraints on the r param-eter using our measurements. If r i refers to the cross-correlation parameter in lens bin i , the constraints weobtain read: r = 1 . ± . , r = 0 . ± . , r = 0 . ± . , r = 1 . ± . , r = 0 . ± . ,shown also in Fig. 15.In addition, it is important to note that the specifiedconstraints on the galaxy bias and the cross-correlationcoefficient are not independent of the assumed cosmol-ogy. The values given above are obtained with the3x2 best-fit cosmological parameters from the DES Y1main cosmological analysis [25], which favours the cross-correlation coefficient being consistent with one, sincethe cosmology is determined assuming the galaxy biasfor galaxy clustering and for galaxy-galaxy lensing isthe same. This is also true for the 2x2 cosmology, from ω ( θ ) + γ t . On the contrary, the cosmological param-eters obtained only from the cosmic shear analysis areindependent of the galaxy bias and the cross-correlationcoefficient and therefore provide a way to test the r = 1 assumption. In Fig. 15, we present the r constraintsfor each of these three cosmologies, which we find all tobe consistent with r = 1 . The r constraints presentedin this section provide further justification for assuming r = 1 in the main DES Y1 cosmological analysis.In the past, different studies have analyzed the con-sistency between different estimates of the galaxy biasof a given galaxy population. In the context of DES,a number of different analyses using galaxy clusteringin [78], CMB lensing in [79], galaxy-galaxy lensing in[29], and projected mass maps in [80] used DES Sci-ence Verification (SV) data to obtain constraints on thegalaxy bias of the main galaxy population (so-calledDES-SV Benchmark sample), finding mild differencesin those estimates that were explored as potential dif-ferences between clustering and lensing. Outside DES,other studies have also examined potential differencesbetween clustering and lensing. In particular, in [81] theauthors perform a galaxy-galaxy lensing measurementaround BOSS CMASS spectroscopic galaxies using datafrom the CFHTLenS and SDSS Stripe 82 surveys, andfind the lensing signal to be lower than that expectedfrom the clustering of lens galaxies and predictions fromstandard models of the galaxy-halo connection. In thisstudy, as expressed in the r values reported above, andmore broadly in the DES Y1 cosmological analysis pre-sented in [25], we find the clustering and lensing sig-nals to be consistent within our uncertainties, thoughwe note that the [81] analysis was done on significantlysmaller scales.0 VIII. CONCLUSIONS
This paper is part of the Dark Energy Survey Year 1(DES Y1) effort to obtain cosmological constraints bycombining three different probes, namely galaxy clus-tering, galaxy-galaxy lensing and cosmic shear. Themain goal of this work is to present and character-ize one of these two-point correlations functions, thegalaxy-galaxy lensing measurement. Besides this prin-cipal task, we use source tomography to put constraintson the mean of the source redshift distributions usingthe geometrical shear-ratio test. Finally, we obtain thegalaxy bias from this probe and we compare it to thecorresponding result from galaxy clustering.Our lens sample is composed of redMaGiC galaxies[31], which are photometrically selected luminous redgalaxies (LRGs) with high-precision photometric red-shifts. This allows us to divide the lens sample intofive equally-spaced tomographic bins between 0.15 and0.9 in redshift. Regarding the source sample, we usetwo independent shear catalogs, namely
