DES-2017-0225 FERMILAB-PUB-17-280-AE Dark Energy Survey Year 1 Results: Galaxy clustering for combined probes
J. Elvin-Poole,1 M. Crocce,2 A. J. Ross,3 T. Giannantonio,4, 5, 6 E. Rozo,7 E. S. Ryko↵,8, 9 S. Avila,10, 11 N. Banik,12, 13 J. Blazek,14, 3 S. L. Bridle,1 R. Cawthon,15 A. Drlica-Wagner,12 O. Friedrich,6, 16 N. Kokron,17, 18 E. Krause,9 N. MacCrann,3, 19 J. Prat,20 C. S´anchez,20 L. F. Secco,21 I. Sevilla-Noarbe,10 M. A. Troxel,3, 19 T. M. C. Abbott,22 F. B. Abdalla,23, 24 S. Allam,12 J. Annis,12 J. Asorey,25, 26 K. Bechtol,27 M. R. Becker,9, 28 A. Benoit-L´evy,23, 29, 30 G. M. Bernstein,21 E. Bertin,30, 29 D. Brooks,23 E. Buckley-Geer,12 D. L. Burke,8, 9 A. Carnero Rosell,31, 18 D. Carollo,26, 32 M. Carrasco Kind,33, 34 J. Carretero,20 F. J. Castander,2 C. E. Cunha,9 C. B. D’Andrea,21 L. N. da Costa,18, 31 T. M. Davis,25, 26 C. Davis,9 S. Desai,35 H. T. Diehl,12 J. P. Dietrich,36, 37 S. Dodelson,12, 15 P. Doel,23 T. F. Eifler,38, 39 A. E. Evrard,40, 41 E. Fernandez,20 B. Flaugher,12 P. Fosalba,2 J. Frieman,12, 15 J. Garc´ıa-Bellido,42 E. Gaztanaga,2 D. W. Gerdes,41, 40 K. Glazebrook,43 D. Gruen,8, 9 R. A. Gruendl,33, 34 J. Gschwend,31, 18 G. Gutierrez,12 W. G. Hartley,23, 44 S. R. Hinton,25 K. Honscheid,3, 19 J. K. Hoormann,25 B. Jain,21 D. J. James,45 M. Jarvis,21 T. Jeltema,46 M. W. G. Johnson,33 M. D. Johnson,33 A. King,25 K. Kuehn,47 S. Kuhlmann,48 N. Kuropatkin,12 O. Lahav,23 G. Lewis,49, 26 T. S. Li,12 C. Lidman,47, 26 M. Lima,17, 18 H. Lin,12 E. Macaulay,25 M. March,21 J. L. Marshall,50 P. Martini,3, 51 P. Melchior,52 F. Menanteau,34, 33 R. Miquel,20, 53 J. J. Mohr,16, 37, 36 A. M¨oller,54, 26 R. C. Nichol,11 B. Nord,12 C. R. O’Neill,25, 26 W.J. Percival,11 D. Petravick,33 A. A. Plazas,39 A. K. Romer,55, 8 M. Sako,21 E. Sanchez,10 V. Scarpine,12 R. Schindler,8 M. Schubnell,41 E. Sheldon,56 M. Smith,57 R. C. Smith,22 M. Soares-Santos,12 F. Sobreira,18, 58 N. E. Sommer,54, 26 E. Suchyta,59 M. E. C. Swanson,33 G. Tarle,41 D. Thomas,11, 19 B. E. Tucker,26, 54 D. L. Tucker,12 S. A. Uddin,26, 60 V. Vikram,48 A. R. Walker,22 R. H. Wechsler,8, 28, 9 J. Weller,36, 16, 6 W. Wester,12 R. C. Wolf,21 F. Yuan,26, 54 B. Zhang,54, 26 and J. Zuntz61 (DES Collaboration) 1Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK 2Institute of Space Sciences, IEEC-CSIC, Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain 3Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA 4Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 5Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 6Universit¨ats-Sternwarte, Fakult¨atf¨urPhysik, Ludwig-Maximilians Universit¨atM¨unchen, Scheinerstr. 1, 81679 M¨unchen, Germany 7Department of Physics, University of Arizona, Tucson, AZ 85721, USA 8SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA 9Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA 10Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas (CIEMAT), Madrid, Spain 11Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK 12Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA 13Department of Physics, University of Florida, Gainesville, Florida 32611, USA 14Institute of Physics, Laboratory of Astrophysics, Ecole´ Polytechnique F´ed´erale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland 15Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA 16Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany 17Departamento de F´ısica Matem´atica, Instituto de F´ısica, Universidade de S˜aoPaulo, CP 66318, S˜aoPaulo, SP, 05314-970, Brazil 18Laborat´orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos´eCristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 19Department of Physics, The Ohio State University, Columbus, OH 43210, USA 20Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain 21Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 22Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile 23Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 24Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa 25School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia 26ARC Centre of Excellence for All-sky Astrophysics (CAASTRO) 27LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA 2
28Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA 29CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 30Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 31Observat´orio Nacional, Rua Gal. Jos´eCristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 32INAF - Osservatorio Astrofisico di Torino, Pino Torinese, Italy 33National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA 34Department of Astronomy, University of Illinois, 1002 W. Green Street, Urbana, IL 61801, USA 35Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India 36Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany 37Faculty of Physics, Ludwig-Maximilians-Universit¨at, Scheinerstr. 1, 81679 Munich, Germany 38Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA 39Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA 40Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA 41Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA 42Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain 43Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Victoria 3122, Australia 44Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland 45Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA 46Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA 47Australian Astronomical Observatory, North Ryde, NSW 2113, Australia 48Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA 49Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW 2006, Australia 50George 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 51Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA 52Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA 53Instituci´oCatalana de Recerca i Estudis Avan¸cats, E-08010 Barcelona, Spain 54The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia 55Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK 56Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA 57School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK 58Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil 59Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 60Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, Jiangshu 210008, China 61Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK (Dated: August 4, 2017) We measure the clustering of DES Year 1 galaxies that are intended to be combined with weak lensing samples in order to produce precise cosmological constraints from the joint analysis of large- scale structure and lensing correlations. Two-point correlation functions are measured for a sample of 6.6 105 luminous red galaxies selected using the redMaGiC algorithm over an area of 1321 square degrees,⇥ in the redshift range 0.15
