Dark Energy Survey Year 1 Results: Galaxy clustering for combined probes
J. Elvin-Poole, M. Crocce, A. J. Ross, T. Giannantonio, E. Rozo, E. S. Rykoff, S. Avila, N. Banik, J. Blazek, S. L. Bridle, R. Cawthon, A. Drlica-Wagner, O. Friedrich, N. Kokron, E. Krause, N. MacCrann, J. Prat, C. Sanchez, L. F. Secco, I. Sevilla-Noarbe, M. A. Troxel, T. M. C. Abbott, F. B. Abdalla, S. Allam, J. Annis, J. Asorey, K. Bechtol, M. R. Becker, A. Benoit-Levy, G. M. Bernstein, E. Bertin, D. Brooks, E. Buckley-Geer, D. L. Burke, A. Carnero Rosell, D. Carollo, M. Carrasco Kind, J. Carretero, F. J. Castander, C. E. Cunha, C. B. DAndrea, L. N. da Costa, T. M. Davis, C. Davis, S. Desai, H. T. Diehl, J. P. Dietrich, S. Dodelson, P. Doel, T. F. Eifler, A. E. Evrard, E. Fernandez, B. Flaugher, P. Fosalba, J. Frieman, J. Garcia-Bellido, E. Gaztanaga, D. W. Gerdes, K. Glazebrook, D. Gruen, R. A. Gruendl, J. Gschwend, G. Gutierrez, W. G. Hartley, S. R. Hinton, K. Honscheid, J. K. Hoormann, B. Jain, D. J. James, M. Jarvis, T. Jeltema, M. W. G. Johnson, M. D. Johnson, A. King, K. Kuehn, S. Kuhlmann, N. Kuropatkin, O. Lahav, G. Lewis, T. S. Li, C. Lidman, M. Lima, H. Lin, E. Macaulay, M. March, J. L. Marshall, P. Martini, P. Melchior, F. Menanteau, R. Miquel, J. J. Mohr, A. Moller, R. C. Nichol, B. Nord, C. R. ONeill, W.J. Percival, D. Petravick, A. A. Plazas, A. K. Romer, M. Sako, et al. (26 additional authors not shown)
DDES-2017-0225FERMILAB-PUB-17-280-AE
Dark Energy Survey Year 1 Results: Galaxy clustering for combined probes
J. Elvin-Poole, M. Crocce, A. J. Ross, T. Giannantonio,
4, 5, 6
E. Rozo, E. S. Rykoff,
8, 9
S. Avila,
10, 11
N. Banik,
12, 13
J. Blazek,
14, 3
S. L. Bridle, R. Cawthon, A. Drlica-Wagner, O. Friedrich,
6, 16
N. Kokron,
17, 18
E. Krause, N. MacCrann,
3, 19
J. Prat, C. S´anchez, L. F. Secco, I. Sevilla-Noarbe, M. A. Troxel,
3, 19
T. M. C. Abbott, F. B. Abdalla,
23, 24
S. Allam, J. Annis, J. Asorey,
25, 26
K. Bechtol, M. R. Becker,
9, 28
A. Benoit-L´evy,
23, 29, 30
G. M. Bernstein, E. Bertin,
30, 29
D. Brooks, E. Buckley-Geer, D. L. Burke,
8, 9
A. Carnero Rosell,
31, 18
D. Carollo,
26, 32
M. Carrasco Kind,
33, 34
J. Carretero, F. J. Castander, C. E. Cunha, C. B. D’Andrea, L. N. da Costa,
18, 31
T. M. Davis,
25, 26
C. Davis, S. Desai, H. T. Diehl, J. P. Dietrich,
36, 37
S. Dodelson,
12, 15
P. Doel, T. F. Eifler,
38, 39
A. E. Evrard,
40, 41
E. Fernandez, B. Flaugher, P. Fosalba, J. Frieman,
12, 15
J. Garc´ıa-Bellido, E. Gaztanaga, D. W. Gerdes,
41, 40
K. Glazebrook, D. Gruen,
8, 9
R. A. Gruendl,
33, 34
J. Gschwend,
31, 18
G. Gutierrez, W. G. Hartley,
23, 44
S. R. Hinton, K. Honscheid,
3, 19
J. K. Hoormann, B. Jain, D. J. James, M. Jarvis, T. Jeltema, M. W. G. Johnson, M. D. Johnson, A. King, K. Kuehn, S. Kuhlmann, N. Kuropatkin, O. Lahav, G. Lewis,
49, 26
T. S. Li, C. Lidman,
47, 26
M. Lima,
17, 18
H. Lin, E. Macaulay, M. March, J. L. Marshall, P. Martini,
3, 51
P. Melchior, F. Menanteau,
34, 33
R. Miquel,
20, 53
J. J. Mohr,
16, 36, 37
A. M¨oller,
54, 26
R. C. Nichol, B. Nord, C. R. O’Neill,
25, 26
W.J. Percival, D. Petravick, A. A. Plazas, A. K. Romer,
55, 8
M. Sako, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, E. Sheldon, M. Smith, R. C. Smith, M. Soares-Santos, F. Sobreira,
18, 58
N. E. Sommer,
54, 26
E. Suchyta, M. E. C. Swanson, G. Tarle, D. Thomas,
11, 19
B. E. Tucker,
26, 54
D. L. Tucker, S. A. Uddin,
26, 60
V. Vikram, A. R. Walker, R. H. Wechsler,
8, 28, 9
J. Weller,
37, 16, 6
W. Wester, R. C. Wolf, F. Yuan,
26, 54
B. Zhang,
54, 26 and J. Zuntz (DES Collaboration) Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,University of Manchester, Oxford Road, Manchester, M13 9PL, UK Institute of Space Sciences, IEEC-CSIC, Campus UAB,Carrer de Can Magrans, s/n, 08193 Barcelona, Spain Center for Cosmology and Astro-Particle Physics,The Ohio State University, Columbus, OH 43210, USA Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Universit¨ats-Sternwarte, Fakult¨at f¨ur Physik, Ludwig-MaximiliansUniversit¨at M¨unchen, Scheinerstr. 1, 81679 M¨unchen, Germany Department of Physics, University of Arizona, Tucson, AZ 85721, USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Kavli Institute for Particle Astrophysics & Cosmology,P. O. Box 2450, Stanford University, Stanford, CA 94305, USA Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas (CIEMAT), Madrid, Spain Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA Department of Physics, University of Florida, Gainesville, Florida 32611, USA Institute of Physics, Laboratory of Astrophysics,´Ecole Polytechnique F´ed´erale de Lausanne (EPFL),Observatoire de Sauverny, 1290 Versoix, Switzerland Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany Departamento de F´ısica Matem´atica, Instituto de F´ısica,Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo, SP, 05314-970, Brazil Laborat´orio Interinstitucional de e-Astronomia - LIneA,Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Department of Physics, The Ohio State University, Columbus, OH 43210, USA Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology,Campus UAB, 08193 Bellaterra (Barcelona) Spain Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia ARC Centre of Excellence for All-sky Astrophysics (CAASTRO) LSST, 933 North Cherry Avenue, Tucson, AZ 85721, USA a r X i v : . [ a s t r o - ph . C O ] A ug Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095,Institut d’Astrophysique de Paris, F-75014, Paris, France Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil INAF - Osservatorio Astrofisico di Torino, Pino Torinese, Italy National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA Department of Astronomy, University of Illinois, 1002 W. Green Street, Urbana, IL 61801, USA Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India Faculty of Physics, Ludwig-Maximilians-Universit¨at, Scheinerstr. 1, 81679 Munich, Germany Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Dr., Pasadena, CA 91109, USA Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Victoria 3122, Australia Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA Australian Astronomical Observatory, North Ryde, NSW 2113, Australia Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA Sydney Institute for Astronomy, School of Physics,A28, The University of Sydney, NSW 2006, Australia George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA Instituci´o Catalana de Recerca i Estudis Avan¸cats, E-08010 Barcelona, Spain The Research School of Astronomy and Astrophysics,Australian National University, ACT 2601, Australia Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, Jiangshu 210008, China Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK (Dated: August 29, 2018)We measure the clustering of DES Year 1 galaxies that are intended to be combined withweak lensing samples in order to produce precise cosmological constraints from the joint anal-ysis of large-scale structure and lensing correlations. Two-point correlation functions are mea-sured for a sample of 6 . × luminous red galaxies selected using the redMaGiC algorithmover an area of 1321 square degrees, in the redshift range 0 . < z < .
9, split into five tomo-graphic redshift bins. The sample has a mean redshift uncertainty of σ z / (1 + z ) = 0 . b ( σ / . | z =0 . = 1 . ± . b ( σ / . | z =0 . = 1 . ± . b ( σ / . | z =0 . = 1 . ± . L/L ∗ > . b ( σ / . | z =0 . = 1 . ± .
04 for
L/L ∗ > b ( σ / . | z =0 . = 1 . ± .