Metacali-bration and im3shape , which are described in detailin [45]. We split the source galaxies into four tomo-graphic bins between 0.2 and 1.3 in redshift using BPZ,a template-based photometric redshift code.In order to characterize the DES Y1 galaxy-galaxylensing measurements, we test them for an extensive setof potential systematic effects. First, we show that thecross-component of the shear is compatible with zero,which should be the case if the shear is only producedby gravitational lensing. Second, PSF residuals are con-sidered and found to leave no imprint on the tangentialshear measurements. Next, we split the source sam-ple into halves of high and low signal-to-noise or size,observing no significant differences between the mea-surements in each half of the split. Finally, we studythe impact of the survey observing conditions, i.e. air-mass, seeing, magnitude limit and sky brightness, onthe galaxy-galaxy lensing signal, finding no significantdependence. To estimate the significance of these testswe use covariance matrices obtained from the jackknifemethod, which we validate using a suite of log-normalsimulations. Overall, we find no significant evidence ofsystematics contamination of the galaxy-galaxy lensingsignal. Besides serving as crucial input and validationfor the DES Y1 cosmological analysis, this set of sys-tematics tests will also be useful for potential futurework relying on DES Y1 galaxy-galaxy lensing mea-surements.In addition to the systematics testing, we apply theshear-ratio test to our source tomographic measure-ments. Given a fixed lens bin, we make use of the ge-ometrical scaling of the tangential shear for differentsource redshift bins to constrain the mean of the sourcetomographic redshift distributions, which is one of thedominant sources of uncertainty in the DES Y1 cosmo-logical analysis. For this test, we restrict the scales tothose ignored in the cosmological analysis, so that it isindependent of the constraints obtained there. Our re-sults are in agreement with other photo- z studies on thesame data sample [34–36], thus showing the robustnessof the photometric redshifts used in the DES Y1 cosmo-logical analysis. We also find this method to be infor- mative of multiplicative shear biases in the data, henceshowing potential as a way of self-calibrating shear bi-ases in future data sets.Finally, restricting to the scales used in the cosmolog-ical analysis, we use the galaxy-galaxy lensing measure-ments in this work to obtain galaxy bias constraints onthe redMaGiC galaxy sample by fixing all the cosmo-logical parameters but leaving free the nuisance param-eters as in [25]. We compare these constraints from theones obtained using the corresponding galaxy clusteringmeasurements in the same lens sample in [37] and usingthe same cosmological model, finding good agreementbetween them. This agreement can also be understoodas a consistency test of the assumption that the galaxy-matter cross-correlation coefficient r = 1 , made in thecosmology analysis. ACKNOWLEDGEMENTS
