First Cosmology Results Using Type Ia Supernovae From the Dark Energy Survey: Analysis, Systematic Uncertainties, and Validation
D. Brout, D. Scolnic, R. Kessler, C. B. D'Andrea, T. M. Davis, R. R. Gupta, S. R. Hinton, A. G. Kim, J. Lasker, C. Lidman, E. Macaulay, A. Möller, R. C. Nichol, M. Sako, M. Smith, M. Sullivan, B. Zhang, P. Andersen, J. Asorey, A. Avelino, B. A. Bassett, P. Brown, J. Calcino, D. Carollo, P. Challis, M. Childress, A. Clocchiatti, A. V. Filippenko, R. J. Foley, L. Galbany, K. Glazebrook, J. K. Hoormann, E. Kasai, R. P. Kirshner, K. Kuehn, S. Kuhlmann, G. F. Lewis, K. S. Mandel, M. March, V. Miranda, E. Morganson, D. Muthukrishna, P. Nugent, A. Palmese, Y.-C. Pan, R. Sharp, N. E. Sommer, E. Swann, R. C. Thomas, B. E. Tucker, S. A. Uddin, W. Wester, T. M. C. Abbott, S. Allam, J. Annis, S. Avila, K. Bechtol, G. M. Bernstein, E. Bertin, D. Brooks, D. L. Burke, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, F. J. Castander, C. E. Cunha, L. N. da Costa, C. Davis, J. De Vicente, D. L. DePoy, S. Desai, H. T. Diehl, P. Doel, A. Drlica-Wagner, T. F. Eifler, J. Estrada, E. Fernandez, B. Flaugher, P. Fosalba, J. Frieman, J. García-Bellido, D. Gruen, R. A. Gruendl, G. Gutierrez, W. G. Hartley, D. L. Hollowood, K. Honscheid, B. Hoyle, D. J. James, M. Jarvis, T. Jeltema, E. Krause, O. Lahav, T. S. Li, M. Lima, M. A. G. Maia, J. Marriner, J. L. Marshall, P. Martini, F. Menanteau, et al. (25 additional authors not shown)
FFERMILAB-PUB-18-541-AEDES-2018-0356
Draft version June 4, 2019
Preprint typeset using L A TEX style emulateapj v. 12/16/11
FIRST COSMOLOGY RESULTS USING TYPE IA SUPERNOVAE FROM THE DARK ENERGY SURVEY:ANALYSIS, SYSTEMATIC UNCERTAINTIES, AND VALIDATION
D. Brout , D. Scolnic , R. Kessler , C. B. D’Andrea , T. M. Davis , R. R. Gupta , S. R. Hinton , A. G. Kim ,J. Lasker , C. Lidman , E. Macaulay , A. M¨oller , R. C. Nichol , M. Sako , M. Smith , M. Sullivan ,B. Zhang , P. Andersen , J. Asorey , A. Avelino , B. A. Bassett , P. Brown , J. Calcino , D. Carollo ,P. Challis , M. Childress , A. Clocchiatti , A. V. Filippenko , R. J. Foley , L. Galbany ,K. Glazebrook , J. K. Hoormann , E. Kasai , R. P. Kirshner , K. Kuehn , S. Kuhlmann , G. F. Lewis ,K. S. Mandel , M. March , V. Miranda , E. Morganson , D. Muthukrishna , P. Nugent , A. Palmese ,Y.-C. Pan , R. Sharp , N. E. Sommer , E. Swann , R. C. Thomas , B. E. Tucker , S. A. Uddin , W. Wester ,T. M. C. Abbott , S. Allam , J. Annis , S. Avila , K. Bechtol , G. M. Bernstein , E. Bertin , D. Brooks ,D. L. Burke , A. Carnero Rosell , M. Carrasco Kind , J. Carretero , F. J. Castander ,C. E. Cunha , L. N. da Costa , C. Davis , J. De Vicente , D. L. DePoy , S. Desai , H. T. Diehl ,P. Doel , A. Drlica-Wagner , T. F. Eifler , J. Estrada , E. Fernandez , B. Flaugher , P. Fosalba ,J. Frieman , J. Garc´ıa-Bellido , D. Gruen , R. A. Gruendl , G. Gutierrez , W. G. Hartley ,D. L. Hollowood , K. Honscheid , B. Hoyle , D. J. James , M. Jarvis , T. Jeltema , E. Krause ,O. Lahav , T. S. Li , M. Lima , M. A. G. Maia , J. Marriner , J. L. Marshall , P. Martini ,F. Menanteau , C. J. Miller , R. Miquel , R. L. C. Ogando , A. A. Plazas , A. K. Romer ,A. Roodman , E. S. Rykoff , E. Sanchez , B. Santiago , V. Scarpine , M. Schubnell , S. Serrano ,I. Sevilla-Noarbe , R. C. Smith , M. Soares-Santos , F. Sobreira , E. Suchyta , M. E. C. Swanson ,G. Tarle , D. Thomas , M. A. Troxel , D. L. Tucker , V. Vikram , A. R. Walker , and Y. Zhang (DES Collaboration) Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK ARC Centre of Excellence for All-sky Astrophysics (CAASTRO) School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK University of Copenhagen, Dark Cosmology Centre, Juliane Maries Vej 30, 2100 Copenhagen O Korea Astronomy and Space Science Institute, Yuseong-gu, Daejeon, 305-348, Korea Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, 7945, South Africa South African Astronomical Observatory, P.O.Box 9, Observatory 7935, South Africa George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics andAstronomy, Texas A&M University, College Station, TX 77843, USA INAF, Astrophysical Observatory of Turin, I-10025 Pino Torinese, Italy Millennium Institute of Astrophysics and Department of Physics and Astronomy, Universidad Cat´olica de Chile, Santiago, Chile Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Miller Senior Fellow, Miller Institute for Basic Research in Science, University of California, Berkeley, CA 94720, USA Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA PITT PACC, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Victoria 3122, Australia Department of Physics, University of Namibia, 340 Mandume Ndemufayo Avenue, Pionierspark, Windhoek, Namibia Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138,USA Gordon and Betty Moore Foundation, 1661 Page Mill Road, Palo Alto, CA 94304,USA Australian Astronomical Optics, Macquarie University, 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 Institute of Astronomy and Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK Department of Astronomy/Steward Observatory, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA Division of Theoretical Astronomy, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile LSST, 933 North Cherry Avenue, Tucson, AZ 85721, 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 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas (CIEMAT), Madrid, Spain Laborat´orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA Institut de F´ısica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra(Barcelona) Spain a r X i v : . [ a s t r o - ph . C O ] J un Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA Department of Physics, The Ohio State University, Columbus, OH 43210, USA Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany Universit¨ats-Sternwarte, Fakult¨at f¨ur Physik, Ludwig-Maximilians Universit¨at M¨unchen, Scheinerstr. 1, 81679 M¨unchen, Germany Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA Departamento de F´ısica Matem´atica, Instituto de F´ısica, Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo, SP, 05314-970, Brazil Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA Instituci´o Catalana de Recerca i Estudis Avan¸cats, E-08010 Barcelona, Spain Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK Instituto de F´ısica, UFRGS, Caixa Postal 15051, Porto Alegre, RS - 91501-970, Brazil Brandeis University, Physics Department, 415 South Street, Waltham MA 02453 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 (Received November 9, 2018; Accepted February 20, 2019)
Draft version June 4, 2019
ABSTRACTWe present the analysis underpinning the measurement of cosmological parameters from 207 spec-troscopically classified type Ia supernovae (SNe Ia) from the first three years of the Dark EnergySurvey Supernova Program (DES-SN), spanning a redshift range of 0 . < z < . z < .
1) SNe Ia, resultingin a “DES-SN3YR” sample of 329 SNe Ia. Our cosmological analyses are blinded: after combiningour DES-SN3YR distances with constraints from the Cosmic Microwave Background (CMB; PlanckCollaboration 2016), our uncertainties in the measurement of the dark energy equation-of-state param-eter, w , are 0.042 (stat) and 0.059 (stat+syst) at 68% confidence. We provide a detailed systematicuncertainty budget, which has nearly equal contributions from photometric calibration, astrophysicalbias corrections, and instrumental bias corrections. We also include several new sources of systematicuncertainty. While our sample is < / w areonly larger by 1 . × , showing the impact of the DES SN Ia light curve quality. We find that thetraditional stretch and color standardization parameters of the DES SNe Ia are in agreement withearlier SN Ia samples such as Pan-STARRS1 and the Supernova Legacy Survey. However, we findsmaller intrinsic scatter about the Hubble diagram (0.077 mag). Interestingly, we find no evidencefor a Hubble residual step (0 . ± .
018 mag) as a function of host galaxy mass for the DES subset,in 2.4 σ tension with previous measurements. We also present novel validation methods of our sampleusing simulated SNe Ia inserted in DECam images and using large catalog-level simulations to testfor biases in our analysis pipelines. Subject headings:
DES INTRODUCTION
The discovery of the accelerating expansion of the uni-verse (Riess et al. 1998; Perlmutter et al. 1999) has moti-vated an era of cosmology surveys with the goal of mea-suring the mysterious properties of dark energy. The useof standardizable type Ia supernovae (SNe Ia) to measuredistances has proven to be a vital tool in constraining thenature of dark energy because they probe the geometryof the universe throughout a large portion of cosmic time.The Dark Energy Survey Supernova Program (here-after DES-SN) has found thousands of photometricallyclassified SNe Ia at redshifts from 0 . < z < . (Bernstein et al.2012). Over the full five years of the survey, DES-SNis expected to obtain the largest single dataset of photo-metrically classified SNe Ia to date. DES-SN has spectro-scopically confirmed a subset of ∼
500 SNe Ia at redshiftsfrom 0 . < z < . z SNe Iafrom CfA3, CfA4, and CSP-1 is hereafter denoted ‘thelow- z subset’ (CfA3-4; Hicken et al. 009a; Hicken et al.2012; CSP-1, Contreras et al. 2010).Over the past two decades, there have been three par-allel and overlapping major developments in using SNe Iato measure cosmological parameters, upon which theDES-SN has made improvements. The first developmentis the order-of-magnitude growth in the number of spec-troscopically confirmed SNe Ia. Original datasets at low-redshift had tens of SNe Ia (e.g., CfA1-CfA2, Riess et al.1999; Jha et al. 2006) and the next generation of low-redshift and high-redshift datasets had hundreds of SNe-Ia (e.g. CfA3-4; CSP-1; ESSENCE: Narayan et al. 2016;SDSS-II: Frieman et al. 2008, Sako et al. 2018); SNLS:Guy et al. 2010; PS1: Rest et al. 2014, Scolnic et al. rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 32017). Today, with the addition of DES-SN, there arenow more than 1500 spectroscopically confirmed SNe Iain total.The second development has been in detector sensi-tivity, which has resulted in improved light curve qual-ity and distance measurement uncertainties. The 570megapixel Dark Energy Camera (DECam; Flaugheret al. 2015), with its fully-depleted CCDs and excellent z -band response, facilitates well-measured optical lightcurves at high redshift (Diehl et al. 2014).The third major development has been the increas-ingly sophisticated analyses of the samples. As SN Iadatasets grow in size, analyses are better able to charac-terize SN Ia populations and expected biases from obser-vational selection and analysis requirements. Improve-ments in the analysis over the last decade have includedscene modeling photometry ( SMP
Holtzman et al. 2008,Astier et al. 2013, B18-SMP: Brout et al. 2019-SMP)instead of classical template subtraction, the modelingand correction of expected biases using large simulations(Perrett et al. 2010, Kessler et al. 2009a, Betoule et al.2014a), and measuring filter transmissions to achieve sub1% calibration uncertainty (Astier et al. 2006, Doi et al.2010, Tonry et al. 2012, Marshall et al. 2013, Burkeet al. 2018). Recent cosmological parameter analyses(B14: Betoule et al. 2014b, S18: Scolnic et al. 2018) havefound that systematic uncertainties are roughly equal tothe statistical uncertainties; this is due to the improv-ing ability to understand and reduce systematic uncer-tainties with larger samples and reduced statistical un-certainties. Each new cosmology analysis (Wood-Vaseyet al. 2007, Kessler et al. 2009a, Sullivan et al. 2011,B14, S14: Scolnic et al. 2014, S18, Jones et al. 2018) hasbuilt on previous analyses in their treatment of system-atic uncertainties. Here we continue in this tradition ofimprovements, and also study several previously unin-vestigated sources of uncertainty.Improvement in understanding of systematic uncer-tainties is crucial to taking advantage of the order-of-magnitude increases in statistics expected in the com-ing years. From DES-SN alone, there is the full sampleof ∼ Fig. 1.—
Analysis flowchart of this paper. Nuisance param-eters, the systematic error budget, and the results of validationare considered the “Results” of this work (Section 5) and the un-blinded cosmological parameter best fit values are presented in Ab-bott et al. (2019). dardization of SNe Ia. Historically α and β , the cor-relation coefficients for stretch and color of supernovalight curves respectively, have been used to standardizeSN Ia luminosities, and σ int has been used to character-ize the scatter in SN Ia luminosities that is not coveredby the measurement uncertainties. Additionally, severalgroups in the last decade have shown that more massivegalaxies tend to host overluminous SNe Ia after color andstretch brightness standardization, suggesting improvedstandardizability of the SNe Ia population (Kelly et al.2010; Sullivan et al. 2010; Lampeitl et al. 2010). Thiseffect has been characterized as a step function in Hub-ble diagram residuals ( γ ) across 10 M (cid:12) . However, thesize of this effect has been seen to vary in different sam-ples and the physical interpretation is not understood.In this paper we discuss our own findings for these nui-sance parameters using DES-SN3YR. The second mainresult is the statistical and systematic uncertainty bud-get from our w CDM cosmological analysis after combin-ing with Planck Collaboration (2016) CMB priors. Us-ing the analysis and results derived here, cosmologicalparameter constraints are shown in Abbott et al. (2019).In order to improve upon the treatment and validation
Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validationof systematic uncertainties from past analyses, we usetwo types of SN Ia simulations to examine biases in ourpipelines and to provide crosschecks of our analysis. Thefirst set of simulations includes hundreds of catalog-levelsimulations with input sources of systematic uncertainty.We analyze the catalog level simulations with steps 3-7above to verify our analysis pipeline and reported statis-tical and systematic uncertainties. These simulations aregenerated by the SuperNova ANAlysis software package ( SNANA : Kessler et al. 2009b), which has been used exten-sively by previous analyses to quantify expected biasesand offers the capability of parallelization for generatingand analyzing large simulations of SNe Ia.For the second set of validation simulations, we gener-ate 100,000 artificial supernova light curves which areinserted as point sources onto DECam images (here-after ‘fakes’). Previous analyses have used artificial pointsources to understand photometric uncertainties (Holtz-man et al. 2008; Perrett et al. 2010). In DES-SN, fake su-pernovae light curves are used for several reasons. Fakesare used to check for biases in photometry (B18-SMP)and in the determination of SN Ia detection efficiency asa function of signal-to-noise (S/N) (Kessler et al. 2018in prep.), thereby modeling subtle pipeline features thatcannot be computed from first principles. Additionally,we present a cosmological analysis of 10,000 fake super-novae that have been recovered by the search pipeline,processed by the photometric pipeline, and processedthrough our cosmological analysis pipeline in the samemanner as the real dataset. This crosscheck is sensitiveto potential un-modeled biases in the image-processingpipelines and their propagation to cosmological distanceand cosmological parameter biases.Unfortunately, neither of the methods above addressthe systematic uncertainty due to calibration. To ad-dress calibration uncertainties, we compare our absolutecalibration with that of the Pan-STARRS survey (Tonryet al. 2012) and SuperCal (Scolnic et al. 2015).The organization of this paper is depicted in Figure 1and is described as follows. In §
2, we introduce thedata samples, a combination of high-redshift SNe Ia fromDES-SN and low-redshift SNe Ia from CfA and CSP-1.In §
3, we discuss analysis procedures and characterizesystematic uncertainties. In §
4, we quantify each sourceof systematic uncertainty. In § § § § § DATASETS
The Dark Energy Survey Supernova Program
DES-SN performed a deep, time-domain survey infour optical bands ( g, r, i, z ) covering ∼
27 deg over5 seasons (2013-2018) using the DECam mounted onthe 4-m Blanco telescope at the Cerro Tololo Inter-American Observatory (CTIO). Exposure processing(Morganson et al. 2018), difference imaging ( DiffImg :Kessler et al. 2015), and automated rejection of sub-traction artifacts (Goldstein et al. 2015) are run on a https://snana.uchicago.edu Fig. 2.—
Histogram of the 251 spectroscopically confirmed SNe Iais shown in green-filled. The sub-sample of SNe Ia used for cos-mological parameter analysis that pass all quality cuts is shown inblack. nightly basis. DES-SN observed in 8 “shallow” fields(C1,C2,X1,X2,E1,E2,S1,S2) with single-epoch 50% com-pleteness depth of ∼ ∼ ∼ ∼ . < z < . . , . , .
