arXiv:2007.08991v1 [astro-ph.CO] 17 Jul 2020 · Corresponding Author: [email protected] 1 University of Edinburgh, Edinburgh EH8 9YL, UK 2 Aix Marseille Univ, CNRS/IN2P3,

Submitted to Physical Review DPreprint typeset using LATEX style emulateapj v 012315

THE COMPLETED SDSS-IV EXTENDED BARYON OSCILLATION SPECTROSCOPIC SURVEYCOSMOLOGICAL IMPLICATIONS FROM TWO DECADES OF SPECTROSCOPIC SURVEYS AT THE

APACHE POINT OBSERVATORY

Shadab Alam1 Marie Aubert2 Santiago Avila34 Christophe Balland5 Julian E Bautista6 Matthew ABershady789 Dmitry Bizyaev1011 Michael R Blanton12 Adam S Bolton1314 Jo Bovy1516 Jonathan

Brinkmann10 Joel R Brownstein14 Etienne Burtin17 Solene Chabanier17 Michael J Chapman1819 PeterDoohyun Choi20 Chia-Hsun Chuang21 Johan Comparat22 Andrei Cuceu23 Kyle S Dawson14 Axel de la

Macorra24 Sylvain de la Torre25 Arnaud de Mattia17 Victoria de Sainte Agathe5Helion du Mas des Bourboux14 Stephanie Escoffier2 Thomas Etourneau17 James Farr23 AndreuFont-Ribera2623 Peter M Frinchaboy27 Sebastien Fromenteau28 Hector Gil-Marın29 Alma X

Gonzalez-Morales3031 Violeta Gonzalez-Perez326 Kathleen Grabowski10 Julien Guy33 Adam J Hawken2Jiamin Hou22 Hui Kong34 Mark Klaene10 Jean-Paul Kneib35 Jean-Marc Le Goff17 Sicheng Lin12 Daniel

Long10 Brad W Lyke36 Marie-Claude Cousinou2 Paul Martini3734 Karen Masters38 Faizan GMohammad1819 Jeongin Moon20 Eva-Maria Mueller396 Andrea Munoz-Gutierrez24 Adam D Myers36

Seshadri Nadathur6 Richard Neveux17 Jeffrey A Newman40 Pasquier Noterdaeme41 Audrey Oravetz10Daniel Oravetz10 Nathalie Palanque-Delabrouille17 Kaike Pan10 James Parker III10 Romain Paviot25 WillJ Percival181942 Ignasi Perez-Rafols5 Patrick Petitjean41 Matthew M Pieri25 Abhishek Prakash43 Anand

Raichoor35 Corentin Ravoux17 Mehdi Rezaie44 James Rich17 Ashley J Ross34 Graziano Rossi20 RossanaRuggeri456 Vanina Ruhlmann-Kleider17 Ariel G Sanchez22 F Javier Sanchez46 Jose R Sanchez-Gallego47Conor Sayres48 Donald P Schneider4950 Hee-Jong Seo44 Arman Shafieloo5152 Anze Slosar53 Alex Smith17Julianna Stermer5 Amelie Tamone35 Jeremy L Tinker54 Rita Tojeiro55 Mariana Vargas-Magana24 Andrei

Variu35 Yuting Wang56 Benjamin A Weaver13 Anne-Marie Weijmans55 Christophe Yeche17 PaulineZarrouk5717 Cheng Zhao35 Gong-Bo Zhao56 and Zheng Zheng14

(eBOSS)Submitted to Physical Review D

ABSTRACT

We present the cosmological implications from final measurements of clustering using galaxiesquasars and Lyα forests from the completed Sloan Digital Sky Survey (SDSS) lineage of experi-ments in large-scale structure These experiments composed of data from SDSS SDSS-II BOSSand eBOSS offer independent measurements of baryon acoustic oscillation (BAO) measurementsof angular-diameter distances and Hubble distances relative to the sound horizon rd from eightdifferent samples and six measurements of the growth rate parameter fσ8 from redshift-space dis-tortions (RSD) This composite sample is the most constraining of its kind and allows us to performa comprehensive assessment of the cosmological model after two decades of dedicated spectroscopicobservation We show that the BAO data alone are able to rule out dark-energy-free models at morethan eight standard deviations in an extension to the flat ΛCDM model that allows for curvatureWhen combined with Planck Cosmic Microwave Background (CMB) measurements of temperatureand polarization under the same model the BAO data provide nearly an order of magnitude improve-ment on curvature constraints relative to primary CMB constraints alone Independent of distancemeasurements the SDSS RSD data complement weak lensing measurements from the Dark EnergySurvey (DES) in demonstrating a preference for a flat ΛCDM cosmological model when combined withPlanck measurements The RSD and lensing measurements indicate a growth rate that is consistentwith predictions from Planck temperature and polarization data and with General Relativity Whencombining the results of SDSS BAO and RSD Planck Pantheon Type Ia supernovae (SNe Ia) andDES weak lensing and clustering measurements all multiple-parameter extensions remain consistentwith a ΛCDM model Regardless of cosmological model the precision on each of the three ΛCDMparameters ΩΛ H0 and σ8 remains at roughly 1 showing changes of less than 06 in the centralvalues between models In a model that allows for free curvature and a time-evolving equation ofstate for dark energy the combined samples produce a constraint Ωk = minus00023plusmn 00022 The darkenergy constraints lead to w0 = minus0912plusmn0081 and wa = minus048+036

minus030 corresponding to an equation ofstate of wp = minus1020plusmn 0032 at a pivot redshift zp = 029 and a Dark Energy Figure of Merit of 92The inverse distance ladder measurement under this model yields H0 = 6820 plusmn 081 km sminus1Mpcminus1remaining in tension with several direct determination methods the BAO data allow Hubble constantestimates that are robust against the assumption of the cosmological model In addition the BAOdata allow estimates of H0 that are independent of the CMB data with similar central values and pre-cision under a ΛCDM model Our most constraining combination of data gives the upper limit on thesum of neutrino masses at

summν lt 0111 eV (95 confidence) Finally we consider the improvements

in cosmology constraints over the last decade by comparing our results to a sample representative ofthe period 2000ndash2010 We compute the relative gain across the five dimensions spanned by w Ωksummν H0 and σ8 and find that the SDSS BAO and RSD data reduce the total posterior volume

2 eBOSS Collaboration

by a factor of 40 relative to the previous generation Adding again the Planck DES and PantheonSN Ia samples leads to an overall contraction in the five-dimensional posterior volume of three ordersof magnitude

1 INTRODUCTION

Understanding the energy content of the Universe thephysical mechanisms behind cosmic expansion and the

PI kdawsonastroutahedu Corresponding Author evamuellerphysicsoxacuk

1 University of Edinburgh Edinburgh EH8 9YL UK2 Aix Marseille Univ CNRSIN2P3 CPPM Marseille France3 Universidad Autonoma de Madrid 28049 Madrid Spain4 Instituto de Fisica Teorica UAMCSIC Universidad Au-

tonoma de Madrid 28049 Madrid Spain5 Sorbonne Universite Universite Paris Diderot

CNRSIN2P3 Laboratoire de Physique Nucleaire et deHautes Energies LPNHE 4 Place Jussieu F-75252 ParisFrance

6 Institute of Cosmology amp Gravitation University ofPortsmouth Dennis Sciama Building Burnaby RoadPortsmouth PO1 3FX UK

7 Department of Astronomy University of Wisconsin-Madison Madison WI 53706 USA

8 South African Astronomical Observatory PO Box 9 Obser-vatory 7935 Cape Town South Africa

9 Department of Astronomy University of Cape TownPrivate Bag X3 Rondebosch 7701 South Africa

10 Apache Point Observatory and New Mexico State Univer-sity PO Box 59 Sunspot NM 88349

11 Sternberg Astronomical Institute Moscow State Univer-sity Moscow 119992 Russia

12 Center for Cosmology and Particle Physics Department ofPhysics New York University 726 Broadway Room 1005 NewYork NY 10003 USA

13 NSFrsquos National Optical-Infrared Astronomy ResearchLaboratory 950 N Cherry Ave Tucson AZ 85719 USA

14 University of Utah Department of Physics and Astronomy115 S 1400 E Salt Lake City UT 84112 USA

15 Department of Astronomy and Astrophysics University ofToronto ON M5S3H4

16 Dunlap Institute for Astronomy and Astrophysics Univer-sity of Toronto ON M5S 3H4 Canada

17 IRFU CEA Universite Paris-Saclay F-91191 Gif-sur-Yvette France

18 Waterloo Centre for Astrophysics University of WaterlooWaterloo ON N2L 3G1 Canada

19 Department of Physics and Astronomy University ofWaterloo 200 University Ave W Waterloo ON N2L 3G1Canada

20 Department of Physics and Astronomy Sejong UniversitySeoul 143-747 Korea

21 Kavli Institute for Particle Astrophysics and CosmologyStanford 94305

22 Max-Planck-Institut fur extraterrestrische Physik (MPE)Giessenbachstrasse 1 D-85748 Garching bei Munchen Germany

23 University College London WC1E 6BT London UnitedKingdom

24 IFUNAM - Instituto de Fısica Universidad NacionalAutonoma de Mexico 04510 CDMX Mexico

25 Aix Marseille Univ CNRS CNES LAM Marseille France26 Institut de Fısica dAltes Energies The Barcelona Institute

of Science and Technology Campus UAB 08193 Bellaterra(Barcelona) Spain

27 Department of Physics amp Astronomy Texas ChristianUniversity Fort Worth TX 76129 USA

28 ICFUNAM - Instituto de Ciencias Fısicas UniversidadNacional Autonoma de Mexico 62210 Cuernavaca MorMexico

29 ICC University of Barcelona IEEC-UB Martı i Franques1 E08028 Barcelona Spain

30 Division de Ciencias e Ingenierıas Universidad de Guana-juato Leon 37150 Mexico

31 Consejo Nacional de Ciencia y Tecnologıa Av InsurgentesSur 1582 Colonia Credito Constructor Del Benito JuarezCP 03940 Mexico DF Mexico

growth of structure are the primary challenges of cos-mology Developmental milestones for the current stan-dard model of these properties the spatially-flat ΛCDMmodel include measurements of the expansion historyusing Type Ia supernovae (SNe Ia) in the 1990rsquos whichprovided the first evidence for cosmic acceleration (Riesset al 1998 Perlmutter et al 1999) and studies of per-turbations in the cosmic microwave background (CMB)which provided the first convincing evidence for a nearlyflat geometry (de Bernardis et al 2000 Balbi et al 2000Jaffe et al 2001) when assuming weak priors and fittingresults from the BOOMERanG (Netterfield et al 2002)and MAXIMA (Hanany et al 2000) CMB experimentsAt around the same time as these observations the firstmeasurements of the baryon and matter densities fromthe shape of the power spectrum from the 2dFGRS (Col-less et al 2001) were published (Percival et al 2001) Thecombination of the galaxy survey data and CMB datais particularly strong for breaking degeneracies inherent

32 Liverpool John Moores University L3 5RF Liverpool United Kingdom

33 Lawrence Berkeley National Laboratory BerkeleyCA 94720 USA

34 Center for Cosmology and AstroParticle Physics The OhioState University Columbus OH 43212

35 Institute of Physics Laboratory of Astrophysics EcolePolytechnique Fdrale de Lausanne (EPFL) Observatoire deSauverny 1290 Versoix Switzerland

36 Department of Physics and Astronomy University ofWyoming Laramie WY 82071 USA

37 The Ohio State University Columbus OH 4321238 Haverford College 370 Lancaster Ave Haverford PA

19041 USA39 Sub-department of Astrophysics Department of Physics

University of Oxford Denys Wilkinson Building Keble RoadOxford OX1 3RH

40 University of Pittsburgh and PITT PACC Pittsburgh PA15260

41 Institut drsquoAstrophysique de Paris CNRS amp SorbonneUniversity UMR 7095 98bis bd Arago 75014 Paris France

42 Perimeter Institute Waterloo ON N2L 2Y5 Canada43 California Institute of Technology Pasadena CA 9112544 Department of Physics and Astronomy Ohio University

Clippinger Labs Athens OH 45701 USA45 Swinburne University of Technology Centre for Astro-

physics and Supercomputing Melbourne VIC 3122 Australia46 Fermi National Accelerator Laboratory Batavia IL 60510

USA47 Department of Astronomy University of Washington Box

351580 Seattle WA 98195 USA48 University of Washington Seattle 9819549 The Pennsylvania State University University Park PA

1680250 Institute for Gravitation and the Cosmos The Pennsylva-

nia State University University Park PA 1680251 Korea Astronomy and Space Science Institute Daejeon

34055 Korea52 University of Science and Technology Daejeon 34113

Korea53 Brookhaven National Laboratory Upton NY 11973 USA54 New York University New York NY 1000355 School of Physics and Astronomy University of St An-

drews North Haugh St Andrews KY16 9SS UK56 National Astronomical Observatories of China Chinese

Academy of Sciences 20A Datun Road Chaoyang DistrictBeijing 100012 China

57 Institute for Computational Cosmology Dept of PhysicsUniv of Durham South Road Durham DH1 3LE UK

Cosmology from eBOSS 3

to either method individually combining early 2dFGRSand CMB data meant that at around the turn of thecentury the physical baryon and cold dark matter den-sities were known to 10 and 8 respectively and theHubble parameter to 7 within the flat ΛCDM model(Percival et al 2002)

The first decade of the 21st century witnessed a strongadvancement in the precision with which the parame-ters of this standard model were known without demon-strating significant tension with this model This camethrough dedicated CMB experiments including ACT(Fowler et al 2007) SPT (Carlstrom et al 2011) and theWilkinson Microwave Anisotropy Probe (WMAP Ben-nett et al 2013) SN Ia observations continued to im-prove in sample size and analysis methodology (Jha et al2006 Riess et al 2007 Frieman et al 2008 Dawson et al2009 Hicken et al 2009 Contreras et al 2010 Guy et al2010 Conley et al 2011 Sullivan et al 2011) and directmeasurements of the local expansion rate using Cepheidvariables and SNe Ia led to estimates of H0 with bet-ter than 4 precision (Riess et al 2009 Freedman et al2012) During this same approximate period the 2dF-GRS and Sloan Digital Sky Survey (SDSS York et al2000) galaxy surveys became sufficiently large to clearlymeasure the Baryon Acoustic Oscillation (BAO) featurein the clustering of galaxies (Eisenstein et al 2005 Coleet al 2005) and use this as a robust cosmological probe(Percival et al 2007) Combined these experiments of-fered strong evidence supporting the simple six parame-ter ΛCDM cosmological model consisting of the baryondensity (Ωb) dark matter density (Ωc) Hubble Con-stant (H0) amplitude of primordial perturbations (As)power-law spectral index of primordial density perturba-tions (ns) and reionization optical depth (τ) The 5-yearWMAP data (Hinshaw et al 2009) combined with theSDSS-II BAO data (Percival et al 2007) and the unionSN sample (Kowalski et al 2008) led to measurementsof the physical baryon and cold dark matter densities to3 and the Hubble parameter to 2 (Komatsu et al2009) within the framework of the ΛCDM model

The last ten years have seen significant advances in cos-mology through CMB observations improved calibrationof systematic errors in SNe Ia studies and large areaspectroscopic surveys Gravitational lensing from theCMB has provided important high signal-to-noise mea-surements of structure growth in the low redshift uni-verse (Planck Collaboration et al 2014 2018c) CMBlensing has been supplemented by increasingly robustand statistically sensitive estimates of weak lensing basedon galaxy shapes including CFHTLenS (Heymans et al2012) KiDS (Kohlinger et al 2017) Dark Energy Sur-vey (DES Zuntz et al 2018 Abbott et al 2018) andHyper Suprime-Cam survey (HSC Mandelbaum et al2018 Hikage et al 2019)

The continuing development of massive spectroscopicsurveys over the last decade is of particular interest tothis study Spectroscopy of galaxies and quasars overwide areas allows precise measurements of cosmic expan-sion history with BAO and measurements of the rate ofstructure growth with redshift space distortions (RSD)The largest spectroscopic survey to date is the BaryonOscillation Spectroscopic Survey (BOSS Dawson et al2013) which was the primary driver for SDSS-III (Eisen-stein et al 2011) In operations spanning 2009ndash2014

BOSS completed spectroscopy on more than 15 milliongalaxies as faint as i = 199 and more than 160000z gt 21 quasars as faint as g = 22 In 2012 the firstBAO measurements from BOSS were published (An-derson et al 2012) just before the final results fromthe WMAP CMB experiment At this point the datawere sufficient to set interesting constraints on modelsthat go beyond ΛCDM For example an analysis un-der a flat ΛCDM model with neutrinos using the finalWMAP data an estimate of H0 = 738 plusmn 24 km sminus1

Mpcminus1 (Riess et al 2011) and the BOSS BAO measure-ments (Anderson et al 2012) together with those fromthe 6dFGS SDSS-II and WiggleZ surveys (Beutler et al2011 Padmanabhan et al 2012 Blake et al 2012) led toa 95 upper limit of 044 eV on the sum of the neutrinomasses (Hinshaw et al 2013) Adding measurements ofluminosity-distance ratios from a large sample of SNe Ia(Guy et al 2010 Conley et al 2011 Sullivan et al 2011)led to constraints consistent with a cosmological con-stant when allowing a Chevallier-Polarski-Linder (CPL)parameterization (Chevallier amp Polarski 2001 Linder2003) of dark energy indicating w0 = minus117+013

minus012 and

wa = 035+050minus049 in a model with a flat universe (Hinshaw

et al 2013) Within the ΛCDM model the errors on thephysical baryon density cold dark matter density werenow at the level of 2 and the Hubble Constant 13

Final measurements of the CMB-calibrated BAO scalefrom the BOSS experiment led to 1 precision mea-surements of the cosmological distance scale for redshiftsz lt 075 (Alam et al 2017) and 2 precision measure-ments at z = 233 (Bautista et al 2017 du Mas desBourboux et al 2017) By the time that the final resultsfrom BOSS were ready the Planck satellite had releasedits 2015 CMB measurements (Planck Collaboration et al2016a) surpassing the precision afforded by WMAP Incombination the 2015 CMB power spectrum measure-ments from the Planck satellite together with BOSS con-strain the rate of structure growth at the level of 6 andconstrain the sum of the neutrino masses to be less than160 meV at 95 confidence (Alam et al 2017) Withthese data the constraints on the physical matter den-sity and Hubble Constant within the ΛCDM model wereboth at the level of 06

So far there have been four generations of SDSS con-ducted from the 25-meter Sloan Telescope (Gunn et al2006) at the Apache Point Observatory The extendedBaryon Oscillation Spectroscopic Survey (eBOSS Daw-son et al 2016)1 is the cosmological survey within SDSS-IV (Blanton et al 2017) Using the same spectrographsused for BOSS (Smee et al 2013) eBOSS concluded 45years of spectroscopic observations of large-scale struc-ture on March 1 2019 eBOSS extends the BOSS anal-ysis using galaxies as direct tracers of the density fieldto measure BAO and RSD to higher redshifts and in-creases the number of quasars used for Lyα forest stud-ies It also marks the last use of the Sloan Telescope forgalaxy redshift surveys designed to measure cosmologicalparameters using BAO and RSD techniques with SDSSnow focusing on other exciting astronomical questions(Kollmeier et al 2017)

In this paper we characterize the advances made in

1 httpswwwsdssorgsurveyseboss

constraining the cosmological model over the last decadefocusing specifically on the impact of the BOSS andeBOSS spectroscopic programs A summary of the keyresults from this work as well as a few additional fig-ures can be found in the SDSS webpages2 The studypresented in this work is part of a coordinated release ofthe final eBOSS measurements of BAO and RSD in theclustering of luminous red galaxies (LRG Bautista et al2020 Gil-Marin et al 2020) emission line galaxies (ELGRaichoor et al 2020 Tamone et al 2020 de Mattia et al2020) and quasars (Hou et al 2020 Neveux et al 2020)At the highest redshifts (z gt 21) the coordinated re-lease of final eBOSS measurements includes measure-ments of BAO in the Lyα forest (du Mas des Bourbouxet al 2020) An essential component of these studiesis the construction of data catalogs (Ross et al 2020Lyke et al 2020) mock catalogs (Lin et al 2020 Zhaoet al 2020 Farr et al 2020) and galaxy mocks based onN-body simulations for assessing theoretical systematicerrors (Alam et al 2020b Avila et al 2020 Rossi et al2020 Smith et al 2020) A summary table of the BAOand RSD measurements with links to supporting studiesand legacy figures describing the measurements can befound in the SDSS webpages3

In all the SDSS BOSS and eBOSS surveys providegalaxy and quasar samples from which BAO can be mea-sured covering all redshifts z lt 22 and Lyα forest ob-servations over 2 lt z lt 35 The aggregate precision ofthe expansion history measurements is 070 at redshiftsz lt 1 and 119 at redshifts z gt 1 while the aggregateprecision of the growth measurements is 477 over theredshift interval 0 lt z lt 15 With this coverage andsensitivity the SDSS experiment is unparalleled in itsability to explore models of dark energy

In Section 2 we present the cosmological backgroundand the signatures in the key observational probes Thissection is intended to provide a high level backgroundthat will put the SDSS spectroscopic surveys into thebroader context for relatively new readers In Section 3we present the data samples for the cosmological analy-ses performed in this work In Section 4 we discuss theimpact of SDSS BAO distance measurements on singleparameter extensions to ΛCDM relative to SNe Ia andCMB probes We also demonstrate the key contributionsfrom BAO measurements in the well-known tensionbetween local measurements of H0 and estimates ex-trapolated from high-redshift observations In Section 5we discuss RSD and weak lensing measurements both inconstraining the relative abundance of dark energy andin testing predictions of growth under an assumptionof General Relativity (GR) In Section 6 we presentthe cosmological model that best describes all of theobservational data used in this work We conclude inSection 7 by presenting the substantial advances in ourunderstanding of the cosmological model that have beenmade in the last decade and the role that the BOSS andeBOSS programs play in those advances

2 httpswwwsdssorgsciencecosmology-results-from-eboss

3 httpswwwsdssorgsciencefinal-bao-and-rsd-measurements

2 COSMOLOGICAL MODEL AND OBSERVABLESIGNATURES

The BOSS and eBOSS surveys have fostered the de-velopment of the BAO technique to percent-level preci-sion over a larger redshift range than any other probe ofthe distance-redshift relation RSD measurements fromBOSS and eBOSS offer constraints on structure growthover nearly as large a redshift range Meanwhile in thelast ten years the CMB maps produced by the Plancksatellite have allowed precise constraints on the condi-tions of the Universe at the time of last scattering andon the angular diameter distance to that epoch Withprobes of the late-time expansion history the evolutionof cosmic expansion can be extrapolated from the CMBto todayrsquos epoch under models with freedom for curva-ture dark energy density dark energy equation of stateand neutrino mass SNe Ia measurements remain themost effective way to constrain expansion history at red-shifts below z lt 03 while the BOSS and eBOSS BAOmeasurements cover redshifts 0 lt z lt 25 and are themost well understood of late-time probes Large weaklensing surveys have measured cosmic shear to constrainthe local matter density and amplitude of fluctuationswhile RSD measure the change in the fluctuation am-plitude with time through measurements of the gravita-tional infall of matter

