Dark Energy Survey Year 3 Results: Multi-Probe Modeling Strategy and Validation

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DES-2020-0554 FERMILAB-PUB-21-240-AE Dark Energy Survey Year 3 Results: Multi-Probe Modeling Strategy and Validation

E. Krause,1, ∗ X. Fang,1 S. Pandey,2 L. F. Secco,2, 3 O. Alves,4, 5, 6 H. Huang,7 J. Blazek,8, 9 J. Prat,10, 3 J. Zuntz,11 T. F. Eifler,1 N. MacCrann,12 J. DeRose,13 M. Crocce,14, 15 A. Porredon,16, 17 B. Jain,2 M. A. Troxel,18 S. Dodelson,19, 20 D. Huterer,4 A. R. Liddle,11, 21, 22 C. D. Leonard,23 A. Amon,24 A. Chen,4 J. Elvin-Poole,16, 17 A. Ferté,25 J. Muir,24 Y. Park,26 S. Samuroff,19 A. Brandao-Souza,27, 6 N. Weaverdyck,4 G. Zacharegkas,3 R. Rosenfeld,28, 6 A. Campos,19 P. Chintalapati,29 A. Choi,16 E. Di Valentino,30 C. Doux,2 K. Herner,29 P. Lemos,31, 32 J. Mena-Fernández,33 Y. Omori,10, 3, 24 M. Paterno,29 M. Rodriguez-Monroy,33 P. Rogozenski,7 R. P. Rollins,30 A. Troja,28, 6 I. Tutusaus,14, 15 R. H. Wechsler,34, 24, 35 T. M. C. Abbott,36 M. Aguena,6 S. Allam,29 F. Andrade-Oliveira,5, 6 J. Annis,29 D. Bacon,37 E. Baxter,38 K. Bechtol,39 G. M. Bernstein,2 D. Brooks,31 E. Buckley-Geer,10, 29 D. L. Burke,24, 35 A. Carnero Rosell,40, 6, 41 M. Carrasco Kind,42, 43 J. Carretero,44 F. J. Castander,14, 15 R. Cawthon,39 C. Chang,10, 3 M. Costanzi,45, 46, 47 L. N. da Costa,6, 48 M. E. S. Pereira,4 J. De Vicente,33 S. Desai,49 H. T. Diehl,29 P. Doel,31 S. Everett,50 A. E. Evrard,51, 4 I. Ferrero,52 B. Flaugher,29 P. Fosalba,14, 15 J. Frieman,29, 3 J. Garc´ıa-Bellido,53 E. Gaztanaga,14, 15 D. W. Gerdes,51, 4 T. Giannantonio,54, 55 D. Gruen,34, 24, 35 R. A. Gruendl,42, 43 J. Gschwend,6, 48 G. Gutierrez,29 W. G. Hartley,56 S. R. Hinton,57 D. L. Hollowood,50 K. Honscheid,16, 17 B. Hoyle,58, 59 E. M. Huff,25 D. J. James,60 K. Kuehn,61, 62 N. Kuropatkin,29 O. Lahav,31 M. Lima,63, 6 M. A. G. Maia,6, 48 J. L. Marshall,64 P. Martini,16, 65, 66 P. Melchior,67 F. Menanteau,42, 43 R. Miquel,68, 44 J. J. Mohr,58, 59 R. Morgan,39 J. Myles,34, 24, 35 A. Palmese,29, 3 F. Paz-Chinchón,42, 54 D. Petravick,42 A. Pieres,6, 48 A. A. Plazas Malagón,67 E. Sanchez,33 V. Scarpine,29 M. Schubnell,4 S. Serrano,14, 15 I. Sevilla-Noarbe,33 M. Smith,69 M. Soares-Santos,4 E. Suchyta,70 G. Tarle,4 D. Thomas,37 C. To,34, 24, 35 T. N. Varga,59, 71 and J. Weller59, 71 (DES Collaboration) 1Department of Astronomy/Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA 2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 3Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA 4Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA 5Instituto de F´ısica Teórica, Universidade Estadual Paulista, SãoPaulo, Brazil 6LaboratórioInterinstitucional de e-Astronomia - LIneA, Rua Gal. JoséCristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 7Department of Physics, University of Arizona, Tucson, AZ 85721, USA 8Department of Physics, Northeastern University, Boston, MA 02115, USA 9Laboratory of Astrophysics, Ecole´ Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland 10Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA 11Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK 12Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK 13Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA 14Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain 15Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain 16Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA 17Department of Physics, The Ohio State University, Columbus, OH 43210, USA 18Department of Physics, Duke University Durham, NC 27708, USA 19Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15312, USA 20NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA 21Instituto de Astrof´ısica e Ciênciasdo Espa¸co,Faculdade de Ciências, Universidade de Lisboa, 1769-016 Lisboa, Portugal 22Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada 23School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, U 24Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA 25Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA 2

26Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 27Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil 28ICTP South American Institute for Fundamental Research Instituto de F´ısica Teórica, Universidade Estadual Paulista, SãoPaulo, Brazil 29Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA 30Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK 31Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 32Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK 33Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain 34Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA 35SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA 36Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile 37Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK 38Institute for Astronomy, University of Hawai’i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA 39Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison, 1150 University Avenue Madison, WI 53706-1390 40Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain 41Universidad de La Laguna, Dpto. Astrof´ısica, E-38206 La Laguna, Tenerife, Spain 42Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA 43Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA 44Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain 45Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy 46INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy 47Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy 48ObservatórioNacional, Rua Gal. JoséCristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 49Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India 50Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA 51Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA 52Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway 53Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain 54Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 55Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 56Department of Astronomy, University of Geneva, ch. d’Ecogia´ 16, CH-1290 Versoix, Switzerland 57School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia 58Faculty of Physics, Ludwig-Maximilians-Universität,Scheinerstr. 1, 81679 Munich, Germany 59Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany 60Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 61Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia 62Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA 63Departamento de F´ısica Matemática, Instituto de F´ısica, Universidade de SãoPaulo, CP 66318, SãoPaulo, SP, 05314-970, Brazil 64George 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 65Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA 66Radcliffe Institute for Advanced Study, Harvard University, Cambridge, MA 02138 67Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA 68InstitucióCatalana de Recerca i Estudis Avan¸cats,E-08010 Barcelona, Spain 69School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK 70Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 71Universitäts-Sternwarte,FakultätfürPhysik, Ludwig-Maximilians UniversitätMünchen,Scheinerstr. 1, 81679 München,Germany (Dated: May 27, 2021) This paper details the modeling pipeline and validates the baseline analysis choices of the DES Year 3 joint analysis of galaxy clustering and weak lensing (a so-called “3×2pt” analysis). These analysis choices include the specific combination of cosmological probes, priors on cosmological and systematics parameters, model parameterizations for systematic effects and related approximations, and angular scales where the model assumptions are validated. We run a large number of simulated 3

likelihood analyses using synthetic data vectors to test the robustness of our baseline analysis. We demonstrate that the DES Year 3 modeling pipeline, including the calibrated scale cuts, is sufficiently accurate relative to the constraining power of the DES Year 3 analyses. Our systematics mitigation strategy accounts for astrophysical systematics, such as galaxy bias, intrinsic alignments, source and lens magnification, baryonic effects, and source clustering, as well as for uncertainties in modeling the matter power spectrum, reduced shear, and estimator effects. We further demonstrate excellent agreement between two independently-developed modeling pipelines, and thus rule out any residual uncertainties due to the numerical implementation.

I. INTRODUCTION and 3 2pt analyses. We systematically define astrophysical× and cosmological modeling choices, and demon- Photometric wide-field imaging surveys, such as the strate the robustness of our adopted baseline parame- Dark Energy Survey (DES1), Hyper Suprime Cam Sub- terization through simulated likelihood analyses. The aru Strategic Project (HSC2), and the Kilo-Degree Sur- resulting model parameterization, analysis choices, and vey (KiDS3) have published exciting results constraining inference pipeline are further validated through analyses the geometry and structure growth of the Universe [1– of detailed DES mock catalogs [13], which are presented 3]. These results are leading up to data sets with ever in DeRose et al. [14]. We note that the DES-Y3 analy- increasing statistical precision that are due to arrive in sis was blinded to minimize experimenter bias, and the the mid 2020s from Rubin Observatory’s Legacy Survey modeling and analysis choices in this paper were final- of Space and Time (LSST4), the Euclid satellite5, and ized before unblinding. We refer to DES Collaboration the Roman Space Telescope6. The statistical precision [10] for a description of the blinding protocol and post- of the data must be matched by the modeling accuracy unblinding analysis choices. on the analysis side, lest cosmological constraints incur In order to identify modeling choices as comprehen- substantial systematic bias. sively as possible, we first summarize the theoretical formalism for calculating angular 3 2pt statistics, avoiding This paper is part of a series describing the method- × ology and results of DES Year 3 (Y3) analysis, which specific model choices to the extent possible, in Sect.II. combines cosmic shear, galaxy-galaxy lensing and galaxy In Sect.III we then motivate modeling choices required to evaluate the general expressions for 3 2pt statistics, clustering measurements (together termed a 3 2pt anal- × ysis) from photometric date covering more than× 4,000 which are summarized in TableI. While each of the sys- square degrees, corresponding to the first 50% of the com- tematics identified in TableI has been studied extensively plete DES data set. Our papers present cosmological con- in isolation before, such principled approaches to identi- straints for cosmic shear [4,5], the combination of galaxy fying relevant systematics will become especially useful clustering [6] and galaxy-galaxy lensing [7], termed 2 2pt with increasing model complexity. We validate the ro- analysis, using two different lens galaxy samples [8,9×], as bustness of the assumed scale dependence and redshift well as the 3 2pt analysis [10]. These cosmological re- dependence of each systematic parameterization through sults are enabled× by extensive methodology developments model stress tests in the form of simulated likelihood at all stages of the analysis; see Appendix A of Ref. [10] analyses (described in Sect.IV). These stress tests are for a summary. detailed in Sect.V and summarized in TableI. Our con- While there is extensive literature characterizing the clusions are presented in Sect.VI. impact of different systematic effects on future surveys in isolation, data analyses require all relevant systematic effects to be identified, modeled and mitigated in com- II. FORMALISM bination. The corresponding analysis choices, which en- compass probes, choice of angular scales analyzed and In this section we derive the theoretical formalism for redshift binning of the measurements/data, choice of computing angular 3 2pt statistics, assuming general parameterizations, and choice of parameter priors, are relativity, a spatially flat× universe, and working to lead- highly survey- and analysis-specific [e.g., 11, 12]. This ing order in the lensing distortion. The astrophysical paper motivates and validates the theoretical modeling model choices required to evaluate these expressions will and analysis choices for the DES-Y3 cosmic shear, 2 2pt be specified in Sect.III, and we specify the galaxy sam- × ples corresponding to the DESY3 analysis in Sect.IV. We use lower-case, italic-type superscripts (i, j) to indicate tomograpic bin indices, and lower-case Greek sub- ∗ [email protected] scripts for vector and shear components on the flat sky 1 www.darkenergysurvey.org/ with respect to the Cartesian coordinate system (e.g., 2 https://hsc.mtk.nao.ac.jp/ssp/ 3 http://www.astro-wise.org/projects/KIDS/ α = 1, 2). Lower-case roman subscripts are used to 4 http://www.lsst.org/lsst denote specific galaxy samples (s for the source galaxy 5 sci.esa.int/euclid/ sample, l for the lens galaxy sample), while the lower- 6 https://roman.gsfc.nasa.gov/ case italic subscript g is used for a generic galaxy sam- 4 ple, g (l, s). When specifying the type of corre- tude of the latter two effects depends on the flux and size lation function∈ or power spectrum, we use upper-case distribution of the galaxy sample, and we introduce the i cursive subscripts to denote a generic tracer field, e.g., proportionality constant Cg to write the magnification , (κ, δm, δg,B,E, IE, IB). For ease of notation, we term as referA B ∈ to angular bins by a representative separation θ, i i i but note that all correlation functions are averaged over δg,µ(nˆ) = Cgκg(nˆ) (3) angular bins [θmin, θmax] (c.f. Eq. 20). where we have introduced the tomographic convergence field A. Field-Level Description Z i i κg(nˆ) = dχ Wκ,g(χ)δm (nˆχ, χ) (4) The DES-Y3 3 2pt analysis is based on the cross- correlation functions× of the observed projected galaxy with δm the 3D matter density contrast, and the tomo- i j graphic lens efficiency density contrast of the lens sample, δl , and shear, γα, in tomographic redshift bins i and j of the lens and source 2 Z ∞ 0 i i 3ΩmH0 0 i 0 χ χ χ galaxy sample respectively. We use ng(z) to denote the Wκ,g(χ) = 2 dχ ng(χ ) −0 . (5) normalized tomographic redshift distribution of galaxy 2c χ a(χ) χ samples.