This paper has gone through internal review by theDES collaboration. It has been assigned DES paper idDES-2016-0210 and FermiLab preprint number PUB-17-277-AE.Funding for the DES Projects has been provided bythe U.S. Department of Energy, the U.S. National Sci-ence Foundation, the Ministry of Science and Educationof Spain, the Science and Technology Facilities Councilof the United Kingdom, the Higher Education FundingCouncil for England, the National Center for Super-computing Applications at the University of Illinois atUrbana-Champaign, the Kavli Institute of Cosmolog-ical Physics at the University of Chicago, the Centerfor Cosmology and Astro-Particle Physics at the OhioState University, the Mitchell Institute for Fundamen-tal Physics and Astronomy at Texas A&M University,Financiadora de Estudos e Projetos, Fundação Car-los Chagas Filho de Amparo à Pesquisa do Estado doRio de Janeiro, Conselho Nacional de DesenvolvimentoCientífico e Tecnológico and the Ministério da Ciência,Tecnologia e Inovação, the Deutsche Forschungsgemein-schaft and the Collaborating Institutions in the DarkEnergy Survey.The Collaborating Institutions are Argonne NationalLaboratory, the University of California at Santa Cruz,the University of Cambridge, Centro de InvestigacionesEnergéticas, Medioambientales y Tecnológicas-Madrid,the University of Chicago, University College Lon-don, the DES-Brazil Consortium, the University ofEdinburgh, the Eidgenössische Technische Hochschule(ETH) Zürich, Fermi National Accelerator Laboratory,the University of Illinois at Urbana-Champaign, the In-stitut de Ciències de l’Espai (IEEC/CSIC), the Insti-tut de Física d’Altes Energies, Lawrence Berkeley Na-tional Laboratory, the Ludwig-Maximilians UniversitätMünchen and the associated Excellence Cluster Uni-verse, the University of Michigan, the National OpticalAstronomy Observatory, the University of Nottingham,The Ohio State University, the University of Pennsylva-nia, the University of Portsmouth, SLAC National Ac-celerator Laboratory, Stanford University, the Univer-sity of Sussex, Texas A&M University, and the OzDESMembership Consortium.1Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical AstronomyObservatory, which is operated by the Association ofUniversities for Research in Astronomy (AURA) un-der a cooperative agreement with the National ScienceFoundation.The DES data management system is supported bythe National Science Foundation under Grant Num-bers AST-1138766 and AST-1536171. The DES partic-ipants from Spanish institutions are partially supportedby MINECO under grants AYA2015-71825, ESP2015-88861, FPA2015-68048, SEV-2012-0234, SEV-2016-0597, and MDM-2015-0509, some of which includeERDF funds from the European Union. IFAE is par-tially funded by the CERCA program of the Generalitatde Catalunya. Research leading to these results hasreceived funding from the European Research Coun-cil under the European Union’s Seventh FrameworkProgram (FP7/2007-2013) including ERC grant agree-ments 240672, 291329, and 306478. We acknowledgesupport from the Australian Research Council Cen-tre of Excellence for All-sky Astrophysics (CAASTRO),through project number CE110001020. This manuscript has been authored by Fermi Re-search Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Officeof Science, Office of High Energy Physics. The