DES-2017-0226 FERMILAB-PUB-17-294-PPD Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing T. M. C. Abbott,1 F. B. Abdalla,2, 3 A. Alarcon,4 J. Aleksić,5 S. Allam,6 S. Allen,7 A. Amara,8 J. Annis,6 J. Asorey,9, 10 S. Avila,11, 12 D. Bacon,11 E. Balbinot,13 M. Banerji,14, 15 N. Banik,6 W. Barkhouse,16 M. Baumer,7, 17, 18 E. Baxter,19 K. Bechtol,20 M. R. Becker,7, 17 A. Benoit-Lévy,3, 21, 22 B. A. Benson,6, 23 G. M. Bernstein,19 E. Bertin,22, 21 J. Blazek,24, 25 S. L. Bridle,26 D. Brooks,3 D. Brout,19 E. Buckley-Geer,6 D. L. Burke,17, 18 M. T. Busha,17 A. Campos,27, 28 D. Capozzi,11 A. Carnero Rosell,28, 29 M. Carrasco Kind,30, 31 J. Carretero,5 F. J. Castander,4 R. Cawthon,23 C. Chang,23 N. Chen,23 M. Childress,32 A. Choi,25 C. Conselice,33 R. Crittenden,11 M. Crocce,4 C. E. Cunha,17 C. B. D’Andrea,19 L. N. da Costa,28, 29 R. Das,34 T. M. Davis,9, 10 C. Davis,17 J. De Vicente,35 D. L. DePoy,36 J. DeRose,7, 17 S. Desai,37 H. T. Diehl,6 J. P. Dietrich,38, 39 S. Dodelson,6, 40 P. Doel,3 A. Drlica-Wagner,6 T. F. Eifler,41, 42 A. E. Elliott,43 F. Elsner,3 J. Elvin-Poole,26 J. Estrada,6 A. E. Evrard,44, 34 Y. Fang,19 E. Fernandez,5 A. Ferté,45 D. A. Finley,6 B. Flaugher,6 P. Fosalba,4 O. Friedrich,46, 47 J. Frieman,23, 6 J. García-Bellido,12 M. Garcia-Fernandez,35 M. Gatti,5 E. Gaztanaga,4 D. W. Gerdes,34, 44 T. Giannantonio,46, 15, 14 M.S.S. Gill,18 K. Glazebrook,48 D. A. Goldstein,49, 50 D. Gruen,51, 17, 18 R. A. Gruendl,31, 30 J. Gschwend,28, 29 G. Gutierrez,6 S. Hamilton,34 W. G. Hartley,3, 8 S. R. Hinton,9 K. Honscheid,25, 43 B. Hoyle,46 D. Huterer,34 B. Jain,19 D. J. James,52 M. Jarvis,19 T. Jeltema,53 M. D. Johnson,30 M. W. G. Johnson,30 T. Kacprzak,8 S. Kent,23, 6 A. G. Kim,50 A. King,9 D. Kirk,3 N. Kokron,54 A. Kovacs,5 E. Krause,17 C. Krawiec,19 A. Kremin,34 K. Kuehn,55 S. Kuhlmann,56 N. Kuropatkin,6 F. Lacasa,27 O. Lahav,3 T. S. Li,6 A. R. Liddle,45 C. Lidman,10, 55 M. Lima,28, 54 H. Lin,6 N. MacCrann,43, 25 M. A. G. Maia,29, 28 M. Makler,57 M. Manera,3 M. March,19 J. L. Marshall,36 P. Martini,58, 25 R. G. McMahon,14, 15 P. Melchior,59 F. Menanteau,30, 31 R. Miquel,5, 60 V. Miranda,19 D. Mudd,58 J. Muir,34 A. Möller,61, 10 E. Neilsen,6 R. C. Nichol,11 B. Nord,6 P. Nugent,50 R. L. C. Ogando,29, 28 A. Palmese,3 J. Peacock,45 H.V. Peiris,3 J. Peoples,6 W.J. Percival,11 D. Petravick,30 A. A. Plazas,42 A. Porredon,4 J. Prat,5 A. Pujol,4 M. M. Rau,46 A. Refregier,8 P. M. Ricker,31, 30 N. Roe,50 R. P. Rollins,26 A. K. Romer,62 A. Roodman,17, 18 R. Rosenfeld,27, 28 A. J. Ross,25 E. Rozo,63 E. S. Rykoff,17, 18 M. Sako,19 A. I. Salvador,35 S. Samuroff,26 C. Sánchez,5 E. Sanchez,35 B. Santiago,64, 28 V. Scarpine,6 R. Schindler,18 D. Scolnic,23 L. F. Secco,19 S. Serrano,4 I. Sevilla-Noarbe,35 E. Sheldon,65 R. C. Smith,1 M. Smith,32 J. Smith,66 M. Soares-Santos,6 F. Sobreira,28, 67 E. Suchyta,68 G. Tarle,34 D. Thomas,11 M. A. Troxel,43, 25 D. L. Tucker,6 B. E. Tucker,10, 61 S. A. Uddin,10, 69 T. N. Varga,47, 46 P. Vielzeuf,5 V. Vikram,56 A. K. Vivas,1 A. R. Walker,1 M. Wang,6 R. H. Wechsler,18, 17, 7 J. Weller,38, 46, 47 W. Wester,6 R. C. Wolf,19 B. Yanny,6 F. Yuan,10, 61 A. Zenteno,1 B. Zhang,61, 10 Y. Zhang,6 and J. Zuntz45 (Dark Energy Survey Collaboration)∗ 1Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile 2Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa 3Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 4Institute of Space Sciences, IEEC-CSIC, Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain 5Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain 6Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA 7Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA 8Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland 9School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia 10ARC Centre of Excellence for All-sky Astrophysics (CAASTRO) 11Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK 12Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain 13Department of Physics, University of Surrey, Guildford GU2 7XH, UK 14Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 15Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 16University of North Dakota, Department of Physics and Astrophysics, Witmer Hall, Grand Forks, ND 58202, USA 17Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA 18SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA 19Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 20LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA 21CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 22Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 23Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA 24Institute of Physics, Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland ar X iv :1 70 8. 01 53 0v 3 [ as tr o- ph .C O ] 1 M ar 2 01 9 2 25Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA 26Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK 27ICTP South American Institute for Fundamental Research Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil 28Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 29Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 30National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA 31Department of Astronomy, University of Illinois, 1002 W. Green Street, Urbana, IL 61801, USA 32School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK 33University of Nottingham, School of Physics and Astronomy, Nottingham NG7 2RD, UK 34Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA 35Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain 36George 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 37Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India 38Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany 39Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany 40Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 41Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA 42Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA 43Department of Physics, The Ohio State University, Columbus, OH 43210, USA 44Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA 45Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK 46Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679 München, Germany 47Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany 48Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Victoria 3122, Australia 49Department of Astronomy, University of California, Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA 50Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA 51Einstein Fellow 52Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA 53Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA 54Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil 55Australian Astronomical Observatory, North Ryde, NSW 2113, Australia 56Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA 57ICRA, Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, CEP 22290-180, Rio de Janeiro, RJ, Brazil 58Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA 59Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA 60Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain 61The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia 62Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK 63Department of Physics, University of Arizona, Tucson, AZ 85721, USA 64Instituto de Física, UFRGS, Caixa Postal 15051, Porto Alegre, RS - 91501-970, Brazil 65Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA 66Austin Peay State University, Dept. Physics-Astronomy, P.O. Box 4608 Clarksville, TN 37044, USA 67Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil 68Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 69Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, Jiangshu 210008, China (Dated: March 4, 2019) We present cosmological results from a combined analysis of galaxy clustering and weak gravitational lens- ing, using 1321 deg2 of griz imaging data from the first year of the Dark Energy Survey (DES Y1). We combine three two-point functions: (i) the cosmic shear correlation function of 26 million source galaxies in four redshift bins, (ii) the galaxy angular autocorrelation function of 650,000 luminous red galaxies in five redshift bins, and (iii) the galaxy-shear cross-correlation of luminous red galaxy positions and source galaxy shears. To demonstrate the robustness of these results, we use independent pairs of galaxy shape, photometric redshift estimation and validation, and likelihood analysis pipelines. To prevent confirmation bias, the bulk of the analysis was carried out while “blind” to the true results; we describe an extensive suite of systemat- ics checks performed and passed during this blinded phase. The data are modeled in flat ΛCDM and wCDM cosmologies, marginalizing over 20 nuisance parameters, varying 6 (for ΛCDM) or 7 (for wCDM) cosmolog- ical parameters including the neutrino mass density and including the 457 × 457 element analytic covariance 3 matrix. We find consistent cosmological results from these three two-point functions, and from their combi- nation obtain S8 ≡ σ8(Ωm/0.3)0.5 = 0.773+0.026 −0.020 and Ωm = 0.267+0.030 −0.017 for ΛCDM; for wCDM, we find S8 = 0.782+0.036 −0.024, Ωm = 0.284+0.033 −0.030, and w = −0.82+0.21 −0.20 at 68% CL. The precision of these DES Y1 constraints rivals that from the Planck cosmic microwave background measurements, allowing a comparison of structure in the very early and late Universe on equal terms. Although the DES Y1 best-fit values for S8 and Ωm are lower than the central values from Planck for both ΛCDM and wCDM, the Bayes factor indicates that the DES Y1 and Planck data sets are consistent with each other in the context of ΛCDM. Combining DES Y1 with Planck, Baryonic Acoustic Oscillation measurements from SDSS, 6dF, and BOSS, and type Ia supernovae from the Joint Lightcurve Analysis (JLA) dataset, we derive very tight constraints on cosmological parameters: S8 = 0.802 ± 0.012 and Ωm = 0.298 ± 0.007 in ΛCDM, and w = −1.00+0.05 −0.04 in wCDM. Upcoming DES analyses will provide more stringent tests of the ΛCDM model and extensions such as a time-varying equation of state of dark energy or modified gravity. I. INTRODUCTION The discovery of cosmic acceleration [1, 2] established the Cosmological Constant (Λ) [3] + Cold Dark Matter (ΛCDM) model as the standard cosmological paradigm that explains a wide variety of phenomena, from the origin and evolution of large-scale structure to the current epoch of accelerated ex- pansion [4, 5]. The successes of ΛCDM, however, must be balanced by its apparent implausibility: three new entities be- yond the Standard Model of particle physics — one that drove an early epoch of inflation; another that serves as dark mat- ter; and a third that is driving the current epoch of acceler- ation — are required, none of them easily connected to the rest of physics [6]. Ongoing and planned cosmic surveys are designed to test ΛCDM and more generally to shed light on the mechanism driving the current epoch of acceleration, be it the vacuum energy associated with the cosmological con- stant, another form of dark energy, a modification of General Relativity, or something more drastic. The Dark Energy Survey (DES1, [7]) is an on-going, five- year survey that, when completed, will map 300 million galax- ies and tens of thousands of galaxy clusters in five filters (grizY ) over 5000 deg2, in addition to discovering several thousand type Ia supernovae in a 27 deg2 time-domain sur- vey. DES will use several cosmological probes to test ΛCDM; galaxy clustering and weak gravitational lensing are two of the most powerful. Jointly, these complementary probes sample the underlying matter density field through the galaxy popula- tion and the distortion of light due to gravitational lensing. In this paper, we use data on this combination from the first year (Y1) of DES to constrain ΛCDM and its simplest extension— wCDM, having a free parameter for the dark energy equation of state. The spatial distribution of galaxies in the Universe, and its temporal evolution, carry important information