1. Galaxy density 2. Weak Lensing

The observed projected galaxy density contrast The two components γα of the observed galaxy shapes i δg,obs(nˆ) of galaxies from sample g and tomographic bin are modeled as gravitational shear (G) and intrinsic el- i at position nˆ can be written as lipticity. The latter is split into a spatially-coherent Z contribution from intrinsic galaxy alignments (IA), and i i (3D) stochastic shape noise δg,obs(nˆ) = dχ Wδ,g (χ) δg (nˆχ, χ) 0 | {z } j j j j i γα(nˆ) = γα,G(nˆ) + γα,IA(nˆ) + α,0(nˆ) . (6) δg,D(nˆ) i i + δg,RSD(nˆ) + δg,µ(nˆ) , (1) As shape noise contributes to the covariance but not to the mean two-point correlation function signal, we do i i with χ the comoving distance, and Wδ,g = ng(z) dz/dχ not include it in the mean model predictions described the normalized selection function of galaxies in tomo- here and defer to Amon et al. [4], Friedrich et al. [20] for graphic bin i. Here the first term (δg,D) is the line-of- details on the covariance modeling. sight projection of the three-dimensional galaxy density The gravitational shear can be calculated as (3D) contrast, δg , of sample g, and the remaining terms are the contributions from linear redshift space distortions γ1,G = (Ψ11 Ψ22) /2 , γ2,G = (Ψ12 + Ψ21) /2 , (7) (RSD) and magnification (µ). − In the linear regime, RSD contribute to the projected with Ψ the lensing distortion tensor, which to leading galaxy density contrast through the apparent large-scale order in the lensing deflection can be calculated as flow, with velocity v, of galaxies across the redshift boundaries of the tomographic bins, which can be mod- Z i ˆ i ˆ eled as [15, 16] Ψαβ(n) = 2 dχWκ,s(χ) χ Φ,αβ(nχ, χ) . (8) Z i i h (3D) i Here Φ are spatial transverse derivatives of the 3D po- δg,obs(nˆ) = dχ Wδ,g (χ+nˆ·v(nˆχ, χ)) 1 + δg (nˆχ, χ) − 1 ,α tential Φ. i + δg,µ(nˆ) Again to leading order in the lensing deflection, in Z ∂ nˆ·v(nˆχ, χ) Fourier space the gravitational shear field with respect ≈δi (nˆ) − dχ W i (χ) + δi (nˆ) , g,D δ,g ∂χ a(χ)H(χ) g,µ to the Cartesian coordinate system can be related to the i (2) convergence κs by

i i with a the scale factor, and H(χ) the Hubble rate at γα,G(`) = Tα(`)κs(`) , (9) redshift z(χ). with T(`) (cos(2φ`), sin(2φ`)), where φ` is the angle of The magnification term describes the change in pro- ≡ jected number density due to geometric dilution as well the ` vector from the `x axis. as magnification effects on galaxy flux [17, 18] and size The intrinsic alignment contribution to the observed [19], which modulate the selection function. The magni- galaxy shear field is a projection of the 3D fieldγ ˜IA 5 weighted by the source galaxy redshift distribution with PAB the corresponding three-dimensional power spectrum, which is specified by the model choices de- Z i i tailed in Sect.III. Ref. [21] demonstrate that using the γ (nˆ) = dχ W (χ)˜γα,IA (nˆχ, χ) . (10) α,IA δ,s Limber approximation for galaxy clustering may cause significant systematic biases in the inferred parameters, The 3D intrinsic alignment field,γ ˜IA, is specified by the and show that this approximation is sufficient for galaxy– choice of intrinsic alignment model, c.f. Sect.IIIA. galaxy lensing and cosmic shear beyond the accuracy of The shear field, including the intrinsic alignment con- the DES-Y3 analysis. The expressions for the galaxy tribution, is decomposed into E- and B-modes as clustering power spectrum without the Limber approxi- i i mation become more tractable with the model parame- γE/B(`) = (δαβ, αβ) Tα(`)γβ(`) , (11) terization choices in Sect.IIIA and we summarize their evaluation in Sect.IIIB. with αβ the two-dimensional Levi-Civita tensor. To leading order, the shear E-mode is given by

i i i 2. Angular 2-pt Correlation Functions γE(`) =Tα(`) Tα(`)κs(`) + γα,IA(`) i i =κ (`) + Tα(`)γ (`) . (12) s α,IA The angular two-point correlation functions for galaxy i ij To leading order, weak lensing does not produce B-modes clustering w (θ), galaxy–galaxy lensing γt (θ), and cos- ij and the B-mode shear γB is given by mic shear ξ+/−(θ), are related to the angular power spectra via the transformations i i γB(`) = αβTα(`)γβ,IA(`) . (13) i X 2` + 1 ii w (θ) = P`(cos θ)C (`) , (17) In the following, we set ` = (`, 0) without loss of gener- 4π δl,obsδl,obs ality. ` X 2` + 1 γij(θ) = P 2(cos θ)Cij (`) , (18) t 4π`(` + 1) ` δl,obsE ` X 2` + 1 B. Angular 2-pt Statistics ξij(θ) = [G+ (cos θ) G− (cos θ)] ± 2π`2(` + 1)2 `,2 ± `,2 ` 1. Angular Power Spectra h i Cij (`) Cij (`) , (19) × EE ± BB In order to evaluate the angular (cross-) power spectra 2 where P` and P` are the Legendre polynomials and of the observed density contrast and shear E/B-modes, +/− we first expand the observed fields into their physical the associated Legendre polynomials, G`,m are given by components. Then the shear power spectra can be writ- Eq. (4.19) of [22]. We calculate the correlation functions ten as within an angular bin [θmin, θmax], by carrying out the average over the angular bin, i.e., replacing P`(cos θ), Cij (`) =Cij (`) + Cij (`) + Cji (`) + Cij (`) , P 2(cos θ) and [G+ (cos θ) G− (cos θ)] with their bin- EE κsκs κsIE κsIE IE IE ` `,2 ± `,2 ij ij 2 + − C (`) =C (`) , (14) averaged versions P`, P` and G`,2 G`,2, defined by BB IB IB ± where we used I as short-hand notation for the E/B- R cos θmax E/B dx P`(x) cos θmin mode decomposition of intrinsic alignments, γE/B,IA. P` (θmin, θmax) ≡ cos θmax cos θmin Similarly, we model the observed galaxy-galaxy lensing − cos θmax [P`+1(x) P`−1(x)] power spectrum as = − cos θmin , (20) (2` + 1)(cos θmax cos θmin) Cij (`) = Cij (`)+Cij (`)+Cij (`)+Cij (`) , − δg,obsE δg,Dκs δg,DIE δg,µκ δg,µIE (15) and analogously for P 2 and G+ G− , for which ana- ` `,2 ± `,2 where we omitted the RSD term, which is negligible for lytic expressions can be found in Ref. [20]. the DES-Y3 tomographic lens bin choices [21]. With the exception of the galaxy clustering power spectrum (Cii (`), c.f. Sect.IIIB), we calculate δg,obsδg,obs III. BASELINE MODEL PARAMETERIZATION the angular cross-power spectrum between two fields , using the Limber approximation A B The formalism for calculating angular 2pt-statistics described in Sect.II makes few model assumptions besides Z W i (χ)W j (χ) ` + 0.5 ij A B a spatially-flat cosmology and working to leading order CAB(`) = dχ 2 PAB k = , z(χ) , χ χ in the lensing deflection. In order to evaluate these ex- (16) pressions for angular two-point statistics, we now specify 6

Model Ingredient Baseline Choice Test of k-dependence Test of z-dependence Validation