UnitedStates Government retains and the publisher, by ac-cepting the article for publication, acknowledges thatthe United States Government retains a non-exclusive,paid-up, irrevocable, world-wide license to publish or re-produce the published form of this manuscript, or allowothers to do so, for United States Government purposes.Support for DG was provided by NASA throughEinstein Postdoctoral Fellowship grant number PF5-160138 awarded by the Chandra X-ray Center, whichis operated by the Smithsonian Astrophysical Observa-tory for NASA under contract NAS8-03060. JB ac-knowledges support from the Swiss National ScienceFoundation.This research used computing resources at SLAC Na-tional Accelerator Laboratory, and at the National En-ergy Research Scientific Computing Center, a DOE Of-fice of Science User Facility supported by the Office ofScience of the U.S. Department of Energy under Con-tract No. DE-AC02-05CH11231. [1] J. A. Tyson, F. Valdes, J. F. Jarvis, and J. Mills, A.P., Astrophys. J. , L59 (1984).[2] T. G. Brainerd, R. D. Blandford, and I. Smail, Astro-phys. J. , 623 (1996).[3] I. P. Dell’Antonio and J. A. Tyson, Astrophys. J. ,L17 (1996).[4] J. F. Navarro, C. S. Frenk, and S. D. M. White, As-trophys. J. , 493 (1997).[5] M. Cacciato, F. C. van den Bosch, S. More, R. Li, H. J.Mo, and X. Yang, Mon. Not. R. Astron. Soc. , 929(2009).[6] R. Mandelbaum, A. Slosar, T. Baldauf, U. Seljak, C. M.Hirata, R. Nakajima, R. Reyes, and R. E. Smith, Mon.Not. R. Astron. Soc. , 1544 (2013).[7] M. Cacciato, F. C. van den Bosch, S. More, H. Mo, andX. Yang, Mon. Not. R. Astron. Soc. , 767 (2013).[8] S. More, H. Miyatake, R. Mandelbaum, M. Takada,D. N. Spergel, J. R. Brownstein, and D. P. Schneider,Astrophys. J. , 2 (2015).[9] J. Kwan, C. Sánchez, et al. , Mon. Not. R. Astron. Soc. , 4045 (2017), 1604.07871.[10] E. van Uitert, B. Joachimi, et al. , Mon. Not. R. Astron.Soc. , 4662 (2018), arXiv:1706.05004.[11] C. Heymans et al. , Mon. Not. R. Astron. Soc. , 146(2012).[12] T. Erben et al. , Mon. Not. R. Astron. Soc. , 2545(2013).[13] B. R. Gillis, M. J. Hudson, T. Erben, C. Heymans,H. Hildebrandt, H. Hoekstra, T. D. Kitching, Y. Mel-lier, L. Miller, L. van Waerbeke, C. Bonnett, J. Coupon,L. Fu, S. Hilbert, B. T. P. Rowe, T. Schrabback, E. Sem-boloni, E. van Uitert, and M. Velander, Mon. Not. R.Astron. Soc. , 1439 (2013).[14] M. Velander et al. , Mon. Not. R. Astron. Soc. , 2111(2014), arXiv:1304.4265.[15] M. J. Hudson et al. , Mon. Not. R. Astron. Soc. ,298 (2015).[16] J. T. A. de Jong, G. A. Verdoes Kleijn, K. H. Kuijken,and E. A. Valentijn, Exp. Astron. , 25 (2013).[17] K. Kuijken et al. , Mon. Not. R. Astron. Soc. , 3500 (2015).[18] C. Sifon, M. Cacciato, H. Hoekstra, M. Brouwer, E. vanUitert, M. Viola, I. Baldry, S. Brough, M. J. I. Brown,A. Choi, S. P. Driver, T. Erben, A. Grado, C. Heymans,H. Hildebrandt, B. Joachimi, J. T. A. de Jong, K. Kui-jken, J. McFarland, L. Miller, R. Nakajima, N. Napoli-tano, P. Norberg, A. S. G. Robotham, P. Schneider,and G. V. Kleijn, Mon. Not. R. Astron. Soc. , 3938(2015).[19] M. Viola et al. , Mon. Not. R. Astron. Soc. , 3529(2015).[20] E. van Uitert et al. , Mon. Not. R. Astron. Soc. ,3251 (2016).[21] S. Joudaki, C. Blake, A. Johnson, A. Amon, M. Asgari,A. Choi, T. Erben, K. Glazebrook, J. Harnois-Déraps,C. Heymans, H. Hildebrandt, H. Hoekstra, D. Klaes,K. Kuijken, C. Lidman, A. Mead, L. Miller, D. Parkin-son, G. B. Poole, P. Schneider, M. Viola, and C. Wolf,Mon. Not. R. Astron. Soc. , 4894 (2018).