DES-2020-0617
FERMILAB-PUB-21-221-AE

Dark Energy Survey Year 3 Results:
Cosmological Constraints from Galaxy Clustering and Weak Lensing
T. M. C. Abbott,1 M. Aguena,2 A. Alarcon,3 S. Allam,4 O. Alves,5, 6, 2 A. Amon,7 F. Andrade-Oliveira,6, 2 J. Annis,4 S. Avila,8D. Bacon,9 E. Baxter,10 K. Bechtol,11 M. R. Becker,3 G. M. Bernstein,12 S. Bhargava,13 S. Birrer,14 J. Blazek,15, 16
A. Brandao-Souza,17, 2 S. L. Bridle,18 D. Brooks,19 E. Buckley-Geer,20, 4 D. L. Burke,7, 21 H. Camacho,6, 2 A. Campos,22
A. Carnero Rosell,23, 2, 24 M. Carrasco Kind,25, 26 J. Carretero,27 F. J. Castander,28, 29 R. Cawthon,11 C. Chang,20, 30 A. Chen,5R. Chen,31 A. Choi,32 C. Conselice,18, 33 J. Cordero,18 M. Costanzi,34, 35, 36 M. Crocce,28, 29 L. N. da Costa,2, 37 M. E. S. Pereira,5
C. Davis,7 T. M. Davis,38 J. De Vicente,39 J. DeRose,40 S. Desai,41 E. Di Valentino,18 H. T. Diehl,4 J. P. Dietrich,42
S. Dodelson,22, 43 P. Doel,19 C. Doux,12 A. Drlica-Wagner,20, 4, 30 K. Eckert,12 T. F. Eifler,44, 45 F. Elsner,19 J. Elvin-Poole,32, 46
S. Everett,47 A. E. Evrard,48, 5 X. Fang,44 A. Farahi,5, 49 E. Fernandez,27 I. Ferrero,50 A. Ferté,45 P. Fosalba,28, 29 O. Friedrich,51
J. Frieman,4, 30 J. García-Bellido,8 M. Gatti,12 E. Gaztanaga,28, 29 D. W. Gerdes,48, 5 T. Giannantonio,52, 51 G. Giannini,27
D. Gruen,53, 7, 21 R. A. Gruendl,25, 26 J. Gschwend,2, 37 G. Gutierrez,4 I. Harrison,54, 18 W. G. Hartley,55 K. Herner,4S. R. Hinton,38 D. L. Hollowood,47 K. Honscheid,32, 46 B. Hoyle,42 E. M. Huff,45 D. Huterer,5 B. Jain,12 D. J. James,56
M. Jarvis,12 N. Jeffrey,19, 57 T. Jeltema,47 A. Kovacs,23, 24 E. Krause,44 R. Kron,4, 30 K. Kuehn,58, 59 N. Kuropatkin,4O. Lahav,19 P.-F. Leget,7 P. Lemos,19, 13 A. R. Liddle,60, 61, 62 C. Lidman,63, 64 M. Lima,65, 2 H. Lin,4 N. MacCrann,66
M. A. G. Maia,2, 37 J. L. Marshall,67 P. Martini,32, 68, 69 J. McCullough,7 P. Melchior,70 J. Mena-Fernández,39
F. Menanteau,25, 26 R. Miquel,71, 27 J. J. Mohr,42, 72 R. Morgan,11 J. Muir,7 J. Myles,53, 7, 21 S. Nadathur,19 A.
Navarro-Alsina,17 R. C. Nichol,9 R. L. C. Ogando,2, 37 Y. Omori,20, 30, 7 A. Palmese,4, 30 S. Pandey,12 Y. Park,73
F. Paz-Chinchón,25, 52 D. Petravick,25 A. Pieres,2, 37 A. A. Plazas Malagón,70 A. Porredon,32, 46 J. Prat,20, 30 M. Raveri,12
M. Rodriguez-Monroy,39 R. P. Rollins,18 A. K. Romer,13 A. Roodman,7, 21 R. Rosenfeld,74, 2 A. J. Ross,32 E. S. Rykoff,7, 21
S. Samuroff,22 C. Sánchez,12 E. Sanchez,39 J. Sanchez,4 D. Sanchez Cid,39 V. Scarpine,4 M. Schubnell,5 D. Scolnic,31
L. F. Secco,12, 30 S. Serrano,28, 29 I. Sevilla-Noarbe,39 E. Sheldon,75 T. Shin,12 M. Smith,76 M. Soares-Santos,5 E. Suchyta,77
M. E. C. Swanson,25 M. Tabbutt,11 G. Tarle,5 D. Thomas,9 C. To,53, 7, 21 A. Troja,74, 2 M. A. Troxel,31 D. L. Tucker,4
I. Tutusaus,28, 29 T. N. Varga,72, 78 A. R. Walker,1 N. Weaverdyck,5 J. Weller,72, 78 B. Yanny,4 B. Yin,22 Y. Zhang,4 and J. Zuntz60(DES Collaboration)
1

Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile
2
Laboratório Interinstitucional de e-Astronomia - LIneA,
Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
3
Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA
4
Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA
5
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
6
Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil
7
Kavli Institute for Particle Astrophysics & Cosmology,
P. O. Box 2450, Stanford University, Stanford, CA 94305, USA
8
Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain
9
Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK
10
Institute for Astronomy, University of Hawai’i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
11
Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison, 1150 University Avenue Madison, WI 53706-1390
12Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
13
Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK
14
Graduate School of Education, Stanford University, 160, 450 Serra Mall, Stanford, CA 94305, USA
15
Department of Physics, Northeastern University, Boston, MA 02115, USA
16
Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
17
Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil
18
Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK
19
Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK
20
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA
21
SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
22
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15312, USA
23
Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain
24
Universidad de La Laguna, Dpto. AstrofÃsica, E-38206 La Laguna, Tenerife, Spain
25
Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA
26
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA
27
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
28
Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

229

Institute of Space Sciences (ICE, CSIC), Campus UAB,
Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
30
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
31
Department of Physics, Duke University Durham, NC 27708, USA
32
Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA
33
University of Nottingham, School of Physics and Astronomy, Nottingham NG7 2RD, UK
34
Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy
35
INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy
36
Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy
37
Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
38
School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
39
Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain40
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
41
Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
42
Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany
43
NSF AI Planning Institute for Physics of the Future,
Carnegie Mellon University, Pittsburgh, PA 15213, USA
44
Department of Astronomy/Steward Observatory, University of Arizona,
933 North Cherry Avenue, Tucson, AZ 85721-0065, USA
45
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA
46
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
47
Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA
48
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
49
Departments of Statistics and Data Science, University of Texas at Austin, Austin, TX 78757, USA
50
Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway
51
Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
52
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
53
Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA
54
Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
55
Department of Astronomy, University of Geneva, ch. d’Écogia 16, CH-1290 Versoix, Switzerland
56
Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
57
Laboratoire de Physique de l’Ecole Normale Supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France
58
Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia
59
Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA
60
Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK
61
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, 1769-016 Lisboa, Portugal
62
Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada
63
Centre for Gravitational Astrophysics, College of Science,
The Australian National University, ACT 2601, Australia
64
The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia
65
Departamento de Física Matemática, Instituto de Física,
Universidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
66
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
67
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
68
Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA
69
Radcliffe Institute for Advanced Study, Harvard University, Cambridge, MA 02138
70
Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA
71
Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
72
Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany
73
Kavli Institute for the Physics and Mathematics of the Universe (WPI),UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan74
ICTP South American Institute for Fundamental ResearchInstituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil
75
Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA
76
School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
77
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
78
Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679 München, Germany(Dated: May 26, 2021)

3
We present the first cosmology results from large-scale structure using the full 5000 deg2 of imaging datafrom the Dark Energy Survey (DES) Data Release 1. We perform an analysis of large-scale structure combining
three two-point correlation functions (3×2pt): (i) cosmic shear using 100 million source galaxies, (ii) galaxy
clustering, and (iii) the cross-correlation of source galaxy shear with lens galaxy positions, galaxy–galaxy lensing. To achieve the cosmological precision enabled by these measurements has required updates to nearly
every part of the analysis from DES Year 1, including the use of two independent galaxy clustering samples,modeling advances, and several novel improvements in the calibration of gravitational shear and photometricredshift inference. The analysis was performed under strict conditions to mitigate confirmation or observer bias;
we describe specific changes made to the lens galaxy sample following unblinding of the results and tests of
the robustness of our results to this decision. We model the data within the flat ΛCDM and wCDM cosmological models, marginalizing over 25 nuisance parameters. We find consistent cosmological results between
the three two-point correlation functions; their combination yields clustering amplitude S8 = 0.776+0.017−0.017+0.026and matter density Ωm = 0.339+0.032−0.031 in ΛCDM, mean with 68% confidence limits; S8 = 0.775−0.024 ,+0.32Ωm = 0.352+0.035−0.041 , and dark energy equation-of-state parameter w = −0.98−0.20 in wCDM. These constraints correspond to an improvement in signal-to-noise of the DES Year 3 3×2pt data relative to DES Year
1 by a factor of 2.1, about 20% more than expected from the increase in observing area alone. This combination of DES data is consistent with the prediction of the model favored by the Planck 2018 cosmic microwave
background (CMB) primary anisotropy data, which is quantified with a probability-to-exceed p = 0.13 to 0.48.
We find better agreement between DES 3×2pt and Planck than in DES Y1, despite the significantly improved
precision of both. When combining DES 3×2pt data with available baryon acoustic oscillation, redshift-spacedistortion, and type Ia supernovae data, we find p = 0.34. Combining all of these data sets with Planck CMB+0.004+0.004
lensing yields joint parameter constraints of S8 = 0.812+0.008−0.008 , Ωm = 0.306−0.005 , h = 0.680−0.003 , andP+0.008+0.006+0.006mν < 0.13 eV (95% CL) in ΛCDM; S8 = 0.812−0.008 , Ωm = 0.302−0.006 , h = 0.687−0.007 , andw = −1.031+0.030−0.027 in wCDM.

I.

INTRODUCTION

The discovery of the accelerated expansion of the universe[1, 2] led to a new standard model of cosmology, which
is dominated by a spatially smooth component with negative pressure called dark energy. Over the intervening two
decades, the evidence for the presence of dark energy has become much stronger thanks to data from an impressive varietyof cosmological probes. Modern cosmological measurements
using type Ia supernovae [3–11], cosmic microwave background (CMB) fluctuations [12–14], galaxy clustering [15–21], and weak gravitational lensing [22–28] are in agreementwith a spatially flat universe with about 30% matter (visible
and dark) and 70% dark energy.
However, the physical nature of the dark energy that causes
accelerated expansion remains unknown. The simplest andbest-known phenomenological model for dark energy is theenergy density of the vacuum, incorporated in the field equations of General Relativity by the cosmological-constant termΛ [29]. The resulting Λ Cold Dark Matter (ΛCDM) model
serves as a benchmark for tests with current and future data.
Beyond ΛCDM, there exists a rich set of other potential
models to explain cosmic acceleration, including evolving
scalar fields, modifications to general relativity, and otherphysically-motivated possibilities. This has spawned an active research area focused on describing and modeling darkenergy and its effects on the expansion rate and the growth ofdensity fluctuations [30, 31].
The quest to understand dark energy has spawned a worldwide effort to better measure the growth and evolution ofcosmic structure in the universe. The current generation ofobservations is spearheaded by the so-called Stage-III darkenergy experiments, which include the Dark Energy Sur-vey (DES)1 [32–34], the Hyper Suprime-Cam Subaru Strategic Program (HSC)2 [27, 35, 36], the Kilo-Degree Survey(KiDS)3 [28, 37], and the Extended Baryon Oscillation Spectroscopic Survey (eBOSS)4 [38]. These surveys have demonstrated the feasibility of ambitious large-scale structure analyses, and featured extensive tests of theory, development ofstate-of-the-art systematics calibration, and new rigor in protecting analyses against observer bias before the results are
revealed. These surveys have, thus far, provided constraints
consistent with the ΛCDM model, and contributed to tightening the constraints on several of the key cosmological parameters related to dark matter and dark energy.
Large-scale structure (LSS) in the universe provides a
powerful set of tools to probe dark energy. The statistics
and temporal growth of cosmic structure complement thelargely geometrical sensitivity to dark energy of type Ia supernovae and the CMB. For nearly half a century, measurements of the galaxy two-point correlation function, a statistic describing the spatial clustering of galaxies, have provided
pioneering cosmological constraints and early evidence forthe ΛCDM model [15, 16, 39–52], as well as recent, highprecision constraints on the cosmological parameters [20, 53–
65]. Another aspect of LSS that is sensitive to both dark matter and dark energy is cosmic shear, slight distortions of the
shapes of distant background galaxies due to weak gravitational lensing of light passing through the structures between
these sources and us. While the interpretation of galaxy clustering is complicated by galaxy bias [66, 67], cosmic shear

1 http://www.darkenergysurvey.org/
2 https://www.naoj.org/Projects/HSC/
3 http://kids.strw.leidenuniv.nl/
4 https://www.sdss.org/surveys/eboss/

4
measurements are more directly related to the distribution ofmass. First detections of cosmic shear [68–71] have been
followed by an impressive maturing of this probe, with increasingly more competitive constraints on cosmological parameters [23–25, 72–78]. Finally, galaxy–galaxy lensing, thecross-correlation of lens galaxy positions and source galaxy
shapes, provides a link between galaxy clustering and cosmic
shear. Galaxy–galaxy lensing measurements have also matured to the point where their combination with galaxy clustering breaks degeneracies between the cosmological parameters and bias, thereby helping to constrain dark energy [79–92]. The combination of galaxy clustering, cosmic shear, andgalaxy–galaxy lensing measurements powerfully constrains
structure formation in the late universe, while strongly selfcalibrating many astrophysical or systematic parameters in themodel.
The stakes have become higher with recent evidence forpossible tensions between parameters as measured by different cosmological probes. These tensions may indicate
new physics beyond ΛCDM — or else could be due tounaccounted-for systematics or an underestimation of uncertainty in some probes. Potentially most significant among
these is the “Hubble tension,” indicated by a ∼4–6σ discrepancy between measurements of the Hubble constant inferred from the primary CMB anisotropies [13] and higher
values measured from a local distance anchor, such as, mostprominently the astronomical distance ladder (e.g., [93, 94])or masers [95], though some measurements also indicate alower value [96, 97] in better agreement with the CMB. The
Hubble tension may indicate new physics, and it is crucial to
improve measurements, revisit assumptions and systematics[e.g., 97, 98] and invest in novel, independent methods andprobes [99, 100].
Additionally, several experiments that are sensitive to thegrowth of structure have historically preferred, on average,
lower values of the parameter S8 ≡ σ8 (Ωm /0.3)0.5 relative to that predicted by the CMB anisotropy, where the amplitude of mass fluctuations σ8 is scaled by the square rootof matter density Ωm . This parameter is predicted to behigher with the CMB [13] than is measured in lensing (e.g.,[27, 33, 101, 102]). The difference has been claimed by otherexperiments to be as large as 2–3σ. Other probes of the late
universe, in particular spectroscopic galaxy clustering [103],
redshift-space distortions (RSD) [21], and the abundance ofgalaxy clusters [104, 105], also tend to favor a lower S8 thanthat measured by the CMB on average (assuming the ΛCDM
model).
Previously, the DES Collaboration analyzed data from its
first year of observations, which covered 1514 deg2 , andconstrained cosmological parameters using galaxy clusteringand gravitational lensing in ΛCDM and wCDM [33], constrained beyond-wCDM models [106], and carried out numerous other tests of the standard cosmological framework[64, 101, 105, 107–114]. Along with the aforementioned
KiDS and HSC observations and analyses, the DES Y1 analysis emphasized redundancy using two shape measurement
methods that are independently calibrated, several photometric redshift estimation and validation techniques, and two in-dependent codes for predicting the measurements and performing a likelihood analysis.
This paper presents key cosmological constraints from thefirst three years of observations (henceforth Y3) of DES. The
DES Y3 data set analyzed here uses images covering nearly
5000 sq. deg., or more than three times the area of Y1. It also
dramatically increases the number of source and lens galaxies,
and introduces new techniques for the analysis and treatmentof statistical and systematic errors. As in Y1, we rely on a
key cosmological probe of photometric LSS surveys, the socalled ‘3×2pt’ analysis, consisting of three two-point correlation functions: (i) w(θ), the angular correlation function of
the lens galaxies; (ii) γt (θ), the correlation of the tangential
shear of sources with lens galaxy positions; and (iii) ξ± (θ),
the correlation functions of different components of the ellipticities of the source galaxies. We use these measurements
only on large angular scales, for which we have verified thata relatively simple model describes the data, although evenwith this restriction we must introduce 25 free parameters to
capture astrophysical and measurement-related systematic uncertainties. The paper is built upon and uses tools and resultsfrom 29 accompanying papers [115–143] that are summarizedin App. A. We summarize in App. B the major updates to theanalysis that are different from the DES Y1 3×2pt analysis.
The cosmological quantity that is best constrained by the
3×2pt analysis is the overall amplitude of matter clustering in
the low redshift universe, parameterized by S8 . The precise
measurement of S8 in this paper allows a powerful test forconsistency between the growth of structure and the expansion history in the broad class of cosmic acceleration models
based on General Relativity (GR) and dark energy. Implementing this test requires a CMB anchor for the matter clustering amplitude at high redshift, and the test becomes sharperand more general when supernova and baryon acoustic oscillation (BAO) data are used to constrain the expansion history.
DES probes matter clustering out to z ≈ 1, so it also constrains dark energy models on its own through the history of
structure growth over this redshift range. The degeneracy between Ωm and σ8 in S8 is broken partly by this redshift evolution and partly by the shape of the correlation functions, andit can be broken more strongly using external data that are
sensitive to Ωm . Lensing measurements depend on the expansion history through the distance-redshift relation. Thisdependence affects our analysis, but the geometric constraints
from DES weak lensing are not as strong as those from current
supernova and BAO data.
In subsequent sections of the paper, we focus first on the
DES data sets in Sec. II and measurements of the three twopoint correlation functions in Sec. III. We describe the modeling and analysis in Sec. IV, then turn to the primary results, tests, and parameter constraints from combining thesemeasurements with additional measurements from DES, theCMB, and other external supernova, BAO, and RSD data in
Sec. V. We conclude in Sec. VIII.

5II.