I. INTRODUCTION lapsed over-densities (dark matter halos; [1]), and the mass of the halo they reside in is known to correlate with Galaxies are a biased tracer of the matter density field. the luminosity and color of the galaxy, with more lumi- In the standard ‘halo model’ paradigm, they form in col- nous and redder galaxies strongly correlated with higher 3 mass. Therefore, the galaxy ‘bias’ depends strongly on statistics and estimate their covariance; Section V sum- the particular sample being studied. Thus, in cosmolog- marizes the results of our observational systematic tests. ical studies the galaxy bias is often treated as a nuisance We present our primary results with galaxy bias mea- parameter — one that is degenerate with the amplitude surements in Section VI and a demonstration of their of the clustering of matter. See, e.g., [2] and references robustness in Section VII before concluding in Section therein. VIII. The degeneracy can be broken with additional ob- In order to avoid confirmation bias, we have performed servables. This includes the weak gravitational lensing this analysis “blind”: we did not measure parameter con- ‘shear’ field, which is induced by the matter density field. straints or plot the correlation function measured from Correlation between the galaxies and the shear field ([3]; the data on the same axis as any theoretical prediction or often referred to as ‘galaxy-galaxy lensing’) contains one simulated clustering measurement until the sample and factor of the galaxy bias and two factors of the matter all measurements in Y1COSMO were finalized. field. The galaxy auto-correlation contains two factors Unless otherwise noted, we use a fiducial ⇤CDM cos- of the galaxy bias and again two factors of the matter mology, fixing cosmological parameters at ⌦m =0.295, 9 field. Thus, the combination of the two measurements As =2.260574 10 ,⌦b =0.0468, h =0.6881, ⇥ can break the degeneracy between the two quantities, ns =0.9676. This is consistent with the latest cosmo- and it is a sensitive probe of the late-time matter field logical data from the Planck mission [18] and is used as (see, e.g., [4, 5]) the fiducial cosmology for all the DES Y1 analyses used The auto-correlation of the shear field alone includes in Y1COSMO. We use this cosmology to generate Gaus- no factors of the galaxy bias and can thus be used directly sian mocks in Section V for systematics testing. as a probe of the matter field. However, its sensitivity to After un-blinding, we re-measure the galaxy bias, fix- many systematic uncertainties related to the estimation ing the cosmological parameters at the mean of the DES of the shear field di↵ers from that of the galaxy-galaxy Y1COSMO posterior, ⌦m =0.2276, As =2.667454 9 ⇥ lensing signal. As shown by [6–9], the impact of such 10 ,⌦b =0.0526, h =0.7619, ns =0.9964 (note that systematic uncertainties can be mitigated by combining we show these values at a greater precision than can be the shear auto-correlation measurements with those of measured by DES). This cosmology was used for all bias galaxy clustering and galaxy-galaxy lensing. Thus there measurements in Section VI. is substantial gain to be obtained from a combined anal- ysis. Such a combined analysis is performed with the Dark II. THEORY Energy Survey (DES[10]; [11]) Year-1 (Y1) data ([12]; hereafter Y1COSMO). DES is an imaging survey cur- Throughout this paper we model redMaGiC cluster- rently amassing data over a 5000 deg2 footprint in ing measurements assuming a local, linear galaxy bias five passbands (grizY ). When completed, it will have model [19], where the galaxy and matter density fluctua- mapped 300 million galaxies and tens of thousands of tions are related by g(x)=b m(x), with density fluctu- galaxy clusters. ations defined by (n(x) n¯)/n¯. The validity of this ⌘ In this work, we study the clustering of red sequence assumption over the scales considered here is provided in galaxies selected from DES Y1 data using the red- [20] and shown in simulations in [21]. MaGiC [13] algorithm. We study the same sample used We consider multiple galaxy redshift bins i, each char- to obtain cosmological results in the Y1COSMO com- acterized by a redMaGiC galaxy redshift distribution i bined analysis. In particular, we study the large-scale ng(z), normalized to unity in redshift, and a galaxy bias clustering amplitude and its sensitivity to observational bi which is assumed to be constant across the redshift bin systematics. Following previous studies [14–17], we use range. Under the Limber [22] and flat-sky approximation angular maps to track the observing conditions of the Y1 the theoretical prediction for the galaxy correlation func- data in order to identify and correct for spurious fluctu- tion w(✓) in a given bin is, ations in the galaxy density field. We further determine 2 dl the e↵ect the corrections have on the covariance matrix of wi(✓)= bi 2⇡J (l✓) d l 0 the angular auto-correlation of the galaxies. We present Z Z i 2 robust measurements of the clustering amplitude of red- ng(z( ) l +1/2 MaGiC galaxies as a function of