[22] E. S. Sheldon, D. E. Johnston, J. A. Frieman, R. Scran-ton, T. A. McKay, A. J. Connolly, T. Budavári, I. Ze-havi, N. A. Bahcall, J. Brinkmann, and M. Fukugita,Astron. J. , 2544 (2004).[23] R. Mandelbaum, U. Seljak, G. Kauffmann, C. M. Hi-rata, and J. Brinkmann, Mon. Not. R. Astron. Soc. , 715 (2006).[24] A. Leauthaud et al. , Astrophys. J. , 159 (2012).[25] T. M. C. Abbott et al. (Dark Energy SurveyCollaboration), Phys. Rev. D , 043526 (2018),arXiv:1708.01530.[26] A. A. Berlind and D. H. Weinberg, Astrophys. J. ,587 (2002).[27] A. Cooray and R. Sheth, Phys. Rep. , 1 (2002).[28] J. Clampitt et al. , Mon. Not. R. Astron. Soc. , 4204(2017), 1603.05790.[29] J. Prat et al. , Mon. Not. R. Astron. Soc. , 1667(2018).[30] Y. Park et al. , Phys. Rev. D , 063533 (2016).[31] E. Rozo et al. (DES Collaboration), Mon. Not. R. As-tron. Soc. , 1431 (2016), arXiv:1507.05460 [astro- ph.IM].[32] E. S. Rykoff, E. Rozo, M. T. Busha, C. E. Cunha,A. Finoguenov, A. Evrard, J. Hao, B. P. Koester,A. Leauthaud, B. Nord, M. Pierre, R. Reddick,T. Sadibekova, E. S. Sheldon, and R. H. Wechsler,Astrophys. J. , 104 (2014).[33] E. S. Rykoff et al. , Astrophys. J. Suppl. Ser. , 1(2016).[34] B. Hoyle, D. Gruen, et al. (DES Collaboration), Mon.Not. R. Astron. Soc. , 592 (2018), arXiv:1708.01532.[35] M. Gatti, P. Vielzeuf, et al. (DES Collabora-tion), Mon. Not. R. Astron. Soc. , 1664 (2018),arXiv:1709.00992.[36] C. Davis et al. (DES Collaboration), ArXiv e-prints(2017), arXiv:1710.02517.[37] J. Elvin-Poole et al. (Dark Energy Survey Col-laboration), Phys. Rev. D , 042006 (2018),arXiv:1708.01536.[38] M. Bartelmann and P. Schneider, Phys. Rep. , 291(2001).[39] B. Flaugher et al. , Astron. J. , 150 (2015).[40] DES Collaboration, Mon. Not. R. Astron. Soc. ,1270 (2016).[41] A. Drlica-Wagner et al. (DES Collaboration), Astro-phys. J. , 33 (2018), arXiv:1708.01531.[42] E. Huff and R. Mandelbaum, arXiv e-prints (2017),arXiv:1702.02600.[43] E. S. Sheldon and E. M. Huff, Astrophys. J. , 24(2017), arXiv:1702.02601.[44] E. S. Sheldon, Mon. Not. R. Astron. Soc. Lett. ,L25 (2014).[45] J. Zuntz, E. Sheldon, et al. (DES Collaboration), Mon.Not. R. Astron. Soc. (2018), 10.1093/mnras/sty2219,arXiv:1708.01533.[46] J. Zuntz, T. Kacprzak, L. Voigt, M. Hirsch, B. Rowe,and S. Bridle, Mon. Not. R. Astron. Soc. , 1604(2013).[47] M. Jarvis et al. , Mon. Not. R. Astron. Soc. , 2245(2016), arXiv:1507.05603.[48] S. Samuroff et al. (DES Collaboration), Mon. Not. R.Astron. Soc. , 4524 (2018), arXiv:1708.01534.[49] C. Sánchez et al. , Mon. Not. R. Astron. Soc. , 1482(2014), arXiv:1406.4407.[50] C. Bonnett et al. , Phys. Rev. D , 042005 (2016),arXiv:1507.05909.[51] N. Benitez, Astrophys. J. , 571 (2000).[52] R. Cawthon et al. (DES Collaboration), ArXiv e-prints(2017), arXiv:1712.07298.[53] J. DeRose et al. , in prep (2018).[54] P. Coles and B. Jones, Mon. Not. R. Astron. Soc. ,1 (1991).[55] I. Kayo, A. Taruya, and Y. Suto, Astrophys. J. ,22 (2001).[56] O. Lahav and Y. Suto, Living Rev. Relativ. , 8 (2004).[57] S. Hilbert, J. Hartlap, and P. Schneider, Astron. As-trophys. , A85 (2011).[58] H. S. Xavier, F. B. Abdalla, and B. Joachimi, Mon.Not. R. Astron. Soc. , 3693 (2016).