Pmm gravity-only baryons+AGN feedback baryons+AGN feedback Sect.VA Pmm halofit fitting function higher accuracy emulators higher accuracy emulators Fig.6 i galaxy bias linear bias b1 perturbative bias passive evolution of b1 Sect.VA,VB per tomographic bin, Eq. 21 IA TATT (Eq. 22) - (extrapolated observations) Sect.VB with power-law z-evolution lensing modeling first-order in distortion next-to-leading order - Fig.8

TABLE I. Summary of model choices and their validation tests presented in this analysis.

model prescriptions for the 3D power spectra PAB that Here we ignored stochastic bias contributions to the include nonlinear structure formation and astrophysics. galaxy power spectrum Pδg δg , which at leading order af- This section describes our model choices for the DES-Y3 fect only to the zero-lag correlation function. Further- baseline analyses in the domain of flat νwCDM cosmolo- more, we model the redshift dependence with one free i gies. parameter b1,g per tomographic bin per galaxy sample g, Calculations of background evolution and transfer neglecting evolution within tomographic bins. functions use the Boltzmann codes CAMB [23, 24] and This model choice neglects the known scale dependence CLASS [25], which are in excellent agreement at the level of galaxy clustering due to higher-order biasing, which we of accuracy of this analysis as demonstrated in Sect.IVA. mitigate through scale cuts (c.f. Sect.VA). We further For the model validation presented here, we do not in- show in Sect.VB that neither neglecting the redshift clude the modeling and marginalization of observational evolution of linear bias within tomographic bins nor ne- systematics, which is a conservative choice given that it glecting scale dependence of galaxy bias due to massive requires a more stringent performance of the theoretical neutrinos biases the DES-Y3 analyses. These bias mod- model. eling assumptions are further validated by applying the baseline analysis model to mock catalogs [14]. Intrinsic Alignments As motivated in detail in A. Theoretical Modeling Choices Ref. [5], we adopt the ‘tidal alignment and tidal torquing’ (TATT) model [29] as the baseline intrinsic alignment Matter power spectrum To model the nonlinear mat- model for the DES-Y3 analyses. Here, we provide a brief summary of the model. We write the intrinsic galaxy ter power spectrum Pmm, we adopt the Takahashi et al. [26] recalibration of the halofit fitting formula [27] for shape field, measured at the location of source galaxies, the gravity-only matter power spectrum, including the as an expansion in the density and tidal tensor sab, which Bird et al. [28] prescription for the impact of massive can be decomposed into components sα as with the cos- neutrinos. This model has two well-known deficiencies: mic shear field:

γ˜α,IA = A1sα + A1δδmsα + A2 (s s) + . (22) The matter power spectrum model does not ac- × α ··· • count for non-gravitational forces, such as the im- In this expansion, the first linear term, with A , corre- pact of baryons. 1 sponds to the well-studied “nonlinear linear alignment” model [NLA, 30–32]. The second term captures the im- As any fitting formula, this model has finite ac- pact of source density weighting [33], and together the • curacy in representing the true gravity-only power two comprise the ‘tidal alignment’ component. The third spectrum. term, quadratic in the tidal field, captures the impact of tidal torquing [30, 34]. We mitigate the model incompleteness through scale cuts that exclude the impact of strong baryonic feedback mod- Based on these alignment processes, TATT prescribes els, as detailed in Sect.VA. In order to quantify the the scale dependence, and parts of the redshift evolu- model accuracy we compare halofit against more re- tion, of the intrinsic alignment E/B-mode power spectra cent matter power spectrum emulators, which are based PIE IE ,PIB IB and the cross-power spectrum between mat- on larger and higher resolution simulations and thus more ter density and intrinsic alignment E-mode PmIE . Ex- accurate (c.f. Sect.VB). pressions for these power spectra are given in Ref. [29]. Galaxy Bias For the baseline analysis, we adopt a lin- At fixed redshift, the TATT power spectra depend on ear bias (b1) prescription relative to the nonlinear mat- three amplitude parameters, NLA amplitude A1, tidal ter density, such that the cross power spectrum between torquing amplitude A2, and an amplitude for the density galaxy density and field is given by weighting term, captured by an effective source bias bta, A such that A1δ = btaA1 in Eq. 22. Note that in the limit Pδ A(k, z) = b1,g(z)PmA(k, z) . (21) A2, bta 0, TATT reduces to NLA. All three parameters g → 7 depend on the source sample selection, and may depend the tangential shear signal of an excess mass B, enclosed on redshift. For the DES-Y3 baseline model we adopt within the transverse scale corresponding to the smallest a five parameter model, assuming bta to be constant in scale at which the correlation function is measured, is redshift and choosing to parameterize the redshift evolu- given by tion of A1 and A2 as power laws with exponents η1,2. In 1 Z Bi Biβij detail, the prefactors are given by ∆γij(θ) = dχW i (χ)W j(χ) t δl κ 2 2 2 ρcritΩm dA(χ) θ ≡ θ η1 ρcritΩm 1 + z (26) A1(z) = a1C¯1 (23) − D(z) 1 + z0 with ρcrit the critical density, and dA(χ) the angular- η2 diameter distance, and where we have neglected the po- ρcritΩm 1 + z A (z) =5a C¯ (24) tential redshift evolution of B within the narrow tomo- 2 2 1 D(z)2 1 + z 0 graphic lens bins of the DES-Y3 analysis7. Analytic marginalization over Bi with a Gaussian prior σ(Bi) with pivot redshift z0 corresponding to the mean redshift modifies the data covariance C as of the source sample, C¯1 a normalization constant, which ¯ −14 −2 2 by convention is fixed to C1 = 5 10 M h Mpc , and Nz,l × X D(z) the linear growth factor. This power law redshift C C + σ2 Bi ti ti . (27) dependence is a common parameterization in recent anal- → ⊗ i=1 yses employing only the NLA-IA model as well [2, 35–37, i though the latter do not include redshift evolution of A1 with Nz,l the number of tomographic lens bins, and t a in their baseline analysis]. We discuss the limitations of vector with length corresponding to the number of data these parameterization choices in Sect.VB. The evalua- points and elements tion of perturbation theory kernels required for the TATT  if a-th element is not γ , or if model, as well as similar calculations for nonlinear biasing 0 t (Sect.VA), were performed using the FAST-PT algorithm  lens-redshift of a-th element = i ti = 6 [38, 39]. a  ij −2 Magnification We model the lensing bias coefficient β θa otherwise i Cg introduced in Eq.3 to include the geometric dilution (28) and the modulation of galaxy flux and size selection [17– where the expression for βij is given in Eq. 26. As de- 19], tailed in Ref. [8], the nonlinear galaxy–matter correlation function in the one- to two-halo transition regime con- i i i ∂ ln ng ∂ ln ng tributes substantially to the enclosed excess mass, and Cg = 5 + 2, (25) i ∂m m ,r ∂ ln r m ,r − the prior on B cannot be informed by the typical mass lim lim lim lim 17 scale of the host halos. We adopt σ(B) = 10 M /h. where the logarithmic derivatives are the slope of the luminosity and size distribution at the sample selection limit. The values of these lensing bias coefficients are B. Evaluation of non-Limber Integrals estimated from the data [40], and are held fixed in the cosmology analysis. The robustness of this assumption, Within the baseline model parameterization specified as well as potential redshift evolution effects within to- above, and restricting to νwCDM cosmologies, we can mographic bins, are discussed in Sect.VB and further now express the computation of galaxy clustering power validated on mock catalogs [14]. spectra in relatively compact form. Nonlocal shear As shear is a nonlocal quantity, the We follow Fang et al. [21] for the non-Limber compu- shear two-point (cross-)correlation functions at separa- tation of galaxy clustering power spectra. Adapting their tion θ also depend on the density distribution at scales notation to the conventions in this paper [also see 45], we smaller than θ. For cosmic shear, this is included in val- define idation of the nonlinear matter power spectrum model- Z ing in Sect.VB. For galaxy-galaxy lensing, the one-halo i i ∆g,D(k, `) = dχ Wδ,g(χ)Tδg (k, z(χ))j`(kχ) , (29) term contribution to the tangential shear signal is non- Z negligible at scales far beyond the projected halo size. i i 00 ∆ (k, `) = dχ f(z(χ))W (χ)Tδ(k, z(χ))j (kχ) , This effect can be mitigated by transforming the tangen- g,RSD − δ,g ` tial shear into statistics that remove small-scale informa- (30) tion [41–43], through scale-cuts [1, 11], or by including it Z i `(` + 1) i dχ in the model. In this analysis, we adopt the point-mass ∆ (k, `) = C Wκ,g(χ)Tδ(k, z(χ))j`(kχ) , g,µ k2 g χ2 marginalization scheme of MacCrann et al. [44], which (31) analytically marginalizes the tangential shear contribution of an enclosed mass and only requires priors on the enclosed mass, but no additional fit parameters. Adapting the original expressions [44] to our notation, 7 But see Ref. [8] for a test of this assumption. 8 where f(z) is the logarithmic growth rate, Tδ(k, z) is the where Pgg(k, χ1, χ2) is the unequal time, nonlinear matter density perturbation transfer function, and Tδg galaxy power spectrum , while components involving the 00 00 00 is the transfer function of galaxy density perturbations, RSD have integrands containing j`j` or j` j` . In order which assuming linear galaxy bias is given by Tδg (k, z) to evaluate the unequal time expressions, we separate the ≈ 2 b1,g(z)Tδ(k, z). linear part b1,gPlin(k, χ1, χ2) and nonlinear contribution 2 The angular galaxy clustering power spectrum can [Pgg b1,gPlin](k, χ1, χ2) of the power spectrum. As the then be written as nonlinear− contribution is significant only on small scales Z where the Limber approximation is sufficiently accurate, ij 2 dk 3 i j C (`) = k PΦ(k)∆ (k, `)∆ (k, `) , we can rewrite Eq. (33) as δg,obsδg,obs π k g,obs g,obs (32) i i i where ∆g,obs(k, `) = ∆g,D(k, `) + ∆g,RSD(k, `) + i ∆g,µ(k, `), and PΦ(k) is the primordial matter power spectrum. The expansion of the product of the different ∆A(`)’s leads to integrals containing two Bessel functions and their derivatives. For example, the “DD” term is Z Z ij 2 i j C (`) = dχ1 W (χ1) dχ2 W (χ2) δg,Dδg,D π δ,g δ,g Z dk k3P (k, χ , χ )j (kχ )j (kχ ) , k gg 1 2 ` 1 ` 2 (33)