In this section we provide an overview of the cosmo-logical model and a pedagogical summary of the observa-tional signatures in BAO RSD CMB SNe Ia and weaklensing surveys that we use to provide new constraints onthat model This section is intended to provide the keydetails of the cosmological models and data sets that areexplored in the remainder of the paper The discussionwill be familiar to the reader experienced in multi-probecosmology constraints and will offer the highlights foradditional study for the less experienced reader

21 Background Models and Notation

Throughout this paper we employ the standard cosmo-logical model based on the Friedmann-Robertson-Walkermetric where the scale factor a is unity today and is re-lated to redshift by a(t) = (1+z)minus1 The evolution of thescale factor with time describes the background expan-sion history of the Universe governed by the Friedmannequation normally written as

H2(a) =8πG

3ρ(a)minus kc2

a2 (1)

H equiv aa is the Hubble parameter and ρ(a) is the to-tal energy density (radiation + matter + dark energy)The curvature constant k parameterizes the global curva-ture of space An open universe is represented by k lt 0and a closed universe by k gt 0 The curvature termcan be expressed in terms of an effective energy densitythrough minuskc2a2 = (8πG3)ρk(a) However we notethat a Universe that is globally flat (k = 0) will appearto have a non-zero mean curvature due to horizon-scalefluctuations in the matter density field These large-scalefluctuations place a fundamental limit on constraints onthe curvature term under inflationary models that bestdescribe CMB fluctuations and the detectable limit isroughly one part in 10000 (Vardanyan et al 2009)

We define the dimensionless density parameter of each

energy component (x) by the ratio

Ωx =ρxρcrit

3H2ρx (2)

so thatsum

Ωx = 1 where the sum is over all energycomponents including the curvature Density param-eters and ρcrit always refer to values at z = 0 un-less a dependence on a or z is stated explicitly egΩx(z) We will frequently refer to the present-day (t0)Hubble parameter H0 through the dimensionless ratioh equiv H0100 km sminus1 Mpcminus1 The dimensionless quantityωx equiv Ωxh

2 is proportional to the physical density ofcomponent x at the present day

The energy components considered in our models arepressureless (cold) dark matter (CDM) baryons pho-tons neutrinos and dark energy The densities of CDMand baryons scale as aminus3 we refer to the density parame-ter of these two components together as Ωcb The energydensity in radiation (Ωr) scales as aminus4 in the standardcosmological model Ωr is dynamically negligible in thelow redshift universe probed by spectroscopic surveysHowever the radiation density is dominant at very highredshift where it receives contributions from the electro-magnetic CMB radiation (ωγ known exquisitely well)and from neutrinos (at temperature T higher than therest energy mν)

ωr(T gt mν) = ωγ + ων =

(3)with Neff = 3045 in the standard case with three neu-trino species (de Salas amp Pastor 2016) (note follow-ing precedent we use Neff = 3046 throughout as inMangano et al 2005) Other potential contributions toradiation energy density are traditionally parameterizedin terms of their change to the effective number of neu-trino species ∆Neff regardless of whether they representextra neutrino species or other light degrees of freedom

While the effect of neutrinos in cosmology has beendetected through their contribution to the radiation en-ergy density in the CMB (Planck Collaboration et al2018b) we have not yet reached the sensitivity to de-tect their mass However the detection of neutrino os-cillations in terrestrial experiments strongly implies thatat least two species are massive and that at least onespecies is now non-relativistic (see de Salas et al 2018for a recent review) The energy density of neutrinoswith non-zero mass scales like radiation at early timeswhen the particles are ultra-relativistic Once cosmic ex-pansion reduces their kinetic energy below the rest massthe neutrinos transition towards dark matter behaviorFor neutrinos with non-degenerate mass eigenstates thetransition to non-relativistic energies will happen at dif-ferent epochs for the three eigenstates In practice forrealistic neutrino masses the transition occurs after theepoch of the last scattering in the CMB but before theepochs observed by spectroscopic surveys Therefore wecan safely assume that the most massive neutrino speciesare ultra-relativistic at epochs relevant for the CMB andact as dark matter at epochs probed by galaxy surveys(Lesgourgues amp Pastor 2006)

At the current level of precision cosmological measure-ments are sensitive only to the sum of neutrino mass

eigenvalues (Slosar 2006 Lesgourgues amp Pastor 2006Font-Ribera et al 2014 de Bernardis et al 2009 Jimenezet al 2010) thus allowing a simple modeling of neutrinomasses with a single parameter

summν We use νΛCDM

to refer to the flat ΛCDM model with this extra free pa-rameter Following standard convention our total mat-ter density at redshift z = 0 therefore includes neutrinosΩm = Ωcb + Ων

Finally the dark energy component is approximatelyconstant in time and thus dominates the late-time evo-lution of the Universe (all the other components scale atleast with aminus2) Conventionally the dark energy compo-nent is parameterized in terms of its pressure-to-densityratio w = pDEρDE (c = 1 units) We consider threebasic phenomenological possibilities for w

w(a) =

minus1

w0 + wa(1minus a)

corresponding to cosmological constant constant equa-tion of state and equation of state in the form of theCPL parameterization respectively

For the three cases in equation (4) the energy densityof dark energy can be analytically integrated into

ρDE(a)

ρDE0=

aminus3(1+w)

aminus3(1+w0+wa) exp[minus3wa(1minus a)]

We describe these models as ΛCDM wCDM andw0waCDM respectively By default these models as-sume a flat geometry but we also consider versions ofthese with free curvature Dark energy models where Ωkis allowed as a free parameter are referred to as oΛCDMowCDM and ow0waCDM All of these models are nestedin the sense that they contain ΛCDM as a special limitwith w = w0 = minus1 wa = 0 and Ωk = 0

22 Growth of Structure in the Cosmological Model

The cosmic expansion history is determined by themean energy densities of the components in the Universeand their evolution with time The structure growthhistory reflects the evolution of density perturbationsagainst the background of cosmic expansion Densityperturbations in the matter are described by their rela-tive perturbations

δ(x t) equiv ρm(x t)minus ρm(t)

ρm(t) (6)

where ρm(t) is the mean matter density of the Universeand x is the comoving coordinate In this paper we ignoretheoretical subtleties related to choice of gauge becauseon the sub-horizon scales of interest the Newtonian de-scription is fully adequate

To the first order in perturbation theory the growthof fluctuations with time is specified by a single scale-independent growth factor D(t)

δ(x t) = D(t)δ(x t0) (7)

where D(t0) = 1 and D(t) satisfies

D + 2H(z)D minus 3

20 (1 + z)3D = 0 (8)

Strictly speaking this equation only holds for a singlefluid However it describes the low-redshift universe verywell since gravitational evolution drives the multiple flu-ids towards a common over-density field Therefore incosmological models consistent with GR the growth ofdensity fluctuations can be predicted uniquely for a givenexpansion history In this work we use growth measure-ments to probe dark energy to measure the amplitudeof the current matter density perturbations to test fortension in the cosmological model as well as to test GRas the model for gravity on cosmological scales

The linear growth rate is often expressed as a differ-ential in the linear growth function with respect to thescale factor

f(z) equiv d lnD

d ln a (9)

In standard cosmological models under GR the growthrate can be approximated as f(z) prop Ωm(z)055 (Wangamp Steinhardt 1998 Linder 2005 Linder amp Cahn 2007)However with the same expansion history theories ofmodified gravity may predict different rates of structuregrowth which motivates a simple parameterization tomodifications to f(z) prop Ωm(z)γ where departures fromγ = 055 correspond to departures from GR Anotherstrong prediction from GR is that the two metric po-tentials Ψ and Φ (corresponding to time and space per-turbations of the metric) are the same (Ψ = Φ) This isnot necessarily so in theories of modified gravity and thedifference in the two potentials (known as gravitationalslip) can affect the difference between the trajectories ofrelativistic and non-relativist particles

In this work we follow the analysis of Abbott et al(2019) to test for more general deviations from GRStarting from scalar metric perturbations in the confor-mal Newtonian Gauge represented as ds2 = a2(τ)[(1 +2Ψ)dτ2 minus (1 minus 2Φ)δijdxidxj ] with conformal time τ de-fined through dτ = dta(t) this phenomenological modelallows modification to the Poisson equations A time-dependent parameter micro(a) plays a similar role to theγ parameter in modifying the growth rate The modelalso allows a perturbation of the potential for masslessparticles relative to matter particles through the time-dependent parameter Σ(a) These two parameters pro-vide linear perturbations to the GR form of gravity ac-cording to the relations

k2Ψ =minus4πGa2(1 + micro(a))ρδ (10)

k2(Ψ + Φ) =minus8πGa2(1 + Σ(a))ρδ (11)

where k is the wavenumber and δ is the comoving-gaugedensity perturbation Both micro(a) and Σ(a) are equal tozero at all redshifts in GR This parameterization hasthe advantage that the Σ term can be constrained in-dependently by weak lensing with only mild degeneracywith micro The RSD measurements probe the response ofmatter to a gravitational potential and therefore provideindependent constraints on the micro term Again followingAbbott et al (2019) we describe the redshift evolutionof micro and Σ as

micro(z) = micro0ΩΛ(z)

ΩΛ Σ(z) = Σ0

ΩΛ(z)

ΩΛ (12)

Finally neutrinos can affect the measured growth of

fluctuations While ultra-relativistic they free-streamout of over-densities and thus suppress growth on scalessmaller than their free-streaming length (eg Lesgour-gues amp Pastor 2006) The dominant effect is a decreasein the amplitude of fluctuations at low redshifts com-pared to extrapolations from the CMB under a modelwith zero neutrino mass

23 Observable Signatures

231 The CMB

The temperature of the CMB is uniform across the skyto one part in 100000 beyond this level anisotropiesappear at all observable scales The angular power spec-trum of the CMB can be predicted to high precisionbased on an inflationary model and an expansion modelThe fluctuation modes corresponding to scales greaterthan one degree were larger than the Hubble distanceat the time of the last scattering and capture the ini-tial conditions imprinted at the end of inflation (ns andits derivative) At smaller scales the sound waves thatpropagate in the ionized universe due to photon-baryoncoupling imprint the characteristic acoustic oscillationsinto the CMB power spectrum The relative amplitudesof the peaks of the oscillations provide information onthe energy contents of the Universe while the spacing ofthe peaks provides a BAO lsquostandard rulerrsquo whose lengthcan be computed using straightforward physics

This BAO feature has a comoving scale of roughly 150Mpc set by the distance rd traveled by sound waves be-tween the end of inflation and the decoupling of baryonsfrom photons after recombination

int infinzd

H(z)dz (13)

where zd is the redshift of the drag epoch and cs is thesound speed Not to be confused with the redshift at thetime of last scattering the drag epoch corresponds tothe time when the baryons decouple from the photonsaround a redshift z = 1020 In the standard cosmologi-cal models explored here rd can be computed given thephysical densities of dark matter (ωc) baryonic matter(ωb) and the radiation content of the Universe Theradiation content can be determined from the tempera-ture of the CMB and the effective number of neutrinospecies (Neff) Combined these abundances determinethe shape and position of the BAO peak in comovingspace that can then be used as a standard ruler Be-cause the CMB provides an image of the oscillations atthe epoch of last photon scattering the BAO scale hasnot reached its maximum size but it can still be mea-sured at very high precision to provide a constraint onthe angular diameter distance to a redshift of z sim 1100

Because the proton-electron plasma does not recom-bine instantaneously the last scattering surface has afinite thickness Photon diffusion also results in damp-ing at the smallest scales leading to a diffusion scalethat depends on the expansion rate and energy densitiesThe effect of damping on the power spectrum thereforeallows constraints on the energy densities of relativis-tic particles primordial helium abundance dark matterand baryon matter at the time of last scattering Finallythe signal from the CMB records the integrated ioniza-tion history of hydrogen and the integrated formation of

structure in the form of polarization and lensing signalsCharacterization of polarization and lensing in the CMBthus provides information about the integrated opticaldepth (τ) to the surface of last scattering and the ef-fects of neutrinos on the growth rate of structure For areview of experimental and analysis methods to extractcosmological information from the CMB see Staggs et al(2018) and Planck Collaboration et al (2018a)

232 BAO measurements from spectroscopic surveys

The same sound waves that appear as acoustic oscil-lations in the CMB appear in the clustering of matterat later times although with a weaker amplitude dueto the coupling of baryonic matter with dark matter(eg Pardo amp Spergel 2020) For this reason surveyvolumes of several Gpc3 are required to reach percent-level precision constraints on the BAO feature The darkmatter distribution that records the BAO feature can-not be probed directly and is instead traced by galax-ies quasars or absorption line systems corresponding toneutral hydrogen or other material in the intergalacticmedium

The cosmological parameters used to calibrate thecharacteristic BAO scale rd are typically derived fromCMB observations The rd scale can also be derived fromBig Bang Nucleosynthesis (BBN) measurements (givingconstraints on ωb) in combination with measurements ofexpansion history (giving constraints on Ωm) if the earlyuniverse is assumed to be a mixture of radiation bary-onic matter and cold dark matter with three neutrinospecies With a calibrated rd the BAO scale can be usedto make absolute distance measurements as a function ofredshift Or rd can be treated as a nuisance parame-ter allowing multiple BAO measurements over a rangeof redshifts to be used for relative measures of the cosmicexpansion history

In a spectroscopic survey the BAO feature appears inboth the line-of-sight direction and the transverse direc-tion Along the line-of-sight direction a measurement ofthe redshift interval ∆z over which the BAO feature ex-tends provides a means to directly measure the Hubbleparameter H(z) = c∆zrd Equivalently it measuresthe Hubble distance at redshift z

DH(z) =c

H(z) (14)

Along the transverse direction the BAO scale corre-sponds to an angle rd = DM (z)∆θ Measuring the angle∆θ subtended by the BAO feature at a given redshift pro-vides a means to estimate the (comoving) angular diam-eter distance DM (z) which depends on the expansionhistory and curvature as

DM (z) =c

(DC(z)

) (15)

Here the line-of-sight comoving distance is

DC(z) =c

dzprimeH0

H(zprime)(16)

Sk(x) =

sin(radicminusΩkx)

radicminusΩk Ωk lt 0

x Ωk = 0

sinh(radic

Ωkx)radic

Ωk Ωk gt 0

When considering the dependence of rd on cosmologythe quantities that the BAO measurements directly con-strain are DM (z)rd and DH(z)rd The BAO mea-surements were also historically summarized by a singlequantity representing the spherically-averaged distance

DV (z) equiv[zD2

M (z)DH(z)]13

or more directly DV (z)rd The powers of 2frasl3 and 1frasl3approximately account for two transverse and one radialdimension and the extra factor of z is a conventional nor-malization Today we almost always specify the trans-verse and radial BAO as two independent measurementswith correlated error bars instead unless the signal-to-noise ratio is low

For measurements using discrete tracers with suffi-ciently high number density the BAO feature in cluster-ing measurements can be sharpened through a processknown as lsquoreconstructionrsquo (Eisenstein et al 2007) Re-construction uses the observed three-dimensional map ofgalaxy positions to infer their peculiar velocities Eachgalaxy tracer is then moved to a position that is approx-imately where the galaxy would reside if there were nobulk flows The process removes the dominant non-lineareffect from the BAO feature which is smearing causedby the large-scale bulk flows Reconstruction recovers al-most all theoretically available information in the BAOIn the SDSS analyses the fitting to reconstructed datais performed with minimal information from the broad-band clustering signal in an attempt to isolate the BAOsignal

A review of BAO as a probe for cosmology is presentedin Section 4 of Weinberg et al (2013) and a discussionon the BAO measurement in practice can be found inAppendix A

233 RSD measurements from galaxy surveys

The galaxy redshifts used in spectroscopic BAO mea-surements can also be used to study anisotropic cluster-ing There are two primary ways in which anisotropyis introduced into the large-scale clustering of matterthe Alcock-Paczynski (AP) effect (Alcock amp Paczynski1979) and the RSD effect from the growth of structure(Kaiser 1987) The AP effect arises in clustering statis-tics as a deviation from physically isotropic signal due toan incorrect translation of angular and radial (redshift)separations to physical ones (see Appendix A) The APeffect thus serves as a way to measure the product ofH(z) and DM (z) offering additional constraints on darkenergy and curvature (eg Nadathur et al 2020)

The RSD effect arises from the growth of structure(Kaiser 1987) and is observed due to the bulk flow ofmatter in response to the gravitational potential of mat-ter overdensities The peculiar velocities introduce addi-tional redshifts on top of those caused by cosmic expan-sion leading to an increase in the measured amplitudeof radial clustering relative to transverse clustering onlarge scales The resulting anisotropy is correlated withthe rate at which structure grows The growth rate f(z)from equation (9) can also be expressed as

f =part lnσ8

part ln a (19)

where σ8(z) describes the amplitude of linear matter fluc-

tuations on a comoving scale of 8hminus1Mpc The RSDmeasurements provide constraints on fσ8 which charac-terizes the amplitude of the velocity power spectrum

The AP and RSD signals are partially degeneratewhich limits the AP signal that can be extracted frommeasurements of clustering (eg Ballinger et al 1996)A review of RSD and AP as a probe for cosmology ispresented in Section 4 of Weinberg et al (2013) whilea discussion of the RSD measurement in practice can befound in Appendix A

234 Weak lensing

As RSD probe the response of matter to a gravitationalpotential gravitational lensing probes the response ofphotons to a gravitational potential Gravitational lens-ing can be observed in several forms in cosmic surveyswe focus on the weak lensing regime in this work Morespecifically we use cosmic shear measurements of weaklensing and galaxy-galaxy lensing measurements in per-forming cosmological constraints

Cosmic shear shows up as distortions on the order of1 that appear in the images of background galaxies dueto lensing by the integrated foreground mass distribu-tion By introducing correlations of neighboring galaxyshapes due to shared foregrounds cosmic shear allowsdirect inference of the gravitational potential gradientsintegrated along the line of sight If these correlationsare computed over discrete intervals over a range of red-shifts a smooth three-dimensional mapping of the mat-ter distribution can be deduced The direct observable inlensing surveys is the cosmic shear power spectrum withan amplitude that scales approximately as Ω2

mσ28 in the

linear regime However weak-lensing measurements areoften in the non-linear regime and also depend on rel-ative distances through the lens equation The relativebalance between Ωm and σ8 in the measurement dependson a number of factors within CDM models as describedin Jain amp Seljak (1997) For the redshifts probed by cur-rent surveys around the benchmark ΛCDM model theredshift evolution of the amplitude of the cosmic shearpower spectrum is best described by the approximatecombination

S8 equiv σ8(Ωm03)05 (20)

A review of cosmic shear methodology and its challengesas a probe for cosmology can be found in Section 5 ofWeinberg et al (2013)

In addition to shear measurements we also use galaxy-galaxy lensing results in Section 6 to provide additionalinformation on the galaxy clustering measurements ob-tained in photometric surveys Galaxy-galaxy lensingmeasurements probe the local gravitational potentialaround specific classes of galaxies For the cosmologystudies presented here these measurements give insightinto mass density profiles thus providing important in-formation on the bias of the galaxies used as tracers inthe photometric clustering measurements

235 Type Ia supernovae

Type Ia supernovae are generally believed to occurwhen a white dwarf approaches the Chandrasekhar masslimit due to mass accretion or merger This class ofSN is easily characterized with spectroscopy due to thestrong calcium and silicon lines and lack of hydrogen

and helium lines While SNe Ia are not perfect stan-dard candles their diversity can be described by the SNlight curve width (hereafter X1) and SN color at max-imum brightness (hereafter C) The distance modulusmicro = 5log10[DL(z)10pc] is then given by

micro = mlowastB minus (MB minus αX1 + βC) (21)

where mlowastB is the observed SN peak magnitude in rest-frame B band (Astier et al 2006) Here DL is theluminosity distance which follows the relation DL =DM (1 + z) The quantity MB characterizes the SN Iaabsolute magnitude while α and β describe the changein magnitude with diversity in width and color respec-tively The linear dependence between SN property andpeak magnitude follows from the empirical observationthat brighter SNe Ia are also slower to rise andor bluerin color (see Hamuy et al 1996 Phillips et al 1999)Beyond those two dominant effects a residual diversityrelated to host galaxy properties was also found (egSullivan et al 2011) with brighter SNe occurring in moremassive galaxies This effect is usually accounted for byconsidering that the SN Ia absolute magnitude is differ-ent depending on the host stellar mass such as in Betouleet al (2014)

MB = M1B if Mstellar lt 1010M

MB = M1B + ∆M otherwise

The model assumes that SNe Ia with identical color lightcurve shape and galactic environment have on averagethe same intrinsic luminosity for all redshifts Note thatthe hypothesis of redshift independence can be checkedwith data for ∆M α and β and so far has been found tobe consistent with observations (eg Scolnic et al 2018)

If the above model is sufficiently accurate the mea-sured SN distance modulus traces the redshift depen-dence of luminosity distance The absolute magnitudecan be calibrated using nearby SNe Ia and Cepheid vari-ables giving a distance ladder from which H0 can becomputed A review of supernova astrophysics and theiruse in cosmology to constrain the dark energy equation-of-state can be found in Goobar amp Leibundgut (2011)

24 Combining measurements

The measurements of the redshift-distance relationthrough BAO AP and SNe Ia provide tests of extendedmodels for dark energy and cosmic expansion that areonly weakly constrained with CMB data alone Gener-ally speaking the SNe Ia data provide a high precisionconstraint of the luminosity distancendashredshift relation inthe dark-energy dominated regime while the BAO andAP measurements sample the matter-dominated regimeand the epoch of matter-dark energy equality Like-wise the measurements of growth of structure throughRSD and weak lensing allow additional tests on thebackground expansion and on whether GR describes therate of structure growth Measurements of the redshift-distance relation and growth of structure allow tests ofthe neutrino mass by constraining the effects on boththe cosmic expansion after the CMB formation and theamplitude of matter fluctuations relative to amplitudeof CMB fluctuations The sensitivity of the latter ap-proach is limited by our knowledge of optical depth τto the last scattering surface Alternative approaches

TABLE 1Symbols and Definitions of Cosmological Parameters

Parameter Definition

Ωm density parameter of matterΩc density parameter of cold dark matterΩb density parameter of baryonsΩΛ density parameter of cosmological constantΩDE density parameter of dark energyΩk curvature parameterωc = Ωch2 physical density parameter of cold dark matterωb = Ωbh