about the physics of the early Universe, as well as details of structure evolution in the late Universe, thereby testing some of the most precise predictions of ΛCDM. Indeed, measurements of the galaxy two-point correlation function, the lowest-order statistic describing the galaxy spatial distribution, provided ∗ For correspondence use des-publication-queries@fnal.gov 1 http://www.darkenergysurvey.org/ early evidence for the ΛCDM model [8–19]. The data– model comparison in this case depends upon uncertainty in the galaxy bias [20], the relation between the galaxy spatial distribution and the theoretically predicted matter distribution. In addition to galaxy clustering, weak gravitational lens- ing has become one of the principal probes of cosmology. While the interpretation of galaxy clustering is complicated by galaxy bias, weak lensing provides direct measurement of the mass distribution via cosmic shear, the correlation of the apparent shapes of pairs of galaxies induced by foreground large-scale structure. Further information on the galaxy bias is provided by galaxy–galaxy lensing, the cross-correlation of lens galaxy positions and source galaxy shapes. The shape distortions produced by gravitational lensing, while cosmologically informative, are extremely difficult to measure, since the induced source galaxy ellipticities are at the percent level, and a number of systematic effects can ob- scure the signal. Indeed, the first detections of weak lens- ing were made by cross-correlating observed shapes of source galaxies with massive foreground lenses [21, 22]. A wa- tershed moment came in the year 2000 when four research groups nearly simultaneously announced the first detections of cosmic shear [23–26]. While these and subsequent weak lensing measurements are also consistent with ΛCDM, only recently have they begun to provide competitive constraints on cosmological parameters [27–36]. Galaxy–galaxy lensing measurements have also matured to the point where their com- bination with galaxy clustering breaks degeneracies between the cosmological parameters and bias, thereby helping to con- strain dark energy [22, 37–48]. The combination of galaxy clustering, cosmic shear, and galaxy–galaxy lensing measure- ments powerfully constrains structure formation in the late universe. As for cosmological analyses of samples of galaxy clusters [see 49, for a review], redshift space distortions in the clustering of galaxies [50, and references therein] and other measurements of late-time structure, a primary test is whether these are consistent, in the framework of ΛCDM, with mea- surements from cosmic microwave background (CMB) exper- iments that are chiefly sensitive to early-universe physics [51– 54] as well as lensing of its photons by the large-scale struc- tures [e.g. 55–57]. The main purpose of this paper is to combine the infor- mation from galaxy clustering and weak lensing, using the galaxy and shear correlation functions as well as the galaxy- shear cross-correlation. It has been recognized for more than a http://www.darkenergysurvey.org/ 4 decade that such a combination contains a tremendous amount of complementary information, as it is remarkably resilient to the presence of nuisance parameters that describe systematic errors and non-cosmological information [58–61]. It is per- haps simplest to see that the combined analysis could sepa- rately solve for galaxy bias and the cosmological parameters; however, it can also internally solve for (or, self-calibrate [62]) the systematics associated with photometric redshifts [63–65], intrinsic alignment [66], and a wide variety of other effects [60]. Such a combined analysis has recently been executed by combining the KiDS 450 deg2 weak lensing survey with two different spectroscopic galaxy surveys [67, 68]. While these multi-probe analyses still rely heavily on prior informa- tion about the nuisance parameters, obtained through a wide variety of physical tests and simulations, this approach does significantly mitigate potential biases due to systematic errors and will likely become even more important as statistical er- rors continue to drop. The multi-probe analyses also extract more precise information about cosmology from the data than any single measurement could. Previously, the DES collaboration analyzed data from the Science Verification (SV) period, which covered 139 deg2, carrying out several pathfinding analyses of galaxy cluster- ing and gravitational lensing, along with numerous others [46, 48, 69–83]. The DES Y1 data set analyzed here cov- ers about ten times more area, albeit shallower, and provides 650,000 lens galaxies and the shapes of 26 million source galaxies, each of them divided into redshift bins. The lens sample comprises bright, red-sequence galaxies, which have secure photometric redshift (photo-z) estimates. We measure three two-point functions from these data: (i) w(θ), the an- gular 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 that a relatively simple model describes the data, although even with this restriction we must introduce twenty parameters to capture astrophysical and measurement- related systematic uncertainties. This paper is built upon, and uses tools and results from, eleven other papers: • Ref. [84], which describes the theory and parameter- fitting methodologies, including the binning and mod- eling of all the two point functions, the marginalization of astrophysical and measurement related uncertainties, and the ways in which we calculate the covariance ma- trix and obtain the ensuing parameter constraints; • Ref. [85], which applies this methodology to image simulations generated to mimic many aspects of the Y1 data sets; • a description of the process by which the value-added galaxy catalog (Y1 Gold) is created from the data and the tests on it to ensure its robustness [86]; • a shape catalog paper, which presents the two shape cat- alogs generated using two independent techniques and the many tests carried out to ensure that residual sys- tematic errors in the inferred shear estimates are suffi- ciently small for Y1 analyses [87]; • Ref. [88], which describes how the redshift distributions of galaxies in these shape catalogs are estimated from their photometry, including a validation of these esti- mates by means of COSMOS multi-band photometry; • three papers [89–91] that describe the use of angular cross-correlation with samples of secure redshifts to in- dependently validate the photometric redshift distribu- tions of lens and source galaxies; • Ref. [92], which measures and derives cosmological constraints from the cosmic shear signal in the DES Y1 data and also addresses the question of whether DES lensing data are consistent with lensing results from other surveys; • Ref. [93], which describes galaxy–galaxy lensing re- sults, including a wide variety of tests for systematic contamination and a cross-check on the redshift distri- butions of source galaxies using the scaling of the lens- ing signal with redshift; • Ref. [94], which describes the galaxy clustering statis- tics, including a series of tests for systematic contam- ination. This paper also describes updates to the red- MaGiC algorithm used to select our lens galaxies and to estimate their photometric redshifts. Armed with the above results, this paper presents the most stringent cosmological constraints from a galaxy imaging sur- vey to date and, combined with external data, the most strin- gent constraints overall. One of the guiding principles of the methods developed in these papers is redundancy: we use two independent shape measurement methods that are independently cali- brated, several photometric redshift estimation and validation techniques, and two independent codes for predicting our sig- nals and performing a likelihood analysis. Comparison of these, as described in the above papers, has been an impor- tant part of the verification of each step of our analysis. The plan of the paper is as follows. §II gives an overview of the data used in the analysis, while §III presents the two-point statistics that contain the relevant information about cosmo- logical parameters. §IV describes the methodology used to compare these statistics to theory, thereby extracting cosmo- logical results. We validated our methodology while remain- ing blinded to the results of the analyses; this process is de- scribed in §V, and some of the tests that convinced us to un- blind are recounted in Appendix A. §VI presents the cosmo- logical results from these three probes as measured by DES in the context of two models, ΛCDM and wCDM, while §VII compares DES results with those from other experiments, of- fering one of the most powerful tests to date of ΛCDM. Then, we combine DES with external data sets with which it is con- sistent to produce the tightest constraints yet on cosmologi- cal parameters. Finally, we conclude in §VIII. Appendix B 5 presents further evidence of the robustness of our results. And Appendix C describes updates in the covariance matrix calcu- lation carried out after the first version of this paper had been posted. II. DATA DES uses the 570-megapixel Dark Energy Camera (DE- Cam [95]), built by the collaboration and deployed on the Cerro Tololo Inter-American Observatory (CTIO) 4m Blanco telescope in Chile, to image the South Galactic Cap in the grizY filters. In this paper, we analyze DECam images taken from August 31, 2013 to February 9, 2014 (“DES Year 1” or Y1), covering 1786 square degrees in griz after coaddition and before masking [86]. The data were processed through the DES Data Management (DESDM) system [96–99], which detrends and calibrates the raw DES images, combines indi- vidual exposures to create coadded images, and detects and catalogs astrophysical objects. Further vetting and subselec- tion of the DESDM data products was performed by [86] to produce a high-quality object catalog (Y1 Gold) augmented by several ancillary data products including a star/galaxy sep- arator. With up to 4 exposures per filter per field in Y1, and individual griz exposures of 90 sec and Y exposures of 45 sec, the characteristic 10σ limiting magnitude for galaxies is g = 23.4, r = 23.2, i = 22.5, z = 21.8, and Y = 20.1 [86]. Additional analyses produced catalogs of red galaxies, photometric-redshift estimates, and galaxy shape estimates, as described below. As noted in §I, we use two samples of galaxies in the cur- rent analysis: lens galaxies, for the angular clustering mea- surement, and source galaxies, whose shapes we estimate and correlate with each other (“cosmic shear”). The tangential shear is measured for the source galaxies about the positions of the lens galaxies (galaxy–galaxy lensing). A. Lens Galaxies We rely on redMaGiC galaxies for all galaxy clustering measurements [94] and as the lens population for the galaxy– galaxy lensing analysis [93]. They have the advantage of be- ing easily identifiable, relatively strongly clustered, and of having relatively small photometric-redshift errors; they are selected using a simple algorithm [100]: 1. Fit every galaxy in the survey to a red-sequence tem- plate and compute the corresponding best-fit redshift zred. 2. Evaluate the goodness-of-fit χ2 of the red-sequence template and the galaxy luminosity, using the assigned photometric redshift. 