Z W i (χ)W j (χ) ij δ,g δ,g ` + 0.5 i j ` + 0.5 C (`) = dχ Pgg , χ b b Plin , χ δg,Dδg,D χ2 χ − 1,g 1,g χ 2 Z Z Z dk + dχ bi W i (χ )D(z )) dχ bj W j (χ )D(z ) k3P (k, 0)j (kχ )j (kχ ) , (34) π 1 1,g δ,g 1 1 2 1,g δ,g 2 2 k lin ` 1 ` 2

i ij ij where we have factorized the time dependence of the lin- M(p) w (θ, p), γt (θ, p), ξ± (θ, p) assuming a Gaus- ear power spectrum. We use the generalized FFTLog al- sian likelihood≡ { [see e.g. 46, for tests of} the validity of this gorithm8 developed in [21] to evaluate the full expression assumption in the context of cosmic shear] for Eq. (32) including RSD and magnification terms. 1 ln (D p) (D M(p))τ C−1 (D M(p)) , L | ∝ −2 − − IV. LIKELIHOOD ANALYSIS SETUP (35) where the covariance is computed using the halo model in [47, 48] and we refer to [20] for a detailed The numerical implementation and scientific accuracy CosmoLike description of the covariance validation. For clarity, we of our baseline model are validated through simulated omit the parameter argument of the model two-point likelihood analyses. We briefly summarize the likelihood functions in the following. The parameter priors assumed analysis methodology in this section. in the likelihood analysis and the fiducial parameter values, used to compute the (synthetic) data vector D, are A. Inference and Pipeline Validation summarized in TableII. The ‘data’ is computed using preliminary DES-Y3-like redshift distributions. The source sample consists of four We set up simulated likelihood analyses using syn- tomographic bins with broad redshift support extending thetic ‘data’ vector D wi(θ), γij(θ), ξij(θ) , gener- ≡ { t ± } to z 1.5 [4, 49]. The DES-Y3 considers two different ated at a fiducial cosmology, and the theoretical baseline lens samples∼ model prediction as a function of model parameters p, redMaGiC [50], a luminosity-threshold sample of red • sequence galaxies with constant comoving density, consisting of five tomographic lens bins with red- 8 https://github.com/xfangcosmo/FFTLog-and-beyond shift boundaries [0.15, 0.35], [0.35, 0.5], [0.5, 0.65], 9

Parameter Prior Fiducial ences in the parameter constraints from both codes (c.f. Fig.1 for an illustration in a subspace of the full parame- p Cosmology ter space; here S8 = σ8 Ωm/0.3 is a derived parameter). Ωm [0.1, 0.9] 0.3 Projection effects on marginalized posteriors Even in −9 U 10 As [0.5, 5.0] 2.19 the idealized case of applying the baseline analysis to a U Ωb [0.03, 0.07] 0.048 synthetic, noiseless data vector generated from the same U ns [0.87, 1.06] 0.97 model, the marginalized parameter posteriors may ap- U h [0.55, 0.91] 0.69 pear biased from the input parameter values due to pa- −4 2 U 10 Ων h [6.0, 64.4] 8.3 rameter volume effects (c.f., Fig.2) while the maximum U w [ 2, 0.33] 1.0 a posteriori point (MAP, which is optimized over the full U − − − parameter space) recovers the input parameter values. Intrinsic Alignment These projection effects can occur when parameters of a1 [ 5.0, 5.0] 0.7 interest are not fully constrained by the data or are de- U − a2 [ 5.0, 5.0] 1.36 generate with other parameters that are prior informed. U − − η1 [ 5.0, 5.0] 1.7 We note that – within a specific set of model and prior U − − η2 [ 5.0, 5.0] 2.5 choices – these projection effects decrease as the data’s U − − bta [0.0, 2.0] 1.0 constraining power increases and are thus not a system- U atic bias. The size and direction of such projection ef- redMaGiC Galaxy Bias fects depends on the parameterization and prior choices, 1···5 b1 [0.8, 3.0] 1.7, 1.7, 1.7, 2.0, 2.0 the underlying parameter values, as well as the data’s U constraining power. This complicates the comparison of redMaGiC Lens Magnification marginalized posteriors between different data sets and 1···5 different analysis choices for the same data (e.g., varia- Cl fixed 0.19, 0.63, 0.69, 1.18, 1.88 − − − tions of scale cuts, or model parameterization variations maglim Galaxy Bias for a specific systematic effect). To indicate possible ef- b1···6 [0.8, 3.0] 1.5, 1.8, 1.8, 1.9, 2.3, 2.3 fects of parameter degeneracies on marginalized param- 1 U eter constraints, 1D-marginalized DES Y3 results on a parameter p are often reported as maglim Lens Magnification 1···6 Cl fixed 0.43, 0.30, 1.75, 1.94, 1.56, 2.96 +upper 34% p = mean value−lower 34% (MAP value) . (36) TABLE II. The parameters varied in simulated analyses presented here, their prior ranges (using U to denote an uniform prior) and the fiducial values used for synthetic data. For ΛCDM analyses, w = −1 is fixed. B. External Data

[0.65, 0.8], [0.8, 0.9] [6]. The most constraining results on cosmological parameters will be obtained by combining the DES 3 2pt anal- maglim [51], defined by a magnitude cut in the i- × • band that depends linearly on the photometric red- ysis with external data sets, assuming these experiments are sufficiently consistent for a combined analysis. As shift [estimated using the algorithm of 52] zphot, detailed in Sect.V, we quantify the accuracy of model i < 4zphot + 18, which is split into six tomographic lens bins with redshift boundaries [0.20, 0.40], choices as bias in inferred parameters relative to the ex- [0.40, 0.55], [0.55, 0.70], [0.85, 0.95], [0.95, 1.05] [9]. pected constraining power, which differ for each analysis (specified by data and parameter spaces). Hence we Angular correlation functions are evaluated in 20 log- also need to validate the DES baseline model choices for spaced angular bins over the range [20.5, 2500] and then the more stringent combined analysis. We simulate the restricted by angular scale cuts (Sect.VA). constraining power of the external data at our fiducial We have developed two independent implementations cosmology, so that parameter biases can be attributed of the baseline analysis within CosmoSIS and Cosmo- to model choices without any contribution from resid- Like, which we compare following the procedure from ual tension between the two experiments. For simulated DES Y1 [11]. These codes use different Boltzmann codes joint DES-Y3+Planck analyses, we add the constrain- (CAMB or CLASS, respectively) and differ in terms of ing power of Planck (TTTEEE+lowE) as a Gaussian their structure, interpolation and integration routines. prior, computed from the parameter covariance of Planck After multiple iterations we achieved an agreement of chains [53], to our simulated DES-Y3 analyses. This re- both pipelines at the level of ∆χ2 < 0.2 when comparing quires approximating the Planck posterior as a Gaus- two model vectors generated at the fiducial cosmology. A sian, which in the parameter dimensions relevant for DES simulated likelihood analysis in the full cosmological and analyses is an acceptable approximation, and shifting the systematics parameter space shows no noticeable differ- Planck best-fit value to match the fiducial parameter val- 10

FIG. 1. 1D-marginalized parameter posteriors for select parameters obtained for baseline 3×2pt-redMaGiC analyses with CosmoLike analyzing an input data vector generated by CosmoSIS (black solid) and CosmoSIS analyzing an input data vector generated by CosmoLike (red dashed). The agreement of the mean posterior values demonstrates that the ∆χ2 < 0.2 diﬀerence between model data vectors obtained from the two codes (at the ﬁducial parameter parameter point) is negligible.