85] respectively.Additional data are acquired using an in-situ calibra-tion process called “DECal” (Marshall et al. 2013). TheBlanco/DECam optical system and filter transmissionfunctions are measured under multi-wavelength illumi-nation. DES-SN also acquires real-time meteorologicaldata using the SUOMINET system to track precipitablewater vapor levels and auxiliary “aTmCAM” instrumen- rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 5tation (Li et al. 2014) to measure atmospheric conditions.
External Low-Redshift Samples
Cosmological constraints from SNe Ia are best ob-tained with samples at both low-redshift and high-redshift. We utilize four publicly available low-redshiftsurveys: CfA3S, CfA3K, CfA4, and CSP-1 (Jha et al.2006; Hicken et al. 009a; Hicken et al. 2012; Contreraset al. 2010) consisting of 303 spectroscopically confirmedSNe Ia in the redshift range 0.01 < z < ANALYSIS
Here we describe the analysis procedures used to mea-sure cosmological parameters. The majority of this sec-tion describes the analysis of the DES subset itself,though we also include our analysis of the low-redshiftsample. The description of systematic uncertainties as-sociated with each step in the analysis is laid out inthis section and each source of systematic uncertaintyis quantified in Section 4. We rely on complementarywork in Kessler et al. (2019), hereafter K18, which de-tails the simulations of DES-SN3YR. These simulationsare used for computing bias corrections in Section 3.7.
Calibration
SN Ia cosmological constraints rely on the ability tointernally transform each SN flux measurement in ADU(Analog/Digital Units) into a ‘top-of-the-galaxy’ bright-ness. This is done in two steps, first via measurementsof Hubble Space Telescope (HST) CalSpec standardstars to obtain a top-of-the-atmosphere brightness, whichis discussed here. Second, we obtain top-of-the-galaxybrightness by accounting for the Milky Way extinctionalong the line of sight, values for which are obtained fromSchlegel et al. (1998) & Schlafly & Finkbeiner (2011a).Measurements of cosmological parameters using SNe Iaare sensitive to filter calibration uncertainties (internal)due to the fact that at higher redshift, constraints of theSN light curve models rely on observed fluxes in a dif-ferent set of filters than at lower redshift. A dependencein SN cosmological distances as a function of redshiftcould arise from differences in the calibration betweenthe low- z and DES subsets (external). Below we discussthe steps taken to both internally and externally cali-brate the DES-SN measurements. Star Catalog
Here we describe the process of calibrating each of theDES-SN images. Photometry of approximately 50 ter-tiary standard stars are used to determine a zero point foreach DECam CCD image. The catalog of tens of thou-sands of tertiary star magnitudes is described in Burkeet al. (2018). These stars are internally calibrated us-ing a ‘Forward Global Calibration Method’ (FGCM) toan RMS of 6 mmag. FGCM models the rate of photonsdetected by the camera by utilizing measurements of in-strument transmission, atmospheric properties, a model Fig. 3.—
Blue: Distribution of observed g − i colors for the DES-SN sample observations. Epochs with S/N >
10 are shown. Black:Distribution of g − i colors for the tertiary standard stars usedfor internal calibration. The validity of chromatic corrections isevaluated over the stellar color range (black) but the correctionsare applied to the DES-SN fluxes (blue). of the atmosphere, and a model of the source. SpectralEnergy Distribution (SED)-dependent chromatic correc-tions are applied to the standard stars which extend the6 mmag calibration uncertainty to be valid over a verywide color range ( − < g − i < g − i color distri-bution of the tertiary standard stars is shown in Figure 3.The color distribution of the DES subset light curves isdifferent from that of the standard stars and is discussedin Section 3.2. AB offsets
The FGCM catalog is calibrated to the AB system(Oke & Gunn 1983) using measurements of the HST Cal-Spec standard C26202. As detailed in Burke et al. (2018),we compute synthetic magnitudes of C26202 by multi-plying the CalSpec spectrum with the standard instru-mental and atmospheric passbands used in the FGCMcalibration DECam filter transmission functions. Thesynthetic magnitudes are compared to the FGCM cat-alog magnitudes of C26202 for each passband, and themagnitude difference is applied to the FGCM catalog sothat the observed and synthetic magnitudes of the stan-dard are in perfect agreement. C26202 was chosen be-cause it is located in ‘C3’, which is one of the deep fieldsand has been observed over 100 times during the courseof the survey. C26202 is sufficiently faint to avoid satu-ration and is observed in a similar range of seeing con-ditions to that of the DES-SN dataset. Other CapSpecstandards in the DES footprint are either saturated, orwere observed with short exposures under twilight con-ditions. We do not find any dependence in the corrected,top-of-the-atmosphere, fluxes of C26202 on airmass, skybrightness, CCD number of the observation, or exposuretime.A secondary method of calibrating the FGCM catalogis to cross-calibrate with catalogs from other surveys thatare also tied to the AB system. Using tertiary standardstars in 8 of the 10 of the DES-SN fields (DES Fields:C1,C2,C3,S1,S2,X1,X2,X3) that overlap with the foot-print of other surveys, we measure the calibrated bright-ness differences for stars observed by both surveys, andcompare these differences to predictions using a spec- Y3A1 passbands from Burke et al. (2018).
Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
Fig. 4.—
The relative offsets in stellar magnitudes when comparing PS1, SDSS and SNLS to overlapping DES fields ( ¯∆ M cal ). Offsets arefurther broken down by field. In each panel, ¯∆ M cal for the HST Calspec magnitude of C26202 is defined to be zero. Each of the points aredetermined from a comparison of DECam and external survey photometry accounting for difference in filter transmission functions. SNLSand SDSS are shown for reference, however it is only PS1 that is used to determine the goodness of the calibration. The vertical red lineis the mean of the PS1-DES overlap (green points) shifted by the PS1 offset to SuperCal. The grey area represents the quadrature sum ofthe uniformity uncertainty and the SuperCal uncertainty in absolute calibration ( § tral library and known filter transmissions. We define∆ M cal as the offset between the predicted and observedbrightness differences for stars with the same color as theCalspec standard C26202. In Figure 4, we examine themean difference ( ¯∆ M cal ) for several groupings of over-lapping calibration stars. For comparison, we examinethe agreement between DES and PS1 (green), DES andSDSS (orange), and DES and SNLS (violet). We also de-fine PS1-SuperCal (red) as the agreement between DES and PS1, if the absolute calibration of PS1 were shiftedby the weighted average of differences between the PS1,SDSS and SNLS calibration (see Scolnic et al. 2015 forexplanation).Burke et al. (2018) apply FGCM to the DECam imagesand achieve a calibration uniformity across the sky of ∼ M cal , which is 4 . , . , . , . rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 7mmags in the g, r, i, z bands respectively. The observedconsistency between PS1 and DES is 2-4 mmag, whichshows that ∼ SN Photometry
The light curves used in this analysis are provided byB18-SMP, which measures SN brightnesses by adoptinga scene modeling approach. In
SMP , a variable transientflux and temporally constant host-galaxy are forwardmodeled simultaneously. B18-SMP test the accuracy ofthe
SMP pipeline by processing a sample of 10,000 realis-tic SNe Ia light curves that were injected as point sourcesonto DECam images (‘fake SNe’). Upon comparison ofinput and measured fake SNe fluxes, B18-SMP find thatbiases in the photometric pipeline are limited to 3 mmag(see Figure 3 of B18-SMP).Analyzing fakes near bright galaxies, B18-SMP alsofind that the photometric scatter increases with the localsurface brightness (denoted “the host SB dependence”).This increase is similar to what was observed in DiffImg(Kessler et al. 2015). The host SB dependence is ac-counted for by scaling our photometric uncertainties offake SNe near bright host-galaxies to match the observedscatter in
SMP flux residuals. This scaling is determinedas a function of host-galaxy surface brightness ( m SB ):ˆ S SMP ( m SB ) = RMS[ ( F true − F SMP ) / σ Ref ] fake (cid:104) σ SMP / σ
Ref (cid:105) fake (1)where RMS is the root-mean-square in a bin of m SB , σ SMP is the
SMP flux uncertainty, (cid:104)(cid:105) indicates an aver-age in the m SB bin, σ Ref is the calculated uncertaintybased on observing conditions (zero point, sky noise,PSF), F SMP is the fit flux from
SMP , and F true is theinput flux of the fake SN. The size of ˆ S SMP ( m SB ) can beseen in Figure 5 of B18-SMP and can be as large as 4 at m SB = 21. These corrections are applied directly to theDES-SN sample.After SMP , there is an additional set of SED-dependentchromatic corrections made to the DES SN Ia fluxes, sim-ilar to the corrections made to the stellar fluxes discussedin Section 3.1.1. The impact of these corrections is pre-sented in Lasker et al. (2019), and is discussed here inSection 4.1. One potential issue is the validity of thechromatic corrections applied to the SN fluxes whosecolor range ( − . < g − i < .
2) is redder than that of themajority of tertiary calibration stars (0 . < g − i < g − i < .
2, there is adrop-off in tertiary standard star counts as the star dis-tribution enters the realm of blue horizontal branch starsand white dwarfs. While we do not have the statisticsto validate the 6 mmag calibration uncertainty for thebluest stars ( − . < g − i < . g − i ∈ [0 ,
3] show nosignificant trends at the bluest colors and thus we haveconfidence in applying the corrections to the fraction of bluest SN Ia epochs in the color range g − i ∈ [ − , Redshifts
Redshifts for the DES subset are presented inD’Andrea et al. (2018). Redshifts of the low-redshiftsample are obtained from their respective surveys towhich we make peculiar velocity corrections. The correc-tions due to coherent flows of SN host galaxies has beenperformed in the same manner as S18. Peculiar velocitiesare calculated using the matter density field calibratedby the 2M++ catalog (Skrutskie et al. 2006) out to z ∼ β = 0.43 and a dipoleas described in Carrick et al. (2015). We adopt the errorin peculiar velocity correction of 250 km / s / Mpc moti-vated by dark matter simulations of Carrick et al. (2015)as well as from the comparison of low-redshift and inter-mediate redshift SNe scatter described in S18.The redshifts of host galaxies used in this analysisare typically reported with an accuracy of ∼ − forlow- z and to ∼ × − for intermediate-redshift. For71 SNe in the DES subset, a host-galaxy redshift wasnot obtained and redshifts were determined from the SNspectrum, resulting in redshift uncertainty ∼ × − .These redshift uncertainties propagate to SN scatter indistance. However, more important than the statisticaluncertainty is the possibility of a systematic shift in red-shift due to cosmological effects. A systematic shift couldbe caused, for example, by a gravitational redshift dueto the density of our local environment (Calcino & Davis2017). Wojtak et al. (2015) show the expected distribu-tions for typical environments in ΛCDM can be describedby a one sigma fluctuation from the mean potential witha shift of ∆ z ≈ × − . Light Curve Fits
In order to obtain distance moduli ( µ ) from SN Ia lightcurves, we fit the light curves with the SALT2 model(Guy et al. 2010) using the trained model parametersfrom B14 over an SED wavelength range of 200 − λ , sat-isfies 280 < ¯ λ/ (1 + z ) < SNANA implementation (Kessler et al. 2009c) based on
MINUIT (James & Roos 1975), and we use the
MINOS option forthe fitted parameter uncertainties. A discussion abouttechniques used to avoid pathological fits is described inAppendix A.Each light curve fit determines parameters color c ,stretch x , the overall amplitude x , with m B ≡− . ( x ), and time of peak brightness t in therest-frame B-band wavelength range. In addition, wecompute light curve fit probability P fit , which is theprobability of finding a light curve data − model χ aslarge or larger assuming Gaussian-distributed flux un-certainties. In Figure 5, three representative DES-SNlight curves are shown with overlaid light curve fits us-ing the SALT2 model. Normalized flux residuals to theSALT2 light curve model for the DES-SN3YR sampleare shown in Figure 6. Both the DES subset and low- z subset SALT2 model fluxes for all rest-frame passbandsare consistent to within < Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
Fig. 5.—
Representative light curves of the DES-SN3YR Spectro-scopic sample with photometric data determined with
SMP (points).SALT2 fits to the light curve, are overlaid (curves) and fitted colorand stretch values are shown. There is no g -band in the bot-tom panel because z=0.829 is beyond the range of the B14 g -bandmodel. Supernovae with C3 (or X3) in the name are found in deepfields, the remaining SNe are found in the shallow fields. Openpoints are excluded from the SALT2 fits. this is described in Section 4.1. All light curve fit param-eters for the DES-SN3YR sample are publicly availablein machine-readable format as described in Appendix Band in Table C.1. Selection Requirements
For this analysis, we require all SNe Ia to have ade-quate light curve coverage in order to reliably constrainlight curve fit parameters and we limit ourselves to amodel-training range of SN Ia properties that limit sys-tematic biases in the recovered distance modulus mea-surement. The sequential loss of SNe Ia from the sampledue to cuts is shown in Table 1. We start by requiring z > .