2 physical density parameter of baryonsH0 current expansion rate (Hubble constant)h H0100 km sminus1Mpcminus1

θMC approximate angular scale of sound horizon (CosmoMC)As power of the primordial curvature perturbations at k = 005 Mpcminus1

σ8 amplitude of matter fluctuation on 8hminus1Mpc comoving scalens power-law index of the scalar spectrumτ Thomson scattering optical depth due to reionizationNeff effective number of neutrino-like relativistic degrees of freedomw (w0) dark energy equation of state w = pDEρDE (c = 1 units)wa time derivative of dark energy equation of state parameter (eq 4)summν sum of neutrino masses

Note mdash Listed are main cosmological parameters in this paper The parame-terization of cosmological models and parameter priors can be found in Table 9

to constrain the neutrino mass rely on measuring theredshift-dependence of growth directly with clusteringdata or scale-dependence of the matter power spectrum(Lesgourgues amp Pastor 2006 Yu et al 2018 Chiang et al2018) but are not explored here

For fitting the measurements model calculationsthroughout this paper are made with CosmoMC (Lewisamp Bridle 2002) Figures are produced with the GetDistPython package (Lewis 2019) The model parametersare summarized in Table 1 while parameterizations andpriors are described in Appendix B We stress that choiceof parameterization is sometimes important ndash the shapeand visual overlap of marginalized contours can be signif-icantly impacted especially in a prior-dominated regimeIn all cases that use information from the shape of thepower spectrum we hold Neff fixed to its baseline valueIn the majority of the studies presented in this paper thepriors we assume on free parameters do not impact theposterior distributions when CMB data are included inthe likelihoods We refer to this series of priors as thosewith the lsquoCMBrsquo parameterization In the cases wherewe study the expansion history without the CMB (Sec-tion 4) we use the lsquobackgroundrsquo parameterization Inall studies the same priors are used for curvature thedark energy equation of state or neutrino masses in thecases that those parameters are fit to the data Thosepriors are reported in the lsquoextendedrsquo portion of the tablein Appendix B

3 DATA AND METHODOLOGY

In this section we provide an overview of the differentmeasurements used in our primary cosmological anal-ysis including Baryon Acoustic Oscillations (BAO)Redshift Space Distortions (RSD) Cosmic MicrowaveBackground (CMB) Supernovae (SN) and Weak Lensing(WL) The samples we use in this work and the namingconventions we choose are summarized in Table 2 Wepresent the state-of-the-art results and discuss how thedifferent probes have evolved during the last decade

31 SDSS BAO and RSD Measurements

The study presented in this work characterizes the im-pact of BAO and RSD measurements from spectroscopicgalaxy and quasar samples obtained over four genera-tions of SDSS A summary of the BAO-only measure-ments is found in Table 3 and in the top panel of Figure 1In these measurements the broadband clustering signalthat carries information on the AP effect or RSD is ef-fectively deweighted to capture only the BAO signatureThese measurements are used to explore the impact ofBAO measurements on models for dark energy in Sec-tion 4 Results from the full-shape fits without informa-tion from reconstructed BAO measurements are foundin the central region of Table 3 These measurementsinclude information from the AP effect and are used toexplore the impact of growth measurements in Section 5A summary of the BAO and RSD measurements includ-ing information from the AP effect and reconstructionis also found in Table 3 and Figure 1 These measure-ments are used to perform the global cosmology fitting inSections 6 and 7 The background to each of these mea-surements is summarized below and described in detailin the relevant references All results in Table 3 reflectthe consensus values in the cases where multiple mea-surements are madeMain Galaxy Sample (MGS) (007 lt z lt 02)

The first two generations of SDSS (SDSS-I and -II) pro-vided redshifts of nearly one million galaxies (Abazajianet al 2009) SDSS galaxies were selected with 145 ltr lt 1764 over a contiguous footprint of 6813 deg2 toperform clustering measurements The sample was fur-ther refined to cover the redshift range 007 lt z lt 02include the bright objects with Mr lt minus212 and in-clude red objects with gminus r gt 08 The resulting samplecontains 63163 galaxies intended to occupy the highestmass halos while providing a roughly uniform numberdensity over the full redshift interval The sample wasused to perform a BAO measurement from the recon-

4 httpsdssphysicsnyueduvagclsshtml

TABLE 2Data sets for cosmology analyses

Name Data Combination Cosmology AnalysisBAO DM (z)rd and DH(z)rd from BAO measurements of all SDSS tracers Section 4RSD fσ8(z) from all SDSS tracers marginalizing over DM (z)rd and DH(z)rd Section 5SDSS DM (z)rd DH(z)rd and fσ8(z) of all SDSS tracers Sections 67CMB TampP Planck TT TE EE and lowE power spectra Sections 45CMB lens Planck lensing measurements Section 5Planck Planck temperature polarization and lensing measurements Sections 67SN Pantheon SNe Ia measurements Sections 467WL DES cosmic shear correlation functions Section 5DES DES 3times2 measurements (cosmic shear galaxy clustering and galaxy-galaxy lensing) Sections 67

DM(z)rdradicz

zDH(z)rdradicz

SDSS MGS

BOSS Galaxy

eBOSS LRG

eBOSS ELG

eBOSS QSO

eBOSS Lyαminus Lyα

eBOSS LyαminusQSO

01 02 05 10 20 30

redshift

Fig 1mdash Top Distance measurements from the SDSS lineage of BAO measurements presented as a function of redshift Measurementsinclude those from SDSS MGS (Ross et al 2015 Howlett et al 2015) BOSS galaxies (Alam et al 2017) eBOSS LRGs (Bautista et al2020 Gil-Marin et al 2020) eBOSS ELGs (Tamone et al 2020 de Mattia et al 2020) eBOSS quasars (Hou et al 2020 Neveux et al2020) the BOSS+eBOSS Lyα auto-correlation and the BOSS+eBOSS Lyα-quasar cross-correlation measurements (du Mas des Bourbouxet al 2020) Red points correspond to transverse BAO while green points to radial BAO The MGS DV measurement is plotted in orangewith a translation to DM assuming a ΛCDM model for illustrative purposes The red and green theory curves are not fit to the BAOdata they are the Planck bestfit predictions for a flat ΛCDM model Bottom Growth rate measurements from the SDSS lineage offσ8 measurements as a function of redshift The measurements match the BAO samples except for z gt 2 where we do not report ameasurement of the growth rate As for the upper panel theory curve is not a fit but a bestfit Planck model

structed correlation function (Ross et al 2015) and anRSD measurement from the anisotropic correlation func-tion (Howlett et al 2015) both at an effective redshiftzeff = 015 The BAO measurement was characterizedwith DV (z)rd and the RSD fit was performed using thepost-reconstruction BAO fit as a prior The likelihoodsfrom this work are found in the Supplementary Data as-sociated with Howlett et al (2015) We refer to thissample as the lsquoMain Galaxy Samplersquo (MGS) in the tableand throughout the paperBOSS DR12 Galaxies (02 lt z lt 06) Over the pe-

riod 2009ndash2014 BOSS performed spectroscopy to mea-sure large-scale structure with galaxies over the redshiftinterval 02 lt z lt 075 BOSS obtained redshifts for1372737 galaxies over 9376 deg2 from which the finalgalaxy catalog was produced for clustering measurements(Reid et al 2016) The sample was divided into threeredshift bins covering 02 lt z lt 05 04 lt z lt 06and 05 lt z lt 075 for studies of BAO and RSD For

each redshift bin seven different measurements of BAOAP and RSD were performed (Ross et al 2017 Vargas-Magana et al 2018 Beutler et al 2017ba Satpathy et al2017 Sanchez et al 2017b Grieb et al 2017) basedon the galaxy correlation function or power spectrumFollowing the methodology of Sanchez et al (2017a)these measurements were combined into a single consen-sus likelihood spanning DM (z)rd and DH(z)rd for theBAO-only measurements and DM (z)rd DH(z)rd andfσ8(z) for the combined BAO and RSD measurementsThese results were computed over all three redshift inter-vals after fully accounting for systematic errors and co-variances between parameters and between redshift bins(Alam et al 2017) We refer to the 02 lt z lt 05 and04 lt z lt 06 samples as the lsquoBOSS GalaxiesrsquoeBOSS Galaxies and Quasars (06 lt z lt 22)

eBOSS began full operations in July 2014 to performspectroscopy on luminous red galaxies (LRGs) emis-sion line galaxies (ELGs) and quasars and concluded

TABLE 3Clustering measurements for each of the BAO and RSD samples used in this paper

Parameter MGS BOSS Galaxy BOSS Galaxy eBOSS LRG eBOSS ELG eBOSS Quasar Lyα-Lyα Lyα-Quasar

Sample Propertiesredshift range 007 lt z lt 02 02 lt z lt 05 04 lt z lt 06 06 lt z lt 10 06 lt z lt 11 08 lt z lt 22 z gt 21 z gt 177Ntracers 63163 604001 686370 377458 173736 343708 210005 341468zeff 015 038 051 070 085 148 233 233Veff (Gpc3) 024 37 42 27 06 06

BAO-Only Measurements (Section 4)

DV (z)rd 447 plusmn 017 1833+057minus062

DM (z)rd 1023 plusmn 017 1336 plusmn 021 1786 plusmn 033 3069 plusmn 080 376 plusmn 19 373 plusmn 17DH(z)rd 2500 plusmn 076 2233 plusmn 058 1933 plusmn 053 1326 plusmn 055 893 plusmn 028 908 plusmn 034

RSD-Only Measurements (Section 5)fσ8(z) 053 plusmn 016 0500 plusmn 0047 0455 plusmn 0039 0448 plusmn 0043 0315 plusmn 0095 0462 plusmn 0045

BAO+RSD Measurements (Sections 6 and 7)DV (z)rd 451 plusmn 014DM (z)rd 1027 plusmn 015 1338 plusmn 018 1765 plusmn 030 195 plusmn 10 3021 plusmn 079 376 plusmn 19 373 plusmn 17DH(z)rd 2489 plusmn 058 2243 plusmn 048 1978 plusmn 046 196 plusmn 21 1323 plusmn 047 893 plusmn 028 908 plusmn 034fσ8(z) 053 plusmn 016 0497 plusmn 0045 0459 plusmn 0038 0473 plusmn 0041 0315 plusmn 0095 0462 plusmn 0045

Note mdash Uncertainties are Gaussian approximations to the likelihoods for each tracer ignoring the correlations between measurementsThe publicly available likelihoods should be used for all cosmology studies In the BAO-only results the measurements for MGS thetwo BOSS galaxy samples eBOSS LRG and eBOSS ELG are performed after reconstruction The BAO+RSD results incorporate thereconstructed BAO measurements for those samples The number of tracers reported for the Lyα-Lyα measurement corresponds to thenumber of sightlines or forests while the number reported for the Lyα-quasar measurements corresponds to the number of tracer quasarsThe effective volume Veff is quoted here in Gpc3 using a flat ΛCDM model with Ωm = 031 and h = 0676

on March 1 2019 eBOSS obtained reliable redshifts for174816 LRGs over the interval 06 lt z lt 1 in an area of4103 deg2 The targets for spectroscopy were selectedfrom SDSS riz imaging data and infrared sky maps fromthe Wide-field Infrared Survey Explorer (WISE Wrightet al 2010) The LRG selection (Prakash et al 2016) wasoptimized to cover 06 lt z lt 1 with a median redshiftz = 072 The sample was supplemented with the galax-ies in the z gt 06 tail of the BOSS DR12 redshift distribu-tion but over the full 9376 deg2 of the BOSS footprintThe addition of BOSS galaxies more than doubles the to-tal sample size to 377458 redshifts while slightly lower-ing the effective redshift This lsquoeBOSS LRGrsquo sample wasused to measure DM (z)rd and DH(z)rd using a cat-alog of reconstructed galaxy positions In addition thesample was used to perform a jointDM (z)rd DH(z)rdand fσ8(z) measurement in both the correlation function(Bautista et al 2020) and the power spectrum (Gil-Marinet al 2020)

Covering an area of 1170 deg2 eBOSS also obtainedreliable redshifts for 173736 ELGs over the redshift range06 lt z lt 11 These targets were identified in grzphotometry from the Dark Energy Camera (DECamFlaugher et al 2015) following the selection algorithmspresented in Raichoor et al (2017) These star-forminggalaxies were spectroscopically confirmed with high ef-ficiency due to their strong emission lines that are eas-ily detectable with the BOSS spectrographs (Smee et al2013) to z = 11 The lsquoeBOSS ELGrsquo sample reaches aneffective redshift zeff = 085 We performed an isotropicBAO fit to measure DV (z)rd (Raichoor et al 2020 deMattia et al 2020) and a combined RSD and BAO anal-ysis to constrain fσ8(z) DH(z)rd and DM (z)rd fromboth the correlation function (Tamone et al 2020) andthe power spectrum (de Mattia et al 2020) Becausethe likelihoods are not well-described by a Gaussian dis-tribution we use the full likelihoods in the cosmology

fittingFinally the lsquoeBOSS quasarrsquo sample includes 343708

reliable redshifts with 08 lt z lt 22 over 4699 deg2 Thesample selection (Myers et al 2015) was derived fromWISE infrared and SDSS optical imaging data 18 ofthese quasars identified by the algorithm had been ob-served in SDSS-I -II or -III The sample was used tomeasure DM (z)rd DH(z)rd and fσ8(z) from both thecorrelation function (Hou et al 2020) and the power spec-trum (Neveux et al 2020) The consensus BAO-only re-sults were determined without reconstruction The full-shape DM (z)rd DH(z)rd and fσ8(z) measurementswere therefore not combined with the BAO-only mea-surementsLyman-α Forest Samples (18 lt z lt 35) The

complete BOSS sample contains the spectra of 157845quasars at 20 lt z lt 35 that are free of significant broadabsorption lines These quasar targets were selected us-ing a variety of techniques (Ross et al 2012) to measurefluctuations in the transmission of the Lyman-α (Lyα)forest due to fluctuations in the density of neutral hy-drogen The auto-correlation of the Lyα forest and itscross-correlation with 217780 quasars at 18 lt z lt 35led to 2 precision measurements of the BAO distancescale at zeff = 233 (Bautista et al 2017 du Mas desBourboux et al 2017)

Several techniques such as those using photometricvariability (Palanque-Delabrouille et al 2016) were usedto select new z gt 21 quasars to observe in eBOSS Inaddition 42859 quasars with low signal-to-noise BOSSspectra were re-observed in eBOSS to better measure thefluctuations in the Lyα forest Finally improvements tothe analysis methods enabled the use of a larger wave-length range for determining the forest The final sam-ple used to trace the Lyα forest has 210005 quasars atz gt 21 consisting of the original sample from BOSS andthe sample from eBOSS A total of 341468 quasars with

z gt 177 were used for cross-correlation studies with theLyα forest

The final eBOSS results are presented in du Mas desBourboux et al (2020) The auto- and cross-correlationmeasurements can also be combined into a single esti-mate of DM (z)rd and DH(z)rd with associated covari-ances (du Mas des Bourboux et al 2020) resulting in a25 reduction in the area of the contours relative to theBOSS DR12 studies The uncertainties quoted in Table 3correspond to a Gaussian approximation of the real like-lihood but in our analysis we use the full (non-Gaussian)likelihoodSummary of SDSS Measurements and System-

atic Errors From the BOSS and eBOSS clusteringanalyses with galaxies and quasars the main systematicerrors in BAO and RSD estimates arise from modeling ofthe two-point statistics the choice of fiducial cosmologytaken as a reference for coordinate transformation andpower spectrum template and from the observationaleffects The systematic errors also have larger effect onthe RSD analyses than the BAO analyses The estima-tion of the systematic errors was done in a similar fashionfor all tracers although some differences in the treatmentremain and are outlined in the following

The modeling systematic errors are studied using ac-curate mocks based on N-body simulations for which thecosmology is known (Rossi et al 2020 Smith et al 2020Alam et al 2020b) Special care is taken to estimate theeffect of having a fixed fiducial cosmology for calculatingdistances and shape of the template for the two-pointstatistics In detail we measure the range of the dif-ferences between true and recovered values obtained byfitting to mocks where the true and fiducial cosmologiesdo not match The distribution of cosmologies spannedby the mocks acts as a prior on lsquoallowed cosmologiesrsquo Allgalaxy and quasar tracers used both blind and non-blindmocks to assess their modeling systematic errors Varia-tions of the Halo Occupation Distribution parameteriza-tions are also taken into account For the BOSS GalaxyELG and LRG samples the modeling systematic error isfurther reduced by scaling the σ8 value according to theisotropic dilation factor measured independently in thedata and in each set of mocks (see Appendix A) For thequasar sample the redshift determination is an order ofmagnitude less precise than for the galaxies and requiresspecial modeling The systematic effect of redshift errorson the two-point statistics is estimated using the N-bodymocks (Smith et al 2020) and is comparable in size tothe systematic errors in modeling

Observational effects are studied using approximatemocks that are modified to account for the observationalconditions (Lin et al 2020 Zhao et al 2020) This in-cludes the dependence of the spectroscopic success rateon the signal-to-noise ratio of the spectra the treatmentof fiber collisions and the variations of the density oftargets for different photometric conditions in the imag-ing data For the ELGs and quasars fiber collisions aretaken into account at the model level and their effect isreduced

More details about the sets of mocks used to estimatethese errors are presented in the papers describing themocks and the papers describing the individual measure-ments (Bautista et al 2020 Gil-Marin et al 2020 Rai-choor et al 2020 Tamone et al 2020 de Mattia et al

2020 Hou et al 2020 Neveux et al 2020) In summaryfor the LRG full-shape analysis the overall systematicerrors amount to about 40 to 60 of the statistical er-ror depending on the parameters The systematic er-rors for the ELG measurement reaches the same levelalthough with different sources of systematic effects Forthe quasars the systematic errors are at the level of 30of the statistical error for all parameters

Several tests for systematic errors were performed forthe Lyα BAO studies such as tests on mock spectramodeling of the broadband signal in the correlation func-tion and assessment of metal and sky contributions tothe Lyα transmission estimates The central values ofthe DMrd and DHrd estimates did not change signif-icantly during these tests and no additional systematicerrors were included in the reported BAO results To ac-count for the somewhat non-Gaussian errors on DMrdand DHrd we generated 1000 realizations to estimatethe translation of the ∆χ2 from each measurement in theparameter space to confidence intervals on the BAO pa-rameters The BAO measurements reported in Table 3include this correctionSummary of SDSS Likelihoods The final

DM (z)rd DH(z)rd and fσ8(z) measurements covereight distinct redshift intervals The systematic errorsand consensus estimates are assessed in the studies thatreport the final measurements and incorporated directlyinto the covariance matrices used in this study Covari-ances between the two BOSS galaxy measurements arepropagated to this study through the same covariancematrix reported in Alam et al (2017)

We find that the expected statistical correlation be-tween clustering measurements derived from the eBOSSsamples is negligibly small and we thus include no covari-ance between them in our cosmological analyses Thisdecision for the covariance between the quasar clusteringmeasurements the Lyα auto-correlation measurementsand the Lyα-quasar cross-correlation measurements wasjustified using mock catalogs that demonstrated negligi-ble correlation For the galaxy and quasar samples thecorrelation within the overlapping volume can be esti-mated as

Co =P1P2

(P1 + 1n1)(P2 + 1n2) (23)

where P represents the power-spectrum amplitude andn is the number density We use the effective P value inRoss et al (2020) and determine an effective 1n valuebased on the effective volume For both the correla-tion between the quasars and the ELGs and between thequasars and the LRGs we find Co is less than 01 imply-ing any correlation with the quasar sample is negligibleWithin their overlapped volume the expected correlationbetween the ELGs and LRGs is higher as each samplehas a peak nP gt 1 However over the full 06 lt z lt 10overlap range we find Co = 024 Accounting for the factthat the ELG footprint is significantly smaller than theLRG footprint again reduces the expected correlation toless than 01

Upon final acceptance for publication the final likeli-hoods for the MGS BOSS galaxy and eBOSS measure-ments will all be found on the public SDSS svn repos-

itory5 and in the Github repository6 The full likeli-hood is reported for BAO-only studies in the MGS ELGand Lyα forest samples The BAO-only results for theBOSS galaxy eBOSS LRG and eBOSS quasar samplesare recorded as a covariance matrix We refer to thecombination of these measurements as the lsquoBAOrsquo mea-surements throughout the paper The combined fits forBAO AP and RSD results are recorded as a full likeli-hood for the MGS and ELG samples while the results forthe BOSS galaxy eBOSS LRG and eBOSS quasar sam-ples are recorded in a single covariance matrix We referto these data samples as the lsquoRSDrsquo samples when no in-formation from reconstruction is used and the likelihoodsare collapsed to a single dimension on fσ8 We refer tothe full analyses of reconstructed BAO and full-shapeAP+RSD fitting as the lsquoSDSSrsquo sample In all cases thelikelihoods include both statistical and systematic errors

32 CMB SNe and WL Measurements

The BAO measurements from the four generations ofSDSS are complemented by relative distance measure-ments from SNe Ia The SDSS RSD measurements arecomplemented by WL measurements from CMB and re-cent imaging programs CMB anisotropies from all-skyspace-based surveys are used throughout to provide abaseline of high redshift cosmological measurements Fi-nally we compare the local value of the Hubble expansionparameter derived from various combinations of CMBBAO SNe Ia and BBN to the most recent results us-ing local measurements Neither the BBN nor the H0

estimates are directly used in any other cosmological fit-ting and are not discussed any further in this sectionalthough the BBN constraints on ωb are used to informpriors in several growth measurements In the remain-der of this section we discuss the results from the CMBSNe Ia and WL studies that we use to assess progressin building the cosmological model

The WMAP satellite launched on June 30 2001 andceased scientific operations on August 19 2010 The cos-mological measurements based only on the final WMAPsample provide constraints of Ωch

2 = 01138plusmn00045 andΩbh

2 = 002264 plusmn 000050 in a flat ΛCDM model ThePlanck satellite (Planck Collaboration et al 2011) oper-ated from 2009ndash2013 to measure CMB temperature andpolarization anisotropies to scales as small as 5prime Thesemeasurements allow very precise constraints on the mat-ter content and early expansion history of the Universeespecially in the limit of a ΛCDM cosmology An analysisunder the assumption of a flat ΛCDM model using onlyPlanck temperature and polarization data leads to con-straints Ωch