[59] M. A. Troxel et al. (Dark Energy Survey Collaboration),Phys. Rev. D , 043528 (2018), arXiv:1708.01538.[60] M. Troxel and M. Ishak, Phys. Rep. , 1 (2015).[61] M. Jarvis, G. Bernstein, and B. Jain, Mon. Not. R.Astron. Soc. , 338 (2004).[62] S. Singh, R. Mandelbaum, U. Seljak, A. Slosar,and J. V. Gonzalez, arXiv:1611.00752 (2016),arXiv:1611.00752.[63] E. Krause, T. F. Eifler, et al. (DES Collaboration),ArXiv e-prints (2017), arXiv:1706.09359.[64] E. Suchyta et al. , Mon. Not. R. Astron. Soc. , 786 (2016).[65] J. Hartlap, P. Simon, and P. Schneider, Astron. Astro-phys. , 399 (2007).[66] E. Bertin, Evans I. N., Accomazzi A., Mink D. J., RotsA. H., eds, Astron. Soc. Pacific Conf. Astron. DataAnal. Softw. Syst. XX. , 435 (2011).[67] K. M. Gorski, E. Hivon, A. J. Banday, B. D. Wandelt,F. K. Hansen, M. Reinecke, and M. Bartelmann, As-trophys. J. , 759 (2005).[68] A. J. Ross et al. , Mon. Not. R. Astron. Soc. , 564(2012).[69] A. J. Ross et al. , Mon. Not. R. Astron. Soc. , 1168(2017).[70] B. Jain and A. Taylor, Phys. Rev. Lett. , 141302(2003).[71] E. Sellentin and A. F. Heavens, Mon. Not. R. Astron.Soc. , L132 (2016).[72] G. Bernstein, Astrophys. J. , 598 (2006).[73] D. Huterer, M. Takada, G. Bernstein, and B. Jain,Mon. Not. R. Astron. Soc. , 101 (2006).[74] G. M. Bernstein, Astrophys. J. , 652 (2009).[75] J. Blazek, R. Mandelbaum, U. Seljak, and R. Naka-jima, J. Cosmol. Astropart. Phys. , 041 (2012).[76] DES Collaboration, Phys. Rev. D , 022001 (2015),arXiv:1507.05552.[77] T. Baldauf, R. E. Smith, U. Seljak, and R. Mandel-baum, Phys. Rev. D , 063531 (2010).[78] M. Crocce et al. , Mon. Not. R. Astron. Soc. , 4301(2016), arXiv:1507.05360.[79] T. Giannantonio et al. , Mon. Not. R. Astron. Soc. ,3213 (2016), arXiv:1507.05551.[80] C. Chang et al. , Mon. Not. R. Astron. Soc. , 3203(2016), arXiv:1601.00405.[81] A. Leauthaud et al. , Mon. Not. R. Astron. Soc. ,3024 (2017). Appendix A: ∆Σ and γ t When we measure the mean tangential alignment ofbackground galaxies around lenses, we need to make achoice as to how we weight each of the lens-source pairs.In this appendix, we discuss the implications of usingeither a uniform weight for all source-lens pairs in agiven combination of source and lens redshift bins, or aweight that takes into account the photometric redshiftestimate of the source to yield a minimum variance es-timate of the surface mass density contrast of the lens.In the first case, and without a shape noise weight-ing of sources, our measurement γ t is simply the arith-metic mean of the tangential components of ellipticitiesof sources i : γ t = N − N (cid:88) i =1 e t,i . (A1)In the second case, we weight each lens-source pairby a weight w e ,i , γ t = (cid:80) Ni =1 w e ,i e t,i (cid:80) Ni =1 w e ,i . (A2)For optimal signal-to-noise ratio and uniform shapenoise of our sample of source galaxies, w e ,i should bechosen to be proportional to the amplitude of the sig-3 . . . . . . z s bin center . . . . . . . . S / N r e l a t i v e t o t r u e z w e i g h t i n g z l = 0 . z l = 0 . z l = 0 . z l = 0 . WeightedUnweighted
FIG. 16. Relative signal-to-noise ratio of lensing signal re-covered when weighting sources uniformly (commonly called γ t , circles) and with " ∆Σ weighting" according to a DES-like photometric redshift point estimate (squares) with σ p = − . z ) + 0 .