redshift and luminos- PNL ,z( ) , (1) ⇥ 2H(z) ity, thus gaining insight into the physical nature of these ⇥ ⇤ ✓ ◆ galaxies and how they compare to other red galaxy sam- where the speed of light has been set to one, (z)isthe ples. The results of this paper are then used for the joint comoving distance to a given redshift (in a flat universe, DES cosmological analysis presented in Y1COSMO. which is assumed throughout); J0 is the Bessel function of This outline of this paper is: we summarize in Section order zero; H(z) is the Hubble expansion rate at redshift II the model we use to describe our galaxy clustering z; and PNL(k; z) is the 3D matter power spectrum at red- measurements; we present in Section III the DES data shift z and wavenumber k (which, in this Limber approx- we use; Section IV presents how we measure clustering imation, is set equal to (l +1/2)/ ). Note that in Eq. 1 4 we have assumed the bias to be constant within each bin, the two probes we use the parameter r which is, see Fig. 8 in [20] for a test of this assumption. Again, i all assumptions and approximations mentioned here have i b r = ⇥ . (3) been shown to be inconsequential in [20, 21]. To model bi cross-correlation between redshift bins we simply change i 2 i j i 2 i j If r = 1, this indicates an inconsistency between the two ng(z) ng(z)ng(z) and (b ) b b in Eq. 1. ! ! bias6 measurements and would thus suggest a breakdown Throughout this paper, we use the CosmoSIS frame- of our simple linear bias model. This test informs the work [23] to compute correlation functions, and to infer choice of fixing r = 1 in the Y1COSMO analysis. cosmological parameters. The evolution of linear density fluctuations is obtained using the Camb module [24] and then converted to a non-linear matter power spectrum III. DATA PNL(k) using the updated Halofit recipe [25]. The theory modeling we use assumes the Limber ap- proximation, and it also neglects redshift space distor- A. Y1 Gold tions. For the samples and redshift binning used in this paper, those e↵ects start to become relevant at scales We use data taken in the first year (Y1) of DES obser- of ✓ & 1 deg [26–28]. In a companion paper [20] it is vations [31]. Photometry and ‘clean’ galaxy samples were explicitly shown that they have negligible impact in de- produced with this data as outlined by [32] (hereafter de- rived cosmological parameters given the statistical error noted ‘Y1GOLD’). The outputs of this process represent bars of DES Y1. Concretely a theory data vector was the Y1 ‘Gold’ catalog. Data were obtained over a total produced with the exact (non-Limber) formula including footprint of 1800 deg2; this footprint is defined by a ⇠ redshift space distortions and was then analyzed with the Healpix [33] map at resolution Nside = 4096 (equiva- baseline pipeline assumed here, and also in Y1COSMO. lent to 0.74 square arcmin) and includes only pixels with Figure 8 in [20] shows that including or not including minimum exposure time of at least 90 seconds in each these non-linear contributions make negligible impact in of the g,r,i, and z-bands, a valid calibration solution, as parameters such as ⌦m and S8 for a LCDM Universe or well as additional constraints (see Y1GOLD for details). w in a wCDM one. We also test the impact of these A series of veto masks, including among others masks e↵ects on the fixed cosmology bias measurements in Sec- for bright stars and the Large Magellanic Cloud, reduce tion VI and find them to be negligible. Hence in what the area by 300 deg2,leaving 1500 deg2 suitable for ⇠ ⇠ follows, such terms are ignored for speed and simplicity. galaxy clustering study. We explain further cuts to the However future data analyses may need to account for angular mask in Section III B. All data described in this these e↵ects due to improved statistical uncertainty. and in subsequent sections are drawn from catalogues We model (and marginalize over) photometric redshift and maps generated as part of the DES Y1 Gold sample bias uncertainties as an additive shift zi in the red- and are fully described in Y1GOLD. i MaGiC redshift distribution nRM(z) for each redshift bin i. B. redMaGiC sample
The galaxy sample we use in this work is generated ni(z)=ni (z zi) (2) RM by the redMaGiC algorithm, run on DES Y1 Gold data. The redMaGiC algorithm selects Luminous Red The priors on the zi nuisance parameters, shown in Galaxies (LRGs) in such a way that photometric red- Table II, are described in [29]. We use the same zi as shift uncertainties are minimized, as is described in [13]. Y1COSMO for all tests of robustness of the parameter This method is able to achieve redshift uncertainties constraints. z/(1 + z) < 0.02 over the redshift range of interest. We also compare the measurements of bi to the same The redMaGiC algorithm produces a redshift predic- quantity measured by galaxy-galaxy lensing using the tion zRM and an uncertainty z which is assumed to be two-point correlation function t (see [30] for definition). Gaussian. We use the notation bi for this measurement. The de- The redMaGiC algorithm makes use of an empiri- tails of this measurement⇥ are described in [30] (hereafter cal red-sequence template generated by the training of Y1GGL). In order to take the o↵-diagonal elements of the redMaPPer cluster finder [34, 35]. As described in the covariance matrix between the two probes into ac- [35], training of the red-sequence template requires over- count, we produce joint constraints from w(✓) and t lapping spectroscopic redshifts, which in this work were at fixed cosmology, using di↵erent bias parameters for obtained from SDSS in the Stripe 82 region [36] and the the two probes, and marginalizing over the same nui- OzDES spectroscopic survey in the DES deep supernova sance parameters as were used in the fiducial analysis of fields [37]. Y1COSMO (intrinsic alignments, source photo z bias, For the redMaGiC samples in this work, we make and shear calibration). To test the consistency between use of two separate versions of the red-sequence training. 5