DARK ENERGY SURVEY DATAThe Dark Energy Survey (DES) was a six-year observing
program using the Dark Energy Camera (DECam [32]) at
the Blanco 4m telescope at the Cerro Tololo Inter-American
Observatory (CTIO) in Chile. The survey covered 5000
deg2 in grizY bandpasses with approximately 10 overlapping dithered exposures in each filter (90 sec in griz, 45 sec
in Y ) covering the survey footprint. In this paper, we utilize data taken during the first three years of DES operations(DES Y3), which made up DES Data Release 1 (DR1 [144]).
This analysis uses imaging data covering the full 5000 deg2
survey footprint for the first time, at approximately half the
full-survey integrated exposure time. Preparing the imaging
data for cosmological analysis is an exacting, multi-year process, and analysis of the final six-year data set is now in its
early stages. The data is processed, calibrated, and coadded to produce a photometric data set of 399 million objects
that is further refined to a ‘Gold’ sample for cosmological use[115, 145, 146]. The Gold sample includes selection requirements (cuts) on minimal image depth and quality, additional
calibration and deblending, and quality flags to identify problematic photometry and regions of the sky with substantial
photometric degradation (e.g., around bright stars). The Goldgalaxy sample extends to a signal-to-noise > 10 (extended)
limiting magnitude of 23 in i-band. The final Gold sample
used in this work after all cuts contains 319 million objects.
In addition to the wide-field Gold sample, we rely on datafrom the DES deep fields [116] covering a subset of the 27
deg2 DES transient search regions and the separate COSMOS
field [147]. These images are taken in ugrizY bandpasses
with DECam, and also have overlapping VIDEO [148] or UltraVISTA [149] imaging for near-IR photometry in Y JHKover most of the area. Coadd images are constructed from thebest images (i.e., with smallest point-spread function (PSF)
full-width half-maximum (FWHM)) with a goal of attaining adepth approximately 10× the typical wide-field coadd image
depth. From these coadd images, we produce a deep catalog
of 2.8 million objects that has 10σ limiting magnitude of 25
in i-band, and photometric variance 0.1× the typical widefield variance. This catalog helps to validate and calibrate
our wide-field data in several ways. First, it is used to create
an input model space for representative objects to draw onto
wide-field-quality images in the Balrog [121] or weak lensing image simulations [120]. Balrog is used to test the survey
selection function, which describes the probability that an object type drawn from a complete galaxy population will be
detected in our wide-field survey, and the weak lensing simulations are used to test our shear calibration. The deep catalog also serves as a stepping-stone in our redshift inference
methodology [123]. It allows us to map available spectroscopic or many-band deep photometric observations into theugrizJHK bandpass space of our deep catalog, for which
we have 1.68 million sources with matched near-infrared photometry covering an area of 5.88 deg2 . This is then mapped
through Balrog onto wide-field galaxy information.

A.

Source Galaxies

1.

ShapesThe DES Y3 shear catalog [117] is derived using the META CALIBRATION pipeline [150, 151], which infers the ellipticity
and similar photometric properties of objects using information from the r,i,z-bands. The pipeline is similar to that usedin the DES Y1 analysis [152], but with a number of updates,
including improved PSF solutions [118], improved astrometric solutions [115], and the inclusion of an inverse-varianceweighting for the galaxies. METACALIBRATION is able toself-calibrate the initial estimate of the shear field from themeasured galaxy shapes, including sample selection biases.
The current METACALIBRATION implementation, however,does not correct for a shear-dependent detection bias [153]that is coupled with object blending effects, which we find tocause a multiplicative bias in the shear at the level of 2-3%.
This residual bias is calibrated using image simulations [120].
Objects are included in the catalog if they pass a number of
selection cuts designed to reduce potential systematic biases[117]. After additional footprint masking to match the lens
catalogs, the final DES Y3 shear catalog yields 100 million
galaxies covering an area of 4143 deg2 , with a weighted effective number density neff = 5.9 per arcmin2 and corresponding
shape noise σe = 0.26.
The catalog has passed a variety of empirical tests [117],
mostly aimed at identifying residual additive biases in theshear estimates. Systematic errors related to PSF modelingwere shown to be negligible for the DES Y3 analysis, due toimproved PSF modeling [118]. The B-mode signal was also
shown to be consistent with zero. Other tests included thedependence of the shear estimates on galaxy and survey properties.
While shear calibration is typically viewed as separablefrom redshift inference, which is described in the following
section, we also account for the first time for how blending
correlates the ensemble shear calibration in each redshift binwith corrections to the effective shape of the n(z) of eachof four redshift bins [120]. These corrections stem from a
blending–detection bias, which biases both the ensemble average shear (some blends will only be detected as separate
objects depending on the shear) and redshift distribution ina potentially correlated way. One way to treat these effects
coherently is to model the multiplicative shear calibration asa scaling of the total number density in each redshift bin,and fit these effects fully in redshift space.
R ∞We would model
then the shear calibration bias as eij = 0 nγ (z)γji (z) + c,
for additive shear bias c, observed ellipticity e, true ellipticity γ, shear component j, and redshift bin i. In practice we continue to separate a scalar multiplicative (m) shear
bias component to be compatible with existing codes, wherenγ (z) ∝ (1 + m)n(z).

Source n(z)

6

Bin 1
Bin 2
Bin 3
Bin 4
Bin 5
Bin 6

4

2

MagLim n(z)

0
1.0

z

0.5

RedMaGiC n(z)

0.0
0.4

z

0.2

0.0
0.0

0.5

1.0
z

1.5

2.0

FIG. 1. The source (top), MagLim lens (middle), and redMaGiC
lens (bottom) redshift distributions. The histograms are normalized
to integrate to the total weighted galaxy density (arcmin−2 ) in eachtomographic bin. The equivalent 1σ uncertainties on the redshift distributions are indicated by the shaded regions. The distributions havebeen corrected by non-zero mean and width offsets derived in the relevant photo-z uncertainty models. We adopt MagLim as our fiducial
lens sample in this work, and use only redshift bins 1–4.

2.

Photometric redshifts

The full redshift inference process for the source galaxies[123] relies on connecting information about deep-field galaxies to those in the wide field that are used for the cosmologicalanalysis [154, 155]. All galaxies in the deep fields with similar
properties are clustered together into different ‘phenotypes’
via a self-organizing map (SOM), while the same is also separately done for all galaxies in the wide field. The deep-fieldgalaxies have much lower photometric noise and additional
wavelength information (i.e., overlapping infrared photometry), so more phenotypes can be uniquely identified than for
the wide-field galaxies. A redshift distribution is inferred foreach of the deep-field galaxy phenotypes using overlapping
spectroscopic [156–160], and photometric COSMOS [147]and PAUS [161, 162] redshift measurements. We then create aprobabilistic mapping between the deep- and wide-field phenotypes using the Balrog simulation [121]. For example, if agiven wide-field galaxy phenotype was mapped uniquely onto

a single deep-field galaxy phenotype, its redshift distribution
would be determined by the available redshift measurementsof the deep-field galaxies that share that particular phenotype.
In practice, the mapping is much more complicated: eachwide-field phenotype has a non-zero probability of comingfrom many deep-field phenotypes, but the algorithm for generating an n(z) for that galaxy phenotype is simply a weighted
average. A given redshift bin is defined by a unique subset ofmany wide-field galaxy phenotypes, and its n(z) then follows
by averaging over these phenotypes. The four source redshiftbins have edges z ∈ [0.0, 0.36, 0.63, 0.87, 2.0].
The process we use to account for uncertainties accumulated in each step of this process is summarized in Ref. [123].
These are due in part to shot-noise and cosmic variance in theredshift samples and deep fields [163], and photometric calibration uncertainty. At low redshift, this uncertainty is primarily due to uncertainties in the photometric calibration, while athigh redshift it is due to a combination of cosmic variance anduncertainties in the redshift samples. Uncertainty in the n(z)due to these effects are modeled or measured, and we generatemany realizations of the redshift distribution, ni (z), that appropriately sample the joint space of this uncertainty without
relying on a simple parameterization like mean and width.
The emerging set of redshift realizations suffer from one
further source of uncertainty that has not been explicitly included before: blending. Galaxies that are nearby one another when projected on the sky can actually be very far apart.
Detection and measurement algorithms can misinterpret theseblends and report not only incorrect shapes but also incorrect
number densities or redshifts. To account for this effect, wecreated realistic simulations [120] and apply the same detection and measurement pipeline used for the DES data to obtain the “observed” number density, shape, and photometricredshift of a known simulated object population that matches
our deep field data, from which the impact of blending can beinferred. The result is a likelihood model describing the impact of blending on the joint shear calibration and n(z) shapethat we add to the ni (z) [120].
We empirically constrain the likelihood of each ni (z) using information from galaxy clustering on small scales that is
not used in the primary 3×2pt observables [125]. We knowthat galaxies are likely to be found near other galaxies due
to gravitational clustering. Therefore, if there is a galaxy ina given direction whose redshift is known, it is likely thatnearby galaxies on the sky are at a similar redshift. We makeuse of multiple galaxy samples with well-determined redshifts
and cross-correlate them with the wide-field source sample,
thereby obtaining a likelihood for each of the ni (z), which is
jointly sampled with the models that produce the ni (z) before
we account for blending effects. We produce several thousand ni (z) samples. We show the final redshift distributions
and their uncertainties in the top panel of Fig. 1.
To sample over ni (z) in a likelihood analysis, we introduce
a set of hyper parameters to our model that rank the ni (z)in multiple dimensions [127]. Ref. [127] has demonstrated
that our constraints on the variation in ni (z) (i.e., shown in
Fig. 1) are sufficiently precise that uncertainty in higher-ordermodes in the ni (z), besides the mean redshift, were not ex-7pected to impact our cosmological constraints at a significant
level in Y3. Thus in practice we simply sample the mean of
the redshift distribution in the four redshift bins within a Gaussian prior based on the measured variance in the mean of eachni (z).
Finally, the measured two-point functions themselves further constrain the possible values of the redshift distributions
via self-calibration in 3×2pt. We further augment this by explicitly using a set of the ratios of the galaxy–galaxy lensing
signal on small scales between source redshift bins sharing
the same lens bin [128], which contains information not usedin the standard 3×2pt analysis. These scales are too difficult
to model robustly in full, but the ratios are to first order independent of cosmological model and depend primarily on theredshift distribution and intrinsic alignment parameters, and toa lesser degree on any redshift dependent bias in shear calibration. This small-scale shear ratio likelihood is jointly sampled
in the cosmological analyses.

B.

Lens Galaxies

We have selected two galaxy populations (MagLim andredMaGiC) that serve as ‘lenses’ in galaxy–galaxy lensing
measurements and for galaxy clustering measurements. The
fiducial results presented in this work use the MagLim sample.
We now describe the two lens samples.

1.

MagLim sample

We have selected a magnitude-limited lens sample [135,141], which results in 10.7 million galaxies. This “MagLim”
sample is defined with a magnitude cut in the i-band that depends linearly on redshift, i < 4z + 18, where z is the photometric redshift estimate from the Directional Neighborhood
Fitting (DNF) algorithm [115, 164]. This selection was optimized for wCDM constraints [135].
The MagLim sample is divided into six tomographic
bins from z = 0.2 to z = 1.05, with bin edges z =[0.20, 0.40, 0.55, 0.70, 0.85, 0.95, 1.05]. The redshift distributions from DNF are shown in the middle panel of Fig. 1 andhave been validated using galaxy clustering cross-correlations[122]. Weights are derived to account for correlations in thenumber density with survey properties [136]. Further validation and characterization of the sample is described in Refs.
[136, 141]. After unblinding, we discovered issues with thesample above z = 0.85, which lead to disagreement between
the galaxy clustering and galaxy–galaxy lensing signal, andcontribute to a substantially poor model fit to any cosmological models considered in this work (i.e., the two right-mostpanels of Fig. 2). This led us to remove these redshift bins in
the fiducial analysis, which is discussed further in Secs. V A& V C.

2.

redMaGiC sample

This sample is selected with the redMaGiC algorithm[165], which results in 2.6 million galaxies. redMaGiC selects
Luminous Red Galaxies (LRGs) according to the magnitude–
color–redshift relation of red sequence galaxies, calibratedusing spectroscopic redshifts. The sample has a luminosity
threshold Lmin and approximately constant comoving density. The redMaGiC sample has approximately 30% narrower
redshift distributions than MagLim, but approximately onefourth the number of objects.
We split the redMaGiC sample into five tomographic bins, selected on the redMaGiC redshift
point estimate quantity.
The bin edges used arez = [0.15, 0.35, 0.50, 0.65, 0.80, 0.90]. The first threebins use a luminosity threshold of Lmin > 0.5L∗ (the ‘highdensity’ sample). The last two redshift bins use a luminosity
threshold of Lmin > 1.0L∗ (the ‘high luminosity’ sample).
The redshift distributions are computed by stacking samples
from a non-Gaussian redshift PDF of each individual redMaGiC galaxy. Each distribution is built from several drawsof the redshift PDF and are shown in the bottom panel ofFig. 1. The mean and RMS width of the redshift distributions
are validated using galaxy clustering cross-correlations inRef. [122].
Weights are derived to account for correlations in the number density with survey properties [136]. Further validation and characterization of the sample is also described inRefs. [136, 140]. We find a potential residual systematic in
the redMaGiC sample at all redshifts, which does not impact
ΛCDM inference and is also discussed in Sec. V C.
III.

TWO-POINT MEASUREMENTS

To extract cosmological information from the lens andsource catalogs, we compute three sets of two-point correlation functions, which each measure information about how
mass in the Universe is clustered. There are two fields representing the matter distribution that we can access with agalaxy survey: 1) the galaxy density field and 2) the weaklensing shear field. These two fields lead to these three sets ofmeasured two-point functions.
Galaxy Clustering: The two-point function between lensgalaxy positions in redshift bins i and j, wij (θ), describes
the excess (over random) number of galaxy pairs separated
by an angular distance θ. The estimator for wij (θ) and its
measurement and validation process are described in detail in
Ref. [136]. We only use the auto-correlations of the measuredwii (θ) in our analysis; these are shown with their uncertainties
in Fig. 2 for MagLim and in App. C for redMaGiC.
Galaxy–Galaxy Lensing: The two-point function between
lens galaxy positions and source galaxy tangential shear inredshift bins i and j, γtij (θ), describes the over-density ofmass around galaxy positions. The matter correlated with thelens galaxy alters the path of the light emitted by the source
galaxy, thereby distorting its shape. The estimator for γtij (θ)
and its measurement and validation process are described in∆w/σw

w

8

10−1
10−2
10−3
10−4

1,1

2,2

3,3

4,4

5,5

6,6

0

101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)∆γt /σγtγt∆γt /σγtγt∆γt /σγtγt∆γt /σγtγtFIG. 2. The measured w(θ) correlation functions for each tomographic bin i of the MagLim lens galaxies (indicated by the i, i label in eachpanel). The best-fit ΛCDM model from the fiducial 3×2pt analysis is plotted as the solid line in the top part of each panel, while the bottom
part of each panel shows the fractional difference between the measurements and the model prediction, (wobs. − wth. )/σw (with y-axis range±5σ). In both the top and bottom part of each panel, 1σ error bars are shown. Small angular scales where the linear galaxy bias assumption
breaks down are not used in the cosmological analysis; these scales are indicated by grey shading. Bins 5 & 6 are not used in the final analysis.

10−2
10−3
10−4
10−5
10−6

1,1

2,1

3,1

4,1

5,1

6,1

1,2

2,2

3,2

4,2

5,2

6,2

1,3

2,3

3,3

4,3

5,3

6,3

1,4

2,4

3,4

4,4

5,4

6,4

0

10−2
10−3
10−4
10−5
10−6
0

10−2
10−3
10−4
10−5
10−6
0

10−2
10−3
10−4
10−5
10−6
0

101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)

FIG. 3. The measured γt (θ) correlation functions for each tomographic bin combination using the MagLim sample. In each panel, the label
i, j refers to MagLim lens tomographic bin i and the source bin j The best-fit ΛCDM model from the fiducial 3×2pt analysis is plotted as thesolid line in the top part of each panel, with dotted curves indicating a negative model fit. The bottom part of each panel shows the fractional
difference between the measurements and the model prediction, (γtobs. − γtth. )/σγt (with y-axis range ±5σ). In both the top and bottom partof each panel, 1σ error bars are included. Small angular scales where the linear galaxy bias assumption breaks down are not used in thecosmological analysis; these scales are indicated by grey shading. Bins 5 & 6 are not used in the final analysis.

Shear Ratio

9

0.75
0.50
0.25

Lens bin 1

Lens bin 2

Lens bin 3

1-4 2-4 3-4 1-4 2-4 3-4 1-4 2-4 3-4
Source-bin Ratio
FIG. 4. The measured small-scale shear ratio values for each tomographic bin combination using the MagLim sample, with 1σ error
bars indicated. The x-axis identifies the two source bins that makeup the measured ratio. The best-fit cosmological model from thefiducial 3×2pt analysis is over-plotted as the solid line for each setof lens-bin shear ratios.

detail in Ref. [137]. The measured γtij (θ) and their uncertainties are shown in Fig. 3 for MagLim and in App. C for redMaGiC. In addition, we include small-scale shear ratio information below the scale cuts used for γt . These ratios are constructed from γt measurements using different source galaxybins, while keeping the lens bin fixed. This effectively erases
their dependence on the galaxy power spectrum, but keeps
information about redshift calibration, shear calibration, andgalaxy intrinsic alignment. A detailed description of shear ratios and their validation can be found in Ref. [128]. Fig. 4
shows the shear-ratio measurement and uncertainties. This
shear ratio data is included when analyzing all combinationsof the three primary two-point functions in our analyses, unless otherwise noted.
Cosmic Shear: The correlation between source galaxy
shears in redshift bins i and j is described by the two functionsijξ±(θ), which are the sum and difference of the products of thetangential- and cross-components of the projected shear. The
ij
estimator for ξ±(θ) and its measurement and validation process are described in detail in Refs. [142, 143]. The measuredijξ±(θ) and their uncertainties are shown in Fig. 5.
The total data vector includes measurements from five or
six lens redshift bins and four source redshift bins, shownin Fig. 1, split into 20 logarithmic angular bins between 2.5
and 250 arcmin, for a total of 1300 elements (not including shear-ratio). After bin pair removal for w(θ), imposing a post-unblinding maximum lens redshift cut, and otherscale cuts, 462 elements remain in the final 3×2pt data vector. The scale cut choices and their validation are describedin Refs. [119, 129, 140, 143], but are generally set to control the impact of unmodeled non-linear effects (e.g., baryonic effects on the matter power spectrum or higher-ordergalaxy bias) to better than 0.3σ in the Ωm –S8 plane in Λand wCDM. A choice of scale cuts optimized for ΛCDM-only(ΛCDM-optimized) leaves 508 data points. All measurements
are made using TreeCorr [166]. We find a total signal-to-noise
S/N = 87 for the 3×2pt data vector
p after fiducial scale cuts,
where S/N ≡ ξdata C −1 ξmodel / ξmodel C −1 ξmodel , withcovariance matrix C and best-fit model ξmodel . This is a factorof 2.1 improvement over the DES Year 1 3×2pt S/N .
All of these measurements are related to the underlying

clustering of matter in the Universe, but in different ways.
The relationship between the galaxy density and the underlying matter density is complex [140] and needs to be modeledwith care. Alternately, the shape distortions depend more directly on the intervening matter, but the measurements themselves — especially of shapes and the redshift distribution —require greater care. Our work on this calibration is summarized in Appendix A. The advantage of using all of these measurements is that the systematic difficulties differ from one toanother, but they all measure the same underlying matter field.
Hence, by comparing the results from each set, we obtain ameasure of consistency and additional ability to self-calibrate
systematics, thereby giving confidence that we are correctly
inferring information about the clustering of matter and thecosmological model.
IV.