3. Include the galaxy in the redMaGiC catalog if and only if it is bright (L ≥ Lmin) and the red-sequence template is a good fit (χ2 ≤ χ2 max). In practice, we do not specify χ2 max but instead demand that the resulting galaxy sample have a constant comoving density as a function of redshift. Consequently, redMaGiC galaxy se- lection depends upon only two parameters: the selected lu- minosity threshold, Lmin, and the comoving density, n̄, of the sample. Of course, not all combinations of parameters are possible: brighter galaxy samples must necessarily be less dense. Three separate redMaGiC samples were generated from the Y1 data, referred to as the high-density, high-luminosity, and higher-luminosity samples. The corresponding lumi- nosity thresholds2 and comoving densities for these sam- ples are, respectively, Lmin = 0.5L∗, L∗, and 1.5L∗, and n̄ = 10−3, 4× 10−4, and 10−4 galaxies/(h−1Mpc)3, where h ≡ H0/(100 km sec−1 Mpc−1) parametrizes the Hubble constant. Naturally, brighter galaxies are easier to map at higher redshifts than are the dimmer galaxies. These galaxies are placed in five nominally disjoint redshift bins. The low- est three bins z = [(0.15 − 0.3), (0.3 − 0.45), (0.45 − 0.6)] are high-density, while the galaxies in the two highest redshift bins ((0.6 − 0.75) and (0.75 − 0.9)) are high-luminosity and higher-luminosity, respectively. The estimated redshift distri- butions of these five binned lens galaxy samples are shown in the upper panel of Figure 1. The clustering properties of these galaxies are an essential part of this combined analysis, so great care is taken in [94] to ensure that the galaxy maps are not contaminated by sys- tematic effects. This requires the shallowest or otherwise ir- regular or patchy regions of the total 1786 deg2 Y1 area to be masked, leaving a contiguous 1321 deg2 as the area for the analysis, the region called “SPT” in [86]. The mask derived for the lens sample is also applied to the source sample. B. Source Galaxies 1. Shapes Gravitational lensing shear is estimated from the statisti- cal alignment of shapes of source galaxies, which are selected from the Y1 Gold catalog [86]. In DES Y1, we measure galaxy shapes and calibrate those measurements by two in- dependent and different algorithms, METACALIBRATION and IM3SHAPE, as described in [87]. METACALIBRATION [101, 102] measures shapes by simul- taneously fitting a 2D Gaussian model for each galaxy to the pixel data for all available r-, i-, and z-band exposures, con- volving with the point-spread functions (PSF) appropriate to each exposure. This procedure is repeated on versions of these images that are artificially sheared, i.e. de-convolved, distorted by a shear operator, and re-convolved by a sym- metrized version of the PSF. By means of these, the response of the shape measurement to gravitational shear is measured 2 Here and throughout, whenever a cosmology is required, we use ΛCDM with the parameters given in Table 1 of [84]. 6 from the images themselves, an approach encoded in META- CALIBRATION. METACALIBRATION also includes an algorithm for calibra- tion of shear-dependent selection effects of galaxies, which could bias shear statistics at the few percent level other- wise, by measuring on both unsheared and sheared images all those galaxy properties that are used to select, bin and weight galaxies in the catalog. Details of the practical appli- cation of these corrections to our lensing estimators are given in [87, 92, 93, 102]. IM3SHAPE estimates a galaxy shape by determining the maximum likelihood set of parameters from fitting either a bulge or a disc model to each object’s r-band observations [103]. The maximum likelihood fit, like the Gaussian fit with METACALIBRATION, provides only a biased estimator of shear. For IM3SHAPE, this bias is calibrated using a large suite of image simulations that resemble the DES Y1 data set closely [87, 104]. Potential biases in the inferred shears are quantified by multiplicative shear-calibration parameters mi in each source redshift bin i, such that the measured shear γmeas = (1 + mi)γtrue. The mi are free parameters in the cosmological in- ferences, using prior constraints on each as determined from the extensive systematic-error analyses in [87]. These shear- calibration priors are listed in Table I. The overall METACAL- IBRATION calibration is accurate at the level of 1.3 percent. This uncertainty is dominated by the impact of neighboring galaxies on shape estimates. For tomographic measurements, the widths of the overall mi prior is increased to yield a per- bin uncertainty in mi, to account conservatively for possible correlations of mi between bins [see appendices of 87, 88]. This yields the 2.3 percent prior per redshift bin shown in Table I. The IM3SHAPE prior is determined with 2.5 percent uncertainty for the overall sample (increased to a 3.5 percent prior per redshift bin), introduced mostly by imperfections in the image simulations. In both catalogs, we have applied conservative cuts, for in- stance on signal-to-noise ratio and size, that reduce the num- ber of galaxies with shape estimates relative to the Y1 Gold input catalog significantly. For METACALIBRATION, we ob- tain 35 million galaxy shape estimates down to an r-band magnitude of ≈ 23. Of these, 26 million are inside the re- stricted area and redshift bins of this analysis. Since its cali- bration is more secure, and its number density is higher than that of IM3SHAPE (see [87] for details on the catalog cuts and methodology details that lead to this difference in num- ber density), we use the METACALIBRATION catalog for our fiducial analysis. 2. Photometric redshifts Redshift probability distributions are also required for source galaxies in cosmological inferences. For each source galaxy, the probability density that it is at redshift z, pBPZ(z), is obtained using a modified version of the BPZ algo- rithm [105], as detailed in [88]. Source galaxies are placed in one of four redshift bins, z = [(0.2 − 0.43), (0.43 − TABLE I. Parameters and priorsa used to describe the measured two- point functions. Flat denotes a flat prior in the range given while Gauss(µ, σ) is a Gaussian prior with mean µ and width σ. Priors for the tomographic nuisance parameters mi and ∆zi have been widened to account for the correlation of calibration errors between bins [88, their appendix A]. The ∆zi priors listed are for METACAL- IBRATION galaxies and BPZ photo-z estimates (see [88] for other combinations). The parameter w is fixed to −1 in the ΛCDM runs. Parameter Prior Cosmology Ωm flat (0.1, 0.9) As flat (5 × 10−10, 5 × 10−9) ns flat (0.87, 1.07) Ωb flat (0.03, 0.07) h flat (0.55, 0.91) Ωνh 2 flat(5 × 10−4,10−2) w flat (−2,−0.33) Lens Galaxy Bias bi(i = 1, 5) flat (0.8, 3.0) Intrinsic Alignment AIA(z) = AIA[(1 + z)/1.62]ηIA AIA flat (−5, 5) ηIA flat (−5, 5) Lens photo-z shift (red sequence) ∆z1l Gauss (0.008, 0.007) ∆z2l Gauss (−0.005, 0.007) ∆z3l Gauss (0.006, 0.006) ∆z4l Gauss (0.000, 0.010) ∆z5l Gauss (0.000, 0.010) Source photo-z shift ∆z1s Gauss (−0.001, 0.016) ∆z2s Gauss (−0.019, 0.013) ∆z3s Gauss (+0.009, 0.011) ∆z4s Gauss (−0.018, 0.022) Shear calibration mi METACALIBRATION(i = 1, 4) Gauss (0.012, 0.023) mi IM3SHAPE(i = 1, 4) Gauss (0.0, 0.035) a The lens photo-z priors changed slightly after unblinding due to changes in the cross-correlation analysis, as described in [90]; we checked that these changes did not impact our results. 0.63), (0.63−0.9), (0.9−1.3)], based upon the mean of their pBPZ(z) distributions. As described in [88], [92] and [93], in the case of METACALIBRATION these bin assignments are based upon photo-z estimates derived using photometric mea- surements made by the METACALIBRATION pipeline in order to allow for correction of selection effects. We denote by niPZ(z) an initial estimate of the redshift dis- tribution of the N i galaxies in bin i produced by randomly drawing a redshift z from the probability distribution pBPZ(z) of each galaxy assigned to the bin, and then bin all these N i redshifts into a histogram. For this step, we use a BPZ esti- mate based on the optimal flux measurements from the multi- epoch multi-object fitting procedure (MOF) described in [86]. For both the source and the lens galaxies, uncertainties in the redshift distribution are quantified by assuming that the true redshift distribution ni(z) in bin i is a shifted version of 7 the photometrically derived distribution: ni(z) = niPZ(z −∆zi), (II.1) with the ∆zi being free parameters in the cosmological anal- yses. Prior constraints on these shift parameters are derived in two ways. First, we constrain ∆zi from a matched sample of galaxies in the COSMOS field, as detailed in [88]. Reliable redshift estimates for nearly all DES-selectable galaxies in the COS- MOS field are available from 30-band imaging [106]. We se- lect and weight a sample of COSMOS galaxies representative of the DES sample with successful shape measurements based on their color, magnitude, and pre-seeing size. The mean red- shift of this COSMOS sample is our estimate of the true mean redshift of the DES source sample, with statistical and system- atic uncertainties detailed in [88]. The sample variance in the best-fit ∆zi from the small COSMOS field is reduced, but not eliminated, by reweighting the COSMOS galaxies to match the multiband flux distribution of the DES source sample. Second, the ∆zi of both lens and source samples are fur- ther constrained by the angular cross-correlation of each with a distinct sample of galaxies with well-determined redshifts. The ∆zil for the three lowest-redshift lens galaxy samples are constrained by cross-correlation of redMaGiC with spec- troscopic redshifts [90] obtained in the overlap of DES Y1 with Stripe 82 of the Sloan Digital Sky Survey. The ∆zis for the three lowest-redshift source galaxy bins are constrained by cross-correlating the sources with the redMaGiC sample, since the redMaGiC photometric redshifts are much more ac- curate and precise than those of the sources [89][91]. The z < 0.85 limit of the redMaGiC sample precludes use of cross-correlation to constrain ∆z4 s , so its prior is determined solely by the reweighted COSMOS galaxies. For the first three source bins, both methods yield an es- timate of ∆zis, and the two estimates are compatible, so we combine them to obtain a joint constraint. The priors derived for both lens and source redshifts are listed in Table I. The re- sulting estimated redshift distributions are shown in Figure 1. Ref. [88] and Figure 20 in Appendix B demonstrate that, at the accuracy attainable in DES Y1, the precise shapes of the ni(z) functions have negligible impact on the in- ferred cosmology as long as the mean redshifts of every bin, parametrized by the ∆zi, are allowed to vary. As a con- sequence, the cosmological inferences are insensitive to the choice of photometric redshift algorithm used to establish the initial niPZ(z) of the bins. III. TWO-POINT MEASUREMENTS We measure three sets of two-point statistics: the auto- correlation of the positions of the redMaGiC lens galaxies, the cross-correlation of the lens positions with the shear of the source galaxies, and the two-point correlation of the source galaxy shear field. Each of the three classes of statistics is measured using treecorr [107] in all pairs of redshift bins of the galaxy samples and in 20 log-spaced bins of angular separation 2.5′ < θ < 250′, although we exclude some of 0 1 2 3 4 5 6 7 8 N or m al iz ed co un ts Lenses redMaGiC 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Redshift 0 1 2 3 4 5 N or m al iz ed co un ts Sources METACALIBRATION IM3SHAPE FIG. 1. Estimated redshift distributions of the lens and source galax- ies used in the Y1 analysis. The shaded vertical regions define the bins: galaxies are placed in the bin spanning their mean photo-z esti- mate. We show both the redshift distributions of galaxies in each bin (colored lines) and their overall redshift distributions (black lines). Note that source galaxies are chosen via two different pipelines IM3SHAPE and METACALIBRATION, so their redshift distributions and total numbers differ (solid vs. dashed lines). the scales and cross-correlations from our fiducial data vector (see section IV). Figures 2 and 3 show these measurements and our best-fit ΛCDM model. A. Galaxy Clustering: w(θ) The inhomogeneous distribution of matter in the Universe is traced by galaxies. The overabundance of pairs at angu- lar separation θ above that expected in a random distribution, w(θ), is one of the simplest measurements of galaxy cluster- ing. It quantifies the strength and scale dependence of the clustering of galaxies, which in turn reflects the clustering of matter. The upper panel of Figure 2 shows the angular