2 = 0120 plusmn 0001 Ωbh2 = 00224 plusmn 00001

ns = 0965 plusmn 0004 and τ = 0054 plusmn 0007 As thelatest generation of CMB experiment Planck thereforeprovides a factor of 45 improvement over WMAP onthe precision of the dark matter density and a factor of5 improvement on the precision of the baryonic matterdensity When computing constraints using the baselinePlanck measurements denoted CMB TampP throughoutwe use the Plik likelihoods for the TT TE EE andlowE power spectra (Planck Collaboration et al 2019)

5 httpssvnsdssorgpublicdataebossmcmctrunklikelihoods

6 httpsgithubcomevamariamCosmoMC_SDSS2020

The data cover multipoles in the range 30 le ` le 2508for the TT power spectrum and 30 le ` le 1996 for thepower spectra that include polarization When includingadditional lensing data from Planck denoted lsquoCMB lensrsquowe use the likelihoods from Planck Collaboration et al(2018c) computed over lensing multipoles 8 le ` le 400When using temperature polarization and lensing datatogether we refer to the sample simply as lsquoPlanckrsquo Thefull likelihoods for Planck and WMAP measurements arefound in the Planck public release of 2018 Cosmologicalparameters and MC chains7 and the WMAP 2013 publicrelease8 respectively

At the time that eBOSS began observations the lead-ing SNe Ia cosmology studies stemmed from the lsquojointlight-curve analysisrsquo (JLA) sample These 740 SNe Ialightcurves were taken from low redshift surveys (Hickenet al 2009 Contreras et al 2010) the SDSS-II Super-nova Survey (2005ndash2007 Frieman et al 2008 Sako et al2018) the Supernova Legacy Survey (SNLS 2003ndash2008Guy et al 2010 Conley et al 2011 Sullivan et al 2011)and high redshift space-based observations with the Hub-ble Space Telescope (Riess et al 2007) A major effortin the analysis focused on reducing systematic uncertain-ties in the photometric calibration of the SNLS and SDSSsurveys For a flat ΛCDM cosmology using only the SNefrom this sample the constraints on the matter contentof the local universe were found to be Ωm = 0295plusmn0034including systematic errors (Betoule et al 2014) Morerecently the lsquoPantheon samplersquo of 1048 SNe Ia was usedin a comprehensive cosmology analysis This sample in-cludes the full set of spectroscopically confirmed SNe Iafrom PanStarrs (Kaiser et al 2010) supplemented bySNe Ia observed at low redshift (Riess et al 1999 Jhaet al 2006 Hicken et al 2009 Contreras et al 2010 Fo-latelli et al 2010 Stritzinger et al 2011) the SDSS andSNLS samples and with HST (Suzuki et al 2012 Riesset al 2007 Rodney et al 2014 Graur et al 2014 Riesset al 2018) While the increase in sample size since theJLA analysis is significant the largest improvement inprecision results from new cross-calibration of all ground-based measurements to the PanStarrs photometric sys-tem Using only this SN sample with the systematicuncertainties leads to a constraint Ωm = 0298plusmn0022 ina flat ΛCDM model Within the basis of a flat ΛCDMthe Pantheon sample therefore offers a factor of 15 im-provement in precision over the JLA sample System-atic errors are still significant and dominated largely byphotometric uncertainties of each sample the calibrationuncertainties of the lightcurve model and the assump-tion of no redshift dependence of MB The statisticaland systematic uncertainties are captured in a covari-ance matrix with an element for each supernova follow-ing the methodology of Conley et al (2011) The sta-tistical component of the uncertainties contributes onlyto the diagonal elements while the off-diagonal elementsare dominated by systematic errors arising from commonuncertainties in bandpass and zeropoint calibration Weprimarily use the Pantheon sample in making cosmolog-

7 httpswikicosmosesaintplanck-legacy-archiveindexphpCosmological_Parameters with a description ofthe CosmoMC implementation at httpscosmologistinfocosmomcreadme_planckhtml

8 httpslambdagsfcnasagovproductmapcurrentlikelihood_getcfm

ical constraints and refer to this as the lsquoSNrsquo sample Thecovariance matrix for both the JLA sample and the Pan-theon sample can be found at the Barbara A MikulskiArchive for Space Telescopes (MAST)9 and are includedwith the CosmoMC installation

Several recent programs (eg Heymans et al 2012Kohlinger et al 2017 Mandelbaum et al 2018 Hikageet al 2019) have reported cosmology constraints frommeasurements of cosmic shear Because we are not ableto account for covariances between these results due toshared systematic errors we do not attempt an analy-sis on the combined weak lensing results Instead asan example of how weak lensing data impact cosmolog-ical constraints we focus here on the results from theDark Energy Survey (DES) conducted with the DarkEnergy Camera (Flaugher et al 2015) DES releasedan analysis of cosmic shear using the first year of datacovering an area exceeding 1000 deg2 with more than 20million galaxy shape measurements Tomographic cos-mic shear measurements were performed after assigningsource galaxies to redshift bins spanning the intervals020 lt z lt 043 043 lt z lt 063 063 lt z lt 090 and090 lt z lt 130 The data are used under an assump-tion of a ΛCDM model to constrain the combination ofΩm and σ8 represented by S8 = 0782 plusmn 0027 at 68confidence (Troxel et al 2018) As in the DES analy-sis we only use scales in the cosmic shear correlationfunctions that are expected to have contributions frombaryonic effects of less than 2 These studies are de-noted lsquoWLrsquo in Section 5 In addition to cosmic shearmeasurements we use the 3times2pt DES Year 1 results inthe analysis presented in Sections 6 and 7 where the ad-ditional correlation functions are computed from galaxyclustering and galaxy-galaxy lensing Following Krauseet al (2017) we only use information from the correla-tion function on comoving scales larger than 8hminus1Mpcfor the galaxy clustering measurements and 12hminus1Mpcfor the galaxy-galaxy lensing We use the CosmoMC im-plementation of the DES likelihood1011 with covariancematrix power spectra measurements and nuisance pa-rameters in agreement with Troxel et al (2018) Krauseet al (2017) and Abbott et al (2018) The combined3times2pt sample is referenced simply as lsquoDESrsquo

4 IMPLICATIONS OF EXPANSION HISTORYMEASUREMENTS

In this section we discuss measurements of the back-ground expansion history with an emphasis on the BAOmeasurements from SDSS We use the Planck temper-ature and polarization data (CMB TampP) the SN datafrom Pantheon and the BAO data from SDSS The BAOdata summarized in the BAO-only section of Table 3 in-clude measurements from galaxy quasar and Lyα forestsamples It is this wide redshift range that enables thetight constraints on cosmological parameters presentedin this section

We start in Section 41 with a discussion on the roleof BAO and SN measurements on single parameter ex-tensions to the ΛCDM By adding measurements of the

9 httpsarchivestscieduprepdsps1cosmo10 httpsgithubcomcmbantCosmoMCblobmasterbatch3

DES_lensingini11 httpsgithubcomcmbantCosmoMCblobmasterbatch3

DESini

expansion history we show that we can break parameterdegeneracies present in the CMB results leaving com-bined fits that are always consistent with a flat ΛCDMmodel The combined probes also offer some of the mostcompetitive constraints on neutrino mass without addingany information from growth of structure In Section 42we show that the BAO data enable estimates of H0 thatare robust against the assumption of cosmological modeland estimates that are independent of CMB anisotropiesaltogether

41 Impact of BAO Measurements on Models for SingleParameter Extensions to ΛCDM

We first report the results in the simplest cosmologythat of a spatially flat universe where dark energy can beexplained by a cosmological constant (ΛCDM) As shownin Table 4 CMB data alone are sufficient to constrain thedark energy density parameter to roughly 1 precisionAdding the BAO and SN data improves this constraintby a factor of 15 for this simplest model of the expansionhistory

Figure 2 shows the residuals of the BAO and SNe Iadistances with respect to the ΛCDM model favored bythe Planck temperature and polarization data Both theBAO and SNe Ia data are in very good agreement withthe predictions from the best-fit ΛCDM model In orderto highlight how BAO and SNe Ia data complement theCMB results in models with a single-parameter exten-sion to ΛCDM in Figure 2 we also show the predictionfor three models that are allowed by Planck but are ruledout by measurements of the low redshift expansion his-tory an oΛCDM model with the Planck-favored valueof Ωk = minus0044 (dashed red) a wCDM model with thePlanck-favored value of w = minus1585 (dot-dashed green)and a ΛCDM model with the Planck 95 upper limitson the sum of the neutrino masses of

summν = 0268 eV

(solid blue)12In the next sub-sections we present in detail how BAO

and SNe Ia can break strong degeneracies present in theCMB data when studying single-parameter extensions tothe ΛCDM model

411 Expansion history and curvature

The Planck temperature and polarization data aloneoffer strong constraints within the oΛCDM model butwith degenerate posteriors as shown in both panels ofFigure 3 The consequences of these degeneracies arequantified in Table 4 where the uncertainty on ΩΛ in thismodel is five times larger than in a flat ΛCDM modelThe preference for a closed universe with a significanceslightly above 95 confidence is discussed in detail inPlanck Collaboration et al (2016b) and Planck Collab-oration et al (2018b) As shown in Figure 2 the pre-dictions from the closed universe favored by the CMB(dashed red lines) are disallowed at high confidence byboth the BAO and the SN data

In an oΛCDM model BAO measurements at differentredshifts constrain different combinations of (Ωm ΩkrdH0c) When we combine BAO results over a wideredshift range we are able to break internal degeneraciesand provide independent constraints on these parameters

12 The slight change in neutrino mass compared to the Planckanalysis is due to our use of an updated version of CosmoMC

CDMo CDM

wCDMCDM 0997

00 05 10 15 20 25 30z

1120 1200

Fig 2mdash Demonstration of BAO SN and CMB constraining power as a function of redshift To construct alternative models wehave fixed to their best-fit ΛCDM values the quantities that are best measured by the CMB Ωbh

2 Ωch2 and the angular acoustic scaleDM (z = 1150)rd Because the sound horizon at decoupling is a function of Ωbh

2 Ωch2 and Neff only the models have the same value ofrd = 14716 Mpc Top The Hubble diagram residuals of BAO DM (z) measurements with DV (z) measurements shown as open circlesCenter The Hubble diagram residuals of BAO DH(z) = cH(z) measurements Bottom The Hubble diagram residuals of SNe Iameasurements with relative normalization of the luminosity distance estimates We display the CMB determination of the angular positionof the acoustic peak as a measurement of transverse BAO and we split the redshift scale to include this data point In each case theresiduals are computed relative to the best-fit ΛCDM model from CMB alone The curves represent the difference between the ΛCDMmodel and single-parameter extensions allowed by the CMB data The oΛCDM model favored by Planck (Ωk = minus0044) is presented indashed red lines the wCDM model favored by Planck (w = minus1585) is presented in dot-dashed green lines and a ΛCDM model withnon-zero neutrino mass is presented in solid blue lines The model with massive neutrinos assumes a summed mass equal to 0268 eVcorresponding to the Planck 95 upper limit

00 02 04 06 08 10

oΛCDM

CMB TampPSNBAO

02 03 04 05 06

minus010

minus005

oΛCDM

CMB TampPCMB TampP+SNCMB TampP+BAO

Fig 3mdash Cosmological constraints under the assumption of a model with a w = minus1 cosmological constant with free curvature (oΛCDMas in Table 4) Left 68 and 95 constraints on ΩmndashΩΛ from the Planck CMB temperature and polarization data (gray) PantheonSNe Ia sample (red) and SDSS BAO-only measurements (blue) The dashed line represents a model with zero curvature Right TheΩmndashΩk constraints for the combination of CMB (gray) CMB + SN (red) and CMB + BAO (blue)

TABLE 4Marginalized values and confidence limits in ΛCDM and one-parameter extensions using only expansion history and CMB temperature

and polarization measurements

ΩDE H0[kmsMpc] Ωk w Σmν [eV]BAO 0701 plusmn 0016 minus minus minus minus

CMB TampP 06836 plusmn 00084 6729 plusmn 061 minus minus minusCMB TampP + BAO 06881 plusmn 00059 6761 plusmn 044 minus minus minusCMB TampP + SN 06856 plusmn 00078 6743 plusmn 057 minus minus minusCMB TampP + BAO + SN 06891 plusmn 00057 6768 plusmn 042 minus minus minusBAO 0637+0084

minus0074 minus 0078+0086minus0099 minus minus

oΛCDM

CMB TampP 0561+0050minus0041 545+33

minus39 minus0044+0019minus0014 minus minus

CMB TampP + BAO 06882 plusmn 00060 6759 plusmn 061 minus00001 plusmn 00018 minus minusCMB TampP + SN 0670 plusmn 0017 652 plusmn 22 minus00061+00062

minus00054 minus minusCMB TampP + BAO + SN 06891 plusmn 00057 6767 plusmn 060 minus00001 plusmn 00018 minus minusBAO 0729+0017

minus0038 minus minus minus069 plusmn 015 minus

CMB TampP 0801+0057minus0022 minus minus minus158+016

minus035 minusCMB TampP + BAO 0694 plusmn 0012 684+14

minus15 minus minus1034+0061minus0053 minus

CMB TampP + SN 0692 plusmn 0010 683 plusmn 11 minus minus1035 plusmn 0037 minusCMB TampP + BAO + SN 06929 plusmn 00075 6821 plusmn 082 minus minus1026 plusmn 0033 minus

νΛCDM

CMB TampP 0680+0016minus00087 670+12

minus067 minus minus lt 0268 (95)CMB TampP + BAO 06890+00069

minus00061 6770+053minus048 minus minus lt 0134 (95)

CMB TampP + SN 0686+0011minus00083 6747+083

minus065 minus minus lt 0174 (95)CMB TampP + BAO + SN 06898 plusmn 00061 6776 plusmn 047 minus minus lt 0125 (95)

Note mdash Reported uncertainties correspond to 68 confidence intervals except forsummν in the νΛCDM model The reportedsum

mν values correspond to the 95 upper limits BAO measure the dimensionless quantity rdH0c and therefore can onlyprovide constraints on H0 when combined with other probes The constraints of CMB TampP in the wCDM model are affectedby the H0 prior of H0 lt 100kmsMpc so no entry is provided here either

00 02 04 06

minus15

minus10

minus05

CMB TampPSNBAO

028 030 032 034 036

νΛCDM

Fig 4mdash Constraints on the wCDM and νΛCDM models as in Table 4 Left wndashΩm constraints under the assumption of a flatwCDM cosmology from the Planck CMB temperature and polarization data (gray) Pantheon SNe Ia sample (red) and SDSS BAO-onlymeasurements (blue) Right

summνndashΩm constraints under the assumption of a flat ΛCDM cosmology where the summed neutrino mass

is allowed as a free parameter for the combination of CMB (grey) CMB + SN (red) and CMB + BAO (blue)

(eg Nadathur et al 2020) Table 4 and the left panelof Figure 3 show that BAO measurements alone lead toΩΛ = 0637+0084

minus0074 an 8σ confidence detection of a cosmo-logical constant without any information from the CMBor SNe Ia data The SNe Ia data alone also favor a flatgeometry but are not as constraining as BAO Usingonly SNe Ia leads to a detection of ΩΛ = 073plusmn 011

The right panel of Figure 3 demonstrates that includ-ing either BAO or SN data reduces the parameter de-generacies in the CMB data The ΩDE results in theCMB TampP + BAO entries in Table 4 are almost thesame in ΛCDM and oΛCDM models The combina-tion of BAO and CMB data favors a flat universe withΩk = minus00001plusmn 00018

412 Expansion history and dark energy

We next consider a flat wCDM model with an extrafree parameter w to describe the equation of state ofdark energy As with the oΛCDM model the left panelof Figure 4 shows that the CMB temperature and polar-ization data leave strong degeneracies between the w andenergy density parameters that determine the expansionhistory Table 4 shows that the constraints on ΩDE areagain degraded by a factor of about five with respect tothe constraints in a ΛCDM model with a shift in thecentral value that is opposite in direction to the shiftin the oΛCDM model The models with very negativevalues of w favored by CMB (dot-dashed green lines inFigure 2) are inconsistent with both the BAO and theSN data

As shown in the left panel of Figure 4 the PlanckΛCDM values (w = minus1 and Ωm = 03164) lie withinthe 95 confidence intervals of both the BAO data aloneand the SN data alone The BAO data alone are able toconstrain the matter density without a strong degener-acy with w Even though the SNe Ia contours have astrong degeneracy in wndashΩm the contours are perpendic-ular to the degeneracy direction of the CMB contoursso the CMB TampP+SN combination results in very tightconstraints on the wCDM model Each of the three com-binations CMB TampP+BAO CMB TampP+SN and evenBAO+SN favor a model with a cosmological constantAs shown in Table 4 the combination of all three datasetsresults in a measurement of the equation of state of darkenergy of w = minus1026plusmn 0033 consistent with a cosmo-logical constant

413 Expansion history and neutrino masses

We now turn our attention to a νΛCDM model wherethe sum of the neutrino masses is considered a free pa-rameter As shown in the right panel of Figure 4 and inTable 4 the Planck temperature and polarization dataoffer a 95 upper bound on the summed neutrino massof 268 meV Neutrinos lighter than sim 500 meV are stillrelativistic at the time of recombination but they impactCMB observables by modifying the late-time expansionin particular the angular diameter distance to the epochof recombination DM (zrec) Neutrino mass constraintsfrom the CMB are therefore degenerate with other cos-mological parameters that modify DM (zrec) like Ωm orH0 Late-time measurements of the expansion can breakthis degeneracy as shown in blue lines of Figure 2 andin the right panel of Figure 4 Adding BAO or SN data

reduces the upper bound on the sum of neutrino massesby a factor of 2 and 15 respectively Combining thethree datasets we obtain a 95 upper limit of 125 meV

In this subsection we have shown that measurementsof the expansion history are very complementary tomeasurements of CMB temperature and polarizationanisotropies As shown in Figure 2 both BAO andSNe Ia are able to constrain single-parameter extensionsto ΛCDM that can not be constrained by CMB alone Asshown in Table 4 adding BAO to the CMB data reducesthe uncertainty on ΩDE in oΛCDM models by a factor ofeight and it excludes models with curvature that wouldotherwise be favored by the CMB Similarly adding SNand BAO to the CMB reduces the uncertainty on ΩDE

in wCDM models by more than a factor of five and itexcludes models with w lt minus1 favored by the CMB Forall models discussed the combination of all three probesresults in a percent measurement of ΩDE consistent withΩDE = 069

42 BAO and the H0 Tension

The present-day expansion rate H0 is one of the basicparameters in the cosmological model because it allowsabsolute estimates of the age and the current energy con-tent of the Universe It is one of the three fundamentalcosmological parameters that are not dimensionless (thetwo other being the temperature of the CMB and theneutrino masses) Moreover as discussed in Hu (2005)and in Weinberg et al (2013) an accurate measurementof H0 would allow a powerful test of dark energy mod-els and tightened constraints on cosmological parame-ters However a statistically-significant tension has beendemonstrated between direct measurements of H0 fromthe local distance ladder and those estimates of H0 in-ferred from the CMB (Riess et al 2016) This tensionhas persisted and even increased in significance despitesignificant effort to identify possible sources of systematicerrors

Measurements of the Hubble constant come in differentflavors as shown in the compilation of studies presentedin the bottom part of Table 5 An example of direct mea-surement referred here as the distance ladder uses par-allaxes from local stars and other techniques to calibratedistances to Cepheid variables which are in turn usedfor absolute luminosity calibration of SNe Ia hosted bynearby galaxies (eg Riess et al 2019) The calibratedluminosity is used to estimate the absolute luminositydistance to a sample of SNe Ia that covers a redshiftrange sufficient to minimize the effect of peculiar veloc-ities relative to the Hubble flow Similar efforts includethe use of other distance indicators such as the tip of thered giant branch (TRGB eg Freedman et al 2020)Tully-Fisher relation in galaxies (TFR eg Neill et al2014) or gravitational waves from neutron star-neutronstar mergers (eg Abbott et al 2017a) These mea-surements typically measure higher values of the Hubbleconstant For example Riess et al (2019) perform astudy using SNe Ia distances calibrated from 70 long-period Cepheids in the Large Magellanic Cloud Theyfind H0 = 7403 plusmn 142 km sminus1Mpcminus1 including system-atic errors

Other measurements of H0 involve data at higher red-shift and need to assume a cosmological model to extrap-

028 030 032 034

CMB TampP+BAO+SN

ow0waCDM

02 03 04 05

Distance Ladder

BAO(z gt 1)+BBN

BAO(z lt 1)+BBNBAO+BBN

Fig 5mdash Left H0 versus Ωm from the inverse distance ladder (CMB+BAO+SN) under two different cosmological models Right H0versus Ωm from the combination of BAO and BBN in a ΛCDM model (blue) The red (gray) contours show the results when using onlyBAO measurements below (above) z = 1 The horizontal shaded area shows the (68 95) measurement of H0 from the distance laddertechnique (SH0ES Riess et al 2019)

TABLE 5Hubble parameter constraints

Dataset Cosmological model H0 (km sminus1Mpcminus1) CommentsCMB TampP+BAO+SN ow0waCDM 6787 plusmn 086 Inverse distance ladderBBN+BAO ΛCDM 6735 plusmn 097 No CMB anisotropiesCMB TampP ΛCDM 6728 plusmn 061 Planck 2018 (a)CMB TampP oΛCDM 545+33

minus39 Planck 2018 (a)Lensing time delays ΛCDM 733 plusmn 18 H0LiCOW (b)Distance ladder - 740 plusmn 14 SH0ES (c)GW sirens - 70 plusmn 10 LIGO (d)TRGB - 696 plusmn 19 LMC anchor (e)TFR - 762 plusmn 43 Cosmicflows (f)

Note mdash The top section shows constraints derived in this paper while the bottom section shows a compilation of resultsfrom the literature (a) CMB anisotropies measured by the Planck satellite (Planck Collaboration et al 2018b) (b) time delaysfrom six gravitationally lensed quasars from H0LiCOW (Wong et al 2020) (c) distance ladder with Cepheids and SNe Ia fromthe SH0ES collaboration (Riess et al 2019) (d) gravitational wave detection of a neutron star binary merger by LIGO (Abbottet al 2017a) (e) tip of the red giant branch (TRGB) calibrated with the LMC distance (Freedman et al 2020) (f) Tully-Fisherrelation (TFR) from the Cosmicflows database of galaxy distances (Tully et al 2016)

olate the constraints to redshift zero One example of thisindirect measurement is that obtained using time delaysin strongly-lensed quasars (eg Birrer et al 2019) Otherindirect measurements of H0 use CMB data under strongassumptions about the model governing the expansionhistory from the last scattering surface to today TheCMB estimates typically give considerably lower valuesof the Hubble constant The final Planck data release forexample finds H0 = 6736 plusmn 054 km sminus1Mpcminus1 (PlanckCollaboration et al 2018b) when assuming the ΛCDMmodel