07 for
L/L ∗ > .
5, broadly consistent with expectations for the red-shift and luminosity dependence of the bias of red galaxies. We show these measurements to beconsistent with the linear bias obtained from tangential shear measurements.
I. INTRODUCTION
Galaxies are a biased tracer of the matter density field.In the standard ‘halo model’ paradigm, they form in col- lapsed over-densities (dark matter halos; [1]), and themass of the halo they reside in is known to correlate withthe luminosity and color of the galaxy, with more lumi-nous and redder galaxies strongly correlated with highermass. Therefore, the galaxy ‘bias’ depends strongly onthe particular sample being studied. Thus, in cosmolog-ical studies the galaxy bias is often treated as a nuisanceparameter — one that is degenerate with the amplitudeof the clustering of matter. See, e.g., [2] and referencestherein.The degeneracy can be broken with additional ob-servables. This includes the weak gravitational lensing‘shear’ field, which is induced by the matter density field.Correlation between the galaxies and the shear field ([3];often referred to as ‘galaxy-galaxy lensing’) contains onefactor of the galaxy bias and two factors of the matterfield. The galaxy auto-correlation contains two factorsof the galaxy bias and again two factors of the matterfield. Thus, the combination of the two measurementscan break the degeneracy between the two quantities,and it is a sensitive probe of the late-time matter field(see, e.g., [4, 5])The auto-correlation of the shear field alone includesno factors of the galaxy bias and can thus be used directlyas a probe of the matter field. However, its sensitivity tomany systematic uncertainties related to the estimationof the shear field differs from that of the galaxy-galaxylensing signal. As shown by [6–9], the impact of suchsystematic uncertainties can be mitigated by combiningthe shear auto-correlation measurements with those ofgalaxy clustering and galaxy-galaxy lensing. Thus thereis substantial gain to be obtained from a combined anal-ysis.Such a combined analysis is performed with the DarkEnergy Survey (DES[10]; [11]) Year-1 (Y1) data ([12];hereafter Y1COSMO). DES is an imaging survey cur-rently amassing data over a 5000 deg footprint infive passbands ( grizY ). When completed, it will havemapped 300 million galaxies and tens of thousands ofgalaxy clusters.In this work, we study the clustering of red sequencegalaxies selected from DES Y1 data using the red-MaGiC [13] algorithm, chosen for its small redshift un-certainty. We study the same sample used to obtaincosmological results in the Y1COSMO combined anal-ysis. In particular, we study the large-scale clusteringamplitude and its sensitivity to observational systemat-ics. Following previous studies [14–17], we use angularmaps to track the observing conditions of the Y1 datain order to identify and correct for spurious fluctuationsin the galaxy density field. We further determine theeffect the corrections have on the covariance matrix ofthe angular auto-correlation of the galaxies. We presentrobust measurements of the clustering amplitude of red-MaGiC galaxies as a function of redshift and luminos-ity, thus gaining insight into the physical nature of thesegalaxies and how they compare to other red galaxy sam-ples. The results of this paper are then used for the jointDES cosmological analysis presented in Y1COSMO.This outline of this paper is: we summarize in SectionII the model we use to describe our galaxy clustering measurements; we present in Section III the DES datawe use; Section IV presents how we measure clusteringstatistics and estimate their covariance; Section V sum-marizes the results of our observational systematic tests.We present our primary results with galaxy bias mea-surements in Section VI and a demonstration of theirrobustness in Section VII before concluding in SectionVIII.In order to avoid confirmation bias, we have performedthis analysis “blind”: we did not measure parameter con-straints or plot the correlation function measured fromthe data on the same axis as any theoretical prediction orsimulated clustering measurement until the sample andall measurements in Y1COSMO were finalized.Unless otherwise noted, we use a fiducial ΛCDM cos-mology, fixing cosmological parameters at Ω m = 0 . A s = 2 . × − , Ω b = 0 . h = 0 . n s = 0 . m = 0 . A s = 2 . × − , Ω b = 0 . h = 0 . n s = 0 . ν =0 . II. THEORY
Throughout this paper we model redMaGiC cluster-ing measurements assuming a local, linear galaxy biasmodel [19], where the galaxy and matter density fluctua-tions are related by δ g ( x ) = bδ m ( x ), with density fluctu-ations defined by δ ≡ ( n ( x ) − ¯ n ) / ¯ n . The validity of thisassumption over the scales considered here is provided in[20] and shown in simulations in [21].The galaxy clustering model used in this papermatches that used in Y1COSMO. This model includes3 neutrinos of degenerate mass.We consider multiple galaxy redshift bins i , each char-acterized by a redMaGiC galaxy redshift distribution n ig ( z ), normalized to unity in redshift, and a galaxy bias b i which is assumed to be constant across the redshift binrange. Under the Limber [22] and flat-sky approximationthe theoretical prediction for the galaxy correlation func-tion w ( θ ) in a given bin is, w i ( θ ) = (cid:0) b i (cid:1) (cid:90) dl l π J ( lθ ) (cid:90) dχ × (cid:2) n i g ( z ( χ )) dz/dχ (cid:3) χ P NL (cid:18) l + 1 / χ , z ( χ ) (cid:19) , (1)where the speed of light has been set to one, χ ( z ) is thecomoving distance to a given redshift (in a flat universe,which is assumed throughout); J is the Bessel function oforder zero; H ( z ) is the Hubble expansion rate at redshift z ; and P NL ( k ; z ) is the 3D matter power spectrum at red-shift z and wavenumber k (which, in this Limber approx-imation, is set equal to ( l + 1 / /χ ). Note that in Eq. 1we have assumed the bias to be constant within each bin,see Fig. 8 in [20] for a test of this assumption. Again,all assumptions and approximations mentioned here havebeen shown to be inconsequential in [20, 21]. To modelcross-correlation between redshift bins we simply change n i g ( z ) → n i g ( z ) n j g ( z ) and ( b i ) → b i b j in Eq. 1.Throughout this paper, we use the CosmoSIS frame-work [23] to compute correlation functions, and to infercosmological parameters. The evolution of linear densityfluctuations is obtained using the
Camb module [24] andthen converted to a non-linear matter power spectrum P NL ( k ) using the updated Halofit recipe [25].The theory modeling we use assumes the Limber ap-proximation, and it also neglects redshift space distor-tions. For the samples and redshift binning used in thispaper, those effects start to become relevant at scalesof θ (cid:38) m and S for a LCDM Universe or w in a wCDM one. We also tested the impact of these effectson the fixed cosmology bias measurements in Section VIand find them to be negligible. Hence in what follows,such terms are ignored for speed and simplicity. How-ever future data analyses may need to account for theseeffects due to improved statistical uncertainty.We model (and marginalize over) photometric redshiftbias uncertainties as an additive shift ∆ z i in the red-MaGiC redshift distribution n i RM ( z ) for each redshiftbin i . n i ( z ) = n i RM ( z − ∆ z i ) (2)The priors on the ∆ z i nuisance parameters, are mea-sured directly using the angular cross correlation betweenthe DES sample and a spectroscopic sample. These val-ues are shown in Table II, and the method is describedin full in [29]. We use the same ∆ z i as Y1COSMO forall tests of robustness of the parameter constraints.We also compare the measurements of b i to the samequantity measured by galaxy-galaxy lensing using thetwo-point correlation function γ t (see [30] for definition).We use the notation b i × for this measurement. The de-tails of this measurement are described in [30] (hereafterY1GGL). In order to take the off-diagonal elements ofthe covariance matrix between the two probes into ac-count, we produce joint constraints from w ( θ ) and γ t at fixed cosmology (the mean of the Y1COSMO poste-rior), using different bias parameters for the two probes,and marginalizing over the same nuisance parameters aswere used in the fiducial analysis of Y1COSMO (intrinsicalignments, source and lens photo − z bias, and shear cali-bration). To test the consistency between the two probeswe use the parameter r which is, r i = b i × b i . (3)If r (cid:54) = 1, this indicates an inconsistency between the twobias measurements and would thus suggest a breakdownof our simple linear bias model. This test informs thechoice of fixing r = 1 in the Y1COSMO analysis.A combination of galaxy clustering and galaxy-galaxylensing, provides a measurement of galaxy bias and σ only if you assume that r = 1. This test provides a mea-surement of r which informs the choice of fixing r = 1in the Y1COSMO analysis. In principle, this test couldalso be performed by including the shear-shear correla-tion which also measures σ . III. DATAA. Y1 Gold
We use data taken in the first year (Y1) of DES obser-vations [31]. Photometry and ‘clean’ galaxy samples wereproduced with this data as outlined by [32] (hereafter de-noted ‘Y1GOLD’). The outputs of this process representthe Y1 ‘Gold’ catalog. Data were obtained over a totalfootprint of ∼ ; this footprint is defined by a Healpix [33] map at resolution N side = 4096 (equiva-lent to 0.74 square arcmin) and includes only pixels withminimum exposure time of at least 90 seconds in eachof the g , r , i , and z -bands, a valid calibration solution, aswell as additional constraints (see Y1GOLD for details).A series of veto masks, including among others masksfor bright stars and the Large Magellanic Cloud, reducethe area by ∼
300 deg , leaving ∼ suitable forgalaxy clustering study. We explain further cuts to theangular mask in Section III B. All data described in thisand in subsequent sections are drawn from cataloguesand maps generated as part of the DES Y1 Gold sampleand are fully described in Y1GOLD. B. redMaGiC sample
The galaxy sample we use in this work is generatedby the redMaGiC algorithm, run on DES Y1 Golddata. The redMaGiC algorithm selects Luminous RedGalaxies (LRGs) in such a way that photometric red-shift uncertainties are minimized, as is described in [13].This method is able to achieve redshift uncertainties σ z / (1 + z ) < .