ANALYSIS

To infer parameters p from the measured two-point functions, we compare data organized in a “data vector” D̂,ij(θ)},
D̂ ≡ {ŵi (θ), γ̂tij (θ), ξˆ±(1)

to a theoretical model prediction organized in a vector TMof two-point correlation functions that are computed using theparameters p of a given model M ,ij(θ, p)},TM (p) ≡ {wi (θ, p), γtij (θ, p), ξ±(2)assuming a Gaussian likelihood,L(D̂|p, M ) ∝ e

− 21

h

T

i

(D̂−TM (p)) C−1 (D̂−TM (p)) .

(3)

Here C is the data covariance, which is obtained through analytic modeling as described and validated in Ref. [130].
We construct a posterior probability distribution for the parameters p of the theoretical model given the data D̂ as
P (p|D̂, M ) ∝ L(D̂|p, M )P (p|M ),(4)where P (p|M ) is a prior probability distribution on the parameters. The proportionality constant is given by the inverseof the Bayesian EvidenceZ
P (D̂|M ) = dp L(D̂|p, M )P (p|M ),(5)which corresponds to the marginalized probability of a dataset
being produced under a given theoretical model.
This section summarizes the theoretical model and parameterization we use for TM (p), which is described in more detail in Ref. [129] and validated in Refs. [119, 129, 139, 140].
For clarity, we drop the parameter argument of the theoretical model predictions, such that, e.g., the predicted clustering
signal is simply denoted as wi (θ).
We report the mean in each parameter, along with the 68%
confidence limit (CL) of posterior volume around the mean.
For completeness, we also report the best-fit maximum posterior values. We have used both a parameter-level and χ2

10

10−5

10−6

10−6

10−7

101

0

10−4

1,3

2,3

3,3

2,2

102
1,2

10−7
0

10−4

10−5

10−5

10−6

10−6

10−7

101

0

10−4

1,2

2,2

3,3

2,3

102
1,3

10−7
0

10−4

10−5

10−5

10−6

10−6

10−7

101

0

10−4

1,1

4,4

3,4

2,4

102
1,4

10−7
0

10−4

10−5

10−5

10−6

10−6

10−7

101

0

101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)

102

10−7
0

∆ξ− /σξ−

10−5

ξ−

10−4

ξ−

1,1

∆ξ− /σξ−

4,4

ξ−

3,4

∆ξ− /σξ−

2,4

ξ−

1,4

∆ξ− /σξ−∆ξ+ /σξ+ξ+∆ξ+ /σξ+ξ+∆ξ+ /σξ+ξ+∆ξ+ /σξ+

ξ+

10−4

101
102
θ (arcmin)

FIG. 5. The measured ξ± (θ) correlation functions for each tomographic bin combination, with labels as described in Fig. 3. The best-fit
ΛCDM model from the fiducial 3×2pt analysis is plotted as the solid line in the top part of each panel, while the bottom part of each panelth.
obs.
)/σξ± (with y-axis range ±5σ). In both− ξ±shows the fractional difference between the measurements and the model prediction, (ξ±the top and bottom part of each panel, 1σ error bars are included. The shaded regions (both light and dark) indicate scales not used in thefiducial analysis, primarily due to uncertainties in the impact of baryonic effects. The lighter shaded regions indicate scales that are used in anΛCDM-optimized analysis, which meets our criterion for scale cuts described in Sec. IV in ΛCDM only.

criterion for limiting the contribution of any systematic error
to bias in the cosmological parameters. The threshold for thiscriterion is intended to limit the expected total bias in the 2D
marginalized Ωm –S8 plane from several independent potential sources of model bias to be contained within the 68% C.L.
region [129] (< 0.3σ for any single contribution). The difference between the mean and best-fit values can give an indication of the magnitude of projection or non-Gaussian effects
in the marginalized parameter posteriors. The estimated impact of projection or volume effects in the DES Year 3 3×2pt
posteriors are tested and summarized in Ref. [129]. We also
provide a 2D figure of merit (FoM) defined for two parame−1/2
ters as FoMp1 ,p2 = (det Cov(p1 , p2 ))[167, 168]. The
FoM is proportional to the inverse area of the confidence region in the space of the two parameters, and can be considereda summary statistic that enables a straightforward comparisonof constraining power of experiments or analysis scenarios.
The analysis was designed and validated without access tothe true cosmological results to protect against confirmation or
observer bias. This process is described in detail in App. D.

A.

Model

We model the observed projected (lens) galaxy density coni
trast δobs(n̂) as a combination of projected galaxy density
contrast and modulation by magnification, δµ ,iδobs(n̂) = δgi (n̂) + δµi (n̂)(6)

for position vector n̂, where i and j represent the redshift bin.
The observed shear signal γ is modeled as the sum of gravitational shear, γG , and intrinsic alignments, I ,jγαj (n̂) = γG,α(n̂) + jI,α (n̂) ,(7)

with α the shear components. While B-modes produced byhigher-order weak lensing effects are negligible for our analysis, it is important to account for B-modes generated by intrinsic alignments in the computation of cosmic shear two-point
correlation functions. In Fourier space, this decomposition
can be written asγEj (`) = κj (`) + jI,E (`) ,jγB(`) = jI,B (`) ,

(8)

11
TABLE I. The model parameters and their priors used in the fiducial
flat ΛCDM and wCDM analyses. The parameter w is fixed to −1 inΛCDM. The parameters are defined in Sec. IV B.
Parameter
CosmologyΩm
109 AsnsΩbh103 Ων h2w
Lens Galaxy Biasbi (i ∈ [1, 4])Lens magnification
Cl1
Cl2
Cl3
Cl4
Lens photo-z∆zl1 × 102∆zl2 × 102∆zl3 × 102∆zl4 × 102
1
σz,l
2
σz,l
3
σz,l
4
σz,l
Intrinsic Alignment
ai (i ∈ [1, 2])
ηi (i ∈ [1, 2])bTA
z0
Source photo-z∆zs1 × 102∆zs2 × 102∆zs3 × 102∆zs4 × 102
Shear calibration
m1 × 102m2 × 102
m3 × 102m4 × 102

Prior

Z

Flat
Flat
Flat
Flat
Flat
Flat
Flat

(0.1, 0.9)
(0.5, 5.0)
(0.87, 1.07)(0.03, 0.07)(0.55, 0.91)(0.60, 6.44)(−2.0, −0.33)

Flat(0.8, 3.0)

Fixed
Fixed
Fixed
Fixed

1.21
1.15
1.88
1.97

3Ωm H022

Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian(−0.9, 0.7)(−3.5, 1.1)(−0.5, 0.6)(−0.7, 0.6)
(0.98, 0.06)(1.31, 0.09)(0.87, 0.05)(0.92, 0.05)

Flat
Flat
Flat
Fixed

(−5, 5)(−5, 5)(0, 2)
0.62

χ

(11)

With the exception of the galaxy clustering power spectraCδx δy , which are evaluated using the method described inRef. [169], we calculate the angular cross-power spectrum between two fields A, B using the Limber approximation` + 12WAi (χ)WBj (χ)
PAB k =dχ, z(χ) ,
χ2
χ
(12)

Z

with PAB the corresponding three-dimensional power spectrum, which is specified by the parameterization choices sumij
correspond to Wκj formarized in IV B. The kernels WA,B
ishear and the lens galaxy density nl for position. The twopoint correlation functions within an angular bin [θmin , θmax ]are related to the projected power spectra asX
wi (θ) =G0 (`, θmin , θmax ) Cδiiobs δobs (`)

ij
ξ±
(θ) =

(−0.6, 0.9)(−2.0, 0.8)(−2.4, 0.8)(−3.7, 0.8)

χ χ0 − χ.
a(χ) χ0Cδiiobs δobs (`) =Cδiig δg (`) + 2Cδiig δµ (`) + Cδiiµ δµ (`) .

X`

Gaussian
Gaussian
Gaussian
Gaussiandχ0 njs (χ0 )Cδijobs E (`) =Cδijg κ (`) + Cδijg IE (`) + Cδijµ κ (`) + Cδijµ IE (`)`(0.0, 1.8)
(0.0, 1.5)
(0.0, 1.1)
(0.0, 1.7)dχ Wκj (χ)δm (n̂χ, χ) ,ij
CBB(`) =CIijB IB

γtij (θ) =Gaussian
Gaussian
Gaussian
GaussianZ χHThe angular power spectra C(`) of these observed fields
can be written as

ij
CAB
(`) =

X
`

G2 (`, θmin , θmax ) Cδijobs E (`)

(13)

h
i
ij
ij
G4,± (`, θmin , θmax ) CEE(`) ± CBB(`) ,with Gn analytic functions detailed in Refs. [129, 130] .

(9)where δm is the 3D matter density contrast. The galaxy density contrast δg is related to δm via a linear galaxy bias bi . The
tomographic lens efficiency isWκj (χ) =Two-point statistics

ij
ij
ji
ij
CEE
(`) =Cκκ(`) + CκI(`) + CκI(`) + CIijE IE (`)
E
Ewith the convergence fieldκj (n̂) =

1.

(10)

χ is the comoving distance, χH the comoving distance to thehorizon, ns (χ) the source galaxy number density distribution,
and a(χ) the scale factor.

B.

Parameterization and Priors

We sample the posterior of these measurements in two cosmological models: flat ΛCDM and wCDM, with the sum of
the three neutrino masses as a free parameter. ΛCDM contains
three energy densities in units of the critical density: the total
matter density Ωm , the baryonic density Ωb , and the massive
neutrino density Ων . We vary Ων h2 , where h is the Hubble
parameter, as a free parameter, while noting that it is often
fixed in other cosmological analyses to be zero or to the minimum mass allowed by oscillation experiments mν = 0.06 eV[170].
The other cosmological parameters we vary within ΛCDM
are the Hubble parameter h, the amplitude of primordial scalardensity perturbations As , and the spectral index ns of thepower spectrum. We assume a flat model, with ΩΛ = 1 − Ωm .
In wCDM, we allow for a free dark energy equation-of-stateparameter w that is constant in time (in ΛCDM, this is fixed
to w = −1, corresponding to a cosmological constant). Thus
ΛCDM includes six free cosmological parameters and wCDM

12contains seven. The prior ranges for cosmological parametersin Table I are either motivated by physical constraints (e.g., an
accelerating universe requires w < −1/3), or for parametersthat are not strongly constrained by the DES data, typicallygiven a range that encompasses five times the 68% C.L. fromrelevant external constraints. In analyses that sample external CMB likelihoods, we include the optical depth τ as a freeparameter.
We will typically refer to the amplitude of density perturbations at z = 0 in terms of the RMS amplitude of mass onscales of 8h−1 Mpc in linear theory, σ8 . The constraints on
the amplitude and density of matter fluctuations are degenerate in our analysis, and we will also refer to the parameter
S8 , which describes the width of the posterior in the direction
roughly orthogonal to the primary degeneracy direction forcosmic shear in the σ8 –Ωm plane [171], though this does nothold exactly for 3×2pt and changes with effective redshift.
In addition to these cosmological parameters, our fiducial
analysis includes an additional 25 free parameters, for a total of 31 (32) parameters in ΛCDM (wCDM). These additional parameters describe astrophysical and systematic contributions to the measured signal. The effective linear galaxy
bias of lens galaxies in each redshift bin is parameterized
by a scalar bi . We also test and apply a nonlinear galaxy
bias model (with one extra free parameter per redshift bin)[119, 140, 141], which is described in App. E 2. The intrinsic alignment of galaxies [172, 173] is modeled with the Tidal
Alignment and Tidal Torquing (TATT) model [174], which isparameterized by an amplitude ai and redshift power-law ηiparameter (with redshift pivot z0 = 0.62), for each of the (1)tidal alignment- and (2) tidal torquing-sourced terms in themodel, as well as an effective source galaxy bias parameterbTA , which is described in further detail in Refs. [129, 174].
The TATT model contains the commonly employed nonlinear linear alignment (NLA) model in the a2 = bTA = 0
subspace. The amplitude of the lens magnification term in
Eq. 6 depends on the slope of the lens sample’s luminosity
and size distribution at the sample detection limit. The corresponding parameter Cli is calibrated from the data, as described in Ref. [139], and held fixed to that value. Nonlocal effects in γt can significantly contaminate larger angular scaleswith nonlinear information due to integration of the projectedmass within a given angular separation from the center of thehalo. This is mitigated by analytically marginalizing over afree point-mass contribution to γt in all analyses [175].
Photometric redshift systematics are parameterized by anadditive shift to the mean redshift of each bin, ∆zli for lenses
and ∆zsi for sources, where the true redshift distribution isrelated to the photometric redshift distribution npz such thatni (z) = nipz (z − ∆z i ).

true shear γj in each bin by
eij = (1 + mi )γji .

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The full set of parameters (cosmological, astrophysical, andsystematic) and their priors are summarized in Table I.
Differences in the redMaGiC analysis are described inApp. C.
C.

Likelihood Analysis

Our likelihood analysis uses two independently developed
analysis and inference pipelines, C OSMO SIS [176] and C OS MO L IKE [177], which have been validated against one anotherto ensure they produce consistent predictions of the observables and final cosmological constraints. A comparison of thetheory predictions from C OSMO SIS and C OSMO L IKE is presented in Ref. [129]. The residual offset of χ2 < 0.2 between
3×2pt model data vectors obtained from both codes in thisanalysis at a reference cosmology is found to have negligible
impact on parameter constraints and we conclude that both
pipelines can be used interchangeably.
1.

CosmoSIS

This pipeline uses the CAMB Boltzmann code [178, 179]
to compute underlying background quantities and the linear
matter power spectrum, and the H ALOFIT [180] version presented in Ref. [181] for the non-linear power spectrum. It
then generates theory predictions following the model described in section IV A, and using the Fast-PT method [182]for non-linear galaxy bias and the TATT model for intrinsicalignments. Non-Limber integrals are computed following
the method of [169]. Accuracy parameters throughout thepipeline are chosen by requiring the log-likelihood to differ
by less than 0.05 from a high precision calculation. For chains
including Planck CMB measurements [13], we use the Planck2018 public likelihood code [183].5
The version of C OSMO SIS6 used for the analysis may befound in the des-y3 branch of the repositories. The C OS MO SIS runs presented here use the PolyChord sampling
method [184, 185], for both posterior samples and Bayesianevidence. The shear calibration values mi are used as fastparameters. The PolyChord parameters we use for our
fiducial runs are: fast_fraction = 0.1, live_points = 500,
num_repeats = 60, tolerance = 0.01, and boost_posteriors= 10.0.

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In addition, differences in the width of the lens redshift distribution are important at DES Y3 precision, which we parameterize by a stretch σzi , such that
ni (z) = σzi nipz σzi [z − hzi] + hzi .
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Finally, uncertainty in the shear calibration bias is parameterized by mi , where the measured ellipticity ej is related to the

5 While running the Planck CLIK lensing likelihood from plc-3.0, discrep-

ancies between the constraints obtained using the likelihood code and the
publicly released chains were found. The disagreement has been identified
to originate from the treatment of the linear correction term to the theoryφφ
CL
spectrum. This has been corrected in the upstream plc-3.01 release.
6 https://bitbucket.org/joezuntz/CosmoSIS
https://bitbucket.org/joezuntz/
CosmoSIS-standard-library13
Analyses with the C OSMO SIS pipeline also use the nonGaussian covariance matrix from C OSMO L IKE described below.
2.

CosmoLike

This pipeline uses the CLASS Boltzmann code [186] tocompute underlying background quantities and linear andnon-linear matter power spectra, using the H ALOFIT version
presented in Ref. [181] for the latter. The theory predictions
are calculated using the model described in section IV A, relying on the FAST-PT method [182, 187] to evaluate integrals
over perturbation theory kernels for non-linear galaxy bias
and the TATT intrinsic alignment model. The computationof non-Limber integrals for galaxy clustering further employs
the FFTLog implementation of Ref. [169].
The evaluation time of angular two-point statistics in
C OSMO L IKE is optimized through a series of interpolation
schemes, for which runtime-optimized accuracy settings arevalidated through comparison to high-accuracy evaluations
with slow runtime. Most DES Y3 likelihood analyses withC OSMO L IKE employ the EMCEE [188] sampler, c.f. [189] fora detailed comparison of sampler configurations for C OSMO L IKE likelihood analyses.
The C OSMO C OV module [190] of C OSMO L IKE is used
to generate covariances for DES Y3 analyses, which include
Gaussian and non-Gaussian terms [177] and account for theeffect of the survey geometry on shape and shot-noise terms[101].
D.

Tests on Simulations

Our model and many other components of our analysis
have been validated end-to-end on a suite of 18 cosmologicalsimulations7 [119]. Validation is performed on the mean ofmeasurements from all 18 of these simulations without shapenoise, including photo-zs and marginalizing over all cosmological and nuisance parameters. We have verified that we canrecover the correct cosmology with our fiducial analysis towithin approximately 0.3σ in the 2D σ8 –Ωm plane (ΛCDM)
and w–Ωm plane (wCDM). We have shown that for a more
stringent test in the absence of photometric redshift and shear
calibration uncertainties (using true redshifts) our model isable to reproduce the mean ξ± and w + γt measurements fromour 18 simulations with a χ2 of 1.4 for cosmic shear (207 data
points), 4.5 for w(θ) (53 data points), and 9.1 for γt (θ) (232data points). These χ2 numbers are relative to the fiducial
covariance for a single DES Y3 realizations, but with a measurement that is the average of 18 realizations without shapenoise. Thus, they represent the potential systematic χ2 contribution due to model inaccuracies, and shouldn’t be interpreted
as a goodness-of-fit metric.

7 Each simulation assumes a flat ΛCDM cosmology with ΩmΩb = 0.046, h = 0.7, ns = 0.96, and σ8 = 0.82.

= 0.286,

In addition to these model tests, we have also investigated
the systematic uncertainty inherent to our redshift inference
process using these simulations. We have shown that thethree independent source redshift n(z) estimates – SOMPZ,
source–lens clustering, and shear ratios – produce consistentconstraints on the redshift distribution. We have also performed our fiducial analysis using a redMaGiC-like lens sample, assuming source redshift distributions that are calibratedusing the same three methods and lens redshift distributions
estimated from redMaGiC. We found that the constraints fromthis analysis are consistent with those that use the true redshift distributions from the simulation. The final constraining
power is similar between redMaGiC and MagLim, so we do
not repeat the simulated analysis twice.

E.

Quantifying internal and external consistency

To quantify consistency of internal and external data sets,
we define a priori a process to guide decisions and conclusions before seeing the cosmological constraints. For internalconsistency, we calculate the Posterior Predictive Distribution(PPD) [132] and derive a (calibrated) probability-to-exceed p.
In short, the idea is to draw realizations of a particular subsetof the data vector for model parameters drawn from the posterior of the same subset (goodness-of-fit tests) or a disjoint
subset (consistency tests). These realizations are then compared to actual observations and a distance metric is computedin data space, which is then used to compute the p-value. We
test the goodness-of-fit for the two combinations of two-point
functions, ξ± and w + γt , and the combination of all three
after confirming they are mutually consistent. In all cases, we
require as part of the unblinding criteria defined in App. D thatp > 0.01. The validation of the use of PPD for these tests is
described in Ref. [132].
To quantify consistency with external experiments, we have
explored a variety of metrics in order to calibrate expectations,which are described in Ref. [133]. We studied the particular case of quantifying consistency between DES 3×2pt andPlanck CMB using both simulated DES Y3 data and real Y1
data. The consistency metrics can be divided into two categories: parameter-based, which measure relative deviations in
the multi-dimensional parameter space, and Evidence-based,
which also account for how well the individual and combined
data sets fit the model. We discuss results in terms of at least
one metric from each category: the parameter difference andSuspiciousness [191], along with the Evidence ratio to compare to DES Y1. These metrics can produce a probability-toexceed, and we require the same criterion of p < 0.01 to conclude there exists evidence for inconsistency between probes.
Detailed results from these consistency studies are shownin App. F.