01 and our light curve fits to converge. We define T rest as the number of days since t in the rest frame ofthe SN. Dai & Wang (2016) showed that poorly sampledlight curves can result in large Hubble residual outlierseven though the fit χ shows no indication of a problem.Thus, we require an observation before peak brightness( T rest < T rest > > − < x < − . < c < . E ( B − V ) MW < .
25. The DES-SN Fields have
Fig. 6.—
Fractional flux residuals to the best fit SALT2 lightcurve model. Top: the DES-SN3YR Spectroscopic sample in thefour DES filter bands [ griz ]. Bottom: the low- z subset where pho-tometric observations have been grouped by filters with similarwavelength coverage [ BV gr ]. F DES and F Lowz are the SN fluxfrom the data, F SALT is the flux of the best-fit SALT2 model.The mean of each distribution is shown in solid curve and the un-certainty on the mean is shown as dashed curves. low MW extinction and thus the E ( B − V ) MW cut hasno effect.S18 placed a P fit > .
001 cut on the low-redshift sam-ple. While decreasing the fit probability cut to agree withPantheon gained us 20 SNe Ia, those additional SNe Iacome in a region of parameter space that is poorly mod-eled by our simulations (see P fit panel of Figure 7). Ad-ditionally, we find that applying a more conservative cutof P fit > .
01 to both the DES and low- z subsets re-sulted in similar statistical constraints on distance. Thedistribution of low- z sample light curve parameters afterquality cuts is shown in the bottom half of Figure 7.In the second to last row of Table 1 (‘Valid BiasCor’), afew SNe are lost due to their SN properties falling withina region of parameter space for which the simulation doesnot have a bias prediction. Bias corrections are discussedin detail in Section 3.8.1.Each SN cosmology analysis that has utilized the his-torical CfA and CSP-1 low- z samples has dealt withthe fact that their Hubble diagram residuals have non-Gaussian tails that are discarded from the cosmological rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 9
TABLE 1
DES-SN Low- z Total SNRequirement a b z > .
01 251 [0] 261 [72] 512 [72]Fit Convergence 244 [7] 257 [4] 501 [11]
S/N > T rest > T rest < E ( B − V ) MW < .
25 230 [0] 243 [5] 473 [5] − . < c < . − < x < σ x < P fit > .
01 208 [3] 127 [23] 335 [26]Valid BiasCor 207 [1] 125 [2] 332 [3]Chauvenets criterion 207 [0] 122 [3] 329 [3]Cosmo. Sample 207 122 329 a Discovered by
DiffImg and spectroscopically confirmed(D’Andrea et al. 2018). b CfA3, CfA4, and CSP-1 samples. fit. In the last row of Table 1 (‘Chauvenets criterion’),we place a final set of cuts before running cosmologicalparameter fits. This is the same cut on Hubble diagramresiduals that was made in S18 of 3 . σ . Host-galaxy Stellar Masses
Previous analyses of large SN Ia samples have founda correlation between standardized SN luminosities andhost-galaxy properties (Gallagher et al. 2008; Kelly et al.2010; Lampeitl et al. 2010; Sullivan et al. 2010, low- z :Childress et al. 2013 and Pan et al. 2014, JLA: B14,PS1: S18). Here we focus on the stellar mass ( M stellar )ratio of the host galaxy R host = log ( M stellar / M (cid:12) ) , (2)as this quantity has been used in SN-cosmology analysesto correct standardized luminosities since Conley et al.(2011).Using catalogs from Science Verification DECam im-ages (Bonnett et al. 2016), the directional light radiusmethod (Sullivan et al. 2006; Gupta et al. 2016) is usedto associate a host galaxy with each SN Ia. The stellarmasses of the DES-SN host galaxies are derived fromfitting SEDs to griz broadband fluxes with ZPEG (LeBorgne & Rocca-Volmerange 2002), where the SEDsare generated with Projet d’Etude des GAlaxies parSynthese Evolutive (
PEGASE : Fioc & Rocca-Volmerange1997).We define δµ host to be a distance modulus correction,often referred to as an SN mag-step correction, betweenSNe with R host < R step and SNe with R host > R step : δµ host = γ × [1 + e ( R host −R step ) / . ] − − γ , (3)where R step = 10. Here, the magnitude of δµ host is de-termined by fitting for γ where δµ host is between [+ γ/ − γ/ R host = 10. Wefind that because we have characterized δµ host as a stepfunction, its dependence on host mass uncertainties is weak, and therefore uncertainties are not accounted forin this calculation. Additionally, because S18 found lit-tle dependence between R step and cosmological param-eters, we fix the location in our cosmology fit. WhileSN Ia host-galaxy properties may change with redshift,we could allow for γ to have a redshift dependence, andthis possibility is discussed in Section 5.For galaxies that ZPEG was not able to determine ahost mass, we first confirm that the hosts are faint andhave not been mis-identified, and then we assign themto the low-mass bin. For the DES subset, there are 116host galaxies with R host <
10 and 91 host galaxies with R host >
10. In Figure 8 we show the distributions ofcolor and stretch as a function of R host . Correlationsbetween SN Ia light curve parameters and R host havebeen reported in previous analyses (B14, S18) and arecharacterized as an average difference (step) for eventswith R host <
10 and R host >
10. As shown in Figure 8,we find steps in stretch, ∆ x = − . ± . c = 0 . ± . x = − . ± .
041 and ∆ c = 0 . ± . Simulations
Here we discuss the use of fakes so that our simulationsincorporate the subtleties of the photometric pipelinethat cannot be computed from first principles. In ad-dition, we describe here the simulations that are used fordetermining bias corrections. Because 11 different typesof simulations are used throughout the analysis and vali-dation, we refer to Table 2, which lists key attributes foreach.
Fakes overlaid on images
Ideally, a large sample of fakes would be used for char-acterizing cosmological distance biases. However, oursample of 10,000 fakes that have been processed with
SMP is insufficient for multiple reasons. First, 10,000fakes is more than an order of magnitude smaller thanwhat is needed for the bias-correction sample used in theBBC method. Second,
SMP (or other similar methods)is far too computationally intensive for the large numberof systematic iterations that are needed to test againstvarying SN properties and assumptions. These tests in-clude multiple iterations of bias corrections, with vary-ing properties, parent populations, and assumptions. Forthe many analysis iterations that are needed, it is vitalto have a rapid method for obtaining simulated catalogphotometry that approximates
SMP . Using the sample offakes processed by
SMP , we tune our catalog simulationsto replicate
SMP flux uncertainties. As shown in Figure 2and Eq. 13 of K18, the SN flux uncertainties of the sim-ulated SNe are scaled ( ˆ S sim ) as a function of host-galaxysurface brightness by the ratio between the observed scat-ter in the fakes relative to the ‘observed’ scatter in thesimulation. As a result we obtain simulations of DES-SNwith the same distribution of photometric uncertaintiesfound in our real dataset and that can be used for rapidanalysis iterations. Simulated light curves for bias corrections Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
DES-SN Sample Data and SimulationsLow- z Sample Data and Simulations
Fig. 7.—
Top: DES subset (black points) is compared to G10 simulations (blue histogram) that are used for bias correction. Thesimulations have ∼ z ), the SALT2 m B , uncertainty in m B , stretch x , uncertainty in x , color c , the uncertaintyin c , the maximum SNR of the light curve, the light curve fit probability ( P fit ), and lastly c as a function of redshift. Bottom: Same as topbut for the external low- z sample. The fractions shown in each panel are χ / ndof. Fig. 8.—
Relations of color, stretch with host-galaxy stellar massfor the DES SN Ia subset before bias corrections have been applied.Steps across log ( M stellar / M sun ) are shown in the dashed lines.Binned data points are also shown. We use catalog-level simulations of large samples ofSNe Ia to model the expected biases in measured dis-tances that follow from the known selection effects andour light curve analysis. The simulations of the DES- SN and low-redshift samples used for this analysis followthe description of K18. For individual events, distancebiases can reach 0.4 mag as shown in Figure 9 of K18,and it is therefore imperative to have accurate simula-tions in order to predict biases. The simulation utilizes
SNANA and, as detailed in Figure 1 of K18, consists of3 main steps: 1) generating a SN source for each epoch(Source model), 2) applying instrumental noise (Noisemodel), and 3) simulating DES-SN observing and selec-tion (Trigger model). Here we discuss each of these stepsbriefly along with specific choices made for this analysis
Source model:
Our simulations first generate restframe SN Ia SEDs with the SALT2 model from B14. Themodel includes SN Ia parent populations of color andstretch, intrinsic luminosity variations, and cosmologicaleffects.For the DES subset, we test the parent distributionsof c and x found in Table 1 of Scolnic & Kessler (2016)(hereafter SK16) and find that the High-z row, represen-tative of the populations of all recent high-z surveys com-bined (SDSS, SNLS, PS1), results in the best agreementin the observed distributions of light curve parameterswhen comparing to our DES dataset.For the low- z subset we follow S18. We do not re- rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 11
TABLE 2Simulations Used in DES-SN3YR
Description Samples Scatter Model Size Used In µ bias a z G10 & C11 ∼ § z G10 & C11 ∼ § µ -bias Cosmo. DES+low z G10 & C11 ∼ § z G10 & C11 ∼ § c , x Parent DES+low z G10 & C11 ∼ § σ int DES+low z G10 & C11 ∼ § Validation b c DES N/A 100,000 SNe § µ bias DES N/A ∼ § z G10 200xDES-SN3YR § d DES+low z G10 200xDES-SN3YR § z G10 & C11 100xDES-SN3YR § a Simulations used to compute distance bias ( µ -bias) corrections (Section 3.7). b Simulations used in the validation of the analysis (Section 6). c Intrinsic scatter set to zero. The simulated fluxes are inserted into DECam images as point sources. d For each band and each sample, a random zero point offset is chosen from Gaussian PDF with σ = 0 .
02 mag. derive x and c parent populations after removal of theCfA1 and CfA2 samples, which compose less than 16% ofthe low-redshift sample, because population parametershave little dependence on selection efficiencies.A model of SN brightness variations, called ‘intrinsicscatter,’ is needed to account for the observed Hubbleresidual scatter that exceeds expectations from measure-ment uncertainties. Most cosmology-fitting likelihoodscharacterize the excess Hubble scatter with an additional σ int term added in quadrature to the measured distanceuncertainty. From an astrophysical perspective, this σ int term is equivalent to an intrinsic scatter model describedby a Gaussian profile where each event undergoes a co-herent fluctuation that is 100% correlated among allphases and wavelengths. Many previous analyses, how-ever, have demonstrated that this simple coherent modeldoes not adequately describe intrinsic scatter. Follow-ing K13, we simulate intrinsic scatter with two differentintrinsic scatter models in order to investigate the sensi-tivity to bias corrections and to the σ int approximationin the cosmology-fitting likelihood.Our intrinsic scatter models include a combinationof coherent (Gaussian σ int ) variations, and wavelength-dependent SALT2 SED variations that introduce scat-ter in the generated SN Ia colors. From K13 the firstmodel, “G10,” is based on Guy et al. (2010) and de-scribes ∼
70% of the excess Hubble scatter from coherentvariations, and the remaining scatter from wavelength-dependent variations. The second model, “C11,” is basedon Chotard et al. (2011) and describes ∼
30% of theexcess Hubble scatter from coherent variations, and theremaining scatter from wavelength-dependent variations.Cosmological effects are applied, which include red-shifting, dimming, lensing, peculiar velocity, and MilkyWay extinction. The simulations used for bias correc-tions are performed in ΛCDM ( w = − .
0, Ω M = 0 . k = 0 . Noise model:
We simulate the DES-SN cadence andobserving conditions (PSF, sky noise, zero point) usingthe catalog of DES-SN images. A sample of simulatedSNe are drawn from 10,000 random sky locations over theDES-SN observing fields and for each epoch, the observ-ing conditions are taken from the corresponding DES-SNimage. For simulations of more than 10,000 events, skylocations are repeated. We assign a host-galaxy surfacebrightness and determine photometric uncertainties fromPSF, sky, and zero point. A photometric uncertaintyscaling as a function of m SB (Sec Sec 5 of K18) is thenapplied. The final product of the noise model is a set ofDECam fluxes and flux uncertainties. Trigger model:
The last step is to apply the DES-SN detection criteria and spectroscopic selection. We re-quire two detections on separate nights within 30 days.The spectroscopic selection function for the DES subset( E spec ) is determined as a function of peak i band mag-nitude (Section 6.1 of K18).The low- z subset trigger model, which is detailed inSection 6 of K18, is based on the procedure developed inB14, S14, and S18, which assume that the low- z subsetis magnitude limited. Separate spectroscopic selectionfunctions are determined for each of the low- z surveys(CfA3, CfA4 and CSP-1). With the assumption of amagnitude limited sample, we are able to obtain goodagreement between simulations and data for the distribu-tion of observed redshifts as shown in Figure 7. However,since it is unclear how selection was done for the low-redshift surveys and that it involved a targeted search ofgalaxies, we simulate as a systematic uncertainty the as-sumption that the low- z subset is in fact volume-limited.The determination of the low- z efficiency function andthe implementation of the volume-limited assumption insimulations is discussed in detail in K18.For a volume-limited low- z subset, redshift evolutionof color and stretch are interpreted as astrophysical ef-2 Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validationfects rather than manifestations of Malmquist bias. Thisallows for the combination of the volume-limited assump-tion and the uncertainty in parent populations of colorand stretch to be analyzed with a single simulation. Theparent populations used for the simulations of the low- z subset are documented in Table 3. Data-Simulation Comparisons
We discuss here the method for evaluating the qualityof our simulations. To characterize the level of agree-ment between data and simulated distributions, we de-fine the χ between the simulation and data for each pop-ulation parameter (p) from the comparison of a binnedlight curve fit parameter distribution of the data and thenormalized binned distribution of the high statistics sim-ulation as follows: χ = (cid:88) i ( N data i − R × N sim i ) /N data i , (4) R = (cid:88) N data i / (cid:88) N sim i , for parameter bins i and simulation normalization R .The simulations have sufficiently high statistics that weignore statistical fluctuations in the simulations and onlyuse the Poisson uncertainties in the dataset.The agreement between simulations and our DES-SNdataset is shown by comparing the distributions of lightcurve fit parameters and uncertainties, redshift, andmaximum S/N among all epochs ( SNRMAX ) in Figure 7.For each subplot in Figure 7 we report [ χ ]/[dof]. Al-though only the simulations using the G10 scatter modelare shown, the distributions using C11 simulations areindistinguishable by eye.We find good agreement between the data and simu-lations for many of the observed parameters, but mostnotably in redshift (Figure 7). In simulating the DESsubset, there was no explicit tuning of the redshift dis-tribution. This gives us confidence in our models used togenerate the simulations.It is important to note that we obtain relatively pooragreement between the DES subset and simulations forthe light curve fit probability ( P fit ) distribution. How-ever, because the agreement for the SNRMAX distributionis good, it is possible that more subtle modeling of pho-tometric uncertainties is needed or that there is variationin the SN population that is not captured by a SALT2model. Agreement between data and simulations for thelow- z subset for SNRMAX and P fit is worse than for theDES subset. This suggests the need for significant im-provements in flux uncertainty modeling. In Section 8.4we discuss the need for improvements to simulations ofSNe Ia datasets. Cosmology
Here we discuss the analysis steps taken to extract cos-mological distances, fit for nuisance parameters, and cor-rect for expected biases. Additionally, we discuss the pro-duction of statistical and systematic distance covariancematrices. Finally, we discuss the cosmological parameterfitting process.