12 (1 + z ) scatter around the true redshift[34]. The point estimate is used to select source bins ofwidth ∆ z = 0 . . nal in each lens-source pair, i.e. w e ,i ∝ D l D ls D s . (A3)We note that, for a given cosmology, the mean shearsof both Eq. (A1) and Eq. (A2) can be converted toan estimate of surface mass density ∆Σ , by multiply-ing with the (weighted) estimate of Σ − , as in Eq. (3).In the case of Eq. (A2) with the weights equal to theexpectation value of Eq. (A3), this is identical to thecommon ∆Σ estimator of [22].The unweighted mean of Eq. (A1) has the consider-able advantage that nuisance parameters describing thesystematic uncertainty of shear and redshift estimatesof the source redshift bins are identical to the ones de-termined for a cosmic shear analysis using the samesamples [34, 45, 59]. This is of particular importancewhen joining cosmic shear and galaxy-galaxy lensingmeasurements into one combined probe [25]. The ques-tion at hand therefore is whether the increase in signal-to-noise ratio (S/N) due to the optimal weighting ofEq. (A2) would warrant the added complication.We make a simple estimate of the loss in S/N in-curred by uniform weighting of sources. To this end,we simulate a source sample with overall Gaussian dis-tribution of true redshifts z t with a mean (cid:104) z t (cid:105) = 0 . andwidth σ t = 0 . . We split sources into redshift bins ofwidth ∆ z p = 0 . by a point estimate z p of their red-shift. For a given source redshift bin centered on z m ,we emulate the latter by adding a Gaussian scatter of σ p = − . z )+0 .
12 (1+ z ) to z t , which is a realisticscatter for DES-Y1 photo- z ’s [34].Figure 16 compares the recovered S/N of the galaxy-galaxy lensing signal to that of weighting each source by the optimal weight using its true redshift for two cases:(1) uniform weighting of all sources in a redshift bin (cir-cles) and (2) weighting each source by Eq. (A3) eval-uated at the source redshift point estimate (squares).Except in the case of source redshift bins overlappingthe lens redshift, uniform weighting does not consider-ably lower the S/N of the measured galaxy-galaxy lens-ing signal. The photo- z resolution results in a biggergain when using optimal weighting compared to uniformweighting. For instance, for z l = 0 . and z s = 0 . ,the gain of using photo- z optimal weighting is 6.4% forthe fiducial photo- z scatter while it goes up to ifwe improve the resolution by a factor of two. In a casewith less overlap between the lens and source redshiftdistributions the improvement is reduced, as expected.For example, for z l = 0 . and z s = 0 . , the gain ofusing photo- z optimal weighting is 1.6% for the fiducialphoto- z scatter while it is 2.1% for a photo- z resolu-tion that is twice as good. Therefore, we conclude that,even though optimal weighting can be important, forthe photo- z precision and the source binning used inthis work, photo- z -dependent weighting of sources doesnot significantly improve the constraining power, anddecide to use uniformly weighted tangential shears inthis analysis. Appendix B: Effect of random point subtraction inthe tangential shear measurement
Our estimator of galaxy-galaxy lensing in Eq. (8)includes subtracting the measurement around randompoints that trace the same survey geometry. This mea-surement, using a set of random points with 10 timesas many points as lens galaxies, is shown in Fig. 17.Even though this is a correction included in the mea-surement, it is nonetheless useful to confirm that it issmall at all scales used in the analysis. The measure-ment tests the importance of systematic shear whichis especially problematic at the survey boundary, andallows us to compare the magnitude of the systematicshear with the magnitude of the signal around actuallens galaxies. We find the tangential shear around ran-dom points to be a small correction, consistent withthe null hypothesis, as it is seen in the top left panel ofFig. 18.Even though the random point subtraction is a mildcorrection to the signal, it has an important effect onthe covariance matrix. Subtracting the measurementaround random points removes a term in the covari-ance due to performing the measurement using the over-density field instead of the density field, as it was stud-ied in detail in [62]. As seen in Fig. 18, we observe thiseffect on scales larger than 20 arcmin., where the co-variance is no longer dominated by shape noise. Whensubtracting the measurement around random points, wedetect both a significant decrease on the uncertainty ofthe tangential shear (top right panel) and a reductionof the correlation between angular bins (lower panels).Finally, another argument that strongly favours ap-plying the random points subtraction is the following.In Sec. IV C we validated the jackknife method usinglog-normal simulations, showing that the uncertainties4on the tangential shear are compatible when using thejackknife method and when using the true variance from1200 independent FLASK simulations (Fig. 4). We haveperformed this comparison both with and without therandom point subtraction, finding that there is onlyagreement between the different methods when the tan-gential shear around random points is removed from thesignal.