14

1.00
Fid. 3×2pt
ΛCDM-Opt. 3×2pt
ξ±γt + w

0.95
0.90

σ8

0.85
0.80
0.75
0.70
0.65
0.60
0.875

Ωm

ΛCDM

0.850
0.825

S8

0.800
0.775
0.750

ing (w + γt ). It is logical to compare these two subsets because each can constrain the ΛCDM parameters to a similar
precision, yet the constraints from the two subsets contain independent information, since they are sensitive to the underlying matter density fluctuations in different ways. Moreover,comparing the two subsets of the 3×2pt measurements provides an internal consistency test.
We must first check that the ξ± and w + γt measurements
are a good fit to the data, mutually consistent, and that their
combination (3×2) is a good fit to the model. For each of
the two measurements individually, we find a PPD result formodel goodness-of-fit p(ξ± ) = 0.21 and p(w + γt ) = 0.02.
The PPD result for consistency between the two model constraints is p(ξ± |w + γt ) = 0.30, meaning that it is appropriate
to combine ξ± and w + γt . The joint 3×2pt goodness-of-fit
is p(ξ± + γt + w) = 0.04. Finally, the shear-ratio data has
goodness-of-fit p = 0.03 in this joint best-fit model. All ofthese p values meet our original criterion of p > 0.01 definedin App. D.
The marginalized constraints from each probe and the
3×2pt combination in the parameter space spanned by σ8 , S8 ,and Ωm are shown in Fig. 6. This is also summarized in Table
II and Fig. 7, which show the numerical constraints on thesethree parameters. The DES Y3 3×2pt constraints on the key
parameters are

0.725

S8 = 0.776+0.017−0.017 (0.776)

0.700Ωm = 0.339+0.032−0.031 (0.372)
0.20

0.25

0.30
0.35
Ωm

0.40

0.45

FIG.
p 6. Marginalized constraints on the three parameters σ8 , S8 =σ8 Ωm /0.3, and Ωm in the ΛCDM model from cosmic shear (ξ± ,blue), galaxy clustering and galaxy–galaxy lensing (γt + w(θ), orange) and their combination (3×2pt, solid black). We also showa ΛCDM-optimized 3×2pt analysis that is valid for ΛCDM using
smaller angular scales in cosmic shear (dashed black). The marginalized contours in this and further figures below show the 68% and
95% confidence levels. The top and side panels show 1D marginalized constraints with the 68% confidence region indicated.
V.

DES Y3 RESULTS: PARAMETER CONSTRAINTS
A. ΛCDM

The principal cosmological test of the DES Y3 data is tocompare our data to the currently favored ΛCDM model. The
model has six cosmological parameters, but also 25 nuisanceparameters for a total of 31 free parameters (listed in Table I)in our fiducial analysis with the MagLim lens sample. Recall
also that, in the fiducial analysis, we use all four source-galaxybins, but only the first four (out of six total) MagLim lensgalaxy bins.
We first concentrate on comparing two subsets of 3×2pt
measurements: those from cosmic shear (ξ± ) and those fromthe combination of galaxy clustering and galaxy–galaxy lens-(17)

σ8 = 0.733+0.039−0.049 (0.696).
The 3×2pt contours in these parameters are not centered on
the overlap of ξ± and γt + w due to degeneracies in the higher
dimensional parameter space.
We also perform two alternative 3×2pt analyses that use
smaller scales. First, we perform a ΛCDM-optimized analysis
that includes smaller-scale information in cosmic shear. This
analysis meets our parameter bias requirements in ΛCDM
(i.e., Sec. IV), but not wCDM. The 3×2pt results from the optimized analysis are shown in the row labeled ‘ΛCDM-Opt.’
in Table II and Fig. 7. The optimized results are consistent
with the fiducial analysis, but are about 30% more constraining in the 2D marginalized Ωm –σ8 plane. The second alternative analysis utilizes a more complicated nonlinear bias modelin order to model smaller scale information in γt + w(θ), andis described in App. E. The 3×2pt results from the nonlinear
analysis, shown in the row labeled ‘NL bias’ in Table II and
Fig. 7, are consistent with the fiducial analysis and lead to anincrease of 15% in constraining power in the Ωm –σ8 plane.
While we found no significant evidence of internal inconsistency with the ΛCDM model using the final MagLim lensselection, we have identified potential systematic modes inthe data at high redshift for the MagLim sample and at all
redshifts for the redMaGiC sample. We had agreed before
seeing any cosmological results that we would pursue potential systematics in the case where the results failed to sufficiently fit any of the models considered in this work (ΛCDM
and wCDM) at p < 0.01. Including MagLim lens bins 5

15
3×2pt
3×2pt (ΛCDM-Opt.)
3×2pt (NL Bias)
3×2pt (Hyperrank)
γt +w(θ)ξ±

DES Y1 3×2pt
3×2pt + BAO + SNe
Ext. Low-z
Planck (no lensing)
3×2pt + Local h
3×2pt + BAO + BBN
3×2pt + Ext. Low-z
3×2pt + Planck
3×2pt + All Ext.
0.7

0.8
0.9
S8 = σ8 (Ωm /0.3)0.5

0.25

0.3
Ωm

0.35

0.7

0.8
σ8

0.9

FIG. 7. Summary of marginalized constraints (mean and 68% CL) and maximum posterior values (crosses) on S8 , Ωm , and σ8 in ΛCDM.
‘Ext. Low-z’ data consists of external SNe Ia, BAO, and RSD, while ‘All Ext.’ data consists of external SNe Ia, BAO, RSD, and Planck
CMB with lensing. The top section shows constraints using only DES data, the middle section only external data, and the bottom section
combinations of DES and external data.

2.5

Maglim (bg )Galaxy bias bg

redMaGiC (bg )

2.0

1.5

Effective ai

10Maglim (a1 )−1Maglim (a2 )redMaGiC (a1 )redMaGiC (a2 )

0.2

0.4

0.6
Redshift z

0.8

1.0

FIG. 8. Constraints on the galaxy bias (bg ) and effective intrinsic alignment (IA) amplitude from tidal alignment (a1 ) and tidal
torquing (a2 ) are shown per redshift bin. Constraints using both lenssamples (MagLim and redMaGiC) are shown. The galaxy bias is
expected to be different for both lens samples, but the IA amplitude
constraints, which are a property of the source galaxy sample, areconsistent. We do not necessarily expect a1 and a2 to be consistent
with one another. We sample over a power-law evolution of the IA
amplitude, so the redshift evolution is forced to be smooth in ai .

and 6 caused a very poor model fit to both models, withp ≈ 5×10−4 . Based on this criterion, we applied a high-z cut
to limit the MagLim sample to approximately the same redshift range of redMaGiC post-unblinding. This change is discussed further in App. D. The two lens samples are comparedand further details of this are discussed in Sec. V C, but allissues that have been uncovered appear to be mostly orthogonal to the 3×2pt ΛCDM parameter dimensions — that is,they do not significantly impact the inferred cosmological parameters, and the cosmological parameters inferred from thetwo lens samples are consistent. This resilience of the 3×2pt
combination of data and its ability to self-calibrate potential
systematics in a subset of the two-point functions is one of
the main motivations for pursuing this cosmological probe forlarge-scale structure.
We find that the DES Y3 3×2pt analysis is able to add information beyond the prior for 15 parameter dimensions in
the model, three of which are cosmological. The cosmological modes that DES 3×2pt most improves with respect to theprior are obtained with the Karhunen-Loève decomposition of
the posterior and prior covariance, and are:p1 = σ8 Ω0.77= 0.317+0.015m−0.014 ,p2 = Ωm σ8−1.16 = 0.49+0.16−0.15 ,

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p3 = hn1.24Ωb−0.39 = 2.11+0.45
s−0.42 .
The combined 3×2pt data is also able to simultaneously constrain a variety of ‘astrophysical’ parameters that encode howgalaxies are connected to the underlying dark matter perturbation field, namely the linear and nonlinear bias parameters andintrinsic alignment of galaxies. Constraints for these model
parameters are shown in Fig. 8. We find slightly higher galaxy
bias constraints for redMaGiC galaxies than in the DES Y1

16

−0.4
−0.6

A value of R greater than unity implies that the wCDM model
is not favored. We find R = 4.3. This indicates that the lateuniverse large-scale structure probed by DES does not show
evidence of needing the more complex dark energy density
scenario of the wCDM model. We discuss further in Sec. VII
more stringent tests of the ΛCDM model that leverage data
across the age of the Universe.

Fid. 3×2pt

wCDM

−0.8

C.

w

−1.0
−1.2
−1.4
−1.6
−1.8
−2.0
0.20

0.25

0.30

0.35
Ωm

0.40

0.45

FIG. 9. Marginalized constraints on the two parameters Ωm and w
in the wCDM model from DES Y3 3×2pt. A dotted line indicatesw = −1 as given by the cosmological constant.

analysis using a similar redMaGiC sample. We find a preference for a slightly smaller intrinsic alignment amplitude than
DES Y1. This value is consistent with the DES Y1 analysis,but is also consistent with zero intrinsic alignment.

B. wCDM

We also fit our data to the wCDM model, in order to test forevidence that the dark energy equation of state departs fromits cosmological-constant value of w = −1. In wCDM, thedark energy density evolves with time with a constant w, suchthat ρDE ∝ (1 + z)3(1+w) . We show marginalized parameter
posteriors for this model in Fig. 9 and parameter values inTable III and Fig. 10.
We find similar levels of agreement between ξ± or γt +w(θ)
as in ΛCDM, and a similarly good fit to the data within thewCDM model, but do not show constraints from these subsets of the data due to increased prior influence and parametervolume effects. The DES Y3 3×2pt constraint on the matter
density and dark energy equation of state parameter are+0.035Ωm = 0.352−0.041(0.339),w = −0.98+0.32−0.20 (−1.03).

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To determine if there is a preference for the wCDM modelover the ΛCDM model, we compute the Bayes factorR=

P (D̂|ΛCDM)
P (D̂|wCDM)

.

(20)

Lens sample comparison

As optical surveys cover larger fractions of the sky andprobe higher redshifts, photometric galaxy clustering becomes both more powerful and more difficult to calibrate. Previous DES analyses used a luminous red sample of galaxies
with constant comoving density, redMaGiC. To ensure robustness, we pursued two lens samples for DES Y3: a magnitudelimited lens sample, MagLim, and the redMaGiC sample. The
redMaGiC sample was optimized for better understood andsmaller photometric redshift errors. The MagLim sample was
optimized for wCDM constraints, balancing increased number density vs. less well-constrained photo-zs, while allowing selection to higher redshifts than possible with redMaGiC.
Comparing the inferred cosmological parameters of our models from these two very different samples, which have fewer
than 20% overlapping objects, allows us to infer potential uncorrected systematics from lens sample selection or photo-z
calibration of the lenses.
Measurements based on the second redshift sample, redMaGiC, also have an acceptable overall model fit to theΛCDM and wCDM models. The cosmic shear in this model isalso consistent with the combination of galaxy–galaxy lensing
and galaxy clustering. These measurements and model fits areshown in App. C. We find a PPD result for model goodnessof-fit p(ξ± ) = 0.25 and p(w + γt ) = 0.04, while the PPD
result for consistency between the two model constraints isp(ξ± |w + γt ) = 0.02. The joint 3×2pt goodness-of-fit isp(ξ± + γt + w) = 0.02.
The marginalized constraints of each individual probe andthe 3×2pt combination on S8 and Ωm in ΛCDM are shown in
Fig. 11, while they are shown for S8 , w, and Ωm in wCDM in
Fig. 12. Both figures compare redMaGiC results to the fiducial 3×2pt using the MagLim sample. As described above,the cosmic shear or γt + w(θ) data alone are consistent withthe 3×2pt model fit, though the γt + w(θ) data on their own
prefer a smaller S8 value. This arises from the strong degeneracy between σ8 and galaxy bias in γt + w(θ). Alone, it
prefers a lower value of σ8 and higher value of galaxy bias.
Adding cosmic shear information effectively fixes the valueof S8 along that degeneracy, which brings the galaxy bias in
3×2pt back down to a value more consistent with the DES
Year 1 redMaGiC galaxy bias constraints.
These results are consistent with the MagLim results andpassed our unblinding requirements, including having a sufficiently good model fit to ΛCDM. However, after unblinding
the results with redMaGiC we found evidence of internal tension in the data. Because w(θ) is not able to constrain cosmology on its own, this has limited impact on the combination

17

TABLE II. Summary of marginalized parameter constraints in ΛCDM. The mean and 68% CL are provided for each cosmological parameter,
followed by the maximum posterior value in parentheses, except for neutrino mass, for which the 95% upper bound is given. Parameters thatare not significantly constrained are indicated by a dash. All data have been re-analyzed with model and prior choices matching the DES Y3
3×2pt analysis.
ΛCDM

S8

Ωm

σ8

Ωb

ns

h

P

mν (eV) FoMσ8 ,Ωm

DES Data
3×2pt+0.032+0.0390.776+0.017−0.017 0.339−0.031 0.733−0.049(0.776)(0.372)(0.696)

–
–

–
–

–
–

–
–

2068

3×2pt (ΛCDM-Opt.)+0.028+0.0340.779+0.014−0.015 0.333−0.029 0.741−0.042(0.781)(0.352)(0.721)

–
–

–
–

–
–

–
–

2765

3×2pt (NL Bias)+0.028+0.0400.785+0.018−0.016 0.327−0.028 0.754−0.044(0.784)(0.324)(0.755)

–
–

–
–

–
–

–

γt +w(θ)+0.034+0.0630.778+0.031−0.037 0.320−0.041 0.758−0.074(0.809)(0.306)(0.801)

–
–

–
–

–
–

–

ξ±

+0.039+0.0730.759+0.025−0.023 0.290−0.063 0.783−0.092(0.755)(0.293)(0.763)

–
–

–
–

–
–

–

DES Y1 3×2pt+0.034+0.0520.747+0.027−0.025 0.303−0.041 0.747−0.068(0.770)(0.253)(0.838)

–
–

–
–

–
–

–

3×2pt + BAO + SNe+0.020+0.037+0.0040.777+0.018−0.017 0.318−0.025 0.756−0.039 0.041−0.010(0.765)(0.333)(0.726)(0.031)

–
–

–
–

–
–

2942

0.049+0.001−0.001(0.050)––0.667+0.009−0.010(0.661)–––

Ext. Low-z+0.012+0.033+0.0140.831+0.042−0.038 0.293−0.012 0.840−0.033 0.053−0.006(0.811)(0.293)(0.820)(0.032)

–
–

–
–

–
–

2325

Planck (no lensing)+0.008+0.024+0.001+0.004+0.0130.827+0.019<0.35−0.017 0.327−0.017 0.793−0.009 0.051−0.002 0.964−0.005 0.665−0.006(0.830)(0.315)(0.810)(0.049)(0.968)(0.674)(95% CL)4217

3×2pt + Local h0+0.026+0.0280.780+0.016−0.018 0.324−0.023 0.752−0.044(0.775)(0.366)(0.702)3×2pt + BAO + BBN

2379
927
740
1085

External Data
Ext. BAO + BBN––

0.295+0.014−0.017(0.293)––

Combined Data––

0.731+0.012−0.013(0.748)––2720+0.011+0.026+0.0010.786+0.017−0.016 0.314−0.014 0.769−0.027 0.048−0.001(0.809)(0.296)(0.815)(0.048)––0.676+0.009−0.009(0.673)––5484

3×2pt + Ext. Low-z+0.007+0.019+0.0120.802+0.014−0.013 0.302−0.009 0.800−0.019 0.050−0.004(0.811)(0.295)(0.817)(0.034)––0.702+0.101−0.056(0.591)––8414

3×2pt + Planck (no lensing)+0.006+0.030+0.001+0.004+0.0150.804+0.013<0.43−0.009 0.320−0.019 0.779−0.008 0.050−0.002 0.967−0.004 0.669−0.005(0.812 b.f.) (0.318 b.f.) (0.788 b.f.) (0.050 b.f.) (0.969 b.f.) (0.670 b.f.) (95% CL)6074

3×2pt + All Ext.

+0.004+0.008+0.0005+0.004+0.0040.812+0.008<0.13−0.008 0.306−0.005 0.804−0.005 0.0487−0.0004 0.969−0.003 0.680−0.003(0.815)(0.306)(0.807)(0.0486)(0.967)(0.681)(95% CL)

34041

–
–

18

3×2pt
3×2pt (NL Bias)
3×2pt + BAO + SNe
Ext. Low-z
Planck (no lensing)
3×2pt + Local h
3×2pt + BAO + BBN
3×2pt + Ext. Low-z
3×2pt + Planck
3×2pt + All Ext.
0.7

0.8
0.9
S8 = σ8 (Ωm /0.3)0.5

0.25

0.3
Ωm

0.35

-1.5

-1.0
w

-0.5

FIG. 10. Summary of marginalized constraints (mean and 68% CL) and maximum posterior values (crosses) on S8 , Ωm , and w in wCDM.
‘Ext. Low-z’ data consists of external SNe Ia, BAO, and RSD, while ‘All Ext.’ data consists of external SNe Ia, BAO, RSD, and Planck
CMB with lensing. The top section shows constraints using only DES data, the middle section only external data, and the bottom section
combinations of DES and external data.