BBC
We use the “BEAMS with Bias Corrections (BBC)”fitting method (Kessler & Scolnic 2017, KS17) to con-vert the light curve fit parameters ( m B , x , c ) into bias-corrected distance modulus values in 20 discrete redshiftbins, and to determine nuisance parameters ( α , β , γ ).This BBC fit uses a modified version of the Tripp formula(Tripp 1998) where the measured distance modulus ( µ )of each SN is determined by µ = m B − M + αx − βc + δµ host + δµ bias . (5) α and β are the correlation coefficients of x and c withluminosity, respectively, and M is the absolute magni-tude of a fiducial SN Ia with x = 0 and c = 0. Follow-ing Conley et al. (2011), we include δµ host (Eq. 3) whichdepends on γ . The bias correction, δµ bias , is determinedfrom large simulations (K18) and is computed from a5-dimensional grid of { z, x , c, α, β } .The BBC likelihood ( L BBC ) is described in detail inEq. 6 of KS17. For the DES-SN3YR sample of spectro-scopically classified events, we set the core collapse SNprobability to zero and L BBC simplifies to − L BBC ) ≡ χ = (cid:88) i (cid:2) ( µ i − µ model ,i − ∆ µ, Z ) /σ µ,i + 2 ln( σ µ,i ) (cid:3) , (6)where the i -summation is over SN Ia events, µ i is thedistance modulus of the i th SN (Eq. 5), µ model ,i is thedistance modulus computed from redshift z i and an ar-bitrary set of reference cosmology parameters (Ω M =0 . , Ω Λ = 0 . , w = − µ, Z is the fitted dis-tance offset in redshift bin-index Z determined from z i .To obtain similar distance constraints in each Z bin, theredshift bin size is proportional to (1 + z ) n with n = 6,and we use 20 Z bins.Dropping the i index in Eq. 6, the distance uncertaintyof each SN is σ µ = C m B ,m B + α C x ,x + β C c,c +2 αC m B ,x − βC m B ,c − αβC x ,c + σ + σ z + σ + σ , (7)where C is the fitted covariance matrix from the lightcurve fit, σ vpec is from the peculiar velocity correction, σ z is from the redshift uncertainty, σ lens is from weakgravitational lensing, and σ int is determined such thatthe reduced χ is 1. Prior to BBC, χ -based analyseshad ignored the 2 ln( σ µ ) term of Eq. 6 because it resultedin large biases (e.g., Appendix B in Conley et al. 2011).However, KS17 found that including the δµ bias term re-moves the previously found biases, and that includingthe 2 ln( σ µ ) is essential within the BBC framework.To fit for cosmological parameters in § (cid:104) z (cid:105) Z = INVERSE( (cid:104) µ model ,i (cid:105) Z ) (8) (cid:104) µ (cid:105) Z = ∆ µ, Z + (cid:104) µ model ,i (cid:105) Z , (9)where INVERSE is a numerical function which computesredshift from the distance modulus, and (cid:104) µ model ,i (cid:105) Z is the rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 13
TABLE 3Parent Populations Parameters table
Description Scatter c peak ( σ + , σ − ) dcdz x ( σ + , σ − ) x ( σ + , σ − ) dx dz ModelDES Nominal G10 − .
054 (0 . , . .
973 (1 . , . .
000 (0 . , . − .
065 (0 . , . .
964 (1 . , . .
000 (0 . , . − .
100 (0 . , . .
964 (1 . , . .
000 (0 . , . − .
112 (0 . , . .
974 (1 . , . .
000 (0 . , . z Nominal G10 − .
055 (0 . , . .
550 (1 . , . − .
500 (0 . , . z Vol. Lim. G10 − .
055 (0 . , . .
200 (1 . , . − .
100 (0 . , . z Nominal C11 − .
069 (0 . , . .
550 (1 . , . − .
500 (0 . , . z Vol. Lim. C11 − .
047 (0 . , . .
200 (1 . , . − .
100 (0 . , . Note . — Parent population parameters of color ( c ) and stretch ( x ) used in SNANA simulations for bias corrections. The low- z x distributions are modeled as two Gaussians with two peaks shown in the table. weighted-average µ model ,i , (cid:104) µ model ,i (cid:105) Z = (cid:34) (cid:88) z i ∈Z µ model ,i /σ µ,i (cid:35) (cid:30) (cid:34) (cid:88) z i ∈Z σ − µ,i (cid:35) (10)where the summations are over the subset of DES-SN3YR events in redshift bin Z . L BBC has 3 types of approximations. The first is thecharacterization of intrinsic scatter with a single σ int term in L BBC , which does not correspond to either of thescatter models. The second approximation in the χ like-lihood is the implicit assumption of symmetric Gaussianuncertainties on the bias-corrected SALT2 fitted parame-ters (March et al. 2011). The final type of approximationis in the modeling for bias corrections, which are deter-mined from simulations that include approximations re-sulting from limited precision in the: SALT2 model, colorand stretch populations, intrinsic scatter model (G10 andC11), estimation of SMP flux uncertainties, and choice ofcosmology parameters.The first two approximations are not included as sys-tematic uncertainties because KS17 performed extensivetesting on nearly one million simulated SNe Ia to demon-strate that the resulting w bias is below 0.01. In addi-tion, we perform our own DES-SN3YR validation testsfor both bias and uncertainty in §
6. Lastly, the thirdset of approximations in simulated bias corrections areincluded as systematic uncertainties.Here we illustrate the BBC method using 100 realiza-tions of DES-SN3YR for both the G10 and C11 scattermodels. The top panels of Figure 9 show the calculated δµ bias as a function of redshift. In the bottom panelsof Figure 9, we show the BBC-fitted distance residualsafter bias corrections have been applied. For our ‘Ideal’analysis (solid lines), the bias corrections have the samescatter model and same selection function as the sim-ulated data, and the BBC-fitted distance residuals areconsistent with zero. While the average µ -bias correc-tion differs by up to 0.08 mag when the wrong model ofintrinsic scatter is used for bias corrections (‘Sys Scat-ter’), the BBC-fitted distance residuals differ by no morethan ∼ .
02 mag. The reduced effect on distance biasesis caused by the different β values from the BBC fit. In summary, χ (Eq. 6) is minimized to determine24 parameters: a distance modulus in each of the 20 red-shift bins (2 of which have no events), 3 nuisance param-eters ( α , β , γ ), and the intrinsic scatter term ( σ int ). Theensemble of 20 [ (cid:104) z (cid:105) Z , (cid:104) µ (cid:105) Z ] pairs is the redshift-binnedHubble diagram used to fit for cosmological parametersin § Covariance Matrix
Following Conley et al. (2011), we compute a system-atic covariance matrix C stat+syst , accounting for both sta-tistical and systematic uncertainties. However instead ofa N × N matrix where N is the number of SNe, here N is the number of redshift bins. C stat is a diagonal ma-trix whose Z th entry is the BBC-fitted µ -uncertainty inthe Z th redshift bin. The statistical uncertainties fromthe binned distance estimates form the diagonal matrix C stat , and C syst is computed from all the systematic un-certainties summarized in Section 4.Using BBC fitted distances, for each source of system-atic uncertainty (‘SYS’) we define distances relative toour nominal analysis (‘NOM’) as follows:∆ (cid:104) µ SYS (cid:105) Z ≡ (cid:104) µ SYS (cid:105) Z − (cid:104) µ NOM (cid:105) Z , (11)for redshift bins Z . For each source of systematicuncertainty (‘SYS’), we compute (cid:104) µ SYS (cid:105) Z by varyingthat source and re-computing bias corrected distances.Groupings of systematic variations are outlined in Ta-ble 4, and there are a total of 74 individual systematicuncertainty contributions that are evaluated.We build our redshift-binned 20 ×
20 systematic covari-ance matrix C syst for all sources (SYS k ), C Z i Z j , syst = K =74 (cid:88) k =1 ∂ ∆ (cid:104) µ SYS (cid:105) Z i ∂ SYS k ∂ ∆ (cid:104) µ SYS (cid:105) Z j ∂ SYS k σ k , (12)which denotes the covariance between the Z thi and Z thj redshift bin summed over the K different sources of sys-tematic uncertainty ( K = 74) with magnitude σ k .The binned covariances and distances are provided inmachine readable format in Appendix C. At the link inAppendix C there is also an un-binned version where thecorrections to individual SNe Ia are computed on a 2D4 Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
Fig. 9.—
Top: bias correction vs. redshift average over 100 DES-SN3YR simulated samples (left: low- z , right: DES-SN). ‘Ideal’corrections have the same scatter model and same selection function in both the simulated data and simulated bias corrections. ‘Sysscatter’ has C11 model for data and G10 model for bias corrections. ‘Sys Vol Lim’ (left) and ‘Sys Spec Eff’ (right) bias corrections arecomputed using the volume-limited low- z subset and the systematic variation on the spectroscopic efficiency function respectively (shortdashed lines) . Bottom:
Hubble diagram residuals after bias corrections are applied. Residuals are consistent with zero for the Ideal biascorrections. C stat+syst = C stat + C syst (13)where C stat is the diagonal matrix of σ µ binned in red-shift and where the indices Z i , Z j have been dropped forconvenience. Fit for Cosmological Parameters
Constraining cosmological parameters with SN datausing χ was first adopted by Riess et al. (1998) andagain by Astier et al. (2006). The systematic covariancetreatment was improved upon by Conley et al. (2011).Here we follow closely the formalism of S18.Cosmological parameters are constrained by minimiz-ing a χ likelihood. χ = (cid:126)D T C − (cid:126)D (14) D Z = (cid:104) µ (cid:105) Z − (cid:104) µ model (cid:105) Z where (cid:126)D is the vector of 20 distances binned in red-shift with each element defined by D Z . In ourcase (cid:104) µ model (cid:105) Z = +5 log( d L / w CDM model d L ( z ) = (1 + z ) c (cid:90) z dz (cid:48) H ( z (cid:48) ) , (15)where for simplicity z ≡ (cid:104) z (cid:105) Z (Eq. 9) and with H ( z (cid:48) ) = H (cid:113) Ω M (1 + z (cid:48) ) + Ω Λ (1 + z (cid:48) ) w ) , (16)where d L ( z ) is calculated at each step of the cosmologicalfitting process and where flatness is assumed in the fitsto determine the systematic error budget.In our analysis we consider two intrinsic scatter mod-els in simulated bias corrections, G10 and C11 (Section3.7.2), to span the range of possibilities in current datasamples. We assign equal probability to each model andcompute (cid:126)D and C stat+syst twice, once for G10 and oncefor C11. We average the binned distance estimates andcovariance matrices for each of the models for intrinsicscatter as follows: (cid:126)D = (cid:126)D G10 + (cid:126)D C11 , (17) C stat+syst = C G10stat+syst + C C11stat+syst , (18)where the superscripts ‘G10’ and ‘C11’ indicate bias cor- rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 15
Fig. 10.—
Residuals to the nominal cosmological analysis forthe DES-SN3YR dataset. Distance residuals are calculated forseveral sources of systematic uncertainty and using bias correctionsimulations of each model of intrinsic scatter (G10 and C11). rections assuming that specific model of intrinsic scatter.The covariances, C G10stat+syst and C C11stat+syst , each includethe covariance to the other model of intrinsic scatter withscaling σ k = 0 . w CDMmodel above to our DES-SN3YR dataset and we com-bine with Planck 2016 priors. The best fit parametersand further extensions to ΛCDM are given in the com-panion key paper (Abbott et al. 2019). In Section 6 wevalidate our analysis and uncertainties and in Section 7we discuss ongoing development of a more complex likeli-hood using a Bayesian hierarchical modeling framework.