correlation function of the redMaGiC galaxies in the five lens redshift bins described above. As described in [94], these correlation functions were computed after quantifying and correcting for spurious clustering induced by each of multiple observational variables. Figure 2 shows the data with the error bars set equal to the square root of the diagonal elements of the covariance matrix, but we note that data points in nearby angular bins are highly correlated. Indeed, as can be seen in Figure 5 of [84], in the lowest redshift bins the correlation coefficient between almost all angular bins is close to unity; at higher redshift, the measurements are highly correlated only over the adja- cent few angular bins. The solid curve in Figure 2 shows the best-fit prediction from ΛCDM after fitting to all three two- 8 point functions. In principle, we could also use the angular cross-correlations between galaxies in different redshift bins in the analysis, but the amount of information in these cross- bin two-point functions is quite small and would require sub- stantially enlarging the covariance matrix, so we use only the auto-correlations. B. Galaxy–galaxy lensing: γt(θ) The shapes of background source galaxies are distorted by the mass associated with foreground lenses. The character- istic distortion is a tangential shear, with the source galaxy ellipticities oriented perpendicular to the line connecting the foreground and background galaxies. This shear, γt(θ), is sensitive to the mass associated with the foreground galax- ies. On scales much larger than the sizes of parent halos of the galaxies, it is proportional to the lens galaxy bias parame- ters bi in each lens bin which quantifies the relative clumping of matter and galaxies. The lower panels of Figure 2 show the measurements of galaxy–galaxy lensing in all pairs of lens- source tomographic bins, including the model prediction for our best-fit parameters. The plots include bin pairs for which the lenses are nominally behind the sources (those towards the upper right), so might be expected to have zero signal. Although the signals for these bins are expected to be small, they can still be useful in constraining the intrinsic alignment parameters in our model (see, e.g., [108]). In [93], we carried out a number of null tests to ensure the robustness of these measurements, none of which showed evi- dence for significant systematic uncertainties besides the ones characterized by the nuisance parameters in this analysis. The model fits the data well. Even the fits that appear quite bad are misleading because of the highly off-diagonal covariance matrix. For the nine data points in the 3–1 bin, for example, χ2 = 14, while χ2 would be 30 if the off-diagonal elements were ignored. C. Cosmic shear: ξ±(θ) The two-point statistics that quantify correlations between the shapes of galaxies are more complex, because they are the products of the components of a spin-2 tensor. Therefore, a pair of two-point functions are used to capture the relevant in- formation: ξ+(θ) and ξ−(θ) are the sum and difference of the products of the tangential and cross components of the shear, measured with respect to the line connecting each galaxy pair. For more details, see [92] or earlier work in Refs [109–116]. Figure 3 shows these functions for different pairs of tomo- graphic bins. As in Figure 2, the best-fit model prediction here includes the impact of intrinsic alignment; the best-fit shifts in the pho- tometric redshift distributions; and the best-fit values of shear calibration. The one-dimensional posteriors on all of these parameters are shown in Figure 19 in Appendix A. IV. ANALYSIS A. Model To extract cosmological information from these two-point functions, we construct a model that depends upon both cosmological parameters and astrophysical and observational nuisance parameters. The cosmological parameters govern the expansion history as well as the evolution and scale depen- dence of the matter clustering amplitude (as quantified, e.g., by the power spectrum). The nuisance parameters account for uncertainties in photometric redshifts, shear calibration, the bias between galaxies and mass, and the contribution of in- trinsic alignment to the shear spectra. §IV B will enumerate these parameters, and our priors on them are listed in Table I. Here, we describe how the two-point functions presented in §III are computed in the model. 1. Galaxy Clustering: w(θ) Following [84], we express the projected (angular) density contrast of redMaGiC galaxies in redshift bin i by δig, the con- vergence field of source tomography bin j as κj , the redshift distribution of the redMaGiC/source galaxy sample in tomog- raphy bin i as nig/κ(z), and the angular number densities of galaxies in this redshift bin as n̄ig/κ = ∫ dz nig/κ(z) . (IV.1) The radial weight function for clustering in terms of the co- moving radial distance χ is qiδg(k, χ) = bi (k, z(χ)) nig(z(χ)) n̄ig dz dχ , (IV.2) with bi(k, z(χ)) the galaxy bias of the redMaGiC galaxies in tomographic bin i, and the lensing efficiency qiκ(χ) = 3H2 0 Ωm 2c2 χ a(χ) ∫ χh χ dχ′ niκ(z(χ′))dz/dχ′ n̄iκ χ′ − χ χ′ , (IV.3) with H0 the Hubble constant, c the speed of light, and a the scale factor. Under the Limber approximation [117–120], the angular correlation function for galaxy clustering can be writ- ten as wi(θ) = ∫ dl l 2π J0(lθ) ∫ dχ qiδg ( l+1/2 χ , χ ) qjδg ( l+1/2 χ , χ ) χ2 ×PNL ( l + 1/2 χ , z(χ) ) (IV.4) with PNL(k, z) the non-linear matter power spectrum at wave vector k and redshift z. The expression in Eq. (IV.4) and the ones in Eqs. (IV.5) and (IV.6) use the “flat-sky” approximation, which was tested against a curved sky implementation in [84] for the case of 9 101 102 0.0 0.5 1.0 11 101 102 21 101 102 31 101 102 41 101 102 51 101 102 0.0 1.0 12 101 102 22 101 102 32 101 102 42 101 102 52 101 102 0.0 1.0 2.0 13 101 102 23 101 102 33 101 102 43 101 102 53 101 102 θ (arcmin) 0.0 1.0 2.0 14 101 102 θ 24 101 102 θ 34 101 102 θ 44 101 102 θ 54 101 102 θ (arcmin) 0.0 1.0 2.0 1 101 102 θ 2 101 102 θ 3 101 102 θ 4 101 102 θ 5 DES Y1 fiducial best-fit model scale cuts θγ t (1 0− 2 ar cm in ) θw (a rc m in ) FIG. 2. Top panels: scaled angular correlation function, θw(θ), of redMaGiC galaxies in the five redshift bins in the top panel of Figure 1, from lowest (left) to highest redshift (right) [94]. The solid lines are predictions from the ΛCDM model that provides the best fit to the combined three two-point functions presented in this paper. Bottom panels: scaled galaxy–galaxy lensing signal, θγt (galaxy-shear correlation), measured in DES Y1 in four source redshift bins induced by lens galaxies in five redMaGiC bins [93]. Columns represent different lens redshift bins while rows represent different source redshift bins, so e.g., bin labeled 12 is the signal from the galaxies in the second source bin lensed by those in the first lens bin. The solid curves are again our best-fit ΛCDM prediction. In all panels, shaded areas display the angular scales that have been excluded from our cosmological analysis (see §IV). 10 101 102 0.0 2.0 11 101 102 0.0 2.0 11 101 102 0.0 2.0 21 101 102 0.0 2.0 21 101 102 0.0 2.0 4.0 31 101 102 0.0 2.0 31 101 102 θ (arcmin) 0.0 5.0 41 101 102 θ (arcmin) 0.0 5.0 41 101 102 22 101 102 22 101 102 32 101 102 32 101 102 θ 42 101 102 θ 42 101 102 33 101 102 33 101 102 θ 43 101 102 θ 43 101 102 θ 44 DES Y1 fiducial best-fit model scale cuts 101 102 θ 44 θ ξ + (1 0− 4 ar cm in ) θ ξ − (1 0− 4 ar cm in ) FIG. 3. The cosmic shear correlation functions ξ+ (top panel) and ξ− (bottom panel) in DES Y1 in four source redshift bins, including cross correlations, measured from the METACALIBRATION shear pipeline (see [92] for the corresponding plot with IM3SHAPE); pairs of numbers in the upper left of each panel indicate the redshift bins. The solid lines show predictions from our best-fit ΛCDM model from the analysis of all three two-point functions, and the shaded areas display the angular scales that are not used in our cosmological analysis (see §IV). 11 galaxy clustering. Ref. [84] uses the more accurate expression that sums over Legendre polynomials, and we find that these two expressions show negligible differences over the scales of interest. The model power spectrum here is the fully nonlinear power spectrum in ΛCDM or wCDM, which we estimate on a grid of (k, z) by first running CAMB [121] or CLASS [122] to obtain the linear spectrum and then HALOFIT [123–125] for the nonlinear spectrum. The smallest angular separations for which the galaxy two-point function measurements are used in the cosmological inference, indicated by the boundaries of the shaded regions in the upper panels of Figure 2, cor- respond to a comoving scale of 8h−1 Mpc; this scale is cho- sen such that modeling uncertainties in the non-linear regime cause negligible impact on the cosmological parameters rela- tive to their statistical errors, as shown in [84] and [92]. As described in §VI of [84], we include the impact of neu- trino bias [126–128] when computing the angular correlation function of galaxies. For Y1 data, this effect is below statis- tical uncertainties, but it is computationally simple to imple- ment and will be relevant for upcoming analyses. 2. Galaxy–galaxy lensing: γt(θ) We model the tangential shear similarly to how we modeled the angular correlation function. Consider the correlation of lens galaxy positions in bin i with source galaxy shear in bin j; on large scales, it can be expressed as an integral over the power spectrum, γijt (θ) = (1 +mj) ∫ dl l 2π J2(lθ) ∫ dχ qiδg ( l+1/2 χ , χ ) qjκ(χ) χ2 ×PNL ( l + 1/2 χ , z(χ) ) (IV.5) where mj is the multiplicative shear bias, and J2 is the 2nd- order Bessel function. The shift parameters characterizing the photo-z uncertainties ∆zjs and ∆zil enter the radial weight functions in Eqs. (IV.2) and (IV.3) via Eqs. (IV.1) and (II.1). The shear signal also depends upon intrinsic alignments of the source shapes with the tidal fields surrounding the lens galax- ies; details of our model for this effect (along with an exami- nation of more complex models) are given in [84] and in [92]. The smallest angular separations for which the galaxy–galaxy lensing measurements are used in the cosmological inference, indicated by the boundaries of the shaded regions in the lower panels of Figure 2, correspond to a comoving scale of 12h−1 Mpc; as above, this scale is chosen such that the model uncer- tainties in the non-linear regime cause insignificant changes to the cosmological parameters relative to the statistical un- certainties, as derived in [84] and verified in [85]. 