TABLE III. Summary of marginalized parameter constraints in wCDM. The mean and 68% CL are provided for each cosmological parameter,
followed by the maximum posterior value in parentheses, except for neutrino mass, for which the 95% upper bound is given. Parameters thatare not significantly constrained are indicated by a dash. All data have been re-analyzed with model and prior choices matching the DES Y3
3×2pt analysis.
wCDM

S8

Ωm

σ8

Ωb

ns

h

w

P

mν (eV) FoMσ8 ,Ωm FoMw,Ωm

DES Data
3×2pt+0.035+0.0370.775+0.026−0.024 0.352−0.041 0.719−0.044(0.780)(0.339)(0.733)

–
–

–
–

–
–

−0.98+0.32−0.20(−1.03)

–
–

1123

115

3×2pt
(NL Bias)+0.036+0.0410.767+0.030−0.023 0.312−0.034 0.756−0.053(0.775)(0.312)(0.760)

–
–

–
–

–
–

−1.24+0.34−0.22(−1.23)

–
–

1159

117

3×2pt
+ BAO + SNe+0.021+0.0350.794+0.020−0.019 0.330−0.022 0.759−0.034(0.771)(0.345)(0.719)

–
–

–
–

–
–

−0.84+0.11−0.10(−0.86)

–
–

2426

455

–
–

–
–

−0.93+0.05−0.04(−0.92)

–
–

2289

1817

<0.46
(95% CL)

957

225

External Data
Ext. Low-z+0.012+0.033+0.0130.832+0.035−0.038 0.287−0.013 0.850−0.037 0.057−0.003(0.821)(0.282)(0.847)(0.059)Planck(no lensing)+0.015+0.071+0.002+0.005+0.089+0.17
0.794+0.025−0.029 0.251−0.056 0.876−0.038 0.039−0.009 0.964−0.004 0.768−0.035 −1.36−0.26(0.779)(0.199)(0.956)(0.031)(0.962)(0.848)(−1.56)

Combined Data
3×2pt
+ Local h0+0.032+0.0290.778+0.025−0.028 0.328−0.029 0.747−0.037(0.803)(0.341)(0.753)––+0.29
0.731+0.013−0.013 −1.00−0.25(0.724)(−0.85)

–
–

1550

167

3×2pt
+ BAO + BBN+0.013+0.027+0.0030.790+0.019−0.020 0.318−0.015 0.768−0.027 0.049−0.003(0.809)(0.314)(0.790)(0.051)––+0.10
0.669+0.017−0.020 −0.97−0.08(0.657)(−0.92)

–
–

3733

919

3×2pt
+ Ext. Low-z+0.008+0.021+0.0110.803+0.016−0.015 0.301−0.008 0.801−0.022 0.051−0.004(0.824)(0.302)(0.821)(0.032)––+0.05
0.709+0.097−0.049 −1.00−0.04(0.576)(−0.97)

–
–

7941

2662

+0.004+0.036+0.128+0.020+0.046+0.004<0.45
3×2pt + Planck 0.800+0.015−0.011 0.303−0.037 0.798−0.026 0.047−0.005 0.966−0.004 0.691−0.031 −1.090−0.113(no lensing) (0.797 b.f.) (0.314 b.f.) (0.779 b.f.) (0.050 b.f.) (0.969 b.f.) (0.674 b.f.) (−1.009 b.f.) (95% CL)

2634

465

+0.006+0.010+0.001+0.003+0.006+0.0300.812+0.008<0.17−0.008 0.302−0.006 0.810−0.009 0.048−0.001 0.968−0.003 0.687−0.007 −1.031−0.027(0.802)(0.298)(0.804)(0.048)(0.972)(0.686)(−1.001)(95% CL)

21216

7421

3×2pt
+ All Ext.

–
–

19
0.95

Fid. 3×2pt
RedMaGiC 3×2pt
RedMaGiC 3×2pt, Free Xlens
RedMaGiC ξ±
RedMaGiC γt + w

ΛCDM
0.90

0.85

Fid. 3×2pt
RedMaGiC 3×2pt
RedMaGiC 3×2pt, Free Xlens

0.90

S8

0.85

S8

0.80

0.80

0.75

0.75

0.70

0.70

0.65
−0.4

0.65

Ωm

wCDM

−0.6

0.60
0.20

0.25

0.30
Ωm

0.35

0.40

−0.8

0.45

FIG. 11. A comparison of the marginalized ΛCDM constraints of thetwo lens samples. Dashed contours show the cosmic shear (blue),galaxy–galaxy lensing and clustering (orange), and 3×2pt (black)constraints based on the redMaGiC lens sample. The 3×2pt redMaGiC constraints marginalizing over a free Xlens parameter are
also shown (dotted black), and the 3×2pt MagLim constraints (solid
black). The inferred cosmological parameters from 3×2pt are consistent in all three cases.

w

−1.0
−1.2
−1.4
−1.6
−1.8
−2.0

of galaxy clustering and galaxy–galaxy lensing and no discernible impact on the 3×2pt combination in ΛCDM. Nevertheless, it is important to understand the source of this internaltension in redMaGiC results and judge its impact on cosmological inference. To do so, we modeled this inconsistencyof the redMaGiC clustering and galaxy–galaxy lensing amplitudes with a systematic parameter Xlens , which is related tothe connection of the galaxy–galaxy lensing and galaxy clustering two-point functions to the matter two-point function:ii
wii (θ) = b2i ξmm(θ)
ij
γtij (θ) = Xlens bi ξmm
(θ)

(21)

where bi is the galaxy bias connecting the observable γt orw(θ) to the matter correlation function (ξmm ) or spectrum and
Xlens is the same for all redshift bins i. We expect Xlens =1 in ΛCDM, if there are no systematic contributions to thesignals.
We show the result of marginalizing over a free Xlens in
the redMaGiC 3×2pt analyses in Figs. 11 and 12. We finda negligible impact on the primary cosmological parametersin ΛCDM, particularly S8 . We find Xlens = 0.877+0.026−0.019 ,strongly inconsistent with Xlens = 1 in ΛCDM. If we fix
Xlens to this value in the redMaGiC γt + w(θ) analysis, thecontour in Fig. 11 shifts upward to agree with cosmic shear in
S8 . The value of Xlens is correlated with the equation-of-state

0.20

0.25

0.30

0.35

0.40

0.45

Ωm

FIG. 12. A comparison of the marginalized wCDM constraints ofthe two lens samples. Dashed black contours show the 3×2pt constraints based on the redMaGiC lens sample. The 3×2pt redMaGiC
constraints marginalizing over a free Xlens parameter (dotted black)and the 3×2pt MagLim constraints (solid black) are also shown. The
inferred cosmological parameters from 3×2pt are generally consistent, but in particular the redMaGiC results are sensitive to the impact
of Xlens in wCDM, showing substantial shifts in the inferred parameter values.

parameter w, so the redMaGiC wCDM constraint is strongly
affected by this potential systematic. Adding the single freeparameter Xlens in ΛCDM leads to an improvement in χ2 of25, while adding a free w leads to an improvement in χ2 of 7.
Thus, Xlens clearly leads to a better model fit.
After unblinding the results with redMaGiC, but before unblinding those with MagLim, we decided to use the MagLim
sample for our fiducial cosmological analysis if it showed no
indication of this scale- and redshift-independent effect thatis present in redMaGiC. This potential systematic was studied at length between the unblinding of the redMaGiC sample
and the MagLim sample. Studies of this effect are discussed
in much more detail in [136, 139, 140]. We have demonstrated
that the effect (and its relative impact vs. the clustering amplitude of the MagLim sample) is roughly independent of redshift, angular scale, or position in the survey footprint.

20
1.

Summary of possible non-systematic causes of Xlens 6= 1

There are several classes of non-systematic explanations for
Xlens , all of which we believe are implausible given our data.
These possible explanations are:
Stochastic Bias: While the effect of Xlens on clusteringand galaxy–galaxy lensing looks very similar to stochasticity,a decorrelation between the galaxy and matter distributions,predictions from galaxy bias models make this interpretation
unlikely. In configuration space, perturbative stochastic terms
are expected to contribute only at small separations r ∼ R?
(the Lagrangian size of halos), and to statistics that involvezero-lag correlators [67].
“Lensing is low”: Ref. [90] reported that the γt signal
around luminous red galaxies is lower than expected from amodel conditioned on their autocorrelation, which resembles
the pattern seen in our redMaGiC sample. But with the possible exception of the large scale results in Ref. [192, 193],
the “lensing is low” result [90] applies to models that fit toscales sensitive to complexities of the small scale dark mattergalaxy connection. There is still debate within the lensingis-low literature as to whether the effect can be accounted forby additional complexity in these small scale models [194].
The DES Y1 results (which also used a redMaGiC sample)
do not support the lensing is low scenario, nor do the DES
Y3 results for MagLim in the redshift range of the redMaGiC
sample. The DES Y3 results for the redMaGiC sample showwhat could be interpreted as galaxy–galaxy lensing being 1015% percent lower than galaxy clustering at fixed cosmology(Planck 2015 in the case of Ref. [90]; DES 3×2pt cosmology
in the DES Y3 results for Xlens ). However, the more plausible cause is that the clustering of the Y3 redMaGiC sample
is anomalously high, as indicated by internal consistency testsof the individual data vectors. While we are still studying theXlens < 1 anomaly, we currently do not believe that it supports a conclusion that galaxy–galaxy lensing is ‘low.’
Fundamental physics: Any dynamical modifications to either the Poisson equation or the shear equation generally
changes the galaxy and matter distributions but their correlation is maintained, i.e. Xlens = 1 is maintained at linear
scales. Beyond this possibility, any separation of the impactof relative ‘bias’ between the two types of matter (apparent inlensing vs. clustering) at the level of 15% would require significant fluctuations in the dark matter field, which would havesubstantial ramifications in other observables that we have notseen. Therefore, we conclude that a fundamental-physics explanation for Xlens < 1 would probably have to be very finetuned.

2.

Potential systematics in w(θ) and γt vs. Xlens 6= 1

We now continue discussing the Xlens 6= 1 anomaly bycomparing redMaGiC to MagLim, and commenting on potential systematics in galaxy clustering and galaxy–galaxy lensing as the cause of the anomaly.
We find the redMaGiC sample shows Xlens < 1 at highsignificance at all scales and redshifts. The highest two red-shift bins of the MagLim sample, which have been removed
from the analysis, also indicate Xlens < 1 at high significance,
which is clearly visible in the model fit in those two bins ofFig. 2. In the redshift range overlapping the redMaGiC sample, we find no evidence of a non-unity Xlens for MagLim.
We discard the two high redshift bins for the MagLim sample as a conservative choice. Based on our investigations so
far and current understanding of theoretical extensions beyond
wCDM, we do not believe these anomalies are indications of
new physics. We have found plausible but unverified indications that the origin may lie in potential systematics, e.g., associated with the photometric uncertainty or background subtraction for large or faint objects, or in the de-reddening process.
These issues are the subject of ongoing investigations,which will be crucial for understanding photometric clustering and its combination with galaxy–galaxy lensing in DES
Y6 and beyond. However, while these measurements are potentially impacted at a level we can measure by some as yet
unidentified systematic, this does not have a significant impact
on ΛCDM cosmology when the three two-point functions are
combined within 3×2pt. For the MagLim sample our tests indicate that both the ΛCDM and wCDM constraints are robust.
This self-calibration effect is one of the primary motivations
for combining these different probes of the same underlying
matter density field into the 3×2pt observable.

VI.

COMPARISON WITH OTHER DES DATA

DES has produced competitive cosmological constraints
using its four primary probes: galaxy clustering and weakgravitational lensing (3×2pt), type Ia supernovae (SNe Ia),
galaxy cluster counts and masses, and BAO. Together, theseprobes have been demonstrated to provide dark energy constraints that can be competitive with the best combined external constraints [107]. We describe each of them briefly below.
Type Ia supernovae: The DES SNe Ia sample has 207 spectroscopically confirmed SNe in the redshift range 0.07 <z < 0.85. The sample-building and analysis pipelines are described in a series of papers that detail the SN search and discovery [146, 195, 196]; simulations [197]; photometry [198];
calibration [145, 199]; spectroscopic follow-up [200]; and selection bias [201–203]. The methodology and systematic uncertainties are found in Ref. [204]. These were used to constrain cosmology [205] and the Hubble constant [206]. These
analyses included additional external low-redshift SNe that
we do not use in this analysis. We compute the SNe likelihoodusing a module [207] implemented in C OSMO SIS, which reproduces the results in Ref. [205]. The constraint from only
DES SNe on Ωm is shown in Fig. 13 (purple).
Galaxy clusters: The DES Y1 redMaPPer catalog consistsof ∼6500 clusters with richness larger than λ = 20 in theredshift range z ∈ [0.2, 0.65]. The first cosmological analysis of DES clusters [105] (‘Clusters 1’: orange contours in
Fig. 13), which combines cluster counts data and mass estimates from the stacked weak lensing analysis of [208], founda larger than 2σ tension with the other DES probes. This

21

1.1
Y3 3×2pt+SNe+BAO
Fid. 3×2pt
Y3 SNe Ia
Y1 3×2pt
Y1 Clusters 1
Y1 Clusters 2

1.0

σ8

0.9

0.8

0.7

0.90

Ωm

ΛCDM

0.85

S8

0.80
0.75
0.70
0.65
0.60
0.55
0.1

0.2

0.3
Ωm

0.4

ies from the DES Y3 ‘Gold’ catalog is selected in the redshift
range 0.6 < z < 1.1 [211] and used to measure the scaleof the BAO feature in the distribution of galaxies at an effective redshift zeff = 0.835 [212]. We use a likelihood from
Ref. [212] for the ratio of the angular diameter distance DA
at zeff and the sound horizon distance at the drag epoch, rd ,which is implemented in C OSMO SIS. The simulated galaxy
catalogs used in the analysis to derive the uncertainty of themeasurement are described in Ref. [213]. While the BAO and3×2pt analyses probe common sky area and redshift range,
and the measurements of this work include scales impacted
by the BAO feature, the overlap in galaxy sample is small andthe method for inferring the BAO distance ratio likelihood isinsensitive to cosmology, so we neglect this non-zero correlation when combining the measurements. This will be further
validated in future work that combines and studies all final
DES Y3 probes.
DES Year 1 3×2pt: We reanalyze the DES Y1 3×2pt datain the Y3 model and prior space, but do not update the scale
cuts or marginalize over a free point mass for galaxy–galaxy
lensing. We also make no changes in priors on systematicparameters (e.g., photo-z or shear calibration parameters).
The comparison of these cosmological constraints in
ΛCDM using the DES probes is shown in Fig. 13. The combination of DES Y3 3×2pt, SNe Ia, and BAO data is also
shown in (blue). While the constraint in S8 is driven primarily
by 3×2pt, there is substantial gain in other parameter dimensions due to the additional data. The marginalized parametervalues are summarized in Tables II & III and Figs. 7 & 10.

0.5

FIG. 13. A comparison of the marginalized constraints on parameters in the ΛCDM model from a variety of DES probes: large-scale
structure and weak lensing (3×2pt; Y3 – black solid, Y1 reanalyzed– black dashed), type Ia supernovae (purple), galaxy cluster number
counts and masses (orange and green), and BAO. The combinationof DES Y3 3×2pt, SNe Ia, and BAO is shown in blue. Going from
Y1 to Y3, we find approximately a factor of two improvement in the
3×2pt constraint in Ωm -S8 plane.

is driven by low-richness systems, and has been interpreted
as unmodeled systematics that affect the stacked weak lensing signal of the optically selected sample. This interpretation is supported by the analysis of [111] (‘Clusters 2’: green
contours in Fig. 13), which recovers results consistent withthe other DES probes by combining cluster abundances withthe large-scale auto-correlations of galaxy and cluster positionand cross-correlations of cluster position with galaxy positionand shear from DES Y1 data (4×2pt+N). The conclusions of[105] are further corroborated by the analysis of [209] which
derive cosmological posteriors consistent with [33] by analyzing the DES Y1 redMaPPer cluster abundances, but replacing
the stacked weak lensing mass estimates of [208] with multiwavelength follow-up data from the SPT-SZ 2500 deg2 survey[210].
Baryon acoustic oscillations: A sample of 7 million galax-

VII.

COMPARISON WITH EXTERNAL DATA

It has been demonstrated that various combinations of lowredshift data and high-redshift data from the CMB can independently fit the ΛCDM model. However, the most stringent
tests of the model will come from combining these data andtesting whether the model can simultaneously fit the diverse
set of cosmological probes available to us at all redshifts simultaneously. These data sets are sensitive to the growth ofdensity perturbations, the expansion and geometry of the Universe, or both, and are sourced from a variety of very different physical processes. The combination of the independent external low-redshift probes with DES Y3 data further
reduces the potential impact of any residual systematic effects
in the low-redshift anchor of the test, while the combined DES
probes have been carefully calibrated from the same data andconsistently protected against confirmation bias. Both of these
considerations give us further confidence, for complementaryreasons, in testing the ΛCDM model across the age of the Universe.

A.

External data sets

The likelihoods from data sets external to DES include:
Type Ia supernovae: The Pantheon sample [214] combines the distance measurements from 1048 SNe ranging from

22
0.01 < z < 2.3, supplementing Pan-STARRS1 measurements with other available samples.
Baryonic acoustic oscillations and redshift-space distortions: We use the constraints from SDSS measurements of
BAO and RSD in eBOSS DR16.8 When using constraints onf (z)σ8 (z), where f (z) is the growth rate, from RSD, we use
the released covariance matrices between the constraints fromthe BAO and RSD measurements. These measurements are
expressed in terms of the Hubble distance DH , sound horizon distance rd , comoving distance DM , and volume-average
2
distance DV = (DMDH z)1/3 . The measurements include(from low to high redshift measurements):• The measurement of DV at an effective redshift ofzeff = 0.15 using the Main Galaxy Sample (MGS)[215] and adding to f σ8 measurement from Ref. [216].
• A re-analysed version of BOSS DR12 measurements ofDM and DH from BAO, and f σ8 from RSD, at zeff =0.38 and 0.51 [217].
• The eBOSS DR16 measurements of DM /rd , DH /rdfrom BAO, and adding f σ8 from the full-shape information, at zeff = 0.698 using Luminous Red Galaxies(LRG) [218, 219].
• The eBOSS DR16 measurements of DV /rd when using
BAO alone and DM /rd , DH /rd , f σ8 when using BAO
and the full-shape information, at zeff = 0.845 using
Emission Line Galaxies (ELG) [220],• The eBOSS DR16 measurements of DM /rd , DH /rdfrom BAO, and adding f σ8 from the full-shape information, at zeff = 1.48 using the Quasar Sample (QSO)[221, 222],• The eBOSS DR16 measurements of DM , DH at zeff =2.33 using the Lyman-α forests [223]. This data set
only has information from BAO.
CMB: We use the likelihoods from the Planck 2018 data
release [13, 183]. Our fiducial combination of Planck likelihoods includes:• The Plik likelihood of the temperature power spectrum C`T T in 30 ≤ ` ≤ 2508 and the E-mode power
spectrum C`EE and the cross power-spectrum between
temperature and E-mode C`T E in the range 30 ≤ ` ≤1996.
• The Commander likelihood of the temperature power
spectrum C`T T in 2 ≤ ` ≤ 29.