≠∫ Marginalized Mean Truth 0.04 ≠∫ Fixed MAP 68% CI Mean Truth 0.02 MAP 68% CI

0.00 8 ± S 0.02 °

0.04 °

0.06 °

0.25 0.30 0.35 0.40 0.25 0.30 0.35 0.40 0.76 0.78 0.80 0.82 0.84 0.86 0.78 0.80 0.82 0.84 0.86 0.5 0.7 ≠m ≠m S8 = æ8(≠m/0.3) S8 = æ8(≠m/0.3) 0.25 0.30 0.35 0.40 0.76 0.78 0.80 0.82 0.84 0.86 0.78 0.80 0.82 0.84 0.86 0.5 0.7 FIG. 2. Marginalized 2D and 1D parameter≠m posteriors from simulatedS8 = æ 38(×≠m2pt/0.3) ΛCDM baseline analysesS8 = æ of8(≠ synthetic,m/0.3) noiseless baseline data vectors. Left: Marginalized (Ωm,S8) posteriors assuming massless neutrinos (blue dashed) and marginalizing over neutrino mass (black solid) using the fiducial value and prior from TableII. The S8 posteriors are shown with respect to its input value in each analysis, which differ due to the difference in input neutrino mass, while the input matter density Ωm = Ωcdm + Ωb + Ων is the same for both analyses (c.f.II). Ω ν is poorly constrained by the 3×2pt analysis and the shift in 2D contours indicates a projection effect from marginalizing over an under-constrained parameter that is correlated with the parameters shown. Center/Right: 1D marginalized posteriors (black curve) in Ωm/S8 for the simulated 3×2pt νΛCDM baseline analysis (black solid line in the left figure). The vertical black/red lines show the input value and MAP estimate, the blue shaded bands indicate the 1D marginalized, symmetric 68% uncertainty regions. The 1D marginalized mean and posterior distribution are biased from the input parameter value due to projection effect by nearly 1σ, while MAP closely recovers the input parameters. ues from TableII. tions, which applies to all modeled systematics. We stress test the baseline parameterization to show its robustness. V. MODEL VALIDATION In practice, we carry out simulated (cosmic shear, We scrutinize the model choices described in Sect.IIIA 2 2pt, 3 2pt, 3 2pt+Planck) analyses in ΛCDM and × × × closely following the DES-Y1 model validation procedure wCDM on contaminated data vectors and quantify the [11], considering two different categories: 2D parameter bias in (Ωm,S8) for ΛCDM and in (Ωm, w) for wCDM. We test the impact of different unmodeled Known, but unmodeled systematic effects, e.g., the systematics in combination by using contaminations at • impact of baryons on the matter power spectrum, the upper limit of credible severity for each effect. For nonlinear galaxy bias. These systematics are miti- the second category, we carry out the parameterization gated through scale cuts that exclude the affected stress test for each systematic effect individually using data points from the analysis. alternative parameterizations as input. Systematics modeled with imperfect parameteriza- To ensure that the total potential systematic bias is • 11 well below 1σ statistical uncertainty, we require the 2D alternative to scale cuts through analytic marginaliza- parameter biases to be smaller than 0.3σ2D; additionally tion of theoretical model uncertainties. However, this ap- we require a residual ∆χ2 < 1 (after fitting the baseline proach requires a model for the distribution of the model model to the contaminated data) for each of these tests in error, an assumption that is much more difficult to val- order to not bias the goodness of fit. If a model variation idate than the choice of one pessimistic contamination changes the data vector by ∆χ2 < 0.2 without refitting, realization for scale cuts. It was recently demonstrated we consider this model variation to be insignificant as the [56] that the model error covariance for redshift-space change in ∆χ2 is less than the residual between the two galaxy power spectrum analyses can be obtained from analysis codes. If a model variation changes the data vec- large mock catalogs. However, an application to bary- tor by ∆χ2 < 1 without refitting, the ∆χ2 threshold after onic feedback effects requires additional research, such refitting is automatically met; hence we use importance as how to fairly sample the space of possible feedback sampling (IS) of the baseline chain to estimate parame- models9. ter biases, which is computationally more efficient than carrying out an independent analysis. We emphasize that all model validation tests in this 1. Baryon Impact Modeling paper were carried out for both lens samples in paral- lel, and both lens samples pass these validation criteria. We bracket the impact of baryonic physics on the For clarity of presentation, we primarily describe quan- DES data vector using measurements of the matter titative results for the redMaGiC sample. Corresponding power spectrum from hydrodynamic simulations with figures and numbers for the maglim sample can be found large AGN feedback. As in the DES-Y1 analysis [1, 11], in Ref. [9]. we use the AGN scenario from the OWLS simulation When small differences in likelihood correspond to a suite [58, 59], which adopts the AGN subgrid physics un- large region in parameter space, e.g., for insufficient or der the prescription of Ref. [60]; black holes inject 1.5% degenerate models, we found MAP parameter estimates of the rest mass energy of the accreted gas into the sur- to be noisy even with repeated numerical optimization rounding matter in the form of heat. using the Nelder-Mead method [54] starting from the We consider an adequate upper limit for best-fit parameter values obtained from likelihood chain. OWLS-AGN the impact of baryonic physics. Huang et al. [61] ranked Hence we determine parameter biases from marginal- various hydrodynamical scenarios based on the amount of ized 2D-parameter posteriors. To account for projec- suppression toward small scales against the dark-matter- tion effects, we evaluate parameter biases relative to only theoretical prediction in the cosmic shear statistics the marginalized 2D-parameter posteriors of an other- (see their Figs. 15 and 16). While more extreme scenar- wise identical analysis of the uncontaminated data vec- ios, most notably the simulation [62], exist, tor, which is subject to the same projection effects (cf. Illustris Il- is known be too extreme in its radio-mode AGN Fig4). While the size and direction of parameter pro- lustris feedback, and underpredicts the amount of baryons in jection effects can be altered by large changes in the in- galaxy groups compared with observations (see Fig.1 of put parameters, we verified their stability with respect Haider et al. 63 or Fig.10 of Genel et al. 64). The succes- to small fluctuations by analyzing 100 noisy data vector sor simulation [e.g., see 65, 66] modifies realizations with noise drawn from the data covariance. IllustrisTNG the original Illustris feedback prescription, resulting in much weaker suppression effect in the summary statistics of power spectrum (see Fig.1 of Ref. [61]). The baseline A. Scale Cuts BAHAMAS simulation [67] is calibrated with observations of the present-day stellar mass function and hot For the DES-Y3 baseline model, the largest unmodeled gas fractions in galaxy groups to ensure the overall mat- systematics are the impact of baryonic feedback on the ter distribution are broadly correct under the effect of matter power spectrum and nonlinear galaxy bias. As baryons. It improves the subgrid models of the OWLS- both of these effects will affect the 3 2pt data, they need AGN simulation, which has overly efficient stellar feed- × 11 to be mitigated in combination. However, in practice back and underpredicts the abundance of < 10 M cosmic shear scale cuts are driven by baryonic feedback galaxies at present day [68]. When performing valida- modifying the matter power spectrum, while nonlinear tion, we thus select the scenario of OWLS-AGN which galaxy biasing is the dominant contamination to galaxy is more extreme than the BAHAMAS, but below the clustering and galaxy-galaxy lensing. suppression level of Illustris10. We optimize scale cuts through a series of simulated analyses of a synthetic data vector contaminated by baryonic feedback effects and nonlinear galaxy bias, using different scale cut proposals. If the contamination model 9 A first exploratory work to incorporate baryonic effects using a assumes a realistic upper boundary for the effect, the theoretical error covariance without a rigorous statistical justifi- analysis will yield unbiased parameter values. The the- cation was recently performed [57] with promising results. oretical model error covariance [55] provides an elegant 10 The high AGN feedback version of BAHAMAS (tagged as BA- 12

We refer to [61, 69] for numerical details on the com- 1.5 Lazeyras et al 2016 putation of contaminated DES model data vectors. Fiducial 1.0

0.5 3 2 4 2

2. Nonlinear Galaxy Bias Modeling b b 0.0

We model the contribution of nonlinear galaxy biasing 0.5 to galaxy clustering and galaxy-galaxy lensing using an ° 1.2 1.4 1.6 1.8 2.0 2.2 1.2 1.4 1.6 1.8 2.0 2.2 effective 1-loop model with renormalized nonlinear bias 3 4 b1 b1 parameters [70–73], b2 (local quadratic bias), bs2 (tidal quadratic bias) and b3nl (third-order nonlocal bias): FIG. 3. Relation between linear bias b1 and local quadratic 1 1 bias b2 for halos [75, dashed line] and galaxies (blue crosses), P (k) = b P (k) + b P (k) + b 2 P 2 (k) gm 1 mm 2 2 b1b2 2 s b1s obtained by averaging the halo relation over HODs con- 1 strained by the redMaGiC abundance and linear bias values + b3nlPb1b3nl (k) , (37) (blue crosses), for tomographic lens bins 3 (left) and 4 (right). 2 The red squares denote the values used to compute the non- 2 2 2 Pgg(k) = b1Pmm(k) + b1b2Pb1b2 (k) + b1bs Pb1s (k) linear bias contamination in Sect.VA. 1 + b b P (k) + b2P (k) 1 3nl b1b3nl 4 2 b2b2 and galaxy-galaxy lensing, we vary the minimum comov- 1 1 2 2 2 2 2 + b2bs Pb2s (k) + bsPs s (k) , (38) ing transverse scale Rmin included in the analysis, corre- 2 4 i i sponding to an angular scale cut θmin = Rmin,w/γt /χ(z ) where we have omitted the subscript denoting the depen- for each tomographic lens bin i. 2 dence of bias parameters on the galaxy sample (e.g., b2,g) For cosmic shear, we choose a threshold ∆χthr value for clarity. Expressions for the power spectrum kernels for the difference between contaminated and baseline cos- Pb b , etc., are given in Saito et al. [73]. mic shear data vectors, and then determine scale cuts 1 2 ij This model was found to describe the clustering of θmin,± for each pair of source bins such that the baryonic redMaGiC-like galaxies in mock catalog at DES-Y3 ac- 2 ij 2 contamination results in ∆χ (ξ± ) < ∆χthr/20 for each curacy down to 4 Mpc/h [74]. As validated in [74], we of the 20 shear 2pt functions (10 different tomographic fix the bias parameters bs2 and b3nl to their co-evolution combinations each for ξ+ and ξ−). value of bs2 = 4(b1 1)/7 and b3nl = b1 1 [73]. For While scale cuts need to be tested for each analysis − − − b2, we use values interpolated from the b2 b1 relation setup (choice of probes, model space, and parameters − measured from mock catalogs of redMaGiC-like galaxies of interest), we require the same scale cuts for cosmic [74], b2(b1 = 1.7) = 0.23, b2(b1 = 2.0) = 0.5. For com- shear, 2 2pt, 3 2pt and 3 2pt+Planck in ΛCDM and parison, b2,halo(b1,halo = 1.7) = 0.51, b2,halo(b1,halo = wCDM× analyses× in order to× enable model comparisons − 2.0) = 0.09 using the fitting function of Lazeyras et al. based on the same data points. We find that scale cuts − [75] for halos. To motivate our choice for b2, we calcu- corresponding to the parameters late the impact of broad halo occupation distributions 2 (HOD) by averaging b2,halo(b1,halo) over HODs that are Rmin,w,Rmin,γt , ∆χthr = 8 Mpc/h, 6 Mpc/h, 0.5 only constrained by the redMaGiC abundance and linear (39) bias values (c.f. discussion of galaxy bias in Sect.V for meet our parameter bias requirements for all these anal- details on these HODs). Figure3 shows that the b2 values yses. Figure4 illustrates the residual ΛCDM parameter measured in mocks are consistent with those expected for biases in cosmic shear, 2 2pt and 3 2pt. The corre- galaxy samples with a broad halo mass distribution. sponding 2D parameter bias× for 3 2pt+×Planck (ΛCDM) × is 0.02σ2D; for wCDM we found parameter biases of 0.1σ2D, 0.16σ2D and 0.05σ2D for 2 2pt, 3 2pt, and 3. Scale Cut Analyses 3 2pt+Planck respectively. × × ×At these fiducial scale cuts, even an extreme contam- We generate data vectors with the baryonic feed- ination such as the Illustris feedback model only results back and nonlinear bias contamination models described in a 0.23σ2D and 0.50σ2D for shear and 3 2pt (ΛCDM), × above, and run simulated likelihood analyses for a fam- respectively, which demonstrates the robustness of our ily of angular scale cut proposals: for galaxy clustering analysis choices. In addition to the fiducial scale cuts described above, Amon et al. [4], Secco et al. [5] introduce scale cuts for an “optimized” cosmic shear analysis, which maxi- HAMAS T8.0 in Huang et al. 61) exhibits feedback strengths mizes the constraining power of cosmic shear, such that between the default BAHAMAS and Illustris; the amount of σ2D,3×2pt = 0.3 in ΛCDM. For the cosmic shear analy- suppression in cosmic shear is similar to OWLS-AGN. sis, these optimized scale cuts result in σ2D,1x2pt = 0.1. 13