02 over the redshift range of interest. D E C n g [ a r c m i n ] 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 thefigure. The area is 1321 square degrees. z n ( z ) × − z < 0.30.3 < z < 0.450.45 < z < 0.60.6 < z < 0.750.75 < z < 0.9 FIG. 2. Redshift distribution of the combined redMaGiC sample in 5 redshift bins. They are calculated by stackingGaussian PDFs with mean equal to the redMaGiC redshiftprediction and standard deviation equal to the redMaGiC redshift error. Each curve is normalized so that the area ofeach curve matches the number of galaxies in its redshift bin.
The redMaGiC algorithm produces a redshift predic-tion z RM and an uncertainty σ z which is assumed to beGaussian. This sample was chosen instead of other DESphotometric samples because of its small redshift uncer-tainty, which is obtained at the expense of number den-sity.The redMaGiC algorithm makes use of an empiri- cal red-sequence template generated by the training ofthe redMaPPer cluster finder [34, 35]. As described in[35], training of the red-sequence template requires over-lapping spectroscopic redshifts, which in this work wereobtained from SDSS in the Stripe 82 region [36] and theOzDES spectroscopic survey in the DES deep supernovafields [37].For the redMaGiC samples in this work, we makeuse of two separate versions of the red-sequence training.The first is based on SExtractor MAG AUTO quantities fromthe Y1 coadd catalogs, as applied to redMaPPer in[38]. The second is based on a simultaneous multi-epoch,multi-band, and multi-object fit (
MOF ) (see Section 6.3of Y1GOLD), as applied to redMaPPer [39]. In gen-eral, due to the careful handling of the point-spread func-tion (PSF) and matched multi-band photometry, the
MOF photometry yields lower color scatter and, hence, smallerscatter in red-sequence photo- z s. For each version of thecatalog, photometric redshifts and uncertainties are pri-marily derived from the fit to the red-sequence template.In addition, an afterburner step is applied (as describedin Section 3.4 of [13]) to ensure that redMaGiC photo- z s and errors are consistent with those derived from theassociated redMaPPer cluster catalog [13].As described in [13], the redMaGiC algorithm com-putes color-cuts necessary to produce a luminosity-thresholded sample of constant co-moving density. Boththe luminosity threshold and desired density are inde-pendently configurable, but in practice higher luminos-ity thresholds require a lower density for good perfor-mance. We note that in [13] the co-moving density wascomputed with the central redshift of each galaxy ( z RM ). FWHM i [pixels] N ga l / › N ga l fi FWHM i [pixels] FWHM i [pixels] FWHM i [pixels] FWHM i [pixels] Exptime i [s] N ga l / › N ga l fi Exptime i [s] Exptime i [s] Exptime i [s] Exptime i [s] FIG. 3. Correlations of volume-limited redMaGiC galaxy number density with seeing FWHM and exposure time before anysurvey property (SP; see text for more details) cuts (illustrated with the red vertical lines) were applied to the mask. In theabsence of systematic correlations, the results obtained from these samples are expected to be consistent with no trend (thereference green dashed line). The cuts removed regions with i -band FWHM > . i -band exposure time > s asthese showed correlations that differed significantly from the mean ( > z range L min /L ∗ n gal (arcmin − ) N gal Photometry0 . < z < . . . < z < .
45 0 . . < z < . . . < z < .
75 1 . . < z < . . L min /L ∗ describes the minimum luminosity threshold of the sample, n gal is the number of galaxies per square degree, and N gal isthe total number of galaxies. z range b i fid ∆ z i . < z < . . , . . < z < .
45 1.55 Gauss ( − . , . . < z < . . , . . < z < .
75 1.8 Gauss (0 . , . . < z < . . , . b i fid is the fiducial lin-ear galaxy bias for bin i applied to the Gaussian mock surveyswe use to construct the covariance matrices. The ∆ z i prioris a Gaussian prior applied to the additive redshift bias un-certainty. These were selected to match the analysis in (DESCollaboration et al. ; Y1COSMO). For this work, the density was computed by samplingfrom a Gaussian distribution z RM ± σ z , which createsa more stable distribution near filter transitions. This isthe only substantial change to the redMaGiC algorithmsince the publication of [13].We use redMaGiC samples split into five redshift binsof width ∆ z = 0 .
15 from z = 0 .
15 to z = 0 .
9. We define our footprint such that the data in each redshiftbin will be complete to its redshift limit across the entirefootprint. To make this possible, we define samples basedon a luminosity threshold. Reference luminosities arecomputed as a function of L ∗ , computed using a Bruzualand Charlot [40] model for a single star-formation burstat z = 3 [See Section 3.2 35]. Naturally, increasing theluminosity threshold provides a higher redshift sample aswell as decreasing the comoving number density. Using adifferent luminosity threshold in each redshift bin allowsus to maximize signal to noise while also providing acomplete sample in each redshift bin. The details of thesebins are given in Table I.The 5 redshift bins were chosen so that the width ofthe bins is significantly wider than the uncertainty onindividual galaxy redshifts, but smaller than the differ-ence between the maximum redshifts of the luminositythresholds used.In addition to the primary redMaGiC selection, wealso apply a cut on the luminosity L/L ∗ < . L ∗ sam-ple, used for z < .
6, are minimized for the
MAG AUTO sample, with a very minor impact on photo- z perfor-mance. For L ∗ ≥ .
0, used for z > .
6, we instead findthat the observation systematic relationships are mini-mized for the
MOF sample, and that the photo- z perfor-mance is also improved. Consequently, we use MAG AUTO for our z < . MOF for z > .
6. See Section Vfor further discussion.We build the area mask for the redMaGiC samplesbased on the depth information produced with the red-
MaGiC catalogs. This information is provided by the z max quantity, which describes the highest redshift atwhich a typical red galaxy of the adopted luminositythreshold (e.g. 0 . L ∗ ) can be detected at 10 σ in the z -band, at 5 σ in the r and i -bands, and at 3 σ in the g -band, as described in Section 3.4 of [35]. The quantity z max varies from point to point in the survey due to ob-serving conditions. Consequently, we construct a z max map, specified on a HealPix map with N side = 4096.In order to obtain a uniform expected number densityof galaxies across the footprint, we only use regions forwhich z max is higher than the upper edge of the redshiftbin under consideration. The footprint is defined as theregions where this condition is met in all redshift bins.Thus, we only use pixels that satisfy each of the condi-tions where the 0 . L ∗ sample is complete to z = 0 .
6, the1 . L ∗ sample is complete to z = 0 .
75, and the 1 . L ∗ sam-ple is complete to z = 0 .