• The SimAll likelihood of the E-mode power spectrum
C`EE in 2 ≤ ` ≤ 29.
We also use the likelihood of the lensing potential φ power
spectrum C`φφ measured by Planck in the range 8 ≤ ` ≤ 400,
either in combination with our fiducial combination of Planck

8 https://svn.sdss.org/public/data/eboss/DR16cosmo/

tags/v1_0_0/likelihoods/

likelihoods described above or alone. In the latter case we use
the likelihood marginalized over the CMB power spectrum.
Planck CMB will refer to the primary CMB anisotropy data(without lensing) unless otherwise stated.
Big Bang nucleosynthesis: We construct an Ωb h2 constraint
based on observations of damped Lyman-α systems [224].
The primordial deuterium-to-hydrogen ratios measured fromthese systems can be translated to constraints on Ωb h2 via
Big Bang nucleosynthesis (BBN) calculations, but different
assumptions on the BBN physics, in particular on the rateof the d(p, γ)3 He nuclear reaction, yield different final constraints on Ωb h2 . Our constraint conservatively incorporates
the two major categories of such assumptions, namely the theoretical approach presented in Ref. [224] and the experimental
measurement-based approach from Ref. [225]. Specifically,
we adopt the mean and the statistical uncertainty on Ωb h2
from Ref. [225], and in addition introduce a systematic uncertainty defined by the difference between 1) the mean from
Ref. [225] and 2) an inverse-variance weighted average of thetwo respective means [224, 225]. This results in our adopted
constraint of 100Ωb h2 = 2.195 ± 0.028.
Local Hubble parameter: We use a local h prior fromSH0ES [93], which constrains h = 0.732 ± 0.013 using a local distance ladder that depends on measurements of Cepheids
and type Ia supernovae.
We use versions of these likelihoods implemented as modules in C OSMO SIS, which are used to obtain the constraints
presented in the following.

B.

High redshift vs. low redshift in ΛCDM

One of the most stringent tests of ΛCDM is to compare
the prediction of the state of the Universe and amplitude of
perturbations from the epoch of recombination, which we canobserve from the CMB, to the current day, which we observewith low-redshift surveys like DES. At the time of the CMB,
the Universe was very hot and dense, and its physics was dominated by radiation. DES is most sensitive to a period in the
Universe approximately eight billion years later, where perturbations have grown by several orders of magnitude and nonlinear growth is important. DES observes a volume of the Universe spanning nearly nine billion years of its evolution. The
volumes probed by large low-redshift surveys provide significant additional information on potential changes to the evolution of perturbations or growth of the Universe over time,allowing them to strongly test the nature of dark energy.
By taking precise measurements of the ΛCDM model from
CMB observations and predicting what we should observe interms of the amplitude of perturbations or matter density inthe late Universe, we can test whether our observations fromsurveys like DES agree with those predictions. If they do notagree at high significance, we have demonstrated that ΛCDM
cannot describe the full evolution of the Universe. There has
been considerable debate about the tendency of late Universe
measurements to prefer slightly lower matter density or amplitude of clustering relative to measurements from the CMB(e.g., [13, 27, 33, 101, 102]). As more powerful data becomes

23

5.0
4.5
4.0

1.05

Fid. 3×2pt
Planck CMB
Ext. SNe+BAO+RSD
3×2pt+Ext. SNe+BAO+RSD
3×2pt + All Ext.

1.00
0.95

3.5

As × 109

Fid. 3×2pt
Planck CMB
Ext. SNe+BAO+RSD
3×2pt+Ext. SNe+BAO+RSD
3×2pt + All Ext.

S8

0.90

3.0

0.85

2.5
0.80
2.0
0.75
1.5
0.70
Ωm

0.900

−0.6

0.875

−0.8

0.850

−1.0

0.825

Ωm

−0.4

ΛCDM

w

S8

0.925

wCDM

−1.2

0.800

−1.4

0.775

−1.6

0.750

−1.8

0.725
0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450Ωm−2.0
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
Ωm

FIG. 14. A comparison of marginalized constraints from three similarly constraining sets of cosmological probes in ΛCDM. Combined
external BAO, RSD, and SNe Ia data (Ext. Low-z) are shown in orange, the combination of DES galaxy clustering and weak lensing
data (3×2pt) is shown in black, and Planck CMB (no lensing) datais shown in green. The three share a common parameter space inthe Ωm –S8 plane at their 68% CL bounds. The combination of Ext.
Low-z data with DES 3×2pt is shown in purple and this combined
additionally with Planck CMB (w/ lensing) is shown in blue.

available, like the current DES Y3 analysis, we can determine
whether these measurements converge towards or away fromthe Planck CMB prediction.
We compare three similarly constraining and complementary subsets of available cosmological probes in ΛCDM andwCDM in Figs. 14 & 15. The external low-redshift SNe Ia,
BAO, and RSD data primarily constrain Ωm and w, while
the DES 3×2pt data adds substantial information on As or
σ8 , which helps to further constrain Ωm and w throughdegeneracy-breaking of correlated parameters. These external
low-redshift data sets complement the DES weak lensing andlarge-scale structure information by probing the growth andgeometry of the cosmological model in fundamentally different ways. The CMB is able to tightly constrain both Ωm andAs or σ8 in ΛCDM, but is comparable in constraining power
to DES 3×2pt in wCDM, since it primarily has access to in-

FIG. 15. A comparison of marginalized parameter constraints fromthree similarly constraining sets of cosmological probes in wCDM.
Combined external BAO, RSD, and SNe Ia data (Ext. Low-z) are
shown in orange, the combination of DES galaxy clustering andweak lensing data (3×2pt) is shown in black, and Planck CMB (nolensing) data is shown in green. The combination of Ext. Low-z datawith DES 3×2pt is shown in purple and this combined additionallywith Planck CMB (w/ lensing) is shown in blue.

formation limited to the surface of last scattering at z ≈ 1100.
The combination of DES 3×2pt with the other low-redshift
data provides substantial gain in As , σ8 , Ωm , and w. We list
marginalized parameter constraints for these probes in Tables
II & III.

1.

Consistency results

We show the comparison of DES 3×2pt and the Planck
CMB data for the ΛCDM and wCDM models in Figs. 16& 17. Visually, we find better agreement in the overlap of
the marginalized Ωm –S8 parameters with the DES Y3 3×2pt
data than found in the DES Y1 analysis [33], despite substantial improvements to the precision of both DES and Planck
predictions. This is qualitatively unchanged when using themore precise, optimized ΛCDM version of the analysis that

24

0.95
Fid. 3×2pt
Planck CMB
3×2pt+Planck CMB

0.90

Fid. 3×2pt
Planck CMB
3×2pt+Planck CMB

0.90

0.85

S8

σ8

0.85
0.80

0.80

0.75
0.70

0.75

0.65
Ωm

0.70
−0.4

ΛCDM

0.86

−0.6

0.84

−0.8

0.82

−1.0

0.80

w

S8

0.88

Ωm

wCDM

−1.2

0.78

−1.4

0.76

−1.6

0.74

−1.8

0.72
0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450
Ωm

−2.0
0.15

0.20

0.25

0.30
Ωm

0.35

0.40

0.45

FIG. 16. A comparison of the marginalized parameter constraints
in the ΛCDM model from the Dark Energy Survey with predictions
from Planck CMB data (no lensing; green). We show the fiducial
3×2pt (solid black) and the combined Y3 3×2pt and Planck (orange)
results.

FIG. 17. A comparison of the marginalized parameter constraints
in the wCDM model from the Dark Energy Survey with predictions
from Planck CMB data (no lensing; green). We show the fiducial
3×2pt (solid black) and the combined Y3 3×2pt and Planck (orange)
results.

uses more small scale information – the DES contour shrinks,
but asymmetrically in the direction of the CMB prediction.
We evaluate the consistency of the DES and Planck data
in several ways, including shifts in parameter space and theBayesian evidence. These are described further in Sec. IV E
and full results are provided in App. F. We find a parameter difference of 1.5σ (p = 0.13) in the cosmological model
space and a Suspiciousness of 0.7 ± 0.1σ, corresponding to
p = 0.48 ± 0.08. This generally leads to the conclusionthat despite substantially increased precision from both experiments, we find no significant evidence against the ΛCDM
model from comparing these data sets. Agreement between
DES and Planck in these metrics has improved relative tothe comparison of DES Y1 3×2pt and earlier Planck results,
which gave a parameter difference of 2.2σ and Suspiciousnessof 2.4 ± 0.2σ [133]. The combined DES and Planck CMB
contour is shown in orange in Figs. 16 & 17.
We repeat this exercise for the full combined low-redshift

data, including DES 3×2pt, all BAO, and external SNe Ia and
RSD data. This comparison is shown in Figs. 14 & 15, andis highly complementary, as the external probes are sensitive
to both growth and geometry in the model in ways the DES
3×2pt data is not, and come from a variety of different experiments. We find better agreement between all of these lowredshift probes and Planck CMB predictions than in the comparison with DES 3×2pt data alone, with a parameter difference of 0.9σ or p = 0.34. These results indicate that we can
combine all these available cosmic probes into a single joint
result in the following subsection.
There are several reasonable motivations for caution in theinterpretation of any strong evidence for or against cosmological consistency in tests like this. It is worth noting that while
we have multiple redundant low-redshift sources of information for each main cosmological probe used, it would be useful
to have a second, blinded large-scale CMB polarization experiment to increase confidence in the test at the high-z limit.

25
While polarization data is required to break degeneracies in
the cosmological parameters with the optical depth τ , we also
repeat the caution from Ref. [13] against over-interpreting the
Planck polarization results and the sensitivity of the final parameter constraints to assumptions made in the constructionof the likelihood, which can lead to a <1σ shift toward the
DES posterior relative to the fiducial Planck likelihood. Similar shifts are seen based on certain analysis choices in the
DES results as well, which are shown in App. E. Neither theshift in Planck posteriors or those from other analysis choices
in DES contribute to a significant change in the final interpretation of the comparisons. Finally, the DES Y3 analysis
has uncovered potential systematics connected to photometry(e.g., Sec. V C). While there is evidence that these do not impact the cosmological results, and thus would not impact thiscomparison of data sets, they have not been connected to a
specific source. However, these are examples of unresolved
uncertainties that call for additional care in interpreting any
statements about the consistency of early- and late-universe
probes in ΛCDM, which should be addressed for future more
precise analyses.

C.

Joint cosmological constraints in ΛCDM and wCDM

We find that external low-redshift (BAO+RSD+SNe Ia),
Planck CMB, and DES 3×2pt data sets are able to provide
three independent, highly complementary, and similarly powerful constraints on parameters related to dark matter and darkenergy in the ΛCDM and wCDM models, as seen in Figs. 14& 15. Given the results of the above consistency tests, detailed in App. F, these data sets are each consistent with one
another, and thus can be combined into a joint constraint on
the models. We present these joint results in Figs. 14 & 15 anda summary in Figs. 7 & 10 and Tables II & III. In the ΛCDM
model, we find
S8 = 0.812+0.008−0.008 (0.815)Ωm = 0.306+0.004−0.005 (0.306)(22)+0.008σ8 = 0.804−0.008(0.807).

In the wCDM model,+0.010σ8 = 0.810−0.009(0.804),+0.006Ωm = 0.302−0.006 (0.298),
w = −1.031+0.030−0.027 (−1.001).

to about 10%;
P and place an upper limit on the sum of neutrino
masses of mν < 0.17 eV (95% CL). Individually, the threesubsets of data constrain σ8 and Ωm in ΛCDM with FoM between 2000 and 4000, while combined, they reach a FoM of
34,000. This clearly demonstrates the highly complementary
nature of these three independent data sets.

D.

We are able to probe the distribution of large-scale structure
via weak gravitational lensing (cosmic shear) in two very different ways, using either the shapes of galaxies or the CMB
photons as tracers for the reconstruction of deflections in thepath of the light. These probe the same physical phenomenon
via independent sources and measurement methods, which aresensitive to different types of systematics. While the effective
kernel of CMB lensing [226] is sensitive to higher redshift
structure than galaxy lensing, their comparison provides a significant validation of the robustness of modern weak lensing
results. We show this comparison in Fig. 18, where we findvery good agreement between the two cosmic shear measurements and the full 3×2pt measurement from DES.
In addition to DES, other concurrent photometric surveys
HSC [27, 36] and KiDS [28, 102] are also pursuing precision weak lensing measurements using galaxy shapes. These
three surveys span a range of depth and survey area tradeoffs,
with HSC being deepest, DES widest, and KiDS using thewidest wavelength coverage. We over-plot recent results fromeach of the surveys with our DES cosmic shear and 3×2pt
results in Fig. 19. Unlike other comparisons in this work,
these external survey data have not been re-analyzed withina consistent model and prior space. Thus, no direct or rigorous comparison can be made about data consistency. We
defer a detailed discussion of the consistency of concurrent
photometric weak lensing surveys and their combination (e.g.,
Ref. [227]) to a future work. The apparent orthogonal direction of the KiDS+BOSS+2dFLenS 3×2pt contours to the DES
3×2pt contours is driven by the very strong constraint comingfrom spectroscopic clustering, similar to the orientation of the
DES γt +w(θ) constraint.

E.

(23)

We find R = 7.8, indicating that there is also no preference
for wCDM over ΛCDM in the full joint data analysis.
These data sets together are able to provide unprecedented
precision on the cosmological parameters of the models. In
ΛCDM, we are able to constrain σ8 , S8 , h, Ωb , and ns to lessthan 1%; Ωm and As to about 1%; τ to about 10%;
P and place
an upper limit on the sum of neutrino masses of mν < 0.13
eV (95% CL). In wCDM, we are able to constrain ns to lessthan 1%; Ωm , Ωb , h, and As to about 1-2%; w to about 3%; τComparison of Lensing Probes

Constraints on the Hubble parameter

There is an interesting disagreement in local measurementof the Hubble parameter h and marginalized constraints on
h from cosmological experiments. Multiple local measurements prefer a higher value of the expansion velocity, suchas, most prominently, the astronomical distance ladder (e.g.,h = 0.732 ± 0.013 [93] with Cepheid variable stars; h =0.733 ± 0.040 [94] with Mira variable stars), or masers (e.g.,h = 0.739 ± 0.030 km/s/Mpc [95]). These local measurements stand in contrast to constraints from the CMB by
Planck, which prefer h = 0.665+0.013−0.006 (when the neutrinomass density is varied) [13]. However, there are also local
measurements with lower values reported (h = 0.696 ± 0.019[96] with tip of the red giant branch distance ladder; h =

26

0.40
Fid. 3×2pt
DES Cosmic Shear
Planck Lensing

0.90

ΛCDM

Fid. 3×2pt+BAO+BBN
DES 3×2pt+All Ext.
BAO+BBN
Planck CMB
SH0ES Local h

ΛCDM
0.38
0.36

0.85

Ωm

S8

0.34
0.80

0.75

0.32
0.30
0.28

0.70

0.26
0.65
0.15

0.20

0.25

0.30
Ωm

0.35

0.40

0.45

FIG. 18. A comparison of weak lensing constraints on the ΛCDM
model. Weak lensing of the CMB is shown in green, weak lensingof galaxies in DES is shown in blue, and the combined DES 3×2pt
data is shown in black.

1.1
DES ΛCDM-Opt. 3×2pt
DES ΛCDM-Opt. ξ±
KiDS+BOSS+2dFLenS Spec. 3×2pt
KiDS COSEBIs
HSC Pseudo-C`
HSC ξ±
Planck CMB

ΛCDM
1.0

S8

0.9

0.8

0.7
0.1

0.2

0.3
Ωm

0.4

0.5

FIG. 19. The DES ΛCDM-optimized 3×2pt and cosmic shear, HSC
and KiDS cosmic shear, and KiDS lensing + BOSS+2dFLenS spectroscopic 3×2pt data results are over-plotted for the ΛCDM model.
Unlike other comparisons in this work, these external survey data
have not been re-analyzed within a consistent model and prior space.
Thus, no direct or rigorous comparison can be made about data consistency.

0.64

0.66

0.68

0.70
h

0.72

0.74

0.76

FIG. 20. Marginalized constraints on h and Ωm in the ΛCDM model
are compared to the SH0ES local determination of h. Planck CMB
data and the combination of BAO and BBN data provide comparable uncertainties on h compared to the local constraint. Adding DES
3×2pt to BAO and BBN improves the constraint on h slightly due
to 3×2pt providing additional information on Ωm , while the combination of DES 3×2pt and all non-local external data provide a constraint on h that is a factor of 3-4 more powerful than the local determination.

0.674+0.041−0.032 with strong lensing when combining the TDCOSMO+SLACS data set [97]). The Hubble tension may
indicate new physics and it is crucial to improve measurements, revisit assumptions [e.g., 97, 98], check for consistencies among different measurements, and invest in novel,
independent methods and probes [99, 100].
We can also constrain the value of h independently of CMB
data using a combination of BAO, BBN constraints on Ωb h2 ,and DES 3×2pt measurements. Constraints on h and Ωm inΛCDM are summarized in Fig. 20. The determination of husing BAO and BBN is of similar constraining power to thatof the CMB and agrees very well with the CMB constraint on
h. Adding DES 3×2pt data slightly improves the constraint
on h and shifts it to higher values by about 1σ. Combining
DES 3×2pt data with BAO, RSD, SNe Ia, and Planck CMB(w/ lensing) leads to a marginalized constraint on hh = 0.680+0.004−0.003 (0.681)(24)that is 3-4 times more powerful than any current local measurement of h. Constraints on other cosmological parameters
are summarized in Tables II & III. We find no significant impact on the other cosmological parameters by adopting thishigh-redshift anchor for the expansion rate vs a local prior on
the expansion rate from Ref. [93]. The final joint constraint
on h is consistent with the Planck- or BAO+BBN-only constraints and slightly less than 4σ offset relative to the local h27PΩm , which we see as an increase in the upper limit on mν
of 23% when combining DES 3×2pt with Planck. Combining the DES 3×2pt data, other low-redshift data, and Planck
CMB, we find an upper limitXmν < 0.13 eV (95% CL),(25)

Fid. 3×2pt
Planck CMB
Fid. 3×2pt + Ext. Low-z
Fid. 3×2pt + All Ext.

nearly a factor of three reduction from the CMB-only constraint.

0.450
0.425

ΛCDM

VIII.

0.400

CONCLUSIONS

Ωm

0.375
0.350
0.325
0.300
0.275
0.250

0.1

0.2

0.3
0.4
P
mν [eV]

0.5

0.6

FIG. 21. Marginalized constraints on the sum of neutrino masses
in the ΛCDM model. We show the DES fiducial 3×2pt constraints(black), DES 3×2pt combined with external BAO, RSD, and SNe
Ia (orange), Planck CMB constraints (green), and DES 3×2pt combined with all of these external data sets. The P
upper panel shows
the one-dimensional marginalized posteriors for mν , with shaded
95% confidence
regions. The lower panel shows 68 and 95% CL forPΩm andmν .

by SH0ES.

F.