Blinding the Analysis
We blind our analysis in two ways simultaneously asthere are a number of steps in the analysis in which onecould infer changes to cosmological parameters. First,we blind the binned distances output by the BBC fit.Additionally, to prevent accidental viewing of results, thecosmological parameter constraints were perturbed withunknown offsets.The cosmological parameters were blinded until pre-liminary results were presented at the 231st meeting ofthe American Astronomical Society in January 2018. Af- ter un-blinding we restored the blinding procedure andmade the following changes. First, we fixed the DECamfilter transmissions after realizing that atmospheric ab-sorption had been mistakenly ignored. Next, we re-tunedsimulations of SMP photometric errors and improved ourhost-galaxy library. Finally, we included several addi-tional sources of systematic uncertainty: a global shiftin our redshifts, two additional calibration systematics(‘1/3 No SuperCal’ and ‘SuperCal Coherent Shift’), anda systematic uncertainty for the use of two σ int .We unblinded again during the internal review pro-cess; w increased by 0.024 and the the total uncertaintyincreased by 4% (0.057 to 0.059). TREATMENT OF SYSTEMATIC UNCERTAINTIES
Here we summarize the treatment and value of eachsystematic uncertainty from the analysis steps in Sec-tion 3 in order to create C sys from Eq. 12. A summarytable of the systematics used is provided in Table 4. InFigure 10 we compare the ∆ (cid:104) µ SYS (cid:105) Z for several system-atics, which allows us to visualize the change in distancesfor some of the major sources of systematic uncertainty.Systematics which have a large change in distance be-tween low and high redshift (i.e. Parent c , x ) are thelargest contributors to the total cosmological parametererror budget which is discussed in Section 5. Calibration
There are several systematic uncertainties related tocalibration which include but are not limited to the un-certainty from the photometry (as discussed in B18-SMP), the calibration to the AB system, and the cali-bration uniformity across the 10 observing fields. Theuncertainty in calibration uniformity across the sky isdefined as σ syst = σ uniformity / √ N where N =3 is the num-ber of DES-SN field groups overlapping PS1 (see C,S,Xin Section 2), and where we adopt σ uniformity = 6 mmagfrom Burke et al. (2018). Within a field group (e.g.,C=C1+C2+C3), we do not count each field (for N ) be-cause the calibration uniformity over 1 degree scales isexpected to be better than the uniformity over the largeseparations between field groups.Uniformity uncertainty due to the location of C26202 isalready accounted for here because C26202 is located inone of our SN fields that overlap with PS1. For DES, wecombine the photometric uncertainty, uniformity uncer-tainty, and statistical uncertainty in the AB calibrationand propagate a single uncertainty in the photometriczero point per band. A final uncertainty is propagatedindependently by band such that there is a separate entryin C syst for each band.To evaluate the agreement of the absolute calibra-tion of the DES-SN fields with the absolute calibrationthat is used for the low- z sample as described in Super-Cal, we utilize the overlap of DES stars with those ofPS1 which have also been calibrated following SuperCal.We compute χ from the difference in absolute cali-bration, ∆ M SuperCal i − DES i , between PS1-SuperCal (red)and DES (grey dashed) shown in (Figure 4) as follows χ = N filter (cid:88) i (cid:104) ∆ M SuperCal i − DES i (cid:105) σ + σ . (19)where ∆ M SuperCal − DES i are the offsets to synthetic mag-6 Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
TABLE 4Sources of Uncertainty
Size a Description Reference
SN Photometry
Calibration / √ σ uniformity Burke et al. (2018)0.6 nm DECam filter curves uncertainty. Abbott et al. (2018)[ − , − , − ,
5] mmag Modeling of C26202 implemented as coherent shift [ g, r, i, z ] Figure 45mmag/700 nm HST Calspec spectrum modeling uncertainty Bohlin et al. (2014)1/3 No SuperCal SuperCal process S18, Scolnic et al. (2015)Following S18 Low- z samples photometric calibration. S18, CfA3-4, CSP-1Following S18 Low- z samples filter curve measurement. S18, CfA3-4, CSP-1Following B14 SALT2 light curve model calibration. B14 Bias Corrections (Astrophysical)
Table 3 c , x Parent populations resulting in ∆ χ = 2 . § − C11) Model of intrinsic scatter variations § σ int Separate fit σ int for each subset § w † Cosmology in which the bias correction sample is simulated. § § Bias Corrections (Survey) σ → σ outlier cut † Low- z Hubble diagram outlier cut. § σ stat Fluctuation Spectroscopic selection function statistical fluctuations. § z Selection Low- z subset magnitude → volume limited survey. § σ phot Underestimation † Incorrect SN photometric uncertainties. § Redshifts × − in z † Coherent z -shift. § . × β bias Peculiar velocity modeling § a Size adopted for each source of systematic uncertainty. † Sources of systematic uncertainty that have not been included in previous analyses. nitudes in each filter (red line of Figure 4) relative to theDES calibrated to C26202, σ SuperCal is the uncertaintyfrom Scolnic et al. (2015) of [3 , , ,
4] mmag in [ g, r, i, z ]bands, and σ syst is the uncertainty in the uniformity ofthe fields used for comparison between PS1 and DES(6 / √ χ = 1 . (cid:104) µ SYS (cid:105) Z for thischoice are propagated in our covariance matrix C syst .The uncertainty in the calibration of the low- z sampleis adopted from SuperCal. Additionally, as was done inS18, we adopt an overall uncertainty associated with theSuperCal itself which Scolnic et al. (2015) characterizedas 1/3 the size of the impact on distances if SuperCalwas not applied.A number of calibration systematics are propagatedseparately from the absolute and relative calibration treatment above. Uncertainty in the DECam filter trans-mission functions propagate to uncertainties in absolutecalibration because FGCM utilizes these transmissionfunctions to predict the flux of C26202. A 0.5nm wave-length uncertainty arises in the determination of the filtertransmission function due to the precision on wavelengthin the measurement. Additionally, there is a 0.3nm effectarising from illumination lamps on the flat field screenthat should be, but are not exactly, on the same opticalaxis. These two wavelength uncertainties are added inquadrature for a total of 0.6nm.We also include the uncertainty in modeling the spec-trum of C26202, which is 5mmag over 700nm. Lastly, wehave not retrained the SALT2 model, and therefore weuse the same SALT2 calibration uncertainty as in B14.We do not include a systematic uncertainty from chro-matic corrections, since we have already included FGCMuncertainties which are based on applying these correc-tions. Furthermore, Lasker et al. (2019) find that if chro-matic corrections are not applied, the change in fit w is0.005. This change in w is consistent with the statisticaluncertainty associated with this correction, and it is wellbelow the systematic uncertainty from our analysis. Intrinsic Scatter Modelrout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 17One of the largest systematic uncertainties results fromthe modeling of intrinsic scatter in the simulations usedto predict bias corrections. We include two intrinsic scat-ter models, G10 and C11, and assign equal probabilityto each model. Because of the parallel treatment of thescatter models (Eqs. 17 & 18), we end up with two setsof nuisance parameters. From here on in this paper, un-less otherwise noted, results and nuisance parameters arestated in the context of the G10 model.As will be shown in Sec 5.1.2, the σ int values show > σ tension when determined separately for the low- z and DES subsets, and this tension persists for bothintrinsic scatter models. In addition, our DES-SN σ int value is the smallest of any rolling SN search, suggestingthat it is a fluctuation. To account for the possibilitythat this σ int difference is real, we include a systematicuncertainty based on an analysis using two σ int values,and compare to the nominal analysis that assumes a sin-gle σ int value. For the “Two σ int ” analysis, we scale thespectral flux variations from the intrinsic scatter model(G10 or C11) so that analyzing the simulation results inthe same σ int values as for the low- z and DES-SN datasubsets. These scaled scatter models are used to gener-ate bias correction simulations, and the BBC fit is modi-fied to constrain the ratio, σ int (low- z )/ σ int (DES-SN), tomatch that of the data. To summarize, there are twouncertainties related to the unknown source of intrinsicscatter. First is the relative contribution of coherent vs.wavelength dependent scatter (G10 vs. C11). Second isthe overall amplitude difference in scatter between thelow- z and DES subsets. Color and Stretch Parent Populations
In order to estimate the uncertainty in parent colorand stretch distributions, we vary the mean and width ofeach parent population in the simulation until we achieve > σ deviations between the observed and simulated dis-tributions. We alter the systematic parent populations ofcolor and stretch in order to increase the ∆ χ p , as definedin Eq. 4, by ∼ . z subset, this is encompassed inthe volume-limited case. In this case, redshift evolutionof color and stretch are interpreted as astrophysical ef-fects rather than manifestations of Malmquist bias (Sec-tion 3.7.2). A different set of parent population parame-ters are determined for the volume-limited case and areshown in Table 3. Spectroscopic Selection
We generate 200 realizations of the DES subset withonly statistical fluctuations. We run our E spec fittingprocedure on each realization and find that biases in re-covering the input E spec are limited to 7% ( E fit / E input -1) across the range 19 < i peak <
24 whilst 1 σ statistical fluctuations are up to 25% at 23 rd mag. Because neitherthe simulation nor BBC fit take into account the statis-tical uncertainty in the E spec , we adopt the 1 σ statisticalfluctuation and propagate it as a systematic uncertainty.We do not include a spectroscopic efficiency system-atic for the low- z subset. Instead, the low- z subset isassumed to be magnitude limited and the systematicuncertainty for simulating this sample is to model it asvolume-limited (see Table 3 and § Cosmology Assumption in Bias Corrections
We include the systematic uncertainty from our choiceto simulate selection biases with a fixed set of w CDMparameters (Ω M =0.3, Ω Λ =0.7, w =-1). Here we re-determine the distance bias after changing the refer-ence cosmological model in our simulations to w ref = w bestfit − .
05, a change that matches the statistical pre-cision of our measurements. The difference in distancebiases for these two reference cosmology values is illus-trated in Figure 10 and is less than 2 mmag across theentire redshift range.
Redshifts
We include two systematic uncertainties for our treat-ment of redshifts. The first is from our modeling of thepeculiar velocities, and following Zhang et al. (2017) wemodify the light-to-matter bias parameter ( β bias ) by 10%and remeasure the redshift corrections. The second is acoherent shift in each redshift of 4 × − as conserva-tively constrained in Calcino & Davis (2017). Low- z Hubble Residual Outliers
We include the systematic uncertainty associated withHubble residual outlier rejection of SNe Ia in the low- z subset. S18 placed a 3.5 σ cut on their sample. Forour dataset of 329 SNe Ia, Chauvenets criterion corre-sponds to a 3 σ cut. We investigate the systematic effectof applying both 3 . σ and 3 σ cuts on Hubble diagramresiduals to the low- z subset. Because this cut dependson the best fit cosmological model, it is discussed laterin Section 5.2. Photometry
For the
SMP pipeline, there is an additional systematicuncertainty beyond the 0.3% biases mentioned in Section3.2. Our
SMP pipeline performs stellar position fits inde-pendently on each night, but uses a globally-fitted posi-tion of the SN across all nights (B18-SMP). Fitting forstellar positions each night independently accounts forthe proper motion of the stars, but B18-SMP find thisdifference in the treatment of the stars and SNe can causea ∼ Milky Way Extinction Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
Fig. 11.—
Residuals in distance to the best fit flat w CDM cos-mology as a function of redshift. Blue: DES subset. Red: Low- z subset. Black: Binned distances used for cosmological fits. BBCfitted distances shown are averaged assuming each model of intrin-sic scatter (G10 and C11). Fig. 12.—
Distance covariance matrix in redshift bins withoutstatistical uncertainties on the diagonal.
Lastly, we account for Milky Way extinction usingmaps from Schlegel et al. (1998), with a scale of 0.86based on Schlafly et al. (2010), and the Milky Way (MW)reddening law from Fitzpatrick (1999). We adopt aglobal 4% uncertainty of E ( B − V ) MW based on the factthat Schlafly & Finkbeiner (2011b), in a re-analysis ofSchlafly et al. (2010), derive smaller values of reddeningby 4%, despite using a very similar SDSS footprint. RESULTS
We perform a cosmological fit to our redshift-binnedand bias-corrected Hubble diagram. The distances ob-tained in this analysis are shown as binned residuals tothe best fit cosmology in Figure 11 after bias correctionshave been applied. The covariance matrix used in ourcosmological fits with each of the systematics compo-nents ( C Cosmosyst ) is shown in Figure 12. In this sectionwe report the fit values for the nuisance parameters inEq. 5 and the systematic error budget on cosmologicalparameters. Several of our results require further dis-cussion which can be found in Section 8. We refer thereader to Abbott et al. (2019) for the unblinded best fit constraints of cosmological parameters.
Nuisance Parameters
The BBC fit output includes 4 nuisance parameters: α , β , σ int , and γ . The values for these parameters are sum-marized in Table 5 along with a comparison with those ofthe PS1 and SNLS samples from S18. Here we describethe values found, their consistency with those of previoussamples, as well as various perturbations to our analysisand the affect on the recovered nuisance parameters. α , β A comparison of α and β , the standardization coeffi-cients of stretch and color, with those of the PS1 andSNLS samples are shown in Table 5. We find that α and β are in agreement with various surveys. We test for α or β dependence with redshift: α = α + z × α , β = β + z × β , (20)and we find that α and β are consistent with zero, withthe possible exception of β in our G10 analysis which wedetect at − . σ . However in our C11 analysis we detect β at − . σ and thus we consider the evolution in theG10 case to be a statistical fluctuation. σ int and σ tot The nominal analysis assumes a single value for theamount of intrinsic scatter needed to bring χ / dof= 1( σ int ). We perform the nominal analysis twice, once foreach model of intrinsic scatter (G10 and C11) and thevalues of σ int are found to be 0.094 and 0.117 respec-tively (Table 5). These are in agreement with the valuesfound by previous analyses (PS1, SNLS). However, wealso examine the σ int for each subset in our analysis. Forthe DES subset we find σ int = 0 .
066 mag for G10 and0 .
088 mag for C11, which are the smallest observed valuesof any rolling supernova survey to date using the SALT2framework. For the low- z subset we find σ int = 0 .
12 magfor G10 and 0 .
14 mag for C11. In analyzing 100 simu-lated statistical realizations of DES-SN3YR, we find thatthe
RM S ( σ int ) for the DES subset is 0.007 and for thelow- z subset it is 0.015. Thus, the σ int values for DES-SN and low- z subsets differ by more than 3 σ . In Section5.2 we discuss the change in fit w if two σ int are used inour analysis ( σ low − z int and σ DESint ).In Table 6 we show the total scatter about the Hub-ble diagram, σ tot , for the subsets in this analysis andwe compare with other surveys. We find the lowest ob-served value of σ tot , 0.129 mag. We also confirm thatthe 5D bias corrections performed in BBC provide im-proved Hubble residual scatter over 1D corrections. 1Dcorrections in this analysis are only used as crosschecksto previous analyses such as B14. Host-galaxy Stellar Mass Step γ Somewhat surprisingly, we find little correlation be-tween host-galaxy stellar mass and Hubble diagramresiduals ( γ = 0 . ± . γ DES = 0 . ± .
018 mag).For the low- z subset we find γ low − z = 0 . ± .