9. We also restrict the analysisto the contiguous region shown in Figure 1.An additional 1.6% of the footprint is vetoed because ithas extreme observing conditions. The selection of thesecuts is detailed in Section V.After masking and additional cuts, we obtain a totalsample of 653,691 objects distributed over an area of 1321square degrees, as shown in Fig. 1. The average redshiftuncertainty of the sample is σ z / (1 + z ) = 0 . z = 0 . IV. ANALYSIS METHODSA. Clustering estimators
We measure the correlation functions w ( θ ) using theLandy & Szalay estimator [41]ˆ w ( θ ) = DD − DR + RRRR , (4)where DD , RR and DR are the number of pairs ofgalaxies from the galaxy sample D and a random cat-alog R . This is calculated in 20 logarithmically sep-arated bins in angle θ between 2.5 arcmin and 250arcmin to match the analysis in Y1COSMO. We use60 times more randoms than data. The pair-countingwas done with the package tree-corr [42] available athttps://github.com/rmjarvis/TreeCorr.We also calculate w ( θ ) on Gaussian random field real-izations which are described in a pixelated map format.For these correlations we use a pixel-based estimator. Us-ing the notation of [17], the correlation between two maps N and N of mean values ¯ N , ¯ N is estimated asˆ w , ( θ ) = N pix (cid:88) i =1 N pix (cid:88) j = i ( N i, − ¯ N )( N j, − ¯ N )¯ N ¯ N Θ i,j , (5) where the sum runs through all pairs of the N pix pixelsin the footprint, N i, is the value of the N map in pixel i , and Θ i,j is unity when the pixels i and j are separatedby an angle θ within the bin size ∆ θ . We have testedthat the difference between the estimators in Equations4 and 5 is negligible for this analysis. B. Covariances
The fiducial covariance matrix we use for the w ( θ )measurement is a theoretical halo model covariance, de-scribed and tested by [20]. The covariance is generatedusing CosmoLike [43], and is computed by calculatingthe four-point correlation functions for galaxy clusteringin the halo model. Additionally, an empirically deter-mined correction for the survey geometry’s effect on theshot-noise component has been added. The presence ofboundaries and holes decrease the effective number ofgalaxy pairs as a function of pair separation, which inturn raises the error budget associated to shot noise overthe standard uniform sky assumptions. This same co-variance is used for the combined probes analysis and isdetailed in Y1COSMO.For the analysis of observational systematics and theircorrelation with the data, we use a set of 1000 mock sur-veys (hereafter ‘mocks’) based on Gaussian random fieldrealizations of the projected density field. These are thenused to obtain an alternative covariance, which includesall the mask effects as in the real data. The mocks weuse were produced using the following method. We firstcalculate, using
Camb [24], the galaxy clustering powerspectrum C ggi ( (cid:96) ), assuming the fiducial cosmology withfixed galaxy bias b i for each redshift bin i ; the galaxy biasvalues are listed in Table II. The angular power spectrumis then used to produce a full-sky Gaussian random fieldof δ g . We apply a mask to this field corresponding tothe Y1 data, as shown in Fig. 1. This is converted into agalaxy number count N gal as a function of sky position,with the same mean as the observed number count ¯ N o ineach redshift bin, using N gal = ¯ N o (1 + δ g ) . (6)Shot noise is finally added to this field by Poisson sam-pling the N gal field.In order to avoid pixels with δ g < −
1, which cannotbe Poisson sampled, we follow the method used by [20]:before Poisson sampling, we multiply the density fieldby a factor α , where α <
1; we then rescale the numberdensity n gal by 1 /α in order to preserve the ratio of shot-noise to sample variance; we then rescale the measured w ( θ ) by 1 /α to obtain the unbiased w ( θ ) for each mock.This procedure is summarized by δ g → αδ g , (7) n gal → n gal /α , (8) w ( θ ) → w ( θ ) /α . (9) n stars (arcmin − ) N ga l / › N ga l fi n stars (arcmin − ) n stars (arcmin − ) n stars (arcmin − ) n stars (arcmin − ) FIG. 4. Galaxy number density divided by the mean number density across the footprint for each redshift bin, split by thenumber density of stars. The points with error-bars display the results for our 3∆ χ (68) weighted sample, the cyan curvesdisplay the results without these weights. For the weighted sample, the χ of the line N gal / (cid:104) N gal (cid:105) = 1 with the data pointsshown for each bin is 24.9, 16.0, 13.1, 6.6 and 10.9 with N d . o . f = 10. The ∆ χ between the null signal and a linear best fit is0.99, 0.95, 0.24, 0.013, and 0.082. This does not meet either of the ∆ χ thresholds used in this analysis. We therefore see noevidence for stellar contamination or obscuration in this sample. FWHM r [pixels] N ga l / › N ga l fi No weightsWeights: airmass-i, FWHM-r
FWHM i [pixels] No weightsWeights: airmass-i, FWHM-r
FIG. 5. Galaxy number density in the highest redshift bin, 0 . < z < .
9, as a function of two example SP maps, FWHM r -band and FWHM i -band. The black points correspond to the 3∆ χ (68) weighted sample, the cyan line is the unweightedsample. In this redshift bin, the SP maps used in the 3∆ χ (68) weights were Airmass i and FWHM r . The left paneldemonstrates the effect of the weights on the FWHM r correlation. The right panel demonstrates that correlations with SPmaps that were not included in the weights are still reduced due to correlations among the SP maps. The full set of SPcorrelations for the maps in Table III are shown in Appendix A. We then use these mocks to estimate statistical errorsin galaxy number density as a function of potential sys-tematics. Alternatively we “contaminate” each of the1000 mocks with survey properties as discussed in Sec-tion V to assess the impact of systematics on the w ( θ )covariance. Note that these mocks would not be fully re-alistic for w ( θ ) covariance and cosmological inference asthey are basically Gaussian realizations. These mocks al-low us to quantify significances (i.e., a ∆ χ ) to null tests,which are a necessary step in our analysis. Further, givensuch a large number of realizations we are able to obtainestimates of both the impact of the systematic correctionon the resulting statistical uncertainty and any bias im-parted by our methodology to well below 1 σ significance(e.g., given 1000 mocks, a 0.1 σ bias can be detected at3 σ significance). V. SYSTEMATICSA. Survey property (SP) maps
The number density of galaxies selected based on theirimaging is likely to fluctuate with the imaging qualitydue to fluctuations in the noise (e.g., Malmquist bias)and limitations in the reduction pipeline. Such fluctua-tions can imprint the structure of certain survey prop-erties onto the galaxy field, thereby producing a non-cosmological signal. In order to quantify the extent ofthese correlations and remove their effect from the two-point function, maps of DES imaging properties were pro-duced using the methods described in Ref. [44]. We con-sider the possibility that depth, seeing, exposure time,sky brightness and airmass, in each band griz , affect thedensity of galaxies we select.In total, we consider 21 survey property maps. Werefer to these as SP maps from here on: • depth: the magnitude limit at which we expect tobe able to detect a galaxy to 10 σ significance; • seeing FWHM: the full width at half maximum ofthe PSF of a point source; • exposure time: total exposure time in a given band; • sky brightness: the brightness of the sky, e.g., dueto background light or the Moon phase; • airmass: the amount of atmosphere a source haspassed through, normalized to be 1 when pointingat zenith.Where relevant, we use the weighted average quantityover all exposures contributing to a given area.We also consider Galactic extinction and stellar con-tamination (or obscuration [14]) as potential systematics.The stellar density map was created by selecting moder-ately bright, high confidence stars. Using the notationof Y1GOLD, this selection is MODEST CLASS = 2 with18 . < i < . FLAGS GOLD = 0, and
BADMASK ≤
2. Wealso include an additional color cut of 0 . < g − i < . g − r >
0. These stars were binned in pixels with N side = 512 (equivalent to 47 square arcmin), and thecorresponding area for each pixel was computed at higherresolution ( N side = 4096) from the Y1 Gold footprint andpixel coverage fraction, as well as the bad region mask.Together, this yields the number of moderately brightstars per square degree that can be used to cross-correlategalaxies with stellar density. Using MODEST CLASS to se-lect stars means this map could potentially contain DESgalaxies. For this reason, we test for correlations withthe astrophysical maps separately to the SP maps. Aswe find no correlation between stellar density and galaxydensity, we do not take this contamination into account.For Galactic extinction, we use the standard map from[45].
B. Systematic corrections
This section describes the method used to identify andcorrect for observational systematics. We also discussthe uncertainty on this correction and its impact on the w ( θ ) covariance. Our approach is to first identify mapsthat are correlated with fluctuations in the galaxy densityfield at a given significance. We then correct for thecontamination using weights to be applied to the galaxycatalog.As demonstrated by [46], when testing a large numberof maps one expects there to be some amount of covari-ance between the maps and the true galaxy density fielddue to chance. Consequently, it is possible to over-correctthe galaxy density field using the type of methods em-ployed in this work. To limit this effect, we do not correctfor all possible maps, and limit ourselves to those mapsthat are detected to be correlated with the galaxy den-sity field at high significance (above a given threshold).We test the robustness of the results on our choice ofthreshold in Section VII A and we test for biases due to z range Maps included in Maps included in3∆ χ (68) weights 2∆ χ (68) weights0 . < z < . r ) Exptime ( i )FHWM ( z )FWHM ( r )Airmass ( z )0 . < z < .
45 Depth ( g ) Depth ( g )0 . < z < . z ) FWHM ( z )Exptime ( g ) Exptime ( g )FWHM ( r ) FWHM ( r )Skybright ( z ) Skybright ( z )Depth ( i )0 . < z < .