Neutrino Mass

Figure 21 shows marginalized constraints on the sum ofneutrino masses,
where neutrino mass and density Ων are reP
lated via mν = 93.14Ων h2 eV. We model massive neutrinos as three degenerate species of equal mass. As expected,
DES does not constrain neutrino mass: whether alone or in
combination with external BAO, RSD, and supernova data,
the marginalized posterior of the sum of neutrino masses isbounded by its prior. Given this, DES data is expected to add
very little direct information on neutrinos.
Beyond constraints on neutrino mass themselves, a motivation for looking at the neutrino mass constraint is to highlighta feature of the relationship between Planck and low-z constraints on Ωm , as shown in e.g. Fig. 14. As has been previously discussed [183], a geometric degeneracy means that
CMB-only
constraints are unable to distinguish between Ωm
Pand mν , but combining CMB data with low redshift expansion history constraints from BAO and/or SNe can break thatdegeneracy. This also illustrates that the parts of the Planck
CMB posterior at higher Ωm also have relatively high neutrino
mass. This is further supported by the behavior of constraints
when neutrino mass is fixed, as shown in App. E.
Lowering the clustering amplitude has a similar impact on

We have described the 3×2pt measurements, calibration,modeling, and analysis from the first three years of DES data.
The substantial improvement in statistical power of the DES
Y3 data, which cover an area of sky about three times that of
DES Y1, has required substantial improvements in almost every part of the data processing, analysis and inference. The
specific improvements relative to the DES Y1 analysis aredetailed in App. B, but we also briefly summarize here. At
the catalog level, some of the most important updates for thisanalysis include improved PSF modeling and two complementary lens sample selections. We have substantially revised
our shear and redshift inference and calibration processes.
This includes more realistic image simulations to derive corrections on shear and redshift bias due to blending and detection, and a redshift inference process that combines spectroscopic and deep, multi-band photometric redshifts from DES
deep-field data, cross-clustering between source and highquality photo-z and spectroscopic samples, and small-scale
galaxy–galaxy lensing shear ratio information. Finally, we
have updated several components of the analysis, including
blinding, modeling of non-Limber and RSD contributions tothe clustering signal, mitigation of nonlocal effects in γt , improved validation of the covariance matrix, and improved, calibrated metrics for evaluating the internal and external consistency of data sets.
The statistical power of the DES Year 3 data set has posed
unique challenges for precision cosmological inference, someof which were unforeseen. We have identified some puzzling
results from our photometric lens samples and galaxy clustering, which have not been identified with a clear source. Thiscan be seen at very large significance as an apparent disagreement in the clustering and lensing amplitudes at all redshifts
and angular scales for redMaGiC, and in the highest redshiftsof MagLim not used in the fiducial analysis. After unblinding
our analysis, it was necessary to make two important revisions
to the fiducial analysis plan: 1) we made the MagLim galaxy
sample our fiducial lens sample owing to the decorrelation of
the lensing and clustering amplitudes inferred from the redMaGiC sample, and 2) we dropped the two highest redshiftbins of the MagLim sample as they contributed to a very poor
fit to all models considered in this paper. Sections V A & V C
provide the detailed rationale. These decisions were made after extensive, careful investigations of possible systematics,
which did not reveal problems that we could address. Further
investigations are already underway and may reveal the source

28of these issues, but our robustness tests so far indicate that any
potential changes to the results will lie well within our quoted
uncertainties.
We have achieved a factor of two improvement in statistical power relative to the DES Y1 3×2pt analysis in theσ8 –Ωm marginalized parameter plane, providing competitive
cosmological constraints relative to both the combination of
all other external non-lensing low-redshift data and the predictions from the Planck CMB data. We find consistentcosmological constraints from cosmic shear and the combination of galaxy clustering and galaxy–galaxy lensing, as
well as consistent cosmological results from 3×2pt utilizing
two lens samples, MagLim and redMaGiC. For the fiducial
3×2pt analysis in ΛCDM, we find constraints on the clustering amplitude S8 = 0.776+0.017−0.017 (0.776) and matter densityΩm = 0.339+0.032(0.372).
In the wCDM model, we find−0.031Ωm = 0.352+0.035(0.339),anddark energy equation of state−0.041+0.32
parameter w = −0.98−0.20 (−1.03).
The low-redshift measurements of the matter clustering amplitude by some galaxy surveys have tended to find lower variance relative to the prediction from Planck CMB anisotropies,
which may indicate some inconsistency between the low- andhigh-redshift Universe within the ΛCDM model, with claimsof up to 2–3σ significance. The DES Y3 3×2pt analysis
is an ideal experiment to test whether this is a real problem
with the ΛCDM model. There have been substantial improvements in constraining power for both DES and Planck since
the DES Y1 3×2pt analysis, yet we continue to find that DES
3×2pt data and the combination of 3×2pt with BAO and external SNe Ia and RSD data that the low-redshift Universe areconsistent with predictions in the ΛCDM model from measurements at the time of the CMB from Planck. We find all
three independent data set combinations (DES 3×2pt; BAO,
RSD, and SNe Ia; and Planck CMB) to be mutually consistent
within ΛCDM. Despite caveats on the precision with which
we can make this statement discussed in Sec. VII B, this is themost powerful test of the standard cosmological model to date,comparing predictions from measurements of acoustic peaksin the early plasma of the Universe when it was 380,000 years
old to measurements of large-scale structure from low-redshift
surveys like DES spanning nearly nine billion years of cosmic
evolution to the current day.
Combining DES 3×2pt, CMB, BAO, RSD, and SNe
Ia data allows us to place the most precise constraints on the ΛCDM and wCDM models to date.
+0.008
We find S8 = 0.812−0.008(0.815) and Ωm =+0.004+0.0080.306−0.005 (0.306) in ΛCDM; σ8 = 0.812−0.008(0.804),+0.006Ωm = 0.302−0.006 (0.298), and w = −1.031+0.030(−1.00)−0.027in wCDM. Additionally, we find an independent constraint
on the Hubble parameter combining DES 3×2pt, BAO, and+0.009BBN data of h = 0.676−0.009(0.673), which is consistent
with the Planck prediction for h. From the combination of
DES 3×2pt, CMB, BAO, RSD, and SNe Ia data, we findh = 0.680+0.004−0.003 (0.681). This is slightly closer to the local h measurement by SH0ES than Planck, but a factor ofthree to four more constraining than either the local or Planckmeasurement of h. We are also able to constrain the sum ofP
neutrino masses to be mν < 0.13 eV (95% CL) in ΛCDM.
While we have shown that the inferred cosmological constraints from the fiducial analysis are robust, there remains
significant work to fully characterize the underlying causes ofthese potential systematics and examine other potential theoretical causes in extended model spaces beyond the dark energy models considered in this work. Further understanding
the potential systematic issues related to differences between
photometric clustering and galaxy–galaxy lensing; improvements in how we deal with shear calibration and redshift inference in the presence of blending; and finding ways to improvesystematic floors in our redshift inference are all important
next steps. This continued followup work will be critical to
the final DES Year 6 analyses and future ‘Stage IV’ photometric surveys like the Euclid Space Telescope,9 the Nancy
G. Roman Space Telescope,10 and the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST).11
The novel advances required for the DES Y3 analyses, summarized in the accompanying 29 papers [115–143], set thestage for these future precision low-redshift large-scale structure and weak lensing studies. DES has utilized only half its
final data set in the DES Y3 3×2pt and BAO analyses, andfuture SNe Ia and galaxy cluster analyses promise even larger
improvements in statistical power. The legacy analyses of thefull DES data will be a focus of the next several years leading
up to the start of Stage IV dark energy experiments.

ACKNOWLEDGMENTS

Funding for the DES Projects has been provided by the U.S.
Department of Energy, the U.S. National Science Foundation,
the Ministry of Science and Education of Spain, the Scienceand Technology Facilities Council of the United Kingdom, theHigher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of
Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics andAstronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo àPesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Científico e Tecnológico and the Ministério
da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark
Energy Survey.
The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas,
Medioambientales y Tecnológicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consor-

9 https://www.euclid-ec.org
10 https://roman.gsfc.nasa.gov
11 https://www.lsst.org

29

The DES data management system is supported by the
National Science Foundation under Grant Numbers AST1138766 and AST-1536171. The DES participants from
Spanish institutions are partially supported by MICINN under grants ESP2017-89838, PGC2018-094773, PGC2018102021, SEV-2016-0588, SEV-2016-0597, and MDM-2015-

0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. Research leading tothese results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We acknowledge support
from the Brazilian Instituto Nacional de Ciência e Tecnologia(INCT) do e-Universo (CNPq grant 465376/2014-2).
This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with
the U.S. Department of Energy, Office of Science, Office ofHigh Energy Physics.
This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the
U.S. Department of Energy under Contract No. DE-AC0205CH11231. This work also used resources on Duke Compute Cluster (DCC), the CCAPP condo of the Ruby Cluster
at the Ohio Supercomputing Center [228], and computing resources at SLAC National Accelerator Laboratory. We also
thank the staff of the Fermilab Computing Sector for theirsupport. Plots in this manuscript were produced partly with
M ATPLOTLIB [229], and it has been prepared using NASA’s
Astrophysics Data System Bibliographic Services.

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Laboratory, the University of Illinois at Urbana-Champaign,
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The Ohio State University, the University of Pennsylvania, theUniversity of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas
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Appendix A: Summary of Associated Papers

This paper is built on the results presented in 29 accompanying papers. A useful way to navigate these is to divide themup into five categories:
Catalog Papers: The link between the raw images and thetwo-point functions from which cosmology is extracted is aset of catalogs. The Gold catalog (Sevilla-Noarbe et al. [115])
uses coadd images to identify galaxies and their properties.
This is a first step to almost all ensuing work. Estimating theredshifts of those galaxies hinges in large part on the much
deeper catalog of galaxies from the DES deep fields with overlapping near-infrared photometry presented in Hartley, Choiet al. [116]. Finally, the shear catalog is presented in Gatti,
Sheldon et al. [117], which uses a new PSF measurement described in Jarvis et al. [118].
Simulations: To test our models and calibration of the data,
we rely on large suites of cosmological and image-level simulations. The Buzzard simulations (DeRose et al. [119]) are
generated from cosmological N-Body simulations, populatedwith realistic galaxy samples, and then used in end-to-end
analyses to stress test our modeling and methods. To calibrate blending and detection biases, we generated multiple
image simulations, described in MacCrann et al. [120], whereshear and redshift biases are evaluated. Balrog (Everett et al.
[121]) is a tool developed to inject realistic images of galaxies
into real DES images to evaluate the survey selection function.
This is critical for multiple purposes, including our photometric redshift inference. The full DES detection and measurement pipelines are run on the simulations described in MacCrann et al. [120] and Everett et al. [121].
Photometric Redshifts: The redshift distributions for the
redMaGiC and MagLim lens galaxy samples are validated using cross-correlations with spectroscopic galaxies in Cawthon
et al. [122], which informs our priors on the uncertainty of
the redshift distribution. The redshift inference process forthe source galaxies is a much more involved process. The
overview of this work is presented in Myles, Alarcon et al.
[123], which builds on the self-organizing map formalism
developed in Buchs, Davis et al. [124], and incorporates
constraints from the cross-correlation with both redMaGiC
and spectroscopic galaxies described in Gatti, Giannini et al.
[125]. A followup analysis applying this methodology to the
MagLim lens redshift distributions is described in Giannini
et al. [126]. The sampling of the resulting realizations of theredshift distribution are described in Cordero, Harrison et al.
[127]. Finally, we also add additional information from theposterior from the shear ratio likelihood, which is most sensitive to photo-z parameters and is described in Sánchez, Prat
et al. [128].
Analysis: The modeling outlined in the text is described andtested in detail in Krause et al. [129]. One of the most important pieces of the analysis is the generation of the covariancematrix of the two-point functions. The way in which this is
generated and tested is described in Friedrich et al. [130]. As
mentioned in the text, we remain blinded to the data throughout the analysis to avoid unconscious bias; this blinding has
multiple levels, but one of the key tiers is described in Muiret al. [131]. Methods to evaluate the internal consistency ofour data and to evaluate consistency with external data sets
were calibrated and described in Doux et al. [132] & Lemos
et al. [133]. Efficient sampling of the likelihood can savemonths of analysis time, and ways to optimally, and more importantly robustly, sample our likelihood are summarized in
Lemos et al. [134]. Finally, the optimization of the MagLim
lens sample is described in Porredon et al. [135].
Results: The galaxy clustering weights and measurement
are presented in Rodríguez-Monroy et al. [136]; galaxy–galaxy lensing in Prat et al. [137] and in Sánchez, Prat et al.
[128] for the smaller-scale shear ratio measurements; and theweak lensing convergence mass map in Jeffrey, Gatti et al.
[138]. The results of combining clustering with galaxy–galaxy lensing are presented in three papers, one of which focuses on magnification (Elvin-Poole, MacCrann et al. [139]),
another on redMaGiC and the bias model (Pandey et al.
[140]), and the last on the MagLim sample (Porredon et al.
[141]). Finally, the cosmic shear results are presented in two
papers, one of which focuses on observational systematics

34(Amon et al. [142]) and the other on biases in theoretical modeling (Secco, Samuroff et al. [143]).

Appendix B: Differences relative to DES Year 1 analysis

In this Appendix, we summarize the major differences inthe DES Y3 analysis relative to DES Y1.
Data Processing: DES Y3 contains significantly more data
than DES Y1 [144], which are processed with an improved
version of the DESDM system [146]. Photometric calibration is performed with the forward global calibration method[145], which significantly improves the relative photometriccalibration compared to the calibration techniques applied in
DES Y1 [230]. The Y3 catalogs also introduce more robustmorphological classification based on multi-epoch fitting andimproved flagging to enhance the quality of the galaxy sample. Further details on improvements to the DES data set andobject catalogs can be found in Sevilla-Noarbe et al. [115].
Catalog-level: We have produced a calibrated deep-fielddata reduction for use in the analysis. We have constructed
only one shape catalog (METACALIBRATION; [117]), but produced two very different lens samples to compare cosmological results [136, 140, 141]. We use a new PSF model (PIFF)[118].
Simulation & Calibration: We have produced the Balrog simulation to characterize the wide-field survey selection
function of objects derived from the deep-field coadd images[121]. We have produced a new suite of image simulations forshear calibration that are more realistic and better match thedata [120]. We have produced a new methodology for shear
calibration that explicitly accounts for the impact of blending and detection biases as a function of redshift, including
modifications to the effective n(z) [120]. We have developed
a new redshift inference and calibration framework, mixing
spectroscopic and photometric redshift information, to produce Bayesian posterior n(z) samples [122–128].
Modeling: We have improved the rigor of constraining potential bias from model approximations, with explicit accuracy goals at the χ2 and parameter levels [129]. We have included non-Limber and redshift-space distortion (RSD) contributions to the clustering theory [129]. We have included
the impact of lens magnification [129, 139]. We have madeupdates to the covariance modeling and improved the rigorof validation tests in both χ2 and parameter space [130]. We
utilize a new intrinsic alignment model that allows for ‘red’,‘blue’, and mixed alignment modes to second order in perturbation theory [174]. We have developed a nonlinear galaxy
bias model for the analysis [140]. We have updated our
methodology for determining scale cuts using explicit accuracy goals in both χ2 and parameter space [129].
Analysis: We have utilized a new summary-statistic-level
blinding scheme [131]. We have introduced a new shearratio likelihood [128]. We have introduced a new crossclustering redshift likelihood [123, 125]. We have implemented a method for sampling over the full-shape n(z) samples [127]. We analytically marginalize over uncertainty inthe lens sample clustering weights [136]. We analytically

TABLE IV. The parameter differences for the redMaGiC analysis
relative to Table I.
Lens Galaxy Bias (redMaGiC )bi (i ∈ [1, 5])
Flat
Lens magnification (redMaGiC )
Cl1
Fixed
Cl2
Fixed
Cl3
Fixed
Cl4
Fixed
Cl5
Fixed
Lens photo-z (redMaGiC )∆zl1
Gaussian∆zl2
Gaussian∆zl3
Gaussian∆zl4
Gaussian∆zl5
Gaussianiσz,l(i ∈ [1, 4])
Fixed
5
σz,l
Gaussian(0.8, 3.0)
0.63
−3.04
−1.33
2.50
1.93
(0.006, 0.004)(0.001, 0.003)(0.004, 0.003)(−0.002, 0.005)(−0.007, 0.010)
1.0(1.23, 0.054)

marginalize over a point-mass contribution to γt to mitigate
non-local effects [129]. We marginalize over the widths of thelens n(z) [122, 136]. We have updated and calibrated a new
set of internal and external consistency tests [132, 133]. We
have updated requirements on sampling and evidence precision and now use the PolyChord sampler [134].

Appendix C: Differences in redMaGiC analysis

The redMaGiC lens analysis differs in several ways due tothe different redshift range, binning, and sample selection relative to the MagLim lens analysis. The parameterization andprior changes relative to the fiducial analysis are listed in Table IV. In particular, we do not vary the width of the redshift
distributions in the first four redMaGiC lens bins, since constraints from clustering on the n(z) agree well with the predictions from the redMaGiC photo-z algorithm [122]. The magnification parameters are also fixed to different values basedon measurements from Balrog [139]. We also show the redMaGiC galaxy clustering and galaxy–galaxy lensing data vectors and best-fit cosmological model in Figs. 22 & 23.

Appendix D: Observer Bias & Validation Process

The role of the observer in determining how to proceed
in an analysis or measurement and when to accept a result
as final has been demonstrated to contribute to uncharacterized bias in results (e.g., [231, 232]). To protect against
this, all cosmologically relevant measurements, calibrations,model validation, and fiducial analysis plans were performed
blinded. This was done in three stages, to provide redundant
protection:
1. The ellipticity in the shape catalog was blinded by arandom factor in b ∈ [0.9, 1.1], transforming ηi → bηi ,∆w/σw

w

35

10−1
10−2
10−3
10−4

1,1

2,2

3,3

4,4

5,5

0

101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)∆γt /σγtγt∆γt /σγtγt∆γt /σγtγt∆γt /σγtγtFIG. 22. The measured w(θ) correlation functions for each tomographic bin combination used in the redMaGiC analysis, which is indicatedby the i, j label in each set of panels. The best-fit ΛCDM model from the analysis using redMaGiC is plotted as the solid line in the top
part of each panel, while the bottom parts of each panel shows the fractional difference between the measurements and the model prediction,(wobs. − wth. )/σw (with y-axis range ±5σ). The best-fit model with fixed Xlens is shown in black, while the best-fit model marginalizing overXlens is shown in blue. Both the top and bottom part of each panel includes 1σ error bars. Small angular scales where the linear galaxy bias
assumption breaking down are not used in the cosmological analysis; these scales are indicated by grey shading.

10−2
10−3
10−4
10−5
10−6

1,1

2,1

3,1

4,1

5,1

1,2

2,2

3,2

4,2

5,2

1,3

2,3

3,3

4,3

5,3

1,4

2,4

3,4

4,4

5,4

0

10−2
10−3
10−4
10−5
10−6
0
−2

10
10−3
10−4
10−5
10−6
0

10−2
10−3
10−4
10−5
10−6
0

101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)101
102
θ (arcmin)

FIG. 23. The measured γt (θ) correlation functions for each tomographic bin combination using the redMaGiC sample, with labels as describedin Fig. 3. The best-fit ΛCDM model from the analysis with fixed Xlens is plotted as the solid line in the top part of each panel, with dottedcurves indicating a negative model fit. The best-fit model marginalizing over Xlens is shown in blue. The bottom part of each panel shows
the fractional differences between the measurements and the model prediction, (γtobs. − γtth. )/σγt (with y-axis range ±5σ). In both panels,
1σ error bars are included. Angular scales not used in the cosmological analysis are indicated by grey shading, which are excluded on small
scales where the linear galaxy bias assumption breaks down.

36

0.84

Fid. CovarianceAlt. CovarianceΛCDMsystematic causes. In addition to this, other validation checks
were also performed at this point on the blinded data vector
and posteriors, but were not required for unblinding.
1. Verify the final independent redshift posteriors are consistent (i.e., from SOMPZ, the shear ratio, and clustering cross-correlations).