0.88 0.88 0.88 1 2pt Peak Contaminated 2 2pt 2 2pt × £ £ 0.87 ΛCDM Peak Baseline 0.87 §CDM 0.87 §CDM redMaGiC maglim 0.3σ Baseline 0.86 0.86 0.3σ Contaminated0.86

0.85 0.85 0.02 0.85 σ 0.27æ 0.17æ 8

8 0.02 æ æ S S S 0.84 0.84 0.84

0.07 σ æ σ 0.14 0.29 0.83 0.83 0.83 0.19 0.24 æ 0.82 0.82 0.82

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0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.29 0.30 0.31 0.32 0.33 0.34 0.29 0.30 0.31 0.32 0.33 0.34 Ωm ≠m ≠m

FIG. 4. Parameter biases for final scale cuts in ΛCDM: the red and blue ellipses show 0.3 σ contours for the 2D marginalized constraints of the baseline and contaminated data vector, respectively. Due to parameter volume effects, the marginalized constraints from the baseline analysis are not centered on the input cosmology, and the parameter bias due to unaccounted systematics is given by the offset between the red and blue contours. Left: Simulated cosmic shear analyses. Center: Simulated 2×2pt-redMaGiC analyses. Right: Simulated 2×2pt-maglim analyses. The dashed horizontal and vertical lines indicate the input parameter values.

0.7 0.88 0.88 − 3 2pt 3 2pt 3 2pt £ £ × 0.87 §CDM 0.87 §CDM 0.8 wCDM redMaGiC maglim − maglim 0.86 0.86 0.9 − 0.04 0.85 0.85 1.0 σ

− 0.23 8

0.34 8 0.28 0.06 σ w

S æ 0.84 æ S 0.84 0.06

æ 0.4 σ æ 1.1 0.12 0.09æ − 0.83 æ 0.83 1.2 − 0.82 0.82 1.3 0.81 0.81 −

1.4 0.28 0.29 0.30 0.31 0.32 0.33 0.340.28 0.29 0.30 0.31 0.32 0.33 0.34 − 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 ≠m ≠m Ωm

FIG. 5. Same as Fig.4 for 3 ×2pt-analyses in ΛCDM and wCDM analyses. Left: Simulated ΛCDM 3×2pt-redMaGiC analyses. Center: Simulated ΛCDM 3×2pt-maglim analyses. Right: Simulated wCDM 3×2pt-maglim analyses.

The corresponding scale cuts were obtained by iteratively thus provide higher model accuracy. These more accurate removing the data point from the tomographic cosmic emulators are designed for a limited range in cosmological shear data vector with the largest contribution to ∆χ2, parameter space, which does not cover the expected 2σ until the baryonic contamination results in a residual parameter uncertainty regions of DES-Y3 cosmic shear ∆χ2 = 1. ΛCDM constraints from a 3 2pt analysis and 2 2pt analyses. Hence current emulators are well based on these optimized cosmic shear scale× cuts are pre- suited× for validation at selected cosmologies, but are not sented in Ref. [10]. We note that this scale cut choice a practicable alternative for the baseline model. does not pass the validation criteria in wCDM and that For completeness, we also compare to the HMCode it was introduced after unblinding of the cosmology refitting function [78]. Figure6 shows the result of an- sults. alyzing simulated data vectors computed from these al- ternate prescriptions for the nonlinear gravity-only power spectrum with halofit. We find insignificant parameter B. Model Parameterization Stress Tests biases relative to the two emulator models (< 0.15σ2D). We note that the comparison of HMCode and halofit 1. Matter Power Spectrum fails our requirements for the 2 2pt analysis. The Mead et al. [78] version of HMCode employed× here is known to In order to bracket potential parameter biases due to overpredict the power spectrum in the quasilinear regime the accuracy limitations of halofit for the nonlinear (k 0.1 ... 0.5h/Mpc, [c.f. Fig. 1 of 12]); the DES 2 2pt gravity-only power spectrum, we compare to the Cos- analysis∼ is particularly sensitive to these scale due× to mic Emu [76] and Euclid Emulator [77] emulators, the clustering scale cut Rmin,w = 8Mpc/h. While these which are based on larger, more recent simulations and comparisons indicate that halofit is sufficient for the 14

DES-Y3 baseline analysis, the accuracy requirements of to test whether TATT is sufficient to model the scale- future weak lensing surveys on matter power spectrum dependence of IA in the context of this analysis. Fur- modeling [e.g., 79] may require a new generation of mod- thermore, we do not test robustness to fully nonlinear els/emulators with sufficient support in parameters. IA contributions, noting that [85] found the 1-halo IA contamination unlikely to significantly bias current generation lensing studies. 2. Galaxy Bias The redshift and galaxy luminosity dependence of NLA have been characterized (for elliptical galaxies) over lim- The scale cuts derived in Sect.VA are designed to en- ited ranges in luminosity and redshift [86, 87]. Assuming able a linear galaxy bias baseline model. It remains, then, that only red galaxies are subject to linear alignments, to validate the assumption that these linear bias parame- one can predict the mean NLA amplitude of the DES ters can be modeled as constant within each tomographic source sample as a function of redshift through extrapo- bin. Two independent mock catalogs [13, 80] display lim- lation (in luminosity and redshift) of the observed NLA ited redshift evolution of the HOD of redMaGiC galaxies scalings down to the limiting magnitude over the redshift within the relatively narrow (∆z = 0.15) tomographic range [88]. This estimate further requires extrapolation bins. Hence we model the potential redshift evolution of of an observed luminosity function for all and red galax- galaxy bias within tomographic bins based on the redshift ies in order to model the red fraction of galaxies [88]. evolution of the halo mass function and halo mass–halo The redshift evolution predicted from extrapolation of bias relation assuming a constant HOD. For each tomo- the NLA amplitude of the MegaZ-Luminous Red Galaxy graphic bin, we generate samples of possible HODs using sample and the DEEP2 luminosity functions [89] is not the parameterization from Ref. [81] for luminosity thresh- monotonic and not well described by a power law. This old samples, with an additional parameter to account for prescription was used in DES Y1 [11] to demonstrate that the incompleteness of central galaxies in a color-selected a power law parameterization for the NLA redshift evolu- sample, constrained by the fiducial linear bias values and tion was sufficient for the DES-Y1 analysis, and predicted observed number density [see8, for details]. Figure7 an NLA amplitude for the DES-Y1 source sample A1(z0) shows that this model for bias redshift evolution results consistent with the constraints obtained in Ref. [1]. in ∆χ2 < 0.2. Hence we expect the redshift evolution of galaxy bias to have negligible impact on parameter con- For DES-Y3, this model for the redshift dependence straints, even if one allowed for redshift evolution of the of A1 results in parameter shifts up to 0.5σ2D for cos- HOD within ∆z = 0.15 tomographic bins. mic shear alone in the context of the TATT model with fiducial a , η , b value. It does not result in significant We also tested that the scale dependence of galaxy 2 2 ta biases for 2 2pt or 3 2pt analyses and passes our re- bias imparted by massive neutrinos [82] has no significant quirements,× except for× the cosmic shear analysis. Hence impact on inferred parameters. we validate the robustness of this parameterization a posteriori on data. This a posteriori validation procedure, as well as several additional IA model robustness tests for 3. Intrinsic Alignments cosmic shear analyses, are presented in Secco, Samuroff et al. [5]. We note that the realism of this model stress The TATT model adopted as the baseline intrinsic test is questionable as it is based on several extrapola- alignment parameterization includes two next-to-leading tions (z and L scaling of observed A1; z, L depen- order effects (density weighting, tidal torquing) which dence of− the luminosity− function for all and red galaxies). are theoretically well-motivated but observationally less Due to the absence of a sharp test or other well-motivated well studied. Samuroff et al. [83] found an indication parameterizations, we proceed with the baseline power- for nonzero values of the associated parameters (bta, a2) law redshift evolution parameterization and a posteriori at around the 1σ level in cosmic shear alone, with the validation on data. less constraining DES-Y1 data. We thus include these parameters in the fiducial setup, which was frozen before unblinding of the DES-Y3 analysis, noting that their redshift evolution has not yet been constrained. Also before unblinding, it was agreed to characterize their redshift 4. Magnification evolution in post-unblinding work [5] if the DES-Y3 analysis found these types of intrinsic alignment effects to be significant. Elvin-Poole et al. [40] demonstrate that DES-Y3 cos- While perturbative IA models that are more complete mology constraints are robust to biases in the estimated i i and extend to higher order than TATT exist [84], there values for Cl , including the extreme scenario of Cl = 0, are no observational constraints on the additional param- i.e., ignoring lens magnification in the analysis. Hence we eters. Due to the lack of realistic amplitudes for higher- leave tests of the redshift evolution of these coefficients to order IA contaminations, we do not use a strategy analo- future analyses, for which lens magnification is forecasted gous to the perturbative bias contamination in Sect.VA to become a significant systematic [90, 91]. 15

Cosmic shear 2x2pt 3x2pt 0.86

Peak Baseline 0.85 Peak CosmicEmu 0.84 Peak EuclidEmu

8 Peak HMCode S 0.83 0.3σ Baseline 0.3σ CosmicEmu 0.82 0.3σ EuclidEmu 0.3σ HMCode 0.81 0.28 0.30 0.32 0.34 0.28 0.30 0.32 0.34 0.28 0.30 0.32 0.34 Ωm Ωm Ωm

FIG. 6. Robustness of parameter constraints to the choice of matter power spectrum model. The black contour shows the 0.3σ2D baseline analysis results (analyzing halofit input with halofit model). The blue/red/green contours illustrate the systematic bias, estimated through importance sampling, when input data from Cosmic Emu/Euclid Emulator/HMCode is analyzed with halofit. All contours levels are 0.3σ2D, and symbols indicate the 2D (marginalized) peak.

angle, respectively), and this mode coupling gives rise 25 w(θ) + γt(θ) to small amounts of shear B-modes. As the resulting B-mode power spectra are negligible even for the next 20 generation of weak lensing surveys, we focus here on the (larger) leading E-mode corrections.