75 FWHM ( gri ) PCA-0 FWHM ( gri ) PCA-0Skybright ( r ) Skybright ( r )FWHM ( z ) FWHM ( z )Exptime ( i )Exptime ( z )0 . < z < . i ) Airmass ( i )FWHM ( r ) FWHM ( r )FWHM ( g )TABLE III. List of the maps used in the SP weights. Eachof these has been determined to impart fluctuations in ourgalaxy sample at > χ (68) or > χ (68) significance.The weights were applied serially for each map in the ordershown, starting from the top of the table. ‘FWHM’ refers tothe full-width-half-maximum size of the PSF. The photomet-ric band of each SP map is in parentheses. over-correction in Section VII C. The end result of ourprocedure is a measurement of w ( θ ) that is free of sys-tematics above a given significance (in our concrete casea galaxy density free of two sigma correlations with SPmaps, as defined below, and visualized in Fig. III) andthat can be directly utilized in combination with weaklensing measurements for cosmological analyses.We identify the most significant SP maps as follows.First, given an SP map of some quantity s , we identify allpixels in some bin s ∈ [ s min , s max ]. We then compute theaverage density of galaxies in these pixels. By scanningacross the whole range of possible s -values for the SPmap, we can directly observe how the galaxy density fieldscales with s . Examples of these type of analyses areshown in Figs. 3, 4 and 5.We first remove regions of the footprint that displayeither especially significant ( > i -bandFWHM > . i -band exposure time > s . Thesecuts remove 1.6% of the Y1 area.After cutting the footprint, we determine which SPmaps most significantly correlate with the data by fittinga linear function to each number density relationship. Weminimize a χ where the model is N gal ∝ A s + B . Wedetermine the significance of a correlation based on thedifference in χ between the best-fit linear parameters,0 M A G A U T O d e p t h r M O F d e p t h i M A G A U T O d e p t h i M O F d e p t h r e x p t i m e i e x p t i m e r e x p t i m e g e x p t i m e z a i r m a ss z f w h m z M A G A U T O d e p t h g f w h m r M O F d e p t h z M O F d e p t h g s k y b r i g h t r s k y b r i g h t g M A G A U T O d e p t h z a i r m a ss r s k y b r i g h t i s k y b r i g h t z a i r m a ss g f w h m i f w h m g a i r m a ss i ∆ χ / ∆ χ ( ) N dof = no weights χ (68) weights χ (68) weights M A G A U T O d e p t h g M A G A U T O d e p t h r M O F d e p t h r M O F d e p t h g e x p t i m e r s k y b r i g h t z s k y b r i g h t i e x p t i m e g a i r m a ss i e x p t i m e z a i r m a ss r e x p t i m e i a i r m a ss g f w h m r M O F d e p t h i a i r m a ss z M A G A U T O d e p t h i f w h m z f w h m g f w h m i M A G A U T O d e p t h z M O F d e p t h z s k y b r i g h t r s k y b r i g h t g ∆ χ / ∆ χ ( ) N dof = no weights χ (68) weights χ (68) weights f w h m z e x p t i m e g M O F d e p t h z f w h m r a i r m a ss z M A G A U T O d e p t h z s k y b r i g h t z e x p t i m e z M A G A U T O d e p t h g e x p t i m e i f w h m g M O F d e p t h g a i r m a ss r f w h m i M A G A U T O d e p t h i s k y b r i g h t i a i r m a ss g s k y b r i g h t g a i r m a ss i s k y b r i g h t r e x p t i m e r M O F d e p t h r M O F d e p t h i M A G A U T O d e p t h r ∆ χ / ∆ χ ( ) N dof = no weights χ (68) weights χ (68) weights f w h m r M A G A U T O d e p t h i f w h m i M O F d e p t h i f w h m g e x p t i m e i M A G A U T O d e p t h r s k y b r i g h t r M O F d e p t h r f w h m z a i r m a ss r a i r m a ss i a i r m a ss g a i r m a ss z s k y b r i g h t i e x p t i m e r M A G A U T O d e p t h z s k y b r i g h t g M O F d e p t h z M O F d e p t h g e x p t i m e z M A G A U T O d e p t h g e x p t i m e g s k y b r i g h t z ∆ χ / ∆ χ ( ) N dof = no weights χ (68) weights χ (68) weights a i r m a ss i f w h m r f w h m g f w h m i a i r m a ss r a i r m a ss g s k y b r i g h t i s k y b r i g h t z M O F d e p t h r M A G A U T O d e p t h r M O F d e p t h g e x p t i m e r M A G A U T O d e p t h g f w h m z a i r m a ss z M O F d e p t h z e x p t i m e i M A G A U T O d e p t h i M A G A U T O d e p t h z M O F d e p t h i s k y b r i g h t r e x p t i m e g s k y b r i g h t g e x p t i m e z ∆ χ / ∆ χ ( ) N dof = no weights χ (68) weights χ (68) weights FIG. 6. The significance of each systematic correlation. The significance is calculated by comparing the ∆ χ measured on thedata to the distribution in the mock realizations. We find the 68th percentile ∆ χ value, labeling it ∆ χ (68), for each mapobtained from the mock realizations. We quote the significance for the relationship obtained on the data as ∆ χ / ∆ χ (68).Weights are applied for the SP map with the largest significance, with the caveat that we do not correct for both depth and thecomponents of depth (e.g. exposure time, PSF FWHM) in the same band. For example, in the bin 0 . < z < .
3, correctingfor r -band depth (the most significant contaminant) did not remove all the r -band correlations with ∆ χ / ∆ χ (68) >
2, so isnot included in the final 2∆ χ / ∆ χ (68) weights. This is repeated iteratively until all maps are below a threshold significance,shown here for thresholds of 2∆ χ / ∆ χ (68) and 3∆ χ / ∆ χ (68). The x axis is shown in order of decreasing significance for theunweighted sample. The labels in bold are the SP maps included in the 2∆ χ / ∆ χ (68) weights. In the second redshift bin,0 . < z < .
45, the 3∆ χ / ∆ χ (68) and 2∆ χ / ∆ χ (68) weights are the same because correcting for only g -band depth removesall correlations with ∆ χ / ∆ χ (68) > and a null test of N gal / (cid:104) N gal (cid:105) = 1,∆ χ = χ − χ . (10)The ∆ χ is then compared to the same quantity mea-sured on the Gaussian random fields described in Sec-tion IV B. We then label each potential systematic tobe significant at 1 σ if the ∆ χ measured on the data is larger than 68% of the mocks respectively. We de-note this threshold as ∆ χ (68) and quote significancesas ∆ χ / ∆ χ (68); the square-root of this number shouldroughly correspond to the significance in terms of σ .Some examples of these tests for the observational sys-tematics can be seen in Fig. 6. The full set of tests canbe seen in Appendix A. We see no significant correlation1 z range b i ( σ / . r i . < z < . . ± .
072 1 . ± . . < z < .
45 1 . ± .
051 0 . ± . . < z < . . ± .
039 0 . ± . . < z < .
75 1 . ± .
045 1 . ± . . < z < . . ± .
070 0 . ± . b i and the ratioof bias from clustering and galaxy-galaxy lensing r i for eachredshift bin i , calculated with cosmological parameters fixedat the mean of the Y1COSMO posterior, varying only biasand nuisance parameters with lens photo − z priors from [29]. with stellar density in the sample, as shown in Fig. 4.Similarly, we find no correlations with Galactic extinc-tion. Thus, our main tests are against SP maps, whichare particular to DES observations.Once we identify the most significant contaminant SPmaps, we define weights to be applied to the galaxysample in order to remove the dependency, following amethod close to that of the latest LSS survey analysis[5, 47–49]. Note however that we identify systematics us-ing a rigorous χ threshold significance criteria, based ona large set of Gaussian realizations, which to our knowl-edge was not done before.For this method we apply the following steps to eachredshift bin separately. The correlation with a systematic s is fitted with a function N gal / (cid:104) N gal (cid:105) = F sys ( s ).For depth and airmass, the function used was a lin-ear fit in s . For exposure time and sky brightness, thefunction was linear in √ s , as this is how these quantitiesenter the depth map. For the seeing correlations, we fitthe model N gal / (cid:104) N gal (cid:105) = F sys ( s FWHM ) F sys ( s FWHM ) = A (cid:20) − erf (cid:18) s FWHM − Bσ (cid:19)(cid:21) , (11)where s FWHM is the seeing full-width half-max value, and A , B and σ are parameters to be fitted. This functionalform matches that applied to BOSS [48, 50]; we believeit is thus the expected form when morphological cutsare applied to reject stars (as this is what causes therelationship for BOSS).Each galaxy i in the sample is then assigned a weight1 /F sys ( s i ) where s i is the value of the systematic at thegalaxy’s location on the sky. This weight is then usedwhen calculating w ( θ ) and in all further null tests.In this sample we find evidence of multiple systemat-ics at a significance of ∆ χ / ∆ χ (68) >
3, some of whichare correlated with each other. To account for this, wefirst apply weights for the systematic with the highest∆ χ / ∆ χ (68). Then, using the weighted sample, we re-measure the significance of each remaining potential sys-tematic and repeat the process until there are no system-atics with a significance greater than a ∆ χ / ∆ χ (68) = 3threshold. The final weights are the product of theweights from each required systematic. We also produce weights using a threshold of ∆ χ / ∆ χ (68) = 2, allowingus to determine if using a greater threshold has any im-pact on our clustering measurements. We refer to theseweights as the 3∆ χ (68) and 2∆ χ (68) weights respec-tively.The final weights used in this sample are described inTable III. The SP maps are either the depth or propertiesthat contribute to the depth (e.g. holding everything elsefixed, a longer exposure time will result in an increaseddepth). Thus, in bins where multiple SP weights wererequired, we avoided correcting for both depth and SPsthat contribute to the depth in the same band. In thesecases, we weight for only the SPs that contribute to thedepth. Fig. A.2 shows the correlation between the sampledensity and the SP maps used in Table III, both with andwithout weights.Fig. 6 summarizes the results of our search for contam-inating SPs, for each redshift bin. The blue points showthe significance for each map, prior to the application ofany weights. The black and red points display the signifi-cance after applying the 3∆ χ (68) and 2∆ χ (68) weightsrespectively. In Section VII, we will test our results withboth choice of weights and whether to expect any biasfrom over-correction from either choice.When F sys ( s ) is a linear function, the method de-scribed above, hereby referred to as the weights method,should be equivalent to the method used in [15, 17]. Thishas been shown in [5] for the DES science verificationredMaGiC sample.The impact of the SP weights on the w ( θ ) measure-ment can be seen in Figure 7. The dashed line displaysthe measurement with no weights applied. One can seethat in all redshift bins, the application of the SP weightsreduces the clustering amplitude and that the effect isgreatest on large scales. This is consistent with expecta-tions (see, e.g. Ref. [14]). VI. RESULTS: GALAXY BIAS ANDSTOCHASTICITY
In this section we present measurements of galaxy bias b i and stochastic bias r i . The amplitude of the galaxyclustering signal is determined by the combination of pa-rameters ( b i σ ) . Equivalently the galaxy-galaxy lensingsignal γ t is sensitive to b i × ( σ ) . In the Y1COSMO com-bined probes analysis, cosmic shear provides a measure-ment of σ meaning that galaxy clustering and galaxygalaxy lensing can each provide an independant measur-ment of galaxy bias (and therefore one could measure r ).In this analysis we fix σ at the mean of the Y1COSMOpostierior ( σ = 0 .