038 mag,which is consistent with previously seen results. TheHubble diagram residual vs. host mass relation for the rout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 19
TABLE 5Nuisance Parameters from BBC Fit
Parameter Description G10 C11 AVG † α DES-SN3YR 0.146 ± ± ± α DES subset 0.151 ± ± ± α Low- z subset 0.145 ± ± ± α PS1 0.167 ± ± ± α SNLS 0.139 ± ± ± β DES-SN3YR 3.03 ± ± ± β DES subset 3.02 ± ± ± β Low- z subset 3.06 ± ± ± β PS1 3.02 ± ± ± β SNLS 3.01 ± ± ± γ DES-SN3YR 0.025 ± ± ± γ DES subset 0 . ± .
018 0 . ± .
017 0.007 ± γ Low- z subset 0.070 ± ± ± γ PS1 0.039 ± ± ± γ SNLS 0.045 ± ± ± * σ int DES-SN3YR 0 . ± .
008 0 . ± .
008 0 . ± . * σ int DES subset 0 . ± .
007 0 . ± .
008 0 . ± . * σ int Low- z subset 0 . ± .
015 0 . ± .
015 0 . ± . σ int PS1 0.08 0.10 0.09 σ int SNLS 0.09 0.10 0.10
Note . — Nuisance parameters and uncertainties for the DES-SN3YR and the DES and low- z subsets with com-parisons to other datasets. The values for PS1 and SNLS are taken from S18 which does not report uncertaintieson σ int . † AVG is presented here solely for comparison purposes and is not used in the analysis. * σ int uncertainty is the RMS from 100 simulated realizations of the dataset. TABLE 6Comparison of σ tot σ tot (G10) σ tot (C11)Dataset
5D [1D] 5D [1D]
DES subset 0.129 [0.156] 0.128 [0.156]Low- z subset 0.156 [0.158] 0.157 [0.158]DES-SN3YR 0.142 [0.155] 0.141 [0.155]PS1 0.14 [0.16] 0.14 [N/A]SNLS 0.14 [0.18] 0.14 [N/A] Note . — Comparison of RMS of Hubblediagram residuals ( σ tot ) for the subsets ofSNe. Comparisons between performing 5Dand 1D bias corrections are also shown. Thevalues for PS1 and SNLS are taken from S18. DES subset are plotted in Figure 13. The DES subsetvalue is 2.4 σ smaller than γ Pantheon found in S18 whichused the same BBC fitting method. As a crosscheck, wehave obtained a second set of host-galaxy stellar mass es-timates using a different set of SED templates (Bruzual &Charlot 2003) and fit the griz magnitudes with Le Phare(Arnouts & Ilbert 2011) spectral fitting code. With theseparate set of mass estimates, we find the γ DES value isstill consistent with zero (Table 7).Another crosscheck is to replace the 5D bias correctionin the BBC fit with a 1D correction that depends onlyon redshift, which is similar to the JLA (B14) analysis.We find that using 1D bias correction in z , analogous tothat of the JLA (B14) analysis, results in a larger γ DES of 0 . ± .
021 mag. This is in agreement with S18 whofind that the 5D bias correction reduces scatter about theHubble diagram and reduces γ DES by ∼ .
02 mag com-
TABLE 7Systematic variations for γ DES
Variation γ [mag] . ± .
018 207 c > . ± .
039 70 c < − . ± .
020 137 x > . ± .
025 119 x < − . ± .
029 88no z band 0 . ± .
021 2021D BiasCorr. 0 . ± .
021 207
DiffImg
Photometry 0 . ± .
020 207 M stellar (cid:54) = null 0 . ± .
020 207 R step = 10 . . ± .
019 20710 z -bins 0 . ± .
018 207Le Phare 0 . ± .
020 207
Note . — Changes in γ for the DES subsetafter perturbations to analysis. Parametervalues are shown for the G10 model of intrinsicscatter only. pared to using the 1D bias correction from B14. This willbe studied in a forthcoming DES-SN analysis (Smith etal. 2018 in prep) of simulations that include correlationsbetween the SN properties and the host in simulations.We note that using 5D bias corrections, S18 find signifi-cant values for γ for each of their subsets of SNe and thatthe value found here for the low- z sample is in agreementwith S18.To examine potential systematics in measuring γ DES ,Table 7 shows several variations in our BBC fitting proce-dure. As DECam has better z -band sensitivity comparedto previous surveys, we ran our analysis without z bandand found a consistent γ DES (0 . ± .
023 mag) with a0
Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
Fig. 13.—
Residuals in distance to the best fit cosmology as afunction of log ( M stellar / M (cid:12) ) for the DES subset only. Correla-tion between residuals and mass is characterized as a step functionat 10 M (cid:12) however we find no clear trend in the DES-SN data. slightly larger uncertainty.Additionally, because color and stretch are both corre-lated with host-galaxy stellar mass (Figure 8), we inves-tigate the effect of various cuts to our dataset on γ DES .Splitting the DES subset into two sub-samples of color,we find that c > c < . σ . Whenperforming the analogous test in stretch, we find x > x < σ . Statistically these measure-ments are self-consistent. As a precautionary check thatthe small γ DES value is not an artifact of our
SMP pipeline,we perform a BBC fit with the
DiffImg photometry andfind that γ DES remains consistent with zero.Since we have included host-galaxies whose mass couldnot be determined (S/N too low), and assigned them tothe R host <
10 bin, we perform the BBC fit with theseevents excluded (‘ M stellar (cid:54) = null’) and still find γ DES consistent with zero. We also test using 10 redshift binsinstead of 20 and again the recovered value for γ DES isconsistent with zero.We perform a separate check for redshift evolution of γ parametrized as γ = γ + γ × z. (21)We find γ is consistent with zero for the DES subset( − . ± .
10 mag).Finally, because specific star formation rate (sSFR) isknown to correlate with host galaxy stellar mass (Rigaultet al. 2015; 2018), we explicitly check for a sSFR stepwith Hubble residuals in the same fashion as Eq. 3 andfind 0 . ± .
025 mag.
Systematic Error Budget
The uncertainties on w are presented in Table 8 forfits to a flat w CDM model with Planck 2016 CMB pri-ors. The systematic uncertainties shown in Table 8 aredefined as σ (cid:48) w = (cid:113) ( σ stat+syst w ) − ( σ stat w ) (22)where σ stat+syst w is the uncertainty when only a specificsystematic uncertainty (or group of uncertainties) is ap-plied such that σ (cid:48) w is the contribution to the total uncer-tainty from the specific systematic alone. Small shifts in w are expected when including systematic uncertaintiesdue to different inverse-variance weights as a function ofredshift from the BBC fit. We characterize this effect in Table 8 by including w − shift = w stat+syst − w stat , (23)which is the difference between including and excludingsystematic uncertainties. Additionally, we show the con-tribution to the uncertainty budget for each systematicgrouping in column σ syst w , and the ratio of systematicuncertainty to statistical uncertainty ( σ syst w / σ stat w ). Wenote that simply summing errors in quadrature from Ta-ble 8 will not result in the uncertainty for ‘ALL’ becauseof redshift-dependent correlations among the systematiceffects.We find that the statistical and systematic uncertain-ties on w for the DES-SN3YR dataset are σ stat w = 0 . ,σ total syst w = 0 . , where σ total syst w is the w uncertainty from all systemat-ics and excluding statistical uncertainties. This indicatesthat our result is equally limited by systematic and sta-tistical uncertainties. In Section 8 we discuss how ad-ditional data may aide in the reduction of systematicuncertainties.In Table 8, we break down the independent contribu-tions to the w -error budget. We also group the system-atic uncertainties into four main categories and find thatnearly equal contributions to the total uncertainty fromthe largest three groupings: 1) photometry and calibra-tion, 2) astrophysical bias corrections, and 3) survey biascorrections, all of which are associated with estimation ofdistances. The final and smallest grouping, 4) describesthe systematic uncertainties associated with the redshiftsin our analysis. Photometry and Calibration:
Because the low-redshift samples are calibrated to the PS1 absolute mag-nitude system and because the DES subset has been cal-ibrated to a single CalSpec standard star, we have in-cluded an additional calibration uncertainty. We assumecoherent offsets to SuperCal to be our systematic uncer-tainty for the potential incorrect modeling of the singleCalSpec standard. We find that this results in an uncer-tainty on w of 0.005. This uncertainty is included in the‘DES’ calibration uncertainty. Astrophysical µ -Bias Corrections: As mentionedin Section 4.2, we run the entire analysis pipeline sepa-rately for G10 and C11 models of intrinsic scatter. Thecontribution to the error budget from intrinsic scattermodel alone is found to be σ w = 0 .
014 While we deriveseparate parent populations associated with each intrin-sic scatter model, we also assess the systematic uncer-tainty in these parent populations. This systematic (‘ c , x Parent Pop’) is as large as that due to the intrinsicscatter model itself.Our nominal analysis assumes that all SNe Ia sampleshave the same amount of intrinsic variation. However,upon examining the σ int of the DES subset, we find thatit is in tension with the value found for the low- z subset.We therefore implement another set of BiasCor simula-tions with separate σ int for each subset and we re-derivedistances allowing for two separate σ int in the nuisanceparameter fitting stage of SALT2mu. This introduces asystematic uncertainty of 0.014 in w . rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 21
TABLE 8 w Uncertainty Contributions for w CDM model a Description b σ (cid:48) w σ (cid:48) w /σ stat w w shiftTotal Stat ( σ stat w ) 0.042 1.00 0.000Total Syst c ( σ total syst w ) 0.042 1.00 -0.006 [Photometry & Calibration] [0.021] [0.50] [-0.005] Low- z d [ µ -Bias Corrections: Survey] [0.023] [0.55] [-0.001] † Low- z σ Cut 0.016 0.38 0.005Low- z Volume Limited 0.010 0.24 0.009Spectroscopic Efficiency 0.007 0.17 0.001 † Flux Err Modeling 0.001 0.02 -0.001 [ µ -Bias Corrections: Astrophysical] [0.026] [0.62] [-0.003] Intrinsic Scatter Model 0.014 0.33 -0.001 c , x Parent Population 0.014 0.33 0.000 † Two σ int † w, Ω M for bias corr. 0.006 0.14 0.001 [Redshift] [0.012] [0.29] [0.003] † z + 0 . a The sample is DES-SN3YR (DES-SN + low- z sample) plus CMB prior. b Items in [bold] are sub-group uncertainty sums. c The quadrature sum of all systematic uncertainties does not equal 0 .
042 because ofredshift-dependent correlations when using the full covariance matrix. d Uncertainty is also included in Photometry & Calibration: DES. † Uncertainty was not included in previous analyses.
Survey µ -Bias Corrections: For our nominal anal-ysis we have followed the treatment in S18 and placeda cut on the Hubble residuals at 3 . σ from the best fitcosmological model. This cut results in a loss of 3 low- z SNe Ia. In addition, we test a 3 σ cut that results inan additional 2 SNe Ia cut from the low- z subset. NoSNe Ia from the DES subset are lost to outlier cuts.The size of the systematic uncertainty in the outlier cutis σ w = 0 . Redshifts: we have included two sources of system-atic uncertainty associated with the redshifts used in thiscosmological analysis. We find that while both the uncer-tainty in the peculiar velocities and a systematic redshiftmeasurement offset must be accounted for, their contri-bution to the w -uncertainty budget is not yet comparableto that of distance uncertainties. New:
We have included several sources of systematicsthat have not been included in previous analyses. Theseare the redshift uncertainty, an uncertainty on the re-ported photometric errors, a change in the reference cos-mology for simulations, outlier cuts to the low- z subset,and separate σ int ’s for each subset. The outlier cut isthe largest single source of uncertainty in our analysisand the separate treatment of σ int is tied for the secondlargest. When all of these new systematic uncertaintiesare combined, we find σ w =0.024, which is comparable to other systematic uncertainty groupings found in in Table8. VALIDATION OF ANALYSIS
Here we describe our validation of the analysis us-ing two separate sets of simulations. The first is basedon 10,000 fake SNe Ia light curves overlaid on images,and processed with
SMP , light curve fitting, BBC andCosmoMC. The second test uses a much larger catalog-level simulation from K18, and is processed as if theywere a catalog produced by
SMP . While these valida-tion tests could have revealed problems leading to addi-tional systematic uncertainties, no such issues were iden-tified and therefore no additional uncertainties are in-cluded. Nonetheless, the validation tests were essentialtools in developing the analysis framework and they pro-vide added confidence in the final analysis. Since thesevalidation tests are not sensitive to errors in calibration,nor to assumptions about SN properties, we caution theirinterpretation.
Fake SNe Ia Overlaid Onto Images
For the DES subset we simulate a sample of fake SNe Ialight curves and insert light curve fluxes onto DES-SNimages at locations near galaxy centers. B18-
SMP usethese fake transients to 1) measure biases associated with
SMP , 2) assess the accuracy of
SMP uncertainties and sub-sequently adjust errors in both data and simulations, and2
Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation3) optimize the photometric pipeline outlier rejection.Here, we take this fake analysis one step further and per-form a cosmology analysis resulting in a measurement of w . The benefit is that we can investigate potential bi-ases that are not correctly modeled in early stages of theanalysis (i.e. the search pipeline), which could propagateto uncorrected biases in distances and fit cosmologicalparameters. While previous analyses (e.g., Astier et al.2006, B14) used fake transients to test their photometrypipelines, our test is the first to validate the cosmologyanalysis with fakes.A sample of 10,000 fake SNe Ia light curves are dis-covered by DiffImg , processed by
SMP , bias correctedwith BBC, and run through our cosmological parame-ter fits with CosmoMC in the same fashion as the realdataset. A detailed description of the analysis of the6,586 fakes that pass quality cuts is found in AppendixD. The agreement between the BiasCor sample used tomodel the fakes dataset, and the fakes processed throughour analysis pipelines is shown in Figure D.1, which isanalogous to Figure 7 for the real data. We analyze thefakes with BBC (Section 3.8.1) to produce a redshift-binned Hubble diagram and the BBC distances residualsto the input ΛCDM distances are shown in Figure 14as a function of redshift. Cosmological fits to the fakeSNe Ia are not performed with Planck 2016 CMB priorsbecause the underlying cosmology of Planck is unknownand therefore we cannot check for cosmological parame-ter biases. Instead, we perform w CDM fits on the binneddistances with a prior on Ω M ∼ N (0 . , . χ / dofin Figure 14 is 2.5, however the amount of additionaldistance uncertainty per SN required to bring χ / dof tounity is 4 mmag, which is much smaller than the intrin-sic scatter in the DES-SN subset ( σ DESint = 0 .
070 mag).Finally, we find w = − . ± .