0.82

0.80
S8

2. Verify no posteriors of systematic parameters concentrate at the edge of their priors in ways that are not understood.

0.78

3. Verify the goodness-of-fit for each subset of thedata (i.e., cosmic shear and galaxy clustering+galaxy–galaxy lensing). We used the posterior predictive distribution (PPD) with a quantitative requirement p > 0.01.

0.76

4. Verify that cosmic shear and galaxy clustering+galaxy–galaxy lensing are consistent with each other, with thesame quantitative requirement.

0.74

0.72
0.20

0.25

0.30

0.35

0.40

0.45

Ωm

FIG. 24. A test of the convergence of the theoretical 3×2pt covariance, showing marginalized parameter constraints using two iterations of the covariance that use the best-fit cosmological parametersfrom the previous analysis.

where ηi = 2ei arctanh(e)/e and e2 = e21 + e22 . The
factor b is generated in a random way by hashing a shortstring phrase, and this phrase was known only to thepeople making the catalog.
2. The two-point correlation functions were coherently
shifted by an unknown vector in cosmological parameter space (Ωm , w), as described in Ref. [131]. This produces a new data vector corresponding to an unknown
wCDM cosmology, and has the benefit of leaving all
three two-point correlation functions internally consistent.
3. All parameter values and the axes of posterior plots inferred from chains were randomly shifted.
These protections were removed one by one as the analysis
matured.
First, the catalog-level blinding was removed when the validation and planned calibration of the catalog were finalized,
to allow null tests and non-cosmological measurements to be
remade for the papers while the data vectors were still blinded.
In practice, this also only occurred after model validation tests
had been completed and other analysis and calibration plans
were also finalized.
Once the data vector, shear and redshift calibration, allmodel validation and choices, and plans for testing internaland external consistency were finalized, we proceeded with apre-determined set of tests on the blinded results. The enumerated tests below were those that would have led us to reconsider whether to proceed with unblinding and pursue potential

5. If any of the previous tests failed in ΛCDM, a passing
condition in wCDM would also be sufficient.
The theoretical covariance matrix associated with the
3×2pt data is calculated with an assumption about the truecosmological and other parameters in our model that may bedifferent from what we find after unblinding our data. Usinga covariance matrix that assumes the wrong cosmology canbias the cosmological inference process. To mitigate this, werecalculate the covariance matrix at the best-fit cosmology of
the initial 3×2pt analysis and run all final chains with this covariance. We confirm that this process has converged by comparing the result of this analysis with an analysis based on yeta third covariance calculated at the second best-fit cosmology.
This is shown in Fig. 24.
We unblinded the redMaGiC sample before the MagLim
sample, finding it to pass all unblinding checks. After unblinding it became apparent from additional internal consistency (PPD) tests that we had statistically significant evidence
for a potential systematic error in the clustering part of thedata vector. This was explored at length in parallel to the final
validation of the MagLim sample leading up to its unblinding. We verified, though did not require, that there was noevidence that MagLim suffered from the same effect before
unblinding it. The MagLim sample, however, did fail the χ2
criterion above for both models after updating the parametervalues for the covariance matrix after the initial unblinding
chains, with an excess χ2 ≈ 100. This was not seen before
unblinding due partly to a poor choice of galaxy bias values
for the initial covariance, leading to the updated covariancematrix being 40-50% smaller in the w(θ) block.
We had agreed that it was implausible that our data shouldbe able to distinguish a non-wCDM model at such a large χ2
before unblinding, and so we had planned to explore potential causes for the poor model fit before proceeding in such acase, correcting any problems that were then discovered before unblinding. Unfortunately, this effect was hidden due tothe initial covariance before parameter updates being weakerin the w(θ) blocks, and so this was a post-unblinding correction. However, the solution (imposing an upper redshift

37

0.84

0.82

0.80

0.78

0.76

0.74

0.72
0.20

1.

Photometric redshifts

In Fig. 25 we show the fiducial 3×2pt analysis comparedto two variations in how we use information related to thephoto-zs of our source sample. First, we simply remove theshear-ratio part of the data vector, which helps to constrain thephoto-z parameters, but does not contribute directly to constraining cosmological parameters. Second, we marginalize
over the full ensemble of n(z) realizations. Before unblinding, we decided to simplify the way we use the redshift information for the sources and marginalize over the more typicalset of parameters that encode shifts in the mean redshift of theaverage n(z) from the ensembles. This choice was driven by
the additional computational expense of marginalizing overthe n(z) realizations directly, but was demonstrated in simulated analyses to produce consistent cosmological results at

0.25

0.30

0.35

0.40

0.45

Ωm

FIG. 25. A test of the impact of alternative redshift analysis choiceson the inferred cosmology from 3×2pt. We compare the fiducial
3×2pt analysis (black) to an analysis where we marginalize over theensemble of n(z) realizations directly via Hyperrank instead of theireffective mean redshifts (blue) and to an analysis where we remove
the shear-ratio data (orange).

0.84

Fid. 3×2pt
Nonlinear Bias
Free Lens Mag.
NLA model

ΛCDM

0.82

Appendix E: Alternative analysis choices and robustness

0.80
S8

In addition to the validation described in the main text of
this work and the associated papers described in App. A, we
discuss several other analysis modifications in this Appendix
that test the robustness of our result.

Fid. 3×2pt
Hyperrank
No Shear Ratio

ΛCDM

S8

cut for the MagLim sample) is something we would almost
surely have pursued before unblinding had this problem been
apparent then. These lens sample issues and related tests are
summarized further in Sec. V C and in Refs. [136, 140, 141].
The list of changes made to the analyses post-unblinding
are as follows. For redMaGiC, we investigated how the lens
weights were calculated due to indications that an alternative method that used a principal component (PC) basis of
the observing condition maps produced a non-trivial shift in
the clustering signal in the direction to correct the observed
excess clustering parameterized by Xlens . However, it wasdetermined that this basis was contaminated by true largescale structure modes. A simplified basis limited to the first 50
PCs to remove any contaminated modes was used in the final
analysis that gives consistent results with the original weights
used at unblinding. An additive component was added to
the w(θ) covariance block of both lens samples that accounts
for potential over-correction and differences between the twoweights methods. The MagLim sample used this final weighting and covariance when unblinded. As discussed previously,
the highest two redshift bins of the MagLim sample were also
removed to resolve a very poor χ2 fit to any dark energy model
considered and a strong indication of inconsistency between
w(θ) and γt approaching Xlens = 0.55 in the highest redshift bin. There was no indication of the impact of Xlens
for MagLim generally within the redshift range overlapping
redMaGiC. This change in redshift limit resulted in a shiftof 0.78σ in the Ωm –S8 plane, almost fully in the directionof increasing Ωm , relative to the parameter values found at
unblinding. This is consistent with shifts observed when removing parts of the redshift range of the data, so not clearly
evidence of systematic impact on the parameter values.

0.78

0.76

0.74

0.72
0.20

0.25

0.30

0.35

0.40

0.45

Ωm

FIG. 26. A test of the impact of alternative analysis choices on
the inferred cosmology from 3×2pt. We compare the fiducial 3×2pt
analysis (black) to an analysis where we marginalize over a nonlinear
bias model using smaller scales in γt and w(θ) in blue, an analysis
that marginalizes over free lens magnification bias parameters in orange, and an analysis that uses the NLA IA model in green.

38
TABLE V. The MagLim galaxy bias constraints (mean with 68% CL) from the fiducial linear bias analysis and the nonlinear bias analysis.
Due to a nontrivial second peak in the posterior of the b24 , we also show in parentheses the 1D marginalized peak value.
b1
b2
b3
b4
b21
b22
b23
b24
+0.10
+0.11
+0.12
+0.11
Fid. Linear Bias 1.49−0.10 1.69−0.11 1.91−0.12 1.79−0.12
–
–
–
–
+0.10
+0.10
+0.11
+0.14
+0.45
+0.34
+0.62
+0.31
Nonlinear Bias 1.44+0.08−0.08 1.60−0.10 1.85−0.11 1.74−0.11 0.12−0.15 0.06−0.26 −0.06−0.27 0.40+0.02 (0.71−0.30 )

0.900

0.950
Synthetic data vector
Buzzard mean data vector

ΛCDM

Fid. 3×2pt

ΛCDM

Fixed Ων h2 3×2pt
Planck CMB
Fixed Ων h2 Planck CMB

0.925

0.875

0.900
0.850
0.875
0.825
S8

S8

0.850
0.800

0.825

0.775

0.800

0.750

0.775
0.750

0.725
0.20

0.25

0.30

0.35

0.40

0.45

Ωm

FIG. 27. The validation of the model and inference pipeline on
the Buzzard simulation suite, where the true cosmology is indicatedby the cross. An analysis of a synthetic data vector (black) at thetrue Buzzard cosmology based on the true redshift distributions with
fixed shear and photo-z parameters is compared to a full analysis of
the mean data vector of 18 simulation realizations (blue) including
all nuisance parameters and n(z) distributions inferred in the same
way we do using the real survey data.

DES Y3 precision. To test this, we run a single chain to show
the potential impact of this choice on the real data. While
all three results are consistent, with a shift of 0.53σ between
Hyperrank and marginalizing over ∆z, it is likely that future
analyses will need to marginalize over the full ensemble ofn(z) realizations directly using something like the Hyperrank
process.

2.

Nonlinear bias modelling

The fiducial analysis of galaxy clustering and galaxy–galaxy lensing assumes a linear galaxy bias model, which
requires removing significant small-scale information in ourdata vector that exhibits substantial nonlinear behavior. This
keeps the analysis simpler with fewer free parameters, but potentially wastes this additional information to further constrain

0.725
0.20

0.25

0.30

0.35

0.40

0.45

Ωm

FIG. 28. A comparison of the fiducial 3×2pt analysis (black solid)
with one that fixes the neutrino mass density to fixed is minimum
value (black dashed). We make a similar comparison for the Planck
CMB data (green solid and dashed).

cosmology on small scales. In our fiducial model, we only
utilize large scales above 8 h−1 Mpc for w(θ) and 6 h−1 Mpc
for γt measurements. To remedy this, we have developed anonlinear galaxy bias model and analysis that can utilize thissmaller scale information from the lens sample at the expenseof marginalizing over additional bias parameters. This is similar to going to smaller scales in cosmic shear and having tomarginalize over additional baryonic effect freedom. In bothcases, the improvements are limited by needing to simultaneously constrain these additional parameters.
Our model is a hybrid 1-Loop effective field theory model,
having five free parameters, as detailed in Refs. [119, 129,
140, 141]. In Ref. [233] we validated this model using 3D
correlation function measurements in redMaGiC and MagLim
mock catalogs and find that this nonlinear bias model agrees
with significantly higher signal-to-noise measurements at better than 2% above scales of 4 h−1 Mpc. For cosmological inferences we fix three of these parameters based on theoreticalconsiderations and validation on these mock catalogs in order to reduce parameter degeneracies and potential parametervolume effects. We have validated the analysis using multiple

39
TABLE VI. Summary of internal goodness-of-fit tests with the posterior predictive distribution for analyses with the redMaGiC and MagLim
samples. The first and second columns indicate the subset of data d considered. Realizations of d are generated for model parameters drawn
from its own posterior and compared to actual observations of d, following the method developed in Ref. [132]. The third column shows the(calibrated) probability-to-exceed for the full data test used for comparison. The other columns indicate the probability-to-exceed for different
subsets used for comparison.
Goodness-of-fit testdp(d|d) p(ξ± |d) p(w + γt |d) p(ξ+ |d) p(ξ− |d) p(γt |d) p(w|d)
MagLim3×2pt
ξ± , γt , w 0.023 0.209
0.209
0.243 0.353 0.009 0.296
2×2pt
γt , w 0.019
–
0.019
–
–
0.013 0.356
Cosmic shearξ±
0.218 0.218–
0.220 0.359––
redMaGiC
3×2pt
ξ± , γt , w 0.005 0.242
0.001
0.260 0.410 0.029 0.041
2×2pt
γt , w 0.038
–
0.038
–
–
0.187 0.072
Cosmic shearξ±
0.255 0.255–
0.253 0.397––

TABLE VII. Summary of internal consistency tests with the posterior
predictive distribution for analyses with the redMaGiC and MagLim
samples. We consider the consistency between different two-point
functions, as listed in the first column. The probability-to-exceedp(a|b) is obtained by generating realizations of a for model parameters drawn from the posterior of b, and comparing these realizations
to the actual observations of a. The third and fourth columns indicate
the probability-to-exceed for the redMaGiC and MagLim samples respectively.
Consistency testp
redMaGiC MagLim
Cosmic shear vs galaxy–galaxy p(ξ± |γt , w) 0.019
0.297
lensing and clusteringgalaxy–galaxy lensing vs cos- p(γt |ξ± , w) 0.016
0.004
mic shear and clustering
Clustering vs cosmic shear and p(w|ξ± , γt ) 0.008
0.526
galaxy–galaxy lensing

buzzard simulation realizations[119, 140, 141] (see Sec. IV D
for details), finding that this model gives a cosmological biasof less than 0.3σ in the Ωm –S8 plane for both the w(θ) + γt
and 3×2pt probes.
In Fig. 26 we show the cosmological constraints when applying this model to our data. The galaxy bias constraints
are summarized in Table V. We find cosmological constraints
in wCDM to be consistent with the fiducial analysis. We
find a similar improvement in constraining power for ΛCDM
with this nonlinear bias analysis as with the ΛCDM-optimized
analysis that instead incorporates additional small scale information in cosmic shear. With DES Year 6 data, it is possible
we could better optimize these choices of how to utilize smallscale information in ΛCDM analyses to substantially improve
constraining power.

3.

Lens magnification bias parameters

In the fiducial analysis, we fix the lens magnification coefficients Cli to values derived in Ref. [139]. In Fig. 26 weshow the impact of freeing these coefficients with a wide flatprior between −6 and 10 in each lens bin. We find the cosmological constraints in ΛCDM to be consistent with the fiducial

analysis. Further investigations into the lens magnification coefficients and their impact on cosmological constraints can befound in Ref. [139].

4.

Intrinsic alignment models

In the fiducial analysis, we use the full TATT intrinsic alignment model. In Fig. 26 we show the impact of limiting the
intrinsic alignment model to the more commonly used NLA
model with free redshift power-law evolution. This is thesame as fixing the parameters A2 = bTA = 0 in the TATT
model. We find consistent cosmological parameters, with again of only 13% relative to TATT in the σ8 –Ωm parameter
plane. The use of NLA was not motivated a priori as the fiducial IA model, but due to the IA amplitude parameters being
constrained to be small in TATT, A2 = bTA = 0 is not a poor
approximation.

5.

Simulation validation tests

We reproduce the fiducial analysis on a suite of 18 Buzzard
simulations described in Sec. IV D. This is shown in Fig. 27,
where the true cosmology is indicated by the cross. We compare two simulated analysis. The first analysis uses a synthetic, noiseless data vector based on the true n(z) from Buzzard and without marginalizing over shear or photo-z parameters. The second reproduces the full analysis on the mean data
vector of all 18 simulation realizations, which marginalizes
over all nuisance parameters and uses an photo-z n(z) that isinferred from the same process we apply to the real survey
data. We find that the two simulated analyses agree very wellwith each other and with the true cosmology for each simulated data vector.

6.

Neutrino mass

Finally, we also show a version of the analysis that fixes
the neutrino mass density at the minimum allowed mass in

40
TABLE VIII. A comparison of metrics testing the consistency of independent data sets within the ΛCDM model. We show results in units ofσ, with a derived probability-to-exceed p shown in parentheses. Overall, we find no significant (defined as p < 0.01) evidence of disagreement
between the DES 3×2pt, external low-redshift, or Planck CMB data sets. The details of the metrics and derivation of p are described fully in
Ref. [133].
Data set 1
Data set 2
Parameter difference
Suspiciousness
Eigentension QUDM
ln(Evidence ratio)
DES 3×2pt
Planck CMB
1.5σ (0.13)(0.7 ± 0.1)σ (0.48 ± 0.08) 1.2σ (0.23) 1.6σ (0.10)
5.9 ± 0.5
DES 3×2pt
Ext. Low-z
1.3σ (0.11)(1.6 ± 0.3)σ (0.13 ± 0.07) 0.9σ (0.40) 1.4σ (0.11)
1.6 ± 0.4
Ext. Low-z
Planck CMB
1.1σ (0.27)–
1.5σ (0.13) 0.6σ (0.54)–
DES 3×2pt + Ext. Low-z Planck CMB
0.9σ (0.34)–
1.2σ (0.25) 0.8σ (0.42)–
DES ΛCDM-Opt. 3×2pt Planck CMB
1.9σ (0.06)–
1.3σ (0.21)––
DES Hyperrank 3×2pt
Planck CMB
2.2σ (0.02)–
2.0σ (0.04)––
DES redMaGiC 3×2pt
Planck CMB
1.5σ (0.13)–
1.0σ (0.30)––

Fig. 28, comparing to a similar fixed neutrino mass density of
the Planck CMB data.

Appendix F: Details of Internal and External Consistency

Results of the final internal consistency tests are reported inTables VI and VII. Table VI reports results of the goodnessof-fit tests where realizations of a subset of the data vector are
generated for model parameters drawn from its own posterior.
We show results for analyses using the MagLim lens sample
and for redMaGiC. Table VII reports extra consistency tests
between the two-point functions that were not included as unblinding criteria. In these consistency tests, the data vector
is split in two, and one part is used to predict the other. For
instance, the first row of Table VII shows the result of a test
where realizations of cosmic shear are generated for model
parameters drawn from the 2 × 2pt posterior, and comparedto observations of cosmic shear. All pre-defined unblinding
requirements were met for these internal data combinations,and both 3×2pt and the cosmologically-constraining subsetsof the data for both lens samples show consistency with eachother and with ΛCDM. However, there persists evidence ofpotential issues with the consistency of some parts of the combination of galaxy clustering and galaxy–galaxy lensing relative to the best-fit model of the full 3×2pt or cosmic shear
data. In particular, for redMaGiC this seems slightly more
likely to be sourced primarily from the clustering data.
We report detailed consistency metrics between external
data set pairs in Table VIII. The details of the metrics andderivation of probability-to-exceed p are described fully in
Ref. [133]. Overall, we find no significant (defined as p <0.01) evidence of disagreement between the DES 3×2pt,
external low-redshift, or Planck CMB data sets shown in
Figs. 14 & 15. We show both a method measuring parameter
differences and a method based on the evidence (‘Suspiciousness’) for the two primary comparisons of DES 3×2pt withexternal data sets. The evidence ratio is also shown, and all
three metrics give qualitatively consistent results.
All data set combinations are largely in agreement, according to all tension metrics. Some differences between methods
are likely due to non-Gaussianity of both posteriors involved,
as well as the different approaches that the methods employ intreating the impact of priors on tension quantifications, but inall cases are below a fraction of a sigma. We also note that the
Eigentension result for the case of DES 3×2pt vs complementary external low-redshift probes may not be capturing the fulltension; the two data sets have comparable constraining powers, for which case Eigentension is not optimized.