81) to measure b i ( σ / .
81) from clus-tering and r i from γ t . This provides a cosmology depen-dent measurement of bias from clustering alone, and testof the assumption r = 1 in Y1COSMO.The w ( θ ) auto-correlation functions of the redMaGiC galaxy sample are shown in Figure 7. We show theauto-correlation calculated with and without a correc-2 [arcmin]0.00.51.01.52.02.5 w () [arcmin]0.00.51.01.5 2,2 10 [arcmin]0.00.51.01.5 3,310 [arcmin]0.00.51.01.5 w () [arcmin]0.00.51.01.5 5,5 FIG. 7. Two-point correlation functions for the fiducial analysis in each of the 5 redshift bins. These panels show the auto-correlation used in Y1COSMO and the galaxy bias measurements presented in this work. A correction for correlations withsurvey properties is applied according to the methodology in Section V. The grey dashed line is the correlation functioncalculated without the SP weights. The black points use the 2∆ χ (68) weights. We show correlations down to θ = 2 . (cid:48) tohighlight the goodness of the fit towards small scales, but data points within grey shaded regions have not been used in biasconstraints or the galaxy clustering part of Y1COSMO. That scale cut has been set in co-moving coordinates at 8 Mpc h − .The solid red curve is the best-fit model using only the w ( θ ) auto-correlations at fixed cosmology, using ∆ z i priors from [29].The solid blue curve is the best-fit model from the full cosmological analysis in Y1COSMO. tion for observational systematics, as described in SecV. A minimum angular scale θ i min has been applied toeach redshift bin i . These were chosen to be θ = 43 (cid:48) , θ = 27 (cid:48) , θ = 20 (cid:48) , θ = 16 (cid:48) , and θ = 14 (cid:48) tomatch the analysis in Y1COSMO. These minimum an-gular scales, varying with redshift, correspond to a sin-gle minimum co-moving scale R = 8 Mpc h − such that θ imin = R/χ ( (cid:104) z i (cid:105) ), where (cid:104) z i (cid:105) is the mean redshift ofgalaxies in bin i [20]. The scale was chosen so that a sig-nificant non-linear galaxy bias or baryonic feedback com-ponent to the Y1COSMO data vector would not bias thecosmological parameter constraints.The angular correlation function has been calculatedon scales below θ i min , but these were removed in all pa-rameter constraints.Fixing all cosmological parameters, including Ω m , atthe Y1COSMO values, we measure the linear bias to be b = 1 . ± . b = 1 . ± . b = 1 . ± . b =1 . ± .
04, and b = 1 . ± .
07. The χ values of thecombined fit and the individual bins are shown in TableV. We note that the bin with the smallest probability isbin 1.The combined goodness-of-fit χ of the bias measure-ments is χ = 67 and the number of degrees of freedomis ν = 54 −
10 (the 10 parameters are b i , ∆ z i ). These values provide a probability to exceed of 1 . χ distribution are not strictly applicable in this casedue to the uncertainty on the estimates of the covari-ance. Further, because the five ∆ z i are nuisance param-eters with tight priors, we also consider ν = 49, whichyields a probability to exceed of 4 . χ is sensitive to the inclusionof the shot-noise correction applied to the covariance de-tailed in Y1COSMO whereas the b i values and uncer-tainty were insensitive to this change.For the L/L ∗ > . σ significance (the cor-relation in the measured bias for bins 1 and 3 is only-0.04, so we can safely ignore it in this discussion). Thedifference between bin 1 and bin 3 is less significant if wedetermine the expectation for a passively evolving sam-ple as in [51, 52], which predicts a bias of 1.52 at z = 0 . z = 0 .
53. The bias increases forthe higher luminosity sample, as expected. The resultsare broadly consistent with previous studies of the biasof red galaxies at low redshift (see, e.g., [53] for a review)3 . . . . . . . r . . . . . . . r . . . . . . . r . . . . . . . r . . . . . . . . r . . . . . . . . Redshift . . . . . . . . . G a l a xy b i a s L min = 0 . L ∗ L ∗ . L ∗ w ( θ ) γ t FIG. 8. Constraints on the ratio, r , of galaxy bias measured on w ( θ ) and measured from the galaxy-galaxy lensing signal (see[30] denoted Y1GGL in the text) in each redshift bin. The histograms show the posterior distributions of r i from an MCMC fitfor each z in i . The bottom-right panel displays the individual measurements for each bin (purple for our w ( θ ) measurementsand orange for those obtained in Y1GGL). All cosmological parameters were fixed at the DES Y1COSMO posterior meanvalues, and all nuisance parameters were varied as in Y1COSMO. The constraints were calculated using the full Y1COSMOcovariance matrix, so the covariance between the two probes has been taken into account. We see no significant evidence for r (cid:54) = 1 within the errors. and BOSS at intermediate redshifts (see, e.g., [54]). Fur-ther study of the details of the redMaGiC samples iswarranted, especially if one wishes to use w ( θ ) at scalessmaller than those studied in Y1COSMO.We compare these bias constraints to those measuredfrom the galaxy-galaxy lensing probe of the same red-MaGiC sample, presented in Y1GGL. We parameter-ize the difference between the two measurements withthe cross-correlation coefficients r i , which are presentedin Figure 8. Beyond linear galaxy bias, r can deviatefrom 1 and acquire scale dependences, and it must beproperly modeled to constrain cosmology with combinedgalaxy clustering and galaxy-galaxy lensing (e.g. [55]).We constrain r i at fixed cosmology using the Y1COSMOcovariance, which includes the covariance between thetwo probes. All the nuisance parameters discussed inY1COSMO are varied for this constraint. With ourchoice of scale cuts, we see no evidence of tension be-tween the two bias measurements. This provides furtherjustification for fixing r = 1 in the Y1COSMO analysis. VII. DEMONSTRATION OF ROBUSTNESS
We apply a number of null tests to our weighted sam-ple to demonstrate its robustness. We do so by obtainingconstraints on the galaxy bias and Ω m . These parametersare sensitive to both multiplicative and additive shiftsin the amplitude of w ( θ ) and we therefore believe theyshould encapsulate any potential systematic bias thatcould affect the cosmological analysis of Y1COSMO. Wethus perform joint fits to the data in each redshift bin toobtain constraints on the five b i and Ω m . For these fits,we marginalize over an additive redshift bias uncertaintydescribed in Table II. All other cosmological parametersare fixed at the Y1COSMO cosmology and as such, thisshould not be interpreted as a measurement of Ω m to beused in further analyses. Results are obtained using theanalysis pipeline described in [20]. We describe how w ( θ )is altered to perform each test throughout the rest of thissection.4 . . . . b . . . . Ω m no weights χ (68) weights χ (68) weights .
50 1 .
65 1 .
80 1 . b no weights χ (68) weights χ (68) weights .