FIG. 1. Galaxy distribution of the redMaGiC Y1 sample used in this analysis. The fluctuations represent the raw counts, without any of the corrections derived in this analysis. We have restricted the analysis to the contiguous region shown in the figure. The area is 1321 square degrees.
2 z range Lmin/L ngal (deg ) Ngal Photometry ⇤ 0.15 TABLE I. Details of the sample in each redshift bin. Lmin/L ⇤ describes the minimum luminosity threshold of the sample, ngal is the number of galaxies per square degree, and Ngal is the total number of galaxies. multi-band photometry, the MOF photometry yields lower color scatter and hence smaller scatter in red-sequence photo-zs. For each version of the catalog, photometric redshifts and uncertainties are primarily derived from the FIG. 2. Redshift distribution of the combined redMaGiC fit to the red-sequence template. In addition, an after- sample in 5 redshift bins. They are calculated by stacking burner step is applied to ensure that redMaGiC photo- Gaussian PDFs with mean equal to the redMaGiC redshift zs and errors are consistent with those derived from the prediction and standard deviation equal to the redMaGiC associated redMaPPer cluster catalog [13]. redshift error. Each curve is normalized so that the area of each curve matches the number of galaxies in its redshift bin. As described in [13], the redMaGiC algorithm com- putes color-cuts necessary to produce a luminosity- thresholded sample of constant co-moving density. Both the luminosity threshold and desired density are inde- The first is based on SExtractor MAG AUTO quantities from pendently configurable, but in practice higher luminos- the Y1 coadd catalogs, as applied to redMaPPer in ity thresholds require a lower density for good perfor- [38]. The second is based on a simultaneous multi-epoch, mance. We note that in [13] the co-moving density was multi-band, and multi-object fit (MOF) (see Section 6.3 of computed with the central redshift of each galaxy (zRM). Y1GOLD), as applied to redMaPPer (Mcclintock et al. For this work, the density was computed by sampling 2017, in preparation). In general, due to the careful han- from a Gaussian distribution zRM z, which creates dling of the point-spread function (PSF) and matched a more stable distribution near filter± transitions. This is 6 FIG. 3. Correlations of volume-limited redMaGiC galaxy number density with seeing FWHM and exposure time before any survey property (SP; see text for more details) cuts (illustrated with the red vertical lines) were applied to the mask. In the absence of systematic correlations, the results obtained from these samples are expected to be consistent with no trend (the reference green dashed line). The cuts removed regions with i-band FWHM > 4.5pixelsandi-band exposure time > 500s as these showed correlations that di↵ered significantly from the mean (> 20%) or were not well fit by a monotonic function. No SP weights were used in this figure. i i i z range bfid z Y1COSMO z [Cawthon et al.] as well as decreasing the comoving number density. The 0.15 1.0L sample is complete to z =0.75, and the 1.5L sam- For the analysis of observational systematics and their ple is⇤ complete to z =0.9. We also restrict the analysis⇤ correlation with the data, we use a set of 1000 mock sur- to the contiguous region shown in Figure 1. veys (hereafter ‘mocks’) based on Gaussian random field An additional 1.6% of the footprint is vetoed because it realizations of the projected density field. These are then has extreme observing conditions. The selection of these used to obtain an alternative covariance, which includes cuts is detailed in Section V. all the mask e↵ects as in the real data. The mocks we After masking and additional cuts, we obtain a total use were produced using the following method. We first sample of 653,691 objects distributed over an area of 1321 calculate, using Camb [24], the galaxy clustering power gg square degrees, as shown in Fig. 1. The average redshift spectrum Ci (`), assuming the fiducial cosmology with i uncertainty of the sample is z/(1 + z)=0.0166. The fixed galaxy bias b for each redshift bin i; the galaxy bias redshift distribution of each bin can be seen in Figure values are listed in Table II. The angular power spectrum 2. The number of objects in each bin increases up to is then used to produce a full-sky Gaussian random field z =0.6 due to the increase in volume, and decreases at of g. We apply a mask to this field corresponding to higher redshift due to the increased luminosity threshold. the Y1 data, as shown in Fig. 1. This is converted into a galaxy number count Ngal as a function of sky position, with the same mean as the observed number count N¯o in IV. ANALYSIS METHODS each redshift bin, using ¯ A. Clustering estimators Ngal = No(1 + g). (6) Shot noise is finally added to this field by Poisson sam- We measure the correlation functions w(✓)usingthe pling the Ngal field. Landy & Szalay estimator [40] In order to avoid pixels with g < 1, which cannot be Poisson sampled, we follow the method used by [20]: DD 2DR + RR wˆ(✓)= , (4) before Poisson sampling, we multiply the density field RR by a factor ↵,where↵<1; we then rescale the number 2 where DD, RR and DR are the number of pairs of density ngal by 1/↵ in order to preserve the ratio of shot- noise to sample variance; we then rescale the measured galaxies from the galaxy sample D and a random cat- 2 alog R. This is calculated in 20 logarithmically sep- w(✓)by1/↵ to obtain the unbiased w(✓) for each mock. arated bins in angle ✓ between 2.5 arcmin and 250 This procedure is summarized by arcmin to match the analysis in Y1COSMO. We use g ↵ g , (7) 60 times more randoms than data. The pair-counting ! n n /↵2 , (8) was done with the package tree-corr [41] available at gal ! gal https://github.com/rmjarvis/TreeCorr. w(✓) w(✓)/↵2 . (9) ! We also calculate cross-correlations between the galaxy We then use these mocks to estimate statistical errors sample and a suite of potential contaminants, including in galaxy number density as a function of potential sys- observational conditions and survey property maps, de- tematics. Alternatively we “contaminate” each of the scribed in Sec. V. As these properties are best described 1000 mocks with survey properties as discussed in Sec- in pixelated map format, for these cross-correlations we tion V to assess the impact of systematics on the w(✓) use a pixel-based estimator. Using the notation of [17], covariance. Note that these mocks would not be fully re- the correlation between two maps N and N of mean 1 2 alistic for w(✓) covariance and cosmological inference as values N¯ , N¯ is estimated as 1 2 they are basically Gaussian realizations. These mocks al- 2 N N low us to quantify significances (i.e., a )tonulltests, pix pix (N N¯ )(N N¯ ) wˆ (✓)= i,1 1 j,2 2 ⇥ , (5) which are a necessary step in our analysis. Further, given 1,2 N¯ N¯ i,j i=1 j=i 1 2 such a large number of realizations we are able to obtain X X estimates of both the impact of the systematic correction where the sum runs through all pairs of the Npix pixels on the resulting statistical uncertainty and any bias im- in the footprint, Ni,1 is the value of the N1 map in pixel parted by our methodology to well below 1 significance i, and ⇥i,j is unity when the pixels i and j are separated (e.g., given 1000 mocks, a 0.1 bias can be detected at by an angle ✓ within the bin size ✓. 3 significance). B. Covariances V. SYSTEMATICS The fiducial covariance matrix we use for the w(✓) A. Survey property (SP) maps measurement is a theoretical halo model covariance, de- scribed and tested by [20]. This same covariance is used The number density of galaxies selected based on their for Y1COSMO. imaging is likely to fluctuate with the imaging quality 8 FIG. 4. Galaxy number density divided by the mean number density across the footprint for each redshift bin, split by the number density of stars. The points with error-bars display the results for our 3 2(68) weighted sample, the cyan curves 2 display the results without these weights. For the weighted sample, the of Ngal/ Ngal = 1 for each bin is 24.9, 16.0, 13.1, 2 h i 6.6 and 10.9, and the is 0.99, 0.95, 0.24, 0.013, and 0.082 with Nd.o.f = 10. We see no evidence for stellar contamination or obscuration in this sample. FIG. 5. Galaxy number density in the highest redshift bin, 0.75 FIG. 6. The significance of each systematic correlation. The significance is calculated by comparing the 2 measured on the data to the distribution in the mock realizations. We find the 68th percentile 2 value, labeling it 2(68), for each map obtained from the mock realizations. We quote the significance for the relationship obtained on the data as 2/ 2(68). Weights are applied for the SP map with the largest significance, with the caveat that we do not correct for both depth and the components of depth (e.g. exposure time, PSF FWHM) in the same band. For example, in the bin 0.15 For depth and airmass, the function used was a lin- the model ear fit in s. For exposure time and sky brightness, the N / N = F (s ) function was linear in ps, as this is how these quantities gal h gali sys FWHM enter the depth map. For the seeing correlations, we fit sFWHM B F (s )=A 1 erf , (11) sys FWHM ✓ ◆ where sFWHM is the seeing full-width half-max value, and A, B and are parameters to be fitted. This functional 11 FIG. 7. Two-point correlation functions for the fiducial analysis in each combination of the 5 redshift bins. The left panel shows the auto-correlation used in Y1COSMO and the galaxy bias measurements. A correction for correlations with survey properties is applied according to the methodology in Section V. The grey dashed line is the correlation function calculated without the SP weights. The black points use the 2 2(68) weights. The right panel shows the cross-correlation of the redshift bins, shown here as a consistency check. These measurements are expected to be non-zero, with a significance related to the degree of overlap in the n(z) displayed in Fig. 2. The numbers in each panel correspond to the redshift bins used in the cross- correlation. The error-bars in the cross-correlations were calculated using the log-normal mock surveys used for Y1COSMO covariance validation [20]. We show correlations down to ✓ =100 to highlight the goodness of the fit towards small scales, but data points within grey shaded regions have not been used in bias constraints or the galaxy clustering part of Y1COSMO. 1 That scale cut has been set in co-moving coordinates at 8Mpc h . The solid red curve is the best-fit model using only the w(✓) auto-correlations at fixed cosmology, using zi priors from [29]. The solid blue curve is the best-fit model from the full cosmological analysis in Y1COSMO. For many of the cross-correlation panels, these predictions are indistinguishable. form matches that applied to BOSS [46, 47]; we believe tematic and repeat the process until there are no system- it is thus the expected form when morphological cuts atics with a significance greater than a 2/ 2(68) = 3 are applied to reject stars (as this is what causes the threshold. The final weights are the product of the relationship for BOSS). weights from each required systematic. We also produce 2 2 Each galaxy i in the sample is then assigned a weight weights using a threshold of / (68) = 2, allowing us to determine if using a greater threshold has any im- 1/Fsys(si)wheresi is the value of the systematic at the pact on our clustering measurements. We refer to these galaxy’s location on the sky. This weight is then used 2 2 when calculating w(✓) and in all further null tests. weights as the 3 (68) and 2 (68) weights respec- tively. In this sample we find evidence of multiple systemat- ics at a significance of 2/ 2(68) > 