Explanations for the tension between direct measure-ments and CMB estimates range from underestimatedsystematic errors or modeling of the primordial powerspectrum (eg Davis et al 2019 Dhawan et al 2020Anderson 2019 Hazra et al 2019) to models for darkenergy (eg Li amp Shafieloo 2019 Alestas et al 2020 DiValentino et al 2020) to unmodeled pre-recombinationphysics that lead to a decreased sound horizon scale (egPoulin et al 2019 Chiang amp Slosar 2018 Beradze amp Gog-berashvili 2020 Vagnozzi 2019 Lin et al 2019 Arendseet al 2019) See Knox amp Millea (2020) for a review ofpossible solutions to the tension

We provide here two alternative analyses to show how

BAO measurements allow estimates of H0 that are ro-bust against the strict assumptions of the CMB-onlyestimates First we combine Planck temperature andpolarization SN and BAO data and allow a very flexi-ble expansion history to demonstrate that the tension inH0 estimates is not due to the assumptions of a ΛCDMmodel Second we present a measurement of H0 thatuses BAO and a BBN prior that is independent of CMBanisotropies to demonstrate that the tension is not dueto systematic errors in the CMB data We finish this sec-tion presenting the combination of the BAO data withthe local distance ladder measurement and we discussthe low value of rd inferred from this analysis

421 H0 and the inverse distance ladder

In this subsection we present a cosmological measure-ment of H0 without an assumption of a flat ΛCDMmodel This approach is often referred as the inversedistance ladder as it relies on a calibrated distance mea-sure at high redshift that is then extrapolated to z = 0Schematically we use information from the CMB to cal-ibrate the BAO distances Those in turn are used tocalibrate the absolute luminosity of SNe Ia

Since the BAO feature follows DH(z)rd = cH(z)rd

and DM (z)rd rather than H(z) directly this mea-surement relies on a calibration of the sound horizon(rd) at the drag epoch to extract the Hubble parame-ter Under the implicit assumption of a smooth expan-sion history standard pre-recombination physics and awell-measured mean temperature of the CMB rd onlydepends on the cold dark matter density (Ωch

2) andthe baryon density (Ωbh

2) Thus rd can be calibratedthrough constraints on Ωch

2 and Ωbh2 arising from the

full CMB temperature and polarization likelihoods ef-fectively fixing rd to its CMB preferred value

The extrapolation of H(z) measurements from BAOto z = 0 can be done using a very flexible cosmologybecause both BAO and SNe Ia relative distance mea-surements constrain the evolving expansion rate Theinclusion of BAO makes the technique robust to the as-sumed properties of dark energy as was demonstrated inearlier BOSS analyses (Aubourg et al 2015)

We choose an ow0waCDM model to allow for a flexi-ble expansion history of the Universe Note that CMBalone can not constrain this model as shown in Table 5the uncertainties on H0 from CMB constraints alreadyincrease by a factor of about six when we consider onlyone parameter extensions such as models with curvatureThe combination of CMB BAO and SN data howeveris able to provide a very precise measurement of H0 evenin this flexible model Our results presented in Table 5and in the left panel of Figure 5 have an uncertaintybetter than 1 km sminus1Mpcminus1 and are consistent with thelow value of H0 measured by the CMB under the strictassumption of ΛCDM

422 H0 independent of CMB anisotropies

In the previous subsection we showed that the value ofH0 measured by the combination of CMB BAO and SNdata is robust under different models for curvature anddark energy equation of state In this section we returnto the ΛCDM model and present a measurement of H0

that is independent of CMB anisotropiesThe combination of BAO measurements at different

redshifts can provide a precise measurement of the di-mensionless quantity rdH0c To translate constraintson this dimensionless quantity to a measurement of H0we use information on ωb by including BBN constraintsωc and H0 are also left free as they can be determinedin the fitting by the BAO data13 We use the resultsof recent high resolution spectroscopic measurements ofseven quasar absorption systems that indicate a primor-dial deuterium abundance DH = (2527plusmn0030)times 10minus5

(Cooke et al 2018) Using the empirically-derived reac-tion cross-section (Adelberger et al 2011) the deuteriumabundances imply ωb = 002235 plusmn 000037 under an as-sumption that Neff = 3046 The 68 confidence intervalreflects the combined deuterium abundance and reactionrate uncertainties

As can be seen in the right panel of Figure 5 we ob-tain a tight constraint on H0 only when we combine BAOmeasurements from a wide redshift range In particularthe line-of-sight BAO measurements above z = 1 (fromquasars and the Lyα forest) provide measurements of the

13 To estimate the radiation density we also use the absoluteCMB temperature measured by FIRAS T0 = 27255 K (Fixsen2009)

60 65 70 75 80

H0 [kmsMpc]

ΛCDM Distance Ladder

BAOBAO+BBNBAO+Distance LadderCMB TampP

Fig 6mdash Cosmological constraints on H0 and rd under the as-sumption of the ΛCDM model using BAO data (blue) in combi-nation with H0 distance ladder measurements (purple) and BBNdata (dark blue) in contrast to CMB measurements (grey) Theshaded band refers to the H0 distance ladder measurement

expansion in the matter-dominated area and their con-tours have different degeneracies in the (Ωm H0) plane

As shown in Table 5 the precision on H0 whencombining BAO measurements with a BBN prior is097 km sminus1Mpcminus1 This result is consistent with thefindings of Addison et al (2018) and Cuceu et al (2019)who used BAO data from SDSS DR12 and DR14 respec-tively The central value remains relatively unchangedfrom the results using CMB BAO and SN data in theow0waCDM model providing further evidence that thetension is not due to peculiarities in the CMB anisotropydata

423 Sound horizon at drag epoch from low redshifts

As shown above the BAO data in combination withinformation on the baryon density from the early Uni-verse can be used to extrapolate late universe expansionhistory to constrain the Hubble constant The BAO datacan also be used to constrain the sound horizon at thedrag epoch when combined with local H0 measurements(eg Cuesta et al 2015)

Figure 6 shows the 2D-contours of H0 and rd for BAOdata in combination with different datasets under theassumption of a ΛCDM model The BAO data aloneare completely degenerate in the H0ndashrd plane howeverthis degeneracy can be broken by either local H0 mea-surements by BBN or by CMB data The local H0

measurements are clearly in tension with early Universemeasurements of the sound horizon BAO and BBN dataprefer a value rd = 1493 plusmn 28 Mpc in good agreementwith the value rd = 14706 plusmn 029 Mpc preferred by theCMB temperature and polarization data alone Theseestimates are much larger than the BAO and distanceladder constraint of rd = 1359 plusmn 32 Mpc These con-straints on rd can also be translated into limits on thebaryon density yielding ωb = 00310 plusmn 00024 for BAOand distance ladder data In comparison the CMB bestfit of ωb = 002236 plusmn 000015 or the BBN best-fit ofωb = 002235plusmn 000037 are much lower

Finally dropping the assumption of a ΛCDM modeland including SN in our analysis of the distance ladderwe find rd = 1351plusmn31 Mpc and ωb = 00377+00052

minus00089 in a

ow0waCDM model This extended distance ladder mea-surement shows that the discrepancy between low andhigh redshift measurements of the sound horizon is in-dependent of the assumption of the cosmological modelHowever we caution that we did not take the correlationbetween the SN data and the local H0 measurement intoaccount for our analysis

To summarize the BAO data allow robust consistentmeasurements of H0 that are insensitive to the strict cos-mological assumptions in CMB-only estimates and insen-sitive to the use of CMB anisotropies altogether if usingthe ΛCDM model In all cases the central values remainbelow H0 = 68 km sminus1Mpcminus1 and the uncertainties re-main at 1 km sminus1Mpcminus1 or better

On the other hand the Cepheid distance ladder orstrong lensing time delays of quasars provide precise es-timates of H0 that favor larger values of H0 or smallervalues of rd if being used to calibrate the BAO scaleCombining their results as independent measurementsproduces an estimate of H0 = 737 plusmn 11 km sminus1Mpcminus1This central value differs from those presented in thiswork by more than four standard deviations whether weuse a multiple parameter model for expansion or the BBNmeasurements of ωb The consistency of the results high-lights that the lsquoH0 tensionrsquo can not be restricted to sys-tematic errors in Planck or to the strict assumptions ofthe ΛCDM model

Both the CMB analysis and those presented hereare sensitive to the assumption of standard pre-recombination physics that sets the scale of rd As sum-marized in Knox amp Millea (2020) there have been manyattempts to reconcile the H0 tension by modifying thevalue of rd with limited success

5 IMPLICATIONS OF GROWTH MEASUREMENTS

A key development of the BOSS and eBOSS surveysis the advancement of RSD as a tool to make high preci-sion measurements of structure growth over a wide red-shift range In this section we assess the impact of thosegrowth measurements on the cosmological model Wefirst compare the RSD measurements to DES weak lens-ing and Planck lensing results to complement the CMBtemperature and polarization data in dark energy mod-els In the second part of this section we explore the useof growth measurements to constrain matter fluctuationsand to test the assumptions of GR in the cosmologicalmodel

51 Impact of RSD Measurements on Models for SingleParameter Extensions to ΛCDM

The constraining power of RSD is illustrated in Fig-ure 7 The low-redshift RSD measurements alone havesensitivity to rule out Einstein-de Sitter (Ωm = 1) modelswhile the higher redshift RSD measurements are sensitiveto variations in the dark energy equation of state Wefirst quantify how these RSD data offer complementaryviews to the WL data on single parameter extensions toa ΛCDM cosmology

We begin by exploring the constraints on a modelwith free curvature (oΛCDM) using growth measure-ments combined with the Planck CMB temperature and

polarization data The marginalized 68 constraints onkey cosmological parameters are shown in the top half ofTable 6 The two-dimensional contours on Ωm and ΩΛ

are shown in the left panel of Figure 8 While the PlanckCMB data alone favor a model with negative curvaturethe combination with all growth measurements (RSDWL and CMB lensing) reduces the Ωk uncertainty by afactor of four and leads to a model consistent with zerocurvature (Ωk = minus00010+00043

minus00038)As shown in the residual diagram of growth measure-

ments (Figure 7) the predictions for growth in a free-curvature oΛCDM universe have the largest deviationsfrom a ΛCDM prediction as redshift approaches z = 0The RSD measurements in this regime are governedlargely by the MGS sample with a precision of 36 onfσ8 As a consequence relative to the CMB-only con-straints on curvature those from the CMB+RSD mea-surements only result in a mild shift with a slightly re-duced uncertainty (Table 6) The DES WL data onthe other hand offer an independent measurement ofthe mass distribution in particular using source galaxiesover the redshift range 020 lt z lt 043 While difficultto visualize in a manner similar to the RSD the WL mea-surements offer significantly higher precision estimates inthe low redshift regime The WL measurements whencombined with the CMB data substantially shift theconstraints on curvature to be consistent with flatness(Ωk = minus00004 plusmn 00048) with a factor of 34 reductionin uncertainty The constraining power of CMB lensinglies in between the RSD and low redshift WL combiningthe Planck lensing with the temperature and polarizationdata leads to a bestfit model consistent with the ΛCDMmodel (Ωk = minus0011plusmn0006 Planck Collaboration et al2018b)

We next explore the constraints on a flat wCDMmodel where the equation of state w for dark energyis constant but allowed to vary The Planck temperatureand polarization data prefer a value of w much more neg-ative than minus1 and adding CMB lensing causes virtuallyno change (Planck Collaboration et al 2018b) As shownin the right panel of Figure 8 the combination of growthmeasurements with CMB data provides constraints onw that enclose the cosmological constant model withinthe 95 contours Contrary to the case of oΛCDM itis the RSD data that have the largest impact in shiftingthe CMB contours As shown in Table 6 combining WLmeasurements with CMB does not significantly improvethe precision on w but RSD measurements are able toimprove the precision by more than a factor of two

The constraining power of RSD on w can be under-stood from Figure 7 A more negative value of w causesincreasingly slower structure growth toward lower red-shifts The fσ8 measurements with BOSS and eBOSSgalaxies sample the growth rate in the redshift range02 lt z lt 11 providing good constraints on the shapeof fσ8(z) around its peak and thus constraints on wThe CMB+RSD data are therefore able to rule out(at 45 standard deviations) the formal central value ofw = minus1585 preferred by CMB alone The combina-tion of all the growth measurements with CMB prefers amodel consistent with w = minus1 at the level of 14σ whenusing the one-dimensional marginalized likelihoods

00 05 10 15 20 25z

CDM o CDM wCDM CDM EdS CDM

Fig 7mdash The SDSS fσ8 measurements as a function of redshift normalized by the Planck 2018 bestfit ΛCDM model (shown in dottedblack) The three colored curves represent the fractional deviations from ΛCDM for an oΛCDM model with Ωk = minus0044 (red) a wCDMmodel with w = minus158 (green) and a νΛCDM model with

summν = 0268 eV (blue) These are the same models as those in Figure 2 An

Einstein de Sitter model (magenta Ωm = 1 ΩΛ = 0 and σ8(z = 0) matching that of fiducial model) is ruled out at high confidence furtherdemonstrating the long-standing preference for growth measurements for models with lower matter densities

TABLE 6Marginalized values and 68 confidence limits on curvature dark energy parameters and the amplitude of density fluctuations

Ωm ΩDE σ8 Ωk w

oΛCDM

CMB TampP 0483+0055minus0069 0561+0050

minus0041 0774+0016minus0014 minus0044+0019

minus0014 minusCMB TampP + RSD 0455+0052

minus0062 0581+0045minus0039 0780 plusmn 0014 minus0036+0017

minus0013 minusCMB TampP + WL 0310 plusmn 0017 0690 plusmn 0013 0806 plusmn 0010 minus00004 plusmn 00048 minus

CMB TampP(+lens)) + RSD + WL 0313 plusmn 0014 0688 plusmn 0011 08069 plusmn 00094 minus00010+00043minus00038 minus

CMB TampP 0199+0022minus0057 0801+0057

minus0022 0970+0096minus0045 minus minus158+016

minus035a

CMB TampP + RSD 0293+0027minus0034 0707+0034

minus0027 0836 plusmn 0030 minus minus109 plusmn 011CMB TampP + WL 0188+0012

minus0046 0812+0046minus0012 0977+0083

minus0037 minus minus161+013minus030

CMB TampP(+lens) + RSD + WL 0275+0023minus0029 0725+0029

minus0023 0846 plusmn 0028 minus minus114 plusmn 010

aThe lower bound on w is affected by the H0 prior

03 04 05 06 07

oΛCDM

CMB TampP+RSD+CMB lensing+WL

02 03 04

minus16

minus14

minus12

minus10

minus08

Fig 8mdash Constraints from CMB temperature and polarization and growth measurements in one-parameter extensions to ΛCDM asin Table 6 Left The ΩmndashΩΛ constraints for a cosmological model under the assumption of a w = minus1 cosmological constant with freecurvature (oΛCDM) Right The wndashΩm constraints for a flat cosmological model where the equation of state is allowed as a constantfree parameter In both cases the gray contours represent the 68 and 95 confidence intervals using only the Planck temperature andpolarization data while the blue contours show the results including RSD data The combination of RSD DES WL Planck lensing andCMB is shown in red

We see that CMB and growth measurements providefactor of 25ndash4 improvements on the precision in extendedΛCDM models when compared to CMB temperature andpolarization data alone The growth data have the neteffect of pulling the CMB data closer to a ΛCDM model

52 RSD Constraints on the Amplitude of MatterFluctuations and Tests of Gravity

Within the ΛCDM model RSD and lensing provide ameans to integrate the rate of structure growth to red-shift zero and estimate the current amplitude of matterfluctuations σ8 This estimate of σ8 can be comparedto the predictions when extrapolating the amplitude ofthe measured CMB power spectrum thus serving as aΛCDM consistency test similar to that of the H0 inversedistance ladder tests in Section 42 Structure growthcan also be used to test the basic assumptions of theΛCDM model through modifications to GR In this casethe redshift evolution of matter density fluctuations andthe interaction of matter and photons with the resultantgravitational potential can be directly compared to pre-dictions of GR

Here we use SDSS RSD measurements DES WL mea-surements and Planck CMB lensing results to assess theamplitude of local matter fluctuations and perform a con-sistency test for GR

521 RSD constraints on the amplitude of matterfluctuations

First we explore the constraints on Ωmndashσ8 from growthmeasurements assuming a ΛCDM cosmology As shownin the left panel of Figure 9 the constraints from any ofthe growth measurements (RSD WL and CMB lensing)are consistent with the predictions of models informedonly by the CMB albeit with much larger contours Notethat we have applied conservative priors on ns and ωb forall contours (see Appendix B)

The degeneracy from growth measurements follows thedirection of lower Ωm (thus slower structure growth) andhigher fluctuation amplitude σ8 The differences in thedegeneracy directions with RSD WL and CMB lensingmeasurements result from their different dependences oncosmology and different redshift sensitivities Given thedifferences we do not seek to present constraints on op-timal combinations of the two parameters Among thethree growth measurements RSD appears to have thelargest contour area but provide the tightest constraintson σ8 with σ8 = 0838plusmn 0059 The WL measurementslead to overall better constraints in the Ωmndashσ8 planewhile the marginalized constraints on Ωm are compara-ble to RSD and those on σ8 (σ8 = 0857plusmn 0147) are notas tight as RSD CMB lensing results in constraints in adirection similar to that of RSD but with a stronger Ωmndashσ8 degeneracy and thus narrower contours than RSD

The combination of RSD WL and CMB lensing isshown in the light purple contours in the left panel ofFigure 9 The resulting constraints are greatly improvedoffering σ8 = 0843 plusmn 0039 and Ωm = 0259 plusmn 0036 Inaddition the 68 confidence intervals overlap the 68confidence intervals from the prediction based on CMBtemperature and polarization data indicating generalconsistency

522 RSD constraints on modified gravity

The difference between the speed of gravity and thespeed of light has been shown to be negligible (Abbottet al 2017b) as predicted by GR fσ8(z) measurementsfrom RSD can be used to further test theories of gravityin the context of structure formation

Here we consider a phenomenological parameterizationof gravity as described in Section 22 allowing for thetwo metric potentials Ψ and Φ to deviate from theirGR prediction independent of the speed of gravitationalwaves The parameter micro0 characterizes the deviationof Ψ which determines the response of matter to thegravitational potential and thus can be probed by RSDThe parameter Σ0 characterizes the deviation of Ψ + Φwhich determines the propagation of light and thus canbe probed by lensing Therefore the combination ofvarying micro0 and Σ0 provides us with a null test of grav-ity along the degeneracy direction of our most potentprobes of modified gravity RSD and lensing (WL andCMB lensing)

Assuming the fiducial cosmological model to be ΛCDMwith the background parameters fixed to the baselinevalues (see Appendix B) we compute the constraintson micro0 and Σ0 (right panel of Figure 9) As expectedWL+CMB lensing mainly constrain Σ0 while RSD isonly sensitive to micro0 A combination of both probes isnecessary to break degeneracies between the two param-eters With the combined RSD and lensing we findmicro0 = minus004 plusmn 025 and Σ0 = minus0024 plusmn 0054 consistentwith the GR prediction of micro0 = Σ0 = 0

6 GLOBAL FITS

After examining the impacts of expansion history andgrowth measurements alone we now proceed to combinethe Planck (including lensing) Pantheon SNe Ia SDSSand DES data to determine the best fitting cosmologicalmodel The SDSS data consist of the combined BAOand RSD measurements found in the bottom section ofTable 3 while the DES data consist of the cosmic sheargalaxy clustering and galaxy-galaxy lensing data (ie3times2pt) We refer to the results from the previous sec-tions where there is guidance on which datasets are pro-viding the critical information

We start by establishing the parameters for the sim-plest cosmology that of a ΛCDM universe with a fixedneutrino mass We examine the distribution of BAO andRSD measurements about this model to assess potentialtension with any of the individual measurements

We then expand the cosmological model to include freeparameters for curvature the dark energy equation ofstate and the neutrino mass In all cases the best fittingvalues of H0 are determined at a precision of better than08 km sminus1Mpcminus1 We then show that the addition ofthese free parameters does not lead to significant changesin any of the ΛCDM parameters and that the resultsremain consistent with a flat ΛCDM universe Finallywe provide a physical interpretation of the cosmologyconstraints on summed neutrino mass in the context ofneutrino oscillation experiments

61 ΛCDM Model

We start by finding the ΛCDM model that best de-scribes the full suite of data As shown in the first rowof Table 7 the dark energy density is constrained at thelevel of 07 This precision is improved by a factor

01 02 03 04 05 06 07

CMB lensWLRSDRSD+CMB lens+WLCMB TampP

minus06 minus04 minus02 00 02 04

minus10

minus05

micro0

RSDWL+CMB lenscombined

Fig 9mdash Left The Ωmndashσ8 constraints for a ΛCDM cosmology The blue contours represent the 68 and 95 constraints when usingonly the RSD sample The red contours represent the same when using only the DES WL results The dark blue contours represent theconstraints from Planck lensing while the light purple represent the combination of RSD DES WL and Planck lensing The gray contoursrepresent the predictions from the Planck temperature and polarization data under an assumption of a ΛCDM cosmological model Notethat the contours for low values of Ωm are affected by the H0 prior Right The Σ0ndashmicro0 constraints for a cosmology with a fixed cosmologicalconstant and perturbations to GR as described in Equations 11 and 12 The blue contours represent the 68 and 95 constraints whenusing only the RSD measurements The red contours represent the same when using the DES WL and Planck lensing data The graycontours represent the constraints when combining RSD DES WL and Planck lensing measurements

TABLE 7Marginalized values and 68 confidence limits for models using Planck Pantheon SNe SDSS BAO+RSD and DES 3times2pt data

ΩDE H0 σ8 Ωk w0 wa Σmν [eV]ΛCDM 06959 plusmn 00047 6819 plusmn 036 08073 plusmn 00056 minus minus minus minusoΛCDM 06958 plusmn 00048 6821 plusmn 055 08076 plusmn 00065 00001 plusmn 00017 minus minus minuswCDM 06992 plusmn 00066 6864 plusmn 073 08128 plusmn 00092 minus minus1020 plusmn 0027 minus minusowCDM 06997 plusmn 00069 6859 plusmn 073 08127 plusmn 00091 minus00004 plusmn 00019 minus1023 plusmn 0030 minus minusw0waCDM 06971 plusmn 00069 6847 plusmn 074 08139 plusmn 00093 minus minus0939 plusmn 0073 minus031+028

minus024 minusow0waCDM 06988 plusmn 00072 6820 plusmn 081 08140 plusmn 00093 minus00023 plusmn 00022 minus0912 plusmn 0081 minus048+036

minus030 minusνΛCDM 06975 plusmn 00053 6834 plusmn 043 08115+00092

minus00068 minus minus minus lt 0111 (95)