3. Cosmic shear ξ±(θ) The cosmic shear signal is independent of galaxy bias but shares the same general form as the other sets of two-point functions. The theoretical predictions for these shear-shear two-point functions are ξij+/−(θ) = (1 +mi)(1 +mj) ∫ dl l 2π J0/4(lθ)∫ dχ qiκ(χ)qjκ(χ) χ2 PNL ( l + 1/2 χ , z(χ) ) (IV.6) where the efficiency functions are defined above, and J0 and J4 are the Bessel functions for ξ+ and ξ−. Intrinsic alignment affects the cosmic shear signal, especially the low-redshift bins, and are modeled as in [84]. Baryons affect the matter power spectrum on small scales, and the cosmic shear sig- nal is potentially sensitive to these uncertain baryonic effects; we restrict our analysis to the unshaded, large-scale regions shown in Figure 3 to reduce uncertainty in these effects below our measurement errors, following the analysis in [92]. B. Parameterization and Priors We use these measurements from the DES Y1 data to es- timate cosmological parameters in the context of two cosmo- logical models, ΛCDM and wCDM. ΛCDM contains three energy densities in units of the critical density: the matter, baryon, and massive neutrino energy densities, Ωm,Ωb, and Ων . The energy density in massive neutrinos is a free param- eter but is often fixed in cosmological analyses to either zero or to a value corresponding to the minimum allowed neutrino mass of 0.06 eV from oscillation experiments [129]. We think it is more appropriate to vary this unknown parameter, and we do so throughout the paper (except in §VII D, where we show that this does not affect our qualitative conclusions). We split the mass equally among the three eigenstates, hence assum- ing a degenerate mass hierarchy for the neutrinos. Since most other survey analyses have fixed Ων , our results for the re- maining parameters will differ slightly from theirs, even when using their data. ΛCDM has three additional free parameters: the Hubble pa- rameter, H0, and the amplitude and spectral index of the pri- mordial scalar density perturbations,As and ns. This model is based on inflation, which fairly generically predicts a flat uni- verse. Further when curvature is allowed to vary in ΛCDM, it is constrained by a number of experiments to be very close to zero. Therefore, although we plan to study the impact of curvature in future work, in this paper we assume the universe is spatially flat, with ΩΛ = 1 − Ωm. It is common to replace As with the RMS amplitude of mass fluctuations on 8 h−1 Mpc scale in linear theory, σ8, which can be derived from the aforementioned parameters. Instead of σ8, in this work we will focus primarily on the related parameter S8 ≡ σ8 ( Ωm 0.3 )0.5 (IV.7) since S8 is better constrained than σ8 and is largely uncorre- lated with Ωm in the DES parameter posterior. We also consider the possibility that the dark energy is not a cosmological constant. Within this wCDM model, the dark 12 energy equation of state parameter, w (not to be confused with the angular correlation function w(θ)), is taken as an addi- tional free parameter instead of being fixed at w = −1 as in ΛCDM. wCDM thus contains 7 cosmological parameters. In future analyses of larger DES data sets, we anticipate con- straining more extended cosmological models, e.g., those in which w is allowed to vary in time. In addition to the cosmological parameters, our model for the data contains 20 nuisance parameters, as indicated in the lower portions of Table I. These are the nine shift parameters, ∆zi, for the source and lens redshift bins, the five redMaGiC bias parameters, bi, the four multiplicative shear biases, mi, and two parameters, AIA and ηIA, that parametrize the intrin- sic alignment model. Table I presents the priors we impose on the cosmological and nuisance parameters in the analysis. For the cosmological parameters, we generally adopt flat priors that span the range of values well beyond the uncertainties reported by recent ex- periments. As an example, although there are currently poten- tially conflicting measurements of h, we choose the lower end of the prior to be 10σ below the lower central value from the Planck cosmic microwave background measurement [53] and the upper end to be 10σ above the higher central value from local measurements [130]. In the case of wCDM, we impose a physical upper bound of w < −0.33, as that is required to obtain cosmic acceleration. As another example, the lower bound of the prior on the massive neutrino density, Ωνh 2, in Table I corresponds to the experimental lower limit on the sum of neutrino masses from oscillation experiments. For the astrophysical parameters bi, AIA, and ηIA that are not well constrained by other analyses, we also adopt conser- vatively wide, flat priors. For all of these relatively uninforma- tive priors, the guiding principle is that they should not impact our final results, and in particular that the tails of the posterior parameter distributions should not lie close to the edges of the priors3. For the remaining nuisance parameters, ∆zi and mi, we adopt Gaussian priors that result from the comprehensive analyses described in Refs. [87–91]. The prior and posterior distributions of these parameters are plotted in Appendix A in Figure 19. In evaluating the likelihood function (§IV C), the param- eters with Gaussian priors are allowed to vary over a range roughly five times wider than the prior; for example, the pa- rameter that accounts for a possible shift in the furthest lens redshift bin, ∆z5 l , has a 1-σ uncertainty of 0.01, so it is al- lowed to vary over |∆z5 l | < 0.05. These sampling ranges conservatively cover the parameter values of interest while avoiding computational problems associated with exploring parameter ranges that are overly broad. Furthermore, overly broad parameter ranges would distort the computation of the Bayesian evidence, which would be problematic as we will use Bayes factors to assess the consistency of the different two-point function measurements, consistency with external 3 The sole exception is the intrinsic-alignment parameter ηIA for which the posterior does hit the edge of the (conservatively selected, given feasible IA evolution) prior; see Figure 19 in the Appendix A. data sets, and the need to introduce additional parameters (such as w) into the analysis. We have verified that our re- sults below are insensitive to the prior ranges chosen. C. Likelihood Analysis For each data set, we sample the likelihood, assumed to be Gaussian, in the many-dimensional parameter space: lnL(~p) = −1 2 ∑ ij [Di − Ti(~p)]C−1 ij [Dj − Tj(~p)] , (IV.8) where ~p is the full set of parameters,Di are the measured two- point function data presented in Figures 2 and 3, and Ti(~p) are the theoretical predictions as given in Eqs. (IV.4, IV.5, IV.6). The likelihood depends upon the covariance matrix C that de- scribes how the measurement in each angular and redshift bin is correlated with every other measurement. Since the DES data vector contains 457 elements, the covariance is a sym- metric 457 × 457 matrix. We generate the covariance matri- ces using CosmoLike [131], which computes the relevant four-point functions in the halo model, as described in [84]. We also describe there how the CosmoLike-generated co- variance matrix is tested with simulations. Eq. (IV.8) leaves out the ln(det(C)) in the prefactor4 and more generally neglects the cosmological dependence of the covariance matrix. Previous work [132] has shown that this dependence is likely to have a small impact on the central value; our rough estimates of the impact of neglecting the de- terminant confirm this; and — as we will show below — our results did not change when we replaced the covariance ma- trix with an updated version based on the best-fit parameters. However, as we will see, the uncertainty in the covariance ma- trix leads to some lingering uncertainty in the error bars. To form the posterior, we multiply the likelihood by the priors, P(~p), as given in Table I. Parallel pipelines, CosmoSIS5 [133] and CosmoLike, are used to compute the theoretical predictions and to gen- erate the Monte Carlo Markov Chain (MCMC) samples that map out the posterior space leading to parameter constraints. The two sets of software use the publicly available samplers MultiNest [134] and emcee [135]. The former provides a powerful way to compute the Bayesian evidence described below so most of the results shown here use CosmoSIS run- ning MultiNest. D. Tests on Simulations The collaboration has produced a number of realistic mock catalogs for the DES Y1 data set, based upon two 4 However, this factor is important for the Bayesian evidence calculations discussed below so is included in those calculations. 5 https://bitbucket.org/joezuntz/cosmosis/ 13 different cosmological N -body simulations (Buzzard [136], MICE [137]), which were analyzed as described in [85]. We applied all the steps of the analysis on the simulations, from measuring the relevant two-point functions to extracting cos- mological parameters. In the case of simulations, the true cosmology is known, and [85] demonstrates that the analysis pipelines we use here do indeed recover the correct cosmolog- ical parameters. V. BLINDING AND VALIDATION The small statistical uncertainties afforded by the Y1 data set present an opportunity to obtain improved precision on cosmological parameters, but also a challenge to avoid con- firmation biases. To preclude such biases, we followed the guiding principle that decisions on whether the data analysis has been successful should not be based upon whether the in- ferred cosmological parameters agreed with our previous ex- pectations. We remained blind to the cosmological parame- ters implied by the data until after the analysis procedure and estimates of uncertainties on various measurement and astro- physical nuisance parameters were frozen. To implement this principle, we first transformed the el- lipticities e in the shear catalogs according to arctanh |e| → λ arctanh |e|, where λ is a fixed blind random number be- tween 0.9 and 1.1. Second, we avoided plotting the measured values and theoretical predictions in the same figure (includ- ing simulation outputs as “theory”). Third, when running codes that derived cosmological parameter constraints from observed statistics, we shifted the resulting parameter values to obscure the best-fit values and/or omitted axis labels on any plots. These measures were all kept in place until the following criteria were satisfied: 1. All non-cosmological systematics tests of the shear measurements were passed, as described in [87], and the priors on the multiplicative biases were finalized. 2. Photo-z catalogs were finalized and passed internal tests, as described in [88–91]. 3. Our analysis pipelines and covariance matrices, as de- scribed in [84, 85], passed all tests, including robustness to intrinsic alignment and bias model assumptions. 4. We checked that the ΛCDM constraints (on, e.g., Ωm, σ8) from the two different cosmic shear pipelines IM3SHAPE and METACALIBRATION agreed. The pipelines were not tuned in any way to force agreement. 5. ΛCDM constraints were stable when dropping the smallest angular bins for METACALIBRATION cosmic shear data. 6. Small-scale METACALIBRATION galaxy–galaxy lens- ing data were consistent between source bins (shear- ratio test, as described in §6 of [93]). We note that while this test is performed in the nominal ΛCDM model, it is close to insensitive to cosmological parameters, and therefore does not introduce confirmation bias. Once the above tests were satisfied, we unblinded the shear catalogs but kept cosmological parameter values blinded while carrying out the following checks, details of which can be found in Appendix A: 7. Consistent results were obtained from the two the- ory/inference pipelines, CosmoSIS and CosmoLike. 8. Consistent results on all cosmological parameters were obtained with the two shear measurement pipelines, METACALIBRATION and IM3SHAPE. 9. Consistent results on the cosmological parameters were obtained when we dropped the smallest-angular-scale components of the data vector, reducing our susceptibil- ity to baryonic effects and departures from linear galaxy biasing. This test uses the combination of the three two- point functions (as opposed to from shear only as in test 5). 