3, some of which The final weights used in this sample are described in are correlated with each other. To account for this, we Table III. The SP maps are either the depth or properties first apply weights for the systematic with the highest that contribute to the depth (e.g. holding everything else 2/ 2(68). Then, using the weighted sample, we re- fixed, a longer exposure time will result in an increased measure the significance of each remaining potential sys- depth). Thus, in bins where multiple SP weights were 12 z range bi ri i on scales below ✓min, but these were removed in all pa- 0.15 8 7 6 5 4 3 2 1 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.4 0.6 0.8 1.0 1.2 1.4 1.6 r1 r2 r3 8 2.8 7 Lmin =0.5L L 1.5L 2.6 bias Galaxy 2.4 6 w( ) 2.2 5 t 2.0 4 1.8 3 1.6 2 1.4 1 1.2 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r4 r5 Redshift FIG. 8. Constraints on the ratio, r, of galaxy bias measured on w(✓) and measured from the galaxy-galaxy lensing signal (see [30] denoted Y1GGL in the text) in each redshift bin (first five panels from left to right, top to bottom). The bottom-right panel then displays the individual measurements for each bin (purple for our w(✓) measurements and orange for those obtained in Y1GGL). All cosmological parameters were fixed at the DES Y1COSMO posterior mean values, using zi priors from [29]. The constraints were calculated using the full Y1COSMO covariance matrix, so the covariance between the two probes has been taken into account. We see no significant evidence for r = 1 within the errors. 6 are sensitive to both multiplicative and additive shifts old, we expect that the only e↵ect from adding extra in the amplitude of w(✓) and we therefore believe they weights would be to bias the measurements (from over- should encapsulate any potential systematic bias that correction) and add greater uncertainty. We test for could a↵ect the cosmological analysis of Y1COSMO. We those e↵ects in the following subsections. Here, in or- thus perform joint fits to the data in each redshift bin to der to demonstrate that our results are insensitive to the i i obtain constraints on the five b and ⌦m. For these fits, choice in threshold, the change in the measured b and we marginalize over an additive redshift bias uncertainty ⌦m must be negligible compared to its uncertainty. described in Table II. All other cosmological parameters Figure 9 shows the di↵erence between the 3 2(68) are fixed at the Y1COSMO cosmology and as such, this and 2 2(68) SP weights. Because the weights correc- should not be interpreted as a true measurement of ⌦m. tion can only decrease the w(✓) signal, applying a stricter Results are obtained using the analysis pipeline described threshold significance is expected to move the contours in [20]. We describe how w(✓) is altered to perform each towards smaller values of bi. Figure 9 shows that this test throughout the rest of this section. impact is very small compared to the overall Y1 un- certainty and we can conclude that the choice between 3 2(68) and 2 2(68) weights will have negligible im- A. Selection of threshold pact on the Y1COSMO parameter constraints (The final weights used in Y1COSMO are the 2 2(68) weights). We test two thresholds used to determine deter- Figure 9 also shows the impact of not including SP mine when to apply weights based on a given SP weights on the parameter constraints. Ignoring the SP map: 3 2(68) and a more restrictive (i.e., more maps correlations would have resulted in significantly bias con- 2 i weighted for) 2 (68). After reaching a certain thresh- straints on b and ⌦m.Ineveryredshiftbin,theshiftis 14 no weights no weights no weights 300 2 2 2 0. 2 (68) weights 2 (68) weights 2 (68) weights 3 2(68) weights 3 2(68) weights 3 2(68) weights 275 0. m 250 0. 225 0. 20 35 50 65 50 65 80 95 50 65 80 95 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. b1 b2 b3 no weights no weights 300 2 2 0. 2 (68) weights 2 (68) weights 3 2(68) weights 3 2(68) weights 275 0. m 250 0. 225 0. 80 95 10 25 8 0 2 4 1. 1. 2. 2. 1. 2. 2. 2. b4 b5 i FIG. 9. Parameter constraints showing the impact of the SP weights, varying ⌦m, 5 linear bias parameters b ,and5nuisance parameters zi. Contours are drawn at 68% and 95% confidence level. These constraints use the same zi priors as Y1COSMO. The blue contour shows the constraints on w(✓) calculated with no SP weights. The gray and red contours use SP weights removing all 2 2(68) and 3 2(68) correlations respectively. In this parameter space, ignoring the correlations with survey properties would have significantly biased the constraints from w(✓). As expected, the best fit when using the 2 2(68) weights is at smaller values of bi than the 3 2(68) weights, although the di↵erence is not significant compared to the size of the contour. greater than 2 along the major axis of the ellipses. the bias in w(✓)tobe, N N B. Estimator bias 1 w = w w (12) est bias N 0 no sys,i weights,j1 i=1 j=1 X X We also test for potential bias in w(✓)inducedbyover- @ A correcting with the weights method and from correlations where N is the total number of realizations. We then add between the SP maps. This was done using the Gaus- i sian mocks described in Section IVB using the following west bias to the measured w(✓) and measure b and ⌦m. method. After the galaxy over-density field has been This test is designed to test for any bias in w(✓)induced generated in each realization, we insert the systematic by the the estimator when using weights. This result can be seen in Figure 10 where it shows correlation using Fsys(s) and the best-fit parameters for each of the systematics in Table III at 2 2(68) signif- negligible impact on the parameter constraints. icance. This is equivalent to dividing each mock galaxy map by a map of the SP weights. We then produced a galaxy number count as before, C. False correlations also adding shot noise. We fit the parameters of Fsys(s) to each realization and apply weights to the maps using Given the large number of SP maps being used in the the same method that is applied to the data. We measure systematics tests, it is possible that chance correlations w(✓) using the pixel estimator in Eqn. 5 on mocks with will appear significant and weights will be applied where systematic contamination and correction, wweights, and no contamination has occurred, biasing the measured sig- on mocks with no systematics added, wno sys.Wedefine nal. To test this, we use the same Gaussian mocks as in 15 fiducial fiducial fiducial fiducial + west fiducial + west fiducial + west .30 0 fiducial + wfalse fiducial + wfalse fiducial + wfalse 28 m 0. 