νwCDM 06993 plusmn 00067 6865 plusmn 073 0813+0011minus00098 minus minus1019+0034

minus0029 minus lt 0161 (95)

of 121 over the value found in Table 4 for a combina-tion of CMB BAO and SN and a factor of 178 over theCMB data alone indicating that the dark energy den-sity constraints are dominated by the expansion historymeasurements

The precision of ΛCDM parameter constraints allowsus to evaluate the distribution of SDSS measurementsabout the model For the purpose here we use Gaussianapproximations to the measurements for the evaluationIn comparison with the bestfit model the 14 BAO-onlymeasurements (DV rd DMrd and DHrd Table 3)give a value of χ2=110 with covariance among mea-surements taken into account Similarly the six RSDmeasurements (Table 3) give χ2=66 Finally we con-sider the full set by combining the 17 BAO+RSD mea-surements with the four BAO measurements from LyαndashLyα and LyαndashQuasar correlations and obtain χ2=237Based on the χ2 distribution with 14 6 and 21 degrees offreedom respectively all sets of measurements are fullyconsistent with the preferred model

To evaluate whether there is any statistically signifi-cant outlier in the measurements we compute the resid-ual between each SDSS measurement and the value pre-dicted by the preferred model In this pull distributionthe residuals are normalized by the measurement uncer-

3 2 1 0 1 2 3Pull

BAORSDBAO+RSD

3 2 1 0 1 2 3Pull

p = 098p = 056p = 071

BAORSDBAO+RSD

Fig 10mdash Distribution of residuals of the SDSS BAO (blue) RSD(red) and BAO+RSD (black) measurements with respect to thebestfit ΛCDM model In all cases the residuals are represented inthe form of pulls ie normalized by the measurement uncertaintyThe left panel shows the probability distribution and the rightpanel the cumulative distribution The p values are from KS testsin comparison with the normal distribution (dotted curves)

tainty so one would expect a Gaussian distribution withunity width if the measurements were distributed accord-ing to the measurement uncertainties We account forthe correlations among measurements by rotating thebasis for each measurement to produce statistically in-dependent pull values The resulting distribution of thepull values is shown in the left panel of Figure 10 For theBAO-only measurements the pull with the largest devia-

tion minus175 comes from the z sim 07 eBOSS LRG sampleFor the RSD measurements the largest deviation 191is from the z sim 148 eBOSS Quasar sample For thefull set (labeled as lsquoBAO+RSDrsquo) the largest deviationis again from the eBOSS Quasar sample After account-ing for the covariance between BAO and RSD measure-ments the measurement differs from the ΛCDM predic-tion by 30 standard deviations which is still acceptablefor the given sample size Furthermore based on theKolmogorovndashSmirnov (KS) test with the cumulative dis-tribution (right panel of Figure 10) the pull distributionsfor BAO RSD and BAO+RSD measurements are foundto be consistent with the normal distribution revealingno unexpected feature in the measurements for a universebest described by the ΛCDM model Additional diagnos-tics using the Hubble parameter can serve as consistencycheck on dark energy constraints (eg Shafieloo et al2012) as has been done to assess the BOSS BAO results(Sahni et al 2014) Preliminary results also indicate thatthe eBOSS data are consistent with a ΛCDM model

Within the ΛCDM model the SDSS DES and Planckdata offer tests of GR predictions on growth rates andmodel-dependent predictions for H0 The left panel ofFigure 11 shows the constraints from these three pro-grams on the amplitude of the (linear) power spectrumwhile the right panel shows the constraints on H0 In aΛCDM cosmology the model describing the DES 3times2ptdata has a strong degeneracy between σ8 and Ωm and astrong degeneracy between H0 and Ωm In both casesthe DES data are described by somewhat lower valuesof Ωm than the model describing the CMB data Themild tension between lensing and CMB estimates of σ8

can therefore be equally explained by the preference inWL estimates for a Ωm value lower than 03 On theother hand the results from the combined SDSS likeli-hoods are in good agreement with both Planck and DES3times2pt The BAO+RSD likelihoods are quite constrainedin both planes and have central values around Ωm = 03without any inference from Planck CMB data

62 Constraints on Dark Energy and Curvature

As was demonstrated in Section 4 the main strength ofthe BAO and SN distance measurements is to constraincosmological models with free curvature and varying darkenergy equation of state respectively As was shownin Section 5 the main strength of the growth measure-ments is in constraining possible deviations from GR Wenow explore the complementarity of distance and growthmeasurements by testing the same single parameter ex-tensions to ΛCDM that were presented in Section 4 fol-lowed by models with increasing degrees of freedom Theresults are found in Table 7

621 ΩDE H0 and σ8 Parameters

First the central values of the three parameters ΩDEH0 and σ8 are all consistent with the prediction fromthe bestfit ΛCDM model (Table 7) at 68 confidence re-gardless of the cosmological model that is assumed Thelargest fractional deviation from the ΛCDM predictionis only 08 in the case of σ8 in the ow0waCDM modelThat measurement is fully consistent with the ΛCDMprediction of σ8 = 08120plusmn00073 from CMB data alone(Planck Collaboration et al 2018b) The robustness of

σ8 measurements to cosmological model provide furtherevidence that the growth of structure can be describedusing GR in a ΛCDM parameterization

In addition the precision on the three parameters doesnot degrade significantly between the ΛCDM model andexpanded models When expanding to the ow0waCDMmodel the precision on the ΩDE and H0 parameters de-grades by factors of 15 and 23 respectively The largestdegradation for σ8 precision occurs with the νwCDMmodel leading to a factor of 19 increase in the uncer-tainties The tight constraints offered in all models area result of the complementarity between observationalprobes

As discussed in Section 4 interesting tensions appearbetween the estimates of the current Hubble expansionrate from local measurements and from extrapolations ofthe calibrated drag scale to z = 0 using the CMB Thoseestimates of a low H0 extrapolated from early times arenot changed with the addition of the growth data Foreven the most flexible ow0waCDM cosmology we findH0 = 6820plusmn081 km sminus1Mpcminus1 consistent with the find-ings in Section 4 The addition of the growth data leadsto a 6 improvement in the precision on H0 comparedto the results using CMB TampP+BAO+SN

622 Curvature and Dark Energy

When comparing the global results in an oΛCDMmodel to those from CMB TampP+BAO+SN in Table 4we find that the addition of the RSD Planck lensingand DES data only provides improvements of 6 on theprecision of curvature constraints The impact of growthmeasurements is larger in the wCDM model the addi-tional data provide improvements of about 22 on theprecision for a constant equation of state As discussedin Section 5 and in Figure 7 the improvement is likelyprimarily from the RSD measurements

When expanding to an evolving dark energy modelwith zero curvature we find that the bestfit models arestill consistent with ΛCDM The w0waCDM model doesnot improve the fit relative to ΛCDM indicating that theadditional free parameter is not providing critical new in-formation Overall we find consistent constraints on w0

with those in a wCDM modelThe complementarity of BAORSD and SNe Ia mea-

surements is best demonstrated in expanded dark energymodels that also allow for free curvature We only findmeaningful prior-independent constraints on the generalow0waCDM model for the combination of all datasets asshown in Figure 12 As shown in Table 7 the uncertain-ties on the two dark energy equation of state parametersin an ow0waCDM model are relatively unchanged whencompared to results under the assumption of a spatiallyflat universe (w0waCDM) The uncertainties on curva-ture are only increased by 30 when compared to thesingle parameter oΛCDM extension

In Figure 12 it can be seen that the w0ndashΩk confidenceintervals from the Planck+SDSS data are orthogonal tothe Planck+SNe contours As was demonstrated in Sec-tion 41 the BAO data best constrain the curvaturewhile the combination of CMB and SNe Ia best con-strains the dark energy equation of state As shown inthe one-dimensional likelihood of Ωk the constraints onΩk are roughly three times better using Planck+SDSSthan those using Planck+SN The Planck+SN data per-

02 03 04

ΛCDMPantheon

DESSDSSPlanck

02 03 04

Distance Ladder

Pantheon

DESSDSSPlanck

Fig 11mdash Left The Ωmndashσ8 constraints for a ΛCDM model A BBN-inspired prior on ωb and a prior of ns = 096 plusmn 002 was assumedfor the SDSS and DES contours Right H0 versus Ωm under a ΛCDM model In both panels the 68 and 95 confidence intervals forthe BAO+RSD data are shown in blue the DES 3times2pt data in red and the Planck CMB and lensing data in gray The faint verticalpurple bands represent the Pantheon constraints of Ωm = 0298plusmn0022 (Scolnic et al 2018) In the right panel the faint brown horizontalbands represent the CepheidSNe Ia measurements from Riess et al (2019) H0 = 7403 plusmn 142 km sminus1Mpcminus1

form slightly better than do the Planck+SDSS data inthe w0ndashwa plane but the net precision on both w0 andwa increases by roughly a factor of two when combin-ing all measurements This statistical increase in con-straining power is much larger than one would expectdue to the contribution of the Planck+BAO data toprovide tight constraints on curvature Most impor-tantly the combination of all cosmological probes revealsagain a preference for the ΛCDM model From the one-dimensional marginalized distributions w0 = minus1 is at11 standard deviations wa = 0 at 13 standard devi-ations and Ωk = 0 almost within the 68 confidenceinterval

In a related CPL parameterization for dark energywe can define a pivot scale factor ap or equivalentlya pivot redshift zp Instead of evaluating the equationof state at z = 0 as is done throughout this paperwe can represent the time-evolving equation of state asw(a) = wp + wa(ap minus a) Note that change of the pivotredshift does not change the model physically becausethe same linear relation can be described by the value andslope at any one point However by choosing the pivotscale appropriate to the redshifts covered by the dataconstraints on wp and wa can be made to be nearly uncor-related In doing so we find constraints in the w0waCDMmodel wp = minus1018 plusmn 0028 and wa = minus031+028

minus024 whenusing a pivot redshift zp = 034 The result demonstratesthat we can constrain the dark energy equation of state to3 precision at an earlier epoch in cosmic history Thisprecision is only degraded by a factor of 104 relative tothe constraint on w in an wCDM model indicating thatthe overall effect of adding the additional parameter fora time-varying equation of state is minimal

The results from joint fits can be used to compute atotal Dark Energy Figure of Merit (FoM Albrecht et al2006) for various sample combinations in a model withtime-varying equation of state Computed as the inverseproduct of wp and wa the FoM associated with the fullSDSS and Planck data is 384 in the w0waCDM modelThe FoM increases by a factor of 35 when adding thePantheon SNe Ia and the DES 3times2pt data Demonstrat-ing the complementarity of the BAO and SNe Ia data in

constraining curvature and the dark energy equation ofstate the Dark Energy FoM for all datasets only de-creases from 134 in the w0waCDM model to 92 in theow0waCDM model

63 Neutrino Mass

The existence of neutrino oscillations has been con-firmed by numerous terrestrial experiments (Abe et al2014 2012 Ahn et al 2012 An et al 2012 Adamsonet al 2008 Araki et al 2005 Ahmad et al 2002 Fukudaet al 1998) These experiments measure the differencebetween the squares of neutrino mass eigenstates lead-ing to two sets of possible solutions for individual masseswhich are referred to as the normal and inverted hierar-chies Both of these two solutions lead to degenerateneutrino masses if

summν amp 015 eV but lead to different

predictions at lower masses For the normal hierarchythe minimum neutrino mass is given by two essentiallymassless neutrinos and one massive neutrino For the in-verted hierarchy the minimum mass is composed of onemassless and two degenerate neutrinos The constraintsfor these two scenarios are (Esteban et al 2019 Choud-hury amp Hannestad 2019)sum

mν gt00588 eV normal hierarchy (24)summν gt00995 eV inverted hierarchy (25)

Throughout this paper we assume the neutrino massesto be at the minimum mass

summν = 006 eV with one

massive and two massless neutrinos When allowing afree parameter to describe the neutrino mass we con-tinue to assume two massless and one massive neutrinoswhich is a good approximation for the masses of interest(Lesgourgues amp Pastor 2006 Fogli et al 2012)

Resolving the hierarchy problem remains a key goal ofground-based neutrino experiments (eg Li et al 2013Akhmedov et al 2013 Abe et al 2015) Likewise thegoal of constraining the absolute mass of neutrinos hasmotivated a series of terrestrial experiments The tight-est constraints of direct measurements arise from the Ka-trin experiment (Aker et al 2019) resulting in a 90upper limit on the effective electron neutrino mass of

minus002 000

minus2

minus1

minus12

minus10

minus08

minus06

minus04

minus02

ow0waCDM

minus10 minus05

minus2 minus1 0

Planck+PantheonPlanck+SDSSall

Fig 12mdash Two dimensional contours on w0 wa and Ωk under the assumption of an ow0waCDM cosmological model The one-dimensionalconstraints on each independent parameter are presented in the top panels The red contours represent the 68 and 95 constraints whenusing the full Planck data (TampP and lensing) and the Pantheon SNe Ia measurements The blue contours represent the constraints fromPlanck and SDSS BAO+RSD while the gray contours represent the combination of all measurements presented in this work

m(νe) lt 11 eVTherefore it is timely to address the status of neu-

trino mass constraints before the advent of Stage IVdark energy experiments We show our results for theνΛCDM and νwCDM cosmological models in Table 8with several quantities The 95 upper limits are de-rived from Markov chains containing a prior

summν gt 0

In requiring only a mass that is positive the cosmologyconstraints assume no prior information from the neu-trino oscillation experiments and offer a fully indepen-dent measurement of neutrino mass Four selected datacombinations are plotted in Figure 13

It is useful to make a Gaussian approximation to bet-ter characterize the central values without influence fromthe prior and to provide a simple compression of the in-formation for other analyses These Gaussian fits aredetermined over the range 0 lt

summν lt 015 eV and are

given in the last column of Table 8 The upper 95 lim-its from the fitted Gaussian posteriors are within 2 ofthe chains for ΛCDM and within 4 when w is free tovary While we see that the preferred neutrino mass im-plied by these fits is negative a solution with

summν = 0

is usually within one standard deviation of the central

value and the minimal mass solution with the normal hi-erarchy is always within the 95 contours Finally theGaussian variances on these fits are essentially secondderivatives of the log-likelihood and are akin to Fishermatrix predictions They can therefore be used to givean insight into the constraining power of various probesthat is free from the vagaries of the most likely position

Because of the ability to break degeneracies with Ωmthe strongest improvement in neutrino mass precisionover CMB-only constraints is caused by the additionof BAO This is due to reasons discussed at length inAubourg et al (2015) and demonstrated in Section 41The RSD which is the canonical neutrino probe forits ability to measure the suppression of growth due tofree-streaming improves the precision by another 23Adding the RSD data is equivalent to an independentmeasurement with an error of about 01 eV in ΛCDMThe ability of BAO to improve upon CMB limits bybreaking degeneracies with matter densities is essentiallyexhausted with the current generation of BAO experi-ments as indicated in the right panel of Figure 4 Whilecurrently not the dominant source of information on neu-trinos RSD should become the main probe with arrival

TABLE 8Constraints on neutrino masses and relative probabilities of neutrino models with νΛCDM and νwCDM cosmological models The 95

upper limits are derived assuming asummν gt 0 prior

Data 95 upper limit [eV] PinvPnorm Punphy Gaussian fit [eV]Planck 0252 064 043Planck + BAO 0129 036 064 minus0026 plusmn 0074Planck + BAO + RSD 0102 024 076 minus0026 plusmn 0060Planck + SN 0170 049 056 minus0076 plusmn 0106Planck + BAO + RSD + SN 0099 022 078 minus0024 plusmn 0057Planck + BAO + RSD + SN + DES 0111 027 071 minus0014 plusmn 0061Planck + BAO + RSD + SN (νwCDM) 0139 040 061 minus0033 plusmn 0082Planck + BAO + RSD + SN + DES (νwCDM) 0161 048 056 minus0048 plusmn 0097

of the new data in the next decadeIn Table 8 we also show several integrated probabilities

defined as

Pnorm =

int infin00588 eV

p(mν)dmν (26)

Pinv =

int infin00995 eV

p(mν)dmν (27)

Punphy =

int 00588 eV

p(mν)dmν (28)

Note that these are not Bayesian evidences because wedo not account for the prior volume Nevertheless theratio of PinvPnorm is the relative probability of the truemass lying in the range allowed by the invertednormalhierarchy and is equivalent to an evidence ratio when thepriors are very wide The quantity Punphy is the probabil-ity of the summed neutrino mass lying in the unphysicalregion with a mass lower than allowed by the normal hi-erarchy We see that these probabilities are always incon-clusive there is no strong evidence from cosmology on apreference for a normal hierarchy an inverted hierarchyor a model where the neutrino mass is anomalously low(with or without allowing extrapolation into the negativesummν) We also note that a 95 upper limit of less than

00995 eV would not constitute a 2σ detection of normalhierarchy because much of that posterior volume belongsto the unphysically low neutrino mass

Evaluating the 95 upper limits the strongest con-straint excluding lensing data is

summν lt 0099 eV which

degrades tosummν lt 0114 eV upon addition of lensing

data This reflects the shift toward a relatively low am-plitude of σ8 in the lensing data with the larger values ofΩm preferred by the other probes

Finally we see that allowing the dark energy equationof state parameter (w) to be free degrades the neutrinomass constraint by a factor of 14 to 16 This effectis due to a known degeneracy direction in the neutrinomass (Hannestad 2005) Nevertheless the effect is notas dramatic as it used to be and with further data it willbecome negligible

7 CONCLUSION

The eight distinct samples of SDSS BAO measure-ments fill a unique niche in their ability to indepen-dently characterize dark energy and curvature in one-parameter extensions to ΛCDM When combined withPlanck temperature and polarization data the BAOmeasurements allow an order of magnitude improvementon curvature constraints when compared to Planck dataalone The BAO data provide strong evidence for anearly flat geometry and allow constraints on curvature

005 000 005 010 015 020m [eV]

abilit

PlanckPlanck + BAOPlanck + BAO + RSDPlanck + BAO + RSD + SN

Fig 13mdash Posterior for sum of neutrino masses for selected com-binations of data with a νΛCDM cosmology Dashed curves showthe implied Gaussian fits Shaded regions correspond to lower lim-its on normal and inverted hiearchies Likelihood curves are nor-malized to have the same area under the curve for

summν gt 0

that are now roughly one order of magnitude within thedetectable limit of σ(Ωk) sim 00001 (Vardanyan et al2009) The SDSS BAO measurements demonstrate thatthe observed cosmic acceleration is best described by adark energy equation of state that is consistent with acosmological constant to better than 6 precision whencombined with the Planck temperature and polarizationdata Finally the SDSS BAO measurements allow robustestimates of the current expansion rate demonstratingH0 lt 70 km sminus1Mpcminus1 at 95 confidence under stan-dard assumptions of pre-recombination physics regard-less of cosmological model These H0 results remain con-sistent even without the Planck CMB data as long asthe ΛCDM model is assumed

Beyond the distance-redshift relation we have alsodemonstrated the complementary role of the six inde-pendent SDSS RSD measurements to DES and Plancklensing measurements The SDSS RSD measurementstighten Planck temperature and polarization constraintson the dark energy equation of state by more than afactor of two the DES WL measurements tighten con-straints on curvature by more than a factor of threeIndependent of any BAO or SNe Ia information on theexpansion history the CMB RSD and WL measure-ments present a history of structure growth that is bestdescribed by a standard ΛCDM cosmology and a GRmodel for gravity

The tightest constraints on the cosmological model arefound when combining current measurements of the ex-

pansion history CMB and growth of structure Thiscombination reveals a dark energy density measured to07 precision under an assumed ΛCDM model Wefind sim1 precision estimates on ΩDE H0 and σ8 withcentral values that barely change under any extensionexplored in Section 6 The best fitting parameters inextended models remain consistent with a ΛCDM cos-mology the most flexible ow0waCDM model indicatesconstraints Ωk = minus00023plusmn00022 w0 = minus0912plusmn0081and wa = minus048+036

minus030 The Dark Energy FoM for the fullcombination in a model that allows for curvature is 92about 38 lower than what was predicted 14 years ago bythe Dark Energy Task Force (Albrecht et al 2006) How-ever the assumptions of the Dark Energy Task Force in-cluded the final DES cosmology results whereas we onlyincluded the results of the first year WL and clusteringstudies If the final DES studies of SNe Ia galaxy clus-ters and WL can provide an additional 60 increase inthe FoM then the Dark Energy Task Force predictionswill be proven accurate The combination of measure-ments also provides an independent constraint on thesummed neutrino mass leading to

summν lt 0111 eV at

95 confidence (νΛCDM) with a slight preference fora normal hierarchy of mass eigenstates over an invertedhierarchy The dominant factors in this neutrino massmeasurement are the constraints from CMB and BAOthus making the result robust against challenges in mod-eling the full-shape of the power spectrum in clusteringand lensing measurements

At the high precision found here cosmic accelerationremains most consistent with predictions from a cosmo-logical constant A deviation from consistency with apure cosmological constant perhaps would have pointedtoward specific dark energy and modified gravity mod-els However since many of these models have parameterchoices that make them indistinguishable from ΛCDMthose models all can be made consistent with our obser-vations Nevertheless the observed consistency with flatΛCDM at the higher precision of this work points increas-ingly towards a pure cosmological constant solution forexample as would be produced by a vacuum energy fine-tuned to have a small value This fine-tuning represents atheoretical difficulty without any agreed-upon resolutionand one that may not be resolvable through fundamen-tal physics considerations alone (Weinberg 1989 Brax ampValageas 2019) This difficulty has been substantiallysharpened by the observations presented here

71 A Decade of Dark Energy

The profound insight offered into the cosmologicalmodel is only possible after several generations of ex-perimental effort Experiments designed to study thenature of dark energy have steadily improved in tech-nique redshift coverage and sample size In particularthe Planck CMB experiment offered a significant boostin spatial coverage and precision over WMAP while theBOSS and eBOSS programs offered vast improvementsin redshift range and statistical precision over the pre-ceding spectroscopic surveys

As a baseline to assess the impact of the current gen-eration of dark energy experiments we first character-ize the dark energy constraints with the analogous pro-grams that were concluding as BOSS was achieving firstlight Representing the approximate period 2000ndash2010

we choose the final WMAP sample (Bennett et al 2013Hinshaw et al 2013) the JLA sample of SNe Ia (Be-toule et al 2014) and the 27 precision measurementof isotropic BAO at z = 0275 (Percival et al 2010) fromSDSS DR7 (Abazajian et al 2009) and the 2-degree FieldGalaxy Redshift Survey (Colless et al 2001) Followingthe convention presented in the report from the Dark En-ergy Task Force (Albrecht et al 2006) we refer to thisdataset as lsquoStage-IIrsquo Although some of these results werereleased as BOSS was nearing its conclusion the data arerepresentative of the previous generation of dark energystudy