15 Reduced Shear As the intrinsic size of source galaxies is unknown, galaxy-shape distortions measure the re- PDF duced shear, g = γ/(1 + κ) rather than γ. The leading 10 correction is

i 5 ˆ i γα,obs(n) i g (nˆ) = γα,obs(nˆ) + γα,obs(nˆ)κ (nˆ) . α,obs 1 κi(nˆ) ≈ − 0 (40) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 ∆χ2 The corresponding correction to the E-mode power spectrum is [92, 93]

Z dχ FIG. 7. Impact of galaxy bias redshift evolution on DES ∆Cij (`) = W i (χ)W j (χ) W i (χ) + W j (χ) 2pt statistics: We compute ∆χ2 between models with galaxy EE χ4 κ,s κ,s κ,s κ,s bias constant within each tomographic bins, and models with Z d2L L ` L ` evolving galaxy bias based on the redshift evolution of the 2 cos(2φL)Bm , − , − , χ halo mass function and halo mass–bias relation, assuming a × (2π) χ χ χ constant HOD within each tomographic bin. The histogram Z d2L 2 χ + cos(2φL) [cos(2φL) + cos(2φL−`)] shows the ∆ distribution when sampling HOD parameters (2π)2 that are constrained only by the ﬁducial linear bias value and observed galaxy number density. Cij (L) Cij ( L ` ) × κκ κκ | − | Z dL + Cii (`) LCjj (L) + (i j) , κκ 2π κκ ↔ 5. Higher-Order Lensing Eﬀects (41)

The calculations of angular statistics described in with Bm(k1, k2, k3, χ) the matter bispectrum at redshift Sect.II included only leading-order lensing effects. The z(χ), and where we have omitted the reduced shear ef- most significant next-to-leading-order contributions to fect on γIA. While the three terms in Eq. (41) are of the observed two-point statistics are reduced shear [92, the same order in the linear matter density contrast, the 93] and source clustering and magnification [94, 95], while last two terms containing products of two angular con- distortions from multiple deflections can safely be ignored vergence power spectra are strongly suppressed as they [96, 97]. We review these effects below, and derive ex- include an additional lens efficiency factor. The B-mode pressions for the impact of source clustering applicable power spectrum from reduced shear is given by convolu- to broad tomographic source bins. tion of two angular convergence power spectra with dif- Each of these effects amounts to weighting the shear ferent phase factor (there is no bispectrum term as the by another field (convergence, galaxy density, deflection first-order B-mode shear is zero); as it is known to be 16 insignificant [95, 97], we will not consider it further here. the modified redshift distribution, For galaxy-galaxy lensing, Z 2 Z " d L i i Z dχ 2 ∆Ψαβ(`) = 2 dχ ns(χ) b1,sδm((` L)/χ, χ) ∆Cij (`) = bi W i (χ) + CiW i (χ) W j (χ) (2π)2 − δlE χ4 1,l δ,l l κ,l κ,s χ # Z 2 Z d L L ` L ` +Ci dχ00W (χ00, z(χ))δ ((` L)/χ00, χ00) cos(2φL)Bm , − , − , χ . s κ m × (2π)2 χ χ χ 0 − (42) Z χ 0 0 LαLβ 0 0 dχ Wκ(χ , z(χ)) 2 δm (L/χ , χ ) ,(45) × 0 L where we omitted the δ3 term as the corresponding O m δ4 power spectrum correction vanishes after carrying O m Source Clustering and Magnification Since the shear out the φL integral. Projection into shear components field is only observed at the positions of source galaxies, (Eqs.7+11) yields we now account for deviations from the mean source red- Z 2 Z " shift distribution along specific lines of sight due to the i d L i i ∆γE(`) = dχ ns(χ) b1,sδm((` L)/χ, χ) physical (3D) clustering of sources and modulation of the (2π)2 − selection function by magnification: Z χ # i 00 00 00 00 h i +Cs dχ Wκ(χ , z(χ))δm((` L)/χ , χ ) i i (3D) i 0 − ns(χ) ns(χ) 1 + δs (nˆχ, χ) 1 + Csκ (nˆ, z) . → Z χ (43) 0 0 0 0 Tα(`)Tα(L) dχ Wκ(χ , z(χ))δm (L/χ , χ ) . × | {z } 0 Here we have introduced the convergence field of a source cos(2φL) plane at redshift z (46) The correction ∆γB is of the same form as Eq. 46, with 3Ω H2 Z z(χ) χ χ(z) χ ˆ m 0 ˆ Tα(l)Tα(L) αβTα(l)Tβ(L) = sin(2φL). κ (n, z) = 2 dχ − δm(nχ, χ) → 2c 0 a(χ) χ(z) The resulting power spectrum corrections are then Z χ(z) straightforward to calculate. While the source cluster- = dχ Wκ(χ, z) δm(nˆχ, χ) , (44) ing fluctuations in Eq. 43 are orders of magnitudes larger 0 than the magnification fluctuations, several of the corresponding power spectrum correction terms are sup- where in the second step we defined Wκ(χ, z), the lens efficiency of a source plane at redshift z. pressed by the lens efficiency being zero at the source plane (Wκ(χ(z), z) = 0). There are three different (22)- type corrections to CEE/BB permitted under the Limber approximation. As for reduced shear, these terms are much smaller than the bispectrum terms, and we show The corresponding corrections to the shear field are here only the dominant term, proportional to the source computed starting from the deflection tensor (Eq.8) with clustering power spectrum,

Z Z χ 0 Z 2 ij i i dχ j 0 2 0 d L L ` L ` 0 ∆CEE(`) = Cs dχ ns(χ) 04 Wκ,s(χ )Wκ (χ , z(χ)) 2 cos(2φL)Bm 0 , −0 , −0 , χ + (i j) 0 χ (2π) χ χ χ ↔ Z i j i j Z 2 Z χ 2 0 b1,sb1,sns(χ)ns (χ) d L 2 ` L 0 Wκ (χ , z(χ)) L 0 + dχ 2 2 cos (2φL)Pm | − |, χ dχ 02 Pm 0 , χ + χ (2π) χ 0 χ χ ··· (47) Z i j i j Z 2 Z χ 2 0 ij b1,sb1,sns(χ)ns (χ) d L 2 ` L 0 Wκ (χ , z(χ)) L 0 ∆CBB(`) = dχ 2 2 sin (2φL)Pm | − |, χ dχ 02 Pm 0 , χ + (48) χ (2π) χ 0 χ χ ··· Z Z χ dχ0 Z d2L L l L l ∆Cij (`) =Cj dχ nj(χ) bi W i (χ0) + CiW i (χ0) W 2 (χ0, z(χ)) cos(2φ )B , , , χ0 . δlE s s 04 1,l δ,l l κ,l κ 2 L m 0 −0 −0 0 χ (2π) χ χ χ (49)

One can read oﬀ two useful observations about source clustering from these expressions: ﬁrst, under the Lim- 17

1···4 ber approximation there notably are no bispectrum-type tion coefficients Cs = ( 1.17, 0.64, 0.55, 0.80), es- corrections from physical source (or lens-source) cluster- timated from data [40], and− modeling− the− matter bispec- (3D) ing, which would originate from the δs term in Eq. 43, trum as tree-level but replacing the linear matter power but are suppressed by the source plane lens efficiency. with the nonlinear matter power spectrum [see 103, for This is in contrast to the projected source clustering a detailed analysis of bispectum models]. The combined impact of higher-order lensing effects on parameter esti- ansatz γi(nˆ) γi(nˆ) 1 + δi (nˆ [95]. For broad to- → s,obs mates is shown in Fig.8 for DES-Y3 ΛCDM analyses; mographic source redshift bins, as applicable to DES-Y3, the two-dimensional parameter bias is below 0.15σ for i i 2D their approach gives rise to a spurious γ δs,D term, which all analyses. would cause substantial parameter biases for the DES-Y3 We note that impact of reduced shear and source clus- analysis. In the limit of narrow tomographic bins [consid- tering on cosmic shear and galaxy-galaxy lensing is sensi- ered by 95], both approaches yield the same power spec- tive to squeezed-limit bispectrum configurations and thus trum corrections. Second, δ4 lens-source clustering O m the nonlinear bispectrum model. Accurate modeling of corrections to the galaxy-galaxy lensing power spectrum these effects will be required for weak lensing analyses of (Eq. 49) vanish under the Limber approximation, which future surveys. is relevant to the discussion of correlation function estimator effects11. The DES-Y3 analysis adopts pair-based estimators VI. SUMMARY AND CONCLUSIONS [98–100] for the 2PCF measurements, with weighting schemes [4,6,7] (for clustering, galaxy-galaxy lensing, and cosmic shear, respectively). These estima- In order to obtain the best possible cosmological pa- tors normalize the galaxy-galaxy lensing/cosmic shear rameter constraints, the accuracy of analysis method- 2PCF measurements by the observed (weighted) lens- ology has to be matched to the precision enabled by source/source-source pair counts in each angular bin, the data set. Hence the required accuracy depends on which introduces a modulation by the projected lens- the survey properties, the combination of cosmological source/source-source clustering correlation functions. probes, and the model/parameter space under investiga- For galaxy-galaxy lensing, this modulation describes the tion. The analysis choices that must be made as part dilution by excess (clustered) lens-source pairs at the lens of the methodology development range from modeling redshift; the DES Y3 galaxy–galaxy lensing analysis [7] that is critical to capturing the underlying physics, such corrects for it at the measurement level through so-called as intrinsic alignments, to extensive validation of scale ij ij cuts imposed on the data vector, to technical decisions Boost factors B (θ) 1+wls (θ)[101, 102]. This effect is separate from the source≈ clustering effect derived here, as such as prior ranges or the precision of integration and 4 interpolation routines that are employed in the model- is evident from the absence of δm terms in Eq. (49). O ing. The limiting factor in some cases is computing time; The measured cosmic shear correlation functions ξ± are not debiased for the modulation of the observed pair while of course the pipeline must be sufficiently accurate ij to match the precision of the data, it is also clear that counts by (1 + wss (θ)) from the pair-based estimator [94, 95]. This effect is partially cancelled by the (22)-type increasing the complexity of the pipeline at a dispropor- tionately high cost in computing time is unwise if the corrections to CEE/BB shown in Eqs. (47,48): for ξ+, in the limit of narrow source redshift bins (and hence improvements are well below the statistical errors. restricting to auto-tomography combinations, i = j), the In this paper we describe the cosmology and astro- sum of these two corrections simplifies to the convolution physics modeling strategy and validation for the DES of the convergence and angular source clustering power Y3 joint weak lensing and galaxy clustering analysis. We spectra demonstrate that our analysis choices are sufficient to mitigate systematic biases in the DES Y3 parameter con- 4 (O(δm)) straints due to theoretical modeling simplifications and ∆C (`) + ∆CBB(`) [Cκκ Cδ δ ](`) , (50) EE ≈ ∗ s s uncertainties.