26 0. 24 0. 20 35 50 65 5 6 7 8 5 6 7 8 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. b1 b2 b3 fiducial fiducial fiducial + west fiducial + west .30 0 fiducial + wfalse fiducial + wfalse 28 m 0. 26 0. 24 0. 8 9 0 1 80 95 10 25 1. 1. 2. 2. 1. 1. 2. 2. b4 b5 FIG. 10. Parameter constraints showing the impact of the estimator bias, west and false correction bias wfalse.Thefiducial 2 data vector and was calculated using the 2 (68) weights on the data. The west and wfalse were measured on Gaussian mock 2 i surveys using a 2 (68) threshold significance. We see no evidence for significant bias in the b ,⌦m plane. These constraints use the same zi priors as Y1COSMO. Section VII B with no added systematic contaminations. D. Impact on covariance We measure the correlation of each mock with each of the 21 SP maps in Section V A, identifying any correlations Correcting for multiple systematic correlations can al- 2 above a 2 (68) threshold significance. ter the covariance of the w(✓) measurement in various ways. We expect that scatter in the best fit parameters should increase the variance, while the removal of some The false correction bias w , is then defined as false bias clustering modes should decrease it. We test the signifi- the average di↵erence between the w(✓) measured with cance of any any changes to the amplitude and structure no corrections, and the w(✓) measured correcting for of the covariance matrix using the Gaussian mocks. all correlations above the threshold using the weights For this test we use the same mocks as in Section VII B method. We then add wfalse bias to the measured w(✓) i which are ‘contaminated’ with the same systematic cor- and test the impact on b and ⌦m constraints. relations found in the data. We fit the Fsys(s) function to each mock and correct using weights. We then mea- This test is designed to test for any bias in w(✓)in- sure the correlation function wweights and calculate the duced by falsely correcting for SP maps that where only covariance matrix of this measurement. We also mea- correlated with the galaxy density by chance. sure the correlation on mocks with no systematics added, wno sys, and calculate the covariance matrix from each i measurement. We calculate the galaxy bias b and ⌦m This result is shown in Figure 10 where wfalse bias for constraints for each covariance matrix and test if the re- the 2 2(68) SP maps has been used. This shows a neg- sulting contours are significantly di↵erent. This test de- ligible impact on the constraints. The wfalse bias for the termines whether this additional uncertainty needs to be 3 2(68) SP maps is not shown as it has an even smaller considered in the Y1COSMO analysis by marginalizing impact. This demonstrates that selecting a 2 2(68) over the fitted parameters. threshold does not induce a bias in the inferred bias pa- The results of this test are shown in Figure 11. We rameters for the set of SP maps used in this analysis. show that for the SP maps selected in this analysis, the 16 30 cov: no sys cov: no sys cov: no sys 0. cov: 2 2(68) sys cov: 2 2(68) sys cov: 2 2(68) sys 28 0. m 26 0. 24 0. 20 35 50 65 52 60 68 76 5 6 7 8 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. b1 b2 b3 30 cov: no sys cov: no sys 0. cov: 2 2(68) sys cov: 2 2(68) sys 28 0. m 26 0. 24 0. 84 92 00 08 8 9 0 1 2 1. 1. 2. 2. 1. 1. 2. 2. 2. b4 b5 FIG. 11. Parameter constraints showing the impact of the systematics correction on the covariance. Both contours use the fiducial theory data vector. The blue contour uses the covariance from mock surveys with no contamination added (labeled ”cov: no sys”). The gray contour uses the covariance determined from mock surveys with the 2 2(68) contaminations added (labeled ”cov: 2 sys”). These constraints use the same zi priors as Y1COSMO. impact on the size of the contours is negligible. We have is insignificant (see Fig. 10). We further demonstrate therefore not included any additional parameters in the that our weighting method imparts negligible changes to MCMC analysis to account for the uncertainty in the the covariance matrix (see Fig. 11). correction. We find the redshift and luminosity dependence of the• bias of redMaGiC galaxies to be broadly consistent with expectations for red galaxies. VIII. CONCLUSIONS We find that the large-scale galaxy bias is consistent with• that determined by the Y1GGL galaxy-galaxy lens- We have presented the 2-point angular galaxy correla- ing measurements. This is consistent with r = 1 at linear tion functions, w(✓), for a sample of luminous red galaxies scales, in agreement with basic galaxy formation theory, in DES Y1 data, selected by the redMaGiC algorithm. and a key assumption in the Y1COSMO analysis. (See This yielded a sample with small redshift uncertainty, a Fig. 8.) wide redshift range, and wide angular area. We split this Our results give an unbiased w(✓) data vector to be • sample into five redshift bins and analyzed its clustering. provided to the Y1COSMO analysis, thereby yielding di- Our findings can be summarized as follows: rect measurements of the amplitude of matter clustering We find that multiple systematic dependencies be- with unprecedented precision. tween• redMaGiC galaxy density and survey properties The methods we have presented, both