The most recent cosmology results are reflected in thePlanck temperature polarization and lensing data thePantheon SNe Ia sample the SDSS BAO+RSD mea-surements and the DES 3times2pt samples This datasetreferred to as lsquoStage-IIIrsquo provides the main constraintspresented in this paper and in Table 7

Finally we isolate the improvements over the Stage-IIconstraints from the SDSS BAO+RSD program We doso by replacing the SDSS DR7 BAO measurements withthe SDSS BAO+RSD measurements while keeping theWMAP CMB and JLA SNe Ia samples intact This com-bination is then referred to as lsquoStage-II+SDSSrsquo In thesame vein we isolate the improvements over the Stage-IIconstraints from recent programs other than SDSS De-noted lsquoStage-III wo SDSSrsquo the constraints are derivedfrom the Stage-II programs Planck temperature polar-ization and lensing data the Pantheon SNe Ia sampleand the DES 3times2pt samples

Although the ow0waCDM model is the most flexibleof all models explored in this work with regards to darkenergy parameterization only the full Stage-III datasetis able to converge without strong priors that exclude un-physical values of wa (eg see Figure 12) On the otherhand the three one-parameter extensions presented inSection 4 demonstrate the complementarity between theprobes in constraining a constant dark energy equationof state curvature and the neutrino mass We there-fore quantify advances of the last decade by comput-ing cosmological constraints in a νowCDM model Themarginalized 68 confidence intervals for each of the keycosmological parameters in this model for each of therelevant Stage-II and Stage-III sample combinations ispresented in Figure 14

The general effect of the Stage-III measurements is topush the Stage-II results closer to a ΛCDM model inboth curvature and the dark energy equation of stateThe Stage-III results also significantly reduce the upperbounds on the neutrino mass without any indication for acentral value that is larger than 0 eV With the exceptionof σ8 the central values of all parameters in the Stage-IIIresults overlap the 68 confidence intervals of the Stage-II results The precision on all parameters has increasedby at least a factor of 25 The largest gains from Stage-II to Stage-III are found in the constraints on Ωk σ8and

summν with improvements in precision by factors of

45 70 and 71 respectivelyWe compute the relative gain across the full volume

of the 68 confidence intervals on w Ωksummν H0

and σ8 We use a figure of merit related to the in-verse of the determinant of the covariance matrix forthese five parameters We define our figure of merit as

0285 0300 0315Ωm

Stage II

Stage II + SDSS

Stage III wo SDSS

Stage III

068 070ΩΛ

0000 0008 0016Ωk

072 080σ8

675 690 705H0

minus108 minus102 minus096w0

00 05 10

Σmν [eV]

Fig 14mdash Central values and 68 contours for each of the parameters describing expansion history and growth of structure in a νowCDMmodel Results are shown for each data set combination presented in the text where Stage-II corresponds to a combination of the WMAPJLA and SDSS DR7 data and Stage-III corresponds to a combination of the SDSS BAO+RSD Planck Pantheon SN Ia and DES 3times2ptdata

FoM = |Cov(pp)|minus1(2N) where N = 5 is the numberof free parameters (represented by p) This form prop-erly tracks the typical gain in the 68 confidence intervalfor each free parameter We find FoM = 11 23 and 44for the Stage-II Stage-II+SDSS and Stage-III resultsrespectively The gain by a factor of 2 when adding theSDSS data to the Stage-II experiments demonstrates thesignificant contribution of BAO and RSD measurementsin advancing the cosmological model The SDSS BAOand RSD data reduce the total volume (within 68 con-fidence) of the five dimensional likelihood surface by afactor of 40

The SDSS BAO+RSD measurements have the mostsignificant impact on the precision of Ωk H0 and

summν

In particular the combination of Stage-II+SDSS leads toestimates ofH0 = 6791plusmn092 km sminus1Mpcminus1 comparableto the tightest constraints on the local expansion ratepresented in Section 42 This result is in disagreementwith the combined Cepheid distance ladder and stronglensing time delay results by more than four standarddeviations further reinforcing one of the biggest surprisesof the last decade of cosmology results

72 Beyond Dark Energy Cosmology from eBOSS

The spectroscopic samples from BOSS and eBOSS al-low for a diverse array of cosmology studies beyond thecosmic expansion history and growth of structure pre-sented in this work These data have already been usedto advance models for the summed neutrino mass and in-flation In addition new techniques have been developedto use combinations of tracers or new tracers for directmeasurements of BAO and RSD

These data have been used to place constraints on neu-trino masses and inflation parameters through measure-ments of the one-dimensional flux power spectrum of theLyα forest (Palanque-Delabrouille et al 2013 Chabanieret al 2019) When combining the recent eBOSS mea-surement with CMB and BAO measurements the sumof the neutrino masses is constrained with a 95 up-per limit

summν lt 009 eV (Palanque-Delabrouille et al

2020) These same Lyα forest power spectrum measure-ments present evidence for a departure from a constant

value for the power-law index of the primordial powerspectrum The model that best describes the Lyα andPlanck data has a running that is non-zero at more than95 confidence αs equiv dnsd ln k = minus0010plusmn 0004

The eBOSS data have been used to further exploreinflationary models through tests for primordial non-Gaussianities of the local form fNL Recent measure-ments of the power spectrum in eBOSS quasars offermeasurements of fNL that are independent of the cur-rent Planck bispectrum limits The measurements findminus51 lt fNL lt 21 at 95 confidence (Castorina et al2019) and indicate that the full eBOSS dataset couldreach σ(fNL) lt 10 using a full range of scales and alarger redshift range

The five-year eBOSS sample also provides an areathat is sampled simultaneously with LRGs ELGs andquasars The overlap in redshifts between samples en-ables techniques to combine multiple tracers and reducethe effects of sample variance (McDonald amp Seljak 2009Seljak 2009) Projections for fNL and RSD from eBOSSfollowing the multi-tracer technique are found in thework by Zhao et al (2016) In an effort to understandthe joint clustering across multiple tracers Alam et al(2020a) detect one-halo conformity between the eBOSSLRG and ELG samples at a significance of more thanthree standard deviations The result presents the chal-lenges of predicting multi-tracer clustering at high preci-sion beyond what is possible with the basic halo modelThe first eBOSS multi-tracer cosmology study is asso-ciated with this final eBOSS release (Wang et al 2020Zhao et al 2020) In the configuration-space study theyfind an improvement in the RSD measurement precisionof approximately 12 over using the LRG samples pre-sented in this work

The eBOSS data have inspired several other advancedtechniques in cosmology Tentative BAO measurementshave been made at z lt 1 using the cross-correlation be-tween the Mg II forest and galaxy and quasar tracers (duMas des Bourboux et al 2019) between the C IV forestand quasars at z gt 2 (Blomqvist et al 2019) and finallyin the cross-correlation between spectroscopic quasarsand high redshift galaxies (Zarrouk et al 2020) selected

from the DESI Legacy Imaging Surveys (Dey et al 2019)Voids in the clustering of galaxies and quasars have longbeen explored as a means to constrain growth of struc-ture and the distance-redshift relation through the APeffect The first void detections in eBOSS are presentedusing DR14 LRGs and quasars (Hawken et al 2020)This analysis has been extended to the eBOSS DR16including the ELG sample for the first time in eBOSSand finding a linear redshift-space distortion parameter(Aubert et al 2020) βLRG(z = 074) = 0415 plusmn 0087βELG(z = 085) = 0665 plusmn 0125 and βQSO(z = 148) =0313plusmn 0134 consistent with other measurements fromeBOSS DR16 using conventional clustering techniquespresented in this paper Ravoux et al (2020) recentlydeveloped a parallel technique for void finding at higherredshifts They derived a three-dimensional map of large-scale matter fluctuations from a region that was denselysampled with Lyα forest quasars Covering a volume of094hminus3Gpc3 with a resolution of 13hminus1Mpc they iden-tify voids and protocluster candidates in the cosmic web

Finally the eBOSS data have enabled new techniquesfor controlling and assessing the selection function fortracers of large-scale structure Among those new tech-niques are those results associated with the release of thispaper such as forward modeling of the selection functionfrom imaging surveys (Kong et al 2020) new modelsfor fiber collisions (Mohammad et al 2020) and N-bodymock catalogs for characterization of the ELG sample(Avila et al 2020)

Having recently completed installation and commis-sioning the Dark Energy Spectroscopic Instrument(DESI DESI Collaboration et al 2016ab) will obtaina sample of nearly 50 million galaxies and quasars span-ning the redshift range 0 lt z lt 35 The techniques de-veloped in eBOSS to use the one-dimensional Lyα forestflux power spectrum large-scale clustering multi-tracerclustering void detection and new models for the se-lection function and halo occupation statistics will beincorporated into the future DESI studies This nextgeneration of the analyses developed within eBOSS willbe an integral part of the final cosmology results at thecompletion of DESI

73 Beyond Cosmology Astrophysics Results andPotential Studies with the eBOSS Spectra

The final eBOSS data sample found in the SDSS Six-teenth Data Release (Ahumada et al 2020) is the resultof nearly two decades of development in the spectral datareduction and redshift classification software pipelinesThese catalogs of more than four million spectra andtheir classifications have been extremely well-vetted andare ripe for further study These data have already beenused to explore a range of astrophysical processes beyondthe cosmology that inspired the program with continuedpotential for studies in galaxy evolution lensing and ab-sorption systems and quasar physics

The high-redshift ELG sample is unique within thefour generations of SDSS and allows systematic studies ofthe internal dynamics composition and environment ofthese star-forming galaxies An example of potential forspectroscopic studies in this large data sample is found inone of the earliest results from eBOSS Zhu et al (2015)constructed a composite spectrum based on 8620 galax-ies over the redshift interval 06 lt z lt 12 This com-

posite spectrum reveals blueshifted lines indicating out-flows driven by star formation This high signal-to-noisespectrum along with smaller aperture emission line mea-surements from Hubble Space Telescope and quasar ab-sorption line observations can all be explained by a self-consistent outflow model The ELG spectra of roughly180000 galaxies were further investigated to constrainthe mass-metallicity relation at high redshift (Huanget al 2019) The results indicate that the 06 lt z lt 105ELGs follow the fundamental metallicity relation that isobserved in the local universe The local environmentof the eBOSS galaxies can also be studied through theillumination of the circumgalactic medium In a studyof SDSS quasar spectra the absorption due to Mg II andFe II in intervening LRGs and ELGs was explored overimpact parameters ranging from 10 kpc to 1 Mpc (Lan ampMo 2018) The metal absorption strengths were strongeralong the minor-axis of the galaxies due to outflowing gasand followed a steeper profile for ELGs than for LRGsindicating more recent enrichment of the circumgalacticmedium due to star formation

The eBOSS spectra have also been used to identifypreviously unknown superpositions of multiple galaxyspectra In a search for serendipitous emission lines inthe spectra of BOSS and eBOSS galaxy targets Talbotet al (2020) were able to identify 1551 strong galaxy-galaxy gravitational lens candidates The full catalog ofthese lens candidates is being released as a value-addedcatalog to enable future studies14 Such a large samplecan be used to study the demographics of backgroundsource galaxies for advanced modeling of the dark mat-ter structure of lens galaxies with a diverse sample andfor calibrating searches for lens galaxies with ground-based imaging programs

Finally eBOSS has unique spectroscopic programs inquasar astrophysics Three dedicated programs were co-ordinated with eBOSS to take advantage of the potentialfor studies in quasar astrophysics

bull The Time Domain Spectroscopic Survey (TDSSMorganson et al 2015 MacLeod et al 2018) char-acterized the spectra of variable stars and quasarsidentified in photometric imaging

bull The Spectroscopic Identification of eROSITASources (SPIDERS Clerc et al 2016 Dwelly et al2017) observed cluster galaxies and active galac-tic nuclei detected in the ROSAT All-Sky Survey(Voges et al 1999 2000) and with XMM-Newton(Jansen et al 2001) observations and

bull The SDSS-RM program monitored a sample of849 quasars at more than 70 epochs over fiveyears The data enable the measurement of moreblack hole masses over a larger range of redshiftthan any previous reverberation mapping program(Shen et al 2015)

Between the clustering quasar sample (Myers et al 2015)and the three quasar programs quasar targets comprisedthe majority of all eBOSS spectra Reverberation map-ping studies have measured lags of broad lines relative

14 Spectroscopic Identification of Lensing Objects (SILO) VAChttpsdatasdssorgsasdr16ebossspectrolensingsilo

to the continuum for Hα (17 quasars Grier et al 2017)Hβ (42 quasars Grier et al 2017) Mg II (57 quasarsHomayouni et al 2020) and C IV (52 quasars in the red-shift range 14 lt z lt 28 Grier et al 2019) Arguablythe biggest surprise in quasar astrophysics from SDSSwas changing-look quasars that change from broad linequasars with strong continua to narrow line systems withweak continua over the course of a few years (LaMassaet al 2015 MacLeod et al 2016 Dexter et al 2019) Thisphenomenon had not previously been seen for luminousAGN

Quasar astrophysics is just one of the topics that mo-tivates the next generation of the Sloan Digital Sky Sur-vey SDSS-V (Kollmeier et al 2017) SDSS-V will pro-vide single-object spectra of more than six million sourcesacross the whole sky with the BOSS spectrographs andthe infrared APOGEE spectrographs (Majewski et al2017 Wilson et al 2019) In addition SDSS-V will per-form spatially-resolved spectroscopy in the Milky Wayand nearby galaxies using new optical spectrographs onseveral small telescopes The SDSS-V program will pro-duce the worldrsquos premier sample of spectra for studies ofMilky Way assembly history origin of the chemical ele-ments mapping the local interstellar medium and time-domain spectroscopy Scheduled for observations over2020ndash2025 the five year program will multiply the sci-ence returns from space-based missions such as Gaia andeROSITA (Merloni et al 2012) while setting the stagefor spectroscopy coordinated with imaging from the VeraRubin Observatory (Stubbs et al 2004)

This paper represents an effort by both the SDSS-IIIand SDSS-IV collaborations Funding for SDSS-III wasprovided by the Alfred P Sloan Foundation the Par-

ticipating Institutions the National Science Foundationand the US Department of Energy Office of ScienceFunding for the Sloan Digital Sky Survey IV has beenprovided by the Alfred P Sloan Foundation the USDepartment of Energy Office of Science and the Partici-pating Institutions SDSS-IV acknowledges support andresources from the Center for High-Performance Com-puting at the University of Utah The SDSS web site iswwwsdssorg

SDSS-IV is managed by the Astrophysical ResearchConsortium for the Participating Institutions of theSDSS Collaboration including the Brazilian Partici-pation Group the Carnegie Institution for ScienceCarnegie Mellon University the Chilean Participa-tion Group the French Participation Group Harvard-Smithsonian Center for Astrophysics Instituto de As-trofısica de Canarias The Johns Hopkins UniversityKavli Institute for the Physics and Mathematics ofthe Universe (IPMU) University of Tokyo LawrenceBerkeley National Laboratory Leibniz Institut fur As-trophysik Potsdam (AIP) Max-Planck-Institut fur As-tronomie (MPIA Heidelberg) Max-Planck-Institut furAstrophysik (MPA Garching) Max-Planck-Institut furExtraterrestrische Physik (MPE) National Astronomi-cal Observatory of China New Mexico State UniversityNew York University University of Notre Dame Obser-vatario Nacional MCTI The Ohio State UniversityPennsylvania State University Shanghai AstronomicalObservatory United Kingdom Participation Group Uni-versidad Nacional Autonoma de Mexico University ofArizona University of Colorado Boulder University ofPortsmouth University of Utah University of VirginiaUniversity of Washington University of Wisconsin Van-derbilt University and Yale University

APPENDIX

A BAO AND RSD MEASUREMENTS AND SYSTEMATIC ERRORS

In galaxy redshift surveys BAO and RSD are usually measured through two-point clustering statistics To calculatethe two-point clustering statistics we convert galaxy redshifts and angular positions into comoving coordinates using afiducial cosmological model denoted by a superscript lsquofidrsquo For a pair of galaxies at effective redshift zeff with a smallseparation the comoving transverse and line-of-sight separations depend on the comoving angular diameter distanceDM (zeff) and the Hubble distance DH(zeff) = cH(zeff) respectively Conversion from radial comoving distance DC to DM depends on the cosmological model (see Section 2) Limiting ourselves to the ΛCDM model and workingin units of hminus1Mpc this conversion depends solely on the value of Ωfid

m Counts of galaxy pairs in the form of thecorrelation function or power spectrum are then fitted with a fixed model in which the BAO feature is located at rfid

d Although not necessary for the methodology we adopt we use the same model for computing rfid

d as we do for theconversion of measured coordinates to comoving coordinates

For the SDSS BAO measurements we parameterize the position of the BAO feature using a dimensionless dilationparameter in the transverse direction (αperp) and in the radial (α) direction The best-fit values and covariance betweenthese parameters are calculated by fitting the template spectrum to the observed BAO positions in the monopolequadrupole and hexadecapole moments of the two-point statistics Information other than the BAO peak positionis removed from the fit by marginalizing over a set of simultaneously fitted parameters that model the shape of themultipole moments

If the true BAO peak is located at rd which can be different from rfidd then both αperp and α will be shifted by

rfidd rd The location of the radial BAO peak in the data depends on DH(zeff)Dfid

H (zeff) while the location of theangular BAO feature depends on DM (zeff)Dfid

M (zeff) Combining these we have

α=DH(zeff)rdDfidH (zeff)rfid

αperp=DM (zeff)rdDfidM (zeff)rfid

We see that αperp and α combine information about the model and distance-redshift relationship in a way that is

perfectly degenerate To demonstrate this we now consider the dependence of the fit on hhfidWorking in units of hminus1Mpc and assuming that the fiducial and true physical densities match so rd = rfid

d in units ofMpc the ratio hhfid enters in the shift of the model to hminus1Mpc and hence via rdr

fidd The ratios Dfid

M (zeff)DM (zeff)and Dfid

H (zeff)DH(zeff) are independent of h and hfid for fixed Ωm Suppose instead that we had worked in unitsof Mpc and measured the two-point functions and model in these units Then we would have to specify hfid beforecalculating the two-point measurements and so the ratio hhfid enters into the calculation of DM (zeff)Dfid

M (zeff) andDH(zeff)Dfid

H (zeff) For models where rd = rfidd there is no h dependence in the theoretical BAO positions

Note that the above thought experiment shows that we should always work in the same basis when calculating thecomponents of both αperp and α That is we should not calculate rdr

fidd in Mpc and Dfid

M (zeff)DM (zeff) in hminus1Mpc

for example which would then ignore the hhfid dependence of the fitAnother way to see this is that the dimensionless quantities rdDM (zeff) and rdDH(zeff) correspond to the size

of the BAO feature in observed quantities namely angular separations measured in radians and radial separationsmeasured in redshift differences As long as we operate in units that make these ratios truly dimensionless (ie withoutresidual dependence on h) we are performing a correct compression of the available information

We present our baseline results as DM (zeff)rd and DH(zeff)rd to reflect what is measured in the data There isno dependence on fiducial values thus removing potential for ambiguity and the exact values assumed in our fiducialmodel

To measure RSD we fit a template power spectrum or correlation function decomposed into multipoles We allowthe template to be shifted in scale by the dimensionless parameters αperp and α and normalized using the parametersbσ8 and fσ8 Use of a template spectrum means that our measurements necessarily depend on the shape of thistemplate and on the fiducial cosmology used to create it As σ8 is defined as the rms fluctuations on comoving scalesof 8hminus1Mpc (ie not the gauge independent angular separations and redshift differences) we also need to considerthe model dependency of the scale on which the normalization parameters are measured

In the analysis procedure in SDSS clustering studies we normalize the template to a predicted σ8 and find the shiftsin scale and normalization required to fit the data One complication is whether the template is shifted in scale by thedilation parameters before or after the normalization of the model is fixed Shifting the template before measuring thenormalization is equivalent to fixing the scale on which we measure fσ8 and bσ8 in the observed two-point clusteringas determined by the fiducial cosmology in units of hminus1Mpc For data at z = 0 this would result in no cosmologicaldependence in the scale chosen However for measurements at higher redshifts such as those from eBOSS we have adependence on the fiducial value of Ωm used to calculate the distance-redshift relationship

If instead we do not shift the template before fixing the normalization of the model then we fix the scale in the unitsof the template The scale from the template can be different from that preferred by the data potentially bringingin a further dependence on hhfid and changing the degeneracies with other cosmological parameters In general wefind a larger systematic error contribution for our measurements in this case due to an increased dependence on thefiducial cosmology

We interpret the dilation parameters as measured from template fits in the same way as those with the aboveBAO measurements assuming that the model dependence arises only through rd and not through other scales in thetheoretical model While the BAO scale provides most of the dilation constraint it is possible that some componentarises from other features and therefore this should be considered an approximation

For our RSD measurements made using the BOSS and eBOSS galaxy samples we have found that rescaling thetemplate before fixing the normalization of the model significantly reduces the dependence on fiducial cosmology andhence the required systematic error

The σ8ndashbased normalization measurements we present and analyze retain a dependence on the fiducial Ωm that setsthe scale on which they are measured and on the shape of the template which links the scales on which fσ8 andbσ8 are defined to those constrained by the data These dependencies are illustrated in Figure 7 of Bautista et al(2020) by comparing recovered measurements (from the LRG correlation function) with mock catalogues where thefiducial cosmology assumed is different from the true cosmology For the power spectrum as measured from the LRGswe show how measurements of fσ8 depend on αiso = (α2

perpα)13 in Figure 15 From these tests we see only a weak

dependence on the fiducial assumptions given our baseline procedure The scatter in the measurements is included inour assumed systematic error

For our MGS and eBOSS quasar measurements the contribution to the systematic error from the fiducial cosmologyis significantly reduced compared to other contributions to the systematic error and we instead adopt the slightlysimpler procedure where we do not rescale the template before normalization This results in slightly larger estimatesof the systematic error induced by the analysis method More details about the systematic errors and the dependencieson the fiducial assumptions in the analysis method can be found in the papers describing the individual measurements

We have shown that the approximations we made to compress BOSS and eBOSS data into combinations of pa-rameters (αperp α fσ8 and bσ8) do not significantly impact the interpretation of the growth measurements Thecompression therefore has minimal impact on our conclusions in testing cosmological models after allowing for appro-priate systematic errors This analysis of the clustering might not be the best approach for future surveys includingDESI and Euclid In order not to compromise the precision available with these future data it may be best to directlyfit models to the two-point multipoles without an intermediate data compression step

Fig 15mdash Dependence of the measured values of fσ8 on αiso = (α2perpα)