030 (yellow) which isconsistent with the ΛCDM cosmology in which the fakeSNe Ia were generated. Since the w bias from analyz-ing the fakes is consistent with zero, we do not assign asystematic uncertainty from this test. Large Catalog-Level
SNANA
Simulations
In contrast to the analysis with fakes, we perform ouranalysis on
SNANA simulations that include systematicvariations in both the DES-SN and low- z samples. Thesesimulations are used to check that our recovered cosmo-logical parameters and their uncertainties are determinedaccurately.We begin by generating 200 data samples of compara-ble size to the DES-SN3YR, each with independent sta-tistical fluctuations, and with no systematic variations.Here we simulate and analyze using the G10 model only.Each sample is processed with light curve fitting, BBCand CosmoMC using an Ω M prior of N (0 . , . w bias consistent with zero ( − . ± . w values with theaverage reported w uncertainty, defined as: R σ ( w ) = (cid:104) σ w (cid:105) / RMS( w statonly ) (24)We find that R σ ( w ) = 1.06 as shown in (r1,c3) of Ta-ble 10, indicating that the average reported errors are inagreement with the RMS of fitted w values ( R σ ( w ) = 1 Fig. 14.—
Hubble residuals from 6586 fake SNe using the sameanalysis procedure as for the DES-SN3YR sample, except the CMBprior is replaced with a Gaussian Ω M prior. Upper: zoomedout showing BBC bins and individual SNe on same y-scale asFig 11.
Lower: zoomed in to show BBC-binned residuals moreclearly. Black horizontal line corresponds to the flat ΛCDM model(Ω M =0.3) used to generate the fakes. Orange line is the best fit w CDM model, and best fit w and Ω M are shown on the lowerpanel. for perfect agreement). In the top panel of Figure 15, wecombine the cosmological parameter posteriors of eachof the 200 BBC fits by adding the χ contours in orderto achieve an “average” contour for the 200 realizationswith size corresponding to the typical statistical uncer-tainty. We also show the best fit parameters for each ofthe 200 statistical realizations, calculated from each ofthe individual posterior peaks, and find that 135 (68%)of the 200 best fits lie within the 1 σ contour.In order to assess the treatment of multiple indepen-dent systematics, we run simulations with systematic bi-ases in the zero point. For each band in each of 200simulated G10 samples, we perturb the calibration with arandomly selected zero point shift from a Gaussian distri-bution with σ = 0.02. This perturbation is for each sam-ple, not each event, and is artificially inflated comparedto our data calibration uncertainties ( ∼ w bias is con-sistent with zero ( − . ± . rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 23
Fig. 15.—
Top : 200 simulated DES-SN3YR datasets with statis-tical only fluctuations. Best fits (red) and average posterior (blackcurve) are shown.
Bottom : 200 simulated DES-SN3YR datasetswith input calibration systematic of 0.02 mag per filter. Thebest fit cosmological parameters for each of the 200 simulationsfrom a BBC+CosmoMC analysis using ( C = C stat ) are shown inred. The average posterior from fits to the 200 simulations using C = C stat + C cal is shown in black. All simulations are generatedin the same input cosmology shown in the grey cross-hairs. All fitshave a tophat prior on Ω M ∈ [0 , using the stat+syst covariance matrix that accounts forzero point systematic uncertainty (black contour). Wefind that 139 (70%) of the 200 best fits (stat-only) fallwithin the averaged one sigma contour (stat+syst), con-sistent with a 1 σ interpretation of the contour. Thisis also shown in (row 2, column 4) of Table 10 wherewe demonstrate that after combining with the Ω M -prior, TABLE 9Bias and Uncertainty Precision for α and β model a α bias β bias R σ ( α ) R σ ( β )G10 − . ± . − . ± .
012 0 .
91 1 . − . ± . − . ± .
016 1 .
05 0 . a Intrinsic scatter model used in simulated samples for data andbias corrections. the RMS in fit w from analyses with C = C stat agreeswith the average output uncertainties on w from analyseswhere C = C stat + C syst : R σ ( w ) = 1.03.In order to validate the treatment of the intrinsic scat-ter model systematic, we generate 100 realizations ofDES-SN3YR using both G10 and C11. When analyz-ing all 200 results from the 100 G10 simulations and 100C11 simulations together using the averaged distancesand covariances of Equations 17 & 18, we find no biases( − . ± . R σ = 1.00(shown in r3,c3 of Table 10). However, because our setof distances used to compute cosmological parameters isaveraged between the best fit distances of each scattermodel, we expect subtle biases when evaluating simula-tions created with a single model of intrinsic scatter. Inanalyzing only the 100 G10 realizations combined withΩ M ∼ N (0 . , . w bias of − .
03, and forthe 100 C11 realizations a w bias of +0 .
03. We note thatcombining SNe with the prior on Ω M is weaker thancombining SNe and Planck Collaboration (2016) CMBconstraints by roughly 50%. The w shift for each scat-ter model when combining with CMB becomes ± . α and β in Table 9. Analogous to R σ ( w ) for w -uncertainties (Eq. 24), we define R σ ( α ) and R σ ( β ) forstatistical-only fits of α and β respectively. For bothintrinsic scatter models, the α bias is consistent withzero. For β , there is a hint of bias at the sub-percentlevel. The uncertainties and RMS ( R σ ( α ) , R σ ( β )) agreeat the 10% level for G10, and at the few percent level forC11.Finally, we generate large simulations of DES-SN3YRwith two separate values of σ int for each subset to exam-ine the biases in our analysis. We analyze with BiasCorsimulations generated with two separate values of σ int and find that (cid:104) w (cid:105) = − . ± .
008 after combining withΩ M ∼ N (0 . , . σ int in BiasCor simulations ensures that ourtreatment of this systematic has been implemented cor-rectly. We also analyze the same realizations our Nom-inal BiasCor, which use a single value for σ int , and find (cid:104) w (cid:105) = − . ± . w whenanalyzing with our nominal analysis justifies the inclu-sion an additional systematic uncertainty. We note againthat combining SNe with the prior on Ω M is weaker thancombining with CMB by roughly 50% and thus the as-sociated systematic uncertainty reported in Table 8 issmaller.4 Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
TABLE 10Summary of Validation Results from Simulations
Column 1 2 3 4Row ¯ w + 1 a RMS( w statonly ) (cid:104) σ w (cid:105) R σ b Description1 − . ± . − . ± . c − . ± . d Note . — 200 “DES Like” realizations with and without input sources of systematic uncertainty. All simulations are fitwith an Ω M = 0 . ± .
01 prior. a Inverse variance weighted average. b R σ , defined in Eq. 24. c ZP Systematic corresponds to a zero point magnitude offset drawn from a Gaussian distribution of width 0.02 mag for eachband independently in each of the 200 simulations. d Intrinsic scatter model systematic corresponds to 200 simulations, 100 with each model of intrinsic scatter (G10 and C11). DEVELOPMENT OF BAYESIAN MODEL FITTING
One of the predominant issues in supernova cosmologyis that color and stretch uncertainties are assumed to beGaussian and symmetric. This assumption is not validwhen the uncertainties are comparable to the intrinsicwidth of the underlying population.This issue has historically been addressed in two dif-ferent ways. The first method, used by BBC, determinesthe true populations of stretch and color (SK16) and ina separate step determines bias corrections with simu-lations. The second method is to construct a model inwhich the true underlying values for color and stretchare parametrized (March et al. 2011). Bayesian Hier-archical Models (BHM) have been developed that bothutilize bias-correct observables (Shariff et al. 2016) andincorporate selection effects directly into the model (Ru-bin et al. 2015). Here we summarize a new methodcalled
Steve (H18: Hinton et al. 2019), which makes useof detailed
SNANA simulations to describe the selectionefficiency as part of the likelihood. In addition,
Steve does not make assumptions about the underlying intrin-sic scatter model, and it uses a parameterized treatmentof systematic uncertainties. Although this method is stillunder development, here we illustrate progress by de-scribing its performance on simulated validation samplesand the DES-SN3YR sample.H18 validate
Steve on the same set of 200 DES-SN3YRsimulations as described in Section 6. For the samplegenerated with the G10 model there is no w bias, whilefor the sample generated with C11 there is a bias of 0.05.When evaluating all 200 validation simulations (G10 andC11 combined), Steve results in an average w bias of+0 .
03 and an average w difference (∆ w ) between Steve and the nominal method (BBC+CosmoMC) is +0 . w is 0 .
06, where this addi-tional scatter comes from the inclusion of fitted parame-ters in
Steve that are fixed in the BBC fit. For example,
Steve allows for redshift dependent populations, that arenot in the BBC fit because we find no evidence for sucha dependence (Section 5.1.1). The extra parameters alsoresult in a larger w uncertainty for Steve in comparisonto BBC.To predict ∆ w for the DES-SN3YR sample, we takethe mean ∆ w from the validation sims. For the RMS,however, the validation sims are fit with a Gaussian Ω M TABLE 11Comparison of
Steve and BBCNuisance Parameters for DES-SN3YR
Steve
BBC(G10) BBC(C11) α . +0 . − . .
146 + ± .
009 0 . ± . β . +0 . − . . ± .
11 3 . ± . γ . +0 . − . . ± .
018 0 . ± . σ int (low- z ) 0 . +0 . − . . ± .
015 0 . ± . σ int (DES-SN) 0 . +0 . − . . ± .
006 0 . ± . Note . — BBC(G10) and BBC(C11) refer to intrinsic scattermodel used to compute bias corrections. prior, N (0 . , . w ) = 0 .
04. On the DES-SN3YR dataset, we finda w -difference of 0 .
07, which is consistent with our sim-ulated prediction of 0 . ± . Steve are com-pared to those from the BBC method in Table 11. The α Steve value is about 0.02 higher than α BBC and β Steve is consistent with β BBC using the C11 intrinsic scattermodel in the bias-correction simulation. γ for both meth-ods is consistent with zero, although γ Steve is more con-sistent with γ BBC using the G10 model. Both methodsshow that the intrinsic scatter term ( σ int ) is significantlydifferent between the low- z and DES subsets, althoughthe σ int agreement between the two methods is marginal. DISCUSSION
Comparison with Other Samples
For the nominal analysis using BBC+CosmoMC, sta-tistical and systematic uncertainties on w from 329 DES-SN3YR SNe Ia are 0.042 (stat) and 0.042 (syst). Pre-vious surveys have also found that their statistical andsystematic uncertainties are roughly equal. S18 analyzedthe PS1 plus low- z subset of the Pantheon sample, andthese 451 events result in a statistical and systematicuncertainties on w of 0.046 (stat) and 0.043 (syst). Ad- rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 25
Fig. 16.—
Distance modulus uncertainty vs. redshift forDES, PS1, SNLS, and SDSS. Distance modulus measurementuncertainties reported by each survey are combined with the σ int from this work (DES) and from S18 (PS1,SNLS,SDSS). Thecolored dots are the individual SNe Ia from the DES Shallow(purple) and Deep (yellow) fields. The solid (DES) and dashed(other) lines are the binned medians of the respective distributions. ditionally, in the Joint Light Curve Analysis (B14) theyreport an uncertainty on w of 0.057 (stat+syst) using 740SNLS+SDSS+low- z +HST SNe Ia. The DES-SN3YR re-sult is a notable improvement in constraining power on w for the given sample size (329 SNe Ia), despite theconsideration of new sources of systematic uncertainty.Much of this improvement is due to the quality of theDECam CCDs which include higher sensitivity to redderwavelengths (Holland et al. 2003, Diehl et al. 2008) re-sulting in improved distance constraints for the most dis-tant supernovae. A comparison of distance uncertaintiesis shown in Figure 16 using the measurement uncertain-ties from each respective survey combined with the σ int for each survey that was derived in S18. We find thatthe DES-SN deep field SNe Ia have smaller uncertaintiesin distance modulus than SNLS, and the DES-SN shal-low field SNe Ia have smaller uncertainties than PS1 butlarger than SNLS. Prospects for Improving Systematic Uncertainties
There are several prospects for future reduction of sys-tematic uncertainties, the largest of which is due to cal-ibration. Multiple improvements are in development forthe calibration of the DES photometric system. In thiswork we used a single HST Calspec standard in oneof the SN fields to link our photometric magnitudes tothe AB system. In the last two seasons of the survey,we measured ugrizY photometry for two other CalSpecstandards (DA White Dwarfs) that are within the DESfootprint. We have identified a large number of hot DAWhite Dwarfs ( ∼ σ w = 0.014.Our low- z subset is redder than the DES-SN and otherhigh- z populations because it was part of a targeted se-lection of host-galaxies. The different color populationof the low- z subset results in increased sensitivity to thechange in scatter model. Additionally, we find that ourdataset is more sensitive to the intrinsic scatter modeluncertainty than S18. This is because the low- z sam-ple is a larger fraction of our cosmological sample (DES-SN3YR) in comparison to S18. The two intrinsic scattermodels are nearly 8 years old and there are currentlymore than ∼ w uncertainty in the parent populationsof color and stretch. There is room for improvement herein two respects: in our analysis methods and in the ex-ternal dataset used.First, in Section 4.3 of our analysis, we employed a sim-ilar ad-hoc procedure as S18 to characterize the uncer-tainty in the 6 parameters describing the parent popula-tions of color and stretch based on estimates from Scolnic& Kessler (2016). A more rigorous method of accountingfor these parameter uncertainties and covariances in theBBC method is needed for future analyses.Second, there is room for improvement from combin-ing with low- z datasets with selection effects that areless severe and better understood. The Foundation su-pernova survey (Foley et al. 2018; hereafter F18) has thepotential to reduce this uncertainty for the low- z sample.Foundation measures light curves for SNe Ia discoveredby other rolling surveys (ASA-SN, ATLAS, etc...) and asa result, obtains a sample with less galaxy-selection biasthan the current low- z sample. The Foundation low- z survey on the Pan-STARRS telescope has released 225low- z SNe Ia in DR1 and they are still collecting datawith the goal of obtaining up to 800 griz light curveswith high quality calibration. They find that the mediancolor ( c = − . x = 0 . c = − .
037 , x = 0 . z sample (i.e. CfA,CSP-1: c = − .