10. An acceptable goodness-of-fit value (χ2) was found be- tween the data and the model produced by the best- fitting parameters. This assured us that the data were consistent with some point in the model space that we are constraining, while not yet revealing which part of parameter space that is. 11. Parameters inferred from cosmic shear (ξ±) were consistent with those inferred from the combination of galaxy–galaxy lensing (γt) and galaxy clustering (w(θ)). Once these tests were satisfied, we unblinded the param- eter inferences. The following minor changes to the analy- sis procedures or priors were made after the unblinding: as planned before unblinding, we re-ran the MCMC chains with a new covariance matrix calculated at the best-fit parameters of the original analysis. This did not noticeably change the constraints (see Figure 21 in Appendix B), as expected from our earlier tests on simulated data [84]. We also agreed be- fore unblinding that we would implement two changes after unblinding: small changes to the photo-z priors referred to in the footnote to Table I, and fixing a bug in IM3SHAPE object blacklisting that affected ≈ 1% of the footprint. All of the above tests passed, most with reassuringly unre- markable results; more details are given in Appendix A. For test 10, we calculated the χ2 (= −2 logL) value of the 457 data points used in the analysis using the full covariance matrix. In ΛCDM, the model used to fit the data has 26 free parameters, so the number of degrees of freedom is ν = 431. The model is calculated at the best-fit parameter values of the posterior distribution (i.e. the point from the posterior sample with lowest χ2). Given the uncertainty on the estimates of the covariance matrix, the formal probabilities of a χ2 distribution are not applicable. We agreed to unblind as long as χ2 was less than 605 (χ2/ν < 1.4). The best-fit value χ2 = 497 14 passes this test 6, with χ2/ν = 1.16. Considering the fact that 13 of the free parameters are nuisance parameters with tight Gaussian priors, we will use ν = 444, giving χ2/ν = 1.12. The best-fit models for the three two-point functions are over-plotted on the data in Figures 2 and 3, from which it is apparent that the χ2 is not dominated by conspicuous out- liers. Figure 4 offers confirmation of this, in the form of a histogram of the differences between the best-fit theory and the data in units of the standard deviation of individual data points. The three probes show similar values of χ2/ν: for ξ±(θ), χ2 = 230 for 227 data points; for γt(θ), χ2 = 185 for 176 data points; and for w(θ), χ2 = 68 for 54 data points. A finer division into each of the 45 individual 2-point func- tions shows no significant concentration of χ2 in particular bin pairs. We also find that removing all data at scales θ > 100′ yields χ2 = 278 for 277 data points (χ2/ν = 1.05), not a sig- nificant reduction, and also yields no significant shift in best- fit parameters. Thus, we find that no particular piece of our data vector dominates our χ2 result. 3 2 1 0 1 2 3 (Data-Theory)/Error 0 10 20 30 40 50 60 FIG. 4. Histogram of the differences between the best-fit ΛCDM model predictions and the 457 data points shown in Figures 2 and 3, in units of the standard deviation of the individual data points. Although the covariance matrix is not diagonal, and thus the diagonal error bars do not tell the whole story, it is clear that there are no large outliers that drive the fits. Finally, for step number 11 in the test list near the beginning of this Section, we examined several measures of consistency between (i) cosmic shear and (ii) γt(θ) + w(θ) in ΛCDM. As an initial test, we computed the mean of the 1D posterior distribution of each of the cosmological parameters and mea- 6 In our original analysis (submitted to the arXiv in August 2017), we originally found χ2 = 572, which passed the aforementioned criterion (χ2 < 605) with proceeding in the analysis. We have since identified a couple of missing ingredients in our computation of the covariance matrix, leading to the present, lower, value χ2 = 497. While the chi squared has significantly decreased, the cosmological constraints are nearly unchanged. Please see Appendix C for more details. sured the shift between (i) and (ii). We then divided this dif- ference by the expected standard deviation of this difference (taking into account the estimated correlation between the ξ± and γt+w inferences), σdiff = [σ2 ξ± +σ2 γtw−2Cov(ξ±, γt+ w)]1/2. For all parameters, these differences had absolute value < 0.4, indicating consistency well within measurement error. As a second consistency check, we compared the posteri- ors for the nuisance parameters from cosmic shear to those from clustering plus galaxy–galaxy lensing, and they agreed well. We found no evidence that any of the nuisance param- eters pushes against the edge of its prior or that the nuisance parameters for cosmic shear and w + γt are pushed to signif- icantly different values. The only mild exceptions are modest shifts in the intrinsic alignment parameters, AIA and ηIA, as well as in the second source redshift bin, ∆z2 s . The full set of posteriors on all 20 nuisance parameters for METACALIBRA- TION is shown in Figure 19 in Appendix A. As a final test of consistency between the two sets of two- point-function measurements, we use the Bayes factor (also called the “evidence ratio”). The Bayes factor is used for dis- criminating between two hypotheses, and is the ratio of the Bayesian evidences, P ( ~D|H) (the probability of observing dataset, ~D, given hypothesis H) for each hypothesis. An ex- ample of such a hypothesis is that dataset ~D can be described by a model M , in which case the Bayesian evidence is P ( ~D|H) = ∫ dNθP ( ~D|~θ,M)P (~θ|M) (V.1) where P ( ~D|~θ,M) is the likelihood of the data given the model M parametrized by its N parameters ~θ, and P (~θ|M) is the prior probability distribution of those model parameters. For two hypotheses H0 and H1, the Bayes factor is given by R = P ( ~D|H0) P ( ~D|H1) = P (H0| ~D)P (H1) P (H1| ~D)P (H0) (V.2) where the second equality follows from Bayes’ theorem and clarifies the meaning of the Bayes factor: if we have equal a priori belief in H0 and H1 (i.e., P (H0) = P (H1)), the Bayes factor is the ratio of the posterior probability of H0 to the posterior probability ofH1. The Bayes factor can be inter- preted in terms of odds, i.e., it implies H0 is favored over H1 with R : 1 odds (or disfavored if R < 1). We will adopt the widely used Jeffreys scale [138] for interpreting Bayes fac- tors: 3.2 < R < 10 and R > 10 are respectively considered substantial and strong evidence for H0 over H1. Conversely, H1 is strongly favored over H0 if R < 0.1, and there is sub- stantial evidence for H1 if 0.1 < R < 0.31. We follow [139] by applying this formalism as a test for consistency between cosmological probes. In this case, the null hypothesis, H0, is that the two datasets were measured from the same universe and therefore share the same model parameters. Two probes would be judged discrepant if they strongly favor the alternative hypothesis, H1, that they are measured from two different universes with different model 15 parameters. So the appropriate Bayes factor for judging con- sistency of two datasets, D1 and D2, is R = P ( ~D1, ~D2|M ) P ( ~D1|M ) P ( ~D2|M ) (V.3) whereM is the model, e.g., ΛCDM orwCDM. The numerator is the evidence for both datasets when model M is fit to both datasets simultaneously. The denominator is the evidence for both datasets when model M is fit to both datasets individu- ally, and therefore each dataset determines its own parameter posteriors. Before the data were unblinded, we decided that we would combine results from these two sets of two-point functions if the Bayes factor defined in Eq. (V.3) did not suggest strong evidence for inconsistency. According to the Jeffreys scale, our condition to combine is therefore that R > 0.1 (since R < 0.1 would imply strong evidence for inconsistency). We find a Bayes factor of R = 583, an indication that DES Y1 cosmic shear and galaxy clustering plus galaxy–galaxy lens- ing are consistent with one another in the context of ΛCDM. The DES Y1 data were thus validated as internally con- sistent and robust to our assumptions before we gained any knowledge of the cosmological parameter values that they im- ply. Any comparisons to external data were, of course, made after the data were unblinded. VI. DES Y1 RESULTS: PARAMETER CONSTRAINTS A. ΛCDM We first consider the ΛCDM model with six cosmological parameters. The DES data are most sensitive to two cosmo- logical parameters, Ωm and S8 as defined in Eq. (IV.7), so for the most part we focus on constraints on these parameters. Given the demonstrated consistency of cosmic shear with clustering plus galaxy–galaxy lensing in the context of ΛCDM as noted above, we proceed to combine the constraints from all three probes. Figure 5 shows the constraints on Ωm and σ8 (bottom panel), and on Ωm and the less degenerate param- eter S8 (top panel). Constraints from cosmic shear, galaxy clustering + galaxy–galaxy lensing, and their combination are shown in these two-dimensional subspaces after marginaliz- ing over the 24 other parameters. The combined results lead to constraints Ωm = 0.267+0.030 −0.017 S8 = 0.773+0.026 −0.020 σ8 = 0.817+0.045 −0.056. (VI.1) The value of Ωm is consistent with the value inferred from either cosmic shear or clustering plus galaxy–galaxy lensing separately. We present the resulting marginalized constraints on the cosmological parameters in the top rows of Table II. The results shown in Figure 5, along with previous anal- yses such as that using KiDS + GAMA data [67], are an 0.70 0.75 0.80 0.85 0.90 S 8 0.2 0.3 0.4 0.5 Ωm 0.60 0.75 0.90 1.05 σ 8 DES Y1 Shear DES Y1 w + γt DES Y1 All 0.70 0.75 0.80 0.85 0.90 S8 FIG. 5. ΛCDM constraints from DES Y1 on Ωm, σ8, and S8 from cosmic shear (green), redMaGiC galaxy clustering plus galaxy– galaxy lensing (red), and their combination (blue). Here, and in all such 2D plots below, the two sets of contours depict the 68% and 95% confidence levels. important step forward in the capability of combined probes from optical surveys to constrain cosmological parameters. These combined constraints transform what has, for the past decade, been a one-dimensional constraint on S8 (which ap- pears banana-shaped in the Ωm − σ8 plane) into tight con- straints on both of these important cosmological parameters. Figure 6 shows the DES Y1 constraints on S8 and Ωm along with some previous results and in combination with exter- nal data sets, as will be discussed below. The sizes of these parameter error bars from the combined DES Y1 probes are comparable to those from the CMB obtained by Planck. In addition to the cosmological parameters, these probes constrain important astrophysical parameters. The intrinsic alignment (IA) signal is modeled to scale as AIA(1 + z)ηIA ; while the data do not constrain the power law well (ηIA = −0.7± 2.2), they are sensitive to the amplitude of the signal: AIA = 0.44+0.38 −0.28 (95% CL). (VI.2) Further strengthening evidence from the recent combined probes analysis of KiDS [67, 68], this result is the strongest evidence to date of IA in a broadly inclusive galaxy sam- ple; previously, significant IA measurements have come from selections of massive elliptical galaxies, usually with spec- troscopic redshifts (e.g. [140]). The ability of DES data to produce such a result without spectroscopic redshifts demon- strates the power of this combined analysis and emphasizes the importance of modeling IA in the pursuit of accurate cos- mology from weak lensing. We are able to rule out AIA = 0 at 99.76% CL with DES alone and at 99.90% CL with the full 16 TABLE II. 68%CL marginalized cosmological constraints in ΛCDM and wCDM using a variety of datasets. “DES Y1 3x2” refers to results from combining all 3 two-point functions in DES Y1. Cells with no entries correspond to posteriors not significantly narrower than the prior widths. The only exception is in wCDM for Planck only, where the posteriors on h are shown to indicate the large values inferred in the model without any data to break the w − h degeneracy. Model Data Sets Ωm S8 ns Ωb h ∑ mν (eV) (95% CL) w ΛCDM DES Y1 ξ±(θ) 0.260+0.065 −0.037 0.782+0.027 −0.027 . . . . . . . . . . . . . . . ΛCDM DES Y1 w(θ) + γt 0.288+0.045 −0.026 0.760+0.033 −0.030 . . . . . . . . . . . . . . . ΛCDM DES Y1 3x2 0.267+0.030 −0.017 0.773+0.026 −0.020 . . . . . . . . . . . . . . . ΛCDM Planck (No Lensing) 0.334+0.037 −0.026 0.841+0.027 −0.025 0.958+0.008 −0.005 0.0503+0.0046 −0.0019 0.658+0.019 −0.027 . . . . . . ΛCDM DES Y1 + Planck (No Lensing) 0.297+0.016 −0.012 0.795+0.020 −0.013 0.972+0.006 −0.004 0.0477+0.0016 −0.0012 0.686+0.009 −0.014 < 0.47 . . . ΛCDM DES Y1 + JLA + BAO 0.295+0.018 −0.014 0.768+0.018 −0.023 1.044+0.019 −0.087 0.0516+0.0050 −0.0080 0.672+0.049 −0.034 . . . . . . ΛCDM Planck + JLA + BAO 0.306+0.007 −0.007 0.815+0.015 −0.013 0.969+0.004 −0.005 0.0483+0.0008 −0.0006 0.678+0.007 −0.005 < 0.22 . . . ΛCDM DES Y1 + Planck + JLA + BAO 0.298+0.007 −0.007 0.802+0.012 −0.012 0.973+0.005 −0.004 