correcting for must be corrected for in order to obtain unbiased clus- systematic dependencies and ensuring the robustness of tering measurements. We correct for these dependencies these corrections, can be used as a guide for future analy- by adding weights to the galaxies, following [45, 46]. ses. Possible improvements to the work include incorpo- We demonstrate both that our methods su ciently rating image simulations [53] and using mode projection remove• systematic contamination (no significant dif- techniques [16]. ferences are found between applying a 2 2(68) and Our galaxy bias results can be extended to study lumi- 3 2(68) threshold; see Fig. 9) and that any bias re- nosity dependence within redshift bins and to use smaller sulting from our method removing true clustering modes scale clustering in order to determine the HOD of red- 17 MaGiC galaxies. Already, our bias measurements can be the associated Excellence Cluster Universe, the Univer- used to inform simulations (e.g., for the support of DES sity of Michigan, the National Optical Astronomy Ob- Y3 analyses) and additional HOD information would be servatory, the University of Nottingham, The Ohio State of further benefit. University, the University of Pennsylvania, the Univer- Finally, the results presented here have been opti- sity of Portsmouth, SLAC National Accelerator Labora- mized for combination with other cosmological probes tory, Stanford University, the University of Sussex, Texas in Y1COSMO and our work has ensured the galaxy clus- A&M University, and the OzDES Membership Consor- tering measurements do not bias the Y1COSMO results. tium. The analysis followed a strict blinding procedure and has Based in part on observations at Cerro Tololo Inter- yielded unprecedentedly precise and robust constraints American Observatory, National Optical Astronomy Ob- when combined with the other 2-point functions. servatory, which is operated by the Association of Univer- sities for Research in Astronomy (AURA) under a coop- erative agreement with the National Science Foundation. The DES data management system is supported by ACKNOWLEDGEMENTS the National Science Foundation under Grant Num- bers AST-1138766 and AST-1536171. The DES partic- Figures 9 to 11 in this paper were produced with ipants from Spanish institutions are partially supported chainconsumer [54]. by MINECO under grants AYA2015-71825, ESP2015- Funding for the DES Projects has been provided by 88861, FPA2015-68048, SEV-2012-0234, SEV-2016-0597, the U.S. Department of Energy, the U.S. National Sci- and MDM-2015-0509, some of which include ERDF funds ence Foundation, the Ministry of Science and Education from the European Union. IFAE is partially funded by of Spain, the Science and Technology Facilities Coun- the CERCA program of the Generalitat de Catalunya. cil of the United Kingdom, the Higher Education Fund- Research leading to these results has received funding ing Council for England, the National Center for Super- from the European Research Council under the Euro- computing Applications at the University of Illinois at pean Union’s Seventh Framework Program (FP7/2007- Urbana-Champaign, the Kavli Institute of Cosmological 2013) including ERC grant agreements 240672, 291329, Physics at the University of Chicago, the Center for Cos- and 306478. We acknowledge support from the Aus- mology and Astro-Particle Physics at the Ohio State Uni- tralian Research Council Centre of Excellence for All- versity, the Mitchell Institute for Fundamental Physics sky Astrophysics (CAASTRO), through project number and Astronomy at Texas A&M University, Financiadora CE110001020. de Estudos e Projetos, Funda¸c˜aoCarlos Chagas Filho This manuscript has been authored by Fermi Research de Amparo `aPesquisa do Estado do Rio de Janeiro, Alliance, LLC under Contract No. DE-AC02-07CH11359 Conselho Nacional de Desenvolvimento Cient´ıfico e Tec- with the U.S. Department of Energy, O ce of Science, nol´ogico and the Minist´erio da Ciˆencia, Tecnologia e In- O ce of High Energy Physics. The United States Gov- ova¸c˜ao, the Deutsche Forschungsgemeinschaft and the ernment retains and the publisher, by accepting the arti- Collaborating Institutions in the Dark Energy Survey. cle for publication, acknowledges that the United States The Collaborating Institutions are Argonne National Government retains a non-exclusive, paid-up, irrevoca- Laboratory, the University of California at Santa Cruz, ble, world-wide license to publish or reproduce the pub- the University of Cambridge, Centro de Investigaciones lished form of this manuscript, or allow others to do so, Energ´eticas, Medioambientales y Tecnol´ogicas-Madrid, for United States Government purposes. the University of Chicago, University College London, MC has been funded by AYA2013-44327-P, AYA2015- the DES-Brazil Consortium, the University of Edin- 71825-P and acknowledges support from the Ramon y burgh, the Eidgen¨ossische Technische Hochschule (ETH) Cajal MICINN program. 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(2017). can be seen in Fig A.1. [33] K. M. G´orski et al., Astrophys. J. 622, 759 (2005), astro- In Fig A.2 we show the correlations between the galaxy ph/0409513. density and all the SP maps listed in Table III. 19 FIG. A.1. Maps of potential sources of systematics. Shown here for i-band only. Maps in other bands show fluctuations on similar scales. Each SP map is shown at Nside =1024.ThestellardensitymapisshownatNside =512. 20 FIG. A.2. Galaxy number density as a function of di↵erent SP maps. We show here only the correlations with SP maps used in the 2 2(68) weights calculation. The cyan line is the correlation of the sample without weights. The black points show the correlation after correction with the 2 2(68) weights. The error bars were calculated by measuring the same correlation on the Gaussian mock surveys described in Section IVB. The significance of these correlations are shown in Figure 6.