021 , x = 0 . z survey may also provide insightinto the distribution of residuals to the Hubble diagramat low redshift. In the DES-SN3YR analysis we find asignificant source of systematic uncertainty ( σ w = 0 . z subset dueto non-Gaussian tails in residuals to the best fit cosmo-logical model. Additional statistics will better allow usto characterize the distribution of low- z SNe Ia aboutthe Hubble diagram. The non-Gaussian Hubble resid-uals could be related to data quality, galaxy selectioneffects, unknown astrophysical effects, or poor SN mod-eling which is more apparent at high S/N. In any case,the Foundation low- z sample will facilitate further studyof this systematic. Host Mass Hubble Residual Step and IntrinsicScatter
For DES-SN3YR, we find small values for both γ and σ int . For the DES subset, γ is consistent with zero, in-dicating no evidence of a correlation between the Hub-6 Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validationble residuals to our best fit cosmology and host-galaxystellar mass. A significant correlation has been seen tovarying degrees in previous analyses (Sullivan et al. 2010,Kelly et al. 2010, Lampeitl et al. 2010, Gupta et al. 2011,D’Andrea et al. 2011, Smith et al. 2012, Wolf et al. 2016,Rigault et al. 2013) and S18 recalculated these quanti-ties within the BBC framework and recovered non-zerosteps of size: SDSS (0 . ± .
015 mag), Pan-STARRS(0 . ± .
016 mag), SNLS (0 . ± .
020 mag), and low- z (0 . ± .
030 mag). In an upcoming work, we plan tosimulate the correlations between color and host-galaxystellar mass, and the host-galaxy stellar mass Hubbleresidual step itself. However, because we recover a non-zero γ value for the low- z sample as seen in previousanalyses, we suspect that the null correlation found forthe DES subset may be the result of selecting a differentpopulation of SNe or host galaxies, but not the result ofour analysis techniques.For future surveys such as LSST and WFIRST, as wellas for low redshift studies of SNe Ia for precision H measurements, it will be important to improve analysistechniques and study selection effects on the host-galaxystellar mass correlation, especially if this effect evolveswith redshift (Childress et al. 2014). Although, in DES-SN3YR, we did not find evidence of evolution of γ as afunction of redshift.Future SN cosmology analyses will also be faced withthe decision whether to include two σ int . We have foundthat the σ int values of the low- z and DES subsets areincompatible. In this work our nominal analysis assumesa single value for σ int for historical reasons, however wefind that the systematic associated with this choice isone of the largest sources of uncertainty in our analysis.Interestingly, looking at recent SNe Ia datasets all an-alyzed with the SALT2 model and BBC 5D formalism,we find a correlation between γ and σ int of the individ-ual samples. Figure 17 shows this correlation for theDES and low- z subsets as well as for the other surveysanalyzed in S18. The incompatibility between the DESsubset and the low- z subset does not appear to be uniqueto the low- z data used in this analysis (CfA and CSP-1).Foley et al. (2018) report in their initial data release anintrinsic scatter of 0.111. The σ int - γ correlation could bea measurement artifact or σ int could have astrophysicaldependence. Future work will be focused on probing thepossibility of a redshift dependent intrinsic scatter term,but will require the use of larger datasets. As uncertaintybudgets shrink with new and larger SNe Ia samples, itwill become important for future analyses to better char-acterize this effect and model it in simulations. Simulating SNe Ia Samples
We have shown that there is still room for improvementin modeling the simulated P fit distributions (Figure 7).We find that the agreement for the DES-SN sample isbetter than that of the low- z sample, especially in therange P fit < .
5. This is in part due to the extensivecare taken to accurately simulate the DES-SN sample asdescribed in Sec 5.1.1 of K18, however it is unclear ifthe lesser agreement in the low- z sample could be the re-sult of unmodeled astrophysics. For the DES-SN sample,there is disagreement between the P fit distributions of thesimulations and the data in the highest bin ( P fit > . Fig. 17.—
Hubble residual step size in mags ( γ ) as a functionof the intrinsic scatter ( σ int ) of SNe Ia samples. The largestrolling surveys (DES, PS1, SNLS, SDSS) are shown in additionto the targeted low- z subset used in this analysis. Values for thenon-DES points are taken from Scolnic et al. (2018) and all arecalculated using 5D bias corrections using BBC fit for consistency. comparing the simulated and fake SN distributions (Fig-ure D.1). Since the same discrepancy is seen with thefakes, we rule out the possibility that this is entirely dueto SN modeling.The P fit agreement between simulations and data forthe low- z sample is poor at both low and high P fit (Fig-ure 7). This disagreement will hopefully be improvedwith the Foundation sample, which will facilitate moreaccurate simulations. In addition, our DES-SN samplehas an additional 90,000 fake supernovae on which we canrun SMP and improve our modeling of flux uncertaintiesin the simulation.
Improvements to the Validation
The validations described in Section 6 are the mostextensive for a SN Ia cosmology analysis pipeline todate. Using fakes we have validated from discovery onDECam images to cosmological parameters, and usingcatalog-level simulations and we have validated the w bias ( < .
01) and treatment of systematic uncertainty.Future work will expand the number of systematics inTable 10. Additionally, because we utilize BBC, whichuses an approximate χ likelihood assuming symmetricGaussian uncertainties, we will validate the BBC confi-dence region for binned distances, and this will eventuallylead to comparing the cosmology likelihoods between theBBC+CosmoMC and Bayesian ( Steve , Section 7) meth-ods. In addition to comparing likelihoods between meth-ods, ideally we would compare our BBC+CosmoMC like-lihood to a true likelihood such as from the Neymanconstruction (Tanabashi et al. 2018). However, such acomparison is computationally challenging. CONCLUSION
We have presented the analysis, cosmological pa-rameter uncertainty budget, and validation of DES-SN3YR sample consisting of of 207 spectroscopically con-firmed Type Ia Supernovae (0 . < z < .
85) discoveredby DES-SN and an external sample of 122 low-redshift rout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 27SNe Ia after quality cuts (0 . < z < . σ w =0.057 (stat+syst). The calibration of the various sam-ples used is the largest source of systematic uncertainty.Additionally we find no correlation between host-galaxystellar mass and Hubble residuals to the best fit cosmol-ogy.Our validation using a population of fake SNe injectedonto real images is the first such test for potentialbiases through the entire SNe Ia discovery, photometry,and analysis pipelines. Resulting biases in distanceare limited to 1% and the fit value of w is consistentwith the cosmology in which the fakes were generated.Additionally, we discuss a rigorous method of validatingthe interpretation of the total uncertainty budget usinghundreds of catalog-level simulations. We find that afteraccounting for sources of systematic uncertainty thereare no significant biases in the cosmological parameteranalysis pipeline and that the RMS( w ) and the averageuncertainty agree to within 6%. The sample fromDES used for this analysis is roughly 10% of the fullDES photometric sample, and treatment and validationof systematic w uncertainties will become even morecrucial with the larger sample.This paper has gone through internal review by theDES collaboration. DB and MS were supported by DOEgrant DE-FOA-0001358 and NSF grant AST-1517742.This research used resources of the National Energy Re-search Scientific Computing Center (NERSC), a DOEOffice of Science User Facility supported by the Officeof Science of the U.S. Department of Energy under Con-tract No. DE-AC02-05CH11231. We are grateful for thesupport of the University of Chicago Research Comput-ing Center for assistance with the calculations carriedout in this work. Part of this research was conductedby the Australian Research Council Centre of Excellencefor All-sky Astrophysics (CAASTRO), through projectnumber CE110001020.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 Uni-versities for Research in Astronomy (AURA) under a co-operative agreement with the National Science Founda-tion.This paper makes use of observations taken us-ing the Anglo-Australian Telescope under programsATAC A/2013B/12 and NOAO 2013B-0317; the Gem-ini Observatory under programs NOAO 2013A-0373/GS-2013B-Q-45, NOAO 2015B-0197/GS-2015B-Q-7, andGS-2015B-Q-8; the Gran Telescopio Canarias un-der programs GTC77-13B, GTC70-14B, and GTC101-15B; the Keck Observatory under programs U063-2013B, U021-2014B, U048-2015B, U038-2016A; the Mag-ellan Observatory under programs CN2015B-89; theMMT under 2014c-SAO-4, 2015a-SAO-12, 2015c-SAO-21; the South African Large Telescope under programs2013-1-RSA OTH-023, 2013-2-RSA OTH-018, 2014-1-RSA OTH-016, 2014-2-SCI-070, 2015-1-SCI-063, and2015-2-SCI-061; and the Very Large Telescope underprograms ESO 093.A-0749(A), 094.A-0310(B), 095.A-0316(A), 096.A-0536(A), 095.D-0797(A).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, ESP2015-66861, FPA2015-68048, SEV-2016-0588, SEV-2016-0597,and MDM-2015-0509, some of which include ERDFfunds from the European Union. IFAE is partiallyfunded by the CERCA program of the Generalitat deCatalunya. Research leading to these results has re-ceived funding from the European Research Councilunder the European Union’s Seventh Framework Pro-gram (FP7/2007-2013) including ERC grant agreements240672, 291329, 306478 and 615929. We acknowledgesupport from the Australian Research Council Cen-tre of Excellence for All-sky Astrophysics (CAASTRO),through project number CE110001020, and the Brazil-ian Instituto Nacional de Ciˆencia e Tecnologia (INCT)e-Universe (CNPq grant 465376/2014-2).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 States8 Brout et al.
First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation
TABLE C.1light curve Fit Parameters
SN-ID z CMB c x m B log( M stellar / M (cid:12) ) µ µ corr ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Note . — SN-ID, redshift, light curve fit parameters, host-galaxy stellar mass, distance moduli, and distance bias correctionsof DES-SNe after quality cuts using the G10 model of intrinsic scatter. A subset of SNe are shown here. The full version of thistable can be found online following the link in Appendix C for both models of intrinsic scatter (G10 and C11) as well additionalinformation including RA, DEC, fit parameter covariances, 5D bias corrections, and more.
Government 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.Based in part on data acquired through theAustralian Astronomical Observatory, under program A/2013B/012. We acknowledge the traditional ownersof the land on which the AAT stands, the Gamilaraaypeople, and pay our respects to elders past and present.
APPENDIX
A. LIGHT CURVE MINIMIZATION ALGORITHMS
Light curve parameter minimization is performed with
SNANA ’s implementation of SALT2 (Guy et al. 2007) basedon CERNLIBs
MINUIT program (James & Roos 1975) using
MINOS minimization. There is an alternative minimizationmethod,
MIGRAD , however we found that it causes pathological errors for 2% of our sample of SNe Ia, resulting inincorrect weighting in the SALT2mu distance fitting process.
MINOS was found to avoid the pathological color errorsalthough it is 2.5x slower than
MIGRAD . MIGRAD ’s speed is useful for development and debugging, however for the finalcosmological analysis we use
MINOS .There are additional fitting anomalies that occur for high-SNR events for both
MIGRAD and
MINOS . These algorithmssometimes fall in false minima, and to avoid these anomalies we add 3% of peak SN flux to all flux uncertainties onthe first of three fit iterations.
B. PUBLIC PRODUCTS USED IN THE ANALYSIS
PEGASE (Fioc & Rocca-Volmerange 1997), Le Phare (Arnouts & Ilbert 2011),
SMP (Brout et al. 2019-SMP),AutoScan (Goldstein et al. 2015), SALT2 models (Guy et al. 2010, B14),
SNANA (Kessler et al. 2009c, K18), CosmoMC(Lewis & Bridle 2002), SNID (Blondin & Tonry 2007), MARZ (Hinton et al. 2016), ZPEG (Le Borgne & Rocca-Volmerange 2002), Superfit (Howell et al. 2005).
C. DATA RELEASE PRODUCTS
DES-SN3YR binned and unbinned distances, measurement uncertainties and covariance are included athttps://des.ncsa.illinois.edu/releases/sn as well as the full Table C.1 in machine readable format.
D. ANALYSIS OF THE FAKES
Here we describe a few details about the cosmology analysis with fake SN light curve fluxes overlaid on DECamimages. To avoid confusion between two sets of
SNANA simulations, we define SIM1 for simulated fluxes overlaid onimages, and SIM2 for the bias-correction simulation used in the BBC fitting stage.For SIM1, SN Ia lightcurve fluxes were generated in a LCDM cosmology over a redshift range from 0.1 to 1.2. Thesefluxes and were inserted as point sources onto DECam images at galaxy locations chosen randomly with probabilityproportional to its surface brightness density. The generation of fake lightcurves and the procedure for image overlaysare described in detail in Section 2 of K18.
DiffImg discovered 40% of the 100,000 fake SNe Ia lightcurves thatwere inserted on the DES-SN images and the
SMP pipeline was run on a representative subset of 10,000 lightcurves.Analysis requirements and SALT2 lightcurve fitting resulted in a sample of 6586 fake SNe Ia that are fit with BBCand CosmoMC.For the BBC fit we create a bias correction sample from
SNANA simulations (SIM2). The underlying SN Ia lightcurve model is identical to that used in SIM1: e.g., color & stretch population, and no intrinsic scatter. In the firstseason (Y1), there was a SIM1 generation bug forcing the same galactic extinction ( E ( B − V = 0 . E spec = 1for both SIM1 and SIM2.Finally, the SIM2 redshift distribution was tuned in each of the ten SN fields to match SIM1 after cuts. Thisfield-dependent redshift tuning was needed because of the subtle way that SIM1 had selected real host-galaxies from rout et al. First Cosmology Results From DES-SN: Analysis, Systematic Uncertainties, and Validation 29the science verification (SV) catalog. Although a single host-galaxy z dependence was specified, the non-uniformdepth of the SV galaxy catalog resulted in a different redshift distribution in each field. To illustrate this feature,consider an extreme example with just two fields (e.g., E1, E2). Next, suppose that the galaxy catalog for E1 onlyincludes redshifts z < . z > .
5. A simulation generating a flat galaxy redshift distributionover 0 < z <
REFERENCES
Fig. D.1.—
Comparison of 6,586 fake supernova light curve fits with simulations used to compute biases in a fake cosmology analysis.Abbott, T. M. C., et al. 2018, ArXiv e-printsAbbott, T. M. C., et al. 2019, ApJ, 872, L30Arnouts, S., & Ilbert, O. 2011, LePHARE: Photometric Analysisfor Redshift Estimate, Astrophysics Source Code LibraryAstier, P., et al. 2013, A&A, 557, A55Astier, P., et al. 2006, A&A, 447, 31Bernstein, G. M., et al. 2017, PASP, 129, 114502Bernstein, J. P., et al. 2012, The Astrophysical Journal, 753, 152Betoule, M., et al. 2014a, A&A, 568, A22Betoule, M., et al. 2014b, A&A, 568, A22Blondin, S., & Tonry, J. L. 2007, ApJ, 666, 1024Bohlin, R. C., Gordon, K. D., & Tremblay, P.-E. 2014, PASP,126, 711Bonnett, C., et al. 2016, Phys. Rev. D, 94, 042005Brout, D. 2018, in American Astronomical Society MeetingAbstracts, Vol. 231, American Astronomical Society MeetingAbstracts Brout et al.