50 1 .
65 1 .
80 1 . b no weights χ (68) weights χ (68) weights .
80 1 .
95 2 .
10 2 . b . . . . Ω m no weights χ (68) weights χ (68) weights . . . . b no weights χ (68) weights χ (68) weights FIG. 9. Parameter constraints showing the impact of the SP weights, varying Ω m , 5 linear bias parameters b i , and 5 nuisanceparameters ∆ z i . Contours are drawn at 68% and 95% confidence level. These constraints use the same ∆ z i priors as Y1COSMO.The blue contour shows the constraints on w ( θ ) calculated with no SP weights. The gray and red contours use SP weightsremoving all 2∆ χ (68) and 3∆ χ (68) correlations respectively. In this parameter space, ignoring the correlations with surveyproperties would have significantly biased the constraints from w ( θ ). As expected, the best fit when using the 2∆ χ (68)weights is at smaller values of b i than the 3∆ χ (68) weights, although the difference is not significant compared to the size ofthe contour. z range χ N data prob0 . < z < . . . . < z < .
45 6 . . < z < . . . . < z < .
75 11 . . . < z < . . . w ( θ ) all bins 67 . . χ , and probability of obtaining a χ exceed-ing this values for each redshift bin and for all bins combined.For the combined χ , the number of free parameters is 10 (5 b i and 5 ∆ z i ). The individual z bin χ values are calculatedusing the best fit to all z bins combined. The covariance be-tween between z bins is sufficiently small that we can treatthese as independent. We have therefore considered each in-dividual bin to have 2 free parameters. It is expected thatmeasuring the bias in each bin separately would have resultedin a smaller χ . A. Selection of threshold
We test two thresholds used to determine when to ap-ply weights based on a given SP map: 3∆ χ (68) and amore restrictive (i.e., more maps weighted for) 2∆ χ (68).After reaching a certain threshold, we expect that theonly effect from adding extra weights would be to biasthe measurements (from over-correction) and add greateruncertainty. We test for those effects in the following sub-sections. Here, in order to demonstrate that our resultsare insensitive to the choice in threshold, the change inthe measured b i and Ω m must be negligible compared toits uncertainty.Figure 9 shows the difference between the 3∆ χ (68)and 2∆ χ (68) SP weights. Because the weights correc-tion can only decrease the w ( θ ) signal, applying a stricterthreshold significance is expected to move the contourstowards smaller values of b i . Figure 9 shows that thisimpact is very small compared to the overall Y1 un-certainty and we can conclude that the choice between3∆ χ (68) and 2∆ χ (68) weights will have negligible im-pact on the Y1COSMO parameter constraints (The final5 .
20 1 .
35 1 .
50 1 . b . . . . Ω m fiducialfiducial + w est fiducial + w false . . . . b fiducialfiducial + w est fiducial + w false . . . . b fiducialfiducial + w est fiducial + w false . . . . b . . . . Ω m fiducialfiducial + w est fiducial + w false .
80 1 .
95 2 .
10 2 . b fiducialfiducial + w est fiducial + w false FIG. 10. Parameter constraints showing the impact of the estimator bias, w est and false correction bias w false . The fiducialdata vector and was calculated using the 2∆ χ (68) weights on the data. The w est and w false were measured on Gaussian mocksurveys using a 2∆ χ (68) threshold significance. We see no evidence for significant bias in the b i , Ω m plane. These constraintsuse the same ∆ z i priors as Y1COSMO. weights used in Y1COSMO are the 2∆ χ (68) weights).Figure 9 also shows the impact of not including SPweights on the parameter constraints. Ignoring the SPcorrelations would have resulted in significantly biasedconstraints on b i and Ω m . In every redshift bin, the shiftis greater than 2 σ along the major axis of the ellipses. B. Estimator bias
We also test for potential bias in w ( θ ) induced by over-correcting with the weights method and from correlationsbetween the SP maps. This was done using the Gaus-sian mocks described in Section IV B using the followingmethod. After the galaxy over-density field has beengenerated in each realization, we insert the systematiccorrelation using F sys ( s ) and the best-fit parameters foreach of the systematics in Table III at 2∆ χ (68) signif-icance. This is equivalent to dividing each mock galaxymap by a map of the SP weights.We then produced a galaxy number count as before,also adding shot noise. We fit the parameters of F sys ( s )to each realization and apply weights to the maps usingthe same method that is applied to the data. We measure w ( θ ) using the pixel estimator in Eqn. 5 on mocks with systematic contamination and correction, w weights , andon mocks with no systematics added, w no sys . We definethe bias in w ( θ ) to be, w est bias = 1 N N (cid:88) i =1 w no sys ,i − N (cid:88) j =1 w weights ,j (12)where N is the total number of realizations. We then add w est bias to the measured w ( θ ) and measure b i and Ω m .This is designed to test for any bias in w ( θ ) induced bythe the estimator when using weights.This result can be seen in Figure 10 where it showsnegligible impact on the parameter constraints. C. False correlations
Given the large number of SP maps being used in thesystematics tests, it is possible that chance correlationswill appear significant and weights will be applied whereno contamination has occurred, biasing the measured sig-nal. To test this, we use the same Gaussian mocks as inSection VII B with no added systematic contaminations.6 . . . . . b . . . . Ω m cov: no syscov: χ (68) sys . . . . b cov: no syscov: χ (68) sys . . . . b cov: no syscov: χ (68) sys . . . . b . . . . Ω m cov: no syscov: χ (68) sys . . . . . b cov: no syscov: χ (68) sys FIG. 11. Parameter constraints showing the impact of the systematics correction on the covariance. Both contours use thefiducial 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∆ χ (68) contaminations added(labeled ”cov: 2 σ sys”). These constraints use the same ∆ z i priors as Y1COSMO. We measure the correlation of each mock with each of the21 SP maps in Section V A, identifying any correlationsabove a 2∆ χ (68) threshold significance.The false correction bias w false bias , is then defined asthe average difference between the w ( θ ) measured withno corrections, and the w ( θ ) measured correcting forall correlations above the threshold using the weightsmethod. We then add w false bias to the measured w ( θ )and test the impact on b i and Ω m constraints.This test is designed to test for any bias in w ( θ ) in-duced by falsely correcting for SP maps that were onlycorrelated with the galaxy density by chance.This result is shown in Figure 10 where w false bias forthe 2∆ χ (68) SP maps has been used. This shows a neg-ligible impact on the constraints. The w false bias for the3∆ χ (68) SP maps is not shown as it has an even smallerimpact. This demonstrates that selecting a 2∆ χ (68)threshold does not induce a bias in the inferred bias pa-rameters for the set of SP maps used in this analysis. D. Impact on covariance
Correcting for multiple systematic correlations can al-ter the covariance of the w ( θ ) measurement in various ways. We expect that scatter in the best fit parametersshould increase the variance, while the removal of someclustering modes should decrease it. We test the signif-icance of any changes to the amplitude and structure ofthe covariance matrix using the Gaussian mocks.For this test we use the same mocks as in Section VII Bwhich are ‘contaminated’ with the same systematic cor-relations found in the data. We fit the F sys ( s ) functionto each mock and correct using weights. We then mea-sure the correlation function w weights and calculate thecovariance matrix of this measurement. We also mea-sure the correlation on mocks with no systematics added, w no sys , and calculate the covariance matrix from eachmeasurement. We calculate the galaxy bias b i and Ω m constraints for each covariance matrix and test if the re-sulting contours are significantly different. This test de-termines whether this additional uncertainty needs to beconsidered in the Y1COSMO analysis by marginalizingover the fitted parameters.The results of this test are shown in Figure 11. Weshow that for the SP maps selected in this analysis, theimpact on the size of the contours is negligible. We havetherefore not included any additional parameters in theMCMC analysis to account for the uncertainty in thecorrection.7 VIII. CONCLUSIONS