0.0479+0.0007 −0.0008 0.685+0.005 −0.007 < 0.26 . . . wCDM DES Y1 ξ±(θ) 0.274+0.073 −0.042 0.777+0.036 −0.038 . . . . . . . . . . . . −0.99+0.33 −0.39 wCDM DES Y1 w(θ) + γt 0.310+0.049 −0.036 0.785+0.040 −0.072 . . . . . . . . . . . . −0.79+0.22 −0.39 wCDM DES Y1 3x2 0.284+0.033 −0.030 0.782+0.036 −0.024 . . . . . . . . . . . . −0.82+0.21 −0.20 wCDM Planck (No Lensing) 0.222+0.069 −0.024 0.810+0.029 −0.036 0.960+0.005 −0.007 0.0334+0.0099 −0.0032 0.801+0.045 −0.097 . . . −1.47+0.31 −0.22 wCDM DES Y1 + Planck (No Lensing) 0.233+0.025 −0.033 0.775+0.021 −0.021 0.971+0.004 −0.006 0.0355+0.0050 −0.0039 0.775+0.056 −0.040 < 0.65 −1.35+0.16 −0.17 wCDM Planck + JLA + BAO 0.303+0.010 −0.008 0.816+0.014 −0.013 0.968+0.004 −0.006 0.0479+0.0016 −0.0014 0.679+0.013 −0.008 < 0.27 −1.02+0.05 −0.05 wCDM DES Y1 + Planck + JLA + BAO 0.301+0.007 −0.010 0.801+0.011 −0.012 0.974+0.005 −0.005 0.0483+0.0014 −0.0016 0.680+0.013 −0.008 < 0.31 −1.00+0.05 −0.04 combination of DES and external data sets. The mean value of AIA is nearly the same when combining with external data sets, suggesting that IA self-calibration has been effective. In- terestingly, the measured amplitude agrees well with a predic- tion made by assuming that only red galaxies contribute to the IA signal, and then extrapolating the IA amplitude measured from spectroscopic samples of luminous galaxies using a re- alistic luminosity function and red galaxy fraction [84]. Our measurement extends the diversity of galaxies with evidence of IA, allowing more precise predictions for the behavior of the expected IA signal. The biases of the redMaGiC galaxy samples in the five lens bins are shown in Figure 7 along with the results with fixed cosmology obtained in [93] and [94]. The biases are mea- sured to be b1 = 1.42+0.13 −0.08, b2 = 1.65+0.08 −0.12, b3 = 1.60+0.11 −0.08, b4 = 1.92+0.14 −0.10, b5 = 2.00+0.13 −0.14. Even when varying a full set of cosmological parameters (including σ8, which is quite de- generate with bias when using galaxy clustering only) and 15 other nuisance parameters, the combined probes in DES Y1 therefore constrain bias at the ten percent level. B. wCDM A variety of theoretical alternatives to the cosmological constant have been proposed [6]. For example, it could be that the cosmological constant vanishes and that another de- gree of freedom, e.g., a very light scalar field, is driving the current epoch of accelerated expansion. Here we restrict our analysis to the simplest class of phenomenological alterna- tives, models in which the dark energy density is not constant, but rather evolves over cosmic history with a constant equa- tion of state parameter, w. We constrain w by adding it as a seventh cosmological parameter. Here, too, DES obtains interesting constraints on only a subset of the seven cosmo- logical parameters, so we show the constraints on the three- dimensional subspace spanned by Ωm, S8, and w. Figure 8 shows the constraints in this 3D space from cosmic shear and from galaxy–galaxy lensing + galaxy clustering. These two sets of probes agree with one another. The consistency in the three-dimensional subspace shown in Figure 8, along with the tests in the previous subsection, is sufficient to combine the two sets of probes. The Bayes factor in this case was equal to 1878. The combined constraint from all three two-point functions is also shown in Figure 8. The marginalized 68% CL constraints onw and on the other two cosmological parameters tightly constrained by DES, S8 17 0.7 0.75 0.8 0.85 S8 8( m/0.3)0.5 0.25 0.3 0.35 m DES Y1 All DES Y1 Shear DES Y1 w + t DES Y1 All + Planck (No Lensing) DES Y1 All + Planck + BAO + JLA DES Y1 All + BAO + JLA DES SV KiDS-450 Planck (No Lensing) Planck + BAO + JLA FIG. 6. 68% confidence levels for ΛCDM on S8 and Ωm from DES Y1 (different subsets considered in the top group, black); DES Y1 with all three probes combined with other experiments (middle group, green); and results from previous experiments (bottom group, purple). Note that neutrino mass has been varied so, e.g., results shown for KiDS-450 were obtained by re-analyzing their data with the neutrino mass left free. The table includes only data sets that are publicly available so that we could re-analyze those using the same assumptions (e.g., free neutrino mass) as are used in our analysis of DES Y1 data. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 z 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 b Lmin = 0.5L∗ L∗ 1.5L∗ w(θ) fixed cosmology γt fixed cosmology DES Y1 − all FIG. 7. The bias of the redMaGiC galaxy samples in the five lens bins from three separate DES Y1 analyses. The two labeled “fixed cosmology” use the galaxy angular correlation function w(θ) and galaxy–galaxy lensing γt respectively, with cosmological parameters fixed at best-fit values from the 3x2 analysis, as described in [93] and [94]. The results labeled “DES Y1 - all” vary all 26 parameters while fitting to all three two-point functions. and ΩM , are shown in Figure 9 and given numerically in Ta- ble II. In the next section, we revisit the question of how con- sistent the DES Y1 results are with other experiments. The marginalized constraint on w from all three DES Y1 probes is w = −0.82+0.21 −0.20. (VI.3) Finally, if one ignores any intuition or prejudice about the mechanism driving cosmic acceleration, studying wCDM translates into adding an additional parameter to describe the data. From a Bayesian point of view, the question of whether wCDM is more likely than ΛCDM can again be addressed by computing the Bayes factor. Here the two models being com- pared are simpler: ΛCDM and wCDM. The Bayes factor is Rw = P ( ~D|wCDM) P ( ~D|ΛCDM) (VI.4) Values of Rw less than unity would imply ΛCDM is favored, while those greater than one argue that the introduction of the additional parameter w is warranted. The Bayes factor is Rw = 0.39 for DES Y1, so although ΛCDM is slightly fa- vored, there is no compelling evidence to favor or disfavor an additional parameter w. It is important to note that, although our result in Eq. (VI.3) is compatible with ΛCDM, the most stringent test of the 18 DES Y1 Shear DES Y1 w + γt DES Y1 All −1 .6 −1 .3 −1 .0 −0 .7 −0 .4 w 0. 16 0. 24 0. 32 0. 40 0. 48 Ωm 0. 66 0. 72 0. 78 0. 84 0. 90 S 8 −1 .6 −1 .3 −1 .0 −0 .7 −0 .4 w 0. 66 0. 72 0. 78 0. 84 0. 90 S8 FIG. 8. Constraints on the three cosmological parameters σ8, Ωm, and w in wCDM from DES Y1 after marginalizing over four other cosmological parameters and ten (cosmic shear only) or 20 (other sets of probes) nuisance parameters. The constraints from cosmic shear only (green); w(θ) + γt(θ) (red); and all three two-point functions (blue) are shown. Here and below, outlying panels show the marginalized 1D posteriors and the corresponding 68% confidence regions. model from DES Y1 is not this parameter, but rather the con- straints on the parameters in the model shown in Figure 5 as compared with constraints on those parameters from the CMB measurements of the universe at high redshift. We turn next to that comparison. VII. COMPARISON WITH EXTERNAL DATA We next explore the cosmological implications of com- parison and combination of DES Y1 results with other ex- periments’ constraints. For the CMB, we take constraints from Planck [53]. In the first subsection below, we use only the temperature and polarization auto- and cross-spectra from Planck, omitting the information due to lensing of the CMB that is contained in the four-point function. The latter de- pends on structure and distances at late times, and we wish in this subsection to segregate late-time information from early- Universe observables. We use the joint TT, EE, BB and TE likelihood for multipoles ` between 2 and 29 and the TT like- lihood for ` between 30 and 2508 (commonly referred to as TT+lowP), provided by Planck.7 In all cases that we have checked, use of WMAP [141] data yields constraints consis- tent with, but weaker than, those obtained with Planck. Recent results from the South Pole Telescope [142] favor a value of σ8 that is 2.6-σ lower than Planck, but we have not yet tried to incorporate these results. We use measured angular diameter distances from the Baryon Acoustic Oscillation (BAO) feature by the 6dF Galaxy 7 Late-universe lensing does smooth the CMB power spectra slightly, so these data sets are not completely independent of low redshift information. 19 0.7 0.8 S8 8( m/0.3)0.5 0.2 0.3 m -1.5 -1.0 -0.5 w DES Y1 All DES Y1 Shear DES Y1 w + t DES Y1 All + Planck (No Lensing) DES Y1 All + Planck + BAO + JLA DES SV KiDS-450 Planck (No Lensing) Planck + BAO + JLA FIG. 9. 68% confidence levels on three cosmological parameters from the joint DES Y1 probes and other experiments for wCDM. Survey [143], the SDSS Data Release 7 Main Galaxy Sam- ple [144], and BOSS Data Release 12 [50], in each case ex- tracting only the BAO constraints. These BAO distances are all measured relative to the physical BAO scale correspond- ing to the sound horizon distance rd; therefore, dependence of rd on cosmological parameters must be included when de- termining the likelihood of any cosmological model (see [50] for details). We also use measures of luminosity distances from observations of distant Type Ia supernovae (SNe) via the Joint Lightcurve Analysis (JLA) data from [145]. This set of BAO and SNe experiments has been shown to be consistent with the ΛCDM andwCDM constraints from the CMB [51, 53], so we can therefore sensibly merge this suite of experiments—BAO, SNe, and Planck—with the DES Y1 results to obtain unprecedented precision on the cosmological parameters. We do not include information about direct mea- surements of the Hubble constant because those are in tension with this bundle of experiments [146]. A. High redshift vs. low redshift in ΛCDM The CMB measures the state of the Universe when it was 380,000 years old, while DES measures the matter distribu- tion in the Universe roughly ten billion years later. Therefore, one obvious question that we can address is: Is the ΛCDM prediction for clustering today, with all cosmological param- eters determined by Planck, consistent with what DES ob- serves? This question, which has of course been addressed by previous surveys (e.g., [31, 35, 67, 68]), is so compelling because (i) of the vast differences in the epochs and condi- tions measured; (ii) the predictions for the DES Y1 values of S8 and Ωm have no free parameters in ΛCDM once the recombination-era parameters are fixed; and (iii) those pre- dictions for what DES should observe are very precise, with S8 and Ωm determined by the CMB to within a few percent. We saw above that S8 and Ωm are constrained by DES Y1 at the few-percent level, so the stage is set for the most stringent test yet of ΛCDM growth predictions. Tension between these two sets of constraints might imply the breakdown of ΛCDM. Figure 10 compares the low-z constraints for ΛCDM from all three DES Y1 probes with the z = 1100 constraints from the Planck anisotropy data. Note that the Planck contours are shifted slightly and widened significantly from those in Fig- ure 18 of [53], because we are marginalizing over the un- known sum of the neutrino masses. We have verified that when the sum of the neutrino masses is fixed as [53] assumed in their fiducial analysis, we recover the constraints shown in their Figure 18. The two-dimensional constraints shown in Figure 10 visu- ally hint at tension between the Planck ΛCDM prediction for RMS mass fluctuations and the matter density of the present- day Universe and the direct determination by DES. The 1D marginal constraints differ by more than 1σ in both S8 and Ωm, as shown in Figure 6. The KiDS survey [35, 67, 68, 147] and, earlier, Canada-France Hawaii Telescope Lensing Sur- vey (CHFTLenS; [31, 148]) also report lower S8 than Planck at marginal significance. However, a more quantitative measure of consistency in the full 26-parameter space is the Bayes factor defined in Eq. (V.3). As mentioned above, a Bayes factor below 0.1 sug- gests strong inconsistency and one above 10 