corresponding to ∆ξ+(θ) = ξ+(θ)wss(θ) in configuration Our modeling systematics mitigation strategy can space 12. be summarized as a combination of (1) modeling and We evaluate the combined effect of reduced shear and marginalizing over systematic-error parameterizations i and (2) imposing scale cuts that exclude data points source clustering assuming b1,s = 1 and source magnifica- where unmodeled systematics are prevalent. In terms of modeling and marginalizing, we choose a baseline analysis parameterization that models the nonlinear matter 11 4 Going beyond linear galaxy clustering, there are O δm correc- power spectrum, galaxy bias, intrinsic alignments, lens tions to galaxy-galaxy lensing; these are contractions of nonlinear magnification, and non-local contributions to galaxy– lens biasing with source magnification corrections, not 3D lens- source clustering. galaxy lensing. While necessarily incomplete, this model 12 This expression is valid on small angular scales, where the trans- is sufficiently complex and robust for the DES Y3 3 2pt × formations for density correlations (Eq. 17) and ξ+ (Eq. 19) have analysis. Specifically, we demonstrate through a large the same limit. number of simulated likelihood analyses that (1) angu- 18

Cosmic shear 2x2pt 3x2pt

0.86

0.85 Peak Baseline

8 Peak Reduced shear S 0.84 Peak Red. shear + source clust. 0.83 0.3σ Baseline 0.3σ Reduced shear 0.82 0.3σ Red. shear + source clust.

0.28 0.30 0.32 0.34 0.28 0.30 0.32 0.34 0.28 0.30 0.32 0.34 Ωm Ωm Ωm

FIG. 8. Robustness of parameter constraints to higher-order lensing effects. The offset between the blue/red contour and black contour illustrates the systematic bias incurred by neglecting reduced shear/reduced shear and source clustering terms. The contours levels are 0.3σ2D, symbols indicate the 2D (marginalized) peak. lar scale cuts mitigate the leading known, but unmod- 107], the application to angular clustering measurements eled, systematic effects, and (2) the baseline parameteri- remains challenging due to parameter degeneracies and zation is sufficiently flexible to marginalize over plausible projection effects. Progress in that particular direction variations from these parameterization choices. We fur- for the DES Y3 analysis is being reported by Pandey ther validate our numerical implementation by compar- et al. [8] and DeRose et al. [14], who validate an exten- ing independently-developed modeling pipelines, and find sion of the baseline analysis described here using next- them to be in excellent mutual agreement. We conclude to-leading-order galaxy biasing; the corresponding gains that the cosmology and astrophysics modeling is robust in constraining power are presented in Ref. [10]. Due within the analysis choices derived here, and that theo- to additional nuisance parameters, the gain in constrain- retical modeling systematics are insignificant for DES Y3 ing power from nonlinear bias modeling is significantly cosmic shear, 2 2pt and 3 2pt analyses, as well as their smaller than a mode counting argument might suggest. combinations with× external× data. To our knowledge, this Upcoming, more constraining, analyses will continue work presents the most detailed validation of a large-scale to balance model accuracy (and hence model complex- structure survey to date. ity) with increased statistical constraining power. One The dominant systematics limiting the constraining of the main limiting factors for such ambitious analy- power of the DES Y3 baseline analyses are astrophys- ses will be the availability of sufficient validation tests, ical effects in the nonlinear regime, in particular non- such as high-resolution, high-volume mock catalogs that linear galaxy bias, baryonic effects on the matter power include plausible variations of the relevant nonlinear as- spectrum, and non-local contributions from the one-halo trophysics, such as baryonic feedback, or variations in and one-to-two halo transition regime to the large-scale the halo–galaxy connection. This issue will be front-and- galaxy–galaxy lensing signal. The former two set the center for future analyses of data from Rubin Observa- angular scale cuts for the baseline analysis, excluding a tory’s LSST, the Euclid mission, and the Roman Space substantial fraction of the measured signal-to-noise in the Telescope, in particular when combining the constraining two-point correlation functions, while the latter degrades power of multiple surveys’ data. the constraining power of small-scale galaxy–galaxy lens- Future analyses will need to evaluate the trade-offs ing. that arise from increasing the number of nuisance parameters in the theoretical models. Model complexifica- The modeling challenges described in this paper have tion can improve the accuracy of cosmological inference long been recognized. Successfully resolving them will by bringing the model space closer to the true Universe. allow us to achieve the ultimate goal of optimally ex- But this may come at the price of increased parameter tracting information from galaxy surveys, particularly degeneracies and reduced cosmological precision, infer- from small spatial scales where a large but challenging- ence instabilities due to multiple minima,13 and biased to-model signal resides. While our work here represents confidence contours due to prior-volume projections. It important developments in mitigating these uncertain- is therefore important to develop the least complex anal- ties, substantial progress remains possible with future ef- ysis that is robust for any given science case and data fort. For instance, suitable modeling or mitigating of set, and to attend closely to priors on additional param- baryonic effects on small scales can be used to extract eters. Mapping out the modeling trade space will require more information from weak lensing surveys, as demonstrated by Asgari et al. [104] and Huang et al. [61] with recent data sets. Another example is nonlinear galaxy bias: while its modeling has become a standard tool in 13 See Appendix B of [4] for a characterization of bimodality in the analyses of galaxy clustering in redshift space [e.g., 105– TATT parameter space due to noise. 19 a large set of simulated analyses customized to the noise versity of Illinois at Urbana-Champaign, the Institut de level, probe combination, and cosmological model space Ciènciesde l’Espai (IEEC/CSIC), the Institut de F´ısica of a given experiment. Advances in theory that restrict d’Altes Energies, Lawrence Berkeley National Labora- rather than expand the model space will be extremely tory, the Ludwig-Maximilians UniversitätMünchen and valuable to to achieve high-accuracy cosmology. the associated Excellence Cluster Universe, the Univer- sity of Michigan, the National Optical Astronomy Ob- servatory, the University of Nottingham, The Ohio State ACKNOWLEDGEMENTS University, the University of Pennsylvania, the Univer- sity of Portsmouth, SLAC National Accelerator Labora- EK thanks Fabian Schmidt and Masahiro Takada for tory, Stanford University, the University of Sussex, Texas helpful discussions, and Fabian Schmidt for feedback on A&M University, and the OzDES Membership Consor- an early draft. EK is supported in part by the Depart- tium. ment of Energy grant DE-SC0020247, the David & Lucile The DES data management system is supported by Packard Foundation and the Alfred P. Sloan Foundation. the National Science Foundation under Grant Num- XF is supported by the Department of Energy grant DE- bers AST-1138766 and AST-1536171. The DES partic- SC0020215. ipants from Spanish institutions are partially supported This work made use of the software packages by MINECO under grants AYA2015-71825, ESP2015- PolyChord [108, 109], GetDist [110], matplotlib [111], 88861, FPA2015-68048, SEV-2012-0234, SEV-2016-0597, and numpy [112]. and MDM-2015-0509, some of which include ERDF funds Funding for the DES Projects has been provided by from the European Union. IFAE is partially funded by the U.S. Department of Energy, the U.S. National Sci- the CERCA program of the Generalitat de Catalunya. ence Foundation, the Ministry of Science and Education Research leading to these results has received funding of Spain, the Science and Technology Facilities Coun- from the European Research Council under the Euro- cil of the United Kingdom, the Higher Education Fund- pean Union’s Seventh Framework Program (FP7/2007- ing Council for England, the National Center for Super- 2013) including ERC grant agreements 240672, 291329, computing Applications at the University of Illinois at and 306478. We acknowledge support from the Aus- Urbana-Champaign, the Kavli Institute of Cosmological tralian Research Council Centre of Excellence for All- Physics at the University of Chicago, the Center for Cos- sky Astrophysics (CAASTRO), through project number mology and Astro-Particle Physics at the Ohio State Uni- CE110001020. versity, the Mitchell Institute for Fundamental Physics This manuscript has been authored by Fermi Research and Astronomy at Texas A&M University, Financiadora Alliance, LLC under Contract No. DE-AC02-07CH11359 de Estudos e Projetos, Funda¸cãoCarlos Chagas Filho with the U.S. Department of Energy, Office of Science, de Amparo àPesquisa do Estado do Rio de Janeiro, Office of High Energy Physics. The United States Gov- Conselho Nacional de Desenvolvimento Cient´ıficoe Tec- ernment retains and the publisher, by accepting the arti- nológicoand the Ministérioda Ciência,Tecnologia e In- cle for publication, acknowledges that the United States ova¸cão, the Deutsche Forschungsgemeinschaft and the Government retains a non-exclusive, paid-up, irrevoca- Collaborating Institutions in the Dark Energy Survey. ble, world-wide license to publish or reproduce the pub- The Collaborating Institutions are Argonne National lished form of this manuscript, or allow others to do so, Laboratory, the University of California at Santa Cruz, for United States Government purposes. the University of Cambridge, Centro de Investigaciones Based in part on observations at Cerro Tololo Inter- Energéticas, Medioambientales y Tecnológicas-Madrid, American Observatory, National Optical Astronomy Ob- the University of Chicago, University College London, servatory, which is operated by the Association of Uni- the DES-Brazil Consortium, the University of Edin- versities for Research in Astronomy (AURA) under a co- burgh, the Eidgenössische Technische Hochschule (ETH) operative agreement with the National Science Founda- Zürich, Fermi National Accelerator Laboratory, the Uni- tion.

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