13 which gives the offset between the template and the truecosmology calculated from the set of Nseries mocks matching the BOSS CMASS NGC LRG sample at an effective redshift zeff = 056Filled ellipses give results fixing the scale at which fσ8 is measured after re-scaling the template allowing for the shift in the best-fit αisothe empty ellipses show results where the template is not rescaled The colors of the ellipses separate fits where different template modelswere used when analyzing the mocks Only a weak dependence is seen when fσ8 is constructed from the re-scaled σ8 significantly smallerthan the statistical errors on the measured values Dotted lines mark a 2 deviation with respect to the expected fσ8 value Furtherdetails of these fits are in Gil-Marin et al (2020)

B MODEL PARAMETERIZATION AND PARAMETER PRIORS

For cosmological models considered in this study it is possible to adopt a single parameterization However itis convenient to choose a parameterization depending on the investigation thus allowing different priors dependingon the constraints provided by the probes being used We employ two sets of parameterization The definitions ofparameters can be found in Table 1

The first parameterization is labeled as the lsquoCMBrsquo parameterization which is used whenever CMB likelihoods areincluded in the analysis It follows the natural degeneracy direction of the CMB constraints The basic parameters inthis parameterization include ωc ωb θMC As ns τ and Neff

The second parameterization is labeled as the lsquobackgroundrsquo parameterization This parameterization is used forchains without a CMB likelihood such as in studying SN-only or BAO-only constraints The basic parameters includeΩm H0 Ωb with the latter two used for BAO constraints

In addition to the basic parameters we also introduce extended parameters for testing extensions to the ΛCDMmodel which include Ωk w or w0 wa and Σmν

In Table 9 we list the parameters for each parameterization along with the priors assigned in the analyses and thesectionssubsections where the priors are adopted The baseline value of a parameter refers to the value used wheneverthe parameter is fixed in an analysis In addition to the flat priors on parameters that are being varied in the analysisthe lsquoCMBrsquo parameterization includes a prior on H0 of 20 km sminus1Mpcminus1 lt H0 lt 100 km sminus1Mpcminus1 We also applied aGaussian prior on ns of ns = 096 plusmn 002 and a BBN-inspired prior of ωb = 00222 plusmn 00005 in all runs that includegrowth information but without CMB data ie in Figure 9 (left) and Figure 11 In Section 422 for H0 constraintswithout CMB data a BBN-motivated Gaussian prior is used for ωb (002235plusmn 000037)

REFERENCES

Abazajian K N Adelman-McCarthy J K Agueros M Aet al 2009 ApJS 182 543

Abbott B P Abbott R Abbott T D et al 2017a Nature551 85

mdash 2017b ApJ 848 L13Abbott T M C Abdalla F B Alarcon A et al 2018

Phys Rev D 98 043526Abbott T M C Abdalla F B Avila S et al 2019

Phys Rev D 99 123505Abe K Adam J Aihara H et al 2014 Phys Rev Lett 112

061802Abe K Aihara H Andreopoulos C et al 2015 Progress of

Theoretical and Experimental Physics 2015 053C02Abe Y Aberle C dos Anjos J C et al 2012 Phys Rev D

86 052008

Adamson P Andreopoulos C Arms K E et al 2008Phys Rev Lett 101 131802

Addison G E Watts D J Bennett C L et al 2018 ApJ853 119

Adelberger E G Garcıa A Robertson R G H et al 2011Reviews of Modern Physics 83 195

Ahmad Q R Allen R C Andersen T C et al 2002Phys Rev Lett 89 011301

Ahn J K Chebotaryov S Choi J H et al 2012Phys Rev Lett 108 191802

Ahumada R Prieto C A Almeida A et al 2020 ApJS 2493

Aker M Altenmuller K Arenz M et al 2019Phys Rev Lett 123 221802

Akhmedov E K Razzaque S amp Smirnov A Y 2013 Journalof High Energy Physics 2013 82

TABLE 9Cosmological Parameters and Priors

Parameterization Parameter Baseline Prior Analysis

CMB ωc 012 (0001 099) whenever CMB likelihood is includedωb 00221 (0005 01) as in Sections 4 5 6100θMC 10411 (05 10) ln(1010As) 305 (161 391) ns 096 (08 12) τ 006 (001 08) Neff 3046 ndash

Background Ωm ndash (01 09) Section 41 (SN-only BAO-only)Ωb ndash (0001 03) Section 41 (BAO-only)H0 (km sminus1Mpcminus1) ndash (20 100) Section 422

Extended Ωk 00 (-08 08) Section 411 Section 62w (w0) -1 (-3 1) Section 412 Section 62wa 0 (-3 075) Section 62summν (eV) 006 (0 5) Section 413 Section 63

Note mdash For each parameter in parentheses is the range of the flat prior The baseline value of a parameteris the one adopted whenever the parameter is fixed in an analysis

Alam S Peacock J A Kraljic K Ross A J amp Comparat J2020a MNRAS arXiv191005095

Alam S Ata M Bailey S et al 2017 MNRAS 470 2617Alam S et al 2020b submittedAlbrecht A Bernstein G Cahn R et al 2006 arXiv

astro-ph0609591Alcock C amp Paczynski B 1979 Nature 281 358Alestas G Kazantzidis L amp Perivolaropoulos L 2020

Phys Rev D 101 123516An F P Bai J Z Balantekin A B et al 2012

Phys Rev Lett 108 171803Anderson L Aubourg E Bailey S et al 2012 MNRAS 427

3435Anderson R I 2019 AampA 631 A165Araki T Eguchi K Enomoto S et al 2005 Phys Rev Lett

94 081801Arendse N Wojtak R J Agnello A et al 2019 ArXiv

e-prints arXiv190907986Astier P Guy J Regnault N et al 2006 AampA 447 31Aubert M et al 2020 submitted

Aubourg E Bailey S Bautista J E et al 2015Phys Rev D 92 123516

Avila S et al 2020 submittedBalbi A Ade P Bock J et al 2000 ApJ 545 L1Ballinger W E Peacock J A amp Heavens A F 1996

MNRAS 282 877Bautista J et al 2020 submittedBautista J E Busca N G Guy J et al 2017 AampA 603 A12Bennett C L Larson D Weiland J L et al 2013 ApJS 208

20Beradze R amp Gogberashvili M 2020 ArXiv e-prints

arXiv200105874Betoule M Kessler R Guy J et al 2014 AampA 568 A22Beutler F Blake C Colless M et al 2011 MNRAS 416 3017Beutler F Seo H-J Saito S et al 2017a MNRAS 466 2242Beutler F Seo H-J Ross A J et al 2017b MNRAS 464

3409Birrer S Treu T Rusu C E et al 2019 MNRAS 484 4726Blake C Brough S Colless M et al 2012 MNRAS 425 405Blanton M R Bershady M A Abolfathi B et al 2017 AJ

154 28Blomqvist M du Mas des Bourboux H Busca N G et al

2019 AampA 629 A86Brax P amp Valageas P 2019 Phys Rev D 99 123506Carlstrom J E Ade P A R Aird K A et al 2011 PASP

123 568Castorina E Hand N Seljak U et al 2019 Journal of

Cosmology and Astroparticle Physics 2019 010

Chabanier S Palanque-Delabrouille N Yeche C et al 2019Journal of Cosmology and Astroparticle Physics 2019 017

Chevallier M amp Polarski D 2001 International Journal ofModern Physics D 10 213

Chiang C-T Hu W Li Y amp LoVerde M 2018Phys Rev D 97 123526

Chiang C-T amp Slosar A 2018 ArXiv e-printsarXiv181103624

Choudhury S R amp Hannestad S 2019 ArXiv e-printsarXiv190712598

Clerc N Merloni A Zhang Y Y et al 2016 MNRAS 4634490

Cole S Percival W J Peacock J A et al 2005 MNRAS362 505

Colless M Dalton G Maddox S et al 2001 MNRAS 3281039

Conley A Guy J Sullivan M et al 2011 ApJS 192 1Contreras C Hamuy M Phillips M M et al 2010 AJ 139

519Cooke R J Pettini M amp Steidel C C 2018 ApJ 855 102Cuceu A Farr J Lemos P amp Font-Ribera A 2019 Journal

of Cosmology and Astroparticle Physics 2019 044Cuesta A J Verde L Riess A amp Jimenez R 2015 Mon

Not Roy Astron Soc 448 3463Davis T M Hinton S R Howlett C amp Calcino J 2019

MNRAS 490 2948Dawson K S Aldering G Amanullah R et al 2009 AJ 138

1271Dawson K S Schlegel D J Ahn C P et al 2013 AJ 145 10Dawson K S Kneib J-P Percival W J et al 2016 AJ 151

44de Bernardis F Kitching T D Heavens A amp Melchiorri A

2009 Phys Rev D 80 123509de Bernardis P Ade P A R Bock J J et al 2000 Nature

404 955de Mattia A et al 2020 submittedde Salas P Forero D Ternes C Trtola M amp Valle J 2018

Physics Letters B 782 633de Salas P F amp Pastor S 2016 Journal of Cosmology and

Astroparticle Physics 2016 051DESI Collaboration Aghamousa A Aguilar J et al 2016a

ArXiv e-prints arXiv161100036mdash 2016b ArXiv e-prints arXiv161100037Dexter J Xin S Shen Y et al 2019 ApJ 885 44Dey A Schlegel D J Lang D et al 2019 AJ 157 168Dhawan S Brout D Scolnic D et al 2020 ApJ 894 54Di Valentino E Melchiorri A Mena O amp Vagnozzi S 2020

Phys Rev D 101 063502

du Mas des Bourboux H Le Goff J-M Blomqvist M et al2017 AampA 608 A130

du Mas des Bourboux H Dawson K S Busca N G et al2019 ApJ 878 47

du Mas des Bourboux H et al 2020 submittedDwelly T Salvato M Merloni A et al 2017 MNRAS 469

1065Eisenstein D J Seo H-J Sirko E amp Spergel D N 2007

ApJ 664 675Eisenstein D J Zehavi I Hogg D W et al 2005 ApJ 633

560Eisenstein D J Weinberg D H Agol E et al 2011 AJ 142

72Esteban I Gonzalez-Garcia M C Hernand ez-Cabezudo A

Maltoni M amp Schwetz T 2019 Journal of High EnergyPhysics 2019 106

Farr J Font-Ribera A du Mas des Bourboux H et al 2020Journal of Cosmology and Astroparticle Physics 2020 068

Fixsen D J 2009 ApJ 707 916Flaugher B Diehl H T Honscheid K et al 2015 AJ 150

150Fogli G L Lisi E Marrone A et al 2012 Phys Rev D 86

013012Folatelli G Phillips M M Burns C R et al 2010 AJ 139

120Font-Ribera A McDonald P Mostek N et al 2014 Journal

of Cosmology and Astroparticle Physics 2014 023Fowler J W Niemack M D Dicker S R et al 2007

Appl Opt 46 3444Freedman W L Madore B F Scowcroft V et al 2012 ApJ

758 24Freedman W L Madore B F Hoyt T et al 2020 ApJ 891

57Frieman J A Bassett B Becker A et al 2008 AJ 135 338Fukuda Y Hayakawa T Ichihara E et al 1998

Phys Rev Lett 81 1562Gil-Marin H et al 2020 submittedGoobar A amp Leibundgut B 2011 Annual Review of Nuclear

and Particle Science 61 251Graur O Rodney S A Maoz D et al 2014 ApJ 783 28Grieb J N Sanchez A G Salazar-Albornoz S et al 2017

MNRAS 467 2085Grier C J Trump J R Shen Y et al 2017 ApJ 851 21Grier C J Shen Y Horne K et al 2019 ApJ 887 38Gunn J E Siegmund W A Mannery E J et al 2006 AJ

131 2332Guy J Sullivan M Conley A et al 2010 AampA 523 A7Hamuy M Phillips M M Suntzeff N B et al 1996 AJ 112

2391Hanany S Ade P Balbi A et al 2000 ApJ 545 L5Hannestad S 2005 Phys Rev Lett 95 221301Hawken A J Aubert M Pisani A et al 2020 Journal of

Cosmology and Astroparticle Physics 2020 012Hazra D K Shafieloo A amp Souradeep T 2019 Journal of

Cosmology and Astroparticle Physics 2019 036Heymans C Van Waerbeke L Miller L et al 2012 MNRAS

427 146Hicken M Wood-Vasey W M Blondin S et al 2009 ApJ

700 1097Hikage C Oguri M Hamana T et al 2019 Publications of

the Astronomical Society of Japan 22Hinshaw G Weiland J L Hill R S et al 2009 ApJS 180

225Hinshaw G Larson D Komatsu E et al 2013 ApJS 208 19Homayouni Y Trump J R Grier C J et al 2020 arXiv

e-prints arXiv200503663Hou J et al 2020 submittedHowlett C Ross A J Samushia L Percival W J amp

Manera M 2015 MNRAS 449 848Hu W 2005 in Astronomical Society of the Pacific Conference

Series Vol 339 Observing Dark Energy ed S C Wolff ampT R Lauer 215

Huang C Zou H Kong X et al 2019 ApJ 886 31Jaffe A H Ade P A Balbi A et al 2001 Physical Review

Letters 86 3475Jain B amp Seljak U 1997 ApJ 484 560Jansen F Lumb D Altieri B et al 2001 AampA 365 L1

Jha S Kirshner R P Challis P et al 2006 AJ 131 527Jimenez R Kitching T Pena-Garay C amp Verde L 2010

Journal of Cosmology and Astroparticle Physics 2010 035Kaiser N 1987 MNRAS 227 1Kaiser N Burgett W Chambers K et al 2010 in Society of

Photo-Optical Instrumentation Engineers (SPIE) ConferenceSeries Vol 7733 Society of Photo-Optical InstrumentationEngineers (SPIE) Conference Series

Knox L amp Millea M 2020 Phys Rev D 101 043533Kohlinger F Viola M Joachimi B et al 2017 MNRAS 471

4412Kollmeier J A Zasowski G Rix H-W et al 2017 arXiv

e-prints arXiv171103234Komatsu E Dunkley J Nolta M R et al 2009 ApJS 180

330Kong H et al 2020 submittedKowalski M Rubin D Aldering G et al 2008 ApJ 686 749Krause E Eifler T F Zuntz J et al 2017 ArXiv e-prints

arXiv170609359LaMassa S M Cales S Moran E C et al 2015 ApJ 800

144Lan T-W amp Mo H 2018 ApJ 866 36Lesgourgues J amp Pastor S 2006 Phys Rep 429 307Lewis A 2019 ArXiv e-prints arXiv191013970Lewis A amp Bridle S 2002 Phys Rev D 66 103511Li X amp Shafieloo A 2019 ApJ 883 L3Li Y-F Cao J Wang Y amp Zhan L 2013 Phys Rev D 88

013008Lin M-X Benevento G Hu W amp Raveri M 2019

Phys Rev D 100 063542Lin S et al 2020 submittedLinder E V 2003 Physical Review Letters 90 091301mdash 2005 Phys Rev D 72 043529Linder E V amp Cahn R N 2007 Astroparticle Physics 28 481Lyke B W et al 2020 submittedMacLeod C L Ross N P Lawrence A et al 2016 MNRAS

457 389MacLeod C L Green P J Anderson S F et al 2018 AJ

155 6Majewski S R Schiavon R P Frinchaboy P M et al 2017

AJ 154 94Mandelbaum R Miyatake H Hamana T et al 2018

Publications of the Astronomical Society of Japan 70 S25Mangano G Miele G Pastor S et al 2005 Nuclear Physics

B 729 221McDonald P amp Seljak U 2009 Journal of Cosmology and

Astroparticle Physics 10 7Merloni A Predehl P Becker W et al 2012 ArXiv e-prints

arXiv12093114Mohammad F et al 2020 submittedMorganson E Green P J Anderson S F et al 2015 ApJ

806 244Myers A D Palanque-Delabrouille N Prakash A et al 2015

ApJS 221 27Nadathur S Percival W J Beutler F amp Winther H A 2020

Phys Rev Lett 124 221301Neill J D Seibert M Tully R B et al 2014 ApJ 792 129Netterfield C B Ade P A R Bock J J et al 2002 ApJ

571 604Neveux R et al 2020 submittedPadmanabhan N Xu X Eisenstein D J et al 2012 MNRAS

427 2132Palanque-Delabrouille N Yeche C Schoneberg N et al 2020

Journal of Cosmology and Astroparticle Physics 2020 038Palanque-Delabrouille N Yeche C Borde A et al 2013

AampA 559 A85Palanque-Delabrouille N Magneville C Yeche C et al 2016

AampA 587 A41Pardo K amp Spergel D N 2020 ArXiv e-prints

arXiv200700555Percival W J Cole S Eisenstein D J et al 2007 MNRAS

381 1053Percival W J Baugh C M Bland-Hawthorn J et al 2001

MNRAS 327 1297Percival W J Sutherland W Peacock J A et al 2002

MNRAS 337 1068

Percival W J Reid B A Eisenstein D J et al 2010MNRAS 401 2148

Perlmutter S Aldering G Goldhaber G et al 1999 ApJ517 565

Phillips M M Lira P Suntzeff N B et al 1999 AJ 118 1766Planck Collaboration Ade P A R Aghanim N et al 2011

AampA 536 A1mdash 2014 AampA 571 A17Planck Collaboration Adam R Ade P A R et al 2016a

AampA 594 A1Planck Collaboration Ade P A R Aghanim N et al 2016b

AampA 594 A13Planck Collaboration Akrami Y Arroja F et al 2018a ArXiv

e-prints arXiv180706205Planck Collaboration Aghanim N Akrami Y et al 2018b

ArXiv e-prints arXiv180706209mdash 2018c ArXiv e-prints arXiv180706210mdash 2019 ArXiv e-prints arXiv190712875Poulin V Smith T L Karwal T amp Kamionkowski M 2019

Phys Rev Lett 122 221301Prakash A Licquia T C Newman J A et al 2016 ApJS

224 34Raichoor A Comparat J Delubac T et al 2017 MNRAS

471 3955Raichoor A et al 2020 submittedRavoux C Armengaud E Walther M et al 2020 ArXiv

e-prints arXiv200401448Reid B Ho S Padmanabhan N et al 2016 MNRAS 455

1553Riess A G Casertano S Yuan W Macri L M amp Scolnic

D 2019 ApJ 876 85Riess A G Filippenko A V Challis P et al 1998 AJ 116

1009Riess A G Kirshner R P Schmidt B P et al 1999 AJ 117

707Riess A G Strolger L-G Casertano S et al 2007 ApJ 659

98Riess A G Macri L Casertano S et al 2009 ApJ 699 539mdash 2011 ApJ 730 119Riess A G Macri L M Hoffmann S L et al 2016 ApJ 826

56Riess A G Rodney S A Scolnic D M et al 2018 ApJ 853

126Rodney S A Riess A G Strolger L-G et al 2014 AJ 148

13Ross A et al 2020 submittedRoss A J Samushia L Howlett C et al 2015 MNRAS 449

835Ross A J Beutler F Chuang C-H et al 2017 MNRAS 464

1168Ross N P Myers A D Sheldon E S et al 2012 ApJS 199 3Rossi G et al 2020 submittedSahni V Shafieloo A amp Starobinsky A A 2014 ApJ 793 L40Sako M Bassett B Becker A C et al 2018 PASP 130

064002

Sanchez A G Grieb J N Salazar-Albornoz S et al 2017aMNRAS 464 1493

Sanchez A G Scoccimarro R Crocce M et al 2017bMNRAS 464 1640

Satpathy S Alam S Ho S et al 2017 MNRAS 469 1369Scolnic D M Jones D O Rest A et al 2018 ApJ 859 101Seljak U 2009 Physical Review Letters 102 021302Shafieloo A Sahni V amp Starobinsky A A 2012

Phys Rev D 86 103527Shen Y Brandt W N Dawson K S et al 2015 ApJS 216 4Slosar A 2006 Phys Rev D 73 123501Smee S A Gunn J E Uomoto A et al 2013 AJ 146 32Smith A et al 2020 submittedStaggs S Dunkley J amp Page L 2018 Reports on Progress in

Physics 81 044901Stritzinger M D Phillips M M Boldt L N et al 2011 AJ

142 156Stubbs C W Sweeney D Tyson J A amp LSST 2004 in

Bulletin of the American Astronomical Society Vol 36 1527ndash+Sullivan M Guy J Conley A et al 2011 ApJ 737 102Suzuki N Rubin D Lidman C et al 2012 ApJ 746 85Talbot M et al 2020 submitted

Tamone A et al 2020 submittedTroxel M A MacCrann N Zuntz J et al 2018

Phys Rev D 98 043528Tully R B Courtois H M amp Sorce J G 2016 AJ 152 50Vagnozzi S 2019 ArXiv e-prints arXiv190707569Vardanyan M Trotta R amp Silk J 2009 MNRAS 397 431Vargas-Magana M Ho S Cuesta A J et al 2018 MNRAS

477 1153Voges W Aschenbach B Boller T et al 1999 AampA 349 389mdash 2000 VizieR Online Data Catalog IX29Wang L amp Steinhardt P J 1998 ApJ 508 483Wang Y et al 2020 submittedWeinberg D H Mortonson M J Eisenstein D J et al 2013

Phys Rep 530 87Weinberg S 1989 Reviews of Modern Physics 61 1Wilson J C Hearty F R Skrutskie M F et al 2019

Publications of the Astronomical Society of the Pacific 131055001

Wong K C Suyu S H Chen G C F et al 2020 MNRASarXiv190704869

Wright E L Eisenhardt P R M Mainzer A K et al 2010AJ 140 1868

York D G Adelman J Anderson J E et al 2000 AJ 1201579

Yu B Knight R Z Sherwin B D et al 2018 ArXiv e-printsarXiv180902120

Zarrouk P et al 2020 submittedZhao C et al 2020 submittedZhao G-B Wang Y Ross A J et al 2016 MNRAS 457

2377Zhu G B Comparat J Kneib J-P et al 2015 ApJ 815 48Zuntz J Sheldon E Samuroff S et al 2018 MNRAS 481

ABSTRACT

1 Introduction

2 Cosmological Model and Observable Signatures




231 The CMB



234 Weak lensing



3 Data and Methodology



4 Implications of Expansion History Measurements

41 Impact of BAO Measurements on Models for Single Parameter Extensions to CDM








5 Implications of Growth Measurements

51 Impact of RSD Measurements on Models for Single Parameter Extensions to CDM



52 RSD Constraints on the Amplitude of Matter Fluctuations and Tests of Gravity

521 RSD constraints on the amplitude of matter fluctuations


6 Global Fits

61 CDM Model


621 DE H0 and 8 Parameters


63 Neutrino Mass

7 Conclusion



73 Beyond Cosmology Astrophysics Results and Potential Studies with the eBOSS Spectra

A A BAO and RSD Measurements and Systematic Errors

B B Model Parameterization and Parameter Priors