Appendix C:
Metacalibration responses scaledependence
As explained in Sec. IV A 1, when applying the
Metacalibration responses we approximate them asbeing scale independent. In this appendix we test the validity of this approximation by measuring the scaledependence of the responses for all the tomographiclens-source bin combinations.In Fig. 19 we display the
Metacalibration re-sponses for all the lens-source redshift bins combina-tions averaged in 20 log-spaced angular bins using the
NK TreeCorr correlation function. Comparing to themean of the responses over the ensemble in each sourceredshift bin, we find the variation with θ to be verysmall compared to the size of our measurement uncer-tainties and thus decide to use a constant value for sim-plicity. Future analyses using Metacalibration onlarger data samples with smaller uncertainties may needto include the scale-dependent responses in their mea-surements.5 − − . < z s < . γ t ( θ ) × − . < z r < .
30 0 . < z r < .
45 0 . < z r < .
60 0 . < z r < .
75 0 . < z r < . M ETACALIBRATIONIM SHAPE − − . < z s < . γ t ( θ ) − − . < z s < . γ t ( θ )
10 100 θ [arcmin] − − . < z s < . γ t ( θ )
10 100 θ [arcmin]
10 100 θ [arcmin]
10 100 θ [arcmin]
10 100 θ [arcmin] FIG. 17. Tangential shear around random points for
Metacalibration and im3shape . − − − − − γ t ( θ ) No RP subtractionRP subtraction − − σ ( γ t ) No RP subtractionRP subtraction θ [arcmin] θ [ a rc m i n ] No RP subtraction θ [arcmin] RP subtraction − . − . . . . . . . . FIG. 18. We show the impact the random point subtraction has on the tangential shear measurement and its correspondingjackknife covariance matrix for an example redshift bin ( . < z l < . and . < z s < . for Metacalibration ). . . . . . . . < z s < . R e s p o n s e s ∆ R/R = 0 . . < z l < .
30 ∆
R/R = 0 . . < z l < .
45 ∆
R/R = 0 . . < z l < .
60 ∆
R/R = 0 . . < z l < .
75 ∆
R/R = 0 . . < z l < . R mean R nk R nk . . . . . . . . . . < z s < . R e s p o n s e s ∆ R/R = 0 .
15% ∆
R/R = 0 .
10% ∆
R/R = 0 .
17% ∆
R/R = 0 .
08% ∆
R/R = 0 . . . . . . . . < z s < . R e s p o n s e s ∆ R/R = − .
00% ∆
R/R = 0 .
05% ∆
R/R = 0 .
04% ∆
R/R = 0 .
14% ∆
R/R = 0 . θ [arcmin] . . . . . . . . . < z s < . R e s p o n s e s ∆ R/R = 0 .
16% 10 100 θ [arcmin] ∆ R/R = 0 .
05% 10 100 θ [arcmin] ∆ R/R = 0 .
17% 10 100 θ [arcmin] ∆ R/R = 0 .
16% 10 100 θ [arcmin] ∆ R/R = 0 . FIG. 19.
Metacalibration responses scale dependence and mean values. We compare the responses averaged in 20 log-spaced angular bins between 2.5 and 250 arcmin in each lens-source redshift bin combination ( R nk ) to the average of theresponses in each source redshift bin ( R mean ). The maximum difference between them is at the .2%