We have presented the 2-point angular galaxy correla-tion functions, w ( θ ), for a sample of luminous red galaxiesin DES Y1 data, selected by the redMaGiC algorithm.This yielded a sample with small redshift uncertainty, awide redshift range, and wide angular area. We split thissample into five redshift bins and analyzed its clustering.Our findings can be summarized as follows: • We find that multiple systematic dependencies be-tween redMaGiC galaxy density and survey propertiesmust be corrected for in order to obtain unbiased clus-tering measurements. We correct for these dependenciesby adding weights to the galaxies, following [47, 48]. • We demonstrate both that our methods sufficientlyremove systematic contamination (no significant dif-ferences are found between applying a 2∆ χ (68) and3∆ χ (68) threshold; see Fig. 9) and that any bias re-sulting from our method removing true clustering modesis insignificant (see Fig. 10). We further demonstratethat our weighting method imparts negligible changes tothe covariance matrix (see Fig. 11). • We find the redshift and luminosity dependence ofthe bias of redMaGiC galaxies to be broadly consistentwith expectations for red galaxies. • We find that the large-scale galaxy bias is consistentwith that determined by the Y1GGL galaxy-galaxy lens-ing measurements. This is consistent with r = 1 at linearscales, in agreement with basic galaxy formation theory,and a key assumption in the Y1COSMO analysis. (SeeFig. 8.) • Our results give an unbiased w ( θ ) data vector to beprovided to the Y1COSMO analysis, and other DES year1 combined proebes analyses.The methods we have presented, both correcting forsystematic dependencies and ensuring the robustness ofthese corrections, can be used as a guide for future analy-ses. Possible improvements to the work include incorpo-rating image simulations [56] and using mode projectiontechniques [16].Our galaxy bias results can be extended to study lumi-nosity dependence within redshift bins and to use smallerscale clustering in order to determine the HOD of red-MaGiC galaxies. Already, our bias measurements can beused to inform simulations (e.g., for the support of DESY3 analyses) and additional HOD information would beof further benefit.Finally, the results presented here have been opti-mized for combination with other cosmological probesin Y1COSMO and our work has ensured the galaxy clus-tering measurements do not bias the Y1COSMO results.The analysis followed a strict blinding procedure and hasyielded cosmological constraints when combined with theother 2-point functions. ACKNOWLEDGEMENTS
Figures 9 to 11 in this paper were produced with chainconsumer [57].Funding for the DES Projects has been provided bythe U.S. Department of Energy, the U.S. National Sci-ence Foundation, the Ministry of Science and Educationof Spain, the Science and Technology Facilities Coun-cil of the United Kingdom, the Higher Education Fund-ing Council for England, the National Center for Super-computing Applications at the University of Illinois atUrbana-Champaign, the Kavli Institute of CosmologicalPhysics at the University of Chicago, the Center for Cos-mology and Astro-Particle Physics at the Ohio State Uni-versity, the Mitchell Institute for Fundamental Physicsand Astronomy at Texas A&M University, Financiadorade Estudos e Projetos, Funda¸c˜ao Carlos Chagas Filhode Amparo `a Pesquisa do Estado do Rio de Janeiro,Conselho Nacional de Desenvolvimento Cient´ıfico e Tec-nol´ogico and the Minist´erio da Ciˆencia, Tecnologia e In-ova¸c˜ao, the Deutsche Forschungsgemeinschaft and theCollaborating Institutions in the Dark Energy Survey.The Collaborating Institutions are Argonne NationalLaboratory, the University of California at Santa Cruz,the University of Cambridge, Centro de InvestigacionesEnerg´eticas, Medioambientales y Tecnol´ogicas-Madrid,the University of Chicago, University College London,the DES-Brazil Consortium, the University of Edin-burgh, the Eidgen¨ossische Technische Hochschule (ETH)Z¨urich, Fermi National Accelerator Laboratory, the Uni-versity of Illinois at Urbana-Champaign, the Institut deCi`encies de l’Espai (IEEC/CSIC), the Institut de F´ısicad’Altes Energies, Lawrence Berkeley National Labora-tory, the Ludwig-Maximilians Universit¨at M¨unchen andthe associated Excellence Cluster Universe, the Univer-sity of Michigan, the National Optical Astronomy Ob-servatory, the University of Nottingham, The Ohio StateUniversity, the University of Pennsylvania, the Univer-sity of Portsmouth, SLAC National Accelerator Labora-tory, Stanford University, the University of Sussex, TexasA&M University, and the OzDES Membership Consor-tium.Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Ob-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 bythe National Science Foundation under Grant Num-bers AST-1138766 and AST-1536171. The DES partic-ipants from Spanish institutions are partially supportedby MINECO under grants AYA2015-71825, ESP2015-88861, FPA2015-68048, SEV-2012-0234, SEV-2016-0597,and MDM-2015-0509, some of which include ERDF fundsfrom the European Union. IFAE is partially funded bythe CERCA program of the Generalitat de Catalunya.Research leading to these results has received fundingfrom the European Research Council under the Euro-8pean Union’s Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329,and 306478. We acknowledge support from the Aus-tralian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project numberCE110001020.This manuscript has been authored by Fermi ResearchAlliance, LLC under Contract No. DE-AC02-07CH11359with the U.S. Department of Energy, Office of Science,Office of High Energy Physics. The United States Gov-ernment retains and the publisher, by accepting the arti-cle for publication, acknowledges that the United StatesGovernment retains a non-exclusive, paid-up, irrevoca- ble, world-wide license to publish or reproduce the pub-lished form of this manuscript, or allow others to do so,for United States Government purposes.MC has been funded by AYA2013-44327-P, AYA2015-71825-P and acknowledges support from the Ramon yCajal MICINN program. ER acknowledges supportby the DOE Early Career Program, DOE grant de-sc0015975, and the Sloan Foundation, grant FG-2016-6443. NB acknowledges the use of University of Flori-das supercomputer HiPerGator 2.0 as well as thanks theUniversity of Floridas Research Computing staff. JBacknowledges support from the Swiss National ScienceFoundation. [1] S. D. M. White and M. J. Rees, MNRAS , 341 (1978).[2] V. Desjacques, D. Jeong, and F. Schmidt, ArXiv e-prints(2016), arXiv:1611.09787.[3] J. A. Tyson, F. Valdes, J. F. Jarvis, and A. P. Mills, Jr.,ApJ , L59 (1984).[4] R. Mandelbaum et al. , MNRAS , 1544 (2013),arXiv:1207.1120.[5] J. Kwan et al. (DES Collaboration), MNRAS , 4045(2017), arXiv:1604.07871.[6] W. Hu and B. Jain, Phys. Rev.
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In this appendix we present some examples of the sur-vey property maps used throughout the analysis. Thesecan be seen in Fig A.1.In Fig A.2 we show the correlations between the galaxydensity and all the SP maps listed in Table III.
Appendix B: Cross correlations
In this appendix we present the galaxy clustering sig-nal between redshift bins, Figure B.1. For these cross-correlations, we use a covariance matrix calculated fromlog-normal simulations described in [20]; the square rootof the diagonal of this covariance matrix yields the error-bars shown in the figure. These are the same simulationsused to validate the Y1COSMO covariance matrix.We overplot the cross-correlation prediction both fromthe best fit bias values from the auto-correlations, and thebest fit cosmology and bias from Y1COSMO. The cross-correlation measurements were not used in the combinedprobes analysis and so the robustness tests were not per-formed on these measurements. We present these resultsto demonstrate that there is a clustering signal in adja-cent redshift bins (2,1), (3,2), (4,3), and (5,4) and not asa robustness test, hence we do not include a goodness-of-fit for this measurement. The amplitude of this signal isdetermined by the overlap in the n ( z ) between redshiftbins (see Figure 2). These correlations could be used infuture analyses to constrain the redshift bias parameters∆ z i .0 F W H M i b a n d [ a r c s e c ] M a g li m i b a n d A i r m a ss i b a n d S k y b r i g h t i b a n d E x p o s u r e t i m e i b a n d [ s ] S t a r s [ a r c m i n ] FIG. A.1. Maps of potential sources of systematics. Shown here for i -band only. Maps in other bands show fluctuations onsimilar scales. Each SP map is shown at N side = 1024. The stellar density map is shown at N side = 512.
100 200 300 400
Exptime i [s] N ga l / › N ga l fi FWHM z [pixels] FWHM r [pixels] airmass z mag lim g -band N ga l / › N ga l fi FWHM z [pixels] N ga l / › N ga l fi
100 200 300 400 500 600
Exptime g [s] FWHM r [pixels] skybright z [ADU] mag lim i -band FWHM gri
PCA − N ga l / › N ga l fi
250 300 350 400 450 skybright r [ADU] FWHM z [pixels]
100 200 300 400
Exptime i [s]
100 200 300 400 500 600
Exptime z [s] airmass i N ga l / › N ga l fi FWHM r [pixels] FWHM g [pixels] FIG. A.2. Galaxy number density as a function of different SP maps. We show here only the correlations with SP maps usedin the 2∆ χ (68) weights calculation. The cyan line is the correlation of the sample without weights. The black points showthe correlation after correction with the 2∆ χ (68) weights. The error bars were calculated by measuring the same correlationon the Gaussian mock surveys described in Section IV B. The significance of these correlations are shown in Figure 6.
024 2,1024 3,1 3,2024 4,1 4,2 4,310
024 5,1 10 w () FIG. B.1. The two-point cross correlations between redshift bins. These measurements are expected to be non-zero, with asignificance related to the degree of overlap in the n ( z ) displayed in Fig. 2. The numbers in each panel correspond to theredshift bins used in the cross-correlation, Adjacent bins are shown on the diagonal. The error-bars were calculated using thelog-normal mock surveys used for Y1COSMO covariance validation [20]. The solid red curve is the best-fit model from theauto-correlation using only w ( θθ