suggests strong evidence for consistency. The Bayes factor for combining DES and Planck (no lensing) in the ΛCDM model isR = 6.6, indicating “substantial” evidence for consistency on the Jef- freys scale, so any inconsistency apparent in Figure 10 is not statistically significant according to this metric. In order to test the sensitivity of this conclusion to the priors used in our analysis, we halve the width of the prior ranges on all cos- mological parameters (the parameters in the first section of Table I). For this case we find R = 0.75; despite dropping by nearly a factor of 10, R it is still above 0.1 and therefore we are still passing the consistency test. The Bayes factor in Eq. (V.3) compares the hypothesis that two datasets can be fit 20 0.24 0.30 0.36 0.42 0.48 Ωm 0.72 0.80 0.88 0.96 S 8 DES Y1 Planck (No Lensing) DES Y1 + Planck (No Lensing) FIG. 10. ΛCDM constraints from the three combined probes in DES Y1 (blue), Planck with no lensing (green), and their combination (red). The agreement between DES and Planck can be quantified via the Bayes factor, which indicates that in the full, multi-dimensional parameter space, the two data sets are consistent (see text). by the same set of N model parameters (the null hypothesis), to the hypothesis that they are each allowed an independent set of the N model parameters (the alternative hypothesis). The alternative hypothesis is naturally penalized in the Bayes fac- tor since the model requires an extra N parameters. We also test an alternative hypothesis where only Ωm and As are al- lowed to be constrained independently by the two datasets; in this case we are introducing only two extra parameters with re- spect to the null hypothesis. For this case, we find R = 0.47, which again indicates that there is no evidence for inconsis- tency between the datasets. We therefore combine the two data sets, resulting in the red contours in Figure 10. This quantitative conclusion that the high– and low– redshift data sets are consistent can even be gleaned by viewing Figure 10 in a slightly different way: if the true parameters lie within the red contours, it is not un- likely for two independent experiments to return the blue and green contour regions. Figure 11 takes the high-z vs. low-z comparison a step fur- ther by combining DES Y1 with results from BAO experi- ments and Type Ia supernovae. While these even tighter low- redshift constraints continue to favor slightly lower values of Ωm and S8 than Planck, the Bayes factor is 0.6, which neither favors nor disfavors the hypothesis that the two sets of data, DES Y1+BAO+JLA on one hand and Planck on the other, are described by the same set of cosmological parameters. The goal of this subsection is to test the ΛCDM prediction for clustering in DES, so we defer the issue of parameter de- termination to the next subsections. However, there is one 0.24 0.30 0.36 0.42 0.48 Ωm 0.72 0.80 0.88 0.96 S 8 High redshift: Planck Low redshift: DES Y1+BAO+JLA FIG. 11. ΛCDM constraints from high redshift (Planck, without lensing) and multiple low redshift experiments (DES Y1+BAO+JLA), see text for references. 0.24 0.30 0.36 0.42 Ωm 0.64 0.72 0.80 0.88 h Planck DES Y1 DES Y1+Planck FIG. 12. ΛCDM constraints from Planck with no lensing (green), DES Y1 (blue) and the two combined (red) in the Ωm, h plane. The positions of the acoustic peaks in the CMB constrain Ωmh 3 ex- tremely well, and the DES determination of Ωm breaks the degen- eracy, leading to a larger value of h than inferred from Planck only (see Table II). 21 aspect of the CMB measurements combined with DES that is worth mentioning here. DES data do not constrain the Hubble constant directly. However, as shown in Figure 12, the DES ΛCDM constraint on Ωm combined with Planck’s measure- ment of Ωmh 3 leads to a shift in the inference of the Hubble constant (in the direction of local measurements [130]). Since Ωm is lower in DES, the inferred value of h moves up. As shown in the figure and quantitatively in Table II, the shift is greater than 1σ. As shown in Table II, this shift in the value of h persists as more data sets are added in. B. Cosmological Parameters in ΛCDM To obtain the most stringent cosmological constraints, we now compare DES Y1 with the bundle of BAO, Planck, and JLA that have been shown to be consistent with one another [53]. Here “Planck” includes the data from the four-point function of the CMB, which captures the effect of lensing due to large-scale structure at late times. Figure 13 shows the con- straints in the Ωm–S8 plane from this bundle of data sets and from DES Y1, in the ΛCDM model. Here the apparent con- sistency of the data sets is borne out by the Bayes factor for dataset consistency (Eq. V.3): P (JLA + Planck + BAO + DES Y1) P (JLA + BAO + Planck)P (DES Y1) = 35. (VII.1) 0.24 0.28 0.32 0.36 Ωm 0.70 0.75 0.80 0.85 0.90 S 8 DES Y1 Planck+BAO+JLA DES Y1+Planck+BAO+JLA FIG. 13. ΛCDM constraints from all three two-point functions within DES and BAO, JLA, and Planck (with lensing) in the Ωm- S8 plane. Combining all of these leads to the tightest constraints yet on ΛCDM parameters, shown in Table II. Highlighting some of these: at 68% C.L., the combination of DES with these external data sets yields Ωm = 0.298± 0.007. (VII.2) This value is about 1σ lower than the value without DES Y1, with comparable error bars. The clustering amplitude is also constrained at the percent level: σ8 = 0.808+0.009 −0.017 S8 = 0.802± 0.012. (VII.3) Note that fortuitously, because Ωm is so close to 0.3, the dif- ference in the central values of σ8 and S8 is negligible. The combined result is about 1σ lower than the inference without DES, and the constraints are tighter by about 20%. As mentioned above, the lower value of Ωm leads to a higher value of the Hubble constant: h = 0.658+0.019 −0.027 (Planck : No Lensing) h = 0.685+0.005 −0.007 (DES Y1 + JLA + BAO + Planck) (VII.4) with neutrino mass varied. C. wCDM Figure 14 shows the results in the extended wCDM param- eter space using Planck alone, DES alone, the two combined, and the two with the addition of BAO+SNe. As discussed in [53], the constraints on the dark energy equation of state from Planck alone are misleading. They stem from the measure- ment of the distance to the last scattering surface, and that distance (in a flat universe) depends upon the Hubble constant as well, so there is a strong w − h degeneracy. The low val- ues of w seen in Figure 14 from Planck alone correspond to very large values of h. Since DES is not sensitive to the Hub- ble constant, it does not break this degeneracy. Additionally, the Bayes factor in Eq. (VI.4) that quantifies whether adding the extra parameter w is warranted is Rw = 0.7. Therefore, opening up the dark energy equation of state is not favored on a formal level for the DES+Planck combination. Finally, the Bayes factor for combining DES and Planck (no lensing) in wCDM is equal to 10.3, indicating “strong” evidence that the two datasets are consistent. DES Y1 and Planck jointly constraint the equation of state to w = −1.35+0.16 −0.17, which is about 2-sigma away from the cosmological-constant value. The addition of BAO, SNe, and Planck lensing data to the DES+Planck combination yields the red contours in Fig- ure 14, shifting the solution substantially along the Planck de- generacy direction, demonstrating (i) the problems mentioned above with the DES+Planck (no lensing) combination and (ii) that these problems are resolved when other data sets are in- troduced that restrict the Hubble parameter to reasonable val- ues. The Bayes factor for combination of Planck (no lensing) with the low-z suite of DES+BAO+SNe in the wCDM model is R = 89 substantially more supportive of the combination of experiments than the case for Planck and DES alone. The 22 Planck No Lensing DES Y1 DES Y1+Planck No Lensing DES Y1+Planck+BAO+JLA 0.56 0.64 0.72 0.80 h −1.8 −1.4 −1.0 −0.6 w 0.24 0.30 0.36 0.42 Ωm 0.70 0.75 0.80 0.85 0.90 S 8 0.56 0.64 0.72 0.80 h −1.8 −1.4 −1.0 −0.6 w 0.70 0.75 0.80 0.85 0.90 S8 FIG. 14. wCDM constraints from the three combined probes in DES Y1 and Planck with no lensing in the Ωm-w-S8-h subspace. Note the strong degeneracy between h and w from Planck data. DES+Planck+BAO+SNe solution shows good consistency in the Ωm–w–S8 subspace and yields our final constraint on the dark energy equation of state: w = −1.00+0.05 −0.04. (VII.5) DES Y1 reduces the width of the allowed 68% region by ten percent. The evidence ratio Rw = 0.1 for this full combina- tion of data sets, disfavoring the introduction of w as a free parameter. D. Neutrino Mass The lower power observed in DES (relative to Planck) has implications for the constraint on the sum of the neutrino masses, as shown in Figure 15. The current most stringent constraint comes from the cosmic microwave background and Lyman-alpha forest [149]. The experiments considered here (DES, JLA, BAO) represent an independent set so offer an al- ternative method for measuring the clustering of matter as a function of scale and redshift, which is one of the key drivers of the neutrino constraints. The 95% C.L. upper limit on the sum of the neutrino masses in ΛCDM becomes less constrain- ing: ∑ mν < 0.26 eV. (VII.6) Adding in DES Y1 loosens the constraint by close to 20% (from 0.22 eV). This is consistent with our finding that the clustering amplitude in DES Y1 is slightly lower than ex- pected in ΛCDM informed by Planck. The three ways of reducing the clustering amplitude are to reduce Ωm, reduce σ8, or increase the sum of the neutrino masses. The best fit cosmology moves all three of these parameters slightly in the direction of less clustering in the present day Universe. We may, conversely, be concerned about the effect of priors on Ωνh 2 on the cosmological inferences in this paper. The re- sults for DES Y1 and Planck depicted in Figure 10 in ΛCDM 23 Planck+BAO+JLA DES Y1+Planck+BAO+JLA 0.1 0.2 0.3 0.4 m ν (e V ) 0.275 0.300 0.325 Ωm 0.775 0.800 0.825 0.850 σ 8 0.1 0.2 0.3 0.4 mν (eV) 0.775 0.800 0.825 0.850 σ8 FIG. 15. ΛCDM constraints on the sum of the neutrino masses from DES and other experiments. The lower power observed in DES can be accommodated either by lowering Ω, or σ8 or by increasing the sum of the neutrino masses. 0.24 0.30 0.36 0.42 Ωm 0.72 0.80 0.88 0.96 S 8 DES Y1, fixed neutrinos DES Y1 Planck, fixed neutrinos Planck FIG. 16. ΛCDM constraints on Ωm and σ8 from Planck without lensing and all three probes in DES. In contrast to all other plots in this paper, the dark contours here show the results when the sum of the neutrino masses was held fixed at its minimum allowed value of 0.06 eV. were obtained when varying the sum of the neutrino masses. Neutrinos have mass [150] and the sum of the masses of the three light neutrinos is indeed unknown, so this parameter does need to be varied. However, many previous analyses have either set the sum to zero or to the minimum value al- lowed by oscillation experiments ( ∑ mν = 0.06 eV), so it is of interest to see if fixing neutrino mass alters any of our conclusions. In particular: does this alter the level of agree- ment between low- and high-redshift probes in ΛCDM? Fig- ure 16 shows the extreme case of fixing the neutrino masses to the lowest value allowed by oscillation data: both the DES and Planck constraints in the Ωm − S8 plane change. The Planck contours shrink toward the low-Ωm side of their con- tours, while the DES constraints shift slightly to lower Ωm and higher S8. The Bayes factor for the combination of DES and Planck in the ΛCDM space changes fromR = 6.6 toR = 3.4 when the minimal neutrino mass is enforced. DES and Planck therefore continue to agree, as seen in Figure 16: when the neutrino mass is fixed, the area in the Ωm − S8 plane allowed by Planck is much smaller than when Ωνh 2 varies, but there remains a substantial overlap between the Planck and DES contours. Finally, fixing the neutrino mass allows us to compare di- rectly to previous analyses that did the same. Although there are other differences in the analyses, such as the